544 Mr. E. Cunningham on the 



Thus in both cases the corrected expression for the longi- 

 tudinal mass as derived from the energy gives the same 

 result as that obtained from the momentum, and no other 

 forces other than electromagnetic come into play. 



On the other hand, since from what has been said above 

 it appears that the electron will naturally retain the spherical 

 shape as measured by the variables associated with the moving- 

 axes, it appears that some extraneous forces would be required 

 to cause it to retain the spherical shape to an observer 

 remaining at rest. 



It is perhaps worth noting that " the principle of relativity " 

 propounded by Biicherer in the Phil. Mag. of April 1907 

 is in essence identical with the statement made in the 

 beginning of this paper. The principle referred to may be 

 stated thus : that in the sequence of electromagnetic pheno- 

 mena the giving of an additional uniform translational 

 velocity v to the whole system of electric and magnetic 

 bodies will not affect the phenomena observed if this velocity 

 v is at the same time given to the observer. The trans- 

 formation of space and time variables mentioned above shows 

 a means of explaining this dependence of the electromagnetic 

 phenomena on relative motion only; and conversely it is a 

 comparatively simple matter to show that it is the only 

 means. For it is required, among other things, to explain 

 how a light-wave travelling outwards in all directions with 

 velocity relative to an observer A, may at the same time 

 be travelling outwards in all directions with the same velocity 

 relative to an observer B moving relative to A with velocity v. 

 This can clearly not be done without some transformation of 

 the space and time variables of the two observers. 



Suppose two, observers A, B, to be situated momentarily in 

 the same spot> and let B be moving relatively to A with 

 velocity v (measured by A in his own system of space and 

 time). Let the direction of motion of B be A's axis of x, 

 and let the instant of coincidence be A's time £ = 0. 



Suppose axes of f rj £ to be B's system of coordinates 

 moving with him with velocity v relative to A's axes of xy z, 

 and coinciding with them at £ = 0. 



Associated with a given point at a given time as marked 

 by the values («£, y, z\ i) will be unique values of (f, rj, £, t), 

 t being Bs measure of the interval elapsed from the time of 

 his coincidence with A, and conversely. There must there- 

 fore be a linear transformation from the variables (x, y, z, t) 

 to (f, v, t, t). 



Consider now points on the axis of x (or f). Then the 



