98 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS, 



and consider T (as we may) as a function of the following form, 



we see that 



I^, = ''i'"'f^ =^3„» (14.) 



and 



and therefore that the general equation (3.) may receive this new transformation, 



^' = ^-fi^> (16.) 



dt ^1i ^ ^ 



If then we introduce, for abridgement, the following expression H, 



H = F - U = F (td-i, ^2, . . . TJTg^, ;?!, r]2, - ' '%J — '^ i^i, 'J2? • • • ^3„). • (17.) 



we are conducted to this new manner of presenting the differential equations of 

 motion of a system of n points, attracting or repelling one another : 



(A.) 



In this view, the problem of mathematical dynamics, for a system of n points, is to 

 integrate a system (A.) of 6n ordinary differential equations of the first order, be- 

 tween the 6 n variables f]. rs. and the time t ; and the solution of the problem must 

 consist in assigning these 6 w variables as functions of the time, and of their own 

 initial values, which we may call e.p.. And all these Qn functions, or Qn relations 

 to determine them, may be expressed, with perfect generality and rigour, by the 

 method of the former Essay, or by the following simplified process. 



Integration of the Equations of Motion, hy means of one Principal Function. 

 4. If we take the variation of the definite integral 



«=X'(2-|^-H)'^' • • ■ • 08.). 



without varying t or dt, we find, by the Calculus of Variations, 



iS=f'lS'.dt, (19.) 



c/o 



in which 



S' = 2.^|^-H, (20.) 



