10 Mr. Ivory on ike Theory of the Astronomical Refractions. 



face : the equations of equilibrium are these two, the radius 

 of the earth being represented by a, viz. 



(-^y 



p> 1+/3t' * p' ' 



The second of these equations has already been noticed; the 

 integral in the first being extended to the top of the atmo- 

 sphere, is equal to the weight of the column of air above the 

 initial height, every infinitesimal mass being urged by a gra- 

 vitation which is equal to unit at the earth's surface, and de- 

 creases in the inverse proportion of the square of the distance 

 from the earth's centre. Qy putting 



1 + - ^ 

 a 



the same two equations will be thus written, viz. 



p z=f!f-d<TC-, 



p — -p' (1— (7) c"". 

 The three quantities u^ q, o-, are severally equal to zero at the 

 earth's surface : and the two values of ^^ will not be identical, 

 unless the same three quantities can be expressed by func- 

 tions of one variable, or, which is equivalent, unless two of 

 them, as q and cr, are each functions of the remaining one u. 

 Now q being a function of u, we shall have, 



_ dq ddq l^ d^ q u^ ^ 



'^ = 1^1 '''^ ~diF '1:2 '^"di/ 't:^:^'^''' 



the differentials being valued when m = 0, that is, the parti- 

 cular values which they have at the earth's surface being 

 taken. According to what was before shown, we have this 

 other series for q, viz. 



q => - (/-/') . ^ + (/-2/' +/'0 . ^3 &c. : 



and as the two series must be identical, it follows that the 

 quantitiosyjy, /", &c., will be known, if we can ascertain the 

 particular values assumed at the surface of the earth by the 

 differentials of q considered as varying with u, or with the 

 density. Thus the coefficients in the formula (7.) are not 

 hypothetical quantities, but such as have a real existence in 

 nature, and which might be determined experimentally, if we 

 had the means of observing the phaenomena of the atmosphere 



