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
Itt' + m (t + t') + n = 0,
we may for convenience speak of the pairs of generators as being in “involution.”
Suppose that two of the generators met by an arbitrary line are in “involution”
we have two roots of the cubic
ats + bt2 + ct + d=0,
connected by the relation
+ m + t2) + n = 0,
where and t2 are the parameters of the two generators and of course roots of
the cubic. Let the third root be t3 and form the product P of the three factors
ltj.2 + m (
then P = + »a (^ + i2) + n]2 [It.2 + 2»ii3 + nJ.
The common lines fall into two groups (1) transversals of the united lines
of the “ Involution ” where the parameters of these united lines satisfy
It2 + imt + 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.
Tn any plane we have belonging to these complexes eight common lines which