106 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



a distance from the province of physics. To this latter point I shall 

 return in discussing later researches of Helmholtz. 



Helniholtz soon became dissatisfied with his hydrodynamic investiga- 

 tions thus far referred to, and in the course of the preparation of his 

 famous handbook of physiological optics he became convinced that, in 

 order to eliminate the discordance between the results of theory and 

 experiment in the investigation of the problems of motion of liquids, it 

 would be necessary to take into consideration the friction between the 

 liquid particles and the sides of the vessel. As the problem (due to 

 some experiment of Bessel) of the vibrations of a pendulum ball under 

 the influence of a surrounding liquid had already been treated, he 

 investigated upon the basis of Bessel's observations, with the help of the 

 already known equations of motion, the condition of the interior of a 

 liquid mass which is subjected to the friction produced by the pro- 

 gressive rotary vibrations about one of its diameters of a pendulum 

 ball consisting of a liquid producing friction. He succeeded in solving 

 the problem mathematically, expressing the wave motions of the liquid 

 produced by friction, and in this way was able to check the experi- 

 mentally derived constants of viscosity for various liquids. 



Two laws, important both theoretically and practically, were discov- 

 ered by him, according to which, under certain circumstances, the flow 

 of viscous liquids through cylindrical tubes is so divided into stationary 

 streams that the loss in kinetic energy caused by the friction is a mini- 

 mum; and in cases of equilibrium of a body swimming in a slow sta- 

 tionary stream the friction itself assumes a minimum value. Following 

 this, in the year 1868, appeared his research, of great interest for the 

 theory of functions, "On discontinuous fluid motions," in which he 

 investigated still further the discharge of fluids and the formation of 

 independent streams, and treated of the discontinuity of motion char- 

 acteristic of fluid discharge and of the formation of vortices. Helm- 

 holtz assumed that from the nature of the problem — the determination 

 of the origin of independent fluid streams — a discontinuity must neces- 

 sarily be met with, and that therefore the fundamental hydrodynamic 

 equations must admit the possibility of a discontinuous relation 

 between the quantities appearing in them. In fact, in the motion of 

 an incompressible fluid the pressure, whose diminution is directly pro- 

 portional to the kinetic energy, becomes negative when the latter 

 exceeds a certain value, and the fluid must be then torn asunder. It 

 was shown that any geometrically completely sharp corner by which 

 the fluid flows must, with fair velocities, cause a parting of the liquid, 

 but that a blunted angle will only cause such a separation when the 

 velocity is considerably greater. With the help of the methods of the 

 theory of functions, the extremely difficult problem of the form of the 

 independent streams was discussed. In this discussion it was assumed 

 that, except for friction, no outside forces are acting, that the streams 

 are stationary, that the velocity potential depends only on two coordi- 



