Ya. I. Nikolin.

Instructor at the Tomsk Technological Institute of Emperor Nicholas II.

Graphical Methods

for Calculating

Water Supply and Sewerage

Part I

Theory and Application of the Method of Conjugate Scales

With 14 tables of drawings.

Tomsk.

Typo-Lithography of the Siberian Association of Printing Works, corner of Yamskoy Lane and Dvoryanskaya Street, own building.

1911.

“Recognizing as permissible and useful, in the design of watercourses, the use of graphical calculations, it is desirable to have graphical tables adapted to the direct solution of the tasks relating to them.”
Resolution of the 17th Russian Water-Supply Congress, 1907.

I. Graphical Methods for Calculating Pipelines.

Problems relating to the movement of water in pipes and watercourses are solved by means of a number of formulas developed by various scholars on the basis of experiments, observations, and theoretical considerations.

The number of such formulas, as is known, is very considerable. Flinn, in his work The Movement of Water in Irrigation Canals, etc., gives 18 formulas used simultaneously both for pipes and for canals, and 19 formulas applied especially to pipes—37 formulas in all. But this list cannot be considered exhaustive, since it does not include some of the formulas that have been used on the continent of Europe, and, of course, the new formulas that appeared after the publication of his book did not enter it. At the present time the number of formulas relating to the movement of water in pipes is undoubtedly greater than fifty.

In the present article we shall deal only with the movement of water in water-supply pipes and watercourses, calling both of these together pipelines. In doing so, in essence, we can speak at first only of movement in pipelines of circular cross-section, i.e., pipes working when completely full; and then, having developed the question in this direction, we shall determine also the changes that are introduced in applying it to pipelines of another cross-section at different degrees of filling.

The basis for all formulas determining the movement of water in pipes is the equation obtained directly from the expression of equilibrium of a liquid undergoing steady uniform motion along a channel of any cross-section,

\[ h-h_o=\frac{\varphi}{\gamma}\cdot\frac{P}{\Omega}\cdot L, \tag{1} \]

where \(\Omega\) is the area of the live cross-section of the channel, in the present case of the pipe;

\(P\) is the wetted perimeter;

\(L\) is the length of the pipe section under consideration;

\(h\) and \(h_o\) are the losses of head at the end and at the beginning of this section;

\(\varphi\) is the magnitude of the hydraulic resistances per unit area of wall;

\(\gamma\) is the weight of a unit volume of water.

Denoting the ratio \(\dfrac{h-h_o}{L}\), the loss of head per unit length, or the hydraulic slope, by \(i\), and the ratio \(\dfrac{\Omega}{P}\), the hydraulic radius, by \(\rho\), we obtain

\[ i=\frac{\varphi}{\gamma\rho}. \tag{2} \]

The magnitude \(\varphi\) varies with changes in several factors characterizing the motion: the velocity of motion, the hydraulic radius and slope, and the condition of the pipe walls. Taking into account the influence of these factors on the basis of experiments and observations, and expressing the magnitude \(\varphi\) in the form of functions of the velocity \(v\) with coefficients either constant or themselves representing functions of other factors, various investigators have developed their own formulas for the motion of water in pipes. But since, in doing so, some strove, for the sake of accuracy, to express the influence of each factor separately, while others, for the sake of simplicity, tried artificially to reduce all influences to the action of a single factor, there resulted from the above single basis a series of formulas, very different in form, in the composition of the variables, in complexity, and in convenience of application.

Formulas for the motion of water in pipes may be classified, on the one hand, according to the composition of the factors on which the magnitude \(\varphi\) is made to depend, and consequently also the magnitude \(i\), which is proportional to it; and, on the other hand, according to the form in which the dependence of these quantities on the indicated factors, especially on velocity, is expressed.

With regard to the composition of the formulas it must be said that the factor whose influence is expressed by all formulas without exception is the velocity \(v\). Therefore all formulas (with the sole exception of Du Buat’s formula, which has had no practical application) may be reduced to the form

\[ \frac{\varphi}{\gamma}=\rho i=bv^2 . \tag{3}\,^{1)} \]

The coefficient \(b\), being a function of the factors that characterize the motion, appears in various formulas as ten different combinations of the composition of these factors, namely:

1) \(b=const.\) (for example, the formulas of Chézy—Eytelwein, Dupuit, Berdmore).

2) \(b=f_1(v)\) (the formulas of Prony, Weisbach, Saint-Venant);

3) \(b=f_2(\rho)\) (Darcy’s formula);

4) \(b=f_3(\rho,n)\) (the formulas of Lévy, Ganguillet-Kutter old and abridged, Knaauff, Manning, Bazin);

5) \(b=f_4(\rho,n_1,n_2)\) (the formulas of Darcy-Bazin, Frank);

6) \(b=f_5(\rho,i)\) (Nevil’s formula);

7) \(b=f_6(\rho,v,n)\) (the formulas of Smith, Lampe, Lampe-Lindley, Flaman);

8) \(b=f_7(\rho,i,n)\) (the complete Ganguillet-Kutter formula);

9) \(b=f_8(\rho,v,\theta)\) (Hagen’s formula);

10) \(b=f_9(\rho,r,n_1,n_2)\) (the formulas of Unwin, Trautwine, Weston).

In these expressions, by \(n\), \(n_1\), \(n_2\) are meant elements of the formulas (coefficients or exponents) expressing the dependence of the motion on the condition of the pipe walls; \(\theta\) denotes the coefficient in Hagen’s formula, dependent on temperature.

Of greater interest to us is another classification of formulas for the motion of water in pipes—according to the form of the function expressing the dependence of the quantities \(\varphi\) and \(i\) on the velocity \(v\). In this respect the given formulas may be divided into five types:

1) \(\varphi\) and the product \(\rho i\) are expressed by a function of \(v\) of the form \(mv^2\), where \(m=const.\) (the formulas of Chézy-Eytelwein, Dupuit, Berdmore); in this case equation (3) takes the form:

\[ \frac{\varphi}{v}=\rho i=mv^2, \tag{3'} \]

whence

\[ v=\sqrt{\frac{1}{m}}\sqrt{\rho i}=k\sqrt{\rho i}, \tag{4} \]

where \(k\) (the velocity coefficient) \(=const.\)

2) \(\varphi\) and \(\rho i\) are expressed by a function of \(v\) of the form \(m_1v^2\), where \(m_1\) is a function of \(\rho\) and \(i\), separately or jointly, and of coefficients of roughness—

\(^{1)}\) This general form of formulas for the motion of water seems to me, in general, more expressive and more convenient for analysis than the Chézy formula.

... (the formulas of Neville, Darcy, Darcy–Bazin, Lévy, Frank, Ganguillet–Kutter, all, Knauff, Manning, Bazin); in this case

\[ \frac{\varphi}{\gamma}=\rho i=m_1 v^2, \tag{3′} \]

\[ v=\sqrt{\frac{1}{m_1}}\sqrt{\rho i}=k_1\sqrt{\rho i}, \tag{4′} \]

where \(k_1=f_1(\rho,i,n)\).

3) \(\varphi\) and \(\rho i\) are expressed by a function of \(v\) of the form \(pv+qv^2\), where \(p=\mathrm{const.}\) or a function of \(\rho\), and \(q=\mathrm{const.}\) (or likewise a function of \(\rho\)—Taton); (the formulas of Prony, Taton).

In this case, from equation (2) one obtains

\[ \frac{\varphi}{\gamma}=\rho i=pv+qv^2, \tag{5} \]

whence

\[ \rho i=\left(\frac{p}{v}+q\right)v^2, \tag{3″} \]

\[ v=\sqrt{\frac{1}{q+\frac{p}{v}}}\sqrt{\rho i}=k_2\sqrt{\rho i}, \tag{4″} \]

where \(k_2=f_2(v,\rho)\).

4) \(\varphi\) and \(\rho i\) are expressed by a function of \(v\) of the form \(pv^{\frac{3}{2}}+qv^2\), where \(p=\mathrm{const.}\) or a function of \(\rho\) (and of the roughness coefficients), and \(q=\mathrm{const.}\) (the formulas of Weisbach, Smith, Weston). Thus

\[ \frac{\varphi}{\gamma}=\rho i=pv^{\frac{3}{2}}+qv^2, \tag{5′} \]

whence

\[ \rho i=\left(\frac{p}{\sqrt{v}}+q\right)v^2, \tag{3‴} \]

\[ v=\sqrt{\frac{1}{q+\frac{p}{\sqrt{v}}}}\sqrt{\rho i}=k_3\sqrt{\rho i}, \tag{4‴} \]

where \(k_3=f_3(v,\rho,n)\).

5) \(\varphi\) and \(\rho i\) are expressed by a function of \(v\) of the form \(m_2 v^z\), where \(m_2\) is a logarithmic function of \(\rho\) (sometimes also of \(i\)—Saint-Venant) and of the roughness coefficient ...

roughness, \(z = const.\) (the formulas of Anvin, Trautwine, Lampe, Lampe-Lindley, Flaman).

From equation (2) one obtains

\[ \frac{\psi}{\gamma}=\rho i=m_2 v^z, \tag{6} \]

\[ v=\frac{(\rho i)^{\frac{1}{z}}}{m_2^{\frac{1}{z}}}. \tag{7} \]

These expressions may be reduced to the form (3) or (4):

\[ \rho i=m_2 v^z=(m_2 v^{z-2})\,v^2, \tag{3'} \]

\[ v=\frac{(\rho i)^{\frac{1}{z}}}{m_2^{\frac{1}{z}}} =\frac{(\rho i)^{\frac{1}{z}-\frac{1}{2}}}{m_2^{\frac{1}{z}}}\cdot(\rho i)^{\frac{1}{2}} =k_4\sqrt{\rho i}, \tag{4'} \]

where \(k_4=f_4(\rho,\ i,\ n)\).

The last subdivision of formulas is characteristic from the standpoint of their simplicity and convenience of application.

The simplest are formulas of the first type. For this reason, they were the first to receive practical application and continue to be used in part up to the present time, especially for preliminary estimates. But, representing the magnitude of hydraulic resistances in the form of a single-valued function of velocity alone and not taking into account the influence of other factors, these formulas cannot express even approximately the general conditions of water motion. They give reliable results only within the narrow limits corresponding to the conditions of the experiments that served to establish the coefficient of one or another formula.

The insufficient accuracy and generality of formulas of the first type, and the serious influence—revealed by experiments—on the magnitude of hydraulic resistances not only of velocity, but also of the hydraulic radius and the condition of the pipe walls, compelled researchers to complicate the formulas of water motion by introducing elements expressing the influence of these latter factors in various combinations. In this way formulas of the following four types were obtained.

Of these, the formulas of the second, third, and fourth types, differing from one another in the composition of their variables, are similar in that, in both the former and the latter, the coefficients of resista-

… and the velocities \(b\) and \(k\) are expressed as functions of one or several variables, and moreover in a polynomial, non-logarithmic form.

The formulas of these three types arose to replace formulas with a constant \(k\). Their number is very considerable, and, as is known, they are still used at the present time with, if not exclusive, then at any rate predominant prevalence in practical applications. These formulas cover the cases of water motion much more broadly and precisely, and at least some of them give results sufficiently close to reality within the limits of practical application. But their use is associated with one inconvenience—the complexity and tediousness of calculating polynomial formulas. In the mass computations inevitable in water-supply and sewerage work, this inconvenience leads to the expenditure of extra time and proves so serious that some formulas, precisely because of this, do not find application (just as, on the other hand, many formulas with a constant coefficient are retained in practice only because they facilitate calculations, even if to the detriment of accuracy).

Formulas of the fifth type differ from the preceding ones in that they have a logarithmic form. This circumstance, of course, greatly facilitates their use. These formulas appeared considerably later; they are not numerous, and their renown and dissemination, owing to the established reputation of the older formulas of polynomial form, are not yet as great as would be desirable. However, their unquestionable advantages with respect to ease of use, combined with sufficient accuracy, draw the serious attention of specialists to them and in recent years have gradually broadened the range of their application. It is precisely these formulas, and their transformations into graphical form, that will constitute the subject of the present article.

We have indicated that the major inconvenience of the formulas of the types most commonly used at present consists in the fact that they have a polynomial form, not amenable to logarithmization. The polynomial form of these formulas is explained by the basic principle laid at the foundation of these formulas. This principle (proved theoretically by Boussinesq in his “Théorie de l’écoulement tourbillonnant et tumultueux des liquides”) consists in the following.

The dependence of the magnitude of hydraulic resistances on velocity may be expressed in the form of equation (3):

\[ \frac{\varphi}{\gamma}=\rho i=bv^{2}, \tag{3} \]

where the coefficient, denoted by \(b\), varies in inverse relation to the quantities \(v\) and \(\rho\), in such a way that it can be expressed quite accurately by a function of the form

\[ b=a\left(1+\frac{\beta}{\rho}+\frac{\beta_1}{v}\right). \tag{8} \]

This principle is equally applicable to the motion of water both in pipes and in watercourses. The hydraulicians of the middle of the last century made this principle the basis for the development of their formulas. In doing so, they understood the significance of the inconveniences introduced by the many-term expression (8) in practical application, but they could do only one thing—reduce the number of terms by excluding one of them. Thus Prony discarded the second term \(\frac{\beta}{\rho}\), while Darcy and Bazin, on the contrary, discarded the third, \(\frac{\beta_1}{v}\). But these original formulas, and the later ones that arose from them by changing the character of the individual terms, still remained two-term formulas, and some assumed an even more complicated form.

Meanwhile, later investigations, as well as Bazin’s experiments, showed that in those cases when the hydraulic radius is small and the walls of the channel are smooth or slightly rough—in particular, in the case of flow through pipes—with a decrease in hydraulic radius and roughness, the law of motion of water approaches Poiseuille’s law, which applies to motion through capillary tubes.

The results of Poiseuille’s experiments, which were published in 1846, amount to the following: when applied to capillary tubes, i.e., to the case of motion through pipes where the hydraulic resistances are reduced entirely or mainly to external friction (against the pipe walls), there exists a very definite dependence between the magnitude of these resistances, the diameter of the pipe (or its hydraulic radius), and the velocity. This dependence was expressed by Poiseuille in the form of the formula:

\[ i=\frac{p}{k}\cdot\frac{v}{D^2}, \tag{9} \]

where

\(i\)—the hydraulic gradient,

\(D\) and \(v\)—the diameter and velocity,

\(k\)—a numerical coefficient,

\(p\)—the coefficient of viscosity of the liquid, depending on the temperature, according to the formula:

\[ p=\frac{1}{1+0.03368\,\theta+0.000221\,\theta^2}, \tag{10} \]

where \(\theta\) is the temperature in \(C\).

Formula (9) may be transformed as follows:

\[ i=\frac{p}{16}\cdot\frac{v}{k}\cdot\frac{aw}{\rho^2}=\frac{aw}{\rho^2}, \]

where

\(\rho\)—as before, the hydraulic radius;

\(a\)—a coefficient proportional to \(p\), at constant temperature \(const.\)

\[ \rho i=\frac{aw}{\rho}; \tag{11} \]

or, comparing with (2),

\[ \frac{\varphi}{\gamma}=\rho i=\frac{aw}{\rho}. \tag{11'} \]

Reducing this formula to the form (3), we obtain

\[ \frac{\varphi}{\gamma}=\rho i=\frac{a}{\rho v}\cdot v^2=b_1 v^2. \tag{12} \]

Thus, for the given particular case of motion, the expression for the coefficient corresponding to \(b\) in formula (3), as also in formula (8), contains the quantities \(\frac{1}{\rho}\) and \(\frac{1}{v}\), but not in the form of a polynomial, rather in the form of the product \(\frac{1}{\rho v}\).

Making observations on a rectangular channel \(0.10\) meter wide, Bazin found that, within the limits of his experiments, the coefficient \(b\) of formula (3) may be considered proportional to the \(1/2\) power of the same product \(\frac{1}{\rho v}\), i.e., may be expressed by the formula

\[ b=\frac{\alpha}{(\rho v)^{\frac12}}. \tag{13} \]

This relation remained valid also for those cases in which the walls of the channel were made rough by covering them with coarse cloth.

Generalizing the results of the experiments mentioned, some investigators (Saint-Venant, Flamant) came to the conclusion that, both for the motion of water in capillary tubes and for motion in pipes in general, the coefficient \(b\) may be expressed as a function of the product \(\frac{1}{\rho v}\), in which this product appears in the form of a certain power, with an exponent varying between 0 (as a limit) and 1 (for capillary tubes). In other words, for all cases of water motion in pipes the following formula may be applied:

\[ b=\frac{a_0}{(\rho v)^z}, \tag{14} \]

where \(0<z<1\).

The roughness that is encountered on the average in water-supply pipes is less than that of the coating of the canal walls mentioned above in Bazin’s experiments; on the other hand, flow in pipes, other conditions being equal, is calmer than in an open canal of the same dimensions. It may therefore be thought that, for cases of water motion in ordinary water-supply pipes, the exponent in the expression

\[ b=\frac{a_0}{(\rho v)^z} \tag{14} \]

must lie within the limits between 0 and \(\frac{1}{2}\) ¹).

Be that as it may, it has turned out that the law of the motion of water in pipes can be expressed not only by formulas (3) and (8), which embrace it together with motion in open channels, but

¹) The type of expression given for \(b\), however, cannot be considered indisputable. Raising \(\frac{1}{v}\) to a power (except the first), in passing to the velocity coefficient \(k\), makes it depend on \(v\), which contradicts Darcy’s experiments and all the theoretical considerations expressed by M. Lévy. Bazin himself, apparently, did not share the opinion of persons who used his experiments to create formula (14) and its derivatives, since his own formula, given by him as early as 1897, preserves a binomial form and makes \(k\) depend only on \(\rho\). Nevertheless, formulas based on the indicated principle cover the experimental data well and, combined with their remarkable simplicity and convenience of use, are highly valuable. In dealing with the question of the limits of applicability of existing formulas for the motion of water in pipes, I arrived at another expression for \(b\), also logarithmic in form, but free from the indicated inconsistencies, and at a very simple formula for velocity that agrees well with reality. This formula, with the tables, diagrams, and considerations relating to it, I intend to publish in the next issue of the present work.

also, and moreover with greater probability, by formulas analogous to those relating to the motion of water in capillary tubes, namely:

\[ \frac{\varphi}{\gamma}=\rho i-b_1 v^2, \tag{3} \]

\[ b_1=\frac{\alpha_0}{(\rho v)^z}, \tag{14} \]

whence the relation is obtained

\[ \frac{\varphi}{\gamma}=\rho i-\frac{\alpha_0}{(\rho v)^z}\cdot v^2 . \tag{15} \]

We see that this relation, in contrast to the equations derived from expression (8), has a monomial and logarithmic form. It also served as the basis for several logarithmic formulas for the motion of water in pipes.

Formulas of the logarithmic form for the solution of hydraulic problems have by the present time found rather wide application in England and America. The first attempts to introduce them into practice appeared almost simultaneously in France, Germany, and England.

Saint-Venant, in his work “Formules et tables nouvelles etc.” (1851), proposed the formula:

\[ v=n\,(Di)^{\frac{7}{12}}, \tag{16} \]

where

\[ D\text{—diameter of the pipe,} \]

\[ n\text{—a constant coefficient,} \]

and developed tables facilitating its application. But his attempt to bring this formula into use did not have the success it deserved. After him, no such attempt was repeated in France until the appearance of Flamant’s formula.

In Germany, Hagen proposed in 1853 a monomial formula similar to Saint-Venant’s formula, but with different exponents for \(i\) and \(D\), and with another coefficient \(n\). But on German soil this attempt likewise had no success, all the more because Hagen himself later went over to a formula of binomial form.

Osborne Reynolds, comparing the results of various experiments, came to the conclusion that the magnitude of hydraulic resistances at

in motion through pipes is proportional not to the second power of \(v\), as is accepted in formula (3), but to some power \(v^z\), whose exponent varies according to the degree of roughness of the walls. In his opinion, this exponent is equal, for example, to 1.7 for pipes with very smooth walls, to 1.722 for lead pipes, and reaches 2 only with very rough walls.

On the basis of these considerations, Prof. Unwin proposed the following formula:

\[ i = n \frac{v^z}{D^{3-z}}, \tag{17} \]

where \(n\) and \(z\) are variables. This latter quantity, appearing in the exponents of \(v\) and \(D\), is taken from 1.79 to 2.00 for cast-iron pipes, generally in accordance with Reynolds’s indications.

It should be noted that Unwin’s formula

\[ i = n \frac{v^z}{D^{3-z}} \tag{17} \]

is fully reducible to the general form expressed by formula (15). Indeed, it may be represented in the form:

\[ Di = n \frac{v^z}{D^{2-z}} = n \frac{1}{(Dv)^{2-z}} \cdot v^2, \tag{17'} \]

which corresponds to (15).

Unwin’s formula found fairly wide application in England, and on its model several new formulas were composed by later investigators, differing from the original in the values of \(n\) and \(z\). Such, for example, is the formula of Thrupp:

\[ i = n \frac{v^z}{D^{0.6616z}}, \tag{18} \]

in which \(z\) varies from 1.70 to 2.00. It coincides with the preceding one when \(z = 1.86\), i.e. when the resistance has approximately a mean value.

In connection with Unwin’s formula there also appeared the formula of Fossa:

\[ Di = \frac{n}{\sqrt[6]{D^2 v}}\, v^2, \tag{18'} \]

or

\[ i = \frac{n v^{1.84}}{D^{1.33}} \tag{18''} \]

and the Manning formula:

\[ Di=\frac{n}{\sqrt[3]{D}}\,v^2 \tag{18''} \]

or

\[ i=\frac{nv^2}{D^{1.5}}. \tag{18^{\mathrm{iv}}} \]

In 1873 Prof. Dr. Lampe, on the basis of a comparison both of his own experiments and of preceding experiments, proposed the logarithmic formula:

\[ D^{1.25}i=n\,v^{1.802}. \tag{19} \]

We shall speak of this formula in greater detail later. For the present, let us point out that it, too, with a slight modification, may be reduced to the form (15). Namely, if instead of the exponent 1.25 we take 1.20, and instead of 1.802 take 1.80, then we obtain:

\[ D^{1.20}i=nv^{1.80}, \tag{19'} \]

\[ D^{\frac{6}{5}}i=nv^{\frac{9}{5}}, \]

\[ Di=n\,\frac{1}{(Dv)^{\frac{1}{5}}}\,v^2, \tag{19'} \]

which fully corresponds to expression (15). Lampe’s formula, as we shall see further on, deserves serious attention. It has had a certain circulation in Germany, and also, thanks to Mr. Lindley, who introduced into it a small modification (the Lampe–Lindley formula), and Prof. N. K. Chizhov, among us in Russia.

Later Meissner proposed the formula

\[ i=\frac{n_1 v^{1.833}}{D^{1.33}}, \tag{19''} \]

which, obviously, was composed on the model of Lampe’s formula, with a slight change in the exponents and the coefficient.

In the 1890s Prof. Flamant found that, instead of the exponent \(\frac{1}{5}\) in formula (19′), the exponent \(1/4\) corresponded better to the results of observations. He found this all the more convenient because a fourth root, if necessary, can be calculated without the aid of tables. Thus there resulted Flamant’s logarithmic formula, having the form:

\[ Di=\frac{n}{(Dv)^{\frac14}}\cdot v^2, \tag{20} \]

of the type of expression (15), also represented by the equations

\[ D^5 i^4=n^4 v^7 \tag{20′} \]

or

\[ D^{\frac54}i=n\,v^{\frac74}. \tag{20″} \]

Flamant’s formula has some circulation in France and, to a greater extent, in America.

In keeping with the content of the present article, we must mention one more logarithmic formula for the motion of water in pipes, namely the Levy–Vallot formula. It was proposed by A. Vallot on the basis of an artificial transformation of M. Levy’s non-logarithmic formula, and has the form:

\[ D=n Q^{\frac38} i^{\frac{3}{16}}, \tag{21} \]

where \(Q\) is the discharge.

Formulas for hydraulic calculation that have a logarithmic form, as was indicated above, already in themselves, owing to their form, offer considerable convenience in use as compared with non-logarithmic formulas. Their application can substantially help save time in mass computations. But the value of logarithmic formulas in this respect is further increased by the circumstance that their logarithmic character provides convenient ways of transforming them into graphical forms, thanks to which the process of calculation is reduced to the utmost simplicity.

It is surprising that logarithmic formulas were not sufficiently appreciated from the very time of their appearance (and even now are only slowly expanding the circle of their application). There may have been several reasons for this. First of all, by the time they appeared, the reputation of certain polynomial formulas and faith in the great accuracy of formulas of this type had already become established. Further, the tendency of most hydraulic engineers to create formulas encompassing the entire range of conditions of water motion played a role; for this purpose polynomial formulas were, or seemed, inevitable. Finally, in former times there was simply not such a frequent and acute need for mass computations, which would have made—

the simplicity of the formulas and of the calculation processes with the question of necessity.

The broad development, in the last decades of the nineteenth century, of the construction of water-supply and sewerage works raised the question of simplifying the methods of hydraulic calculation, with a view to saving labor and time, in a most urgent manner. This circumstance affected, among other things, those formulas of the usual, non-logarithmic form which enjoyed the widest application in practice. The new tendency was expressed here in the striving, insofar as possible without detriment to practical accuracy, to simplify these formulas, to develop auxiliary tables extensively, and finally to apply methods of graphical representation of formulas. On the other hand, the same circumstance again brought to the fore the question of hydraulic formulas of logarithmic form. Under the pressure of practical necessity, the attempts of certain specialists to introduce formulas of this kind into use met with a more favorable reception and were crowned with considerable success. It goes without saying that a number of measures were also applied to logarithmic formulas with the aim of further simplifying calculation, all the more since the form of such formulas makes them especially convenient for this purpose. These measures, apart from the preparation of numerical tables, came down to various ways of transforming formulas into graphical form, namely to representing them in the form of diagrams of curved or straight lines, or of special logarithmic scales (including among these the preparation of special logarithmic slide rules for hydraulic calculation).

