42 IRROTATIONAL MOTION. [CHAP. III. 



If however the original region throughout which the irrota- 

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

 therefore all the higher) derivatives of &amp;lt; 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 (13), equal to the mean 



value of - 7 - over the surface. Since ~- vanishes at infinity, we 

 ax dx 



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

 as small as we please. Hence C is not altered by a displacement 

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

 see that -C is not altered by a displacement parallel to y or z; 

 i. e. it is absolutely constant. 



If the internal boundaries 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 &amp;lt;f&amp;gt; over any spherical surface enclosing 

 them all is the same. 



47. (a) If &amp;lt;/&amp;gt; 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 mini 

 mum value at some point of the region. 



Otherwise: we have seen in Arts. 42, 44 that the stream-lines 

 cannot begin or end at any point of the region, and that they 

 cannot form closed curves lying wholly within it. They must 

 therefore traverse the region, beginning and ending on its bound 

 ary. In our case this is however impossible, for a stream-line 

 always proceeds from places where &amp;lt;j&amp;gt; is less to places where it is 

 greater, whereas &amp;lt; is, by hypothesis, constant over the boundary. 

 Hence there can be no motion, i.e. 



dj d&amp;lt;j&amp;gt; = ^ 



dx dy dz 



and therefore &amp;lt;/&amp;gt; is constant and equal to its value at the boundary. 



applies are reconcileable (in a sense analogous to that of Art. 41) with one 

 another. 



* This step is taken from Kirchhoff, Vorlesungen ilber Math. Physik. Mechanik, 

 p. 191. 



