and on Astronomical Refractions. 503 



Laplace assumes the relation between <o and x. 



u 



"7" 

 c 



or in the notation of this treatise 



I X 



if v 

 -fxc 



f and /' being arbitrary quantities, such that 



/= '4-9042 /' = -00074,1816. 



A table, similar to that which I have given in p. 4-98, showing the 

 constitution of the atmosphere, which Laplace has assumed, would 

 be instructive, and would enable us to judge of the admissibility 

 of the conditions attributed to the higher regions of the atmosphere 

 by that great philosopher. 



In this treatise I have obtained an expression* for the altitude 

 in terms of the pressure, founded upon the conditions of elastic va- 

 pours generally ; this gives the relation between u and co (see p. 4-71) 

 from which a relation between x and u must afterwards be sought. 

 When on the contrary the relation between co and x is assumed 

 (as was done by Laplace) an advantage may be gained in the cal- 

 culation of the refraction, at the expense, however, of a simple and 

 intelligible definition of the constitution of the atmosphere ; and 

 such a relation is of course also unconnected with any considera- 

 tions founded upon the nature of caloric. 



Mr. Ivory assumes the relation 



4JL L 



p> " 9 g' 9 g' 2 



p' denoting the pressure, and g' the density of the atmosphere at 

 the earth's surface. From this relation it follows that (see p. 473) 



7A(l + 

 a i u = i - 



9g 

 lc, a, 9' are L, & T', in Mr. Ivory's notation. 



, ' , **(! + 0Q /. 8\ 

 log -- H --- *~z - - 1 1 ~r J 



' 9 g \ 4' 



= -ailog(l- 



