may be either cylindrical, of about a quarter of an inch in diameter,
or a flat bar, three eighths of an inch wide, and one eighth of an inch
thick: its length from S to F, that is, from the bottom of the spring
to the bottom of the rod at F, should be 34 inches. The lower part of
this rod, which passes through the top of the stirrup, and about half
an inch above and below the same, must be formed into a _coarse_ and
_deep_ screw, about two tenths of an inch in diameter, and having about
thirty turns in an inch. A steel nut with a milled head must be placed
at the end of the rod, in order to support the stirrup; and a similar
nut must also be placed on the rod _above_ the head of the stirrup,
in order to screw firmly down on the same, and thus secure it in its
position, after it has been adjusted _nearly_ to the required rate.
These nuts are represented at B and C. A small slit is cut in the rod,
where it passes through the head of the stirrup, through which a steel
pin E is screwed, in order to keep the stirrup from turning round on
the rod. The stirrup itself is also made of steel, and the side pieces
should be of the same form as the rod, in order that they may readily
acquire the same temperature. The top of the stirrup consists of a flat
piece of steel, shaped as in the drawing, somewhat more than three
eighths of an inch thick. Through the middle of the top (which at this
part is about one inch deep) a hole must be drilled sufficiently large
to enable the screw of the rod to pass _freely_, but without _shaking_.
The inside height of the stirrup from A to D may be 8-1/2 inches, and
the inside width between the bars about three inches. The bottom piece
should be about three eighths of an inch thick, and hollowed out nearly
a quarter of an inch deep, so as to admit the glass cylinder freely.
This glass cylinder should have a brass or iron cover G, which should
fit the mouth of it freely, with a shoulder projecting on each side, by
means of which it should be screwed to the side bars of the stirrup,
and thus be secured always in the same position. This cap should not
_press_ on the glass cylinder, so as to prevent its expansion. The
measures above given may require a slight modification, according to
the weight of the mercury employed, and the magnitude of the cylinder:
the final adjustment, however, may be safely left to the artist. Some
persons have recommended that a circular piece of thick plate glass
should float on the mercury, in order to preserve its surface uniformly
level.[7] The part at the bottom marked H is a piece of brass fastened
with screws to the front of the bottom of the stirrup, through a small
hole, in which a steel wire or common needle is passed, in order to
indicate (on a scale affixed to the case of the clock) the arc of
vibration. This wire should merely rest in the hole, whereby it may
be easily removed when it is required to detach the pendulum from the
clock, in order that the stirrup might then stand securely on its base.
One of the screw holes should be rather larger than the body of the
screw, in order to admit of a small adjustment, in case the steel wire
should not stand exactly perpendicular to the axis of motion. The scale
should be divided into _degrees_, and not _inches_, observing that with
a radius of 44 inches (the estimated distance from the bend of the
spring to the end of the steel wire) the length of each degree on the
scale must be 0·768 inch.”
[7] The variation produced in the height of the column of mercury
(supposed to be 6-1/2 inches high) by an alteration of ± 16° in the
temperature will be only ± 1/100 of an inch, or in other words, 1/100
of an inch will be the total variation from its _mean_ state, by an
alteration of 32° in the temperature. It is therefore probable that, in
most cases of moderate alteration in the temperature, the _centre_ only
of the column of mercury is subject to elevation and depression, whilst
the exterior parts remain attached to the sides of the glass vessel. It
was with a view to obviate this inconvenience that Henry Browne, Esq.
of Portland Place (I believe) first suggested the piece of floating
glass.
In order to determine the length of the mercurial column necessary
to form the compensation for this pendulum, we must proceed in the
following manner:--
Let us suppose the length of the steel rod and stirrup together to be
42 inches. The absolute expansion of the mercury is ·00010010; but it
is not the absolute expansion, but the vertical expansion in a glass
cylinder, which is required, and this will evidently be influenced by
the expansion of the base of this cylinder. It is easily demonstrable
that, if we multiply the linear expansion of any substance (always
supposed to be a very small part of its length) by 3, we may in all
cases take the result for the cubical or absolute expansion of such
substance. In like manner, if we multiply the linear expansion by 2, we
shall have the superficial expansion.
