260 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



ST ST 



it 



that T is expressed (as it may) as a function of its own coefficients g^. ^^.^ 



which will always be, with respect to these, homogeneous of the second dimension, 

 and may also involve explicitly the quantities Vi^2 ■ - • ^sn'^ and that Tq is expressed 



as a similar function of its coefficients y^, y^, . . . g-p-^ ; so that 



3n 



/ST VT IT X 



T - P (Ho STp iloX . [ 



(25.) 



and that then these coefficients of T and To are changed to their values (R.) and (S.), 

 so as to give, instead of (F.) and (G.), two other transformed equations, namely, 



'^eW^.-S^^ + H, ex.) 



and, on account of the homogeneity and dimension of Tq, 



^8V 8V 8V 



^C^> 



g /» J • • • 8^ 



) = Uo + H (U.) 



8. Nor is there any difficulty in deducing analogous transformations for the known 

 diffisrential equations of motion of the second order, of any system of free points, by 

 taking the variation of the new form (T.) of the law of living force, and by attending 

 to the dynamical meanings of the coefficients of our characteristic function. For if 

 we observe that the final living force 2 T, when considered as a function of pj^ pjg • • • ^3n 

 and of Vi ^2 • • • ^'sni ^^ necessarily homogeneous of the second dimension with respect 

 to the latter set of variables^ and must therefore satisfy the condition 



ST 



, ST 



we shall perceive that its total variation. 



ST 



3n 



(26.) 



ST 



ST 



ST 



may be put under the form 



(27.) 



IT 



'^i^i^.-^'^2^f^ + 



ST. ST. 



"H^^'i" ■87/^2-"-. 



V ;v^T ^ STv 



11 



ST 1 



3n 



ST . 



'3n 



(28.) 



