302 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



function of relative motion V^, along with the principal part or approximate value V^i- 

 The equations (X^) (Y^) of the twelfth number, give rigorously 





and 



— m.^a.'^m ^' lu,' ^i ~ m, 8/3, ^ m^ ^ 



S_..'. = i-^Ji+l- 



m. 8/3. "f~ ?;^^ "'^ 8^.' ^i — ?». Sy. ^" wz^ ^/ 8y.' 



(U«.) 

 (V«.) 



(W«.) 



(X".) 



the sign of summation 2„ referring only to the disturbing masses nij^, to the exclusion 

 of Mi andm„ ; and these equations (W^.) (X^.) are the rigorous formulae, corresponding 

 to the approximate relations (P.) (K^.)- I^^ ^i^^ manner, the formula (L^.) for the time 

 of motion in a binary system, which is only an approximation when the system is con- 

 sidered as multiple, may be rigorously corrected for perturbation by adding to it an 

 analogous term deduced from the disturbing part V^g ^^ the whole characteristic 

 function ; that is, by changing it to the following : 

 8w^^^ . 8V,. 



' = i^ + ra;. (Y^-) 



which gives, for this other coefficient of w^"^, the corrected and rigorous expression 



8w(*^ 8 v., 



i7) = '-nf= (ZM 



V,2 being here supposed so chosen as to be rigorously the correction of V^j. If therefore 

 by the theory of binary systems, or by eliminating g^^ between the four equations 

 (K^.) {JJ.), we have deduced expressions for the three varying relative coordinates 

 ^i 'Jj ^ as functions of the time t, and of the six initial quantities a^ /S^ y,- a',- /S'^ y',-, which 

 may be thus denoted. 



