222 
Proceedings of Royal Society of Edinburgh. [sess. 
Ax + bz az + cx 
ny +mz Aly + lz 
i.e. (bl - am)z 2 + (MM - an)yz + (A l - cm)zx + (AM - cn)xy = 0 , 
which is the result desired. 
(4) Taking now the three original and three derived equations, 
we have 
A . . a b c 
M , l m n 
R p q r 
ng - mr . . MR - Ip riR - mp Mg' - Ir 
pc - ra . R c -([a RA - qb pA - rb 
. . bl - am 5M - an A l - cm AM - cn ; 
and this we proceed to investigate with the object of attaining 
further simplicity, and at the same time arriving at a form having 
that appearance of symmetry which may reasonably he expected 
from a study of the structure of the original equations. 
(5) Expressing the determinant in terms of products of minors 
formed from the first three columns and the last three columns, 
we have 
MR - Ip ?iR - mp Mg - Ir 
p q r 
AMR 
Rc - qa RA - qb pA - rb 
&M -an Al -cm AAl-cn 
- AM [bl - am) 
MR - Ip riR - mp Mg - Ir 
Rc -qa'RA-qb pA - rb 
l 
m 
n 
+ A {pc - ra)(bl - am) 
l m n 
+ AR(g>c - ar) 
MR- Ip 
?iR - mp 
Mg - Ir 
p q r 
JM- an 
Al - cm 
AM - cn 
MR - Ip riR - mp Mg - Ir 
- MR(?ig - mr) 
a 
b 
c 
- M [nq - mr) ( bl - am) 
a b c 
Re - qa 
RA - qb 
pA -rb 
p q r 
&M - an 
Al - cm 
AM - cn 
Rc - qa RA - qb pA - rb 
+ R(?ig - rm) {pc - ra) 
a b c 
- {nq - mr) {pc - ra) {bl - am) 
a 
b c 
l m n 
l 
m n 
6M -an Al-cm AAL-cn 
p 
q r 
(6) Again, if we multiply the determinant columnwise by 
nq - mr . 1 
. pc - ra . . 1 
bl - am . . 1 
- A . . . . 
-M .... 
6 I k ... 
