96 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



of a certain Function of Elements, which may be variously chosen, and may either 

 be rigorously determined, or at least approached to, with an indefinite accuracy, 

 by a corollary of the general method. And to illustrate all these new general 

 processes, but especially those which are connected with problems of perturbation, 

 they are applied in this Essay to a very simple example, suggested by the motions of 

 projectiles, the parabolic path being treated as the undisturbed. As a more important 

 example, the problem of determining the motions of a ternary or multiple system, 

 with any laws of attraction or repulsion, and with one predominant mass, which was 

 touched upon in the former Essay, is here resumed in a new way, by forming and inte- 

 grating the differential equations of a new set of vaiying elements, entirely distinct 

 in theory (though little differing in practice) from the elements conceived by La- 

 grange, and having this advantage, that the differentials of all the new elements for 

 both the disturbed and disturbing masses may be expressed by the coefficients of one 

 disturbing function. 



Transformations of the Differential Equations of Motion of an Attracting or Repelling 



Si/stem. 



1 . It is well known to mathematicians, that the differential equations of motion of 

 any system of free points, attracting or repelling one another according to any func- 

 tions of their distances, and not disturbed by any foreign force, may be comprised in 

 the following formula : 



2.m{a^'^x+y"l7/ + z"^z)=z^V: . ......... (1.) 



the sign of summation 2 extending to all the points of the system ; m being, for any 

 one such point, the constant called its mass, and a?7/z being its rectangular coordi- 

 nates ; while jc"i/"z" are the accelerations, or second differential coefficients taken 

 with respect to the time, and Ix, ly, I z are any arbitrary infinitesimal variations of 

 those coordinates, and U is a certain force-function, introduced into dynamics by La- 

 grange, and involving the masses and mutual distances of the several points of the 

 system. If the number of those points be w, the formula (1.) may be decomposed into 

 3 n ordinary differential equations of the second order, between the coordinates and 

 the time, 



^»^» = FZ' '^yi = F^i ^i^i=^IT'' (2.) 



and to integrate these differential equations of motion of an attracting or repelling 

 system, or some transformations of these, is the chief and perhaps ultimately the only 

 problem of mathematical dynamics. 



2. To facilitate and generalize the solution of this problem, it is useful to express 

 previously the 3w rectangular coordinates j?y ^ as functions of 3w other and more 

 general marks of position ??i ^2 • • • ''s n 5 ^^^ ^^^n the differential equations of motion 

 take this more general form, discovered by Lagrange, 



