72 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



the integration extending round the whole boundary. If this boundary be a 

 circle, and if r, 6 be polar co-ordinates referred to the centre P of this circle 

 as origin, the last equation may be written 



- or 



Hence the integral 5- I 



27T./0 



e. the mean-value of $ 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 < at P. 



If the region occupied by the fluid be periphractic, and if we apply (i) 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 



(ii); 



where the integration in the first member extends over the circle only, and 

 2jrM denotes the flux into the region across the internal boundary. Hence 



d 1 *' M 



which gives on integration 



i.e. the mean value of $ over a circle with centre P and radius r is equal to 



- M log r + C, where C is independent of r but may vary with the position of P. 

 This formula holds of course only so far as the circle embraces the same 

 internal boundary, and lies itself wholly in the fluid. 



If the region be unlimited externally, and if the circle embrace the whole 

 of the internal boundaries, and if further the velocity be everywhere zero at 

 infinity, then C is an absolute constant; as is seen by reasoning similar to 

 that of Art. 41. It may then be shewn that the value of at a very great 

 distance r from the internal boundary tends to the value Mlog r+C. In the 

 particular case of M=0 the limit to which $ tends at infinity is finite; in 

 all other cases it is infinite, and of the opposite sign to M. We infer, as before, 

 that there is only one single- valued function cf> which 1 satisfies the equation 

 (2) at every point of the plane xy external to a given system of closed curves, 

 2 makes the value of d<p/dn equal to an arbitrarily given quantity at every 

 point of these curves, and 3 has its first differential coefficients all zero at 

 infinity. 



If we imagine point-sources, of the type explained in Art. 56, to be distri- 

 buted uniformly along the axis of z, it is readily found that the velocity at a 

 distance r from this axis will be in the direction of r, and equal to m/r t where 

 m is a certain constant. This arrangement constitutes what may be called a 



* line-source,' and its velocity -potential may be taken to be 



0= wilogr ................................. (iv). 



