262 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 

 1 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&quot;} and the radii (or , or&quot;) of the vortex- 

 ring and its image are connected by the relation 



(i). 



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&quot; 



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



du dv dw a , 

 dx dx doc 



Hence, if as in Art. 143 we put 



du dv dw a , ,, _ , _ c?O _ 1 dp 

 dx dx doc dx p dx 



we have 



It follows that 



dx d ) 



dx dy ^ dz 



* &quot;On the Images of Vortices in a Spherical Vessel,&quot; Quart. Journ. Math., 

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



t &quot; Electro-magnetic and other Images in Spheres and Planes,&quot; Quart. Journ. 

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