Thus we see, in the last decades of the nineteenth century, in the field of calculation for water supply and sewerage, a new tendency, very interesting and promising much, namely the striving, with a view to simplifying this calculation, toward diverse applications of graphical methods. The application of these methods also affected formulas of the usual, polynomial form, but it received especially broad development with respect to logarithmic formulas.

Not having the possibility, in the present article, of dwelling even briefly on all the applications of graphical methods to hydraulic calculation, I consider it necessary, at least, to give a list of these applications.

These include:

1) Diagrams for determining various coefficients of hydraulic-

…formulas. It is applied to formulas of a non-logarithmic type and is sufficiently well known.

2) Graphical tables for representing relations between quantities in the form of a system of straight and curved lines (diagrams of isopleth curves). It is applicable to formulas of both non-logarithmic and logarithmic types. Such are the diagrams of Gobrecht¹, Gerhardt, Gurtin, Colignon, D’Ocagne and Willers, Coffin, E. and G. Taylor, and P. F. Gorbachev.

3) Logarithmographic tables for representing relations between the logarithms of quantities in the form of a system of straight lines (diagrams of isopleth straight lines). It is applied only to formulas of logarithmic type. Such are the diagrams of Timé, Frank, Fromm, V. A. Satkevich, and M. S. Yasyukovich.

4) Representation of relations between the logarithms of quantities in the form of diagrams of conjugate scales, constructed by the method of functional scales. They are used only with respect to logarithmic formulas. Such are the diagrams according to Lampe’s formula for metric measures; for the Wiener measures, for Russian measures—those of Prof. N. K. Chizhov.

5) The use of slide logarithmic rules for hydraulic calculation (chiefly in England), which represent one of the types of conjugate scales.

6) Diagrams of conjugate scales, constructed by the method of points of rectilinear intersection (diagrams, or “abaci,” of isopleth points). The latter are most conveniently applied to formulas of logarithmic type². Such diagrams are: according to the Lévy-Vallo formula for metric measures of Daries; according to the Flamant formula for metric measures of Bertrand; for English (and Russian) measures, the one placed in Prof. M. M. Cherepashinsky’s course.

7) Applications of graphical methods to the calculation of a closed water-supply network, for example, the method proposed by M. S. Yasyukovich.

This new current, and all these methods of applying graphical methods, proceeding by different paths toward one goal—the simplification of the process of hydraulic calculation and the saving of labor and time—deserve the close attention of specialist technicians, in the sense of study and further development.

And indeed, in recent times these applications have begun to attract noticeable attention. However, the technical literature devoted to this question, both in Russian and in foreign languages, is still limited almost exclusively to more or less brief reports, scattered through various journals, concerning individual methods of graphical calculation. Materials that might serve to elucidate the question, both theoretical and practical (tables, albums), are still more scattered and fragmentary, as is evident, for example, from the number of sources that had to be used in compiling the proposed part of the present work. These circumstances, and especially the theoretical interest and practical importance of this new tendency, compel one to regard as timely the study of the question of the application of graphical methods of every kind to the calculation of water supply and sewerage, with a view, on the one hand, to systematizing and elucidating these methods with their theoretical premises, mathematical and hydraulic, and with the generalizations and conclusions following from them; and, on the other hand, to popularizing these methods among specialists, which may facilitate their practical application and further development.

The article offered here is the result of such a study with respect to one element of the question. In it I touch upon the application of graphical methods to hydraulic calculation only as regards formulae of logarithmic form. Incidentally, these formulae are in themselves of special interest. And since they possess outstanding merits and, undoubtedly, deserve to be better known, I consider it not superfluous, in passing, to give rather complete information about them. Of the three methods of graphical calculation applicable to these formulae (the methods of isopleth lines, logarithmographic tables, and conjugate scales), here the discussion will deal exclusively with the method of conjugate scales. The choice of precisely this method for the beginning of the work is explained, besides its merits, also by the fact that it is the least illuminated in the technical literature. With respect to this latter method, the present article aims to give, as far as possible, complete and systematic information, embracing its theoretical foundations and the cases, methods, and results of its application to various formulae of hydraulic calculation.

It may be not superfluous, however, for purposes of comparison, to give readers an idea also of diagrams relating to

to the second and third types. For this purpose, independently of the references to the literature, there are appended, as examples, one diagram constructed by the method of isopleth curves, from the album of E. B. & G. M. Taylor (the Ganguillet–Kutter formula, modified by Jackson) (drawing 1), and a logarithmic table belonging to V. A. Satkevich (the Lampe formula) (drawing 2).

II. Development and principles of the method of conjugate scales.

The applications of graphical methods to hydraulic calculation, which represent various forms of the method of conjugate scales, are among the newest applications of mathematics. They are based on the principles of the mathematical science known as nomography¹, namely, on the one hand—on the principle of scales of functions, and on the other—on the principle of points of rectilinear intersection, or, otherwise, isopleth points.

If, for all the variables that are connected by a known equation, one constructs a system of geometrical elements (points or lines), graduated in accordance with the values of these variables—

On the basis of the considerations set forth, it would be more exact to assign to the methods of hydraulic calculation here presented the name nomographic. But we shall continue to call them simply graphical, since the distinction in terms in the present case is of no importance.

...variables, and if the relation between the variables, established by the equation, is expressed geometrically by the easily determined relative position of the corresponding geometric elements, then the aggregate of the latter constitutes the diagram of the given equation. By the name nomography is meant the theory of such diagrams, i.e., the theory of the graphical representation of mathematical laws expressed by equations with any number of variables.

The principle of the geometric representation of equations with two variables goes back to the time when Descartes laid the foundation of analytic geometry. With respect to equations with three variables, the first example of a graphical representation was given by Pouchet, who in his Arithmétique linéaire (1795) applied a system of curves arranged on a rectangular coordinate net. The same idea was applied in the works of Obenheim, Piombert, Bellencontre, and Allix, dating from 1814–1840. Terquem, in 1830, in connection with the works of Obenheim and Bellencontre, drew attention to the generality of this method of representing equations with 3 variables, and also pointed out its connection with certain ideas of mathematicians of preceding centuries.

However, in the first 50 years of the method’s existence, the use of graphical tables was very limited. The development, in the 1840s, of the railway network in France brought to the fore the problem of rapidly calculating large quantities of earthworks. To solve this problem, the engineers Lalanne (Léon-Louis-Chrétien Lalanne, 1811–1892) and Devaine conceived the idea of applying graphical methods. To this circumstance we owe the fact that Lalanne discovered (in 1843) a new principle for the graphical expression of equations, which he called the principle of anamorphosis. Lalanne’s discovery was published in the form of a memoir in Annales des Ponts et Chaussées for the year 1846 under the title “Mémoire sur les tables graphiques et sur la géométrie anamorphose appliquée à diverses questions qui se rattachent à l’art de l’ingénieur.” This work became the basis for the further development of nomography and, in itself, provided the ground for a whole series of applications to technical calculation. Lalanne himself was extremely interested in the practical application of his theory and developed several manuals for graphical calculation (Tables nouvelles pour abréger divers calculs, Tables graphiques à l’usage des chemins de fer, Description et usage de l’abaque ou compteur universel, Instruction sur les règles de ...

calcul and others); he also invented a special device for the mechanical performance of calculations, balance à calcul, balance algébrique.

In 1884 Prof. Massau, in his Mémoire sur l’intégration graphique, brought the principle of anamorphosis to the highest degree of generality. Lallemand gave anamorphic diagrams a special form, to which he assigned the name of hexagonal diagrams. But the special merit in the continuation of Lalanne’s work and in the development of nomography belongs to d’Ocagne (Maurice d’Ocagne, professor at the Paris École des Ponts et Chaussées). He devoted a whole series of works to the question of the graphical representation of equations, and in 1884 published a new general principle that he had found in this field, to which he arrived by transforming Lalanne’s diagrams, and which was made the basis of the method of aligned points (méthode des points alignés). D’Ocagne originally gave the latter the name method of isopleth points (méthode des points isoplèthes), wishing thereby to emphasize that the systems of points that play a role in this method correspond to the systems of isopleth straight lines that appear in diagrams constructed by Lalanne’s method.

The foundations of this method were first set forth in d’Ocagne’s memoir “Procédé nouveau de calcul graphique” (Annales des Ponts et Chaussées, 1884). Then the same question was subjected to broader development in d’Ocagne’s subsequent article, which appeared in 1890, “Méthode de calcul graphique fondée sur l’emploi des coordonnées parallèles” (Génie Civil, 1890), and, finally, received full development in his extensive monographs “Nomographie” (1891) and “Traité de nomographie” (1899).

Through these works d’Ocagne succeeded in arousing interest among representatives of various branches of technology, and the new method he discovered provided a basis for many applications to different fields of technical calculation.

The principles laid down by Lalanne as the basis of nomography received further development not only in the homeland of this science. Beyond France he has followers who continue the development of nomography in new directions; such are, for example, Prof. Mehmke, Goedseels, and others.

Turning now specifically to the method of conjugate scales, we shall recall that, as indicated above, this method belongs to the field of nomography and is based on two nomographic principles: the principle of scales of functions and the principle of points

rectilinear intersection. The first of these principles, which, together with that, is the fundamental proposition of all nomography, serves for constructing the graduations of the separate scales in diagrams. Its essence reduces to the following.

Let there be some function \(f(x)\) of the independent variable \(x\), within such limits that for each value of the variable \(x\) there is only one definite value of the function.

We shall lay off on the axis \(OX\), from the origin of coordinates \(O\) (Fig. 3), the lengths:

\[ \begin{aligned} l_1&=\lambda f(x_1),\\ l_2&=\lambda f(x_2),\\ l_3&=\lambda f(x_3),\\ &\ldots \end{aligned} \tag{22} \]

where \(\lambda\) is an arbitrarily chosen length, and we shall write above the points denoting the ends of the segments \(l_1, l_2, l_3,\ldots\) the corresponding values \(x_1, x_2, x_3,\ldots\) of the variable.

The aggregate of the points thus obtained, with numerical markings, constitutes the scale of the function \(f(x)\). The length \(\lambda\) is called the modulus of this scale.

If the scale of the function is to be bounded by two particular values of the variable, e.g. \(x_0\) and \(x_n\), then it may be constructed, beginning with the lower limit \(x_0\), without involving the origin of coordinates \(O\). To obtain the points \(x_1, x_2, x_3,\ldots,x_n\), it is necessary to lay off on the axis, beginning from an arbitrarily chosen point marked \(x_0\), the segments:

\[ \begin{aligned} l_1&=\lambda [f(x_1)-f(x_0)],\\ l_2&=\lambda [f(x_2)-f(x_1)],\\ l_3&=\lambda [f(x_3)-f(x_2)],\\ &\ldots\\ L&=\lambda [f(x_n)-f(x_0)], \end{aligned} \tag{23} \]

where \(L\) is the length of the scale.

Taking

\[ f(x)=x \tag{24} \]

and varying it by an equal and round number of units of one or another decimal order, we obtain, by the indicated construction, a normal scale. Depending on the problems subject to graphical solution, it is sometimes necessary to use construc-

…to other functions, and then scales of functions of another character are obtained, for example logarithmic, segmental, isographic¹).

If we take

\[ f(x)=\log x, \tag{25} \]

then the construction will give a logarithmic scale of the function. Its model may be provided by the divisions of a logarithmic slide rule (Fig. 4). Such a scale is used for constructing all logarithmic diagrams, including diagrams of conjugate scales.

With regard to this logarithmic scale it is necessary to note that, if it is continued beyond 10, then in the interval from 10 to 100 it will have the same length and the same divisions as from 0 to 10, while the divisions will correspond to quantities 10 times greater. The same would be true from 100 to 1000, only with marks still 10 times greater, and so on. From this it follows, first, that on a logarithmic scale constructed within the limits from 0 to 10, the units may refer to any decimal order whatever; secondly, that the degree of relative accuracy of reading, when applied to any order, remains constant.

The first follows from the fact that:

\[ \lambda \log(10^n \cdot x)=\lambda(n+\log x). \tag{26} \]

Thus, for numbers within the limits from \(10^n\) to \(10^{n+1}\), the logarithmic scale remains the same as from 1 to 10; it is only necessary to imagine its beginning displaced by a distance \(n\lambda\) to the left of the actual one.

That the relative accuracy (or percentage of accuracy) of calculation when using the logarithmic scale is everywhere the same is evident from the following reasoning. At any point of the scale, the greatest magnitude of the error in reading by eye is, generally speaking, one and the same in length. This magnitude represents the difference of two logarithms of certain values, between which the true value lies. Thus, denoting the limiting values by \(x_n\) and \(x_{n+1}\), we may say that:

\[ \log x_{n+1}-\log x_n = const. \tag{27} \]

But

\[ \log x_{n+1}-\log x_n=\log \frac{x_{n+1}}{x_n}, \]

whence

\[ \log \frac{x_{n+1}}{x_n}=const., \]

\[ \frac{x_{n+1}}{x_n}=const., \]

\[ \frac{x_{n+1}-x_n}{x_n}=const. \tag{28} \]

Thus, if, when reading on a logarithmic scale, we assign to its numerical marks values 10 times greater, then the numerical values of the error limits will also increase 10-fold, while the relative magnitude of the error and the percentage of accuracy of the calculation will remain one and the same.

Another nomographic principle that is of interest to us is d’Ocagne’s developed method of points of rectilinear intersection. It is precisely on this that the construction of one type of diagram with conjugate scales is based. To clarify this method, and also by way of comparing it with other methods used for graphical calculation, it is very convenient to trace, in the most general outline, the logical thread to which the various methods reduce in their successive development and historical change.

Suppose that we have an equation with 3 variables of the form

\[ F'(x,y,z)=0, \tag{29} \]

which we wish to represent in graphical form. The simplest method applicable for this purpose reduces to the following.

Assign to one of the variables (preferably the one that is most often expressed as a function of the other two), for example \(z\), a definite value. Then we obtain one equation with two variables. Such an equation is easily represented in the form of a curve drawn on a network of rectangular coordinates, determined by the equalities:

\[ \begin{aligned} x'&=\lambda_1 x,\\ y'&=\lambda_2 y, \end{aligned} \tag{30} \]

where \(\lambda_1\) and \(\lambda_2\) are, respectively, the chosen scale moduli for plotting the quantities \(x\) and \(y\) along the coordinate axes (cf. drawing 5).

The equation of this curve will be

\[ F\left(\frac{x'}{\lambda_1},\,\frac{y'}{\lambda_2},\, z\right)=0 . \tag{29′} \]

Such a curve, along the length of which the element \(z\) retains one and the same value, was called by Lalanne a curve of equal element (courbe d’égale élément), and later by the German author Vogler—an isometric curve (\(\ἴσος\)—equal, \(\muέτρον\)—magnitude). This latter term was subsequently adopted by Lalanne himself.

Let us construct, in the same way, curves corresponding to an entire series of values of \(z\), increasing by definite intervals, and let us write beside each curve the value of \(z\) corresponding to it. In doing so, of course, it is sufficient to draw the part of each curve within the rectangle formed by two pairs of perpendiculars drawn to the axes \(OX\) and \(OY\) through the points corresponding to the limiting values of \(x\) and \(y\).

In this way we obtain a system of curves within a rectangle (drawing 6), divided by rows of coordinates into cells. This system represents graphically our variables within the assigned limits. Diagrams of this type, and by analogy with them other diagrams with graduation and numerical markings, were given by Lalanne and his French pupils the name abaques (les abaques des lignes isoplèthes). We shall call them simply diagrams, in the present case—diagrams of isometric curves¹.

Prof. M. M. Cheremshansky, speaking of the method to which we, in agreement with d’Ocagne, give the new name method of points of rectilinear intersection, uses for its designation, proceeding from the old terminology (la méthode des points isoplèthes, les abaques des points isoplèthes), the expression method of isometric abaci. Such a term appears to us not entirely fortunate in general, since it does not express the distinction of the given method from the methods underlying diagrams of isopleth curves and straight lines; and therefore we replace it with the expression given above, which is more definite. As regards the word abacus in particular, it must be said that it has not obtained citizenship in the Russian language in the application under consideration. Moreover, this word in Greek as well as in all other languages is masculine. These latter considerations make it preferable and sufficient to use instead of it, in this and analogous cases, the term diagram.

If we agree to designate, by the terms horizontal and vertical, lines parallel respectively to \(OX\) and \(OY\), then the method of using such a diagram for determining the value of \(z\), from given \(x\) and \(y\), may be formulated as follows: read the marking of the curve passing through the point of intersection of the vertical with marking \(x\) and the horizontal with marking \(y\). It is self-evident that, when reading, interpolation is used if necessary. The same diagram makes it possible to determine \(x\) or \(y\), if one of these variables is given, as well as \(z\). For example, if \(y\) and \(z\) are given, then \(x\) can be obtained by reading the marking of the vertical passing through the point of intersection of the horizontal with marking \(y\) and the isopleth curve with marking \(z\).

It is not difficult to see that, in diagrams of isopleth curves, the graphical representation of an equation is created by horizontal sections of the surface determined by equation (29), in which \(x\), \(y\), and \(z\) are taken as the Cartesian coordinates of space.

In constructing the diagrams just discussed, there may be special cases in which the curves \(z\) become straight lines. Returning to the preceding discussion and considering equations (29), \((29')\), and (30), we see that this occurs when equation (29) can be reduced to the form

\[ \frac{x'}{\lambda_1} f(z) + \frac{y'}{\lambda_2} \varphi(z) + \psi(z) = 0 \tag{31} \]

or, by virtue of (30),

\[ x f(z) + y \varphi(z) + \psi(z) = 0 . \tag{31'} \]

In the more general case, when, for the construction of equation (29), it proves more convenient to lay off along the coordinate axes not \(x\) and \(y\), but \(f_1(x)\) and \(f_2(y)\), so that

\[ \begin{aligned} x' &= \lambda_1 f_1(x),\\ y' &= \lambda_2 f_2(y), \end{aligned} \tag{30'} \]

then, in order that the curve \(z\) of the diagram should become a straight line, equation (29) must have the form:

\[ \frac{x'}{\lambda_1} f(z) + \frac{y'}{\lambda_2} \varphi(z) + \psi(z) = 0 \tag{32} \]

or

\[ f_1(x) f(z) + f_2(y) \varphi(z) + \psi(z) = 0 . \tag{32'} \]

Nomographic diagrams distinguished by the indicated property are called diagrams of isopleth straight lines.

The construction of such diagrams is, of course, considerably easier than that of diagrams with curved lines. It is therefore understandable that in practice diagrams of this type are much more readily used, provided that the equation defining the relation between the elements of the problem can be reduced to the form (31′) or (32′).

Lalanne, Lallemand, and later investigators showed in various ways that diagrams containing curved lines (diagrams of isopleth curves), in certain cases, thanks to certain preliminary operations (geometric or logarithmic anamorphosis), can be artificially replaced by diagrams consisting exclusively of straight lines, parallel or nonparallel to one another, i.e., by diagrams of isopleth straight lines. The matter then reduces to transforming the equations subject to graphical representation into the form (31′) or (32′), which is usually accompanied by a change in the scales of the functions along the coordinate axes. Sometimes it is necessary to resort to the introduction of a new coordinate system or of a special third scale of functions (Lallemand’s hexagonal diagrams).

Among the equations that represent particular cases of type (32′) are equations of the form

\[ f_1(x)+f_2(y)+f_3(z)=0 \tag{33} \]

which are very often encountered in practice. In this case, applying the former method of laying off on the coordinate axes, we have

\[ \begin{aligned} x'&=\lambda_1 f_1(x),\\ y'&=\lambda_2 f_2(y), \end{aligned} \tag{34} \]

whence the equation of the curves \(z\) is obtained:

\[ \frac{x'}{\lambda_1}+\frac{y'}{\lambda_2}+f_3(z)=0 . \tag{35} \]

Equations of this form yield a diagram of isopleth straight lines parallel to one another.

It must be said that equations of the form (33) also allow the possibility of graphical representation by the method of the so-called hexagonal diagrams of Lallemand, and also by the method of rectilinear-intersection points that is of interest to us.

Another form of equations, very useful in practice, has the form

\[ f_1(x)\, f_2(y)=f_3(z). \tag{36} \]

It can be reduced to the preceding type by taking logarithms of both sides:

\[ \log f_1(x)+\log f_2(y)=\log f_3(z). \tag{37} \]

In this form the equation can be represented graphically in the form of a diagram of parallel straight lines (on a logarithmic grid).

This simple and convenient method of transformation has very great importance. It belongs to Lalanne, and was the first method of anamorphosis discovered by him.

The diagram of isoplethic straight lines obtained in this way, by virtue of the logarithmic character of the transformation and plotting, has acquired in practice the name of logarithmo-graphical tables.

The manner of using the diagram, when passing from isoplethic curves to isoplethic straight lines, changes only slightly. This method, however, is not without certain inconveniences, which chiefly amount to the fact that, in order to obtain readings, one has to follow the lines with the eyes and interpolate between them, which is tiring and does not sufficiently guard against errors. Without yet entering into a discussion of the relative significance of these inconveniences, which will be dealt with later, it is enough to note their existence. d’Ocagne drew attention to these inconveniences and found that it is more desirable to use, as elements with numerical markings for the graphical representation of equations, as far as possible, only points. He also proposed a method that makes it possible to achieve this aim for certain equations—namely, the method of points of rectilinear alignment.

The transition to this latter method is based on the principle of geometric dualism. This principle, as applied to the present case, is expressed in the fact that for every system of intersecting straight lines there exists such a system of points that three mutually intersecting straight lines of the first system correspond in the second to three points lying on one straight line. Thus we have—

…have here to do with one of the geometric transformations that are called dualistic¹.

Let us suppose that we have applied such a transformation to a diagram consisting of three systems of some straight lines, preserving, of course, in the passage from one figure to another, the numerical mark of each element. Then we shall obtain a new diagram (Drawing 7), on which each of the variables \(x\), \(y\), and \(z\) will correspond to a system of points with numerical marks arranged along a line, in the general case—a curve. These three systems of points with numerical marks will represent curvilinear scales.

Just as, in the first diagram, three straight lines with numerical marks of the values \(x\), \(y\), and \(z\), satisfying the equation, converged with one another, so here three corresponding points will lie on one straight line intersecting all three scales. This new diagram represents the most general form of a diagram of points of rectilinear intersection. Hence the manner of using a diagram of this kind is: the straight line joining the points with marks \(x\) and \(y\) on two curvilinear scales meets the third scale at the point with mark \(z\).

In practice there is used, under the name of d’Ocagne’s method of points of rectilinear intersection, a special case of the above-mentioned diagrams, when the curvilinear scales of the functions, owing to the properties of the equations subject to graphical representation, turn into straight lines parallel to one another. Such a case is obtained when the equations have the form (33) or (36–37). In this case the diagram has the form of three (or several) scales of functions, arranged parallel to one another and standing in such a mutual relation that any straight line intersecting these scales meets them at points with such numerical values of the variables as, when substituted simultaneously, satisfy the equation represented by the diagram.

The method in question was originally given by its inventor, d’Ocagne, the name method of isolated points (méthode des

points isoplèthes, by analogy with Lalanne’s method of isoplethic straight lines. But this name was later recognized by the author as unsuccessful, and for some time he used a new term—méthode des points cotés (the method of points with numerical marks), wishing to emphasize that in his diagrams the means of reading are precisely points supplied with numerical marks, and not lines. However, here there is the consideration that one may construct diagrams in which there also enter only points with numerical marks, but which by no means possess the indicated property with respect to a secant straight line. Therefore, in the end d’Ocagne settled on the term—méthode des points alignés, characterizing the relative position of points read simultaneously. We shall translate this term by the expression—method of points of rectilinear intersection.

The manner of applying this method to the solution of hydraulic problems we call the method of conjugate scales, and the diagrams serving for this purpose—diagrams of conjugate scales. The meaning of these terms is dictated by the external form of the diagrams and by the existing connection between the separate scales, which permits only their joint use. Their introduction into the classification of methods of hydraulic calculation seemed desirable for two reasons. First, these terms are more convenient and clearer than an expression indicating the mathematical method of construction. Secondly, there exists another analogous type of diagrams, which likewise represents an application of scales parallel to one another and logarithmic, as in the present case, but constructed without applying the method of points of rectilinear intersection, simply according to the principle of functional scales. These diagrams, in external appearance and partly in the manner of use, are close to the diagrams of conjugate scales constructed by d’Ocagne’s method. Therefore it seemed convenient, in the classification of graphical methods and, while distinguishing the methods of applying functional scales from the method of logarithmetic tables, to bring both of the above-mentioned similar methods under the general name—methods of conjugate scales. In cases where it is necessary to distinguish one method of conjugate scales from the other, one may, of course, add an indication of the method of construction.

The method of points of rectilinear intersection is applicable, as was indicated, to the graphical representation of equations having the form

\[ f_1(x)+f_2(y)=f_3(z). \tag{33} \]

Recall that equations of the form

\[ f_1(x)\cdot f_2(y)=f_3(z) \tag{36} \]

are reduced, by transformation, to

\[ \log f_1(x)+\log f_2(y)=\log f_3(z). \tag{37} \]

The possibility of converting equation (33) into the form of an alignment-chart diagram can be proved in various ways. The proof given by d’Ocagne is carried out with the aid of the so-called parallel coordinates. In view of the unusual application of this distinctive coordinate system, one might have avoided discussing this method of proof. However, in view of the remarkable simplicity attained by this application, and also the interest of the system of parallel coordinates itself, we do not consider it possible to pass over this method, and we shall set it forth, introducing on our part a certain simplification by eliminating superfluous elements (the auxiliary system of coordinate axes \(xOy\) and the quantity \(\delta\) connected with them—the half-distance between the origins of the parallel coordinates). For this purpose we must first explain, in a few words, the essence and the special features of the system of parallel coordinates.

