PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 301 



or thus : 



> . (P^.) 



(Q^.) 



1 / Sto^'^ Sw^"^ c^ Sto^^ Sto^^^ 8ty^^^ STO^'^ 



or fin ally. 



In general, for a multiple system, we may put 



V,, = ^,.m,m,W^'''^', (R«.) 



and approximately, 



or 



~ m \^k 8^. + ^k 8>). + ^/t 8^. + ^/c S«. + ft S/3. + ^'a Sy. A 



1 



> . (S6.) 



(T6.) 



Rigorous transition from the theory of Binary to that of Multiple Systems, hy means of 

 the disturbing part of the ivhole Characteristic Function; and approximate ex- 

 pressions for the perturbations, 



21. The three equations (K^'.) when the auxiliary constant g^'^ is eliminated by the 

 form.ula (L<^.), are rigorously (by our theory) the three final integrals of the three 

 known equations of the second order (M^.), for the relative motion of the binary 

 system (m^ ?«J ; and give, for such a system, the three varying relative coordinates 

 i» ^i K,ii ^s functions of their initial values and initial rates of increase a^ /3j y^ cc'^ (^'^ y\^ 

 and of the time t. In like manner the three equations (F.), when g^^^ is eliminated 

 by (L^.)j t^^^ rigorously the three intermediate integrals of the same known differential 

 equations of motion of the same binary system. These integrals, however, cease to 

 be rigorous when we introduce the perturbations of the relative motion of this partial 

 or binary system (m,- mj, arising from the attractions or repulsions of the other 

 points w^, of the whole proposed multiple system ; but they may be corrected and 

 rendered rigorous by employing the remaining part V,2 ^^ ^^ whole characteristic 



MDCCCXXXIV. 2 R 



