498 
Proceedings of Royal Society of Edinhurgh. 
^11 “'12 • • • “i5 “21 “22 • • • “25 . . . • ot.io-1 “io>2 • • • “10,5 
Al ^12 • • • /^15 > /^21 ^22 • * • ^25 » .... ^ 10 J ^10,2 * ' * Ao-5 * 
In the corresponding identity of Cauchy, there are only 50 diherent 
elements, viz., the elements of the two square arrays — 
ail 
a^2 • ' 
• • «15 
“11 “12 • 
a2i 
“22 * ■ 
, . a25 
“21 “22 • 
• • “25 
“51 
^52 • • 
■ • «55» 
“51 “52 • 
• • “55 
Indeed, — and it is this which brings the comparison to a point, — 
if from the first of these square arrays we form 10 rectangular 
arrays by taking every possible pair of rows, thus using each row 
4 times over, viz., 
^11 ^12 • * • ^15 ^11 ^12 • • • ^41 ^^42 • ' • 
^22 • • • ^25 ) %1 %2 • • • %5 > ’ ^51 ^52 * * * %5 ’ 
and similarly from the as form a second set of 10 arrays, viz.. 
“11 “12 • • • “15 5 “11 “12 • • • “15 “41 “42 • • • “45 
“21 “22 • • * “25 ’ “31 “32 • • • “35 » 5 “51 “52 • • • “55 i 
and then to these two special sets of arrays apply Binet’s theorem, 
we obtain Cauchy’s theorem. Kegarding the two theorems in all 
their generality, the decision we have reached may therefore be 
expressed by saying that Binet’s is a theorem concerning '2s7mi 
quantities, where s, m, 7i are any positive integers, and Cauchy’s is a 
case of it in which s = . . . (??2 - + 1)/1 . 2 . 3 . . . 
and in which, further, the number of different quantities involved 
is not 
m{m-\) . . . (??^-?^^-l) 
1 . 2 
X mn , 
but by reason of repetitions is only 
2m2. 
Although this decision is against Cauchy’s claim as put by him- 
self, it deserves to be noticed that, apparently by oversight, he 
failed to make his case as strong as he might have done. It will 
be remembered that Binet made two advances in the generalisation 
