129-130] COMPONENTS OF IMPULSE. 195 



Hence, since & T + T lf we have 



By virtue of Lord Kelvin's extension of Green's theorem, al- 

 ready referred to, these may be written in the alternative forms 



(4). 

 dp ' ^JJ dn - r ^J/^" 



Adding to these the terms due to the impulsive pressures 

 applied to the barriers, we have, finally, for the components of the 

 total impulse of the motion, 



m 



MS, ^ d 



x= dp- +x " *-^r 



where, for example, 



f^d* + p*[((l + 



dnj JJ\ dn 



X = pK II I ny - mz + -% j da -f p K ' I (ny -mz+ -^M do-' 4- . . . 



It is evident that the constants f , 770 , f , X , /^ OJ .^o are the 

 components of the impulse of the cyclic fluid motion which 

 remains when the solid is, by forces applied to it alone, brought 

 to rest. 



By the argument of Art. 116, the total impulse is subject to 

 the same laws as the momentum of a finite dynamical system. 

 Hence the equations of motion of the solid are obtained by substi- 

 tuting from (5) in the equations (1) of Art. 117*. 



* This conclusion may be verified by direct calculation from the pressure- 

 formula of Art. 21 ; see Bryan, " Hydrodynamical Proof of the Equations of Motion 



of a Perforated Solid, ," Phil. Mag., May, 1893. 



132 



