482 



THOMAS MUIR ON BIPARTITE FUNCTIONS. 



46. A continuant is expressible as a bipartite. 

 Thus 





5^.„ + i 



= 



V 



-1 



1 1 



2 \, 







p 1 



2 1 



r = W+P+*" 



— 



2> 



1 







1 







-1 



1 



1 









• 



-1 



r 



> 



p 1 



- , , = pqrs + rs 

 q 1 r 1 r * 



j ] +pq+ps 



= 1 p 

 -1 



1 



1 



• 



, i + 1 





. 



-1 



r 



1 









, 





-1 



s 



and more generally 



1> 1 



q b 



1 











9' 



1 





C 







S 



rf 





p 

 -1 



2 c 

 -1 r d 



. -1 .9 



This mode of expression seems more natural than the determinant form, a 

 continuant consisting essentially of positive terms, if the elements be positive. 



All the known theorems regarding continuants flow with the utmost 

 readiness from the properties of bipartites. 



