Ray of Light in the Solar Gravitational Field. 583 



and then allowing the ends to vary, we get the additional 

 end condition 



av 



If a particular x r does not appear in F, the corresponding 

 Euler-Lagrangian equation gives at once ^- — 7 = const.; 



QX r 



and so, if the' other co-ordinates of the end-points are kept 



fixed, we find S# r | =0, provided the constant ^ — -, is not 



zero. This may be conveniently written Si dx r = 0. This 

 is the explanation of Eddington's statement * that a light-ray 

 '• is determined by stationary values of \dt instead of jefo." 

 In the special case of*the solar gravitational field 



F.(Sry= (ds) 2 = (l-2mlr)(dt) 2 -(l-2m/r)- 1 (dr) 2 



- T \d9y-r 2 sin 2 (d$)* 



r^F 

 does not involve t, and -^-7 = 2(1 — 2m/r)t r zfi>() las t is not 



ot 



constant along the light-ray ; so that 8wdt = 0, which is 



known as a Fermat or Huygens's Principle of Least Time. 



The space co-ordinates r, 0, cj> of the end-points remain fixed : 



it may be observed that since </> does not occur in F, we 



have equally well ojj dcf) = along the world line of a light 



pulse, the co-ordinates r, 6 of the end-points being fixed, as 



also the time of departure and arrival of the pulse. We 



shall not make any use of this Fermat Least-Time Principle, 



but think it worth while to point out its exact significance. 



§ 2. Determination of the Elliptic Integral giving 

 the Displacement. 



A first integral of the Euler-Lagrangian equations is 

 immediately found. F being homogeneous of degree 2 



* Report, p. 55. The writer does not see how this " end- condition " 

 can of itself replace the Euler-La^range differential equations. If, for 

 example, we are investigating- the shortest distance from a point to a 

 curve in the ordinary Euclidean plane, the Euler-Lagrangian equations 

 tell us that the curve must be linear (using Cartesian co-ordinates) , 

 whilst the end-condition says it must be at right angles to the original 

 curve : quite a different kind of statement altogether. 