If we want the apparent expansion of mercury, the absolute or cubical
expansion of the glass vessel must be deducted from the absolute
expansion of the mercury, which will leave its excess or apparent
expansion. In like manner, deducting the superficial expansion of glass
from the absolute expansion of mercury, we shall have its relative
vertical expansion. Now, taking the rate of expansion of glass to be
·00000479, and multiplying it by 2, the relative vertical expansion
of the mercury in the glass cylinder will be ·00010010 - ·00000958 =
·00009052.
The expansion of a steel rod, according to our table, is ·0000063596;
which, divided by ·00009052, gives ·0703 for the length of a column of
mercury, the expansion of which is equal to that of a steel rod whose
length is unity.
We have now to multiply 42 inches by ·0703, which gives 2·95 inches;
and this, deducted from 42, leaves 39·1 inches; so that the length
of rod we have chosen is sufficiently near the truth. Now, double
2·95 inches, and add one tenth of its product, and we shall have 6·49
inches for the length of the mercurial column forming the requisite
compensation. Mr. Baily’s more accurate calculation gives 6·31 inches.
A mercurial compensation pendulum may be formed, having a cylinder
of steel or iron, with its top constructed in the same manner as the
top of the stirrup, so as to receive the screw of the rod. To find
the length of the mercurial column necessary in a pendulum of this
description (that is, with a cylinder made of steel), we must double
the linear expansion of steel, and take it from the absolute expansion
of mercury to obtain the relative vertical expansion of the mercury.
This will be ·00010010 - ·00001272 = ·00008738; and, proceeding as
before, we have ·0000063596/·00008738 = ·07279.
Let the length of the steel rod be, as before, 42 inches. Multiplying
this by ·07279, we have 3·057, which being doubled, and one tenth
of the product added, we obtain 6·72 inches for the length of the
compensating mercurial column; which Mr. Baily states to be 6·59.
A mercurial compensation pendulum having a rod of glass has been
employed by the writer of this article, who has had reason to think
well of its performance. Its cheapness and simplicity much recommend
it. It is merely a cylinder of glass of about 7 inches in depth, and
2-1/2 inches diameter, terminated by a long neck, which forms the
rod of the pendulum, the whole blown in one piece. A cap of brass is
clamped by means of screws to the top of the rod, and to this the
pendulum spring is pinned.
We have unquestionable authority for saying, that the mercurial
pendulum of the usual construction, that is, with a steel rod and glass
cylinder, is not affected by a change of temperature simultaneously in
all its parts. Now, the pendulum of which we are treating being formed
throughout of the same material in a single piece, and in every part
of the same thickness, it is presumed it cannot expand in a linear
direction, until the temperature has penetrated to the whole interior
surface of the glass, when it is rapidly diffused through the mass of
mercury. M. Biot mentions that a pendulum of this kind was formerly
used in France, and expresses his surprise that it was no longer
employed, as he had heard it very highly spoken of. The writer of this
article has also used a pendulum with a glass rod, which differs from
that we have just mentioned, in having the lower end of the rod firmly
fixed in a socket attached to the centre of a circular iron plate, on
the circumference of which a screw is cut, which fits into a collar of
iron, supporting the cylinder (to which it is cemented) by means of a
circular lip.
This arrangement, though perhaps less perfect than that we have just
described, the pendulum not being in one piece, has the advantage of
allowing a circular plate of glass to be placed upon the surface of
the mercury, as practised by Mr. Browne. To determine the length of
a column of mercury for a glass pendulum, let us suppose the glass,
including the cylinder, to be 41 inches in length. Multiplying this
by ·0529, the number taken from Table II. for a glass rod and mercury
in a glass cylinder, we have 2·17 inches for the uncorrected length
of mercury, which compensates 41 inches of glass. Suppose the steel
spring to be one inch and a half long: multiplying this by ·0703, the
appropriate decimal taken from Table II., we have 0·1, the length of
mercury due to the steel, making with the former 2·27 inches, which,
being doubled, and the product increased by its one-tenth part, we
obtain five inches for the length of the required column of mercury.
