454 
Mr. Ivory on the astronomical refractions. 
It appears, therefore, that in all the atmospheres compre- 
hended in the assumed formula, the density corresponding to 
a given pressure and temperature coincides very nearly with 
what is actually found by experiment. But although this 
be admitted, it may still be questioned, whether the height at 
which any proposed pressure takes place will agree equally 
well with observation. Now, in the real atmosphere, the 
height belonging to any pressure is usually deduced from the 
formula for barometrical measurements ; and it will be suf- 
ficient to show, that the same formula is true in all the atmo- 
spheres we are considering. 
In the first place, when m = 4, we have 
p =(>-i) ! 
I 4 s_ 
I +0t' 5 
From the first of these equations we get 
Log- T = 5 log. ~= s (1 + t • y)> 
I— T 
neglecting the cube and the higher powers of s : and hence, 
s = (i — i-y) 1°6’ T • 
But, from the other equation, we get 
1 + 
1 + gr s 
1 + /3 t ' ' " 5 
5 
and hence, 
X + 0 . 
T -j- t' 
1+0T “ Z 5 
Now substitute this value in the foregoing equation, and 
likewise, for 5, write the equivalent quantity ; and 
we shall obtain, 
