PROCEEDINGS
of the Tomsk Technological Institute
of Emperor Nicholas II

Vol. 19. 1910. No. 3.

I.

S. A. Vvedensky

THE INFLUENCE OF LOW MOUNTAIN PASSES ON VIRTUAL CALCULATIONS.

1–28.

The Influence of Low Passes on Virtual Calculations

S. A. Vvedensky.

One of the most fundamental tasks of railway operation is the question of the most advantageous weight of a train and its most advantageous speed.

For a chosen type of locomotive, both of these depend on the conditions of the plan and profile; they also influence the very choice of the locomotive. Hence the reversibility of the problem: that is, the surveyor and the designer of a line are faced with the question of connecting the given points in such a way that the cost of movement will be the least.

Let us note here as well that the possible reduction in speed by a small amount will have an extremely slight effect on railway management, since the turnover of cars and the average commercial speed depend to a far greater degree on station operations, especially transfers at junctions, than on the average speed of movement on the runs.

Thus, theoretically, what is obtained is only the necessity of a certain ratio between construction and operating—more precisely, traction—expenses, under which the given line proves to be the most advantageous.

Such a general formulation of the question for all railways logically led to the establishment of a definite, universally unchanging scale for comparing variants, for which the length of a straight horizontal track, or virtual length, was chosen. According to it, the sum of the work of movement and the time required to perform it are calculated. In the first case, the run is divided into sections characterized by the constancy of a uniform resistance depending only on the plan and profile; conversion coefficients, the so-called traction virtual coefficients, are calculated, and in this way the work of movement on the given run is reduced to an equal amount of work on a straight horizontal track of the sought length.

For calculating the running time, in the same way a running virtual length is found from the condition of equality of the time for running

along the virtual, i.e. straight horizontal, path of the required length, and for a specified section of that length.

In the calculations, the motion is taken to be uniform from the beginning of each section with an unchanging resistance (depending on the type of line) to its end, at a speed taken from the tractive-force diagram for the given locomotive and the given train, constructed on the assumption of continuity of the functions:

\[ Z=\frac{75\,N}{v}; \]

\[ W=A+B\,V+C\,V^{2}. \]

In such a simple form of applying the scale, it becomes possible to compile a table of coefficients of influence for various grades and radii of curves, and very quickly to compute virtual lengths. In fact, however, at the foundation of the entire system lie the incorrect assumptions of continuity of the functions \(W=A+BV+CV^{2}\) and of the instantaneous change of speeds at the break points of the profile and plan; furthermore, the reversibility of energy into work is not taken into account. The last two facts, while simplifying the calculations, erase the individual differences between lines. If, as the basis of the calculation, we take at least the work of the locomotive when passing over passes, we obtain the very simple formula:

\[ T=(L+Q)H=(L+Q)\sum h, \]

where it is immaterial whether it was one pass of height \(H\), or whether this same \(H\) was made up of a whole series of passes arranged in different ways—in the formula there are no changes.

Meanwhile, in these two cases, mathematically, an enormous difference may be found, which in fact constitutes the aim of the present investigation.

As the starting point of the reasoning I take the following mechanical principle: no sudden change by a finite amount in the velocity of any moving body can occur without impact. No impacts occur in the normal motion of a train; consequently, a change of velocity requires for itself a definite, calculable interval of time and distance traveled. If, over this length, the character of the constant resistance, i.e. the fictitious grade, changes, then the resisting force also changes; that is, in the motion, before it reaches the calculated speed, a change will again occur.

This kind of accounting for nonuniform motion will always require taking into consideration the individual properties inherent in the given variant.

On the other hand, it will make it possible to find variables indicating the total reserve of energy, kinetic and potential, possessed at different places along the route by the moving train; and, on the basis of its convertibility into work, it will be possible to dispose of it methodically within certain limits. Nevertheless, in the present examination, all the necessary rigor in the correspondence of the conditions of motion with those adopted in the formulas cannot be maintained, owing to the excessive variability of the resistance forces.

It was assumed in 1) that the resistance force is expressed by three terms of an expansion in powers of \(V\), \(\frac{\mathrm{km}}{\text{hour}}\), i.e.

\[ W = A + BV + CV^2, \]

where \(A\), \(B\), and \(C\) are constants and do not depend on \(V\).

In fact, this would correspond to a constant make-up and weight of the train, invariability of the mechanical properties of the track and rolling stock, a constant temperature, a constant wind force, and a constant angle of its direction with the direction of motion.

If the first assumptions do not contradict reality within certain limits, then the question of wind does not correspond to it at all. The inconstant element of the air possesses such variability and suddenness in these changes that, theoretically, consideration of its influence is inconceivable except under one or another approximate supposition. On existing railways, by means of many years of observations, one may still derive the average speed and average direction of the wind for definite zones and different seasons; but this average force and direction are not known at all on lines under construction. Moreover, during its run a train, passing along curves, itself changes the direction of its motion, which further complicates taking account of the influence of the wind. Considering such a problem beyond the strength of a general solution of the question under examination, and taking into account that for different places the influence will numerically differ depending not only on the outlines and number of crews, but also on changes in the direction and force of the wind, I have assumed the coefficients \(A\), \(B\), \(C\) for each section of railway and train to be constant and numerically determined.

This assumption is all the more permissible because the whole investigation relates exclusively to low passes, i.e. short sections of varying inclination to the horizon. At a significant speed

railway trains cover short distances in a few minutes, and the assumption that accidental causes, i.e., the force and direction of the wind, do not change over such periods of time, is hardly especially erroneous.

In 2) it is assumed that, when passing over breaks in the profile, it is not a train that moves, but only a single locomotive, possessing all the properties of a train. Until now, so far as I know, the influence on motion of curves in the vertical plane has not been taken into account at all and has not been tested. In reality, owing to the great length of the train and the flexible coupling of the cars, the passage of a train over a break in the profile places it in a very complex position, where one component of the force of gravity acts on one part of it, another on another part, and the resistance force itself from the influence of the profile at the locomotive’s coupling hook is variable throughout the passage of the curve. The investigation of this interesting question is still subject to separate development; at present, for the sake of simplicity, a jump in this force is admitted, which, of course, will be reflected only in the length of the period of changes in speeds. Moreover, in the sense of expenditure of time and work, there is here a certain compensation, since all the phenomena that occur when passing into the beginning of the ascent, i.e., when passing from a more gentle section to a steep one, are repeated in reverse order at its end—when passing over the summit of the pass from a steep section to a gentle one.

