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XXV. — On Bipartite Functions. By Thomas Mum, LL.D. 



(Read 16th February 1885.) 



1. If a row of n elements be taken, and closely following this array, but 

 separated by a bar from it, we write n rows of n elements each ; and closely 

 following either outside column of this square array, but separated by a bar 

 from it, we write n columns of n elements each ; and closely following an 

 outside row of this second square array, but separated by a bar from it, we 

 write n roivs of n elements each ; and so on, passing from the rows or columns 

 of one array to the columns or rows of the next, and ending not with a square 

 array, but, as we began, with a single line of elements, we have the matrix 

 representation of a bipartite function. 



For example, when n — 2 and the number of square arrays is 4, the repre- 

 sentation is 



or 









} h 



\ 



k 



r i 



T 2 



r 3 









K 



K 



h 



n i 



n 2 



n z 



«1 



«2 



«3 



*i 



K 



k 



Wlj 



»h 



m 3 



a 2 



d 1 ie l 



rf sl/i f-2 /a 

 <** ' ffi 9i <h 



d 



h 



e 2 



h 



A 



A 



fs 



9i 



92 



ffs 



K 



\ 



h 



K 



fCn 



h 



h 3 



h 



h 



111, 



3/1 



n 2 



■in. 



2. The ordinary algebraical expression of the function is obtained from the 

 matrix representation by forming every possible term containing as a factor 

 one, and only one, element from each array, subject to the condition that 

 the element to be taken from any one array must be in the same row or 

 column with the element taken from the preceding array, and in the same 

 column or row with the element taken from the following array ; and then 

 connecting, by means of plus signs, the terms thus formed. 



VOL. XXXII. PART III. 4 F 



