resulting from Hamilton's Theory of Motion. 279 



repelling, but subject to no other forces, so that the soliciting 

 forces can be represented by the negative partial differential 

 quotients of a function of the coordinates of all the points, the 

 force-function U. This function contains as variable quantities 

 at all events the coordinates q t of the points in motion, of which 

 it is here always presupposed that they identically satisfy at any 

 moment the equations of condition, of whatever form 5 and there- 

 fore, if m such equations are given, occur to the number of 

 3n — ?n=/jL. Moreover the time t may appear explicitly in the 

 force- function, as well as other quantities c k) which change only 

 when a transition takes place from one motion to another. For 

 motions of this general sort, Hamilton's method for gaining the 

 general symbolical equation of motion which refers to the varia- 

 tion of the motion is to be extended. If the vis viva of the 

 point-system be denoted by T, and the primary function V de- 

 lined by 



.in 



the problem is nearer to that of finding the variation of this inte- 

 gral on the hypotheses made. 



In forming this variation, the time t is first regarded as an 

 independent variable which is not variated. All the quantities 

 present in the primary function are therefore regarded as func- 

 tions of t and a number of arbitrary constants ; and from the 

 variation of these constants alone will the variation of those 

 quantities, and hence that of the primary function, result. Of 

 such arbitrary quantities there will always be 2/u, in the quantities 

 mentioned, which can be supposed to arise from the integration 

 of the /j, differential equations of the second order of the motion ; 

 but since a variation of the force-function on the transition from 

 one motion to another is presupposed, to those 2/jl constants any 

 number of others may be added; these latter, which at all events 

 are assumed to be independent of one another, are the quantities 

 c k . If, then, these 2/m-\-v constants change, but t be supposed 

 unchanged, we obtain 



Jo Jo 



and we have only to do with the variation of the quantity (T — U). 

 Since the equations of condition of the system may explicitly 

 contain the time t, the vis viva T will in general, as well as the 

 force-function U, likewise explicitly contain it; but since the 

 time is not variated, in the formation of the total variation 8V 



