104 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



and based upon the negative issue of numerous futile attempts to con- 

 struct a perpetual motion. This great general law governing the quan- 

 titative relations which must subsist during all transformations, does 

 not, however, determine whether work can be changed into heat with- 

 out reserve, and vice versa, and the same uncertainty exists with 

 regard to light, electricity, and other forms of energy. These are 

 questions whose answer shall later exhibit the deep and comprehensive 

 significance of the energy conception in mathematical physics. 



After Helmholtz had investigated the physical aspects of this funda- 

 mental principle of mechanics from most varied points of view, he 

 turned his attention to physiological researches growing out of his notice 

 on the " Theory of acoustics," and from this to very general mechanical 

 problems and special hydrodynamic investigations. In the year 1858 

 appeared his famous memoir " Upon the integrals of the hydrodynamic 

 equations which correspond to wave motions." This research formed 

 the foundation for an entirely new conception of the motions of fluids, 

 which was later made fruitful in various branches of physics, notably 

 by W. Thomson (Lord Kelvin) in his theory of vortex atoms, and by 

 other physicists as well. Upon the assumption that for a perfect fluid — 

 that is, one in which there is no friction between the particles — the 

 pressure is equal in all directions, Euler and Lagrange had already 

 obtained analytical relations between the pressure in the fluid, its 

 density, the time, the coordinates of the particle under consideration, 

 and, on the one hand, the velocity components; on the other, the 

 position of this particle at the beginning of the motion. Further, they 

 had inferred the so-called continuity equation, which required that the 

 mass of a given particle of the fluid should not change with the time, 

 therefore that the surface of the liquid should be continually com- 

 posed of the same particles. 



All these equations form for the perfect fluid the analogue of the 

 principle of d'Alembert and lead to the determination of the variable 

 quantities through the time and the original situation as a mathemat- 

 ical problem whose solution, to use an expression of Kirchhoff, would 

 describe the motion. The problem can be solved for some particular 

 cases in which the components of the velocity of each fluid particle 

 may be placed equal to the differential coefficients of a determined 

 function, which Helmholtz called the velocity potential, along the corre- 

 sponding directions. This function, for incompressible fluids at least, 

 has the same form as the potential of gravitating masses for points out- 

 side of them. But such a velocity potential does not always exist, and 

 so Helmholtz attacked the extremely difficult problem of the forms of 

 motion with complete generality in the memoir above referred to, 

 which appeared in the year of his coming to Heidelberg. 



First of all, he recognized that the change which an indefinitely 

 small volume of fluid undergoes in an indefinitely small interval of 

 time is composed of three different motions — a displacement of the 



