OF WHICH IS MOVING ROT ATION ALLY AND PART IRROTATIONALLY. 393 
• c 'l 
potential of the density ——throughout the cylinder =1, and which is given 
in Example I. 
V 
These values make - -j-V continuous at the surface. 
P 
12. Example III. Supposing the relation found in the last article between f and ci 
not satisfied, it is required to find a solution if possible. 
Using the elliptic coordinates e, v which satisfy the equations 
x 
n 
+ 
y 
cr -f e b 2 + e 
X' 2 y' 2 
a? + v b 2 -\-v 
= 1 
where 
Hence 
and putting 
x' 2 = 
— b 2 < e < go 
— a 2 <u< — b 2 
(a 2 + e)(a 2 + v) ,, _ (b 2 + e)(-b 2 - v) 
a 2 — b 2 >y~— o 2 -b 2 
a— log (\/a 2 +e+ \/6 2 H-e) 
/?= — tan -1 aJ - 
-b 2 -v 
+ a 2 + u 
PY d 9 Y 
Laplace’s equation becomes —=0. 
da. 2 dp 2 
If Y be a function of e only, then V=Ca+C / . Similarly V=C/3-|-C / is a solution. 
But if Y=E.U where E is a function of e only, U a function of v only, then 
TT d 2 E , „d 2 U 
U 3?+ E 4*=° 
d 2 ~Ej 
Suppose there exist a value of E such that =p 2 E. 
Then 
Therefore if 
E=Ae^ a +Be~^“ 
U=A' cos pfi +B' si n p/3 
d 2 U 
d/3 
\+pr U = 0. 
E=A(\/crd-e-f \Zb 2 -\-e)P-\- B(v/a 2 +e+ \Zb 2 -{- e) p 
U = A' cos jp tan- 1 aJ -B' sin j p tan" 1 aJ 
3 E 
MDCCCLXX XIV. 
