262 PROFESSOR C4. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
\^Tpdv = ^M[~J + + . . ( 10 ). 
Retuming now to the deteimination of the mass of + R, and observing that the 
mass of the pear is equal to that oft/ — R, we have 
h \3 
Therefore 
ilf = M ( : [I + eVo + 
I~R = \ + e-T, + + c*8. 
A term e‘^§ of the fourth order has been introduced, but it will appear that it is 
unnecessary to evaluate it. 
There will be frequent occasion to express in terms of /q®. Now 
'' k V 
= 1 - f - I(^e)']- 
But this will only Ije needed explicitly as far as e^, and to that order 
/ k 
h 
5 9 , 
( 11 ). 
' J‘ ' 2 / /■ ^ 
It is, however, necessary to determine f ( ) — f to the fourth order. 
. ^••0 / 
Now 
- ' f)' = i 1 - I [eV. + iteMt + c-->8 - |c‘ (o-,)']}. 
k 
= i (1 - I [c'V, + 22cy:-./,/ + c‘S - |a(o-;)'-]!- 
Hence to the fourth order 
(II). 
It will be observed that the i/> integrals and S have both disa])peaied. 
5. The Energies hJJ and JR. 
It «!, /q, C] are the semi-axes of a Jacobian ellipsoid of mass J/j and angular 
velocity w, its lost energy, inclusive of rotation, is 
3-AT ~ 
where T' is the usual auxiliary function. 
w 4 . ; 
3.1/. " 
