576 GEOMETRY 



dependent. 1 It is just this that makes the subject difficult as com- 

 pared with the theory of curves, in which the defining equations are 

 entirely arbitrary. The questions demanding treatment fall under 

 these two headings: first, the derivation of the natural equations 

 of the familiar types of surfaces, and second, the study of the new 

 types that correspond to equations of simple form. The natural 

 geometry of the Weingarten class of surfaces requires a distinct basis. 



The fact that intrinsic* coordinates are, at bottom, differential 

 invariants with respect to the group of motions, suggests the exten- 

 sion of the same idea to the other groups. Thus in the projective 

 geometry of arbitrary (algebraic or transcendental) curves, coor- 

 dinates are required which, unlike the distances and angles ordin- 

 arily used, are invariant under projection. These might, for exam- 

 ple, be introduced as follows. At each point of the general curve C, 

 there is a unique osculating cubic and a unique osculating W (self- 

 projective) curve. Connected with each of these osculating curves 

 is an absolute projective invariant defined as an anharmonic ratio. 

 These ratios may then be taken as natural projective coordinates 

 y and o>, and the natural equation on the curve is of the form 

 y=f( w }. The principal advantage of such a representation is that 

 the necessary and sufficient condition for the equivalence of two 

 curves under projective transformations is simply the identity of the 

 corresponding equations. 



Returning to the theory of surfaces, natural coordinates may 

 be introduced so as to fit into the so-called geometry of a flexible 

 but inextensible surface, originated by Gauss, in which the criterion 

 of equivalence is applicability, or, according to the more accurate 

 phraseology of Voss, isometry. Intrinsic coordinates must then be 

 invariant with respect to bending (Biegungsinvariante) . This pro- 

 perty is fulfilled, for example, by the Gaussian curvature K and the 

 differential parameters connected with it X=A (K, K), /t=A(/c, X), 

 ^=A(X, X), all capable of simple geometric interpretation. The 

 intrinsic equations are then of the form p,=(j)(K, X), V=(J>(K, X). 



A pair of equations of this kind thus represent, not so much a 

 single surface S, as the totality of all surfaces applicable on S (or 

 into which S may be bent) a totality which is termed a complete 

 group G, since no additional surfaces are obtained when the same 

 process is applied to any member of the totality. The discussion of 

 such groups is ordinarily based on the first fundamental form (repre- 

 senting the squared element of length), since this is the same for 

 isometric surfaces; though of course it changes on the introduction 

 of new parameters. 



The simplest example of a complete isometric group is the group 



1 The three relations connecting the functions / n , /, f n , f n have been worked 

 out recently by S. Heller, Math. Annalen, vol. LVIII (1904). 



