352 



then 



MAJOR P. A. MxcMAHON: MEMOIR ON THE 



which is of the same form as the original equation. 

 Hence, if IL ( oo ; a, 6) be a solution of the equs 



ar-{IL(a,;a,&Ha+l)IL(oo;a-l,&)} 

 is also a solution. 



exhibiting the new solution as the result of the performance of a certain operation 

 upon the original one. 



Again put, in the original equation, 



a-{IL(cD ; a,fc)-(b)IL( a ; a, 6-1)} = V 2 (o ; a, 1) = O.JL ( ; a, 6), 



and we find 



so that another solution is 



V,( oo ;,&) = 0IL(; a,&). 

 I write further 



O- 1L ( oo ; a, 6) = a;--*{IL ( ; a, />)-(a+l) IL ( oo ; a-1, &) 



; a, &-l) + 



so that, from the functional equation itself, 



and it is easy to verify that 



O O(, = O ai . 



Art, 10. Since we know one solution of the equation 



H.^*>i^3, OT more conveniently (l) + a^ +1 ( a b )> 



or better still (a+l)-* (b), we may at once apply the operators O , O 6 . Operating 

 81 times successively with O a , I find 



O.- 1 { 

 and 