_Compensation Pendulum of Wood and Lead, on the Principle of the
Mercurial Pendulum._
If by any contrivance wood could be rendered impervious to moisture,
it would afford one of the most convenient substances known for a
compensation pendulum. It does not appear that sufficient experiments
have been made upon this subject to decide the question. Mr. Browne
of Portland Place, who has devoted much of his time and attention to
the most delicate enquiries of this kind, has, we believe, found that
if a teak rod is well gilded, it will not afterwards be affected by
moisture. At all events, it makes a far superior pendulum, when thus
prepared, to what it does when such preparation is omitted.
Mr. Baily, in the paper we have before alluded to, proposes an
economical pendulum to be constructed by means of a leaden cylinder and
a deal rod. He prefers lead to zinc, on account of its inferior price,
and the ease with which it may be formed into the required shape; and
as there is no considerable difference in their rates of expansion, it
is equally applicable to the purpose.
Let the length of the deal rod be taken at 46 inches. Then, to find the
length of the cylinder of lead to compensate this, we have, in Table
II., ·1427 for such a pendulum; which, being multiplied by 46, the
product doubled, and one tenth of the result added to it, gives 14·44
inches for the length of the leaden cylinder. Mr. Baily’s compensation
gives 14·3 inches.
[Illustration: _Captn. Kater, del._ _H. Adlard, sc._
_London, Pubd. by Longman & Co._]
The rod is recommended to be made of about three eighths of an inch
in diameter: the leaden cylinder is to be cast with a hole through
its centre, which will admit with perfect freedom the cylindrical end
of the rod. The cylinder is supported upon a nut, which screws on the
end of the rod in the usual manner. This pendulum is represented at
_fig. 224._
Mr. Baily proposes that the pendulum should be adjusted nearly to the
given rate by means of the screw at the bottom, and that the final
adjustment be made by means of a slider moving along the rod. Indeed,
this is a means of adjustment which we would recommend to be employed
in every pendulum.
_Smeaton’s Pendulum._
We shall conclude our account of compensation pendulums with a
description of that invented by Mr. Smeaton. The compensation for
temperature in this pendulum is effected by combining the two modes,
which have been so fully described in the preceding part of this
article.
The pendulum rod is of solid glass, and is furnished with a steel screw
and nut at the bottom in the usual manner. Upon the glass rod a hollow
cylinder of zinc, about the eighth of an inch thick, and about 12
inches long, passes freely, and rests upon the nut at the bottom of the
pendulum rod.
Over the zinc cylinder passes a tube made of sheet-iron. The edge of
this tube at the top is turned inwards, and is notched so as to allow
of this being effected. A flanche is thus formed, by which the iron
tube is supported, upon the zinc cylinder. The lower edge of the iron
tube is turned outwards, so as to form a base destined to support a
leaden cylinder, which we are about to describe.
A cylinder of lead, rather more than 12 inches long, is cast with a
hole through its axis, of such a diameter as to allow of its sliding
freely, but without shake, upon the iron tube over which it passes, and
by the lower extremity of which it is supported.
Now the zinc, resting upon the nut and expanding upwards, will raise
the whole of the remaining part of the compensation. This expansion
upwards will be slightly counteracted by the lesser expansion downwards
of the iron tube, which carries with it the leaden cylinder. The
cylinder of lead now acts upon the principle of the mercurial pendulum,
and, expanding upwards, contributes that which was wanting to restore
the centre of oscillation to its proper distance from the point of
suspension.
This pendulum, we have been informed, does well in practice, and we are
not aware that any description of it has been before published.
The method of calculating the length of the tubes required to form the
compensation is very simple; nothing more is necessary than to find the
length of zinc, the expansion of which is equal to that of the pendulum
rod.
Let the pendulum rod be composed of 43 inches of glass, the spring
being an inch and a half long, and the screw between the end of the
glass rod and the nut half an inch, making in the whole two inches of
steel and 43 inches of glass.
