PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 249 



at the publication of that theory. And besides the idea itself, the manner of calcu- 

 lation also, which has been thus exemplified in the sciences of optics and dynamics, 

 seems not confined to those two sciences, but capable of other applications ; and the 

 peculiar combination which it involves, of the principles of variations with those of 

 partial differentials, for the determination and use of an important class of integrals, 

 may constitute, when it shall be matured by the future labours of mathematicians, a 

 separate branch of analysis. 



William R. Hamilton. 

 Observatory, Dublin, 

 March 1834. 



Integration of the Equations of Motion of a System, characteristic Function of such 



Motion, and Law of varying Action. 



1. The known differential equations of motion of a system of free points, repelling 

 or attracting one another according to any functions of their distances, and not dis- 

 turbed by any foreign force, may be comprised in the following formula : 



2.m{x"lx+y"ly + z"hz) =:hV (1.) 



In this formula the sign of summation 2 extends to all the points of the system ; m is, 

 for any one such point, the constant called its mass ; x", y", z", are its component ac- 

 celerations, or the second differential coefficients of its rectangular coordinates x,y, z, 

 taken with respect to the time; I x, hy, h z, are any arbitrary infinitesimal displace- 

 ments which the point can be imagined to receive in the same three rectangular 

 directions ; and ^ U is the infinitesimal variation corresponding, of a function U of 

 the masses and mutual distances of the several points of the system^ of which the 

 form depends on the laws of their mutual actions, by the equation 



V = 'Z,mmJ{r), (2.) 



r being the distance between any two points m, m^, and the function / (r) being such 

 that its derivative or differential coefficient/' (r) expresses the law of their repulsion, 

 being negative in the case of attraction. The function which has been here called U, 

 may be named the force-function of a system : it is of great utility in theoretical 

 mechanics, into which it was introduced by Lagrange, and it furnishes the following 

 elegant forms for the differential equations of motion, included in the formula (1.) : 

 „ _ UJ „ _ 8U /, _ ^ - 



„ 8 U „ S U „ 8 U . . . 



„ 8U „ 8U „ 8U 



m,z, = ^-; m,z^=^^; ...m^z^^ = ^j 



the second members of these equations being the partial differential coeflicients of 



