( 550 ) 

 d'^; 2{v 4- w) du dv 



' After substitution we get : 



D' = -^F and X' ^ Y' + Z' = 1, 



so that really all the conditions of the problem prove to be satisfied by 

 the equations {VII). Thus only the solution of {VII) is left to be 

 found. 



§ 4. We already know, that for the coordinates % an<i i] 



H 

 V — .F 



2 



must be satisfied. 



But moreover follows from the equations of Codazzi^): 



— r=0 and -Y^ = 



So 



!>=/,(§) and J?" =/,(7j), .... (F///) 



where /i and ƒ, are respectively functions of è and 7j only. 



The case that either D or Z>" is equal to zero offers no difficul- 

 ties, but nothing remarkable either. 



The case that D and D' are both equal to zero, leads, as 

 is immediately clear, to the sphere as the simplest form of a surface 

 with constant mean curvature. We can namely write down the 

 condition for umbilical points, wliicli is as follows with the omission 

 of infinitesimals of higher order : *) 



E _F _ G 



When for each point of the surface E =: G ^ then each point 

 is an umbilical point, as soon as always Z)^Z)" = 0; and these 

 surfaces are (in as far as it concerns the real solution) spheres only. 



^ 5. We shall now take the matter a little more generally. 

 Let us regard the total cur\'ature of a surface as a simultaneous 

 differential-invariant of both grouiidforms, we then find'): 



1) BiANCHi, I.e. p. 91. In using the coordinates ^ and v the Ghristoffel 



(1 1) 12 2) ^ ^, H^ 



symbols are all zero, except and -By making use ot D = — — F, 



we prove what was said in the text. 



^) See e. g. V. and K. Kommerell, Allgemeine Theorie der Raurakurven und 

 Flachen, II, p. 21. 



8) BlANGHI, I.e. p. 68. 



