FIGURE AND STABILITY OF A LIQUID SATELLITE. 
163 
enounced by him. The attempt is given in an appendix to my paper, but 
unfortunately I failed to understand his work completely, and my investigation, 
as it stands, is erroneous from the fact that one term in the energy is omitted. # 
I think, however, that the defect may easily be made good. 
The analysis of the present paper is carried out by means of ellipsoidal harmonic 
analysis. In the course of the work it becomes necessary to refer to previous papers 
by myself, all published in the ‘ Philosophical Transactions ? ; they are : “ Ellipsoidal 
Harmonic Analysis,” vol. 197 (1901) A, pp. 461-557 ; “ The Pear-Shaped Figure of 
Equilibrium of a Rotating Mass of Liquid,” vol. 198 (1902) A, pp. 301-331 ; “The 
Stability of the Pear-Shaped Figure of Equilibrium, &c.,” vol. 200 (1903) A, 
pp. 251-314; “The Integrals of the Squares of Ellipsoidal Surface Harmonic 
Functions,” vol. 203 (1904) A, pp. 111-137. These papers are hereafter referred to 
by the abridged titles “ Harmonics,” “ The Pear-Shaped Figure,” “ Stability,” and 
“ Integrals.” 
The analysis involved in the investigation is unfortunately long and complicated, 
but the subject itself is not an easy one, and the complication was perhaps 
unavoidable. 
The principal inducement to attack this problem was the hope that it might throw 
further light on the form of the pear-shaped figure in an advanced stage of develop¬ 
ment when it might be supposed to consist of two bulbs of liquid joined by a very 
thin neck. The arguments adduced below seem to show that such a figure must be 
unstable. 
M. Liapounoff has recently published a paper in which he states that he is able 
to prove the instability of the pear-shaped figure even when only infinitesimally 
furrowed.! In view of my previous work on the stability of this figure, and from 
other considerations it seems very difficult to accept the correctness of this result. 
At the end a summary is given of the conclusions arrived at, and this last subject 
is discussed amongst others. 
Part I.— Analysis. 
§ 1. The Stability of Liquid Satellites. 
This paper deals with two problems concerning liquid satellites which possess so 
much resemblance that I did not for some time perceive that there is an essential 
difference between them. One of these is the determination of the figures and of the 
secular stability of two masses of liquid revolving about one another in a circular 
orbit without relative motion of their parts. We may refer to this as the problem of 
* This is the term denoted -Jw 2 (S/) 2 // below. 
t “Sur un probleme de Tchebychef,” ‘Acad. Imp. des Sci. de St. Petersbourg,’ vol. 17, No. 3 (1905). 
Y 2 