Let two points \(A\) and \(B\) (Fig. 8) be given, which we shall call the origins of the parallel coordinates. Through these points draw two parallel straight lines \(AU\) and \(BV\)—the axes of the parallel coordinates \(^{1}\). Lay off on the axes the segments

\[ AM=u \quad \text{and} \quad BN=v . \tag{38} \]

We shall agree to regard such segments as positive in the directions from \(A\) to \(U\) and from \(B\) to \(V\), and negative in the opposite direction. The ends of the segments \(AM\) and \(BN\) determine points \(M\) and \(N\) on the coordinate axes, belonging to the straight line \(MN\). The lengths \(u\) and \(v\), taken with their corresponding signs, are called the parallel coordinates of the straight line \(MN\). Here \(AU, BV, u, v\) are the usual designations of axes and coordinates in the given system, just as \(OX, OY, x, y\) are in rectangular coordinates.

If we consider a fixed point \(P\) (Fig. 9) and a variable straight line \(MN\) passing through this point, then one can derive the corresponding—

\(^{1}\) D’Ocagne, Coordonnées parallèles et axiales, p. 4, 9.

relation between the coordinates of such a straight line, which will be the equation of the point \(P\).

Draw the lines \(AP\) and \(BP\), with extensions \(PB'\) and \(PA'\), and let

\[ AA'=\alpha,\quad BB'=\beta . \tag{39} \]

For any position of the straight line \(MN\) we shall have

\[ \frac{BN}{MA'}=\frac{NB'}{AM} \]

or

\[ \frac{v}{\alpha-u}=\frac{\beta-v}{u}. \]

Hence one obtains

\[ \frac{u}{\alpha}+\frac{v}{\beta}=1, \tag{40} \]

which will be the equation of the point \(P\) in parallel coordinates.

Conversely, every first-degree equation denotes a point in parallel coordinates. Suppose, for example, that there is an equation

\[ au+bv+c=0, \tag{41} \]

which is satisfied by the coordinates of a straight line. Reducing equation (41) to the form (40), where

\[ \alpha=-\frac{c}{a};\quad \beta=-\frac{c}{b}, \]

it is easy to see that the straight line satisfying equation (41) constantly passes through some point \(P\). If through this point one draws the line \(CP\) parallel to the coordinate axes until it intersects \(AB\), and denotes the length \(CP\) by \(w\), \(AB\) by \(s\), and the distances of the point \(C\) from the origins of coordinates by \(s_1\) and \(s_2\) (which must be considered positive or negative, depending on the direction), then the following relations are obtained:

\[ \frac{s_1}{-s_2}=\frac{AC}{CB}=\frac{AP}{PB'}=\frac{\alpha}{\beta}=\frac{b}{a}\,^{1}), \]

\(^{1}\) For the case when the point \(P\) lies between the coordinate axes. When it lies outside the space between the axes (Fig. 9a), \(s_1\) and \(s_2\) have the same signs, but the signs of \(\alpha\) and \(\beta\) in equation (40), and of \(a\) and \(b\) in equation (41), are different. Thus, for this latter case one obtains:

\[ \frac{s_1}{s_2}=\frac{AC}{CB}=-\frac{\alpha}{\beta}=-\frac{b}{a}, \tag{42} \]

i.e., the former relation.

$$ \frac{s_1}{s_2}=-\frac{\alpha}{\beta}=-\frac{b}{a}; \tag{42} $$

$$ \frac{w}{\alpha}=-\frac{s_2}{s}, \quad \frac{w}{\beta}=\frac{s_1}{s}; $$

$$ w\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)=\frac{s_1-s_2}{s}=1; $$

$$ w=\frac{\alpha\beta}{\alpha+\beta}=-\frac{c}{a+b}. \tag{42′} $$

Now let us turn to the question of transforming into graphical form an equation of the type

$$ f_1(x)+f_2(y)=f_3(z). \tag{33} $$

Take an arbitrary straight line \(AB\) and mark on it two points \(A\) and \(B\) (Fig. 10), located at a distance \(s\) from one another, from the origin of parallel coordinates. Through these points draw two parallel straight lines \(AU\) and \(BV\), which will be the coordinate axes.

Let us lay off the functions \(f_1(x)\) and \(f_2(y)\) along the coordinate axes \(AU\) and \(BV\) in the form of function scales, with moduli \(\lambda_1\) and \(\lambda_2\). Then, denoting the segments on the axes \(AU\) and \(BV\), respectively, by \(u\) and \(v\), we obtain

$$ u=\lambda_1 f_1(x), \\ v=\lambda_2 f_2(y). \tag{43} $$

We have the right to regard these segments as parallel coordinates of some variable straight line \(MN\), intersecting the coordinate axes at the points \(M\) and \(N\), located at distances \(u\) and \(v\) from the origins of coordinates.

Substituting the expressions for \(f_1(x)\) and \(f_2(y)\) following from equalities (43) into the fundamental equation (33), we obtain the new equation

$$ \frac{u}{\lambda_1}+\frac{v}{\lambda_2}=f_3(z) \tag{44} $$

between the parallel coordinates \(u\) and \(v\).

Let us assign to \(f_3(z)\) some constant value \(C\). Then equation (44) will take the form

$$ \frac{u}{\lambda_1}+\frac{v}{\lambda_2}=C. \tag{44′} $$

Ya. I. Nikolin.

Comparing this equation with (41), we see that equation (44) determines a certain point \(P\), lying on the line \(MN\), for arbitrary values of the coordinates \(u\) and \(v\). This means that, when the values of \(u\) and \(v\) change, the straight line \(MN\) moves, rotating about the point \(P\). The position of this point will be determined if, through it, we draw the straight line \(CP \parallel AU\) and \(BV\) up to its intersection with \(AB\) at the point \(C\), with the lengths of the segments \(AC\) and \(CP\). On the basis of (42), denoting, as before, \(AB\) by \(s\), and \(AC\) and \(CB\) by \(s_1\) and \(s_2\),

\[ \frac{s_1}{s_2} = -\frac{\frac{1}{\lambda_2}}{\frac{1}{\lambda_1}} = -\frac{\lambda_1}{\lambda_2}; \tag{45} \]

moreover,

\[ s_1 - s_2 = s; \tag{46} \]

whence

\[ \frac{s_1}{s_1 - s} = -\frac{\lambda_1}{\lambda_2} \]

and

\[ s_1 = s \frac{\lambda_1}{\lambda_1 + \lambda_2}. \tag{47} \]

On the other hand, denoting the length \(CP\) by \(w\), on the basis of the 2nd equality of (42) and (44),

\[ w = \frac{C}{\frac{1}{\lambda_1} + \frac{1}{\lambda_2}} = C \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2}. \tag{48} \]

The quantities \(s_1\) and \(w\) completely determine the position of the point \(P\) for the given value of \(f_3(z)\). With a change in the value of \(f_3(z)\), a new point \(P'\) is obtained (and, depending on the values of \(u\) and \(v\), a new series of straight lines invariably passing through this new point).

At the same time, considering formula (47), we see that the value of \(s_1\) does not depend on changes in \(f_3(z)\) and is, in general, constant for definite, arbitrarily chosen \(s\), \(\lambda_1\), and \(\lambda_2\). This means that the points \(P, P'\), etc., determined by equation (44) for different values of \(f_3(z)\), are necessarily situated on one line, parallel to the coordinate axes, whose position is determined by the choice of the quantities \(s\), \(\lambda_1\), and \(\lambda_2\).

This property of these points makes it possible to find them graphically for given values of \(u\) and \(v\) in the equation

or, equivalently, \(f_1(x)\) and \(f_2(y)\) in equation (33). For this, after the above-indicated construction of the coordinate axes and of the function scales, it is necessary to mark on the line of origins \(AB\) the point \(C\) at a distance from \(A\)

\[ s_1=s\frac{\lambda_1}{\lambda_1+\lambda_2}, \tag{47} \]

to draw through it the line \(CP\), parallel to the axes, and then to connect by the straight line \(MN\) the points of the function scales corresponding to \(u\) and \(v\). The point of intersection of the straight lines \(MN\) and \(CP\) will be the desired one.

Considering, on the other hand, formula (48), we observe that the distance of the points \(P, P'\), etc., from the point \(C\), which satisfies equality (47), for given values of \(\lambda_1\) and \(\lambda_2\), depends entirely on the change in the value of \(f_1(z)\), and is directly proportional to this value and is obtained from it by multiplication by

\[ \frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}. \]

This means that, if we were to represent \(f_3(z)\) in the form of a function scale laid off from the point \(C\) along the line \(CP\), with modulus equal to

\[ \frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}, \]

then the points inscribed with the values \(f_3(z)\) must coincide with the points \(P, P'\), etc., which are determined by equation (44) for these values of \(f_3(z)\).

This connection between the points satisfying equation (44) and the corresponding values of \(f_3(z)\) makes it possible to extend the graphical method of determination directly to the values \(f_3(z)\). In fact, let us continue the line \(CP\) in the form of a third function scale, denote it by \(CW\), and lay off on it from the point \(C\) the lengths \(w\), determined by the equality

\[ w=\lambda_3 f_3(z)=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}\,f_3(z), \tag{50} \]

where

\[ \lambda_3=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2} \tag{51} \]

is the modulus of the scale of \(f_3(z)\).

Having now three function scales \(AU\), \(BV\), and \(CW\), we can, for given values of \(f_1(x)\) and \(f_2(y)\), find by the same graphical method simultaneously the position of the point satisfying equation (44), and the value \(f_3(z)\) satisfying equation (33). For this, through the points on the scales \(AU\) and \(BV\) corresponding to the values \(f_1(x)\) and \(f_2(y)\), we draw a straight line, and the latter in

the intersection with the scale \(CW\) gives the desired point and indicates the corresponding value \(f_3(z)\).

Thus, the three scales \(AU\), \(BV\), and \(CW\) graphically represent the relation between the quantities \(f_1(x)\), \(f_2(y)\), \(f_3(z)\), which enter into an equation of the form (33), i.e., they are a graphical representation of equations of this kind. As is clear from the preceding, to each combination of values of the variables in equation (33) there corresponds, on the diagram, a system of three points with numerical marks, situated on one straight line cutting the scales of the functions. This characterizes the method of representing equations as a method of points of rectilinear intersection.

The method of constructing the three scales of functions, which we call conjugate scales, is clear from the foregoing. It is only necessary to note that the relation between the modulus of the scale \(f_3(z)\) and the moduli of the functions \(f_1(x)\) and \(f_2(y)\) can be put into a more expressive form. Indeed, the formula

\[ \lambda_3 = \frac{\lambda_1 \lambda_2}{\lambda_1 + \lambda_2}, \tag{51} \]

by transformation, assumes the following form:

\[ \frac{1}{\lambda_1} + \frac{1}{\lambda_2} = \frac{1}{\lambda_3}. \tag{51'} \]

Thus, in brief, for constructing the diagram of an equation of the form (33), having chosen the moduli of the scales \(\lambda_1\) and \(\lambda_2\) and having determined the modulus \(\lambda_3\) by formula (51'), it is sufficient to take on the line \(AB\) a point \(C\) satisfying equality (47), to draw through the points \(A\), \(B\), and \(C\) three parallel axes \(AU\), \(BV\), \(CW\), and to lay off on these axes, beginning from the points \(A\), \(B\), and \(C\), the corresponding scales of the functions, determined by formulas (43) and (50).

With regard to the use of diagrams of conjugate scales of this kind, it must be added to the preceding that the relation established between the three functions entering into equation (33), represented by the diagram, makes it possible to determine the value of any one of these functions if the other two are known, by the same method that we derived for \(f_3(z)\). In general, any straight line that intersects the three conjugate scales of the diagram, at the points of intersection with them, gives values of the functions laid off on the scales, which simultaneously satisfy the equation represented by the diagram.

The principle of diagrams constructed by the method of points of rectilinear intersection can also be proved in other ways, without resorting to systems of parallel coordinates. For example, with respect to equations with three variables, which have been discussed up to now, one may reason as follows.

Let us have, as before, an equation of the form

\[ f_1(x)+f_2(y)=f_3(z). \tag{33} \]

Take on an arbitrary straight line (Fig. 11) points \(A\) and \(B\), and draw through them two parallel lines \(AX\) and \(BY\). On the latter lay off \(f_1(x)\) and \(f_2(y)\) in the form of functional scales, with moduli \(\lambda_1\) and \(\lambda_2\), on the basis of the equalities:

\[ \begin{aligned} \lambda_1 f_1(x)&=x',\\ \lambda_2 f_2(y)&=y', \end{aligned} \tag{52} \]

where \(x'\) and \(y'\) are the lengths of the corresponding segments on the lines \(AX\) and \(BY\).

Substituting in equation (33) \(f_1(x)\) and \(f_2(y)\) according to (52), we obtain the equation

\[ \frac{x}{\lambda_1}+\frac{y}{\lambda_2}=f_3(z). \tag{53} \]

Give \(f_3(z)\) the constant value \(f_3(z_0)\).

Then equation (53) becomes

\[ \frac{x}{\lambda_1}+\frac{y}{\lambda_2}=f_3(z_0). \tag{53'} \]

Let this last equation be satisfied by certain values \(x_0\) and \(y_0\), i.e.

\[ \frac{x_0}{\lambda_1}+\frac{y_0}{\lambda_2}=f_3(z_0). \tag{53''} \]

The quantities \(x_0\) and \(y_0\) in Fig. 11 determine a certain straight line \(X_0Y_0\). This straight line, depending on changes in the values of \(x\) and \(y\), changes its position, rotating about a fixed point whose position is easily determined.

Indeed, suppose that equation (53′) is satisfied by another pair of values \(x_1\) and \(y_1\), i.e.

\[ \frac{x_1}{\lambda_1}+\frac{y_1}{\lambda_2}=f_3(z_0), \tag{53'''} \]

whereby the other straight line \(X_1Y_1\) is determined. Let us call the point of intersection of the straight lines \(X_0Y_0\) and \(X_1Y_1\) \(P\), and determine its position.

From equations \((53'')\) and \((53''')\) it follows that

\[ \frac{x_0}{\lambda_1}+\frac{y_0}{\lambda_2} = \frac{x_1}{\lambda_1}+\frac{y_1}{\lambda_2}, \tag{54} \]

\[ \frac{1}{\lambda_1}(x_0-x_1)=\frac{1}{\lambda_2}(y_1-y_0), \]

\[ \frac{x_0-x_1}{y_1-y_0}=\frac{\lambda_1}{\lambda_2} \tag{55} \]

or, according to the drawing,

\[ \frac{X_0X_1}{Y_0Y_1}=\frac{\lambda_1}{\lambda_2}. \tag{55'} \]

Draw through the point \(P\) the straight line \(CZ\), parallel to \(AX\) and \(BY\), up to its intersection with \(AB\) at the point \(C\), and the straight line \(QR\), parallel to \(AB\).

The triangles \(X_0PX_1\) and \(Y_0PY_1\) are similar to one another. Therefore

\[ \frac{PQ}{PR}=\frac{X_0X_1}{Y_0Y_1}=\frac{\lambda_1}{\lambda_2}, \tag{56} \]

or, what is the same,

\[ \frac{AC}{CB}=\frac{\lambda_1}{\lambda_2}. \tag{56'} \]

On the other hand, the similarity of the triangles \(X_0PQ\) and \(Y_0PR\) gives:

\[ \frac{QX_0}{RY_0}=\frac{PQ}{PR}=\frac{\lambda_1}{\lambda_2}. \tag{57} \]

Taking into account that

\[ \begin{aligned} QX_0&=x_0-CP,\\ RY_0&=CP-y_0, \end{aligned} \tag{58} \]

we have

\[ \frac{x_0-CP}{CP-y_0}=\frac{\lambda_1}{\lambda_2}, \tag{57} \]

whence

\[ CP=\frac{\lambda_2x_0+\lambda_1y_0}{\lambda_1+\lambda_2}. \tag{59} \]

But equation (53′) may be represented in the form

\[ \lambda_2 x_0+\lambda_1 y_0=\lambda_1\lambda_2 f_3(z_0). \tag{53^{\mathrm{IV}}} \]

Whence the final value for \(CP\) is obtained in the form

\[ CP=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}\, f_3(z_0). \tag{60} \]

Relations (56′) and (60), for a known length of the line \(AB\), completely determine the point \(P\). Comparing them with (45) and (50), we see that our method of proof leads to the same results as the use of parallel coordinates.

Relation (56′) shows that the point of intersection of the straight lines \(X_0Y_0\) and \(X_1Y_1\) lies on the fixed axis \(CZ\). Relation (60), on the other hand, proves that the distance \(CP\) of the point \(P\) from \(AB\), for a constant value of \(f_3(z_0)\), is constant, and consequently the point \(P\) also remains fixed, as was required to be proved. The position of the point \(P\) is determined if we draw the straight line \(CZ \parallel AX\) and \(BY\), and at such a distance between them that the relation

\[ \frac{AC}{CB}=\frac{\lambda_1}{\lambda_2}, \tag{56′} \]

is satisfied, and on this straight line lay off from \(AB\) the length

\[ CP=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}\, f_3(z_0) \tag{60} \]

On this basis, if on \(AX\), \(BY\), \(CZ\) one respectively lays out \(f_1(x)\), \(f_2(y)\), \(f_3(z)\), entering into equation (33), in the form of functional scales, with moduli \(\lambda_1\), \(\lambda_2\) and \(\dfrac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}\), then the divisions corresponding to values of the variables satisfying the equation simultaneously will lie on one straight line intersecting all three scales. From this it follows, obviously, that if the values of two of the mentioned variables are given, it will be sufficient to draw the straight line connecting the corresponding points of two scales in order to read, at its intersection with the third scale, the value of the third variable.

The method of graphical construction by means of conjugate scales may be applied to equations of a form analogous to (33), not only with three variables, but with any number of them. We shall prove this in the most general form as follows.

Suppose that we have an equation with \(n\) variables, of the form

\[ f_1(x)+f_2(y)+f_3(z)+\cdots+f_{n-2}(u)+f_{n-1}(v)=f_n(w), \tag{61} \]

which we wish to solve with respect to the variable \(w\).

Take the line \(AN\) (Fig. 12) and mark on it \(n-1\) points \(A, B, C, \ldots M, N\), at arbitrary distances from one another.

Through these points draw a series of parallel straight lines, for convenience perpendicular to the line \(AN\), namely \(AX, BI, CZ, \ldots MU, NV\), and on these straight lines, starting from \(AN\), lay off the quantities \(f_1(x), f_2(y), \ldots f_{n-1}(v)\) in the form of functional scales, with moduli \(\lambda_1, \lambda_2, \ldots \lambda_{n-1}\), on the basis of the equalities

\[ \lambda_1 f_1(x)=x', \]

\[ \lambda_2 f_2(y)=y', \]

\[ \lambda_3 f_3(z)=z', \tag{62} \]

\[ \ldots \]

\[ \lambda_{n-1} f_{n-1}(v)=v'. \]

Expressing \(f_1(x), f_2(y), \ldots f_{n-1}(v)\) by means of \(\lambda_1, \lambda_2, \ldots\) and \(x', y', \ldots\), and substituting into equation (61), we obtain a new equation:

\[ \frac{x'}{\lambda_1}+\frac{y'}{\lambda_2}+\frac{z'}{\lambda_3}+\cdots+\frac{u'}{\lambda_{n-2}}+\frac{v'}{\lambda_{n-1}}=f_n(w). \tag{63} \]

If we represent this latter equation in the form

\[ \frac{1}{\lambda_1}x' + \frac{1}{\lambda_2}y' + \frac{1}{\lambda_3}z' + \cdots + \frac{1}{\lambda_{n-2}}u' + \frac{1}{\lambda_{n-1}}v' = f_n(w), \tag{63'} \]

then we shall see that its first part may be considered as

the sum of the moments of parallel forces equal to \(\frac{1}{\lambda_1}, \frac{1}{\lambda_2}, \ldots, \frac{1}{\lambda_{n-1}}\), about the axis represented by the line \(AN\), the quantities \(x', y', z', \ldots, v'\) serving as the arms of the moments.

By virtue of this, the quantity \(f_n(w)\) may be represented in the form of the moment of the resultant of these forces, and the procedures appropriate to the relations between the moment of the resultant and the moments of the components may be applied to the quantities entering into equation \((63')\).

Let \(f_n(w)\) have some particular value \(f_n(w_0)\), and let equation \((63')\) be satisfied, for this value, by the values \(x'_0, y'_0, z'_0, \ldots, v'_0\), which are laid off along the lines \(AX, BU, \ldots, NV\) as lengths \(AX_0, BY_0, \ldots, NV_0\). We refer all the elements constructed by us to the rectangular coordinate system \(\Xi \Omega YZ\), in which the straight line \(AN\) is taken as the axis of abscissas, and the axis of ordinates is parallel to the constructed scales of the functions; and let the distances from the origin of coordinates to the points \(A, B, \ldots, N\) be \(s_1, s_2, \ldots, s_{n-1}\). Thus, in the plane \(\Xi \Omega Y\), we have \(n-1\) points of application of the parallel forces

\[ \frac{1}{\lambda_1},\ \frac{1}{\lambda_2},\ \frac{1}{\lambda_3},\ \ldots,\ \frac{1}{\lambda_{n-1}}, \]

directed perpendicular to this plane, namely \(X_0, Y_0, Z_0, \ldots, U_0, V_0\), whose abscissas will be \(s_1, s_2, s_3, \ldots, s_{n-2}, s_{n-1}\), and whose ordinates will be the lengths \(x'_0, y'_0, z'_0, \ldots, u'_0, v'_0\). Taking the moments of the given forces and of their resultant with respect to the coordinate axes \(\Omega \Xi\) and \(\Omega Y\), it is not difficult to see that the coordinates of the point of application of the resultant are determined analytically by the formulas:

\[ s_n = \frac{ \frac{1}{\lambda_1}s_1 + \frac{1}{\lambda_2}s_2 + \frac{1}{\lambda_3}s_3 + \cdots + \frac{1}{\lambda_{n-2}}s_{n-2} + \frac{1}{\lambda_{n-1}}s_{n-1} }{ \frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3} + \cdots + \frac{1}{\lambda_{n-2}} + \frac{1}{\lambda_{n-1}} } = \frac{ \displaystyle \sum_{m=1}^{m=n-1} \frac{s_m}{\lambda_m} }{ \displaystyle \sum_{m=1}^{m=n-1} \frac{1}{\lambda_m} } \tag{64} \]

\[ w'_0 = \frac{ \frac{1}{\lambda_1}x'_0 + \frac{1}{\lambda_2}y'_0 + \frac{1}{\lambda_3}z'_0 + \cdots + \frac{1}{\lambda_{n-2}}u'_0 + \frac{1}{\lambda_{n-1}}v'_0 }{ \frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3} + \cdots + \frac{1}{\lambda_{n-2}} + \frac{1}{\lambda_{n-1}} } = \frac{ f_n(w_0) }{ \displaystyle \sum_{m=1}^{m=n-1} \frac{1}{\lambda_m} } \tag{65} \]

The expressions we have obtained are extremely interesting. Considering them, we see, first of all, that the magnitude of the abscissa of the point of application of the resultant \(s_n\) does not depend on any of the variables entering into equation (63′), but is dependent only on the quantities \(\lambda_1, \lambda_2, \ldots \lambda_{n-1}\) and \(s_1, s_2, \ldots s_{n-1}\). Since the quantities \(\lambda_m\) and \(s_m\), by assumption, are chosen arbitrarily and for the given diagram are constants, the quantity \(s_n\) may also be regarded as constant. Thus the point of application of the resultant of which we are speaking, for given and arbitrarily chosen \(\lambda_m\) and \(s_m\), is always situated on one and the same ordinate, whose abscissa \(s_n\) is determined by formula (64).

On the other hand, formula (65) shows that the ordinate of the point of application of our resultant, for given \(\lambda_m\), depends exclusively on the change in the value of \(f_n(w)\) and is proportional to this value. Hence it follows, conversely, that if this point of application can be found in some way, it is sufficient to measure its ordinate in order to obtain the desired value \(f_n(w_0)\). To find the point of application of the resultant of parallel forces acting at specified points, it is easy to use known graphical methods. Applying them, we can, thanks to the property of expression (65), obtain a graphical method for finding \(f_n(w_0)\).

It is known that the geometrical determination of the point of application of the resultant of any number of parallel forces reduces to determining the point of application of the resultant of two parallel forces. The resultant of any two forces, when added to a third force, gives the resultant of the system of these three forces and its point of application; this latter, when added to a fourth force, gives the resultant of four forces, and so on in successive order, up to the common resultant of all the given forces. In doing this, of course, one may proceed in any order. It is also known that the point of application of the resultant of two parallel forces lies on the line joining the points of application of the components, and divides this line into parts inversely proportional to the magnitudes of the components.

