THOMAS MITIR ON BIPARTITE FUNCTIONS. 



469 



22. The theorem of § 15 is the case where the compound bipartite is of the 

 third degree, and its square array is a square array of the original bipartite. Thus 

 the identity (a) there given may be written also in the form 



where 







A 



92 



\ 



k 2 







a x 



<h 



A 



ffl 



K 



h 2 



Pi 



Si 



hi 



°i 



*i 



d-2 



h 



k 



m 1 



m 2 



h 



c* 



«i 



H 



H 



h 



n i 



n 2 





h 



2 f) — x 





Qs 



A 92 R< 



A ffi s 4 



_ *1 



k 2 



Pi 



ft 



'J, — 



k 



m l 



m 2 



*2 



h 



n, 



7l 2 



*i 



^ 2 



ft 



ft 



*i 



^1 



mj 



?»._, 



l 2 



* 



w i 



n 2 



23. The theorem of § 19 is the rather important case where the compound 

 bipartite is again of the third degree, but having for its initial and final lines 

 the initial and final lines of the original bipartite. 



Thus 



where 







A 



92 



K 



h 







"1 



a-2 



fl 



ffl 



*i 



h 2 



Pi 



ffi 



hi 



c i 



*l 



d 2 



h 



k 



m l 



m 2 



h 



C 2 



h 



«2 



h 



k 



71, 



n 2 



u 6 = 



a., 



^6 X 6 



V 6 Y 6 





A 



92 



h 



k 2 







A 



ffl 



K 



K 



h 



*. 



d 2 



H 



k 



vi 1 



K 



e i 



«i 



h 



h 



n x 







A 



92 



K 



k 2 





x 6 = 



c i 



A 



ffl 



*1 



h 2 





<h 



d 2 



h 



k 



m x 





H 



e i 



H 



h 



h. 



n i 



v.= 





A 



92 



*. 



k 2 





A 



ffl 



*d 



K 



hi 



d, 



d 2 



*i 



k 



\ 



h 



e 2 



t 2 



k 



m c 



. T 



/l 



.92 



h 



k 2 





/l 



ft 



K 



h 2 





<*! 



d 2 



h 



k 



m 2 



«1 



«2 



h 



h 



n 2 



24. There are, however, other modes of expressing a bipartite of a degree 

 higher than the third as a compound bipartite of the third degree. These we 

 obtain by making use of both § 15 and § 17. 



For example, by § 17 (or § 13) we have 





A 92 



rC-\ fi/ct 





«, a 2 



fi 9i 



K h 2 



Vi ft 



6, c, 



dj d 2 



h k 



m-t «? 2 



}) 2 C 2 



e 1 e 2 



*2 H 



n l 7t,. 2 



VOL. XXXTI. PART ITT. 





A 



92 



h 



K 









A 



ffi 



*i 



K 



Pi 



ffl 



hi 



d x 



d 2 



h 



k 



m 1 



m 2 



h 



«i 



H 



H 



h 



"l 



n. 2 



+rt 2 . 





A 92 



fC-^ A.*2 







fl ffl 



K h 



Pi ft 



c l 



dy d 2 



h k 



m x m 2 



H 



*1 e 2 



h h 



n x tin 



4g 



(*) 



