129 131.] FLUID LIMITED BY FINITE SURFACE. 153 



of Su t Sv, Siv is perpendicular to the plane containing the direction 

 of the vortex-line at (x, y t z ) - and the line r, and also that its 

 sense is that in which the point (x, y, z) would move if it were rigidly 

 attached to a body rotating with the fluid element at (x, y, z). 

 The magnitude of the resultant is 



.......... (16), 



(by an elementary formula of solid geometry), where ^ is the 

 angle which r makes with the direction of the vortex-line at 



(* , y &amp;gt; * ) 



A relation of exactly the same form, as that here developed 

 obtains between the magnetic force and the electric currents in 

 any electro-magnetic field. If we suppose a system of electric 

 currents arranged in exactly the same manner as the vortex-fila 

 ments, the components of the current at (x, y , z) being f, 77, the 

 components of the magnetic force at (x, y, z) due to these currents 

 will be u, v, w. 



In the general case (i.e. when 6 is not everywhere zero) we 

 must add to the values of u t v, w obtained by integrating (15) 



the terms -^ , -j- ,. -7-, respectively, where P has the value (10). 



These are the components of the force at (ar, y } z) produced by a 

 distribution of imaginary magnetic matter with density 6. 



131. Let us revise the investigation of Art. 129 with a view 

 to adapting it to the case where the region occupied by the fluid 

 is not infinite, but is limited by surfaces at which the value of the 

 normal velocity X is given. The equations to be satisfied by u, v, w 

 are (3), (4) and (5). The integrals (10) and (11) being supposed 

 to refer to this limited region, the surface-integral in the last 

 member of (12) will not in general vanish unless all the vortices 

 present form closed filaments lying wholly in the region. If on 

 the other hand the vortex-lines traverse the region, beginning and 

 ending on the boundary, we may suppose them continued outside 

 the region, or along its surface, in such a manner that they form 

 closed curves. We thus obtain a larger region in which all the 

 vortex-filaments are closed, and if we now suppose the integrals 

 in (10) and (11) to refer to this extended region, the surface-inte- 



