APPENDIX I. 491 



Conversely if we are given a, b, c, d we have a cubic equation in t which 

 on solution determines the three generators of the cylindroid which a given line 

 intersects. 



If the generators are connected in pairs by a one-to-one relation of the type 



lit + m (t + t ) + n 0, 

 we may for convenience speak of the pairs of generators as being in &quot;involution.&quot; 



Suppose that two of the generators met by an arbitrary line are in &quot;involution&quot; 

 we have two roots of the cubic 



at 3 + bt? + ct + d=Q, 



connected by the relation 



ItJz + m (^ + t. 2 ) + n = 0, 



where ^ and t z are the parameters of the two generators and of course roots of 

 the cubic. Let the third root be t 3 and form the product P of the three factors 



lt- + m t + + n 



t^ + m t s + ^ + n. 



If we replace the symmetric functions of the roots by their values we find that 

 P is a homogeneous function of a, b, c, d in the second degree. 



The equation P = represents the complex of transversals intersecting corre 

 sponding generators of the involution. This complex is of the second order 

 and the transversals in a plane therefore envelop a conic and those through 

 a point lie on a quadric cone. 



In like manner the discriminant of the cubic itself when equated to zero 

 represents a complex Q of the fourth order which consists of all the tangents to 

 the cylindroid. The lines in a plane envelop a curve of the fourth class (the 

 section of the cylindroid) and the lines through a point are generators of the 

 tangent cone of the fourth order. 



Let us now consider the lines common to the two complexes P and Q. 

 If we suppose two roots of the cubic equal, for example 



*=**, 



then P = \ltJz + m (^ + t 2 ) + n] 2 [lt. 2 2 + 2mt 2 + n]. 



The common lines fall into two groups (1) transversals of the united lines 

 of the &quot; Involution &quot; where the parameters of these united lines satisfy 

 IP + 2mt + n = 0, and (2) where the odd point on the transversal coincides with 

 one of the points in which the transversal meets the conjugate generators. The 

 occurrence of the square factor shows that these latter lines are to be counted 

 twice. 



In any plane we have belonging to these complexes eight common lines which 



