ON CALCULATION OF MATHEMATICAL TABLES. 251 



Calculation of Mathematical Tables— Report of Committee 



(Prof. J. W. NiCHOLSOX, Chairman ; Dr. J. R. Airey, Secretary ; 



Drs. L. J. CoMRiE and A. T. Doodson ; Prof. L. N. G. Filon, 



Drs. R. A. Fisher and J. Henderson ; Prof. E. W. Hobson ; 



Mr. J. 0. Irwin, Profs. A. Lodge, A. E. H. Love and H. M. 



Macdonald; Drs. A. J. Thompson and J. F. Tocher; Mr. T. 



Whitwell and Dr. J. Wishart). 



(Note. — The Committee of Section A reports that owing to the special circumstances 



of the South African Meeting the full report of the Committee on the Calculation 



of Mathematical Tables could not be received. The following consists of such 



portions of the report as are at present available for publication.) 



NEW TABLES IN COURSE OF CONSTRUCTION. 



The present Report contains Prof. A. Lodge's tables of Harmonic Series, 9 (x), 

 which extend over the range x from 00 to 60-3 by 0-1 intervals to sixteen 

 places and x from 5000 to 51-00 by 0-01 intervals to ten places with first and 



second differences. This function is equivalent to Y+ j log^ P(l + x), where 



y is Euler's constant, the value of which is given below to twenty decimal places. 



Some other tables have been completed and others are nearing completion, which 



will, in due course, be submitted to the Committee for consideration. These 



include Zonal Harmonics with first derivative, P„ (cos 0) and — P„ (cos 0), 



n from 1 to 20, from 0° to 90° by 5° intervals, to twelve places. 



Gaussian functions G"'((a), n and m from 11 to 20, [l from 0-00 to 1-00 by 0-05 

 intervals, eight or nine places. 



Besself unction product, I.^G^ argument x v~, real part Yr(x), imaginary partV„(a;), 

 X from 0-00 to 10-00 by 0-02 intervals and higher values of x, with the first 

 ten zeros of these functions, six places. 



Bessel functions and first and second derivatives, J^{x), ^ Jvi^), -g^j J. (x)> 



X from 1 to 20, and v from a;— 1 to a;+ 1 by 0- 1 intervals, six, five and four places. 

 Bessel functions of imaginary order, Jaii^)- 



THE TABULATION OF THE SUMMATION FUNCTION FOR 



HARMONIC SERIES. 

 By Prof. Alfred Lodge. 



in which 9(a;+l)=——+9(a;) 



(1) 



When X is an integer, (f{x)=l + -^+ „+ • • • +-• 



-a-i-H- ■ • • ) 



= 2(l-log,2) (2) 



For small values of x the following formula is available : 



9W=^+Aa-A3xHA,-r^ • • • -^-'^y'~'\^"+^i^{^) + ¥^^:^)-^ ' ' " J ' ^^^' 



