408 
PROFESSOR M. J. M. HILL OH THE MOTION OF FLUID, PART 
Therefore the potential for the whole cylinder at an external point is 
a~ + ¥ + 2P \ [(« 2 + P) 2 1 (b 
2v/(« 2 + P)(6 2 + P) 
« 2 + 6 2 + 2P 
-£—\ 
b 2 + P) 2 J 
, f l 2 
« 2 + P ' 5 2 + PJ 
dP 
cab 2 \/(a 2 + P)(& 3 + P) , ccib _ 1 
b 2 —a 2 
xy 
cab 
^ , r 
« 2 +p 5 2 +p 
« 2 + & 2 + 2e 
' 2 v/(a 2 + e)(Z> 2 + e) 
_ tan 
2 iy 
- A 2 + P ] 
v V « 2 + PJ 
+ 
c«5 ( « 2 + 5 2 + 2e \ , cab / . « 
= 52 _ a? xl J[ 1 “ 2 y (a 2 + e)(b 2 + e)/ + T ( Sm V / ^ + < 
sin 1 
v/^ + J 2 
To find the potential at an internal point x, y. 
x 2 y 2 
Suppose that this point lies on the elliptic cylinder t~ -f-rrr;= 1 
(/*»)" o & )~ 
Then the potential at x, y — potential of matter inside this cylinder 
+ potential of matter outside it. 
x 2 y 2 . . 
The potential of the matter inside - 1 —4--^—=1 is obtamed by taking the same 
1 (/^) 2 ( yb ) 2 J & 
x 2 y 2 
integral as in finding the potential of the cylinder — +—=1 at an external point, but 
x 2 y 2 
the limits for m are now 0 and y, i.e., P= 00 to P=a root of the equation ^r^y,+^ + j , — P' 
which makes a 2 +P, 6 3 +P both positive, but this root is zero, since 7—w:+-r - rn;=l. 
This gives 
{yaf 1 (yb ) 2 
cab xy ( (a — b) 2 \ , cab/, ,bx , ,x 
- ---f - — f- +— tan" 1 -tan" 1 
b“ —a~y\ 2 ab ) 2 \ ay y 
i.e., 
c a —b xy , cab 
---2 _L—P 
2 a + b y 2 ^ 2 1 
sm 
V o o 
sin 1 
\/x 2 +y 2 
The potential of the matter outside the cylinder -y,+|- 1 y-;= 1 is the integral of 
—2c8m^yt -- X y between the limits m—y and m—]. 
m' * a + b J n 
