122 Integration of Differential Equation of the Second Order. 

 and substituting this in (12), observing that 



/'M =/»-£(!)•*> 



we get 



o-t./; W+ s./>)-{t.'® + s. £(§)}* 



+(q- ?)(/«- J-*)+ N *+ M - 



Hence, equating to zero the coefficient of #, since/* contains a? 

 and y only, we must have simultaneously 



= T./ a '(2/) + S./» + (Q-|T)./ e + M, . (15) 



-**(S)+'-»(i)+(«-w*)-S+»-w 



And, finally, in order that the equation 



0=Br + S* + T* + Pp + Q§r+"N* + M 



(where It, S, T, P, Q, M, N are functions of x and ?/ only) may 

 be derivable from a single partial differential equation of the 

 first order not containing p, we must have satisfied two condi- 

 tions. First, we must have R = 0; second, the other coeffi- 

 cients of (1) must satisfy (15). When these are satisfied, the 

 required first integral will be 



P 



where f a is determined by (14). 



The mode of treatment when the equation of the first order 

 from which (1) is derivable does not contain q is, mutatis mu- 

 tandis, precisely similar. 



6 New Square, Lincoln's Inn, 

 January 7, 1868. 



