The application of the pendulum to clock-work was one of the great steps of human progress. The motion of timekeepers had previously been regulated by means of a balanced arm, which was made to vibrate backwards and forwards by the teeth of a wheel, called, from its peculiar shape, a crown-wheel. The time of vibration of this balance depended on the extent of its motion, and on the propelling force; and therefore clocks of this kind could hardly go with regularity; yet their performance was so satisfactory that they came into general use, and were made so small as to be portable.

Galileo, happening to observe that the oscillation of the chandelier in a cathedral was performed in the same time when the motion was dying away, as when it was extensive, was led to the idea of employing pendulums for the measurement of time. But the discovery of the true theory of isochronous vibration was made by Christian Huygens, who showed that the time of an oscillation is independent of its extent when the tendency of the body to return to its point of rest is exactly proportional to its distance from that point, and applied pendulums to regulate the motion of clocks. Now, in the case of a pendulum, the tendency to return is proportional to the sine of the arc, whereas the distance is proportional to the arc itself, and therefore the oscillations of a pendulum through large and through small arcs cannot be performed in times exactly equal to each other.

Let two bodies, A and B (fig. 1), of exactly equal weights,

be drawn to their points of rest, O and Q, by tendencies exactly proportional to their distances from them; suppose that A is drawn twice as far from O as B is from Q, and that both are released at the same instant. The tendency of A to return is double of the tendency of B, and therefore, in some minute interval of time, when A has moved over the distance AC, B will have moved over BD, the half of AC, and will have acquired a velocity half of that of A; wherefore, at every succeeding interval of time, A must have moved twice as far as B, so that A must reach O at the very instant when B reaches Q. Proceeding on the other side of their respective points of rest, their velocities will be gradually diminished, that of A always being double that of B; and the two velocities will be extinguished simultaneously when the two bodies have reached distances equal severally, to OA and QB. Thus it is clear that the two oscillations would go on together, and, as the same argument would hold for any other ratio of OA to QB, that the oscillations are performed in the same time, whatever may be their extents.

In order to obtain a faultless time-keeper, Huygens saw that some contrivance must be made for rendering the redressing tendency exactly proportional to the distance from the mean position. Now, the pressure required to bend a spring is proportional to the extent of flexure; and so, by attaching one end of a slender spring to the axis of the balance, the other end being fixed to the frame-work, Huygens at once brought the construction of watches to perfection as far as this part of their theory goes.

For the purpose of rendering the oscillations of a pendulum isochronous, he remarked that the tendency of a body to descend along a curve varies as the sine of the angle which the curve makes with the horizon; so that if a curve can be found such that the length of its arc, counted from the lowest point, is proportional to the sine of its obliquity, the oscillations of a heavy body describing that curve would be isochronous. Now, the cycloid, or curve described by a point in the circumference of a wheel which is rolled along a straight line, possesses this property; so that it only remained to contrive some mechanical arrangement which might produce a cycloidal motion. Huygens availed himself of the fact, that the evolute of a cycloid is another cycloid of equal dimensions. Let BOC (fig. 2) be the cycloid generated by rolling the circle of which DO is the diameter along the straight line BDC, and having produced OD to the equal distance DA, place BA and AC, two half cycloids, equal to OC and OB. Then, if a tangent be applied at any point T in the half cycloid AC, and be continued to meet OC in P, the length of the tangent TP is exactly equal to that of the cycloidal arc TC. Hence, if a plate of wood or metal be shaped into the form Pendulum. ATC, and if a string fastened at A, and equal in length to AO, be partially wound upon the edge AT of the plate, and G a minute heavy particle, constrained by the thread Pendulum, to move in the circular arc QOQ'. If the weight be brought aside to the point Q, and then let go, it will descend, gaining speed until it reach the lowest point O; then rising along the arc QOQ' its velocity will gradually diminish until, at a distance OQ', equal to OQ, it become zero, when the body will begin immediately to descend again towards O. If there were no resistance from friction or from the air, this oscillation would continue for ever.

For the purpose of computing the time in which the pendulum performs its oscillation, we observe that, according to the well-known laws of mechanics, the velocity which the body acquires in descending from Q to G along the curve, is exactly equal to that which it would have acquired in falling from B to C. There is thus no difficulty in telling the velocity of the pendulum at each point of its arc; but it is by no means so easy to compute the time during which, with this ever-changing velocity, the pendulum will pass from Q to O.

If g represent the intensity of gravitation as measured by the velocity which a falling body acquires in one second of time, and if h denote the height through which the body has fallen, its velocity is given by the well known formula

\[ v = \sqrt{2gh} \]

so that if h be taken for BC, the same expression gives the velocity which the pendulum has as it passes the point G.

If we put L for the length of the thread AG, A for the angle OAQ, a for OAG, we have BC=L (cos a - cos A), and therefore

\[ v = \sqrt{2gL} [\cos a - \cos A]^{\frac{1}{2}} \]

Assume now some small arc GH, and the time which the heavy body would take to pass from G to H would be \( \frac{GH}{v} \), if its velocity did not change during that time. Actually the time must be somewhat less than this, because the velocity at H is somewhat greater than that at G; but the smaller this distance GH is taken, the less change there will be in the velocity, and therefore the more nearly will the fraction \( \frac{GH}{v} \) express the time occupied in passing from the one point to the other. In order to compute the whole time, we must suppose the arc QO to be divided into a great number of parts; compute the time for each, and add all these times together; and, that this computation may have any pretension to exactitude, the parts must be made very numerous. It is impossible to accomplish such a calculation without the aid of the infinitesimal method.

Let then \( \delta a \) represent the minute angle GAH, and \( \delta t \) the corresponding element of time; then, since GH=L\( \delta a \), we have

\[ v = \frac{L\delta a}{\delta t}, \quad \text{and} \quad \delta t = \sqrt{\left(\frac{L}{g}\right)\left[\cos a - \cos A\right]}^{-1} \delta a \]

which gives the differential of the time, and which has to be summed or integrated in order to give the whole time.

Considerable artifice is required in order to effect the integration of this formula. Putting A=2B, a=2\( \beta \), whence \( \cos A = 1 - 2 \sin^2 B \), \( \cos a = 1 - 2 \sin^2 \beta \); it becomes changed into

\[ \delta t = \sqrt{\left(\frac{L}{g}\right)\left[1 - \sin^2 B \sin^2 \gamma\right]}^{-1} \delta \gamma \]

Assume now an angle \( \gamma \), such that

\[ \sin \beta = \sin B \cdot \sin \gamma \]

and equation (4) becomes transformed into this one—

\[ \delta t = \sqrt{\left(\frac{L}{g}\right)\left[1 - \sin^2 B \sin^2 \gamma\right]}^{-1} \delta \gamma \]

or, developing by help of the binomial theorem, \[ \delta t = \sqrt{\frac{L}{g}} \left[ 1 + \frac{1}{2} \sin B^2 \sin \gamma^2 + \frac{1}{8} \sin B^4 \sin \gamma^4 + \cdots \right] \delta y, \]

each separate term of which can be integrated by help of the formula

\[ \int \sin \gamma^2 \cdot \delta y = \frac{n-1}{n} \int \sin \gamma^{n-2} \delta y - \frac{1}{n} \sin \gamma^{n-1} \cos \gamma. \]

The result becomes, when arranged,

\[ t = \sqrt{\frac{L}{g}} \left[ \gamma \left[ 1 + \left( \frac{1}{2} \right)^2 \sin B^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \right)^2 \sin B^4 + \cdots \right] - \cos \gamma \sin \gamma \left[ \left( \frac{1}{2} \right)^2 \sin B^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \right)^2 \sin B^4 + \cdots \right] - \frac{3}{4} \cos \gamma \sin \gamma \left[ \left( \frac{1}{2} \cdot \frac{3}{4} \right)^2 \sin B^4 + \cdots \right] + \cdots \right]. \]

This complex formula gives us the interval of time during which the pendulum describes the arc OG, corresponding to the angle OAG = 2β. In order to find the entire time of describing the arc OQ, we must put γ = 90°, in which case β becomes equal to B; this gives cos γ = 0, and

\[ \frac{1}{2} T = \sqrt{\frac{L}{g}} \left[ \frac{1}{2} \left( 1 + \left( \frac{1}{2} \right)^2 \sin B^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \right)^2 \sin B^4 + \cdots \right) \right]; \]

or, denoting by T the whole time of a beat,

\[ T = \sqrt{\frac{L}{g}} \left[ 1 + \left( \frac{1}{2} \right)^2 \sin B^2 + \left( \frac{1}{2} \cdot \frac{3}{4} \right)^2 \sin B^4 + \cdots \right]. \]

for the time of passing from Q to Q'. From this the time of an oscillation can be very readily computed, since, in all practical cases, the angle B is very small.

