•588 Deflexion of a Ray of Light in Solar Gravitational Field. 



here it will be observed that it is necessary to use the second 

 approximation for u i} since it is to be divided by u ± — u 2 — ..., 

 a small quantity of the first order. This may be written 



sin" 1 — pJ 1 — — J, and expanding sin/— + ej by the Taylor 



theorem, we see that the lower limit is -r r-. Using 



the well-known result that J sin 2 6 d6 = -J-(# — sin 6 cos 6), 

 we find 



D + 7T = i{l—ma)[0 + ±k 2 (0 -sin cos 0) . . .j* 7 ^ 



> 



where, to a first approximation, P— 4tmot. In the term 

 multiplied by k 2 it is sufficient to use for the lower limit 



the rough approximation j, and we have 



D + ,r = 4(l- m «)g+^ + m *(! + i)] 



= (1 — ?tta)[7T-f ???a(4 + 7r)] 

 = 7r + 4tmcL -{-. . . . 



Hence the deflexion D is 4??2«, where a is to a first 

 approximation the largest value u 2 of n : that is, the 

 reciprocal of a is the smallest value of the distance r from 

 the attracting centre to the light-ray. The closeness of the 

 approximation is easily seen by taking the second approxi- 

 mation u 2 — a + ma 2 , giving, on solving this quadratic for a, 



2ma = — 1 + (1 + ^mu 2 ^ = 2mu 2 — 2m 2 u 2 2 , 



whence 1 / m\ 



a = u 2 — mu</ = g-l 1 — j£ ) ; 



U = — being the sun's radius. 



u 2 ft 



Hence writing « = p gives us an accuracy which allows 



an error of about 1 in 5. 10 5 , since ??i = l'47 and R= 697,000. 



On substituting these values in D = -p- and converting radian 



measure into seconds of arc, we recover Einstein's prediction 

 of a deflexion of about 1"*73 for a ray which just grazes 

 the sun. 



Johns Hopkins University, Baltimore, 

 July 11th, 1921. 