Likewise, as before, a simultaneous change of cut-off, coinciding with the passage over the break, is admitted, which already depends on the attentiveness of the engine driver.

Without at all intending to revise the modern virtual calculation, the present investigation, as is evident, leaves a number of former assumptions in place.

Every crossing over a watershed is a pass. Theoretically, three types of crossings are conceivable: 1) the train rises from a given horizon to a higher one and then descends to the original one (Fig. 1); 2) the train rises to a higher one and then descends to a horizon lower than the original one (Fig. 2); 3) the train rises from a given horizon to a higher one and then

Fig. 1.          Fig. 2.          Fig. 3.

descends to a horizon higher than the original one (Fig. 3).

descends to a horizon situated above the original one (Fig. 3).

In all cases there may be the most diverse combinations of the lengths of ascents, descents, and level stretches; likewise, immediately before the pass and behind it there may be either level stretches, or ascents and descents with a very slight gradient. Of all these possibilities, only the gradients are of interest. If the component of the force of gravity on an ascent or descent is greater than the resistance encountered by the train when moving on a horizontal track of the same curvature and at the same speed, then the ascent is called harmful:

\[ (L+Q)\operatorname{tg}\alpha > W_{v_i}+C, \]

where \(W_{v_i}\) is the resistance on a straight horizontal track at the speed of motion on the ascent, or the speed permitted on the descent, \(i\); \(C\) is the resistance due to the curve. On a descent with such a gradient, with the regulator closed, the train will move with acceleration, since the applied force

\[ (L+Q)\operatorname{tg}\alpha - W_{v_i}-C > 0. \]

Taking into account the requirement of safety, the speed of such motion is limited by calculated limiting values, and the work of the driving force is absorbed by braking. Under the condition

\[ (L+Q)\operatorname{tg}\alpha < W_{v_i}+C \]

motion with the regulator closed becomes retarded.

Gradients of the first kind are called harmful; gradients of the second kind are called harmless. This name is assigned to them on the basis of calculating the work in traveling up and down an ascent of a given length \(l\). The work of the tractive force when moving upward is expressed as:

\[ (L+Q)\operatorname{tg}\alpha \cdot l + (W+C)\cdot l = T_1. \]

When moving downward on a harmful gradient, \(T_2=0\); on a harmless one,

\[ T'_2=(W+C)l-(L+Q)\operatorname{tg}\alpha \cdot l. \]

The sum of the works on a harmful gradient is

\[ T_1+T_2=(L+Q)\operatorname{tg}\alpha \cdot l+(W+C)l, \]

and on a harmless one,

\[ T'_1+T'_2=2(W+C)l; \]

and since \((L+Q)\operatorname{tg}\alpha\) on a harmful gradient is greater than \(W+C\), the first sum is greater than the second, i.e.

\[ T_1+T_2 > T'_1+T'_2. \]

It follows from the preceding consideration that, when passing over passes, braking is performed on all steep, harmful descents; meanwhile, precisely at this moment the train occupies the highest position, i.e., possesses the greatest potential energy.

Fig. 4.

[Figure text:]

Change in kinetic energy \(K\) when moving from \(A\) to \(Z\)

Change in potential energy \(P\)

Change in current energy \(P + K\)

Work of braking

\(ABC\ldots JZ\)—profile
\(BC\) and \(HJ\)—harmless ascents.
\(DE\) and \(FG\)—harmful ascents.

\[ K = a \frac{L+Q}{g}\cdot\frac{v^{2}}{2} \]

When calculated by virtual coefficients, at that same moment it has also a considerable kinetic energy

\[ a(L+Q)\frac{v^{2}}{2g}\ \mathrm{tm}, \]

where \(\alpha = 1.06—1.10\) and the influence of the rotating slopes is taken into account. Let us now trace the change in the total sum of energies in the train corresponding to the calculation by virtual coefficients.

From Fig. 4 it is seen that, for continuity in the change of the total reserve of energy, it is necessary first of all that the speeds vary over a certain distance; then the ledges are replaced by smooth curves. In addition, it is necessary that the work of the braking forces \(PQ\) be equal to the loss in kinetic energy plus the work of the motive force \([(L+Q)\tg \alpha - W - C]\,l\), i.e. the train at point \(D\) acquires neither more nor less than the harmful reserve of energy; the sum \(P+K\) is excessive, and it has to be reduced artificially by the work of the harmful braking forces. Meanwhile this sum is composed of two components

\[ P = (L+Q)\,h,\quad K = \alpha\,(L+Q)\,\frac{v^2}{2g}, \]

expressed in ton-meters; the latter is in essence variable, since it includes a quantity varying within fairly wide limits—the speed at point \(D\). If the speed at point \(D\) is reduced to the limit established by the calculation, then, for continuity of the sum of energies, it is necessary that the speed be reduced also at the preceding points; in other words, the ascent \(BC\) must be taken partly from acceleration, with a loss of speed, in order to make up for the loss on the descent \(DE\), i.e. to reduce the locomotive’s work on the ascent. As a result there will be some loss of time and a gain in the work of the tractive force, and also, if not complete elimination, then a reduction in the wear of the rolling stock and track from braking on the descent \(DE\).

For the planned use of the live force theoretically expended on the work of the braking forces, it is necessary above all to be able to find it precisely, i.e. to have before oneself a picture of the actual change in running speeds at all points of the route. Here, precisely, the special features of variants stand out clearly in the sense of the character of the passage across watersheds.

When considering the influence of a particular factor on the course of one or another phenomenon, subject to a whole network of influences independent of one another, it is necessary to consider it in isolation, i.e. on the assumption that all the other influences remain constant throughout the entire process. Then a picture emerges of the significance of the factor under consideration, but the numerical conclusions obtained from the formulas are practically inapplicable, since in real life no correspondence with the assumptions is encountered. In striving to give a practically applicable

the result, I shall, as far as possible, come closer to the ordinary conditions of a train’s motion.

It was already indicated above that the formula of the trinomial expansion

\[ W = A + BV + CV^{2} \]

cannot be regarded as a function continuous for the entire running time, since the resistance of a train, as of a body moving in a moving medium, depends not only on its absolute speed, but also on the relative speed, i.e., on the speed \(v'\) and the direction \(\alpha\) of the wind; in other words, the coefficients \(A\), \(B\), and \(C\) are not constant, but are functions of \(v'\) and \(\alpha\). From this point of view, what is obtained is no longer a curve, but

\[ W = f(V, v', \alpha), \]

i.e., a certain curvilinear surface. However, for the case \(v' = \mathrm{const.}\) and \(\alpha = \mathrm{const.}\), there remains only \(W = f_{1}(V)\), geometrically represented by a section of the surface by a definite plane, given by the force and direction of the wind. This function of one independent variable, during the period of invariability of \(v'\) and \(\alpha\), may already be recognized as continuous,

\[ W = A + BV + CV^{2}, \]

and the coefficients \(A\), \(B\), and \(C\), under the stated conditions, are constant and can be found numerically. As indicated above, only the change of speeds over a short interval of time, of several minutes, is considered, during which the assumption made does not contradict reality.

