PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



269 



after this preparation, therefore, of the function V^, the six multipliers determined by 

 (58.) and (59.) will vanish, so that we shall have 



Xi = 0, X2 = 0, X3 = 0, Ai = 0, A2 = 0, A3 = 0, (64.) 



and the groups (M'.) and (N'.) will reduce themselves to the two following : 





8V, 



^n ^ in 5 



11/ _ 



5V 



' = mi z' 



8V, 



and 



1 * '1 ' 82. — ^2^/2' 



8V, 





= ^nl/in> 



m„z 



n in^ 



(Q'.) 



^ = - mi «\i ; y— *- = — m., a 



a^2 



8a 



ii 



/2» 



8V 

 8 a, 



= — m„a 





In 

 8V 



n^'m ' 



^2 ^ /2 5 • • • 8 6 



' = -^n^'/n; 





wi^c 



n /n ' 



(R'.) 



analogous in all respects to the groups (C.) and (D.). We find, therefore, for the re- 

 lative motion of ja system about its own centre of gravity, equations of the same form 

 as those which we had obtained before for the absolute motion of the same system of 

 points in space. And ^e see that in investigating such relative motion only, it is 

 useful to confine ourselves to the part V^ of our whole characteristic function, that is, 

 to the relative action of the system, or accumulated living force of the motion about 

 the centre of gravity ; and to consider this part as the characteristic function of such 

 relative motion, in a sense analogous to that which has been already explained. 



This relative action, or part V^ may, however, be otherwise expressed, and even in 

 an infinite variety of ways, on account of the six equations of condition which con- 

 nect the 6 n centrobaric coordinates ; and every different preparation of its form will 

 give a different set of values for the six multipliers X^ X2 ^3 -^i -^2 -^s- ^^^ example, 

 we might eliminate, by a previous preparation, the six centrobaric coordinates of the 

 point m„ from the expression of V^ so as to make this expression involve only the 

 centrobaric coordinates of the other n — I points of the system, and then we should 

 have 



8 a:, 



^^In ^^In 



0, Y-^ = 0, 



' a, ' 



8 6, 



8 c,. 



0, 



(SK) 



'In "In In In In In 



and therefore, by the six last equations of the groups (M'.) and (N'.), the multipliers 

 would take the values 



and would reduce, by (60.) and (61.), the preceding 6n — 6 equations of the same 

 groups (M*.) and (N\), to the forms 



MDCCCXXXIV. 2 N 



