2 
SIR G. H. DARWIN : FURTHER CONSIDERATION OF THE 
term involves under the integral sign the factor 1 (~2 + r^- 2 ] — 3G, all the other 
\I i Ac/ 
factors being positive. If we attribute to 6 and to 4 > various values between and 
zero, we see that in part of the range the factor is positive and in other parts 
negative. A general inspection does not suffice to determine whether the positive 
portion outweighs the negative, as in fact it does. Therefore, in order to feel 
abundantly sure that no gross mistake had been made, I computed by quadratures 
the eight constituent integrals involved in the final result, and confirmed the correct¬ 
ness of the value found by the rigorous evaluation. 
The analysis of the investigation has been carefully examined throughout, and I 
have, besides, applied the same method to the investigation of Maclaurin’s spheroid, 
where the solution can be verified by the known exact result.* 
As a further check, the formulae of the paper on the Pear-shaped Figure have been 
examined on the hypothesis that the ellipsoid of reference reduces to a sphere. The 
several terms correctly reproduce the analogous terms in the paper on Maclaurin’s 
spheroid, but in effecting the comparison it is necessary to note that the variable r of 
the Pear-shaped Figure reduces to |Yl — ' l —\ whereas in the paper on Maclaurin’s 
spheroid the correspondin 
variable r denotes (1 — 3 
\ CIj 
1 
.3\ 
where r is radius vector and 
a the radius of the sphere. 
Dissent from so distinguished a mathematician as M. Liapounoff is not to be 
undertaken lightly, and I have, as explained, taken especial pains to ensure correct¬ 
ness. Having made my revision, and completed the computations as set forth here¬ 
after, I feel a conviction that the source of our disagreement will be found in some 
matter of principle, and not in the neglected residue of this series. I can now only 
express a hope that some one else will take up the question. 
In the revision of the computations, the methods now used are much better than 
the old ones. In as far as this paper is a mere repetition of the former work with 
improved methods, the results will only be stated in outline, but I now show how 
any of the ellipsoidal harmonic functions may be computed without approximation, 
and how the functions of the second kind may be found rigorously. 
The Cambridge University Press is now bringing out a collection of my mathe¬ 
matical papers, and when we come to the paper on the stability of the pear-shaped 
figure, the new methods of computation will be substituted for the old. 
This paper is supplementary to the former one on Stability,! and it will only be 
intelligible in connection therewith. As before, I refer to the papers in the 
4 Philosophical Transactions } as “ Harmonics” and the ££ Pear-shaped Figure.” 
* 1 Amer. Matt. Soc. Trans.,’ 1903, vol. 4, p. 113, on “The Approximate Determination of the Form of 
Maclaurin’s Spheroid,” and a further note on the same subject, recently sent to the same Society, 
t ‘ Phil. Trans.,’ A, vol. 200, p. 251. 
I Vol. 197, A, p. 461, and vol. 198, A, p. 301. 
