Trowbridge.] *^<^^ LNov. 3, 



in which R„ = 4.948 miles. These equations give 



gll' „ P_ Ac _ i+^t^ (22). 



(l+Ato'R, 







g 



Ro " Po A 



In Eq. (10) we must therefoi-e substitute for R the value R' 



"which gives 



Ro (1 + /to)Mog. ^ = - gz ri+_^"| (23;. 



A L r+z J 



Equation (23) is independent of K, which depends on the mass of the 

 planets' atmosphere, as it should be since we have neglected the attractive 

 influence of the atmosphere. This seems allowable since the mass of the 

 Earth's atmosphere is but a small fraction of the mass of the Earth. 



4. Now let to = in (23) and the resulting equation will apply to an at- 

 mosphere of 32° Fah. 



Rolog. -^ = -gz (24). 



To make some application of these formulas we shall take g = 1 and 

 ^ = 2/y, and since N log. 2 = 0.6931472 we find z^ — 3.4297 miles ; and 

 for /\^ = 4/> we have z^ = 6.8594 miles, and so on, A = 2°^' giving z„=nzi. 



Now let us suppose g = 1 and Atj = 1, then, A = ^p> ^i =^ '^ + — r— 



r+z 



_ z' + 2rz ^^ 7.1= — (r— 2zi)± Jr^ + 4z,-^. 

 — r + z 1 ^ 1/ ^ I 



"We must reject the negative value, and for the other we have 



2z^ 



z' = 2zi + r — very nearly, = 6.8649 miles. 



1 ^' 



In a similar manner we find z' = 13.7425, or z' = 2z' + 0.0127 miles. 



2 2 1 



We should also find z' = 3z', very nearly, for A = 3p • and so on. 



3 1 ' 



We thus see that if the Earth were 490° Fah. warmer than it now is 



/since it is found that A = ); or in other Avords, if its own heat 



\ 490 ' 



was such as to heat the air in contact with it to 490° more than 32° (sup- 

 posing it to be 32° now), and to vary according to the law which we have 

 supposed, the density of the atmosphere would be one-half what it is at the 

 surface at the hight of about 7 miles, instead of 3^ as at present ; and one- 

 fourth the density would be reached at about 14 miles, and so on. The 

 hight of the atmosphere under such conditions would be more than double 

 what it now is. If we suppose the density at the surface of the Earth to 

 be about a million times as great as at the surface of the atmosphere, or 

 A = 2'"*p, we shall find z = 139, nearly, under the conditions of high tem- 

 perature ; while in the other case It will be but 69 miles. 



For the temperature which we have supposed the Earth to have, it would 

 scarcely give out any light. To suppose the body self-luminous, it will be 

 necessary to make to about twice what we have supposed. We shall now 

 make an application of our formuhe to the planet Jupiter, and since that 



