518 FOURTH KEPORT 1834. 



and considering the time as given, he finds now the formula of 

 variation 



gS = 2 ( OTgr) - /jSej, (16.) 



and therefore the 6 n separate equations 



8 »ji d d 



which are forms for the sought relations. 



Professor Hamilton thinks that these two formulae of va- 

 riation, (13.) and (16.) namely 



H = S (r)' 3 ra- — sr' e rj), (A.) 



and 



SS = 2 (ot^jj -fjSe), (B.) 



are worthy of attention, as expressing, under concise and 

 simple forms, the one the differential and the other the inte- 

 gral equations of motion, of an attracting or repelling system. 

 They may be extended to other problems of dynamics, be- 

 sides this capital problem. The expression H can always 

 easily be found, and the function S can be determined with in- 

 definite accuracy by a method of successive approximation of 

 the kind already explained. 



These properties of his Principal Function are treated of 

 more fully in his " Second Essay on a General Method in Dy- 

 namics*"; in which he has introduced several forms of a cer- 

 tain Function of Elements, connected with the Principal Func- 

 tion, and with each other, and adapted to questions of per- 

 turbation ; and has shown that for the perturbations of a 

 ternary or multiple system with any laws of attraction or re- 

 pulsion, and with one predominant mass, the differential equa- 

 tions of the varying elements of all the smaller masses may be 

 expressed together, and as simply as in the usual way, by the 

 coefficients of one disturbing function, (namely, the disturbing 

 part of the whole expression H,) and may be integrated rigor- 

 ously by a corollary of his general method. 



• This Essa)' will be found in the Philosophical Transactions for 1835. 



