50 53.] FINITE STREAM-LINES IMPOSSIBLE. 47 



in the mean values can arise from this cause. On the other hand, 

 when Q, Q fall without S, the corresponding values of &amp;lt;/&amp;gt; cannot 

 differ by so much as e . Q Qf, for e is by definition a superior limit 

 to the rate of variation of &amp;lt;f&amp;gt;. Hence, the mean values of &amp;lt;/&amp;gt; over 

 the two spherical surfaces must differ by less than e . QQ . Since 

 QQ is finite, whilst e may by taking S large enough be made as 

 small as we please, the difference of the mean values may, by 

 taking P sufficiently distant, be made infinitely small. 



Now we have seen in Art. 46 that the mean value of &amp;lt;/&amp;gt; over 

 the inner sphere is equal to its value at P, and that the mean 

 value over the outer sphere is (since J/=0) equal to a constant 

 quantity G. Hence, ultimately, the value of &amp;lt; at infinity tends 

 everywhere to the constant value C. 



The same result holds even if the normal velocity ~ be not 



J dn 



zero over the internal boundary ; for in the theorem of Art. 46 

 M is divided by r, which is in our case infinite. 



52. The theorem stated at the end of Art. 50 is now obvious. 

 For, under the conditions there stated, no stream-lines can begin 

 or end on the internal boundary. Hence, any stream-lines which 

 exist must come from an infinite distance, traverse the region, and 

 pass off again to infinity; i.e. they perform infinitely long courses 

 between places where &amp;lt; has, within an infinitely small amount, the 

 same value C, which is impossible. Hence no stream-lines exist, 

 or in other words there is no motion. 



We derive, exactly as in Art. 49, the important theorem that 



if j?r- be given at every point of the internal boundary, and if the 

 velocity be zero at infinity, the motion is everywhere determinate. 



53. Before discussing the properties of irrotational motion in 

 multiply-connected regions we must examine more in detail the 

 nature and classification of such regions. In the following synopsis 

 of this branch of the geometry of position we recapitulate for the 

 sake of completeness one or two definitions already given. 



On Multiply-Connected Regions. 



We consider any connected region of space, enclosed by bound 

 aries. A region is connected when it is possible to pass from 



