Chapter VII: Part 7

Since g is constant, and the length of the pendulum does not vary, it follows that when a pendulum is drawn aside through a small arc the force tending to bring it back to rest is proportional to the displacement (approximately). Thus the pendulum bob under the influence of gravity, if the arc of swing is small, acts as though instead of being acted on by gravity it was acted on by a spring tending to drag it towards D, and therefore is isochronous. The qualification "If the arc of swing is small" is introduced because, as was discovered by Christiaan Huygens, the arc of vibration of a truly isochronous pendulum should not be a circle with centre O, but a cycloid DM, generated by the rolling of a circle with diameter DQ = 1/2OD, upon a straight line QM. However, for a short distance near the bottom, the circle so nearly coincides with the cycloid that a pendulum swinging in the usual circular path is, for small arcs, isochronous for practical purposes.

The formula representing the time of oscillation of a pendulum, in a
circular arc, is thus found:--Let OB (fig. 6) be the pendulum, B be
the position from which the bob is let go, and P be its position at
some period during its swing. Put FC = h, and MC = x, and OB = l. Now
when a body is allowed to move under the force of gravity in any path
from a height h, the velocity it attains is the same as a body would
attain falling freely vertically through the distance h. Whence if v
be the velocity of the bob at P, v = sqrt(2gFM) = sqrt(2g(h - x)). Let
Pp = ds, and the vertical distance of p below P = dx, then Pp =
velocity at P X dt; that is, dt = ds/v.

ds l l
Also -- = -- = ---------------,
dx MP sqrt(x(2l - x))

ds ldx 1
whence dt = -- = --------------- . ---------------
v sqrt(x(2l - x)) sqrt(2g(h - x))

1 / l dx 1
= --- / --- . -------------- . ----------------
2 \/ g sqrt(x(p - x)) sqrt(1 - (x/2l))

Expanding the second part we have

1 / l dx / x \
dt = --- / --- . -------------- . ( 1 + --- + ... ).
2 \/ g sqrt(x(h - x)) \ 4l /

If this is integrated between the limits of 0 and h, we have

/ l / h \
t = [pi] / --- . ( 1 + --- + ... ),
\/ g \ 8l /

where t is the time of swing from B to A. The terms after the second
may be neglected. The first term, [pi] sqrt(l/g), is the time of swing
in a cycloid. The second part represents the addition necessary if the
swing is circular and not cycloidal, and therefore expresses the
"circular error." Now h = BC^2/l = 2[pi]^2[theta]^2l / 360^2, where
[theta] is half the angle of swing expressed in degrees; hence h/(8l)
= [theta]^2/52520, and the formula becomes

/ l / [theta]^2 \
t = [pi] / --- ( 1 + -------- ).
\/ g \ 52520 /

Hence the ratio of the time of swing of an ordinary pendulum of any
length, with a semiarc of swing = [theta] degrees is to the time of
swing of a corresponding cycloidal pendulum as 1 + [theta]^2/52520 : 1.
Also the difference of time of swing caused by a small increase
[theta]' in the semiarc of swing = 2[theta][theta]' / 52520 second per
second, or 3.3[theta][theta]' seconds per day. Hence in the case of a
seconds pendulum whose semiarc of swing is 2 deg. an increase of .1 deg. in
this semiarc of 2 deg. would cause the clock to lose 3.3 X 2 X 0.1 = .66
second a day.

Huygens proposed to apply his discovery to clocks, and since the
evolute of a cycloid is an equal cycloid, he suggested the use of a
flexible pendulum swinging between cycloidal cheeks. But this was only
an example of theory pushed too far, because the friction on the
cycloidal cheeks involves more error than they correct, and other
disturbances of a higher degree of importance are left uncorrected. In
fact the application of pendulums to clocks, though governed in the
abstract by theory, has to be modified by experiment.

Neglecting the circular error, if L be the length of a pendulum and g
the acceleration of gravity at the place where the pendulum is, then
T, the time of a single vibration = [pi] sqrt(L/g). From this formula
it follows that the times of vibration of pendulums are directly
proportional to the square root of their lengths, and inversely
proportional to the square root of the acceleration of gravity at the
place where the pendulum is swinging. The value of g for London is
32.2 ft. per second per second, whence it results that the length of a
pendulum for London to beat seconds of mean solar time = 39.14 in.
nearly, the length of an astronomical pendulum to beat seconds of
sidereal time being 38.87 in.

