274 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAxVIICS. 



that is, with the following formula, 



in which r^ is the initial distance 



ro = >s/(«i - «2)' + (^ - *2)' + (^1 - c^)\ {77.) 



and H is a constant quantity, introduced by integration ; we could, by the combina- 

 tion of these seven relations, determine the time t, and the six initial components of 

 velocity a\ h\ c\ a! 2 ^'2 (^2^ ^^ functions of the twelve final and initial coordinates 

 '^1 y\ ^1 ^2 .^2 ^2 ^1 ^1 ^1 ^2 h ^2? ^^^ ^^ t^^ quantity H, (involving also the masses :) 

 we could therefore determine whatever else depends on the manner and time of 

 motion of this system of two points, as a function of the same extreme coordinates 

 and of the same quantity H. In particular, we could determine the action, or accu- 

 mulated living force of the system, namely, 



V^m,r (^'i2 + y ^2 + .,'^2) dt-{-?n2 f W + 2/7 + ^7) dt, . . ( A^.) 



c/ *>' 



as a function of those thirteen quantities x^ y^ z^ cc^ y^ z^ a^ h^ q «2 ^2 ^2 ^'- ^^^ 

 might then calculate the variation of this function, 



1 



8V 8V 8V 8V 8V 8V 



sv 



8V 





8 V 



8V 



5 Co 



+ 8^^«i + 8A^^i'^S7:^^i + 8^^^2 + 0;^*2+8,^ —2 

 + g^ § H. 



(B2.) 



But the essence of our method consists in forming previoicsly the expression of this 

 variation, hy our law of varying action, namely, 



S V = »^l (t^/j Ixi — a\ ha^ -\- y\ ^yi — h\ ^ ^i + ^\ ^ Zj — c'j § Cj) "| 



+ m^ {a^2 ^ '^2 — ^'2 ^ «2 + y'2 ^^2 — ^'2 ^ ^2 + ^^'2 ^ Z2 — ^2 ^ <^2) ^ • (C^-) 



and in considering V a* « characteristic function of the motion, from the form of which 

 may be deduced all the intermediate and all the final integrals of the known differen- 

 tial equations, by resolving the expression (C^.) into the following separate groups, 

 (included in (C.) and (D.),) 



SV ,8V ,8V ,1 



8 V 

 and 



ij 



Ixc, — ^-^^^' ly^ 



8V ,8V , 



= ^2^25 87" = ^'>2; 



}> 



"2 



2-*2» 



(D2.) 



8V ,8V ,, 8V , -) 



8^ = - ^1 «i' ST; = ~ ^1 * P 87; = - '''i ^1' I 



8v__ ^ !Z_ 7.' IZ 



8 ^2 — "" ^2 « 2J 8 ^2 — — ^2 ^ 2> 8 C2 



7^2 ^2i 



(E^) 



