PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 289 



And to diminish the difficulty of thus determining the function V, which depends on 

 18 coordinates, we may separate it, by principles already explained, into a part V^, 

 depending only on the motion of the centre of gravity of the system, and determined 

 by the formula (H^), and another part V^, depending only on the relative motions of 

 the points of the system about this internal centre, and equal to the accumulated 

 living force, connected with this relative motion only. In this manner the difficulty 

 is reduced to determining the relative action V^ ; and if we introduce the relative co- 

 ordinates 



d2 ^^ -^2 — -^S' ^2 ^^ y2 3^3» ^2 



= ^1 - %.") , , 



— 252 — Z3,J 



and 



«1 = «1 — «3J ^1 = ^1 — K 7\ = 



^2 = «2 — %? ^2 = h - K y-i 



'I''"'^"} .... (12;.) 



'2 — ^2 ^3» ^ 



we easily find, by the principles of the tenth and following numbers, that the function 

 V, may be considered as depending only on these relative coordinates, and on a quan- 

 tity H, analogous to H (besides the masses of the system) ; and that it must satisfy 

 two partial differential equations, analogous to (F*.) and (G^.), namely, 



+^{(ii'+gy+(5'+S'r+(if'+ifr} 



and 



1 / C^JiS^ 4. ("il'V . /^iX/V 1 4. J- / /'^'V 4. /"^'V 4. ("^-Z/V 1 "^ 



= mi /Wg/J'' ^^ + mi mg/J'' '^ + m^ m^f^' ^^ + H^ : 

 the law of the variation of this function being, by (Z'.), 



IV, = an, + mi (ri^li - «'i^ai + ^'i^ni - fB'i^ft + C'l^Ci - r'i^7i) ' 



+ 7/^2 (1 2 ^ I2 — "''2 ^ <^2 + '^'2 ^ ^^2 — ^'2 ^ /32 + C'2 ^ ^2 — ^'2 ^ 72) 



[(mil'i+mg^y (mi^li+mg^y — (mia'i+y/Zaay (mi^ai+mg^ag) 

 ^ +(mi;j'i + m2;7'2) (»ii^;?i+m2S;72) — (^iP'1+^23'2) (miSft + m2^/32) 



KK^.) 



' ' H +(miC'i+m2?2) K^Ci+m2^C2)-K7'i+^2y'2)K^yi+^2^y2) 



which resolves itself in the same manner as before into the six intermediate and six 

 final integrals of relative motion, namely, into the following equations : 



