160 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



At a great distance from the origin the ellipsoids X become 

 spheres of radius X*, and the velocity is therefore ultimately equal 

 to 2(7/r 2 , where r denotes the distance from the origin. Over any 

 particular equipotential surface X, the velocity varies as the 

 perpendicular from the centre on the tangent plane. 



To find the distribution of sources over the surface X = which 

 would produce the actual motion in the external space, we 

 substitute for < the value (1), in the formula (11) of Art. 58, and 

 for </>' (which refers to the internal space) the constant value 



t w- 



The formula referred to then gives, for the surface-density of the 

 required distribution, 



n 



i (5)- 



The solution (1) may also be interpreted as representing the 

 motion due to a change in dimensions of the ellipsoid, such that 

 the ellipsoid remains similar to itself, and retains the directions of 

 its axes unchanged in space. If we put 



a/a = b/b = c/c, = k, say, 

 the surface-condition Art. 107 (4) becomes 



which is identical with (3), if we put C = 



A particular case of (5) is where the sources are distributed 

 over the elliptic disk for which X = c 2 , and therefore z* = 0. This 

 is important in Electrostatics, but a more interesting application 

 from the present point of view is to the flow through an elliptic 

 aperture, viz. if the plane xy be occupied by a thin rigid partition 

 with the exception of the part included by the ellipse 



we have, putting c = in the previous formulae, 



