308 PROFESSOR G. 11. DxlPaVIN ON THE STABILITY OF THE PEAR-SHAPED 
of stal^ility is then completely determined by means of the angular momentum of the 
pear ; if it is greater than that of the critical Jacobian the pear is stable, and, 
if less, unstable. 
It suffices then to determine the figure by means of the variations of E vith 
constant angular velocity, and afterwards to evaluate the angular momentum. 
It was i^roved by M. Poincare, and rejDeated by me in my previous paper, that 
the first approximation to the pear-shaped figure is given by the third zonal 
harmonic inequality of the critical Jacobian ellipsoid—zonal with re.spect to its longest 
axis. In 2 :)roceeding to the higher approximation I siij^pose that the amplitude ot 
the third zonal harmonic is measured by a parameter e, which is to be regarded as a 
(pmntity of the first order. We must now also supj)ose the ellipsoid to be deformed 
liy all and any other harmonics, but with amplitudes of order eb In the first 
approximation the lost energy TP is proportional to but it now becomes necessary 
to determine TT^ as far as the order e^. A chano-e in the sie'n of e means that the 
figure of equilibrium is rotated in azimuth through 180°. Such a rotation cannot 
afiect tlie value of the energy, and it thus becomes obvious that the odd powers of e 
must Ije absent from the exjDression for W. We have further to find the moment of 
inertia of the l:)ody as far as the terms of order e”, and thence to find the kinetic 
energy T. The function E is equal to W + T. 
In order to attain the requisite degree of accuracy, it is convenient to regard the 
pear as being l)uilt up in an artificial manner. I construct an ellipsoid similar to and 
concentric witli the critical Jacobian, and therefore itself possessing the same 
character. The size of this new Jacobian, which I call J, is undefined, and is subject 
only to the condition that it shall be large enough to enclose the whole pear. The 
regions between J and the pear being called /A I suppose the pear to consist of 
positive density tliroughout J and negative density tliroughout R (§ 1). 
The lost energy of tlie pear consists of that of J with itself, say \JJ ; of J with 
7?, which is filled with negative density, say — JPi ; and of R with itself, say \RR. 
Tliis last contriliution to the energy must be lu’oken into several portions. It was 
the evaluation of \RR which baffled me, until M. Poincare’s solution came to 
my help. 
If we imagine the ellipsoid J to be intersected by a family of orthogonal quadrics, 
and if we supjiose for the moment that the region R is filled with positive density, 
we may further imagine the matter lying inside any orthogonal tube to be transported 
along the tulie, and to be deposited on the surface of J in the form of a concentration 
of positive surfiice density + 0. The mass of + C is equal to tliat of + P, but it 
is differently arranged. In tlie actual system R is filled with negative volume 
density, and we may clearly add to this two equal and opposite surface densities 
+ C and — C on J. 
Thus the matter lying in the region R may be regarded as consisting of negative 
surface density — C on J, together with a double system, namely negative volume 
