192 MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 



c-'^ .du^ 



-=Ri-2 + 2R,_, + R, 



c-n^t^ = — (R,. _ 3 + 3 Rj _ 2 + 3 Ri _ 1 + R,), 



&c. 

 3rdly. n being less than i, 



These things being premised, the temperature of an atmosphere in equilibrium will 

 have for its complete expression this formula, - 



L±ll_i /-R f, d.c--R, d\c-^R, 



the coefficients/,/',/", &c. being indeterminate constant quantities. A little atten- 

 tion will show that this expression is equivalent to a series of the powers of u ; for, 

 first, let the differential operations in the several terms be performed, which will 

 bring out 



\^ = 1 -/R, +/ (R^ + R3) -/' (R3 + 2 R4 + R5) + &c. i 



next, expand Rj, Rg, &c., and the result will be. 



-(/-2/+/') 



# 



1.2.3 



+ (/- 2/ + 3/' -/") j-^^^ 



— &c. 



The intention of assuming the formula (7.) is to express the temperature in terms of 

 such a form as will produce, in the refraction, independent parts that decrease rapidly. 

 In order to elucidate what is said, and more especially to prove that the analysis 

 here followed comprehends all atmospheres, whether of dry air or of air mixed with 

 aqueous vapour ; let p', f', r' denote, as before, the pressure, the density, and the 

 temperature, at the surface of the earth ; and put p, f, r for the like quantities at 

 the elevation z above the surface : the equations of equilibrium are these two, the 

 radius of the earth being represented by a, viz. 



/— dz.e 

 



(' -:r 



p 1 +/3t j_ 



p' — 1 +/3t' ' §'' 



