543-546] Hamilton's Principle 477 



If we sum equation (493) for all the particles of the system, replacing the 

 terms on the left by their values as just obtained, we arrive at the equation 



^- 2 m^ (M! &P! + Vj, %i + w 1 &Zi) 2 m l (u Bu^ + v 1 Sv^ + w l S w x ) 



= 2 (X^x, + Y^ + ZJzJ (494). 



Let T denote the kinetic energy of the actual motion, and T-\- ST that of 

 the slightly varied motion, then 



so that ST=^m l (u^ Si^ + v^ ^ + w l Sw^, 



and this is the value of the second term in equation (494). 



If W and W + 8 W are the potential energies of the two configurations 

 (assuming the forces to form a conservative system), we have 



TT = -2 



and 8W= - 2 (X 1 Bx 1 +-Y l By 1 + Z^\ 



so that the value of the right-hand member of equation (494) is SW. 

 We may now rewrite equation (494) in the form 



8 (T- W ) = j t Swi! (u^cc, + v^y, + wjzj. 



This equation is true at every instant of the motion. Let us integrate it 

 throughout the whole of the motion, say from t = to t = r. We obtain 



8 (\T- W)dt= IZm^u^ + v^ + w^zJ ] (495). 



Jo [_ Jt=o 



The displaced motion has been supposed to be any motion which 

 differs only slightly from the actual motion. Let us now limit it by the 

 restriction that the configurations at the beginning and end of the motion 

 are to coincide with those of the actual motion, so that the displaced motion 

 is now to be one in which the system starts from the same configuration as in 

 the actual motion at time t = 0, and, after passing through a series of con- 

 figurations slightly different from those of the actual motion, finally ends in 

 the same configuration at time t = r as that of the actual motion. Mathe- 

 matically this new restriction is expressed by saying that at times t = and 

 t r we must have $x=Sy = 8z = for each particle. Equation (495) now 

 becomes 



.(496). 



546. Speaking of the two parts of the mechanism under discussion 

 as the " accessible " and " concealed " parts, let us suppose that the kinetic 

 and potential energies T and W depend only on the configuration of the 