Thus, for the present case, the finding of the point of application of the resultant \(P\), lying on the ordinate \(SW\), can be carried out in the following order. Let us begin with finding the point of application of the resultant of the forces \(\dfrac{1}{\lambda_1}\) and \(\left(\dfrac{1}{\lambda_2}\right)\), applied at \(X_0\) and \(Y_0\),

which we shall call \(P_{1,2}\). To find the abscissa of this point, according to (64), we must lay off from \(O\) the length

\[ s_{1,2}= \frac{\dfrac{1}{\lambda_1}s_1+\dfrac{1}{\lambda_2}s_2} {\dfrac{1}{\lambda_1}+\dfrac{1}{\lambda_2}} = \frac{s_1\lambda_2+s_2\lambda_1}{\lambda_1+\lambda_2}. \tag{64′} \]

Computing, on the basis of this expression, the lengths of the segments from the point \(S_{1,2}\) to \(A\) and \(B\), and then their ratio, we obtain:

\[ AS_{1,2}=s_{1,2}-s_1=\frac{\lambda_1(s_2-s_1)}{\lambda_1+\lambda_2}, \]

\[ S_{1,2}B=s_2-s_{1,2}=\frac{\lambda_2(s_2-s_1)}{\lambda_1+\lambda_2}, \tag{66} \]

\[ \frac{AS_{1,2}}{S_{1,2}B} = \frac{\lambda_1}{\lambda_2} = \frac{\dfrac{1}{\lambda_2}}{\dfrac{1}{\lambda_1}}, \]

i.e. the end of the abscissa of the required point of application of the resultant of the forces \(\dfrac{1}{\lambda_1}\) and \(\dfrac{1}{\lambda_2}\) divides the distance between the points \(A\) and \(B\) into parts inversely proportional to the quantities \(\dfrac{1}{\lambda_1}\) and \(\dfrac{1}{\lambda_2}\), as we should have expected on the basis of the preceding. Thus the position of the ordinate \(S_{1,2}W_{1,2}\), on which the point \(P_{1,2}\) lies, is determined very simply by dividing the segment \(AB\) between the lines of ordinates of the points of application of the components \(\dfrac{1}{\lambda_1}\) and \(\dfrac{1}{\lambda_2}\) inversely proportional to the latter, or, in the present case, proportionally to the moduli of the corresponding scales.

To find the ordinate of the point \(P_{1,2}\), it is sufficient to connect by a straight line the points of application of the components \(X_0\) and \(Y_0\). The intersection of the line \(X_0Y_0\) with the direction of the ordinate will determine for us the position of the point \(P_{1,2}\) and the length of the ordinate \(S_{1,2}P_{1,2}\).

Taking then only the point \(P_{1,2}\) thus found and the point of application of the next component \(Z_0\), in the same way one can find the point of application \(P_{1,2,3}\) of the next partial resultant, and so

further, in consecutive order. By this graphical process we shall ultimately find the point of application of the resultant \(P\) of all the forces; its abscissa and ordinate satisfy formulas (64) and (65).

Formula (65) shows that between the ordinate of the point \(P\) and the quantity \(f_n(w_0)\) there is an unbroken connection. Namely, the ordinate \(w'_0\) is directly proportional to the quantity \(f_n(w_0)\) and is obtained from the latter by multiplying by

\[ \frac{1}{\displaystyle \sum_{m=1}^{m=n-1}\frac{1}{\lambda_m}}. \]

This means that, if we lay off \(f_n(w_0)\) on the ordinate of the point \(P\) from its origin in the form of the scale of functions, with modulus

\[ \frac{1}{\displaystyle \sum_{m=1}^{m=n-1}\frac{1}{\lambda_m}}, \]

then the position of the point \(P\) on the ordinate will be determined by the mark on the scale of functions corresponding to the value \(f_n(w_0)\).

Thus the entire process leading to the determination of \(f_n(w_0)\) satisfying equations (63′) and (61) reduces to a series of graphical constructions on a diagram representing a series of parallel axes, namely the scales of the functions \(AX, BV, CZ \ldots MU, NV\) and \(SW\), and the auxiliary axes \(S_{1,2}W_{1,2},\ S_{1,2,3}W_{1,2,3}\ldots\). These elements completely determine the relation between the variables of equation (61), and therefore together constitute a graphical representation of this equation.

Finally, let us note that the modulus of the scale of the sought function \(f_n(w)\) satisfies the following relations:

\[ \lambda_n = \frac{1}{\displaystyle \sum_{m=1}^{m=n}\frac{1}{\lambda_m}} = \frac{1}{\displaystyle \frac{1}{\lambda_1}+\frac{1}{\lambda_2}+\cdots+\frac{1}{\lambda_{n-1}}}. \tag{67} \]

or

\[ \frac{1}{\lambda_n} = \frac{1}{\lambda_1} + \frac{1}{\lambda_2} + \frac{1}{\lambda_3} +\cdots+ \frac{1}{\lambda_{n-2}} + \frac{1}{\lambda_{n-1}}. \tag{67′} \]

The latter relation fully corresponds to expression (51′) for the case of an equation with three variables, derived earlier by means of parallel coordinates.

Thus, in order to construct a diagram of an equation with \(n\) variables \(x, y, \ldots v, w\) of the form (61), in the form of determining the variable \(w\), it is necessary, first of all, from arbitrary points lying on one straight line, to construct \(n - 1\) parallel scales corresponding to the variables \(x, y, \ldots v\), with arbitrarily chosen moduli \(\lambda_1, \lambda_2, \ldots \lambda_{n-1}\); then, by successively dividing the distances between the scales proportionally to the moduli (principal and auxiliary), to draw the auxiliary axes \(S_{1,2}W_{1,2}\), \(S_{1,2,3}W_{1,2,3}\), … and the scale axis of the unknown variable \(SW\), and, finally, to graduate the scale of this last variable according to equality (67′). It is clear that the construction of the auxiliary axes may be carried out in any order of the variables.

To find the value of the variable \(w_0\) from given values of the other variables \(x_0, y_0, z_0, \ldots v_0\), one must, having drawn a straight line through the points \(X_0\) and \(Y_0\), find its intersection with the auxiliary axis \(S_{1,2}W_{1,2}\); then through this point of intersection and \(Z_0\) draw a straight line and find its intersection with the axis \(S_{1,2,3}W_{1,2,3}\), and so on, until at the intersection of the last secant with the scale \(SW\) we obtain the desired value \(w_0\).

To find another variable satisfying the same equation, it is, of course, possible to use the same set of principal scales, but with a different combination of auxiliary axes.

Let us take, for example, an equation with four variables of the form

\[ f_1(x) + f_2(y) + f_3(z) = f_4(u), \tag{61} \]

which we wish to solve with respect to \(f_4(u)\). Choose on the line \(AC\) (Fig. 13) three arbitrary points \(A, B\), and \(C\), and draw through them three parallel lines \(AX, BY, CZ\). On these last lay off \(f_1(x)\), \(f_2(y)\), and \(f_3(z)\) in the form of function scales, with arbitrary moduli \(\lambda_1, \lambda_2, \lambda_3\). Take certain particular values of the variables \(x_0\) and \(y_0\). For these values the points \(X_0, Y_0\) are obtained on the scales. Divide the distance \(AB\) into parts proportional to the ratio \(\lambda_1 : \lambda_2\) (inversely proportional to \(\frac{1}{\lambda_1} : \frac{1}{\lambda_2}\)), and through the point \(S_{1,2}\) thus obtained draw the line \(S_{1,2}U_{1,2} \parallel AX\) and \(BY\). Join the points \(X_0\) and \(Y_0\) by a straight line. The intersection \(X_0Y_0\) with \(S_{1,2}U_{1,2}\) will give us

point \(P_{1,2}\). From the preceding, the significance of this point is clear. If we had the equation

\[ f_1(x_0)+f_1(y_0)=f_{1,2}(u_{1,2}), \tag{69} \]

then, laying off along the line \(S_{1,2}U_{1,2}\) \(f_{1,2}(u_{1,2})\) in the form of a functional scale, with modulus \(\lambda_{1,2}\) determined by the relation

\[ \frac{1}{\lambda_{1,2}}=\frac{1}{\lambda_1}+\frac{1}{\lambda_2}, \tag{67'} \]

the point \(P_{1,2}\) would give us, by the numerical mark of the scale, the value \(f_{1,2}(u_{1,2})\) for the given \(x_0, y_0\).

Now divide the distance \(S_{1,2}C\) into parts proportional to the ratio \(\lambda_{1,2}:\lambda_3\), and through the point \(S\) thus obtained draw a straight line \(SU\) parallel to the other scales. Connect the point \(P_{1,2}\) with \(Z_0\). The straight line \(P_{1,2}Z_0\), in its intersection with \(SU\), will give the point \(P_0\). If on the line \(SU\) we lay off \(f_4(u)\) in the form of a functional scale, with modulus \(\lambda_4\) satisfying the expression

\[ \frac{1}{\lambda_4}=\frac{1}{\lambda_1}+\frac{1}{\lambda_2}+\frac{1}{\lambda_3}, \tag{67'''} \]

then the point \(P_0\) will give us, by the numerical mark of the scale, the value \(f_4(u)\) for the given values \(x_0, y_0\), and \(z_0\). The four conjugate functional scales \(AX\), \(BY\), \(CZ\), and \(SU\), constructed in this way, make it possible, in exactly the same manner, to determine the value \(f_4(u)\) for any prescribed values \(f_1(x), f_2(y), f_3(z)\).

The same construction could be carried out in another order of variables, beginning, for example, with the variables \(x\) and \(z\). Then one would first obtain (Fig. 13) the lines \(S_{1,3}U_{1,3}\) and \(X_0Z_0\) and the point \(P_{1,3}\), and then, in connection with the point \(Y_0\), finally the same line \(SU\) and the point \(P_0\).

If in equation (68) the quantities \(f_4(u_0)\) and two arbitrary variables, for example \(f_1(x_0)\) and \(f_3(z_0)\), were known, then, having the four conjugate scales constructed, it is easy to find the value \(f_2(y)\). For this it is enough to construct the straight lines \(S_{1,3}U_{1,3}\) and \(X_0Z_0\), and to find the point \(P_{1,3}\). Joining the latter with the point \(P_0\) and extending to intersection with the scale \(BY\), we find the point \(Y_0\), determining the value \(f_2(y_0)\).

In constructing the four principal scales \(AX, BV, CZ\) and \(SU\), it is necessary to resort to the construction of auxiliary lines \(S_{1,2}U_{1,2}\), \(S_{2,3}U_{2,3}\), \(S_{1,3}U_{1,3}\) (Fig. 13). Separate points of these lines, \(P_{1,2}\), \(P_{2,3}\), \(P_{1,3}\), as we have seen in application to one case, represent, like the points \(X_0, Y_0, Z_0, P\), the values of certain functions \(f_{1,2}(u_{1,2})\), \(f_{2,3}(u_{2,3})\), \(f_{1,3}(u_{1,3})\), which satisfy the equations

\[ f_1(x) + f_2(y) = f_{1,2}(u_{1,2}), \]

\[ f_2(y) + f_3(z) = f_{2,3}(u_{2,3}), \tag{69} \]

\[ f_1(x) + f_3(z) = f_{1,3}(u_{1,3}). \]

When these latter functions themselves have a real meaning, they can be determined according to the general rule, if we place them on the lines \(S_{1,2}U_{1,2}\), \(S_{2,3}U_{2,3}\), \(S_{1,2}U_{1,3}\), in the form of function scales, with correspondingly determined scale moduli. Under this condition the diagram of conjugate scales, constructed analogously to Fig. 13, would turn into an organic combination of seven conjugate scales, which make possible the graphical solution of the four equations

\[ f_1(x) + f_2(y) + f_3(z) = f_4(w), \tag{68} \]

\[ f_1(x) + f_2(y) = f_{1,2}(u_{1,2}), \]

\[ f_2(y) + f_3(z) = f_{2,3}(u_{2,3}), \tag{69'} \]

\[ f_1(x) + f_3(z) = f_{2,3}(u_{2,3}), \]

and the graphical determination of seven variables, if in each case of solution the values of three principal variables are given.

In the same way one could be convinced that it is possible to construct a diagram of conjugate scales containing one scale for each variable, which will give the values of the function \(f_n(w)\) satisfying the equation

\[ f_1(x) + f_2(y) + f_3(z) + \ldots + f_{n-2}(u) + f_{n-1}(w) = f_n(w), \tag{67} \]

as well as the values of functions satisfying equations formed by analogy with equations (69′). It is also clear that such a diagram could give the value of any of the functions entering into

equation (61), as a function of \(n-2\) other functions of the first part of the equation and \(f_n(w)\).

In contrast to the organic combinations of conjugate scales indicated above, which proceed from one basic equation of the form (61) and from the equations (69′) formed from it, other combinations of such scales are possible and are encountered in practice; these represent a purely mechanical joining of groups of conjugate scales proceeding from several equations of a simple three-member type.

Suppose, for example, that we have several equations intended for the solution of analogous or successive problems, of the following form:

\[ f_1(x)+f_2(y)=f_3(z), \tag{70} \]

\[ f_1(x)+f_4(u)=f_5(v), \tag{70′} \]

\[ f_2(y)+f_4(u)=f_6(w). \tag{70″} \]

Let us represent equation (70), according to the usual rules, in the form of a diagram of three conjugate scales. Nothing prevents us, making use of the presence of the scale \(f_1(x)\), from constructing on the same diagram equation (70′), leaving unchanged the position and modulus of this scale and adding, in the corresponding manner, the scales \(f_4(u)\) and \(f_5(v)\). Then, in the same way, the scales \(f_2(y)\) and \(f_4(u)\) may enter into the diagram of equation (70″), and so on. The difference between such a combination of scales and the combination discussed earlier is self-evident. Here we are dealing with a purely mechanical joining on one drawing of scales whose groups represent independent diagrams of separate equations. The connection between them is reduced only to the common scales of functions entering into several equations. These diagrams are constructed and function quite independently of one another and may at any time be broken up into several independent diagrams.

An equation of the form

\[ f_1(x)\cdot f_2(y)\cdot f_3(z)\cdot \ldots \cdot f_{n-2}(u)\cdot f_{n-1}(v)=f_n(w), \tag{71} \]

whose first part represents a product of functions, is reduced to the above-considered (61) by taking logarithms of both sides of the equation. Then we have, in fact:

\[ \log f_1(x)+\log f_2(y)+\log f_3(z)+\ldots+\log f_{n-1}(v)=\log f_n(w). \tag{71′} \]

After this, in order to find graphically the value \(f_n(w)\), or likewise one of the other functions, as a function of the remaining \(n - 1\), it is sufficient to construct a diagram similar to the one described, plotting along the ordinates, as the scales of the functions, not the functions themselves, but their logarithms, which is easily done by using a logarithmic scale.

In conclusion, a few words must be said about the practical methods of constructing diagrams of conjugate scales. It is clear, first of all, that the freedom in choosing the moduli \(\lambda_1\) and \(\lambda_2\) makes the given method very convenient in application. Owing to this circumstance, it is always possible to take both principal scales of equal length. Let us first examine the method of constructing three conjugate scales for the equation

\[ f_1(x) + f_2(y) = f_3(z) \tag{33} \]

in the ideal case, in order subsequently to indicate the deviations required by practice.

First of all, one may arbitrarily specify the length of the scales. The moduli \(\lambda_1\) and \(\lambda_2\) are obtained according to the usual rules for constructing scales¹, on the basis of the formula

\[ \lambda \,[f(x_n) - f(x_1)] = L, \tag{72} \]

where \(\lambda\) is the required value of the modulus,

\(f(x_1)\) and \(f(x_n)\) are the extreme values of the function,

\(L\) is the adopted length of the scale,

and the modulus \(\lambda_3\) according to formula \((51')\).

After this we lay off, on two parallel axes, the scale \(f_1(x)\), beginning from point \(A\) to \(X\) (Fig. 14), and the scale \(f_2(y)\) from \(B\) to \(Y\). Since the inclination of the line of origins to the axes in parallel coordinates may be arbitrary, we choose the points \(A\) and \(B\) on a line perpendicular to the direction \(AX\) and \(BY\). The distance between the scales \(AX\) and \(BY\) is arbitrary. Therefore we may choose it in such a way that the smallest angle formed, when reading with a ruler or with the index line of a transparent sheet, with the direction of the scales is not too small and has the value desired by us (for example, \(\pi/4\)).

Now draw the line \(CZ\), parallel to the other two scales, in such a way that

\[ \frac{AC}{BC}=\frac{\lambda_1}{\lambda_2}, \tag{45} \]

and choose a pair of values of the variables \(x\) and \(y\), for which the values of the variable \(z\) would be obtained from the basic equation in the most convenient manner. Having drawn on the diagram the corresponding lines and found the corresponding values of \(z\), it is not difficult to construct the scale of the function \(f_3(z)\), within the limits of the straight lines \(AB\) and \(XY\).

Such is the theoretical type of the diagram of conjugate scales. In practice one may depart from it to a greater or lesser degree.

It is clear, for example, that if the computed values of the moduli \(\lambda_1\) and \(\lambda_2\) are not expressed by simple numbers, they must be rounded, which introduces inequality in the lengths of the scales. On the other hand, the format of the diagram may make it necessary to reduce the calculated limiting angle of inclination of the transparent index to the scale lines. It happens that in practice the limiting values of one or another variable refer only to part of the scale of another variable. Then the scale \(Z\) may be reduced accordingly.

III. Application of the method of conjugate scales to Lampe’s formula.

Lampe’s formula was the result of well-known experiments carried out by its author, Professor Dr. Lampe, in 1869–1871, on a water conduit supplying water for the water supply of the city of Danzig from Lake Prangenau. Its type was worked out on the basis of a consideration of the experiments of Poiseuille, Jacobson, and Hagen on the motion of water in pipes of small diameter, and of a critical discussion of the conclusions and formulas obtained by Hagen from Darcy’s experiments and from his own,\(^{1}\)

Hagen, in conducting the investigation, on the basis of the results of observations known at that time,\(^{2}\) of such relations of the form

\[ i = m v^z, \tag{73} \]

\[ i = m v + p v^2, \tag{74} \]

\[ i = m_1 v^2 \tag{75} \]

\(^{1}\) Dr. Lampe. Untersuchungen über die Bewegung des Wassers in Röhren. Civiling. 1873.

\(^{2}\) Hagen, G. Ueber die Bewegung des Wassers in cylindrischen, nahe horizontalen Leitungen. P. 1870.

best suited for expressing \(i\), at first stopped at a logarithmic formula of the type

\[ \frac{i}{v} = m v^z, \tag{73′} \]

where

\[ m = \frac{n}{D^{1.25}}, \]

\[ n = const., \]

\(z\) is a decimal fraction less than 1.

However, he then rejected a formula of this type and found the more suitable form (74), the result of which was his binomial formula

\[ i = \frac{n}{D^2} v + \frac{n_1}{D} v^2. \tag{76} \]

In this formula, initially both coefficients \(n\) and \(n_1\) were constant, and later it was assumed that \(n\) is a function (three-term) of the temperature \(\theta\), while \(n_1 = const.\)

Lampe, having applied formula (76), in its final version, to the experiments of Poiseuille and Jacobson and to his own experiments on the Danzig water-supply system, found that the coefficients \(n\) and \(n_1\), which (at one and the same temperature) should have been constant, turn out to be different for each case. This forced Lampe, in order to express the results of his experiments, to occupy himself with developing a new, more perfect formula.

At first Lampe made an attempt, while retaining the general form of Hagen’s formula (76), to limit himself to changing the coefficients \(n\) and \(n_1\), whereby he excluded dependence on temperature and took both coefficients as constant. Thus the first Lampe formula, of non-logarithmic form, was obtained,

\[ \frac{i}{v} = m + pv, \tag{77} \]

where \(m\) and \(p\) are functions of the diameter.

However, this formula did not satisfy Lampe, and he, making use of Hagen’s precedent, attempted to express the results of his experiments analytically by means of a formula of logarithmic form. Thus the second Lampe formula, logarithmic, with a constant coefficient, was obtained, of the form

\[ \frac{i}{v} = m v^z, \tag{78} \]

in which \(m = const. = 0.0027716\),

\[ z = 0.80186. \]

But it then turned out that formula (78) with a constant coefficient \(m\) does not in any way cover the experimental data, and Lampe made \(m\) a function of the diameter raised to a fractional power, the exponent \(1.25\) being borrowed from the old Hagen formula.

Thus the final, third formula of Lampe was obtained. It has the form

\[ \frac{i}{v} = m_1 v^z , \tag{79} \]

where

\[ m_1 = \frac{n}{D^{1.25}}, \]

\[ n = \text{const.}, \]

\[ z = 0.802. \]

For new pipes Lampe takes\(^1\) the coefficient \(n\) as equal to \(0.0007555\).

Thus in this case the formula assumes the form

\[ i = n \frac{v^{1.802}}{D^{1.25}} = 0.0007555 \frac{v^{1.802}}{D^{1.25}} . \tag{80} \]

It turned out that this formula, while covering the results of the author’s experiments on the Danzig water-supply system, agrees fairly well with Darcy’s experiments, although it is also less applicable to pipes of small diameter.

Such agreement with the phenomena of reality, within the limits of dimensions used in practice, together with the generally recognized value of the author’s experiments, singled out Lampe’s formula from the numerous series of hydraulic formulas and secured for it the attention of specialists. This is demonstrated, for example, by the opinions of the American hydraulic engineer H. Smith,\(^2\) Wehner,\(^3\) Prof. N. K. Chizhov, A. A. Satkevich, Engineer Lindley, and also by its considerable spread in practice, among other places in our country, Russia.\(^4\) H. Smith, who worked specially on the question of the comparative merit of various hydraulic experiments and formulas, holds the very highest opinion of the accuracy and practical value of Lampe’s experiments.\(^5\) The results of the investigation carried out by him

\(^1\) A. Frank, Die Formeln über die Bewegung des Wassers in Röhren. Civiling, 1881.

\(^2\) H. Smith, Hydraulics.

\(^3\) H. Wehner, Ein Betrag zur Berechnung des Rohrwiederstandes in der Praxis. Gesundheits-Ing., 1897.

\(^4\) Lampe’s formula is sufficiently well known in Russia and is often applied to the design of water mains, on the initiative of Engineer Lindley and Prof. N. K. Chizhov and A. A. Satkevich. It was also adopted by Engineer Lindley in his projects for the sewerage of the cities of Warsaw and Petersburg.

\(^5\) “In all probability the most accurate experiments of the flow through a long pipe, where the loess of head from frict on et cet. has been measured by piezometer, by Prof. D-r C. J. H. Lampe”. (H. Smith, Hydraulics, p. 231).

for the comparative study of formulas also make it possible to judge the merit of Lampe’s formula. From a series of experimental data, Smith determined the most probable values of the velocity for pipes of various diameters at various hydraulic gradients, on the assumption that the pipes were new asphalted pipes, and compiled a table in which such normal velocities are shown, as well as the velocities obtained by calculation according to 11 different formulas. This table is cited by Prof. F. E. Maksimenko in his course on Hydraulics, with the addition of a new formula proposed by the author¹.

The table marks the results of calculation that come closer than the others to the data adopted as normal (deviating by no more than 10% in one direction or the other), and by their number evaluates the merit of the various formulas. From this table it is evident that the most satisfactory results are obtained by two formulas: those of Lampe and F. E. Maksimenko. These results differ from the experimental data by more than 10% in only 2 cases out of 15.²

In practical application Lampe’s formula (80) was somewhat simplified by taking the exponent of \(v\), instead of 1.802, to be equal to 1.80. This does not introduce serious changes into the results of the calculation.³ Owing to this change, the formula finally assumed the following form:

\[ i = n \frac{v^{1.8}}{\rho^{1.25}} . \tag{81} \]

When transformed according to the type of formula (6), it has the form

\[ \frac{\varrho}{r} = \rho i = n \frac{v^{1.8}}{\rho^{0.25}}, \tag{82} \]

or, introducing the value of the diameter,

\[ \frac{Di}{4} = 1.414 n \frac{v^{1.8}}{D^{0.25}} = n' \frac{v^{1.8}}{D^{0.25}} . \tag{83} \]

Finally, in a transformation of the type of the Chézy formula, one obtains

\[ v=\sqrt{\frac{\rho^{0.25}v^{0.20}}{n}}\sqrt{\rho i}; \tag{84} \]

\[ k=\sqrt{\frac{\rho^{0.25}v^{0.20}}{n}}. \]

The roughness coefficient \(n\) in formula (82) has the following values for measurements in meters (according to Lampe and Lindley¹):

\[ \begin{aligned} n_1&=0.000134 &&\text{— for new clean pipes;}\\ n_2&=0.00018 &&\text{— for water-supply pipes;}\\ n_3&=0.00025 &&\text{— for ordinary drains;}\\ n_4&=0.00030 &&\text{— Lindley recommends using for drains laid at very small gradients.} \end{aligned} \]

Accordingly, the values of the coefficients \(n' = 1.414\,n\), for formula (83) with diameter \(D\), should be:

\[ n'_1=0.00019; \]

\[ n'_2=0.00025; \]

\[ n'_3=0.00035; \]

\[ n'_4=0.00042. \]

The dependence of the roughness coefficient on the gradient, recommended by Lindley, is a certain peculiarity. This may be explained by the fact that, at small gradients, the drains themselves, being in unfavorable conditions, become fouled, form an uneven surface, and their roughness increases; consequently, the gradient in this case acts only indirectly, as an indication of the condition of the drain.

A comparison between the values of the coefficient \(k\) according to the formulas of Lampe, Darcy–Bazin, and Ganguillet–Kutter for approximately equal degrees of roughness shows that the increase of \(k\) according to Lampe’s formula is close to the results obtained from the two other formulas, but the value of \(k\) for small diameters (\(6''\) and especially \(3''\)) is somewhat exaggerated.

For measurements in feet, Prof. N. K. Chizhov, on the basis of prac-

practical guidance of engineers who use the Lampe formula in their calculations, it may be considered possible to adopt the following values of the roughness coefficient \(n\) in formula (82):

\(n_1 = 0.0000675\)—for perfectly new asphalted cast-iron pipes carrying clean water;

\(n_2 = 0.00007\text{—}0.00008\)—for cast-iron pipes in somewhat poorer condition or carrying turbid water;

\(n_3 = 0.00009\text{—}0.0001\)—for cast-iron pipes covered with encrusting deposits (this value is especially suitable for the calculation of municipal water mains);

\(n_4 = 0.00015\)—for poor and very old cast-iron pipes;

\(n_{\mathrm{I}} = 0.000095\)—for ceramic drainpipes;

\(n_{\mathrm{II}} = 0.0001\text{—}0.0002\)—for drainage canals.

