522 ALGEBRA AND ANALYSIS 



been called by Weingarten and Knoblauch, who were among the first 

 writers emphasizing and developing to a certain extent the invari- 

 antive side of differential geometry, in the case of invariants proper, 

 "Biegungsinvarianten," in the case of differential parameters, " Bie- 

 gungscovarianten," and this notation has been more or less generally 

 adopted. The notation " Biegungscovarianten " does not agree with 

 the definition of a covariant given above, but a differential para- 

 meter of ds 2 can easily be modified into a covariantive form by 

 replacing according to the differential equation of the curve 



U(u,v) = const. 



dU dU 



the derivatives - - and by pdv and p.du. 

 ou ov 



A surface is completely defined, apart from its location in space, 

 when in addition to the quadratic form ds 2 also 



ds 2 



= Ldu 2 + 2Mdudv + Ndv 2 

 P 



is given, where p denotes the radius of curvature along ds, a the- 

 orem which was proved (1867) by Bonnet. 



With these two differential quantics given, we can now at once 

 form simultaneous invariants and differential parameters. The six 

 coefficients, E, F, G, L, M , N are, however, not independent; they 

 are related by three partial differential equations, - - the Gaussian 

 relation and the two Codazzi-Mainardi equations. These three 

 relations are expressible in an invariantive form. The Gaussian re- 

 lation is 



tiff =<f > 



while the two Codazzi formulas are given by the identical vanishing 

 of one simultaneous linear covariant. 



As examples of simultaneous differential parameters and covariants 

 I mention the expressions which, when set equal to zero, represent 

 the differential equations of conjugate lines, asymptotic lines, and 

 lines of curvature. The differential equation of lines of curvature, for 

 instance, if written in terms of du, dv represents a linear simultaneous 

 covariant; if written as a partial differential equation derived from 



U(u,v) = const. 



it represents a simultaneous differential parameter involving the 

 arbitrary function U. The differential equation of conjugate lines, 

 if written in two sets of differentials du, dv and du, dv represents 

 a bilinear simultaneous covariant; if written as a partial differential 

 equation it represents a differential parameter involving two arbi- 

 trary functions U and V. 



The theory of invariants of the above two differential quadratics, 



