Mr. Ivory on the Theory of the Astronomical 'Refractions. 1 1 



with sufficient exactness, so as to be able to determine q when 

 u is given. It is further to be observed, that the same formula 

 is general for all atmospheres, whether the air be entirely dry, 

 or mixed with aqueous vapour: for it has been investigated 

 from equations common to all atmospheres in equilibrium, 

 without any consideration of a particular state of the air. 

 By substituting the series for q in the equation 



^ = (l-y)c-S 



we obtain, 



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



■^ = A /— d(xc-% 

 we shall find 



f-d,rc-' = ^.J^^'' -/. .- K, -/' i^^^" - &c.}. 



Now, let this expression be differentiated ; then divided by 

 C-"; and, finally, integrated, attending to the nature of the 

 functions concerned; and the following result will be ob- 

 tained : 



0" = = "^{n—f. -r- — / —, — ^ — &c. 1(9.) 



, , z p \ '^ c-^.du ^ c-^.duc^ r ' 



a 



The equations (7.)j (8.)> (9') contain the theoretical ex- 

 planation of the properties of the atmosphere. What is said 

 may easily be proved by applying them to such phasnomena 

 as have been ascertained in a satisfactory manner. This ap- 

 plication is besides necessary for determining the numerical 

 values of the coefficients 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 ii. 



By performing the differentiations in the equation (9.), 

 there will be obtained, 



cr= ^{?^ + /(R, + R2)-/'(R2+2R3+R4)+&c.: 

 and, by expanding the functions, 



0- = -^' . { (1 +/) e.-(2/-/'). |-+ (2/-3/'+/")y^-&c. 

 Now, by reverting the series for <7, we get 