If we imagine the extent of an oscillation to be exceedingly minute, the terms containing the powers of sin B may be neglected, and we have the usual formula

\[ T = \pi \sqrt{\frac{L}{g}}, \quad \text{or} \quad gT^2 = \pi^2 L. \]

which gives the time of oscillation of an ideal pendulum, consisting of a single heavy point supported by a thread having no weight, and making oscillations imperceptible in extent.

From this equation it follows, that the lengths of simple pendulums are proportional to the squares of their times: for example, a pendulum swinging thirty times per minute must be four times as long as the common seconds pendulum, while a half-seconds pendulum must only have one-fourth part of the length.

According to observations made on moving bodies by means of Attwood's machine, the value of \( g \) comes out to be about 32 English feet, or 384 inches; wherefore the length of a simple pendulum vibrating in seconds should be

\[ \frac{384}{(3^2 \cdot 16)^2} = 39 \text{ inches nearly}. \]

But neither observations made directly on falling bodies, nor those made by help of Attwood's machine, are susceptible of precision, while, as we shall see, measurements of the pendulum can be made with very great nicety; and thus, instead of deducing the length of the pendulum from the value of \( g \), we derive an accurate knowledge of the intensity of gravitation from experiments made with the pendulum.

How minute soever the arc of vibration may be, the time of describing it must be longer than the result obtained from equation (10); and our first business, in attempting to deduce any accurate results from experiments made with the pendulum is, to determine the effect of the amplitude of the arc.

By help of equation (9), the following table has been computed, showing the time in which a pendulum will oscillate through arcs of 2°, 4°, 6°...up to 20°, that of making an infinitely small oscillation being taken as the unit:

| Half Arc | Time of Oscillation | |---------|---------------------| | 0° | 1:00000 00000 | | 1 | 1:00001 90389 | | 2 | 1:00007 617208 | | 3 | 1:00017 57412 | | 4 | 1:00039 47024 | | 5 | 1:00047 61725 | | 6 | 1:00058 58201 | | 7 | 1:00063 38894 | | 8 | 1:00121 98324 | | 9 | 1:00154 43092 | | 10 | 1:00199 71881 | | 11 | 1:00230 83457 | | 12 | 1:00274 84654 | | 13 | 1:00374 70434 | | 14 | 1:00474 55777 | | 15 | 1:00430 05792 | | 16 | 1:00489 57658 | | 17 | 1:00533 00642 | | 18 | 1:00620 30066 | | 19 | 1:00691 65459 | | 20 | 1:00766 90258 |

The first column in each of these tables contains the half arc of vibration, or the extreme angular distance from the vertical line. The second column of the first table contains the time of an oscillation, with the first and second differences for the purpose of interpolating; and that of the second, the excess of the apparent day (or twenty-four hours as shown by the clock) over the true day (or that which would be shown by a clock of which the oscillations are imperceptible).

From these tables we see the importance of having a clock pendulum arranged to make small oscillations. If, to take an extreme example, we had a pendulum kept swinging at a distance of 20° on each side of the vertical line, and had it regulated to go to true time; and if, by the thickening of the oil, or some analogous change in the maintaining power, the oscillations were reduced to 1°, the clock would gain upon true time by 64°567, since the number of beats per day would be increased in the ratio of 1:00691 65459 : 1:00766 90258. But if, by augmenting the load on the pendulum, and, of course, properly modifying the escapement, the arc of vibration were reduced to only 1° on each side of the vertical line, and the clock again adjusted to go in true time; and thereafter, if a change in the maintaining power were to take place, so as to reduce the arc by one-twentieth part as before, the beats

Pendulum of the clock would be increased in the ratio of \(1:0000172491\) to \(1:0000190389\), and there would be a daily gain of \(0:154636\). Thus a variation in the intensity of the maintaining power would occasion four hundred times as great an error on the pendulum with the long sweep as on that with the short one; the unsteadiness arising from this source being proportional nearly to the square of the amplitude.

"This may be confirmed by a very simple but beautiful experiment. Having suspended a leaden ball by means of a slender thread, let this simple pendulum be put in motion, so that the ball may describe a curve known to bear a considerable resemblance to the ellipse. If the times of vibration along the two axes of this curve were exactly equal to each other, the ball would repeatedly retrace the same orbit; but these times of vibration are different, and during the passage from end to end of the long axis the ball has more than returned to its position in reference to the short one, so that the axes of the orbit are gradually displaced in the direction of the movement of the ball." (Edin. New Phil. Jour., vol. xv., p. 140.)

Having now investigated the law of the motion of an imaginary simple pendulum, we have to examine the case of a real one, consisting of parts, each moving with its own relative velocity. We shall first consider it as moving in a vacuum, and examine afterwards the effect of the air.

The weights of the various parts of the compound pendulum produce a tendency to turn it upon its axis, exactly equal to that which would be produced by the whole weight of the pendulum acting at its centre of gravity. But the quantity of motion existing in a moving body is greater than that which would have existed in it if all concentrated at its centre of gravity, by the motion of rotation which it would have had if turning simply with the same angular velocity on an axis passing through the centre of gravity; wherefore, in every possible case, the oscillations of a pendulum must be slower than those of a simple pendulum represented by the centre of gravity of the compound one.

If \(A\) (fig. 4) be the axis of motion supposed perpendicular to the plane of the paper, and \(G\) the centre of gravity of a compound pendulum, its motion may be represented thus:

From \(G\), with the distance \(GR\) equal to the mean distance of gyration of the mass, describe a circle; then if we suppose the whole weight of the pendulum to be distributed uniformly round the circumference of this circle, the motions of this imaginary ring will represent those of the compound pendulum.

When the point \(G\) is moving along the arc \(QGQ'\), with the velocity \(v\), the force of translation is \(v^2W\), \(W\) being the whole weight of the pendulum; but the velocity of rotation of the ring \(R\) is \(\frac{v}{AG}\), or, symbolically, \(\frac{v}{T}\); putting \(GR = R\), \(AG = l\); therefore the rotatory motion is

\[ \frac{v^2R^2}{l} W; \]

and therefore the whole motion existing in the moving mass is

\[ We^{\frac{l^2 + R^2}{l}} = F. \quad \ldots \quad (11.) \]

Now the total quantity of motion in any system, as measured by combining the weight of each part with the square of its velocity, is proportional to the quantity of work which has produced that motion, the work being estimated by combining each pressure with the distance through which it has acted, so that in any system whatever

\[ 2g \cdot \text{work} = \text{motion}. \]

But the pressure in this case is \(W\), and the distance through which it has acted is \(l (\cos \alpha - \cos A)\); so that \(WI (\cos \alpha - \cos A)\) represents the quantity of work expended in producing the motion \(F\). Therefore

\[ 2g WI (\cos \alpha - \cos A) = We^{\frac{l^2 + R^2}{l}}; \quad (12.) \]

or, putting for \(v\) its equivalent \(l \frac{\delta \alpha}{\delta t}\),

\[ \delta t = \sqrt{\left(\frac{l^2 + R^2}{2g}\right) [\cos \alpha - \cos A]^{-1} \delta \alpha}. \quad (12.) \]

If, then, we make \(GS\) a third proportional to \(AG\) and \(GR\), and put \(AS = L\), or

\[ L = l + \frac{R^2}{l}, \quad \ldots \quad (13.) \]

we shall have

\[ \delta t = \sqrt{\left(\frac{1}{2g}\right) [\cos \alpha - \cos A]^{-1} \delta \alpha}, \]

which is an exact copy of equation (3); and therefore we conclude that the compound pendulum, of which \(GR\) is the mean distance of gyration, will oscillate in exactly the same time as a simple pendulum of which the length is \(AS\).

