Ya. I. Nikolin

Professor of the Tomsk Technological Institute of Emperor Nicholas II.

TRANSITIONAL SCALE

AS APPLIED

TO D’OCAGNE DIAGRAMS

FOR CALCULATING WATER PIPELINES

With 3 plates of drawings.

Tomsk.

Typo-lithography of the Siberian Association of Printing, Dvoranskaya St. and Yamskoy Lane, own building.

1913.

Printed by order of the Director of the Tomsk Technological
Institute of Emperor Nicholas II.

The graphical calculation of pipelines is carried out by means of diagrams which, depending on the nomographic principles underlying them, are called diagrams of isopleth curves or straight lines, logarithmic tables, and diagrams of conjugate scales.^1) Of the large number of such diagrams, the newest and most convenient in application are the diagrams of conjugate scales, constructed by the method of points of rectilinear intersection. Since both the discovery of this method and its first applications to questions of graphical calculation in engineering belong to M. d’Ocagne (Maurice d’Ocagne, professor of the Paris École des Ponts et Chaussées), I shall call the indicated diagrams, as a distinction from another type of conjugate-scale diagrams, d’Ocagne diagrams. In this method, which is applicable and applied to hydraulic formulae of logarithmic form, thanks to the almost ideal simplicity of handling the diagrams, we have an extremely valuable means that reduces to a minimum the labor and time required in the calculation of water supply and sewerage systems, without any detriment to the accuracy of the results obtained.

Leaving aside the theoretical foundations of the method underlying d’Ocagne diagrams, we shall explain their essence and application by a particular example, namely by the diagram for calculating circular-section water pipes according to Prof. Flamant’s formula, especially since we shall have occasion to use this diagram in the further exposition as well.

D’Ocagne diagrams serve for the graphical representation of equations having the form

\[ f_1(x) + f_2(y) = f_3(z). \tag{1} \]

Equations of the form

\[ f_1(x) \cdot f_2(y) = f_3(z), \tag{2} \]

are reduced to them by taking logarithms of both sides and transforming into the form

\[ \log f_1(x) + \log f_2(y) = \log f_3(z). \tag{3} \]

^1) For details on this question, see my work “On the Question of Graphical Methods for Calculating Water Supply and Sewerage. Issue II. Classification and Theoretical Premises of Nomographic Methods for Calculating Water Pipelines.” (Proceedings of the Tomsk Technological Institute, 1913, and separate offprint).

These diagrams have the form of three (or a larger number, when there is a greater number of variables in the equation) logarithmic scales, arranged parallel to one another and standing in such 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.

Before proceeding to explain the method of constructing equations of the form (1) as d’Ocagne diagrams, it is necessary to give an idea of one fundamental device of nomography, namely the construction of scales of functions.

Let there be a certain 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. 1), the lengths

\[ \begin{aligned} l_1 &= \lambda f(x_1),\\ l_2 &= \lambda f(x_2),\\ l_3 &= \lambda f(x_3),\\ &\cdots \end{aligned} \tag{4} \]

where \(\lambda\) is an arbitrarily chosen length, and we shall write above the points marking 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 their numerical markings, will constitute 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 partial values of the variable, for example \(x_0\) and \(x_n\), then it may be constructed starting from the lower limit \(x_0\), without use of the origin of coordinates \(O\). To obtain the points \(x_1, x_2, x_3,\ldots\), one must lay off on the axis, beginning from an arbitrarily chosen point marked \(x_0\), the segments

\[ \begin{aligned} l_1 &= \lambda [f(x_1)-f(x_0)]\\ l_2 &= \lambda [f(x_2)-f(x_0)]\\ l_3 &= \lambda [f(x_3)-f(x_0)]\\ &\cdots\\ L &= \lambda [f(x_n)-f(x_0)], \end{aligned} \tag{5} \]

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

Taking

\[ f(x)=x \]

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, one sometimes has to apply the construction to other functions; then scales of functions of another character are obtained, for example logarithmic, segmental, isogradual.

Of the various scales that are obtained as a result of plotting functions, the most important is the logarithmic scale. If in equalities (4) we take

\[ f(x)=\log x, \tag{6} \]

and apply to this case the method of construction indicated above, then the logarithmic scale of the function is obtained.

For constructing a d’Ocagne diagram representing the equation

\[ f_1(x)+f_2(y)=f_3(z), \tag{1} \]

one must choose on an arbitrary straight line two points \(A\) and \(B\), at any distance from one another, and through these points draw the straight lines \(AX\) and \(BY\), parallel to one another. Along these straight lines \(f_1(x)\) and \(f_2(y)\) are laid off in the form of function scales, with arbitrarily chosen moduli \(\lambda_1\) and \(\lambda_2\), on the basis of the equalities

\[ \begin{aligned} x'&=\lambda_1 f_1(x),\\ y'&=\lambda_2 f_2(y), \end{aligned} \tag{7} \]

where \(x'\) and \(y'\) are the distances of the graduation points of the scales from the points \(A\) and \(B\). Next, on the line \(AB\) a point \(C\) is chosen which divides the distance between the points \(A\) and \(B\) into parts \(AC\) and \(CB\) proportional to the chosen moduli of the scales \(\lambda_1\) and \(\lambda_2\), i.e. on the basis of the proportion

\[ AC:CB=\lambda_1:\lambda_2. \tag{7′} \]

Through the point \(C\) a straight line is drawn parallel to \(AX\) and \(BY\), which is taken as the axis of the scale of the function \(f_3(z)\). The modulus of the scale of this function \(\lambda_3\) is determined on the basis of the equality

\[ \lambda_3=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}, \tag{8} \]

or, otherwise,

\[ \frac{1}{\lambda_3}=\frac{1}{\lambda_1}+\frac{1}{\lambda_2}. \tag{8′} \]

Finally, along the axis \(CZ\), from the point \(C\), \(f_3(z)\) is laid off in the form of a function scale, with modulus \(\lambda_3\). The system of three scales thus obtained will represent a d’Ocagne diagram for equation (1).

Such a diagram makes it possible to determine the value of any one of the functions entering into equation (1), if the other two functions are known. For this it is sufficient to intersect the scale of the sought function by a straight line connecting the points of the other two scales corresponding to the assigned values of the functions. In general, any straight line intersecting the three conjugate scales of the diagram gives, at the points of intersection with them, values of the functions laid off on the scales that simultaneously satisfy the equation represented by the diagram.

Ya. I. Nikolin.

