432 M. C. Szily on Hamilton's Dynamic 



of points in a perfectly analogous form by the equation 



2SE=2^p (10) 



But this modification is truly so slight, and follows so natu- 

 rally from the notion of the system, that we may regard equa- 

 tion (10) as the necessary consequence of Hamilton's equation. 

 As, however, this equation does not occur in Hamilton, and he 

 limited the employment of the summation-symbol to the case of 

 the points all accomplishing their paths in the same time, it will 

 perhaps not be unadvisable to effect a special demonstration, 

 after Hamilton's method, of equation (10). It is moreover to 

 be remarked that this equation is first found in Boltzmann, and 

 that the demonstration here given deviates essentially from his 

 only at the commencement, and afterwards only with respect to 

 the arrangement. 



Let m be the mass of any point whatever of a system, x, y, z 

 its rectangular coordinates, v the velocity, T the vis viva, U the 

 force-function, E the energy, and i the period of a revolution. 

 T and U have different values at different places in the path ; 

 but their sum (that is, E) will remain constant so long as the 

 point describes the same path. Hence this quantity is indepen- 

 dent of the time. Let the motion of the point undergo a change : 

 instead of its former closed path and periodic motion, let it fol- 

 low a new closed path infinitesimally different from the previous 

 one. In this new path the mass of the point alone remains as 

 before; all the rest of the quantities are in general changed. 

 Let the variations springing from the change of path be denoted 

 by 8. It follows, from the conception of the energy, that 



SE = ST + SU. 

 If we multiply this equation by the time-element and integrate 

 it from to i, and at the same time take into account that both 

 E and SE are independent of the time, then 



8V=l(\8T + B\J)dt. 



Let us form analogous expressions for the other points of the 

 system, and sum them : 



S3E = sir(8T + SU)^, 

 which can also be written thus : — 



