470 



THOMAS MTJIR ON BIPARTITE FUNCTIONS. 



and by § 15 the cof actor of a x is equal to 





A 



Hi 



*, 





A 



9i 



K 



\ 



4 



d.. 





h 



e i 



s 



l 





ft 



Si 



h 



m l 



m 2 



s 



n x 



n 2 



and the cofactor of « 2 is equal to 



+ 





A 



<Ji 



k 2 





A 



<7, 



h 2 



h 



<k 



u 2 





h 



H 



H 





ft Si 



VI, 



m. 



w 



Hence 



where 





/. 



#2 





/, 



#1 



Ci 



^ 



rf 2 



«* 



«i 



«2 





7'! 



ft 



h 



w 2 



m 2 



«» 



% 



n 2 



+ 



/l 



#2 



K 









/l 



0] 



h 2 



Pi 



ft 



<*1 



d 2 





k 



m x 



m 2 



«1 



H 





h 



n i 



n 2 







A 



9 2 



K 



k 2 







a 1 



a -2 



A 



ff< 



K 



h 2 



ft 



Si 



h 



C l 



d x 



d 2 



h 



h 



m x 



m 2 



h 



C-2 



e i 



C 2 



*2 



32 



ih 



n 2 





A 



92 





A 



9i 



h 



*i 



d 2 



\ 



Cl 



H 



K 



M 4 = 



a 



P 4 M 4 



Q, N 4 



E, 





A 



92 



h 





A 



.ft 



\ 



c l 



<h 



d 2 





C2 



e i 



e 2 





R,= 





Pi 



Si 



h 



m i 



m 2 



h 



n x 



n 2 



(&) 



Q* 



A 



92 



A 



9i 



d x 



d 2 



e i 



e 2 



"'2 



N 4 = 



/■ 



92 



/l 



9i 



d 2 



e l 



e 2 



q _ ft 



72 



»l 



», 



Si 



m, 



It will be observed that, in using § 15 the second time here, it is necessary 

 to do so in such a way that each term of (/3 2 ) shall have a factor common to the 

 corresponding term of (/3j). 



25. When we note that in using the theorem of § 17 in the preceding para- 

 graph, several other identities might have been got in place of (a), and that in 

 using the theorem of § 15 we might have chosen several other pairs of identities 

 in place of (y8j)(y3 2 ), it is clear that we have not by any means exhausted the 

 possible ways of expressing the given bipartite as a compound bipartite of the 

 3rd degree. 



There is little difficulty in seeing what the theorem is which includes these 

 different forms in the same way as theorems of § 17 and § 15 include all the 

 forms of §§ 21, 22; but the formulating of it would at the present stage be 

 troublesome [see § 42]. 



26. The expression of a bipartite of a degree higher than the 4th as a 

 compound bipartite of the 4th degree, and generally the expression of a bipar- 



