328 Patli of Ray of Light in Gravitation Held of Sun. 



If we write M 1 = l/r 1 , so that neglecting squares of ra, 

 uz=u 1 (l — mu-i) and substitute in (3), it is readily seen that 

 to the same degree of approximation the last term disappears 

 and we are left with the ordinary Newtonian equation giving 

 no motion of the perihelion. This is Prof. Anderson's 

 result. It would seem, however, that criticism might be 

 directed against the introduction of the co-ordinate 7\ into 

 the first problem rather than against its omission in the 

 second. It is, therefore, of interest to inquire what the 

 effect of the gravitational field would be upon the path of a 

 ray of light if we regard r as the actual measured radius 

 vector. 



If we write Q' = dO/dr, then tan % = ?'#' and (2) becomes 



f l + r 2 0' 2 |t 



and the time between two points is 



t=\ry- 1 {l + yr 2 ,2 }idr, 



where r, are given at both limits. 



By Huyghens' principle t is stationary for small variations 

 of the path. We have 



t^** '* 



{l + 7 r^}H Jdr \ (l + 7 r 2 <9')* ( 



dr. 



The first term taken between limits vanishes, and the 

 vanishing of the integral for 80 arbitrary gives 



r 4 0' 2 = c 2 (l + yr 2 /2 ), 



where c is a constant. 



Writing u=l/r, we have after differentiating 



§+« = 3>m< 2 (4) 



It is interesting to compare this with the differential 

 equation of planetary orbits. The first term on the right- 

 hand side of (3) represents the ordinary Newtonian attrac- 

 tion, and the second the small Einstein correction. Equation 

 (4) might then be interpreted by saying that a ray of light 

 is not subject to the Newtonian attraction, but that the 

 Einstein effect is the same as for a planet. The advance of 



