342 Sir Joseph Larmor 



piecing together a cognate analytical scheme on a symmetrical 

 fourfold basis which tries to make no difference between them. 



It is not improbable that these remarks merely turn over 

 ground that has already been explored by cultivators of hyper- 

 geometry. But it may be claimed that the interest of this range 

 of ideas extends far beyond the analytical technique, and that their 

 naive expression in a form of language outside its conventions may 

 prove to be helpful in other regions of speculation. 



The argument above has been based on the supposition that the 

 mathematical analysis must establish a complete correspondence, 

 element for element, between the activities in the new space-time 

 and in the Newtonian space and time. That however is not the case. 

 There is a gravitational correspondence into which radiation and 

 its rays do not enter. As regards the latter no conclusions could be 

 drawn at all, except in the special circumstances in which the 

 coordinate X/^ that stands nearest to time * does not enter explicitly 

 into the quadratic expression determining the space. If that is 

 postulated the equations of propagation of radiation have their 

 solutions periodic as regards x^^^ treated as a quasi-tim.Q, therefore 

 every beam of radiation carries with it a scale of X/^^ throughout 

 its course |. Moreover, if the spacial quadratic contained hx^ in a 

 product term, the velocities of the waves of radiation in forward 

 and backward directions would not be the same : their half difference 

 would thus be the local velocity of the frame of reference in that 

 direction. Where hx/^^ does not occur in the first power, the frame of 

 reference is thus fixed locally with respect to the waves of light 

 and their assumed underlying uniform fourfold extension with 

 regard to which they are propagated. 



Thus, under these postulated circumstances of x^ not occurring 

 explicitly in So-^, the mere fact that isotropic vibratory radiation 

 exists with its absolute velocity c is sufficient, not merely to de- 

 termine absolute measurements both in space and time, at every 

 locality in the extension, but also to determine the rate of motional 

 change of the coordinates as referred to the uniform space-time of 

 the radiation. It is gravitational correspondence, subject to this 

 general control of the whole range of space-time by observations 

 of light, with its isotropic and uniform qualities, that has led to 

 verifiable conclusions. Cf. letter in Nature, Jan. 22, 1920: also 

 Monthly Notices R. Astron. Soc. 



* That is the one coordinate the square of whose differential is affected in dcr- 

 with a negative sign, which marks it off from the others. 



t It is the alleged measurement of this abstract coordinate x^ by a travelling 

 clock, which connotes a physical system, that is a main source of confusion. 



