All- 101 19 



rectangular coordinates considerably simplifies the analysis, 

 we shall first transform the equations of motion from spherical 

 to rectangular coordinates in such a manner that the equilibrium 

 free surface which establishes itself in the spherical system 

 as a result of gravity and centripetal acceleration corresponds 

 to the x-y plane of the rectangular system. The apparent gravi- 

 tational force, i.e., the force v;hich is the resultant of true 

 gravity and centripetal acceleration, acts in a direction normal 

 to this equilibrium surface. 



In Appendix 1, it is shown that our original equations 

 reduce to 



Sul 



at 



+ u' |i^ + v» Ml - 2Qv' sinil) = - -i Mu(VA.V)u' (l) 

 3x ay R p ax 5 1 



Ml + u' erLl + v' .^ + 2Qu» sin(Z) = - I 9^ +4(V A.V)v' (2) 



at ax ay R p ay i i 



-ifE-g (3) 



p oz 



.Qui + avi + mil = ih) 



ax ay az 



where 



x,u' denote the east-west coordinate and velocity 

 respectively (x is positive eastward), 



y,v' denote the north-south coordinate ajid velocity 

 respectively (y is positive northward), 



z,w' denote the vertical coordinate and velocity 

 respectively (z is positive upv/ard), 



R is the mean radius of the earth, 



g is the apparent gravitational acceleration on 



the earth's surface, 



2Qsin(— ) is the radial component of the angular velocity 

 ^ vector of the earth. 