From this equation (12) it appears that the oscillations of a compound are similar to those of a simple pendulum, and that therefore the tables which we have given apply to actual clocks.

But the investigation also shows clearly that the oscillations of a compound cycloidal pendulum cannot be isochronous; for when the pendulum is at the middle, the quantity of motion is augmented in the ratio of \(AO^2 + R^2 : AO^2\), whereas, when the pendulum is in the direction \(TP\), the augmentation is in the ratio of \(TP^2 + R^2 : TP^2\), which is a higher ratio; so that this circumstance ought to be taken into account when we investigate the isochronism of a compound pendulum.

The point \(S\) is called the centre of oscillation, or that point at which, if the whole mass of the pendulum were supposed to be concentrated, the time of oscillation would not be changed; and we have this property, that the rectangle under the distances of the point of suspension, and of the centre of oscillation from the centre of gravity, is equivalent to the square of the mean distance of gyration, and is therefore constant for the same pendulum; that is \(AG \cdot GS = GR^2\).

From this law Huygens concluded that the point of suspension and the centre of oscillation are interchangeable; in other words, if the pendulum were suspended on an axis passing through the point \(S\), its time of oscillation would be the same as when suspended from \(A\).

Captain Kater proposed to utilize this property by employing it to determine the exact length of a simple pendulum vibrating in the same time with a compound one. For this purpose he placed two knife-edges exactly parallel to each other, one at \(A\), the other at \(S\), and carefully adjusted the weights of the parts by repeated trials, until he found that the two times of oscillation were alike; then, measuring the distance between the two knife-edges, he obtained the length of the corresponding simple pendulum; from which the intensity of gravitation can be computed.

This beautiful process is the only one available for determining with great nicety the length of the seconds pendulum, since it avoids the difficult and unsatisfactory operation of measuring with great precision the dimensions of the various parts; it is, however, subject to several sources of minute error, which have either to be guarded against or allowed for.

The first of these sources of error is the wearing of the knife-edges, which gradually changes the points of suspension, and renders a new adjustment necessary; and it is a very important question, whether the distance between two blunted knife-edges on which the oscillations are performed in equal times be truly the length of the corresponding simple pendulum. M. Laplace demonstrated that the blunting of the edges does not impair the accuracy of the convertible pendulum.

For the purpose of examining thoroughly into this matter, let the pendulum be suspended on a cylinder, of which the axis is A (fig. 5), and the radius AM = p; this cylinder rolling upon a horizontal plane KL. In this arrangement the point G must describe a convoluted cycloid, while A moves in a horizontal line.

The linear motion of the point G is composed of two motions, one perpendicular and proportional to AG, represented by l, δa, and the other horizontally, on account of the motion of the centre A, represented by ρ, δa. The angle of these two motions being α, it follows that the square of the actual linear motion of G is \( \frac{1}{2} (l^2 + \rho^2) \cos^2 \alpha + \rho^2 \delta a^2 \); therefore the motion of translation is

\[ W \left[ \frac{\partial}{\partial t} \left( \frac{l^2 - 2l\rho \cos \alpha + \rho^2}{g} \right) \right]^2 \]

and the entire quantity of motion in the system

\[ W \left[ \frac{\partial}{\partial t} \left( \frac{l^2 - 2l\rho \cos \alpha + \rho^2 + R^2}{g} \right) \right]^2; \]

while the descent of the centre of gravity is as before, \( l(\cos A - \cos a) \), so that the equation of motion becomes, putting, as before, \( A = 2B, a = 2\beta \),

\[ \frac{\partial}{\partial t} \left( \frac{1}{gl} \right) \left[ \frac{(l-\rho)^2 + R^2 + 4l\rho \sin \beta}{\sin B^2 - \sin \beta^2} \right] \frac{\partial}{\partial \beta} \]

or, making \( \sin \beta = \sin B \cdot \sin \gamma \), and putting, for shortness sake,

\[ \frac{4l\rho}{(l-\rho)^2 + R^2} = k \]

\[ \frac{\partial}{\partial t} \left( \frac{1}{gl} \right) \left[ \frac{1 + k \sin B^2 \sin \gamma^2}{1 - \sin B^2 \sin \gamma^2} \right] \frac{\partial}{\partial \gamma}. \]

Expanding the variable part of this expression according to the powers of \( \sin \gamma \), integrating each term, and taking the integral between the limits \( \gamma = -\frac{\pi}{2}, \gamma = \frac{\pi}{2} \), we have for the time of an oscillation—

\[ T = \sqrt{\frac{(l-\rho)^2 + R^2}{gl}} \left[ 1 + \left( \frac{1}{2} \right)^2 \sin B^2 \left[ 1 + \frac{1}{2} \cdot \frac{1}{2} \cdot k \right] \right. \]

\[ + \left( \frac{1}{2} \cdot \frac{3}{2} \right)^2 \sin B^2 \left[ 1 + \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot k - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot k \right] \]

\[ + \left( \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{5}{2} \right)^2 \sin B^2 \left[ 1 + \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot k - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot k \right] \]

\[ + \left( \frac{1}{2} \cdot \frac{3}{2} \cdot \frac{5}{2} \cdot \frac{7}{2} \right)^2 \sin B^2 \left[ 1 + \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot k - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot k \right] \]

\[ + \cdots \text{etc.} \]

the law of progression of which is obvious.

This formula gives the time of oscillation of a pendulum suspended on a cylinder, without any restriction as to the radius of curvature. From it we see that the correction for amplitude is not the same as for a pendulum hung by a perfect knife-edge, and that therefore the motions of no simple pendulum can strictly represent those of this one.

When the value of \( k \) is small,—that is to say, when AM is minute in comparison with AG, and when the angle B also is very small,—the expression becomes

\[ T = \sqrt{\frac{l^2 - 2l\rho + R^2}{gl}} \left[ 1 + \frac{1}{2} \sin B^2 (1 + k) \right] \]

and, if the oscillations be supposed to be infinitesimally minute,

\[ T = \sqrt{\frac{l^2 - 2l\rho + R^2 + \rho^2}{gl}} \]

which agrees with those of a simple pendulum of which the length is

\[ L = l + \frac{R^2 + \rho^2}{l} - 2\rho. \]

If, then, it were proposed to place the cylinder A at such a distance from the centre of gravity as that the time of a minute oscillation may be equal to that of a simple pendulum having the length \( L \), we should have to solve the quadratic equation—

\[ P = (L + 2\rho)l + R^2 + \rho^2 = 0. \]

Now, according to the properties of such equations, if \( l \) and \( \lambda \) be the two roots, their sum must be \( L + 2\rho \), and their product \( R^2 + \rho^2 \), or

\[ l + \lambda = L + 2\rho; \quad lA = R^2 + \rho^2. \]

Now \( l + \lambda \) represents the distance between the centres of two cylinders placed on opposite sides of the centre of gravity; and thus, since

\[ l + \lambda - 2\rho = L, \]

the distance between the surfaces of the cylinders is equal to the length of the corresponding simple pendulum.

This investigation is free from all limitations as to the magnitude of the cylinders; so that if carefully-turned and polished cylinders were substituted for the knife-edges, and if the corrections for the amplitudes of the arcs of vibration were made according to the formula (16), the results would be rigorously exact.

