Property of Vulgar Fractions. 325 



It is evident that the new fractions make their appearance 

 at the beginning and end, and between each pair of fractions 

 which are such that the sum of their denominators is equal to 7; 

 and, generally, in the series of fractions whose numerators 

 and denominators do not exceed n, the fractions with deno- 

 minator n + 1 make their appearance between e?ch pair of 

 fractions which are such that the sum of their denominators is 

 equal to n + 1. If, therefore, (f>(n) denote the number of 

 numbers less than n and prime to it (unity included), there 

 will be <£ (n + 1) — 2 such pairs of fractions ; and the corre- 

 sponding sums of numerators will be the {(/>(n + l) — 2} num- 

 bers which are less than n + 1 and prime to it, unity and n being 



n 

 excluded, as these correspond to .. and — -^, which appear 



at the beginning and end of the series. 



At the beginning of the series of fractions there will be 

 several with unit numerators ; if the greatest numerator or 

 denominator be uneven, = 2ti+1, then the series will be 



11 1 2 1 



2?i + l 2n , '"n + l 2n + l n'"' 



viz. the first fraction that has not a unit numerator will be 



2 11 



and this will appear between =- and -; so that there 



2n + l' rr n + 1 



will be n + 1 fractions with unit numerators at the beginning ; 



and, of course, since the second half of the series of fractions 



is complementary to the first half, the last n + 1 fractions will 



have numerators differing from their denominators by unity. 



If the greatest numerator or denominator be even, =2n, the 



2 

 first fraction whose numerator is not unity will be ~ = , and 



11 2,n ~ 



this will appear between - and T : so that in this case 



XJ - n 7i — l 



there will be at the beginning n + 1 fractions with unit deno- 

 minators, and at the end the same number of fractions with 

 numerators differing by unity from their denominators. 



§ 4. The first property may also be proved directly, by means 

 of the first part of the lemma, without the intervention of the 

 property relating to the difference of two consecutive fractions, 

 in the following manner. 



It is convenient to have a name for the two fractions nearest 



to 7, below and above, having denominators less than b ; and 

 these will therefore be referred to in this section as respectively 



