S3 35.] ROTATING FLUID. 29 



The free surface, p = const., is therefore a paraboloid of revol 

 ution about the axis of z, having its concavity upwards, and its 



20 



latus rectum = ^ . 

 co 



Since j- -y- = 2o&amp;gt;, a velocity-potential does not exist. A 



motion of this kind could not be generated in a perfect fluid, 

 i.e. in one unable to sustain tangential stress. The fact that it 

 can be realized with actual fluids shews that these are not 

 perfect. 



35. Example 4. Instead of supposing the angular velocity co 

 to be uniform, let us suppose it to be a function of the distance r 

 from that axis, and let us inquire what form must be assigned 

 to this function in order that a velocity-potential may exist for the 

 motion. We find 



dv du . dco 



3 -- j- = 2&amp;gt;+r-j-&amp;gt; 

 dx dy dr 



and that this may vanish we must have cor 2 = p, a constant. The 

 velocity at any point is = - , so that the equation (9) becomes 



if we suppose, for simplicity, that no external forces act. To find 

 the velocity-potential &amp;lt;, let us introduce polar co-ordinates r, 6. 

 By Art. 27 



d6 



-~- velocity along r 0, 



-^ = velocity perpendicular to r = - , 

 so that 



&amp;lt;f&amp;gt; = /j,6 + const. =fji arc tan - + const ........... (19). 



x 



We have here an instance of a many-valued or cyclic function. 

 A function is said to be single-valued throughout any region of 

 space when it is possible to assign to every point of that region a 

 definite value of the function, in such a way that these values shall 

 form a continuous series. This is not the case with the function 



