PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 259 



system, but being now considered as a function of rli . . . rl^^, involving also the 

 masses, and in general ?7i • . . ?73„, and obtained by substituting for the quantities ^2/' z' 

 their values of the form (20.) in the equation of definition 



T = i2.m(a/2 + y2^sj'2) (4.) 



In like manner we find this transformation for the sum (18.), 



2.m(a'3« + A'56 + c'Sc) = |j5e, + gSe2 + ... + gSe3„. . (24.) 



The law of varying action, or the formula (A.), becomes therefore, when expressed 

 by the present more general coordinates or marks of position, 



SV=2.^$p;-2.^^e + ^m; ...... (Q.) 



and instead of the groups (C.) and (D.), into which, along with the equation (E.), 

 this law resolved itself before, it gives now these other groups, 



SV ST 8V ST SV 8T 



SiJi SVi' S>j2 SVa' * ' * ^*J3n ^Vsra 



and 



(R.) 



i_Y_ nTo sv;_ sTo iZ._-_lIo .ox 



3n 



The quantities ^j ^2 ' ' * ^3 ^^^ ^1 ^2" ' ^'sn ^^^ ^^^ ^^ initial data respecting 

 the manner of motion of the system ; and the 3 n final integrals, connecting these 6 n 

 initial data, and the n masses, with the time t, and with the 3 n final or varying quan- 

 tities f]if]2 . . . >?3^, which mark the varying positions of the n moving points of the 



system, are now to be obtained by eliminating the auxiliary constant H between the 

 3n -\- I equations (S.) and (E.) ; while the 3 n intermediate integrals, or integrals of 

 the first order, which connect the same varying marks of position and their first dif- 

 ferential coefficients with the time, the masses, and the initial marks of position, are 

 the result of elimination of the same auxiliary constant H between the equations (R.) 

 and (E.). Our fundamental formula, and intermediate and final integrals, can there- 

 fore be very simply expressed with any new sets of coordinates ; and the partial dif- 

 ferential equations (F.) (G.), which our characteristic function V must satisfy, and 

 which are, as we have said, essential in the theory of that function, can also easily be 

 expressed with any such transformed coordinates, by merely combining the final and 

 initial expressions of the law of living force, 



T = U + H, (6.) 



To = Uo + H, (7.) 



with the new groups (R.) and (S.). For this purpose we must now consider the func- 

 tion U, of the masses and mutual distances of the several points of the system, as 

 depending on the new marks of position ri^vj^ . . , ri^^'^ ^^^ ^^^ analogous function Uq, 

 as depending similarly on the initial quantities ej 62 . . . eg^ ; we must also suppose 



