Ray of Light in the Solar Gravitational Field,, 581 



§ 1. The minimal Geodesies or Light-Rays of a Gravitational 

 Space : the Fermat-Hnygens Principle of Least Time, 



The metrical geometry of the Space-Time continuum 

 being characterized by 



(ds) 2 = g mn dx m dx R 



(in, n dummy or umbral symbols), 



it is postulated that along a ray of light (ds) 2 =0. Further, 

 by (ds) is understood the positive square root of the 

 expression for (ds) 2 , so that the arc-length integral I=\ds 

 has an extreme or minimum value (zero) when extended over 

 a light-ray. For this reason, these lines of zero length are 

 known as minimal geodesies. However, the usual method 

 of the Calculus of Variations for determining the differential 

 equations of the geodesies is not immediately applicable. In 

 this method we express the co-ordinates of a point in terms of 

 two independent parameters r and a : 



x r = X r (r, a) (r= 1 ... 4) . 



The value of a being assigned, we have on varying r 

 & particular curve C a , and when the integral 1(a) extended 

 over this curve, a = let us say, has an extreme value, 



SI /which is defined as =(^— ) dot\ is zero, — granting 



that 1(a) has a Taylor development near a = which requires 



the existence of the derivative {^—] • However, in the 



\(Wa=o 



particular problem confronting us, 



1(a) = ^Vgmnocj xj dr (m, n umbral), 



where x m = & m (r, a) and primes denote differentiations 

 with respect to t, and 



d 1 



a = ) (.2 ^(9>nn%J Xn)IV gmnXrrl X* \ fc 



is not defined at a = on account of the zero factor 

 (v^mn^'*n')a=o occurring in the denominator *. This 



* This is the reason for Eddington's remark (Report, p. 55) that " the 

 notion of a geodesic fails for motion with the speed of light." This 

 statement seems unfortunate, since the curves are uniformly referred to 

 by geometers as " minimal geodesies." 



