300 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



racteristic function V^ to those of its first part V,i, and by attending to the foregoing 

 equations, 



and consequently 



The general transformation of the foregoing number gives therefore, rigorously, for 

 the remaining part V^ of the characteristic function V^ of relative motion of the mul- 

 tiple system, the equation 



B'w^^l'w^''^ div^hw^^^ dw^^hw^^^' 



Jo ^0 _^m^^m,){m^ + m,) 



n 



J 



and, approximately, the expression 



with which last expression we may combine the following approximate formulae be- 

 longing in rigour to binary systems only. 



^i—~JJ7>''i—'J^'^i'~' 8?.' ^ ^^ 



t 'I • t 



and 



^ = r-(o- • (^•) 



We have also, rigorously, for binary systems, the following differential equations of 

 motion of the second order, 



8 /-CO 8 /■('■) 8 /•(»■) 



which enable us to transform in various ways the approximate expression (H^.). Thus, 

 in the case of a ternary system, with any laws of attraction or repulsion, but with one 

 predominant mass W3, the disturbing part V^ of the characteristic function V^ of re- 

 lative motion, may be put under the form 



V^rrrmimgW, (N«.) 



in which the coefficient W may approximately be expressed as follows : 



