DEVELOPMENT OF GEOMETRIC METHODS 551 



get from its formulas any mode of generation of these surfaces, nor 

 even any particular surface. We will not here retrace the detailed 

 history which we have presented in our Lecons sur la theorie des 

 surfaces ; but it is proper to recall the fundamental researches of 

 Bonnet which have given us, in particular, the notion of surfaces 

 associated with a given surface, the formulas of Weierstrass which 

 establish a close bond between the minimal surfaces and the functions 

 of a complex variable, the researches of Lie by which it was estab- 

 lished that just the formulas of Monge can to-day serve as founda- 

 tion for a fruitful study of minimal surfaces. 



In seeking to determine the minimal surfaces of smallest classes 

 or degrees, we were led to the notion of double minimal surfaces 

 which is dependent on analysis situs. 



Three problems of unequal importance have been studied in this 

 theory. 



The first, relative to the determination of minimal surfaces in- 

 scribed along a given contour in a developable equally given, was 

 solved by celebrated formulas which have led to a great number of 

 propositions. For example, every straight traced on such a surface 

 is an axis of symmetry. 



The second, set by S. Lie, concerns the determination of all the 

 algebraic minimal surfaces inscribed, in an algebraic developable, 

 without the curve of contact being given. It also has been entirely 

 elucidated. 



The third and the most difficult is what the physicists solve experi- 

 mentally, by plunging a closed contour into a solution of glycerine. 

 It concerns the determination of the minimal surface passing through 

 a given contour. 



The solution of this problem evidently surpasses the resources of 

 geometry. Thanks to the resources of the highest analysis, it has 

 been solved for particular contours in the celebrated memoir of 

 Riemann and in the profound researches which have followed or 

 accompanied this memoir. 



For the most general contour, its study has been brilliantly begun; 

 it will be continued by our successors. 



After the minimal surfaces, the surfaces of constant curvature at- 

 tracted the attention of geometers. An ingenious remark of Bonnet 

 connects with each other the surfaces of which one or the other of the 

 two curvatures, mean curvature or total curvature, is constant. 



Bour announced that the partial differential equation of surfaces 

 of constant curvature could be completely integrated. This result 

 has not been secured; it would seem even very doubtful if we con- 

 sider a research where S. Lie has essayed in vain to apply a general 

 method of integration of partial differential equations to the particu- 

 lar equation of surfaces of constant curvature. 



