94 



By the well known analogy between the sum and difF. of 

 the sines of two arcs and the tangents of the i sum, and 

 T difF, the equat. (3) gives 



Tan. ^ R = i-( Mzzlj b (J) ta„. {J - -£=1 R) (« 

 orR=i-(,)tan.(*-(j|l,-i-(R) = 



From equation (5) we may obtain the same conclusions as 

 in art. 3. ' 



For if the surface of tlie earth were a plane, equation (5) 

 would become 



Q = {m — 1) tan. (^+ \ Q) nearly 



Also because R and Q are very nearly equal at all zenith 

 distances less than 80°. By equat. (4) 



R = (m— 1) tan. (^ + iQ— /Q). 



From this equation it readily appears that 



R= (m-l)tan. (^+i-Q) ^\IS— 



Therefore R = Q (^-i)^J°"-fl , as before in art. 3. 



^ a COS. * S 



^ ■-.'■. . 



^ ^The formula used by Bradley \% R=.k tan. (9 — nR). He determined n from 

 the comparison of the horizontal refraction, and the refraction at a given altitude. This 

 would be exact if the density of the atmosphere decreased uniformly. But k and 

 thence n may be determined by direct experiments on the refractive force of air, and 

 also by observations of circumpolar stars at zenith distances not greater than 80°. With 

 these values oik and n the refractions at the horizon and low altitudes may be computed, 

 and are not found to agree with observations, therefore the density of the atmosphere 

 does not decrease uniformly. 



