Gravitation and Light 329 



is given that, passing to the general problem, the demands of the 

 universal gravitational correspondence (to be evolved immediately, 

 infra) require that the apparent space of the observers must be 

 constructed so that S/^ — c"^ht^ where c is a function of r shall be 

 invariant. This requires slight warping of the fourfold space, so 

 that the section in the plane r, t is curved away from its tangent 

 plane. But is the warped element of extension ^r' .c'ht thereby 

 altered only to the second order from its corresponding previous 

 normal value Sr.cS^? If that be so, the scale of t must be altered 

 in the inverse ratio to the scale of velocity c' or (what is the same 

 in another aspect) of time t : and in fact it is partly this secondary 

 change of scale of r that modifies the astronomical gravitation, as 

 will presently appear. 



The answer to this question might at first be imagined to be as 

 follows : any change in the element of surface may be made in two 

 stages, a stretching on the original plane and a displacement along 

 the direction normal to that tangent plane: it is only the former 

 that can produce a first-order effect: but this is only an apparent 

 change, a mere alteration of coordinates, because in it the curvature 

 of the plane is conserved, so it cannot affect the concatenation of 

 relations or events which alone counts : the latter does affect them, 

 e.g. disturb the law of gravitation, but only to the second order. 



But as will appear presently this relation of conservation of 

 extent is between coordinate systems that most closely correspond, 

 so is a real imposed condition which cannot be adjusted by 

 change to another set in the fiat. It is the expression of, or at any 

 rate is involved in, a restriction that in the containing fivefold 

 the distance between corresponding points on the two systems is 

 everywhere small, so that approximate methods can apply con- 

 sistently throughout, of which otherv/ise, in making continuations 

 in an uncharted extension, there would be no guarantee. 



3. Now let us survey this problem of transcending gravitation 

 from the other side, on which it originated. With Minkowski the 

 very incomplete relativity of electrodynamics, referring only to 

 uniform translatory convection, crystallised into the complete pro- 

 position that events occur in a uniform fourfold of mixed space 

 and time, determined by the consstitutive spacial equation 



Here c has nothing to do with the velocity of radiation : it is simply 

 the dimensional factor, prescribing a scale of measurement, that 

 is needed to make time homogeneous with length and may be 

 taken as unity. Gravitation remains outside this electrodynamic 

 scheme, being formulated in the different Newtonian reckonings of 

 space and time. Can it be forced in, either exactly or approxi- 

 mately? 



