MR. IVORY ON THE THEORY OF ASTRONOMICAL REFRACTIONS. 193 



The second of these equations has already been noticed : the integral in the first 

 being extended to the top of the atmosphere, is equal to the weight of the column of 

 air above the initial height, every infinitesimal mass being urged by a gravitation 

 which is equal to unit at the earth's surface, and decreases in the inverse proportion 

 of the square of the distance from the earth's centre. By putting 



a 

 the same two equations will be thus written, viz. 



p=p'(l-q)c-\ 



The three quantities u, q, <r, are severally equal to zero at the earth's surface : and 

 the two values oi p will not be identical, unless the same three quantities can be ex- 

 pressed by functions of one variable, or, which is equivalent, unless two of them, as 

 q and ff, are each functions of the remaining one u. Now q being a function of u, 

 we shall have, 



dq ddq u^ d^ q z^ „ 



9 — dH'^'^ TIF ' T72 + dl^ ' 1.2.3 ^^-^ 



the differentials being valued when m = 0, that is, the particular 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. 



? =/« - (/-/) -T^ + (/- 2/ +/') -rr^g &c.: 



and as the two series must be identical, it follows that the quantities f,fif', &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 phenomena of the atmosphere 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-?)c-«, 

 we obtain, 



7 = c-''-/c-«R,-/^-^^'-/'.^i^^-&c (8.) 



MDCCCXXXVIII. 2 C 



