The equations for the problem we are about to address were given by Poisson in his 'Theory of Heat', Chapter X. The integration procedure to be used is indicated there; but the solution is developed only for the case where the sphere is homogeneous.

When the sphere is heterogeneous, but nevertheless such that the temperature is the same throughout the entire extent of each spherical layer, the problem depends on the integration of a linear, second-order partial difference equation in two variables. This integration leads, using ordinary methods, to a differential equation that can only be integrated in a small number of particular cases, and even when the expression for the function can be obtained, it is difficult to discover its progression and its principal properties.

It was therefore necessary to find, through the direct consideration of this differential equation, a means of recognising the nature of the function it implicitly determines. This is what M Sturm did in his fine 'Memoir on Linear Second-Order Differential Equations', where, using a method as elegant as it was fruitful, he established the curious properties of these integrals. The same geometer demonstrated, in his 'Memoir on a Class of Partial Differential Equations', the use that can be made of these properties. We should also mention several very interesting memoirs by M Liouville on the same subject.

Based on these various works, we will study the distribution of heat in a sphere, where the temperature depends at each instant only on the distance from the centre. We will then deduce from the general formulas those relating to the homogeneous sphere, and we will indicate their use in the question of the movement of heat within and on the surface of the Earth.

The search for equilibrium figures for a homogeneous fluid mass, endowed with rotational motion, is a difficult problem, which can only be completely resolved when this figure is assumed to be little different from that of a sphere. In this case, there is only one equilibrium figure, which is that of an ellipsoid of revolution flattened at the poles.

To demonstrate this important proposition, Laplace used two methods found in the third book of 'Celestial Mechanics'. The first is based on a series development, which he introduced into analysis, and which recurs repeatedly in the problem of the attraction of spheroids, in that of tides, and in the theory of heat. The illustrious geometer admits the possibility of representing, by this type of series, any function of two variables, a theorem that has since been demonstrated a priori by Poisson.

For fear that this assumption might cast doubt on the results of his analysis, Laplace believed it necessary to demonstrate the proposition in question, independently of the use of series. This second demonstration is incomplete, as noted by M Liouville, who provided a rigorous one. Poisson provided another but this one is still inexact, as we will see.

We will assume here that, in addition to the mutual attraction of its molecules, the fluid mass is subject to the action of a force placed at its centre and reciprocal to the square of the distances; which implicitly includes the case where the fluid mass covers a sphere of density different from its own.

... from then on, the feelings of the most tender affection for a mother, a father and a brother worthy of him were alone to divert the solitude of his heart. It was, I believe, from that day that he ceased to be able to laugh, while retaining the habit of that amiable smile, an expression of his somewhat melancholy goodness, which all his pupils knew.

My studies on the shape of celestial bodies and the arrangement of the level layers in the atmospheres surrounding them have led me to some results applicable to the Sun, its atmosphere, and the progressive condensation by which, as we know, the various planets were formed. I propose to bring together here the consequences of my previous work relating to these questions, and to show how they confirm, clarify, or modify Laplace's cosmogonic theory.

Various problems concerning the solar system will be addressed in this Memoir; but, given the difficulty of the subject, the reader will not be surprised to find, on many points, only mere overviews.

Roche devoted himself to topics that lay generally outside of mainstream astronomical research in his time. He studied the equilibrium figure of a rotating fluid body that was subjected to an external gravitational force caused by another body. This condition is extremely important for determining the equipotential surfaces around a pair of bodies like a binary star. Where one (or both) of the components assumes a tear-drop shape, the space(s) so filled is called the Roche lobe.