This length is calculated on the supposition that the arc of swing is
cycloidal and that the whole mass of the pendulum is concentrated at a
point whose distance, called the radius of oscillation, from the point
of suspension of the pendulum is 39.14 in. From this it might be
imagined that if a sphere, say of iron, were suspended from a light
rod, so that its centre were 39.14 in. below its point of support, it
would vibrate once per second. This, however, is not the case. For as
the pendulum swings, the ball also tends to turn in space to and fro
round a horizontal axis perpendicular to the direction of its motion.
Hence the force stored up in the pendulum is expended, not only in
making it swing, but also in causing the ball to oscillate to and fro
through a small angle about a horizontal axis. We have therefore to
consider not merely the vibrations of the rod, but the oscillations of
the bob. The moment of the momentum of the system round the point of
suspension, called its moment of inertia, is composed of the sum of
the mass of each particle multiplied into the square of its distance
from the axis of rotation. Hence the moment of inertia of the body I
= [Sigma](ma^2). If k be defined by the relation [Sigma](ma^2) =
[Sigma](m) X k^2, then k is called the radius of gyration. If k be the
radius of gyration of a bob round a horizontal axis through its centre
of gravity, h the distance of its centre of gravity below its point of
suspension, and k' the radius of gyration of the bob round the centre
of suspension, then k'^2 = h^2 + k^2. If l be the length of a simple
pendulum that oscillates in the same time, then lh = k'^2 = h^2 + k^2.
Now k can be calculated if we know the form of the bob, and l is the
length of the simple pendulum = 39.14 in.; hence h, the distance of
the centre of gravity of the bob below the point of suspension, can be
found.

In an ordinary pendulum, with a thin rod and a bob, this distance h is
not very different from the theoretical length, l = 39.14 in., of a
simple theoretical pendulum in which the rod has no weight and the bob
is only a single heavy point. For the effect of the weight of the rod
is to throw the centre of oscillation a little above the centre of
gravity of the bob, while the effect of the size of the bob is to
throw the centre of oscillation a little down. In ordinary practice it
is usual to make the pendulum so that the centre of gravity is about
39 in. below the upper free end of the suspension spring and leave the
exact length to be determined by trial.

Regulation.

Since T = [pi]sqrt(L/g), we have, by differentiating, dL/L = 2dT/T,
that is, any small percentage of increase in L will correspond to
double the percentage of increase in T. Therefore with a seconds
pendulum, in order to make a second's difference in a day, equivalent
to 1/86,400 of the pendulum's rate of vibration, since there are
86,400 seconds in 24 hours, we must have a difference of length
amounting to 2/86,400 = 1/43,200 of the length of the rod. This is
39.138/43,200 = .000906 in. Hence if under the pendulum bob be put a
nut working a screw of 32 threads to the inch and having its head
divided into 30 parts, a turn of this nut through one division will
alter the length of the pendulum by .0009 in. and change the rate of
the clock by about a second a day. To accelerate the clock the nut has
always to be turned to the right, or as you would drive in a corkscrew
and vice versa. But in astronomical and in large turret clocks, it is
desirable to avoid stopping or in any way disturbing the pendulum; and
for the finer adjustments other methods of regulation are adopted. The
best is that of fixing a collar, as shown in fig. 7 at C, about midway
down the rod, capable of having very small weights laid upon it, this
being the place where the addition of any small weight produces the
greatest effect, and where, it may be added, any moving of that weight
up or down on the rod produces the least effect. If M is the weight of
the pendulum and l its length (down to the centre of oscillation), and
m a small weight added at the distance n below the centre of
suspension or above the c.o. (since they are reciprocal), t the time
of vibration, and -dt the acceleration due to adding m; then

-dt m / n n^2 \
--- = --- ( --- - ---- ):
t 2M \ l l^2 /

from which it is evident that if n = l/2, then = dt/t = m/8M. But as
there are 86400 seconds in a day, -dT, the daily acceleration, = 86400
dt, or 10800 m/M, or if m is the 10800th of the weight of the pendulum
it will accelerate the clock a second a day, or 10 grains will do that
on a pendulum of 15 lb weight (7000 gr. being = 1 lb.), or an ounce on
a pendulum of 6 cwt. In like manner if n = l/3 from either top or
bottom, m must = M/7200 to accelerate the clock a second a day. The
higher up the collar the less is the risk of disturbing the pendulum
in putting on or taking off the regulating weights, but the bigger the
weight required to produce the effect. The weights should be made in a
series, and marked 1/4, 1/2, 1, 2, according to the number of seconds a
day by which they will accelerate; and the pendulum adjusted at first
to lose a little, perhaps a second a day, when there are no weights on
the collar, so that it may always have some weight on, which can be
diminished or increased from time to time with certainty, as the rate
may vary.

Compensation.

The length of pendulum rods is also affected by temperature and also,
if they are made of wood, by damp. Hence, to ensure good time-keeping
qualities in a clock, it is necessary (1) to make the rods of
materials that are as little affected by such influences as possible,
and (2) to provide means of compensation by which the effective length
of the rod is kept constant in spite of expansion or contraction in
the material of which it is composed. Fairly good pendulums for
ordinary use may be made out of very well dried wood, soaked in a thin
solution of shellac in spirits of wine, or in melted paraffin wax; but
wood shrinks in so uncertain a manner that such pendulums are not
admissible for clocks of high exactitude. Steel is an excellent
material for pendulum rods, for the metal is strong, is not stretched
by the weight of the bob, and does not suffer great changes in
molecular structure in the course of time. But a steel rod expands on
the average lineally by .0000064 of its length for each degree F. by
which its temperature rises; hence an expansion of .00009 in. on a
pendulum rod of 39.14 in., that is .000023 of its length, will be
caused by an increase of temperature of about 4 deg. F., and that is
sufficient to make the clock lose a second a day. Since the summer and
winter temperatures of a room may differ by as much as 50 deg. F., the
going of a clock may thus be affected by an error of 12 seconds a day.
With a pendulum rod of brass, which has a coefficient of expansion of
.00001, a clock might gain one-third of a minute daily in winter as
compared with its rate in summer. The coefficients of linear expansion
per degree F. of some other materials used in making pendulums are as
follows: white deal, .0000024; flint glass, .0000048; iron, .000007;
lead, .000016; zinc, .000016; and mercury, .000033. The solid or
cubical expansions of these bodies are three times the above
quantities respectively.

