Prof. Helmholtz on Integrals expressing Vortex-motion. 491 



Every motion of a fluid enclosed in a singly-connected space, 

 when a velocity-potential exists, is therefore dependent on the 

 motion of the free surface of the fluid. When the latter (i. e. 



—) is given, the whole motion of the enclosed fluid is com- 

 an / 



pletely determined. For suppose that two functions <£, and <f> n 



could satisfy the condition 



!n-=* 

 at the surface of S, and also 



dx* + dy* * dz* ~ 

 through the whole interior of S. The function tf> k — <j> u satisfies 



the latter: but at the free surface 1 7 ll = 0, and therefore, 



dn 



as before, we must have for the whole interior 



<#/ — £//) _p <</>y-0//) _ o <ft f -ft iy ) _ 

 dx -^ dy~ ' dz ~ U} 



and the component velocities at every point in the fluid are 

 therefore alike for both cases. 



Thus it is only when there is no velocity-potential that some 

 fluid elements can rotate, and that others can move round along 

 a closed curve, in a simply-connected closed space. We may 

 therefore call the motions which have no velocity-potential, gene- 

 rally, vortex-motions. 



§2. ^ 



We must next determine the variations of the angular veloci- 

 ties f , 7], £ during the motion, when the only external forces are 

 such as have a potential. Let us note once for all, that if yjr be 

 a function of x, y, z, and t, and if it increase by Sty when these 

 increase by Sx, tiy, Sz, St respectively, then 



1 at ax ay az 



If we now take the change of ty in the time St for the same 

 elementary volume of fluid, we must give Sx, Sy, Sz the values 

 which they have for the moving element — that is, 



Sx = uSt } Sy == vSt, Sz == ivSt ; 

 and we obtain 



^t = ^t +u ^L + v < tt +lv d t. 

 St dt dx dy dz 



