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of the pendulum PN, in the arch A B, is to the tion which determines the length of the pendutime of vibration of the pendulum PO in the lum; on the contrary the centre of oscillation similar arc C D in the subduplicate ratio of A N will not occupy the same place in the given body, to CO: and since the radii PN, PO, are pro- when describing different parts of the tract it portional to the similar arcs AN, CO, therefore moves through, but will continually be moved in the time of vibration of the pendulum P N will respect of the pendulum itself during its vibrabe to the time of vibration of the pendulum PO tion. This circumstance has prevented any gein a subduplicate ratio of P N to PO. If the neral determination of the time of vibration in a length of a pendulum vibrating seconds be cycloidal arc, except in the imaginary case re39.174 inches, then the length of a pendulum ferred to. There are many other obstacles which vibrating half seconds will be 9.793 inches. For concur in rendering the application of this curve

to the vibration of pendulums designed for the 1":"":: N 39.174 : N 2'; and 1:1:: 39.174

measures of time the source of errors far greater 39.174 Hence r = :9.793.

than those which by its peculiar property it is

intended to obviate ; and it is now wholly disProp. VI.---The length ! of pendulums vi- used in practice. Although the times of vibrabrating in the same time, in different places, tion of a pendulum in different arches be nearly will be as the forces of gravity. For the ve- equal, yet, from what has been said, it will aplocity generated in any given time is directly as

pear that, if the ratio of the least of these arches the force of gravity, and inversely as the quan, to the greatest be considerable, the vibrations tity of matter. Now, the matter being supposed will be performed in different times; and the the same in both pendulums, the velocity is as difference, though small, will become sensible in the force of gravity; and the space passed the course of one or more days. In clocks used through in a given time will be as the velocity; for astronomical purposes it will therefore be that is, as the gravity. Cor. Since the length of necessary to observe the arc of vibration; which pendulums vibrating in the same time in small if different from that described by the pendulum arcs are as the gravitating forces, and as gravity when the clock keeps time, there a correction increases with the latitude on account of the must be applied to the time shown by the clock. spheroidal figure of the earth and its rotation This correction, expressed in seconds of time, about its axis; hence the length of a pendulum will be equal to the half of three times the differvibrating in a given time will be variable with ence of the square of the given arc, and of that the latitude, and the same pendulum will vi- of the arc described by the pendulum when the brate slower the nearer it is carried to the clock keeps time, these are being expressed in equator.

degrees; and so much will the clock gain or lose PROP. VII.—The time of vibrations of pen- according as the first of these arches is less or dulums of the same length, acted upon by dif- greater than the second. Thus, if the clock keeps ferent forces of gravity, are reciprocally as the time when the pendulum vibrates in an arch of 3, square roots of the forces. For, when the matter it will lose 104" daily in an arch of 4o. For is given, the velocity is as the force and time; 42_32 x 1 =7x1 = 103". The length of a and the space described by any given force, is as the force and square of the time. Hence the pendulum rod increases with heat ; and the lengths of pendulums are as the forces and the quantity of expansion answering to any given squares of the times of falling through them. But degree of heat is experimentally found by means these times are in a given ratio to the times of of a pyrometer (see PyromeTER); but the devibration; whence the lengths of pendulums gree of heat at any given time is shown by a are as the forces and the squares of the times of thermometer : hence that instrument should be vibration. Therefore, when the lengths are given, placed within the clock-case at a height nearly the forces will be reciprocally as the square of equal to that of the middle of the pendulum; the times, and the times of vibration reciprocally nined at least once a day. "Now, by a table con

and its height, for this purpose, should be exaas the square roots of the forces. Cor. Let p= structed to exhibit the daily quantity of acceleength of pendulum, g = force of gravity, and t=time of vibration. Then since 1 = g'* t. ration or retardation of the clock, answering 10

every probable height of the thermometer, the Hence g =px;'ar

and t =V

PX -
That corresponding correction may be obtained. It

is also necessary to observe that the mean height .is, the forces in different places are directly as of the thermometer during the interval ought to the lengths of the pendulums, and inversely as be used. In Six's thermometer this height may the square roots of the times of vibration; and be easily obtained; but in the thermometers of the times of vibration are directly as the square the common construction it will be more difficult roots of the lengths of the pendulums, and in- to find this mean. It has been found, by reversely as the square roots of the gravitating peated experiments, that a brass rod equal in forces.