When the curvatures of the two edges are not alike, an error is introduced, because the one distance is then the major root of the equation

\[ P = (L + 2\rho)l + R^2 + \rho^2 = 0, \]

while the other is the minor root of another equation

\[ P = (L + 2\rho)l + R^2 + \rho^2 = 0. \]

The sum of these roots, when \( \rho \) and \( \rho' \) are minute, approaches sensibly to

\[ l + \rho + \rho', \]

so that in practical cases the fact of the edges being blunted does not vitiate the results.

The next source of error which we shall consider is the buoyancy of the air in which the experiments are made.

If the pendulum were composed entirely of one kind of material, its centre of gravity would coincide with the centre of buoyancy, and the coefficient \( W \) in the first member of equation (12) would have to be replaced by \( W - A \), \( A \) being the weight of the displaced air. But when the pendulum is made of various materials, the inquiry becomes a little more difficult. It will be enough to consider those cases in which the centre of buoyancy, the centre of gravity, and the points of suspension, lie all in one straight line. Let then B (fig. 6) be the centre of buoyancy, its distance from G being represented by \( e \); then the motion which the system has acquired being due to the descent of the weight \( W \) through the distance \( l(\cos a - \cos A) \) less that of \( A \) through \( (l - e)(\cos a - \cos A) \), we must have \[2g\left(W - A\right) \cos a = \frac{d^2}{dt^2} \left(\cos a - \cos A\right)\]

wherefore the length of a corresponding simple pendulum vibrating in vacuo is

\[L = \frac{W \left(F + R^2\right)}{\left(W - A\right) t + Ae}\]

(19)

If then we desire to place \(A\) so that the oscillations may agree with those of the simple pendulum \(L\), we must solve the quadratic equation

\[F - \left(1 - \frac{A}{W}\right)L + R^2 - \frac{A}{W}Le = 0.\]

(20)

The two roots of this equation will not represent the one \(GA\) on the one side and the other \(GA'\) on the other side of the centre of gravity, but two distances on one side, either of which will satisfy the condition of oscillation.

When the pendulum is inverted to be suspended from the point \(A'\), the equation becomes, putting \(A\) for \(GA'\),

\[\lambda^2 - \left(1 - \frac{A}{W}\right)L + R^2 + \frac{A}{W}Le = 0.\]

(21)

The major root of the one and the minor root of the other being taken, we have the proper values of \(l\) and \(\lambda\), viz.,

\[l = \frac{W - A}{2W}L \pm \sqrt{\left(\frac{W - A}{2W}L\right)^2 - R^2 + \frac{A}{W}Le}\]

\[\lambda = \frac{W - A}{2W}L \mp \sqrt{\left(\frac{W - A}{2W}L\right)^2 - R^2 - \frac{A}{W}Le}\]

the sum of which is no longer independent of the specialties of construction of the pendulum.

If the centre of buoyancy coincide with the centre of gravity, \(e\) becomes zero, and

\[l + \lambda = \frac{W - A}{W}L;\]

(22)

from which the length of the simple pendulum can be easily deduced; but in no other case can the reversible pendulum vibrating in air give a result dependent only on the measurement of the distance between the knife-edges.

While the air modifies the vibrations of a pendulum by its buoyancy, it also influences them by its resistance to motion. The law of this resistance, like all the laws connected with the motion of fluid bodies, is very imperfectly known; no method of analysis has been discovered which can at all approach the difficulties of the subject; nor has any glimpse yet been obtained of the internal arrangement of the parts of fluids, so that we have not even a foundation on which to build a train of reasoning.

The resistance which the air presents to a moving body is usually supposed to be proportional to the square of the velocity, and also, with similar solids, to the extent of surface; but there is some reason to believe that both proportions are only approximate. Even when assisted by this supposition, the powers of the higher calculus fail in discovering what effect this resistance has on the going of a clock. The equation of motion, when expressed in the notation of Leibnitz, takes the form

\[\frac{d^2a}{dt^2} = P \sin a - Q \left(\frac{da}{dt}\right)^2\]

which is not integrable by any known process.

Attempts have been made to overcome this difficulty by assuming that the resistance is exceedingly small in comparison with the tendency of the pendulum to descend along the arc; and it has been shown that, in this case, the extent, but not the time, of an oscillation is sensibly affected. Yet all such demonstrations want conclusiveness, since the result extracted may be only a disguised form of that assumption which has been arbitrarily made in order to bring Pendulum the matter apparently within the power of the calculus.

Although we cannot ascertain the precise effects of the air's resistance, we may, by a general view of the subject, arrive at some useful conclusions.

On comparing the motion of a pendulum describing a large arc \(QQ'\) (fig. 7), with that of another of the same dimensions oscillating through the smaller arc \(qq'\), we perceive that, generally speaking, the velocities of the two are proportional to the lengths of the arcs; wherefore the intensities of the air's resistance must be nearly proportional to the squares of those lengths. Now, the redressing tendencies are proportional to the sines of the arcs; wherefore it follows that the air's resistance bears a less proportion to the redressing tendency in the case of the small arc than in that of the large one, which forms another argument in favour of a small arc of vibration.

While the intensity of the resistance is proportional to the square of the length of the arc, the distance through which it acts is proportional to that length, wherefore the quantity of motion destroyed, or the loss of force, is proportional to the cube of the length. Now, if the pendulum, after having been dropped from \(Q\), rise only on the other side to the distance \(OR\), the loss of force is proportional to the distance \(SR\), by which \(R\) is lower than \(Q\); and therefore, we have, approximately

\[SR : sr :: QR^3 : qr^3.\]

Again, if we assume that the arcs \(QR, qr\) are so small as to be undistinguishable from their tangents, the trigons \(QSR, qsr\) are similar to \(QBA, qba\); so that if the arcs \(OQ, oq\) themselves be small,

\[QR : qr :: OQ^3 : oq^3\]

approximately.

If, then, the oscillations of a pendulum be slowly arrested by the air's resistance alone, the successive amplitudes must form a progression in which the differences are proportional to the squares of the terms; that is, they must form a harmonic series, and the number of beats occurring between the amplitudes \(OQ = A\) and \(oq = a\) must be proportional to the difference of the inverse powers of \(a\) and \(A\), or to

\[\frac{1}{a} - \frac{1}{A}.\]

It has been erroneously asserted that the arcs should form a decreasing geometrical progression. In virtue merely of the air's resistance, then, the pendulum would never be brought absolutely to rest.

In addition to this resistance there is the friction on the knife-edge, which is nearly constant. When the pendulum is inclined at the angle \(a\), its pressure upon the support is \(W \cdot \cos a\), and therefore if \(f\) be the coefficient of friction for the particular knife-edge, the redressing tendency is \(W (\sin a - f \cos a)\); and if we put \(f = \tan \phi\), \(\phi\) being then the angle of repose, or the minute angle through which the pendulum may be drawn aside from the vertical position without tending to return, the redressing tendency becomes

\[\frac{W}{\cos \phi} (\cos \phi \sin a - \sin \phi \cos a) = W \sec \phi \cdot \sin (a - \phi).\]

From this it is apparent that each oscillation will be performed on either side of a line inclined to the vertical line by the angle \(\phi\), so that the amplitude of the oscillations will form a decreasing arithmetical progression, of which the common difference is \(2 \phi\). From the same expression it follows, that the time of an oscillation will be reduced in the ratio of \( \sqrt{\cos \phi} \) to 1; or that, in order to make it vibrate in true time, the pendulum must be lengthened in the ratio of sec \( \phi \) to unit.

In all practical cases, however, the angle \( \phi \) is exceedingly minute, so that its cosine and secant cannot differ from the radius by any appreciable quantity; in other words, we may hold that the friction on the knife-edge has no perceptible influence on the time-keeping.

When both the air's resistance and the friction act together, the amplitudes must form a complex progression, approaching more and more nearly to an arithmetical one as the arcs become shorter.

