PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 257 





J 



(P.) 



and if we change the final coefficients of V to the final components of momentum, 

 and the initial coefficients to the initial components taken negatively, according to 

 the dynamical properties of this function expressed by the integrals (C.) and (D.), wc 

 shall change these partial differential equations (O.) (P.), to the following, 



2 . w 0?' = 2 . /« a' ; 2 . m y = 2 . w &' ; 2 . m x' = 2 . m c' ; . . . (15.) 

 and 



2 .m {xi/' — 7/x') = "^^ .m (a b* — b a') ; "] 



2.m{7/z' ^ z?/') = 2.m{bc' - cb'); > (16.) 



1, . m {z a/ — X z') = 2 . m (c a' — a d). J 



In this manner, therefore, we can deduce from the properties of our characteristic 

 function the six other known integrals above mentioned, in addition to that seventh 

 which contains the law of living force, and which assisted in the discovery of our 

 method. 



Introduction of relative or polar Coordinates, or other marks of position of a System. 



7. The property of our characteristic function, by which it depends only on the 

 internal or mutual relations between the positions initial and final of the points of an 

 attracting or repelling system, suggests an advantage in employing internal or relative 

 coordinates ; and from the analogy of other applications of algebraical methods to 

 researches of a geometrical kind, it may be expected that polar and other marks of 

 position will also often be found useful. Supposing, therefore, that the 3 n final coordi- 

 nates x-^yiZ-^ . . . X y^ z^ have been expressed as functions of 3 w other variables, 



^i*i2 • ' ' ^3n^ ^^^ ^^^^ ^^^ ^^ initial coordinates have in like manner been expressed 

 as functions of 3 w similar quantities, which we shall call e^ eg . . . e^^, we shall pro- 

 ceed to assign a general method for introducing these new marks of position into the 

 expressions of our fundamental relations. 



For this purpose we have only to transform the law of varying action, or the fun- 

 damental formula (A.), by transforming the two sums, 



2.m{a^^x+y'ly + z'lz),Sindl.m(a'ha-\-b'hb + c'hc), 



which it involves, and which are respectively equivalent to the following more deve- 

 loped expressions. 