The first method of compensating a pendulum was invented in 1722 by
George Graham, who proposed to use a bob of mercury, taking advantage
of the high coefficient of expansion of that metal. As now employed,
the mercurial pendulum consists of a rod of steel terminating in a
stirrup of the same metal on which rests a glass vessel full of
mercury, having its centre of gravity about 39 in. below the point of
suspension of the pendulum. For each Fahrenheit degree of temperature
the centre of gravity of the bob is lowered by the expansion of the
rod about 1/4000 of an inch. The glass vessel and the mercury in it
have therefore to be so contrived, that their centre of gravity will
rise 1/4000 in. per degree F. The glass having a small coefficient of
expansion, the lateral expansion of the mercury will be checked by it,
and this will help to raise the column. For the linear coefficient of
expansion of glass is .0000048 per degree F., whence the sectional
area of a glass vessel increases by .0000096 per degree F., and
therefore the coefficient of vertical expansion of a column of mercury
whose volumetric expansion coefficient is .0001 per degree F. is
(.0001 - .0000096) = .0000904. Let x be the height of the vessel
necessary to compensate a steel rod upon the bottom of which it rests.
Then, the coefficient of expansion of steel being .0000066 per degree
F., we have

x
--- (.0000904 - .0000066) = .0000066 X 39.14, whence x = 6-1/4 in.
2

It must, however, be remembered that the glass jar has some weight and
that it does not rise by anything like the amount of the mercury. This
tends to keep the centre of gravity down. So that the height of
mercury of 6-1/4 in. will not be sufficient to effect the compensation,
and about 6-3/4 to 7 in. will be required. Some authors specify 7 in.;
this is when the diameter of the jar is small. A certain amount of
negative compensation must also be deducted to allow for the changes
of temperature in the air, as will presently be seen; this amounts in
the case of mercury to about 1/5 in.

In consequence of the complication of all these calculations it is
usual to allow about 6-3/4 to 7 in. of mercury in the glass vessel and to
adjust the exact amount of mercury by trial.

Another very good form of mercurial pendulum was proposed by E. J.
Dent; it consists of a cast-iron jar into the top of which the steel
pendulum rod is screwed, having its end plunged into the mercury
contained in the jar. By this means the mercury, jar and rod rapidly
acquire the same temperature. This pendulum is less likely to break
than the form just described. The depth of mercury required in an iron
jar is stated by Lord Grimthorpe to be 8-1/2 to 9 in. The reason why it
is greater than it is when a glass jar is employed is that iron has a
larger coefficient of expansion than glass, and that it is also
heavier. In all cases, however, of mercury pendulums experiment seems
to be the only ultimate test of the quantity of mercury required, for
the results are so complicated by the behaviour of the oil and the
barometric errors that at its best the regulation of a clock can only
be ultimately a matter of scientifically guided compromise. A small
amount of compensation of a purely experimental character is also
allowed to compensate the changes which temperature effects on the
suspension spring. This is sometimes made as much as 1/6 of the length
correction.

As an alternative to the mercurial pendulum other systems have been
employed. The "gridiron" pendulum consists of a group of alternate
rods of steel and brass, so arranged that the expansion of the brass
acts upwards and counteracts that of the steel downwards. It was
invented in 1726 by John Harrison. Assuming that 9 rods are used--5 of
steel and 4 of brass--their lengths may be as follows from pin to
pin:--Centre steel rod 31.5 in.; 2 steel rods next the centre 24.5
in.; 2 steel rods farthest from centre 29.5 in.; from the lower end of
outside steel rods to centre of bob 3 in.; total 89.5 in. Of the 4
brass rods the 2 outside ones are 26.87 in.; and the two inside ones
22.25 in.; total 49.12 in. Thus the expansion of 88-1/2 in. of steel is
counteracted by the expansion of 49 1/8 in. of brass. Everything
depends, however, on the expansion coefficient of the steel and brass
employed, the requirement in every case being that of total lengths of
the brass and iron should be in proportion to the linear coefficients
of expansion of those metals. The above figures are for a very soft
brass and steel. Thos. Reid, with more ordinary steel and brass,
prescribed a ratio of 112 to 71, Lord Grimthorpe a ratio of 100 to 61.
It is absolutely necessary to put the actual rods to be used for
making the pendulum in a hot water bath, and measure their expansions
with a microscope.

