Advance of the Perihelion of a Planet.- 627 



and we get, remembering that i\ = r when m = 0, 



4r 1= :(^ + (r-2m)*) 2 , 

 which gives r = (2r ] +???) 2 /4?' 1 . 



Writing r for r 1} we now get 



velocity of light = (l - ™ ) ; (l + 0, 



or 



the index of refrac 



; tion^=(l+^,)7(l-^,). 



The p and r equation of the path of a ray of light is 

 consequently 



where a is the distance of the sun's centre from the apse, 

 and m is assumed to have the same meaning as before. 



-If squares of — are neglected, yu,= l-f- — , and the equation 



of the path is 



We may remark, though perhaps the assumption is very 

 violent, that if the mass of the sun were concentrated in a 

 sphere of diameter 1*47 kilometres, the index of refraction 

 near it would become infinitely great, and we should have a 

 very powerf a\ condensing lens, too powerful indeed, for the 

 light emitted by the sun itself would have no velocity at its 

 surface. Thus if, in accordance with the suggestion of 

 Helmholtz, the body of the sun should go on contracting, 

 there will come a time when it will be shrouded in darkness, 

 not because it has no light to emit, but because its gravita- 

 tional field will become impermeable to light. 



Taking the p and r equation of the path of a ray of light 



2c6 

 we find that the angle between its asymptotes is - — , where 



~ 2m(fc 2 — 2nia) — /c(a 2 K 2 -\-4:ma/c 2 — /c 4 )^ 



S1U * = f\ 2 , 2\ ■ > 



and k 2 — c" — hn 2 . If w r e neglect squares of m, this becomes 



.-. . _, 2m 4m 



'J. sin L or . 



a a 



