A DEVELOPMENT OF A DETERMINANT OF THE MN m ORDEK. 



627 



pairs will be always even whatever r may be ; and, when m is odd, the number of 

 additional inverted-pairs will be odd or even according as r is odd or even. From this 

 there is immediately deduced the following theorem : — 



If the first mn integers he separated into n groups of m each, say the groups A x , 

 A 2 , A,, . • • , A n , so that 



A x stands for any permutation of I, 2, 3, ... , m 



A., ... ... ... m -f- 1, m + 2, ... , 2m 



mn — m + 1 , mn — m + 2, 



mn, 



then, when m is even, any one of the n ! permutations of A^A- . . . A n has an even 

 number of inverted-pairs of the mn integers more than the standard permutation 

 k x k 2 K 3 . • . A n has ; and, when m is odd, has an odd or even number of inverted-pairs 

 more, according as the suffixes of the A's have an odd or even number of inverted-pairs. 



(7) Returning now to the consideration of the terms of the original determinant of 

 the (mn) th order, which are obtainable from the term M^.M^M,,, .... of the compound 

 determinant, Ave see that, in order to tell the sign in any case, we have to take into 

 account the sign of each of the contributing terms in its own minor, and, in addition to 

 this, the sign which ought to precede M /d M p<1 M rs . . . Now, we have just seen that when 

 m is even, this sign must be a + in every case, and that when m is odd its sign must be 

 the sign which it bears in the compound determinant to which it belongs. This is 

 equivalent to saying that when m is even the compound determinants with which we 

 started should be all viewed as permanents, and that this change is not necessary when 

 m is odd. 



(8) When the given determinant of the (win)" 1 order is an alternant, interesting 

 changes are possible in connection with each term of the development. 



Thus in the case of the simplest form of alternant of the 4th order we have 



aWd 3 



= 



| a°b l \ | a 2 b 3 | 1 1 | «V | 

 | cW | | c 2 # 1 J 1 | IPcP- | 



| a 2 c s | 

 | l 2 d s | 



+ 



| 6V | 



| a 2 d s | 

 | 8W | 



\l-a aW-aW- 1 



| d — c c 2 d 3 — c s d 2 \ 





.., 













= 





- . 





1 







which agrees with a result obtained by Jacobi in his paper, " De functionibus alter- 

 nantibus" (Crelles Journ., xxii. p. 363). 



VOL. XXXIX. PAET IIT. (NO. 24). 5 D 



