260 Proceedings of Royal Society of Edinburgh. [sess* 
There is nothing to suggest that the numerators of all these 
expressions are determinants, and still less that in the case of 
a aa 
P T* 
aa 
~F’ 
aa 
PT 
a a a a 
X* PT 
a a a 
PT 
h 
the numerators are* the ten principal minors of 
* For the modern reader the following substitute for the missing demonstra- 
tion will suffice : — 
If the cofactors of the elements in the four-line determinant above given]be 
denoted by [11], [12], . . . , then from the equations 
— (a + G r )a + b'a + b" a' + V"a! n = 0 
— b f a + (a! — Gr)a f + d" a" + c" a!" = 0 
— b" a + c'V + (a"-G')a" + d «" = 0 
+ c'V + c' a" + (a'"-G')«"' = G- 
we have 
a _ a _ a" _ a" 
m ~ m ~ [is] ” iu]» 
[21] [22] ’ • • * ’ 
a 
[31] ’ 
[41] 
Multiplying in these lines by «, a, a", a" respectively we see that 
2 2 2 2 
a _ a _ a" _ a!" 
[TT] ” [22] “ [33] “ [iT] 
and therefore that each of them is equal to 
a — a 
r / 2 #// 2 
a -a 
and thus equal to 
[11] -[22] -[33] -[44] 
- 1c 
[11] - [22] - [33] - [44] 
But by the rule for differentiating a determinant the denominator here is the 
differential -quotient of the determinant with respect to G' ; and this because 
of the theorem 
clx 
{{x-r x )(x- r 2 )(x-r 3 ) ... } 
(?\ - r 2 )(r 1 - r 3 ) . . . 
is equal to - (G' - G")(G' - G'")(G' + G) : consequently 
k _ a 2 
(G'-G")(G'-G'")(G' + G) “ [11] _ 
