( 555 ) 



The equation 



du du d^u 



d§ d»j ^ ^ dèdri 



is satisfied by u = x{^). 



So there remains to be integrated 



when w = X (§)• 

 We find: 



dv dv d^v 



|=^(^-x(^))'-/(a 



with /(§) as arbitrary function of §. 



The solution z< = / (§) furnishes (see {VI)) the value zero for 



d.v dy dz dx dy . Ö0 



r- , ^- and -- ; whilst for ^ , ^ and -- the wellknown formulae are 



found back for the minimal curves. 



Entirely the same (with exchange of ii and v, § and i]) is found 

 by putting v = x, (>;)• 



This solution therefore shows what relations there are between 

 the minimal surfaces and those under consideration. For the former 

 we have but to join the two solutions found to get the complete 

 solution with, two arbitrary functions. So we see that the minimal 

 surfaces are translation surfaces, generated by moving a minimal 

 curve out of a set along the various points of a curve out of the 

 second set ; i. 0. w. we have found back the integration of the minimal 

 surfaces and in the usual form too. 



Because of H tending to zero there is in this case no fear of 

 F becoming 0. 



§ 9. Now that the special cases of sphere (plane), cylinder and 

 minimal surfaces are excluded, the integration of the equations (VII) 

 would remain. I have not been able to attain more than the lowering 

 of the order of the two differential equations, which is perhaps a 

 step onward to a complete solution or to solutions for definite 

 series of surfaces. 



To this end we put : 



bv 1 w, du 1 w. 



üi {v—uy 2 dï]{v—uy 



where iv^^ and 10^ are functions of ^ and i^. 



