292 THE MECHANICS OF THE EARTh's ATMOSPHERE. 



SO that 



2^72 ' \^'^) 



then the solution of (12) is 



11- = Ae""'^ -\- Be">2^ ^ (14) 



A and B denoting arbitrary constants in which the factor e"'f may be 

 supposed to be included. 



The case already considered corresponds to the neglect of g- in the 



radical of (13), so that 



(7 ±2 nai 

 ni='^ ^— 



2 tt2 



and 



we "" =Ae ^ "^-^Be ^ "^ (^"^^ 



A wave propagated upwards is thus 



kit 

 w=e cos H ( t j (IG) 



and there is nothing of the nature of reflection from the upper atmos- 

 phere. 

 A stationery wave would be of type 



is? 



w=e cosjitsin — (17) 



7C being supposed to vanish with z. According to (17), the energy of 

 vibration is the same in every wave length, not diminishing witti ele- 

 vation. The viscosity of the rarefied air in the upper regions wouhl 

 suffice to put a stop to such a motion, which can not therefore be taken 

 to represent anything that could actually happen. 



When 2 na <(/, the values of m from (13) are real, and are both posi- 

 tive. We will suppose that mi is greater than ^2. If ?r vanish with z, 

 we have from (14) as the expression of the stationar}' vibration 



e —e Ji (18) 



which shows that w is of one sign throughout. Again by (8) 



a^p=n sin nt )^ ^ > (19) 



( Wi ni2 ) 



Hence /*, proportional to ic, is of one sign throughout; p itself is 

 ciz 



negative for small values of z, and positive for large values, vanishing 



once when 



r'""'=^^' ........ (20) 