Correspondingly, the values of \(n'\) for formula (83) should be the following:

\[ n'_1 = 0.000095, \]

\[ n'_2 = 0.00010\text{—}0.00011, \]

\[ n'_3 = 0.00013\text{—}0.00014, \]

\[ n'_4 = 0.00021; \]

\[ n'_{\mathrm{I}} = 0.000135, \]

\[ n'_{\mathrm{II}} = 0.00014\text{—}0.00028. \]

The use of the Lampe formula in calculations of water supply and sewerage is considerably more convenient than the use of polynomial formulas, since by its form it is well suited to logarithmic computation.

However, with the great mass of calculations of head loss that is inevitable in designing entire networks, calculations even by this formula prove to be very painstaking and, to the highest degree, fatiguing. Fortunately, the same logarithmic form of the Lampe formula makes it possible to avoid the indicated tiresome arithmetical operations and to reduce to a minimum the useless loss of time, while the final results of the calculations, thanks to the merits of the formula with respect to accuracy, are not only entirely sufficient for all practical purposes, but also correspond to the best of the available formulas and are distinguished by their closeness to reality.

This possibility is achieved by a graphical representation of the Lampe formula in the form of a series of scales.

The Lampe formula can be transformed into the form of conjugate scales also by the method of points of rectilinear intersection, as we shall show shortly with respect to other, more refined—

but similar to it in formulae. Here, however, we shall indicate another, very ingenious method of graphical representation, applied to this formula first by H. Wehner, and then by Prof. N. K. Chizhov. The basis of this method is only the method of scales of functions.

The method, set forth here, of graphical representation of Lampe’s formula was published in 1897 by G. Wehner in Gesundheits-Ingenieur, in the article Ein Beitrag zur Berechnung des Rohrwiderstandes in der Praxis, the diagram being drawn up for measurements in meters. In the same year Prof. N. K. Chizhov applied the same method to Lampe’s formula, but, in accordance with the needs of Russian practice, drew up his diagram for measurements in feet. This diagram appeared in the journal Stroitel’ in an article entitled A Mechanical Method for Calculating Head Loss. Not having been able to obtain G. Wehner’s article up to now,¹ we append (Drawing 15) only N. K. Chizhov’s diagram. It is clear that, for explaining the method of construction and use, it is immaterial to what measures the diagram refers; but in the sense of practical application, a diagram drawn up in foot measures is more interesting and suitable for Russian conditions.

The method of constructing the diagram is as follows. Lampe’s formula has the form

\[ i = n \frac{v^{1.8}}{\rho^{1.25}} . \tag{81} \]

Taking logarithms of this formula, we obtain

\[ \log i - \log n = 1.8 \log v - 1.25 \log \rho . \tag{85} \]

On an arbitrary straight line (see the lower scale of the diagram²) \(i - n\) (Drawing 15), we mark \(\log i\) and \(\log n\) in the form of scales of functions, i.e., we set off, in any scale of lengths, quantities proportional to \(\log i\) and \(\log n\), and over the divisions we write the corresponding values of \(i\) and \(n\).

On another arbitrary straight line we mark, in the same way (see the upper scale of the diagram, \(\rho\) and \(v\)), on one side \(1.8 \log v\), on the other side \(1.25 \log \rho\). In this case the scale modulus for these functions must be the same as for the scale \(\log i\) and \(\log n\). The absolute lengths of the new two scales between identical divisions will be different, namely 1.8 and 1.25 times greater than the corresponding lengths on the scale \(\log i\) and \(\log n\).

The setting-off must begin from a common point, denoted by the number—

¹ Wehner’s diagram was obtained after the completion of the work and is shown in Drawing 15a.
² In reproducing the diagram it proved inconvenient to change the notations of the original; therefore, in our Drawing 15, \(\rho\) and \(n\) correspond to \(R\) and \(k\).

mark 1.0 both for \(v\) and for \(\rho\). This follows from the following consideration.

For the case when

\[ \log i_0 - \log n_0 = 0, \tag{86} \]

equation (85) becomes

\[ 1.8 \log v_0 - 1.25 \log \rho_0 = 0. \tag{85'} \]

If at the same time

\[ \rho = 1.0, \]

then

\[ 1.8 \log v_0 = 0, \]

\[ v_0 = 1.0. \]

Having marked on the ordinates the corresponding values of \(\rho\) and \(v\), we obtain a double logarithmic scale of hydraulic radii and velocities.

The two constructed scales are sufficient for determining any of the quantities \(i\), \(v\), and \(\rho\) entering into the Lamé formula (81), if two of them are given. But, in order also to introduce into the calculation the value of the discharge \(Q\), logarithmic scales of discharges (in cubic feet) have also been constructed on the diagram for 20 different diameters, beginning with \(2''\) and up to \(30''\). These scales are constructed on lines parallel to the scale of velocities, and moreover in such a way that the divisions of the discharge scales corresponding to one and the same velocity for different diameters lie on the same vertical line with the division denoting that velocity. The modulus of the scale satisfying such a condition is determined on the basis of the relation

\[ l_0\left(\log \frac{\pi D_0^2 v_0}{4} - \log \frac{\pi D_0^2}{4}\cdot 1\right) = 1.8\, l \log v_0, \tag{87} \]

where \(l\) is the modulus of the scale of \(\log i\) and \(n\).

\(l_0\) is the modulus of the scale of the corresponding discharge \(Q\).

Hence \(l_0 = 1.8\,l\).

This means that the intervals between like-valued divisions on the discharge scale will be the same as on the velocity scale.

This is understandable if one takes into consideration that

\[ \log \frac{\pi D_0^2 v_0}{4} = \log \frac{\pi D_0^2}{4} + \log v_0, \tag{88} \]

i.e., the logarithm of the discharge may be represented as the sum of two segments, of which the latter is proportional to \(\log v_0\), and in the present case must be equal to the length corresponding to \(\log v_0\) on the velocity scale.

As for the construction of each such scale, it is carried out in the following manner. Having constructed, on the basis of the normal logarithmic scale, a scale with modulus \(l_0\), on the line marked for the scale of discharges of the given diameter one marks a point corresponding to some velocity. Then, having calculated the discharge from the diameter and the selected velocity, the constructed scale is applied in such a way that the marked point coincides with the place corresponding to the calculated discharge, and the other divisions are marked off.

It should be noted that the discharge scales for different diameters terminate on the left at points (indicated on the diagram by circles) which lie on the same vertical with the divisions marking on the scale \(\rho\) the values of the hydraulic radii for the given diameters. The significance of this circumstance will become clear when the use of the diagram is described.

In order to determine, for a given roughness coefficient \(n\), the velocity from given values, for example, of the hydraulic slope and the hydraulic radius (or diameter), one must take with a compass, on the \(i—n\) scale, the distance between the divisions corresponding to the given \(i\) and \(n\). In this way we obtain a length corresponding to \(\log i - \log n\). Then we set off the obtained length along the upper combined scale from the division denoting the given \(\rho\). Then the other end of the compass will indicate to us the division corresponding to the desired velocity \(v\).

That this is indeed so is clear from the fact that, setting off from the mark \(\rho\) the length taken with the compass, we obtain graphically the value of the expression

\[ \log i - \log n + 1.25 \log \rho = 1.8 \log v, \tag{85′} \]

and from the value of the resulting \(\log\) we read off the value of \(v\).

In exactly the same way, from the given slope and velocity, the hydraulic radius of the section can be determined. The determination of the required slope from a given hydraulic radius (or diameter) and velocity is performed in an analogous manner, only in the reverse order.

When, from a known diameter and hydraulic slope (with a specified roughness coefficient), it is required to find the discharge, one could first determine the velocity, entirely according to what has been indicated, and then, drawing through the corresponding division a vertical line to its intersection with the discharge scale for the given diameter, read the required discharge at the point of intersection. But the operation is simplified by the fact that the beginnings of the discharge scale for the various

diameters are located on one vertical line with the marks of the corresponding hydraulic radii. It is therefore more convenient, taking the compass opening on the \(i—n\) scale equal to the distance between the corresponding divisions, to set it off from the beginning of the discharge scale for the given diameter. Then the other leg of the compass will indicate to us directly the desired discharge.

Determining the slope from a known diameter and discharge is done just as simply, in the reverse order. Determining the diameter from a given discharge and hydraulic slope is carried out by successive trials, laying off, according to scale, the \(i—n\) length corresponding to \(\log i—\log n\) along the discharge scales. That diameter on whose scale the free leg of the compass indicates the discharge nearest to, and at the same time greater than, the given one will be the required diameter.

From the foregoing it is clear that N. K. Chizhov’s diagram can be used for calculations not only within the limits of the 20 diameters of round pipes that have special scales on it. The principal scales of the diagram make it possible to use it also for other diameters or for conduits of a cross-section other than round.

The great convenience of this diagram consists in the fact that it permits the use of any coefficient of roughness. We prefer to speak separately about the special features of applying it in cases of different degrees of filling and at different slopes, considering these features as applied to all diagrams of conjugate scales in general.

Examining the method of using a diagram of the type of Venner or N. K. Chizhov, it is not difficult to see that the matter reduces to operations with logarithmic scales drawn on paper, similar to those used when employing generally known calculating logarithmic rules.

It must be said that the application of the principle of the logarithmic rule to hydraulic calculation, in its most concrete form, appeared considerably earlier. Similar logarithmic rules, with divisions placed on two fixed scales and on a movable middle part, have long been manufactured in England; but, unfortunately, they are still very little known among engineers in general, and in Russia such rules cannot be obtained for sale.

This is explained in part also by the fact that these rules are not entirely convenient for us Russians, since the divisions corresponding to dis—

the flow of water \(Q\), are marked in the unused part of the dead gallons¹.

IV. Application of the method of conjugate scales to the formula of Lévy-Vallot.

The scientific analysis of the results obtained by Darcy in his well-known experiments on pipes, carried out in 1849–1751, gave the well-known mathematician and engineer M. Lévy (Maurice Lévy) the idea of obtaining a formula defining rectilinear motion, and motion close to it, in pipes by a purely theoretical route, without resorting to hypothetical premises, by considering the mechanical conditions of motion of a particle of liquid. This idea was realized by him in a work published in 1867², and there resulted a formula bearing his name, of which we must speak.

Reflecting on the question of the insufficient scientific clarity of matters relating to the motion of water, and on the diversity in the practical expressions seeking to define this motion, Lévy came to the idea that perhaps our ignorance arises from the fact that investigators have given themselves over too much to hypotheses and have not paid sufficient attention to the mechanical conditions of the motion being studied—conditions knowledge of which would eliminate many assumptions. Thus, at the foundation of Darcy’s empirical formula lies the hypothesis that the friction between two layers of liquid is proportional to the square of the relative velocity. Darcy’s formula successfully represents his experiments. However, Lévy found incidentally in his work that the law of friction adopted by Darcy, and which at first glance presents nothing inadmissible, nevertheless contradicts the mechanical definition of the motion to which it is applied. Lévy came to the conclusion that if it is admitted that the friction between two molecules depends exclusively on the relative velocity, then it can be proportional only to the first power of this velocity; that is, that the hypothesis adopted by Newton,

by Poiseuille and Navier, is the only possible one in the present case. But this latter hypothesis too, as is known, does not correspond to the facts. These circumstances led Lévy, first of all, to the conclusion that both hypotheses do not correspond to reality, and that friction in any case depends not on relative velocity alone. The latter was indirectly confirmed by the experiments of Bazin, who arrived at the idea that friction also depends on absolute velocity, although in this respect he indicated no exact law. Another conclusion at which Lévy arrived was that, in determining the laws of motion of water, it is desirable not to make any hypotheses a priori, but to proceed from consideration of the purely mechanical conditions of this motion, beginning with the simplest case, namely, rectilinear steady motion.

Setting himself the task of carrying out his investigation precisely in this way, Lévy made an assumption only with respect to that general fact which is, by agreement, recognized by all investigators, namely, that the elementary friction between two particles of a liquid is independent of pressure, and that, consequently, it can depend only on the absolute and relative velocities of these two particles. But he admitted a priori no hypothesis as to what function of absolute and relative velocity this friction represents. Relying only on the mechanical conditions of the steady motion of liquids, he showed first of all¹ that this function cannot be arbitrary, and that it necessarily represents the product of the relative velocity by a function of the absolute velocity, i.e., that if by \(\varphi\) we denote the friction referred to a unit of surface, instead of having

\[ \varphi = F\left(u \frac{du}{dR}\right), \tag{88} \]

where \(F\) denotes some function, one necessarily obtains the relation

\[ \varphi = \frac{du}{dR} f(u), \tag{89} \]

where \(f\) is likewise the notation for a function.

Further, Lévy, without assigning in advance any particular form to the function \(f\), found theoretically a general property of the steady rectilinear motion of liquids, which consists in the fact that, whatever the form of the wetted perimeter may be in such motion, the curv—

¹ Maurice Lévy, l. c.

...of equal velocities (isotachs) in the cross-section are parallel to one another. This theorem also finds confirmation in Bazin’s experiments on pipes of rectangular section.

This proposition gave Lévy the possibility of very easily (and again without defining the function \(f\)) solving the problem of the distribution of velocities at the points of the cross-section, whatever the curvilinear or rectangular form of the wetted perimeter.

The differential equation which he obtained contains the unknown function of velocity, whose determination depends on the determination of the function \(f\). This equation is

\[ dF(u)=\gamma i\rho\,dl, \tag{90} \]

where \(l\) is the distance of the points of the isotach of velocity \(u\) from a certain isotach taken as the initial one, measured along the normal to the isotachs;

\(\rho\)—the hydraulic radius of the part of the section bounded by the isotach \(u\);

\(i\)—the slope or loss of head per unit length;

\(\gamma\)—the weight of a cubic meter of water,

on integration, becomes

\[ F(u)-F(u_0)=\gamma i\int_{l_0}^{l}\rho\,dl, \tag{91} \]

and then, upon determining the quantity \(\rho\), into

\[ F(u)-F(u_0)=\gamma i\int_{l_0}^{l} \frac{\Omega_0+(l-l_0)P_0+\frac{1}{2}\delta(l-l_0)} {P_0+\delta(l-l_0)}\,dl. \tag{92} \]

The integral is easily calculated, and as a result one obtains

\[ F(u)-F(u_0)=\gamma i\left[ \frac{1}{4}(l-l_0)^2+\frac{P_0}{2\delta}(l-l_0)+ \right. \]

\[ \left. +\left(\frac{\Omega_0}{\delta}-\frac{P_0^2}{2\delta^2}\right) \lg\left(\frac{P_0+\delta(l-l_0)}{P_0}\right) \right], \tag{93} \]

where \(u_0\) and \(u\) are the velocities of particles of the liquid situated on two isotachs;

\(l_0\) and \(l\)—the distances of the points of the isotachs \(u_0\) and \(u\) from the isotach taken as the initial one;

\(\Omega_0\)—the area of the section bounded by the isotach \(u_0\);

\(P_0\)—the wetted perimeter corresponding to the isotach \(u_0\);

\(\delta\)—the angle formed by the normals to the ends of the isotach \(u\) (in the case of unclosed isotachs);

\(F\)—a function depending on the nature of the function \(f\), which determines the magnitude of friction.

Between the functions \(f\) and \(F\) there exists the relation

\[ f(u)=\frac{dF(u)}{du}. \tag{94} \]

Formula (93), at which Lévy arrived, appears complicated so long as it is given in its most general form, but it is greatly simplified in application to practice.

In the case of pipes of circular cross-section, if the center of the pipe is taken as the initial isotach \(u_0\), when, consequently,

\[ Q_0=P_0=l_0=\delta_0=0, \tag{95} \]

then, in place of (93), the relation obtained is

\[ F(u)-F(u_0)=\frac{\gamma i r^2}{4}, \tag{96} \]

where \(u_0\) is the velocity of the central filament;
\(r\) is the distance from the center of the isotach with velocity \(u\).

If this equation is compared with the results of Darcy’s experiments, it turns out that the function \(F(u)\) which best satisfies the experiments will be

\[ F(u)=-\alpha u^2. \tag{97} \]

Hence the preceding formula takes the form

\[ \alpha\left(u_0^2-u^2\right)=\frac{\gamma i r^2}{4}. \tag{98} \]

But it turned out, in this case, that the coefficient \(\alpha\) is not quite constant, and that no function \(F(u)\) with constant coefficients can cover, in a sufficiently complete manner, the results of all experiments. Since, however, the formula is exact and is not founded on any hypothesis when rectilinear motion is in question, Lévy concluded from this that the cause of the inconstancy of the coefficient \(\alpha\) in motion in pipes is the deviations, in this case, from rectilinear motion.

Formula (98), found for rectilinear motion, nevertheless applies very well to motion in pipes by means of a correction coefficient, the meaning of which is explained very simply.

Changes in the coefficient \(\alpha\) occur solely because motion in pipes is not quite rectilinear. If this assertion

correct, then \(\alpha\) should depend only on the curvature of the trajectories described by the particles of the liquid. But this curvature must vanish along the axis of the pipe and increase as the particles of the liquid move away from the axis. Lévy came to the conclusion that these changes depend neither on the slope, nor on the diameter, nor on the material of the pipe, but exclusively on the distance of the particle whose motion is being considered from the center of the pipe, i.e., on the magnitude \(r\).

He found that, if one sets

\[ \alpha=\beta\sqrt{r}, \tag{99} \]

then formula (98) satisfies all of Darcy’s experiments. It follows from this that the law of distribution of velocities in pipes is expressed by the formula

\[ \beta\sqrt{r}\,(u_0^2-u^2)=\frac{\gamma i r^2}{4} \tag{100} \]

or

\[ u_0^2-u^2=\frac{\gamma}{4\beta}\,i r^{\frac{3}{2}}, \tag{100'} \]

where the coefficient \(\dfrac{\gamma}{4\beta}\), in Lévy’s opinion, is perfectly constant for a given liquid and characterizes it from the point of view of its density. For water

\[ \frac{\gamma}{4\beta}=2640, \]

and formula \((100')\) takes the form

\[ u_0^2-u^2=2640\,i r^{\frac{3}{2}}. \tag{100''} \]

Differentiating formulas (98) and \((100'')\), Lévy arrived at expressions for the law of friction: on the one hand, for perfectly rectilinear motion; on the other, for motion in pipes. From (98) and \((100'')\) one obtains:

\[ -\alpha u\frac{du}{dr}=\frac{\gamma}{4}\,ri; \tag{101} \]

\[ -r^{\frac{1}{2}}u\frac{du}{dr}=\frac{3}{4}\cdot 2640\,ri. \tag{102} \]

Since \(ri\) is proportional to the friction over the surface of two concentric cylinders of liquid, it follows from this that:

a) when the motion of the liquid is perfectly rectilinear, the friction between two adjacent layers is proportional to the product of the absolute velocity and the relative velocity;

b) in pipes of circular cross-section, the friction between two concentric cylinders is, in addition, proportional to the square root of the radius.

The laws which Lévy formulated in this way, as well as the formulas from which they were derived, are noteworthy in that they contain no dependence on the diameter of the pipe.

Having derived, in the manner indicated, a formula for internal friction, Lévy also expressed the law of friction against the wall. He found that this friction is approximately proportional to the square of the velocity of motion near the wall, i.e.

\[ w^2=\alpha' Ri, \tag{103} \]

where \(R\) is the radius of the pipe,
\(\alpha'\) is a coefficient depending on the properties of the wall.

But this is only a first approximation, and a more exact expression is

\[ w^2=Ri\left(\alpha' + \beta'\sqrt{R}\right). \tag{104} \]

Finally, using expressions (100), (103), and (104), Lévy arrived at a general expression for the mean velocity of motion in a pipe. Taking the value of \(w\) from (103) or from (104), in every case this expression has one and the same form, namely

\[ v^2=Ri\left(p+q\sqrt{R}\right), \tag{105} \]

where \(p\) and \(q\) are coefficients, both of which depend on the properties of the wall, if expression (104) is taken for \(w\); but if expression (103) is adopted, then one of them varies depending on the properties of the wall, while the other depends exclusively on the properties of the liquid.

On the basis of comparison with the results of Darcy’s experiments, Lévy gave values of \(p\) and \(q\) for the cases of new cast-iron pipes and cast-iron pipes with deposits. Thus the Lévy formula was obtained, which the author gave in the following general form:

\[ \left(\frac{v}{n}\right)^2 = Ri\left(a+b\sqrt{R}\right), \tag{106} \]

where \(R\) is the (geometric) radius of the pipe;

\(n, a, b\) are numerical coefficients.

The coefficients have the following values:

for new cast-iron pipes

\[ n_1 = 36.4, \]

\[ a_1 = 1, \]

\[ b_1 = 1; \]

for pipes with deposits

\[ n_2 = 20.5, \]

\[ a_2 = 1, \]

\[ b_2 = 3. \]

Thus Lévy’s formula will have the numerical form, according to (106):

for new cast-iron pipes

\[ \left(\frac{v}{36.4}\right)^2 = Ri \left(1+\sqrt{R}\right), \tag{106a} \]

for pipes with deposits

\[ \left(\frac{v}{20.5}\right)^2 = Ri \left(1+3\sqrt{R}\right). \tag{106b} \]

In calculating water mains Lévy recommends using, of course, the formula relating to pipes covered with deposits, i.e. (106b).

If we put

\[ \mu = 20.5 \sqrt{R \left(1+3\sqrt{R}\right)}, \tag{107} \]

then formula (106b) assumes the form

\[ v = \mu \sqrt{i}, \tag{108} \]

where \(\mu\) depends only on the radius \(R\) of the pipe. Therefore the values of \(\mu\) can be computed in advance. Making use of this, Lévy, for convenience of computation, compiled a table¹ which makes it possible to solve quickly all problems relating to water-supply pipes. It applies to pipes of sizes beginning with 1 centimeter up to 1 meter. The table gives the diameters, geometric radii and cross-sectional areas of the pipes, the values of the quantity \(\mu\), and the values of the products \(\pi R^2 \cdot \mu\), the area multiplied by the coefficient \(\mu\).

Reducing Lévy’s formula to the form of expression (6), we obtain: in the general form

\[ Ri=\frac{v^2}{n^2\left(a+b\sqrt{R}\right)}, \tag{107} \]

for new pipes

\[ Ri=\frac{v^2}{1325\left(1+3\sqrt{R}\right)}, \tag{107a} \]

for old pipes

\[ Ri=\frac{v^2}{420{,}2\left(1+3\sqrt{R}\right)}. \tag{107b} \]

Sometimes it is also used in another form, which is obtained by replacing \(R\) by the diameter \(D\), namely:
for new pipes

\[ Di=\frac{v^2}{662{,}5+468{,}5\sqrt{D}}. \tag{107'a} \]

for old pipes

\[ Di=\frac{v^2}{210{,}1+445{,}7\sqrt{D}}, \tag{107'b} \]

Reducing Lévy’s formula to the Chezy form, we obtain, respectively, the general form

\[ v=n\sqrt{a+b\sqrt{R}}\,\sqrt{Ri}, \tag{108} \]

for new pipes

\[ v=36{,}4\sqrt{\,1+\sqrt{R}\,}\sqrt{Ri}, \tag{108a} \]

\[ v=\sqrt{2650+3747\sqrt{\rho}}\,\sqrt{\rho i}, \tag{108a} \]

where \(\rho\) is the hydraulic radius;
for old pipes

\[ v=20{,}5\sqrt{\,1+3\sqrt{R}\,}\sqrt{Ri}, \tag{108b} \]

\[ v=\sqrt{840+3566\sqrt{\rho}}\cdot \sqrt{\rho i}. \tag{108b'} \]

Both of Lévy’s formulas, for new and for old pipes, were compared by him with Darcy’s experiments. According to the author, they cover the results of the experiments in the most satisfactory manner, and coincide with Darcy’s empirical formulas within the limits of the experiments; outside those limits, however, they give somewhat different results, which the author considers deserving of greater confidence, as having theoretical foundations.¹)

M. Lévy’s formula in its original form presents, however, considerable inconvenience for practical application, as do all other formulas of polynomial type, since they are not suitable for logarithming.

Owing to this inconvenience of Lévy’s original formula, Henri Vallot, while studying the question of applying this formula, began to seek an equivalent expression for it of monomial form. After several trials A. Vallot settled on the formula:

\[ D=0.324\,\frac{Q^{3/8}}{i^{5/16}} =0.324\left(\frac{Q}{\sqrt{i}}\right)^{3/8}. \tag{109} \]

We shall call this formula the Lévy–Vallot formula. The Lévy–Vallot formula gives diameters somewhat higher in comparison with Lévy’s original formula (for pipes with deposits). But the difference is very small; it does not exceed 2 mm for diameters between 0.01 m and 1.00 m, reaches 3 mm at a diameter of 2.00 m, and 15 mm at a diameter of 3.00 m. Such a monomial formula offers great conveniences, in comparison with a polynomial one, with respect to operating with it.

The Lévy–Vallot formula attracted the attention of hydraulic engineers, especially of persons working in the service of the Paris water supply, and there it came into considerable use. For some time it was used in the form given above. Later G. Dariès drew attention to the fact that the form of this formula, being amenable to logarithming, thereby makes it possible to represent it graphically by the method of points of rectilinear intersection.

¹) M. Lévy, l. c., p. 245.

It must be said, however, that in doing so he based his reasoning on a special case of a function of the form (33), namely a linear equation of the form

\[ ax+by=c. \tag{110} \]

In application to this narrow purpose this is, of course, immaterial and even somewhat simplifies the matter; but such a formulation of the question needlessly excludes any conception of the breadth of scope and the theoretical importance of the method he employs.

Be that as it may, the matter reduces to the fact that the Lévy-Valleau formula

\[ D=0.324\,\frac{Q^{3/8}}{i^{3/16}} \tag{109} \]

represents a special case of a function of the form

\[ f_1(x)f_2(y)f_3(z), \tag{36} \]

which, as was shown in its proper place, admits transformation into a diagram of conjugate scales by the method of points of rectilinear intersection.

It was also indicated above that, in order to make possible the graphical representation of an equation of the form (36), it must first be reduced to the form (33) by taking logarithms.