Prof. Flamant’s formula¹, of whose diagram we shall now have to speak, is usually applied in the form

\[ D^{5/4} i = a v^{7/4} \tag{9} \]

or

\[ D^5 i^4 = a^4 v^7, \tag{9'} \]

where \(D\) is the diameter of the pipe,
\(i\) is the hydraulic gradient,
\(v\) is the velocity,
\(a\) is the coefficient of roughness.

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 covered with deposits, such as is usual after several years of service of a water-supply pipe, on the average

\[ a_2 = 0.00092. \]

Flamant’s formula, in whatever form it may be taken, represents a special case of an equation of the form

\[ f_1(x)\cdot f_2(y) = f_3(z) \tag{2} \]

and therefore can be represented in the form of a diagram of conjugate scales, by d’Ocagne’s method.

To represent this formula in the form of a diagram with scales of diameters, hydraulic gradients, and discharges, it is necessary to transform the formula

\[ D^5 i^4 = a^4 v^7 \tag{9'} \]

on the basis of the relation

\[ v = \frac{4Q}{\pi D^2}. \tag{10} \]

Then, for the coefficient \(a\) for pipes that have been in service, we obtain

\[ D = 0.251 \frac{Q^{7/19}}{i^{4/19}}. \tag{11} \]

Applying logarithms to formula (11), we obtain:

\[ \log D = \log 0.251 + \frac{7}{19}\log Q - \frac{4}{19}\log i. \tag{12} \]

To represent this equation in the form of a diagram, guided by the rules set forth above, we shall take two parallel

the axes \(Q\) and \(J\) (Fig. 2), 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, on each of them respectively, by means of a logarithmic ruler, we lay off the scales of the functions \(Q\) and \(i\).

For choosing the modulus of laying off, we shall take the following into consideration: if we were to adopt, for both scales without alteration, the modulus \(\lambda\) of the logarithmic ruler, then, in order to lay off the functions \(\dfrac{7}{19}\log Q\) and \(\dfrac{4}{19}\log i\), we should have either to multiply all \(\log\) by \(\dfrac{7}{19}\) and \(\dfrac{4}{19}\), or to change the scale of the logarithmic ruler in the ratios \(\dfrac{7}{19}\) and \(\dfrac{4}{19}\).

In order not to do this, it is more convenient to take the moduli

\[ \text{for the scale } Q\ —\ \lambda_1=\frac{19}{7}\lambda \]

and for the scale \(i\) —

\[ \lambda_2=\frac{19}{4}\lambda \]

(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 scales \(Q\) and \(i\), in order to express the functions \(\dfrac{7}{19}\log Q\) and \(\dfrac{4}{19}\log i\), the quantities \(\log Q\) and \(\log i\) on the scale of the logarithmic ruler.

Since the function \(\dfrac{4}{19}\log i\) has a negative sign, while \(\dfrac{7}{19}\log Q\) is positive, the increase of the numerical values of the divisions on the scales \(Q\) and \(J\) must proceed in different 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 carry the divisions from them in both directions.

Having thus constructed the scales of discharges and slopes, it is necessary, on the basis of relations (7) and (8), to determine the position, modulus, and divisions of the scale of diameters \(D\).

Since, according to the preceding, the moduli of the scales \(Q\) and \(i\) are

\[ \lambda_1=\frac{19}{7}\lambda, \]

\[ \lambda_2=\frac{19}{4}\lambda, \]

then, denoting the distance from the axis \(Q\) to the axis \(D\) by \(x\), we must have, by (7),

\[ \frac{x}{s-x}=\frac{19}{7}:\frac{19}{4}=4:7, \tag{13} \]

whence

\[ x=\frac{4}{11}s. \tag{13'} \]

The modulus of the scale of the function \((\log D - \log 0.251)\) is determined on the basis of (8)

\[ \lambda_3=\frac{\lambda_1\lambda_2}{\lambda_1+\lambda_2}=\frac{19}{11}\lambda, \tag{14} \]

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

To lay off the divisions on the diameter scale, we connect the previously chosen points \(O\) and \(O'\) (or any others) of the scales \(Q\) and \(i\). The intersection of the line \(OO'\) with the scale \(D\) gives the point \(O''\), whose mark, corresponding to the values of \(Q\) and \(i\) at the points \(O\) and \(O'\), is determined by calculation. Then, having constructed a logarithmic scale with modulus \(\lambda_3\) and applying it, with the proper division, to the point \(O''\), we mark out the other divisions of the diameter scale, extending it in both directions as far as necessary.

If, for convenience, other moduli (multiples of \(\lambda_1\) and \(\lambda_2\)) have been selected, then the position of the axis \(D\) must be changed accordingly. For example, with the scale of discharges three times greater than the scale of slopes, from equations \((7')\) we obtain:

\[ \frac{x}{s-x}=\frac{19.3}{7}:\frac{19}{4}=12:7, \tag{15} \]

whence

\[ x=\frac{12}{19}s. \]

Usually, on diagrams of hydraulic formulas constructed by d’Ocagne’s method, another axis is drawn, serving as a 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

\[ v=\frac{4Q}{\pi D^2}, \tag{10} \]

taking logarithms of this expression and representing the equation thus obtained in the form of a system of conjugate scales in which the scales \(Q\) and \(D\) coincide with those constructed earlier. But there is no need to resort to this, since the required positions and divisions can be determined by a purely graphical method. For this it is only necessary to note 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 line and thus find the image point of the axis \(v\), labeled \(1.00\) meter. 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 the velocity \(0.10\) 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 m and to continue the divisions in both directions.

Drawing 3 represents a d’Ocagne diagram for the graphical calculation of pipelines of circular cross-section by Flamant’s formula, constructed by Daries (more correctly, reduced from Bertrand’s diagram). It consists of four scales, running in the following order, counting from the left: the scale of discharges, the scale of diameters, the scale of head losses, and the scale of velocities. This diagram is constructed in the following manner. The distance between the scale of discharges and the scale of hydraulic gradients is chosen with such a calculation that the logarithms of the discharges are laid off on a scale three times larger than the logarithms of the gradients. In this case, if we denote the distance between the scales of discharges and gradients by \(l\), and the distance between the scale of discharges and the scale of diameters by \(x\), then, by (15), the relation must be satisfied

\[ x = \frac{12}{19} l . \tag{16} \]

The value \(x\) is taken equal to 70 mm. Therefore

\[ l = \frac{70 \cdot 19}{12} = 110.8 \text{ mm.} \]

The modulus of the scale of hydraulic gradients is taken in such a way that one logarithmic unit corresponds to 20 mm. The scale of discharges, as stated, is 3 times larger, i.e. 60 mm per logarithmic unit. Proceeding from this ratio, on the lines placed at the above-mentioned distance the logarithms of the numbers are laid off: for discharges from 1 to 1000, and for gradients from 1 to 0.000001. Thus the scales of discharges and hydraulic gradients are obtained. To obtain the scale of diameters, there is chosen (on the basis of the table for Flamant’s formula) such a relation of \(Q\) and \(i\) that with it \(D\) is equal to 1.00 m, and the points corresponding to these values of \(Q\) and \(i\) are connected by a straight line. Its intersection with the line of the diameter scale gives the point marked 1.00. In exactly the same way a point corresponding to \(D = 0.10\) m is found. The distance between these points is divided proportionally to the divisions of the logarithmic ruler, and the divisions are continued in both directions. Thus the scale of diameters is obtained. The scale of velocities is obtained in a similar manner.

