dy

dy dx'

i.e. condition (4) is fulfilled

for the functions X and Y; which is contrary to the assumption.

In this case then the integration is impossible, so long as x and y are considered as independent variables. If however we assume any fixed relation to hold between these two quantities, so that one may be expressed as a function of the other, the integration again becomes possible. For if we put

in which ƒ expresses any function whatever, then by means of this equation we can eliminate one of the variables and its differential from the differential equation. (The general form in which equation (6) is given of course comprises the special case in which one of the variables is taken as constant; its differential then becomes zero, and the variable itself only appears as part of the constant coefficient). Supposing y to be the variable eliminated, the equation (3) takes the form dW = p (x) dx, which is a simple differential equation, and gives on integration an equation of the form

The two equations (6) and (7) may thus be treated as forming together a solution of the differential equation. As the form of the function f(xy) may be anything whatever, it is clear that the number of solutions thus to be obtained is infinite.

The form of equation (7) may of course be modified. Thus if we had expressed x in terms of y by means of equation (6), and then eliminated x and do from the differential equation, this latter would then have taken the form

dw=4, (y) dy,

and on integrating we should have had an equation

W= F(y)+const........

.(7a).

This same equation can be obtained from equation (7) by substituting y for x in that equation by means of equation (6). Or, instead of completely eliminating x from (7), we may

prefer a partial elimination. For if the function F(x) contains x several times over in different terms, (and if this does not hold in the original form of the equation, it can be easily introduced into it by writing instead of x an expression such 2+1 as (1 − a) x + ax, &c.) then it is possible to substitute

y for a in some of these expressions, and to let a remain in others. The equation then takes the form

which is a more general form, embracing the other two as special cases. It is of course understood that the three equations (7), (7a), (7b), each of which has no meaning except when combined with equation (6), are not different solutions, but different expressions for one and the same solution of the differential equation.

Instead of equation (6), we may also employ, to integrate the differential equation (3), another equation of less simple form, which in addition to the two variables x and y also contains W, and which may itself be a differential equation; the simpler form however suffices for our present purpose, and with this restriction we may sum up the results of this section as follows.

When the condition of immediate integrability, expressed by equation (4), is fulfilled, then we can obtain directly an integral equation of the form:

W = F(x, y) + const........

..(A).

When this condition is not fulfilled, we must first assume some relation between the variables, in order to make integration possible; and we shall thereby obtain a system of two equations of the following form:

f(x, y) = 0,

W= F(x, y) + const.

}

(B);

in which the form of the function F depends not only on that of the original differential equation, but also on that of the function f, which may be assumed at pleasure,

§ 4. Geometrical interpretation of the foregoing results, and observations on partial differential coefficients.

The important difference between the results in the two cases mentioned above is rendered more clear by treating them geometrically. In so doing we shall for the sake of simplicity assume that the function F(x, y) in equation (A) is such that it has only a single value for any one point in the plane of co-ordinates. We shall also assume that in the movement of the point p its original and final positions are known, and given by the co-ordinates x, y, and a1, y, respectively. Then in the first case we can find an expression for the work done by the effective force during the motion, without needing to know the actual path traversed. For it is clear, that this work will be expressed, according to condition (A), by the difference F(x,y)- F(x,y). Thus, while the moving point may pass from one position to the other by very different paths, the amount of work done by the force is wholly independent of these, and is completely known as soon as the original and final positions are given.

In the second case it is otherwise. In the system of equations (B), which belongs to this case, the first equation must be treated as the equation to a curve; and (since the form of the second depends upon it) the relation between them may be geometrically expressed by saying that the work done by the effective force during the motion of the point p can only be determined, when the whole of the curve, on which the point moves, is known. If the original and final positions are given, the first equation must indeed be so chosen, that the curve which corresponds to it may pass through those two points; but the number of such possible curves is infinite, and accordingly, in spite of their coincidence at their extremities, they will give an infinite number of possible quantities of work done during the motion.

If we assume that the point p describes a closed curve, so that the final and initial positions coincide, and thus the coordinates x, y, have the same value as x, y, then in the first case the total work done is equal to zero: in the second case, on the other hand, it need not equal zero, but may have any value positive or negative.

The case here examined also illustrates the fact that a

y

quantity, which cannot be expressed as a function of x and (so long as these are taken as independent variables), may yet have partial differential coefficients according to x and y, which are expressed by determinate functions of those variables. For it. is manifest that, in the strict sense of the words, the components. X and Y must be termed the partial differential coefficients of the work W according to x and y: since, when increases by dx, y remaining constant, the work increases by Xdx; and when y increases by dy, a remaining constant, the work increases by Ydy. Now whether W be a quantity generally expressible as a function of x and y, or one which can only be determined on knowing the path described by the moving point, we may always employ the ordinary notation for the partial differential coefficients of W, and write

Using this notation we may also write the condition (4), the fulfilment or non-fulfilment of which causes the distinction between the two modes of treating the differential equation, in the following form:

dy (dW) = d(dW)

dy

(9).

Thus we may say that the distinction which has to be drawn in reference to the quantity W depends on whether

dW

d

the difference

dy dx

a finite value.

d

dx dy

(dly)

is equal to zero, or has

§ 5. Extension of the above to three dimensions.

If the point p be not restricted in its movement to one plane, but left free in space, we then obtain for the element of work an expression very similar to that given in equation (3). Let a, b, c be the cosines of the angles which the direction of the force P, acting on the point, makes with three

rectangular axes of co-ordinates; then the three components X, Y, Z of this force will be given by the equations

X=aP, Y=kP, Z=cP.

Again, let α, B, y be the cosines of the angles, which the element of space ds makes with the axes; then the three projections dx, dy, dz of this element on those axes are given by the equations

Hence we have

then

dx= ads, dy = ẞds, dz=yds.

Xd +Ydy+Zdz = (a+bB+ cy) Pds.

But if o be the angle between the direction of P and ds,

hence

ax+bB+cy = cos :

Xd + Ydy+Zdz = cos ¢ × Pds.

Comparing this with equation (2), we obtain

d W = Xd + Ydy+Zdz...

.(10).

This is the differential equation for determining the work done. The quantities X, Y, Z may be any functions whatever of the co-ordinates x, y, z; since whatever may be the values of these three components at different points in space, a resultant force P may always be derived from them.

In treating this equation, we must first consider the following three conditions:

and must enquire whether or not the functions X, Y, Z satisfy them.

If these three conditions are satisfied, then the expression on the right-hand side of (11) is the complete differential of a function of x, y, z, in which these may all be treated as independent variables. The integration may therefore be at once effected, and we obtain an equation of the form

W = F(xyz)+const.......

..(12).