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



Now, from (8) and (10), 



by (9). Again 



S(X A+rtoi + Z &Q = Q l t *q l + Qj*qs + ............... (iv), 



where, for example, 



l i 



It is evident, on comparison with (12), that $/, $ 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^, A&amp;lt;? 2 v? 



dT dT 



a ft&amp;lt; 



The quantities 



dT dT 



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

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

 homogeneous quadratic function of q lt q 2 ,..., it follows that 



In terms of the generalized coordinates q 1} q. 2) ... the equation 

 (5) becomes 



[ tl {&T+Q 1 bq l + Q,bq 9 + ...}dt = ......... (14), 



Jto 



where 



A ^ dT ^ dT A . dT A dT . , 1C . 



^T=^r A&amp;lt;7!+ ^.-Ag 2 + ... + -7- Ag^ T-Aft-f ... (15). 

 dq, dq, dq, dq 2 



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

 thesis, Ag 1? A^ 2 ,... all vanish at the limits t , t l} we find 



dT dT \ . ddT dT 



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



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



