[ 671 ] 



LXX. Figures of Equilibrium of Rotating Fluid under the 

 restriction that the Figure is to be a Surface of Revolution. 

 By J. R. Wilton, M.A., B.Sc, Assistant Lecturer in 

 Mathematics at the University of Sheffield*. 



THE following paper was begun rather more than a year 

 ago. It was then put aside until a more convenient 

 season, and now, owing to other work which I have under- 

 taken, its completion has been rendered impossible for a long- 

 time to come. I believe, however, tiiat there is some interest 

 in the paper as it stands, and I venture to publish it in its un- 

 finished state. There is a great deal of heavy, but straight- 

 forward, arithmetic required to complete it. It will be seen 

 that the whole paper presents very striking analogies with 

 that of Mr. J. H. Jeans on the " Equilibrium of Rotating 

 Liquid Cylinders "f. 



In the case of a surface of revolution the potential can be 

 very simply written down without a knowledge of the form 

 of the surface. For the potential of a uniform circular disk 

 at a point on its axis is 



whore &% is the thickness, «■ the radius, p the density of the 

 disk, and z is the distance of the point where the potential 

 is measured from the plane of the disk; — if z is negative its 

 sign must be changed in the above expression. Whence it 

 follows that the potential of a surface of revolution at a 

 point on its axis, within the surface, is 



V = 2* P J f W^ + (i - W ~ (-- - 1 # 



K J 



tsr being a function of f ; and the potential at any point 

 within the surface is therefore, by a well-known theorem, 



+ f (2+ *Rcos\- ?)<??} dX, 



*, R being the cylindrical coordinates of the point. 



* Communicated bv the Author. 



t Phil. Trans. A. cc. (1002) pp. G7-104. 



