490 Prof. HelmhoJtz on Integrals expressing Vortex -motion. 

 rise to the component velocities 



{y-m, ~{x-v% 0; 



and the velocities of the element whose coordinates are x, y } z 

 are now 



u=:A + a(x-x) + (y + Q(y-\))+(/3- V ){z-%), 



z; = B + ( 7 -^-ri+%-9) + (« + ?)(*-^ 

 w = C + (!3-l- V )(x-p) + {u-Z){7,-\)) + c(z-i). 

 From these, by differentiation, 



dv dw _ ofc "1 



dz d v 



dw du _ 9 



dx dz ~ 



du dv __ny 



dy dx 



(2) 



The quantities on the left, which are zero by (1 c) when there 

 is a velocity-potential, are now the doubles of the angular velo- 

 cities of the fluid element about the coordinate axes. The exist- 

 ence of a velocity-potential, then, is inconsistent with the exist- 

 ence of a rotation of the fluid element. 



Asa further characteristic property of motion with a velocity- 

 potential, we may adduce that no such motion can exist in a 

 rigid and unmoved closed vessel full of fluid, whose interior is a 

 simply-connected space S. For if n be the normal to the bounding 

 surface drawn inwards, the velocity-component perpendicular to 



the surface is -~, and must of course vanish. But Green* has 

 an 



shown that 



j'JJKi )"♦ GY+ e»**- jvjs*. 



where the first integral extends to the whole space S, and the 

 second to the whole bounding surface of S, an element of which 



is denoted by dw. As in this case ~ is identically zero over 



the whole surface, the triple integral must vanish, and therefore 

 throughout the whole space S we must have 



lx~^ dy ' dz - U ' 

 so that there cannot exist such a motion of the fluid. 



* This, as before remarked, is not true for complexly-connected spaces. 



