656 Proceedings of the Royal Irish Academij. 



Then it is readily shown that 



{n+\, p-] + ln, p-v\'] = D{Z\), (i) 



which is the only Extension in the case of two general indices. 

 Again, by (4), we have 



\n,p,q-] = D{Z2\), 



from which it is deduced by (i), that 



[n+l, p,q]^[n, p+l, q]-^ln,p, (?+ 1] = i)(421), (ii) 

 and 



[n + 1, i?+l, $-] +[«+!, p, ^ + l] + [w, p-\-\, ^ + l]=i)(431), (iii) 



which two results are the Extensions in the case of three general 

 indices. 



And again, using four general indices, we have, by (5), 



[n,p,q,r-] = L{'iZ2l), 



from which it is deduced by (ii) that 



2 [w + 1, p, q, r], 



having four times in each of which there is one index of the form 

 A + 1, is equal to i) (5321 ) + Z) (4421), the latter of which, being zero, 

 gives 



%{n+l, p,q,r'] = D {5^21). (iy) 



Also, it is in like manner shown by the application of (iii) that 



S[w+1, i)+l, ^ + 1, r], 



having four terms, in each of which there are three indices of the 

 form A + 1, is equal to I) (5431) + D (2542) ; and thus 



2[w+l, i? + l, ^+ l] = i)(5431). (v) 



And lastly, to find the corresponding value of S [n + 1, j9 + 1, $', r], 

 which consists of six terms, each of which has two indices of the form 

 X + 1, it is shown by considering separately the terms involving atT 

 from those having a'^"^, that by (ii) and (iii) we get 



i) (5421) + Z> (2532); 



and thus 



2[n+l, iJ + 1, q, r-\ = B{5i2l). (vi) 



These results, (iv), (v), and (vi), are the Extensions in the case of four 

 general indices. 



