Convection Currents in the Atmosphere. 451 



Now in the undisturbed state u = v = w = 0, 8 = 0. Hence 



1 3Po 1 



— G)*X = — 



po 0% 



2 1 <^0 



Po O- 



y 



(5) 



Hence 



Pi,=ipo°> 2 {x 2 + /)-9PoZ *- const. 



(6) 



Then putting p = p +p' r p — po + p\ and substituting for 

 p 03 we have, remembering that p', p\ and 8 are small 

 quantities, 



0& Po 0^ 



5* 





^' -***>■ 



(7) 



V i a 



Po Po 





; 



In the case we are at present considering, p varies only 

 on account of temperature variations ; so that if V is the 

 temperature and a the coefficient of volume expansion, 

 we have 



p' = -p aY and B = cftYlbt. ... (8) 



From equations (?) we find, by differentiating with 

 regard to #, y ; and z respectively, adding and substituting 

 for 6, 



Po* 



v 2 (y-jw 



=^'vf-^^(|-|). . « 



Transform to cylindrical co-ordinates r, c£ ; 0, and let the 

 corresponding velocities be (R, <I>, io). Then 



= R cos (j> — <& sine/), v = R sin <j> + <I> cos 0. . (10) 



