a Theorem in Continuants. 



b a (b a -a) , ,, , 



- b ' + w+b^b 3 +^hz4 



* + «»-*.+ .. fl. ( fi._ a) 



365 

 /3F+«*i . (0) 



therefore also, as before, 

 b *+ 8 b\h l ^ 3 ~") 



Hence we have the curious theorem 

 b 2 — u 



PF + ubj 

 F-b 1 



(y) 



b n (b n —g) 

 0-b n + * 



1 + 



h{h—a.) 



b n - x {b n — a) 



•.+ 



1 ^ 2 — a 



= 1 + — -g— b 3 (b 3 —ct) 



/3 + 



/3 



/3 + 



*• , b n _ x (b n —ct) 



(D) 



Again, denoting the continued fractions in (/3) and (7) by/ 

 and/' respectively, we obtain from (C), 



(te +/ )(«? + /o=fi?- , ! 



whence 



£=V+^/+//' =0, 



and 



«— /3 a + /3 



= 2. 



/' / (E) 



It would be hard perhaps to find a better illustration than is 

 afforded by the foregoing of the great assistance which is de- 

 rived from continuants in the investigation of the subject of 

 continued fractions. Wallis's theorem was left by him un- 

 proved; and the demonstration of it, as Professor Bauer points 



