Mr. Ivory on the astronomical refractions. 437 
we have been able to ascend. The mean elevation for one 
degree of depression of the centigrade thermometer is very 
nearly 90 English fathoms ; and in the great height ascended 
by Gay Lussac, rather more than 4^ miles, the same quan- 
tity comes out equal to 95 fathoms. To this great extent 
the law of a uniform decrease of temperature holds good, 
without much deviation from the truth. It therefore seems 
to be the assumption most likely to guide us aright in ap- 
proximating to the true constitution of the atmosphere. 
The law we have mentioned is expressed by this equation, 
viz. 
1 + s # 
1 •+ (3 t m + 1 ’ 
m -j- 1 being a constant quantity which, in the case of nature, 
will be determined by equation (B). Now if we substitute 
this value of 1 + f * in the formula (A), and then equate the 
I -j- p T v 
two values of P, we shall get, 
0 — ib) • (1 — «)=/— ds(i — u) ; 
and, hence, 
d o) ds m 
_! v • 
I — a m -\- 1 s ’ 
m + 1 
consequently, 
Thus we obtain, 
p=(i 
s 
m - f 1 
m+ 1 
I -h (3 T S 
; = 1 
I + gr m -\- 1 
In these equations, the hypothesis of Cassini corresponds to 
m =2= 0 ; that of a density decreasing uniformly as the altitude 