Now to find the length of zinc that will compensate the glass, we have,
from Table II., for glass and zinc ·2773, which, multiplied by 43,
gives 11·92 inches. In like manner we obtain as a compensation for two
inches of steel 0·74 of zinc, which, added to 11·92, gives 12·66 inches
for the total length of the zinc cylinder.
Now if the iron tube and the lead cylinder be each made of the same
length as the zinc, and arranged as we have described, the compensation
will be perfect.
To prove this, find, by means of the expansions given in Table I., the
actual expansion of each of the substances employed in the pendulum,
and we shall have the following results:--
The expansion of 12·66 inches of zinc expanding
upwards is ·0002186
Deduct that of 12·66 inches of iron expanding
downwards ·0000869
--------
Remaining effect of expansion upwards, referred
to the lower extremity of the iron tube ·0001317
Now, for the lead.--On the principle of the
mercurial compensation, subtract one tenth part
of the length of the cylinder, and take half
the remainder, and we shall have six inches of
lead, the expansion of which upwards is ·0000955
--------
Total expansion of the compensation upwards ·0002272
--------
To find the expansion of the rod, we have
the expansion of 43 inches of glass ·0002059
Of two inches of steel ·0000127
--------
Total expansion of the pendulum rod ·0002186
Agreeing near enough with that of the compensation before found.
As we conceive we have been sufficiently explicit in our description
of this pendulum, in the construction of which no difficulty presents
itself, we think an engraved representation of it would be superfluous.
We have hitherto treated only of compensations for temperature; but
there is another kind of error, which has been sometimes insisted upon,
arising from a variation in the density of the atmosphere. If the
density of the atmosphere be increased, the pendulum will experience
a greater resistance, the arc of vibration will in consequence be
diminished, and the pendulum will vibrate faster. This, however, is in
some measure counteracted by the increased buoyancy of the atmosphere,
which, acting in opposition to gravity, occasions the pendulum to
vibrate slower. If the one effect exactly equalled the other, it is
evident no error would arise; and in a paper by Mr. Davies Gilbert,
President of the Royal Society of London, published in the Quarterly
Journal for 1826, he has proved that, by a happy chance, the arc in
which pendulums of clocks are usually made to vibrate is the arc at
which this compensation of error takes place. This arc, for a pendulum
having a brass bob, is 1° 56′ 30″ on each side of the perpendicular;
and for a mercurial pendulum, 1° 31′ 44″, or about one degree and a
half.
It is well known that, if a pendulum vibrates in a circular arc, the
times of vibration will vary nearly as the squares of the arcs; but
if the pendulum could be made to vibrate in a cycloid, the time of
its vibration in arcs of different extent would then remain the same.
Huygens and others, therefore, endeavoured to effect this by placing
the spring of the pendulum between cheeks of a cycloidal form.
When escapements are employed which do not insure an unvarying impulse
to the pendulum, the force may be unequally transmitted through the
train of the clock in consequence of unavoidable imperfections of
workmanship, and the arc of vibration may suffer some increase or
diminution from this cause. To discover a remedy for this is certainly
desirable.
The writer of this article some years ago imagined a mode, which he
believes has also been suggested by others, by which he conceived a
pendulum might be made to describe an arc approaching in form to that
of a cycloid. The pendulum spring was of a triangular form, and the
point or vertex was pinned into the top of the pendulum rod, the base
of the triangle forming the axis of suspension. Now it is evident that
when the pendulum is in motion, the spring will resist bending at the
axis of suspension, with a force in some sort proportionate to the base
of the triangle.
Suppose the pendulum to have arrived at the extent of its vibrations;
the spring will present a curved appearance; and if the distance from
the point of suspension to the centre of oscillation be then measured,
it will evidently, in consequence of the curvature of the spring, be
shorter than the distance from the point of suspension to the centre of
oscillation, measured when the pendulum is in a perpendicular position,
and consequently when the spring is perfectly straight.
The base of the triangle may be diminished, or the spring be made
thinner; either of which will lessen its effect. We cannot say how this
plan might answer upon further trial, as sufficient experiments were
not made at the time to authorize a decisive conclusion.
We have thus completed our account of compensation pendulums; but
before we conclude, it may not be unacceptable if we offer a few
remarks on some points which may be found of practical utility.
