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 ^<k. The velocity at any point of the 

 aperture (=0) is 



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

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



Again, the motion due to a planetary ellipsoid ( = *<,) moving 

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



given by 



(3), 



-cot->? ...... (4), 



where A=-ku + - - cot- 1 



IbO T 1 



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



so that the eccentricity e of the meridian section is 



e = (# + !)-* 

 In terms of these quantities 



............ (5). 



The forms of the lines of motion, for equidistant values of ty, 

 are shewn on the opposite page. 



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

 which e = l, and A = 2uc/?r. The value (3) of </> for the two sides 

 of the disk becomes equal to + Ap, or + A (1 -cT 2 /c 2 )*, and the 

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



2 .................................... (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. 



