108 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



we find 



p&M (j Af + rj AT; + A f)l - f 2m (f Af + ^Ar) + ?A) eft 



f 2 (XA + FAT; + A?) (ft. 

 ./* 



If we put, as usual, 



s ) ..................... (3), 



this may be written 



T + 2 (ZAf + FAr; + 



(4). 



If we now introduce the condition that in the varied motion 

 the initial and final positions (at times and ^) shall be respec 

 tively the same for each particle as in the actual motion, the 

 quantities Af, AT;, Af vanish at both limits, and the above 

 equation reduces to 



= ...... (5). 



This formula is especially valuable in the case of a system 

 whose freedom is limited more or less by constraints. If 

 the variations Af, AT;, Af be such as are consistent with these 

 constraints, some of the internal forces of the system disappear as 

 a rule from the sum 



for example, all the internal reactions between the particles of a 

 rigid body, and (as we shall prove presently) the mutual pressures 

 between the elements of an incompressible perfect fluid. 



In the case of a * conservative system, we have 



(6), 



where V is the potential energy, and the equation (5) takes the 

 form 



(7)*. 



* Sir W. E, Hamilton, &quot; Oil a General Method in Dynamics,&quot; Phil. Trans. 

 1834, 1835. 



