A Treatise on the Theory of Screws

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 (<x + i2) + n, lt2t3 + m (t2 + t3) + n, + m. (t3 + Zj) + 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 = 0 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 h = ^2> 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