ON THE CONGRUENCES OF TWISTED CURVES* 



Arnold Emch 



1. Two surfaces with the equations 



F {^, y, -'. «, ^)=0, (1) 



<^ (.'?', y, z, a, 'b)=0, 

 where a and h are arbitrary parameters, define a congruence of curves. 

 Any relation between a and h, 



h=f{a) (2) 



gives rise to a system of curves which generate a surface of the con,, 

 gruence. The condition that among these surfaces there shall be 

 surfaces being the envelope of its generating curves is 



^a hh da ''ha^hljda 

 Eliminating ,r, y, s, between (1) and (3), this condition may be 

 expressed by a relation of the form 



1^ 



(«, h ^)=0, (4) 



i. e,, by a differential equation of the first order between the param- 

 eters a and h (i). Every solution of this equation is equivalent 

 with a certain surface of the congruence having an envelope. 



2. I shall now assume that (4) has a singular solution, so that 

 the result of the elimination of ^> between 



and ^ =0,^ ^^ 



satisfies (4). In this case the theorem holds: 



* Presented at the 1900, December, meeting of the American Mathematical Society (Chicaeo 



Section). 

 (') See Darboux, Th§orie G6n6rale des Surfaces, Vol. II, p. 7. 



