80 
Proceedings of Royal Society of Edinburgh. [sess. 
result 0 for each of them, unless the determinant of their coefficients 
vanishes. Now, the determinant of their coefficients is the com- 
pound determinant whose elements are the minors of the 4th order 
formed from , and this by a well-known theorem is 
equal to the 10th power (• i.e ., C 5 , 3 ) of |X 1 /x 2 v 3 p 4 cr 5 r 6 | . The theorem 
is thus established. 
Cayley’s last theorem closely resembles his second, being to the 
effect that if 
<h 
a 2 
a 3 
a, 
a 5 
a 6 
h 
h 
Cl 
c 2 
C 3 
c 4 
C 5 
C 6 
d 1 
d 2 
d 3 
d± 
d 5 
d 6 
then 
a. 
a 2 
• • • • a G 
hfi + /x 1 C 1 + v x d x 
\b 2 + PjC 2 + vf 2 • 
• • • \b 6 + ^Cq + v-^q 
A. 2^1 + p 2 c x + v f^i 
^ + ^2 + ^2 ' 
A 2 &6 + P'2 C 6 + v 2 d§ 
h 3 \ + /x 3 Cx + v z d x 
+ p 3 C 2 + 
■ ■ * * h 3 b ( . + p 3 c 6 + v/l G 
This amounts to saying that so far as multiplication by 
1 . . . 
• Xi X X 
• Pi P 2 H 
. V 1 V 2 V 3 
is concerned, the given rectangular array may he viewed as a single 
entity. The proof here is exactly similar to that of the analogous 
theorem, hut is simpler; for the fifteen determinants of the new 
array are manifestly equal to 
» 
\a 3 \c 5 df\\n 2 v 3 \ , 
that is to say, are merely multiples of the fifteen determinants of 
the original array, and must therefore vanish along with them. 