Applying logarithms to formula (109), we obtain:

\[ \log D=\log 0.324+\frac{3}{8}\log Q-\frac{3}{16}\log i. \tag{109'} \]

This expression represents an equation of the form

\[ f_1(x)+f_2(y)=f_3(z), \tag{33} \]

and we can represent it in the form of a diagram, guided by the rules set forth above.

For this purpose we take two parallel axes \(Q\) and \(I\) (Fig. 16), situated at an arbitrarily chosen distance \(s\) from one another, and take the first of them as the scale of discharges, and the second as the scale of slopes. Then we lay off, respectively on each of them, with the aid of a logarithmic ruler, the scales of the functions \(Q\) and \(i\).

To choose the plotting modulus, we shall take the following into consideration. If we were to adopt, for both scales, the modulus of the logarithmic scale without alteration, then, in order to plot the functions \(\frac{3}{8}\log Q\) and \(\frac{3}{16}\log i\), we would have either to multiply the entire \(\log\) by \(\frac{3}{8}\) and \(\frac{3}{16}\), or to change the scale of the logarithmic ruler in the ratios \(\frac{3}{8}\) and \(\frac{3}{16}\). To avoid doing this, it is more convenient to adopt moduli for the scale \(Q l_1=\frac{8}{3}l\) and for the scale \(i l_2=\frac{16}{3}l\) (or multiples of them, depending on the dimensions of the diagram and the mutual arrangement of the scales). Then we shall have to lay off, on the \(Q\) and \(i\) scales, in order to express the functions \(\frac{3}{8}\log Q\) and \(\frac{3}{16}\log i\), the quantities \(\log Q\) and \(\log i\) in the scale of the ruler.

Since the function \(\frac{3}{16}\log i\) has a negative sign, while \(\frac{3}{8}\log Q\) has a positive one, the increase of the numerical values of the divisions on the \(Q\) and \(I\) scales must proceed in opposite directions. This must be borne in mind when choosing the initial points from which the divisions are laid off. In the present case it is more convenient to place, approximately in the middle of the diagram, the divisions \(O\) and \(O'\), corresponding to certain mean values of \(Q\) and \(i\), and then to draw the divisions from them in both directions.

Since, as before, the moduli of the scales \(Q\) and \(i\) are

\[ l_1=\frac{8}{3}l, \]

\[ l_2=\frac{16}{3}l, \]

then, denoting by \(x\) the distance from the axis \(Q\) to the axis \(D\), we must have, by (45),

\[ \frac{x}{s-x}=\frac{8}{3}:\frac{16}{3}=\frac{1}{2}:1, \tag{111} \]

whence

\[ x=\frac{s}{3}. \tag{111'} \]

The modulus of the scale of the function \((\log D-\log 0.324)\) is determined, on the basis of (47),

\[ l_3=\frac{l_1 l_2}{l_1+l_2}=\frac{16}{9}l, \tag{112} \]

where \(l\), as before, is the modulus of the logarithmic ruler.

To put the divisions on the diameter scale, we join the previously chosen points \(O\) and \(O'\) (or any other points) of the scales \(Q\) and \(i\). The intersection of the line \(OO'\) with the scales \(D\) gives the point \(O''\), the mark of which, corresponding to the values of \(Q\) and \(i\) at the points \(O\) and \(O'\), is determined by calculation. Having then constructed a logarithmic scale with modulus \(l_3\) and applied its corresponding division to the point \(O''\), we mark the other divisions of the diameter scale, extending it in both directions as far as necessary.

If, for convenience, other moduli are chosen (multiples of \(l_1\) and \(l_2\)), then the position of the axis \(D\) must be changed accordingly. For example, for \(s=0.10\) m, with the discharge scale 3 times larger than the slope scale, from equation (45) we obtain:

\[ \frac{x}{0.10-x}=8:\frac{16}{3}=3:2, \]

whence \(x=0.06\) m. The modulus of the diameter scale in this case is equal to

\[ \frac{8\cdot\frac{16}{3}}{8+\frac{16}{3}}\,l=\frac{16}{5}l. \]

Usually, on diagrams of hydraulic formulas constructed by the method of points of rectilinear intersection, one more axis is drawn, serving as the scale for determining velocities \(v\).

To determine the position and divisions of the velocity scale, on the basis of the positions of the scales \(Q\) and \(D\), one could proceed in the same way, starting from the relation:

\[ v=\frac{4Q}{\pi D^2}, \tag{113} \]

taking logarithms of this expression and representing the resulting equation in the form of a system of conjugate scales, in which the scales \(Q\) and \(D\) coincide with those constructed earlier.

But it is not necessary to resort to this, since the desired position and division can be determined by a purely graphical method. For this it is only necessary to take into account that a velocity of 1.00 m is developed in a pipeline of diameter 0.30 m at a discharge of 70.7 liters, and in a pipeline of diameter 0.60 m at a discharge of 282.7 liters. On this basis we can draw the corresponding intersecting straight lines and thus find the point of the \(v\)-axis marked 1.00 meters. Repeating the same operation with the data \(Q = 7.07\) liters, \(D = 0.30\) m and \(Q = 28.27\) liters, \(D = 0.60\) m, we obtain the point corresponding to a velocity of 0.10 m. These two points must lie on a line parallel to the other scales. It remains only to graduate the distance between 0.10 m and 1.00 and to continue the divisions in both directions.

Fig. 17 represents a diagram of conjugate scales for the graphical calculation of circular-section pipelines according to the Lévy-Valot formula. It consists of four scales, proceeding in the following order from left to right: the scale of discharges (in meters per second), the scale of diameters (in meters), the scale of head losses (in the form of decimal fractions defining the ratio of the height of head loss to the length of the pipeline), and the scale of velocities (in meters per second). The discharge scale includes discharges from 1.00 liter to 3000 liters per second, the numbers increasing from bottom to top. The diameter scale covers diameters from 0.01 meter to 3.00 meters, the numbers likewise increasing from bottom to top. The scale of slopes (head losses) contains slopes from 1:1000000 to 1:1, the numbers increasing from top to bottom. Finally, the velocity scale includes velocities within the limits from 0.05 meter to 10.00 meters.

This diagram is constructed in the following manner. The distance between the discharge scale and the hydraulic-slope scale is chosen so that the logarithms of the discharges are laid off on a scale 3 times larger than the logarithms of the slopes. In this case, if the distance between the discharge and slope scales is denoted by \(l\), and the distance between the discharge and diameter scales by \(x\), then the relation must be satisfied

\[ \frac{x}{l - x} = \frac{3.8}{3} : \frac{16}{3} = 3 : 2, \]

whence

\[ x = 0.6\,l. \]

The value of \(x\) has been taken equal to 95 mm.

Therefore,

\[ l = \frac{95}{0.6} = 158.3 \text{ mm}. \]

The scale modulus of the hydraulic gradients is to be taken such that one unit of logarithms corresponds to 30 mm. The scale of discharges, as stated, is 3 times larger, i.e. 90 mm per unit of logarithms. Starting from this ratio, on the lines placed at the above-indicated distance there are plotted the logarithms of the numbers: for discharges from 1 to 3000, and for gradients from 1 to 0.000001. In this way the scales of discharges and hydraulic gradients are obtained.

To obtain the scale of diameters, such a ratio of \(Q\) and \(i\) was chosen that with it \(D\) would be equal to 1.00 m, and the points corresponding to these values of \(Q\) and \(i\) were connected by a straight line. Its intersection with the line of the diameter scale gives the point marked 1.00. In the same way the point for 0.10 m was found. The distance between these points was divided proportionally to the divisions of the logarithmic line, and the divisions were continued in both directions. In this way the scale of diameters is obtained. The scale of velocities is obtained in the same manner.

The method of using the diagram for the Lévy-Valot formula, presented in Fig. 17, follows naturally from the preceding. The corresponding values of the four elements determining the flow—discharge, diameter, hydraulic gradient, and velocity—lie on one secant straight line of the scale, which is determined by two of these given quantities. The other two unknowns are read at the points where the secant line meets the corresponding scales.

In practice one avoids drawing secant lines on the drawing itself, since this would entail its rapid soiling and damage. Instead, it is much simpler, faster, and more convenient to use either a stretched thread, or a transparent strip of paper, celluloid, etc., on which a straight line has previously been drawn (called a transparent). In this case such a strip may be shifted either directly, or, better, by means of two pins with points fastened at the ends. In the latter case it is convenient, after placing one point on a known division, to rotate the straightedge about the axis of the point, without danger of slipping or displacement.

V. Application of the method of conjugate scales to Flaman’s formula.

Professor Flaman (A. Flamant), engaged in a detailed study of a large number of results of experiments and observations relating—

…pertaining to the motion of water in pipes, paid particular attention to formulas of a binomial logarithmic form. Working in this direction, he ultimately gave his own formula of the logarithmic type, the merits of which are acknowledged by specialists and which is applied not only in the author’s native country, France, but also, converted into English measures, has become widespread in the United States of America.

Flamant came to the conclusion that, if the basic formula for the motion of water is taken in the form (3),

\[ \frac{\varphi}{r}=\rho i=\frac{Di}{4}=bv^{2}, \tag{3} \]

then the coefficient \(b\) may be represented in the form

\[ b=\frac{n'}{\sqrt[4]{Dv}}. \tag{114} \]

Thus the general form of Flamant’s formula is obtained:

\[ \frac{Di}{4}=n'\frac{v^{7/4}}{D^{1/4}}. \tag{115} \]

This formula, however, is more often applied in the form

\[ D^{5/4}=a\,v^{7/4} \tag{115'} \]

or

\[ D^{5}i^{4}=a^{4}v^{7} \tag{115''} \]

where \(a=4n'\).

For the coefficient \(a\), Flamant gives two values: for pipes with an entirely smooth inner surface, or coated inside with a composition that smooths their irregularities,

\[ a_{1}=0,00074, \]

and for pipes slightly coated with deposits, such as water-supply pipes usually are after several years of service, on the average

\[ a_{2}=0,00092. \]

Flamant notes, moreover, that if the pipes are heavily covered with deposits, then no formula is directly applicable, since one has to allow for the reduction of the cross-section.

Pipes with an entirely smooth surface are, in practice, an exception, and it very rarely happens that they long retain their original smoothness. It may be thought, from a comparison of the results of observations made by Darcy, that a barely perceptible deposit, with a thickness expressed in small fractions of a millimeter, is sufficient to change the condition of flow by increasing the hydraulic resistances. Therefore, of course, for practical cases formula (115) should be applied in the form

\[ Di=a_2\frac{v^{7/4}}{D^{1/4}}=0.000921\sqrt{\frac{v^7}{D}}. \tag{115'''} \]

To facilitate calculations by his formula, Flamant compiled special tables.

Flamant’s formula is very close, both in outward form and in the values of its exponents and coefficients, to Lampe’s formula. Indeed, replacing in expression (115) the diameter \(D\) by the hydraulic radius \(\rho\), we obtain

\[ \rho i=n\frac{v^{7/4}}{\rho^{1/4}}=n\frac{v^{1.75}}{\rho^{0.25}}. \tag{119} \]

The coefficient \(n\) in this formula proves to be equal to:

\[ n_1=0.00013 \]

for pipes with a smooth internal surface,

\[ n_2=0.00016 \]

for pipes with deposits.

Lampe’s formula itself, as we have seen, has the form

\[ \rho i=n\frac{v^{1.80}}{\rho^{0.25}}, \tag{82} \]

i.e. it differs in outward form only by having the exponent \(1.80\) instead of \(1.75\). The corresponding values of the coefficients in Lampe’s formula are:

\[ n_1=0.00013, \]

\[ n_2 = 0.00018 \]

Flamant’s formula is also close to Prof. Anvin’s formula

\[ i=\frac{n v^2}{D^{3-z}}, \tag{17} \]

if in the latter one takes

\[ z=1.75, \]

(i.e., close to the lower limit indicated by Anvin, 1.79). In fact, substituting this value for \(z\), we obtain the expression

\[ i=\frac{a v^{1.75}}{D^{1.25}}, \tag{120} \]

which is a modification of (115).

A comparison of Flamant’s formula with other formulas¹ shows, as was to be expected, that the values of \(k\), according to Flamant at \(a_2\), are close to the values obtained from Lampe’s formula when \(k_2=0.00018\), but somewhat higher than the latter. Judging from the fact that the mean values of \(k\) (at \(i=0.003\)) come closest to the results obtained from the Ganguillet–Kutter formula with a roughness coefficient \(n=0.011\), Flamant’s formula is most applicable to water mains that are in good condition.

Flamant’s formula for foot measures is used in the United States of America for the calculation of water mains in the following form²:

\[ v=76.28\,D^{5/7} i^{4/7}. \tag{121} \]

Flamant’s formula, in whatever form it may be presented, is a particular case of an equation of the form

\[ f_1(x)\cdot f_2(y)=f_3(z), \tag{36} \]

and therefore can be transformed into the form of a diagram of conjugate scales, by the method of points of rectilinear intersection.

The first very interesting attempt at a graphical representation of this formula, for metric measures, was made by the French military—

¹ Such a comparison was made in my work “Logarithmic-Type Formulas for the Calculation of Water Mains.”

² M. M. Cherepashinsky, Water Supply.

... by the engineer Bertrand (L. Bertrand). Then the same method was applied to the Flamant formula, likewise for foot measures.

The method of handling this formula (as well as its derivatives, with which we shall have to deal when describing the transformation carried out by Bertrand) is exactly the same as that which we explained above in a general form and in application to the Lévy–Vallot formula.

Taking, for example, the basic formula

\[ D^{5/4} i = a v^{7/4} \tag{115} \]

(or a similar basic derivative), we take its logarithm. In the present case we obtain

\[ \frac{5}{4}\log D \log a = \frac{7}{4}\log v \cdot \log i . \tag{123} \]

Constructing parallel logarithmic scales of the functions \(\left(\frac{5}{4}\log D - \log a\right)\) and \(\frac{7}{4}\log v\) at an arbitrary distance from one another, with suitably chosen moduli, and then determining, according to the known rules, the position and modulus of the scale of the function \(\log i\) and laying it out, we ultimately obtain a system of three conjugate scales, applied in accordance with the fundamental property of the straight lines that intersect them.

Such a system may be supplemented by scales of other functions which are connected with those already included in the diagram by equations of the same-membered form.

Thus, for example, using the relation

\[ Q = \frac{\pi D^2 v}{4}, \tag{124} \]

we may construct an additional scale of discharges \(Q\). For this purpose, taking the logarithm of equation (124), we obtain

\[ \log Q - \log \frac{\pi}{4} = 2\log D + \log v, \tag{124'} \]

and construct a system of conjugate scales of the functions \(\left(\log Q - \log \frac{\pi}{4}\right)\), \(2\log D\), and \(\log v\), taking the arrangement and graduation of the scales \(D\) and \(v\) (by the corresponding choice of moduli) as those which are already available.

on the diagram. Then, as a result of the construction, we obtain the fourth conjugate scale \(Q\).

In this way the number of conjugate scales of the diagram may be further increased, depending on the number of relations that it is desirable to represent and solve graphically.

The most interesting of the diagrams with conjugate scales is the diagram constructed by Bertrand for graphical calculation by Flamant’s formula in metric measures¹.

Bertrand took the formula proposed by Flamant in the form

\[ D^5 i^4 = a^4 v^7 \tag{115''} \]

and applied it first of all to the construction of a diagram of three conjugate scales, which gives one of the three quantities \(D\), \(i\), or \(v\) from the two others specified.

But Bertrand rightly observed here that other quantities depending on those indicated can and should be determined simultaneously by the same graphical method.

These new variables are, first of all, the following.

1) Discharge \(Q\). It is connected with the velocity \(v\) and the diameter \(D\) by the relation

\[ 4Q = \pi D^2 v, \tag{124'} \]

which makes it possible to represent the preceding formula \((115'')\) in the form

\[ D^{19} i^4 = \left(\frac{4}{\pi}\right)^7 a^4 Q^7. \tag{125} \]

To simplify this expression, one may replace by a single letter \(c\) the constant product

\[ \left(\frac{4}{\pi}\right)^7 a^4 = c \tag{126} \]

and write

\[ D^{19} i^4 = c Q^7. \tag{125} \]

This expression determines the relation between \(D\), \(i\), and \(Q\) and makes it possible to construct, on the same diagram, with respect to the scales \(D\) and \(i\), the scale \(Q\).

2) The total length of the pipeline \(L\) and the total head loss \(H\). They are related to the unit head loss \(i\) by the relation

\[ H=Li. \tag{127} \]

With respect to these new quantities, formula (125) will take the form

\[ D^{19}H^{4}=cL^{4}Q^{7}, \tag{128} \]

or

\[ H=c^{\frac14}LQ^{\frac74}D^{-\frac{19}{4}}. \tag{128'} \]

The quantities \(L\) and \(H\), being represented in the same way in the form of logarithmic scales, together with the preceding ones, give a diagram of six conjugate scales, forming two independently operating systems:

\[ \text{1) } Q,\ D,\ i,\ v; \]

\[ \text{2) } L,\ i,\ H. \]

It should be noted that scales are often graduated on two sides. Thus, for example, on the scale of discharges, on one side divisions are marked corresponding to liters per second, and on the other to liters per minute; on the scale of diameters—on one side diameters, and on the other wetted cross-sections; on the scale of velocities—on one side velocities, and on the other the values of \(\dfrac{v^{2}}{2g}\).

In constructing the diagram, Bertrand sought to apply Flamant’s formula and its graphical transformation to all quantities characterizing a pipeline. In doing so, it occurred to him, by analogy with the elements that determine not the external dimensions of a pipeline but characterize, so to speak, its inner essence, such as the hydraulic gradient and the velocity of motion, to introduce new concepts for other elements capable of characterizing the pipeline in various respects and representing functions of quantities that usually determine a pipeline. Proceeding from this, he introduces the concept of four elements of this kind, which he calls the resistance (la résistance), the conveyance capacity (le débouché), the power (la puissance) and the coefficient of variation of cost (la variation du prix) of the pipeline. These elements, besides their practical usefulness in certain cases of calculation,

are of considerable interest from the point of view, if one may so express it, of the philosophy of water-supply hydraulics. Their significance will be clear from the following exposition.

The quantities of resistance, carrying capacity, power, and the coefficient of change in cost, on the basis of equations expressing them as functions of other quantities determining the pipeline, are also represented on Bertrand’s diagram in the form of three more conjugate scales, one of which is common to resistance and carrying capacity. Thus Bertrand’s diagram consists of nine conjugate scales.

3) Resistance of a pipeline. By this term (la résistance) Bertrand denotes the function \(\dfrac{H}{Q^{7/4}}\), which we shall denote by the letter \(r\).

Thus,

\[ r = H Q^{-\frac{7}{4}} = c^{\frac{1}{4}} L D^{-\frac{19}{4}} \tag{129} \]

or

\[ H^4 = r^4 Q^7, \tag{129'} \]

\[ c L^4 = r^4 D^{19}. \tag{129''} \]

If we take a pipeline composed of several pipes having different diameters \(D_1, D_2 \ldots\) and lengths respectively \(L_1, L_2 \ldots\), which are laid one after another and carry one and the same discharge \(Q\), then the total loss of head will be, by (128′),

\[ H = c^{\frac{1}{4}} Q^{\frac{7}{4}} \left( L_1 D_1^{-\frac{19}{4}} + L_2 D_2^{-\frac{19}{4}} + \ldots \right) \tag{130} \]

whence

\[ \frac{H}{Q^{7/4}} = c^{\frac{1}{4}} \left( L_1 D_2^{-\frac{19}{4}} + L_2 D_2^{-\frac{19}{4}} + \ldots \right) \tag{130'} \]

and, by (129),

\[ r = r_1 + r_2 + \ldots \tag{131} \]

The total resistance will thus be the sum of the resistances \(r_1 + r_2 + \ldots\) of the separate parts of the pipeline. From this it is evident that what we understand in the present case by the term resistance is analogous to electrical resistance in conductors.

4) Conveyance capacity of a pipeline. By this term (le débouché) Bertrand understands the property inverse to resistance. He denotes by this name the function

\[ \left(\frac{1}{r}\right)^{\frac{4}{7}}=\frac{Q}{H^{\frac{4}{7}}}, \tag{132} \]

which we shall call by the letter \(d\):

\[ d=QH^{-\frac{4}{7}}=c^{-\frac{1}{7}}L^{-\frac{4}{7}}D^{\frac{19}{7}}. \tag{133} \]

or

\[ Q^7=d^7H^4 \tag{133'} \]

and

\[ D^{19}=c^4d^7L^4. \tag{133''} \]

If two reservoirs are connected with one another by a known number of pipes, which have diameters \(D_1, D_2\ldots\) and, correspondingly, lengths \(L_1, L_2\ldots\), in such a way that the total loss of head \(H\) is the same for all of them, then the total discharge \(Q\) (according to 128) will have the expression

\[ Q=c^{-\frac{1}{7}}H^{\frac{4}{7}}\left(L_1^{-\frac{7}{4}}D_1^{\frac{19}{7}}+L_2^{-\frac{4}{7}}D_2^{\frac{19}{7}}+\ldots\right) \tag{134} \]

\[ \frac{Q}{H^{\frac{4}{7}}}=c^{-\frac{1}{7}}\left(L_1^{-\frac{4}{7}}D_1^{\frac{19}{7}}+L_2^{-\frac{7}{4}}D_2^{\frac{19}{7}}+\ldots\right) \tag{134'} \]

\[ d=d_1+d_2\ldots \tag{135} \]

Thus the total conveyance capacity \(d\) will be equal to the sum of the conveyance capacities \(d_1, d_2,\ldots\) of the individual pipes. Thus, conveyance capacity proves to be analogous to the electrical conductivity of conductors.

5) Power of a pipeline. If one has to study various combinations of diameter and loss of head that may occur in a pipeline to be constructed, in which only the total length \(L\) and discharge \(Q\) are known, then, in Bertrand’s opinion, it is useful to determine in advance the function

\[ f = Q^{\frac{7}{4}} L . \tag{136} \]

This function is called by them the power (la puissance) of the pipeline, and we shall denote it by the letter \(f\). On the basis of (128′)

\[ f = Q^{\frac{7}{4}} L = c^{-\frac{1}{4}} D^{\frac{19}{4}} H \tag{137} \]

6) Coefficient of change in cost. Bertrand designates by this name (variation du prix) the expression \(\dfrac{4}{19}\dfrac{i}{D}\), and for the following reason.

If a pipeline of a definite length is given, then, while leaving its discharge constant, one may vary in opposite directions, within known limits, the diameter and the loss of head; in other words, the cost and the loss of head, since for a given length the cost is approximately proportional to the diameter.

The equation given above,

\[ D^{19} H^4 = c L^1 Q^7 \tag{121} \]

is easily reduced, by taking logarithms and differentiating, taking the second term as constant, to the form

\[ \frac{dD}{D} = -\frac{4}{19}\frac{dH}{H} \tag{138} \]

or

\[ d(LD) = -\frac{4}{19}\frac{LD}{H}\,dH = -\frac{4}{19}\frac{D}{i}\,dH . \tag{137} \]

Thus the change in cost, proportional to the change in the product \(LD\), is equal to the change in loss of head multiplied by the function \(\dfrac{4}{19}\dfrac{D}{i}\). It is positive if the loss of head decreases, and negative if the latter increases.

We shall denote this function, called the coefficient of change in cost, by \(\varepsilon\).

Thus

\[ \varepsilon = \frac{4}{19}\frac{D}{i}, \tag{140} \]

or

\[ 19\varepsilon i = 4D. \tag{140′} \]

From the preceding it is evident that all the quantities which, in any given case, may enter into the calculation of water supply on the basis of Flamant’s formula are connected with one another by monomial expressions; this made it possible for Bertrand to represent them graphically in the form of a general diagram of conjugate scales.

Bertrand’s diagram is shown in Drawing 18.

It comprises nine parallel scales with logarithmic divisions, corresponding, in order from left to right, to the quantities \(Q\), \(r\) (and \(d\)), \(H\), \(D\), \(f\), \(\varepsilon\), \(i\), \(v\), and \(L\). These scales have in their headings the designations of the quantities which they represent, and whose definitions and algebraic expressions we have given above. Taking into account these definitions and the general theory of conjugate scales set forth at the beginning, one can see that the diagram represents several independent systems of conjugate scales, in which the lines of rectilinear intersection may serve to determine the following groups of quantities:

1) discharge \(Q\), diameter \(D\), hydraulic slope \(i\), velocity \(v\), and coefficient of change in cost \(\varepsilon\);

2) diameter \(D\), length \(L\), resistance \(r\), and conducting capacity \(d\);

3) discharge \(Q\), resistance \(r\), conducting capacity \(d\), and total head loss \(H\);

4) length \(L\), discharge \(Q\), and power \(D\);

5) diameter \(D\), head loss \(H\), and power \(D\);

6) head loss \(H\), length \(L\), and slope \(i\).

It follows from this that if two quantities are known, the corresponding secant is determined and in turn determines the remaining element or other elements of the same group. If two elements of one group and a third element from another group are known, then all the remaining elements may be obtained \(^{1}\).

In fact, the polygon drawn at the top of the diagram is determined when three of its vertices not lying on a single straight line are known. Suppose, for example, that \(Q\), \(r\), and \(D\) are known.

\(Q\) and \(r\) give \(H\).

\(Q\) and \(D\) give \(i\).

\(H\) and \(D\) give \(f\).

\(r\) and \(D\), or \(H\) and \(i\), or \(Q\) and \(f\), give \(L\).

\(^{1}\) However, it is assumed that, among the given two elements of the first group, velocity and the coefficient of change in cost cannot be counted.

A polygon drawn over the scales of the diagram is the graphical expression of the relations that were mentioned.

