220 REPORTS ON THE STATE OF SCIENCE, ETC. 



Calculation of Mathematical Tables- — Report of Committee (Prof. 

 J. W. Nicholson, Chairman ; Dr. J. E. Airey, Secretary ; Dr. D. 

 Wrinch-Nicholson, Mr. T. W. Chaundy, Dr. A. T. Doodson, 

 Prof. L. N. G. Filon, Dr. R. A. Fisher, Profs. E. W. Hobson, Alfred 

 Lodge, A. E. H. Love, and H. M. Macdonald). 



Reference was made in previous Reports to the desirability of publishing various 

 tables of functions. The tables in this Report include the Confluent Hypergeometric 

 Function, M (a . y . x), y=l, 2, 3, 4 and <x= — 4 to +4 by - 5 intervals and further 

 values of the function for Y == i4i i-l l the Exponential, Sine and Cosine Integrals, 

 considerably extending the tables calculated by Dr. Glaisher (Phil. Trans., 160, pp. 

 367-3S7, 1870) : Zeros of Bessel functions of small fractional order and the Ber, Bei 

 and other functions. 



For next year it is proposed to publish tables of 



(a) The Integral I (.r)= e~~* . dt and functions derived by repeated integration 

 of To(^), x from - to 7-0 by 0-1 intervals to ten decimal places. 



lb) The Derivatives of Bessel Functions, - J v (x) and J_,,(a;), where v = — L-Z 



Sv Sv 2 



x from O'O to 10'0 by - l intervals to six places of decimals. 



g 



(c ) The first derivative of the Zonal Harmonics, - P n (cos 0) for large values of 



oo 

 the order, to six places of decimals. A table of P„(cos0) to Pi O (cos0) has been 

 calculated by Prof. A. Lodge (Phil. Trans., 203 A, 1904). 



(d) The hyperbolic sines and cosines, Sinh tzx and Cosh t.x, x from 0-0 to 4-0 

 by 0-01 intervals to fifteen places of decimals. 



A list has been prepared of the tables which have appeared in the Reports of the 

 Committee. The functions tabulated include the Circular and Hyperbolic functions, 

 Gamma functions, the Exponential, Sine and Cosine Integrals, the Integrals of Fresnel, 

 Zonal Harmonics, Riccati-Bessel functions, Bessel and other functions with real, 

 imaginary and complex arguments, Lommel- Weber functions, and the Confluent 

 Hypergeometric function. In a few cases prefatory notes to the tables give the 

 properties of the functions and their applications to physical and engineering problems. 

 Some tables from other sources are also included in the list. Before publishing in 

 book form it will be necessary to rearrange the tables and remove a number of errors 

 which have been discovered. 



The Confluent Hypergeometric Function, M (a . y . x). 



In the construction of the tables for y=+£, y= + f, two calculations were made 

 to ten decimal places for each value of the arguments, M(— \ . £ .x) and M( — if . \ . x). 

 Since M(£ . J .x)=er, the three values could be checked by the recurrence formula, 



«M(a+l . y .z) = (a:+2oc-Y)M(a. y.:r) + (Y-a)M(a-l, y ■ x). 



The remaining values were obtained from the recurrence formula given in the intro- 

 ductory note to the tables published last year. 



Similarly, when a is a positive integer, M(l . \ . x) and M(l . ij .x) were calculated 

 for each value of x and the results checked by the formula 



-M(a+1 . y + 1 . a:)=M(a+l . y . *)-M(a . y . *). 



The tables are a continuation of those given in last year's Report. Differential 

 equations of the second order which can be solved in terms of the function M (a . y . x) 

 are also set out in the 1926 Report. 



