Tables o/ber and bei and ker and kei Functions. 49 



The roughness of the approximation to the diurnal pressure 

 variation makes it useless to proceed further in the deter- 

 mination of the upward motion. Indeed the results for the 

 horizontal motion will be affected so much by convection 

 that it is doubtful if they will show any agreement with the 

 results of observations taken near the earth's surface. 



The phase of the motion changes with height in the same 

 way as that of the pressure variation. 



IV . Tables of the ber and bei and ker and kei Functions, with 

 further Formula for their Computation. By Hakold 

 G. Savidgtc*. 



SHORTLY after the publication of Dr. A. Russell's paper 

 on the ber and bei and allied functions (Phil. Mag. 

 April 1909), Professor Lees suggested that tables of the 

 ker and kei functions, as defined by Dr. Russell, and tables 

 of composite functions would be of great use to electricians 

 and others ; and the subjoined tables are the outcome of this 

 suggestion. Other suggestions have been given in the 

 course of the work by Professor Lees and Dr. Russell, for 

 which the author desires to take this opportunity of expressing 

 bis thanks. 



Definitions. 

 The functions tabulated may be defined as follows : — 



ber x + £ bei x = I (x s/ 1) | = J (ix \/ 1), 

 ber x — i bei x = J (xVl) =1 (lx^/i), 

 ker x + 1 kei x — K (<^ V0 = Y (w? \/t), 

 ker x — tkeix — Y (x V ' i) = K (cx\/' i), 



X(a') = ber 2 x + bei 2 <r, 

 Y(x) = hev' 2 x + bei' 2 x, 

 Z(x) — ber x ber' x + bei x bei' x, 

 W(a?) = ber x bei'<2? — bei x ber' x. 



* Communicated by the Physical Society : read November 12, 1909. 



f See Gray and Matthews, ' Bessel's Functions.' J and Y are the 

 Bessel functions of the first aud second kind respectively, and I and K 

 the same functions for unreal values of the argument. Dr. Russell's 

 function Y(.r) is here called V(x). her' x and bei' a; stand for the differ- 

 ential coefficients of ber x and bei x with respect to x. 



Phil. Mag. S. 6. Vol. 19. No. 109. Jan. 1910. E 