John Smeaton, taking advantage of a far greater expansion coefficient
of zinc as compared with brass, proposed to use a steel rod with a
collar at the bottom, on which rested a hard drawn zinc rod. From this
rod hung a steel tube to which the bob was attached. The total length
of the steel rod and of the steel tube down to the centre of the bob
was made to the total length of the zinc tube, in the ratio of 5 to 2
(being the ratio of the expansions of zinc and steel); for a 39.14 in.
pendulum we should therefore want a zinc tube equal in length to 2/3
(39.14) = 26-1/4 in. In practice the zinc tube is made about 27 in. long,
and then gradually cut down by trial. In fact the weight of a heavy
pendulum squeezes the zinc, and it is impossible by mere theory to
determine what will be its behaviour. The zinc tube must be of rolled
zinc, hard drawn through a die, and must not be cast. Ventilating
holes must be made in suitable places in the steel tube and the collar
on which it rests, to ensure that changes of temperature are rapidly
communicated throughout the system.

A pendulum with a rod of dry varnished deal is tolerably compensated
by a bob of lead or of zinc 10-1/2 to 13 in. in height, resting on a nut
at the bottom of the rod.

Invar.

The old methods of pendulum compensation for heat may now be
considered as superseded by the invention of "invar," a combination of
nickel and steel, due to Charles E. Guillaume, of the International
Office of Weights and Measures at Sevres near Paris. This alloy has a
linear coefficient of expansion on the average of .000001 per degree
centigrade, that is to say, only about 1/11 that of ordinary steel.
Hence it can be easily compensated by means of brass, lead or any
other suitable metal. Brass is usually employed. In the invar pendulum
introduced into Great Britain by Mr Agar Baugh a departure is made
from the previous practice of merely calculating the length of the
compensator, fastening it to the lower part of the pendulum, and
attaching it to the centre of the bob. In the case of these pendulums,
accurate computations are made of the moments of inertia of every
separate individual part. Thus, for instance, since an addition of
volume due to the effect of heat to the upper part of the bob has a
different effect upon the moment of inertia from that of an equal
quantity added to the lower part of the bob, the bob is suspended not
from its centre, but from a point about 1/10 in. below it, the
distance varying according to the shape of the bob, so that the heat
expansion of the bob may cause its centre of gravity to rise and
compensate the effect of its increased moment of inertia. Again the
suspension spring is measured for isochronism, and an alloy of steel
prepared for it which does not alter its elasticity with change of
temperature. Moreover, since rods of invar steel subjected to strain
do not acquire their final coefficients of expansion and elasticity
for some time, the invar is artificially "aged" by exposure to strain
and heat.

These considerations serve as a guide in arranging for the
compensation of the expansion of the rod and bob due to change of
temperature. But they are not the only ones required; we have also to
deal with changes due to the density of the air in which the pendulum
is moving. A body suspended in a fluid loses in weight by an amount
equal to the weight of the fluid displaced, whence it follows that a
pendulum suspended in air has not the weight which ought truly to
correspond to its mass. M remains constant while M_g_ is less than in
a vacuum. If the density of the air remained constant, this loss of
weight, being constant, could be allowed for and would make no
difference to the time-keeping. The period of swing would only be a
little increased over what it would be _in vacuo_. But the weight of a
given volume of air varies both with the barometric pressure and also
with temperature. If the bob be of type metal it weighs less in air
than in a vacuum by about .000103 part, and for each 1 deg. F. rise in
temperature (the barometer remaining constant and therefore the
pressure remaining the same), the variation of density causes the bob
to gain .00000024 of its weight. This, of course, makes the pendulum
go quicker. Since the time of vibration varies as the inverse square
root of _g_, it follows that a small increment of weight, the mass
remaining constant, produces a diminution of one half that increment
in time of swing. Hence, then, a rise of temperature of 1 deg. F. will
produce a diminution in the time of swing of .00000012th part or .0104
second in a day. But in making this calculation it has been assumed
that the mass moved remains unaltered by the temperature. This is not
so. A pendulum when swinging sets in motion a volume of air dependent
on the size of the bob, but in a 10 lb bob nearly equal to its own
volume. Hence while the rise of 1 deg. of temperature increases the weight
by .00000012th part, it also decreases the mass by about the same
proportion, and therefore the increase of period due to a rise of
temperature of 1 deg. F. will, instead of being .0104 second a day, be
about .02 second. This must be compensated negatively by lengthening
the pendulum by about .02/1000 in. for each degree of rise of
temperature, which will require a piece of brass about 2 in. long. It
follows, therefore, that with an invar rod having a linear expansion
coefficient of .0000002 per degree F., which requires a piece of brass
about .8 in. long to compensate it, the compensation which is to
regulate both the expansion of the rod and also that of the air must
be .8 in. - 2 in., or -1.2 in.; so that the bob must be hung downwards
from a piece of brass nearly 1-1/5 in. in length. If the coefficient
of expansion of the invar were .00000053 per degree F., then the two
corrections, one for the expansion of the rod and the other for the
expansion of the air, would just neutralize one another, and the
pendulum rod would require no compensator at all. There are a number
of other refinements which might be added, but which are too long for
insertion here. By taking in all the sources of error of higher
orders, it has been possible to calculate a pendulum so accurately
that, when the clock is loaded with the weight sufficient to give the
pendulum the arc of swing for which it is designed, a rate of error
has been produced of only half a minute in a year. These refinements,
however, are only required for clocks of precision; for ordinary
clocks an invar pendulum with a lead bob and brass compensator is
quite sufficient.

