III. 



ON DISCONTINUOUS MOTIONS IN LIQUIDS.* 



By Prof. H. VON Helmholtz. 



It is well known that the hydro-dynamic equations give precisely the 

 same partial differential equations for the interior of an incompressible 

 fluid that is not subject to friction and whose particles have no mo- 

 tion of rotation, as obtain for stationary currents of electricity or heat 

 in conductors of uniform conductivity. One might therefore expect 

 that for the same external form of the space traversed by the cur- 

 rent and for the same boundary conditions the form of the current (ex- 

 cept for differences depending on small incidental conditions), would be 

 the same for liquids, for electricity, and for heat. In reality however 

 in many cases there exist easily recognizable and very fundamental 

 differences between the currents in a liquid aud the above mentioned 

 imponderables. 



Such differences are especially notable when the currents flowing 

 through an opening with sharp edges enter into a wider space. In such 

 cases the stream lines of electricity radiate from the openiug outwards 

 immediately towards all directions, while a flowing fluid, water as 

 well as air, moves from the opening at first forward iu a compact stream 

 which at a less or greater distance then ordinarily resolves itself into a 

 whirl. The portions of the fluid iu the larger receiving vessel lying 

 near the opening but outside the stream can, on the other hand, remain 

 almost at perfect rest. Everyone is familiar with this mode of motion, 

 especially as a current of air impregnated with smoke shows it very 

 plainly. In fact the compressibility of the air does not come much into 

 consideration in these processes, and with slight variations air sho'ws 

 the same forms of motion as does water. 



On account of the great differences between the facts as observed 

 aud the results of theoretical analysis as hitherto achieved the hydro- 

 dynamic equations must necessarily appear to the physicist as a prac- 



* From the MounisJxrichic of the Royal Academy of Science, Berlin. 1868, April 

 2:?, pp. 215-228. Helmholtz JVhsenschaftliche Abhandlungen, vol. I, pp. 146-157. Ber- 

 lin, 1832. 

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