58
THE THEORY or SCREWS. [63-
If, therefore, AX, BY, OG be perpendiculars on CC, we have, from the
principle of the centre of gravity,
Pl P-2 \p! pj
°1’ paAX +plBY= + yj2) ()(]■
but, by a well-known property of the circle, if m be the radius,
2mAX=AC. AC- 2mBY = BC.BC-
whence
or
P1BC. BC + p.,AG. AC = 2 m (P1 + Pt) OG = m (§ 60),
™ AC AC OG
P1AB- AB+p<AB- ÄB=mÖS-
But, from the expressions for screw co-ordinates (§ 57), this reduces to
Pi^ißi +p2^ß2 = m .
U/S
The required expression has thus been demonstrated.
We can give another proof of this theorem as follows
If the two screws of reference be reciprocal, and if P1 and p2 be the co-
ordinates of another screw, then it is known, from the theory of the co-
oidinates, that the virtual coefficients of this screw, with respect to the screws
oi reference, are p1P1 and p2f>2, respectively (§ 37).
Thus (Fig. 15) the virtual coefficient of X and A must be (§ 57),
BX
P1 AB ’