402 THOMAS MUIR ON BIPARTITE FUNCTIONS. 



For example, 

 a b 



c d 

 * f 



9 * 

 * 3 



k I 



and 



= acyk + acil + aehk + acjl + bdgk + bdil + b/hk + bfjl ; 



1 ==ae+bf+cg+dh, 



e f g h 



= (a b c d\c f g h) . 



3. If the number of elements in a row or column be n, the bipartite is said 

 to be of the n th order : if the number of arrays, square or not, be in, it is 

 evidently of the m** degree ; and combining these we may speak of such a 

 bipartite as being of the deg-order (m, n). 



4. The number of terms in the final expansion of a bipartite of deg-order 

 (m, n) is if' 1 . 



For the deg-order (2, n) the number is evidently n, i.e.,n l ~ x : for the deg- 

 order (3, ii) there must be one term, and one only, for every element in the 

 square array, and therefore in all n 2 terms, i.e., w 3_1 ; and if the number of 

 terms in a bipartite of deg-order (p, n) be n p ~ l , it is readily made evident that 

 the number in the bipartite of deg-order (p + 1, n) is n p : hence the statement 

 is established. 



.">. Each element of any one of the square arrays of a bipartite of deg-order 

 (m, it) occurs n m ~ 1 -^n 2 , i.e., n m ~ 3 times in the final expansion; and each element 

 of either of the other arrays occurs n" l ~ 1 +-n, i.e., n m ~' 2 times. 



For, one of the former, and only one, must occur in each term, and there are 

 a 1 of them ; and one of the latter, and only one, must occur in each term, and 

 there are 11 of them. 



6. The elements of the square arrays may therefore be called secondary 

 elements, and the others primary. 



7. The two lines of primary elements may be distinguished as initial and 

 final. Strictly speaking, however, either is at the beginning, and the other at 

 the end; for the definition shows that the order of writing the arrays may be 

 reversed without affecting the final expansion. Thus 





a b 



I k 





c d y i 



j h 



/ • 



e f h, j 



i .'/ 



d c 



k I b a 



