THOMAS MUIR ON BIPARTITE FUNCTIONS. 



467 



given as an example in § 15, viz. 



./;• 



a l 



a 2 



d, 



h 



K 



Pi 



m x 



t + - 



a i 



a 2 



d 2 



K 



K 



Pi 



<li 



h 



c i 



h 



k 



h 



Cl 



h 



k 



7Tl 1 



m 2 





h 



c 2 



e i 



h 



3-i 



*i 



n 2 



h 



C 2 



e 2 



h 



h 



n x 



n 2 



f 





a 2 



d X ' 



\ 



h 2 



Pi 



?1 1 a 



a x 



a 9 



d 





k 2 

 k 



Pi 

 m x 



gi 



J2- 



h 



h 



'"i 



m 2 + 9>1 - 



h 



c i 



m 2 





h 



C 2 



h 



H 



h 



n x 



n 2 



\ 



c 2 



H 



h 



h 



n x 



n 2 



we observe that the first two terms have the common factor 



h k 



vi. m., 



n. 



Tlo 



the full cofactor being 





a. 2 a x 



r \ d ~*~9i- h 



c i "i ( 'i 



Co e, b., 



which, we know from § 13, is equal to 



/l 



9i 



di 



d 2 



«i 



C-2 



Similarly, the cofactor of the factor common to the last two terms is seen 

 to be 



«1 



a 2 



A 



92 



h 



°1 



di 



d 2 



\ 



c 2 



e i 



C 2 



Hence we have as an example of the present theorem — 



Ht 



"l «2 



;nce we have 



A 92 \ K 



A Pi ! ^i K 



h, c 2 



di d 2 

 «i H 



h k 



H h 



Pi gl . 



m x m 2 



n i n 2 



a x a 2 



fi 



9i 



b 2 c 2 



di 

 e i 



d 2 

 e 2 



K 



h 2 



h 



h 



i 2 



k 



Vi Si 



n, n» 



€ti tvo 



h 



A 



92 



k x k 2 



Pi <Li 



d\ 



d 2 



h Ji 



m x m 2 



«i 



e 2 



h h 



n x n 2 



(a) 



Had we combined the first and third terms of the same development, and 

 then the second and fourth, we should have obtained the example — 





fi 9i 



k x k 2 







a x a 2 



/i 9i 



h x h 



Pi gi _ 



. «1 «2 



\ c i 



d x d 2 



h h 



m x m 2 



b x c x 



b 2 c 2 



e i e 2 



h k 



n x n 2 



b 2 c 2 



/i K K 



d i' \h k 

 e i I h k 



Pi gi + « 



m x m 2 v x 

 n x n 2 b 2 





92 



hi k 2 





9i 



K h 2 



1 d 2 





h k 



1 e 2 





h 32 



Pi gl 



m x m 2 

 n, n. 



