FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 117 



These matters formed the contents of memoirs which Helmholtz pub- 

 lished in 1886 and 1887 with the titles, "On the physical significance of 

 the principle of least action," and "The history of the principle of least 

 action." Hertz regarded these works at the time as marking the fur- 

 thest advance of physics. Defining, after Leibnitz, the quantitative 

 measure of the action following from the inertia of a moving mass as 

 the product of the mass into the space traversed into the velocity, or 

 as the product of the kinetic energy into the time, then the principle of 

 least action requires that the total amount of the action shall have a 

 limiting value in the passage from a given position of starting to a 

 given position of rest. In performing the variation the coordinates of 

 the points corresponding to intermediate positions of the system are 

 varied simultaneously with the time in such a manner that the total 

 energy of the system is not changed. This latter requirement can be 

 satisfied by the condition that the energy at a given instant shall be 

 the same for all variations as at the same instant in the unvaried 

 motion, without regard to the magnitude of this latter, which it is pos- 

 sible might change in the course of a normal motion. In this way 

 Lagrange and Hamilton have treated the problem. Jacobi, however, 

 made the preliminary condition that the potential energy is independent 

 of the time, and this requires that the amount of energy shall retain a 

 definite value, in which case this relation may be used to eliminate the 

 time increment from the action. Physically, Jacobi's restricting con- 

 dition holds for a completely determined and closed system, while the 

 Lagrange-Hamilton form of the equations of motion also holds true for 

 an incompletely closed system, upon which variable outside influences 

 are at work independent of the reaction of the moving system. 



Hamilton, keeping the Lagrange conditions, has given the principle 

 of least action another form in which it is called " Hamilton's principle." 

 Defining the principal function of Hamilton as the difference between 

 potential energy and the kinetic energy of the system, then the principle 

 which bears his name requires that the negative mean values of the 

 principal function, reckoned for equal time elements, shall have a 

 definite value for a normal motion between given points. 



But Lagrange, Hamilton, and Jacobi had proved the principle (first 

 stated, but not proved by Maupertuis in 1744) only under the physical 

 assumption of Newton's laws ; and the motion of the points of a material 

 system had been deduced from it under the condition of a rigid con- 

 nection of the points, and with the express assumption of the principle 

 of the constancy of energy. When Helmholtz had showed the general 

 validity of the law of the constancy of energy, this last hypothesis 

 remained a limitation no longer for cases in which all the forms are 

 known in which energy equivalents are transformed during the progress 

 of the change. It now remained to decide whether i>hysical processes 

 which depend not simply on the motions of determinate masses for 

 which Newton's laws are applicable, but in which quantities of energy 



