516 
Proceedings of Royal Society of Edinburgh. [july 18, 
This result, afterwards so well known, Cauchy translates into words 
as follows (p. 82) 
. ... le determinant du systeme (b Vn ) adjoint au systeme (a Vn ) 
est egal d la, (n— 1 ) me puissance du determinant de ce dernier 
systeme . (xxi. 2). 
Again, by returning to the identity, 
n 
and substituting the value of B n just obtained, there is deduced the 
result 
= ( xxxix.) 
or, in words, 
. . . etant donne un ter me quelconque a Y . v du systeme (a r «), 
pour obtenir le ter me correspondant du systhne adjoint du 
second ordre (c Vv ) il sujjira de multiplier le terme donne par la 
in — 2) me puissance du determinant du premier systeme. (xxxix.) 
A considerable amount of space (pp. 82-92) is devoted to the 
consideration of the adjugate systems of 
(«!■») 3 (®1*«) J ( m rn) 3 
and the adjugates of these adjugates ; but nothing new is elicited. 
The section closes with the manifest identity 
(a rl 4- a 2ll 4- . . 
• + a n-i) 4- a 2< i 4 - • 
. 4 -a n .j) 
+ (cip2 "h a 2'2 • 
• • Ta„. 2 ) (a 1 . 2 + a 2 . 2 + . . 
. + a n . 2 ) 
4-&C. ....... 
4- (a 1 . n 4- a. 2 . n + . , 
• T ®- n .ii) (U]. w 4" a 2 . n 4- . • 
. + ct . ) 
niy i 4* m 2 .^ • ■ 
. 4- m n . T 
4* my 2 4“ m 2 . 2 4- . . 
, + m v . 2 
4- 
d" m i •« + %«+ • ■ • + m n' n 3 
which, using later technical terms, we may express as follows : — 
If there be two determinants, and the sum of the elements of one 
first column be midtiplied by the sum of the elements of the other 
first column, the sum of the elements of one second column by the sum 
of the elements of the other second column, and so on, then the sum 
of these products is equal to the sum of the elements of the product 
of the two determinants. (xl.) 