The method of using the diagram for Flamant’s formula, presented in Drawing 3, follows naturally from the preceding. The corresponding values of the four elements determining the flow—discharge, diameter, hydraulic gradient, and velocity—lie on one straight scale line crossing the scales, which is determined

two specified among these quantities. The other two unknowns are read at the points of intersection of the secant line with 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 far simpler, quicker, and more convenient to use either a stretched thread or a transparent strip of paper, celluloid, etc., on which a straight line has been drawn beforehand (the so-called transparent straightedge). In doing so, one may move this strip either by hand or, better, by means of two pointed pins fastened at its ends. In the latter case it is convenient, having placed one point on a known division, to rotate the straight line about the axis of the point, without danger of slipping or displacement.

In addition to the d’Ocagne diagram described by us for the Flamant formula, constructed by Daries, two more diagrams of this type are used for the hydraulic computation of pipelines, namely Bertrand’s diagram for the Flamant formula and Daries’s diagram for the Lévy–Valot formula. The latter is in all respects similar to the diagram described for the Flamant formula. Bertrand’s diagram differs from the other two in that it contains not four, but nine scales, representing a series of quantities derived from $Q$, $i$, $D$, and $v$.¹ The method of using all the diagrams is, in principle, one and the same, although it is somewhat more complicated when applied to Bertrand’s diagram, owing to the multiplicity of scales, which in essence constitute several systems of conjugate scales.

It must be said, however, that all the enumerated diagrams of conjugate scales, constructed according to d’Ocagne’s method, also possess one common shortcoming. The point is that they are all intended exclusively for the calculation of water-supply pipes under definite working conditions, i.e. for cast-iron pipes of circular cross-section, running completely full, and with one (for each diagram) definite degree of roughness.

Meanwhile, in the calculation of water-supply systems there may arise problems requiring the determination of velocity and discharge at another degree of roughness, depending on the material and the character of operation of the pipelines; and in the calculation of sewerage systems, as is well known, one must constantly take account of different degrees of filling. Moreover, in sewerage, and in other cases also in water supply, it is necessary to employ, besides the circular section, other types of cross-sections. For all these cases the diagrams listed are not adapted, and thus the range of their application is at present, of necessity, limited to the calculation of water-supply pipelines under ordinary average working conditions.

¹ Details are in my work: “Classification and Theoretical Premises of Nomographic Methods for the Calculation of Water Pipelines.”

The indicated shortcoming, which narrows the field of application of d’Ocagne diagrams, is not, however, irremediable. First of all, it is, of course, possible to construct new diagrams for other cases of calculation, just as common as the calculations of water conduits; i.e., diagrams for circular sections with half-full flow, used in sewerage, with the corresponding roughness coefficients; for ovoid sections with filling up to five-sixths; and for other commonly used types of sections, with the corresponding degrees of filling and roughness. The construction of such new independent diagrams, with a view to applying the given graphical method to all possible cases of calculation, is highly desirable and timely.

However, the question is not thereby finally settled. The point is that cases may arise in which it is of interest to know the velocities and discharges under the condition of increasing or decreasing the degree of roughness to a norm not provided for by the ready-made diagrams. Furthermore, it is quite impossible to have ready-made diagrams for given sections at all the various degrees of filling. The diagrams must provide for the possibility of such cases, and a means must be found that makes it possible easily and quickly to pass from the data of a ready-made diagram, corresponding to a definite filling and roughness, to various degrees of filling and roughness determined by other operating conditions of pipelines.

It is precisely such a means that I propose in the form of special graphs, which I call transitional scales. Such graphs, in my opinion, satisfying the stated aims, should supplement all diagrams of conjugate scales; and, if they are not present on the diagrams, they can easily be prepared separately. It should be added to this that such transitional scales make it possible, among other things, to use d’Ocagne diagrams constructed for sections of one type for the purpose of calculating sections of other types, which may also sometimes be useful.

From the foregoing it is clear that the question of transitional scales for d’Ocagne diagrams naturally falls into three parts, corresponding to the possible cases of transition: first, to new degrees of roughness; second—to different degrees of filling; third—to other forms of cross-sections.

The method that is proposed we shall consider as applied to the diagram of the Flamant formula, constructed by Daries, with which we have already dealt. It is applicable in the same measure and in the very same form to the diagram of the Levy-Valot formula. As regards—

Ya. I. Nikolit.

If the Flamant–Bertrand diagram is used, then, theoretically speaking, it fully permits the application of the given method. However, the considerable complexity of the diagram itself makes it practically inadmissible to apply still further operations to it, in order to avoid errors, the probability of which under these conditions increases greatly.

Turning, then, to the Flamant–Dariès diagram and to the question of passing from it to other roughness coefficients, we must stipulate that, practically speaking, the question of applying other roughness coefficients to this diagram is of no significance, since these coefficients have not been worked out for the formula itself (not counting the coefficient for pipes with a smooth internal surface), and it is intended for solving problems as applied to one definite case of water motion. This does not prevent us, however, from showing, by the example of the given diagram, the possibility of passing to other degrees of roughness, since d’Ocagne’s method of conjugate scales can, and probably will, be applied to logarithmic formulas providing for a wider range of roughness coefficients.

