248 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



number of the attracting or repelling points^ unless we previously diminish by unity 

 this latter number, by considering only relative motions. Thus, in the solar system, 

 when we consider only the mutual attractions of the sun and of the ten known 

 planets, the determination of the motions of the latter about the former is reduced, 

 by the usual methods, to the integration of a system of thirty ordinary differential 

 equations of the second order, between the coordinates and the time ; or, by a trans- 

 formation of Lagrange, to the integration of a system of sixty ordinary differential 

 equations of the first order, between the time and the elliptic elements : by which 

 integrations, the thirty varying coordinates, or the sixty varying elements, are to be 

 found as functions of the time. In the method of the present essay, this problem is 

 reduced to the search and differentiation of a single function, which satisfies two 

 partial differential equations of the first order and of the second degree : and every 

 other dynamical problem, respecting the motions of any system, however numerous, 

 of attracting or repelling points, (even if we suppose those points restricted by any 

 conditions of connexion consistent with the law of living force,) is reduced, in like 

 manner, to the study of one central function, of which the form marks out and cha- 

 racterizes the properties of the moving system, and is to be determined by a pair of 

 partial differential equations of the first order, combined with some simple considera- 

 tions. The difficulty is therefore at least transferred from the integration of many 

 equations of one class to the integration of two of another : and even if it should be 

 thought that no practical facility is gained, yet an intellectual pleasure may result 

 from the reduction of the most complex and, probably, of all researches respecting 

 the forces and motions of body, to the study of one characteristic function*, the un- 

 folding of one central relation. 



The present essay does not pretend to treat fully of this extensive subject, — a task 

 which may require the labours of many years and many minds ; but only to suggest 

 the thought and propose the path to others. Although, therefore, the method may be 

 used in the most varied dynamical researches, it is at present only applied to the 

 orbits and perturbations of a system with any laws of attraction or repulsion, and 

 with one predominant mass or centre of predominant energy ; and only so far, even 

 in this one research, as appears sufficient to make the principle itself understood. It 

 may be mentioned here, that this dynamical principle is only another form of that 

 idea which has already been applied to optics in the Theory of systems of rays, and 

 that an intention of applying it to the motions of systems of bodies was announced -f- 



* Lagrange and, after him, Laplace and others, have employed a single function to express the different 

 forces of a system, and so to form in an elegant manner the differential equations of its motion. By this con- 

 ception, great simplicity has been given to the statement of the problem of dynamics ; but the solution of that 

 problem, or the expression of the motions themselves, and of their integrals, depends on a very different and 

 hitherto unimagined function, as it is the purpose of this essay to show. 



t Transactions of the Royal Irish Academy, vol. xv. page 80. A notice of this dynamical principle was also 

 lately given in an article " On a general Method of expressing the Paths of Light and of the Planets," pub- 

 lished in the Dublin University Review for October 1833. 



