78 
Proceedings of Royal Society of Edinburgh. [sess. 
a x a 2 a 3 a 4 
b\ ^2 ^3 ^4 
C 1 C 2 C 3 G 1 
d x c^2 <^3 d ^ d^f d^y 
vanish, Cayley wrote 
a l 
a 2 
a t 
<% 
a 6 
h 
h 
h 
«1 
C 2 
C 3 
«4 
C 5 
C 6 
d. 
d 2 
d 3 
d 4 
< 
d 6 
and the theorem referred to is that if this group of equations holds 
it follows that the similar group got by the quasi-multiplication of 
both sides by the determinant |A. 1 /a 2 v 3 p 4 ct 5 t 6 ! holds also ; in other 
words, that so far as multiplication by |A. 1 ju, 2 v 3 p 4 o- 5 r 6 | is concerned, we 
may view the rectangular array as if it denoted a single entity. 
Taking the first of the fifteen determinants of the new group, 
viz. : 
Ajdq + X 2 a 2 + . . . + 
Ai^i + A 2 & 2 + . . . 4- V 6 
AiCi + A 2 c 2 + . . . + A 6 c 6 
AjC^i + A 2 c? 2 + . . . + Xfl g 
Pi a i + . . . + n 6 a 6 
Pfl + * • • + P&^6 
P \ G \ + • • • + Pq c q 
P\d\ + . . . + fi 6 d 6 
v i a i + . . . +v 6 % 
v 1 b 1 + .. .+v 6 b Q 
v 1 c 1 + ...+v 6 Cq 
V\d x + . . . + Vq d§ 
Pi a i + • • • + PQ a Q 
Pl\ + • • • 
P 1 C 1 + . . .+pqCq 
P\ d x + • . • + Pftdft , 
we see that it is equal to the sum of products usually represented by 
a i 
a 2 
a 3 
a 4 
a 5 
a 6 
A 2 
A 3 
K 
A 6 
h 
h 
be 
Pi 
P2 
H 
Pi 
Pb 
pQ 
c i 
C 2 
C S 
C 4 
C 5 
«6 
V l 
V 2 
V 3 
v i 
V b 
d x 
d 2 
d 3 
d 6 
that is to say, it is equal to 
2 |ai& 2 c 3 dy-|Ai//, 2 i/ 3 p 4 | , 
and consequently must vanish, because the first factor of every one 
of these products vanishes. The same is readily seen to be true of 
any other one of the fifteen determinants; in fact, the equivalents 
of the fifteen are 
