476 



THOMAS MTJIR ON BIPARTITE FUNCTIONS. 



instead of being squares, may be merely rectangles, the length of the first 

 rectangle being the same as that of the initial line, the length of the second the 

 same as the breadth of the first, the length of the third the same as the breadth 

 of the second, and so on. Thus, starting with a line of m elements, we draw 

 the separating bar and write n rows of m elements each, then r columns of 

 n elements each, then .9 rows of r elements each, and end with a line of 

 5 elements. The case of this where m — 2, n = 3, r = 4, 5 = 1 is represented by 



a, «., 



b c 



4 



d 2 



<h 



d, 



h C 2 



g i 



H 



e s 



f i 



h C 3 



A 



A 



A 



A 





9x 



*I 



\ 



l x | m 



38. A little consideration serves to show that almost every theorem we 

 have given can be extended so as to hold true of bipartites with rectangular 

 arrays whose length and breadth are different. Indeed, this extended defini- 

 tion was not adopted from the first, only because it was seen that by doing so 

 the difficulties of exposition would have been considerably increased. 



39. Of course any bipartite with arrays that are merely rectangular may be 

 expressed as a bipartite with square arrays by the introduction of a sufficient 

 number of zero elements ; and in this way what we have called the more 

 general form of bipartite may also be looked upon as a degeneration of the 

 particular form. Thus the square-arrayed bipartite 



a-, a 



\ Cj 







4 



d. 2 



d. 



d t 



\ C 2 







<h 



e 2 



''3 



h 



h c 3 







A 



A 



A 



A 











<h 



ff-z 



3% 



9a 





*I 



h 



h 



m 



is evidently equal to the merely rectangular bipartite 



«i « a 



\ 



1 



d x d 2 d 3 d 4 



2 



h H ^ « 4 



8 



J\ fi Ji A 



>h h \ h 



wi, 



and has only 2 • 3 • 4 terms instead of 4 • 4 • 4. Fewer zeros than twelve, be it 

 also remarked, would make it assume the latter form; for, in a square-arrayed 

 bipartite, if any line of any square array be a line of zeros, the line collinear 

 with it in one of the adjacent squares may be made a line of zeros also. 



