ON CALCULATION OF MATHEMATICAL TABLES. 263 



Calculation of Mathematical Tables. — Report of Committee 

 (Professor J. W. Nicholsox, Chairman; Dr. J. R. Airey, Secre- 

 tary; Mr. T. W. Chaundy, Professor L. N. G. Filon, Colonel 

 HippiSLEY, Professor E. W. Hobson, Mr. G. Kennedy, and 

 Professors Alfred Lodge, A. E. H. Love, H. M. Macdonald, 

 G. N. Watson, and A. G. Webster). 



Since the last report of the Committee in 1919, a number of tables of functions 

 have been computed, including Bessel-Clifford and Lommel-Weber functions of zero 

 and unit orders and Lommel-Weber and other related functions of equal order and 

 argument. 



Dr. Doodson contributes in Part I of this report a set of tables of the Riccati-Bessel 

 functions in continuation of the tables already published in the 1914 and 1916 reports, 

 M'here they are incorrectly described as Bessel functions of half integral order. 



Part II contains a table of the zeros of Bessel functions of high order to which 

 reference was made in the 1916 Report. 



It is proposed to defer the publication of the following tables which have been 

 calculated without the assistance of a grant from the Committee : — 



Sin and Cos 6 to fifteen places of decimals for 6 in circular measure from 1 to 

 100 radians. These were originally computed to twenty-four places of decimals. 



Bessel-ClifEord functions, Co(.x') and Ci(x) to six places of decimals for a;=0'00 to 

 20-00 by intervals of 0-02. 



Lommel-Weber functions, Qo{^) and Q.i{z) to six places of decimals for a;=0-00 to 

 16-00 by intervals of 0-02. 



Bessel, Neumann, Sohlafli and Lommel-Weber functions to six places of decimals 

 where the order and argument are equal or differ by unity, the values of the order 

 ranging from to 10 by intervals of 0-25. 



Part I. 

 Riccati-Bessel Functions. 



{X =0-1 to a; =0-9.) 



These functions are defined in the Reports of the British Association for the 

 Advancement of Science, 1914 and 1916 .which contain tables for x=l, 2 ... 10, 

 and a;=l-l, 1-2, . . . . 1-9, respectively. 



Note on a Method of Interpolation. 

 The Riccati-Bessel functions are subject to the following relations : — 

 E„(x) = C„{x)-v'^.S„{x) 



X 



E„'(x) = E„_,(.r)-"-E„(a;) 



X 

 X 



I . . I. 



From the first and second equations we may obtain respectively 



E«(X)=^' {E„(.)-(ji)E„.,(^)+2',(0>-.(-)-- • • . 



E„(X)=g{E,.W+(2^).E„_,(a:)+^,Qy':E„_,(x)+.... I • • - H, 

 whore X'^=x'^+m'^. 