A more serious cause of error in the determination of the length of the pendulum, is to be found in the yieldingness of the support. In our investigations, the point of suspension has been supposed to be absolutely immovable; but there is no substance which does not yield to the pressure applied to it, and therefore, as the pendulum swings from side to side, the point of suspension oscillates also; and the whole framework becomes truly a part of the vibrating mass.

Hence the propriety of securing the suspending plate to a strong wall; and hence also the impropriety of the too common practice of making the house-clock rest on the bottom of its case.

When a pressure is applied to some point in an elastic structure, the form of the structure is deranged, and the point to which the pressure is applied is displaced; but the displacement is not necessarily in the direction of the pressure. Every elastic structure has, in reference to any given point in it, three directions at right angles to each other, in one of which the flexibility is a maximum, in a second a minimum, and in the third a maximum or minimum. And it is only when the pressure is applied in one of these three directions that the displacement is in the direction of the pressure. Hence, a general inquiry into the effect which the flexibility of the support may have upon the motion of the pendulum would be excessively complicated. In order to simplify it, we shall assume that the three directions alluded to are one vertical, one horizontal in the plane of the oscillation, and the third also horizontal, but perpendicular to the plane of oscillation; and this supposition is admissible, because of the symmetry of the parts.

The pressure, \( W \cos a \), which the knife-edge exerts against the suspending plate may be decomposed into two pressures, \( W \cos a^2 \) in a vertical, and \( W \cos a \sin a \) in a horizontal direction; so that if \( \mu \) and \( m \) be the coefficients of flexibility in those two directions, \( \mu W \cos a^2 \) will be the depression, and \( mW \cos a \sin a \) the horizontal displacement of the point of suspension in virtue of this pressure.

But the point of suspension does not take up truly the position indicated by these quantities, because the matter of the framework has to be moved; and the motion imparted to it forms part of the sum total. Yet, as the quantity of motion is measured by combining the square of the velocity with the mass, and as the whole displacement is microscopic, the quantity of motion in the framework must be excessively minute, and we may assume that the above expressions represent truly the position of the support corresponding to the inclination \( a \).

The differentials of these are

\[ \mu W \sin 2a \cdot \delta a, \quad \text{and } mW \cos 2a \cdot \delta a; \]

wherefore the motion of the centre of gravity of the pendulum is composed of three motions,—\( \delta a \) at an inclination of \( a \) to the horizon, \( \mu W \sin 2a \cdot \delta a \) upwards, and \( mW \cos 2a \cdot \delta a \) horizontally outwards; so that the square of its actual motion is, neglecting those terms which contain the squares of \( m \) and \( \mu \),

\[ \left( \frac{R^2 + P + 2lW(m \cos 2a + \mu \sin a \cdot \sin 2a)}{2gl(\cos a - \cos A)} \right)^2 \delta a^2; \]

and therefore the differential equation of motion is

\[ \ddot{a} = \frac{R^2 + P + 2lW(m \cos 2a + \mu \sin a \cdot \sin 2a)}{2gl(\cos a - \cos A)} \delta a, \]

and the integration of this expression would give the value of the time \( t \).

When the arc of vibration is minute, the length of a simple pendulum vibrating in the same time becomes

\[ L = \frac{R^2 + P + 2lW}{l}; \]

and therefore, if we wish to determine that value of \( l \) which will produce a given time of oscillation, we must resolve the quadratic equation

\[ P - (L - 2mW)l + R^2 = 0. \]

Now, if \( l \) and \( \lambda \) be the two roots of this equation, we must have

\[ l + \lambda = L - 2mW; \]

and thus the distance between the knife-edges of a reversible pendulum is less than the true length of the simple pendulum by \( 2mW \), \( m \) being a coefficient depending on the flexibility of the supports.

This equation suggests a method of discovering the amount of this inaccuracy; for if two pendulums of different weights be swung in succession from the same support, the values of \( L \) deduced from them will differ by the quantity \( 2m(W - W') \), and that difference will give the value of the coefficient \( m \).

We have all along supposed the parts of the pendulum to be perfectly rigid, and have now to inquire whether the flexibility of the rod be not another source of error. This part of the subject has not received an adequate share of attention; and, indeed, the method employed by Borda for determining the length of the pendulum seems to have been contrived in neglect of the errors occasioned by the flexibility of the parts.

In order to put this matter in a clear light, let us construct a pendulum AXY (fig. 8), composed of a fine inflexible line carrying weights X, Y, &c., at various distances, AS being the length of the corresponding simple pendulum. Then, as this system oscillates, the changes in the motion of a particle placed at S are exactly those which gravitation would induce on that particle if suspended separately from the point A; the whole of its tendency to descend along the arc is expended in accelerating or retarding its motion. But it is different with a particle placed at X, above S; the actual motions of this particle are less rapid than they would have been if it were attached to A by a separate thread, in the ratio of AX to AS; and therefore only a part of the weight X, represented by \( X \cdot \frac{AX}{AS} \), is employed in accelerating or retarding its motion, the remainder \( X \cdot \frac{XS}{AS} \) being resisted by the stiffness of the pendulum-rod. That is to say, the body X tends to bend the pendulum-rod inwards by a pressure \( X \cdot \frac{XS}{AS} \sin a \).

In the same way, a body placed at Y, below S, has its motions made more rapid than they would have been if it were separately suspended, and therefore it presses outwards on the rod with an intensity of \( Y \cdot \frac{YS}{AS} \sin a \).

It thus seems that the structure of the pendulum is subjected to variable internal strains, caused by the motions of Every body has three principal axes passing through the Pendulum centre of gravity at right angles to each other, on which its motion of rotation, with a given angular velocity, is maximum or minimum; and the amount of rotation on any other axis varies according to its position in reference to these three. On account of the symmetry of the parts in all actual cases, one of the principal axes may be assumed as lying along the length of the pendulum; one of the other two being in the plane of oscillation, the third at right angles to it.

The bob, or heavy part of the pendulum, is sometimes made flat or lenticular, the rod and other parts partaking of the same shape. In such case, the amount of rotation on the axis parallel to the knife-edge is greater than on that perpendicular to it, and any twist on the pendulum-rod causes the rate of the clock to vary; and, particularly when the rod is of wood, this form of bob causes great irregularity. In order to avoid this source of error, and to render innocuous any deviations of the knife-edges in azimuth, the mass of the pendulum should be distributed uniformly round its vertical axis, so as to render the amount of rotation constant for every horizontal axis passing through the centre of gravity.

The convertible pendulum is not kept in motion by machinery, like the pendulum of a clock, but having been set in motion, is allowed gradually to come to rest; and as it continues to oscillate for but a short while, the number of oscillations which it would make per day has to be ascertained by comparing its movements with those of a clock pendulum of which the rate is known.

In order to make this comparison, the convertible pendulum is erected in front of the clock, a telescope of low power being fixed at a little distance. A distinct white mark is made on the clock pendulum, of size sufficient to be just hid from view by a part of the convertible pendulum when both are at rest. If now the two pendulums were to oscillate in the same direction, keeping true time with each other, the white mark would be always concealed; but, if, as is purposely arranged, the one pendulum oscillate a little more slowly than the other, they will gradually separate from each other, and the white mark will become visible. As the separation increases to become one beat, the pendulums pass each other in different directions, and the mark is covered for too short a time to prevent its being visible; but as the difference approaches to be two beats, the motions again agree, and the spot is hid. Thus, by noticing the successive instants of the complete disappearance of the mark, the observer can discover the difference between the rates of the two pendulums with great precision.

On account, however, of the diminution of the amplitude, the rate of the free pendulum slowly accelerates; and thus the intervals between the successive disappearances are unequal. Besides this, the velocities at the middles of the arcs become different, and the disappearance is not complete; the most favourable time for observation being when the apparent velocities of the parts, as seen from the telescope, are exactly equal to each other. It is thus a matter of very considerable difficulty to determine with precision the ratio of the beat of the free pendulum to that of the clock; and, in addition, there is the error to which the rate of the clock itself is subject.

