1903-4.] Dr Muir on General Determinants. 81 
bipartites being thus seen to be 
X 
y 
z 
-1 
1 1 a Y"- • • 
\ - \ a'/3". . . 1 .... 
£ 
1 -lay'... 
| |a/r...| .... 
V 
-\M~- 
1 |ay... 
| -\ap..:\ .... 
e 
Now the properties of this which are investigated by Cayley are 
properties possessed by the more general bipartite 
* y * .... 
«i 
a 2 
a 3 
. . . . | 
»3. 
. . . . i 
c i 
c 2 
C 3 
1 
. . . . 1 
which is not expressible in the form of a determinant. So far, 
therefore, as this section of the memoir is concerned, it is evident 
that the title is somewhat misleading, and it is unnecessary to enter 
into detail regarding the properties in question. 
In the course of the section, however, having occasion to use 
Jacobi’s theorem regarding a minor of the adjugate, Cayley gives 
at the outset a formal proof which it is most important to note, as 
it is the natural generalisation of Cauchy’s proof for the ultimate 
case, and consequently has since become the standard proof given 
in text-books. The passage is 
“ Let A , B , . . . . , A' , B' , . . . . be given by the equations 
A = 
P y ■ ■ ■ 
B = ± 
y S' ... | 
/3" y" ... 
y" 8" ... 1 
A' = ± 
P" 7" • • • 
B' = 
y' • • • 
P" y" ■ ■ ■ 
> 
/// §rrr 
y s . . . 
the upper or lower signs being taken according as n is odd or 
even. 
PROC. ROY. SOC. EDIN. — YOL. XXV. 
6 
