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 



-j -- T-= 2&) + r ^j-> 

 dx dy dr 



and in order that this may vanish we must have wr^/*, a 

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

 equation (2) of Art. 22 becomes 



i ...................... (1), 



if no extraneous forces act. To find the value of <f> we have 



dr~ 

 whence </> = - pO + const. = p tan" 1 - + const .......... (2). 



oc 



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 <, if it vary continuously, changes by ZTT/J, 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 o>r from r = to r = a, and to wa?/r for 

 r > a. The corresponding forms of the free surface are then 

 given by 