Besides all these sources of minute error, which are special to the subject, there is that which attends all attempts at minute measurement, and which arises from thermal expansion. The comparison of the distance between the two knife-edges with the divisions on a standard scale, at a given temperature, requires an accurate knowledge of the rates of expansion of the two rods, and during the motion of the pendulum an account must be taken of the temperature. Now, the thermometer only can give the temperature

of the surrounding air, which only agrees with that of the pendulum when the changes are very slow; and therefore it is of the utmost importance, for the exactitude of the conclusions, that all experiments be carried on in a place not liable to fluctuations of temperature.

On reviewing all these sources of error, we need not be surprised to find disagreements among the results obtained by different observers, particularly when we consider that the expressions for the lengths of the pendulum have been carried to the ninth decimal place, that for the thousand-millionth part of a metre. It is matter of astonishment to find the results agree so well.

The determination of the length of the pendulum in different latitudes throws a great light upon certain departments of physical astronomy; and it is this circumstance which gives importance to the present subject. Among the experiments which have been recorded, there are many close coincidences, and not a few discrepancies, in the last-place figures. From these latter, attempts have been made to estimate irregularities in the structure of the earth; while from the former the amount of its oblateness is inferred. In this way, the pendulum becomes an important and valuable instrument for astronomical research.

For the purpose of discovering the variation of gravity at different parts of the earth's surface, it is sufficient to compare the number of oscillations which a pendulum makes per day at one place with the number which the same pendulum makes at another place; the intensities of gravitation at the two places are inversely as the square of these numbers.

Whichever of these two pendulums may be used, the trustworthiness of the result depends mainly upon the means which are taken to compare the oscillations with the length of the day. Though it be customary to use the mean solar day as the standard, the comparisons are really made with the sidereal day, that interval of time being, of all others, the most susceptible of exact and ready measurement. Now, with a four-foot portable transit instrument, the instant of the passage of a star over the meridian can hardly be determined to within one-tenth part of a second, the distance passed over by the image of an equatorial star in that time being little more than the three-thousandth of an inch upon the wires, and the position of the instrument being liable to errors of collimation, of leveling, and of azimuth. We cannot therefore venture to say that the clock's rate in a given day can be determined certainly to within one-tenth part of a second, although the comparisons have been made at an interval of twenty-four hours. Seeing, then, that the free pendulum is compared with the clock only over a small fraction of the day, it is a great deal to expect that its daily rate can be ascertained to within one second of time.

A change of one second per day in the rate of a clock corresponds to a change of \( \frac{1}{365} \) th in the length of the pendulum, which is about \( \frac{1}{1000} \) th of an inch, or \( \frac{1}{400} \) th of a millimetre; and therefore we may regard this distance as indicating the probable limit of exactitude.

From careful measurements made at the instance of the British government, the length of the seconds pendulum at London, and at the level of the sea, is 39°1393 inches; while the length of the decimal pendulum, obtained at Paris with equal care, is 7419076 metre. Now, by discussions on measurements made at various places, the length of the pendulum is found to be

\[ 1 - 0.0275 \cos 2 \text{ lat}, \]

according to some, and

\[ 1 - 0.0268 \cos 2 \text{ lat}, \]

according to other investigators; and therefore we may assume, with Vince, that

\[ 1 - 0.0270 \cos 2 \text{ lat}. \]

represents the length of the pendulum at any latitude, when the pendulum at latitude 45° is taken as the unit.

According to this formula, the pendulums at Paris and London should be in the ratio of 1:0000560 to 1:0000605. Multiplying the length of the decimal pendulum by the square of the fraction \( \frac{1}{1000} \), we obtain 9938534 for the length of the common seconds pendulum at Paris, and augmenting this in the above ratio, 9940967 metre for the length of the pendulum at London; wherefore the ratio of the metre to the inch, as determined by help of the pendulum, is 39°1393 to 9940967, from which the length of the metre comes out 39°37172 inches. Direct comparison has given 39°37073, and there is a discrepancy of very nearly one-thousandth part of an inch.

Adopting the authorized value 39°1393 for the length of the pendulum at London, and using Vince's formula for reduction, the length of the pendulum at 45° is 39°1156 inches, and at any other latitude,

\[ 39°1156 (1 - 0.0027 \cos 2 \text{ lat}). \]

From this, by help of equation 10, we obtain the intensity of gravitation—

\[ 386°05 (1 - 0.0027 \cos 2 \text{ lat}), \]

as measured in English inches; that is to say, a heavy body falls during the first second of time through 193 inches, or 16 feet 1 inch, in latitude 45°, and in a vacuum.

The determination of the oblateness of the earth from observations on the length of the pendulum belongs to physical astronomy. It is enough to remark here, that the irregularities of two, three, or even five seconds per day which occur among the observations, can scarcely support a speculation on the irregularity of the earth's structure, seeing that in the immediate vicinity of powerful stationary transit instruments, and with every appliance and convenience for insuring precision, there has been a disagreement of a whole second.

ATTACHED PENDULUMS.

The great utility of the pendulum lies in its application to time-keeping. This part of the subject has already been adverted to under the head CLOCK-WORK; in the present place, we shall consider a little more in detail the abstract principles which are involved.

As has been stated, clock pendulums are never suspended on knife-edges; they are hung by means of thin flat springs. The elasticity of these, and the form which they take on being bent, cause a deviation from the circular motion, the effect of which has not been rigorously investigated, but which evidently tends toward the condition of cycloidal motion; so that the errors caused by changes in the amplitudes must be less than in the case of the knife-edge suspension.

The wheel-work of the clock performs the double function of recording the number of oscillations and of maintaining the motion; and the perfection of its action requires that it never miss count, and that it preserve a constant amplitude. Now, nothing short of perfect workmanship can secure the attainment of the latter condition; and therefore the primary object in contriving the parts of the movement is to arrange them so that a variation in the maintaining power may produce no perceptible change in the daily rate, for which purpose the natural motion of the pendulum must be interfered with as little as possible.

The progress of the wheels cannot be continuous, for then the extent of the oscillation would depend on the dimensions of the parts; the wheel-work must then move by steps; and therefore its action must consist of three distinct parts,—viz., the detention, the release, and the impulse. The apparatus for accomplishing these three actions constitute what is called an escapement. In some escapements the detention is effected by allowing a tooth to rest upon some part of the pendulum; in such cases the detaining surface must be exactly or nearly concentric with the motion of the pendulum; if it be truly concentric, the wheels remain perfectly stationary during the detention, and the escapement is called dead-beat; but if, as in the earlier clocks, the detaining surface be inclined, the wheels move backwards and forwards during the detention, and the escapement is called recoil. The recoil escapement is so obviously faulty that one is puzzled to account for its having remained so long in use, particularly when we reflect that a very slight change in the form of the parts converts it into a dead-beat.

In these escapements the release is effected by allowing the tooth to slip off one end of the detaining surface. In the case of the simple dead-beat the end of the detent is sloped a little from the radial direction, in order that the tooth in descending along it may give the impulse; or the tooth itself is so shaped that it may produce the same effect. When the impulse has been given, another detaining surface comes into such a position as to oppose the motion of another tooth, and the blow produced by this contact beats the second. For counting, this kind of escapement is decidedly the best.

In the duplex escapement, when the long tooth slips off the detent, the wheel turns rapidly through a distance which the clockmaker arranges to be as small as possible, until the impulse-tooth fall upon, or rather overtake, the pallet; this produces a slight sound. When the impulse has been given, the impulse-tooth is relieved from the pallet, and the escape-wheel turns rapidly, but again only through a small distance, until the alternate detent stop the next long tooth. This last action is accompanied by a more distinct beat, indicating the second; but the astronomer, unless placed at such a distance as not to hear the preceding slight sound, finds it more difficult to estimate the fractions of a second with this than with the former escapement.