The general form of Flamant’s formula, as was indicated above, with the roughness coefficient \(a_1\), will be

\[ D^\alpha i = a_1 v^\beta . \tag{17} \]

To represent it graphically, the formula is logarithmized

\[ \alpha \log D - \log a_1 = \beta \log v - \log i, \tag{17'} \]

and is then constructed, according to known rules, in the form of scales of the functions \((\alpha \log D - \log a_1)\), \(\beta \log v\), and \(\log i\). With such a method of construction it is clear that the transition from the roughness coefficient \(a_1\) to \(a_2\) introduces no changes into the diagram with respect to the scales of slope \(i\) and velocity \(v\); the change in the scale of diameters reduces to the fact that, instead of the function \((\alpha \log D - \log a_1)\), the function \((\alpha \log D - \log a_2)\) is laid off from the former initial point and with the same modulus. It is not difficult to see (Fig. 3) how such a change will be expressed graphically. Owing to the preservation of the modulus, the divisions of the scale will remain the same, but the entire scale will be shifted by a length representing \((\log a_2 - \log a_1)\) toward the origin. As a result of this, the numerical mark corresponding to any point of the scale axis must change, namely increase with an increase of the roughness coefficient, i.e. when

\[ \frac{a_2}{a_1} > 1;\quad \log a_2 - \log a_1 > 0, \tag{18} \]

and decrease in the opposite case. In doing this, the change must be made in such a way that the distance between the points which correspond

give, for one and the same reading on the scales of diameters constructed for coefficient \(a_1\) and for coefficient \(a_2\), a value that remains constant and is 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 on the axis of the scale corresponding to the scale for another degree of roughness. For this it is only necessary to lay off from this point, in the appropriate direction, a length corresponding, with the modulus of the scale, to the quantity \(\pm(\log a_2-\log a_1)\), and the mark of the new point thus obtained, read on the existing scale, will be the desired one.

Consequently, one and the same scale of diameters and one Flamant–Darcy diagram, if desired, could serve for calculations under the assumption of different degrees of roughness. Thus, for example, with this diagram one can solve problems concerning the motion of water in pipes with a smooth inner surface, when the coefficient \(a=0.00074\). For this, operating in the usual way with all the axes of the diagram, one need only remember that the marks of the points of the diameter scale in the present case must be read below the points themselves at a distance corresponding to

\[ \log a_1-\log a_2=0.09456, \]

with the modulus of the diameter scale.

In such cases it is more convenient to take readings by means of a transitional scale prepared in advance, giving the required lengths for different coefficients of roughness. It is clear that the indicated operation could also be replaced by solving the problem with reference to the principal coefficient, and then changing the result in proportion to the ratio of \(a_2\) to \(a_1\). The form of such a scale, with the inclusion also of divisions for other cases of transition, will be given below.

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 coefficient of roughness and the other for the other coefficient, and to use one or the other scale depending on the conditions of the problem.

It was indicated above that the Flamant–Bertrand diagram, like others of the same type, is adapted only for the calculation of pipes working under complete filling. This case is predominant in the calculation of water-supply systems. Quite the opposite is seen in the calculation of sewerage: here complete filling is an exceptional case, while incomplete filling is the normal one. In fact, the fundamental calculation of a sewer network is carried out for partial filling. Moreover, the quantity of wastewater entering

into a sewer system is subject to considerable fluctuations, and therefore a drain calculated for the maximum quantity of liquid under complete filling 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 stream of liquid in its middle part to the total internal height of the drain in the same part.

Thus in practice it is very often necessary to calculate drains under their incomplete filling, and also to determine the depth of flow in pipes. Here one must distinguish two cases, 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 one and the same diameter \(D\) and slope \(i\) in a drain, under half filling the velocity \(v\) is the same as under complete filling, while the discharge is two times smaller. Problems concerning, in general, any degrees of filling are reduced to determining the relation between the discharge and velocity under complete filling and the discharge and velocity under various degrees of filling.

Let us consider this question, by way of example, with respect to drains of circular cross-section. Let us imagine (Fig. 4) a drain of circular cross-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 of it. Computing for each degree of filling the value of the hydraulic radius \(\rho\) and specifying some slope \(i\), we determine the discharge \(Q\).

Having the value \(Q\) for complete filling and \(Q_1, Q_2\), and \(Q_3 \ldots Q_9\) for fillings from 0.1 to 0.9, one can construct a curve showing how, for a given diameter and slope, \(Q\) changes with the change in filling. To construct this curve, take \(Q\) as unity and set it off, on some scale, to the right of the line \(AB\) along the horizontal line \(AC\), corresponding to complete filling. Since we have taken \(Q\), corresponding to complete 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 set off the values of the ratios

\[ z_1=\frac{Q_1}{Q}, \]

\[ z_2=\frac{Q_2}{Q}, \]

\[ \ldots \tag{19} \]

The points obtained by plotting are connected by a curved line,

If we take another diameter \(D\), and another slope \(i_1\), we shall obtain another curve \(Q\). Carrying out the construction for various \(D\) and \(i\), we can satisfy ourselves that all the curves \(Q\) coincide so closely with one another that, for practical purposes, one may adopt a single common curve \(Q\), the same for all \(D\) and \(i\).

Having such a curve for the variation of discharges as a function of the degree of filling, it is easy, from the quantity \(Q\) corresponding to complete filling of the conduit for given \(D\) and \(i\), to find \(Q\) for any filling. For this it is only necessary, after determining the magnitude \(\alpha\) corresponding to the desired filling, to multiply \(Q\) by \(\alpha\), or to divide it by

\[ \frac{1}{\alpha}. \]

In solving problems relating to incomplete filling, such a transition from the discharge at complete filling to the discharge at various degrees of filling can be made directly on the discharge scale. Indeed, equalities (19) may be represented in the form

\[ \begin{aligned} Q_1 &= \frac{Q}{\dfrac{1}{\alpha_1}} \\ Q_2 &= \frac{Q}{\dfrac{1}{\alpha_2}} \\ &\ldots \end{aligned} \tag{19'} \]

Taking logarithms of the separate equalities (19′), we obtain

\[ \begin{aligned} \log Q_1 &= \log Q - \log \frac{1}{\alpha_1} \\ \log Q_2 &= \log Q - \log \frac{1}{\alpha_2} \\ &\ldots \end{aligned} \tag{20} \]

The expression \(\frac{1}{\alpha_m}\) is introduced because its \(\log\) is positive in most cases. Equations (20) show that, in order to obtain on the diagram’s discharge scale the discharge values \(Q_1, Q_2 \ldots\) 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 scale by a length which, at the scale modulus, represents \(\log \frac{1}{\alpha_1}, \log \frac{1}{\alpha_2}, \ldots\), and the reading at the point thus obtained will give the desired value of the discharge at incomplete filling.

