A DEVELOPMENT OF A DETERMINANT OF THE MN TB ORDER. 



625 



as roughly representing a compound determinant of the n th order, each of whose 

 dements is a determinant of the m w order and a minor of the original determinant, 

 and each term of the compound determinant thus arising will produce (m!)" terms, 

 differing at most only in sign from terms of the original determinant. 



The original determinant being denoted by 



' a n "22 'h 



the compound determinant referred to will be 



mn,mn 



'1,3 



a, 



->.i 



a.,. 



H,m 



a,,,., ... a, 



a., 



l,m+l 



a., 



•2,m+l 



H,in + 2 



h,m+2 



a l,-2m 



a 2:2m 



1 in,2iii, > 



i/i+l,l a m+l,2 • 



*m+2,l ""m+2,2 



" , m+l,<m 



a m+2,m 



a 2m,\ a 2m,2 • ■ • a 2h 



d.y,,.' 



+2,j«+l a m,+2,m+2 



a m+l,2m 



iii+2,2iii 



a. 



2m,m+l w 2m,m+2 



«., 



• ^Slm. 9 



2m,2m 



or say, for shortness' sake, 



M u M 12 

 M 21 M 22 



iV1 i/i 

 M 



M nl M, l2 . . . M 1; 



any term of which will, of course, be of the form 



Mm M p9 M,, 



where no two of the M's, by the definition of a determinant, belong to the same set of 

 m rows or m columns of the original. Now, each term of the development of the minor 

 M hk being in this way the product of m elements taken from m rows and m columns 

 of the original determinant, and each term of the development of the minor M m being 

 the product of m elements taken from m other rows and m other columns, and each 

 term of the development of the minor M,. g being the product of m elements taken from 

 a third set of m rows and a third set of m columns, and so on, it follows that each term 

 got by multiplying together a term of each of the n minors M hk , M pq , "M w '. ... will 

 contain an element from each of the mn rows and mh columns, and therefore by defini- 

 tion be a term of that determinant. Further, since the number of terms in each minor 

 is m !, the number of terms of the original determinant which arise from the term 

 MjjMpjM,, .... of the compound determinant is (m !)", and therefore the number 

 of those which arise from all the terms of the compound determinant is n ! (m !)*.' 



