284 Prof. M. J. M. Hill. On the Motion of Fluid. [Feb. 7, 



One set of the surfaces which always contain the same vortex lines 

 is given by ar 2 (z — Z) 2 + 6(r 2 — a 2 ) 2 =const. 



If the constant be<&a 4 , the equation represents ring-shaped 

 surfaces. 



The above expressions for the components of the velocity determine 

 a possible case of rotational motion inside the hollow smooth rigid 

 surface of annular form whose equation is ar*(z — Z) 2 + b(r 2 — o?Y — 

 const., and which moves parallel to the axis of z with velocity Z. 



The values of X, ifr of Clebsch's forms of expression for the com- 

 ponents of the velocity are calculated, but the writer has not succeeded 

 in finding an ir rotational motion outside one of the above annular 

 vortex sheets continuous with the rotational motion inside it. 



Could the solution be completed it would amount to a discussion 

 of the motion of an annular vortex, all of whose parts are of finite 

 dimensions. 



Appendix. 



The paper concludes with an Appendix, in which has been placed an 

 account of Examples 1 and 2, the calculation of the potential of the 



Q 2 • 



elliptic cylinder _-|-?L=l, the density of which at the point x, y, is 



' a? W 



(Although this density is infinite along the axis of the cylinder, yet 

 the total amount of matter in any elliptic cylinder surrounding the 



2 2 



axis whose equation is —+^~= const., however small the constant 



a 2 b' 2 



may be, vanishes. Hence it is not singular that the potential is 

 finite.) 



The potential inside is— 



c a — b , cab 

 2 a+b J 2 



/ \ 



x 



sin- 1 -- — sm" 1 



% + 1 



Vaj 2 + < 



The potential outside is — 



cab ( a 2 + 6 2 + 2e -A . cab: . _, x x \ 

 . — . — xyi . — ! L _1 )+ — i sm 1 — — sm 1 — - } 



-i IV u 



where — — + — £ — = 



a 2 + e b 2 + e 



