PRESENT PROBLEMS OF GEOMETRY 577 



typified by the plane, consisting of all the developable surfaces. In 

 this case the equations of the group may be obtained explicitly, in 

 terms of eliminations, differentiations, and quadratures. This is, 

 however, quite exceptional; thus, even in the case of the surfaces 

 applicable on the unit sphere (surfaces of constant Gaussian curv- 

 ature + 1), the differential equation of the group has not been 

 integrated explicitly. In fact, until the year 1866, not a single case 

 analogous to that of the developable surfaces was discovered. Wein- 

 garten, by means of his theory -of e volutes, then succeeded in deter- 

 mining the complete group of the catenoid and of the paraboloid 

 of revolution, and, some twenty years later, a fourth group defined 

 in terms of minimal surfaces. 



During the past decade, the French geometers have concentrated 

 their efforts in this field mainly on the arbitrary paraboloid (and to 

 some extent on the arbitrary quadric). The difficulties even in this 

 extremely restricted and apparently simple case are great, and are 

 only gradually being conquered by the use of almost the whole 

 wealth of modern analysis and the invention of new methods which 

 undoubtedly have wider fields of application. The results obtained 

 exhibit, for example, connections with the theories of surfaces of 

 constant curvature, isometric surfaces, Backlund transformations, 

 and motions with two degrees of freedom. The principal workers 

 are Darboux, Goursat, Bianchi, Thybaut, Cosserat, Servant, Gui- 

 chard, and Raffy. 



Geometry im Grossen 



The questions we have just been considering, in common with 

 almost all the developments of general or infinitesimal geometry, 

 deal with the properties of the figure studied im Kleinen, that is, 

 in the sufficiently small neighborhood of a given point. Algebraic 

 geometry, on the other hand, deals with curves and surfaces in their 

 entirety. This distinction, however, is not inherent in the subject- 

 matter, but is rather a subjective one due to the limitations of our 

 analysis: our results being obtained by the use of power series are 

 valid only in the region of convergence. The properties of a curve 

 or surface (assumed analytic) considered as a whole are represented 

 not by means of function elements, but by means of the entire func- 

 tions obtained say by analytic continuation. 



Only the merest traces of such a transcendental geometry im 

 Grossen are in existence, but the interest of many investigators is 

 undoubtedly tending in this direction. The difficulty of the problems 

 which arise (in spite of their simple and natural character) and the 

 delicacy of method necessary in their treatment may be compared 

 to the corresponding problems and methods of celestial mechanics. 

 The calculation of the ephemeris of a planet for a limited time is 



