6-6 DK THOMAS MUIK ON 



(5) In the preceding the mn rows of the original determinant were separated in the 

 ■simplest way into n sets of m rows each, but it is clear that if they had been 

 separated in a different way into n sets of m rows each, the same mode of reasoning 

 would have led to the same result. Now the number of different ways of breaking up 

 mn things into n sets of m each is 



C C C C 1 



mn,in ' mn-m,m' nm-2m,m m,m 



n ! 

 or (what is the same thing, since C,.„. = — C,.-i, s -i), 



C C P 



mn-\,m 1 ' mn-m- l.m- I ' ^mn -2m - l.m-1 



(inn) ! (mn — m) ! (mn — 2m) ! m 



i.e., m\(mn — w)[ m\(mn — 2m)\ m \\m~n — 3m) ! ' ?«Tu" 



i.e., 



(mn) ! 

 Jm})" nl 



If, therefore, we form this number of different compound determinants, the total 

 number of terms of the original determinant which we shall thus obtain is 



o 



(mn) ! ,. .. 

 , ,. ,x n Um I)" 

 (m l)"n ! v 



i.e. (mn) I 



which is exactly the full number of terms in the original determinant. It is conse- 

 quently manifest that the terms of the original determinant can be represented, so far 

 as magnitude is concerned, by a sum of compound determinants obtainable in the, 

 manner indicated. 



(6) As the consideration of the question of sign necessitates the use of a theorem 

 regarding inverted-pairs, it is desirable to digress for a moment in order to enunciate 

 and prove this theorem. It is to the following effect : — 



If, in the natural series of integers 1, 2, 3, a group o/'m consecutive members, 



taken in any order, be passed backwards over r groups of s consecutive numbers each, 

 the members of each group) being arranged in any order, the number of additional 

 inverted-pairs in the permutation thus obtained is mrs. 



This is self-evident, for the number of members passed over is clearly rs, and eacli 

 of the m members to which precedence has been given will thus give rise to r.s- inverted- 

 pairs, making mrs inverted-pairs in all. 



When the number of members in the group moved is the same as the number in 

 each of the groups passed over, — that is, when s = m, — the number of additional inverted- 

 pairs is, of course, mi 2 . Consequently, when m is even, the number of additional inverted 



