24 RECIPROCAL SCREWS. 



this ellipse is the locus of the feet of the perpendiculars 

 let fall from O 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 gene- 

 rator of the cylindroid which it meets at V. This generator 

 is, therefore, perpendicular to the plane of OT and VT y 

 and, therefore, to the line O V. It follows that V must 

 be the foot of the perpendicular from O on the generator 

 through V y and that, therefore, the cone drawn from 

 O to the ellipse TL VM is the cone required. 



We hence deduce the following construction for the 

 cone of reciprocal screws which can be drawn to a cylin- 

 droid from any point O. 



Draw through O a line parallel to the nodal line of 

 the cylindroid, and let T be the one real point in which 

 this line cuts the surface. Find the second screw L M 

 on the cylindroid which has a pitch equal to the pitch 

 of the screw which passes through T. A plane drawn 

 through the point T and the straight line L M will cut 

 the cylindroid in an ellipse, the various points of which 

 joined to O give the cone required. 



We may further remark that as the plane TLM 

 passes through a generator it must be a tangent plane to 

 the cylindroid at the point Z, while at the point M the 

 line LM must intersect another generator.* It follows 

 that L must be the foot of the perpendicular from Tupon 

 L M y and that M must be a point upon the nodal line. 



26. Locus of a Screw Reciprocal to Four Screws. 

 Since a screw is determined by five quantities, it is 

 clear that when the four conditions of .reciprocity are 

 fulfilled the screw must be confined to a certain ruled 

 surface. Now this surface can be no other than a cylin- 



* Salmon, "Analytic Geometry of Three Dimensions," 2nd Ed., p. 348. 



