Theory of Stellar Scintillation. 133 



" Draw normal planes to a ray at two consecutive points 

 of its path. Then the distance of their intersection from either 

 point will be p, the radius of curvature. But these normal 

 planes are tangential to the wave -front in its two con- 

 secutive positions. Hence it is easily shown by similar 

 triangles that a very short line dN drawn from either of the 

 points towards the centre of curvature is to the whole length 

 p y of which it forms part, as dv the difference of the velocities 

 of light at its two ends is to v the velocity at either end. 

 That is 



dN/p=—dv/v, 

 the negative sign being used because the velocity diminishes 

 in approaching the centre of curvature. But, since v varies 

 inversely as //,, we have 



—dv/v = dp,/p,. 



Hence the curvature 1/p is given by any of the four following 

 expressions : — 



1 _ 1 dv d log v _ 1 dfju d log pu n . 



p~ vdN~ ~d^ ~ Ji dN = ~M~' ' (l ' 



" The curvatures of different rays at the same point are 

 directly as the rates of increase of p, in travelling along their 

 respective normals." If denote the angle which the ray 

 makes with the direction of most rapid increase of index, 

 the curvatures will be directly as the values of sin 0. In fact, 

 if dfi/dr denote the rate at which p, increases in a direction 

 normal to the surfaces of equal index, we have 



dfi dp, . a 

 -™t — t- sin u, 



<iN dr ' 



and therefore 



4— =--psm0= — r^-smfl (2) 



~~p p, dr dr 



Everett shows how the well-known equation 



p,p= const (3) 



can be deduced from (2), p being the perpendicular upon the 

 ray from the centre of spherical surfaces of equal index. 

 In general, 



1 ldp . Q p 



p r dr 7 r 1 



and thus 



1 dp pd log p, 



r dr r dr ' 

 giving (3) on integration. 