For the mathematical analysis itself there remains the usual assumption for virtual calculations of the equality of tractive force and resistance in steady motion, i.e., on long sections with unchanging resistance.

Character of Motion over a Pass

When crossing a pass, four phases are observed: 1) the transition from a level section or a slight ascent to a steeper ascent, 2) the transition from the latter to a level section, 3) the transition from a level section to a gradient, and 4) the transition from a gradient to a level section or to a gentler gradient.

The third phase is of two kinds: a) the transition from a level section to an adverse gradient, and b) the transition from a level section to a harmless gradient. In what follows, all these phases are considered separately.

First thesis.

Let the following assumption be made: the train approaches the break in the profile in steady motion, i.e.

\[ Z_0 = W_0 = A + BV_0 + CV_0^2 \tag{1} \]

When passing over the vertex of the angle, a change of tractive force is made with such a calculation that

\[ Z_i = W_i = A + BV_i + CV_i^2 + i (L + Q) 1000; \tag{2} \]

\(i\) characterizes the grade in thousandths, and \(V_i\) is the speed of uniform motion on the ascent for the given values \(A, B, C\), i.e. for the given meteorological conditions.

With a view to subsequently integrating the equation of motion, we shall recalculate the resistance formulas so that the speed enters them not in \(V \dfrac{\mathrm{klm}}{\text{hour}}\), but in \(v \dfrac{\mathrm{mtr.}}{\mathrm{sec.}}\), whence only the numerical value of the coefficients \(A, B, C\) will change into \(A', B', C'\).

Since the speed cannot pass instantaneously from the value \(v_0\) to the value \(v_i\), the equation of motion for the initial moment is obtained as

\[ Z_i - W_0 - i (L + Q) \cdot 1000 = \alpha \cdot 1000 \frac{L + Q}{g}\,\gamma, \tag{3} \]

where \(\gamma\) is the variable acceleration imparted to the train by the excess of tractive force over resistance; \(\alpha\) is a coefficient allowing for the influence of the rotating masses on the train’s mass of inertia and varying from 1.06 to 1.10. Substituting the values of \(Z_i\) and \(W_0\) from formulas (2) and (1) into formula (3), we obtain

\[ \alpha \cdot 1000 \frac{L + Q}{g}\,\gamma = A' + B'v_i + C'v_i^2 + i (L + Q)\cdot 1000 - A' - B'v_0 - \]

\[ - C'v_0^2 - i (L + Q)\cdot 1000; \]

after reduction we obtain

\[ \alpha \cdot 1000 \frac{L + Q}{g}\,\gamma = B'(v_i - v_0) + C'(v_i^2 - v_0^2); \]

\(v_i\) is in general less than \(v_0\), since the locomotive can haul the same train uniformly, under the same conditions, on the level at a speed greater than on an ascent; thus equation (3) characterizes the motion

retarded and moreover non-uniform, since the resistance varies according to the law

\[ W = A' + B'v + C'v^2 \]

Then the equation of motion will take the form

\[ \tag{4} \alpha \cdot 1000 \frac{L+Q}{g}\,\gamma = B'(v_i - v) + C'(v_i^2 - v^2), \]

where \(v\) varies within the limits from the initial value \(v_0\) to the value, for uniform motion along the ascent, \(v_i\).

Introducing the notation

\[ L+Q=P,\quad \frac{g}{\alpha \cdot 1000 (L+Q)}=D,\quad B'D=2HK,\quad C'D=K^2, \]

we obtain

\[ \gamma = 2HK(v_i-v)+K^2(v_i^2-v^2); \]

if we denote

\[ 2HKv_i + K^2v_i^2 = G, \]

then formula (4) will take the form

\[ \tag{5} \gamma=\frac{dv}{dt}=G-2HKv-K^2v^2; \]

adding and subtracting \(H^2\) on the right-hand side, we obtain

\[ \frac{dv}{dt}=(G+H^2)-(Kv+H)^2; \]

calling \(G+H^2=M^2\), we shall have

\[ \frac{dv}{dt}=M^2-(Kv+H)^2; \]

by separation of variables

\[ \tag{6} \frac{dv}{M^2-(Kv+H^2)}=dt; \]

here

\[ M^2-(Kv+H)^2<0, \]

since, after substituting into this difference the values

\[ M^2=G+H^2=2HKv_i+K^2v_i^2+H^2, \]

we obtain

\[ 2HKv_i+K^2v^2+H^2-K^2v^2-2HKv-H^2 \]

\[ =2HK(v_i-v)+K^2(v_i^2-v^2); \]

in this case the numbers H and K are positive, while \(v\) always remains greater than \(v_i\).

Multiplying both parts of equation (6) by \(-1\), we obtain

\[ \frac{dv}{(Kv+\mathrm{H})^2-\mathrm{M}^2}=-dt \]

or

\[ \frac{1}{2\mathrm{M}}\left[\frac{dv}{Kv+\mathrm{H}-\mathrm{M}}-\frac{dv}{Kv+\mathrm{H}+\mathrm{M}}\right] = -\,dt. \tag{7} \]

Integrating equation (7), we come to the conclusion that

\[ \frac{1}{2\mathrm{M}K}\left[\lg(Kv+\mathrm{H}-\mathrm{M})-\lg(Kv+\mathrm{H}+\mathrm{M})\right] = C-t; \tag{8} \]

for \(t=0\)

\[ C=\frac{1}{2\mathrm{M}K}\lg\frac{Kv_0+\mathrm{H}-\mathrm{M}}{Kv_0+\mathrm{H}+\mathrm{M}}. \]

