DEVELOPMENT OF GEOMETRIC METHODS. 429 



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 ex- 

 tended 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 neces- 

 sarily linear, its employment in all researches gives precious light on 

 the properties of the proposed system and on the possibility of obtain- 

 ing 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 

 Lame; but also, in developing Moutard's results relative to a par- 

 ticular 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 com- 

 pletely 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 com- 

 menced 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 con- 

 gruences and, in particular, the congruences called isotropes, which 

 intervene in the happiest way in the study of minimal surfaces. 



The triply orthogonal systems which Lame 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 



