272 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



+ v^ (2^ m a' . 2^ m ^ a -|- 2^ m /3' . 2^ m S |3 + 2^ m y' . 2, m § y), 



> • (Z'.) 



and might have been otherwise deduced by our rule, from this other known trans- 

 formation of T^, 



And to obtain, with any set of internal or relative marks of position, the two partial 

 differential equations which the characteristic function V^ of relative motion must 

 satisfy, and which offer (as we shall find) the chief means of discovering its form, 

 namely, the equations analogous to those marked (F.) and (G.), we have only to eli- 

 minate the rates of increase of the marks of position of the system, which determine 

 the final and initial components of the relative velocities of its points, by the law of 

 varying relative action, from the final and initial expressions of the law of relative 

 living force ; namely, from the following equations : 



T, = U + H„ (50.) 



and 



T^ = Uo + H, (70.) 



The law of areas, or the property respecting rotation which was expressed by the 

 partial differential equations (P.), will also always admit of being expressed in rela- 

 tive coordinates, and will assist in discovering the form of the characteristic function 

 V^ ; by showing that this function involves only such internal coordinates (in number 

 6 w — 9) as do not alter by any common rotation of all points final and initial, round 

 the centre of gravity, or round any other internal origin ; that origin being treated as 

 fixed, and the quantity H^ as constant, in determining the effects of this rotation. The 

 general problem of dynamics, respecting the motions of a free system of n points 

 attracting or repelling one another, is therefore reduced, in the last analysis, by the 

 method of the present essay, to the research and differentiation of a function V^, 

 depending on Gn— 9 internal or relative coordinates, and on the quantity H^, and 

 satisfying a pair of partial differential equations of the first order and second degree ; 

 in integrating which equations, we are to observe, that at the assumed origin of the 

 motion, namely at the moment when # = 0, the final or variable coordinates are equal 



to their initial values, and the partial differential coeflScient ^^ vanishes ; and, that 



at a moment infinitely little distant, the differential alterations of the coordinates have 

 ratios connected with the other partial differential coefl&cients of the characteristic 

 function V^ by the law of varying relative action. It may be here observed, that. 



