362 MAJOR P. A. MAcMAHON: MEMOIR ON THE 



This expression was found to satisfy the functional equation, so that certainly 



IL (I ;,&) = 1 +oj* +1 , y 1 , ' y; ; 



and then, observing that we may write 



l (a+1), (b) 



IL ' "(1) (l+a+1) x( fi } (l + 1) 



and remembering the nature of solution when I = oo , it became clear that we should 

 seek solutions for the order 2 of the forms 



(a+1) v (b) F . 



' * r " (l+a+1) 



where F, is a function of I to be determined in each case. 



I therefore substitute fo"*" 1 ) . F, for IL (I ; a, I) in the functional equation and 



(l+a+1) 



arrive at the relation 



(l)F, = (l+l)F,_ i; 

 from which I deduce 



F, = 0+1), 



yielding for me the fundamental solution 



(l+a+1) 

 Similarly I find that another fundamental solution is 



0>) (I) 

 (l+a+1)' 



and, in terms of these two solutions, I find 



:o ' 6) = (i)t 



Art. 24. This simple exposition for the second order clearly points out the path of 

 investigation for the third order. For, guided by the six fundamental solutions 

 when / = oo, it is natural to seek for solutions of the functional equation of the six 

 types 



(l+a+1) (l+a+2) (l+b+1)' . (l+ a +l) (l+a+2) (l+b+1)' 



(c-l)(c)(a+2)F, 



(l+a+1) (l+a+2) (l+b+1) ' (l+a+1) (l+a+2) (l+b+1) ' 



(l+a+1) (l+a+2) (l+b+1) ' (l+a+1) (l+a+2) (l+b + 1) ' 

 where F f is a function of / to be determined in each case. 



