THOMAS MUIR ON BIPARTITE FUNCTIONS. 



475 



34. The product of two bipartites of the third degree may be expressed 

 as a bipartite of the third degree. 

 For example — 



a l 



a 2 



« 3 



h 



c i 



d x 



\ 



C 2 



d 2 



h 



h 



d 3 



x Ul " 2 



«3 _ 



K \ h 



«i A 9i 



«i /l /7i 



rf T rf 2 d 3 



e i & 7i 



/l A 72 

 <7l & 73 



$2 



s 3 



•l 



Xi 





«2 e l 

 tf„rA. 



«8«1 



aj a 2 a 3 



«1%1 «2%1 «3%1 



The law of formation of the product is the same for all orders. 



A 



7i 



Si 



«i 



a 2 



as 



& 



72 



s 2 



ai 



a 2 



«3 



& 73 S 8 



35. The product of a bipartite of deg-order (3, n) by the (w - 2) th power of 

 the determinant of its square array is expressible as a determinant of the 

 (w + l) th order. 



Thus denoting the determinant \b 1 c 2 d 3 e i \ by A, and its adjugate by 

 | B x C 2 D 3 E 4 1 , we have 



K C 2 d Z e i\ X 





% 



a 2 



a, 



a 4 



A 



\ 



c i 



<k 



h 



ffi 



h 



C 2 



d 2 



H 



K 



h 



H 



d 3 



«3 



*l 



h 



^ 



d± 



H 





«i 



a 2 



«3 



«4 



A 



hA 



CiA 



^A 



e l A 



9x 



b 2 A 



c 2 A 



d 2 A 



e. 2 A 



*i 



b,A 



c sA 



d z A 



e 3 A 



h 



KA 



c 4 A 



d A A 



e 4 A 





a l 



<l 2 



a% 



«4 



/i 



|C 2 D 3 E 4 | 



-|B 2 D 3 E 4 | 



1 B 2 C 3 E 4 1 



— 1 B 2 3 D 4 1 



0i 



-|CiD 8 E 4 | 



IB^EJ 



-|B X C 3 E 4 | 



IBA^I 



*1 



IC^EJ 



-IB.D.E, 



|BAE 4 I 



-IBAB*, 



* 



-ICAE3I 



IBjD^I 



-IBAE3! 



IB^D, 



A B, 



£1 B 2 



Ai B 3 



&i B 4 



tt 2 

 0, 



a 



B>i 

 D 9 



■a 4 



E x 

 E, 



C 3 D 3 E 3 



a d, e 4 



36. § 22 makes it evident that the foregoing theorem is quite generally true 

 — that is to say, is true when the bipartite is of deg-order (m, n) («/. being of 

 course greater than 3), and when the determinant taken is the determinant of 

 any one of the square arrays of the bipartite. 



37. A very much wider definition of a bipartite may be given than that 

 with which we started. The arrays lying between the initial and final lines, 



