PAPER BY PROF. HELMHOLTZ. 41 



whence it follows that the product of the velocity of rotation by the area 

 of the section is constant throughout the ichole length of the vortex fila- 

 ment. It has already been shown that this product does not change 

 during the progressive motion of the filament. 



It follows from this that a vortex filament can not possibly end any- 

 where within the fluid, but must either return into itself, like a ring 

 within the fluid, or must continue on to the boundaries of the fluid. 

 For in case a filament ended auy where within the fluid it would be pos- 

 sible to construct a closed surface for which the integral fa cos 6 doo 

 is not zero. 



III. INTEGRATION BY VOLUME. 



When we can determine the motions of the vortex filaments present 

 in the fluid we can, by means of the above established propositions, 

 also determine completely the quantities <?, ?/, aud C- We will now 

 consider the problem to determine the velocities u, v, and w from the 

 quantities «§, ;/, and £. 



Within a mass of liquid that fills the region S let values of 5, 7, and Z 

 be given, which quantities should satisfy the condition that — 



dZ dH 2C =0 (2a). 



dx^dy d% 



Such values of w, v, and w are to be found as may, throughout the 

 whole region 8, satisfy the conditions [of Eq. (1 4 ) and (2) viz.] 



du dv W (1 4 ) 



dx + dy^d* u 







=2r ? } 



(2) 





du_jv 



}y ~dx ^ 3 



to which is still to be added the condition demanded by the boundary 

 of the region IS, according to the nature of the specific problem in hand. 

 According to the distribution of B, i h C, as above specifically given, 

 there can occur on the one hand such vortex lines as shall return into 

 themselves within the limits of the region 8 and on the other hand such 

 as extend to the boundary and there suddenly break off. When this 

 latter is the case then we can certainly prolong these [fragments olj 

 vortex lines either along the surface of 8 or beyond 8 until they re- 

 turn into themselves, so that a larger space 8 { exists that contains only 

 closed vortex lines and for whose whole surface both £, 7, C and their 

 resultant itself are all equal to zero or at least 



£ cos a+t/ cos p+Z cos y=0 cos 0=0. 



