POINTS OF THE INTEGRAL CALCULUS. [127] 



in which the constant of integration is combined with the old ones in such manner that only 

 three distinct arbitrary constants appear. 



The integrating factor of a common differential equation is any solution of a certain 

 partial differential equation, and therefore contains an arbitrary function, which is the equi- 

 valent of an infinite number of arbitrary constants. It must therefore be shewn that the 

 primitive of a common differential equation does not contain an arbitrary function. In the 

 case of the first order this is easy, for if Pdx + Qdy be made dV by multiplication by M, 

 the general integrating factor is (pV.M, and the primitive is of the form \j/ V ■= const., 

 which, were >J/ V to contain a million of arbitrary constants, amounts to no more than 

 V= const. But no such proof applies to equations of a higher order than the first; on 

 the contrary, I think it can be shewn that all the primitives of such an equation, except 

 only the general primitive, contain effective arbitrary functions. 



Let us take y" ' = 0. What are called the primitives of the second order are y" + a = 0^ 

 xy" — y' + b = 0, x*y" — 2xy + 2y + c = 0. But it is obvious that y" = is the equation 

 of the third order, and the only one, which follows from 



/ {y" + a, xy" -y + b, tfy" - 2zy' + 2y + c) = 0. 



x 2 

 And the complete integral of this is y = A 1- Bx + C, subject to the condition / (,4 + a, 



Neither can there be any other equation of the second order, dedu'cible from, or deducing, 

 y" = 0, and not contained in the above. For, let F (*, y, y\ y") = be the most general 

 solution of y" = 0, and let y" + a = P, xy" - &c. = Q, x 2 y" - &c. = R ; and, substituting 

 the values of y, y\ y", thence obtained in terms of P, Q, R, x, let F = be thus reduced 

 to (p (P, Q, R, x) = 0. Then we have (<p P + Q . x + (p R . x 2 ) y" + <p x = 0, which, since 

 y" = is to be an equation of the third order deducible, gives <p x = for all values of x, 

 so that (p does not contain x explicitly. 



The following extension of the theorem involved in the method of treating y = y x +fy' 

 may now be easily proved. Let (f> {x, y, A^...A^) = be the complete integral of an equation 

 of the rath order, and let U x + A t = 0,...U n + A n = 0, be the fundamental integrals of this 

 equation of the order ra — 1. Then the general preceding primitive of the equation of the 

 rath order is / (t^ + A x ,...U n + A n ) = 0, / being any form whatsoever ; and the original 

 primitive of this equation, independently considered, is (p (x, y, R i ,...B„) in which the ra new 

 constants B l ,...B n are subject to the condition / (A x - B t ,...A n - B n ) = 0. This process 

 must be repeated on the last primitive in order to produce the most general equation of 

 the order ra — 2 ; and so on. 



For illustration, I return to the case of y" = 0, and indicate the mode of finding the 

 general primitive of the first order. Let y = ax 2 +bx + c be first taken as a primitive: its 

 general equation of the third order is y" = 0, of which the immediate primitive is 



x s y" - 2 xy' + 2y - 2c =f(xy" - y + b, y" - 2a), 



which however, strictly belongs to y = ax 2 + bx + c only when =/(0, 0). The complete 



