PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



267 



a^i bji Cji being- the initial coordinates of the centre of gravity, so that 



2, .m a 

 ^'1 ~ Xm ' 



, S .mb 



^11 ~ Sm ' 



X .mc 

 X m 



(54.) 



And for the variation ^ V of the whole function V, the rule of the last number gives 



SV = 2 . m{x^,loc^ — c^^la^-^y\ly,-~ b','6b, + %',lz, — c',^c;) -^ 



+ i^n ^ ^^// - «'y/ ^ «// + 3/'// ^y// - *'/> ^ ^// + ^'// ^ ^// - C„ ^ O 2^1 . (II.) 

 4- ^ ^ H + ^1 2 . m ^ jc, + ?i2 2 . m ^ i/^ + X3 2 . m ^ ;s^ f 



+ Aj 2 . m ^ «^ -f A2 2 . m H^ + A3 2 . m S c^ ; 



while the variation of the part V^^ determined by the equation (H'.), is easily shown 

 to be equivalent to the part 



^ V,^ = (af„ I x„ — a\, I a„ + 1/',, li/„ - b'„ h b„ + z'„ I z„ — d,, § c,) ^m + tl H^, 



the variation of the other part V^ may therefore be thus expressed, 



^Y I = 2 . m {x', h x^ — a\ ^ «/ + y\ ^y, — b\ ^b^-\- z', hzi — c\ I c) 1 



+ ^ ^ H^ + Xj 2 . m ^ 0?, + X2 2 . m ^ 1/^ + X3 2 . 7n^z^ ^ . 



+ Aj 2 . m § «^ + A2 2 . m § i, + A3 2 . w ^ c^ : -^ 



and it resolves itself into the following separate expressions, in which the part V^ is 

 considered as a function of the 6n -{- I quantities x^. y^^ z^. a ^. b,. c^ H^, of which, how- 

 ever, only 6 w — 5 are really independent : 



first group. 



(Ki.) 



(L'.) 





8V 



■ = f'hy'a + h^i ; 



second group. 



dz. 



8V 



w«i ^ a + h ^^1 ; 







m. 



(M'.) 



g^' = - mi «'^i + Ai mi ; 

 — mi c'ji + A3 mi ; 



8V 



-^ = - m„ «'^„ -h Ai 7^„ ;" 









(Ni.) 



and finally, 



8V 



8H, 



.^ = /. 



(01.) 



With respect to the six multipliers X^ X2 ^j ^i ^2 A^ which were introduced by the 

 3 final equations of condition (45.), and by the 3 analogous initial equations of con- 

 dition. 



^ . mOi = 0, 2 . m ^^ = 0, 2 . 



ffl c, 



0; (55.) 



