174 VORTEX MOTION. [CHAP. VI. 



in Chapter iv.) must be constant along a stream-line, i.e. by 

 Art. 69 (5), 



where /(^) is an arbitrary function of 

 A particular solution of (54) is 



in which case the stream-lines are a system of similar and coaxial 

 conic sections. The velocities at corresponding points are pro 

 portional to the linear dimensions of the curves; the angular 

 velocity at any point is A + C, and is therefore uniform. 



In the case of motion symmetrical about an axis (say that 

 of x), considered in Art. 103, we have Zwdnq constant along 

 a stream-line. The conditions for steady motion reduce then to 

 this, that the ratio : OT must be constant along a stream-line. 

 If T/T be the stream-function, we have 



so that the condition is 



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



* Conditions (54), (56) were given by Stokes, Camb. Trans. Vol. vn. (1842). 



