1899—1900.] Dr Muir on the Theory of Shew Determinants. 213 
so that there is obtained 
A = ^/A rr -h- ss ) 
and since in this aggregate the values possible for r are exactly 
those possible for s, he concludes (without knowing the signs of 
the terms of the aggregate, be it observed) that it is resolvable 
into two factors, viz. 
Cj^j a ir \/ A a ls JA s ^j . 
It is then argued that the two factors are identical even in the 
signs of their various terms “ da durch das Zeichen einer Wurzel 
die Zeichen der tibrigen bestimmt sind ” ; and that therefore 
- an aggregate of n - 1 terms, since the values to be given to 
r are 2, 3, . . . , n. The next step consists in pointing out 
that A' rr being a determinant similar to A but of order n- 2, 
it must follow that JA' rr can in the same way be expressed as 
an aggregate of n - 3 terms, and that this process can be continued 
until the minor under the root-sign is of the 2nd order, when 
manifestly its value is the square of one of its elements. The 
final result thus is that J A is expressible as an aggregate of (n- 1) 
{n — 3) . . . 3.1 terms each of which is the product of \n elements 
whose collected suffixes form a permutation of 1, 2, . . . , n. 
By way of corollary to this it is pointed out that 
±a 10 a Q 
l n—l, n 
is one of the terms of the aggregate, and the same is proved by 
showing that the square of this is a term of A, the reasoning 
being as follows : — Since in every case a rs — - a sr we have 
(^12^34 ’ ’ ‘ a n— 1. nf“ ~ (^12%4 ‘ * * a n- l,n) ’ ( — (^21^43 ’ * * a n,n-l)i 
and . . = ( “ )^ W (o&i 2 ®21 ^34 ^43 ' " * a n—l,n a n, n - 1 )> 
which clearly contains n elements, one from every row and one 
