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CHAP, vj                                 FAKBY SERIES.                                       157
equal to the corresponding terms of /.   Such a series has the properties noted by Farey and Cauchy for Farey series.
E. Lucas269 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 2n~1+l, their sum is S^+l, the greatest two terms (of rank 2n~2+ld=2n~1) are
Changing n to p, we obtain the value of certain other terms.
J. W. L. Glaisher260 gave some of the above facts on the history of Farey series. Glaisher261 treated the history more fully and proved (p. 328) that the properties noted by Farey and Cauchy hold also for the series of irreducible fractions of numerators ^m and denominators ^n.
Edward Sang262 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. Minine263 considered the number /S(a, N) of irreducible fractions a/6 such that b+aa^N. Let <K&)P denote the number of integers ^p which are prune to 6. Then, fora >0,
N-a                         r-vr _ t-i
S(a, AT) =S <*>(&)„          p- \^~ \>
6«»1                                     L.     **    J
since for each denominator b there are <t>(b)p integers prime to b for which b+aa^N and hence that number of fractions.
A. F. Pullich264 proved Farey's theorem by induction, using continued fractions.
G. Airy266 gave the 3043 irreducible fractions with numerator and denominator ^ 100.
J. J. Sylvester266 showed how to deduce the number of fractions in a Farey series by means of a functional equation.
Sylvester,65'56 Cesaro,65 Vahlen,83 Axer,115 and Lehmer218 investigated the number of fractions in a Farey series.
Sylvester2660 discussed the fractions x/y for which x<n, y<n, x+y^n.
M. d'Ocagne267 prolonged Farey's series by adding 1/1 in the pth place, where p =</>(!)+ . - .+</>(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 a;/&; in order of magnitude such that bi+aa^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.                »°Proc. Carnbr. Phil. Soc., 3, 1878, 194.
""London Ed. Dub. Phil. Mag., (5), 7, 1879, 321-336.
""Trans. Roy. Soc. Edinburgh, 28, 1879, 287.
2MJour. de math. e*16m. et sp6c., 1880, 278.    Math. Soc. Moscow, 1880.
'"Mathesis, 1, 1881, 161-3.                M6Trans. Inst. Civil Engineers; cf. Phil. Mag., 1881, 175.
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