22 RECIPROCAL SCREWS. 



24. Screw Reciprocal to Cylindroid. If a screw T? be- 

 reciprocal to two given screws 6 and 0, then rj is recipro- 

 cal to every screw on the cylindroid (0, 0). 



For a body only free to twist about rj would be undis- 

 turbed by wrenches on 9 and ; but a wrench on any 

 screw i// of the cylindroid can be resolved into wrenches 

 on 9 and ; therefore a wrench on i/> cannot disturb a 

 body only free to twist about rj; therefore i// and rj are 

 reciprocal. We may say for brevity that rj is reciprocal 

 to the cylindroid. 



TJ cuts the cylindroid in three points,* and one screw 

 of the cylindroid passes through each of these three 

 points ; these three screws must, of course, be reciprocal 

 to 17. Now two intersecting screws can only be reciprocal 

 when they are at right angles, or when the sum of their 

 pitches is zero. The pitch of the screw upon the cylin- 

 droid which makes an angle / with the axis of x is 

 p a cos 2 / + pp sin 2 /. 



This is also the pitch of the screw TT - I. There are, 

 therefore, two screws of any given pitch ; but there can- 

 not be more than two. It follows that TJ can at most in- 

 tersect two screws upon the cylindroid of pitch equal and 

 opposite to its own ; and, therefore, r\ must be perpendi- 

 cular to the third screw.f Hence any screw reciprocal 

 to a cylindroid must intersect one of the generators at 

 right angles. We easily infer, also, that a line intersect- 

 ing one screw of a cylindroid at right angles, must cut 

 the surface again in two points, the screws passing 

 through which have equal pitch. 



25. Reciprocal Cone. From any point P perpen- 



* Every right line meets a surface of the third degree in three points.. 

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



+ The writer may, perhaps, be excused for adding that it was the percep- 

 tion of this point which first gave him clear views on the subject of the present 

 volume. 



