498 Prof. Helmholtz on Integrals expressing Vortex-motion. 



Am= i ; (y-6)£-(«-cK & db dc> 



2ir r 3 



Aw=± ^- a)f >- (3 '-m dadbdc 



2?r r 3 , 



and it follows from these that 



Au(x— a)+Av(y — b)+Aw(z— c) = 0; 



hence the resultant of Aw, Av, Aw is at right angles to r. 

 Further, 



£Aw + ^ + £^ = 0. 



Hence this resultant is also at right angles to the resultant axis 

 of rotation at a, b, c. Lastly, 



Ap = \/(Au) 2 + (Av) 2 + (AwY= a ° q sin v, 



where q is the resultant of f tfj rj , £ a > an< ^ v tbe angle it makes 

 with r, which is found from 



qr cos v= {x- a)% a + (y-b)y a + (*-c)? a . 



-E«c/« rotating element of fluid (a) implies in each other element 

 (b) o/ //ze same ^mVJ wzass a velocity whose direction is perpendi- 

 cular to the plane through (b) and the axis of rotation of (a). The 

 magnitude of this velocity is directly proportional to the volume of 

 (a), its angular velocity, and the sine of the angle between the line 

 (a) (b) and that axis of rotation, and inversely proportional to the 

 square of the distance between (a) and (b) . 



The same law holds for the force exerted by an element of an 

 electric current at (a), parallel to its axis of rotation, on a particle 

 of magnetism at (b) . 



The mathematical connexion of these phenomena consists in 

 this — that in the fluid vortices, for any element of the fluid 

 which has no rotation, a velocity-potential cf> exists satisfying the 

 equation 



** + li + S-n- 

 &fi-&f + dz* ' 



and this holds everywhere but within the vortex-filaments. If we 

 consider the latter as always reentrant either within or without the 

 fluid, the space for which the above equation for <£ holds is com- 

 plexly connected, since it remains single if we conceive surfaces 

 of separation through it, each of which is completely bounded by 

 a vortex-filament. In such complexly connected spaces a function 



