492 
Proceedings of Royal Society of Rdiiibu7''gli. 
credit of the generalisatioa of the multiplication-theorem. The 
identities on which the claim is based are — 
S”(%.v)} =S^S^(m^.v) (50) 
S’*{S^(/?^.,) (51) 
S” { (70) 
The first of these, given formerly (p. 516) in the uncontracted form 
(«l-l + «-2-l+ • • • 
+ a«-i) (ai-i + ag-ib . . 
. +«n*l) 
-f {gy2 “f” • ' • 
+ ««-2) (^l-2 + S-2+ • • 
. +«,r2) 
+ 
+ (^i-M + 0,2-, j + . . . 
+ Otrt-n) • • 
= my-^ +^^2'1 • • 
+ my2 +’%2 + • • 
. +m„.2 
+ 
+ my,, + m2.n + . . 
where v = cv- lO^v. i + «/ 
UL’2^V'2 "T . . • + , 
may he at once left out of consideration ; it is not even a case of 
the multiplication-theorem. Cauchy, we may he sure, mentioned it 
only because it is the first of the series to which (51) and (70) 
belong. The next concerns the systems 
(/^l•9^) J (^l*w) ) (^l*w) 
ad jugate to the systems 
(®Tw) ) (^l*w) 5 (pel'll) 
dealt with in (50). It indeed is comparable with Binet’s theorem ; 
but as it is only a case of (70), — the minors in (70) being of any 
order whatever, whereas in (51) they are the principal minors, — we 
may without loss pass it over. Directing our attention, then, to 
(70) let us for the sake of greater definiteness take the case where 
5 4 
^^ = 5 and^ = 2, and where consequently P = — ^ — =10. The theorem 
1 • z 
then becomes 
For the purpose of comparison with Binet’s result, it is absolutely 
necessary, however, to depart from this exceedingly condensed mode 
