6 Dr. Harold Jeffreys on 



their third powers to be neglected in the present approxi- 

 mation. Then 



2(0 oy' 2(o ~d 



_? + 7 JLf|^ & ... (12) 



X 2(0 p J; QX 



to the first order, and substituting from these icto (11) we 

 find 



_ *w> 3/wb? si _ a? _sh_ _ jjtt vz , ar a 3 ? \ H 



- GrX- + P) A ft >*+ - ^fTIW 



\3a?% W/po\jz ox pooyJ () J z ox 6 



HBfVi 2 ?'^'; 



die 



re 



(13) 



1 f H j' H 

 ?=?+-h ft&cfo, .... (14) 



B 2 , O' 2 



and Vr=^ + 



^^ 2 "dy 



Now <7/Oof i s the mean of the excesses of the pressure 

 above normal at all points of a vertical column. Also by 

 hypothesis op/ot — 0, and therefore if V=gp £, P satisfies 

 the equation 



8 a .' Po 3P_ 3(V 1 »P,P) „ 



<,H & ~ 3(«,y) ■ • • • I 10 ' 



From this several interesting consequences can be deduced 

 at once. If for instance this mean pressure anomaly is- 

 constant over concentric circles, it is a function of the 

 distance from the common centre of these circles, and 

 therefore Vi 2 P is a function of P and the Jacobian vanishes. 

 Thus if a depression is perfectly symmetrical the pressure 

 cannot vary with the time, and therefore all pressure changes 

 must be caused by departures from circular symmetry. 

 Again, if the curves P = constant are symmetrical with re- 

 spect to two perpendicular axes, let us take these to be the axes 

 of x and y ; then P is an even function of x and y, and therefore 

 so is Vi 2 P ; hence the Jacobian is an even function of x 

 and y and the same must apply to the pressure changes. 



