4 Dr. Harold Jeffreys on 



Suppose that the speed of translation of the cyclone is 

 small, o£ the same order of magnitude at least as that of the 

 winds themselves. Then "dfdt is of the order udfdx and the 

 only first order terms in the equations of motion are those 

 in the equations 



P *"\> (6) 



p^y J 



Thus the geostrophic relation still holds as a first approxi- 

 mation. When, however, the second powers of the velocities 

 are negligible, no disturbance can travel. For the rate of 

 increase of the surface pressure at any point is given by 



1° -f 'Ss*- -*f {I o») + 1 ™ + M* (7) 



by the equation of continuity. 



Now pw is zero when 2 = and when z is infinite, since 

 there is no vertical velocity on the ground and the density 

 tends to zero at a great height. Hence 



%—>£ {h<*> + %™}*- ■ ■ (8) 



Substituting in this from the equations (6) we have 

 identically 



|r= w 



Thus to this order the pressure distribution is not varying. 

 Hence if it is changing at all it must depend on powers of 

 the pressure gradient higher than the first. The assumption 

 that this kind of approximation is possible is therefore so 

 far justified. 



A second approximation may now be obtained by substi- 

 tuting the values of u and v given by (6) into the terms in 

 (1) depending on the squares of the velocities, and again 

 determining u and v as far as the second powers of the 

 pressure gradients. As "dpfdt is of higher order than udpfda r 

 it follows that "duj^t and "dvfdt are of higher order than 

 udufdx ; thus they contain no terms of the second order 

 and therefore must be neglected. The vertical velocity also 

 must be zero unless the distribution is changing; hence to 



