ON THE CONGRUENCES OF TWISTED CURVES 31 



all cones having their vertices on one of the given 'curves and pass- 

 ing through the other curves. In case that equation (4) has a sin- 

 gular solution, the system of cones are tangent to a developable surface 

 D containing the given curves. Every tangent plane to D is also 

 tangent to the two given curves, so that the singular developable sur- 

 face may also be considered as generated by all tangent planes com- 

 mon to both of the given cu.rves. If ui and n are the classes of the 

 given curves, the developable surface of its common tangent planes 

 is of the class m n. Of particular interest is the case where one of 

 the given curves is an infinitely distant circle (given by a cone- 

 director) so that the generating planes of the developable surface all 

 have the same inclination towards a fixed plane (perpendicular to the 

 axis of cone-director.) Assuming the plane XOY parallel to the 

 plane of the infinite circle, this problem is equivalent with a particu- 

 lar solution of Monge's e(|uation 



^.'»2+</y2_/.2^.2=0^ ^^^0) (7) 



or of the partial differential equation 



^- m- ^- ^' ''' 



1 



where -jr "= tawy is the tangent of the constant angle of inclination 



of the generating planes with the plane X^O Y(y ^ ) . The class of the 

 developable surface is now 2m and its order or rank 



V = 2m{2m—l)~2h—3^, (9) 



where h is the number of double tangents in a plane section of the 

 developable surface and ^ the number of stationary planes of the 

 latter. (^) 



4. I shall now consider the curve of intersections , ,6' , of the de- 

 velopable surface with the plane XOY, whose order is given by (9). 

 The developable surfaces of the congruence are right cones whose 

 elements include constant angles with (^XOY) and whose base-circles 



(') Lie. ^oc.crt. p. 262. 



(') Fiedler. Geometrie der Lage, Vol. II, pp. 132-142. 



