152 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



This solution represents the flow of a liquid through a circular 

 aperture in an infinite plane wall, viz. the aperture is the portion 

 of the plane yz for which or &amp;lt; k. The velocity at any point of the 



aperture (f = 0) is 



_ !_ cty _ A 



~vd~ (#--BJ a )* 



since, over the aperture, kfjb = (& 2 ^ 2 )i The velocity is therefore 

 infinite at the edge. Compare Art. 66, 1. 



Again, the motion due to a planetary ellipsoid (f=?o) moving 

 with velocity u parallel to its axis in an infinite mass of liquid is 



given by 



(3), 



-cot-^ 



where A=-ku + \j~j - cot- 1 U - 



IbO + - 1 - j 



Denoting the polar and equatorial radii by a and c, we have 



so that the eccentricity e of the meridian section is 



=(?. +!)-. 



In terms of these quantities 



(5). 



The forms of the lines of motion, for equidistant values of o/r, 

 are shewn on the opposite page. 



The most interesting case is that of the circular disk, for 

 which e = 1, and A = 2uc/?r. The value (3) of cf&amp;gt; for the two sides 

 of the disk becomes equal to Ap, or ^.(1 r 2 /c 2 )*, and the 

 normal velocity + u. Hence the formula (4) of Art. 44 gives 



=-2p [ 



Jo 



.................................... (6). 



The effective addition to the inertia of the disk is therefore 

 2/7T (= 6365) times the mass of a spherical portion of the fluid, of 

 the same radius. 



