EEPOETS 



ON THE 



STATE OF SCIENCE, 



The Further Tabulation of Bessel Functions. — Report of the Committee, 

 consisting of Professor M. J. M. Hill (Chairman), Dr. J. "W. 

 Nicholson (Secretary), Professor Alfred Lodge, and Dr. L. N. Gr. 

 Filon. 



The Committee, having now practically completed their proposed calcula- 

 tions of the Bessel Functions of type J„ (x), are proceeding to a calculation 

 on similar lines of the functions I„ (x) and K K (x) for various values of 

 n and x. The former is the function of which, for the cases n=0 and 

 n=l, extensive tables were published by the British Association in 1889, 

 1893, and 1896, and is denned by 



The function K„ (x) satisfies the same differential equation as I„ (x) 

 and vanishes for an infinite argument. It may be conveniently defined 

 by the integral 



K » (a ° = for+ |)Q J d * sinh 2 " * e "' c ° 8h *• 



Writing 



*« -(=£>> 



where T„, t n are new functions, then it may be shown that T„ is the 

 function R hitherto used by the Committee, with its alternate signs 

 changed to negative. As the material for the calculation of B is still at 

 hand, that of T„ will thus present no difficulty. It is proposed to 

 tabulate T„ for all the available values and thence to deduce t n from the 

 tables for I„ (x), checking the results by a recurrence formula for t n valid 

 when n is half an odd. integer. The tabulation of K„ (x) will then 

 follow. 



