208] PLANE REPRESENTATION OF THE THIRD ORDER. 207
A pair of reciprocal screws are conjugate points with respect to the zero-
pitch conic.
From this theorem we can at once draw the following conclusions:—
All the screws of the system reciprocal to a given screw 0 lie upon a
cylindroid.
For the locus of points conjugate to 6 is, of course, the polar of 6
with respect to the zero-pitch conic, and this polar will correspond to a
cylindroid.
On any cylindroid one screw can always be found reciprocal to a given
screw 6. For this will be the intersection of the polar of 6 with the line
corresponding to the given cylindroid.
A triad of co-reciprocal screws will correspond to a self-conjugate triangle
of the conic of zero-pitch.
208. The Principal Screws of the System.
Draw the conic of zero-pitch A, and the conic of infinite pitch B, which
intersect in the four screws of indeterminate pitch, Py,P2> Ps, Pt (see fig- 39).
Draw the diagonals of the complete quadrilateral, and let them intersect in
Fig. 39.
the points X, Y, Z. These three points are significant. Take any pair of
them, X and Y; then, by the known properties of conics, X and Y are con-
jugate points with respect to both of the conics A and B. The screws X