374 Astronomical and Nautical Collections. 



the value of tlie particular fluents of the several terms from consi- 

 derations nearly resembling those which Laplace has employed in 

 the Celestial Mechanics, so that the fluent of e~"df becomes a par- 



ticular case of his solution. Then taking y r= z*, he computes 

 the horizontal refraction 34' 1".3, instead of 33' 51".5, which is 

 the result commonly adopted, and he finds at all altitudes the for- 

 mula differs a few seconds only from the French tables. But this 

 equation supposes the whole height of the atmosphere to be no 



more than about 25 miles, since dy = — z* dz, and dx = ^ 



4 mz 



*; _3. _ 5 1 5 1 



== — —z *dz and/dj = z^ + — , and — is a little more 



4 m m VI 17 1 



than 5 miles. Mr. Ivory therefore inquires, what would be the effect of 

 an atmosphere in which y ^fz* + (l — f)z "^j/ being "~ 



471— 4 



1 113 



so that when — = 0, and n = co,f:^ — , andw" — z-+ — z, 

 n 4 4 4 



the height being thus made to vary from 25 miles to infinity. 

 But in all these cases he observes, that the rate at which the heat 

 decreases, becomes slower at greater heights than at smaller. 



" When n is less than 4, /becomes negative ; but these cases 

 are excluded, since they belong to atmospheres still less elevated 

 than when « = 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 con- 

 trary to experience." Among these excluded atmospheres is that 



33. \ 



which supposes 2/ := — z- — — z^ : and no doubt the first ob- 

 jection against its pneuraatical accuracy is valid ; and Mr. Ivory's 

 expression is more accurate at extreme heights ; but there is no 

 ground whatever from experience to deny that the rate of decre- 

 ment of temperature initially increases : the observations of Hum- 

 boldt, Leslie, and others, on mountains, have sufficiently shown 

 that the rate increases for the earth's surface, and it will be there- 

 fore difficult to show that it must be otherwise for the atmosphere ; 



