Royal Society, 299 



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, w^hich was first invented by Lagrange, 

 and was afterwards improved by Poisson, is considered in this second 

 essay under a form rather more general j 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, whea 

 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, itsuggests 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 especially those connected 

 with problems of perturbation, they are applied, in this essay, first, 

 to a very simple example, suggested by the motions of projectiles, 

 the parabolic path being treated as the undisturbed; and secondly, to 

 the problem of determining the motions of a ternary or multiple 

 system, with any laws of attraction or repulsion, and with one pre- 

 dominant mass. This latter problem, which was touched upon in the 

 former es.say, is here resumed in a new manner, by forming and in- 

 tegrating the differential equations of a new set of varying elements, 

 entirely distinct in theory (though little differing in practice) from 

 the elements conceived by Lagrange; 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. 



An Account of the Eruption of Mount Etna in the year 1536, from 

 an original cotemporary document, communicated in a letter to J. G. 

 Children, Esq., Secretary of the Royal Society. By Sir Francis 

 Palgrave, K.G.H., F.R.S. 



Record Office of the Treasury, Chapter House, 

 Poets' Corner, Westminster, Jan. 14, 1835. 

 Amongst various shreds and fragments of the correspondence from 

 Italy dqring the period that Henry VIIL was negotiating with the 

 Italian princes, is a document of a very different nature from the rest, 

 being an extract from a letter written by the Barone di Burgis, dated 

 at Palermo, 1 0th of April 1536, and giving an account of the then re- 

 cent eruption of Mount Etna. 



** Die xxiij. Martii, M. D. xxxvi., nocte, Mons Elhna qui nunc 

 Mongibellus vocatur; facto, orientem versus, ostio, emisit materiam 

 igneam, quae ad instar fluminis vagata est per octo miliaria in longi- 

 tudine, et per unum miliare in latitudine ; ejus vero altitudo erat 

 palmarum duodecim. Eadem nocte ignis extinctus est, et ubique 



2 Q2 



