206 Proceedings of Royal Society of Edinburgh. [sess. 
use of the notation of differential-quotients in specifying the 
minors of the determinant. Denoting the determinant of even 
order by P, he starts with the development — 
Pi -a s 
8 2 p 
dob lr 0a ri 
8 2 P 
da ls da sl 
!a lr a 
0 2 P 
lr 15 da ir da is 
Then as a previously obtained general identity, originally due to 
Jacobi, viz., 
p 8 2 P 0P 0P 0P 0P 
da rs 1 a pq da rs ' da pq da ps ' da rq ’ 
gives in this special case the identities 
p 8 2 P 0P_ 0P p 8 2 P _0P 0P^ 
ba xr da r i da ir 0a ri ’ 0a 15 da sl da ls da sl 5 
p 8 2 P 0P dF 
da ir da sl da ir da sl ’ 
because the cofactor, awkwardly denoted by 0P /da a , of any vanish- 
ing element a ti in the principal diagonal is zero in accordance with 
the preceding theorem of Cayley’s. Prom the first two of these 
we have 
™ 0 2 P 0 2 P 0P 0P 0P 0P 
bci -^ i 0oq^ s\ 0oq^» 0 t? 2 ^ da sl 
the right side of which can be changed into 
/0P 0P\ 2 
\da ir ' da sl J 
by reason of the fact that for a determinant such as P we have 
in every case 
0P = _ 0P 
da r da sr 
But from the third identity above, by squaring, we obtain on 
the right the same expression; so that there thus results 
/ 8 2 P \ 2 _ 0 2 P _ 8 2 P 
\da ir da sl J da ir da rl da ls ba sl ’ 
— an equation which exactly expresses the property that the 
