11-12] 



Hence, 



or 



Similarly, 



IMPULSIVE MOTION. 



u f u) = 



, v , 1 dvr 



u u = X -y- 



p dx 



. TT/ 1 d^r 



v ' - v = Y' - - -r- 

 p dy 



, , 1 d 



w w= Z 3- 



p dz 



13 



dtp 



These equations might also have been deduced from (2) of Art. 6, by 

 multiplying the latter by dt, integrating between the limits and r, putting 



X'= [ r Xdt, Y' = ( T Ydt, Z'= ( T Zdt, vr= [ pdt, 



Jo Jo yo jo 



and then making r vanish. 



In a liquid an instantaneous change of motion can be produced 

 by the action of impulsive pressures only, even when no impulsive 

 forces act bodily on the mass. In this case we have X', Y', Z' = 0, 

 so that 



u ' u -- -j- 

 p dx 



v --- =- 



p d y 



1 d'ST 



= -- -j- 

 p dz 



(2). 



If we differentiate these equations with respect to x, y, z, 

 respectively, and add, and if we further suppose the density to 

 be uniform, we find by Art. 9 (1) that 



dx* 



_ 



dz* ~ 



The problem then, in any given case, is to determine a value of OT 

 satisfying this equation and the proper boundary conditions* ; the 

 instantaneous change of motion is then given by (2). 



* It will appear in Chapter in. that the value of w is thus determinate, save as 

 to an additive constant. 



