PAPER BY PROF. HELMHOLTZ. 47 



located within the fluid are to be determined, taking account of their 

 mutual influences and of the boundary conditions whereby the problem 

 becomes much more complicated. However, for certain simple cases 

 even this problem can be solved, especially in those cases where the 

 rotations of the liquid particles take place only on certain surfaces or 

 lines and the forms of these surfaces and lines remain unchanged dur- 

 ing the translatory motions. 



The properties of surfaces that adjoin an indefinitely thin layer of 

 rotating fluid particles are easily seen from the equations (5a). When 

 <?, 77, and C differ from zero only within an infinitely thin layer, then, ac- 

 cording to well-known propositions, the potential functions L, M, and 

 N will have equal values on both sides of the layer,* but the partial 

 differential coefficients of these functions for the direction normal to the 

 layer will be different on the two sides of the layer. Imagine the 

 coordinate axes so placed that at the point of the vortex sheet under 

 consideration the axis of z corresponds to the normal to the sheet, the 

 axis of x to the axis of rotation of the liquid particles situated in the 

 sheet, so that at this point we have ?/=?:=0; then will the potentials 

 M and N, as also their partial differential coefficients, have the same 



values on both sides of the sheet, similarly L and — and — ; but —- 



jx L )y jz 



will have two different values whose difference is equal to 2£je, when 

 e indicates the thickness of the stratum. Corresponding to this the 

 equation (4) shows that u and w have the same values on each side of 

 the vortex sheet, but v has values that differ from each other by 2gs. 

 Therefore, that component of the velocity that is perpendicular to the 

 vortex line and tangent to the vortex sheet has different values on 

 either side of tbe vortex sheet. Within the layer of rotating liquid 

 particles we must imagine the respective components of the velocity 

 as uniformly increasing from the value that obtains on one side of the 

 surface to that which obtains on the other side. For when, as here, g 

 is constant through the whole thickness of the layer, and we indicate 

 by a a proper fraction, by v 1 the value oft? on one side, by i\ its value 

 on the other side, by v a its value within the layer itself at a distance 

 as from the former side ; then, as we saw before, 



because a layer of the thickness £ and the rotatory velocity B, lies be- 

 tween the two sides. For the same reasons we must have 



v 1 — v OL =2%£a=a (v l —Vi), 



which covers the proposition just enunciated. Since we must think of 

 the rotating liquid particles as themselves moving forward and since 

 the change of distribution on the surface depends on their own motion, 

 therefore we must, through the whole thickness of the layer, attribute 



* [This is the "vortex sheet" of English writers.] 



