THOMAS MUIR ON BIPARTITE FUNCTIONS. 



465 



Thus in 



the cofactor of k l is 



and the cofactor of/ 3 is 



or 







h 3 



fc 3 



h 



Pi Vi P* 









h 2 



K 



h 



n x n 2 n 3 



«1 «2 



a 3 



K 



h 



h 



m-L m 2 m s 



\ c x 



di 



«i 



H 



H 



»'i h «! 



x x 



h c 2 



d 2 



A 



A 



A 



? o So Ivo 



Vi 



h C 3 



d 3 



ffi 



9i 



9z 



r 3 s 3 tc 3 



z i 



«i 



a 2 



«, 



h 



«1 



d x 



h 



c 2 



d 2 



h 



c z 



d 3 



A 



9i 



"h 



m 2 



m 3 



r l 



s i 



u x 



r 2 



«2 



u 2 



r 3 



*S 



«3 



h Pi Pi Pz 



L 



7h ?lo 



a 2 a * h I 7)l l m 2 m 3 



rf. 



r l 



S l 



«1 



^ 



'"2 



s 2 



w 2 



yi 



»S 



S 3 



* ( 3 



«1 



a„ a 



L l l 2 



h 



*i 



.'/1 



*T 



m x n x 



Pi 



r i 



?' 2 



^3 



m 2 n 2 



P-i 



h 



S 2 



S 3 



m 3 n 3 



Pi 



"1 



»2 



u 3 



15. A bipartite of deg-order (m, n) is tlius expressible as a sum of 11 2 products 

 of three factors each, the first factors being elements all taken from any one of 

 the square arrays, the p th say, the second factors being minor bipartites of deg- 

 order (p— 1, n), and the third factors being minors of deg-order (n—p, n). 



Thus 







A 



9i 



ft, 



h 







a l 



a 2 



A 



9i 



K 



h 2 



Pi 



2i 



h 



°i 



A 



d 2 i x 



k 



m 1 



m 2 



h 



Co 



<h 



e 2 



i 2 



A 



n i 



n 2 



if we decide on taking the elements of its fourth array, is equal to 



A 



+A 



a l 



«2 



h 



C l 



\ 



C 2 



«1 



«2 



\ 



C l 



\ 



C 2 ' 



K 



K 



Vi 



ft 



h 



A 



m x 



m 



h 



h 



n i 



7i 2 



K 



K 



Vi 



f /l 



H 



A 



m l 



m 2 



% 



3i 



«1 



n 2 



+ 9i 



+ 92 



\ 



c i 



A ' 



h 



A 



\ 



'■2 



e 2 



h 



A 



a x 



Cl 



d 2 



K 



k 2 



h 



h 



A 



h 



£2 



e 2 



%. 2 



A 



V 



t\ 



ra, m. 2 



Pi <li 



n, n. 



(a) 



