232 



VORTEX MOTION. 



[CHAP, vii 



where o>' is the angular velocity of the fluid. Hence the formulas 

 (6) of Art. 145 become 



_ m [dx' r ,_m' (dy' _ m [dz' 



~ STT J V ' ~ ZTT] T ' ~ **) r 





where m, = o>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' (f d 1 , , dl ,\ 

 U = ^j(dyr' dz -dzr' d y)> 



with similar results for v, w. We thus find 



f z z' dz' x x\ ds 



_ m 

 ~ 



.(2)*. 



_ m Udz' x x' dx' y y' 

 S ZirJ\d8' r "df~r 



_ m fi dx' y y dy' z z'\ ds' 

 " f ~r ds 7 r ) ^ 



If &u, Aw, Aty denote the parts of these expressions which corre- 

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

 resultant of Aw, Aw, Aw is a velocity perpendicular to the plane 

 containing the direction of the vortex- line at (#', 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 (at ', y', /). For the magnitude of the resultant 

 we have 



where % is the angle which r makes with the vortex-line at (x f , 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 polef. 



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



t Ampere, Theorie mathematique des phenoimnes electro-dynamiques, Paris, 1826. 



