excess of driving over driven E.M.F., namely (E− e), is diminished, and the current and horse-power delivered are diminished unless the decrease so effected be neutralised by increase in e, and, therefore, also in E, or by decrease in R+r. With small resistances, however, combined with high voltages, large horse-powers can be transmitted with high electric efficiency. The necessary voltages and resistances required to deliver any required horse-power with any given efficiency are easy to calculate, but the calculations do not illustrate the subject of this chapter.

A large number of electric transmission calculations of values giving maximum economy, etc,, under various conditions, may be found in a series of articles by the author in Industries, 1889.

190. Maxima of Function of Two Independent Variables.If a function of two independent variables be represented by the height of a surface with the two variables as horizontal co-ordinates, then any plane vertical section will give a curve which will rise to maximum height or fall to minimum height where the partial first gradient is zero, and the partial second gradient is not zero. If two such sections cross each other at a surface point where the partial first gradients are zero in both sections, then the surface is level throughout a small extent all round the intersection point. There are six cases to be distinguished.

(1) The partial second gradient is positive in both sections; then the point is at the bottom of a hollow in the surface, all neighbouring points being higher; so that the value of the function represented by the height is here a minimum. (2) The partial second gradients in the two sections are both negative; then the point is at the top of a convexity or globular part of the surface, all neighbouring points being lower; so that the value of the height function is here a maximum.

(3) The one partial second gradient is positive, while the other is negative. Here the surface is anticlinal, or saddleshaped; it is hollow in one direction, and round in the other. This gives neither minimum nor maximum value to the height function.

(4) The one partial second gradient is positive, while the other is zero, passing from positive to negative. Here the part of the surface is the junction between a hollow portion lying on one side of a vertical plane and an anticlinal portion lying on the other side of the same; and the height function is again at neither maximum nor minimum value.

(5) The one may be negative, while, as in (4), the other is zero.

There is here indicated the junction between a round portion and an anticlinal portion; and again neither maximum nor minimum occurs here.

(6) Again both may be zero.

Here two semi-anticlinal surfaces

join together, and the point gives neither maximum nor minimum.

The first two cases alone are important as regards finding maximum and minimum values. In these cases both first gradients are zero, and the second gradients are of the same sign. To find the maximum or minimum values of a function of two independent variables, the process is to equate the two partial first gradients to zero, and combine these two equations as simultaneous ones. Afterwards the second gradients should be examined as to sign if the physical character of the problem is not so plain as to make this unnecessary.

191. Most Economic Location of Junction of Three Branch Railways. As an illustration of this process, take an elementary problem in the theory of railway location. Three centres of traffic are supposed situated in a plain across which the construction of the railway is equally easy in all directions. The three centres are to be joined by three branch lines radiating from a junction, the finding of the best position for which junction is the problem proposed. The traffic issuing from and entering each centre is the sum of the two traffics between it and the other two (inclusive possibly of traffic going through these to more distant points). The importance of the three traffics to and from the three centres being properly measured, and being here symbolised by A, B, and C; and the distances of the three centres from the junction being called a, b, c; the correct solution of this problem means the location of the junction so as to give a minimum value to

Aa + Bb+Cc

A, B, C being given, while a, b, c are to be found.

Referring to the notation of fig. 29, in which the distance between the points of traffic A and C is called L, and the projection on L of the distance between the points of traffic A and B is called λ, while the projection of the same perpendicular to L is called H; the problem may be conveniently stated to be to find the co-ordinates / and h of the best junction.

We have with this notation

a={12+h2}1

The two partial

`b = { (λ − 1)2 + (H − h)2} 3

c = {(L − 1)2 + h2 } ‡.

and h gradients of (Aa + Bb+Cc) are taken;

thus

Therefore equating both partial gradients to zero, we obtain the simultaneous equations

Eliminating the terms in B by cross-multiplication, and again similarly eliminating the terms in C, there are obtained the two ratios

By reference to the figure it will be seen that the three angles entering into these ratios, namely, (180°+ß − y), (180°+ a − ß),

and (ya), are the three angles between b and c, a and b, c and a. Since the sides of a triangle are proportional to the sines of the opposite angles, it follows immediately that these three angles

Λ Λ

Λ

bc, ab, and ca, to which the three branches are to be adjusted, are the exterior angles of a triangle whose sides are made equal (to any convenient scale) to A, C, and B. By constructing this triangle these angles can be found, and by drawing upon two of the lines joining two pairs of centres of traffic two arcs of circles containing these angles, the proper junction is located as the intersection of these arcs.

This result may be perceived more directly by noticing that the two partial gradients to be equated to zero are, one the sum of the projections on L, and the other the sum of the projections on H, of A, B, C, measured along the lines a, b, c, outwards from the junction.

CHAPTER X.

INTEGRATION OF DIFFERENTIAL EQUATIONS.

192. Explicit and Implicit Relations between Gradients and Variables. The utility of the art of integration arises from the fact that in the investigation of phenomena it often happens that the discovery of the ratio between simultaneous increments (or gradient) of mutually related quantities is effected more easily by direct observation than is the discovery of the main complete relation between these same quantities. This complete relation is then logically deduced by the help of integration, combined with the further observation of special or "limiting" values. complete relation being less general than the differential relation, there appears in it an 'arbitrary constant" whose value is not given by the differential relation, and which value must be discovered by examination of the " limiting conditions." The differential relation applies equally well to a whole "family" of integral relations which differ among themselves in respect of these limiting conditions.

The

If, when the differential relation is expressed as an equation, the gradient can be placed alone on one side of the equation, while on the other appears a function of one only of the mutually dependent variables; then, in order to establish the integral relation, nothing

more has to be done than to integrate this latter function directly according to one or other of the methods explained in previous chapters, or given in the appended Classified Reference List of Integrals. This is the case of the gradient being expressed as an explicit function of one of the variables.

If, however, the differential equation involve the gradient and both variables in such a way that the above simple separation does not appear; that is, if the gradient appear as an implicit function of the variables; then either the implicitness of the relation must be got rid of by some algebraic process, or else a special method of integration must be employed. This process is called the "solution" of the differential equation.

193. Degree and Order of an Equation. Nomenclature.-An ordinary algebraic equation is said to be of the nth power or degree when it involves the nth power of the "unknown" quantity or of either of the "variables." In a differential equation, the gradients being the quantities to be got rid of by integration, the equation is said to be of the nth degree when the nth power of the highest occurring gradient is involved in it.

If it involve a second gradient (2) it is said to be of the

second order. If it involve an nth gradient, it is said to be of the nth order.

The integral relation deduced from the differential equation may be called the integral equation, but is also called the primitive equation.

194. X' =ƒ(x).

dx

The form of differential equation simplest to integrate is the

explicit relation,

where f(x) is a function

This is equivalent to

X' = f(x)

involving the variables x alone, and not X.

dX = f(x)dx

and if f() is integrable by help of any of the formulas in the Reference List, the integration can be effected at once; thus:—