PRESENT PROBLEMS OF GEOMETRY 575 



The application to surfaces is not so evident. Thus, in Cesaro's 

 standard work, while the discussion of curves is consistently in- 

 trinsic, this is true to only a slight extent in the treatment of surfaces. 

 The natural geometry of surfaces is in fact only in process of forma- 

 tion. Bianchi proposes as intrinsic the familiar representation by 

 means of the two fundamental quadratic differential forms; but, 

 although it is true that the surfaces corresponding to a given pair 

 of forms are necessarily congruent, there is the disadvantage, arising 

 from the presence of arbitrary parameters, that the same surface 

 may be represented by distinct pairs of forms. One way of over- 

 coming this difficulty is to introduce the common feature of all pairs 

 corresponding to a surface, that is, the invariants of the forms: in 

 this direction we may cite Ricci's principle of co variant differentia- 

 tion and Maschke's recent application of symbolic methods. 



The basis of natural geometry is, essentially, the theory of differ- 

 ential invariants. Under the group of motions, a given configuration 

 assumes co r positions, where r is in general 6, but may be smaller 

 in certain cases. The r parameters which thus enter in the analytic 

 representation may be eliminated by the formation of differential 

 equations. The aim of natural geometry is to express these differ- 

 ential equations in terms of the simplest geometric elements of the 

 given configuration. 



The beginning of such a discussion of surfaces was given by Sophus 

 Lie in 1896 and his work has been somewhat simplified by Scheffers. 

 As natural coordinates we may take the principal radii of curvature 

 R\, R 2 , at a point of the surface, and their derivatives 



dR, dR l dR 2 dR, 



*" = "&; 12= ds 2 S " = ds, 22= ds 2 



taken in the principal directions. For a given surface (excluding 

 the Weingarten class) the radii are independent, and there are four 

 relations of the form 



Conversely, these equations are not satisfied by any surfaces except 

 those congruent or symmetric to the given surface. 



It is to be noticed that four equations thus appear to be necessary 

 to define a surface, although two are sufficient for a twisted curve. 

 If a single equation in the above-mentioned natural coordinates is 

 considered, it is not, as in the case of ordinary coordinates, charac- 

 teristic: surfaces not congruent or symmetric to the given surface 

 would satisfy the equation. The apparent inconsistency which arises 

 is removed, however, by the fact that the four natural equations are 