For convenience in carrying out such an operation, one may plot the quantities \(\log \frac{1}{\alpha_1}, \log \frac{1}{\alpha_2}\), etc., on the scale adopted for the quantity \(Q\), on

with the transitional scale, separate or common with other transitional quantities (the form of which will be given below), having inscribed opposite the divisions the corresponding degrees of filling.

The values of \(a\) and \(\dfrac{1}{a}\) necessary for constructing the scale are given in the tables (p. 16) as applied to the ordinary circular, ovoid, and trough-shaped sections (with two centers, with the ratio of span to trough radius shown in Drawing 5).

The curve \(Q\) (Drawing 4) 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 at complete filling. At a filling of about \(0.8\), the discharge is equal to the discharge corresponding to complete filling, while at lower degrees of filling it is less than the latter. Therefore, on the transitional scale the mark for filling \(0.8\) coincides with the mark \(1.0\), the mark \(0.9\) will lie higher, and the remaining ones below unity. It is clear that for degrees of filling at which \(a_m>1\), \(\log \dfrac{1}{a_m}\) will be negative, and therefore one must lay off on the scale of velocities the quantity \(-\log \dfrac{1}{a_m}\) in the direction of increasing divisions. In constructing the transitional scale, this occurs of itself. Having such a scale cut out and placing it on the diagram so that the point marked

\[ \frac{H}{D}=1.0-0.8 \]

coincides with the point corresponding to the conditions of complete filling, we shall find \(q_m\) for this filling. Conversely, from \(q_m\), \(i\), and \(D\) one may find the corresponding degree of filling \(\dfrac{H}{D}\).

The velocity of flow \(v\), as is known, also varies depending on the degree of filling, and in sewer calculations it is sometimes necessary to determine velocities at different degrees of filling. As in the case of discharge, the question here reduces to passing from the velocity at complete filling to the velocity at various degrees of filling, on the basis of the previously determined ratio between these velocities. Let us again take, for example, a circular section and as before adopt 10 degrees of filling corresponding to Drawing 4. Determining, for each degree of filling at mean values of diameter and slope, the velocities of flow \(v_1\), \(v_2\ldots\), and comparing them with the velocity at complete filling, we obtain the ratios

\[ \begin{aligned} \beta &= \frac{v_1}{v} \\ \beta &= \frac{v_2}{v} \\ &\ldots \end{aligned} \tag{21} \]

which, like the preceding ones, may be considered practically corresponding for all diameters and slopes. Plotting the values \(\beta_1, \beta_2, \ldots\) on the same lines as \(\alpha_1, \alpha_2, \ldots\), we obtain a curve determining the variation of velocities as a function of the degree of filling. This curve makes it possible, from a known velocity at complete filling, to determine velocities at various degrees of filling on the basis of the relation

\[ v_m=\beta_m v. \tag{21'} \]

To apply the same method in d’Ocagne’s diagram, we take the logarithms of equalities of type (21′) in the form

\[ \log v_m=\log v-\log \frac{1}{\beta}. \tag{22} \]

Equation (22) shows that the flow velocity at any degree of filling \(v_m\) may be obtained from the velocity scale of the diagram if a reading is taken at a distance corresponding to \(\log \frac{1}{\beta_m}\), in the direction from the origin of the scale from the division determining the velocity \(v\) at complete filling.

It is most convenient here also to put the quantities \(\log \frac{1}{\beta_1}, \log \frac{1}{\beta_2}, \ldots\) on the transitional scale. The values of \(\beta\) and \(\frac{1}{\beta}\) needed for this are likewise given in the table, as applied to sections of the three types. The use of the scale is in all respects analogous to the use of the above-mentioned transitional scale for discharges.

From the form of the curve of variation of velocities at different degrees of filling it may be concluded 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. The arrangement of the divisions on the transitional scale relative to \(\frac{H}{D}=1.0\) will correspond to this.

The method of transitional scales, which provides a convenient means for passing, in diagrams constructed by d’Ocagne’s method, to new degrees of roughness and to other degrees of filling, makes it possible to extend the range of application of each diagram in yet another direction. It allows, when necessary, the use of diagrams compiled for one type of section—for example, circular sections—for calculating sections of another form.

TABLE

of values of \(\alpha,\ \dfrac{1}{\alpha},\ \beta,\ \dfrac{1}{\beta}\) for various degrees of filling.

Degree of filling	Quantity	1	0,9	0,8	0,7	0,6	0,5	0,4	0,3	0,2	0,1
Circular section.	\(\alpha\)	1	1,07	1	0,85	0,67	0,5	0,33	0,19	0,09	0,02
Circular section.	\(\log \dfrac{1}{\alpha}\)	0	−0,03	0	0,07	0,17	0,3	0,48	0,72	1,05	1,7
Circular section.	\(\beta\)	1	1,14	1,15	1,13	1,08	1	0,9	0,77	0,59	0,35
Circular section.	\(\log \dfrac{1}{\beta}\)	0	−0,05	−0,06	−0,05	−0,03	0	0,05	0,11	0,23	0,46
Ovoid section.	\(\alpha\)	1	1,05	0,9	0,75	0,58	0,42	0,26	0,15	0,07	0,02
Ovoid section.	\(\log \dfrac{1}{\alpha}\)	0	−0,02	0,05	0,13	0,24	0,38	0,59	0,82	1,2	1,7
Ovoid section.	\(\beta\)	1	1,12	1,12	1,08	1,03	0,94	0,85	0,75	0,61	0,41
Ovoid section.	\(\log \dfrac{1}{\beta}\)	0	−0,05	−0,05	−0,03	−0,01	0,03	0,07	0,13	0,23	0,39
Trough section.	\(\alpha\)	1	1,07	1	0,88	0,7	0,52	0,35	0,19	0,07	0,01
Trough section.	\(\log \dfrac{1}{\alpha}\)	0	−0,03	0	0,06	0,16	0,3	0,46	0,72	1,16	2
Trough section.	\(\beta\)	1	1,15	1,16	1,13	1,07	1	0,89	0,73	0,52	0,28
Trough section.	\(\log \dfrac{1}{\beta}\)	0	−0,06	−0,06	−0,05	0,03	0	0,05	0,14	0,29	0,45

It must be stipulated at the very outset that the artificial calculation indicated here of sections of one form by means of diagrams of another can, of course, be recommended only for sporadic cases. However simple the proposed method may be in principle, it nevertheless complicates the reading, requires a considerable expenditure of attention, and can easily give rise to errors. Thus, when it is necessary to solve a whole series of problems relating to a section of a new type, it is preferable to construct a special diagram for that type of section. It is further clear that the question of applying this to the calculation of pipelines of arbitrary section can arise only with regard to diagrams with four scales, since, for example, in Flaman’s complete diagram Bertrand’s part of the additional scales essentially relates only to pressure pipelines, and the complexity of the diagram itself makes the introduction of still further manipulations practically inconvenient.

