31]
SCREW CO-ORDINATES.
33
30. The Intensity of the Resultant may be expressed in terms of the
intensities of its components on the six screws of reference.
Let a be any screw of pitch pa, let p„ p2, Szc. p6 be the pitches of the
six screws of reference &>1; co,, ... coe; then taking each of the screws of refer-
ence in succession, for y in § 29, and remembering that the virtual coefficient
of two coincident screws is simply equal to the pitch, we have the following
equations:—
= a/'pi + a2"OTi2 + ... + a6"’w16,
a"wB8 = + ... + «6"wM4- ae"pe.
But taking the screw p in place of y we have
Ct pa Ot j ^"«6 •
Substituting for wal ... wall from the former equations, we deduce
paa"s = 2 (jpi«/'2) + 22 (ai"«2"w12).
This result may recall, the well-known expression for the square of a force
acting at a point in terms of its components along three axes passing through
the point. This expression is of course greatly simplified when the three
axes are rectangular, and we shall now show how by a special disposition
°f the screws of reference, a corresponding simplification can be made in the
formula just written.
31. Co-Reciprocal Screws.
We have hitherto chosen the six screws of reference quite arbitrarily;
we now proceed in a different manner. Take for co,, any screw; for w2, any
screw reciprocal to cop, for