OF WHICH IS MOVING ROTATIONALLY AND PART IRROTATIONALLY. 405 
The values of the components of the velocity found however completely solve the 
following problem :— 
A hollow, smooth, rigid surface of annular form, whose ecpiation is 
ar 2 (z— Z) 2 +5(r 3 — of) 2 — const. 
moves parallel to the axis of z with arbitrary velocity Z, to find a possible rotational 
motion inside it. 
Appendix. 
X 2 if 
15. The density inside the elliptic cylinder —-f-—= L is 
1 1 
4t r\f a*l J 
x 1 y 
it is required (on account of Examples I., II., and III.) to calculate the potential inside 
and outside. 
(It may be noticed that although the density is very great near the axis of the 
elliptic cylinder, yet the total mass of matter inside the cylinder = const., 
however small the constant may be, vanishes. Hence it is not singular that the 
potential should be finite). 
The density varies as xy on every ellipse whose equation is < , + / , = const. 
It will be well to commence by finding the potential of a cylindric shell bounded by 
the cylinders 
-,+%= landT^ + -^=l 
a 2 /3 2 (may (m/3) 2 
where m is a little > 1, and where the density varies as xy. 
The following are suitable values for the potential 
and 
where 
V'=Cu“-Sw inside 
2 « + /3 J 
a 3 + /3 3 + 2e 
V—r —— - (] - 
2 v /(« 2 + 6 )(^ + 6 ) i 
xy outside 
a~ + e /3 3 + e 
and C has to be suitably determined. 
