118 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



come to consideration, may be treated by the principle of least action. 

 As the forces of heat had already been referred to the hidden motion 

 of conceivable masses, and as Maxwell had recognized the source of 

 electrodynamic actions in the motion of unseen masses, Helmholtz 

 wished to introduce the motion and energy of such hidden masses gen- 

 erally in physical problems. He recognized as antecedent to obscure 

 phenomena motion and masses, which are to our senses invisible. 



He chose Hamilton's principle for the expression of all motion, since 

 it admits that upon a mechanical system whose inner forces may be 

 determined as differential coefficients of force functions independent of 

 the time, external forces may be exerted depending on the time. The 

 work done by such forces is to be independently computed as not 

 belonging to the conservative process, but dependent on other physical 

 events. 



Since, as Lagrange has showed, the outwardly directed forces of the 

 moving system may be expressed through the principal function, 

 Helmholtz called this the '• kinetic potential," and thus by the principle 

 of least action there follows this general characteristic of the progress 

 of all physical phenomena: The negative mean value of the kinetic 

 potential reckoned for equal time elements along the path is a minimum, 

 or when longer intervals are considered it has a limiting value in com- 

 parison with all other neighboring paths which lead from the starting 

 point to the end point in the same time. The kinetic potential goes 

 over into potential energy for the case of a body at rest, and from the 

 Hamilton principle it follows that for equilibrium the potential energy 

 is a minimum. It was already known that when certain coordinates 

 are represented in the value of the principal function only by their 

 differential coefficients, and the corresponding forces are equal to zero, 

 the Lagrange expression for the forces acting along the other coordi- 

 nates becomes, analytically, exactly as in the general cases, a trans- 

 formed principal function, which no longer as before contains the deriv- 

 itives of the coordinates only in the second, but contains them also in 

 the first degree. Thus, forms of the kinetic potential may appear in 

 which the separation of the two forms of energy can not be recognized. 

 Indeed, the kinetic potential may be any function whatever of the 

 general coordinates and of the corresponding velocities. These facts 

 led Helmholtz to inquire what form the principal function must take in 

 order that the Lagrange expression for the external forces should 

 remain unchanged. He found at once that this condition is satisfied 

 when this function is increased by the sum of the products of the coor- 

 dinates into the exterior forces expressed as a function of the time and 

 resolved along these coordinates. This expanded expression for the 

 law of least action gives the Lagrange formula for the forces immedi- 

 ately on performing the variation. 



The importance of Lagrange's form of the equations of motion had 

 already been shown by Helmholtz, since it is applicable to cases where 



