199-200] ROTATING SHEET OF WATER. 333 



If we write ? = -*% .............................. ( 6 )&amp;gt; 



these become 



The equation of continuity has the same form as in Art. 189 

 viz. 



4(fe) 4(ftv) 

 cft~ cte &amp;lt;fy 



where ^ denotes the depth, from the free surface to the bottom, in 

 the undisturbed condition. This depth will not, of course, be 

 uniform unless the bottom follows the curvature of the free 

 surface as given by (3). 



If we eliminate from the equations (7), by cross-differentiation, 

 we find 



d 



or, writing u d^dt^ v = drj/dt, 



and integrating with respect to t, 



dv du fd d\ 



- + 2n l-f+ )=const (ii). 



dx dy \dx dy) ^ 



This is merely the expression of von Helmholtz theorem that the product of 

 the angular velocity 



. (dv du\ 



rc + i -y-- ^-1 



z \dx dy) 



and the cross-section 



of a vortex-filament, is constant. 



In the case of a simple-harmonic disturbance, the time-factor 

 being e*, the equations (7) and (8) become 



-, 



and f V?=--- ...... (10). 



dx dy 



