Reciprocals.—Barlow's tables give the reciprocals to 7 places of numbers to to,000. The most useful table of reciprocals is Oake's Table of Reciprocals of Numbers from z to zoo,000 (Lon don, 1865), giving the reciprocals to 7 significant figures. First differences and proportional parts are shown, so that the reciprocals of all numbers to io,000,000 can be immediately written down to 7 significant figures. A similar table is Cotsworth's Direct Reci procals (M'Corquodaie, Leeds). A few other tables give re ciprocals, but for a very small range. Picarte, La Division reduite a une addition (Paris, 1860, in his table shows the reciprocals of '000(0 io,000 to io significant figures. The table actually gives the first nine multiples of these reciprocals to be used in a method for simplifying division; by this method the process of division is converted into a simple one of addition.
(Factorials).—The first table exhibiting the logarithms of the P-function was given by Gauss in 1813; he gives log iz to 20 decimal places for z =0.00(0.01)1.0o brz= r(z-1- I)]. The largest and best known table until recent times is that of Legendre, published in Exercises de Calcul Integral (Paris, 1817), and in Traite des fonctions elliptiques (Paris, 1825). This gives log I'z to 12 decimal places for z= t•000(o.00i)2.000 with first, second and third differences. This table was reproduced by Schlomilch in Analytische Studien (Leipzig, 1848), pp. 183 209. A 7-figure abridgement is given in Smithsonian Physical Tables (1920), and a 6-figure abridgement in Integral Calculus (1884) by Williamson. In Brit. Assoc. Report, 1916, pp.
Watson gives a small table showing
to io places for x=0.005(0.005)1.000. A 7-figure table also appears in Tables for Statisticians and Biometricians (Cambridge University Press, 1914). A photographic reprint of Legendre's table was published in 1921, Table of the Logarithms of the Complete I' function to 12 figures, Tracts for Computers, No. IV. (Cambridge University Press). In the Tracts for Computers series, No. VIII. (1922) gives Table of the Logarithms of the Complete r-function to zo decimal places for argument 2 to 1,200 beyond Legendre's Range (Argument z to 2) by E. S. Pearson, while No. IX. gives Log 1' (x) from x= 1 to 50.9 by intervals of o.or, by Brownlee. A very extensive table of the Incomplete Gamma-Function has been published by the Biometric Laboratory, University of Lon don, under the supervision of Professor Pearson.
is tabu lated by Degen in Tabularum Enneas (Copenhagen, 1824) to 18 decimal places for n=1(1) I 200.
sink x,
cosh x.—Gudermann, Theorie der potenzial- oder cyklisch hyperbolischen Functionen (Berlin, 1833), gave tables for the quadrant at intervals of o-oi of a grade to 7-places and also a 9-place table for x=2.500(0.001)5.000 and a io-place table for x= 5.00 (o.o i ) i 2.00. Ligowski, Tafeln der Hyperbolfunctionen (Berlin, 1890) supplies the gap x=0.000 to 2.000 using 5 places
and also x= 2.00 to 9.0o. Becker and Van Orstrand, Smithsonian Mathematical Tables (Washington, 1909) give 5-place logarithms for x=0.0000(0.000' )0.too(o.00i )3.00(0.006-0o.
Ligowski gives these functions to 6 places for x=0.00(0.01)8.00; Burrau (Berlin, 1907) to 5 places for x=o.00(o•oI)10.00; Dale, Five-figure Tables (London, Arnold, 1903), to 5 places for x=04)0(04302.0(0.06.0; and Becker and Van Orstrand for the same arguments and to the same accuracy as the logarithms.
The most extensive table is that of Glaisher, Comb. Phil. Trans. XIII., 1883, which exhibits the values to 10 places for (i.) x=o.000(o.00i)o.ioo, (ii.) 0.00(0.002.00, (iii.) o.o(o.i) io.o(i.o)5oo. Becker and Van Orstrand give 7-place values for x=o.000(o.00i)3.00(o.oi)6.00.
is given in an extensive table by Newman, Camb. Phil. Trans. XIII. (1883) to 18 places for x=o.000(o.00i )15.350; to 14 places for x=15.35o(o.002)17.300 (0.0°5)27.635. It is given by Becker and Van Orstrand for the same range and accuracy as log ex. In the Camb. Phil. Trans. (1883) Glaisher also gives ex to 9 figures for the same arguments as log ex. In the Tables of the Exponential Functions (Wash ington, 1913) Van Orstrand gives ex and Cs to 20 places for x = 0.0(0.032.0.
Tables of Some of the Higher Mathematical Functions.— The majority of such tables are of limited range and have gen erally been calculated for some special purpose. They appear in a few collections of tables to 4 or 5 places, but more usually in the journals of scientific societies to a larger number of places.
There are a considerable number of small tables of this integral, particularly in collections of tables for statisticians ; there is a certain amount of variation in the actual form of the integral tabulated. Burgess, On the Definite Integral --f
di (Edinburgh, 1898) gives a number of refer Air o ences to existing tables. His main table exhibits the 9-decimal values of the function for t=o.000 (0.0001.25o with first and sec 1
and differences and Air 0 to 9 decimals; also 2
to 15 places with first four differences for t= i.000(o.00r)i.5oo (o.002)3.000(o.1)5(o.5)6.o. When great accuracy is not required, the 4-place tables of Jahnke and Emde, Funktiontafeln (Leipzig, 1909) may be used. Useful tables are given in Tables for Statis ticians (Pearson). The first table of this integral is given in the form f
dt by Kramp in his Analyse des Refractions (Strass burg, 1798), to 8 places for t=o.00(o.oi)2.00 and to II places for t=2.00(o.oi)3.00. A similar table appears in De Morgan's Theory of Probabilities, where there is also a 7-place table for
dt for t=o-oo(o.o1)2.00. Markoff in Tables des valeurs Air 0 co de l'integrale f
dt (St. Petersburg, 1888), gives 11-place values with first four differences for x=o.00(o.oi)4.80.