﻿66 Dr. W. F. G. Swann on the Effects of Uniform Motion 



point charge, and X , Y 0> Z is the pari of X, Y, Z which is 

 derivable from a potential <t> of the form 



the point charge being taken momentarily as origin and e 

 being written for I 1 — -^ J *. Thus 



, T Y 7 x «CV** ^C»e-^y *CW 2 c m 



If we have a unit charge moving in company with the 

 charge e, and with the same velocity, X , Y , Z of course 

 represents the force on it. 



Suppose we consider the force on our moving unit when 

 placed, not at the point x, y, r, but at the point e~ 1/2 x, y, z ; 

 its components X l5 Y u Z { are from (1) 



■ eC 2 x eWe-Vty eC 2 e~^z 2) 



K *' lJ l) ~ (x 2 + y 2 + z 2 fl 2 > {x 2 ~+y 2 +z 2 f 2 ' [x< -\ y 2 + z 2 fl 2 ' V ' 



Now consider any electron a of our system at rest. Let e v 

 be the strength of the electron. The component forces on it, 

 due to an electron b of strength e 2 , whose a?, y, and s com- 

 ponents differ by x l9 yi, z lJ are P, Q, R, where 



(V O -ox _ e&C-Xx e^e 2 Q 2 y x e J^^l__ 



(T,^ ^-(^+^2 + ^2)3/2' (tf + tf+zW (*i 1 +yi , - + ft , )^ r 



Since a is in equilibrium, the a? components due to the action 

 of all the electrons on it, balance out, and a similar remark 

 applies to the y and z components. If the system is set in 

 motion, however, the components will not balance out in this 

 way unless the system alters its dimensions, for the com- 

 ponents due to each electron are altered by the motion to an 

 extent which depends on the coordinates of that electron with 

 respect to a. Suppose, however, that the matter shrinks in 

 the x direction so that the x components of any pair of electrons 

 are reduced in the ratio e -1 ' 2 , then it follows from (2) that the 

 x component forces on a due to the electron b, for example, 

 are 



gifgC 2 ^ e&QU-W^ e 1 e 2 C 2 e-V 2 z 1 



W+tf + Zi 2 ) 31 " (tf+.tf +*!*)** U'i 2 -r-*/i 2 + £i 2 ) 3/2 ' 

 or 



P, €" 12 Q, e-VSR. 



* ' Recent Researches in Electricity and Magnetism,' p. 17. 



