638 Mr W. HOPKINS, ON THE EXTERNAL TEMPERATURE OF THE EARTH, 



Poisson has worked out the problem of which the fundamental equations are (l), (2), 

 and (3), assuming k to vary as — , which gives 



1 d? 



p 2 d% 



and has obtained the complete expressions for p, p, and £ in terms of z. The condition 

 p = then gives the height of the atmosphere and its density at its upper limit. The 

 author does not attach any practical value to this solution, but merely gives it as an 

 example of the manner in which the problem would have to be treated to obtain the 

 actual values of p, p, and £, if we had more perfect knowledge of the law by which the 

 atmosphere does actually transmit heat from the surface of the Earth. He obtains the 

 result that the density of the atmosphere at its upper limit would, under the conditions 

 assumed, be finite; and consequently that the condition p = at that limit could only 

 be satisfied by 



1 + a£=0, 



or r = - - = 273° (C.) 

 a 



It should be remarked that this very low temperature would not be that of surrounding space, 

 but that of the constituent particles of the atmosphere at its upper limit, where the air would 

 consequently lose all its elasticity, and be reduced to the state of a fluid. This value of £ 

 would be that which I have designated by t 2 , and it would follow, according to this solution, 

 that the temperature heretofore denoted by t must be less than 273° (C.) But the result could 

 give us no knowledge of what I have termed t u or consequently of t + t t the proper tempera- 

 ture of surrounding space. 



13. Another method of treating this problem consists in making use of an assumed 

 expression for 1 + a£ in terms of the density p, instead of equation (l) (Art. 12). Thus we 

 may assume 



1 + et£ = M + M ip m ' + M 2 p m * + &c. ; 



and substituting in (3), we have 



p = a 5 {M a p + M lP m * +1 + &c.} (4), 



and therefore 



-^=a 2 {ilf + (m 1 + l)J/ 1 n"'. + &C.J. 

 dp 



Equation (2) then becomes 



«« {*? + (», + 1) M lP *>-> + &c.l *£--g r^ ; 



[ p r )d% ° (r + zy 



.: o* \m log p + ^JtJ M.p\ + ...1 = C + gr* — ; 



