THOMAS MUIR ON BIPARTITE FUNCTIONS. 



471 



tite of a degree higher than the m th as a compound bipartite of the m th degree, 

 can be obtained in a manner similar to that of § 24 by repeated use of the 

 theorems of §§ 15, 17. 



27. A more expeditious and more instructive method, however, is to make 

 use of the already obtained transformations into compound bipartites of the 

 2nd and 3rd degrees. 



Suppose, for example, that it is required to express the bipartite of the 6th 

 degree — 







K 



h 



h 



r x r 2 r 3 













K 



h 



h 



n x n 2 n 3 









<h C( 2 



a 3 



K 



K 



h 



m x m 2 m z 



or 



& 



say, 



h l C l 



*i 



e i 



H 



h 



«i Vx «i 



h h 



d 2 



A 



A 



A 











h C 3 



d 3 



ffi 



9<2 



9s 











as a compound bipartite of the 4th degree. A probable form for the latter 

 would be that in which the initial and final lines were the same as those of the 

 original, and the elements of both its square arrays bipartites of the 2nd degree. 

 Let us denote this form by 



or B 4 



Then, changing both /3 6 and B 4 into bipartites of the 3rd degree by means of 

 the theorem of § 23, we obtain on comparison of the two results — 



a l 



a 2 



a 3 



x i 



Vx z x 



Pi 



Qi 



R 



S a 



S 2 S 3 



p 2 



Q 2 



R 2 



T: 



T 2 T 3 



Ps 



Q 3 



R. ( 



u, 



U 2 U 3 , 



P t P 2 P s 

 Si T x V\ 



Pi_APs 



S 2 T 2 IJ 2 



h \ h 



m x n x r x 



b x b 2 b z 



m x n x r x 



e i fx 9x 

 H A 9 1 

 H A 9s 



h x h 2 h 3 ~ 

 #1 K 2 "'a 



'x ^2 ^3 



A 2 



A, 



say. 



\ h h 



™. 2 n 2 r a 



A 3 



and seven other similar equations. But by the theorem of § 21 these may be 

 transformed into 



&i h h 



h x \ h 



\ b 2 b z 



P lt P 2 , P 3 , _ e i A 9x ' H A 9i ' % A 9z 

 S„T X> \5 X 



m x n x r i 



IV-, fb.y it it A--1 fCn /Co l-t On &> 



h h 2 h iiMs 



\ b 2 b 3 



p p p 



e x A ffi 



e 2 A 9l 



g 3 A 9i 



^2» *-2> ^2 



m 2 n 2 r 2 



m 2 n 2 r 2 



K-, fCn /Co 



m 2 n, r 2 



'l "2 '3 



