On Expansion in BesseVs Functions. 547 



The two List equations reducing to those above for y = 

 and c = 0. 



But our hypothesis requires that the equation 



p + 7? 2 + £ 2 = 6 .2 T 2 



shall be a result o£ the equation 



x 2 + i/ 2 + z 2 = eH 2 . 



This is so if, and only if, c 1? c 2 , d x , d% are all zero. 



Thus we have arrived exactly at the transformation as given 

 above, and, as Einstein has shown (loc. cit.), in order that the 

 electromagnetic equations may be invariant under this trans- 

 formation the electric and magnetic vectors in the two systems 

 must be correlated in the manner done in this paper. 



Biicherer in the paper referred to does not take into account 

 this necessary modification of coordinates, and therefore when 

 in the latter part of it he evaluates the electromagnetic mass 

 of the electron on the assumption that it is spherical he is in 

 reality considering the Abraham electron, and so obtains 

 Abraham's express-ion for its mass. 



LTI. On Expansion in BesseVs Functions. 

 By Andrew Stephenson *. 



TN the ordinary Fourier expansion in sine series 



-*- f[x) = X A sin ax 



for the range o£. x from to c the determination of the 



coefficients depends upon the vanishing of the integral 



Jain a/cX sin a m xdx when Jc^rrL I have shown, however, 

 o 

 that the coefficients can also be found readily in the more 

 general case when 



[ sin u k x sin a m xdx=p sin a k c sin a TO c, 

 Jo 



where l^m and p is some constant f. Similarly the cosine 

 expulsion can be effected if 



j: 



cos x k x cos a m xdx = p cos a 7 .c cos a ni c. 



* Communicated by the Author 



t "An Extension of the Fourier method of Expansion in Sine Series," 

 Messenger of Mathematics, vol. xxxiii. pp. 70-77 (1903). A more general 

 discussion is given in a second paper in the same volume. 



