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



Ag 2 ,...=0 for t = t Q and t = t l} the fluid is left to take its own 

 course in consequence of this. The varied motion of the fluid 

 will now be irrotational, and therefore T+AT will be the same 

 function of the varied coordinates q + Ag, and the varied velocities 

 q + Ag, that T is of q and q. Hence we may write, in (7), 



dT . . dT . dT . dT . /o\*. 



The derivation of the Lagrangian equations then follows 

 exactly as before. 



It is a simple consequence of Lagrange s equations, thus established for 

 the present case, that the generalized components of the impulse by which 

 the actual motion at any instant could be generated instantaneously from 

 rest are 



dT dT 



1 2&quot;&quot; 



If we put 7 T =T + T 1 , we infer that the terms 



dT dT 

 W d&~ 



must represent the impulsive pressures which would be exerted by the solids 

 on the fluid in contact with them. 



This may be verified as follows. If A, Ar/, A denote arbitrary variations 

 subject only to the condition of incompressibility, and to the condition that 

 the fluid is to remain in contact with the solids, it is found as above that, 

 considering the fluid only, 



(i). 

 Now by the kinematical condition to be satisfied at the surface, we have 



jA + mAi7 + wA=- jk A?1 _ Jb Ag , 2 _ ............... (ii), 



\AjJlt (.If ll 



and therefore 



^ dn 



2 + ---)A2 i 

 dT 



by (1), (2), (3) above. This proves the statement. 



With the help of equation (iii) the reader may easily construct a proof of 

 Lagrange s equations, for the present case, analogous to that usually given in 

 text-books of Dynamics. 



* This investigation is amplified from Kirchhoff, I.e. ante p. 167. 



