474 



THOMAS MUIR ON BIPARTITE FUNCTIONS. 



31. To every general theorem which takes the form of an identical relation 

 between a number of the minors of a bipartite or between the bipartite itself 

 and a number of its minors, there corresponds another theorem derivable from 

 the former by merely substituting for every minor its cofactor in the bipartite, 

 and then multiplying any term by such a power of the bipartite as will make 

 all the terms of the same degree. 



This is the important Law of Complementaries already known to hold good 

 in regard to Determinants and Pfaffians. Any class, indeed, of algebraical com 

 binatory functions, concerning which we can assert the truth of two theorems 

 like those of §§ 29, 30, is ruled by the Law of Complementaries. The mode of 

 establishing the law is literally the same for all. (See my Theory of Deter- 

 minants, pp. 141, 142.) 



32. If all the rows or all the columns of any square array of a bipartite be 

 identical the bipartite is resolvable into two factors. 



For by the theorem of § 17 the bipartite is expressible as a sum of products 

 of pairs of factors, and all these products by the datum have one factor in 

 common. Thus 



«j a 2 a 



e i 



H 





*i c i d \ 



h 



h c 2 d 2 



h 



e 2 



«S 



h h d i 



e i 



e 2 



«S , 





A 



ffl 



K 





A 



9-2 



h 





A 



9 3 



K 



1 a l a 2 a s a-^ a 2 a 

 ~ 1 b 1 c, c?! b 2 c 2 d 



a x 



a 2 



a 3 



»! aa s 



h 



H 



d 2 



h c 3 d 3 



«i 



H 



e s 





A 



ffi 



K 



*. 



A 



92 



K 



h 



A 



9z 



K 



m x 



33. Any power of a bipartite of the second degree may be expressed as a 

 bipartite. 



For example — 



a 



b 



c 



ax 



ay 



az 



bx 



h J 



bz 



ex 



c 9 



cz 



a 



b 



c 



X 



y 



z 



X 



y 



z 



X 



y 



Z 1 



ax 

 by 



- ( a ° c \. 

 \x y z J ' 



and 



a 



b 



c 



X 



y 



z 



ax 



ay 



az 



ax 



bx 



ex 



bx 



by 



A: 



ay 



l >y 



cy 



ex 



'.'/ 



CZ 



az 



bz 



r: 



a 



b 



c 



x y 



z 



x 



y 



z 



a 2 x abx 



acx 



X 



y 



z 



aby b 2 y 



bey 



X 



y 



z 



acz bcz 



eh 



a 



b 



c 



ax 



by 



CZ 



X 



y 



z 



ax 



ax 



ax 



X 



y 



z 



by 



by 



by 



X 



y 



z 



cz 



rz 



cz 



= ( a I ° Y by ^ 32. 



V* y z J J > 



