290 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 





TWj + Wg + TTIq ' 7//2 8 ))2 

 ^1 + % + ^3 ' 5^2 ^ ?2 



'J2 



^'2 



^1 ^\ + ?»2 ^^2 . 1 



wiVl±jv5!2. 



"h ^'1 + ^2 ^2 



. (U) 



and 



— 1 8V, __ , ^1 g^i + m^ u'cj 



•" ^ « 1 Wi + ^2 + W3 ' 



-1 8V, __ 



1 S«i 



m 



2 ^ «2 



«o — 



18V, __, m,^\ + m^^'^ -18 V, 



m, 



8^1 



= (3'i- 



?«! + »?2 + ^3 ' 



OTc 



8^, 



= ^'2 



:^ ^' = y _ ^1/1 + ^2^2 . zlI ?v, ^ 



Wl 8 yj '1 TWi + ^2 + Wg * ^2 8 72 ■ 



which must be combined with our old formula. 



m^ a\ + m^ «^2 . "] 



Wj + 7^2 + W3 ' 



mi + ?»2 + W3 ' 



2 »2i + W2 + OT3 ' - 



(M\) 



8H, 



(O'.) 



18. The quantity H, in V„ and the analogous quantity H,, in V,„ are indeed inde- 

 pendent of the time, and do not vary in the course of the motion ; but it is required 

 by the spirit of our method, that in deducing the absolute action or original character- 

 istic function V from the two parts V, and V^,, we should consider these two parts 

 H and H„ of the original quantity H, as functions involving each the nine initial and 

 nme nnal coordinates of the points of the ternary system ; the forms of these two 

 tunctions, of the eighteen coordinates and of H, being determined by the two con- 

 ditions. 



8V, 



8V. 



SH, — 8 h",' ^' + ^" — ^• 



(N^.) 



However, it results from these conditions, that in taking the variation of the whole 

 original function V, of the first order, with respect to the eighteen coordinates, we may 

 treat the two auxiliary quantities H, and H„ as constant ; and therefore that we have 

 the following expressions for the partial differential coefficients of the first order of V, 

 taken with respect to the coordinates parallel to .r, 



8V_8V, m, 8V„ 8V 8V, . m, 8V„ 



8^ — xt + 



S.^i 



m^ 8V„ 8V __ ^/ , m, ^ 



TWi + 7W2 + Wg 8 Xii ^ 8 aj ¥al '*' m^ + ntc^ + jtIq Z a„ » 



8V _8V, 



— X£ "T 



mn 



mo 



8V, 



^'■^2 SI2 "^ ^1 + »^2 + ^3 8a^„ ' 8 fltg 8 a,^2 "*" Wi + W2 + ^3 ^ ^ti ' 



(O*-) 



8V__ 8V, 8V, 



— X£ X£ I" 



Wo 



8x 



8V„ 8V 



8^, 8^2 »»i + ^2 + ^3 ^ -^/z ' 8 «a 



8V, 

 8 a. 



ma 



8V, 



8 a2 ^1 + »^2 + ^3 ^ «// ' 



together with analogous expressions for the partial differential coefficients of the same 

 order, taken with respect to the other coordinates. Substituting these expressions in 

 the equations of the form (O.), namely, in the following. 