length to a second pendulum will expand or Prop. VIII.-A pendulum which vibrates in contract one 1000th part of an inch by a change the arch of a cycloid describes the greatest and of temperature of 1° in Fahrenheit's thermomeleast vibrations in the same time. This property ter; and, since the times of vibration are in a is demonstrated only on a supposition that the subduplicate ratio of the lengths of the pendulum, whole mass of the pendulum is concentrated in bence an expansion or contraction of one 1000dth a point : but this cannot take place in any really part of an inch will answer nearly to 1" daily; vibrating body; and, when the pendulum is of therefore a change of 1° in the thermometer will knite magnitude, there is no point given in posi- occasion a difference in the rate of the clock

1

equal to 1" daily. Whence, if the clock be so length of the common half second pendulum. adjusted as to keep time when the thermometer Let O be the focus of the parabola M E C, and is at 55°, it will lose 10" daily when the thermo- MC the latus rectum; and make A E=MO meter is at 65°, and gain as much when it is at = 4MC = the length of a common half se45°. Hence the daily variation of the rate of the cond pendulum. At the point A of the verge clock from summer to winter will be very consi- let a thin plate A B be fixed at one end, and at derable. It is true indeed that most pendulums "the other end B let it be fastened to a bar or arm have a nut or regulator at the lower end, by B D perpendicular to D H, and to which it is which the bob may be raised or lowered a de- fixed at the point D. The figure of the plate terminate quantity: and therefore, while the A B is that of the evolute of the given parabola height of the thermometer is the same, the rate of MEC. The equation of this evolute, being also the clock will be uniform. But since the state of

27 the weather is ever variable, and as it is impossi

that of the semicubical parabola, is px? = y.

16 ble to be raising or lowering the bob of the pen

27 dulum at every change of the thermometer, there – Let p=P; then P.x* =yo, and in the focus

16 fore the correction formerly mentioned is to be P=2y. In this case 2 x2 = y*=£P?: hence applied. This correction, however, is in some measure liable to a small degree of uncertainty ; x = } P, and r = PVT= LP = the and, in order to avoid it altogether, several contrivances have been proposed, by constructing a distance of the focus from the vertex A.-By aspendulum of different materials, and so disposing suming the value of x, the ordinates of the curve them that their effects may be in opposite direc- may be found; and hence it may be easily tions, and thereby counterbalance each other; and 'drawn. The string of the pendulum must be of thus the pendulum will continue of the same length. such a length that, when one end is fixed at B, it

PENDULUM, Angular, is formed of two pieces may lie over the plate A B, and then hang peror legs like a sector, and is suspended by the an- pendicular from it, so that the centre of the bob gular point. This pendulum was invented with may be at E when at rest. Now, the verge K H a view to diminish the length of the common being put in motion, the ball of the pendulum pendulum, but at the same time to preserve or

will begin to gyrate, and thereby contrive a ceneven increase the time of vibration. In this pen- trifugal force which will carry it out from the dulum, the time of vibration depends on the axis to some point F, where it will circulate selength of the legs, and on the angle contained conds or half seconds, according as the line A E between them conjointly, the duration of the is 9.8 inches, or two inches and a quarter, time of vibration increasing with the angle. and A B answerable to it. One advantage posHence a pendulum of this construction may be sessed by a clock having a pendulum of this conmade to oscillate in any given time. At the structior

. is, that the second hand moves in a lower extremity of each leg of the pendulum is a regular and uniform manner, without being sulyball or bob as usual. It may be easily shown, ject to those jerks or starts as in common clocks; that, in this kind of a pendulum, the squares of and the pendulum is entirely silent. the times of vibration are as the secants of half PENDULUM, Fır. The expansion or contracthe angle contained by the legs: hence, if a pen- tion of straight-grained fir wood lengthwise, by dulum of this construction vibrates half seconds change of temperature, is so small that it is when its legs are close, it will vibrate whole se- fonnd to make very good pendulum rods. The conds when the legs are opened, so as to contain wood called sapadillo is said to be still better. an angle equal to 151° 2' 30".

