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



it follows that if A ~ be the nearest fractions to 7-, below and 

 q q V 



above, having denominators less than b, then 



a p _ 1 vL_ a _ 1 

 b q~~ bq q 1 b q'b 



which is the second part of the lemma. 



§ 3. The properties enunciated in § 1 are direct conse- 

 quences of the lemma ; for if the series of fractions whose 

 numerators and denominators do not exceed n be written down 

 in order of magnitude, and if the fractions with the denomi- 

 nator n + 1 be inserted in their proper places in the series, then, 



if the fraction T be inserted between 7^ and ~, we have, by 



w + 1 61 b 2 ' ' J 



the second part of the lemma, 



m % 1 a 2 m 1 



? 7- 



71 + I b x (n + l)b x b 2 7i+l b 2 (n + l)' 



for ~j 7^ are the nearest fractions, below and above, to T , 



b-i b 2 ; n + V 



1 n 

 having less denominators than tz + 1. Also T and =- 



71+1 71 + 1 



will appear at the beginning and end of the series, and 



1 1_ _1 _n n-1 1 



n Ti + 1 71(71 +1), Ti+1 n (n+'l)n 



Thus if the second property be true for all the fractions in the 

 original series, it still remains true after the fractions with 

 denominator n + 1 have been inserted. It can be at once veri- 

 fied that the property is true for ti = 2, 3, &c; and therefore it 

 is true generally. Thus the second property is proved ; and 

 the first property, which is a consequence of it, is therefore 

 proved also. For example, if n = 6, the series of fractions is 



11112132345 



6' 5' 4' 3' 5' 2 ? 5 ? 3' 4' 5' 6 ; 



1 



when the fractions with denominator 7 are introduced, = ap- 



2 1 13 2 1 



pears at the beginning, = between ^ and ~, = between -= and ~> 



- between ^ and -=. ■& between 75 and -., and = at the end. 



Since 7 is the largest denominator, the second property is true 

 (by the lemma) for all fractions in which this denominator is 

 involved; so that if the property is true for n = 6, it is true 

 for 7i = 7. 



