442 
Proceedings of Royal Society of Edinburgh. [sess. 
systems of equations, we come to two sections devoted to the multi- 
plication-theorem. Of the five formally enunciated propositions 
which they contain, two, the second and fourth, need not be more 
than referred to, as their substance comes from Binet and Cauchy, 
and as the mode in which they are established will be sufficiently 
understood from the treatment of one of the others. The general 
problem of the two sections is the investigation of the determinant 
2±ce lC2 "....c« 
where 
(fc) 
(0 (*) 
.(OJfr) 
a a + cq a x +....+ a p a p 
Taking a single term of the determinant, we have of course 
cci'cf . . . c ( ^= (a a +a 1 a 1 + . . . . +%a p ) 
x (a! a! + oq'aq' + . . . . + a p af) 
x-(.V + + ««<£>), 
and we see that if the multiplications indicated on the right be 
performed there must arise a series of (p+ l) n+1 terms of the type 
a r a r • a s a s * a t" a i" 
or by alteration of the order of the factors 
(n) , f 
a w • a r a s a t 
W (») 
w 
In) 
r s t 
where each of the inferior indices r,s,t, . . . , w may be any member 
of the series 0,1,2, . . . , p. If we bear in mind the meaning which 
we thereby assign to the summatory symbol S we may write this in 
the form 
ccJcf .... = S(a a V" . . . . a a 'a " . . . a^ n) ) . 
1 ^ n \ r s t w r s t w ' 
The next point to consider is the transition from the single term 
cc{cf . . . c^ to the full aggregate 2 ± ccfcf . . . c^. A glance 
at the sum of terms denoted by cj® shows that by permuting the 
superior indices of cc{cf . . . , the superior indices of the a’s 
are subjected to the same permutation, and that, on the other hand, 
when we permute the inferior indices of cc{cf . . . it is the a’s 
