288 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



But the law of varying action gives, previously, the following expression for this 

 variation : 



+ m^ (^2 ^ -^2 - «'2 ^ «2 + y'2 ^ 3/2 - ^'2 ^ ^2 + ^'2 ^ ^2 - ^'2 ^ ^2) I .^4 



+ m3 (^'3 ^ ^3 - a'3 ^ «3 + ^3 ^ 3/3 - ^'3 ^ ^3 + ^'3 ^ % - V3 S C3) I 



and shows, therefore, that the research of all the intermediate and all the final integral 

 equations, of motion of the system, may be reduced, reciprocally, to the search and 

 differentiation of this one characteristic function V ; because if we knew this one 

 function, we should have the nine intermediate integrals of the known differential 

 equations, under the forms 



8V 



8^2 "" ^'*2 •^2» 8^2 — ^'^23/25 g2„ — »*2 ^ 2» 



8V 

 8d;„ 



'2 

 8V 



"2 



8V 



— ^% "^'S' S w_ ~~ ''^S y 3> 8 r — ^3 ^ 35 



3' 8 J/3 



(D^.) 



and the nine final integrals under the forms 



8V 



8V 



8fli — ■" ^1^1' 8^1 



= — niib' 



rv_ 



8^2 



8V 

 8aq 



8V 



8V 

 l'8ci 



8V 



- ^1 c 1, 



— //Jo I* 95 S> Z — "^ Wio O 95 J. — ~~ ?/i( 



''2 "• 2J 8 J, 



2 - 2» 8 c. 



'^2 ^2j 



> 



= — Wo a 



SV 



3"^3J8Zi. 



= — Mob 



8V 



3''3» 8c^ 



= — m, c 



3 ^3> 



(E'.) 



*3 ""'a ' ""3 



the auxiliary constant H being to be eliminated, and the time t introduced, by this 

 other equation, which has often occurred in this essay, 



8 V 

 ' = 8H (E.) 



The same law of varying action suggests also a method of investigating the form 

 of this characteristic function V, not requiring the previous integration of the known 

 equations of motion ; namely, the integration of a pair of partial differential equations 

 connected with the law of living force ; which are, 



+ 2^{ (y + (873)^+ W^T] =^1^2/'' '^^1^3/'' '^+ m^my' '^ + H, J 

 and 



2^ { \8tJ + KfTj + xfvj j + 2^ { V8^) + \rbj + V8^/ j 



;>(P). 



2) „(1, 3) (2, 3) 



(G4.) 



