340 Prof. Helmholtz on Discontinuous Movements of Fluids. 



surface the air is then in motion with constant velocity ; but 

 outside the stream, even quite close to it, the air is not at all, 

 or scarcely at all, moved. This sharp separation may also be 

 seen very distinctly if a calmly flowing cylindrical current of air 

 be led through the apex of a flame. A sharply bordered piece 

 of the flame is cut out, while the rest of it remains quite undis- 

 turbed, or at most a very thin sheet, corresponding to the boun- 

 dary layers caused by the friction, is carried along a little way 

 with the air-current. 



With regard to the mathematical theory of these motions, I 

 have already given the limiting conditions for an internal surface 

 of separation of the liquid. These conditions are, that the pres- 

 sure on both sides of the surface must be the same, as also must 

 be the components of the velocity in a direction perpendicular to 

 the surface of separation. Since, now, the motion throughout 

 the interior of an incompressible fluid, the particles of which 

 have no rotary motion, is fully known when the motion of its 

 entire surface and its interior discontinuities are given, it follows 

 generally that when a fluid is rigidly enclosed, we have only to 

 consider the motion of the surface of separation and the changes 

 in the discontinuity. 



Such a surface of separation can be treated mathematically 

 exactly as if it were a surface of gyration — that is, as if it were 

 completely covered with lines of gyration (Wirbelf'dden) of infi- 

 nitely small mass, but finite moments of rotation. In every sur- 

 face unit of such a surface of gyration there is one direction in 

 which the components of the tangental velocities are equal. This 

 gives at once the direction of the lines of gyration at the corre- 

 sponding place. The moment of these lines must be made pro- 

 portional to the difference shown by the perpendicular compo- 

 nents of the tangental velocity on both sides of the surface. 



The existence of such lines of gyration for an ideal fluid with- 

 out friction is a mathematical fiction which facilitates the inte- 

 gration. In an actual liquid subjected to friction this fiction is 

 quickly realized, because the bordering particles are set in rota- 

 tion by the friction, and then immediately are formed lines of 

 gyration of finite gradually increasing mass, while the disconti- 

 nuity of the motion is thereby at the same time equalized. 



The motion of a surface of gyration, and of the lines of gyra- 

 tion lying in it, are to be determined according to the rules 

 established in my work on gyratory motion. The mathematical 

 difficulties of this problem can only, it is true, be overcome in a 

 few of the more simple cases. But in many other cases we may 

 at least draw conclusions concerning the direction of the changes 

 by means of the method of viewing the matter just described. 



It is especially to be remarked that, according to the laws 



