VORTEX MOTION. [CHAP. VII 



where the rectilinear axis of the ring passes through the centre of 

 the sphere, has been investigated by Lewis*, by the method of 

 ' images.' 



The following simplified proof is due to Larmorf. The vortex- ring is 

 equivalent (Art. 148) to a spherical sheet of double-sources of uniform 

 density, concentric with the fixed sphere. The 'image' of this sheet will, 

 by Art. 95, be another uniform concentric double-sheet, which is, again, 

 equivalent to a vortex-ring coaxial with the first. It easily follows from the 

 Art. last cited that the strengths (m', m"} and the radii (or', tzr") of the vortex- 

 ring and its image are connected by the relation 



The argument obviously applies to the case of a reentrant vortex of any 

 form, provided it lie on a sphere concentric with the boundary. 



On the Conditions for Steady Motion. 



164. In steady motion, i.e. when 



du_ dv_ dw_ 

 dt~ ' dt~ dt~ 



the equations (2) of Art. 6 may be written 



du dv dw , 



U J- + V IT + w j -- 2 (v 

 dx dx dx 



Hence, if as in Art. 143 we put 



du dv dw , x dl 1 dp 

 U J- + V IT + w j -- 2 (v wf]) = - -; --- -/->> 

 dx dx dx dx p dx 



-i-tt+n ..................... (i), 



we have 



It follows that 



tt y + ^' 



dx dy 



* dx^ '' dy^ b dz ~ 



* " On the Images of Vortices in a Spherical Vessel," Quart. Journ. Math., 

 t. xvi., p. 338 (1879). 



t "Electro-magnetic and other Images in Spheres and Planes," Quart. Journ. 

 Math., t. xxiii., p. 94 (1889). 



