Theory of Stellar Scintillation. 135 



The origin of s is arbitrary, but we may conveniently take 

 it at the point (A) where the ray strikes the earth's surface. 



We will now consider also a second ray, of another colour, 

 deviating from the line 6 by the distance rj + 8rj, and corre- 

 sponding to a change of //. to fi + 8/jl. The distance between 

 the two rays at any point y is 



^-otJ '***+**«*# 0) 



In this equation 8/3 denotes the separation of the rays at A, 

 where y = 0, s = 0. And 8u denotes the angle between the 

 rays when outside the atmosphere. 



Equation (9) may be applied at once to Montigny's 

 problem, that is to determine the separation of two rays of 

 different colours, both coming from the same star, and both 

 arriving at the same point A. The first condition gives 

 8u = 0, and the second gives 8/3=0. Accordingly, 



*«£4J>* (10 > 



is the solution of the question. 



The integral in (10) may be otherwise expressed by means 

 of the principle that (//,— 1) and 8/jl are proportional to the 

 density. Thus, if I denote the " height of the homogeneous 

 atmosphere/' and h the elevation in such an atmosphere 

 determined by the condition that there shall be as much air 

 below it as actually exists below y, 



8/j,dy = 8p h, (11) 



8fi being the value of 8/j, at the surface of the earth. Equa- 

 tion (10) thus becomes 



^w 



COS^tf v ' 



At the limits of the atmosphere and beyond, h=l, and the 

 separation there is 



g Wgi0 



cos 2 6 v ' 



These results are applicable to all altitudes higher than 

 about 10°. 



The formulae given by Montigny (loc. cit.) are quite dif- 

 ferent from the above. That corresponding to (13) is 



87j = 8jju asiii0, (14) 



