resulting from Hamilton's Theory of Motion. 283 



mogeneous function of the second degree of the variable q\ ; 

 therefore 



consequently we have simply 



and after substitution in the above equation, 



-SV=%S?,.-2 ft %»-£s|^S C/e *-E&. . . (4) 



This form, connecting itself with Hamilton's equation*, is the 

 starting-point for the following. At the same time it is signifi- 

 cant that the ordinary equations of motion of Lagrange are re- 

 garded as satisfied only in the motion itself, and not during the 

 change of motion. The system must therefore, in the motion, 

 always be a closed one, subject to no action from without ; on 

 the contrary, during the variation of motion such an action from 

 without must take place. Meanwhile the energy of the system 

 may remain constant or vary ; whether the one or the other, has 

 no influence on the validity of equation (4). This independence 

 of Hamilton's equation upon the nature of the variation of the 

 motion has the same signification as that of Lagrange's equation 

 of motion upon the nature of the variation of the configuration. 

 If, therefore, Lagrange's method reaches to systems with and 

 without conditions, Hamilton's equation (4) extends to systems 

 which with the alteration of their motion retain the energy con- 

 stant or even receive energy from without. 



§2. 



Hamilton's symbolic equation of motion plays in the treat- 

 ment of the mechanical problem a part like that of the symbolic 

 equation of motion of Lagrange, only with the difference that it 

 refers to the variation of the motion, while the latter concerns 

 the variation of the configuration in a motion. If, now, in La- 

 grange's method from the equation of motion a series of princi- 

 ples result which have partly the purely analytical signification 

 of integrals of the differential equations, and partly the essen- 

 tially physical meaning of general propositions valid for motion 

 generally, the question arises whether similar principles do not 

 connect themselves with Hamilton's equation. Tins shall be 

 investigated especially in regard to the proposition concerning 



* Phil. Trans. 1834, p. 30/. 



