G4 G5.] KINETIC ENERGY OF A LIQUID. 61 



motion. It is a proposition in Dynamics* that the work done by 

 an impulse is measured by the product of the impulse into half 

 the sum of the initial and final velocities, resolved in the direction 

 of the impulse, of the point to which it is applied. Hence the 

 right-hand side of (24) when modified as described, expresses the 

 work done by the system of impulsive pressures which, applied to 

 to the surface of S, generate the actual motion ; whilst the left- 

 hand side gives the kinetic energy of this motion. The equation 

 (24) asserts that these two quantities are equal, thus verifying for 

 our particular case the principle of energy. 



Hence if T denote the total kinetic energy of the liquid, we 

 have 



Corollary 5. In (24) instead of c/&amp;gt; let us write -^-, which will of 



CLX 



course satisfy (10) as &amp;lt; does; and let us apply the resulting theorem 

 to the region included within a spherical surface of radius r having 

 any point (#, y, z) as centre. With the same notation as in Art. 46, 

 we have 



$&\ , ( 



d *4&amp;gt; \* l d *&amp;lt;t&amp;gt; VI i j j 



T-J- + TJ f dxdydz, 

 dxdy) \dxdz) } 



dx 2 J \dxdy 



an essentially positive quantity. Hence, writing &amp;lt;f = u* + v 2 + w*, 

 we see that 



d [r 



37-JJ 2 dv 



is positive; i.e. the mean value of ^ 2 , taken over a sphere having 

 any point as centre, increases with the radius of the sphere. Hence 

 q cannot be a maximum at any point of the fluid, as was otherwise 

 proved in Art. 45. 



65. We shall require to know, hereafter, the form assumed by 

 the expression (25) for the kinetic energy when the fluid extends 



* Thomson and Tait, Natural Philosophy, Art. 308. 