As for diagrams with four scales, such as the diagram of the Lévy-Valot formula and the diagram of the Flaman formula compiled by Dariès, they admit application to the calculation of pipe-

Transitional Scale for d’Ocagne Diagrams

...of pipes of any cross-section by means of the method of transitional scales, the matter being reduced to the transition to an equivalent diagram constructed on the basis of the hydraulic radius.

Let us consider, from the point of view of the question that interests us, for example, the Flamant–Dariès diagram.

The fundamental equation, represented graphically in the form of this diagram, has the general form

\[ D^{\alpha} i = a v^{\beta}. \tag{23} \]

For the transition to a graphical representation, it was put into logarithmic form

\[ \alpha \log D - \log a = \beta \log v - \log i. \tag{23′} \]

Equation (23) may be represented, by introducing the hydraulic radius \(\rho\) in place of the quantity \(D\), in the form

\[ \rho^{\alpha} i = b v^{\beta}. \tag{24} \]

where

\[ b = \frac{a}{4^{\alpha}}. \tag{25} \]

If we wished to represent equation (24) in the same graphical form as equation (23), then, taking logarithms of (24), we obtain

\[ \alpha \log \rho - \log a + \log 4^{\alpha} = \beta \log v - \log i. \tag{24′} \]

Comparing equations (23′) and (24′), we see that in constructing, on the basis of (24′), a diagram of conjugate scales including \(\rho\) instead of \(D\), of the three scales of the diagram the scales of velocities \(v\) and slopes \(i\) must have the same position and the same modulus as in the existing diagram. As for the scale of hydraulic radii \(\rho\), its distance from the other scales, as well as its modulus, and consequently its graduation, will remain the same as for the scale \(D\), but the entire scale will be shifted in the direction opposite to the origin by a length representing \(\log 4^{\alpha}\).

Indeed, let us take the scheme of the diagram of conjugate scales \(D\), \(i\), and \(v\) (Fig. 6) and denote by \(O\) the point from which the scales of the functions \((\alpha \log D - \log a)\) and \((\alpha \log \rho - \log a + \log 4^{\alpha})\) are laid off, by \(D\text{—}\rho\) an arbitrary point of the axis of the scales, whose value according to the scale of diameters \(D\), and according to the scale of hydraulic radii \(\rho\), and by \(A\) and \(A_1\) the unit (i.e. corresponding to the unit of that order to which the quantities \(D\) and \(\rho\) belong) points of the scales. Then, by construction, for the point \(D\)

\[ OD = \alpha \log D - \log a, \]

and for the unit point \(A\)

\[ OA=\alpha \log 10^n-\log a=n\alpha-\log a, \]

for the point \(\rho\), coinciding with \(D\),

\[ O\rho=\alpha \log \rho-\log a+\log 4^\alpha, \]

for the unit point \(A_1\)

\[ OA_1=n\alpha-\log a+\log 4^\alpha. \]

Thus the distance between the unit points of the scales \(D\) and \(\rho\) will be

\[ AA_1=OA_1-OA=\log 4^\alpha, \tag{25} \]

i.e. the unit point of the scale \(\rho\), and together with it the others as well, are shifted, in comparison with the corresponding points of the scale \(D\), by the length \(4^\alpha\) in the positive direction, i.e. in the direction opposite to the origin.

The indicated relation between the scales of hydraulic radii and the scales of diameters makes it possible, for given values of \(v\) and \(i\), to determine, in the absence of the scale \(\rho\), the magnitudes of the hydraulic radii from the scale \(D\). For this it is only necessary to read the required value on the scale not at the point of intersection with the intersecting line, but nearer to the origin of the scale by a length corresponding to \(\log 4^\alpha=\alpha\log 4\). For convenience this length may also be marked on the transitional scale.

Since equation (24) is valid for any form of cross-section, the relation we have derived between the scales of diameters and hydraulic radii, and the method of transition from \(D\) to \(\rho\), is suitable for transition to the hydraulic radius of a cross-section of any form in a diagram of circular cross-section. If we had a diagram for a cross-section of another form, it could be used for transition from its parameter, span \(D\) or height \(H\), to the hydraulic radius of all cross-sections in an analogous manner, by introducing other quantities instead of \(\log 4^\alpha\). These quantities are easily determined for each type of cross-section. Thus, for example, for the diagram of an ordinary ovoid cross-section with parameter \(D\) (span), we have

\[ \Omega=1.148\,D^2 \]
\[ P=3.965\,D \tag{26} \]
\[ \rho=0.2896\,D. \]

Consequently, instead of \(\alpha\log 4\), one would have to take

\[ \alpha\log\left(\frac{1}{0.2896}\right). \]

It is possible, of course, also to mark the divisions of the scale of hydraulic radii on the axis of the scale \(D\) on the other side, if it is not intended for the divisions corresponding to another coefficient of roughness.

Until now, in discussing the question of calculation by means of the Flament–Bertrand diagram of arbitrary cross-sections, we have taken into consideration only the three scales of the diagram \(i\), \(D\), and \(v\). The introduction into the calculation of the discharge \(Q\), and the use of the scale of discharges, is likewise possible as applied to such a calculation.

Let us begin the examination of the question again with the fundamental equation of the Flament–Bertrand diagram

\[ D^\alpha i = a v^\beta . \tag{23} \]

This equation may be transformed by introducing, instead of the velocity \(v\), its expression as a function of \(Q\) and \(D\),

\[ v=\frac{4Q}{\pi D^2}. \tag{10} \]

Then there is obtained

\[ D^{\alpha+2\beta} i = a\left(\frac{4}{\pi}\right)^\beta Q^\beta . \tag{27} \]

No matter how the scale of discharges \(Q\) is constructed, the system of the three scales of the diagram \(D\), \(i\), and \(Q\) constitutes an independent diagram, which is the graphical representation of equation (27), and may be obtained from it by the usual method of construction, after reduction to logarithmic form,

\[ (\alpha+2\beta)\log D-\log a=\beta\log Q+\log\left(\frac{4}{\pi}\right)^\beta-\log i. \tag{27'} \]

If we wished to construct a diagram representing the relation between \(\rho\), \(i\), and \(Q\), we should have to begin with the equation

\[ \rho^\alpha i = b v^\beta, \tag{24} \]

where

\[ n=\frac{a}{4^\alpha}. \tag{25} \]

Substitute into equation (24) the expression

\[ v=\frac{4Q}{\pi D^2}=\frac{Q}{4\pi\rho^2}. \tag{10'} \]

This latter has meaning only for a circular cross-section. Consequently, the method of transition from the scale of discharges of the diagram for circular pipelines with the parameter \(D\) to the scale of discharges with the parameter \(\rho\), which we obtain on the basis of (10′), will be applicable only for the calculation of circular cross-sections.

