4IO Leigh Page, 



where the primes denote differentiation with respect to r, and 

 similar expressions in y and s for Bnn and B^^. Also 



B,=v"+2'^ = o. (6i) 



All other components of B vanish. As (60) is true for all 

 values of .^% these four equations reduce to three independent 

 relations, to wit : 



r 



7' 



n , "' 

 V -\-2 - — O , 



from which it follows that A' = - v , and as A and v both vanish 

 at infinity, A = - r. 



Integrating the last of the three equations above, 



m 



where the constant of integration m, as will appear later, is the 

 mass of the attracting body (i. e. the sun) in astronomical units 

 divided by the square of the velocity of light. 

 The linear element, then, is given by 



;;/ tn 



2— —2 



ds' = e '' {dx'' + df + dz') + e '' dr (62) 



to the first order of approximation. 



{d) PHENOMENA IN A RADIAL FIELD. 



The equivalence hypothesis requires that the motion of a par- 

 ticle in a gravitational field shall be given by 



Zfds = o, 

 where ds may be expressed in polar coordinates by 



ds^ = h^ (dr^ + r'de^ + r^sm^6 d<f>^) + k^dl\ 



Hence, if 



H^yJ k^-\. /iV"2 + h-r-6- + hH^sin^ecj,'' (63) 



where • _ '^-^ h ^''^^ 1 ^4> 



''-dr ^ dr "^^lir 



it follows that 