The cock of the pendulum should be firmly fixed either to the wall or
to the case of the clock, and not to the clock itself, as is sometimes
done, and which has occasioned much irregularity in its rate, from the
motion communicated to the point of suspension. We prefer a bracket or
shelf of cast iron or brass, upon which the clock may be fixed, and the
cock carrying the pendulum attached to its perpendicular back. This
bracket may either be screwed to the back of the clock-case, or, which
is the better mode, securely fixed to the wall; and if the latter be
adopted, the whole may be defended from the atmosphere, or from dust,
by the clock-case, which thus has no connection either with the clock
or with the pendulum.
The point of suspension should be distinctly defined and immovable.
This may be readily effected, after the pendulum shall have taken the
direction of gravity, by means of a strong screw entering the cock
(which should be very stout) on one side, and pressing a flat piece of
brass into firm contact with the spring.
The impulse should be given in that plane of the rod which coincides
with the plane of vibration passing through the axis of the rod. If the
impulse be given at any point either before or behind this plane, the
probable result will be a tremulous unsteady motion of the pendulum.
A few rough trials, and moving the weight, will bring the pendulum near
its intended time of vibration, which should be left a little too slow;
when the bob should be firmly fixed to the rod, if the form of the
pendulum will admit of it, by a pin or screw passing through its centre.
The more delicate adjustment may be completed by shifting the place of
the slider with which the pendulum is supposed to be furnished on the
rod.
Mr. Browne (of whom we have before spoken) practises the following very
delicate mode of adjustment for rate, which will be found extremely
convenient, as it is not necessary to stop the pendulum in order to
make the required alteration. Having ascertained, by experiment, the
effect produced on the rate of the clock, by placing a weight upon the
bob equal to a given number of grains, he prepares certain smaller
weights of sheet-lead, which are turned up at the corners, that they
may be conveniently laid hold of by a pair of forceps, and the effect
of these small weights on the rate of the clock will be, of course,
known by proportion. The rate being supposed to be in defect, the
weights necessary to correct this may be deposited, without difficulty,
upon the bob of the pendulum, or upon some convenient plane surface,
placed in order to receive them: and should it be necessary to remove
any one of the weights, this may readily be done by employing a
delicate pair of forceps, without producing the slightest disturbance
in the motion of the pendulum.
INDEX.
A.
Action and reaction, 34.
Aeriform fluids, 26.
Animalcules, 12.
Atmosphere, impenetrability of, 22.
Compressibility and elasticity of, 23.
Atoms, 6.
Coherence of, 7.
Attraction, magnetic, of gravitation, 8, 50, 64.
Molecular or atomic, 69.
Cohesion, 70.
Attwood, machine of, 92.
Axes, principal, 138.
Axis, mechanical properties of, 128.
B.
Balance, 279.
Of Bates, 288.
Use of, 289.
Danish, 299.
Bent-lever of Brady, 301.
Bodies, 2.
Lines, surfaces, edges, area, length of, 4.
Figure, volume, shape of, 5.
Porosity of, 17.
Compressibility of, 18.
Elasticity, dilatibility of, 19.
Inertia of, 27.
Rule for determining velocity of; motion of two bodies after
impact, 38.
C.
Capillary attraction, 73.
Capstan, 179.
Cause and effect, 7.
Circle of curvature, 99.
Cog, hunting, 191.
Components, 51.
Cord, 163.
Cordage, friction and rigidity of, 260.
Crank, 241.
Crystallisation, 14.
Cycloid, 158.
D.
Damper, self-acting, 234.
Deparcieux’s compensation pendulum, 319.
Diagonal, 51.
Dynamics, 160.
Dynamometer, 305.
E.
Electricity, 76.
Electro-magnetism, 76.
Equilibrium, neutral, instable, and stable, 118.
F.
Figure, 5.
Fly-wheel, 239.
Force, 6.
Composition and resolution of, 49.
Centrifugal, 98.
Moment of; leverage of, 135.
Regulation and accumulation of, 224.
Friction, effects of, 96.
Laws of, 264.
G.
Governor, 227.
