THOMAS MUIR ON BIPARTITE FUNCTIONS. 



46, 



8. Also, it may be remarked, the law of formation of the terms would give 

 the same result if the initial row of any bipartite were made into a column, and 

 at the same time all the other rows and columns altered accordingly. Thus 

 the bipartite of § 7 may also be written 



a c e 



b[df 



gh 



i 3 



or 



(J h 

 ace 

 b df 



if on any occasion there be convenience in so doing. 



9. If any two rows or two columns of a square array be interchanged, and, 

 at the same time, the two collinear rows or columns in one of the adjacent 

 arrays, the bipartite is in substance unaltered. 



Thus 







VI 



n 



9 



a 



b 



k 



I 



V 



c 



d 



9 



i 





e 



f 



h 



3 









k 



I 



P 



a 



b 



m 



n 



1 



c 



d 



9 



% 





e 



f 



h 



3 





10. A bipartite is multiplied by any quantity if each of the elements of any 

 one of its arrays be multiplied by that quantity. 



11. A bipartite having every element of one of its square arrays a sum of 

 p terms may be expressed as the sum of p bipartites, the first of which is got 

 from the original by deleting all the terms of each of the £>-termed elements 

 except the first term, the second by deleting all the terms of each of the 

 y/-termed elements except the second term, and so on. 



12. The cofactor of any one of the principal elements of a bipartite of 

 deg-order (»?., n) is expressible as a bipartite of deg-order (m — 1, ii), which is 

 obtained from the original bipartite by deleting, first, the line to which the said 

 principal element belongs, and then the elements of the adjacent square array 

 which are not collinear with the said principal element. 



