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



we find 



[5i (f Af + *}AT; + ?A)7 - j* 1 %m (f Af + rj^ + A?) dt 



= f ' 2 (ZAf + FAT; + ZAf) d*. 

 Jt. 



If we put, as usual, 



2r = 2m( 

 this may be written 



{AT + 2 (ZAf + FAT; + 



............... (4). 



If we now introduce the condition that in the varied motion 

 the initial and final positions (at times t and ti) 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 



t = ...... (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. K, Hamilton, "On a General Method in Dynamics," Phil. Trans. 

 1834, 1835. 