For the purpose of avoiding entirely the rubbing during the detention, a separate detent is provided, on which the tooth may fall. This detent and the wheel-work remain stationary until the release be effected by some part of the pendulum coming in contact with and lifting the detent, so as to permit the impulse to be given: escapements of this class are called detached. The now very common detached lever and the chronometer escapements are characteristic though dissimilar examples of this class. Because of the number of abrupt actions, each accompanied by a slight sound, neither of them is well adapted for minute observations, unless the ear be so far removed as not to catch the lesser sounds.

In discussing the comparative merits of different kinds of escapements, it would be tedious to enter into the peculiarities of their construction; it will be enough to examine their actions under the general heads of detention, release, and impulse.

In the common dead-beat, and also in the duplex escapement, the friction of the detained tooth upon the circular arc is nearly constant; but the influence of a constant friction upon the motion of a pendulum has not been strictly investigated; so that we have nothing better than approximation to guide us. It can easily be shown that a constant friction acting upon the balance of a watch, in which the redressing tendency is proportional to the distance from the point of rest, produces no change in the time of the oscillation, and only displaces, first to one side and then to the other, the middle of the arc; and we have shown above that a friction varying as the cosine of the obliquity produces both a displacement of the arc and an imperceptible acceleration on the pendulum; wherefore we may conclude that, with a small arc of oscillation, the effect of friction on the clock's rate must be very trifling. The most serious effect is, the diminution of the amplitude, which necessitates a more powerful impulse. In respect of this, the duplex is decidedly preferable to the simple dead-beat; but the detached escapements have the advantage over both, since they occasion no friction at all.

The release in the dead-beat and duplex escapements is merely the end of the detention; the tooth slips off the detaining arc. But in detached escapements the release is a distinct action; the detent has to be removed from before the tooth. In the chronometer escapement the detent is kept in its place by a spring, which must be bent backwards when the detent is raised, thus causing a retardation through a small part of the arc. The comparative value of the two classes of escapements must depend to some extent on whether this resistance of the detent or the friction on the pallet be most injurious to the time-keeping. The watchmaker gives a motion as extensive as possible to the balance, while the clockmaker's aim is to render the arc as small as he can; and thus the advantages of the detached escapement are greater in watch than in clock work.

When the release has been accomplished, the wheels are impelled forward, but not instantaneously. They acquire velocity gradually, until arrested by the stroke of the impulse-tooth upon the pallet.

In a common dead-beat escapement, let A (fig. 10) be the axis of the crutch, C that of the escape-wheel, and T the tooth, which, having been detained upon the surface EF, is just released by the motion of the pendulum towards the right. The pallet continues its motion, and the tooth begins to descend according to the ordinary law, its distances being proportional to the squares of the elapsed times. In consequence of these two motions, T would trace upon a surface attached to the crutch a paraboloid curve FG, tangent to the arc EF at F; and if the end of the pallet were exactly shaped to this curve, the tooth would merely graze it, without communicating any impulse.

When the end of the pallet is bevelled, as FH, making a sharp corner at F, the tooth falls behind at first, gains speed, and overtakes the pallet at I; so that the impulse is only given from I to H. Although the interval FI be so small as to escape observation, it is not the less real, and as its tendency is to cause a slight uncertainty in the action, it ought always to be obviated by rounding a little the corners of the ruby. It is completely removed in those escapements which have the tooth T part of a cylinder.

But this interval exists unavoidably in the duplex and in the detached escapement. Its extent in all cases is augmented by an increase in the weight of the escape-wheel, for which reason that wheel ought to be made as light as is consistent with accuracy and permanency of form.

The great derangement of the time-keeping is from the impulse. Friction, imperfect elasticity, and the air's resistance, are injurious chiefly because they render an impulse necessary; and the important matter in contriving an escapement is to place the arc of impulse so as not to influence the time of the oscillation.

The effect of the impulse can be strictly investigated when the redressing tendency is proportional to the derangement; and it can be shown (Edin. New Phil. Jour. for July 1835) that the proper time for the impulse is nearly at the middle of the oscillation; but when, as in the Pendulum case of the pendulum, the redressing tendency is proportional to the sine of the inclination, the inquiry transcends the power of the calculus. We may, however, satisfy ourselves that the impulse ought to be given when the pendulum is near its lowest point by considerations which do not require any elaborate formulas.

The impulse is to restore the motion that has been lost by friction and the air's resistance; and therefore the product of the impelling pressure, by the distance through which it acts, must be equal to that of the resistance by its distance. Its effect upon the pendulum is to augment the square of the velocity by a certain quantity, and to shorten the time in which the arc of impulse is passed over. If $V$ be the velocity which the pendulum would have had at the end of the arc of impulse if let alone, and $V'$ the velocity which it has after having received the impulse, we have

$$V'^2 - V^2 = I,$$

or

$$(V' + V)(V' - V) = Pt.$$

When the impulse is given near the middle of the arc, $V' + V$ is large, and therefore $V'^2 - V^2$ small; but when it is given near the limit of the motion, $V' + V$ is small, and therefore $V'^2 - V^2$ proportionally great. Now $i$ being the extent of the arc of impact, $\frac{i}{V}$ is the time during which it would have been described with the velocity $V$, and $\frac{i}{V'}$ that with the velocity $V'$; hence $\frac{i}{V} - \frac{i}{V'}$ is proportional, approximately, to the direct influence of the impulse upon the time of an oscillation. This may be put under the form $\frac{V'^2 - V^2}{VV'}$; and it is to be observed of this fraction that its numerator is inversely proportional to $V' + V$, whence $\frac{V'^2 - V^2}{VV'}(V' + V)$ represents the error in time-keeping caused by the impact.

From this we conclude, that when the change $V'^2 - V^2$ is not great, the error caused by the impulse is inversely as the cube of the velocity of the pendulum at the time of receiving it; and also that the shorter the arc, the pressure being proportionally augmented, the less the derangement is.

Now, neither the detached escapement nor the duplex can be used with a very small arc of impulse, as the risk of missing would be great; while the simple dead-beat can have that arc reduced by an alteration on the forms of the tooth and pallet; so that, considering its smooth, quiet action, and the facility which it gives for reducing the direct error of the impulse, we need not wonder that it is regarded as, practically, the best escapement.

REMONTOIRES.

As the maintaining power is communicated to the pallets through a long train of wheels, it is liable to periodical variations from malformation of the teeth, and to more permanent changes from dust, and from the thickening of the oil. Also in spring time-keepers the form of the fusee does not compensate perfectly for the varying tension of the mainspring. Hence the utility of what are called remontoires, or re-winders.

This name is given to various arrangements by which a small weight or slender spring, as the case may be, is wound to a determinate height at short intervals of time, this weight or spring becoming virtually the maintaining power. If the remontoire be made to act every minute, the seconds hand proceeds as usual, but the minute and hour hands spring forward by one minute at a time. The periodical inequalities are in this case repeated every minute, and cannot accumulate to become perceptible; while, as all the parts above the remontoire can be highly finished or jewelled, the variations of friction are little felt.

When the re-winding occurs every second, the arrangement becomes what is known under the name of the gravity escapement. In the greater number of gravity escapements the impulse weight is held by the clock-work at a fixed height until the pendulum relieve it and carry it to the extremity of the oscillation. During this part of the action the weight retards the motion; but when the pendulum begins to descend, the weight descends also, and as it is not arrested till it has reached a point lower than that from which it started, it accelerates more than it retards the pendulum. When the impulse-weight has reached its lowest point, the detent is unlocked, and the wheel-work raises the weight to its first place, there to await the pendulum.

In electric clocks, galvanic contact is made or broken at the proper instant, and the impulse-weight is lifted by the electro-magnet.

These escapements are essentially recoil escapements, and have all the defects which are inseparable from the class. An extraneous influence cannot be brought to bear on the pendulum at a worse place than at the extremity of the motion, where the velocity is zero. And seeing that, for good time-keeping, the arc of impulse must be very short, it cannot but be detrimental to lengthen it out over a half oscillation. These escapements, in fact, are in direct opposition to the principles which have just been explained.

