PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



299 



respectively with the initial distances corresponding, namely, Tq^^^ . . . Tq^" ^\ The 

 variation of the sum V,i is, by (S\), 



(1) 



gy _. ^1 m^^ti^^ ^ m^ m^ S w 



„(2) 



+ 



nic, + ?«. 



+ ...+ 





{Z\) 



in which, by the equations of condition, we may treat all the auxiliary quantities 

 ^(1)^0) ^ ^ ^("- )^(«-i) ^g constant, if H^ be considered as given: so that the part 

 of this variation S V^, which depends on the variations of the final relative coordinates, 

 may be put under the form, 



+ ••• 



By the equations (T^.) (U^.) ? or by the theory of binary systems, we have, rigorously. 



(C«.) 



and the rigorous law of relative living force for the whole multiple system, is 



T, = U + H„ (50.) 



in which 



U = m„ (mi/" + m,/« + ...+»«„_,/"-") + 2, . m^ »«,/"•« 



and 



+--^i(=CT+i){(^;+(e;)'+0'} 



2 /8V,8V 8V,8V, 8V,8VA 

 + m//V8^. 8^,+ 8,. 8,,-»-8?. 8?,/- 



We have therefore, by changing in this last expression the coefficients of the cha- 



(D«.) 



