114-116] SURFACE CONDITIONS. 169 



whence, substituting the value (2) of <, we find 



j 



y- = I, - -f = ny - mz 

 dn dn 



2 7 



- -f* = m , -* Iz - nx 

 dn dn 



.(3). 



_ = _ =mx _ L 



dn dn 



Since these functions must also satisfy (1), and have their deri- 

 vatives zero at infinity, they are completely determinate, by 

 Art. 41*. 



116. Now whatever the motion of the solid and fluid at any 

 instant, it might have been generated instantaneously from rest by 

 a properly adjusted impulsive 'wrench ' applied to the solid. This 

 wrench is in fact that which would be required to counteract the 

 impulsive pressures p<p on the surface, and, in addition, to generate 

 the actual momentum of the solid. It is called by Lord Kelvin 

 the ' impulse ' of the system at the moment under consideration. 

 It is to be noted that the impulse, as thus defined, cannot be 

 asserted to be equivalent to the total momentum of the system, 

 which is indeed in the present problem indeterminate. We 

 proceed to shew however that the impulse varies, in consequence 

 of extraneous forces acting on the solid, in exactly the same way as 

 the momentum of a finite dynamical system. 



Let us in the first instance consider any actual motion of a 

 solid, from time to time ^, under any given forces applied to it, 

 in SL finite mass of liquid enclosed by a fixed envelope of any form. 

 Let us imagine the motion to have been generated from rest, 

 previously to the time t , by forces (whether gradual or impulsive) 

 applied to the solid, and to be arrested, in like manner, by 

 forces applied to the solid after the time t t . Since the momentum 

 of the system is null both at the beginning and at the end of this 

 process, the time-integrals of the forces applied to the solid, to- 

 gether with the time-integral of the pressures exerted on the fluid 



* For the particular case of an ellipsoidal surface, their values may be written 

 down from the results of Arts. Ill, 112. 



