CHAPTER XXIX. ON THE SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION Rr + Ss + Tt + U (s3 — rt) = V, IN WHICH R, S, T, U, V ARE GIVEN FUNCTIONS OF X, Y, Z, P, Q. 1. THE equation, the theory of the solution of which I propose to consider in this paper, is remarkable from its connexion with Geometry. If the equation of a surface contain three constants which vary as parameters in subjection to any two conditions connecting them, the generated envelope will satisfy a partial differential equation of the above form. In other words any envelope of the surface F(x, y, z, a, b, c) = 0 formed by the variation of a, b, c in subjection to two connecting conditions P1 (a, b, c) = 0, 42 (a, b, c) = 0 is necessarily an integral of a partial differential equation of the form given above. Now this. theorem is the more important, because it is only when three parameters in the equation of a surface vary in subjection to two relations that the envelope possesses, irrespectively of the form of the connecting relations, any definite character. If there be but one connecting relation it is possible to determine that relation so as to make the envelope assume the form of any surface whatever, and therefore the possible system of envelopes is in such case B.D. E. II. 10 unlimited. If there be three connecting relations the parameters become absolutely constant and no envelope exists. The partial differential equation Rr + Ss+ Tt+U (s2 — rt) = V is remarkable also as including all the cases in which a partial differential equation of the second order admits a first integral of the form u = f(v), u and v being definite functions of x, y, z, p, q, and ƒ (v) arbitrary in form. Neither of these statements is sufficiently general to constitute a theory of the genesis of the partial differential equation under consideration, but the second one is more general than the first, and is indeed sufficiently so to serve as the ground of an investigation which connects the solution of the equation in all cases with the satisfaction of a system of simultaneous ordinary differential equations of the first order and degree. And this is the ground upon which the method of the paper will rest. I propose to shew, 1st that the solution of the given equation on the assumption that a first integral of the form u=f (v) exists requires the satisfaction of a system of two partial differential equations of the first order and second degree; 2ndly that this system may be resolved into four systems, each consisting of two partial differential equations of the first order and first degree, two of which systems are irrelevant and the other two relevant; 3rdly that the solution of the two relevant systems ultimately depends on the solution of a system of ordinary differential equations of the first order, and that from these ordinary differential equations the given equation of the second order may be deduced independently of the assumption above mentioned. I shall also discuss the theory of the second integration. And I shall exemplify another method of solution connected by a remarkable law of reciprocity with the above method. First Investigation. 2. PROP. I. If u=ƒ (v) be a first integral of the equation Rr + Ss+ Tt + U (s2 — rt) = V............................(1), then will u and v, considered as functions of x, y, z, p, q, each satisfy two partial differential equations of the form R du 2 dx Sdu (du du /du dp dx + = 0 dq dy (2), dy dp To demonstrate this proposition we shall form directly the partial differential equation of the second order of which u=f(v) is an integral and, comparing that equation with (1), deduce the conditions for the determination of u and v. Differentiating u=ƒ(v), first with respect to x and secondly with respect to y, we have du du dz, du dp + + Eliminating f'(v) we arrive at the partial differential equation of the second order, It is seen that as respects the mode in which the quantities r, s, t are involved this equation is of the same form as the given equation (1). That it may be equivalent, its coefficients must stand to those of (1) in a common ratio μ. This gives As we have here five equations which are homogeneous with respect to the four differential coefficients of v and to μ, it is clear that we can, by the elimination of these quantities, obtain a relation connecting the differential coefficients of u with R, S, T, &c. But the peculiar cyclical form of the functions in the first members of the above system enables us to effect this elimination so as to lead to two final equations independent of v and μ. Thus multiplying (a) by (du) du, dq' (c) by du du dy dp and adding, we find, on Again, multiplying (a) by (du)", (b) by (du) (b) by (du) (du), (c) by |