490 Proceedings of Boyal Society of Ediiiburgli. 
The enunciation of this in modern phraseology would he — 
Any compound of a product-determinant is equal to the product 
of the corresponding compounds of the tivo factors. (xliii.) 
The next deduction is stated equally succinctly (p. 109) — 
“ Si Ton ajoute entre elles les equations (63) on aura la 
suivante, 
(66) j } -■= , (xxx. 2) 
le premier signe S, c’est-k-dire le signe exterieur, etant relatif 
a Tindice v, et les autres, c’est-a-dire les signes interieurs, etant 
relatif s a I’indice 
This (66) corresponds to (xl.) as (65) corresponds to the multipli- 
cation theorem 
the transition from the general to the particular being effected in 
both cases by putting p — \. 
With these deductions, the 4th Section practically comes to an 
end ; but one or two results, intentionally omitted in the account of 
the 2nd Section because they seemed to belong naturally to the 4th, 
fall now to be noted. 
The first is very simple. It arises (p. 91) from observing that 
(D.)-^x(8J-^ = (DAr\ 
and .-. =(M,)"-i 
by the multiplication-theorem. The result (xxi. 2) above (p. 110), 
is then thrice applied, and a theorem at once takes shape, which in 
later times we find enunciated as follows : — 
The adjugate of the product-determinant is equal to the product 
of the adjugates of the two factors. (xliii. 2.) 
It is not noted, however, by Cauchy that this is but a case of xliii., 
viz., where p=^n-\. 
The next is 
S 6i.^) = D„a„. J , 
or S = (xliv.) 
It is nothing more than the result of solving the n.n equations 
(33) % [S”(avi Vi) = m^.v] 
