in the Theory of Determinants. 423 



This could scarcely be improved upon. The letters on the 

 one side are in order exactly the same as those on the other; 

 the only difference, in fact, between the sides being in the 

 position of three of the straight lines. 



10. When in the generating gnomon one of the incomplete 

 rows is identical with the corresponding part of one of the 

 complete rows, or when more than one such pair of partially 

 identical rows exist, a number of the products on the left-hand 

 side vanish; and the theorem then becomes : — 



2± 



=2± 



a x . . . a p+3 &! . . .b 



% . . . a p+i 

 a, . . . oc n+ . 



bq+l • • • Vq+k a l • • • a p 

 ^p+s+q I Pi Rp 



0\ . . . O q °q+l • • • Oq + k a l ' • ' a p 

 Kp+s+l • • • Up+s+g p\ ftp 



This is the first special case which Schweins gives. 



The next case is virtually the same, as his notation leads 

 him to consider as different two theorems which are derivable 

 the one from the other by the interchange of rows and columns. 



The third case arises when in the generating gnomon 

 there exist, in addition to the partially identical rows just 

 referred to, one or more pairs of columns having the like partial 

 identity : the theorem then is 



d\ a p+s 



h 



bq+i • • bh+k-p+q a x . ,a t 



lp + s-h + 



l • • &p+s-h + q Pi Ph+k 



«1 



h 



fr 



''p + s 



Ip+s-h &p+8-h-\-l • • CLp+s— h + q 



bq+i • . ^+i-p+ 7 «i- . a p 

 A A + * 



The first two of these three special cases he then further 

 specializes, putting in the first k = 0, and in the result thence 

 obtained g = l, and so on. 



11. It is rather strange that Schweins did not observe what 

 is perhaps the neatest, and certainly is now the best known 

 special case of all, viz. that which is got from the general 

 theorem by putting 



m =q + n -\ 



and /3 b . . .,/3 n = « 1 , . . .,a n y 



In this case all the determinants on the right-hand side 

 vanish, and the theorem takes the form 



ax . . . a n -q #i • . • b I I b q+l . , . b q + n ' 



2± 



= 0, 



