234 
Proceedings of Royal Society of Edinburgh . [sess. 
where A and A' are the third-order determinants of § 1, and 
o 
consequently A + A' - a = a 4- cr' - A .p.l. 
Expressing this determinant in terms of the elements of the last 
row and their complementary minors, we have a form similar to 
that of § 16, viz. 
(A + A'-</) 
A 
a 
• 
• 
• 
c 
a 
• 
• 
• 
c 
b 
. 
M 
m 
• 
• 
- 22A Ir 
M 
m 
l 
• 
• 
n 
r 
• 
R 
• 
<1 
• 
• 
R 
• 
Q. 
• 
P 
• 
• 
b 
a 
A 
• 
• 
b 
a 
A 
• 
c 
n 
• 
• 
. 
m 
M 
• 
• 
. 
m 
M 
l 
• 
P 
• 
R 
• 
r 
P 
• 
R 
• 
r 
< 2 
(19) There can he little doubt that this is the best form yet 
found. On putting either A = M = R = 0 or a = m = r= 0, the 
o p . 
terms prefaced by 22 all vanish, the coefficient A + A'-<r 
becomes either A or A', and the first determinant becomes the 
product of the coaxial minors of A or A '. 
