A Treatise on the Theory of Screws

159] THE GEOMETRY OF THE CYLINDROID. 149 belonging to the other system on the paraboloid must also be parallel to a plane, being that defined by the other generator I', in which the plane at infinity cuts the quadric. Let PQ be a common perpendicular to a and 7, then since it intersects y at right angles, it must also intersect I; and since PQ cuts the three generators of the paraboloid a, 7 and I, it must be itself a generator, and therefore intersects ß. But a, ß, y are all parallel to the same plane, and hence the common perpendicular to a and 7 must be also perpendicular to ß. We hence deduce the important result, that all the screws on the surface 8 must intersect the common perpendicular to a and ß, and be at right angles thereto. The geometrical construction of Ä is then as follows:—Draw two rays a and ß, and also their common perpendicular X. Draw any third ray d, subject only to the condition that it shall intersect both a. and ß. Then the common perpendicular p to both 3 and X will be one of the required generators, and as 3 varies this perpendicular will trace out the surface. It might at first appear that there should be a doubly infinite series of common perpendiculars p to X and to 3. Were this so, of course £ would not be a surface. The difficulty is removed by the consideration that every trans- versal across p, a, ß is perpendicular to p. Each p thus corresponds to a singly infinite number of screws 3, and all the rays p form only a singly infinite series, i.e. a surface. A simple geometrical relation can now be proved. Let the perpendicular distance between p and a be dlt and the angle between p and a be A; let d2 and be the similar quantities for p and ß, then it will be obvious that d1 : d2:: tan 71x : tan A..,; or d1 + d2:d1- d2:: sin (A + A.ß : sin ( dj - yl2), iffz be the distance of p from the central point of the perpendicular h between a and 3; and if^e be the angle between a and ß, andetø be the angle made by p with a parallel to the bisector of the angle e, then we have from the above z : h :: sin 2<£ : sin 2e. The equation of the surface >S is now deduced for cc tan <b = -\ y whence we obtain the equation of the cylindroid in the well-known form •z (®2 + y) = xy- The law of the distribution of pitch upon the cylindroid can also be deduced