The purpose of this work is to study some properties of functions defined by a linear homogeneous equation with second-order partial derivatives, of Laplace form. It is divided into three Parts: In the first, I demonstrated the existence of an infinity of particular integrals from which we can deduce more general solutions by quadratures with variable limits relating to an arbitrary function. I called them 'principal integrals'. I have studied the series developments of these functions and of some of the integrals which follow from them. The second Part is devoted to the study of places of what I call 'accidental singular points', namely those which derive from the initial data defining the integrals and not from the particular form of the coefficients of the equation. I have defined normal integrals and demonstrated that they cannot have accidental singular curves other than characteristics. After having studied the form of the integrals in the vicinity of the critical points, I have shown how one can integrate the equation starting from it, particular solutions which admit singular moveable characteristics. In the third Part, I applied the previous theories to some simple equations. I have endeavoured, as far as possible, to deduce all the results with a general and uniform method.

The marriage took place on 18 October 1897 and the ceremony was the occasion to see famous people arriving at Callac by the new means of transport put into service by the Breton Network [railway] in 1893; let us quote Gabriel Koenigs, famous professor of the Faculty of Paris, and friend of Jean Marie Le Roux: "Rose Marie was not left on her own because she was accompanied by her older brother, Pierre, a merchant in Lannion and her younger brother, Joseph Gilles, polytechnician and Lieutenant of the Marine Artillery garrison in Cherbourg. Her father, François Marie, previously a 1st class surveyor of roads, then retired from the post for health reasons in 1880, had become a trader, timber merchant and municipal councillor." The very happy couple soon travelled to the distant city of Montpellier, passing through Paris and travelling first on the Compagnie de L"Ouest [early French railway company] then on the Paris-Lyon-Marseille (PLM) railway. In the following year, in July, they went the opposite way for the birth of their daughter Gabrielle Marie Laurence Le Roux on 12 August 1898 in Callac. Jean Marie Le Roux had just been appointed a Lecturer in Mathematics at the Faculty of Science at the University of Rennes.

In his famous Memoirs on torsion and bending of prisms, Barré de Saint-Venant placed himself above all others from the point of view of statics, seeking the deformations which result from certain distributions of forces. The object of this work is quite different. I only take care of the geometric study of deformations without any consideration of static or dynamic. Torsion and bending do not exist in homogeneous deformation. Their analytical representation in the vicinity of a point depends on the second derivatives of displacements. They are consequently differential elements of the second order of the deformation, whose role can be compared to that of curvature in the theory of surfaces. On the contrary the dilation and the average rotation are elements of the first order, like the tangent plane and the linear element in geometry. It seemed to me that knowing the necessary laws of the distribution of second order deformations could be as useful an introduction to the study of the mechanics of continuous media as knowing the elements of curvature when studying mechanics of a point. It has been found, moreover, that this theory, in addition to its practical utility, presents its own interest by the simple manner in which the results are grouped and are coordinated. I mainly had in mind the infinitesimal strains, but most of the calculations and methods apply without great modifications to the case of finite strains.

An interesting extension of [Saint-Venant's] theory, involving the introduction of secondary elements of strain, has been made by J Le Roux ... The secondary elements of strain are the curvature and twist of slender filaments, and the curvature of thin sheets of the material, the filaments and sheets being straight and plane in the unstrained state.

The discovery of a law of gravitation capable of explaining the movement of the perihelion of Mercury was considered to be a vivid confirmation of the theory of relativity. Judicious criticism observes that this result has been obtained with regard to the theory of relativity, but that it is not a consequence of it and does not even constitute an argument in its favour.

The results provided by Einstein's theory of gravitation seemed, at first glance, to agree remarkably well with observation, particularly in the case of the secular movement of the perihelion of Mercury. However, to reach this conclusion, we must admit that the perturbations due to the mutual actions of the planets keep in Einstein's theory the same values as in classical mechanics. If the disturbances are removed, the concordance disappears. Now it happens that Einstein's fundamental hypothesis is incompatible with the existence of mutual actions and disturbances as they are considered in classical mechanics. ... The confrontation with experience in the particular case of the movement of Mercury gives rise to the following observations. The secular advance observed is 574''. Newton's theory, which causes the disturbances, provides a satisfactory explanation up to a maximum limit of 536'', with an unexplained minimum residue of 38''. In Einstein's theory, the movement deduced from the $ds^{2}$ calculated by Schwarzschild would give Mercury a secular advance of 43''. But, as this theory excludes disturbances due to mutual actions, there remains an unexplained residue of 531''. ... In the meantime, it should be noted that Einstein's theory, in its current state, neither makes it possible to explain or predict, even with the roughest approximation, the secular movement of Mercury.

Everyone knows that Einstein's theory of gravitation includes Newton's as a first approximation. The Note from M Le Roux shows that it is nevertheless useful to recall it. ... The criticisms, of M Le Roux, are completely unfounded.

The very clear conclusion we come to is this. The principle of special relativity, in the sense of Einstein, is sometimes superficial and sometimes absurd, depending on the field to which it is applied.

The work is violently anti-relativistic. But the statements of the author break open doors when he shows again in long calculations that for the theory of a wave process with constant propagation speed due to the wave equation a relativity principle of the Lorentzian type applies. The actual difficulties, which the special theory of relativity overcame, for example the different behaviour of sound and light waves, he could have made clear but he did not understand.

It says at the beginning of the second part, after in the first part the group theoretical foundations have been explored, "The purpose of invariant dynamics is to express the general laws of dynamics, and in particular of gravitation, in a form valid for any reference system." One would like to expect a general theory of relativity, but that's not the point here. On the contrary, the theory of relativity is not invariant in the sense of the theory presented here. ... If the sentence quoted were to be accepted absolutely, the mechanics of the author would not exist ...

The author starts from the requirement of Einstein that the laws of physics are to be expressed in a form valid for all arbitrarily moving systems of reference, but his theory is not in agreement with Einstein's relativity because he accepts an absolute simultaneity. The time is a parameter as in classical mechanics, and is not one of the coordinates subject to transformation. In fact, the author accepts the Newtonian kinematics of rigid bodies, but with a generalisation.

In the transcription of the Breton text I have departed on a few points from the common usage of modern writers. It seemed to me that the orthographic system used since the Le Gouidec reform has various drawbacks which oppose the unification of the written language.