276 PROFESSOR G. II. DARWIN ON THE STABIEITY OF THE PEAR-SHAPED 
When this is integi'ated we may put equal to unity. In the integral the 
M~ siir yS 
first term vanishes, and the second term gives — i --r-In the 
” “ /.’o cos /3 CO.S 7 ‘ 
third term we substitute for $ its value and have 
3 cos^ /3 cos^ 7 sin ^ C 1 
- n-TV- 
TT 
Sill" 7 
A 3-p 2 \ T"' 
. _il 1 1 1 
A,=){Krr'*'Ari 
rv ) — SG 
(W i?(f) 
AF 
. which is equal'to 
^ i/" ^ 6 cos-^ cos® 7 sill /3 
IT 
Sill" 7 
- / 1 
1 \ 
. J1 if,“A,* r,n,' 
-rl 1 
" ' r.‘Ar 
1 \ ] iWrlci> 
Fi^Ad/J AF^ 
By the definition ( 8 ) this is equal to — ^ — e^cr, 
Hence the required term in the energy is 
M 
A 
k 
0 L 
sin- /3 ^ 
cos /3 cos 7 
(24). 
§ 12 . The Energy {p 
dV 
e~ , - da. 
an 
It is first necessary to determine elVjdn. 
Siq:)pose that the ellipsoid J is coated with surface density S, and that a second 
surface is drawn inside -/ at an infinitesimal distance r, and coated with negative 
surface density — S', so that the two form a double layer. Then tS being a function 
of the trvo angular co-ordinates on the ellipsoid may be expanded in surface 
harmonics ; suppose then that 
TS=^ihtSE 
0 
Consider the two functions 
Tb = S (v — 1 )" ( V — 1 ^ {^) foi’ external sjmce, 
/3I dv^ 
(^) Si, for internal space. 
UVq 
. Since these functions are solid harmonics, the matter of which Ib and lb are the 
potentials is entirely confined to the surface of the ellipsoid, and since they are not 
continuous with one another, the ellipsoid must be a double layer. 
Now 
= W(^) 
dv 
and therefore 
