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VII. Second Essay on a General Method in Dynamics. By William Rowan Hamilton, 

 Member of several Scientific Societies in Great Britain and in Foreign Countries, 

 Andrews Professor of Astronomy in the University of Dublin, and Royal Astro- 

 nomer of Ireland. Communicated by Captain Beaufort, R.N. F.R.S. 



Received October 29, 1834,— Read January 15, 1835. 



Introductory Remarks. 

 1 HE former Essay* contained a general method for reducing all the most important 

 problems of dynamics to the study of one characteristic function, one central or ra- 

 dical relation. It was remarked at the close of that Essay, that many eliminations 

 required by this method in its first conception, might be avoided by a general trans- 

 formation, introducing the time explicitly into a part S of the whole characteristic 

 function V ; and it is now proposed to fix the attention chiefly on this part S, and to 

 call it the Principal Function. The properties of this part or function S, which were 

 noticed briefly in the former Essay, are now more fully set forth ; and especially its 

 uses in questions of perturbation, in which it dispenses with many laborious and cir- 

 cuitous processes, and enables us to express accurately the disturbed configuration of 

 a system by the rules of undisturbed motion, if only the initial components of veloci- 

 ties be changed in a suitable manner. Another manner of extending rigorously to 

 disturbed motion the rules of undisturbed, by the gradual variation of elements, in 

 number double the number of the coordinates or other marks of position of the 

 system, which was first invented by Lagrange, and was afterwards improved by 

 PoissoN, is considered in this Second Essay under a form perhaps a little more ge- 

 neral ; and the general method of calculation which has already been applied to 

 other analogous questions in optics and in dynamics by the author of the present 

 Essay, is now applied to the integration of the equations which determine these ele- 

 ments. This general method is founded chiefly on a combination of the principles of 

 variations with those of partial diff*erentials, and may furnish, when it shall be ma- 

 tured by the labours of other analysts, a separate branch of algebra, which may be 

 called perhaps the Calculus of Principal Functions ; because, in all the chief applica- 

 tions of algebra to physics, and in a, very extensive class of purely mathematical 

 questions, it reduces the determination of many mutually connected functions to the 

 search and study of one principal or central relation. When applied to the integration 

 of the equations of varying elements, it suggests, as is now shown, the consideration 



* Philosophical Transactions for the year 1834, Second Part. 



