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of the pendulum PN, in the arch A B, is to the time of vibration of the pendulum PO in the similar arc C D in the subduplicate ratio of A N to CO and since the radii PN, PO, are proportional to the similar arcs AN, CO, therefore the time of vibration of the pendulum PN will be to the time of vibration of the pendulum PO in a subduplicate ratio of PN to PO. If the length of a pendulum vibrating seconds be 39-174 inches, then the length of a pendulum vibrating half seconds will be 9-793 inches. For ::: 39-174 1":"/39-174: ; and Hence r

: x.

39.174 4

9.793.

PROP. VI. The length of pendulums vibrating in the same time, in different places, will be as the forces of gravity. For the velocity generated in any given time is directly as the force of gravity, and inversely as the quantity of matter. Now, the matter being supposed the same in both pendulums, the velocity is as the force of gravity; and the space passed through in a given time will be as the velocity; that is, as the gravity. Cor. Since the length of pendulums vibrating in the same time in small arcs are as the gravitating forces, and as gravity increases with the latitude on account of the spheroidal figure of the earth and its rotation about its axis; hence the length of a pendulum vibrating in a given time will be variable with the latitude, and the same pendulum will vibrate slower the nearer it is carried to the

equator.

PROP. VII.-The time of vibrations of pendulums of the same length, acted upon by different forces of gravity, are reciprocally as the square roots of the forces. For, when the matter is given, the velocity is as the force and time; and the space described by any given force, is as the force and square of the time. Hence the lengths of pendulums are as the forces and the squares of the times of falling through them. But these times are in a given ratio to the times of vibration; whence the lengths of pendulums are as the forces and the squares of the times of vibration. Therefore, when the lengths are given, the forces will be reciprocally as the square of the times, and the times of vibration reciprocally as the square roots of the forces. Cor. Let p = ength of pendulum, g = force of gravity, and t=time of vibration. Then since gx t.

Hence g=px;'and t√px

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is, the forces in different places are directly as the lengths of the pendulums, and inversely as the square roots of the times of vibration; and the times of vibration are directly as the square roots of the lengths of the pendulums, and inversely as the square roots of the gravitating forces.

PROP. VIII.-A pendulum which vibrates in the arch of a cycloid describes the greatest and least vibrations in the same time. This property is demonstrated only on a supposition that the whole mass of the pendulum is concentrated in a point but this cannot take place in any really vibrating body; and, when the pendulum is of finite magnitude, there is no point given in posi

tion which determines the length of the pendulum; on the contrary the centre of oscillation will not occupy the same place in the given body, when describing different parts of the tract it moves through, but will continually be moved in respect of the pendulum itself during its vibration. This circumstance has prevented any general determination of the time of vibration in a cycloidal arc, except in the imaginary case referred to. There are many other obstacles which concur in rendering the application of this curve to the vibration of pendulums designed for the measures of time the source of errors far greater than those which by its peculiar property it is intended to obviate; and it is now wholly disused in practice. Although the times of vibration of a pendulum in different arches be nearly equal, yet, from what has been said, it will appear that, if the ratio of the least of these arches to the greatest be considerable, the vibrations will be performed in different times; and the difference, though small, will become sensible in the course of one or more days. In clocks used for astronomical purposes it will therefore be necessary to observe the arc of vibration; which if different from that described by the pendulum when the clock keeps time, there a correction must be applied to the time shown by the clock. This correction, expressed in seconds of time, will be equal to the half of three times the difference of the square of the given arc, and of that of the arc described by the pendulum when the clock keeps time, these arcs being expressed in degrees; and so much will the clock gain or lose according as the first of these arches is less or greater than the second. Thus, if the clock keeps time when the pendulum vibrates in an arch of 3o, it will lose 101 daily in an arch of 4°. For 42-3x=7× = 101". The length of a pendulum rod increases with heat; and the quantity of expansion answering to any given degree of heat is experimentally found by means of a pyrometer (see PYROMETER); but the degree of heat at any given time is shown by a thermometer: hence that instrument should be placed within the clock-case at a height nearly equal to that of the middle of the pendulum; mined at least once a day. Now, by a table conand its height, for this purpose, should be exastructed to exhibit the daily quantity of accele ration or retardation of the clock, answering to every probable height of the thermometer, the corresponding correction may be obtained. It is also necessary to observe that the mean height of the thermometer during the interval ought to be used. In Six's thermometer this height may be easily obtained; but in the thermometers of the common construction it will be more difficult to find this mean. It has been found, by repeated experiments, that a brass rod equal in length to a second pendulum will expand or contract one 1000th part of an inch by a change of temperature of 1° in Fahrenheit's thermometer; and, since the times of vibration are in a subduplicate ratio of the lengths of the pendulum, hence an expansion or contraction of one 1000dth part of an inch will answer nearly to 1′′ daily; therefore a change of 1o in the thermometer will occasion a difference in the rate of the clock

