Manchester Men, oirs, Vol. xlviii. (1904), No. 10- 



XIX. Tables of the Bessel Functions for pure 

 imaginary values of the arg-ument. 



By J. G. I SHERWOOD, B.Sc. 



{Communicated by Dr. C. H. Lees). 



Received and read April sdf/i, igo4. 



The complete integral of the equation 



d^y I dy 

 dx 



I dy / n'\ 



X dx \ x'r ~ 



may be written : — 



y = A/„{x)i-BK„{x) 



where n is an integer, A, B are constants, and IJx) and 



K^lpc) are the functions to which the Bessel Functions 



reduce when the argument becomes imaginary. /„ and 



Kn obey the sequence law 



K^ and K^ have already been calculated by W. S. Aldis, 

 M.A. (v. Proc. Roy. Soc, Vol. LXIV. p. 219), and it was 

 suggested to me by Dr. C. H. Lees that the values of 

 K^, K^, &c., correct to 5 figures would be useful for 

 Physical work. I give below the results for values of .r as 

 far as 5"0, and of n as far as 10 inclusive, obtained by 

 successive calculation by means of the sequence law. 



The corresponding values of /„, /, /,, &c., are for 

 convenience also reproduced from the tables at the end 

 of Gray and Matthews' Treatise on Bessel Functions. 



The values obtained for K^^ have been checked by 

 means of the relation 



I„^-i'K„ - I„K„j^.-i = -cosnir. 



X 



^n-\-l **■« -'9z-'*-«+l " 



{vide Gray and Matthews, p. 68.) 



If the values in the last two columns satisfy this 

 relation, it is obvious that every previous pair must do so. 



May 2^th, igo4. 



