80 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



Assume that the fluid adheres to the earth's surface where z = 0, 

 therefore for this surface we have 



*>^ 



u = <Uo (2a 



At the upper boundary surface where z = h the fluid experieuces no 

 friction, therefore for that surface we have 



? = <Uh (26) 



jz 



Of the special integrals of the equation (2) that fulfill the boundary 

 condition (2a), namely: 



u = Ae~ nt sin (qx) 



W 2 



n = — <r 

 s 



the one that also fulfills the condition (2 b ) and is the most slowly dimin- 

 ishing is given by the value 



7t 

 9 = 2l 



Hence follows 



¥ 7T 2 



€ Ah 1 



The factor e~ nt becomes 1 at the time t=U: in order that this factor 

 may be equal to one-half we must have 



nt = nat. log. 2 = 0.69315. 



According to Maxwell's determinations (Theory of Heat, London 



ft 2 

 1871, p. 279, where j is expressed by v and ¥ by /*), we have 



¥ = 0.13417 [1 + 0.003660.1 . L centim ^ 

 L J second 



s 



where d c indicates the temperature centigrade. From this there re- 

 sults, for the temperature 0° O., 



t = 42747 years. 



If we distribute the same mass of air throughout a thicker stratum 

 with less density so that e . h, as also the A; 2 which is independent of e, 

 retains its value unchanged, then t must increase with h. Hence it 

 follows that in the upper thinner strata of the atmosphere the effect of 

 viscosity propagates itself through atmospheric strata of equal mass 

 more slowly than through the lower denser strata. 



On the other hand an increase of the absolute temperature 6 will 



cause the time t to diminish as - - . The lower temperature of the upper 



