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XV. On a Genetml Method m Dynamics ; hy which the Study of the Motions of all free 

 Systems of attracting or repelling Points is reduced to the Search and Differentiation 

 of one central Relation, or characteristic Function. By William Rowan Hamilton, 

 Member of several scientific Societies in the British Dominions, and of the American 

 Academy of Arts and Sciences, Andrews Professor of Astronomy in the University 

 of Dublin, and Royal Astronomer of Ireland. Communicated by Captain Beaufort, 

 R.N.F.R.S. 



Received April 1,— Read April 10, 1834. 



Introductory Remarks. 



A HE theoretical development of the laws of motion of bodies is a problem Qf such 

 interest and importance, that it has engaged the attention of all the most eminent 

 mathematicians, since the invention of dynamics as a mathematical science by 

 Galileo, and especially since the wonderful extension which was given to that science 

 by Newton. Among the successors of those illustrious men, Lagrange has perhaps 

 done more than any other analyst, to give extent and harmony to such deductive 

 researches, by showing that the most varied consequences respecting the motions of 

 systems of bodies may be derived from one radical formula; the beauty of the 

 method so suiting the dignity of the results, as to make of his great work a kind of 

 scientific poem. But the science of force, or of power acting by law in space and 

 time, has undergone already another revolution, and has become already more dyna- 

 mic, by having almost dismissed the conceptions of solidity and cohesion, and those 

 other material ties, or geometrically imaginable conditions, which Lagrange so hap- 

 pily reasoned on, and by tending more and more to resolve all connexions and 

 actions of bodies into attractions and repulsions of points : and while the science is 

 advancing thus in one direction by the improvement of physical views, it may 

 advance in another direction also by the invention of mathematical methods. And 

 the method proposed in the present essay, for the deductive study of the motions of 

 attracting or repelling systems, will perhaps be received with indulgence, as an 

 attempt to assist in carrying forward so high an inquiry. 



In the methods commonly employed, the determination of the motion of a free 

 point in space, under the influence of accelerating forces, depends on the integration 

 of three equations in ordinary differentials of the second order ; and the determina- 

 tion of the motions of a system of free points, attracting or repelling one another, 

 depends on the integration of a system of such equations, in number threefold the 



2 K 2 



