resulting from Hamilton's Theory of Motion. 287 



introduced into the equation 



we get 



and from this, 



H l 



dV=~2Ldt + Tudt; 

 V=f'(E-2L)^ ...... ( 7 ) 



* 



a relation which is immediately obtained from the earlier defini- 

 tion-equation of V by making use of the proposition of the 

 energy, was used by Hamilton without hesitation as a definition 

 of the function V, and expresses the proper mechanical meaning 

 of the primary function. 



As, then, in (y) the primary function appears to be formed 

 analogous to the earlier quantity W, it will be convenient to in- 

 dicate this by a similar notation ; and the names of potential 

 and kinetic action may commend themselves for V and W. If 

 we put, moreover, 



A = V + W=E*, 



and name this simply the action of the system, equation (7) 

 changes into 



£-Bf]=°. « 



and the proposition reads : — That alteration of the action which 

 is conditioned by the valuation of the initial and final coordinates 

 alone, vanishes with the change of every motion that presupposes a 

 force-function and does not contain the time explicitly either in this 

 or in the limitations. In this form it may be designated as the 

 principle of the action. 



Here potential and kinetic action are quantities characterizing 

 the given motion in like manner as the potential and kinetic 

 energy the corresponding configuration. If we imagine the 

 whole series of constantly altered motions to be run through, 

 they will in general be distinguished by different values of these 

 quantities : in proportion as, by the mere alteration of the coor- 

 dinates, the one diminishes, the other increases through the 

 same alteration ; so that in this new view perfect correspondence 

 exists with the proposition of the energy. 



In regard to the limitations under which the proposition of 

 the action has been obtained, it is to be remarked that the for- 

 mation of I 2T dt, like that of T, remains possible even without 



Jo m 

 a force-function. Now the question, what in these cases becomes 



