252 
riJOFESSOK (I. H. DARWIN ON THE STABHTTY OF THE PEAR-SHAPED 
Inteoduction. 
By aid of the methods of a paper on “Ellipsoidal Harmonic Analysis” (‘Phil. Trans., 
A, vol. 197, })p. 4G1-557), I here resume the subject of a previous paper (‘ Phil. Trans.,’ 
A, vol. ] 98, pp. 301-331). Tliese papers will he referred to hereafter by the abridged 
titles of “ Harmonics” and “The Pear-shaped Figure.”' 
At the end of the latter of these it was stated that the stahilitv of the fio’ure coidd 
«y o 
not he proved definitely without approximation of a higlier order of accuracy. After 
some correspondeiice witli M. Poincare; during the course of my work, I made an 
attemjit to carry out this further approximation, hut found that the expression 
for a certain portion of tlie energy entirely foiled me. Meanwhile he liad turned his 
attentioii to the subject, and lie has shown (‘Phil. Tran.s.,’A, vol. 198, pp. 333-373) 
by a method of the greatest ingenuity and skill how the prolilem ma}^ lie solved. He 
has not, however, pursued the arduous task of converting his analytical results into 
numbers, so that he left the question as to the stability of the pear still unanswered. 
M. Poincare was so kind as to allow me to detain his manuscript on its way to the 
Royal Society tor two or three days, and I devoted that time almost entirely to 
understanding the method of his attack on the key of the position—namely, the 
method of doulile layers, exjDounded in my own language in § 9 below. Being thus 
furnished with the means, T was able to resume my attempt under favourable 
conditions, and this paper is the result. 
The substance of the analysis of this paper is, of course, essentially the same as his, 
lint the arrangement and notation are so different that the two present but little 
superficial resemblance. This difference arises partly from the fact that I desired to 
use my own notation for the ellijisoidal harmonics, and partly liecause during the time 
tliat I was working at the analysis his paper was still unprinted and therefore 
inaccessible to me. But it is, perhaps, Avell that the trvo iuA'estigations of so 
complicated a suliject should lie nearly independent of one another. 
It is rather unf u'tunate that I did not feel myself sufficiently expert in the use of 
the methods of Weierstrass and Schaaavrz to ewduate tlie elliptic integrals after the 
methods suggested liy M. Poincare, Imt every exertion has been taken to insure 
correctness in the arithmetical results, on Avhich the proof of stability depends. My 
choice of antiquated methods of computation leavms tlie Avay open for some one else to 
verify the conclusions liy wholly independent and more elegant calculations. It is 
lilghly desiralfie that such a Aerification should he made. 
As the liody of this paper Avill hardly he studied by any one unless they should be 
actually Avorking at the sidiject, I give a summary at the end. EA’en the mathe¬ 
matician AAdio desires to study the subject in detail may find it adA’antageous to read 
the summary before looking at the analytical investigation. 
