ON THE PRINCIPLE OF RELATIVITY. 230 



and «' the acceleration of the same point wfcen referred to %' y' z' V, wc 



have 



a', = «,.//*> a/ = « ; ,//3 2 , «/ = ajp. 



3. 

 To discuss, however, only the equations of free space will lead us to 

 no tangible result. It is with material phenomena that we are concerned. 

 The propagation of light is only observed by means of phenomena occurring 

 when light meets matter. The theory of matter which at present holds 

 the field is the electron theory. We pass therefore at once to the extension 

 of the above theorem to the equations of that theory in the most commonly 

 adopted form — viz., that of Lorentz. 



c \dt X ' ) cdt (6) 



div e = p ; div b = ; 



where p is the density of electricity and w the velocity with which it is 

 moving relative to the frame of reference with respect to which these 

 equations are assumed to hold. 



We have already seen the transformation applicable to velocity. The 

 only remaining quantity is i>, and we find that if we add to (i) and (ii) the 

 further equation 



R = /? P (l-^'j, . . . (Hi) 



then the equations (b) above transform into identical equations connect- 

 ing XYZTE HP W. 

 *" The relation between corresponding elements of volume cS and cs is 



tS = Ss//3 (1 - vw. c ), 



so that RtS = pcs, 



i.e., corresponding elements of change are equal. 



But we cannot at once proceed to argue as above that all effects 

 of a uniform translation will be obscured, for the reason that equations 

 (6) are not complete, as the scheme for the free aether is. It is not 

 sufficient to add, after Lorentz, the statement that the moving force is 

 e', because that requires some assumed relation between force and accelera- 

 tion — i.e., some assumption about mass or intrinsic inertia of electricity. 

 This extra equation of Lorentz belongs really to the stage of the derivation 

 of the mechanical equations from the electromagnetic theory. That 

 theory is in the above form incomplete. 



A possible way out of this difficulty is to adopt the modification of the 

 theory employed by Larmor, in which electricity is conceived as consist- 

 ing of so-called isolated point singularities in the aether, i.e., points at 

 which E and B become infinite in a certain manner, and in which the 

 original equations are assumed to be the whole scheme — holding at al 

 points excepting those singular points — and to be sufficient to determine 

 the motion of those points. The mathematical discussion of this aspect 

 is, however, incomplete. 



