512 
PROFESSOR Ct. H. DARWIN ON ELLIPSOIDAL HARMONIC ANALYSIS. 
Hence 
+ 1 • . C8). 
It follows from (55) that 
I/' 2 
krv' 
Whence 
.... (59). 
iif + o = 
Jc^ 
IfA^/ ,B + 2/3 
+ n4> r+/3 
^.2(p)Co((f)) + 2h' 
1 . . (60). 
o ,.o 
O-C'^ 
This is needed to express the rotation potential {f + ^)- If we add to this 
* /t" 
we have 
+ f + z^) = ^ 115 
k 
1 - /3 
l^L ^ B ^ _p 3 j,-3 _ ^ ^ _ _ (61)^ 
This expression will be needed hereafter. 
Returning now to the formation of the expression for I find 
3^2 1 r 1 + y^B + 1- {B+1 - 13) 
f ~ -1) (/- - 14|) L“ 2A' ‘ 1 - ^ 
1 + B vHB -1 + S^)-(B-1 + /3) 
+ 
4:B 
1 -/d 
-n a 4 n_ 1 
+ •S" - r~^ + r_-^j • 
()n considering the forms of the functions found that this result 
mav be written thus : 
t/ 
/d ^ 1 + B 
3rRi.^-l)(.2-;A?)-3(l-/3) 
Therefore, writing Po(p)^o(*^) f'^-'r unity, the surface density of the focaloid shell, for 
which V — v^, is 
1 + A’ 
. 2B ""f o(z0 ^" 
^ 2B 
+ 1 
]■ 
PR 
~1 + B 
6B 
n 
