•254 PUOFERflOR (4. II. DARAMN ON THE STABIETTY OF THE PEAR-SHAPED 
it is clear that if 2e5^/ is a constant, say c, the surface defined is an ellipsoid similar 
to the surface of reference, with semi-axes augmented in the proportion of (1 cf to 
unity. 
T now replace the variable v by a new one, namely. 
" = - h - A) 
(i). 
The negative sign is taken Ijecause the points to be specified will lie inside J. 
Then r = c, a constant, defines an inteilor ellq^soid similar to and concentric witli ./. 
The ecjuation to the j^ear may now be written 
00 
1 
Th only condition wliich lias been imposed on c is that it shall he great enough to 
make r always positive. 
In order to solve our problem it is necessary to determine the energy lost in the 
process of concentration from a condition of infinite dispersion into the final con¬ 
figuration. This involves the use of the formula for the gravity of J, inclusive of 
rotation. It is well known that this forimda is simple for the inside of J and more 
coinjilicated for the outside. Since the whole region B lies inside J there is no 
necessity in the present case to use the more complicated formula. 
The final expression for the lost energy cannot involve the size of J, the exterior 
ellipsoid of reference, and therefore the arbitrary constant c must ultimately disappear. 
It is therefore legitimate to make c zero from the beginning. 
It is clear that we might with ecpial justice have discussed the problem by means 
of an ellipsoid which shmdd lie entirely inside the pear, the region between the pear 
and the ellipsoid would then have l^een filled with positive density, and the formula 
for external gravity would have been needed. The same argument as before would 
then have justified our putting the constant c equal to zero. 
We thus arrive at the same conclusion as does M. Poincare, namely, that it is 
immaterial whether the formula for external or internal gravity be used. 
I now revert to my first hypothesis of the enveloping ellipsoid, but put c equal 
to zero from the first. In order, however, to aftbrd clearness to our conceptions, I 
shall continue to discuss the problem as though c were not zero and as though J 
enclosed the whole pear. With this explanation, we may write the equation to the 
jiear in the form 
00 
T = — cS^ — .(2). 
§ 2. The Lost Energy of the System. 
If the negative density in B is transported along tubes formed by a famil}^ ot 
orthogonal curves and deposited as surface density on J, we may refer to such a 
