326 Mr. J. W. L. Glaisher on a 



the inferior and superior convergent of t, or as the conver- 



gents of r. If in the series of fractions the two next to % on 

 b o } 



each side of it, have denominators less than b, then by the 



lemma thev are the two convergents -» *--, of 7-? 



q q b 



and 



p+p' = a, q + q'~b. 



In the general case, let ■*-, -2- be respectively the inferior and 



superior convergents of j , and let -; — be the two fractions 

 which stand next to j-, below and above, in the series of frac- 

 tions, v and v r being supposed greater than b ; so that the 

 series of fractions is 



p u a u' p f 

 ^ 7 V v J q n 



By the first part of the lemma the convergents of - are ■= and 

 ^ : if v — b>b, the convergents or T are T and ^-: 



if v — 2b>b. the convergents of ^7 are T and ^-, &c. 



j & v — 2b b v — ob' 



Suppose that v — (m — 1)6 > 6, but that v—mb<b (i. e. suppose 

 that v>mb and < (m + ±)b) ; then we must have v—mb = q and 

 u—ma=p; for v—mb is the first denominator less than b 



which occurs in the descending series of fractions from 7 ; and 



this is in fact the definition of q. 



Similarly in the ascending series, we have q'=v r —nb, 

 p'=zu'— na if i/>nbm& <(n + l)b. Aho p+p f =a, q + q f =b, 

 and therefore 



(11 — ma) + (y! — na) =a, 



whence 



so that 



(v — mb) + (v f — nb) = b : 



u + u f = (m + n + l)a, 

 v + v f =(m + n + V)b ; 



u-ru r a 

 v + v f = b 