Passing from logarithms to functions, we obtain

\[ e^{2\mathrm{M}KC}=\frac{Kv_0+\mathrm{H}-\mathrm{M}}{Kv_0+\mathrm{H}+\mathrm{M}}; \tag{9} \]

since in the right-hand side of formula (9) all terms are constant, let us call

\[ e^{2\mathrm{M}KC}=\mathrm{N}; \]

then from (8) successively we obtain

\[ \frac{\mathrm{N}}{e^{2\mathrm{M}Kt}}=\frac{Kv+\mathrm{H}-\mathrm{M}}{Kv+\mathrm{H}+\mathrm{M}}, \tag{10} \]

\[ \frac{Kv+\mathrm{H}-\mathrm{M}}{2\mathrm{M}}=\frac{\mathrm{N}}{e^{2\mathrm{M}Kt}-\mathrm{N}}, \]

\[ Kv+\mathrm{H}-\mathrm{M}=\frac{2\mathrm{M}\mathrm{N}}{e^{2\mathrm{M}Kt}-\mathrm{N}}, \]

\[ v=\frac{2\mathrm{M}\mathrm{N}+(\mathrm{M}-\mathrm{H})\left(e^{2\mathrm{M}Kt}-\mathrm{N}\right)}{K\left(e^{2\mathrm{M}Kt}-\mathrm{N}\right)}. \tag{11} \]

Formula (11) is somewhat cumbersome; nevertheless, from it one can construct the curve (Fig. 5)

Fig. 5.

\[ v=\psi(t) \]

\[ v=\varphi(t). \]

The distance over which the change of \(v\) from \(v_0\) to \(v_i\) takes place is determined as follows:

\[ s=\int_0^t v\,dt =\int_0^t \frac{2MN+(M-H)\left(e^{2MKt}-N\right)} {K\left(e^{2MKt}-N\right)}\,dt \]

\[ =\frac{M-H}{K}\,t+\frac{2MN}{K}\int_0^t \frac{dt}{e^{2MKt}-N} \]

\[ =\frac{M-H}{K}\,t-\frac{2M}{K}\int_0^t \frac{dt}{1-\dfrac{e^{2MKt}}{N}} \]

\[ =\frac{M-H}{K}\,t-\frac{2M}{K}\int_0^t \frac{1-\dfrac{e^{2MKt}}{N}+\dfrac{e^{2MKt}}{N}} {1-\dfrac{e^{2MKt}}{N}}\,dt \]

\[ =\frac{M-H}{K}\,t-\frac{2M}{K}\,t +\frac{1}{K^2}\left[ \lg\left(\left(1-\frac{e^{2MKt}}{N}\right)\right) -\lg\left(1-\frac{1}{N}\right) \right] \]

\[ \tag{12} =\frac{1}{K^2}\lg\frac{N-e^{2MKt}}{N-1} -\frac{M+H}{K}\,t. \]

By formula (12) there can be constructed

\[ s=\psi(t) \]

—the curve of the change of distance with time (Fig. 6). If \(t\) is eliminated from formulas (11) and (12), then a certain dependence is obtained

\[ v = \chi(s). \]

Because of the great difficulty of such an elimination, one may use the constructed curves

\[ v = \varphi(t),\quad s = \psi(t), \]

taking from them, every 5–10 seconds, the corresponding values, and construct (Fig. 7) the curve

\[ v = \chi(s). \]

Fig. 6.
Fig. 7.

OC shows the length of the run at \(v=\mathrm{const}=v_i=t\varphi g\), as is assumed in calculations by the virtual-coefficient method.

From it one sees that the mean speed of the run of the ascent depends to a large degree on its length; if this length is greater than \(s_1\), the value obtained from the curve

\[ v = \chi(s), \]

at which \(v = v_i\), then at the highest point of the ascent the speed is equal to \(v_i\); otherwise it is greater than this value.

From Fig. 6 it is seen that, according to the proposed calculation, the path traveled in the time \(t_1\) is greater than by the method of the virtual coefficient by the amount \(AB\); this excess is credited to the loss of kinetic energy

\[ \alpha \frac{P}{g}\cdot 1000\,\frac{v_0^2-v_i^2}{2} \]

with account taken of the change of resistance.

Phase Two.

Transition from a steep ascent to the level stretch at the highest point.

To take account of the phenomena occurring in this case, let us assume that the train approaches the break with uniform speed

\[ Z_i = W_i = A' + B'v_i + C'v_i^2 + 1000.\, i.\, P; \tag{13} \]

where \(A'\), \(B'\), \(C'\), depending on the curves in plan, may also have other values, since the angle \(\alpha\) changes; likewise, if ...

...or other causes of resistance have changed, then accounting for them will enter into the values of the coefficients. If, however, there are no grounds for a discontinuity of the function

\[ A' + B'v + C'v^2, \]

then the coefficients remain the same as in the first case.

The train may approach the break with a speed unequal to \(v_i\), if the length of the ascent is shorter than the value \(s_1\) obtained from the curve

\[ v = \chi(s), \]

derived in considering the first phase; the latter will affect only the limits of integration: for the initial speed one must take not \(v_i\), but the one found from the above-mentioned curve.

The assumption of continuity of the function during the transition remains in force: whether \(A'\), \(B'\), \(C'\) are the former or different ones, they must be constant over this short interval of time.

At the moment of the change of tractive force from \(Z_i\) to \(Z_0\), since there is a level section at the summit, the equation of motion of the train takes the form:

\[ \begin{aligned} Z_0 - W_i - z_0 - A' - B'v_i - C'v_i^2 &= A' + B'v_0 + C'v_0^2 - A' - B'v_i - C'v_i^2 \\ &= B'(v_0 - v_i) + C'(v_0^2 - v_i^2) \end{aligned} \tag{14} \]

Here the excess of tractive force over resistance is apparent, since \(v_0\) is greater than \(v_i\); the motion is nonuniformly accelerated, and \(v\) will vary from \(v_i\) to \(v_0\).

The equation of motion, with the former notation, becomes

\[ \alpha \cdot 1000 \frac{P}{g}\,\gamma = B'(v_0 - v) + C'(v_0^2 - v^2), \]

\[ \gamma = \frac{dv}{dt} = 2HK\,(v_0 - v) + K^2\,(v_0^2 - v^2). \tag{15} \]

Denoting

\[ 2HKv_0 + K^2v_0^2 = G', \]

we obtain formula (15) in the form

\[ \frac{dv}{dt} = G' - 2HKv - K^2v^2. \]

or, after adding and subtracting \(H^2\) from the right-hand side,

\[ \frac{dv}{dt}=(G' + H^2)-(Kv+H)^2; \]

then, putting

\[ G' + H^2 = M_1^2, \]

we obtain

\[ \frac{dv}{dt}=M_1^2-(Kv+H)^2 . \tag{16} \]

In formula (16)

\[ M_1^2-(Kv+H)^2>0, \]

since

\[ M_1^2=G' + H^2 = 2HKv_0 + K^2v_0^2 + H^2; \]

therefore, replacing \(M_1^2\) by its value, we obtain

\[ 2HKv_0 + K^2v_0^2 + H^2 - K^2v^2 - 2HKv - H^2 = \]

\[ = 2HK(v_0-v) + K^2(v_0^2-v^2); \]

and since \(v\) varies within the limits from \(v_i\) to \(v_0\), the difference is positive.

