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158                   HISTORY OF THE THEORY OF NUMBERS.               [CHAP, v
5(0, N) by use of
o,(a, A0=a>(0, N),          b,(a, tf) = &<(0, AO-aa^O, N).
Thus a&t-.-i a,--i&i=l, so that the area of OA^-.]. is 1/2 if the point J.> has the coordinates ai} &*. All points representing terms of the same rank in all the series of the same base lie at equally spaced intervals on a parallel to the o^axis, and the distance between adjacent points is the number of units between this parallel and the re-axis.
A. Hurwitz268 applied Farey series to the approximation of numbers by rational fractions and to the reduction of binary quadratic forms.
J. Hermes269 designated as numbers of Farey the numbers r^l, r2 = 2, T3=T4 = 3j r5 = 4, T6=r7 = 5, r8 = 4, . . . with the recursion formula
and connected with the representation of numbers to base 2. The ratios of the r's give the Farey fractions.
K. Th. Vahlen269a noted that the formation of the convergents to a fraction w by Farey's series coincides with the development of w into a continued fraction whose numerators are ==1, and made an application to the composition of linear fractional substitutions.
H. Made270 applied Hurwitz's method to numbers a+bi.
E. Busche271 applied geometrically the series of irreducible fractions of denominators ^a and numerators ^6, and noted that the properties of Farey series (a = Z>) hold [Glaisher261].
W. Sierpinski272 used consecutive fractions of Farey series of order m
of the theory of Farey series were given by E. Lucas,278 E. Cahen,274 Bachmann.275
An anonymous writer,276 starting with the irreducible fractions <1, arranged in order of magnitude, with the denominators fg 10, inserted the fractions with denominator 11 by listing the pairs of fractions 0/1, 1/10; 1/6, 1/5; 1/4, 2/7;..., the sum of whose denominators is 11, and noting that between the two of each pair lies a fraction with denominator 11 and numerator equal the sum of their numerators.
268Math. Annalen, 44, 1894, 4-17-436; 39, 1891, 279; 45, 1894, 85; Math. Papers of the Chicago Congress, 1896, 125. Cf. F. Klein, Ausgewahlte Kapitel der Zahlentheorie, I, 1896, 196-210. Cf. G. Humbert, Jour, de Math., (7), 2, 1916, 116-7.
269Math. Annalen, 45, 1894, 371.    Cf. L. von Schrutka, 71, 1912, 574, 583.
169<Mour. fur Math., 115, 1895, 221-233.
270Ueber Fareysche Doppelreihen, Diss. Giesaen, Darmstadt, 1903.
271Math. Annalen, 60, 1905, 288.
272BulL Inter. Acad. Sc. Cracovie, 1909, II, 725-7.