478 



THOMAS MUIR ON BIPARTITE FUNCTIONS. 



The truth of any individual case of this follows at once from the preceding 

 case by use of § 1 3. 



42. The existence of a notation for the elements of a determinantal product 

 and a knowledge of the properties of the elements facilitate very much the in- 

 vestigation of the laws of repeated determinantal multiplication. Some results 

 of an investigation of this kind are intended to form the subject of a future 

 paper : the matter is therefore not now entered on. Suffice it merely to 

 draw attention to the fact that, when using the notation of bipartites, the Jirst 

 element of a determinantal product is all that need be given. For example, the 

 last instance of § 41 is quite fully stated when we write 



a, b 2 c 3 \ • | a, /3, y 3 



ys*s 



h, k L I = 



"1 ^2 "8 



1 9 1 



K 



K 



h 



<*i A 7i 



x x 



x i 



x s 



a 2 j3 2 y 2 



h 



y-i 



y$ 



«3 & Ys 



h 



Z ; 



«s 



The element given on the right hand side is the element of the place (1, 1), and 

 the element of the place (r, s) is got by substituting for the 1st row a lt a 2 , a z 

 the r th row of the same determinant, and for the 1st column h u k lf l x the s tt 

 column of the same determinant. 



43. This relation between bipartites and determinants is of considerable 

 importance to the bipartite theory itself. Thus, we have seen (§ 23) that 



■nh x n^ 7\ 



where 



(*■■. €>Jk) t'o 



a i ft 7i 



\ \ b 3 



«2 A> y> 



C l C 2 C 8 



as A y 3 





1 Q -/'o 





y-i y-2 v% 





Z \ Z 2 *S 



Pr 



Q, 



x 1 x 2 x 3 



«i ft 7i " i 



a s B 2 y 2 & i 



«3 ft y s r i 



_ V\ !h y% 



o a ft y 2 ft, 



a 8 ft Vs r '\ 



m l 



n 



L*l 



Pi 



P 2 



P 3 



Qi 



Q 2 



Q 3 



Ri 



K 2 



Bs 



T> ,7 ' l ^2 ^3 



«, ft y x \a. 2 



«2 ft 72 h 



as ft 73 I c 2 



, &c. 



1 2 3 



<*i ft y, 



«2 ^2 72 



a 8 ft y s h 



But these bipartites are in order the elements of the determinant which is the 