From equation (16), by separation of variables, we get

\[ \frac{dv}{M_1^2-(Kv+H)^2}=dt; \]

on integrating, we shall have

\[ \frac{1}{2M_1K}\left\{-\lg\left[M_1-(Kv+H)\right]+\lg(M_1+Kv+H)\right\}=t+C; \tag{17} \]

for \(t=0\), \(v\) is equal either to \(v_i\), or to the value taken from the curve \(v=\chi(s)\); then

\[ C=\frac{1}{2M_1K}\lg\frac{M_1+Kv_0+H}{M_1-Kv_0-H}, \]

\[ e^{2M_1KC}=\frac{M_1+Kv_0+H}{M_1-(Kv_0+H)}=N_1; \]

from equation (17) it follows that

\[ N_1 e^{2M_1Kt}=\frac{M_1+Kv+H}{M_1-Kv-H}, \]

\[ \frac{2M_1}{M_1-Kv-H}=N_1 e^{2M_1Kt}+1 \]

\[ M_1-Kv-H=\frac{2M_1}{1+N e^{2M_1Kt}}. \]

\[ \tag{18} v=\frac{(M_1-H)\left(1+N_1 e^{2M_1Kt}\right)-2M_1} {K\left(1+N_1 e^{2M_1Kt}\right)} \]

Fig. 8.

Formula (18) expresses the relation between the change of velocity and time, and makes it possible to construct the corresponding curve (Fig. 8)

\[ v=\varphi_1(t). \]

To determine the dependence of distance on time, we shall use the formula

\[ s=\int_0^t v\,dt =\int_0^t \frac{(M_1-H)\left(1+N_1 e^{2M_1Kt}\right)-2M_1} {K\left(1+N_1 e^{2M_1Kt}\right)} \,dt \]

\[ =\frac{M_1-H}{K}\,t-\frac{2M_1}{K}\int_0^t \frac{dt}{N_1 e^{2M_1Kt}+1} \]

\[ =\frac{M_1-H}{K}\,t-\frac{2M_1}{K}\,t +\frac{2M_1}{K}\int_0^t \frac{N_1 e^{2M_1Kt}}{1+N_1 e^{2M_1Kt}}\,dt \]

\[ =\frac{M_1-H}{K}\,t-\frac{2M_1}{K}\,t +\frac{1}{K^2} \left[ \lg\left(1+N_1 e^{2M_1Kt}\right)-\lg(N_1+1) \right] \]

\[ \tag{91} =\frac{1}{K^2}\lg\frac{1+N_1 e^{2_1MKt}}{1+N_1} -\frac{M_1+H}{K}\,t. \]

Here the dependence between distance and time is expressed; from points at intervals of 5–10 seconds one can construct the curve (Fig. 9).

\[ s=\psi_1(t). \]

Along the curves

\[ v=\varphi_1(t),\quad s=\psi_1(t) \]

one may construct (Fig. 10) the curve

\[ v=\chi_1(s). \]

Fig. 9.  Fig. 10.

From Fig. 9 it is seen that, with the segment of time \(t_1\) adopted for the calculation, the train traverses along \(DE\) a shorter distance than by the method of virtual coefficients; as is known, this loss goes toward increasing the live force up to the value

\[ \alpha\,1000\,\frac{P}{g}\,\frac{v_0^{2}}{2}. \]

The segment \(OC\) shows the length of the run at \(v=\mathrm{const}=v_0=\operatorname{tg}\varphi\), as is adopted in the calculation by virtual coefficients.

Although, as is seen from the computation performed, the gain in run in the lower point is compensated by the loss in the upper one, whether this compensation is complete or partial on low passes depends on the lengths of the sections considered.

In other words, all the differences encountered in the variants are taken into account here, which is not the case in the method of virtual coefficients; the latter establishes always and everywhere complete compensation, which occurs only for long sections.

From the consideration of the further phases it will be seen how such a generalization is reflected on low passes; for the present one can point out only the following considerations. If the actual length of the ascent is insufficient for changing the speed from \(v_0\) to \(v_i\), then the train will approach the upper platform with a speed greater than \(v_i\); but if the length of the connecting platform is sufficiently small, as often occurs, then by its end the speed \(v_0\) will not be obtained, and the train already passes onto the grade, along which the motion, for reasons of safety, must almost always be carried out at a speed less than \(v_0\). To take into account the influence of this factor, let us consider the third phase.

Phase Three

Transition from the adjoining level stretch at the summit of the watershed to a harmless grade.

The considerations concerning the discontinuity of the function

\[ A' + B'v + C'v^2 \]

remain the same as those given in the examination of the second phase.
With a sufficiently long level stretch the train approaches the break in the grade with speed \(v_0\), i.e. its resistance is

\[ W = A' + B'v_0 + C'v_0^2; \]

if, however, the length of the level stretch is insufficient, then the speed will be less than \(v_0\) and can be found from the curve

\[ v = \chi_1(s). \]

If such a cutoff is possible that

\[ Z_{-i} = W_{-i} - 1000\, i\, P \]

\[ = A' + B'v_{-i} + C'v_{-i}^{2} - 1000\, i\, P, \]

where the value \(Z_{-i}\) may be very small, depending on the magnitude of \(i\)—an assumption taken from the calculation along virtual lengths—then, at the cutoff variable, the equation of motion of the train will be

\[ \tag{20} a \cdot 1000\,\frac{P}{g}\,\gamma = Z_{-i} - A' - B'v - C'v^2 + 1000\, i\, P. \]

Substituting into formula (20) the value of \(Z_{-i}\), as defined above, and after reduction, we obtain

\[ \tag{21} 1000\, a\,\frac{P}{g}\,\gamma = B'(v_{-i}-v) + C'(v_{-i}^{2}-v^{2}); \]

\(v_{-i}\), from the condition of the possibility of braking the train over a distance of 250 sazhen on grades, is almost always less than \(v_0\). This condition, of course, depends to the greatest degree on the percentage of braked cars, and also on the magnitude of the grade, although only for very slight grades is it not required.

