552 GEOMETRY 



But, if it is impossible to determine in finite terms all these sur- 

 faces, it has at least been possible to obtain certain of them, char- 

 acterized by special properties, such as that of having their lines of 

 curvature plane or spheric; and it has been shown, by employing a 

 method which succeeds in many other problems, that from every sur- 

 face of constant curvature may be derived an infinity of other surfaces 

 of the same nature, by employing operations clearly defined which 

 require only quadratures. 



The theory of the deformation of surfaces in the sense of Gauss 

 has been also much enriched. We owe to Minding and to Bour the 

 detailed study of that special deformation of ruled surfaces which 

 leaves the generators rectilineal. If we have not been able, as has 

 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 advantage- 

 ous 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 

 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 pro- 

 blem of spheric representation. 



Just so has been solved the correlative problem relative to asymp- 

 totic 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 



