PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 307 



General introduction of the Time, into the expression of the Characteristic Function in 



any dynamical problem. 



23. Before we conclude this sketch of our general method in dynamics, it will be 

 proper to notice briefly a transformation of the characteristic function, which may be 

 used in all applications. This transformation consists in putting, generally, 



V = ^H + S, (Ug 



and considering the part 8, namely, the definite integral 



s=y^'(T + u)^/, (\7.) 



as a function of the initial and final coordinates and of the time, of which the varia- 

 tion is, by our law of varying action, 



SS= -lidt-{-^.m{a^lx — a!la-\-y'ly-'h'lb-\-z'lz — dlc). . (W^.) 



The partial differential coefficients of the first order of this auxiliary function S, are 

 hence, 



'rf = -H; (X'.) 



and 



8 S , S S , 8 S , ^_ 



These last expressions (Z^.),are forms for the final integrals of motion of any system, 

 corresponding to the result of elimination of H between the equations (D.) and (E.) ; 

 and the expressions (Y^.) are forms for the intermediate integrals, more convenient 

 in many respects than the forms already employed. 



24. The limits of the present essay do not permit us here to develope the conse- 

 quences of these new expressions. We can only observe, that the auxiliary function S 

 must satisfy the two following equations, in partial differentials of the first order, 

 analogous to, and deduced from, the equations (F.) and (G.) : 



8S , ^ 1 r/8S\2 , /8S\2 , /8S\21 



and 



_8_S 

 It 



and that to correct an approximate value Sj of S, in the integration of these equations, 

 or to find the remaining part Sg, if 



s = Si 4-82, ' (c^O 



we may employ the symbolic equation 



