150 Proceedings of the Royal Society of Edinburgh. [Sess. 
determinants A , B , and the minors of their product C ; or, as he un- 
fortunately feels himself compelled to put it, “ wie die partiellen Differential - 
quotienten der Determinante C nach ihren Elementen c genommen durch 
die partiellen Differential-quotienten der Factoren A und B nach ihren 
Elementen genommen sich ausdriicken lassen.” What follows thereafter 
may be described as the establishment of the simple identity 
u u 
U 12 
a l 
u l 1 
u 12 
7i 
“n 
U 12 
a 3 
2 
U l\ 
U \2 
a l 
a 2 
U 2l 
U 22 
a 2 
U 21 
u 22 
72 
— 
U 21 
u 22 
a 2 
= 
c 
U 2\ 
U 22 
7i 
72 
a l 
a 2 
0 
7i 
72 
0 
7i 
72 
0 
where u 12 = u 12 , by multiplying together 
a l 
a 2 1 
u n 
U \2 
~ a 2 
a l 
7i 
-72 If 
U 2\ 
U 22 
» "72 
7i 
in row-by-row fashion ; and then the generalisation of this identity in two 
different directions. 
The first generalisation consists in the proposition that the two-line 
axisymmetric determinant 
u n 
U\2 
. . . u ln 
a i 
u n 
M 12 
. . . 
U ln 
7l 
Ui i 
w 12 . . 
• u ln 
a i 
U 2l 
M 2 2 
. . . u 2n 
a 2 
u n 
^22 
U<in 
72 
%22 • • 
U 2n 
a 2 
««1 
. . . u nn 
a n 
Uni 
U n 2 
U n n 
7 » 
M„i 
U n‘l • ■ 
• • u nn 
<*n 
tti 
a 2 
. . . a n 
0 
7i 
72 
7 n 
0 
7i 
72 • 
• • 7» 
0 
where u Kk = u kK , contains 
Mil 
m 12 . ■ 
■ • w ln 
U 2 2 • • 
W 2 n 
M„i 
• 
• • 
as a factor, and that the cofactor is an integral homogeneous function of 
the a’s and likewise of the y’s. The case where n is equal to 3 is treated 
as follows. The determinant 
M 11 
2^12 
W I3 
a l 
W 21 
M 22 
U 23 
a 2 
W 31 
% 32 
a 3 
7l 
72 
73 
.0 
5 
having u Kk = w kK , is introduced and denoted by B , with the result that the 
two-line determinant in question is representable by 
