﻿relative to a Rotating Earth. 57 



By means of these relations we can readily transform (16) 

 and obtain the corresponding equations for the rates of 

 change of the circulation components of an element of fluid 

 referred to the rotating axes ; in this way we find 



Df <-B« . B« . V B" f./'d" , Bw . "dw\ 



d# B// Z B^ \B.i' By Bs/ 



with the corresponding equations in rj and f. Now the 

 hydrodynamical theorem that relates to the permanence 

 of a velocity potential for the motion of a given portion of 

 fluid and the theorem of the permanence of the circulation 

 of an element of fluid depend on equations (16). The 

 equations which we have obtained for the relative circula- 

 tions reduce to these equations exactly when the conditions 

 expressed in (12) are fulfilled ; and these conditions must 

 accordingly be fulfilled in order that the theorems referred 

 to may apply to the relative motion, in the same way as they 

 apply to the actual motion. 



Particular Cases of Motion Relative to the Earth, 



We shall now discuss one or two particular cases of fluid 

 motion relative to the Earth, and we shall, to begin with, 

 take the reference point O as a fixed point on the surface of 

 the Earth at latitude (j> degrees North. The angular velocity 

 components w x , a> v , co z have then the values 0, fl cos <£, 

 12 sin </>, respectively, where fl represents the angular 

 velocity of rotation of the Earth about its axis. If we now 

 let zr represent the perpendicular distance from any point 

 #, y 9 z to the axis of the Earth, we can write the equations 

 of motion (1), (2), (3), in the form 



where V'=Y— JDV. V is, in fact, the potential function 



