[ 360 ] 



XLIX. Extension of a Theorem in Continuants, with an im- 

 portant application. By Thomas Muir, M.A., F.R.S.E* 



IN the Philosophical Magazine for February there appeared, 

 with a demonstration, a solitary theorem in continuants, 

 which was thought worth placing on record on account of its 

 intrinsic neatness, and not because of any known application 

 to the subject on which continuants bear. Now, however, I. 

 am enabled not only to make a generalization of it, but to 

 apply it in demonstrating an extension of an important 

 theorem of Professor Bauer's regarding the product of two 

 continued fractions. 



Beginning with the continuant 



8 + 



hci 











-1 



r 



c^r 



b 2 c 2 























-1 





r 



-c 2 r 





hc 3 





















-1 





8 + 



r 



c z r 





b±c 4 





















-1 





8 + 



l ^-c A r 



i transform it first into 















r 



h 

























—0i 



r 



-Cir 



h 























—c 2 





8+*- 3 - 

 r 



c 2 r 





h 





















—Cs 





8 + 



r 



c B r 





h 





















—c 4 





84 



b ->-w 



Now, as before, increasing the elements of the first, second, . . . 

 rows by r times the corresponding elements of the second, 

 third, . . . rows, we obtain for the continuant an expression in 

 non-continuant form ; then, in this third form, diminishing the 

 elements of the second column by r times the corresponding 

 elements of the first column, diminishing the elements of the 

 third column by r times the corresponding elements of the 

 new second column, and so on, there results 



* Communicated by the Author. 



