658 Proceedings of the Royal Irish Academy. 



Again, to find the value of 



%[n+\, p+\, q, ...y'], 



having m-iCT, terms, in each of which there are two indices of the form 

 X + 1, — it will be convenient to divide the series into two parts, con- 

 sidering separately those terms in which the index y + 1 occurs from 

 the rest. 



There are /a - 3 terms having this index, the sum of which is shown 

 by (a') to be equal to 



i)(124 . . . m-1). 



Also, the sum of the remaining m-z^^i terms in which the index y 

 occurs, is shown by (^') to be equal to D (2235 . . . m-\), that is, ; 

 and thus we have 



%[n+\, p + \, q, ...2/] = Z>(124...m-l). (/?) 



In like manner, to find the value of 



2[w + l, i^ + l, $- + 1, r, ... y], 



having m-oC^ terms, in each of which there are three indices of the 

 form A + 1 — it is shown by (^') that the sum of the m-zC-y, terms in 

 which the index y + 1 occurs is equal to 



i)(1235 ...m-1); 



while by (y') the sum of the remaining m-^C^ terms involving the 

 index y is equal to i) (22346 ... m - 1), that is, 0; and thus we 

 have 



2[w+l, p+l, q+\, r, ... y] = Z)(1235 ... m-1). (y) - 



Similarly, when each tenn in 2 has four indices of the form A + 1 , 

 it is deduced that 



^4 -i> (12346 ..'. w-1), (S) 



and generally, if indices of the form A + 1 occur /x, at a time in each 

 term of ^S, it is shown that 



2^ =i)(123 .../A, /x + 2, .../x-1). (/a) 



"While lastly, if there are »i - 3 indices of this form in each term of %, 

 we shall get 



2[?J + 1, i?+ 1, q + l, . .. x+\, y] =Z>(123 ... m-^, m-l), 



... (m - 3) 



the number of terms in S being m-2, which completes the Extensions 

 generally, the law of formation of the successive results being obvious 

 on inspection. 



