/' 



Mr. Ivory on the Theory of the Astronomical Refractions. 9 



f—Ridu— R,+i, 

 the integral being taken equal to zero, when ii = 0. 



2ndly. d.(r-R, __ , 



-^Ti,;^ --(Hi-i+K.), 



d d . c~" Ri „ ^ „ T. 



-^^-rf«^'=^-= + 2R._, + R, 



4^^'= - (Ri-3+3R,_2+3R,_. + R,), 



&c. 

 3rdly, n being less than /, 



These things being premised, the temperature of an atmo- 

 sphere in equilibrium will have for its complete expression 

 this formula, 



the coefficients yj /', /", &c. being indeterminate constant 

 quantities. A little attention 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 



i±|-^ = l-/R,+/(R2+R3)~/"(R3 + 2R4+R5) + &c.; 



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



-(/-2/'+/"). 



1.2.3.4 



1.2.3 

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



-&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 atmo- 

 spheres, whether of dry air or of air mixed with aqueous va- 

 pour ; lety, q\ r' denote, as before, the pressure, the density, 

 and the temperature, at the surface of the earth ; and put 

 Pi §i T for the like quantities at the elevation z above the sur- 



