47 2 THOMAS MUIR ON BIPARTITE FUNCTIONS, 



which are evidently all satisfied by 



P _ b l b 2 K P _ h l b 2 \ p _ h K K 



«i A 9x e 2 A 92 % A 9s 



Q - C ' ;* ° 3 , &c. 



e , /] 9i 

 The problem is thus solved. 



28. Of compound bipartites, some interest attaches to that special form, 

 each of whose elements is the cofactor of the corresponding element in another 

 bipartite, and which, in reference to an analogue in the theory of determinants, 

 we may term the bipartite adjugate to the parent bipartite. 



"29. The bipartite adjugate to a bipartite of the m th degree is equal to the 

 (in — l) th power of the latter. The bipartite adjugate to 



(t-, #2 ^Q 

 &! Cj d 1 ' 



or fi 2 , 



is ? } ? , or B 2 : 



hence evidently 



The bipartite adjugate to 



«! &1 «3 



B 2 =A 



is 



1 9 ^5 







&j Cj d 1 



"2 C 2 ^2 



b 3 c 3 d 3 

 Aj A 2 A 



A 



9i 



'3 



> 



or £ 3 , 



a i C l a 2 g l tt 3 



Ox/, a^ a 8 

 a i9j a z9i « 3 





F, 



or B 3 , 



if the capital letters be used to denote the cofactors of the corresponding small 

 letters in /3 3 . Now (§ 13) 



^ 8 =o 1 Ai+a 2 A 2 +a 8 A 8 , 

 and 



and multiplying together the two dexter members we obtain nine terms, which 

 are exactly the nine terms of B 3 . Hence 



B 3 = #. 



