264 VORTEX MOTION. [CHAP. VII 



This condition is satisfied in all cases of motion in concentric circles 

 about the origin. Another obvious solution of (3) is 



in which case the stream-lines are similar and coaxial conies. The angular 

 velocity at any point is \ (A + ), and is therefore uniform. 



Again, if we put / (*//) = & 2 \^, where k is a constant, and transform to 

 polar coordinates r, 0, we get 



which is satisfied by 



sm 



where J 8 is a 'Bessel's Function.' This gives various solutions consistent 

 with a fixed circular boundary of radius a, the admissible values of Ic being 

 determined by 



....................................... (iv). 



The character of these solutions will be understood from the properties of 

 Bessel's Functions, of which some indication will be given in Chapter vm. 



In the case of motion symmetrical about an axis (x), we have 

 q . 27TOT&1 constant along a stream-line, CT- denoting as in Art. 93 

 the distance of any point from the axis of symmetry. The con- 

 dition for steady motion then is that the ratio O>/CT must be 

 constant along any stream-line. Hence, if i/r be the stream- 

 function, we must have, by Art. 161 (2), 



where /(i|r) denotes an arbitrary function of ^r. 



An interesting example of (4) is furnished by the case of Hill's ' Spherical 

 Vortex f.' If we assume 



where r 2 =# 2 -}-OT 2 , for all points within the sphere r = a, the formula (2) 

 of Art. 161 makes 



<u= f Am, 



so that the condition of steady motion is satisfied. Again it is evident, on 

 reference to Arts. 95, 96 that the irrotational flow of a stream with the 



* This result is due to Stokes, "On the Steady Motion of Incompressible 

 Fluids," Camb. Trans., t. vii. (1842), Math, and Phys. Papers, t. i., p. 15. 

 t " On a Spherical Vortex," Phil Trans., 1894, A. 



