46 IRROTATIONAL MOTION. [CHAP. III. 



present, that this region is simply-connected, and that &amp;lt; is there 

 fore single-valued. 



If &amp;lt;/&amp;gt; be constant over the internal boundary of the region, and 

 tend everywhere to the same constant value at an infinite distance 

 from the internal boundary, it is constant throughout the region. 

 For otherwise &amp;lt;/&amp;gt; would be a maximum or a minimum at some 

 point of the region. 



We infer, exactly as in Art. 49, that if &amp;lt; be given arbitrarily 

 over the internal boundary, and have a given constant value at 

 infinity, its value is everywhere determinate. 



Of more importance in our present subject is the theorem that 



if the normal velocity -J- be zero at every point of the internal 



boundary, and if the fluid be at rest at infinity, then &amp;lt;j&amp;gt; is every 

 where constant. We cannot however infer this at once from the 

 proof of the corresponding theorem in Art. 47. It is true that we 

 may suppose the region limited externally by an infinitely large 



surface at every point of which -^ is infinitely small; but it is 



conceivable that the integral 1 1 -=* dS taken over a portion of this 



surface might be finite, in which case the investigation referred to 

 would fail. We proceed, therefore, somewhat indirectly, as follows. 



51. Since the velocity tends to the limit zero at an infinite 

 distance from the internal boundary ($, say), it must be possible 

 to draw a closed surface 2, completely enclosing 8, beyond .which 

 the velocity is everywhere less than a certain small value e, which 

 value may, by making 2 large enough, be made as small as we 

 please. Now in any direction from S let us take a point P at such 

 a distance beyond 2 that the solid angle which 2 subtends at it is 

 infinitely small ; and with P as centre describe two spheres, one 

 just excluding, the other just including 8. We shall prove that 

 the mean value of &amp;lt;j&amp;gt; over each of these spheres is, within an 

 infinitely small amount, the same. For if Q, Q be points of 

 these spheres on a common radius PQ Q , then if Q, Q fall within 

 2 the corresponding values of &amp;lt;/&amp;gt; may differ by a finite amount; 

 but since the portion of either spherical surface which falls within 

 2 is an infinitely small fraction of the whole, no finite difference 