length of the common half second pendulum. Let O be the focus of the parabola M E C, and M C the latus rectum; and make AE=MO

MC the length of a common half second pendulum. At the point A of the verge let a thin plate A B be fixed at one end, and at the other end B let it be fastened to a bar or arm BD perpendicular to DH, and to which it is fixed at the point D. The figure of the plate AB is that of the evolute of the given parabola M EC. The equation of this evolute, being also that of the semicubical parabola, is 27 px2=y3.

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p=P; then Pay3, and in the focus P-2y. In this case 2x2=y2 = 4 P2; hence a2 = { P2, and x = P√T=

equal to 1" daily. Whence, if the clock be so
adjusted as to keep time when the thermometer
is at 55°, it will lose 10" daily when the thermo-
meter is at 65°, and gain as much when it is at
45°. Hence the daily variation of the rate of the
clock from summer to winter will be very consi-
derable. It is true indeed that most pendulums
have a nut or regulator at the lower end, by
which the bob may be raised or lowered a de-
terminate quantity: and therefore, while the
height of the thermometer is the same, the rate of
the clock will be uniform. But since the state of
the weather is ever variable, and as it is impossi-
ble to be raising or lowering the bob of the pen-
dulum at every change of the thermometer, there--Let
fore the correction formerly mentioned is to be
applied. This correction, however, is in some
measure liable to a small degree of uncertainty;
and, in order to avoid it altogether, several con-
trivances have been proposed, by constructing a
pendulum of different materials, and so disposing
them that their effects may be in opposite direc-
tions, and thereby counterbalance each other; and
thus the pendulum will continue of the same length.
PENDULUM, ANGULAR, is formed of two pieces
or legs like a sector, and is suspended by the an-
gular point. This pendulum was invented with
a view to diminish the length of the common
pendulum, but at the same time to preserve or
even increase the time of vibration. In this pen-
dulum, the time of vibration depends on the
length of the legs, and on the angle contained
between them conjointly, the duration of the
time of vibration increasing with the angle.
Hence a pendulum of this construction may be
made to oscillate in any given time. At the
lower extremity of each leg of the pendulum is a
ball or bob as usual. It may be easily shown,
that, in this kind of a pendulum, the squares of
the times of vibration are as the secants of half
the angle contained by the legs: hence, if a pen-
dulum of this construction vibrates half seconds
when its legs are close, it will vibrate whole se-
conds when the legs are opened, so as to contain
an angle equal to 151° 2′ 30′′.

