660 



that is, 



Proceedings of the Royal Irish Academy. 



1 1111 

 r - m + b, 

 JD q-vi + 5, (4321) 

 p - 7)1 + 5, 

 n - m + 5, 



(m.) 



the total number of terms in the left-hand side being 1+4 + 6 + 44 1, 

 *'. e. 2* ; and where we see that to obtain this result we simply border 

 (4321), the equivalent of [n, p, q, r], as indicated by (in.). 

 Generally, in the case of m - 2 general indices, we have 



\ih'g,q, ... y] = i)(234...w-l); 



and the complete sum, involving Extensions, arising out of this, 

 namely, 



[w, p, q, ... 9/']-'^[n+l, p, cj, . . . y) + :S [w + 1, ^ + 1, q, . . . fj 

 -... + (-l)'»--[w+l, p+1, ^+1, ...y + 1] 

 = i)(234 ...«i-l)-i?(134 . .. m-l) + J){12i ...m-l) 



that is, 



D 







. +(-l)'^2Z)(123 . 



..m-2), 



1, 1 



. . 1 







y-l.\ 











x-1, 











s-h 



[(234. 



..m-l) 



} 



(Jf-3) 



p-h 











n-l, J 











in which we see that to obtain this result we simply border (234 . . . 

 m- I), the equivalent of [n,p, q, . . . y], as shown by {M-3); and 

 the total number of terms in the left-hand series is 



1 + m-2 C'l + m-2 C^ + • • • + m-i Cm-3 + 1 } 



that is, the total number of combinations of »i- 2 things + 1, i.e. 2"^-^ 



