506 



science: 



[N. S. Vol. XLI. No. 1057 



D. Kellogg; " Note on the potential and the 

 antipotential group of a given group," by G. 

 A. Miller ; " The equation of Picard-Fuchs 

 for an algebraic surface with arbitrary singu- 

 larities," by S. Lefschetz; Eeview of Man- 

 ning's Geometry of Four Dimensions, by J. L. 

 Coolidge; "Shorter Notices"; Schroder's 

 Entwicklung des mathematischen TJnterrichts 

 an den hoheren Madchenschulen Deutsehlands, 

 by E. B. Cowley; de Montessus and d'Ad- 

 hemar's Calcul numerique and Dickson's Ele- 

 mentary Theory of Equations, by E. D. Car- 

 michael; Smith's Teaching of Geometry and 

 Smith and Mikami's History of Japanese 

 Mathematics, by J. V. McKelvey; Study's Die 

 realistische Weltansicht und die Lehre vom 

 Eaume and Jordan and Fiedler's Contribution 

 a I'Etude des Courbes convexes fermees et de 

 eertaines Courbes qui s'y rattaehent, by Arnold 

 Emch; Mrs. GifPord's Natural Sines to Every 

 Second of Arc, and Eight Places of Decimals, 

 by D. E. Smith; Cobb's Applied Mathematics, 

 by E. B. Lytle; von Sanden's Praktische 

 Analysis and Hjelmslev's Darstellende Geo- 

 metric, by Virgil Snyder ; " Notes " ; and 

 "New Publications." 



The March number of the Bulletin con- 

 tains: Report of the twenty-first annual 

 meeting of the society, by F. N. Cole; 

 Report of the winter meeting of the society at 

 Chicago, by H. E. Slaught ; " The structure of 

 the ether," by Harry Bateman; "Shorter 

 Notices " : Killing and Hovestadt's Handbuoh 

 des mathematischen TJnterrichts, Band II, by 

 D. D. Leib; Cahen's Theorie des Nombres, 

 Tome premier, and Darboux's Theorie gen- 

 erale des Surfaces, premiere Partie, by T. H. 

 Gronwall ; " Notes " ; and " New Publica- 

 tions." 



SPECIAL ABTICLES 



INTERPOLATION AS A MEANS OF APPROXIMATION TO 

 THE GAMMA FDNCTION FOR HIGH VALUES OF n ^ 



Various approximations to the value of r(n) 

 when n is large have been suggested by differ- 

 ent workers and are in every-day use. In 



1 Papers from the Biological Laboratory of the 

 Maine Agricultural Experiment Station, No. 80. 



actual statistical practise the one which has 

 appealed to the writer as most satisfactory, 

 having regard to ease of calculation and degree 

 of accuracy of result, is that of rorsyth,^ 

 which is 



r{n + 



1)=V2^( ^"'+/ + ^ ) 



This is in error (in defect) in the proportion 

 of l/240n3. 



It lately occurred to me that possibly a 

 further saving of labor in computation, with- 

 out loss of accuracy, could be made by inter- 

 polating in a table of log \n to get log r(n). 

 Tables of the sums of the logarithms of the 

 natural numbers have recently been made 

 readily available to statistical workers from 

 different sources.^ Such tables all proceed, of 

 course, by integral steps of the argument n. 



The question then is to determine what the 

 order of magnitude of the error will be if one 

 interpolates from such a table proceeding by 

 integral steps, in order to determine T(n). 

 The relation 



r(«H-l)=|w (i) 



is exact when n is an integer. How great is 

 the inequality when n is not integral but 

 fairly large? 



To test this matter I asked Mr. John Rice 

 Miner, the staff computer of the laboratory, to 

 carry through the computations for a short 

 series of representative values of n. This he 

 has done, with the results set forth in Table I., 

 for which I am greatly obliged. It should be 

 said that in all the computations seven-place 

 logarithms only have been used. The first 

 column, headed " exact value," gives the result 

 obtained by using the value of log T(x) for 

 a; ^1.123 from Legendre's tables, and then 

 summing the logarithms up to n — 1 for each 

 desired value. This is the usual process, de- 

 pending on the relation 



2 Forsyth, Brit. Assoc. Eept. for 1883, p. 47. 



3 Cf. Pearl and McPheters, Amer. Nat., Vol. 

 XLV., 1911, p. 756. More recently a longer table 

 of sums of logarithms has been published in Pear- 

 son 's "Tables for Statisticians and Biometri- 

 cians, " Cambridge, 1914. 



