180 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



The first of the equations of motion now becomes 



du =ll-aT) GR> \- - c 2 ^ f p + k4u+26V. 

 at J x o® 



If the temperature of the atmosphere depended only on the altitude 

 above the earth's surface and were therefore ouly a function of r, then 

 would these equations be fulfilled by putting w, v, w respectively = ; 

 the atmosphere would then be in equilibrium. Therefore put 



T = To + T, 



wherein T is a function of r ouly, but T, is also a function of the longi- 

 tude and latitude ; therefore 



? l 



dx~~ W r' 



dr 



Finally one may put 



1 Jx ' d$ r 3x 



p =_p, . (1 + v). 



The quantity v in this latter equation expresses those changes of press- 

 ure that are caused by the phenomena of motion. Since v is small in 

 comparison with unity, therefore instead of log (1 + v) the quantity v 

 itself cau be substituted. By this means the first principal equation 

 becomes 



= GR> d - { 1 — °^1 + a f T : dr \ - c*^-^ -c^ + uAu + 2 

 dx\ r J r 2 S d% dx 



After transforming the two other principal equations in the same man- 

 ner we can put 



cHogpt = constant + GR 2 \ 1 °^ + a f~dr J 



(2) 



This equation gives the diminution of pressure at larger altitudes 

 above the earth's surface, and can for smaller differences of altitude 

 easily be transformed into the ordinary equation of barometric hyp- 

 sometry. 