Thus, from (24) and (10′) we obtain:

\[ \rho^{\alpha+2\beta} i = b\left(\frac{1}{4\pi}\right)^\beta Q^\beta; \tag{28} \]

or, taking into account (25″),

\[ \rho^{\alpha+2\beta} i=\frac{a}{4^\alpha}\left(\frac{1}{4\pi}\right)^\beta Q^\beta . \tag{28'} \]

To represent this relation in graphical form, we must take its logarithm, whereby we obtain

\[ (\alpha+2\beta)\log \rho-\log a+\log 4^\alpha = \beta \log Q+\log \left(\frac{1}{4\pi}\right)^\beta-\log i. \tag{28''} \]

Comparing equation (28'') with (27′), we see that if we were to construct a diagram according to equation (28''), it would coincide with the existing diagram corresponding to (27′) as regards the mutual arrangement of the scales. Further, the graduation of the slope scale remains unchanged, while the graduation of the scale of diameters and the method of reading from it change in accordance with the transition to the parameter \(\rho\), entirely in agreement with what was stated above, which is quite natural. As for the scale of discharges \(Q\), here, with the same modulus as before, instead of the function \(\beta\log Q+\log\left(\dfrac{4}{\pi}\right)^\beta\), it is necessary to lay off the function \(\beta\log Q+\log\left(\dfrac{1}{4\pi}\right)^\beta\). It is not difficult to see, following the example of the preceding case, what this means. The graduation of the scale of discharges remains unchanged, but the value of all divisions is changed in such a way as if the entire scale were shifted in the direction of increasing \(Q\) by a length corresponding to the quantity \(\log\left(\dfrac{1}{4\pi}\right)^\beta-\log\left(\dfrac{4}{\pi}\right)^\beta\), which is equal to

\[ \log\left(\frac{1}{4\pi}\right)^\beta-\log\left(\frac{4}{\pi}\right)^\beta = \beta\log\frac{1}{16}. \tag{29} \]

Since the value of the logarithmic difference has proved to be negative, the mentioned displacement in fact takes place in the direction toward the origin.

Thus, in order to obtain from the \(Q\) scale of the given diagram the value of \(Q\) corresponding to the diagram with parameter \(\rho\), for a circular cross-section, the reading must be taken not at the point of intersection of the \(Q\) scale with the intersecting line, but at a length \(-\beta\log\dfrac{1}{16}\) in the direction of increasing values of \(Q\). This distance, for convenience, may be marked on the transitional scale, which must, as far as possible, be common to all transitional quantities, both those already mentioned and those that follow.

In a completely analogous way it can be shown that, with the aid of a diagram with parameter \(D\) for circular cross-sections, it is possible to solve problems involving the discharge \(Q\), for arbitrary cross-sections, by means of the same method. The difference will be only in the numerical values of the logarithmic difference (29) and the corresponding lengths of the transitional scale.

At the present point, we shall consider in parallel, from this point of view, three forms of cross-section:

1) circular, of diameter \(D\);

2) ordinary ovoid, with span \(D\) and height \(H=\dfrac{3D}{2}\) (Fig. 5), for the parameter of which we take the span \(D^1\);

3) a tray-shaped section with two centers, with span and radius of the tray \(=D\) (Fig. 5), which we also take as the parameter.

For all these sections the following relations hold in general form:

\[ \rho=\frac{D}{m}, \tag{30} \]

\[ D=m\rho, \]

\[ Q=pD^2v, \tag{31} \]

\[ v=\frac{1}{p}\frac{Q}{D^2}, \tag{32} \]

\[ v=\frac{1}{m^2p}\frac{Q}{\rho^2}. \tag{33} \]

Here, for the different forms of cross-section, the quantities \(m\) and \(p\) have the following numerical values:

1) circular section:

\[ m=4, \]

\[ p=\frac{\pi}{4}, \]

\[ \frac{1}{p}=\frac{4}{\pi}, \]

\[ \frac{1}{m^2p}=\frac{1}{4\pi}; \]

2) ovoid section:

\[ m=3{,}453, \]

\[ p=1{,}148; \]

3) tray-shaped section:

\[ m=5{,}464, \]

\[ p=0{,}48; \]

Substitute in expression (29), in place of \(\dfrac{1}{4\pi}\) and \(\dfrac{4}{\pi}\), the corresponding general expressions relating to sections of all three types, i.e. \(\dfrac{1}{m^2p}\) and \(\dfrac{1}{p}\). Then we obtain the general expression replacing (29):

\[ \log\left(\frac{1}{m^2p}\right)^\beta-\log\left(\frac{1}{p}\right)^\beta =2\beta\log\frac{1}{m}. \tag{34} \]

\(^{1}\) Cf. Yasyukovich. Calculation of Watercourses by Means of Logarithmic-Graphical Tables.

Thus the length of the transitional scale for reading on the given diagram the value \(Q\) for the parameter \(\rho\) must correspond:

1) for an ovoid section, \(3.5 \log 0.2896\);
2) for a trough-shaped section, \(3.5 \log 0.183\).

All the foregoing considerations, relating to the Flamant–Bertrand diagram with four conjugate scales, are also applicable to the analogous Lévy–Larès diagram.

From what precedes we see that, with a single diagram of conjugate scales for circular-section water conduits, one can solve problems relating not only to circular sections, but also to any regular sections whatever in which there is a dependence of the area and the hydraulic radius on one or another parameter, in the form:

\[ \begin{aligned} \rho &= mD,\\ \Omega &= pD^2, \end{aligned} \tag{35} \]

where \(D\) is the parameter.

Such a diagram, common to all sections, will consist of the following elements:

1) a scale of discharges \(Q\) (possibly double, for seconds and minutes);
2) a scale of diameters \(D\) (possibly for two degrees of roughness);
3) a scale of slopes \(i\);
4) a scale of velocities \(v\);
5) a transitional scale, the full composition of which will be determined below.

