.318 
PROFESSOR G. H. DARWIN ON THE PEAR-SHAPED FIGURE 
where 
if 
P 
■2 P 
dW UP 
Now clianofe the notation and write 
<j - - |il/a _ - 
¥ 
M fhF 
a -y- 
p da 
Then 
Now 
and 
o 70/ n TO yn/o t\ 0 700 
«' = A- ( Vcd — ^ , />- = /^T2^o' ~ 1) 5 <^' = » 
dv 
fPF 
a- 
■> y 
i1)*’ 
■ - - 'If ■’_ 1 +/3'r 'h 
l-yo 
i9o (>') = 1. Pi‘ (><) = (— id . 
fOO 
Pd (*^o) (^ 0 ) = [Pd (Gj)? _ ; 
so 
that 
and 
a 
= J. Po (Gj) €0 (G)) ^ 
f =-|p‘WQ.'W, 
? = d Pi‘ (-0) Qi' {’'«) 
plo 
(36). 
We may note in passing that the condition for a Jacobian ellipsoid (the last 
equation of (34)) is reducible to the form 
/cA- G siii'^-vp 
siid7 J ^ A=^ 
7 , tan-ip 
clip =. K cot' y — 7 — 
J A -A 
chh. 
On examining the series of functions given in (25), we see that it may be written 
P/ (^^o) €2^ i^o) = (^0) Qi^ (^)o • 
This agrees with M. Poincake’s equation (l) on p. 341 of his memoir. 
We will now suppose that the body, instead of being an ellijjsoid, is an ellipsoidal 
harmonic deformation of an elli23Soid, which is itself a figure of equilibrium for 
rotation co. 
The addition to E will consist of three parts; first that due to the mutual 
