Gravitation and Light 335 



is the unchanging element of absolute time divided by c' Ic, or that 

 the scale of apparent time is variable with locality in the ratio cjc' : 

 also that the scale of apparent radial length is variable in the 

 ratio c jc: and therefore the scale of radial velocity is variable as 

 their quotient c^jc"^. How then with respect to the velocity of 

 rays of light whose absolute value is the same as the dimensional 

 constant c? Referred to these variable scales its apparent value 

 along any element of arc ought to be changed at the same rate 

 as any other velocity along that element of arc would be changed, 

 if rays are not to remain outside the correspondence between 

 hx, 8y, Sz, St representing time-space in the apparent gravitational 

 world and the same quantities, now elements of mere coordinates in 

 a difEerentially given world in a curved space-time which has 

 absorbed gravitation. This maintenance of correspondence is 

 secured if we determine the ray- velocity along any element of arc 

 by making Scr = : and the modified theory of radiation for the 

 apparent space of gravitation must be such as can accept this 

 value of the velocity of propagation *. The correspondence takes 

 over the same values of the coordinate differential elements. In 

 the apparent gravitational world they represent its space and time, 

 in the new world differentially specified, they belong to mere 

 coordinates: absolute elements of space and of time are there ex- 

 pressed by 8o-, but a relation of scales can be established from the 

 formula which expresses Sct^. 



The transformation which changes orbits into geodesies in the 

 difEerentially given space-time does not turn rays into rays: their 

 velocity is too great and moreover their minimum property is 

 relative to their locus 8a = 0. But if the ray is supposed to have 

 a constant underlying absolute period of pulsation and a constant 

 absolute wave-length (and therefore to be a straight line in an 

 auxiliary uniform fivefold) its apparent period in the gra^dtational 

 world must vary with locality as (c'/c)"^, also its apparent element 

 of length inversely as the scale of length pertaining to its direction 

 on that locality, and its apparent velocity as before specified. Its 

 apparent path in the gravitational world will correspond to the 

 true absolute path Sfda/X^ == 0, therefore will be given by 



Sjds/X = 0, 



complications being avoided as fortunately t is not involved ex- 

 plicitly in these equations. But at the same place the scales of 

 apparent 8s and apparent A would alter on the same ratio owing 

 to the presence of gravitation : therefore its influence is eliminated 

 in the quotient, and the path is not affected by the gravitation, 

 is the same whatever be its intensity. A ray passing near the Sun 

 ought not to be deflected on this view: an observed deflection, 



* On this and the following paragraphs, cf. however the end of the paper. 

 VOL. XIX. PART VI. 23 



