450 
Mr. Ivory on the astronomical refractions. 
evanescent at the top of the atmosphere, however great m is 
supposed to be. We likewise get 
T. (making s = o) = - + , )/ = f ; 
wherefore 
/ i m — 4 
4 771 — • I ' 
In this formula,/ = o, when m = 4, which is the case already 
considered ; and/= ~, when w is infinitely great. Between 
these two extreme cases, there are contained an infinite num- 
ber of atmospheres gradually extending higher above the 
earth's surface, till the total height from being about 25 miles 
becomes unlimited. In all these different atmospheres ^ has 
the same value when s = 0 ; and therefore they all agree 
with one another, and with nature, in having the same grada- 
tion of heat at the earth's surface. But the rate at which the 
heat decreases is different in every one ; being equable only 
when m = 4, and in all the rest becoming slower as the height 
increases. As all this is easily made out from the foregoing 
equations, it will not be necessary to enter into any detail on 
the subject. 
When m is less than 4, f becomes negative : but these 
cases are excluded, since they belong to atmospheres still 
less elevated than when m == 4. They are excluded too for 
another reason : for, although the rate of the decrease of 
heat at the earth's surface agrees with nature, yet it increases 
in ascending, which is contrary to experience. 
It remains to determine the refractions in the different at- 
mospheres included in the formula. As we already know the 
horizontal refraction in one extreme case, it will be sufficient 
to seek its amount in the other extreme case. Now, if we put 