If \(v_{-i}\) is less than \(v_0\), then equation (21) expresses nonuniformly retarded motion, of course only under the assumption of a sufficient length of the coupling section, over which the speed \(v_0\) has time to develop. It is integrated in exactly the same formulas as equation (4), i.e.

\[ v = \frac{ 2 M_2 N_2 + (M_2 - H)\left(e^{2M_2 K t} - N_2\right) }{ K\left(e^{2M_2 K t} - N_2\right) }, \tag{22} \]

\[ s = \frac{1}{K_2}\lg \frac{N_2 - e^{2M_2 K t}}{N_2 - 1} - \frac{M_2 + H}{K}\,t, \tag{23} \]

where

\[ M_2^2 = 2HKv_{-i} + K^2v_{-i}^2 + H^2, \]

\[ N_2 = \frac{Kv_0 + H - M_2}{Kv_0 + H + M_2}. \]

The curves expressing these functions are quite analogous to those shown in Figs. 5, 6, 7, and show a certain gain in the run. If, however, the section is not sufficiently long and the train on it does not develop the speed \(v_0\), but only some \(v\) less than \(v_{-i}\), or if, because of the number of braking cars and the magnitude of the grade, \(v_{-i}\) is greater than \(v_0\), then equation (21) expresses nonuniformly accelerated motion and is integrated as equation (15).

Then functions are obtained analogous to Figs. 8, 9, and 10, and they express a certain loss in the run for increasing the living force.

Case of a harmful grade.

Since the component force of gravity is greater than the resistance depending on speed, the train proceeds with acceleration and with the regulator closed; so that a speed does not develop at which it becomes impossible to brake the train over a distance of 250 sazhens; it is braked, maintaining a certain calculated speed \(v_{-i}\), at which the resistance is expressed in the form

\[ A' + B'v_{-i} + C'v_{-i}^{2}. \]

At the moment of passing over the break of grade, the equation of motion assumes the form

\[ \tag{24} 1000\,\alpha \frac{P}{g}\,\gamma = 1000\,iP+3L' - A' - B'v_0 - C'v_0^2, \]

where

\(1000\,iP\) is the component of gravity parallel to the grade;

\(3L'\) is, according to Villemin, the reduction of resistance in the locomotive with the regulator closed, where \(L'\) is the weight of the locomotive in tons;

\(v_0\) is the speed with which the train approaches the break in grade; if the level stretch is not sufficiently long, it is not equal to \(v_0\), but is taken from the curve

\[ v=\chi_1(s). \]

Since the magnitude of the grade is such that the component of gravity is greater than the forces of resistance, the motion is accelerated, and moreover nonuniform, owing to the change in the resistance

\[ A' + B'v + C'v^2. \]

Meanwhile, from the safety condition—the braking of the train over a distance of 250 sazhen (530 metr)—it is clear that the speed in such motion cannot be equal to the speed on a straight horizontal line. Otherwise, at the moment of danger the train would have the same kinetic energy,

\[ \alpha \cdot 1000\,\frac{P}{g}\,\frac{v_0^2}{2}, \]

and even on a straight horizontal track, on which this energy is absorbed by the work of the available brakes over a distance of 250 sazhen, on a grade the same brakes over the same distance must also absorb the work of the moving force

\[ \tag{A} 1000\,iP - A' - B'v - C'v^2 + 3L'. \]

In formula (A), \(v\) changes from \(v_0\) to 0. Obviously, for safety \(v_i\) is less than \(v_0\); in other words, the speed of approach to the break in grade is excessive, and it must be reduced.

The calculation of braking presents enormous difficulties for introducing it; moreover, the process itself depends on a mass of contingencies. Let us assume that, in order to maintain a constant speed on the descent, \(v_i\), the required braking effort is also constant; since in uniform motion the force imparting acceleration is equal to 0, the braking effort will take the following form:

\[ \tag{25} T=1000\,iP+3L' - A' - B'v_i - C'v_i^2. \]

its work is equal to \(Ts\), if \(s\) is the length of the harmful gradient. To it one must also add the work of reinforced braking in order to reduce the kinetic energy by the amount

\[ \alpha \cdot 1000 \frac{P}{g}\,\frac{v_0^{2}-v_i^{2}}{2}\ \mathrm{klgmt}. \]

If the descent or ascent is nonuniform, but consists of sections with different inclinations to the horizon, then the derived propositions must be applied to each homogeneous section and the results summed.

Phase Four.

Transition from a gradient to a level section or to a gentler gradient.

As a general rule, the speed of approach to the corresponding break \(v_i\) is less than \(v_0\), the speed on the level section. With a very gentle slope it may sometimes also be the case that \(v_i\) is greater than \(v_0\).

After deducting the tractive force,

\[ Z_0 = A' + B'v_0 + C'v_0^{2}, \]

where the coefficients \(A'\), \(B'\), \(C'\), as already indicated, in the general case may have values different from those in the preceding phase, the equation of motion takes the form

\[ 1000\,\alpha \frac{P}{g}\,\gamma = A' + B'v_0 + C'v_0^{2} - A' - B'v - C'v^{2}, \]

after reduction,

\[ 1000\,\alpha \frac{P}{g}\,\gamma = B'(v_0-v) + C'(v_0^{2}-v). \tag{25} \]

Equation (25) is integrated in exactly the same way as equation (14); the form of the functions

\[ v=\varphi_3(t),\quad s=\psi_3(t),\quad v=\chi_3(s) \]

will be analogous to those indicated in Figs. 8, 9, 10. As for the second case, equation (25) remains in the same form, but is integrated according to the type of equation (4) and gives the functions

\[ v=\varphi_4(t),\quad s=\psi_4(t),\quad v=\chi_4(s) \]

by analogy with Figures 5, 6, 7.

In the phase under consideration, a very interesting question is that of when one must begin braking in order, without violating safety requirements, to approach the break with a speed corresponding to the following section.

Owing to the special nature of the investigation of braking, this question lies outside the scope of the article.

From the preceding consideration it is clear that, when a train descends from a pass along a steep gradient, at the initial point of the slope it has an excessive speed and, for safety reasons, is subjected to braking; in this case the speed is artificially reduced by the work of undoubtedly harmful braking forces. Meanwhile, if the speed at the highest point of the descent is lowered beforehand to the required standard, then in the first phase a smaller tractive force can be developed. The point is that the resistance, whatever the numerical values of the coefficients \(A'\), \(B'\), and \(C'\), increases with speed, and if the speed is reduced, then the work of the tractive force over the same length will be smaller.

