232 



Mr. R. Moon on the Integration of the General Linear 

 - 7 u t <?A m _! 



= A, 



fp — Af. 



dy 

 du 



dy 



^~^ m - l dy 



If the values of m v m Q are constant, I gather that the integral 

 in the cases where it occurs will be of the form 



-//(2/)^ (m - 2) W+/^ (w " 1} M, 



in which it may be conjectured that the number of terms in the 

 series of derivatives of </> will in general be defined by the fact 

 of the derivatives of/p beyond a certain number taking the value 

 zero. 



To return to the cases in which the foregoing process requires 

 modification, one of which, it has been already shown, occurs 

 when u contains x or y only. 



Suppose u to contain y only, and that 



p=¥(xyzV^), 



q=f{xyzV_ 1 \J), 

 where, as before, U_! or ^c»»— i) j s the highest derivative of <f> in 

 the integral equation. Then 



dp 



dx 



=F,+F(U_,).0=P. 



|=F, + F(U_,).U, 



% =/.+/'(U-.) • U . 0+/'(U) . u +1 . 0=f„ 



dx 

 dq 

 dy 



=/.+/'(U- 1 ).U+/'(U).U 



+i' 



Substitute in the equations (5a) these values, observing that 

 since U +1 occurs in ~ we must have Tj = 0, and we get 



Assume V = Pjo + Q^ + N^ + M, and that U, S, P, Q, N, M are 

 functions of xy only. The last equations become, if we assume 

 F'(U_,)=0, 



0= E{F(«) +F(*)F} + S{F(y) +FW}\ (25) 



+ PF + Q/+N* + M, . . . . J ' K ' 



0=/'(*)+/'(*)F-F'( 2 ,)-F'(*)/. .... (26) 



