509]
179
509.
PLAN OF A CURVE-TRACING APPARATUS.
[From the Proceedings of the London Mathematical Society, vol. iv. (1871—1873),
pp. 345—347. Read May 8, 1873.]
I have devised a curve-tracing apparatus on the following plan :
Imagine two planes II, IT moving in the same horizontal plane, and above the
two planes respectively the two points P, P' moving in the same or a parallel plane.
To fix the ideas, suppose that the two planes each move according to a law (that
is, let the motion of each of them depend on a single variable parameter; for instance,
they may each of them rotate about an axis); but let the motion of the two points
be free.
Suppose, next, that the planes are connected together in such manner that the
motion of one of them determines the motion of the other (e.g. by a train of wheel-
work) ; and that the two points are also connected together in such manner that the
motion of one of them determines the motion of the other (e.g. by a pentagraph; or
by a slotted rod, the slot of which works on an axle, so as to allow the rod to
move lengthways as well as rotate).
Suppose, finally, that one of the points, say P', is attached to a point of the
plane IT; then the plane II being set in motion, this determines the motion of IT,
consequently of P', consequently of P ; and the moving point P, or say the pencil P,
will describe on the moving plane II a curve, the nature of which will of course depend
on the nature of the motions of II, IT, and on that of the connection between these
planes and of the connections between the points P and P'.
I propose to describe the apparatus as nearly as I have actually constructed it.
(See sketch-plan Fig. 1, and side-elevation Fig. 2.)
The framework of the instrument consists of two longitudinal bars (B) each about
three feet long, one inch thick, and three inches broad, supported edgewise at a
distance of about eighteen inches on the cross-pieces G, G; and half-way between
them, supported by the same cross-pieces, is an axis carrying at each extremity two
mitre wheels. The bars B support three cross-pieces D, D, D, and between these are
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