PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



271 



. (68.) 



But to give an example of its application, let us resume the problem already men- 

 tioned, namely to express ^ V^ by means of the 6n — 5 independent variations S |. I)]. 



^ ^. S a. ^ /3. S 7. ^ H^. For this purpose we shall employ a known transformation of the 

 relative living force 2Tp multiplied by the sum of the masses of the system, namely 

 the following : 



2T,2m = 2.m.m^{ix'.- a/^)^ + (3/'. - y^^ + (^., _ ^>^)2^ : . . . (67.) 



the sign of summation 2 extending, in the second member, to all the combinations of 

 points two by two, which can be formed without repetition. This transformation 

 gives, by {66.), 



2T,^m = m^l,.m (|'2 4- ^,'2 + ^'2) 



+ ^r m, m^ { (r, - r,)^ + «. - n\Y + (?, - O'} 



the sign of summation 2^ extending only to the first n— I points of the system. Ap- 

 plying, therefore, our general rule or law of varying relative action, and observing 

 that the 6n — 6 internal coordinates Ivi ^a, ^ y are independent, we find the follow- 

 ing new expression : 



w 



+ ^^.2,.mmj(r,-r,)(^i,-ay+(<.-.'j(H-H)+(CW?*)(H-^y 



-^«-2r^,^4K-<)(^«-^«,) + (ft-ft)(¥-W + (y'~y';(^y-^n)}: 



which gives, besides the equation (O'.), the following groups : 



TE" = ^ • 2 . m (f. - I') = m. i i'. — — ir-^ ), 



^' = ^;.2..,-.') = ..(<-|^'> . 



(X».) 



and 



K— ' = —-*. 2 . m (a'. -- a' = - m. (ex!. - -^), 

 8«^ Xm ' \ ^m ) 



|^' = Z^-.2.m(/. - /) = - m. (y\-^) ; 



(Y^) 



results which may be thus summed up : 



2 N 2 



