THOMAS MUIR ON BIPARTITE FUNCTIONS. 



473 



Thirdly, denoting the bipartite adjugate to 



a, a. 2 a 3 



\ 



h 



h 



b l c x d x 

 b 2 c 2 d 2 

 b 3 c 3 d 3 



A 



e 2 



A 



9z 



e s 



A 



9s 



or 



by 



A, 



A 2 



A3 



H x 



K, 



Li 



Bi 



0, 



Bl 



A 



E 2 



K 3 



B 2 



c 2 



r> 2 



*\ 



F 2 



F 3 



B 3 



c 3 



D 3 



Qi 



^"2 



G 3 , 



or 



A, 



B + 



and, proceeding on the same lines as in the foregoing case, we have 



(§13) ft = a 1 A 1 + tf 2 A 2 + a 8 A3 , 



(§13) /8 4 =A 1 H 1 + * l k 1 + tL 1 . 



Also, by an extension of the same theorem (§ 17), 



A= 



Cfcrt t/q 



*1 



*i 



6 3 & 2 



6i 2 « 3 



' A A A ' *s 



«., 



«3 



*I 



h 



h 



«1 



H 



2 2 



c 3 



rf 3 



ffl 



02 



9% 



b x c x d x 



Multiplying together the three dexter members of these equations we obtain an 

 expression of twenty-seven terms which are exactly the twenty-seven terms of 

 B, Hence ^^ 



The same mode of demonstration is evidently applicable when the bipartites 

 are of any higher degree. 



30. If the bipartite adjugate to a given bipartite be formed, any minor of it 

 of the r th degree is equal to the product obtained by multiplying the cofactor 

 of the corresponding minor in the original bipartite by the (■/*— l) th power 

 of the latter. 



For example, in the third case of the preceding paragraph, the minor 



1 1 1 1 1 1 1 1 1 



Pi D2 D 3 

 Ej Fj Gj 



A / 2 /3' 3 ffi 92 9s 





a s hi 



1 h C-2 



A, 



di a 



b, c 3 d 3 



Ct"t CLi) Ujn 



h x k x l x 



\ c x d x 



2 2 9 



e l e 2 e 3 



A A A 



h c s d s ! 9\ 9-i 9s 



and a z h x is the cofactor of that minor of fi 4 which corresponds with the minor 

 1), D 2 D, 



«, F, G, 



of B 4 



