﻿relative to a Rotating Earth. 55 



beginning and ending at the point A, we obtain 



D C C 



jj-l (uds+vdy+wdz)= — \ (d l dx + e 2 dy+6 z dz), (10) 



v s * s 



where the suffix S indicates that the integration is to be 

 taken along a definite curve S. We have assumed in 

 obtaining (10) from (9) that V is a single valued function 

 of («r, i/, z), and that p is a f miction of p. It now appears 

 that the rate of change of the relative circulation in any 

 closed circuit which consists of the same fluid particles at 

 all times is not zero unless, in addition to the above con- 

 ditions, we have 



B01_d03 ^2_Bfl ^3_^2 nn 



When these conditions are not fulfilled, the relative vorticity 

 does not move with the fluid itself, and if a velocity potential 

 exists for a certain portion of fluid at a given instant, a 

 velocity potential will not exist for that portion of fluid at 

 a later instant. 



The first case of importance of the above conditions in 

 relation to problems relating to the atmosphere is that 

 in which the angular velocities co x , o> y , co z of the axes are 

 constants. In this case, the conditions given above take the 

 form 



1 * 1 o 1 £ "bit "dv "dw n9 N 



(D r Wy co, o<v oy 0% 



where 8 represents an operator defined by 



'-(•4, + +fr-h)- • • • (13) 



These equations have a solution of the form 



u v ic 



,-^= «=/(•*+-#+-/)> ■ ' (14) 



where /denotes any arbitrary function. If we draw an axis 

 to coincide with the axis defined by the resultant of the 

 three component rotations co X) co y , co z , then [co x x + co y y + co z z) 

 is equal to URcos^), where fl is the resultant of (a> x , a> y , (*> z ) 

 and R is the line joining the origin to the point a, y, z. 

 That is, u, v, w are functions of p the perpendicular from 

 the origin to a plane through the point (<r, ?/, z) perpen- 

 dicular to the axis of the resultant rotation. 



