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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162
THE THEORY OF SCREWS.
[164,
Let a, ß, 7 be the co-ordinates of the point. Then the plane through
this point, and the generator of the cylindroid defined by the equations
y = æ tan 3,
z — m sin 20,
15 (y — x tan 0) (y — m sin 23) = (9 — a tan 3) (z — m sin 23);
or , if we arrange in powers of tan 3, we obtain
A tan3 3 + B tan2 3 + G tan 3 + D = 0,
in which
A = az — yx; I) = yy — ßz,
B — yy — ßz + 2mx — 2ma ; G = az — yx + 2mß — 2my.
If the same transversal also crosses the generator defined by 3', then,
A tan3 3' + B tan2 3' + G tan 3' + D = 0.
When the two screws defined by 3 and 3' are reciprocal,
tan 3 tan 3' = H,
when II is a constant.
By eliminating and rejecting the factor
tan 3 — tan 3'
we obtain
- AW + 71 GIB - BDH + = 0.
And as this is of the second degree in x, y, z, the required theorem has been
proved.
All these cones must pass through the centre of the cylindroid, inasmuch
as the two principal screws of the cylindroid are reciprocal. If a constant
be added to the pitches of all the screws on the cylindroid, then the pairs
of reciprocals alter, inasmuch as II alters. The cone changes accordingly,
and thus there would be through each point a family of cones, all of which,
however, agree in having, as a generator, the ray from the vertex to the
centre of the cylindroid. Thus, even when the cylindroid is given, we must
further have the pitch of a stated screw given before the cone becomes
definite. This state of things may be contrasted with that presented by the
cone of reciprocal screws which may be drawn through a point. The latter
depends only upon the cylindroid itself, and is not altered if all the pitches
be modified by a constant increment.