1884.] which is moving Rotationally and part Irrotationally. 283 



The components of the velocity of this rotationally moving liquid 

 are — 



y f (^t- — oj \ and —x'(^£-—u\ 



The components of the velocity of the irrotationally moving liquid 

 are — 



a 3_fc2|_ I J \a* &*/ SVtf + e lab ah J J 



az-tfl V\a? h*J J V a 2 + e lab ab J J 



Example 4. The vortex sheets are coaxial circular cylinders, and 

 the motion is everywhere perpendicular to the radius vector from the 

 axis, and a function of the distance. 



Thus radial velocity =0, velocity perpendicular radius vector = 

 -F(r). 



This example is given merely to illustrate the expression of the com- 

 ponents of the velocity in Clebsch's forms. 

 It is shown that 



X=J(r). 



+ JF(r)F"(r)-l(F(r))2 + iF(r)F(r)-f*(F'0-))A 

 I r J r J 



If F(a) = 0, and the rotational motion be supposed confined to the 

 interior of the cylinder r=a, then a suitable value of the velocity 

 potential at external points is — 



0=0(_ a F(a)) + ^|-l(F / (a))2-|^(F f («)) 2 j 

 Example 5. r, z being cylindric co-ordinates. 



Z, an arbitrary function of the time ; Z, its differential coefficient 

 with regard to the time, 

 a, a positive constants. 



t the velocity from the axis of cylindric co-ordinates in the direction 

 of r. 



w the velocity parallel the axis of cylindric co-ordinates. 

 It is shown that T=2ar(«-Z); w = Z-2a(z~Zy~~U(rZ-^) 

 satisfy the equations of motion. 



