4S8 Mr. Ivory on the astronomical refractions. 
increases, to m— 1 ; and, when m is infinitely great, the 
same equations become, 
P= c~ s 
1 — a = c,~ s 
I + 0T 
c being the base of the hyperbolic logarithms ; and they now 
belong to an atmosphere in which the density is proportional 
to the pressure, and the heat is the same in every part. These 
three suppositions, with some modifications of them, are the 
foundations of all the theories that have been advanced with 
regard to the variations of density in the atmosphere. They 
are the simplest cases that come under the foregoing formula;, 
and likewise those that are suggested by the most obvious 
physical hypotheses. But in reality these considerations afford 
no good ground of preference; since, whatever value we give 
to m, the general laws relating to the heat and pressure of the 
air, are equally well represented. The refractions near the 
zenith will likewise be the same, whatever number m stands 
for. We may therefore adopt that value of m which will 
give the true refractions near the horizon ; or that one, which 
will satisfy equation ( B ) , in which case the gradation of heat 
will coincide with that actually observed at the surface of the 
earth. More especially if, by the same value of m, we can 
conciliate both the above-mentioned conditions, we may con- 
clude that the solution of the problem must agree well with 
observation. But, in order to continue this research, it is 
necessary to find a method that will enable us to compute the 
refractions for any proposed value of m. 
If we make z = then 
m- J- 1 ’ 
