652 Proceedings of the Royal Irish Academy. 



Again, in the case of three general indices, it is shown by (3') that 



«". 



fl«, &c. 



=^a" {hcd . . .1) 



q-m -v^, q-m + 2 

 p- 7)1 + 3, p-m + 2 



- &c. 



a, 

 1, 



Changing those terms involving q from k' to h by the relation {£), 

 the right-hand side becomes, by (2'), 



a", 



a"'-^, &c. 



a, 

 1, 



l^o-m+i ~ 



''q-m+3 



a", 



a"^^, &c. 



a, 



a"'-3, &c. 



a, 



1, 



q-m+2' 



a"*-^, &c. 



1, 



and this, by (3), and an extension of it, is equal to {ahe . . . I) multi- 

 plied by 



p~m + 2, p-m + 1 

 n-m + 2, n-m + 1 



'q-m+3 ' 



p - m + 3, p-m+1 

 n - m + 3, n-m+1 



''q-m+2 



p-m+ 3, p-m + 2 

 n-m + 3, n -m + 2 



q-^n+lf 



that is, by 



q-m + S, q-m + 2, q-m + 1 

 p-m+3, p-m + 2, p-m+1 

 ii-m + 3, 71- m+2, n-m+l 



(4) 



