Anglin — On some Theorems in Determinants. 



659 



6. The foregoing Extensions are necessary to establish the main 

 proposition of the Paper. The following results, deducible from them, 

 are however interesting, and may be worthy of notice. 



Using m letters, a, b, c ... I, we have, in the case of two general 

 indices n and p, 



[_7i, p'] = D {m-2, m-1), 



the complete sum arising out of which, namely, 



[n,p']-ln+l, p']-[n, p + l^ + ln+1, p+l'] 



is, by (i), equal to 



D{m-2, m-l)-D{m-3, m-l)+D{m-3, m-2), 



that is, 



1 1 1 



D 



p-m + 3, p-m + 2, p-m+l 

 n-m + 3, n-m+ 2, n-m + 1 



(!■) 



the number of terms being 2^. 



Again, in the case of three general indices, w, p, q, we have 



[n,p, q^j = J){321), 



the complete sum, involving the extensions, arising out of which, 

 namely, 



[n,p, q']-:$[_n+l, p, q'] + '^{n + l, p + 1, q) - {n+ 1, p + 1, q + 1] 



is, by (ii) and (iii), equal to 



i)(321) - D{421) + i)(431) -D(432), 



that is, 



D 



1111 



q-tn + i, q-m + 3, q- m + 2, q-m + 1 

 p-m + 4, p-m + 3, p-m + 2, p-m+l 

 n-m + 4, n-m + 3, n-m + 2, n-m+1 



(II.) 



the total number of terms being 1 + 3 + 3 + 1, i.e. 2^. 



Likewise, the complete sum, involving Extensions, arising out of 



ln,p,q,r^=JD {4321), 

 namely, 



ln,p,q, r']-'^[_n + l,p,q,r^ + 'S,[n+l,p + l,q,r']-'^ln + l,p + l,q+l,r'] 



+ [n+l, p + l, q+ 1, r + 1] 



= D (4321) - D (5321) + D (5421) - J) (5431) + J) (5432), 



