212 "vviscoJ5si]sr academy sciences, arts, ajtd letters. 



In magnetization, the rate at which I vanes along a line, determ- 

 ines the intensity of the magnetization, at any given point 

 within the magnet, in the direction of that line. 



In fluid motion, the rate at which Q varies along a line, determ- 

 ines the velocity of the fluid, at any given point, in the direction 

 of that line. 



In the case of ordinarj' fluid motion, moreover, if the conditions 



dQ dQ dQ 



dz"! dij ^ dz 



hold true, then also we have the conditions 



du dv d V dw dio die 



TjT d^ ^5 ~d~z Jy' ^' Ix ~dz ^' 



as is well known. 



We have seen [Eqs. (D) and (E] that precise!}'' similar conditions 

 obtain in the case of ordinary distributions of electricity and mag- 

 netism, so long as we confine ourselves to the space outside of that 

 which contains the so-called magnetic or electric matter. 



The motions of fluids heretofore discussed in treatises on the 

 dynamics af fluids are such as fulfill the conditions imposed by 

 equations (0) and (P.) They are motions of translation, or of expansion 

 and contraction; oscillatory movements being merely periodic 

 movements of translation, of greater or less extent. All such mo- 

 tions have assumed for i\i&xn.^ velocity potential^ the differential coef- 

 ficients of which with reference to the coordinates, are the com- 

 ponent velocities of the fluid in the direction of the 30ordinate3. 



The assumption of a velocity potential necessitates the set of 

 conditions given above in equations (0) and (P.) 



But Helmholtz in a remarkable memoir on " Integrals Express- 

 ing Vortex Motion " to be found translated by Prof. Tait, in the 

 Philosophical Mag., for 1867, has shown that these conditions do not 

 hold if there be some of ihe elements of the fluid in rotation. In such 

 cases if ^o^^ v?^ lo^, represent the angular velocities of the rotating 

 fluid element about the coordinate axes, then we have 



du dv 



dw du ^ 



d V dw 

 d z ay 



