Prof. Helmholtz on Integrals expressing Vortex-motion. 501 



element, so that r) = %=0 } the potentials M and N and their 

 differential coefficients have the same value at both sides. Such 



is also the case with L, -=— > and -7—: but -7— has two distinct 

 dx dy dz 



values, whose difference is 2fe, if e denote the thickness of the 

 sheet. Consequently equations (4) show that u and w have 

 the same values on both sides of the vortex-surface, but the 

 values of v differ by 2fe. Hence the values of that component of 

 the velocity which is a tangent to the vortex-surface and at right 

 angles to the vortex-lines differ on opposite sides of the surface. 

 Within the sheet of revolving elements we must take this com- 

 ponent of the velocity as gradually and equably varying from 

 one value to the other. For if f is here constant through the 

 whole thickness of the shell, and a represent a proper fraction, 

 1/, Vj the values of v at the sides, v x the value in the shell at a 

 distance ae from the first side, we saw that v' — v i = 2^e, while 

 between was a sheet of thickness e and angular velocity f . We 

 have in the same way v , —v x = 2^eoi=:u(v' — v 1 ) ) which expresses 

 the above result. As we must consider the revolving elements 

 as being themselves moved, and the change of their distribu- 

 tion on the surface depends on their motion, we must assign as 

 their mean velocity along the surface for the whole thickness of 

 the sheet the arithmetical mean of v' and v v 



Such a vortex-sheet will be produced if two separate moving 

 masses of fluid come in contact. At the surface of contact the 

 velocity perpendicular to this must be the same for both, but the 

 tangential velocities will in general be different in the two. Thus 

 the surface of contact will have the properties of a vortex-sheet. 



Hence in general isolated vortex-filaments cannot be supposed 

 indefinitely thin, since otherwise the velocities at opposite sides 

 would be indefinitely great and in opposite directions, and the 

 proper velocity of the filament would remain undetermined. To 

 obtain, therefore, certain general conclusions about the motion 

 of very fine filaments of any section, we must make use of the prin- 

 ciple of the conservation of vis viva. 



Before we proceed to treat of separate examples, we will first 



write the expression for the vis viva K of the moving mass of fluid, 



K=ihffi(tP+*+&)d*tb&. ... (6) 



We now from equations (4) substitute in this integral 



- U \dx + dy dz)' 



V ~ V \dy + dz dx )> 

 , /dY dM dL\ 



^"y^+^—dj) 



Phil. Mag. S. 4. No. 226. Suppl. Vol. 33. 2 L 



u* 



