1887.] Mr A. H. Anglin on Summation of Alternants. 195 
and every possible determinant of the third degree be formed from 
this matrix by deleting the last column of a row and the first element 
of the other rows, the sum of these determinants is equal to the deter- 
minant got from the matrix by deleting the second column : thus 
cq 
a 2 
«3 
a 2 
C H 
a 4 
a 2 
a 2 
^4 
a i 
co 
e 
^4 
h 
h 
+ 
h 
+ 
b 2 
h 
K 
= 
\ 
h 
h 
C 2 
C 3 
C 4 
C 2 
c ?> 
C 4 
C l 
C 2 
C 3 
C 1 
C 3 
0 
and a like series equal to the determinant got by deleting the third 
column, namely 
a 1 
a 2 
a 3 
cq 
a 2 
«3 
a 2 
CO 
« 4 
a 1 
a 2 
a i 
\ 
h 
h 
+ 
h 
h 
+ 
\ 
h 
h 
= 
h 
h 
h 
C 2 
C 3 
c i 
c i 
C 2 
C 3 
c i 
C 2 
C 3 
c i 
0-2 
C 4 
2. The proof of any case of this latter theorem depends on the 
case before it. Thus, taking the matrix 
the identity for the case of determinants of the second order is 
a i a 2 
h h 
+ 
a 2 «3 
&i b 2 
a. 
a. 
a i ^3 
• ( 1 ), 
the truth of it being self-evident. 
Turning to the case for determinants of the third order, and 
expanding each determinant in the left-hand side in terms of the 
elements of the first column, we see that the coefficient of a 1 is 
| b 3 c 4 | , while the coefficient of a 2 consists of the sum of two deter- 
minants which by (1) is equal to | b 2 c 4 | ; and likewise for the 
coefficients of the elements involving b’s and c’s. Hence we get 
4 
H 
« 4 
a 2 
a 2 
a 4 
\ 
^4 
+ 
h 
h 
c i 
C 3 
C 4 
C 2 
C 2 
C 4 
and thus 
“1 
a 2 
a 3 
cq 
a 4 
2 
h 
h 
= 
\ 
h 
\ 
= 
cq 
0 2 
C 3 
C 4 
A 
C 4 
• • ( 2 ); 
