In this way the discontinuity in the coordinates on the plane a? = is displaced to the 

 outlying part where 75 > A; = c, which is now a rigid boundary, the two halves of each coordi- 

 nate ellipsoid meet there with opposite values of ^. The central part of the plane, on which 

 7o<k - c, represents a circular aperture. In the last section, discontinuities of /x were allow- 

 ed to occur on the central disk of radius k, but this part was there enclosed in a rigid body. 

 Thus in each case continuity of coordinates is preserved throughout the entire space occupied 

 by fluid. 



It will be found that the differential Equations [138u] and [138v, w] are satisfied by 



cf) ^ A cot~^ C 4/ = Ac 11. [139c, d] 



The velocity is in the (^-direction, so that q = \qA^ and from Equations [I38r, n] 



^^ = i (^2^ l)-i/2 (^2+^,2^1/2 [139e] 



On either face of the plane boundary, /i = 0, j- = and To = c(i^ + 1) ^ or 

 C - - v'^ ~'^ /<^; hence q = \q^\ and 



q^^tq Ai^2^,yU2^^^±^^_ [I39f] 



The sign i is to be taken the same as the sign of C 



In the plane of the opening 4 = 0, a- = 0, W = c(l - ji^)^^'^, q - \u\, and 



A A 



u = q.-— = . . [139g] 



On the axis of symmetry or a'-axis, /i = 1, a; = c^, q = \u\ and 



A CA r >, . 



= . [fS9h] 



c{c} + l) a'^ + c2 

 The velocity is thus infinite at the edge of the opening. 



i% 