The scales adopted are as follows. For the length \(L\), the loss of head \(H\), and the diameter \(D\)—the meter with its subdivisions. The velocity \(v\) is expressed in meters per second. On the same axis another scale gives the values \(\frac{v^2}{2g}\), likewise in metric meters. The discharge \(Q\) is expressed by two scales: to the left, the discharge per second; to the right, the discharge per minute, in liters, and, for large discharges, in cubic meters. The slope \(i\) is the ratio of two lengths, \(H\) and \(L\), and therefore is an abstract number, expressed in units or in decimal fractions of a unit. The same applies to the other quantities: resistance, carrying capacity, power, and change in cost. As the unit of resistance there has been adopted that resistance which corresponds to a discharge of 1 cubic meter per second with a loss of head of 1 meter; as the unit of carrying capacity—the carrying capacity of such a pipe as gives 1 cubic meter per second with a loss of head of 1 meter; for the coefficient of change in cost, the unit corresponds to such values of \(D\) and \(i\) for which the function \(\frac{4}{19}\frac{D}{i}\) is equal to 1.

A certain inconvenience of Bertrand’s diagram is its complexity, which arises from the large number of scales and may make reading difficult. This circumstance is very important, since the chief advantage of diagrams in their practical use is simplicity. Therefore one should reduce in them the number of lines and, in general, remove everything that is not very necessary and that may complicate reading, tire the eye, and thereby lead to errors when attention is weakened. Proceeding from this idea, G. Dariès reduced Bertrand’s diagram, leaving in it only those elements which are necessary from the standpoint of everyday practice, i.e., discharge, diameter, hydraulic slope, and velocity. In this form the diagram is reproduced in drawing 19. The method of using it is clear from the preceding material.

Drawing 20 presents a diagram of conjugate scales for calculating water-supply pipes by the formula of Flaman, adapted for use with the measures employed in the United States of America, namely

\[ v = 76{,}28\,D^{\frac{5}{7}} i^{\frac{4}{7}}. \tag{122} \]

The diagram is constructed by the method of points of rectilinear intersection for the quantities: diameters \(D\), slopes \(i\), velocities \(v\), and discharges \(Q\). Its construction has been carried out in the order set forth above: for the quantities \(D\), \(i\), and \(v\) on the basis of expression (122), and for \(Q\) also on the basis of the relation

\[ Q = \frac{\pi D^{2}v}{4}. \tag{124} \]

Diagram drawing 20, like the diagram of the Lévy-Wallo formula, consists of 4 scales. On the first scale are represented, on the left, the discharges \(Q\) in cubic feet per second, and on the right, in American gallons per minute (1 gallon \(=0.1605\) cubic ft. \(=231\) cubic in.); on the second scale, the diameters \(D\) in inches; on the third, the head losses \(i\) in feet over a distance of 1000 ft.; and on the fourth, the velocities \(v\) in feet per second. The scale \(Q\) includes discharges beginning with 0.10 cubic ft. to 20 cubic ft. per second, and from 50 gallons to 9000 gallons per minute. The scale \(D\) comprises diameters from 1.5 inches to 72 inches \(=6\) ft. The scale of head losses \(i\) begins with a slope of 0.03 ft. per 1000 ft. and extends to 400 ft. per 1000 ft. Finally, the scale \(v\) includes velocities within the limits from 0.7 to 10 ft. per second.

This diagram is used in the same way as the diagram of the Lévy-Wallo formula. By the property of the diagram, if an arbitrary straight line is drawn, it cuts off on the scales corresponding to one another values for the quantities \(Q\), \(D\), \(v\), and \(i\). Therefore, when two of them are given, the straight line drawn through the corresponding points of the scales gives, at its intersections with the other two scales, the remaining two sought quantities.

VI. Limits and Features of the Application of the Method of Conjugate Scales and Its General Evaluation.

Turning to an evaluation of the method of hydraulic calculation of sanitary-engineering structures by means of diagrams of conjugate scales, from the standpoint of the breadth and convenience of its application in comparison with other methods of calculation, we must first of all consider the question of the limits of application of this method with respect to the various sizes of water conduits under different conditions of their operation, to sections of non-circular form, to various co-

coefficients of roughness and, finally, to the various degrees of filling of conduits.

The limits of application of the method of conjugate scales to pipelines of various sizes and to the conditions of their service depend, of course, on the character and breadth of application of those hydraulic formulas which are represented graphically by this method. Thus the question is reduced to determining the most suitable limits of application of the formulas of Lampe, Lévy-Vallot, and Flaman, by comparison with formulas whose character is regarded as sufficiently well known. Such a comparison was made in another work of mine¹, where the corresponding tables and information relating to the individual formulas of logarithmic form are given. These data indicate that all three formulas give results sufficiently close, under certain operating conditions of pipelines, to the formulas of Darcy-Bazin and Ganguillet-Kutter, which enjoy deserved authority and which are used, with increasing success, in more or less broad practical application; and in general they may be counted among the formulas deserving the attention of specialists, even in themselves, irrespective of the special convenience of using them. To this one may add only one further reference.

Prof. F. E. Maksimenko, in his course on Hydraulics, gives, on the basis of Smith’s work, comparative tables of velocities for pipes of various diameters, computed by different formulas. It turns out that, among the formulas that have come into use, the best results—in the sense of agreement with reality—were given, for new asphalted pipes, by the formulas of Lampe, Ganguillet-Kutter, Weisbach, and Lévy; for old pipes, by the formulas of Frank, Darcy-Bazin, and Lévy. Flaman’s formula is not included in both of these comparative tables, and Lampe’s formula is included in the latter.

Not to mention many less perfect formulas, the limits of whose application are very narrow, even Darcy’s formula, for example, with respect to pipes with deposits, gives correct diameters only between 0.05 and 0.50 m; below 0.04 m it is noticeably inaccurate, and above 0.50 m it gives exaggerated results, this exaggeration being especially noticeable beginning with 0.80 m. The formulas of Lampe, Lévy-Vallot, and Flaman, depending on those theoretical and practical foundations from which they proceed, are likewise each especially suitable for definite conditions of water motion, which also indicate the limits of their applicability. These conditions are brought out quite clearly on the basis of the comparative

Table I. This table contains the values of the velocity coefficient \(k\), which are obtained from the formulas under consideration (after transforming them according to the Chézy type) as applied to the case of water flow in water-supply pipes covered with deposits, and, according to the Lampe formulas, also to the case of ordinary gutters. For comparison, the values of the coefficient \(k\) according to the Darcy–Bazin and Ganguillet–Kutter formulas have been added, for such magnitudes of roughness as give values suited to the formulas of interest to us. The values of \(k\) refer to six diameters (3, 6, 12, 18, 30, and 48 in.) and are arranged, as far as possible, in decreasing order of these values. Such an arrangement makes it possible easily to establish the position of the formulas being studied in relation to the Darcy–Bazin and Ganguillet–Kutter formulas under different conditions, and to determine their character.

TABLE I.

Comparative values of the velocity coefficient \(k\) according to the formulas of Lampe, Flamant, Levy-Valot, Darcy–Bazin, and Ganguillet–Kutter.

Pipe diameter	Ganguillet–Kutter \(n = 0{,}011\)	Flamant \(\alpha_2 = 0{,}00092\), \(i = 0{,}003\)	Ganguillet–Kutter \(n = 0{,}012\)	Lampe \(n_2 = 0{,}00018\), \(i = 0{,}003\)	Darcy–Bazin \(\gamma = 0{,}00019\), \(i = 0{,}000133\)	Ganguillet–Kutter \(n = 0{,}013\)	Lampe \(n_3 = 0{,}00025\), \(i = 0{,}003\)	Ganguillet–Kutter \(n = 0{,}014\)	Levy-Valot
3″	40,2	40,7	35,3	40,5	33,6	31,6	33,9	28,2	36,5
6″	49,6	47,8	43,9	46,0	43,1	39,9	38,4	35,4	39,2
12″	59,5	55,7	53,0	52,8	52,1	48,0	44,0	43,4	42,6
18″	65,2	60,8	58,3	57,1	57,1	53,0	47,6	47,4	45,2
30″	72,2	67,9	64,9	63,1	62,1	59,3	52,5	54,1	49,0
48″	78,9	74,9	70,7	69,3	65,4	64,9	57,7	59,4	53,1

Examination of the table presented leads to the following conclusions regarding the limits of applicability of the logarithmic formulas under consideration and of the diagrams representing them.

The Lampe formula, with a coefficient for water-supply pipes of 0,00018, at an average value of the slope, for the majority of diameters of urban water supply, almost coincides with the Darcy–Bazin formula for pipes with deposits and with the Ganguillet–Kutter formula at a coefficient of 0,012, which are very close to one another. For pipes of small diameter (3″), the coefficient \(k\) according to Lampe’s formula is somewhat higher, which in general corresponds to reality. Thus, Lampe’s formula and the Lampe–Chizhov diagram are very suitable for calculating

pipes of municipal and domestic water supply over the entire range of diameters of such pipes, on the assumption of a quite satisfactory internal condition.

The Lampe formula, with the coefficient for ordinary water conduits equal to 0.00025, gives coefficients lying in the interval between the results obtained by the Ganguillet–Kutter formula with roughness coefficients 0.013 and 0.014. This shows that, as applied to water conduits, the Lampe formula is calculated for conditions in which they are not particularly well maintained.

The Flamant formula, with the coefficient for pipes with deposits, in Table I occupies a place between the columns for the Ganguillet–Kutter formula with coefficients 0.011 and 0.012, in general (with the exception of the 3″ caliber) closer to the values for \(n = 0.012\). At the same time the values of \(k\) according to the Flamant formula are close to the corresponding values according to Lampe, but somewhat higher than them. Thus, with regard to the Flamant formula and the Flamant–Bertrand diagram, almost the same must be said as of the Lampe formula. They are entirely suitable for calculating pipes of municipal water supply on the assumption of a good internal condition of the pipes and of water producing little deposit. In cases where the conveyance of unfiltered water is expected, or water with a large quantity of known salts—in general, where the formation of a considerable quantity of deposits is presumed—Flamant recommends adding directly to the diameters obtained by calculation. Such advice, in view of the insufficient definiteness of the size of the increment, is inconvenient to carry out; and from this point of view the Lampe formula, which allows a larger margin in the roughness coefficient, appears more practical. In Darcy’s opinion, the Flamant formula gives excellent results from 0.01 m to 1.30 m; for 0.05 m the figures are still satisfactory; the same is true above 1.30 m, although here the dimensions of the diameters obtained are somewhat small. Flamant, as was indicated, gives no coefficient for other cases of practice, and his coefficient for pipes with smooth surfaces has no great practical significance.

The Lévy–Vallot formula gives, for small diameters, values of \(k\) close to the Darcy–Bazin formula. For the calibers, however, corresponding to municipal water supply, this formula gives results approaching the Ganguillet–Kutter formula with coefficient 0.014. Thus the Lévy–Vallot formula and its diagram, when applied to problems of municipal water supply, are suitab-

... for those cases when considerable sediment and fouling of the pipeline are expected. Such, for example, is the calculation of large suburban mains conveying unfiltered water.

For the calculation of an urban network this formula is applicable on the assumption of long service, with a large amount of deposits and, in general, a not entirely satisfactory condition of the inner surface of the pipes. French specialists consider the Levy–Vallot formula suitable for the whole range of diameters, from 0.05 to 3.00 meters (one must suppose, under the assumptions mentioned), and especially for large diameters, beginning with 1.30 meters, i.e. precisely for suburban mains. This formula also has no coefficient relating to watercourses, while the coefficient for new and clean pipes has no practical application.

Turning to the question of the limits of applicability of the above-described diagrams of conjugate scales with respect to various coefficients of roughness, we see first of all that the Lampe–Chizhov diagram includes in its construction, among other things, a scale of roughness coefficients and thereby permits a free variation of these coefficients by the very method of using the diagram. The Flamant–Bertrand diagram is constructed for one definite roughness coefficient, namely \(n_2 = 0.00092\). Practically speaking, the question of applying other roughness coefficients to this diagram is of no importance, since these coefficients have not been developed for the formula itself (not counting the coefficient for pipes with a smooth inner surface), and it is intended for solving problems as applied to one definite case of water motion. However, the Flamant–Bertrand diagram (and similar ones) in itself allows the use of other roughness coefficients quite freely. We shall show this, for example, in application to the reduced Flamant–Bertrand diagram with four scales.

The general form of Flamant’s formula with roughness coefficient \(a_1\) will be

\[ D^{\alpha} i = a_1 v^{\beta}. \tag{141} \]

To represent it in graphical form, we take the logarithm of the formula and obtain

\[ \alpha \log D - \log a_1 = \beta \log v - \log i \tag{141'} \]

and then construct, according to the known rules, in the form of scales...

functions \((\alpha \log D - \log a_1)\), \(\beta \log v\), and \(\log i\). With such a method of construction it is clear that passing from the roughness coefficient \(a_1\) to \(a_2\) introduces no changes into the diagram as regards the scales of slope \(i\) and velocity \(v\); the change in the scale of diameters reduces to the fact that, instead of the function \((\alpha \log D - \log a_1)\), there is laid off from the former initial point, and with the same modulus, the function \((\alpha \log D - \log a_2)\).

It is not difficult to see (Fig. 21) how such a change will be expressed graphically. Owing to the preservation of the modulus, the divisions of the scale remain the same, but the whole scale is displaced by a length representing \((\log a_2 - \log a_1)\), toward the origin. Consequently the numerical mark corresponding to any point of the axis of the scale must change, namely increase with an increase in the roughness coefficient, i.e. when

\[ \frac{a_2}{a_1} > 1,\ \log a_2 - \log a_1 > 0, \]

and decrease in the opposite case. In this case the change takes place in such a way that the distance between the points which correspond to one and the same reading on the diameter scales constructed with coefficient \(a_1\) and with coefficient \(a_2\) remains constant and equal to \(\pm(\log a_2 - \log a_1)\).

Thus, having a scale of diameters for one degree of roughness, one can easily determine the reading of any point of the scale axis corresponding to the scale for another degree of roughness. To do this it is only necessary to lay off from this point, in the appropriate direction, a length corresponding, for the modulus of the scale, to the value \(\pm(\log a_2 - \log a_1)\), and the mark of the new point obtained, read on the existing scale, will be the desired one. Thus one and the same scale of diameters and one Flamant–Bertrand diagram, if desired, could serve for calculations for various assumed degrees of roughness. Thus, for example, with this diagram one may solve problems concerning the motion of water in pipes with a smooth internal surface, when the coefficient \(n = 0.00074\). For this purpose, operating in the usual manner with all the scales of the diagram, it is only necessary to remember that the marks of the points of the diameter scale in the given case must be read below the points themselves at a distance corresponding to

\[ \log a_1 - \log a_2 = 0.09456 \]

for the modulus of the diameter scale.

In such cases it is more convenient to take readings by means of a previously prepared transfer scale, giving the required lengths for different roughness coefficients.

It is clear that the indicated operation could also be replaced by solving the problem as applied to the principal coefficient, with subsequent alteration of the result in proportion to the ratio of \(a_2\) to \(a_1\).

In the case where the number of roughness coefficients with which one has to deal does not exceed two, it is possible to construct two scales of diameters on the two sides of one and the same axis, one for one roughness coefficient and the other for the other roughness coefficient, and to use one scale or the other depending on the conditions of the problem.

It would be possible, theoretically speaking, to extend the conclusion we have drawn also to the complete Flamant–Bertrand diagram, applying the same considerations and method to the individual systems of component conjugate scales. But the quite definite purpose, and moreover the complexity, of this diagram naturally preclude the application of such an operation to it.

The Lévy-Dariès diagram, being in construction entirely similar to the reduced Flamant–Bertrand diagram, of course permits questions to be solved, when applied to different degrees of roughness, in the same manner and by the same method.

All the above-mentioned diagrams of conjugate scales, constructed by d’Ocagne’s method, as has been indicated more than once, are intended for the calculation of conduits of circular cross-section. But the limits of their application are by no means restricted only to this type of cross-section. On the contrary, all of them can likewise be applied to the calculation of conduits of any cross-section whatever.

Beginning with the Lampe–Chizhov diagram, we see that its principal scales of slopes, roughness coefficients, hydraulic radii, and velocities are suitable for the calculation of all kinds of sections, and only the discharge scales for different diameters serve exclusively for circular pipelines. Therefore the Lampe–Chizhov diagram permits the direct solution of all problems relating to conduits of any cross-section whatever, the operation being carried out by means of only the principal scales; and the element determining the size of the section is, of course, the hydraulic radius, from which, depending on the shape of the section, the span or the height of the latter is determined.

The Lévy-Dariès and Flamant–Bertrand diagrams, by their construction, are adapted directly to the calculation of pipelines. The question of their application to the calculation of conduits of arbitrary cross-section can, of course, arise only with respect to diagrams with four scales, since part of the supplementary scales of the complete dia-

The Flamant–Bertrand diagrams, in essence, apply only to pressure pipelines, and the complexity of the diagram makes it practically inconvenient to introduce still further manipulations. As for diagrams with four scales, the Lévy–Vallot diagrams and the reduced Flamant–Bertrand diagram, they allow application to the calculation of conduits of any cross-section by means of a method which, of course, reduces to passing to an equivalent diagram constructed on the basis of the hydraulic radius. Such a method, with respect to the Flamant–Bertrand diagram (the reduced one), was proposed by me in a separate work¹.

Here it must be stipulated that this method, simple and practical though it is, nevertheless somewhat complicates the reading, and can be recommended only for sporadic cases of calculating other cross-sections; when it is necessary to solve a whole series of problems relating to a section of one type or another, it is more convenient to construct, or to have, a special diagram for this type of section.

In the preceding discussion we have been speaking throughout of the calculation of pipes working under complete filling. Such a case is predominant in the calculation of water-supply systems. The exact opposite is seen in the calculation of sewerage: here complete filling is an exceptional case, while incomplete filling is normal.

In fact, the principal calculation of a sewerage network is made for half filling. Moreover, the quantity of wastewater entering the sewerage system is subject to considerable fluctuations, and therefore a sewer, calculated for the maximum quantity of liquid when filled to its full height, will, when a smaller quantity enters it, be filled only in part, and in it there will occur what is called incomplete filling. The degree of this filling is usually determined by the ratio of the depth of the liquid stream in its middle part to the full internal height of the sewer in that same part.

Thus, in practice it is very often necessary to calculate sewers under incomplete filling, and also to determine the depth of flow in pipes.

Here two cases must be distinguished, namely half filling and filling to an arbitrary degree.

Problems relating to the first are easily reduced to complete filling on the basis of the consideration that, for the same diameter \(D\) and slope \(i\), in a sewer at half filling the velocity \(v\) is the same as under complete filling, while the discharge is twice smaller.

Problems concerning all degrees of filling in general reduce to determining the relation between the discharge and the velocity at full filling, and the discharge and the velocity at various degrees of filling.

Let us consider this question, for example, with respect to drains of circular section.

Let us imagine (Fig. 22) a drain of circular section with diameter \(D\). Divide its vertical diameter into 10 equal parts and draw through the points of division horizontal straight lines representing the surfaces of the liquid flowing through the drain at the corresponding fillings.

Calculating for each degree of filling the magnitude of the hydraulic radius \(\rho\), and assuming some slope \(i\), we shall determine the discharge \(Q\).

Having the value \(Q\) for full filling and \(Q_1, Q_2, Q_3 \ldots Q_9\) for fillings from 0.1 to 0.9, one may construct a curve showing how, for a given diameter and slope, \(Q\) changes with the change in filling. To construct this curve we take \(Q\) as unity and lay it off, on some scale, to the right of the line \(AB\) along the horizontal straight line \(AC\), corresponding to full filling. Since we have taken \(Q\), corresponding to full filling, as 1, then instead of the quantities \(Q_1, Q_2 \ldots\) it is necessary, on the lines corresponding to the other fillings, to lay off the values of the ratios

\[ \mu_1 = \frac{Q_1}{Q}, \]

\[ \mu_2 = \frac{Q_2}{Q}, \tag{154} \]

\[ \ldots\ldots \]

The points obtained by this laying-off are joined by a curved line.

If we take another diameter \(D_1\) and another slope \(i_1\), we obtain another curve \(Q\). Carrying out the construction for various \(D\) and \(i\), we can convince ourselves that all the curves \(Q\) so nearly coincide with one another that, for practical purposes, one common curve \(Q\), the same for all \(D\) and \(i\), may be adopted.

Having such a curve of the change of discharges as a function of the degree of filling, it is easy, from the quantity \(Q\) corresponding to full filling of the drain for the given \(D\) and \(i\), to find \(Q_m\) for any filling. For this it is only necessary, after determining the value \(\mu\) corresponding to the desired filling, to multiply \(Q\) by \(\mu\), or to divide it by

\[ \frac{1}{\mu}. \]

In the case of solving problems relating to incomplete filling, the same transition from the discharge at complete filling to the discharge at different degrees of filling can be carried out directly on the discharge scale. Indeed, equality (154) may be represented in the form

\[ Q_1=\frac{Q}{\frac{1}{\mu_1}}, \tag{154′} \]

\[ Q_2=\frac{Q}{\frac{1}{\mu_2}}, \]

\[ \cdots\cdots\cdots \]

Taking logarithms of the separate equalities (154′), we obtain

\[ \log Q_1=\log Q-\log \frac{1}{\mu_1}, \]

\[ \log Q_2=\log Q-\log \frac{1}{\mu_2}, \tag{155} \]

\[ \cdots\cdots\cdots \]

The expression \(\frac{1}{\mu_m}\) is introduced because its \(\log\) is positive for the majority of cases. Equations (155) show that, in order to obtain, on the diagram’s discharge scale, the magnitudes of the discharges \(Q_1, Q_2\ldots\) at different degrees of filling of the section, one must, from the division corresponding to the discharge \(Q\) at complete filling, step back toward the beginning of the scales by a length representing, with the scale modulus, \(\log \frac{1}{\mu_1}, \log \frac{1}{\mu_2}\ldots\), and the reading at the point thus obtained will give the required value of the discharge at incomplete filling.

For the sake of convenience in such an operation, the quantities \(\log \frac{1}{\mu_1}, \log \frac{1}{\mu_2}\ldots\), on the scale adopted for the quantity \(Q\), may be plotted on a special transition scale, separate or common with other transition quantities (such a scale will be presented below), with the degrees of filling corresponding to them inscribed opposite the divisions. The values of \(\mu\) and \(\frac{1}{\mu}\) needed for constructing the scale are given in Table II, as applied to a circular, ordinary ovoid, and trough-shaped section (with two centers), of such a type that, if the span is denoted by \(D\), then the radius of the trough is also equal to \(D\), while the radius of the overlapping semicircular arch is \(= \frac{D}{2}\).

The curve \(\ell\) shows that the quantity of liquid flowing through a drain of any section at a given slope will be greatest at a degree of filling approximately equal to 0.9. This quantity is greater than with complete filling. At a filling of about 0.8 the discharge is the same as the discharge corresponding to complete filling, and at lower degrees of filling it is less than the latter. Therefore, on the transition scale the mark for filling 0.8 coincides with the mark 1.0; the mark 0.9 will lie above, and the remaining ones below unity. It is clear that for degrees of filling for which

\[ \mu > 1,\ \log \frac{1}{\mu_m} \]

will be negative, and therefore one must lay off on the discharge scale the quantity

\[ -\log \frac{1}{\mu_m} \]

in the direction of increasing divisions. In constructing the transition scale, this occurs automatically.

Having such a scale cut out, and placing it on the diagram so that the point marked \(\frac{H}{D} = 1.0\) and 0.8 coincides with the point corresponding to the conditions of complete filling, at the point bearing the inscription of the given filling we find \(Q_m\) for that filling. Conversely, from \(Q_m\), \(i\), and \(D\) one can find the degree of filling \(\frac{H}{D}\) corresponding to them.

The velocity of flow \(v\), as is known, also varies depending on the degree of filling, and in sewerage calculations it is sometimes necessary to determine velocities at different degrees of filling. As in the case of discharge, the question here is reduced to passing from the velocity at complete filling to the velocity at various degrees of filling, on the basis of a previously determined ratio between these velocities. Let us again take, for example, a circular section and apply as before 10 degrees of filling, corresponding to Drawing 22. Determining, for each degree of filling at the mean value of the diameter and slope, the flow velocities \(v_1, v_2 \ldots\) and comparing them with the velocity at complete filling, we obtain the ratios

\[ \nu_1 = \frac{v_1}{v}, \]

\[ \nu_2 = \frac{v_2}{v}, \tag{156} \]

\[ \ldots \]

which, like the preceding ones, may be regarded as practically corresponding to all diameters and slopes.

By laying off the quantities \(\nu_1, \nu_2 \ldots\) on the same lines where \(\mu_1, \mu_2 \ldots\) were laid off, we obtain a curve determining the variation of velocities as a function—

dependence on the degrees of filling. This curve makes it possible, from the known velocity at complete filling, to determine the velocities at various degrees of filling, on the basis of the relation

\[ v_m=\nu_m v . \tag{156′} \]

To apply the same method in a diagram of conjugate scales, we take the logarithms of equalities of type (156′):

\[ \log v_m=\log v-\log \frac{1}{\nu_m}. \tag{157} \]

Equation (157) shows that the velocity of flow at any degree of filling \(v_m\) can be obtained on the velocity scale of the diagram if, from the division determining the velocity \(v\) at complete filling, one takes a reading at a distance corresponding to \(\log \dfrac{1}{\nu_m}\) in the direction toward the beginning of the scale.

Here also it is most convenient to put the quantities \(\log \dfrac{1}{\nu_1}\), \(\log \dfrac{1}{\nu_2}\) ... on a transfer scale. The values of \(\nu\) and \(\dfrac{1}{\nu}\) necessary for this are also given in Table II, as applied to sections of three types. The use of the scale is in all respects similar to the use of the transfer scale for discharges mentioned above.

TABLE II.

Values of \(\mu\), \(\dfrac{1}{\mu}\), \(\nu\), \(\dfrac{1}{\nu}\) for different degrees of filling.

