ON THE CALCULATION OF MATHEMATICAL TABLES. 97 



Pakt III. 

 Bessel Functions of Half-Integral Order. 



Some progress has been made with the calculation of the functions 

 S„(a;), C„(x), &c., tables of which for integral values of x from 1 to 10 

 appear in the Keport for 1914. 



The tables now presented continue the work for a;=l*l, . . . , 1'9. 

 It is hoped that the Tables for a;=0-l .... 0*9 will be presented in the 

 next Eeport. In addition the initial functions So(a;)=sinx, Co(a;)=cos x 

 have been calculated to 15 decimal places for x=0'l, 02, . . . , 10 in the 

 preceding table (Table III.). 



The functions S„(x), G„{x) are solutions of the differential equation 



rf/^ 



{,_«J±^},,=0, 



These have been calculated with S'„(a;), C'„(*). t^^eir derivatives with 

 respect to x, and also the important functions |E„(a5)P, \'^'nix)\^ where 



E„(x) =C„(a;) -iQ„{x) 

 E',Xx) = C'n{x)-i3'„{x). 



The logarithms of the functions tabulated are given for the whole range 

 of n — in the previous tables it was not possible to do this for all values 

 of n. As before |E„(x)|2 and \E'„{x)\^ are given until {S„(a;)} ^ and 

 {S'„(a^)} ^ become negligible. 



Several misprints occur in the tables published in the Report for 

 1914. The most serious are |E, 3(9)12 and 1E,4(9)|S which should be 

 respectively 334-4745 and 2189-467. The following are correct:— 

 S'6(3) = -0253461, C'o(9) = --4121185, C, (9) = --9456727. The 

 logarithms of the following functions should have negative charac- 

 teristics : — 



The functions Q„{x), C„(a;) are connected with Bessel Functions oi 

 Half- Integral Order as follows : — 



S„(x)=v/i^J«+i(a') 



C„(a;)=(-l)Vi'^J— i(*)- 



They are not really ' Bessel Functions of Half -Integral Order,' and it is 

 suggested that a more appropriate name for them is that of ' Riccati- 

 Bessel Functions.' 



1916 



