PAPER BY MAX MARGULES. 307 



where P, £1, A are the components of the forces in the directions of the 



new coordinates dr, rdco, r sin go d A. If the velocities are so small 

 that we can neglect their squares and products, then only the first term 

 will remain on the right-hand side of each of these equations. If we 

 put 



we have 



§? dco dX 



dt~ a ' r ~M= h ^ rsinGO dt= G 



*—%' n ~w A= dt 



Therefore the equations of motion of a fluid that is only under the in- 

 fluence of a constant force of gravity positive in the direction of the 

 diminishing radius, are 



(8) 



These equations are applicable to the motion on a sphere at rest. In 

 order to investigate the relative motion on the rotating terrestrial sphere, 

 we modify equation (7) in that we put vt + X in place of X where v is 



the velocity of rotatiou of the earth. In place of -j-. in equation (7) 

 there now occurs — iff~- !£ again, we put c in place of the new r sin 



gt-jt, if we retain the products ra, vb, re, and if on the other hand we 



omit the terms in v 2 , which indicate only a slight change in the force of 

 gravity, then we obtain the equations for the motion of a fluid on a 

 rotating sphere. On the right-hand sides of the equations (8) the 

 terms — 2vc sin co, —2rc cos go and -\-2vasiu. Go+2vb cos go are to be 

 added respectively. 



The equation of continuity has the same form for the sphere at rest 

 as for the rotating sphere. 



jgj. ?(^ r!8ft ) ,i d (t* b siD ^ I d(/*c) __ , 9) 



dt r 2 dr r sin go doo r sin go d X .... \ > 



Introducing the notation 



p=p (l+s), T=T„(l + r) 



allied to that above used, we obtain the following differential equations 



