748 On the Einstein Spectral Line Effect. 



If the last term in (4) be transformed by means of (5), the 

 equation (4) becomes 



-Further, by the Principle of Equivalence, 



(£)"Mf)W ■•■->) 



along the ray. 



It follows from (6) and (7) that the constant k must be 

 infinite for a light pulse. In that case h is also infinite and 

 (5) can be written 



dt 



ay, ...... (8) 



where a is the ratio of the two infinite constants. 

 Elimination of t from (7) and (8) leads to 



du 2 „ _„ 



where w = r -1 , 



and, on differentiation, this equation becomes 



d 2 u 



j~+u==.-3mii 2 . ...... (9) 



Since m = l*47 km. and u cannot be greater than (697000) -1 , 

 the term on the right-hand side of (9) is small compared with u. 

 Hence (9) can be solved by approximate methods. 



If i£ = Acos</> is a first approximation, the second is 

 given by 



w = Acos</> + fmA 2 [l-J-cos20]. . . (10) 



The directions of the asymptotes are given by u = 0, or 



2mA 2 cos 2 $ - 2 A cos <f>- 4mA 2 = 0. 



If second powers of mA be neglected, the solution of this 

 equation is 



cos<£= —2mA, 



,£= + [!+ 2mA]. 



Hence the angle between the asymptotes of the light patli 

 is approximately 4?nA. It follows from (10) that A = R -1 , 

 where H is the distance of perihelion. 