Section	Quantity	1	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1
Circular section	\(\mu\)	1.00	1.07	1.00	0.85	0.67	0.50	0.33	0.19	0.09	0.02
Circular section	\(\log \dfrac{1}{\mu}\)	0	−0.03	0	0.07	0.17	0.30	0.48	0.72	1.05	1.70
Circular section	\(\nu\)	1.00	1.14	1.15	1.13	1.08	1.00	0.90	0.77	0.59	0.35
Circular section	\(\log \dfrac{1}{\nu}\)	0	−0.05	−0.06	−0.05	−0.03	0	0.05	0.11	0.23	0.46
Ovoid section	\(\mu\)	1.00	1.05	0.90	0.75	0.58	0.42	0.26	0.15	0.07	0.02
Ovoid section	\(\log \dfrac{1}{\mu}\)	0	−0.02	0.05	0.13	0.24	0.38	0.59	0.82	1.20	1.70
Ovoid section	\(\nu\)	1.00	1.12	1.12	1.08	1.03	0.94	0.85	0.75	0.61	0.41
Ovoid section	\(\log \dfrac{1}{\nu}\)	0	−0.05	−0.05	0.03	−0.01	0.03	0.07	0.13	0.23	0.39
Flume section	\(\mu\)	1.00	1.07	1.00	0.88	0.70	0.52	0.35	0.19	0.07	0.01
Flume section	\(\log \dfrac{1}{\mu}\)	0	−0.03	0	0.06	0.16	0.30	0.46	0.72	1.16	2.00
Flume section	\(\nu\)	1.00	1.15	1.16	1.13	1.07	1.00	0.89	0.73	0.54	0.28
Flume section	\(\log \dfrac{1}{\nu}\)	0	−0.06	−0.06	−0.05	0.03	0	0.05	0.14	0.29	0.44

From the form of the curve of variation of velocities at different degrees of filling, one may conclude that the greatest velocity is obtained approximately at

\[ \frac{H}{D}=0.8; \]

at a degree of filling of \(0.5\) the velocity is the same as at complete filling; when \(\frac{H}{D}\) is greater than \(0.5\) it is greater, and when \(\frac{H}{D}\) is less than \(0.5\) it is less than at complete filling. This will also correspond to the arrangement of the divisions on the conversion scale relative to

\[ \frac{H}{D}=1.0 \]

From the foregoing it is clear that diagrams of conjugate scales, whether constructed specially for each form of cross-section or even intended for the calculation of only one form of cross-section, equally make it possible to solve all problems relating to every cross-section, transverse dimension, degree of filling, and roughness coefficient. The limits of applicability of each such diagram embrace, in essence, all cases that may arise in the practice of calculations for water supply and sewerage; even the introduction of separate diagrams for different cross-sections is dictated not by necessity, but only by considerations of practical convenience.

All operations relating to the calculation are, as we know, very simple manipulations with the scales of the diagram; and in the case of a change in those basic conditions for which a separate diagram is intended, the matter reduces to transferring readings by means of the conversion scale.

Knowing now all the cases and problems for which this repeatedly mentioned conversion scale can serve, we may finally dwell on its composition and form. The composition of such a scale depends on the composition and number of diagrams. We shall consider the conversion scale in the form corresponding to the preceding exposition, i.e., a scale intended for transition from a diagram for a definite form of cross-section (namely, circular, which is predominantly likely to occur in practice), at one degree of roughness and at complete filling, to other roughness coefficients and degrees of filling. Such, for example, might be the conversion scale for Flamant–Bertrand’s diagram of conjugate scales with four scales, which we shall also construct.

Summarizing, on the basis of the foregoing, the cases of application of the conversion scale, we see that it must give the following distances:

1) distances serving for the transition from the roughness coefficient for which the diagram is constructed to the calculation with other roughness coefficients; they correspond to \((\log a_m - \log a)\), with the modulus of the scale of diameters, where \(a\) is the principal roughness coefficient, and \(a_m\) is the general designation of other roughness coefficients to which a transition may be required;

2) distances serving for the transition from the velocity \(v\) at full filling to the velocities \(v_1, v_2 \ldots\) at different degrees of filling, which represent the quantities \(\log \frac{1}{\nu_1}, \log \frac{1}{\nu_2} \ldots\), laid off with the modulus of the velocity scale;

3) distances serving for the transition from the discharge \(Q\) at full filling to the discharges at other degrees of filling \(Q_1, Q_2 \ldots\), which correspond to the quantities \(\log \frac{1}{\mu_1}, \log \frac{1}{\mu_2} \ldots\), with the modulus of the discharge scale.

These four elements of the transition scale are the most important. To them one might also add:

1) the distance serving for the transition from a calculation with the parameter \(D\) and the system of scales \(v, i, D\) to the parameter \(\rho\), i.e. to the diagram \(v, i, \rho\);

2) distances for determining, by hydraulic radii, the parameters of sections of a form other than circular, i.e. the span \(D\) or the height \(H\);

3) distances for the transition from calculating discharges with the parameter \(D\) to calculating discharges with the parameter \(\rho\) for various forms of section.

These latter quantities are needed only for passing from the diagram of a circular section to the calculation of cross-sections of other forms. The data relating to them, together with the correspondingly modified transition scale, will form the subject of the special article mentioned above. Therefore I consider it possible here to exclude these quantities from consideration.

Drawing 26 represents a transition scale, compiled by us, as an example, for the Flamant–Bertrand diagram of conjugate scales. It consists of two scales. On the left scale, from the point \(D\) downward, there is first laid off the distance for the transition from the roughness coefficient \(a = 0.00092\) to \(a_1 = 0.00074\). This distance, with the modulus of the scale of diameters of the diagram shown in Drawing 19, proves to be equal to 6.8 mm, and its end is marked by the letter \(D'\). Farther, from the same point \(D\), there is laid off downward the distance for

of transition from the scale of diameters to the scale of hydraulic radii. Its length in the present case is 54.2 mm, and its end is denoted by the letter $\rho$. The line corresponding to $\rho$ is also denoted by the letter $v$ and serves as the origin of distances for passing from the velocity at complete filling to the velocities at degrees of filling 0.1, 0.2...0.9, the ends of such distances being denoted respectively by the figures 1, 2, 3...9. On the right-hand scale of the scale, from the initial line denoted by the letter $Q$, distances are laid off upward and downward for passing from the discharge at complete filling to the discharges at degrees of filling 0.1—0.9, the ends of which are denoted by the figures 1—9 and, for the sake of clarity, are connected with the divisions for the velocities at the corresponding degrees of filling.

The application of such a transitional scale is known from the foregoing. It may be used with the aid of dividers, or, better still, by pasting it onto cardboard and applying it each time to the corresponding scale of the diagram.

In thus concluding the exposition of the questions relating to hydraulic calculation by means of diagrams of coupled scales and to individual applications of this method, we have sufficient material for comparing it with other methods of calculation and for a comparative appraisal of its merits. From the foregoing it is evident that the method of coupled scales represents a valuable means for facilitating the calculation of sanitary-engineering structures. A comparative appraisal of it should indicate to us its place among other means serving the same purpose.

The value of all means that accelerate and facilitate the performance of numerous and complex computations connected with engineering calculations is universally recognized. All aids directed to this purpose in the field of hydraulic calculation strive to eliminate completely the need to carry out any calculations whatever and to make it possible to perform the entire calculation of pipe networks with only this aid and a form for recording the results obtained at hand. Such aids are various kinds of tables, numerical or graphical. The method of coupled scales, while achieving to perfection the aim set by all methods in general that are directed toward simplifying hydraulic calculation, brings the calculation process, one may say, to ideal simplicity and speed, without any detriment to practical accuracy.

These merits of the method arise from the successful combination of the advantages that are inherently characteristic of graphical tables in substance, and that are obtained thanks to the introduction of a logarithmic character into the subdivision of the tables, as well as from the properties of the method that lies at the basis of the present method. Indeed, comparing the method of calculation by means of diagrams of conjugate scales with calculation by means of numerical tables, we can point out a number of features that emphasize the advantages of graphical tables and, in particular, of diagrams of conjugate scales.

Numerical tables, allowing the possibility of having only two entering variable quantities along two coordinates, compel one to break up a calculation formula, which usually contains a larger number of variables, into component parts, to prepare a separate table for each such part, and to use several tables jointly. In this way both the number of tables and the time required for the work increase, as does the possibility of error in using them. In diagrams of conjugate scales, the scales for all variables are placed on one and the same sheet, in the form of one or several systems. Moreover, even in the case of passing from one system to another, the operation is carried out very easily, without memorizing complicated numbers, merely by the simple movement of a ruler or transparent straightedge.

Numerical tables give, in calculation, a series of numbers that say little to the mind and, moreover, do not permit confidence in their freedom from error, if one takes into account the difficulty of correcting numerical tables of all kinds. Thus, when using them, one has to rely on the fortunate chance that no error of any sort will be encountered in any of the figures taken, without any possibility of easily detecting the error. For a graphical table, which gives results in an evident regular order to which the eye quickly becomes accustomed, the error is limited only by the bounds of the accuracy of reading (if one does not count the possibility of error in writing down the result obtained). A gross error, however, cannot occur when the diagram is used correctly.

Numerical tables, offering a direct answer only for numbers written along the coordinates and inevitably differing from one another by greater or lesser amounts, require, for intermediate values, either interpolation (not always exact when the intervals are large) or only an approximate calculation; this, with formulas broken up into several parts, is undesirable, especially in ...

to calculate the first of these parts, which serve as the basis for calculating the following ones. A graphical table, by giving a continuous series of values embodying the dependence between variable quantities, offers an immediate answer for any given values.

The advantages just enumerated are characteristic of graphical tables in general. But, as we mentioned at the very beginning, among graphical tables two types must be distinguished. In some, in order to construct the lines expressing the equations of calculation, the variables themselves are plotted along the coordinate axes (diagrams of isopleth curves); in others, along axes arranged in one way or another, the logarithms of the variables are laid off (logarithmic tables and diagrams of conjugate scales). It must be said that the method of conjugate scales, by adopting the logarithmic character of the divisions, thereby provides new advantages distinguishing it from diagrams with normal graduation of the axes. These advantages consist chiefly in the following.

Owing to the law of variation of the logarithms of consecutive numbers, the range of a table, i.e. the limits of the values entering into it, may be very broad, while the divisions remain sufficiently accurate. Indeed, in order to fit into a normal scale quantities from 0.00001 to 1, while still making it possible to read them off, either an enormous drawing size would be required, or the marks of the divisions would become very widely separated, or the divisions would be imperceptible to the naked eye. In diagrams of the logarithmic type, for the same purpose extremely limited drawing dimensions are required, with entirely sufficient clarity and accuracy. This is the valuable quality of logarithmic diagrams.

Further, the percentage accuracy of calculations when readings are taken from diagrams with logarithmic subdivisions is always one and the same. In operations with small numbers the absolute accuracy is greater; in operations with large numbers it is smaller, but the magnitude of the possible error relative to the reading remains the same. This has been proved elsewhere, and here we shall only repeat that such a property of our diagrams fully corresponds to the requirements of practice.

Finally, the graphical representation of all formulas involving multiplication and division of variables and raising them to any integral or fractional, positive or negative power, on diagrams of the logarithmic type is reduced to the shifting of straight parallel lines or of points to one or another

another distance, which greatly facilitates the construction of diagrams, easily permits their supplementation with new variables, and transitions from one set of variables to another.

Only one disadvantageous aspect of diagrams of the logarithmic type could be pointed out. They express very complicatedly formulas that include the operations of addition and subtraction. However, in application to the calculation of water supply and sewerage this disadvantage disappears, since for this purpose we have formulas of a binomial logarithmic type, which provide no less a guarantee of calculation accuracy than polynomial formulas.

If one compares the two kinds of logarithmo-graphic calculation—namely, the method of logarithmo-graphic tables and the method of conjugate scales—then it must first of all be said that both of these kinds are distinguished by such general merits and bring hydraulic calculation to such simplicity and convenience that it is difficult to draw a serious distinction between them. Nevertheless, in the method of logarithmo-graphic tables one may note certain shortcomings that are avoided in the method of conjugate scales.

In fact, the method of using logarithmo-graphic tables reduces each time to taking the point of intersection of two lines and reading the graduation of the straight line passing through this point. Since the graduations, of necessity, are written either at the ends of the lines or, in any case, at some usually considerable distance from one another, one must, whether starting from the graduation and seeking the point of intersection, or conversely, follow along the line for a certain length. When this operation is applied to three lines at each reading, there is always the danger, at the slightest inattention, of passing from the given line to a neighboring one and thus making an error in the reading. This inconvenience is mitigated, in compiling logarithmo-graphic tables, by various measures, but cannot be avoided completely. In this respect, use of a diagram of conjugate scales, where the sought point and its reading are indicated by the intersection of a ruler or a line on a transparent sheet with the corresponding scale, appears somewhat simpler.

Further, if the values of the variables with which one has to deal are not those to which the lines actually drawn on the drawing correspond, then, when using a logarithmo-graphic table, one has to make a mental interpolation between these lines. This operation, for all its ease, is still not as simple as when it is necessary

to mark by eye an intermediate point between the divisions placed on one of the scales of our diagrams.

The number and mutual intersection of lines in logarithmic-graphic tables do not, of course, present any especially serious inconvenience, since the eye quickly becomes accustomed to this. However, this circumstance, as well as the considerable attention required in the work, may in the end tire the eye; and the more systems of lines a table contains, the sooner this will happen. In a diagram of conjugate scales, in its usual composition, there can be no question of one system of scales obscuring another; and even with several systems of scales there is no need to speak of the darkening that is quite possible in a logarithmic-graphic table. It should also be supposed that, for this reason, the use of a diagram of conjugate scales is less tiring.

Finally, in connection with what has just been said, it must be added that if it is necessary to introduce into the diagram new variables which, for some reason, were not taken into account when it was prepared, then although this is equally easy in diagrams of both types, each new variable contributes to the darkening of the logarithmic-graphic table, something that need not be feared with respect to the diagram of conjugate scales.

Here, incidentally, it must be emphasized, with regard to both types of diagrams, that their chief advantage lies in their simplicity. Therefore one should avoid in them everything that may be superfluous and that may make reading difficult. In addition, they must be drawn on a sufficiently large scale; otherwise the approximation may be insufficient.

The foregoing does not prevent us, while recommending to the attention of readers the method of conjugate scales, from considering calculation by means of logarithmic-graphic tables to be very convenient and valuable.

There is no need to say much about comparing the merits of the individual applications of the method of conjugate scales with one another after what has been said earlier. We have seen that all the diagrams we have indicated—Lampe–Chizhov, Flamant–Bertrand, and Lévy–Darcy—are fully adapted to solving hydraulic problems relating to those dimensions of cross-sections and conditions of water motion that are covered by the formulas on which they are based. We have also seen that all these diagrams are applicable either directly or with the aid of transition scales to all degrees of filling, roughness coefficients, and cross-sectional forms. Such breadth of application is limited, however, by the condition that, for these applications,

roughness coefficients must be developed that correspond to the respective conditions of water motion. If this circumstance is taken into account, the broad applicability of the diagrams of Flamant-Bertrand and Lévy-Darcy is only theoretical, since the absence of roughness coefficients relating to the motion of wastewater makes their practical application possible only to circular sections. However, the roughness coefficients for Flamant’s formula can be determined by comparison with Lampe’s formula and its coefficients.

Here it is also appropriate to note that the Lampe-Chizhov diagram may be compared with a logarithmic slide rule for hydraulic calculation, constructed according to the same Lampe formula. It must be said that although the Lampe-Chizhov diagram has no movable parts, unlike a rule, nevertheless using it with the aid of a pair of compasses is just as convenient as using a rule. This diagram is very convenient because of its portability, and also because of the possibility of directly using different roughness coefficients.

The data just presented on the advantages of the method of calculation by means of diagrams of conjugate scales show with sufficient clarity that in this method we have an extremely valuable means for the hydraulic calculation of water-supply and sewerage systems. From this point of view it is useful and desirable that the method of conjugate scales be applied as widely as possible in this country, following the example of other countries, in the practice of sanitary-engineering work.

It is also generally desirable to apply formulas of logarithmic form to the calculation of sanitary-engineering structures; part of the present issue has been devoted to consideration of them, in view of their greater simplicity of use, their sufficient practical accuracy, and the possibility of transforming them into graphical form, which further facilitates the calculation process. If technical life should bring forward new formulas for calculating pipelines that go beyond the aforementioned logarithmic formulas in their correspondence to reality and in the breadth of the limits of application (since, in terms of practicality, they cannot be surpassed), then one may wish that they be cast in the same logarithmic form and be applied in practice in the form of diagrams of logarithmic type, in particular diagrams of conjugate scales.

1908.

Literature

Akulov. Service of old water-supply pipes and application of the graphical method to the solution of hydraulic problems (Proceedings of the V Water-Supply Congress, 1901).

D’Aubrivé et Villerupt. L’album des abaques pour le calcul des conduites d’eau. 1891.

Baumeister. Städtische Strassenwesen und Städtereinigung. 1887.

Bechmann. Salubrité urbaine. II. Assainissement. 1899.

Wehner. Ein Betrag zur Berechnung des Rohrwiderstandes in der Praxis. (Gesundheits-Ingenieur, 1897).

Vrubevsky. Graphical method for calculating drains (News of the Society of Civil Engineers, 1907).

Hobrecht. Die Kanalisation von Berlin. 1882.

Gorbachev N. F. On the calculation of flow velocities and conveyance capacity in water-supply lines and drains. 1904.

Daries. Calcul des conduites d’eau. 1900.

Evnevich I. A. Course of Hydraulics. 1897.

Coffin. The graphical solution of hydraulic problems. 1900.

Collignon. Cours de mécanique. II. Hydraulique.

Lalanne. Mémoire sur les tables graphiques et sur la géométrie anamorphose appliquée à diverses questions qui se rattachent à l’art de l’ingénieur (Annales des Ponts et Chaussées, 1846).

Lévy. Théorie d’un courant liquide à filets rectilignes et parallèles de forme traversale quelconque. Applications aux tuyaux de conduites. (Annales des Ponts et Chaussées, 1867.)

Lampe. Untersuchungen über die Bewegung des Wassers in Röhren (Civil-Ing., 1873. Bd., XIX.)

• Maksimenko, F. E. Course in Hydraulics. 1902.

• Van-Muyden. Abaque logarithmique pour le calcul des conduites d’eau sous pression. 1905.

• D’Ocagne. Coordonnées parallèles et axiales. 1885.

• D’Ocagne. Nomographie. 1891.

• D’Ocagne. Traité de Nomographie. 1899.

• D’Ocagne. Exposé synthétique des principes fondamentaux de la Nomographie. 1903.

• Satkevich, A. A. Calculation of central-heating and ventilation systems by means of graphical tables with a logarithmic grid. 1906.

• Satkevich, B. A. Calculation of a water-supply pipe network by means of a logarithmic-graphical table (Stroitel’, 1898).

• Schilling. Ueber die Nomographie de M. d’Ocagne.

• E. B. and G. M. Taylor’s diagrams of the discharge of pipes in accordance with Kutter’s formula. 1891.

• Thiem. Ueber graphische Durchmesserbestimmung bei Wasserleitung (Journ. für Gasbeleucht. und Wasserversorg., 1885).

• Flynn. The motion of water in irrigation canals, ditches, flumes, water-supply pipes, drains, etc. 1893.

• Flamant. Hydraulique. 1900.

• Frank. Die Formeln über die Bewegung des Wassers in Röhren (Civil—Ingenieur, 1881).

• Frank. Berechnung der Kanäle und Rohrleitungen.

• Cherepashinsky, M. M. Water Supply of Cities. 1905.

• Chizhov, N. K. Water Supply Systems (lithographed course).

• Chizhov, N. K. A mechanical method for calculating head loss (Stroitel’, 1897).

• Yaskovich, M. Calculation of drains by means of logarithmic-nomographic tables. 1906.

• Yaskovich, M. A graphical method for calculating a network of water-supply pipes. 1905.

Table 1. Diagram of the applicability of curves for calculating water-supply pipes (Bazin–[[unclear: second name]] formula).

Hydraulic gradients

Loss of head, in meters

Pipe length, in meters

Curve labels shown on the diagram:

• \(d = \frac{1}{5}\)
• \(d = \frac{1}{6}\)
• \(d = \frac{1}{7}\)
• \(d = \frac{1}{8}\)
• \(d = \frac{1}{9}\)
• \(d = \frac{1}{10}\)
• \(d = \frac{1}{11}\)
• \(d = \frac{1}{12}\)

Table II

Fig. 9. Logarithmic table for calculating water-supply pipes (Lampe formula).

Pipe diameter \(D\), in meters.

Length \(l\), in meters, plotted on the \(n\)-axis.

Head loss \(h\), in meters.

Metal pipes

Cast-iron pipes

Inset:

Diagram of the graphical method of calculation.

[[unclear: small handwritten explanatory labels and formula inside the inset diagram]]

The main field of the plate is a logarithmic calculation chart with families of oblique and vertical guide lines, numerical scales, and plotted construction lines for determining the quantities \(D\), \(l\), and \(h\) in water-supply pipe calculations according to Lampe’s formula.

Table III

Fig. 3.

[Diagram: a horizontal axis from \(0\) to \(X\), with marked points \(x_0, x_1, x_2, x_3, \ldots, x_n\).]

Fig. 4.

[Diagram: a graduated scale from \(1\) to \(10\).]

Fig. 5.

[Graph: coordinate grid with vertical axis \(y\) from \(0\) to \(800\), horizontal axis \(x\) from \(0\) to \(70\), and a curve labeled \(z=a\).]

Plate IV.

Figure 6.

Figure 7.

Figure 8.

Table V.

Fig. 9.

Fig. 9a.

Fig. 10.

Fig. 11.

Plate VI.

Fig. 12.

Visible labels in the diagram: \(Y\), \(x\), \(w_{1,2}\), \(y\), \(w_{2,3}\), \(z\), \(w\), \(u\), \(w_{u,v}\), \(v\); \(x_0\), \(y_0\), \(z_0\), \(u_0\), \(v_0\); \(P_{1,2}\), \(P_{2,3}\), \(P\), \(P_{u,v}\); \(\Omega\), \(E\); \(A\), \(S_{1,2}\), \(B\), \(S_{2,3}\), \(C\), \(J\), \(M\), \(S_{u,v}\), \(N\); \(2\).

Fig. 13.

Visible labels in the diagram: \(x\), \(w_{1,2}\), \(y\), \(u\), \(u_{1,3}\), \(w_{2,3}\), \(z\); \(x_0\), \(y_0\), \(z_0\); \(P_{1,2}\), \(P_0\), \(P_{1,3}\), \(P_{2,3}\); \(A\), \(S_{1,2}\), \(B\), \(S\), \(S_{1,3}\), \(S_{2,3}\), \(C\).

Plate VIII.

Fig. 5. Diagram by Prof. N. K. Chizhov for calculating water-supply pipes
(Lampe formula).

\(Q\) — discharge of water per second \([in cubic feet]\), on the scale for a circular pipe filled with water, with diameter from \(2\) to \(30''\).

\(R\) — hydraulic radius:

\[ R=\frac{\text{area of the live cross-section}}{\text{wetted perimeter}} \]

\(V\) — mean velocity of flow.

\(J\) — slope of the piezometric line of the pipe.

\(k\) — roughness coefficient of the Lampe formula of the form

\[ J=k\frac{V^{1.8}}{R^{1.25}} . \]

\(L_{\mathrm{mp}}\) — type of pipe.

Values of \(k\) for — smooth conduits; canals, rivers.

Table VIII

Drawing 156. Wener diagram for calculating water-supply pipes (Lampe formula).

Visible labels in the diagram:

• \(Q\) — quantity of water.
• \(D = 80\) to \(D = 600\) — pipe diameters.
• \(V\) — velocity.
• \(J\) — loss of pressure head.
• \(K\) — coefficient.

\[ J = K\,\frac{V^{[[unclear: exponent]]}}{R^{[[unclear: exponent]]}} \]

Additional visible multilingual labels include:

• Geschwindigkeit — velocity.
• Druckhöhenverlust — loss of pressure head.
• Coefficient / Koefficient — coefficient.
• La vitesse — velocity.
• Le coefficient — coefficient.

Plate IX.

Drawing 14

Drawing 16

Drawing 22.

Table X.

Drawing 17. Diagram for calculating Darius water-supply pipes (Lévy–Vallot formula).

Diagram labels:

• Flow rate in liters per second
• Diameters
• Velocity in meters per second
• Pipe materials:
• Bituminized cast iron
• New cast iron
• Ordinary cast iron
• Cement
• Riveted
• [[unclear: pipe material label]]

Table XI.

Drawing 18. Diagram for the calculation of Bertrand water-supply pipes (Flamant formula).

Table XII.

Drawing 19. Reduced Bertrand diagram for calculating water-supply pipes (Flamant formula).

Flow rates
(liters per sec.).

Diameters
(meters).

Hydraulic
gradients.

Velocities
(meters per sec.).

Plate XIII.

Drawing 20. Diagram for calculating water-supply pipes according to Flamant’s formula (foot measures).

Discharge. Cub. ft. per sec.
Gallons per min.

Diameters in inches.

Head losses in feet per 1000 ft.

Velocities in feet per sec.

Table XIV.

Drawing 21.

Drawing 23.

1. “The transitional scale and its application to the calculation of conduits.” (To be printed in the Proceedings of the Tomsk Technological Institute for 1911.) ↩↩↩↩↩↩↩↩↩↩↩↩↩

2. Théorie d’un courant liquide à filets rectilignes et parallèles de forme transversale quelconque. Applications aux tuyaux de conduites. Annales des Ponts et Chaussées. 1867. ↩↩↩

3. P. F. Gorbachev, in his work “On the Calculation of Flow Velocities, etc.,” shows this by means of a table giving comparative values of the velocity coefficient \(k\) (p. 50). ↩