44 IRROTATIONAL MOTION. [CHAP. Ill 



the surface-integral extending over the sphere only. This may be 

 written 



1 d , M 



whence ^dS^^^C ............... (4). 



That is, the mean value of <f> over any spherical surface drawn 

 under the above-mentioned conditions is equal to M/r 4- C, where 

 r is the radius, M an absolute constant, and C a quantity which is 

 independent of the radius but may vary with the position of the 

 centre *. 



If however the original region throughout which the irrotational 

 motion holds be unlimited externally, and if the first derivative (and 

 therefore all the higher derivatives) of </> vanish at infinity, then C 

 is the same for all spherical surfaces enclosing the whole of the 

 internal boundaries. For if such a sphere be displaced parallel 

 to #(-, without alteration of size, the rate at which C varies in 

 consequence of this displacement is, by (4), equal to the mean 

 value of d<j)/dx over the surface. Since d^jdx vanishes at infinity, 

 we can by taking the sphere large enough make the latter mean 

 value as small as we please. Hence C is not altered by a displace- 

 ment of the centre of the sphere parallel to x. In the same way 

 we see that G is not altered by a displacement parallel to y or z ; 

 i.e. it is absolutely constant. 



If the internal boundaries of the region considered be such 

 that the total flux across them is zero, e.g. if they be the surfaces 

 of solids, or of portions of incompressible fluid whose motion is 

 rotational, we have M=0, so that the mean value of <f> over any 

 spherical surface enclosing them all is the same. 



40. (a) If < be constant over the boundary of any simply- 

 connected region occupied by liquid moving irrotationally, it has 

 the same constant value throughout the interior of that region. 

 For if not constant it would necessarily have a maximum or a 

 minimum value at some point of the region. 



* It is understood, of course, that the spherical surfaces to which this statement 

 applies are reconcileable with one another, in a sense analogous to that of Art. 35. 

 f Kirchhoff, Mechanik, p. 191. 



