RECIPROCAL SCREWS. 



diculars can be let fall upon the generators of the 

 cylindroid, and if to these perpendiculars pitches are 

 assigned which are equal in magnitude and opposite 

 in sign to the pitches of the two remaining screws on 

 the cylindroid intersected by the perpendicular, then the 

 perpendiculars 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 O be the point from which the cone is to be 

 drawn, and through O let a line 07" be drawn which is 

 parallel to the nodal line, and, therefore, perpendicular 

 to all the generators. This line will cut the cylindroid 

 in one real point T (Fig. 2), the two other points of inter- 

 section coalescing into the infinitely distant point in 

 which OT intersects the nodal line. 



Draw a plane through T and through the screw L M 

 which, lying on the cylin- 

 droid, has the same pitch 

 as the screw through T. 

 Now this plane must cut 

 the cylindroid in a conic 

 section, for the line LM 

 and the conic will then 

 make up the curve of the 

 third degree, in which the 

 plane must cut the sur- 

 face.* Also since the entire 

 cylindroid (or at least its 

 curved portion) is included 

 between two parallel planes, 

 19, it follows, that this 

 conic must be an ellipse. 

 We shall now prove that 



Fig. 2. 



Salmon, "Analytic Geometry of Three Dimensions," 2nd Ed., p. 14. 



