12 15.] IMPULSIVE MOTION. 11 



planes meeting in P, and the surface F=Q, is ultimately 



| p (ufiy + vya. + wa/3) ; 

 and the rate of increase of the mass included in this space is ultim 



ately -j- dpo^y). Equating these expressions, substituting for 



a, /3, 7 their values, and omitting infinitesimals of higher order 

 than the second, we readily find 



dF dF dF dF 



U-T- +V -j +W-j- = -JT, 



ax ay dz at 



which agrees with (11). 



Impulsive Generation of Motion. 



15. If at any instant impulsive forces act on the mass of the 

 fluid, or if the boundary conditions suddenly change, a sudden 

 alteration in the motion may take place. The latter case may 

 arise, for instance, when a solid immersed in the fluid is suddenly 

 set in motion. 



Let p be the density, u, v, w the component velocities immedi 

 ately before, u 9 v, w those immediately after the impulse, X , Y , 

 Z the components of the external impulsive forces per unit mass, 

 w the impulsive pressure, at the point (#, y, z). The change of 

 momentum parallel to x of the element defined in Art. 6 is then 

 pdxdydz(uu); the ^-component of the external impulsive forces 

 is pdxdydzX j and the resultant impulsive pressure in the same 



direction is -^-dxdydz. Since an impulse is to be regarded as 



CLOG 



an infinitely great force acting for an infinitely short time (r, say), 

 the effects of all finite forces during this interval are neglected. 



Hence, 



pdxdydz(u u) = pdxdydzX -j- dxdydz, 



, v/ 1 dvr 



or u u = A j 



p ax 



C1 1 1 T T7 1 ClTJT 



Similarly, v v = Y 7 



p dy 



~, 1 dw 



w w = Z j- . 



p dz 



,(19). 



