488 Proceedings of Royal Soeiety of Edinburgh. 
%•! 
my 2 • ■ 
mg-i 
77^2*2 • ■ 
■ . ni^-n 
m„.2 . . 
and the corresponding “ derived systems ” of the factors 
^1-1 
^1-2 
. . . 
a^n 
a\'2 • • 
^2-1 
^2*2 
a2>n 
®2-l 
®2-2 • ” 
. tt2.^ 
^n*2 
^n'n 
) 
an-i 
a„. 2 
■ • Otn' n 
in other words, the relations which must connect the systems 
. (“S) . (««) 
by reason of the relations 
5[S ^M*i) “ 
(given in full above on p. 513, Vol. XIV.) which connect the systems 
(«!•«) , («!•«) . (”*!•«) • 
First of all, attention is concentrated on a single “ terme ” of the 
system 
(K5), 
or, as we should nowadays say, on a minor of the product-deter- 
minant. The process of reasoning, which occupies about four 
quarto pages, is exactly analogous to that previously followed in 
dealing with the product-determinant itself ; and the result obtained 
is 
jtl.V j/.l J i 
(xviii. 5), 
where is meant to indicate that the terms on the right-hand side 
are got by changing the second suffixes into 2, 3, 4, . . . , P in 
succession. Speaking roughly and in modern phraseology, we may 
say that this means that 
Any minor of a product-determinant is expressible as a sum of 
products of minors of the two factors. (xviii. 5.) 
Cauchy then proceeds (p. 107) — 
“ Si dans cette equation [xviii. 5] on donne successivement a 