Let us try to approach the question of how much the work of the harmful forces can be weakened, and how much can be gained in the work of the locomotive, by making use of the convertibility of the energy reserve into work. From Fig. 4 it is seen that, at point \(D\), at the beginning of the descent, the train possesses the energy

\[ 1000 \cdot \frac{P}{g}\left(\frac{\alpha v_0^2}{2}+gh\right)\ \text{klgmtr.}, \]

whereas at its lowest point it has the energy

\[ 1000 \cdot \frac{P}{g}\left(\frac{\alpha v_0^2}{2}\right)\ \text{klgmtr.}; \]

if the engine driver succeeds in practically solving the problem of releasing the brakes, then in this ideal case all the potential energy \(Ph\) ton. met. disappears; whereas for moving the train along the slope only work equal to

\[ \tag{26} \int_{0}^{s_1} (A' + B'v + C'v^2)\,ds + (A' + B'v_i + C'v_i^2)(s - s_1 - s_2) \]

\[ + \int_{0}^{s_2} (A' + B'v + C'v^2)\,ds. \]

In formula (26), \(s\) denotes the length of the slope, \(s_1\) that portion of it on which braking reduces the speed from \(v_0\) to \(v_i\), and \(s_2\) the length,

...on which, after braking, the speed increased from \(v_{-i}\) to the value \(v_0\). Having the curve \(v=\chi(s)\), the quadrature

\[ \int_0^{s_1}\left(A' + B'v + C'v^2\right)\,ds \]

is easy to perform, since \(A'\), \(B'\), \(C'\) have, in each case, definite numerical values.

If, however, one sets oneself the problem of letting the train descend uniformly with the greatest permissible speed \(v_{-i}\) on the grade, then the work for this operation is equal to

\[ \left(A' + B'v_{-i} + C'v_{-i}^2\right)(s-s_1) + \int_0^{s_1}\left(A' + B'v + C'v^2\right)\,ds; \tag{27} \]

in formula (27), \(s\) is the length of the grade, \(s_1\) the length of that part of it on which the speed falls from the value \(v_0\) to the value \(v_{-i}\); but in this case, at the lower point of the grade, the store of energy of the train is equal to

\[ 1000\,\alpha\,P\,\frac{v_{-i}^2}{2g}\ \mathrm{klg\ mtr.}, \]

and its loss during the descent is equal to

\[ 1000\,\frac{P}{g}\left[\frac{\alpha\left(v_0^2-v_{-i}^2\right)}{2}+gh\right]\ \mathrm{klgmtr.}, \tag{28} \]

which is also expended on braking, excluding the value of the work according to formula (27).

Having adopted this simpler case, let us see how much the detrimental work can be reduced. The second term in formula (28) depends only on the configuration of the profile and cannot be subject to any changes; the first, however, can be varied within wide limits, for which it is only necessary to lower the approach speed to the grade to the minimum practically advantageous value \(v'\). Then the decrease of energy at the upper point of the grade will be expressed by the value

\[ 1000\,\alpha\,P\,\frac{v_0^2-v'^2}{2g}\ \mathrm{klgmtr.}; \tag{29} \]

the work on the descent of the train will receive the value

\[ \int_0^{s_1}\left(A' + B'v + C'v^2\right)\,ds + \left(A' + B'v_{-i} + C'v_{-i}^2\right)(s-s_1), \tag{30} \]

where \(s_1\) is the length over which the speed increased from the value \(v'\) to \(v_{-i}\).

The work determined by formula (30) is not so very difficult to compute, knowing even only approximately the form of the function \(v=\chi_4(s)\),

since there is no braking here. When comparing the numerical value of the loss of energy by the train during the descent, which, in the present notation, is equal to

\[ \tag{31} 1000\,P\left[\frac{\alpha\,(v'^2-v_i^2)}{2g}+h\right]\ \mathrm{klgmtr.}, \]

with the numerical value of the work on the descent, expressed by formula (30), there may be three cases:

1) the numerical values of formulas (30) and (31) are equal to one another,

2) the numerical value of formula (30) is greater than the numerical value of formula (31),

3) the numerical value of formula (30) is less than the numerical value of formula (31).

In the first case there will be no braking at all, and it is possible in advance to use for useful work

\[ 1000\,\alpha\,P\,\frac{v_0^2-v'^2}{2g}\ \mathrm{klgmtr.} \]

—the decrease of kinetic energy at point D.

In the second case a decrease of the speed at the highest point of the incline to the value \(v'\) is not permissible. Its minimum permissible value is obtained from the equation

\[ \tag{32} \int_{0}^{s_x}\left(A' + B'v + C'v^2\right)\,ds +\left(A' + B'v_i + C'v_i^2\right)(s-s_x) \]

\[ =1000\,P\left[\frac{\alpha\,(v_x^2-v_i^2)}{2g}+h\right]*), \]

and, with complete abolition of braking, at point D there is freed for the preliminary expenditure on the ascent

\[ 1000\,\alpha\,\frac{P}{g}\,\frac{v_0^2-v_x^2}{2}\ \mathrm{klgmtr.} \]

of work.

Finally, the third case shows that braking in general cannot be avoided, and the same amount of work is freed as in the first case.

Whatever the value of the energy freed, it can be used by reducing the tractive force on the ascent

\[ \tag{33} 1000\,\alpha\,P\,\frac{v_0^2-v_x^2}{2g}=(Z_i-Z_x)\,s. \]

*) \(s_x\) is the length over which the speed changes from the value \(v_x\) to the value \(v_i\).

Equation (33) makes it possible to find that tractive force \(Z_x\) at which the pass is traversed with the least work of the braking forces, i.e., in the most advantageous manner.

To determine the actual distribution of speeds and the magnitude of the inevitable loss of time, it is necessary to recalculate phases one and two, substituting \(Z_x\).

From formula (33) one obtains

\[ Z_x = Z_i - \alpha \cdot 1000 \frac{\mathrm{P}}{q} \frac{v_0^2 - v_x^2}{2s}, \tag{34} \]

whence the enormous significance of the length of the ascent is clearly seen: for a large length \(s\), the second term on the right-hand side will have a very small value, whereas, by lowering the average speed over a large section, one can obtain a considerable loss of time.

Consequences arising from the preceding considerations.

By the method of virtual tractive calculations, one finds the total amount of work for movement and the expenditures of water and fuel necessary for accomplishing it. If one sets as the objective the possible reduction of braking work, then over the whole haul one can save

\[ \mathrm{T} = \sum (Z_i - Z_x)\, s \ \mathrm{klgmtr.}, \tag{35} \]

of work, carrying out the summation over all passes to be taken into account. For lines passing through terrain cut by watercourses, in other words, for lines running along the valleys of large rivers, the number of watersheds crossed is considerable, while their heights for secondary watercourses are insignificant. Thus, for roads of this character the numerical value of \(\mathrm{T}\) may be very large.

