A Treatise on the Theory of Screws
Forfatter: Sir Robert Stawell Ball
År: 1900
Forlag: The University Press
Sted: Cambride
Sider: 544
UDK: 531.1
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492
THE THEORY OF SCREWS.
are the common tangents of a curve of the fourth and a curve of the second class.
Two and only two of these intersect the united lines of the involution. The
occurrence of the square factor indicates that the remaining six coincide in pairs
and hence we have the general result that the conic has triple contact with the
cubic.
NOTE V.
Remarks on § 210.
Piofessor C. J. Joly has communicated to me the following theorems with regard
to the cubic which is the locus of the points corresponding to the screws of a
3-system which intersect a given screw of the system (§ 210).
Let 0 be the double point on the cubic and P,, P2 the two points correspond-
ing to a pair of screws of equal pitch which intersect 0. Then all the chords P^
for different pitches are concurrent.
For the cylindroid defined by the screws corresponding to must be cut by
the screw corresponding to 0 in a third point which lies on the generator O' of the
cyhndroid such that 0 and O' are at right angles (§ 22). As there is only one screw
of the 3-system intersecting 0 at right angles it follows that all the chords P,P2
will be concurrent. The point corresponding to O' is that whose co-ordinates are
given on p. 213, viz.
Ps-Pi Pi-Ps Pi~Pl
ai ’ a2 ’ a3 ’
where an a2) a3 are the co-ordinates of 0.
There is also to be noted the construction for the tangents at the double point
of the cubic. They are the lines to the points in which the pitch-conic through
the double point is met by the polar of the double point with respect to the conic
of infinite pitch.
Let Sp = 0 be the conic of pitch p. Let Pp = 0 be the polar of 0 with respect
to the conic of pitch p and let Sp = 0 be the result of substituting the co-ordinates
of 0 in the equation of the conic Sp = 0.
As all the conics have four points common, suppose
— kSp + ,
where k and I are certain constants.
Likewise P. = kPp + IP* ■ S’ = kSJ + IS'*,
whence after a few steps (§ 210) we have the new form for the cubic
(SPS’* + SXSP') = 0.
If = 0 passes through the double point then Sp = 0 and the cubic is
-7*^ = 0,