MOVEMENTS OF ATMOSPHERE GULDBERG AND MOIIN 



193 



Considering the special case, where we have 



dw dv du dw dv du 

 dy dz dz dx' dx dy 



equations (1) are' reduced to a single one. Denoting the absolute 

 velocity by V, we have 



and 



V 2 = u 2 + v 2 + w 2 



d p = — V dV - gdz, 



and consequently 



P = Po +$p(V i ? - V 2 ) + g P (z -z) 



(3) 



(4) 



where p denotes the pressure for V = V and z — z . 



We designate the distance from the origin of coordinates by R 

 and its horizontal projection by r, the horizontal velocity by U , 

 and consequently we have 



x 2 + y 2 + z 2 = r 2 + z 2 = R 2 

 U 2 = u 2 + v 2 



First example. The trajectories are straight lines directed toward 

 a fixed center. 



We take the fixed center (see fig. 22) at the origin of coordinates 

 and put the equations of the trajectories under the form 



y 



y 



R 

 R n 



1 + 



3oAi 



R 3 



(5) 



Calculation shows that these equations satisfy the conditions which 

 we have introduced above. For the time t = o, we have R = R , 

 from which we conclude that all the particles of air are found at first 

 on the surface of a sphere whose radius is R . Differentiating x f 



