145-147] ELECTRO-MAGNETIC ANALOGY. 231 



146. There is a remarkable analogy between the analytical 

 relations above developed and those which obtain in the theory of 

 Electro-magnetism. If, in the equations (1) and (2) of Art. 144, 

 we write 



a, y8, 7, p, p } q, r 



for U, V, W, 0/47T, /27T, T;/27r, J/27T, 



respectively, we obtain 



da d/3 dy \ 



IT + ~T~ + ~J- = 47TP, 



dx dy dz I 



dy dj3 da dy d@ da {&quot; 



~T~ ~ ~T = ^Trp, ~J -T~ ^q, ~J ~J~ = ^ 7rr 



dy dz dz dx dx dy J 



which are the fundamental relations of the subject referred to ; 

 viz. a, /:?, 7 are the components of magnetic force, p, q, r those of 

 electric current, and p is the volume-density of the imaginary 

 magnetic matter by which any magnetization present in the field 

 may be represented. Hence, if we disregard constant factors, the 

 vortex-filaments correspond to electric circuits, the strengths of 

 the vortices to the strengths of the currents in these circuits, 

 sources and sinks to positive and negative magnetic poles, and, 

 finally, fluid velocity to magnetic force f. 



The analogy will of course extend to all results deduced from 

 the fundamental relations ; thus, in equations (8) of the preceding 

 Art., &amp;lt;l&amp;gt; corresponds to the magnetic potential and F, G, H to the 

 components of electro-magnetic momentum. 



147. To interpret the result contained in Art. 145 (8), we 

 may calculate the values of u, v, w due to an isolated re-entrant 

 vortex-filament situate in an infinite mass of incompressible fluid 

 which is at rest at infinity. 



Since 6 = 0, we shall have &amp;lt;3&amp;gt; = 0. Again, to calculate the 

 values of F, G, H, we may replace the volume-element So/Sy S* 

 by v Ss , where &s is an element of the length of the filament, and 

 &amp;lt;r its cross-section. Also, we have 



c., , dx , , dy .,, , dz 

 * =(a ds&quot; &quot; =&amp;lt;a ds&quot; ?=W d? 



* Cf. Maxwell, Electricity and Magnetism, Art. 607. 



f This analogy was first pointed out by von Helmholtz ; it has been extensively 

 Utilized by Lord Kelvin in his papers on Electrostatics and Magnetism. 



