OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 403 
Differentiate the last equation with regard to r. 
Therefore 
W(r) 
Therefore 
x= ^F(r)-rFW+ r i|^} + <{F(r)F>)-i{F'(r)}H S ^ ) -[{F'(0} 3 f 
\=F(r) 
1 F<r) 
If the rotationally-moving liquid be bounded by the cylinder \=0, and its radius 
be r—-a; then F (a) = 0 
Therefore when 
r=a, x =0(-aF»)+<{ -*(F'(a)r-J'(F'(a))^} 
A suitable value for the velocity potential at points outside the cylinder r=a is 
^=^-«F'(«))+«{-i(F'(«)) s -{(F'(«)) 3 * 
This will make the velocity and pressure continuous at the surface of the rotation- 
ally-moving liquid. Also the velocity at infinity will be infinitely small. 
14. Example Y. This case is of interest, because one set of the vortex sheets, viz., 
ar z (z—Z) 2 -\-b(r 2 —or) 2 = const., consists in part of ring-shaped surfaces. The results 
only are given. 
If a and b are positive, and the constant <Z>a 4 ', then this represents ring-shaped 
surfaces. 
The equation in A of Art. 9, includes as a special case 
A particular integral is 
ltd? 1 d d?\ 
r 2 \dr 2 r dr^dz^ C ° nSt ’ 
\= ar 2 (z —Z) 3 -f 6 (r 3 —a 2 ) 3 
T=2ar(z— Z) and w=Z — 2a(z — Z) 2 —46(r 2 —a 2 ) 
3 F 2 
giving 
