134] APPLICATION TO HYDRODYNAMICS. 20o 



and in the case of a moving solid it is cancelled by the terms due 

 to the pressure exerted by the fluid on the solid. Hence the 

 symbols X, Y, Z may be taken to refer only to the extraneous 

 forces acting on the system, and we may write 



......... (6), 



where Q lt Q 2 ,... now denote the generalized components of ex 

 traneous force. 



We have still to consider the right-hand side of (5). Let us 

 suppose that in the arbitrarily varied motion the initial and final 

 positions of the solids are respectively the same as in the actual 

 motion. For every particle of the solids we shall then have 



Af=0, A77 = 0, A? = 0, 



at both limits, but the same will not hold as a rule with regard to 

 the particles of the fluid. The corresponding part of the sum 



will however vanish ; viz. we have 



^^ S I 7 t */ I 7 fc- 



v ay dz 



= pff(t&amp;gt; (l^ + m&Tj + jiAJ) dflf 



of which the second term vanishes by the condition of incom- 

 pressibility, and the first term vanishes at the limits and t l} 

 since we then have, by hypothesis, 



at the surfaces of the solids. Hence, under the above conditions, 

 the right-hand side of (5) vanishes, and therefore 



....}ftd ............ (7). 



The varied motion of the fluid has still a high degree of 

 generality. We will now farther limit it by supposing that 

 whilst the solids are, by suitable forces applied to them, made to 

 execute an arbitrary motion, subject to the conditions that Ag 1} 



