4(H) 



THOMAS MTJIR ON BIPARTITE FUNCTIONS. 



or, if we decide on taking the elements of its 3rd array, is equal to 





+ d. 



A 



A 



9i 

 9x 



h, K 



Pi ?i 



m, 



h h "i 



k 1 k\ 2 



h i K 



h h 



Vl„ 



n. 



+ «i 





Pi 



2i 



m x 



m 2 



n i 



n 2 



+ e 2 . 



/» 



K 



h 







/l 





k 



Pi 



Qi 





m x 



m 2 





H 



h 



n i 



n. 2 



#2 



K 



k 2 







ffl 



K 



K 



Vi 

 m 1 



ii 





h 



h 



m 2 





H 



k 



ih 



» 2 



8 



16. Since a bipartite function is linear with respect to the elements of any 

 one of its arrays, the cofactor of any of the elements (which has been shown 

 above to be expressible as a minor bipartite or as a product of minors) is 

 expressible also as the first differential coefficient of the function with respect 

 to the element in question. 



Hence, B denoting the bipartite whose initial line is a x , a 2 , a 3 , . . . . , a n , 

 the theorem of § 13 may be alternatively stated in symbols thus — 



B = 2tf, 



SB 



8a,. 



(r-l, 2, 



») 



and the elements of any square array of B being the elements of the deter- 

 minant \a ln \, the theorem of § 15 is 



B = 2« r 



SB 



/r = l, 2, . . ., n\ 



[s=l, 2, . . ., n) 



17. A bipartite of deg-order (m, n) is expressible as the sum of n products 

 of two factors each, viz., a minor bipartite of any degree less than m, say of 

 the degree p, and a minor of the degree m—p, the former being obtained from 

 the first p arrays by deleting all the lines of the p th array except one, and the 

 latter being obtained from the last n — p arrays by deleting all the lines of the 

 (//— //) th array except the line collinear with that formerly undeleted in the 

 //'' array from the beginning. 



This theorem is deduced from the theorem of § 15 by combining those 

 terms of the development there obtained which have a common factor. Thus, 

 taking the first development of 







A 



02 



*. 



K 







«1 



a, 



A 



ffl 



*i 



K 



Pi 



Si 



6, 



c i 



*x 



d, 



h 



ji 



III »j 



m 2 



b. 



H 



e i 



e 2 



s 



J2 



n i 



«2 



