406 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
For V' is finite and continuous inside the cylinder and satisfies Laplace’s equation. 
It is continuous with V at the surface of the cylinder. 
Also V satisfies Laplace’s equation outside the cylinder, and —, — both vanish 
at infinity. 
To find the volume density p of the shell, let Sn be the thickness of the shell, dn 
an element of normal drawn outwards. 
Also 
dY dY' , t . 
—- 7 -p &Trpon = 0 
dn dn ' 
dY x 2 y 2 
Calculating the value of — at the surface —+ —= 1,fie., at the surface 
« 2 +e /3 2 + e 
where e=0 
dY _ n 0.-/3 xy 
dn 
1 
p being the perpendicular from the centre on the tangent to —+^= 1 at the point x, y. 
Similarly at the same point 
therefore 
dn U ' 2 A + /3\a- /3V V 
dY dY' pCxy o 2 -/3 2 
dn dn 2 a 2 /3 2 
Also —= — where afi-Sa is semi-major axis of external boundary of shell. 
Hence 
C xy o~ — /3 2 o 
^ 87T « 2 yS 2 So 
Now consider the cylindric shell bounded by the two surfaces 
or y x~ t ?/'- 
(«j’ 2 ^pj’ 2=1 allC [(m + Sm )«] 2 “*~[{m + 8m)b] 2 
The density at the point x, y inside it is 
c 
'HV 
