42 IRROTATIONAL MOTION. [CHAP. Ill 



38. The function </> cannot be a maximum or minimum at a 

 point in the interior of the fluid ; for, if it were, we should have 

 d^/dn everywhere positive, or everywhere negative, over a small 

 closed surface surrounding the point in question. Either of these 

 suppositions is inconsistent with (2). 



Further, the absolute value of 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 equa- 

 tion (1), and therefore also the equation (2), is satisfied if we write 

 d<f>/dx for </>. The above argument then shews that d$/dx cannot 

 be a maximum or a minimum at P. Hence there must be some 

 point in the immediate neighbourhood of P for which dfyjdx has 

 a numerically greater value, and therefore a fortiori, for which 



is numerically greater than d^jdx, i.e. the velocity of the fluid at 

 some neighbouring point is greater than at P*. 



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

 fluid. The simplest case is that of a zero velocity ; see, for example, the figure 

 of Art. 69, below. 



39. Let us apply (2) 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, SOT the elementary solid angle subtended 

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



d<f>/dn = d<j)/dr, 

 and SS = r 2 $m. Omitting the factor r 2 , (2) becomes 



1 w w \y . 



dr 

 or Til ^d'sr = (3). 



Since l/4?r . //<COT or l/4?rr 2 .fftydS measures the mean value of 

 $ over the surface of the sphere, (3) shews that this mean value is 

 independent of the radius. It is therefore the same for any sphere, 

 concentric with the former one, which can be made to coincide 



* This theorem was enunciated, in another connection, by Lord Kelvin, Phil. 

 Mag., Oct. 1850; Eeprint of Papers on Electrostatics, &c., London, 1872, Art. 665. 

 The above demonstration is due to Kirchhoff, Vorlesungen iiber mathematische 

 Phijsik, Mechanik, Leipzig, 1876, p. 186. For another proof see Art. 44 below. 