Gravitation, attraction of, 77.
Terrestrial, 84.
Gravity, centre of, 107.
Gyration, radius of, centre of, 137.
H.
Hooke’s universal joint, 252.
Hydrophane, porosity of, 18.
I.
Impact, 39.
Impulse, 65.
Inclined plane, 163–209.
Inclined roads, 211.
Inertia, 27.
Laws of, 32.
Moment of, 137.
J.
Julien le Roy, compensation tube of, 319.
L.
Lever, 163.
Fulcrum of; three kinds of, 167.
Equivalent, 176.
Line of direction, 110.
Liquids, compressibility of, 24.
Loadstone, 68.
M.
Machines, simple, 160.
Power of, 175.
Regulation of, 225.
Magnet, 68.
Magnetic attraction, 8.
Magnetism, 76.
Magnitude, 4.
Marriott’s patent weighing machine, 305.
Materials, strength of, 272.
Matter, properties of, 2.
Impenetrability of, 4.
Atoms of; molecules of, 6.
Divisibility of, 9.
Examples of the subtilty of, 12.
Limit to the divisibility of, 13.
Porosity of; density of, 17.
Compressibility of, 18.
Elasticity and dilatability of, 19.
Impenetrability of, 22.
Inertia of, 27.
Mechanical science, foundation of, 16.
Metronomes, principles of, 153.
Molecules, 6.
Motion, laws of, 46.
Uniformly accelerated, 87.
Table illustrative of, 90.
Retarded; of bodies on inclined planes and curves, 94.
Rotary and progressive, 127.
Mechanical contrivances for the modification of, 245.
Continued rectilinear; reciprocatory rectilinear; continued
circular; reciprocating circular, 246.
N.
Newton, method of, for determining the thickness of transparent
substances, 10.
Laws of motion of, 46.
O.
Oscillation, 129.
Of the pendulum, 145.
Centre of, 152.
P.
Parallelogram, 51.
Particle, 6.
Pendulum, oscillation or vibration of, 145.
Isochronism of, 147.
Centre of oscillation of, 152.
Of Troughton, 284.
Compensation, 307.
Of Harrison, 313.
Tubular, of Troughton, 314.
Of Benzenberg, 316.
Ward’s compensation, 318.
Captain Kater’s compensation, 320.
Reid’s; Ellicott’s compensation, 322.
Steel and brass compensation, 324.
Mercurial, 326.
Graham’s mercurial, 329.
Wood and lead, 334.
Smeaton’s, 335.
Percussion, 130.
Centre of, 144.
Planes of cleavage, 15.
Porosity, 17.
Power, 161.
Properties, 2.
Projectiles, curvilinear path of, 82.
Pulley, 164.
Tackle; fixed, 198.
Single moveable, 200.
Called a runner; Spanish bartons, 205.
R.
Rail-roads, 213.
Regulating damper, 233.
Regulators, 227.
Repulsion, 8.
Molecular, 74.
Resultant, 51.
Rose-engine, 250.
S.
Salters, spring balance of, 305.
Screw, 209.
Concave, 217.
Micrometer, 223.
Shape, 5.
Siphon, capillary, 73.
Spring, 304.
Statics, 160.
Steelyard, 294.
C. Paul’s, 296.
Chinese, 299.
T.
Table, whirling, 99.
Tachometer, 234.
Tread-mill, 179.
V.
Velocity, angular, 99.
Vibration, 129.
Of the pendulum, 145.
Centre of, 152.
Volume, 5–17.
W.
Watch, mainspring of; balance wheel of, 195.
Water regulator, 229.
Wedge, 209.
Use of, 215.
Weight, 161–291.
Weighing machines, 278.
For turnpike roads, 302.
By means of a spring, 303.
Wheels, spur, crown, bevelled, 189.
Escapement, 194.
Wheel and axle, 177.
Wheel-work, 176.
Winch, 179.
Windlass, 178.
Wollaston’s wire, 10.
Z.
Zureda, apparatus of; Leupold’s application of, 251.
END OF MECHANICS.
LONDON:
SPOTTISWOODES and SHAW
New-street-Square.