80 INTEGRATION OF THE EQUATIONS IN SPECIAL CASES. [CHAP. II. 



in (19) ; for the value of &amp;lt; there given, if it vary continuously, 

 changes by STT/A as the point to which it refers describes a complete 

 circuit round the origin, whereas a single-valued function would 

 under the same circumstances return to its original value. 



A function which like the above experiences a finite change of 

 value when the point to which it refers describes a closed curve, 

 returning to the point whence it started, is said to be many-valued 

 or cyclic. The theory of many-valued velocity-potentials will be 

 discussed in the next chapter. 



36. Example 5. A mass of liquid filling a right circular cy 

 linder moves from rest under the action of the forces 



the axis of z being that of the cylinder. 



Let us assume u^ vy, v = ax, w = 0, where &&amp;gt; is a function 

 of t only. These values satisfy the equation of continuity and 

 the boundary conditions. The dynamical equations become 



dco 2 , D I dp 



y, -- (ox Ax + By -/-, 



* P dx (20) 



dco 



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

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



dco B - B 

 dt~ 2 



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

 uniformly increasing angular velocity, except in the particular case 



when B = B . To find p, we substitute the value of ~ in (20) 

 and integrate ; thus we get 



P = io&amp;gt; 2 (x* + f] 4- i (Ax* + 2&xy + O/) + const, 

 P 



where 2/3 = 5 + H. 



37. Example 6. Let X=-^- } Y=^-, Z=0; the other 

 circumstances remaining the same as in the preceding example. 



