194 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



Further, if this value of ^ be substituted in the equation 



we shall find 



/_rf.c-» = ^-{c-'-/.c-»R,-/^^^^-&c.}. 



Now, let this expression be differentiated ; then divided by c~ " ; and, finally, inte- 

 grated, attending to the nature of the functions concerned ; and the following result 

 will be obtained : 





1+'- 



a 





The equations {7.), (8.), (9.) contain the theoretical explanation of the properties of 

 the atmosphere. What is said may easily be proved by applying them to such phe- 

 nomena as have been ascertained in a satisfactory manner. This application is be- 

 sides necessary for determining the numerical values of the coefiicients f, f, f\ &c., 

 which enter into the expression of the refraction. For this purpose it is requisite to 

 find the relations that subsist between the pressure, the temperature, and the height 

 above the earth's surface, by combining the equations so as to exterminate u. 



By performing the differentiations in the equation (9.), there will be obtained, 



(T = ^ {i. -|-/(Ri -I- R,) -/ (R2 + 2 R3 + R4) + &c.: 

 and, by expanding the functions, 



' = 7 • { (1 +/) « - (2/-/) 4 + (2/- 3/ +/')t:^ - &c. } . 



Now, by reverting the series for q, we get 



,, _ g , f-r f , ^r - ^ff + 3f' -ff" ^ , o_^ 



and, by substituting this value of u, the following formula will be obtained : 



a 



a = 



This equation between the perpendicular elevation %, and the difference of tempera- 

 ture 



contains the law according to which the heat decreases as the height above the 

 earth's surface increases. 

 Further, from the equation 



