i86 HERMANN VON HELMHOLTZ 



Piotrowski, the latter having done the experimental work under 

 Helmholtz's direction, and presented it on April 12, 1860, 

 under the title, ' The Friction of Liquids/ to the Academy of 

 Vienna. 



The equations of motion within a non-viscous fluid mass 

 subject to friction had been developed earlier by Poisson, 

 Navier, and Stokes, and confirmed by experiments conducted 

 in very long, narrow tubes, but it had proved impracticable to 

 reconcile theory with experiment when the tubes were wide. 

 Helmholtz now undertook to investigate a second case of 

 motion in fluids (the theory of which can be derived com- 

 pletely from the hydrodynamic equations for fluids exerting 

 friction), in order to obtain a new determination of the constant 

 for the internal friction of water, so far derived only from 

 Poiseuille's observations, and to compare the same with the 

 observations. He succeeded in proving this for the movement 

 of water in a sphere, polished and gilded inside, by throwing 

 the spherical vessel into vibration, round a perpendicular axis, 

 by means of a special apparatus, while the lag in the vibrations 

 in the fluid was measured with a reflecting mirror and tele- 

 scope. In this case, the force exerted by the fluid within the 

 vessel upon its walls was experimentally determined, and com- 

 pared with the force calculated from the mathematical theory 

 of the motions of fluids. 



He simplified the hydrodynamic equations by making the 

 vibrations of the sphere so small that the squares of the velo- 

 city vanished as compared with its first power, and thereby 

 succeeded, on the assumption that gravity was the sole external 

 force, in finding particular integral equations, by which the 

 components of the velocity at any moment of the water, present 

 at a given point, could be expressed as the product of the co- 

 ordinates, multiplied by a function of the time and of the 

 distance of the point from the origin of the co-ordinates. This 

 function satisfies a differential equation, analogous to the 

 known potential equation of a sphere, which here contains 

 a further factor, involving the friction-constant for the interior 

 of the fluid ; the form of the motion corresponding with this 

 integral equation may be described by saying that the mass 

 of water splits into concentric spherical layers, each of which 

 performs a rotary movement like that of a thin hollow sphere 



