202 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



^ ^ V 1 + 2 <f> (/9) 



the quantity S & being supposed to increase from the surface 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 <p (f) is proportional to the density f ; so that 



^ (^) = K X f , 

 K being a constant. Adverting to the mode of expression before used, we have 



and hence 



/ 1 + 2 K p' 

 y = asin^Xv/]^:^K7fzi;: 



and by introducing new symbols in order to abridge expressions, 



^— 1 + 2Kp" 



a sin d 

 " -/l — 2 aco 



Let this value oiy be substituted in the differential of the refraction ; then 

 r2 = (« + %Y = a2 (l + ^)' = 



('-rf 



c^.§^ = sin^X iTZ^X 



v/7 



(■-t) 



2 a ctf . a 



In further transforming this expression, it is to be observed that a is a very small 

 fraction less than -0003 ; and if the atmosphere extend fifty miles above the earth's 



surface, — or — when greatest will not exceed '012. If we now put 

 we shall have 



the quantities rejected being plainly of no account relatively to those retained. Further, 

 because u is always less than 1, ^ _o^ is contained between a and« (1 -j- 2 a) ; and 



