PAPER BY PROF. HELMHOLTZ. Q\ 



p being negative. Under this condition the tearing asunder of the 

 mass of air would not be necessary. 



It is possible to convince one's self of the actual existence of such 

 discontinuities when we allow a stream of air impregnated with smoke 

 to issue from a round opening or a cylindrical tube with moderate 

 velocity so'that no hissing occurs. Under favorable circumstances one 

 obtains thin rays or jets of this kind of a few lines diameter and a 

 length of many feet. Within the cylindrical surface the air is iii mo- 

 tion with constant velocity, but outside it, on the other hand, in the 

 immediate neighborhood of the jet it moves not at all or very slightly. 

 One sees this very sharp separation clearly when we conduct a steadily 

 flowiug cylindrical jet of air through the point of a flame, out of which 

 it cuts a sharply defined piece, while the rest of the flame remains en- 

 tirely undisturbed, and at most a very thin stratum of flame, which 

 corresponds to the bouudary layer of the jet influenced by friction, is 

 carried along a little way. 



As concerns the mathematical theory of this motion I have already 

 given the boundary conditions for the existence of an interior surface 

 of separation within the liquid. They consist in this that the pressures 

 on both sides the surface must be equal and equally so the components 

 of the velocity normal to the discontinuous surface. Since now the 

 movement throughout the entire interior of a liquid whose particles 

 have no motion of rotation is wholly determined when the motion of 

 its entire exterior surface and its interior discontinuities are given, 

 therefore in general for a liquid whose exterior boundary is fixed, it is 

 only necessary to know the movement of the surfaces of separation and 

 the variations of the discontinuity. 



Now such a discontinuous surface can be treated mathematically pre- 

 cisely as if it were a vortex sheet, that is to say, as if it were continu- 

 ously enveloped by vortex filaments of indefinitely small mass but 

 finite moments of rotation. For each element of such a vortex sheet 

 there is a direction for which the components of the tangential veloci- 

 ties are equal. This gives at once the direction of the vortex filaments 

 at the corresponding place. The moment of this filament is to be put 

 proportional to the difference existiug between the components, taken 

 perpendicular to it, of the tangential velocity on both sides of the 

 surface. 



The existeuce of such vortex filaments in an ideal frictiouless liquid 

 is a mathematical fiction that facilitates the integration. In a real 

 liquid subject to friction, this fiction becomes at once a reality inasmuch 

 as by the friction the boundary particles are set in rotation, and thus 

 vortex filaments originate there having finite gradually increasing 

 masses, while the discontinuity of the motion is thereby at the same 

 time compensated. 



The motion of a vortex sheet and the vortex filaments lying in it is 

 to be determined by the rules established in my Memoir on Vortex 



