274 PROFESSOE G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
We may now revert to the Gaussian notation with single integral sign, and we 
see that the lost energy of the system is 
\{W,-U,)-{W-U)-\pdv. 
But W — U is the potential of the double system at Q, and is therefore Th; and 
Wq — C/q is the potential of the double system at P, and is therefore Vq. 
Accordingly the lost energy 
},DD = ^ {V,-V^)pdv 
= |7rp3 |(e3 _ _p y 1^3 ^dcr .(21) 
10. Determination of e and X. 
fc- is the arc of the orthogonal curve from J to the pear. 
The arc of outward normal is connected with and our variable r by the equation 
vdv-^^^dr. 
P V 
It follows that 
— pQ dr, integrated from J to the pear. 
By (50) of “ Harmonics,” with the notation of § 3 of this paper 
1 — jd cos -pd 
k 
f 2 2h 
o\i / o 
1 -d 
sin d 
d* 
B i> (v~ — !)'■ [ — 
\ 1 - d 
Therefore 
1 + dy cos d cos 7 (1 — Tj)^ (1 — Tj sec- d)" (1 “ D sec- 7 )“ 
do 
P 
1 - dJ 
Ad 
1 - d 
IV 
(1 — rd- (1 — sec- d)" (1 — rj see- 7 )^ 
= 1 
Xji 
1 
1 
Ad Fd 
= 1 — r 
cos- d eos® 7/1 
0 + 7^ 
AdFd VAd ■ Fd 
Integrating this from J to the pear 
2G 
e = - Vo 
eS., + i + y - ggj ■ (22)- 
We have, moreover, by the formula before integration 
