36 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



17. Now it may happen that the simultaneous system of equations admits of no 

 new common solution in either case ; the inference then is that no equation of the 

 second order containing a single arbitrary function can be associated with, or is 

 compatible with, the given differential equation. But it may then be that some new 

 equation of the third order new, that is, in the sense that it is not one of the 

 immediate derivatives of the given equation containing an arbitrary function can be 

 associated with the given equation ; and this may occur with each of the linear 

 factors of A = 0. And so on, precisely as in DAKBOUX'S method for dealing with 

 partial differential equations in two independent variables ; we seek to obtain one 

 equation (or, it may be, two equations) of finite order which ai'e compatible with the 

 given equation, contain one arbitrary function, and are not mere derivatives from 

 that given equation. 



We have been proceeding on the supposition that the equation possesses no 

 intermediary integral. If no other equation of finite order is compatible with 

 the given equation,* then the method ceases to be effective. In that case, the only 

 result generally attainable seems at present to be that which occurs in the establish- 

 ment of CATJCHY'S existence-theorem ; the integral certainly contains two arbitrary 

 functions, but its expression (in the form of a converging series) is not finite. 



18. Suppose that the conditions for the existence of a new common solution 

 are satisfied for neither of the systems in 15, 16, so that no new equation of the 

 second order, containing only a single arbitrary function, is compatible with the 

 given equation. We proceed to construct the system of subsidiary equations which 

 determine an equation (if any) of the third order containing only one arbitrary 

 function, and compatible with the given equation 



F (a, b, c,f, g, h, I, m, n, v, x, y, z) = 0. 



On account of this equation, we have three derived equations of the third order, 

 viz., with the former notation 



X + Aa + H/3 + Ga x + By + Fft + C 2 = 0" 



Y + A/3 + H 7o + Gft + B8 + F 7l + C& = 

 Z + A ai + Hft + G 3 + B 7l + F& + C 3 = . 



and the new equation (if any) must be compatible with these. 



19. The process is an amplification of that used in 2. When the proper value 

 of v is substituted in F == 0, the latter becomes an identity, so that, when it is 



* A simple instance is given by 



y + 



where X is a positive constant other than a.n integer. 



