﻿deduced from the Electrical Theory of Matter. 81 



Again, i£ X 2 , Y 2 , Z 2 and X 2 ' , Y 2 ', Z 2 ' are the component 

 forces due to the electron a on the unit electron b in the fixed 

 and moving systems respectively, and i£ p u q u r x and p\. q L ', 

 ri are the corresponding component velocities of b in the two 

 systems, we see from equations (8) that 



(X,\ Y,', Z 2 ') = X' + 2l V-r//3', Y' + nV-^y, 



Using the values of pi, q x ', r/ in terms of p 1} q u r x as 

 given by (9) and the values, of X', Y', Z', a', fi', y' given 

 by (12) and (13), we find that it is only by neglect of quan- 



W / v 2 \ 

 tities of the order — ( ns) that we can ex press the force at 



a, point in the moving system in terms of the force at the 

 corresponding point in the fixed system, without explicitly 

 introducing the velocities of the electrons relative to the 

 system as a whole. We shall discuss this restriction on 

 page 85, but accepting it for the present, we find that at 

 corresponding points and times 



(X S ',Y,\ Z 3 ')=X + ?l7 -r 1 /9, «-W(Y + r,«-f> l7 ). 



e-V»('Z.+ Pl /8-j 1 «J; 



or (X/, Y 2 ', ZJ) = (X 2 , e-" 2 Y 2 , e-/ 2 Z 2 ). 



In view of the fact that if a set of instantaneous electronic 

 motions in the system at rest give rise to simultaneous 

 effects at a point, the corresponding motions give rise to 

 simultaneous effects at the corresponding point and time in 

 the moving system, we may extend what has been proved 

 above for the vectors due to the motion of a single electron, 

 to the total values of the vectors due to the motions of all 

 the electrons. Thus, if P, Q, R and P', Q' P' represent the 

 component forces on a unit electron in the fixed and moving 

 systems respectively, we have at corresponding points and 

 times , . , 



(P', Q', R') = (P, ,-WQ, e-" 2 R), 



which is, of course, equivalent to the statement that P', 

 e l2 Q', e l/2 R' must be the same functions of 



y 



and e l/2 t — e 1/2 



r./ 



u 



in the moving system, as in the fixed system P, Q, R are 

 Phil. 'Mag-'S. 6. Vol. 23. No. 133. Jan. 1912. G 



