451 
Mr. Ivory on the astronomical refractions. 
u '= m z, then, supposing m infinitely great, we get 
i — «==(!— z) m = c - u i 
5= (•!—/) M + s/(i — ~ u ) : 
consequently, 
r= a ( 1 a) Sin. 0 x 
V i 
Cos* 0 + 2j’a.M + 2za.A(i— c / 
Applying to this expression the method already employed 
in No. 9, we shall get, in the case of the horizontal refraction. 
Now,/=i; x = 0.21909; k = 0.37455 : with these num- 
bers r comes out equal to 2059". 7 when all the terms of the 
series that are set down, are taken in except the last ; and to 
2057". 4, when all the terms are taken in: the more correct 
value of ris therefore 2058". 5, which is just 17". 2 more than 
in the other extreme case of m = 4. Thus the refractions 
undergo hardly any change in all the atmospheres compre- 
hended in the formula ; although their height increases from 
about 25 miles to be infinitely great ; and although the rate 
of the decrease of heat, which has always the same initial 
value, varies differently in each. 
Reflecting on what has just been proved, it is extremely 
probable that, for every value of m between the two extreme 
cases, the densities and pressures will be found, at least to a 
great height, very nearly the same as in the real atmosphere. 
We can hardly account, on any other supposition, for the 
0.(1 +g)A/7T 
ji— A (,) . k+ A (2) . k 2 — A ( %+A (4) .r— A (S V} 
