POINTS OF THE INTEGRAL CALCULUS. [137] 



To take another such instance, let us have Xx + Yy + Zz = 1, X + Zp = 0, Y + Zq = 0, 

 giving 



X P , Y 1 , Z=- 



px + qy — x px + qy — z px + qy — z 



„ * ~ V 



z z 



in which the great and small letters are interchangeable. 



M. Chasles himself has gone so far as to extend his case, by trial, to every instance in 

 which X, Y, and Z are fractions having numerators and denominators which are linear func- 

 tions of p, q, and px + qy — z. The preceding is therefore not a new case: but we might 

 get one if we took X*x* + Y*y* + Z"z 2 = 1, or the like, to say nothing of the unlimited number 

 of cases in which the transforming functions are not reciprocal. 



Section 3. In confirmation of the intermediate primitives of a differential equation 

 containing arbitrary functions, it may be shown that an equation of the nth order is not 

 distinguishable, so far as one integration is concerned, from a partial differential equation with 

 n independent variables. Let y" = d> (x, y, y',y"), or, making y =p, y" = q, &c, 



r = <p(x,y,p,q) (l) 



the first integral of this must give q in terms of x, y, p. When this integral is found, it must 

 satisfy the partial differential equation 

 dq dq dq 

 dx + dy- p+ Tp- q = ^ y ' P > q) (2) 



Obviously, then, the first integral of (l) is contained in the complete solution of (2): if my 

 conclusion be true, the second is not more extensive than the first. The first and chief objec- 

 tion to this would seem to be that (l) has in it implied conditions which do not accompany (2). 

 In (l), p is understood to be a function of x and y, and y a function of x: while in (2), though 

 these relations are recognized in the formation of the first side from q, they do not exist as part 

 of the connexion between the equation and its solution. To which I reply that though in the 

 course of the progress towards the original primitive of (l), we recognize the connexion 

 between p, y, and x, yet in the first step of that transition, it will be found that we do 

 not further recognize it than was done in the substitution of q x + q„-p +q p <q instead of r. 

 For if q = \f/(x, y, p) be the first integral, we return to (l) by combining it with 



r = ^ x +^ y .p + ^ p .q, 

 without any other consideration whatever. 



Now let q = \j/(x, y,p) satisfy (2) : x, y, p, being independent. The following equation 

 then is identically true, 



»f'. + , rVP + , rV , r"-0(*.SM>i , r') ( 3 ) 



and being identically true, is true if for p we write y. And in this equation is all that is 

 necessary for deducing r = <p(x, y, p, q) from q = yfs(x, y, p) on the supposition that p, q, and 

 r, are y, y", y": which shows that r=<p(x, y, p, q) follows from q = ^(#, y, p), when p = y, &c. 

 provided only that r = (p(x, y,p, q) satisfies (2) when x, y,p are independent. The whole of 

 this part of the argument may be summed up as follows : — 



Vol. IX. Paet II. 42 



