28 



THE THEORY OF SCREWS. 



[23, 



screws on the cylindroid intersected by the perpendicular, then the perpen 

 diculars form a cone of reciprocal screws. 



We shall now prove that this cone is of the second order, and we shall 

 show how it can be constructed. 



Let be the point from which the cone is to be drawn, and through let 

 a line OT be drawn which is parallel to the nodal line, and, therefore, perpen 

 dicular to all the generators. This line will cut the cylindroid in one real 

 point T (Fig. 4), the two other points of intersection coalescing into the in 

 finitely distant point in which OT intersects the nodal line. 



Draw a plane through T and through the screw LM which, lying on the 

 cylindroid, has the same pitch as the screw through T. This plane can cut 

 the cylindroid in a conic section only, for the line LM and the conic will then 



Fig. 4. 



make up the curve of the third degree, in which the plane must intersect the 

 surface. Also since the entire cylindroid (or at least its curved portion) is 

 included between two parallel planes ( 17), it follows that this conic must be 

 an ellipse. 



We shall now prove that this ellipse is the locus of the feet of the per 

 pendiculars let fall from on the generators of the cylindroid. Draw in the 

 plane of the ellipse any line TUV through T ; then, since this line intersects 

 two screws of equal pitch in T and U, it must be perpendicular to that 

 generator of the cylindroid which it meets at V. This generator is, therefore, 

 perpendicular to the plane of OT and VT, and, therefore, to the line 0V. 

 It follows that V must be the foot of the perpendicular from on the 



