4(54 



THOMAS MUIR ON BIPARTITE FUNCTIONS. 



Thus in 



the cofactor of a 2 is 



and the cofactor of h x is 



1 9 -I 



K 



h 



h 



b 2 c 2 d 2 

 h c 3 d i 



A 



9i 



«2 



A 



9i 



H 



A 



9i 





ft, 



*i 



' 2 



«1 



«i 



e 2 



«3 



H 



/i 



/ 2 



/ 3 



c 3 



#i 



#2 



^3 



a., ti, 



h 



Cl 



d, 



h 



h 



c 2 



d 2 



A 



h 



% 



d 3 



9i 



13. A bipartite of cleg-order (m, n) is thus expressible as a sum of n products 

 of two factors each, the first factors being elements taken either all from the 

 initial line or all from the final line, and the second factors being bipartites of 

 deg-order (m — 1, n). 



Thus 



tti Ctn CVr> 



*! 



h 



k 



&! 6'j (l x 



h 



H 



h 



b 2 c 2 d 2 



A 



A 



A 



U t > Co '(-<» 



f/i 



92 



9s 



ft, li\ l x 



h 



e 1 e 2 



H 



h 



A A 



A 



h 



9\ 9 2 



9s 



1 1 1 



1 ^2 *t 



A A A 



9\ 92 9z 





*! 



^ 



h 



ch 



e l 



«2 



h 



d 2 



/l 



/ 2 



A 



^3 



01 



92 



9a 



This recurrent law of formation of a bipartite might of course have been 

 adopted as the definition. 



14. The cofactor of any one of the secondary elements belonging to the p tu 

 array of a bipartite of deg-order (m, n) is expressible as the product of two 

 bipartites, one of deg-order (p—l, n) and the other of deg-order (n—p, n), 

 the first being got from the first p — l arrays by deleting from the (p— 1)"' 

 array all the elements not collinear with the element in question, and the 

 second being got from the last n— j> arrays in the same way. 



