Hence substituting

whence

or

dy=-4(C) dx,

y+4(C) x=C',

y+x$(ax+by+ cz) =C'.

Thus the final integral is

y + xp (ax+by+cz) = ↓ (ax+by+cz).

This solution may also be expressed in the form

z=x¤1(ax+by+cz)+y¥, (ax+by+cz),

in which it is in fact presented by Monge, (Application de l Analyse à la Géométrie, Liouville's edition, p. 79). The equation solved is that of surfaces formed by the motion of a straight line which is always parallel to a given plane, and always passes through two given curves.

7. In the above examples V is equal to 0, and this always facilitates the application of Monge's method. The following is an example in which V is not equal to 0.

There is also another system, but it is not integrable in the form u = a, v=

= b.

From the first of the above equations we get

y-x=a, dp-dq+

4pdx 2y α

0,

the latter of which may, since dz = pdx + qdx, be reduced to the form

d (2ya) (p-q) + 2dz = 0,

(2y-a) (p −q) + 2z = b,

whence

or, replacing a by y-x,

(x + y) (p − q) + 2z = b.

Hence a first integral of the proposed equation will be (x+y) (p −q) + 2x = f(y-x).

Now this being linear, we have, by Lagrange's method, the auxiliary system

The equation of the first two members gives y+x=a, and this reduces the equation of the second and third to the form

The final integral will therefore be found by substituting in the above, after integration, y +x for a, and F(y + x) for b.

8. Monge's method fails in so many cases, owing to the non-existence of a first integral of the assumed form u=. =f(v), that it becomes important to inquire how its defects may be supplied. And various methods, all of limited generality, have been discovered. Thus Laplace has developed a method applicable to all equations of the form

Rr + Ss+ Tt + Pp + Qq+Zz=U;

R, S, T, P, Q, Z, and U being functions of x and y only,which consists in a series of transformations, each of which has the effect of reducing the equation to the form

8+ Pp+Qq+Z2 = U,

P, Q, Z and U being functions of x and y, to which each transformation gives new forms. It may be that among these successive forms, some one will be found which will admit of resolution into two linear equations of the first order. But there are probably no instances in which this method has been applied in which the solution may not be effected with far greater elegance, and with far greater simplicity, by the symbolical methods of the following Chapters. And even Laplace's method is better exhibited in a symbolical form. The subject will be resumed. See Chap. XVII. Art. 14. The following sections contain miscellaneous but important additions.

Miscellaneous Theorems.

9. Poisson has shewn how to deduce a particular integral of any partial differential equation of the form

where P is a function of p, q, r, s, t, homogeneous with respect to the three last, n a positive index, and Q any function of x, y, z, and the differential coefficients of z of any order, which does not become infinite when rt — s2 = 0.

Assuming q = (p), we have

s='(p)r, t='(p) s = {p′ (p)}2 r .......................(46), values which make rt-s2=0. Hence, substituting in (45), the second member vanishes, while in the first, which is homogeneous with respect to r, s, t, some power of r only will remain as a common factor. Dividing by that factor, we shall have an equation involving only p, (p), and p'(p), i. e. p, q, and dq Integrating this as an ordinary differential equation we obtain a relation between p, q and an arbitrary constant; and this, integrated as a partial differential equation of the first order, gives the solution in question.

dp

Ex. Given 2 — t2 = rt — 82.

Proceeding as above, we find 1- {$' (p)}' = 0 ;

The above method is applicable to all equations of the second order which are homogeneous with respect to r, s, t, for then we have only to suppose Q=0.

10. There exists in partial differential equations a remarkable duality, in virtue of which each equation stands connected with some other equation of the same order by relations of a perfectly reciprocal character. As respects equations of the first order the principle may be thus stated.

Suppose that in the given equation

we interchange x and p, y and q, and change z into px+qy-z, giving

$(p, q, px+qy − z, x, y) = 0........(48); then, if either of these equations can be integrated in the form z=(x, y), the solution of the other will be found by eliminating X and Y between the equations

For, since dz=pdx + qdy, we have

z=px+qy-(xdp+ydq)...(50).

Hence xdp+ydq is an exact differential. Represent it by dZ, and assume Z for dependent variable. Assume also two new independent variables X and Y, connected with the former ones by the relations X=p, Y=q. Then

dZ=xdp+ydq=xdX + yd Y.

therefore

Z = f(xdp+ydq) = px+qy− z by (50) ;

z=px+qy− Z=xX+yY− Z.

On examining the above equations we see that x, y, z, and X, Y, Z are reciprocally related. Writing, side by side, the equations which are conjugate to each other, we have

We see too that the equations (49) which express one set of the relations suffice to convert any relation found by integration between X, Y, Z, where Z stands for f (X, Y), into a corresponding relation between x, y, z.

and we have to eliminate X, Y, between the equations