110-111] FLOW THROUGH AN ELLIPTIC APERTURE. 161 



where the upper limit is the positive root of 



and the negative or the positive sign is to be taken according as the 

 point for which $ is required lies on the positive or the negative side 

 of the plane xy. The two values of (f&amp;gt; are continuous at the aperture, 

 where X = 0. As before, the velocity at a great distance is equal to 

 2 A /r*, nearly. For points in the aperture the velocity may be 

 found immediately from (6) and (7) ; thus we may put 



approximately, since \ is small, whence 

 dt_2A ( x- . 



~ 



This becomes infinite, as we should expect, at the edge. The 

 particular case of a circular aperture has already been solved 

 otherwise in Art. 105. 



111. We proceed to investigate the solution of V 2 &amp;lt; = 0, finite 

 at infinity, which corresponds, for the space external to the ellipsoid, 

 to the solution &amp;lt;/&amp;gt; = x for the internal space. Following the analogy 

 of spherical harmonics we may assume for trial 



&amp;lt;}&amp;gt;=*x .............................. (i). 



which gives V 2 ^ + --^=0 ........................ (2), 



X d/X 



and inquire whether this can be satisfied by making ^ equal 

 to some function of X only. On this supposition we shall have, by 

 Art. 108 (3), 



and therefore, by Art. 109 (4), (6), 



xdx (\ 

 L. 11 



