Page:Popular Science Monthly Volume 66.djvu/433

Rh been said, to determine the surfaces applicable on the sphere, other surfaces of the second degree have been attacked with more success, and, in particular, the paraboloid of revolution.

The systematic study of the deformation of general surfaces of the second degree is already entered upon; it is one of those which will give shortly the most important results.

The theory of infinitesimal deformation constitutes to-day one of the most finished chapters of geometry. It is the first somewhat extended application of a general method which seems to have a great future.

Being given a system of differential or partial differential equations, suitable to determine a certain number of unknowns, it is advantageous to associate with it a system of equations which we have called auxiliary system and which determines the systems of solutions infinitely near any given system of solutions. The auxiliary system being necessarily linear, its employment in all researches gives precious light on the properties of the proposed system and on the possibility of obtaining its integration.

The theory of lines of curvature and of asymptotic lines has been notably extended. Not only have been determined these two series of lines for particular surfaces such as the tetrahedral surfaces of Lamé; but also, in developing Moutard's results relative to a particular class of linear partial differential equations of the second order, it proved possible to generalize all that had been obtained for surfaces with lines of curvature plane or spheric, in determining completely all the classes of surfaces for which could be solved the problem of spheric representation.

Just so has been solved the correlative problem relative to asymptotic lines in making known all the surfaces of which the infinitesimal deformation can be determined in finite terms. Here is a vast field for research whose exploration is scarcely begun.

The infinitesimal study of rectilinear congruences, already commenced long ago by Dupin, Bertrand, Hamilton, Kummer, has come to intermingle in all these researches. Bibaucour, who has taken in it a preponderant part, studied particular classes of rectilinear congruences and, in particular, the congruences called isotropes, which intervene in the happiest way in the study of minimal surfaces.

The triply orthogonal systems which Lamé used in mathematical physics have become the object of systematic researches. Cayley was the first to form the partial differential equation of the third order on which the general solution of this problem was made to depend.

The system of homofocal surfaces of the second degree has been generalized and has given birth to that theory of general cyclides in which may be employed at the same time the resources of metric