For roads under construction, the comparison of alternatives is carried out in the following manner*).

First alternative: length \(N\) actual versts, construction cost \(n\) rubles per actual verst, virtual operating length \(Z\) versts, operating expense \(s\) rubles per virtual verst. Second alternative: actual length \(N + U\) versts, virtual length \(Z - t\) versts, construction expense \(n - y\) rubles per actual verst, operating expense \(s\) rubles per virtual verst.

Capitalizing the operating expense at \(5\%\) per annum, we obtain the total sums of expense:

for the first alternative

\[ Nn + 20 Zs = K_1, \]

for the second alternative

\[ (N + U)(n - y) + 20 (Z - t) s = K_2, \]

* Kraevskii, “Railway Surveys,” 1902, vol. I, p. 297.

the difference will prove to be equal to

\[ \begin{aligned} K_1 - K_2 &= Nn + 20 Zs - Nn - 20 Zs - nU + (Ny + Uy + 20ts),\\ \tag{36} K_1 - K_2 &= Ny + Uy + 20ts - Un. \end{aligned} \]

The question of whether \(K_1 - K_2\) is greater or less than zero determines the advantage of the first or the second variant.

The proposed considerations apply only to the value \(20ts\), since only in this term is the economy taken into account. Leaving \(s\)—the operating expense per verst in the direction—aside, let us turn our attention to \(t\)—the reduction in the number of virtual versts in the second variant.

The calculation of the virtual length in surveys is carried out according to the formula

\[ \tag{37} L = l_0 + \sum l_n (1 + \alpha) + l_c + \sum l_k \beta \]

where \(l_0\) is the sum of the lengths of the horizontal sections,

\(l_n\) is the length of the equal ascents on the section,

\(l_c\) is the length of all descents on the section,

\(l_k\) is the length of curves of one radius,

\(1+\alpha\) is the virtual coefficient for an ascent,

\(\beta\) is the virtual coefficient for curves,

This formula was given by the Ministry of Ways of Communication in the circular of the Department of Railways of July 31, 1891, No. 9817.

In it, the virtual coefficient \(1+\alpha\) is calculated from the conditions of equality of work on an ascent of a given length and on the corresponding virtual length; the values of the coefficients are given in the following table:

Number of thousandths of ascent	\(\alpha\)	Number of thousandths of ascent	\(\alpha\)	Number of thousandths of ascent	\(\alpha\)	Number of thousandths of ascent	\(\alpha\)	Number of thousandths of ascent	\(\alpha\)
0,50	0,164	4,00	1,303	7,50	2,419	11,00	3,416	14 50	4,594
1,00	0,328	4 50	1,462	8,00	2,571	11,50	3,671	15,00	4,745
1,50	0,491	5,00	1,623	8,50	2,734	12,00	3,826	15,50	4,898
2,00	0,654	5,50	1,783	9.00	2,891	12,50	3,980	16,00	5,050
2,50	0,817	6,00	1,943	9,50	3,048	13,00	4,140	16,50	5,201
3,00	0 979	6,50	2,102	10,00	3,205	13,50	4,288	17.00	5,353
3,50	1,140	7,00	2 260	10,50	3,365	14,00	4,439	17,242	5,426

If one examines this table, one will see that for the grade \(i=0.006\) its length is almost tripled, \(1+\alpha=2.943\); the length of the grade \(i=0.007\) is multiplied by \(1+\alpha=3.26\); and so on.

According to the proposed considerations, it is evident that, for the corresponding cases on low passes, the tractive force on the ascent is reduced from the calculated value \(Z_i\) to the computed value \(Z_x\), and in the same ratio the work required to carry out the transportation is also reduced. The consequence will be a reduction of the virtual coefficient for such ascents in the ratio

\[ \frac{Z_x}{Z_i}, \]

i.e., a reduction also of the virtual operating length, which will be reflected in the value \(20\,ts\) in formula (36).

On the first, shorter variant crossing the locality, the ascents may be steeper and the number of passes greater; if their height does not exceed the limits at which the reduction of \(Z_x\) is already only weakly perceptible, then what has been said about the significance of passes will have direct application to it. On the second variant, which goes around with gentle slopes, the influence of the passes will be felt considerably less.

In conclusion, it may be pointed out that there are such heights of passes at which, under the proposed method of travel, the virtual coefficient is equal to unity.

In the presence of uniform motion,

\[ Z_i = 1000\, i\, P + A' + B'v_i + C'v_i^2, \]

\[ Z_0 = A' + B'v_0 + C'v_0^2. \]

If the tractive force on the ascent is reduced to the value \(Z_0\), i.e., if the cut-off is not changed when entering it, then the equation of motion

\[ Z_0 - 1000\, i\, P - (A' + B'v + C'v^2) = \alpha\, 1000\, \frac{P}{g}\,\gamma \tag{38} \]

expresses nonuniformly retarded motion.

Let us assume, as a reserve, that the fall in speed is not reflected in the resistances; then the saving in work from leaving the cut-off at \(Z_0\) will be expressed as

\[ (Z_i - Z_0)s = 1000\,\alpha\,\frac{P}{g}\,\frac{v_0^2 - v_x^2}{2}\ \mathrm{klgmtr.}, \tag{39} \]

where \(s\) is the length of the ascent, equal to

\[ \frac{h}{i}, \]

\(h\) is its height,

\(v_x\) has the value derived earlier.

S. A. Vvedensky.

In formula (39), the second term expresses the energy released as a result of the decrease, at the end of the upper level stretch, of the speed from the value \(v_0\) to the value \(v_x\); this released energy is then used to reduce the work of the tractive force on the ascent:

\[ \frac{h}{i}\left[1000\, i P - B'(v_0 - v_i) - C'(v_0^2 - v_i^2)\right] = 1000\,\alpha\,\frac{P}{g}\,\frac{v_0^2 - v_x^2}{2}. \]

\[ \tag{40} h = \frac{1000\,\alpha\, i P\,(v_0^2 - v_x^2)} {\left[1000\, i P - B'(v_0 - v_i) - C'(v_0^2 - v_i^2)\right]\,2g}; \]

Such a height of the pass, found from formula (40), reduces, with a margin, the tractive work on the ascent to the work on the level stretch; that is, the virtual coefficient of ascent is equal to unity.