PAPER BY PROF. HELMHOLTZ. 59 



tically very imperfect approximation to the reality. The cause of this 

 might be suspected to lieiu the internal friction or viscosity of the fluid, 

 although all forms of infreqent and sudden irregularities (with which 

 certainly everyone has to contend who has instituted observations on 

 the movements of fluids) can evidently never be explained as the eflect 

 of the steadily and uniformly acting friction. 



The investigation of cases where periodical movements are excited 

 by a continuous current of air, as, for example, in organ pipes, showed 

 me that such an effect could only be produced by a discontinuous motion 

 of the air, or at least by a hind of motion coming very near to it, and 

 this has lead me to the discovery of a condition that must betaken into 

 consideration in the integration of the hydro-dynamic equations, aud 

 that, so far as I know, has been overlooked hitherto, whose considera- 

 tion on the other hand, in those cases where the computation can be 

 carried out, really gives, in fact, forms of motions such as those that are 

 actually observed. This condition is due to the following circumstance : 



In the hydro-dynamic equations the velocity and the pressure of the 

 flowing particles are treated as continuous functions of the coordinates. 

 On the other hand, there is no reason in the nature of a liquid, if we 

 consider it as perfectly fluid, therefore not subject to viscosity, why two 

 contiguous layers of liquid should not glide past each other with defi- 

 nite velocities. At least those properties of fluids that are considered 

 in the hydro-dynamic equations, namely, the constancy of the mass in 

 each element of space and the uniformity of pressure in all directions, 

 evidently furnish no reasons why tangential velocities of finite differ- 

 ence in magnitude should not exist on both sides of a surface located in 

 the interior. On the other hand, the components of velocity and of pres- 

 sure perpendicular to the surface must of course be equal on both sides of 

 such a surface. I have already in my memoir on vortex motions called 

 attention to the fact that such a case must occur when two moving 

 masses of liquid previously separate and having different motions come 

 to have their surfaces in contact. In that memoir I was led to the idea 

 of such a surface of separation,* or vortex surface as I there called itf 

 through the fact that I imagined a system of parallel vortex filaments 

 arranged continuously over the surface whose mass was indefinitely 

 small without losing their moment of rotation. 



Now, in a liquid at first quiet or in continuous motion a definite dif- 

 ference in the movement of immediately adjoining particles of liquid 

 can only be brought about through moving forces acting discontinu- 

 ously. Among the external forces the only oue that can here come 

 into consideration is impact. 



But in the interior of liquids there is also a cause present that can 



['Ordinarily called surface of discontinuity or " a discontinuous surface " by English 



[t That is, an infinitely thin layer of parallel vortex filaments, the " vortex sheet " ol 

 English writers.] 



