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



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 



-; --- ^-=20) + r-r- , 

 dx dy dr 



and in order that this may vanish we must have o&amp;gt;r 2 = //,, a 

 constant. The velocity at any point is then = /tt/r, so that the 

 equation (2) of Art. 22 becomes 



2 = const. -i ...................... (1), 



if no extraneous forces act. To find the value of &amp;lt; we have 



- _ 



dr~ rd6~ r 



whence (/&amp;gt; = - p6 + const. = fi tan&quot; 1 - 4- const .......... (2). 



x 



We have here an instance of a cyclic function. A function 

 is said to be single-valued throughout any region of space when 

 we can assign to every point of that region a definite value of the 

 function in such a way that these values shall form a continuous 

 system. This is not possible with the function (2) ; for the value 

 of (/&amp;gt;, if it vary continuously, changes by - 2-TTyu, as the point to 

 which it refers describes a complete circuit round the origin. The 

 general theory of cyclic velocity-potentials will be given in the 

 next chapter. 



If gravity act, and if the axis of z be vertical, we must add to 

 (1) the term gz. The form of the free surface is therefore that 

 generated by the revolution of the hyperbolic curve a?z = const. 

 about the axis of z. 



By properly fitting together the two preceding solutions we 

 obtain the case of Rankine s combined vortex. Thus the 

 motion being everywhere in coaxial circles, let us suppose the 

 velocity to be equal to wr from r = to r = a, and to wa?/r for 

 r &amp;gt; a. The corresponding forms of the free surface are then 

 given by 





