412 Mr green, ON THE DETERMINATION OF THE 



(j) = Acpr. 



01 being a determinate function of ^,, ^a, E<- 



We thus see that when we consider functions of the form (20) 

 only, the most general solution that the equation 



= v'^ - *o'0 (19') 



admits is 



or, (p = 0; or, (p = atp; 



a being a quantity independent of ^,, ^2, ^„ and (p any function 



which satisfies for <p to the equation (19'). But by affecting both sides 

 of the equation 



with the symbol v, we get 



= V • v' - *o' . V ^ ; 



and we shall afterwards prove the operations indicated by v and v' 

 to be such, that whatever may be, 



V v' = V' V 0- 



Hence, the last equation becomes 



v' (v ^) - k„' V (p; 



and as V like (p is of the form (20), it follows from what has just 

 been shewn, that 



either = v cp, or, \7 (p = acp, 



a being a quantity independent of ^i, ^2, ?«• 



The first is inadmissible, since it would give ^ = 0; therefore when 

 (p satisfies (19'), we have 



V 0' = a(p, i.e. = V — "0- 



But since a is independent of ^1, ^2, Bs, this last equation is 



evidently identical with (18), since the equation (18) merely requires that 

 K should be independent of fi, ^2, ^s- 



