43-45] KINETIC ENERGY. 51 



these two quantities are equal. Hence if T denote the total 

 kinetic energy of the liquid, we have the very important result 



-j^dS (4). 



If in (3), in place of &amp;lt;/&amp;gt;, we write d&amp;lt;f)/dx, which will of course satisfy 

 = Q, and apply the resulting theorem to the region included within a 

 spherical surface of radius r having any point (#, y, z) as centre, then with the 

 same notation as in Art. 39, we have 



d (&amp;lt;kt&amp;gt;\ 7 

 ~ -T- (-=-*- dS 

 dx dn dx 



1 &amp;lt;i d [f *j [f du jo f 



2 r T I I u d = I / u -j- dS = - I I 



dr J J J J dr J J 



M 



Hence, writing q 2 = 



Since this latter expression is essentially positive, the mean value of q 2 , taken 

 over a sphere having any given point as centre, increases with the radius of 

 the sphere. Hence q cannot be a maximum at any point of the fluid, as was 

 proved otherwise in Art. 38. 



Moreover, recalling the formula for the pressure in any case of irrotational 

 motion of a liquid, viz. 



we infer that, provided the potential Q. of the external forces satisfy the 

 condition 



V 2 Q = ....................................... (iii), 



the mean value of p over a sphere described with any point in the interior of 

 the fluid as centre will diminish as the radius increases. The place of least 

 pressure will therefore be somewhere on the boundary of the fluid. This has 

 a bearing on the point discussed in Art. 24. 



45. In this connection we may note a remarkable theorem 

 discovered by Lord Kelvin*, and afterwards generalized by him 

 into an universal property of dynamical systems started impulsively 

 from rest under prescribed velocity-conditions -f-. 



The irrotational motion of a liquid occupying a simply-con 

 nected region has less kinetic energy than any other motion 

 consistent with the same normal motion of the boundary. 



* (W. Thomson) &quot;On the Vis- Viva of a Liquid in Motion,&quot; Carrib. and Dub. 

 Math. Journ., 1849; Mathematical and Physical Papers, t. i., p. 10.7. 

 t Thomson and Tait, Natural Philosophy, Art. 312. 



42 



