196 Proceedings of Royal Society of Edinburgh, [april 18 
and in like manner, 
a 1 
a 2 
CO 
& 
a x 
«2 
a 4 
h 
b 2 
h 
= 
\ 
\ 
C 3 
e i 
C 2 
C 4 
a i h 2 C 4 I • ‘ ( 3 )« 
3. Taking the next case, arising out of tlie matrix 
C l C 2 C 3 C 4 C 5 
cl 4 d. 2 do d A , 
consisting of four rows of five quantities each : we have to find the 
value of 
5 
c 
2 
C b 
d 2 d ?) d 4 d b 
or 
where the suffix-order (1 2 3 4) occurs once in each determinant and 
consequently consists of four terms. Expanding the determinants 
as before, we see that the coefficient of a Y is \ b 3 c 4 d b | , while the 
coefficient of a 2 consists of three determinants whose sum by (2) is 
equal to | b 2 c 4 d 5 | ; and similarly for the coefficients of the elements 
involving b, c , cl. Hence we get 
and thus 
a Y 
a s 
«4 
a B 
a 2 
<H 
« 4 
«5 
h 
h 
^4 
h 
+ 
h 
h 
h 
h 
c i 
C 3 
C 4 
C 5 
C 2 
C 4 
C 5 
C?1 
d 3 
d 4 
d 5 
<h 
d 2 
d 4 
c h 
= ! 
«i 
lj 3 C 4 
• 
the value of 
a i 
a 2 
% 
ci 4 
K 
b. 
K 
b. 
2 
l 
z 
3 
4 
or 
V 
C 2 
C S 
C 4 
C 5 
h j 2 
d 3 
d 4 
• w 
consisting of six terms, in each of which the suffix-order (1 2 3 4) 
occurs twice. Expanding in the same manner as before, it will be 
