60] THE REPRESENTATION OF THE CYL1NDROID BY A CIRCLE. 53 



intersect BT at E. Then, since is the pole of PT, the line PT bisects the 

 angle ATE ( 58), and therefore AE must be bisected at P. 



From similar triangles, 



OB:AB:: OT : AE; 

 whence, if p a be the pitch of A, arid, of course, equal to AP, or \AE, 



_AB.OT 

 IP*- QB . 



But since the quadrilateral ASBT is inscribable in a circle, 



OT.OS = OA.OB; 

 whence, eliminating OT, we have, finally, 



_ AO.AB 



Pa m&amp;lt;r : 



as OS is constant, we see that p a varies as AO . AB, whence the following 

 theorem : 



If AB be any chord passing through 0, the pole of the axis of pitch, then 

 the pitch of the screw A is proportional to the product AO . AB. 



60. Pitches of Reciprocal Screws. 



It is known that the sum of the reciprocals of the pitches of a pair of 

 reciprocal screws on the cylindroid is constant ( 40). This is also plain 

 from the geometrical representation. For, since the triangles APT and BQT 

 (Fig. 10) are similar, we have 



AP-.BQ:: TP : TQ :: OA :OB; 