Invar pendulum rods are often made of steel with coefficients of
expansion of about .0000012 linear per 1 deg. C.; such a bob as this would
require about 6.7 cm. of brass to compensate it, and, deducting 5 cm.
of brass for the air compensation, this leaves about 1.7 cm. of
positive compensation for the pendulum. But as has been said, the
exact deduction depends on the shape and size of the bob, and the
metal of which it is made. The diameters of the rods are 8 mm. for a
15 lb bob, 5 mm. for a 4 lb bob, and 12 to 15 mm. for a 60 lb bob. The
bob is either a single cylinder or two cylinders with the rod between
them. Lenticular and spherical bobs are not used. The great object is
to allow the air ready access to all parts of the rod and compensator,
so that they are all heated or cooled simultaneously. The bobs are
usually made of a compound of lead, antimony, and tin, which forms a
hard metal, free from bubbles and with a specific gravity of about 10.
The usual weight of the bobs of the best pendulums for an ordinary
astronomical clock is about 15 lb. A greater weight than this is found
liable to make the support of the pendulum rock and to put an undue
strain on the parts, without any corresponding advantage. The rods
used are all artificially aged, and have their heat expansion
measured. No adjusting screw at the bottom is provided, the regulation
being done by the addition of weights half way up the rod. An
adjusting screw at the bottom has the disadvantage that it is
impossible to know on which of the threads the rod is really resting;
hence extra compensation may be introduced when not required. It is
considered better that the supports of the bob should be rigid and
invariable.

Barometrical error.

The effect of changes in the pressure of the air as shown by a
barometer is too important to be omitted in the design of a good
clock. But we do not propose to give more than a mere indication of
the principles which govern compensation for this effect, since the
full discussion of the problem would be too protracted. We have seen
that the action of the air in affecting the time of oscillation of a
pendulum depends chiefly on the fact that its buoyancy makes the
pendulum lighter, so that while the mass of the bob which has to be
moved remains the same or nearly the same, the acceleration of gravity
on it has less effect. A volume of air at ordinary temperature and
pressure has, as has been said, .000103 the weight of an equal volume
of type metal, whence it follows that the acceleration of gravity on a
type metal bob in air is .999897 of the acceleration of gravity on the
bob _in vacuo_. If, therefore, we diminish the value of g in the
formula T = [pi]sqrt(L/g) by .000103, we shall have the difference of
time of vibration of a type metal bob in air, as compared with its
time _in vacuo_, and this, by virtue of the principle used when
discussing the increase of time of oscillation due to increased
pendulum lengths, is 1/2(.000103) second in one second, or about 4-1/2
seconds in a day of 86,400 seconds. It follows that a barometric
pressure of 30 in. causes a loss of 4-1/2 seconds in the day, equivalent
to .15 second per day for each inch of difference of the barometer.
But, as has already been explained, the effect of the mass of the air
transported with the pendulum must also be taken into account and
therefore the above figures must be doubled or nearly doubled. A
difference of 30 in. of barometric pressure would thus make a
difference of 9 seconds per day in the rate of the pendulum, and the
clock would lose about 1/3 of a second a day for each inch of rise of
the barometer, the result being of the same magnitude as would be
produced by a fall of temperature of 15 deg. F. in the air. Either of
these effects would require a shortening of the pendulum of 1/3000 in.
This estimate is not far from the truth, for observations taken at
various European observatories on various clocks, and collected by
Jakob Hilfiker, give a mean of .15 second of retardation per day per
centimetre of barometric pressure, or .37 second per day for each inch
rise of the barometer.

In order to counteract variations in going which must thus obviously
be produced by variations of barometrical pressure, attempts have been
made purposely to disturb the isochronism of the pendulum, by making
the arcs of vibration abnormally large. Again, the bob has been fitted
with a piece of iron, which is subjected to the attraction of a piece
of magnetized steel floating on the mercury in the open end of a
barometer tube, so that when the barometer falls the attraction is
increased and the pendulum retarded. Again, mercury barometers have
been attached to pendulums. A simple method is to fix an aneroid
barometer with about seven compartments on the pendulum about 5 to 6
in. below the suspension spring, and to attach to the top of it a
suitable weight which is lowered as the barometric pressure increases.
One of the best methods of neutralizing the effects of variations of
barometric pressure is to enclose the whole clock in an air-tight
case, which may either be a large glass cylinder or a square case with
a stout plate-glass front. This renders it independent of outside
variations, whether of temperature or pressure, and keeps the density
of the air inside the case uniform. If the case could be completely,
or almost completely, exhausted of air, and kept so exhausted, of
course the pendulum would experience the minimum of resistance and
would have to be lengthened a little. But in practice it is impossible
to secure the maintenance of a good vacuum without sealing up the case
in such a way as to render repairs very difficult, and this plan is
therefore rarely resorted to. What is usually done is to put the clock
in a metal case covered with a thick sheet of plate glass bedded in
india-rubber strips, and held down by an iron flanged lid or frame
firmly fixed by means of small bolts. An air-pump is attached to the
case, a turn-off tap being inserted, and by a few strokes the pressure
of the air inside the case can be lowered to (say) 29 in., or a little
below the usual barometric height at the place where the clock is. The
difference of pressure being small, the tendency of air from outside
to leak in is also small, and if the workmanship is good the inside
pressure will remain unaltered for many days. In any case the
difference produced by leakage will be small, and will not greatly
affect the going of the clock. With care, and a daily or weekly touch
of the pump, the pressure inside can be kept practically constant, and
hence the atmospheric error will be eliminated. The cover has also
incidentally the effect of keeping damp and fumes from the clock and
thus preserving it from rust, especially if a vessel with quicklime or
some hygroscopic material be put in the case.

