408 



THOMAS MTJIR ON BIPARTITE FUNCTIONS. 



Again, by combining the first and third terms of the second development in 

 § 1."), we should have the case where the one factor is of the second degree, and 

 the other of the sixth, viz. — 



% n.. 





A ffi 



h K 



"l a -2 



A ffi 



\ K 



1>1 h 



di d 2 



h. h 



!, 2 ''■• 



e x *-i 



h J-2 



l\ 7i.. = «i_ 



a % 



m x m. z b x 



c x 



n 1 ra, 





f. 



ffa 



h 



K 







fx 



9x 



K 



K 



Pi 9x 



«1 «2 



*! 



d 2 



h 



h 



m l m 2 



h Ci 







h 



3i 



n i n 2 





A 



A 



9-> 

 9\ 



e -2 





e i 



The case where the one factor is of the 1st degree and the other of the 7th 

 falls under the theorem of § 13, which may thus be looked on as a particular 

 case of the present theorem. 



18. Two minors such as those of each term of the development in the 

 preceding paragraph — that is to say, minors which, when multiplied, give terms 

 that are all terms of the parent bipartite — may be called complementary minors. 



19. A bipartite of deg- order (m, n) is expressible as a sum of n 2 products of 

 three factors each, the first being an element of the initial line, the second an 

 element of the final line, and the third the minor bipartite of deg-order [m — 2, n), 

 which is obtained from the original by deleting the initial and final lines, and 

 those lines of the first and last square arrays which are not collinear with one 

 of the said pair of elements. 



Thus 





Co 



A 9i 



H fh-Jl 



+ «x9x 



d x do 



+ a J\T- 



+ «<l9l 



do 



This of course is but the result of a double application of the theorem of § 13, 



20. A bipartite function may be expressed as a bipartite of lower degree, 

 in which elements occur that are themselves bipartites, and to which, on that 

 account, the name compound bipartite may be given. 



21. The theorem of § 17 is the case of this where the compound bipartite is 

 of the second degree. Thus the identity (a) there given may be written also in 

 the form 





A 



9-i 



/■-, 



k.. 







"■ 



A 



9x 



*. 



h, 



Vx 



<?, 



C] 



d, 



do 



h 



h 



m x 



m 2 



Co 



<h 



e 2 



Xo 



h 



n x 



n.. 



"1 



6, 



Co 



A 



d, 



"., 



&i 



bo 



A 

 d, 

 e, e, 



d a 





h h 



111, 



VI., 



"1 



h 



A 



lh Qx 



m. 



lllo 



?lo 



and so of t lie others. 



