134 Lord Bayleigh on the 



At a first application of (2) we may find by means of it a 

 first approximation to the law of atmospheric refraction, on 

 the supposition that the whole refraction is small and that the 

 curvature of the earth may be neglected. Under these limi- 

 tations in (2) may be treated as constant along the whole 

 path of the ray ; and if dyjr be the angle through which 

 the ray turns in describing the element of arc ds, we have 



dyfr= — -jM-sin#ds = tan0 d\og jjl. 



If we integrate this along the whole course of the ray 

 through the atmosphere, that is from //, = 1 to A fc= A t o^ we g e *> 

 as the whole refraction, 



Tjr=logft o tan0=(ju, o --1) tan 0, ... (4) 



for to the order of approximation in question log fi may be 

 identified with (yu- — 1). 



If S^jr denote the chromatic variation of i/r corresponding 

 to S/ub Q , we have from (4) 



8^=8^/(^-1) (5) 



According to Mascart * the value of the right-hand member 

 of (5) in the case of air and of the lines B and H is 



$/*o/(/*o-l) = -024 (6) 



We will now take a step further and calculate the linear 

 deviation of a ray from a straight course, still upon the 

 supposition that the whole refraction is small. If 97 denote 

 the linear deviation (reckoned perpendicularly) at any point 

 defined by the length s measured along the ray 0, we have 



^ 2 _ -tanff— _ ■ 



so that 



-5?= I tan d log /*= tan #(//,— l)+a, 



a being a constant of integration. A second integration now 

 gives 



(fi — l)ds + *8 + /3, .... (7) 



?7=tan0 I 



which determines the path of the ray. If y be the height of 

 any point above the surface of the earth, ds = dy sec 6 ; so 

 that (7) may also be written 



* Ererett's C.G.S. System of Units. 