Cases have considerable effect on the air, which moves with a pendulum
and is flung off from it at each vibration; the going rate of a
chronometer can be altered by removing the case. It is therefore
desirable that cases enclosing pendulums should be roomy. Many people
prefer to omit the air-tight case, and to keep a record of barometric,
thermometric and hygrometric changes, applying corrections based on
these to the times shown by the clock.

Suspension of pendulums.

It was formerly usual to suspend pendulums by means of a single spring
about 1/2 in. wide riveted with chops of metal. The upper chop had a pin
driven through it, which rested in grooves so as to allow the pendulum
to hang vertically. The best modern pendulums are now made with two
parallel springs put a little less than an inch apart. The edges of
the chops where the springs enter are slightly rounded so as to avoid
too sharp bending of the springs. Suspension of pendulums on knife
edges was tried by B. L. Vulliamy and others, but did not prove a
success.

It was once thought that lenticular pendulum bobs resisted the air
less than those of other shapes, but it was forgotten that their large
surface offered more "skin friction." They are now no longer used, nor
are spheres on account of difficulty of construction. A cylinder is
the best form of bob; it is sometimes rounded at the top and bottom.

_Escapements._--The term escapement is applied to any arrangement by which, as the wheels rotate, periodic impulses are given to the pendulum, while at the same time the motion of the wheels is arrested until the vibration of the pendulum has been completed. It thus serves as a mechanism for both counting and impelling. Since the vibrations of a pendulum through small arcs are performed in times independent of the length of the arc, it follows that if a pendulum hanging at rest receive an impulse it will swing out and in again, and the time of its excursion outwards and of its return will remain the same whatever (within limits) be the arc of the swing, and whatever be the impulse given to it. If the impulse is big, it starts with a high velocity, but makes a larger excursion outwards, and the distance it has to travel counteracts its increase of speed, so that its time remains the same. Hence a pendulum, if free to swing outwards and in again, without impediment, will adapt the length of its swing to the impulse it has received, and any interference with it, as by the locking or unlocking of the escapement, will be far less deleterious to its isochronism when such interference occurs at the middle of its path rather than at the ends. It follows that the best escapement will be one which gives an impulse to the pendulum for a short period at the lowest point of its path, and then leaves it quite free to move as it chooses until the time comes for the next impulse.

But a pendulum is not quite truly isochronous, and has its time slightly affected by an increase of its arc; it is therefore desirable that the impulses given to it shall always be equal. If the escapement forms the termination of a clock-train impelled by a weight, the driving force of the escapement is apt to vary according to the friction of the wheels, while every change in temperature causes a difference in the thickness of the oil. It is therefore desirable, if possible, to secure uniformity of impulse--say, by causing the train of wheels to lift up a certain specified weight, and let it drop on the pendulum at regular intervals, or by some equivalent method.

The two requirements above stated have given rise respectively to what are known as detached escapements, and remontoires, which will be described presently. In the first place, however, it is desirable to describe the principal forms of escapement in ordinary use.

Balance escapement.

The balance escapement, which has been already mentioned, was in use
before the days of pendulums. It was to a balance escapement that
Huygens applied the pendulum, by removing the weight from one arm and
increasing the length of the other arm.

Anchor escapement.

Very shortly afterwards R. Hooke invented the anchor or recoil
escapement. This is represented in fig. 8, where a tooth of the
escape-wheel is just escaping from the right pallet, and another tooth
at the same time falls upon the left-hand pallet at some distance from
its point. As the pendulum moves on in the same direction, the tooth
slides farther up the pallet, thus producing a recoil, as in the
crown-wheel escapement. The acting faces of the pallets should be
convex. For when they are flat, and of course still more when they are
concave, the points of the teeth always wear a hole in the pallets at
the extremity of their usual swing, and the motion is obviously easier
and therefore better when the pallets are made convex; in fact, they
then approach more nearly to the "dead" escapement, which will be
described presently. The effect of some escapements is not only to
counteract the circular error, or the natural increase of the time of
a pendulum as the arc increases, but to over-balance it by an error of
the contrary kind. The recoil escapement does so; for it is almost
invariably found that whatever may be the shape of these pallets, the
clock loses as the arc of the pendulum falls off, and vice versa. It
is unfortunately impossible so to arrange the pallets that the
circular error may be thus exactly neutralized, because the escapement
error depends, in a manner reducible to no law, upon variations in
friction of the pallets themselves and of the clock train, which
produce different effects; and the result is that it is impossible to
obtain very accurate time-keeping from any clock of this construction.
The point in which the anchor escapement was superior to all that had
gone before, was that it would work well with a small arc of swing of
the pendulum. The balance escapement, even when adapted to a pendulum,
necessitated a swing of some 20 deg., and hence the circular error, that
is to say, the deviation of the path from a true cycloid, was
considerable. But with an anchor escapement the pendulum swing need be
only 3 deg. or 4 deg. On the other hand, it violates the conditions above
laid down for a perfect escapement, inasmuch as the pendulum is never
free, but at the end of its swing is still operated on by the
escapement, which it causes to recoil.

