HERMANN VON HELMHOLTZ 



value, and thus eliminated the increment of time 

 from the action. Hamilton stated the principle of 

 least action in yet another form. 1 He showed that 

 the complete solution of any kinetical problem, as 

 regards the action of any conservative system of 

 forces and constraint depending on the reaction of 

 smooth surfaces or curves, is reducible to the deter- 

 mination of a single quantity, which he called the 

 characteristic function of the motion. This quantity 

 is found from a partial differential equation of the 

 first order and second degree, and from the com- 

 plete integral of this equation all the circumstances 

 of the motion may be deduced by differentiation. 

 This has been called the method or principle of 

 varying action. 2 



The principle of least action may be thus briefly 

 stated : Given a conservative system in any con- 

 figuration, and different paths by which it could be 

 guided to any other definite configuration, under the 

 condition that the sum of its potential and kinetic 

 energies is constant, then the path for which the 

 action is least is the one along which the system 

 would move unguided if the proper initial velocities 

 were given to it. The term action* as applied to a 



' W. R. Hamilton on a General Method of Dynamics. Phil. Trans. 

 1834,1835. 



2 Thomson and Tail's Natural Philosophy, vol. i., pt. insect. 333 ; also 

 Tait on the Application of 'Hamilton 's Characteristic Function to Special Cases 

 of Constraint. Trans. Roy. Sec. Edin., vol. xxiv., and Scientific Papers, vol. 

 i., p. 54. 



244 



