PAPER BY PROF. HELMHOLTZ. 71 



even in extensive spaces, the compressibility loses its influence. Under 

 such circumstances gases move like cohesive incompressible fluids 

 [viz, liquids], as is well known practically from many examples. 



If the velocities of the material parts are in general very small, as in 

 the case of exceedingly small oscillations, so that the course of the 

 movement remains sensibly unchanged for a uniform increase in these 

 velocities, then it will only be the velocity of sound that changes, and 

 our proposition will take the following form : The sonorous vibrations 

 of a compressible fluid can, in larger spaces, behave mechanically the 

 same as more rapid oscillations of a less compressible fluid in smaller 

 spaces. An example of the utilization of the similarity here spoken of 

 is found in my investigations on the acoustic movement at the ends 

 of open organ pipes.* In that study the possibility of replacing the 

 analytical conditions of the motion of the air by the simpler ones of 

 the motion of water depended on the principle that the dimensions of 

 the given spaces must be very small in comparison to the wave lengths 

 of the existing acoustic vibrations. 



On the other hand the viscosity also shows itself less influential in the 

 movements of fluids in large spaces. If we let n remain unchanged 

 while q increases we obtain the same ratio between the frictional forces 

 and the pressure forces. That is to say, if we increase the dimensions 

 and the friction constants in the same ratio, then the movements in the 

 enlarged system remain similar so long as the velocities do not change. 

 Hence it follows that in such an enlarged model, when the friction con- 

 stant is not increased in the same ratio, but remains unchanged, the 

 friction loses in influence for the same velocity. That which holds 

 good for greater dimensions with unchanged velocities also obtains for 

 increased velocities with unchanged dimensions. For one can also 

 simultaneously let n increase proportional to q. 



In fact, in most practical experiments in extended fluid masses, the 

 resistance that arises from the accelerations of the fluid,t and especially 

 in consequence of the formation of surfaces of discontinuity is by far 

 the most important. Its magnitude increases proportionally to the 

 square of the velocity, whereas the resistance depending upon the fric- 

 tion proper (internal friction or viscosity and surface-hesiou), which 

 increases simply in proportion to the velocity, becomes appreciable only 

 in experiments in very narrow tubes and vessels. 



Neglecting the friction, that is to say, if in the above equations we 



put the constants 



h=K=0 



then will the constant q also become arbitrary, and we can change the 

 dimensions and velocities in any ratio whatever. 

 If however the force of gravity comes into consideration as in the 



* Borchardt's Journal fur Mathematik, 1859, vol. lvii, pp. 1-72- 

 t [These resistances are those that I have called " collective" in my Treatise on 

 Meteorological Methods and Apparatus.— C. A.~\ 



