l'.>18] on Gravitation and the Principle of Relativity 221) 



In four dimensions, the formula is generalized to — 



rW = dx- + dif + dz' - dt' (2)* 



Here, again, ds is a measured quantity (partly by scales and partly by 

 clocks), and the right-hand side refers to the inferred locations. The 

 units are chosen so that the velocity of light is unity. According to the 

 old theory of relativity the measured "distance," ds, between two events 

 is not affected by any uniform motion of the observer. 



If four new co-ordinates x^, a;^, x.^, x^, which are arbitrary functions 

 of X, y, z and f , are introduced and substituted in the right-hand side of 

 (2), we obtain an equation of the general form — 



ds' - ^11 dx-^- + goo dx.2- + . . . + ^g^., dx^ dx., + 2^13 dx^ dx.^ + . . . (3) 



where the ^'s are functions of the co-ordinates. Cubes and higher 

 powers of the infinitesimals can be neglected. 



In natural rectangular co-ordinates the path of a particle under no 

 forces is a straight line described with uniform velocity, or more briefly 

 a straight line in four dimensions. This may be expressed in a form 

 which is independent of the choice of co-ordinates, viz. fds is a minimum. 

 By substituting under the integral the value of ds from (3) and applying 

 the calculus of variations, we obtain the general equations of motion 

 under no forces applicable to any system of co-ordinates. The ^'s and 

 their derivatives will occur in these equations. 



In particular, by taking x^, x^, x.^, x^ to be rectangular axes rotating 

 with the earth, we should obtain the equations' of motion of a particle 

 under no "real" forces referred to those axes— in other words, the 

 equations of motion of a particle in a field of centrifugal force. The 

 centrifugal force enters into the equations through the intermediary of 

 the corresponding ^'s ; and we thus get the notion of a field of force as 

 defined by a set of values of the ^'s. Our hypothesis of the complete 

 equivalence of gravitation to forces like the centrifugal force arising 

 from a transformation of the axes of reference shows that we may also 

 define the gravitational field by a set of values of the ^'s. In the case 

 of the centrifugal force the values of the ^'s are such that by a trans- 

 formation of the co-ordinates we can transform (3) to (2). It does not 

 necessarily follow that this can be done when the ^'s have values 

 corresponding to a gravitational field ; and in fact we cannot do it for a 

 finite region of space, although, of course, in an infinitesimal element 

 gravitation may be made to disappear by an appropriate transformation. 



The ^"s defining the gravitational field may be regarded as the ten 

 components of a generalized gravitational potential. In fact, in rect- 

 angular co-ordinates one of them, g_^^, corresponds to twice the New- 

 tonian potential. t Newton's law is therefore expressed by Laplace's 

 equation — 



VV44 = (4) 



in free space. It is impossible to accept this as a general law satisfied 

 by the g's, because, for example, it is not satisfied when the ^'s repre- 



* This formula is usually given with the reversed sign. 



t It is an easy iUustration to work out the transformation of (2) to rotating 

 axes, when it will be found that g^^ is twice the potential of the corresponding 

 centrifugal force. 



Vol. XXII. (No. 112) r 



