THOMAS MUIR ON BIPARTITE FUNCTIONS. 



477 



40. A few examples will now be given of the occurrence of bipartite func- 

 tions in mathematical investigations. These will partially indicate the bearing 

 which the preceding theory has on cognate branches of analysis. 



41. The elements of the determinant which is the product of m determinants 

 of the 11 th order are bipartites of the degree-order (m, n). 



Thus 



"., «., 



«1 a i a s 



«1 «2 « 3 



b, \ b, | x fa fa fa 



' i 

 <h h c s I 7i 72 7 3 



<h 



fii 



7i 



a,. 



fa 



72 



\ 



b. 



h 



h 



h 



h 



a, 



ft 



7i 



a.. 



fa 



72 



( 'i 



<-'o 



h 



c j 



Ca 



( ':( 



and 



a x a 2 a 3 



h h h 



C-i Cn C(j 



a, a., a„ 



x ft & fa 

 7i 7s 7s 



'' 1 ''•.' "*3 



Z/i ^ ft 



•"2 *3 



«i A 7i x i 



«2 A 72 i ft 



«3 /^s Va ? i 



h h h 



«i ft 7i k 



«2 & y 2 ft 



a 3 # 3 73 1*1 



«J 



a a 



«S 



«1 



fa 



7i 



a 2 



fl 2 



72 



«3 



A 



7s 



6 1 & 2 ^3 



«1 A 7l !«2 



« 2 & y 2 2/2 



«S & 73 k 



f, Co (', 



aj jSj y, » 8 

 «2 & 72 ft 



"3 ^3 73 Z 3 



h h h 



a : fa yj 



«2 & 72 

 «3 ^3 73 





ft 



«i ft 7i 



•''i 



«i A 7i 



x 2 



ai fa 7i 



',■. 



a 2 /3 2 y 2 



V\ 



«2 & 72 



y-i 



«2 Ai 72 



ft 



«s A y 3 



*i 



aa & y s 



*■> 



«S A 7s 



*3 



and the determinant which is the equivalent of 



I a i \ c 3 I • K A> 7 8 1 • I x \ Hi z z 



<*>\ 



u 2 



a n 



a 3 



fa 



73 



W 



\ 



h 



a s 



fa 



7b 



c i 



H 



C 3 



«1 & 7 X ' «2 fa 72 ' 



"1 ^2 ^3 



has for its first element 



Gj a 2 ft s I h x k x l x 



«i A yi j «i ar, x s 



<*2 A y 2 ' ft ft 2/3 



«3 A ys *1 *-> *S • 



A 73 



VOL. XXXII. PART III. 



4h 



