388] THE THEORY OF PERMANENT SCREWS. 425 



We now examine the points on the cylindroid intersected by the axis of 

 the screw 



F=pX+yZ -zY, 



G=pY+zX-xZ, 

 H=pZ + xY-yX. 



We write the equations of the cylindroid in the form 



x R cos &amp;lt;f&amp;gt; ; y = R sin &amp;lt; ; z = m sin2&amp;lt;/&amp;gt;; 

 then, eliminating p and R, and making 



V = FX+GY+HZ, 

 we find, after a few reductions, 



tan 3 ^(YV-GU) + tan 2 &amp;lt;f&amp;gt; (X V - FU + 2mX U) 



This cubic corresponds, of course, to the three generators of the cylindroid 

 which the ray intersects. 



If we put 



then the cubic becomes, by eliminating m, 



The factor Ftan^ + Z simply means that the restraining screw cuts the 

 instantaneous screw at right angles. 



The two other screws in which 77 intersects the cylindroid are given by 

 the equation 



(XYV-XGU) tan 2 + (XYV- FUY) = 0. 



These two screws are of equal pitch, and the value of the pitch is 



Pl (XYV-XGU)+p 2 (FUY-XYV} 

 U(FY-GX) 



where p^ and p. 2 are the pitches of the two principal screws on the cylindroid. 

 After a few reductions the expression becomes 



V (I* - x opl ) sin + (I* + y opa ) cos 6 

 U x Q sin y cos 6 



This is the pitch of the two equal pitch screws on the cylindroid which rj 

 intersects. If 77 is to be reciprocal to the cylindroid, then, of course, the 

 pitch of 77 itself should be equal and opposite in value to this expression. 

 Hence the permanent screw on the cylindroid is given by 

 (li* - XopJ sin 6 + (7 2 2 + y p 2 ) cos 6 = 0. 



