30 UNIVERSITY OF COLORADO STUDIES 



All surfaces defined hy (i) are tangent to the surface corre- 

 sponding to the singular solution of (4). 



To prove this, let 5^ be a fixed value of 2, while x and y are in- 

 dependent variables, and write 



x=f, (a, b, -,), (6) 



?/=(/), («, I), ^j), 



in accordance with (1). Equations (6) represent a contact-transfor- 

 mation in the plane 5;=5'j, when a and h are interpreted as Cartesian 

 co-ordinates in this plane, in the same manner as x and 2/(0- ^ ^iii©- 

 element is transformed into another line-element, a system of line- 

 elements into a system of line-elements (Elementverein), two curves 

 in contact into two curves in contact of the same order. To every 

 solution of (4) belongs a curve C with an equation of the form 

 g (a, J)=0, which is tangent to the curve S representing the sin- 

 gular solution (5). Applying the contact-transformation (6) to the 

 curves C and S, a new system of curves C is obtained which are all 

 tangent to the transformed singular curve S'. Now, to every solu- 

 tion g {a, J)=Oof (4), by (1), corresponds a surface of the congru- 

 ence having an envelope, and it is evident that its trace on the plane 

 z^=s^ is the curve C. From this it is further seen that system of 

 surfaces with an envelope contained in (1) intersect the plane z=^z^ 

 in a system of curves C having S' as an envelope. The same holds 

 true for all values of 2^. As every value of s defines a curve S' it 

 follows that the system of curves S' forms a surface which, accord- 

 ing to (3) and (4) has an envelope. The system of surfaces of the 

 congruence having an envelope is therefore tangent to a surface with 

 an envelope, and corresponding to the singular solution of (4), as was 

 to be proved. 



3. Without entering into further details I shall apply the 

 previous result to the case of a congruence of straight lines inter- 

 secting singly each of two given plane curves in space. As is well- 

 known, the two parts of the focal surface of the congruence are the 

 curves themselves, and the system of developable surfaces consists of 



(') See Sophus Lie, Geometrie der Beruhrungs-Transformationen, Vol. I, p. 10 and pp. 43-67. 



