40 IRROTATIONAL MOTION. [CHAP. III. 



Further, the velocity cannot be a maximum at a point in the 

 interior of the fluid. For let the axis of x be taken parallel to the 

 direction of the velocity at any point P. The equation (10), 

 and therefore also the equation (11), is satisfied if we write 



~ for (f&amp;gt;. The above argument then shews that ~ cannot be a 

 cLoc dx 



maximum at P. Hence there must be some point in the imme 

 diate neighbourhood of P for which -~*-- h as a greater value, and 



.1 c /-, i i f/^Y . fdd&amp;gt;\* fdd&amp;gt;\ 2 }$ . 



therefore a fortiori, for which s-p -f [ -r 1 ft*)&quot;? 1S greater 



(\dxj \dyj \dz) } 



than -~ t i.e. the velocity of the fluid at some neighbouring point 

 is greater than that at P*. 



On the other hand, the velocity may be a minimum at some 

 point of the fluid. In fact, taking any case of fluid motion, let us 

 impress on the whole mass a velocity equal and opposite to that 

 at any point P of it. In the resulting motion the velocity at P 

 will be zero, and therefore a minimum. 



46. Let us apply (11) to the boundary of a finite spherical 

 portion of the liquid. If r denote the distance of any point from 

 the centre of the sphere, dW the elementary solid angle subtended 

 at the centre by an element dS of the surface, we have 



dn dr 

 and dS = r d-ar. Omitting the constant factor r 3 , (11) becomes 



2^=o, 



or 



Since j- 1 1 &amp;lt;f&amp;gt;d&, or - 2 II &amp;lt;f&amp;gt;dS, is the mean value of &amp;lt; over 

 the surface of the sphere, (12) shews that this mean value is inde- 



* This theorem was set by Prof. Maxwell as a question in the Mathematical 

 Tripos, 1873. The ahove proof is taken from Kirchhoff, Vorlesungen iiber Mathema- 

 tische Phynik. Mechanik, p. 186. Another proof is given below, Art. 64. 



