428 POPULAR SCIENCE MONTHLY. 



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, with- 

 out 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 Eiemann 

 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 recovered; it would seem even very doubtful if we consider 

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

 of integration of partial differential equations to the particular equa- 

 tion of surfaces of constant curvature. 



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 



