528 ALGEBRA AND ANALYSIS 



tics was given by myself. I applied a symbolic method to the theory 

 which consists chiefly in identifying the fundamental quadratic 



with the square of a linear expression 



Widxy 



by setting fifk=dik- This is strictly analogous to the introduction of 

 symbols in the algebraic theory. The difference, of course, comes 

 in at once when we have to consider also the derivatives of a,*. 



A systematic development leads to expressions and formulas 

 which with respect to simplicity and shortness are as superior to 

 the formulas of the ordinary notation as the formulas of the so- 

 called symbolic notation in the algebraic theory are superior to the 

 non-symbolic expressions. 



As examples I give the most important invariant expressions 

 for the case n=2. 



Let us introduce the abbreviation 



(Pi Q* -P, Qi ) =(PQ), where P*= etc.; 



a 2 i 2 



let further /, <p, $ . . .be symbols of .A, so that 



and let f/, F ... be arbitrary functions of x l} x r 

 Then we have 



=Ai U, 



= 2K (Gaussian curvature), 

 <p) (<pU) ((//)/) : (Ji t/) 3 =Geodesic curvature of curve U=const. 

 To give also some examples of simultaneous invariant expressions 

 let /, <p, . . . be as before symbols of 



Edu 2 +2Fdudv+Gdv 2 

 and F, d> . . . symbols of 



Ldu z +2Mdudv +Ndv 2 , 

 Then: 



(/F) 2 = mean curvature. 

 The differential equations 



of asymptotic curves U =c are (Ft/) 2 =0, 

 of conjugate curves U =c, V=c: (FU)(FV)=Q, 

 of lines of curvature U = c : (fF) (fU) (FU) = 0. 

 The equation (fo) (<pF) ( (/^) C7) =0 gives the two Cadazzi formulas 

 by setting the coefficients of C7, and U? separately equal to zero. 



In these examples the invariant expressions always appear as 

 products of factors of the type (P.R). The general theorem holds 

 that any product of factors of this type represents always an in- 



