Mr. Ivory on ike Theory of the Astronomical Refractions. 389 



of the earth to the top of the atmosphere. For the sake of 

 perspicuity, we shall, in the first place, confine our attention 

 to an atmosphere of dry air, in which case it is known by 

 experiment that the refractive power (^ (g) is proportional to 

 the density p ; so that 



<l> (?) = K X p, 



K being a constant. Adverting to the mode of expression 

 before used, we have 



p — p c , 



and hence 



and by introducing new symbols in order to abridge expres- 

 sions, 



- Kg^ 



CO = 1 — c-", 



a sin d 

 1/ = — - 



Vl — 2aw 



Let this value of j/ be substituted in the differential of the 

 refraction; then 



d.dd = sin fl X 



doo 



3 » 



In further transforming this expression, it is to be observed 

 that a is a very small fraction less than -0003; and if the at- 

 mosphere extend fifty miles above the earth's surface, — or— 



a a 

 when greatest will not exceed -012. If we now put 



(T S 



we shall have 