Roche ([12], [13], [14]) had shown some interest in the problems associated with the deformation and fission of rotating fluids, but the work which has become associated with his name as the Roche limit was concerned with the formation of Saturn's rings through mass loss from a hypothetical satellite ([15], [16]). The satellite is taken to be small in size, not deforming Saturn and not in contact in contrast to the assumed situation in the Earth-Moon problem. The gravitational field of both Saturn and the satellite are taken to be that of point masses and Roche consider the loss of a particle from the surface of the satellite at the point nearest Saturn. The satellite is also effectively assumed to be face-on to Saturn, though Roche actually assumed nothing of the satellite, only that the third body that was to be lost stayed on the line joining Saturn and the satellite until lost. ... The Roche limit actually places a limit for the closest distance that a secondary can approach a primary and remain stable under a very idealised set of conditions. In particular the secondary is assumed to move on a circular orbit and maintain the face-on configuration. In reality virtually no secondary moves on a circular orbit and the face-on configuration is far from common.

Long before the [Roche problems] emerged to become almost a byword in double-star astronomy in the second half of the present century, astronomers were well aware of their existence which can be traced back - and hence its name - to the French mathematician Édouard Albert Roche (1820-1883). Their roots, to be sure, go much further back in the history of our subject; for - through their connection with the energy integral of the problem of three bodies - many of their properties were known to the German mathematician C G J Jacobi (1804-1851) with whose name they are still sometimes associated. In reality, however, its properties go back to the 18th century to the work of J L Lagrange (1736-1813) and, in fact, to Leonard Euler (1707-1789) - and thus to the early applications of the infinitesimal calculus to dynamical astronomy in the days of Isaac Newton (1642-1727). Yet all these grand men of our science were concerned with certain problems ... solely from the viewpoint of particle mechanics (i.e., the motion of individual particles in the course of time); their relevance to the problems of double-star astronomy was glossed over mainly on account of the fact that no binary stars (or, at least, close binaries) were known to exist in the sky at that time. However, a study of the particular case of equipotential surfaces associated with gravitational dipoles (to which the energy associated with close gravitational dipoles - and to which the energy integral reduces to surfaces of zero velocity (or, more precisely, to surfaces over which the squares of the velocity components are small enough to be ignorable) - has remained the historical task of E A Roche; and it is, therefore, eminently appropriate that these be permanently associated with his name.

... the rest of the academicians (trying no doubt to avoid the injustice of electing him as one of their equals) voted for others whose names are all forgotten, and their work as dead as mutton. ... Perhaps the main cause of delay to recognise the fundamental nature of Roche's contributions to modern astrophysics was the extent of the ignorance of his age of the internal structure of the stars; for even by the time when Roche died, scholars of the calibre of Henri Poincaré did not find it unreasonable to regard the stars as homogeneous (and sometimes incompressible!) self-gravitating globes for the sake of mathematical simplifications permissible on such a basis. That the actual structure of the stars must be very far from such a model did not begin to transpire until the beginning of the 20th century ...

Death did not surprise him. For a long time he had found, in the exercise of the Christian virtues and in the religious practices of a simple faith, all the consolations attached to the hope of a better life and of a more complete light than that which both rejoices and disturbs our wavering reason here below. There he drew the strength to endure with resignation the sufferings of the obstinate cough which was gradually exhausting him. Provided with the last rites of the Church, he saw without faltering the approach of his supreme hour.

... had, in October and November 1859, completed my preparation for the baccalaureate in science with a few lessons.

May the expression of the unanimous regrets [offered by Roche's colleagues on his death] soften the pain of his brother, to whom you will allow me to offer, in closing, these few pages, because, an excellent professor himself, although in a more modest sphere, he has also acquired, in this capacity, the right to my gratitude.

As a student of the Faculty of Sciences of Montpellier, I had the happy experience of following, in 1860 and 1861, Roche's courses in Infinitesimal Analysis and Mechanics, whose listeners unanimously appreciated his clarity and communicative enthusiasm. I was able above all to appreciate, on various occasions, the excellence of his kindness. I find it impossible to forget, for example, the kindness with which he welcomed me and encouraged my first efforts, when, still having only a slight tinge of special Mathematics, I was introduced to him by his younger brother, M Arthur Roche and I expressed to him the desire to devote my life to the study of science.