Axisymmetric Determinants . 



453 



and so on. In the first place all the elements of the last row 

 and of the last column of each compound determinant become 

 resolvable into two factors, one of which is the minor common 

 to all the elements of the said determinant. The division indicated 

 can thus in part be performed, namely, in the first identity we can 

 divide by a 2 IIt in the second by \a XI a 22 \ 2 , and so on, with the result 

 that the row and column in question become, as far as may be, a 

 repetition of the corresponding row and column on the left-hand 

 side. Thus, in the case where n = 6 we have 



a I 



a 2 



a 3 



a 4 



a 5 



. 



K 



K 





K 



b 5 



be 



c I 



c 2 



C 3 



C 4 



C 5 



c& 



d T 



d 2 



d 3 



d 4 



d 5 



d 6 



e x 



e 2 



e 3 



e 4 



e 5 



e 6 



• 



/- 



h 



A 



/s 



• 



CLj 



a. 



K 



b 



Cj 



c. 



d x 



ci 



&i 



e. 



Ctn 



a. 



d 3 d 4 d 5 



e 3 e 4 e 5 



J3 J 4 J 5 



ce 

 d 6 



e 6 



a 1 a 2 



b x b 2 



c x c 2 



dj d 2 



e x e 2 



a z 



a, 



a 



(X o CVj CI r 



e 3 e 4 e 5 



de 

 e 6 



a T b 2 j 



CL X C 2 | 



a x d 2 \ 

 a z e 2 | 



/- 



a 1 b 2 c 3 

 a x b 2 d 3 

 a x b 2 e 3 



U 



a x b 2 c 3 d 4 

 a 1 b 2 c 3 e 4 



u 



a x b 3 

 a x c 3 

 a T d 3 

 a x e 3 



h 



a t b 4 | | a 1 b 5 

 a z c 5 

 a x d 5 

 a T e 5 



is 



'4 



a x c 4 

 a x d 4 

 a x e 4 



h 



b 6 

 ce 

 d 6 

 ee 



rft;, 



a x b 2 c 4 

 a x b 2 d 4 

 a x b 2 e 4 



a x b 2 c 5 

 a z b z d 5 



a I b 2 e 5 



Ce 

 d 6 



e 6 



a x b 2 \, 



a x b 2 c 3 d 5 

 a x b 2 c 3 e 5 



fs 



d 6 

 e 6 



(X.) 



9. Let us consider now the case where none of the three non- 

 zero elements is in the place 11, A being 



Here, as before, we have 



where 



U = 



. . • 



d 



e 



f 







. g h 



i 



j k 







. h I 



m 



n o 







dim 



V 



q r 







e j n 



2 



s t 







f h o 



r 



t u 







n 







. d 2 



e 2 



JE+f JV) 



= 



d 2 . 

 e 2 Q 



U 









f 2 



S 



P 



f 2 



s 



p 



u 



m 



, 



d 





y 

 h 



h 



I 



i 

 m 



d 



i 



m 



V 



e 



j 



n 



9. 