There is good reason to believe that the previPENDULUM, Conical, or circular, is so call ous baking, varnishing, gilding, or soaking of ed from the figure described by the string or ball these woods in any melted matter, only tends to of the pendulum. This pendulum was invented impair the property that renders them valuable. by Mr. Huygens, and also claimed by Dr. They should be simply rubbed on the outside Hook. To understand its principles it will be with wax and a cloth. In pendulums of this necessary to premise the following lemma, viz. construction the error is greatly diminished, but the times of all the circular revolutions of a heavy not taken away. globular body, revolving within an inverted hol PENDULUM, GRIDIRON, or Harrison's, is an low paraboloid, will be equal, whatever be the ingenious contrivance for the purpose aboveradii of the circles described by that body. To mentioned. Instead of one rod, this pendulum construct the pendulum, therefore, so that its ball is composed of any convenient odd number of may always describe its revolutions in a parabo- rods, as five, seven, or nine; being so connected loid surface, it will be necessary that the rod of that the effect of one set of them counteracts that the pendulum be flexible, and that it be suspended of the other set; and therefore, if they are proin such a manner as to form the evolute of the perly adjusted to each other, the centres of susgiven parabola. Hence, let K H (fig. 9) be an pension and oscillation will always be equidistant. axis perpendicular to the horizon, having a pi- Fig. 7 represents a gridiron pendulum composed nion at K moved by the last wheel in the train of nine rods, steel and brass alternately. The of the clock; and a hardened steel point at H two outer rods, A B,C D, which are of steel, are moving in an ayate pivot, to render ihe mution fastened to the cross pieces A C, B D, by means as free as possible. Now, let it be required that of pins. The next two rods, E F, G H, are of the pendulum shall perform each revolution in a brass, and are fastened to the lower bar B D, and second ; then the paraboloid surface it moves in to the second upper bar E G. The two followmust be such whose latus rectum is double the ing rods are of steel, and are fastened to the cross

bars E G and I K. The two rods adjacent to the have the following description in the Transactions central rod, being of brass, are fastened to the of the Royal Irish Academy, 1788 :— A and cross pieces I K and L M; and the central rod, B, fig. 8, are two rods of steel forged out of the to which the ball of the pendulum is attached, is same bar, at the same time, of the same temper, suspended from the cross piece L M, and passes and in every respect similar. On the top of B freely through a perforation in each of the cross is formed a gibbet C; this rod is firmly supbars I K, BD. From this disposition of the, ported by a steel bracket D, fixed on a large rods, it is evident that, by the expansion of the piece of marble E, firmly set into the wall F, extreme rods, the cross piece B D, and the two and having liberty to move freely upwards berods attached to it, will descend : but, since these tween cross staples of brass, 1, 2, 3, 4, which rods are expanded by the same heat, the cross touch only in a point in front and rear (the stapiece E G will consequently be raised, and there- ples having been carefully formed for that purfore also the two next rods; but, because these pose); to the other rod is firmly fixed by its rods are also expanded, the cross bar I K will centre the lens G, of twenty-four pounds weight, descend; and, by the expansion of the two next although it should in strictness be a little below rods, the piece L M will be raised a quantity suf- it. This pendulum is suspended by a short steel ficient to counteract the expansion of the central spring on the gibbet at C; all which is entirely rod. Whence it is obvious that the effect of the independent of the clock. To the back of the steel rods is to increase the length of the pendu- clock-plate I are firmly screwed two cheeks luna in hot weather, and to diminish it in cold nearly cycloidal at K, exactly in a line with the weather, and that the brass rods bave a contrary centre of the verge L. The maintaining power effect upon the pendulum. The effect of the brass is applied by a cylindrical steel stud, in the usual rods must, however, be equivalent, not only to way of regulators, at M. Now it is very evithat of the steel rods, but also to the part above dent that any expansion or contraction that takes the frame and spring, which connects it with the place in either of these exactly similar rods is clock, and to that part between the lower part of instantly counteracted by the other; whereas in the frame and the centre of the ball.

