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



other property of the fractions arranged in order of magnitude 

 as above, viz. that the difference of any two consecutive frac- 

 tions is equal to the reciprocal of the product of their denomi- 

 nators; thus, for example, 



6 7~ r 6.7' 5 6~5.6'4 5~4.5' 



The first property follows at once from this : for if -^ =J> ~ 

 1 L J u / h h h 



be any three consecutive fractions of the series, so that 



a 2 % __ 1 « 3 a 2 1 



then 

 whence 

 therefore 

 viz. 



ajb 1 — a x b 2 = 1, a B b 2 — a 2 b z = 1 ; 

 a 2 bi — a x b 2 = a B b 2 — a 2 b % : 



a 2 _ % + a% 

 b 2 ~bi + b* 



which is the first property. 



In the next two sections I give an elementary demonstration 

 of these properties: §§4 and 5 contain an independent proof 

 of the first property ; § 6 contains an extension of the circum- 

 stances in which the properties are true ; and §§ 7-13 are de- 

 voted chiefly to their history. 



§ 2. Lemma. — If j be a proper fraction in its lowest terms, 



and if -> *-j be the nearest fractions, below and above, to r, 



q q 7 J b 7 



having denominators less than 5, then 



p+p'=za f q + q f = b; 



and also 



a p __ 1 p r at 



b q bq q' b q f b 



a 7) 



For let T be converted into a continued fraction, and let - be 

 b q 



the last convergent to ^-, and suppose that - < y, then 



a p _ 1 . 

 b q~~ bq 

 whence 



aq—hp = l. 



