52 



THE THEORY OF SCREWS. 



[58- 



Since SO . ST = &amp;gt;SM 2 = SB 2 , we have Z STA = Z SAB, and z 

 whence Z A TB is bisected by ST, and therefore 



Z A TP = \ Z 4 SB = 6 = Z B TQ. 



= z 



It follows that .4Pcos0 = PTs mO, since each is equal to the perpen 

 dicular from P on A T. 



Similarly, 



BQ cos = QT sin ; 



whence ( A P + BQ) cos0-(PT + QT) sin = 0, 



which reduces to 



(p a + pi) cos 6 daft sin 6 = 0. 

 The theorem has thus been proved. 



We have, therefore, a simple construction for finding the screw B reci 

 procal to a given screw A. It is only necessary to join A to 0, the pole 

 of the axis of pitch, and the point in which this cuts the circle again gives 

 B the required reciprocal screw. 



We also notice that the two principal screws of the cylindroid are reci 

 procal, inasmuch as their chord passes through 0. 



59. Another Representation of the Pitch. 



We can obtain another geometrical expression for the pitch, which will be 

 often more convenient than the perpendicular distance from the point to the 

 axis of pitch. 



Let A (Fig. 11) be the point of which the pitch is required. Join 

 AOB, draw AP perpendicular to the axis of pitch PT, and produce AP to 



