AND THE OTHER PLANETS OF THE SOLAR SYSTEM. 639 



and if p = p Q at the Earth's surface, 



aMilf logA)+— Jf,o ">...} "C + gr*-; 



/o w, + 1 



.-. a 8 {itf„ log ^ + -J if, fa"' - p"') +...}- «T 



p m l ' ' r + z 



Now since p = at the upper limit of the atmosphere, we must also have by equation (4), 



p = for that limit. But the first side of the last equation would then become infinite, 



whereas no value of z will render the second side infinite ; and therefore the equation cannot 



be satisfied unless 



Jf,-0, 



and we then have 



*^w-$+ ")-**£-. (5) - 



If % 1 be the limiting value of z, we thus obtain 



*/*+!*> + ...l-«r-*- 



= gz l nearly ; 



1 g [ rn x ir J 



a finite quantity. 



We have also (since M = 0) 



1 + a£ = Mtf^ + il/ap'" 2 + &c. ... ; 



but this gives us for the determination of £ at the upper atmospheric limit (or Tjj of our 



previous notation) 



1 + ot 2 = 0, 



r 2 = - I = - 273" (C), 

 a 



the same as that determined by Poisson. 



In this method the conductive power of the atmosphere is tacitly assumed to be such as is 

 consistent with the upward transmission of a determinate quantity of heat, whether by con- 

 duction or limited radiation through the atmosphere at rest, the law of density being such as 

 is expressed by equation (5). The condition of the emission of the same quantity of heat 

 through the upper surface into the surrounding space is also assumed to be satisfied. 



Another method consists in combining, with equations (2) and (3), an equation which 

 expresses a relation between £ and z. Thus we may assume 



l + a£= c - Ci* + c^z* + ... + e t x*, 



and since £ = t when z = 0, 



1 + a t = c ; 

 Vol. IX. Paet IV. 82 



