216 "WISCOJSrSIN' ACADEMY SCIENCES, ARTS, AND LETTERS. 



Helmholtz further shoYrs,'in the memoir above alluded to, how 

 to find the velocities, m, f>, u\ of any portion or element of the fluid 

 when we know «;^, 7V^, iv^, the angular velocities of a vortex-fila- 

 ment established in it, b}^ the following method: 



Let there be given within a mass of fluid which includes the 

 space S, the values of «'\ tv~, id'\ satisfying the "equation of conti- 

 nuity " 



Ih 1^ '^ "^""^- (1) 



Also, «, V, and ?r, must satisfy a similar equation , 



du dv dw r\ 



— -4- ^ -I = 0. 



dx ^ dy ^ dz (2) 



And likewise also, these conditions, 



dv dw du> du „ du dv 



dz dy ' dx dz dy dx \^) 



The conditions for the bounding surface S are supposed to be 

 given according to the particular problem, a, b, c, can be taken as 

 the three angles made with a normal to the surface 5"; q as the re- 

 sultant angular velocity of the three components tv'^, t'/, w^; t, the 

 angle between the normal'to the surface S and the axis of the ro- 

 tating filament; then we shall have 



w'l cos a -\- w" cos b -}- ir^ C03 c = q cos. t= 0. 



over the Avhole of this surface S, or if this surface S cuts any of 

 the vortex-filaments, over the whole of some larger surface 5", which 

 includes all the filaments, and their continuations, if there be any 

 in the first surface S. 



Now we can find values of ?;, v, and w, satisfying equations (2) 

 and (3), if 



(O 



dP dM dL 

 dz dx dy 



