﻿relative to a Rotating Earth, 53 



One object o£ the present paper is to investigate the con- 

 ditions under which the circulation theorem may be applied 

 to atmospheric motions relative to the Earth's surface ; or 

 more generally to motions relative to any three rectangular 

 axes which are themselves rotating about each other, with a 

 fixed origin. In the later part of the paper one or two 

 additional cases of motion of the atmosphere are discussed 

 and the system of isobars corresponding to each motion 

 determined. 



In view of the problems to be considered, we shall begin 

 by specifying the system of rotating axes most convenient 

 in dealing with fluid motion in the neighbourhood of any 

 point of reference on the Earth's surface. The axis OZ 

 is drawn upwards along the apparent vertical at 0, and 

 line OZ continued downwards meets the axis of the Earth 

 at a point 0' which is taken as origin of coordinates. Then 

 axes O'X and O'Y are drawn parallel to horizontal lines 

 through the reference point in directions due East and 

 due North respectively. In the most general case to be 

 considered the reference point may be in motion relative 

 to the Earth's surface, and this involves also a motion of the 

 origin 0' if point moves either North or South. But 

 the motion of 0' corresponding to any moderate motion 

 of is very small, and for our present purpose we may 

 regard the origin 0' as a fixed point, very near to the centre 

 of the Earth. We shall denote by (.r, y, z') the coordinates 

 of any point referred to origin 0', and by (#, ?/, z) the co- 

 ordinates of the same point referred to parallel axes through 

 0. This makes z r = z + H, where R represents approximately 

 the radius of the Earth. The components of the velocity of 

 any particle relative to the axes at any instant are repre- 

 sented by u, r, w, and the angular velocities of the axes 

 themselves, that is, of each two axes about the third, are 

 represented by ft> z , ay^ a) z , respectively. We shall introduce 

 the particular values of ao x , w y , co z corresponding to a reference 

 point fixed in position on the Earth, or moving relative 

 to the Earth, when we come to deal with special problems. 

 Referred to the above system of axes, the equations of motion 

 of any fluid particle take the form : — 



Dz < , a - <* V 1 "dp ( i n 





