31(5 
PROFESSOR G. H. DARWIN ON THE PEAR-SHAPED FIGURE 
It 
= f. 
chi 
0 + C") 
.2\i ’ 
the internal potential of an ellipsoid of mass M and semi-axes a, 6, c is 
w 
■ , 'J? cW , 7/2 ,-j2 tW 
'Vj/' - - _1— - 
a da b db c dc 
Hence 
r 
j -^12 0 
'P d-7~ + 
a cm 
dm. 
Now if A, B, C denote the principal moments of inertia of the ellipsoid about 
x, y, z, 
I x\hn = ^{C + B - A) = 
and similar formulae hold for the two other axes. 
Therefore 
I [ *"/’»? = p/. ■ 
J -^13 ^ 
^ 1 . (/'P , thi' 
' cla db dc , 
But since ’^P is a homogeneous function of degree — 1 in a, h, c, the sum of the 
three ditferential terms is equal to — Hence this expression is equal to 
Since 
iAco^^ = + c^) coh 
we have 
E = 
If E be varied, whilst ahc and w are constant, it is stationary if 
d'P /d'P 2h A /c/^P 2c o w _ 
+ ‘AiF " ) ^^ + ( .7. + 
da 
dM / ' \ fZc ' 33/ 
ha ^ hh he 
N + T + T = 0. 
a 0 c 
Eliminating S«, 86, 8c we have the well-known conditions for Jacobi’s ellipsoid 
2o3~b- J^P , d^f 
XV,p ■= a ~ -6 
33/ cla db 
O 2 
Zco'^c'' 
cl'¥ 
33/ — Act ~ d€ ’ 
1 / fZ^P , /'P\ 1 / /^P fZTA 
/2 da ^ da) ^ c2 fZa ~ ^ dc) ‘ 
(34) 
