it is very desirable to keep the force which acts on the pendulum uniform. This in fact is the great object of the best escapements. Inasmuch as the impulse on the pendulum, derived from the work done by a falling weight or an unwinding spring, is transmitted through a train of wheels, it is desirable that that transmission should be as free from friction and as regular as possible. This involves care in the shaping of the teeth. The object to be aimed at is that as the wheel turns round the ratio of the power of the driver to that of the driven wheel (“runner” or “follower”) should never vary. That is to say, whether the back part of the tooth of the driver is acting on the tip of the tooth of the follower, or the tip of the driver is acting on the back part of the tooth of the follower, the leverage ratio shall always be uniform. For simplicity of manufacture the pinion wheels are always constructed with radial leaves, so that the surface of each tooth is a plane passing through the axis of the wheel. The semicircular rounding of the end of the tooth is merely ornamental. The question therefore is, suppose that it is desired by means of a tooth on a wheel to push a plane round an axis, what is the shape that must be given to that tooth in order that the leverage ratio may remain unaltered?

Fig. 22.—Cam and Plane.Fig. 23. |

If a curved surface, known as a “cam,” press upon a plane one,
both being hinged or centred upon pivots A and B respectively
(fig. 22), then the line of action and reaction at D, the
point where they touch, will be perpendicular to their
Epicy-

cloidal teeth.
surfaces at the point of contact—that is perpendicular
to BD, and the ratio of leverage will obviously be
AE : BD, or AC : CB. Hence to cause the leverage ratio of
the cam to the plane always to remain unaltered, the cam must
be so shaped that in any position the ratio AC : CB will remain
unchanged. In other words the shape of the cam must be such
that, as it moves and pushes BD before it, the normal at the
point of contact must always pass through the fixed point C.

Fig. 24. |

If a circle PMB roll upon another circle SPT (fig. 23) any point M on it will generate an epicycloid MN. The radius of curvature of the curve at M will always be MP, for the part at M is being produced by rotation round the point P. It follows that a line from B to M will always be tangential to the epicycloid. If the epicycloid be a cam moving as a centre round the centre R (not shown in the figure) of the circle SPT, the leverage it will exert upon a plane surface BM moving round a parallel axis at B, will always be as BP to PR, that is, a constant; whence MN is the proper shape of a tooth to act on a pinion with radial arms and centred at B. In designing a pair of wheels to transmit motion, which is to be multiplied say 6 times in the transmission (about the usual ratio for clock wheels), if we take two circles (called the “pitch circles”) touching one another with radii as 1:6, then the circumference of the smaller will roll 6 times round that of the larger. The smaller wheel will have a number of teeth, say 8 to 16, each of them being sectors of the circle (fig. 24). If there are 16 teeth, then on the surface of the driving wheel there will be 96 teeth. Each of these teeth will be shaped as the curve of an epicycloid formed by the rolling on the big circle of a circle whose diameter is the radius of the pitch circle of the pinion. Points of the teeth so formed are cut off, so as to allow of the pinion having a solid core to support it, and gaps are made into the pitch circle to admit the rounded ends of the leaves of the pinon wheel. Thus a cog-wheel is shaped out.

Clock wheels are made of hard hammered brass cut out by a wheel cutting machine. This machine consists of a vertical spindle on the top of which the wheel to be cut is fixed on a firmly resisting plate of metal of slightly smaller diameter, so as to allow the wheel to overlap. A cutter with the edges most delicately ground to the exact shape of the gap between two teeth is caused to rotate 3000-4000 times a minute, and brought down upon the edge of the wheel. The shavings that come off are like fine dust, but the cutter is pushed on so as to plunge right through the rim of the wheel in a direction parallel to the axis. In this way one gap is cut. The vertical spindle is now rotated one division, by means of a dividing plate, and another tooth is cut, and so the operation goes on round the wheel.

It is not desirable in clocks that the pinion wheels which are driven should have too few teeth, for this throws all the work on a pair of surfaces before the centres and is apt to produce a grinding motion. Theoretically the more leaves a pinion has the better. Pinions can be made with leaves of thin steel watch-spring. In this case quite small pinions can have 20 leaves or more. The teeth in the driving wheels then become mere notches for which great accuracy of shape is not necessary. Such wheels are easy to make and run well. Lantern pinions are also excellent and are much used in American clocks. They are easy to make in an ordinary lathe. The cog-wheels must, however, be specially shaped to fit them. They consist of a number of round pins arranged in a circle round the axis of the wheel and parallel to it. The ends are secured in flanges like the wires of a squirrel cage. The teeth of cog-wheels engage them and thus drive the wheel round. They were much used at one time but are now falling out of favour again.

It is possible to make toothed wheels that drive with perfect
uniformity by using for the curve of the teeth involutes of circles.
These involutes are traced out by a point on a string
that is gradually unwound from a circle. They are
Involute teeth.
in fact epicycloids traced by a rolling circle of infinite
radius, *i.e.* a straight line. Involute teeth have the advantage
that they roll on one another instead of sliding. When badly
made they put considerable strain on the axes or shafts that
carry them. Hence they have not been regarded with great
favour by clockmakers.

By the pitch of a wheel is meant the number of teeth to the inch of circumference or diameter of the wheel; the former is called the circumferential pitch, the latter the diametral pitch. Thus if we say that a wheel has 40 diametral pitch we mean that it has 40 teeth to each inch of diameter. The circumferential pitch is of course got by dividing Pitch. the diametral pitch by π. Wheel-cutters are made for all sizes of pitches. If it were needed to make a pair of wheels the ratio of whose motion was say 6:1 and we determined to use a diametral pitch of 30 to the inch, that is teeth about 110 in. wide at the base, and if the smaller circle were to have 20 teeth, we should need a blank of a diameter of 2030 + 230 = 2230 in. for the smaller wheel, and one of 12030 + 230 = 12230 in. for the larger wheel which would have 120 teeth to the inch and be 4.06 in diameter to the tips of the teeth. The smaller toothed wheel would be .73 of an inch in diameter over all. The pitch circles of the wheels would be 23 and 4 in. respectively. For fine wheel work, where the driver is always much larger than the driven wheel, the epicycloidal tooth appears preferable, as it is generally considered to put less side strain on the pinion wheel. But the relative merits of the two systems have never been properly tested for clock work.

*Going Barrels.*—A clock which is capable of going accurately must have some contrivance to keep it going while it is being wound up. In the old-fashioned house clocks, which were wound up by merely pulling one of the strings, and in which one such winding served for both the going and striking parts, this was done by what is called the endless chain of Huygens, which consists of a string or chain with the ends joined together, and passing over two pulleys on the arbors of the great wheels,