60] THE REl’RESENTATION OF THE CYLINDKOIIJ BY A CIRCLE. 53
intersect BT at E. Then, since 0 is the pole of PT, the line PT bisects the
angle ATE (§ 58), and therefoi’e AE must be bisected at P.
From similar triangles,
OB: AB:: 0T:AE;
whence, if pa be the pitch of A, and, of course, equal to AP, or ^AE,
„ AB.OT
2pa 0B .
But since the quadrilateral ASBT is inscribable in a circle,
0T.0S = 0A.0B;
whence, eliminating OT, we have, finally,
AO. AB
pa~ 20S :
as OS is constant, we see that pa 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. 1 or, since the triangles APT and BQT
(Fig. 10) are similar, we have
AP : BQ :: TP : TQ :: OA : OB;