PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 275 



besides this other equation, which had occurred before, 



ra = ^ (E) 



By this new method, the difficulty of integrating the six known equations of motion 

 of the second order (74.), is reduced to the search and differentiation of a single 

 function V ; and to find the form of this function, we are to employ the following 

 pair of partial differential equations of the first order : 



= mim2/(r)+H, (F2.) 



= mim2/(ro)+H, . (G2.) 



combined with some simple considerations. And it easily results from the principles 

 already laid down, that the integral of this pair of equations, adapted to the present 

 question, is 



in which x^, ?/^, z^, a^ b„ c^^, denote the coordinates, final and initial, of the centre of 

 gravity of the system. 





in, a, + m^ 

 a — 





(78.) 



and ^ is the angle between the final and initial distances r, Tq : we have also put for 

 abridgement 



? = ±\/2 K + m,) (/(,•) +^j-^, (79.) 



the upper or the lower sign to be used, according as the distance r is increasing or 

 decreasing ; and have introduced three auxiliary quantities h, H^, H^^ to be deter- 

 mined by this condition, 



''=^+X8-l'^'-' (''•> 



combined with the two following, 



