ON THE PRINCIPLE OF RELATIVITY. 241 



correct and R, J, P, M are expressible in terms of the position and motions 

 of charges, these should follow by mere application of the geometrical 

 transformations for space and velocity, and this has in fact been shown 

 to be so. 



Generalised Relativity. 



We may note at this point that we have not examined how far the 

 possibilities of this relativity extend. We have only investigated space- 

 time transformations of one special form. Are these the only possible 

 ones ? Let us take the fact that light is propagated equally in all directions 

 with velocity c, and examine what possible changes in the measures of 

 time and space would be allowable in order that this property might be 

 preserved. We are to have 



3 S = chT 

 a consequence of 



cs = col 



for all directions of £s through all points. 



We must, therefore, allow only of such transformations as give 



tV + hf + cz"- - c*-U- = f (x, y, z, t) (c>- + I if + cz 2 - c'T-'). 



Putting ict = u, icT = U, we must have 



aX 2 + gY 8 +SZ 2 +3tJ 2 _ 



independently of the ratios of &c : cy : vz: ?u, i.e., the transformation 

 would be a conformal one in space of four dimensions in which x, y, z, u 

 were co-ordinates. It has been proved that such transformations are 

 compounded of two kinds only, viz. : 



(i) Generalised rigid body motions, i.e., translations and rotations, 



or motions which leave all lengths unchanged, 

 (ii) Inversions in what we may call four dimension spheres. 



Of these a translation is merely a changing of origin, while a rotation 

 leads to the above transformations. 



The simplest transformation of the form (ii) is 



Va ley ¥z_ m 



^-r'-cH*' I -V_ ( ft9> 6- 1 z_ ( w> l- r 2_ c ^> 



and a transformation has been developed for the electromagnetic equations 

 which is at all points analogous to that above outlined. 



We have reason, therefore, to suppose that a uniform motion of trans- 

 lation is not the only one which would be concealed in an electromagnetic 

 field. The only reason that attention has been mostly confined to the 

 simpler case, is that nothing in Newtonian dynamics could suggest the 

 more complex one. 



The chief point at the present moment is that we have exhausted the 

 possible chances of complete relativity without arriving at any transfor- 

 mations corresponding to a motion of rotation or accelerated motion of a 

 1911. B 



