315 



appears to be very different in different places : no trace of such a 

 change is found in the South of Scania. In those places where its 

 amount was ascertained with greatest accuracy, it appears to be about 

 three feet in a century. The phenomenon in question having ex- 

 cited increasing interest among the philosophers of Sweden, and having 

 especially excited the attention of Professor Berzelius, it is to be hoped 

 that the means of accurate determination will be greatly multiplied. 



January 15, 1835. 



JOHN WILLIAM LUBBOCK, Esq., M.A., V.P. and Treasurer, in 



the Chair. 



Second Essay on a general Method in Dynamics. By William 

 Rowan Hamilton, Esq., Andrew's Professor of Astronomy in the Uni- 

 versity of Dublin, and Royal Astronomer of Ireland. Communicated 

 by Captain Beaufort, R.N., F.R.S. 



This essay is a sequel of the one which appeared in the last volume 

 of the Philosophical Transactions, and which contained a general me- 

 thod for reducing all the most important problems of dynamics to the 

 study of one characteristic function, or one central or radical relation. 

 It was there remarked that many eliminations required by this me- 

 thod might be avoided by a general transformation, introducing the 

 lime explicitly into a part (S) of the whole characteristic function (V) ; 

 and the first object of the present essay is to examine and develope 

 the properties of this part (S), which the author designates by the 

 term Principal Function. This function is applied by the author to 

 problems of perturbation, in which he finds it dispenses with many 

 laborious and circuitous processes, and furnishes accurate expressions 

 of the disturbed configurations of a system by the rules of undisturbed 

 motion, if only the initial components of velocities be changed in a 

 suitable manner. Another manner of extending rigorously to dis- 

 turbed, the rules of undisturbed motion, 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 rather more general 3 and the general method of 

 calculation which has already been applied by the author to other 

 ' analogous questions in optics and in dynamics, is now applied to the 

 integration of the equations which determine these elements. This 

 general method is founded chiefly on a combination of the principle 

 of variations with those of partial differentials, and may furnish, when 

 matured, a separate branch of analysis, which may be denominated 

 the Calculus of Principal Functions. When applied to the integra- 

 tion of the equations of varying elements, it suggests the consideration 

 of a certain Function of Elements, capable of being variously trans- 

 formed, and which may be either rigorously determined, or at least 

 approached to, by a corollary of the general method. With a view to 

 illustrate these new principles, and more esoecially those connected 



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