54
THE THEORY OP SCREWS.
[60-
whence 0 is the centre of gravity of particles of masses - and - placed at
. , „ Pa Pß
A and B, respectively.
From the known property of the centre of gravity,
AP- + BQ- = 0T.(- +
Pa Pß \Pa PßJ
but each of the terms on the left-hand side is unity, whence, as required,
1 + 1=A
Pa Pß GT'
representing the pitch also verifies this theorem.
The second mode of
For since (§ 59)
we have
AO. AB
pa 2OS ’
BO. BA
Pß 208 ’
from which
AB’
pr'+pß~20S’
AB2. AO. BO
papß 40S^~
1
Pa pß
2. OS
OA.OB’
constant for every chord through 0; and, as OS is constant,
the sum of the reciprocals of the pitches of two reciprocal
but OA . OB is
it follows that
screws on any cylindroid must be constant.
61. The Virtual Coefficient.
Let A and B (Fig. 12) be the two screws. Let, as usual, 0 be the pole of
the axis of pitch PT. Let O’ be the point in which the chord AB intersects
01 the perpendicular drawn from 0 to the axis of pitch, and let P T be the
polar of O’, which is easily shown to be perpendicular to SO. From T let fall
the perpendicular TF upon 4 7", and from 0 let fall the perpendicular OG
upon AB.
As before (§ 58), we have Z ATP’ = z TTF = 3; also, since
Z SA O’= Z A TO', and A SAO = /. A TO,
we must have z SA O'-z SA0 = a ATO' - z ATO, or AOAG = aTAF-
whence the triangles OAG and 7^ are similar, and, consequently,
TF-°°-io-OG^