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



Now, from (8) and (10), 



_ dT dT 



by (9). Again 



where, for example, 



It is evident, on comparison with (12), that $/, Q 2 ',... are the time- 

 integrals of Q lt Q 2 ,... taken over the infinitely short duration of the impulse, 

 in other words they are the generalized components of the impulse. Equating 

 the right-hand sides of (iii) and (v) we have, on account of the independence 

 of the variations A< 



dT n , dT 



ar ft ' s 



The quantities 



dT dT 



are therefore called the * generalized components of momentum ' of the system, 

 they are usually denoted by the symbols p lt p 2 ,.... Since T is, by (9), a 

 homogeneous quadratic function of g 15 q 2 ,..., it follows that 



(vii). 



In terms of the generalized coordinates q lt g 2 ,... the equation 

 (5) becomes 



g 1 + Q 2 Ag 2 + ...)^ = ......... (14), 



to 



where 



AT=^ T Ag 1 + ^Ag 2 +...+|^A^ 1 

 dq l dq 2 dq l 



Hence, by a partial integration, and remembering that, by hypo- 

 thesis, A^j, A^a,... all vanish at the limits t , t l} we find 



dT n \ A fddT dT 



i -- ft) Aft + (^j^-- 3 -- 



dq l **) ^ \cttdq 2 dq 2 



......... (16). 



Since the values of A^, A^,... within the limits of integration 



