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XXXVI. On the Path of a Ray of Light in the Gravitation 

 Field of the Sun. By Gr. B. Jeffery, M.A., B.Sc, Fellow 

 of University College, London *. 



IN a paper published in the May number of the \ Philo- 

 sophical Magazine,' Prof. A. Anderson gives a modi- 

 fication of the usual theory of the motion of a planet under 

 Einstein's theory from which he concludes that there should 

 be no advance of the perihelion. 



The particular integral of Einstein's contracted tensor 

 -equations for a particle of mass m at rest at the origin of 

 polar co-ordinates gives for the line element in a plane 

 through the origin 



ds 2 =-ry-\ir*-r 2 d0 2 + ydt 2 , . . . . (1) 



where 7=1 — 2m/r. 



For the propagation of light we have ds = 0, which gives 



1 \di) tr U) =Y ' 



so that for a ray of light making an angle ^ with the radius 

 vector the velocity of propagation is v where 



r = 7 (cos 2 x + Ysin 2 x)-* (2) 



In order to avoid this dependence of the velocity of 

 propagation upon the direction ^, it is usual to write 

 r=r 1 +m and to interpret r x as the actual measured radius 

 vector f. Then, neglecting the square of m/r, the velocity 

 of propagation becomes l-r'2mjri, and is independent of the 

 direction. The problem thus becomes identical with that of 

 the determination of the path of a ray of light in a medium 

 of variable refractive index u = l + 2mjr A . 



This leads quite simply to the conclusion that a ray of 

 light passing at a distance B, from the Sun would be 

 deflected through an angle 4?n/R. 



Einstein's equations of motion for a planet give 



d 2 u m n „ 



d6 2+u= T* +Bmu > ( 3 ) 



where w = l/r and h is a constant. 



* Communicated by the Author. 



t The introduction of r s is also defended on the ground that to the 

 first order in m/r it reduces (1) to the form ds 2 = — y~ l {dr 1 2 ■}- r l 2 d8 2 )-\-ydi- L , 

 so that the scales of measurement along and perpeudicular to the radius 

 vector are the same. 



