Chap. V] FaREY SeRIES. 157 
equal to the corresponding terms of /. Such a series has the properties 
noted by Farey and Cauchy for Farey series. 
E. Lucas^^^ considered series 1, 1 and 1, 2, 1, etc., formed as by Stern. 
For the nth series it is stated that the number of terms is 2"~-^ + l, their 
sum is 3""^ + l, the greatest two terms (of rank 2""^+l=±=2"~^) are 
(i+V5r+^-(i-\/5r+^ 
2"+V5 
Changing n to p, we obtain the value of certain other terms. 
J. W. L. Glaisher^^° gave some of the above facts on the history of Farey 
series. Glaisher^" treated the history more fully and proved (p. 328) that 
the properties noted by Farey and Cauchy hold also for the series of irre- 
ducible fractions of numerators ^ m and denominators ^ n. 
Edward Sang^^" proved that any fraction between A/ a and C/y is'of 
the form {'pA-[-qC)/{'pa-{-qy), where p and q are integers, and is irreducible 
if p, q are relatively prime. 
A. Minine^®^ considered the number S{a, N) of irreducible fractions a/h 
such that h-\-aa^N. Let 0(6)p denote the number of integers ^p which 
are prime to h. Then, for a > 0, 
>S(a,iV)= S0(6)p, P=L a J' 
since for each denominator h there are (/)(6)p integers prime to h for which 
h+aa-^N and hence that number of fractions. 
A. F. Pullich^^^ proved Farey's theorem by induction, using continued 
fractions. 
G. Airy^^^ gave the 3043 irreducible fractions with numerator and denom- 
inator ^ 100. 
J. J. Sylvester^^^ showed how to deduce the number of fractions in a 
Farey series by means of a functional equation. 
Sylvester,^^' ^^ Cesaro,^^ Vahlen,^^ Axer,^^^ and Lehmer^^^ investigated the 
number of fractions in a Farey series. 
Sylvester^^^" discussed the fractions x/y for which x<n^ y<n, x-\-y^n. 
M. d'Ocagne^^'^ prolonged Farey's series by adding 1/1 in the pth place, 
where p=<^(l)+ . . . +(j>{n). From the first p terms we obtain the next p 
by adding unity, then the next p by adding unity, etc. Consider a series 
S{a, N) of irreducible fractions Ui/hi in order of magnitude such that 
bi+atti^N, where a is any fixed integer called the characteristic. All 
the series S(_a, N) with a given base N may be derived from Farey's series 
««Bull. Soc. Math. France, 6, 1877-8, 118-9. ^oProc. Cambr. Phil. Soc, 3, 1878, 194. 
MiLondon Ed. Dub. Phil. Mag., (5), 7, 1879, 321-336. 
'"Trans. Roy. Soc. Edinburgh, 28, 1879, 287. 
"'Jour, de math. el6m. et spec, 1880, 278. Math. Soc. Moscow, 1880. 
'"Mathesis, 1, 1881, 161-3. '"Trans. Inst. Civil Engineers; cf. Phil. Mag., 1881, 175. 
'"Johns Hopkins Univ. Circulars, 2, 1883, 44-5, 143; Coll. Math. Papers, 3, 672-6, 687-8. 
266aAmer. Jour. Math., 5, 1882, 303-7, 327-330; Coll. Math. Papers, IV, 55-9, 78-81. 
'"Annales Soc. Sc. Bruxelles, 10, 1885-6, II, 90. Extract in Bull. Soc. Math. France, 14, 1885-6, 
93-7. 
