Oscillations of Rotating Water. Ill 



From (1) or (4) we find, by differentiation &c, 



d (dv du\ (du dv\ n ,.. 



which is the equation of vortex motion in the circumstances. 



These equations reduced to polar coordinates, with the fol- 

 lowing notation, 



become 



x = r cos 6, y=r sin 0, 



u—%cosd — rsintf, v = fsin^ + TC0S^ 



r ■ dr rdd dt' 



j+> \* ) 



.... (4') 



dK c dh 



dt y dr 



dr dh 



In these equations D may be any function of the coordinates. 

 Cases of special interest in connexion with Laplace's tidal 

 equations are had by supposing D to be a function of r alone. 

 For the present, however, we shall suppose D to be constant. 

 Then (2) used in (5) or (2 X ) in (5') gives, after integration 

 with respect to t, 



dv du a h /n . 



Tx-dy= 2( °W < 6 > 



or, in polar coordinates, 



t dr dt, It 



r + 7h--^W =2m B W 



These equations (6), (6') are instructive and convenient, 

 though they contain nothing more than is contained in (2) or 

 (20, and (4) or (4'). _ 



Separating u and v in (4), or £ and t in (4'), we find 



d 2 u . „ / d dh _ dh\ 



de 



and cPv . , , /„ dh 



dtdv/ J 



2„, / JL rl ,77, \ C ' ' V 



