69 71.] CIRCULAR BOUNDARY. f&amp;gt;9 



71. If ds be an element of the boundary of any portion of the 

 plane xy which is occupied wholly by moving liquid, and if dr be 

 an element of the normal to ds drawn inwards, we have, by 

 Art. 44, 



the integration extending round the whole boundary. If this 

 boundary be a circle, and if r, be polar co-ordinates referred to 

 the centre P of this circle as origin, the equation (8) may be 

 written 





1 f 2 &quot;&quot; 



Hence the value of the integral -=- I 6dO, i.e. the mean- value of 



ZTTJ Q 



(f&amp;gt; over a circle of centre P, and radius r, is independent of the 

 value of r, and therefore remains unaltered when r is diminished 

 without limit, in which case it becomes the value of &amp;lt;f&amp;gt; at P. 



If the region occupied by the fluid be periphractic, and we 

 apply (8) to the space enclosed between one of the internal 

 boundaries and a circle with centre P and radius r surrounding 

 this boundary, and lying wholly in the fluid, we have 



dr 



where the integration in the first member extends over the circle 

 only, and 27rlf denotes the flux into the region across the internal 

 boundary. Hence 



d 1 /*- - M 

 3- 3 I 6dO = , 



dr 27rJo r 



which gives on integration 



c 0); 



i.e. the mean value of &amp;lt;/&amp;gt; over a circle with centre P and radius r 

 is equal to J/logr + G, where C is independent of r but may vary 



