520 



ALGEBRA AND ANALYSIS 



investigation. He also gave, in what has been called after him the 

 Gaussian Curvature 



oE 



K = (E,F,G,> )) 

 ou 



the first example of an invariant. Gauss defines this curvature 

 geometrically and finds for it the analytic expression 



LN-M 2 

 EG-F 2 



which is a simultaneous invariant of two differential quantics, 



ds 2 



namely, of ds 2 and of- - =-Ldu 2 +2Mdudv+Ndv 2 . 



P 



This shows that K is independent of the w^-system on the 

 surface. And now Gauss expresses K in terms of E, F, G and the 

 first and second derivatives of these quantities alone. A direct 

 demonstration that K is an invariant proper of the differential 

 quantic ds 2 alone, - - that is, without passing through the second 



ds 2 



differential quantic , is of course desirable. 1 Each one of the 



P 



general methods of treating the theory of invariants, which will be 

 discussed in the latter part of this paper, furnishes such a direct 

 proof. In particular, the aspect of the formula for K, on p. 528, 

 deduced by the symbolic method, shows immediately the invariant 

 character of K. 



Differential parameters were introduced into differential geometry 

 by Beltrami in 1863. These are the well-known expressions 



EG-F 2 



F(*,0 = 



ri L ' E 1 I ~ ' i L. ' 



dv dv \du dv dv duj " du du 

 EG-F 2 



where <p and <f> are the arbitrary functions which take the place of 

 U, V in our general definition of differential parameters. Beltrami 

 adopted the name "differential parameters" and also the notation 



1 Cf . on this subject the interesting paper by Knoblauch : " Der Gauss'sche Satz 

 vom Kriimmungsmass," Sitzungsberichte der Berliner Mathem. Gesellschaft. April 

 27, 1904. 



