270 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 





m, 



8V, 



° -^in - 1 



1 Sn _ 15 





= »^«-i^n-i. r ('^^•) 



8«n 



sv,__ 



2;o 



8V, 



''n-l 



= m„ 



^» — 1j 



and 



8V, 



-5 — '- = — m, a 1, y- 



== — JWo a 



8V, 



/2 



2? 



8a 



= — m 



/»- 1 



n- 1 '*'n- 1' 



8V, , SV, , 



T7f = -^i>'i'T7r = -^2y2, 



-^^ = - m 3' 



/n- 1 



8V, 



8 c 



'n - 1 



W^n-l/n-l- 



(U>.) 



12. We might also express the relative action V^ not as a function of the centro- 

 baric, but of some other internal coordinates, or marks of relative position. We might, 

 for instance, express it and its variation as functions of the 671 — 6 independent in- 

 ternal coordinates ^ 71 ^ a ^ y already mentioned, and of their variations, defining these 

 without any reference to the centre of gravity, by the equations 



(66.) 



di 



= «i - On, Pi = h— K 71 = ^i 





For all such transformations of ^ V, it is easy to establish a rule or law, which may be 

 called the law of varying relative action (exactly analogous to the rule (B*.)), namely, 

 the following : 



which implies that we are to express the half T^ of the relative living force of the 

 system as a function of the rates of increase rl^ of any marks of relative position ; and 

 after taking its variation with respect to these rates, to change their variations to the 

 variations of the marks of position themselves ; then to subtract the initial from the 

 final value of the result, and to add the variations of the final and initial functions 

 (p^ O^, which enter into the equations of condition, if any, of the form <p^ = 0, O^ = 0, 

 (connecting the final and initial marks of relative position,) multiplied respectively 

 by undetermined factors X^ A^ ; and lastly, to equate the whole result to ^ V^ — ^ ^ H^, 

 H^ being the quantity independent of the time in the equation (50.) of relative living 

 force, and V, being the relative action, of which we desired to express the variation. 

 It is not necessary to dwell here on the demonstration of this new rule (V.), which 

 may easily be deduced from the principles already laid down ; or by the calculus of 

 variations from the law of relative living force, combined with the differential equa- 

 tions of the second order of relative motion. 