PENDULUM, CONICAL, or circular, is so called from the figure described by the string or ball of the pendulum. This pendulum was invented by Mr. Huygens, and also claimed by Dr. Hook. To understand its principles it will be necessary to premise the following lemma, viz. the times of all the circular revolutions of a heavy globular body, revolving within an inverted hollow paraboloid, will be equal, whatever be the radii of the circles described by that body. To construct the pendulum, therefore, so that its ball may always describe its revolutions in a paraboloid surface, it will be necessary that the rod of the pendulum be flexible, and that it be suspended in such a manner as to form the evolute of the given parabola. Hence, let K H (fig. 9) be an axis perpendicular to the horizon, having a pinion at K moved by the last wheel in the train of the clock; and a hardened steel point at H moving in an agate pivot, to render the motion as free as possible. Now, let it be required that the pendulum shall perform each revolution in a second; then the paraboloid surface it moves in must be such whose latus rectum is double the

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16.

Pthe distance of the focus from the vertex A.-By assuming the value of x, the ordinates of the curve may be found; and hence it may be easily drawn. The string of the pendulum must be of such a length that, when one end is fixed at B, it may lie over the plate A B, and then hang perpendicular from it, so that the centre of the bob may be at E when at rest. Now, the verge K H being put in motion, the ball of the pendulum will begin to gyrate, and thereby contrive a centrifugal force which will carry it out from the axis to some point F, where it will circulate seconds or half seconds, according as the line A E is 9.8 inches, or two inches and a quarter, and A B answerable to it. One advantage possessed by a clock having a pendulum of this construction. is, that the second hand moves in a regular and uniform manner, without being subject to those jerks or starts as in common clocks; and the pendulum is entirely silent.

PENDULUM, FIR. The expansion or contraction of straight-grained fir wood lengthwise, by change of temperature, is so small that it is fonnd to make very good pendulum rods. The wood called sapadillo is said to be still better. There is good reason to believe that the previous baking, varnishing, gilding, or soaking of these woods in any melted matter, only tends to impair the property that renders them valuable. They should be simply rubbed on the outside with wax and a cloth. In pendulums of this construction the error is greatly diminished, but not taken away.

PENDULUM, GRIDIRON, Or Harrison's, is an ingenious contrivance for the purpose abovementioned. Instead of one rod, this pendulum is composed of any convenient odd number of rods, as five, seven, or nine; being so connected that the effect of one set of them counteracts that of the other set; and therefore, if they are properly adjusted to each other, the centres of suspension and oscillation will always be equidistant. Fig. 7 represents a gridiron pendulum composed of nine rods, steel and brass alternately. The two outer rods, A B, C D, which are of steel, are fastened to the cross pieces A C, B D, by means of pins. The next two rods, E F, GH, are of brass, and are fastened to the lower bar BD, and to the second upper bar E G. The two following rods are of steel, and are fastened to the cross

bars EG and I K. The two rods adjacent to the central rod, being of brass, are fastened to the cross pieces I K and L M; and the central rod, to which the ball of the pendulum is attached, is suspended from the cross piece L M, and passes freely through a perforation in each of the cross bars IK, BD. From this disposition of the, rods, it is evident that, by the expansion of the extreme rods, the cross piece BD, and the two rods attached to it, will descend: but, since these rods are expanded by the same heat, the cross piece E G will consequently be raised, and therefore also the two next rods; but, because these rods are also expanded, the cross bar I K will descend; and, by the expansion of the two next rods, the piece LM will be raised a quantity sufficient to counteract the expansion of the central rod. Whence it is obvious that the effect of the steel rods is to increase the length of the pendulum in hot weather, and to diminish it in cold weather, and that the brass rods have a contrary effect upon the pendulum. The effect of the brass rods must, however, be equivalent, not only to that of the steel rods, but also to the part above the frame and spring, which connects it with the clock, and to that part between the lower part of the frame and the centre of the ball.

