310 PROFESSOR G. H. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
Although the constant expressive of the size of J is put equal to zero—v-hich 
means that the pear is really partly protuberant above the ellipsoid—I have found 
that a considerable amount of mental convenience results from always discussing the 
subject as though the constant were not zero, so that the ellipsoid envelopes the 
pear, and I shall continue to do so here. 
^Vhen an ellipsoid is deformed by an harmonic inequality, the volume of the 
deformed body is only equal to that of the ellipsoid to the first order of small 
quantities. In the case of the pear, all the inequalities, excepting the third zonal 
one, are of the second order, and as far as concerns them the volumes of J and 
of the pear are the same. But it is otherwise as regards the third zonal harmonic 
term, and the first task is to find the volume of such an inequality as far as c". 
When tins is done we can express the volume of J in terms of that of the j)ear, 
which is, of course, a constant (§§ 3, 4). 
By aid of ellipsoidal harmonic analysis we may now express the first four terms 
of E in terms of the mass of the pear, and of certain definite integrals which depend 
on the shape of the critical Jacobian elli]3Soid (§§ 5, 6, 7). 
The energy h ED presents much more difficulty, and it is especially in this that 
M. Poincare’s insight and skill have been shown. Tlie system D consists of a layer 
of negative volume density, coated on its outer surface with a layer of surface 
density of equal and opposite mass. 
Two surfaces, infinitely near to one another, coated with equal and opposite surface 
densities, form together a magnetic layer or a layer of doublets. The change of 
jiotential in crossing such a layer is Itt times the magnetic moment at the point of 
crossing, and is independent of the form of surface. To find the difference between 
the potential at two points at a finite distance apart, one being on one side and the 
other on the other side of the layer, we have to add to the preceding difference 
a term equal to the force on either side of the magnetic layer multiplied by the 
distance between the two points. This additional term is small compared with that 
involving the magnetic moment, provided that the distance is small. If the magnetic 
layer coincided with the surface of an ellipsoid the force in question would be exactly 
calculable, and if it lies on the surface of a slightly deformed ellipsoid the force 
remains unchanged by the deformation as a first approximation. 
Thus it follows that it is possible to calcidate the difference of potential at two 
points lying on a curve orthogonal to an ellipsoid, when one point is on one side 
and tlie other on the other side of a magnetic layer residing on a deformation of the 
ellipsoid. Further, if the two points lie on the same side of the magnetic layer the 
term dependent on magnetic moment (which would represent the crossing of the 
layer) disappears, and only the term dependent on the force remains. 
Two equal and opjiosite layers of matter at a finite distance apart may be built up 
from an infinite munber of magnetic layers Interposed between the two surfaces. 
Hence by the integration of the result for a magnetic layer we may find the change 
