TRANSACTIONS OF THK SECTIONS. 517 



useful to express the 3 n rectangular coordinates x\ ij^ s;,... 

 ^n Vn ^n j as fuuctions of 3 ti other marks of position, which may 

 be thus denoted, )j, >)2...>!3„; and if 3 ?« other new variables, 

 ziT, ro-2...ro-3„, be introduced, and defined as follows, 



^, =2.,«(^.'if + y^^ + «'-") . . (10.) 



it is, in general, possible to express, reciprocally, the 6 ?i va- 

 riables a.' 1/ s: x' y s:' as functions of these 6 « new variables ij w ; 

 it is, therefore, possible to express, as such a function, the 

 quantity 



H = 2. |(.r'2 + y'^ + s'2) - U, . . . . (U.) 



under the form 



H = F (CTj,...ra-3„, iJi,... >).,„) — U (rii,...%n), . (12.) 



in which the part F is rational, integer, and homogeneous of 

 the second dimension with respect to the variables w. Now 

 Mr. Hamilton has found that when the quantity H is expressed 

 in this last way, as a function of these 6 n new variables, >] w, its 

 variation may be put under this form, 



2H = 2 (ij'^w- tir'ar,), (13.) 



»j' ct' denoting the first differential coefficients of these new va- 

 riables >) cr, considered as functions of the time. The 3 n dif- 

 ferential equations of motion of the second order, (4.), between 

 the rectangular coordinates and the time, for any attracting or 

 repelling system, may therefore be generally transformed into 

 twice that number of equations of the first order, between these 

 6 n variables and the time, of the forms 



( 2H , SH /I J X 



v. = s— , ra-'i = - ^ — (I*-) 



To integrate this system of equations, is to assign, from them, 

 6 n relations between the time t, the 6 n variables rji ra-j- , and 

 their 6 « initial values which may be called e,- pi. Mr. Hamilton 

 resolves the problem, under this more general form, by the 

 same principal fit nction S as before, regarding it, however, as 

 depending now on the new marks >] e of final and initial posi- 

 tions of the various points of the system. For, putting, in this 

 new notation, 



S= r {^-^K^ - H) rf^ . . . . (15.) 



