PAPER BY MAX MARGULES. 307 



where P, il, A are the compoueuts of the forces in the directious of the 

 new coordinates dr, rdco, r sin oo d X. If the velocities are so small 

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

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

 put 



we have 



^^ ^ r^(^ , . dX 



dV ^^-W ^=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 A where v is 



the velocity of rotation of the earth. In place of -ji in equation (7) 

 there now occurs tt^- ^^i again, we put c in place of the new r sin 



cj?-tt, if we retain the jiroducts ra, vb, vc, and if on the other hand we 



omit the terms in v^, 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, —2vc cos &? and +2K«sin Gj-\-2vb cos &? are to be 

 added respectively. 



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

 as for the rotating sphere. 



Jt r'^Jr ~^ r siu GO Joj r sin 00 ;) X .... { ) 



Introducing the notation 



p=po{l + e), T=To{1+t) 



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



