35 37.] ROTATING FLUID. 31 



Assuming u coy, v = cox, w = 0, where w is a function of 

 r (= Jx* + if) and t only, we find 



dco , 2 LLX 1 dp 



x 7 - ft) y = ^-5- --- f- . 

 dt r 2 p dy 



Eliminating p, we obtain 



The solution of this is 



where F and /denote arbitrary functions. Since w = when t = 0, 

 we have 



and therefore 



where X is a function of t which vanishes for t = 0. Substituting 

 in (21), and integrating, we find 



Since p is essentially a single-valued function, we must have 

 -j- = fj&amp;gt;, or X = fjit. Hence the fluid rotates with an angular velocity 



which varies inversely as the square of the distance from the axis, 

 and increases uniformly with the time. 



