265]
EMAN ANTS AND PITCH INVARIANTS.
277
If three screws, a, ß, y, are all parallel to the same plane, and if 6 be a
sci’ew normal to that plane, then we must have
dR> n dR n
0! 37 ■ da.!
a dR 1 dß i ■ 1 a dR _ „ "+9tdßt- ’
a dR .. + 06^ = o.
264. Conditions that Three Screws shall be parallel to a
Plane.
Since a screw of a three-system can be drawn parallel to any direction,
it will be possible to make any three of the quantities ... equal to zero.
Hence, we have as the condition that the three screws, a, ß, y shall be
all parallel to a plane the evanescence of all the determinants of the type
dR d«! ’ dR da2 ’ dR da3
dR dR dR
dßi dß2 ’ dß3
dR dR dR
dyr ’ dy2 ’ dy3
265. Screws on the same Axis.
The locus of the screws 0 perpendicular to a is represented by the
equation
/> i ü dR ß
If we assume that the screws of reference are co-reciprocal, then the
equation just written can only denote all the screws reciprocal to the one
screw whose co-ordinates are
LdR l_dR
Pi dat’ ■” peda6'
It is manifest that all the screws perpendicular to a given line cannot be
reciprocal to a single screw unless the pitch of that screw be infinite, other-
wise the condition
(pa +Pe) cos </> — d! sin </> = 0
could not be fulfilled. We therefore see that the co-ordinates just written
can only denote those of a screw of infinite pitch parallel to a.