312 PROFESSOR G. 11. DARWIN ON THE STABILITY OF THE PEAR-SHAPED 
reducing the analytical results to numbers, and it may suffice to say that the task 
proved to be a very laborious one. 
The harmonic terms included in the computation were those of degree 2 and ranks 
0 and 2, of degree 4 and ranks 0, 2, 4, and of degree 6 and ranks 0, 2, 4. The sixth 
sectorial harmonic was omitted because its contribution would certainly prove 
neo'licrible. 
The expression for Sco^ was found in the form of a fraction, of which the denominator 
is determinate and the numerator consists of the sum of an infinite series. Nine 
terms of this series were computed, namely, a constant term and the contribution of 
the eight harmonic terms sjDecified above. I found, in fact, that it would only change 
the numerator by about one-twentieth part of itself, if all the harmonics excepting 
the zonal ones of degrees 2, 4, 6 had been dropped. 
The result shows that the sc|uare of the angular velocity of the pear is less than 
that of the critical Jacobian ellipsoid in about the proportion to 1 — -g-e" to 1 . On 
the other hand the angular momentum of the pear is greater than that of the 
ellipsoid in about the proportion of 1 + to 1 . If this last result were based 
on a rigorous summation of the infinite series, it would, in accordance with the 
principle explained above, absolutely j^rove the stability of the pear. The inclusion 
of the uncomputed residue of the series would undoubtedly tend in the direction 
of reducing the coefficient given in round numbers as ^^ 5 -, and if it were to reduce it 
to a negative quantity, we should conclude that the pear was unstable after all. 
The apparently rapid convergence of the series seemed to render it almost incredible 
that the inclusion of the residue could bring about such a reversal of our conclusion, 
yet I thought it advisable to make a rough estimate of the amount of change which 
would arise from the contribution of the eighth zonal harmonic. 
The contribution of the sixth zonal harmonic to the series above referred to was 
about ’00006, and I find that if the contribution of the uncomputed residue should 
amount to ’00014, the apparent stability of the pear would be just reversed. Now 
my estimate of the contribution of the eighth zonal harmonic to the same sum is 
’0000008, or only of fh© critical amount. 
Since the convergency of the series is obviously very rapid, it is wholly incredible 
that the inclusion of the uncomputed residue could materially alter, much less 
reverse our result. I regard it then as proved, but by something short of an 
absolute algebraic argument, that the pear-shaped figure is stable. 
The numbers obtained in the course of the determination of the stability aftbrd 
the means of giving a second approximation to the form of the pear. The result is 
shown graphically in the figure of § 20 , where the largest value of e is adopted 
which seemed to secure a fair degree of approximation in the result. I originally 
called the figure “ pear-shaped,” because M. Poincare’s conjectural sketch in the 
‘’ Acta Mathematica ” was very like a pear. In the first approximation, shown in my 
former paper, the resemblance to a pear was not striking, and it needs some imagina- 
