28 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OV 



Let us consider, first, the case when 



p + 1 = 



so that q remains arbitrary. Evidently 



=0 



<fy 



will not be the equation to be satisfied ; so that the effective combination must arise 

 in the form 



when p + I = Q. In other words, we must have the equation 

 30 ?-0 , 26 M fc6 cd v6 Bt) 



a7 + a aT + h ^i + <J& ~ (j~: + 9 a +f a - + ' ^ 



r^ ili_ ,c6_db_ ,dO_<hi,rO_<y_, M_ % , o0_ <lh_ 

 + Sa ( b:. + 36 >...- + 8c (f. + a/ <lx + % <Jt- + 8/6 rf^! ' ' ' 



(where the value p = 1 has been inserted), satisfied in virtue of the system of 

 subsidiary equations. The relations of identity all involve q, and therefore cannot 

 be useful for the purpose. Hence the above equation must be a linear combination of 



da <U< i Y - 



~T~ ~ T + . = 



ax ax 



KM ax 



ill/ Jf 

 d^"^ + /l = ^ 



where 



-y - n ^ .'/ 



A. = ~ 



;'/ + s 



-. '2h b f '11 ;. 



y + a (y + ^) 



7 _ 



y + z (y+ s)3 



these being the subsidiary equations particular to the present case when the value 

 j-> = 1 is inserted. 



When therefore we substitute 



da _ cUi_ <tt_ (U^ dj_ d a 



dx " dc ' ~ A> dx ~ dx + Y) dx ~ fa + A 



