110-111] FLOW THROUGH AN ELLIPTIC APERTURE. 161 



where the upper limit is the positive root of 



** y* ,* 2 _-i m 



a' + X + F+X + X~ 



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> are continuous at the aperture, 

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

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

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



approximately, since A, is small, whence 



_ 



dn~ab' 





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 < = 0, finite 

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

 to the solution < = x for the internal space. Following the analogy 

 of spherical harmonics we may assume for trial 



*=* .............................. (1), 



which gives V2 % + ? = ........................ ( 2 ), 



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), 



2dx =li (fr' 

 xdx (\ 



L. 11 