PENDULUM, MERCURIAL, was invented by the celebrated Mr. George Graham, and is considered as the compensating pendulum. In this the rod of the pendulum is a hollow tube, in which a sufficient quantity of mercury is put. Mr. Graham first used a glass tube, and the clock to which it was applied was placed in the most exposed part of the house. It was kept constantly going, without having the hands or pendulum altered, from the 9th of June 1722 to the 14th of October 1725, and its rate was determined by transits of fixed stars. Another clock made with extraordinary care, having a pendulum about sixty pounds weight, and not vibrating above 1° 30′ from the perpendicular, was placed beside the former, the more readily to compare them with each other, and that they might both be equally exposed. The result of all the observations was this, that the irregularity of the clock with the quicksilver pendulum exceeded not, when greatest, sixth part of that of the other clock with the common pendulum, but for the greatest part of the year not above an eighth or ninth part; and even this quantity would have been lessened, had the column of mercury been a little shorter: for it differed a little the contrary way from the other clock, going faster with heat and slower with cold. To confirm this experiment more, about the beginning of July 1723 Mr. Graham took off the heavy pendulum from the other clock, and made another with mercury, but with this difference, that instead of a glass tube he used a brass one, and varnished the inside to secure it from being injured by the mercury. This pendulum he used afterwards, and found it about the same degree of exactness as the other.

M. Thiout's Pendulum.-Another excellent contrivance for the same purpose is described by M. Thiout, a French author on clock-making. Of this pendulum, somewhat improved by Mr. Crosthwaite, watch and clock maker, Dublin, we

have the following description in the Transactions of the Royal Irish Academy, 1788:- A and B, fig. 8, are two rods of steel forged out of the same bar, at the same time, of the same temper, and in every respect similar. On the top of B is formed a gibbet C; this rod is firmly supported by a steel bracket D, fixed on a large piece of marble E, firmly set into the wall F, and having liberty to move freely upwards between cross staples of brass, 1, 2, 3, 4, which touch only in a point in front and rear (the sta ples having been carefully formed for that purpose); to the other rod is firmly fixed by its centre the lens G, of twenty-four pounds weight, although it should in strictness be a little below it. This pendulum is suspended by a short steel spring on the gibbet at C; all which is entirely independent of the clock. To the back of the clock-plate I are firmly screwed two cheeks nearly cycloidal at K, exactly in a line with the centre of the verge L. The maintaining power is applied by a cylindrical steel stud, in the usual way of regulators, at M. Now it is very evident that any expansion or contraction that takes place in either of these exactly similar rods is instantly counteracted by the other; whereas in all compensation pendulums composed of different materials, however just calculation may seem to be, that can never be the case, as not only different metals, but also different bars of the same metal that are not manufactured at the same time, and exactly in the same manner, are found by a good pyrometer to differ materially in their degrees of expansion and contraction, a very small change affecting one and not the other.' Theory has pointed out several other pendulums, known by the names of elliptic, horizontal, rotulary, &c., pendulums. We can only select two or three of the more modern inventions of this kind.

Elliott's compensating pendulum.-The adjustment of the rods for the temperature in the Gridiron pendulum of Harrison being found in convenient, and accompanied sometimes by a considerable change in the rate: in the pendu lum of Elliott two levers are adopted instead o. one, and they are applied at the bob instead of at the superior end of the verge.

Fig. 1, plate II., PENDULUMS, represents this pendulum; ab is a bar of brass made quite fast at the upper end by pins, and held contiguous at several equal distances, by the screws, 1, 2, &c., to the rod of the pendulum, which is a bar of iron; and, so far as the brass bar reaches, is filed of the same size and shape, though it does not appear so in the figure, but, a little below the end of the brass, the iron is left broader, as at dd, for the conveniency of fixing the work to it, and is made of a sufficient length to pass quite through the ball of the pendulum to c. The holes 1, 2, &c., in the brass, through which the screws pass into the iron rod of the pendulum, are filed of a sufficient length to suffer the brass to contract and dilate freely by heat and cold under the heads of the screws: ee ee represent the ball of the pendulum; f,f, two strong pieces of steel, or levers, whose inner centres, or pivots, turn in two holes drilled in the broad part of the pendulum rod, and their outer ones in a strong

PENDULUMS.

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