Dead escapements.

To get rid of this defect the dead escapement, or, as the French call
it, _l'echappement a repos_, was invented by G. Graham. It is
represented in fig. 9. It will be observed that the teeth of the
scape-wheel have their points set the opposite way to those of the
recoil escapement. The tooth B is here represented in the act of
dropping on to the right-hand pallet as the tooth A escapes from the
left pallet. But instead of the pallet having a continuous face as in
the recoil escapement, it is divided into two, of which BE on the
right pallet, and FA on the left, are called the impulse faces, and
BD, FG, the dead faces. The dead faces are portions of circles (not
necessarily of the same circle), having the axis of the pallets C for
their centre; and the consequence evidently is, that as the pendulum
goes on, carrying the pallet still nearer to the wheel than the
position in which a tooth falls on to the corner A or B of the impulse
and the dead faces, the tooth still rests on the dead faces without
any recoil, until the pendulum returns and lets the tooth slide down
the impulse face, giving the impulse to the pendulum as it goes. In
order to diminish the friction and the necessity for using oil as far
as possible, the best clocks are made with jewels (sapphires are the
best for the purpose) let into the pallets.

The pallets are generally made to embrace about one-third of the
circumference of the wheel, and it is not at all desirable that they
should embrace more; for the longer they are, the longer is the run of
the teeth upon them, and the greater the friction. In some clocks the
seconds hand moves very slowly and rests a very short time; this shows
that the impulse is long in proportion to the arc of swing. In others
the contrary is the case. A not uncommon proportion is that out of a
total arc of swing of 3 deg., 2 deg., or about one degree on each side
of the vertical, are occupied in receiving the impulse. In other
words, the points F and A should subtend an angle of 2 deg. at the
centre C. It is not to be forgotten that the scape-wheel tooth does
not overtake the face of the pallet immediately, on account of the
moment of inertia of the wheel. The wheels of astronomical clocks, and
indeed of all English house clocks, are generally made too heavy,
especially the scape-wheel, which, by increasing the moment of
inertia, causes a part of the work to be lost in giving blows, instead
of being all used up in gentle pushes.

A very useful form of the dead escapement, which is adopted in many of
the best turret clocks, is called the "pin-wheel escapement." Fig. 10
will sufficiently explain its action and construction. Its advantages
are--that it does not require so much accuracy as the other; if a pin
gets broken it is easily replaced, whereas in the other the wheel is
ruined if the point of a tooth is injured; a wheel of given size will
work with more pins than teeth, and therefore a train of less velocity
will do, and that sometimes amounts to a saving of one wheel in the
train, and a good deal of friction; and the blow on both pallets being
downwards, instead of one up and the other down, the action is more
steady; all which things are of more consequence in the heavy and
rough work of a turret clock than in an astronomical one. It has been
found expedient to make the dead faces not quite dead, but with a very
slight recoil, which rather tends to check the variations of arc, and
also the general disposition to lose time if the arc is increased;
when so made the escapement is generally called "half-dead."

In the dead escapement, during each excursion of the pendulum the
repose surface of the pallets rubs against the points of the teeth of
the scape-wheel. Thus the pendulum is subject to a constant
retardation by friction. Curiously enough, this friction, which at
first sight might appear a defect, is an advantage, and to a large
extent accounts for the excellence of the escapement. For if the
driving force of the clock is increased so that the impulse on the
pallets is greater, the velocity of the pendulum is increased. But
this very increase of the driving force causes a greater pressure of
the teeth of the scape-wheel on the rest-faces of the pallets, and
hence counteracts the increased drive of the pendulum by an increased
frictional retardation. If the clock weight be enormously increased,
the frictional retardation becomes increased relatively in a greater
proportion than the drive, so that as the weight of the clock is
increased the pendulum's time of vibration is first diminished, until
at last a neutral point is reached and finally the increased loading
of the clock weight begins to make the time of vibration increase
again. It is the neutral point which it is desirable to arrange for,
and only trial and experience can so fit the shape and size of the
pallets, scape-wheel and clock weight to one another, as to secure
that a moderate variation of the driving power neither accelerates nor
retards the motion of the pendulum, while at the same time such an arc
of vibration is secured as shall be least subject to barometric error,
and not have too great a circular error. The celebrated clockmaker B.
L. Vulliamy (1780-1854) greatly improved Graham's escapement by
careful experiment, and other makers introduced further improvements
into the shape of the scape-wheel and pallets, so that the best form
of the deadbeat escapement is now fairly well determined and is given
in books upon horology. For small clocks a little slope is given to
the rest-faces so as to diminish the friction retardation. This is
known as the half-dead escapement. The pin-wheel escapement, if
properly constructed, is also "dead," that is to say, the outward
swing of the pendulum is unfettered except by the slight friction of
the teeth against the dead faces of the pallets.

