DEVELOPMENT OF GEOMETRIC METHODS 553 



to intermingle in all these researches. Ribaucour, 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 

 geometry, of protective geometry, and of infinitesimal geometry. 

 Many other orthogonal systems have been made known. Among 

 these it is proper to signalize the cyclic systems of Ribaucour, for 

 which one of the three families admits circles as orthogonal trajecto- 

 ries and the more general systems for which these orthogonal trajec- 

 tories are simply plane curves. 



The systematic employment of imaginaries, which we must be 

 careful not to exclude from geometry, has permitted the connection 

 of all these determinations with the study of the finite deformation 

 of a particular surface. 



Among the methods which have permitted the establishment of 

 all these results, it is proper to note the systematic employment of 

 linear partial differential equations of the second order and of systems 

 formed of such equations. The most recent researches show that this 

 employment is destined to renovate most of the theories. 



Infinitesimal geometry could not neglect the study of the two 

 fundamental problems set it by the calculus of variations. 



The problem of the shortest path on a surface was the object of 

 masterly studies by Jacobi and by Ossian Bonnet. The study of 

 geodesic lines has been followed up; we have learned to determine 

 them for new surfaces. The theory of ensembles has come to permit 

 the following of these lines in their course on a given surface. 



The solution of a problem relative to the representation of two 

 surfaces one on the other has greatly increased the interest of dis- 

 coveries of Jacobi and of Liouville relative to a particular class of 

 surfaces of which the geodesic lines could be determined. The results 

 concerning this particular case led to the examination of a new ques- 

 tion: to investigate all the problems of the calculus of variations of 

 which the solution is given by curves satisfying a given differential 

 equation. 



Finally, the methods of Jacobi have been extended to space of 

 three dimensions and applied to the solution of a question which 

 presented the greatest difficulties: the study of properties of mini- 



