46 IRROTATIONAL MOTION. [CHAP. Ill 



41. A class of cases of great importance, but not strictly in- 

 cluded in the scope of the foregoing theorems, occurs when the 

 region occupied by the irrotationally moving liquid extends to 

 infinity, but is bounded internally by one or more closed surfaces. 

 We assume, for the present, that this region is simply-connected, 

 and that <f> is therefore single- valued. 



If </> 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 $ would be a maximum or a minimum at some 

 point. 



We infer, exactly as in Art. 40, that if </> 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 be zero at every point of the internal 

 boundary, and if the fluid be at rest at infinity, then <f) is every- 

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

 proof of the corresponding theorem in Art. 40. It is true that we 

 may suppose the region limited externally by an infinitely large 

 surface at every point of which d<f>/dn is infinitely small ; but it is 

 conceivable that the integral ffdcfr/dn . dS, taken over a portion of 

 this surface, might still be finite, in which case the investigation 

 referred to would fail. We proceed therefore as follows. 



Since the velocity tends to the limit zero at an infinite 

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

 to draw a closed surface 2, completely enclosing S, beyond which 

 the velocity is everywhere less than a certain 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 let us describe two spheres, 

 one just excluding, the other just including S. We shall prove 

 that the mean value of < 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 PQQ\ then if Q, Q' fall within 

 2 the corresponding values of <j) 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 



