338 Prof. Helmholtz on Discontinuous Movements of Fluids, 



tion which must be taken into account in the integration of the 

 hydrodynamic equations, and which has, as far as I am aware, 

 been hitherto neglected. By taking this condition into account, 

 on the other hand, where the calculation can be carried out, 

 forms of motion are obtained such as are observed in reality. 

 The circumstance in point is the following : — 



In the hydrodynamic equations the velocity and the pressure 

 of the current-particles are treated as continuous functions of 

 the coordinates. But, on the other hand, if we consider a per- 

 fect liquid (that is, one not subject to friction), there is nothing 

 essential in its nature to prevent two immediately neighbouring 

 liquid layers from slipping past one another with finite velocity. 

 At all events, those properties of fluids which are considered in 

 the hydrodynamic equations, namely the constancy of the mass 

 in all units of volume and the equality of the pressure in all di- 

 rections, clearly offer no obstacle to the occurrence of tangental 

 motions of finite difference of magnitude on both sides of a plane 

 passing through the interior. The components perpendicular to 

 the surface of the velocity and the pressure must, on the other 

 hand, of course be equal on both sides of such a surface. In my 

 work on gyratory movements*, I have already pointed out that 

 such a case must occur if two previously separated masses of 

 water come into superficial contact when they have different 

 motions. In my above-mentioned work I was led to the idea 

 of such a surface of separation, or, as it was then called, "gyra- 

 tion-surface" by the conception of lines of gyration arranged 

 continuously along a surface, the mass of which may be vanish- 

 ingly small, while its moment of rotation remains finite. 



Now, in a fluid which at first is at rest, or in continuous mo- 

 tion, a finite difference of motion between immediately neigh- 

 bouring particles can only be brought about by a moving force 

 which acts discontinuously. Of external forces, we here only 

 consider the force of impact. 



But there exists also in the interior of the fluids a cause which 

 may give rise to discontinuity in the motion. Pressure, indeed, 

 may assume any positive value, and the density of the fluid will 

 vary continuously with the pressure. But the moment the pres- 

 sure passes zero and commences to become negative, a disconti- 

 nuous change in the density takes place, the fluid is broken 

 asunder. 



Now the magnitude of the pressure in a moving fluid depends 

 upon the velocity, and, indeed, in incompressible fluids the dimi- 

 nution of the pressure under otherwise similar circumstances is 

 directly proportional to the vis viva of the moving particles. If 

 the vis viva exceeds a certain amount, the pressure must in fact 

 * Journal fiir reine una* angewandte Mathematik, vol. lx. 



