20-23] RECIPROCAL SCREWS. 27 



For a body only free to twist about 77 would be undisturbed by wrenches 

 on 6 and &amp;lt; ; but a wrench on any screw ty of the cylindroid can be resolved 

 into wrenches on 6 and &amp;lt;/&amp;gt; ; therefore a wrench on -^r cannot disturb a body 

 only free to twist about 77 ; therefore *fy and rj are reciprocal. We may say 

 for brevity that 77 is reciprocal to the cylindroid. 



77 cuts the cylindroid in three points because the surface is of the third 

 degree, and one screw of the cylindroid passes through each of these three 

 points ; these three screws must, of course, be reciprocal to 77. But 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 I with the axis of x is 



p a cos 3 / +pp sin 2 /. 



This is also the pitch of the screw TT I. There are, therefore, two screws 

 of any given pitch ; but there cannot be more than two. It follows that 77 

 can at most intersect two screws upon the cylindroid of pitch equal and 

 opposite to its own ; and, therefore, 77 must be perpendicular to the third 

 screw. Hence any screw reciprocal to a cylindroid must intersect one of the 

 generators at right angles. We easily infer, also, that a line intersecting one 

 screw of a cyliudroid at right angles must cut the surface again in two 

 points, and the screws passing through these points have equal pitch. 



These important results can be otherwise proved as follows. A wrench 

 can always be expressed by a force at any point 0, and a couple in a 

 plane L through that point but not of course in general normal to the force. 



For wrenches on the several screws of a cylindroid, the forces at any 

 point all lie on a plane and the couples all intersect in a ray. 



The first part of this statement is obvious since all the screws on the 

 cylindroid are parallel to a plane. 



To prove the second it is only necessary to note that any wrench on the 

 cylindroid can be decomposed into forces along the two screws of zero pitch. 

 Their moments will be in the planes drawn through arid the two screws of 

 zero pitch. The transversal across the two screws of zero pitch drawn from 

 must therefore lie in every plane L. 



We hence see that the third screw on the cylindroid which is crossed by 

 such a transversal must be perpendicular to that transversal. 



23. Reciprocal Cone. 



From any point P perpendiculars 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 



