28-29] 



ROTATING FLUID. 



31 



these being continuous when r = a. The depth of the central 

 depression below the general level of the surface is therefore 



29. To illustrate, by way of contrast, the case of external 

 forces not having a potential, let us suppose that a mass of liquid 

 filling a right circular cylinder moves from rest under the action 

 of the forces 



the axis of z being that of the cylinder. 



If we assume u= - o>y, v = <ox, w=0, where o> is a function of t only, these 

 values satisfy the equation of continuity and the boundary conditions. The 

 dynamical equations become 



-y -A= 



dt 



Differentiating the first of these with respect to y, and the second with 

 respect to x and subtracting, we eliminate p, and find 



The fluid therefore rotates as a whole about the axis of z with constantly 

 accelerated angular velocity, except in the particular case when B=B. To 

 find p, we substitute the value of datjdt in (i) and integrate ; we thus get 



where 2 = 



