1887.] Mr A. H. Anglin on Summation of Alternants. 197 
seen that the coefficient of a Y consists of the sum of three deter- 
minants which by (2) is equal to \b 2 c A d b \, while the coefficient 
of a 2 likewise involves three determinants whose sum by (3) is 
equal to \b 2 c 3 d 5 | . Similar expressions holding for the coefficients 
of the other elements in the first columns, we get 
I a \ ^2 C 4 ^5 I "b I a 2 ^2 C 3 ^5 I > 
and thus 
% 2 = \ a i b 2 c A df (5). 
Lastly, it may be shown in like manner that, in the case of 
2 
a. 
d n 
a 2 a ?> a A 
b 2 b 3 b A 
d s rf 4 a ? b 
or S 3 , 
consisting of four terms, in each of which Ahe suffix-order (1 2 3 4) 
occurs three times, 
^3 = i a i ^2 C 3 ^5 I (^)* 
The results (4), (5), and (6) constitute the theorem in the case of 
determinants of the fourth order, and are formed by deleting suc- 
cessively the second, third, and fourth columns from the matrix out 
of which they arise. 
4. Thus, assuming the truth of the identities for the case of deter- 
minants of the (n— l)th order, we can establish the corresponding 
results for the case of determinants of the ?dh order ; that is to say, 
taking the matrix 
a A 
a 2 
«3 ' ' 
. a n _| 
h 
h 
h ■ 
■ • 
c i 
C 2 
c 3 . 
• * ^ n- (-1 
h 
h 
h ■ 
• • l'n + 1 : 
consisting of n rows of n+ 1 quantities each, we can obtain the 
following n - 1 results involving determinants of the wth order, 
and which are formed by deleting successively the second, third, 
fourth, . . . , ni\\ columns from the matrix, viz. : — 