But such a general diagram, as was indicated above, can be considered convenient for calculating sections of forms other than circular only when there is a small simultaneous number of such calculations, since the certain complication introduced into the work by the use of the transitional scale paralyzes the chief advantage of diagrams of conjugate scales—the simplicity of the operations. Therefore, for systematic calculations of different sections, it is much more convenient to use several diagrams compiled specially for each type of section. The composition of these diagrams is the same as indicated, but the composition of the transitional scale may be correspondingly reduced. For calculating water supply and sewerage it is sufficient to have, for example, diagrams for the following most commonly used types of cross-sections: circular, ovoid (one or two), trough-shaped (one or two), and circular with a trough at the bottom¹.

Having now learned all the cases and problems for which the transitional scale recommended by us may serve, we shall finally turn to its composition and form. The composition of such a scale depends on the composition of the basic diagrams. We shall consider the transitional scale in its most complex form, i.e., a scale intended for transition from one diagram for a definite form of section (namely, circular, which is predominantly possible in practice), at one degree of roughness and complete filling, to other forms of section, roughness coefficients, and degrees of filling. Such, for example, might be a transitional scale for a diagram of conjugate Flaman–Daries scales with four scales, which we shall construct.

Summarizing, on the basis of the preceding, all the cases of application of the transitional scale, we see that it must give the following distances:

1) Distances serving for the transition from the roughness coefficient for which the diagram was constructed to calculation with other roughness coefficients; they correspond to $(\log a_m - \log a)$ with the modulus of the diameter scale, where

$a$ — the basic roughness coefficient;

$a_m$ — a general designation for other roughness coefficients to which transition may be required.

2) A distance serving for the transition from calculation with the parameter $D$ and the system of scales $v$, $i$, $D$ to the parameter $\rho$, i.e., to the diagram $v$, $i$, $\rho$, which corresponds to the quantity $\alpha \log 4$, laid off with the modulus of the diameter scale.

3) Distances serving for the transition from the velocity $v$ at complete filling to velocities $v_1, v_2 \ldots$ at various degrees of filling, which represent the quantities $\log \dfrac{1}{\alpha_1}$, $\log \dfrac{1}{\alpha_2}\ldots$, laid off with the modulus of the velocity scale.

4) Distances serving for the transition from the discharge $Q$ at complete filling to discharges at other degrees of filling $Q_1, Q_2 \ldots$, which correspond to the quantities $\log \dfrac{1}{\beta_1}$, $\log \dfrac{1}{\beta_2}\ldots$ with the modulus of the discharge scale.

These four elements of the transitional scale are the most important. To them may also be added:

5) Distances for determining, by hydraulic radii, the parameters of sections of another form besides circular, i.e., the span $D$ or height $H$; it represents the quantity $\alpha \log \dfrac{1}{\lambda}$ with the modulus

of the scale of diameters, where

\[ \lambda=\frac{\rho}{D}, \tag{36} \]

or

\[ \lambda=\frac{\rho}{H}, \tag{37} \]

depending on what is taken as the parameter of one or the other form of section;

6) Distances, for passing from the calculation of discharges with parameter \(D\) to the calculation of discharges with parameter \(\rho\) for the various forms of sections; it corresponds to the value \(2\beta \log \frac{1}{\lambda}\) for the modulus of the scale of discharges.

Drawing 7 represents a transitional scale, prepared by us, by way of example, for the diagram of conjugate scales of Flamant–Bertrand.

It consists of two scales. On the left scale, from point D downward, there is first laid off the distance for passing 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 represented in drawing 3, proves equal to 4.3 mm, and its end is marked by the letter D′.

Further, from the same point D, there is laid off downward the distance for passing from the scale of diameters to the scale of hydraulic radii. Its length in the present case is 33.9 mm, and its end is denoted by the letter \(\rho\).

From this last point upward there are laid off the distances for passing from the hydraulic radius to the span \((D_1)\) and height \((H_1)\) of an ordinary ovoidal section, and to the span \((D_2)\) of a trough-shaped section with two centers.

The line corresponding to \(\rho\) is also designated by the letter \(v\) and serves as the origin of the distances for passing from the velocity at complete filling to the velocities at degrees of filling 0.1, 0.2, …, the ends of such distances being denoted respectively by the numerals 1, 2, 3, …

On the right scale of the scale, from the initial line, designated by the letter Q, there are laid off upward and downward the distances 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 numerals 1–9 and, for clarity, are connected with the divisions for the velocities at the corresponding degrees of filling.

Finally, on the same scale, from the line Q upward there are laid off distances for passing from the calculation of discharges with parameter \(D\) to the calculation with parameter \(\rho\) for sections of circular and ordinary ovoidal—

further, their ends being denoted respectively by the letters \(Q^\rho\) and \(Q_1^\rho\).

The application of such a transitional scale is evident from the preceding. 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.

1912.

Literature

Bechmann. Salubrité urbaine. II. Assainissement. 1899.

Brublevskii. A graphical method for calculating drains (Transactions of the Society of Civil Engineers, 1907).

Dariès. Calcul des conduites d’eau. 1900.

Kashkarov N. A. Calculation of pipelines by Bertrand’s graphical method. 1907.

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).

Nikolin Ya. I. Logarithmic formulas for the calculation of water pipelines. 1910.

Nikolin Ya. I. Graphical methods for calculating water supply and sewerage. Issue I. Theory and applications of the method of conjugate scales. 1911.

Idem. Issue II. Classification and theoretical prerequisites of nomographic methods for calculating water pipelines. 1913.

D’Ocagne. Nomographie. 1891.

D’Ocagne. Traité de Nomographie. 1899.

D’Ocagne. Exposé synthétique des principes fondamentaux de la Nomographie. 1903.

Schilling. Ueber die Nomographie de M. d’Ocagne.

Cherepashinskii M. M. Water supply of cities. 1905.

Yasyukovich M. Calculation of drains with the aid of logarithmic-graphical tables. 1906.

Table 1.

Drawing 1.

Drawing 2.

Drawing 4.

Plate II.

Drawing 3. Daries’s diagram (Bertrand’s shortened version)
for calculating water-supply pipes (Flamant formula).

Diagram labels:

• Discharges
  (cubic meters per sec.)

• Diameters
  (meters)

• Hydraulic gradients

• Velocities
  (meters per sec.)

Table III

Drawing 5.

Drawing 6.

Drawing 7.

1. Owing to a certain haste in completing the present issue, it was not possible to finish the diagrams already begun, composed of conjugate scales for the indicated types of sections, as applied to the Lampe formula. These diagrams are intended to be given subsequently. ↩↩