PROFESSOE Ct. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
509 
Therefore 
VnSy 
pvQ> 
+ 
2 _ 1+3 I 2 
L) 1_;3 *^0 
+i) 
or 
p^v^Sv = 
pv,^ - 1) {v‘^ - i^l) 
ov, 
4 _ r 1 + 3 
0 1 - 3 1 - 3 
(53). 
If Sn be the thickness of the shell at the point where p is the perpendicular on 
the tangent })lane, 
Sn = VnSp . 
V 
k^ 
If we multiply both sides of (53) by —, we see that the surfiice density of the 
focaloid shell is 
P 
- i-)(v - rii-3) ^ 
C,. 4 _ ■^‘'0^ 4. 1 nr 
< 52^0 — + 1-3 
7'2 
/v • 
If therefore we can express — in the form of surface harmonics, it will be easy 
to write down the external potential of the ellipsoid by means of the formula (51). 
Before doing this I will, however, take one other step. 
It is easy to see that 
q 4 "^^0^ I \ + _ q / 
5-0 -1_;3+ 1-13- 
■ 2 + 1 ! 
2 - B 
^0^ - 
3(1 -/3)/V ^ 3(1 -/3), 
. . (54), 
where for brevity = (1 + 3y8^)h 
Now on referring to 7, (17) and (23), we see that 
i. (0 = P. (0 + P./ (O. f / (0 = - P. (0 + P/ (0 , 
where Bo (i/) = By (i^) = 3 (w — 1). 
If then we put ^3 {p) = ocp- -f- y, 
\^.^(p) = — a'p- — y , or (j^) = ct'/x' y', 
it is clear that 
B — 1 + 3/3 
a 
a. = 
and 
7 
2/3 
:3( /j_ 1 - /3) 
/3 
B-2 
y = 
y = 
7 
— B + 1 — p . 
2/3 
— -R + 1 + 3/3 . 
-B-2 
* 3 (1 — yS) 
B 3(1 -/3) 
