1G2 
SIR G. II. DARWIN ON THE 
Preface. ' 
More than half a century ago Edouard Roche wrote his celebrated paper on the 
form assumed by a liquid satellite when revolving, without relative motion, about- a 
solid planet. # In consequence of the singular modesty of Roche’s style, and also 
because the publication was made at Montpellier, this paper seems to have remained 
almost unnoticed for many years, but it has ultimately attained its due position as a 
classical memoir. 
The laborious computations necessary for obtaining numerical results were carried 
out, partly at least, by graphical methods. Verification of the calculations, which as 
far as I know have never been repeated, forms part of the work of the present paper. 
The distance from a spherical planet which has been called “Roche’s limit” is 
expressed by the number of planetary radii in the radius vector of the nearest 
possible infinitesimal liquid satellite, of the same density as the planet, revolving so 
as always to present the same aspect to the planet. Our moon, if it were homo¬ 
geneous, would have the form of one of Roche’s ellipsoids ; but its present radius 
vector is of course far greater than the limit. Roche assigned to the limit in 
question the numerical value 2‘44 ; in the present paper I show that the true value 
is 2’455, and the closeness of the agreement with the previously accejDted value 
affords a remarkable testimony to the accuracy with which he must have drawn his 
figures. 
He made no attempt to obtain numerical solutions except in the case of the 
infinitely small satellite. In this case the figure is rigorously ellipsoidal, but for 
a finite satellite this is no longer the case ; nor do his equations afford the means of 
determining exactly the ellipsoid which most nearly represents the truth. These 
deficiencies are made good below, and we find that even in the extreme case of two 
equal masses in limiting stability the ellipsoid is a much closer approximation to 
accuracy than might have been expected. 
It is natural that Roche, writing as he did half a century ago, should not have 
been in a position to discuss the stability of his solutions with completeness, and 
although he did much in that direction he necessarily left a good deal unsettled. 
In 1887 I attempted the discussion of some of the problems to which this paper is 
devoted, by means of spherical harmonic analysis. ! Poincare’s celebrated memoir 
on figures of equilibriumJ was published just when my work was finished, and I kept 
my paper back for a year in order to apply to my solutions the principles of stability 
* “ La figure cl’une masse fluide soumise a l’attraction d’un point eloigne,” ‘ Acad, des Sci. de 
Montpellier,’ vol. 1, 1847-50, p. 243. 
t “Figures of Equilibrium of Rotating Masses of Fluid,” ‘Phil. Trans. Roy. Soc.,’ vol. 178 (18S7) A, 
pp. 379-428. 
t “ Sur l’equilibre d’une masse fluide animee d’un mouvement de rotation,” 
pp. 259-380. 
Acta. Math.’ 7 : 3, 4 (1885), 
