PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



251 



is evidently connected with the analogous increments of the functions T and U, by 

 the relation 



AT = AU + AH, . .- (8.) 



which, for the case of infinitesimal variations, may conveniently be written thus, 



ST = ^U + ^H; (9.) 



and this last relation, when multiplied by dt, and integrated, conducts to an import- 

 ant result. For it thus becomes, by (4.) and (1.), 



yS . m {dx ,lx^ -\- dy .^y^ •{• d z .1 z') •=■ 



/2.m{dx' .^x+ dy'.ly + dz' ,^z)-{-/lH.dt, (10.) 



that is, by the principles of the calculus of variations, 



lV = 2.m{a/la:+y'hy + z'hz)-2.m{a'^a-{-b'lb + clc)-\-t^n,. . (A.) 

 if we denote by V the integral 



y-f2.m{j/dx-\-y'dy + z'dz')=/o2Tdt, (B.) 



namely, the accumulated living force, called often the action of the system, from its 

 initial to its final position. 



If, then, we consider (as it is easy to see that we may) the action V as a function of 

 the initial and final coordinates, and of the quantity H, we shall have, by (A.), the 

 following groups of equations ; first, the group, 



8^1 — ^1*^1' 8a;2 ~^2-^2; 



— = m 07 ; 



n n 



= rn^y\'-> 



Secondly, the group. 



8V 

 8V 



= wigy'a; 



n 



sv 



. K — =. m y \ 



; y 



z=z m^ z 



=1 nioZ 



2 •*2 ' • 



8V 

 . . s — = ?n z ; 



Z n n 



(C.) 



8V 

 8V 



rv 



?n,a,', ^ 



:= — nil h\ 



8V 



2 



8V 

 8 6, 



»2i c 1 ; g ^ 



2 



8V 



'2 



»«2 ^ 2 ; 



^2 ^1 5 



8V 

 8«„ 



— m a \ 



n n 



8V 

 lb. 



V , , 



r-r- = — m^b \ 



8V 



J — z=. — m c \ 



■ ■ (P-) 



and finally, the equation, 



.8V 



8H 



= t 



(E.) 



So that if this function V were known, it would only remain to eliminate H between 

 the 3w+ 1 equations (C.) and (E.), in order to obtain all the 3?? intermediate inte- 

 grals, or between (D.) and (E.) to obtain all the 3 n final integrals of the differential 

 equations of motion ; that is, ultimately, to obtain the 3 n sought relations between 



