52
THE THEORY OF SCREWS. [58-
Since SO . ST = SA2 = SB2, we have Z ST A = ^SAB, and z STB = z SB A
whence z ATB is bisected by ST, and therefore
z A TP = I z A SB = 0 = z BTQ.
It follows that ,4Pcos0=PT«n0, since each is equal to the perpen-
dicular from P on A T. 11
Similarly,
BQ cos 0 = QT sin 0 ;
whence (^P+2?Q)cos 0-(PT+QT)sin 0 = 0,
which reduces to
(p<< + pß) cos 0 — daß sin 0 = ((.
The theorem has thus been proved.
We have, therefore, a simple construction for finding the screw B reci
of°the axis SV(>S reW/u ™ t0 j°in A t0 °’ the Pole
of the axis of pitch, and the point in which this cuts the circle again gives
B the required reciprocal screw. S
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.
Wo can obtain another geometrical expression for the pitch, which will bo
åS of',°7 r™1"““ th“” the »""■ the point to the
«■Alb UI plLCn.
yin«6,] A 11'> bf- the point of whlch the Pitch is required. Join
, draw Al perpendwular to the axis of pitch PT, and produce AP to