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PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



and must therefore be still true, when, in passing to a multiple system, we change 

 the coefficients of w^'^ to their rigorous values (W^.) (Z^.)- The three intermediate 

 integrals (E''.) of the motion of a binary system may therefore be adapted rigorously 

 to the case of a multiple system, by first adding to the time t the perturbational term 

 (D^), and afterwards adding to the resulting values of the final components of rela- 

 tive velocity the terms 



= 2 



mk 





(*) 





sv,. 



Hi 



^^i-^,-m^ + m^ 8,, -^ m. 8,. "^ m^^> 



^A 8 to^*^ 1 8 V 1 



^^i— ^u- m^ + m 8?, ^ m. 8 r. ^ m ^' 8^ 



8 v. 



(G^) 



22. To derive now, from these rigorous results, some useful approximate expres- 

 sions, we shall neglect, in the perturbations, the terms which are of the second order, 

 with respect to the small masses of the system, and with respect to the constant 2 H^ 

 of relative living force, which is easily seen to be small of the same order as the 

 masses : and then the perturbations of the coordinates, deduced by the method that 

 has been explained, become 





8«', 



8)j. 

 'I 



8/3' 



A,, = ^Aa', + ^Ap', + ^Ay, + -^A^, 



8/ 



8^ 

 8/ 



8 ?. , 8 ^. 



A £. = — -^ A a'. + — i- A Q'. 4- 



8«' 



A/< + ^A^, 



8* 



it. 



(H-.) 



in which we may employ, instead of the rigorous values (C''.) for A a'., Aj3'-, A/-, the 

 following approximate values : 



A a'. = 2 ^l^^'^ . ^ ^V/2 



nir 



+ 



8«; ' 





w. 



^' " m„ 8v^ "^ m. 8v.* 



(F.) 



To calculate the four coefficients 



!Z/i iZ^ ?v^ 8v^ 



^««' 8/3£ ' 8yi ^ 8H,^ 



which enter into the values (F.) (D^.), we may consider V,2» ^1 (R^-) (T^-)j and by the 

 theory of binary systems, as a function of the initial and final relative coordinates, and 

 initial components of relative velocities, involving also expressly the time t, and the 



