568 Proceedings of Royal Society of Edinburgh. [sess. 
from the original determinant D by suppressing the g i]x row and 
i th column. 
The second is that — 
“ Pour tout determinant nul on a 
[g, £/]•[*> *] = 1 . ?]•[>> *'] 
et par consequent pour un determinant a la fois nul et 
symetrique 
[g,g~\- [*> *'] = [*> g] 2 = [g, *] 2 -” 
This is proved independently, hut, of course, it is nowadays 
best viewed as a special case of Jacobi’s theorem (1833) regarding 
a minor of the adjugate. The third and fourth propositions 
combined are to the effect that in every perfectly general 
determinant 
<#D 
while in a symmetric determinant 
dP 
dA nf 
= [g, g ] > 
d D 
d A, 
-(-!)*+« 2 [f.rf 
A proof of the last of these is given,* the starting-point being 
the identity 
P = A n<n \ [n,n\ - \n,n - 1] + A n>TC _ 2 [n,n - 2] - • • • • 
where D is expressed in terms of the elements of the last row and 
their cofactors. By differentiating both sides of this with respect 
to the particular non-diagonal element A n>n _! there is obtained 
dP 
dA n>n _ i 
= 0 - 
d[n,n - 1] ^ A d[n,n - 2] 
dA^ i + n ’ n ~ 2 dA n<n _-y 
The differentiands on the right of this, viz. \n,n - 1] , \n,n - 2], . . . 
although not involving A w>w _ 1 do involve A n _ hn which is the same 
as A Bi w _i : consequently their differential coefficients are other 
than zero and have to be found, — that is, we have to find 
lil where i < n. 
“A-l, n 
* There are several misprints in the original, and the paging of the volume 
is hereabouts all wrong. 