In order to diminish the effect of the impact of the scape-wheel on
the pallets, and of the crutch on the pendulum rod, the plan has been
tried of making the crutch into an elastic spring. In theory this of
course would not destroy the isochronism of the pendulum, for it would
only be to apply upon the pendulum a force at right angles to the rod,
and varying as the displacement. Hence any acceleration given by such
a spring would, like the action of gravity, be harmonic, and it is an
analytical principle that harmonic motions superposed on one another
still remain harmonic. Hence, then, the action of a spring superadded
upon the action of gravity on a pendulum still leaves the motion
harmonic. But changes of temperature would affect the spring
considerably. In the case of such a spring the repose faces of
Graham's escapement might be minimized and the escapement checked each
side by a stop, so as to prevent the pallets from rubbing on the
points of the scape-wheel. Graham's escapement can, if well made, be
arranged so as not to vary more than an average of 1/30 of a second
from its mean daily rate, and this is so good a result that many
people doubt whether further effort in the direction of inventing new
escapements will result in any better form. Two adaptations of
Graham's escapement have been made, one by Clemens Riefler of
Nesselwang, and the other by L. Strasser of Glashutte, Saxony, which
give good results in practice. Riefler's scheme is to mount the upper
block, into which the suspension spring is fastened, upon knife edges,
and rock it to and fro by the action of a modified Graham's
escapement, thus giving impulses to the pendulum. Fig. 11 shows the
arrangement. PP are the agates upon which the knife edges CC rest. A
is the anchor, RH the scape-wheels, and S the pallets.

Strasser's clock is arranged on the same idea as that of Riefler, only
that the rocking motion is given, not to the springs that carry the
pendulum, but to a second pair of springs placed outside of them and
parallel to them. The weight of the pendulum is therefore carried by
an upper stationary block, but above that a second block is subjected
to the rocking motion of the anchor. The general design is shown in
fig. 12. The pallets are each formed of two stones, so contrived as to
minimize the banging of the teeth of the scape-wheel. Both Riefler's
and Strasser's clocks aim at haying a virtually free pendulum; in
fact, they are in reality adaptations of the principle of the
spring-clutch to Graham's escapement. The weak point in both is the
tampering with the suspension.

Detached escapement.

The dead escapement is not, however, truly free. In order to make a
free escapement it would be necessary to provide that as soon as the
pendulum approached its centre position, some pin or projecting point
upon it should free the escapement wheel, a tooth of which should thus
be enabled to leap upon the back of the pendulum, give it a short
push, and then be locked until the pendulum had returned and again
swung forward. An arrangement of this kind is shown in fig. 13. Let A
be a block of metal fixed on the lower end of a pendulum rod. On the
block let a small pall B be fastened, free to move round a centre C
and resting against a stop D. Let E be a 4-leaved scape-wheel, the
teeth of which as they come round rest against the bent pall GFL at G.
The pall is prevented from flying too far back by a pin H, and kept up
to position by a very delicate spring K. As soon as the pendulum rod,
moving from left to right, has arrived at the position shown in the
figure, the pall B will engage the arm FL, force it forwards, and by
raising G will liberate the scape-wheel, a tooth of which, M, will
thus close upon the heel N of the block A, and urge it forward. As
soon, however, as N has arrived at G the tooth M will slip off the
block A and rest on the pall G, and the impulse will cease. The
pendulum is now perfectly free or "detached," and can swing on
unimpeded as far as it chooses. On its return from right to left, the
pall B slips over the pall L without disturbing it, and the pendulum
is still free to make an excursion towards the left. On its return
journey from left to right the process is again repeated. Such an
escapement operates once every 2 seconds. One made on a somewhat
similar plan was applied to a clock by Robert-Houdin, about 1830, and
afterwards by Mr Haswell, and another by Sir George Airy. But the
principle was already an old one, as may be seen from fig. 14, which
was the work of an anonymous maker in the 18th century. A
consideration of this escapement will show that it is only the
application of the detached chronometer escapement to a clock.

Even detached escapements, however, are not perfect. In order that an
escapement should be perfect, the impulse given to the pendulum should
be always exactly the same. It may be asked why, if the time of
oscillation of the pendulum be independent of the amplitude of the arc
of vibration, and hence of the impulse, it is necessary that the
impulse should be uniform. The answer is that the arc of vibration not
being a true cycloid, as it should be if true isochronism is to be
secured, but being the arc of a circle, any change of amplitude of
vibration produces a change of time in the swing given by the formula
(3/2)(a^2 - b^2) = loss in seconds per day, where a and b are the
semi-arcs of vibration estimated in degrees. Thus 10' increase of arc
in a swing of 4 deg., that is to say, .1 in. increase of arc in a total
arc of 2-1/2 in., produces an error of about a second a day. Now cold
weather, by making the oil thick and thus clogging the wheels, will
easily produce such a change of arc; dust will also make a change even
though the clock weight, acted on by gravity, still exerts a uniform
pull. Besides, if the clock has work to do of a varying amount--as
when the hands of a turret clock are acted on by a heavy wind pressure
tending sometimes to retard them, sometimes to drive them on--then it
is clear that the impulses given by the scape-wheel to the pendulum
may be very unequal, and that the arc of vibration of the pendulum may
thus be seriously affected and its isochronism disturbed.

Remontoire.