all compensation pendulums composed of difPENDULUM, MERCURIAL, was invented by ferent materials, however just calculation may the celebrated Mr. George Graham, and is con seem to be, that can never be the case, as not sidered as the compensating pendulum. In this only different metals, but also different bars of the rod of the pendulum is a hollow tube, in the same metal that are not manufactured at the which a sufficient quantity of mercury is put. same time, and exactly in the same manner, are Mr. Graham first used a glass tube, and the clock found by a good pyrometer to differ materially to which it was applied was placed in the most in their degrees of expansion and contraction, a exposed part of the house. It was kept constant- very small change affecting one and not the ly going, without having the hands or pendulum other.' Theory has pointed out several other altered, from the 9th of June 1722 to the 14th of pendulums, known by the names of elliptic, hoOctober 1725, and its rate was determined by rizontal, rotulary, &c., pendulums. We can only transits of fixed stars. Another clock made with select two or three of the more modern invenextraordinary care, having a pendulum about tions of this kind. sixty pounds weight, and not vibrating above 1° Elliott's compensating pendulum.—The adjust30' from the perpendicular, was placed beside ment of the rods for the temperature in the Gridthe former, the more readily to compare them iron pendulum of Harrison being found in with each other, and that they might both be convenient, and accompanied sometimes by a equally exposed. The result of all the observa- considerable change in the rate : in the pendu tions was this, that the irregularity of the clock lum of Elliott two levers are adopted instead o. with the quicksilver pendulum exceeded not, one, and they are applied at the bob instead of when greatest, a sixth part of that of the other at the superior end of the verge. clock with the common pendulum, but for the Fig. 1, plate II., Pendulums, represents this greatest part of the year not above an eighth or pendulum; ab is a bar of brass made quite fast ninth part; and even this quantity would have at the upper end by pins, and held contiguous at been lessened, had the column of mercury been several equal distances, by the screws, 1, 2, &c., a little shorter: for it differed a little the contrary to the rod of the pendulum, which is a bar of way from the other clock, going faster with heat iron ; and, so far as the brass bar reaches, is filed and slower with cold. To confirm this experi- of the same size and shape, though it does not ment more, about the beginning of July 1723 appear so in the figure, but, a little below the Mr. Graham took off the heavy pendulum from end of the brass, the iron is left broader, as at the other clock, and made another with mercury, dd, for the conveniency of fixing the work to it, but with this difference, that instead of a glass and is made of a sufficient length to pass quite tube he used a brass one, and varnished the in- through the ba!l of the pendulum to c. The side to secure it from being injured by the mer- holes 1, 2, &c., in the brass, through which the cury. This pendulum he used afterwards, and screws pass into the iron rod of the pendulum, found it about the same degree of exactness as are filed of a sufficient length to suffer the brass the other.

to contract and dilate freely by heat and cold M. Thiout's Pendulum.-Another excellent under the heads of the screws : eeee represent contrivance for the same purpose is described the ball of the pendulum ; f,f, two strong pieces by M. Thiout, a French anthor on clock-making. of steel, or levers, whose inner centres, or pirots, Of this pendulum, somewhat improved Mr. turn in two boles drilled in the broad part of the Crosthwaite, watch and clock maker, Dublin, we pendulum rod, and their outer ones in a strong

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