232 VORTEX MOTION. [CHAP. VII 



where o&amp;gt; is the angular velocity of the fluid. Hence the formal* 

 (6) of Art. 145 become 



m dot m dy m dz 



~ 



where m , = a&amp;gt;V , measures the strength of the vortex, and the 

 integrals are to be taken along the whole length of the filament. 



Hence, by Art. 145 (8), we have 



m (t 



u = o~ I 

 27rJ \ 



d\ d 1 



j .dz--j--. 

 dy r dz r 



with similar results for v, w. We thus find 



_ m ffdy z z dz x x \ ds 



.(2)*. 



m ((dz x x dx y y ^ 



^ If 7 / 



27rJ\ds r ds r 



m [/ dx y y dy z z \ ds 



nn ^-^- I I ^L ~L~ ^_ *^ I 



27rJ\ds r ds r ) r~ 



If &u, Av, A^6 denote the parts of these expressions which corre 

 spond to the element Ss of the filament, it appears that the 

 resultant of Aw, Av, At6&amp;gt; is a velocity perpendicular to the plane 

 containing the direction of the vortex- line at (x, y , z } and the 

 line r, and that its sense is that in which the point (x, y, z) would 

 be carried if it were attached to a rigid body rotating with the 

 fluid element at (x , y , z ). For the magnitude of the resultant 

 we have 



&quot;&quot;&quot;&quot; &quot; (3), 



where % is the angle which r makes with the vortex-line at (# , y , z). 



With the change of symbols indicated in the preceding Art. 

 this result becomes identical with the law of action of an electric 

 current on a magnetic pole*)*. 



* These are equivalent to the forms obtained by Stokes, 1. c. ante p. 228. 



t Ampere, Theorie matheniatique des plienomenes electro-dynamiques, Paris, 1826. 



