Prof. Everett on the Optics of Mirage, 169 



a is about two miles, or when the focal length ira is about six 

 miles. 



VII. We now proceed to the consideration of the physical 

 circumstances on which the variation of index in the atmosphere 

 depends. 



The experiments of Biot and Arago have shown that, for varia- 

 tions of density due either to change of temperature or change 

 of pressure, fi — 1 varies directly as the density; and it further 

 appears from the experiments of Jamin, that at ordinary tempe- 

 ratures the value of /u,— 1 is sensibly the same for dry as for 

 saturated air at the same density. If a denote the coefficient of 

 expansion -00366 or ^i^i and h the pressure expressed in milli- 

 metres of mercury, the formula for jjl — 1 is 



, _ -0002943 h 

 * 1 + at '760 ; 



and this may also be regarded as the value of log/A, since the 

 difference j(/u<— l) 3 — &c. is too small to be appreciable. Hence, 



for horizontal or nearly horizontal rays, the curvature- or —^ 



p dy 



at any point is 



1 _ -0002943 f dh 1 dt ha \ 



p~ 760 L dy l + ut * dy (l + a*)«J ' 



But — -j-, being the fall of the barometric column per unit of 



ascent, is equal to p, where H denotes the height of the homo- 

 geneous atmosphere, which height, if we neglect variations of 

 gravity, is 26200 (1 + a/) feet. We have therefore, if we make 

 the foot the unit of measurement for y and p, 



= -0002943 . ~L • „. 1 - , g i^T^ + -^ ^f\ 

 + ut) 2 126200 273 dy J 



_„ h 



760 (1 + *tf L26200 ' 273 dy 



1 



rw{' +9 4;> 



"89000000 760 (l+at)* [_ dy 



This expression vanishes when -7-' = — ^jp« Hence, when the 



diminution of temperature per foot of ascent is -^ of a degree 

 Centigrade, the density of the air is uniform and rays are straight. 

 When the decrease of temperature upwards is more rapid than 

 this, the upper air is the denser, and rays are bent upwards, in 

 other words, their curvature is opposite to that of the earth. 