METRONOME PENDULUMS.

We have seen that the length of simple pendulums vibrating in different times are proportional to the squares of those times, and that these lengths are fixed by the intensity of gravitation; thus a simple pendulum vibrating once a second must be about 39 inches long. But it does not follow that a compound pendulum must have this length.

For the purpose of marking time, musicians employ an instrument called a metronome (fig. 11), of which the principal part is a small pendulum capable of being regulated to suit the character of the opera, B being the principal weight of this pendulum, and A the axis of motion, the rod BA is prolonged above the fulcrum, and on the prolongation AD an adjusting weight C is made to slide; divisions also are engraved and numbered on AD, to indicate the number of beats which the pendulum makes per minute.

If $m_1, m_2, m_3, \ldots$ represent the masses of which a pendulum is made up, $r_1, r_2, r_3, \ldots$ their distances from the axis of motion, and if AG be the distance of the centre of gravity from that axis,

$$AG = \frac{m_1r_1^2 + m_2r_2^2 + m_3r_3^2 + \ldots}{m_1 + m_2 + m_3 + \ldots}$$

expresses the length of the simple pendulum vibrating in the same time.

Now, by raising the weight C, we augment the numerator of this fraction, and at the same time lessen the denominator, since the centre of gravity is also raised, so that we lengthen the time of the oscillation. If C were raised so high as to bring the centre of gravity up to A, the mean distance of oscillation would become infinite; so that there is no degree of slowness that cannot be attained by this arrangement.

If M be the weight of the permanent part of the metronome pendulum, including the axis and pallets, R its mean distance of gyration round the axis, and l the distance of its centre of gravity, then

$$I_g = \frac{MR^2}{Ml}.$$

Pendulum, is the mean distance of oscillation of the unloaded metronome. And if \( m \) be the weight of the sliding-piece, \( p \) its mean distance of gyration round an axis passing through its own centre of gravity parallel to the axis of the pendulum, and \( \lambda \) the distance of its centre of gravity above that axis,

\[ L = \frac{MR^2 + mp^2 + m\lambda^2}{M - m\lambda} \]

is the mean distance of oscillation of the loaded metronome. The weights \( M \) and \( m \) can be determined directly, and the length \( L_0 \) can be obtained by observing the oscillations of the pendulum when unloaded, using the equation

\[ l = l' \times 39.1393; \]

by this means we can eliminate \( MR^2 \), and the equation

\[ \frac{M}{m}L - \left( \frac{M}{m}L_0 + p^2 \right) = \lambda^2 - L\lambda \]

expresses the relation between \( L \) and \( \lambda \). In this there remain still the quantities \( l \) and \( p \) to be determined by experiment. By placing the weight \( C \) at three different positions on the stem, and observing the times of oscillation, these quantities \( l \) and \( p \) can be found, and thence the graduation of the instrument.

However, as the matter does not call for extreme precision, the ordinary method of finding by trial three or four points on the scale, and then graduating between, is sufficient for the purpose.

COMPENSATION PENDULUMS.

The expansions and contractions of the pendulum-rod, consequent on changes of temperature, are sources of great irregularity in the going of clocks. Thus, iron wire expands, according to M. Biot, \( \frac{3}{4} \)th from freezing to boiling water, and therefore this change of temperature would cause an alteration of 53 seconds in the daily rate; that is, very nearly 3 seconds for every 10 degrees of Fahrenheit's thermometer. A piece of white deal expands only about the third part as much as iron; and hence the irregularity of a pendulum with a deal rod is only about 1 second per day for a change of 10 degrees Fahrenheit. A well-made clock, with a deal-rod pendulum, is sufficiently accurate for the ordinary purposes of life; but where great nicety is required, means are taken to correct the effects of expansion.

As this part of the subject has already been adverted to under the head CLOCK-WORK, it is enough, here, to indicate the general features of the investigation.

A very simple and inexpensive compensation pendulum may be made of a cylindric deal rod carrying a long leaden cylinder; and if the rod be composed of several slips, well glued together, and thoroughly varnished, to prevent the effects of the moisture, there is perhaps no pendulum yet contrived which will keep better time.

The strict investigation into the effects of a change of temperature upon a clock furnished with such a pendulum would take into account—1st, The weight, form, and expansion of the crutch and its appendages, since these form an essential part of the oscillating mass; 2d, The expansion and change of stiffness of the slender steel spring AB (fig. 12); 3d, The expansion, lateral as well as longitudinal, of the deal rod BC; and, lastly, The expansion of the perforated leaden bob CD, and the thermal compensation, will be effected when the various dimensions are arranged; so that the numerator and denominator of the fraction

\[ \frac{\sum m_i r_i}{AG \cdot 2m} \]

receive proportional augmentations on an increase of temperature. Now it is evident that the square of the term \( \frac{m_i r_i}{AG \cdot 2m} \) will be found in the new numerator, while only its first power can enter into the denominator; and thus we see that no compensation can be perfect for every change of temperature. However, as the expansions are all minute, the second powers of the fractions which represent them may be neglected without causing any appreciable error.

There is a great uncertainty as to the rates of expansion of different bodies, arising partly from the difficulty of making the measurements, and partly from irregularities among different specimens of the same substances. On this account, it would be a waste of labour to compute the conditions of compensation strictly—the more so, that the fine adjustment must be made by trial; we shall content ourselves with an analysis on the supposition that the parts are merely linear, and that the pendulum is swung from a knife-edge.

If \( m_r \) and \( m_d \) be the weight of the deal rod AC, and of the leaden bob CD, the mean distance of oscillation is

\[ l = \frac{m_r \cdot AC^2 + m_d (AC + AD)^2}{m_r \cdot AC + m_d (AC + AD)} \]

for the temperature at which the measurements are made; and if \( e_r \) and \( e_d \) be the rates of expansion for one degree, while AC and CD are denoted by \( H \) and \( h \), the condition of compensation is expressed by the equation

\[ \frac{m_r H^2 + m_d (3H^2 - 3Hh + h^2)}{m_r H + m_d (2H - h)} = \frac{m_r (2He_r) + m_d (6H^2e_r - 3HA(e_r + e_d) + 2h^2e_r)}{m_r (He_r) + m_d (2H(e_r - e_d))} \]

Again, if we suppose that the weight \( m_r \) of the rod is insignificant in comparison with \( m_d \), the equation becomes much simpler, and reduces itself to the form

\[ e_r H^3 - (4e_r + e_d)H^2 + (3e_r + 6e_d)Hh - 6e_dH^2 = 0; \]

so that, even in this most simple case, the determination of \( h \) to \( H \) requires the resolution of a cubic equation.

Supposing that for deal \( e_r = \frac{1}{4} \)th, and that for lead \( e_d = \frac{3}{4} \)th, we have \( e_r : e_d :: 7 : 48 \), and the equation becomes

\[ 48H^3 - 1994H^2 + 1864HP - 42H^2 = 0; \]

whence very nearly \( H = 34 \); that is, the column of lead must be one-third part as long as the deal-rod; and for a seconds pendulum the whole length must be about 46.35 inches; the leaden bob being 15.45 long.

ATMOSPHERIC INFLUENCE.

The effect of the buoyancy of the air upon the going of a clock is considerable: thus lead is about 9000 times heavier than air of the same volume, and therefore a change of one inch in the barometer will alter the apparent weight of the leaden bob by one part in 270,000, which corresponds to an alteration in the daily rate of the clock of \( \frac{1}{100} \)ths of a second, or rather more than one second per week; and as the whole of the pendulum is not made of such weighty materials, the actual derangement must be somewhat more than this. It is thus impossible, in a climate so variable as ours, to construct a clock that shall keep true astronomical time, unless it be cased and protected entirely from changes in the buoyancy of the air, or unless some means be fallen upon to compensate for atmospheric changes.

In the delicate operation of determining the apparent right ascension of a star or planet, the time of transit over the wires of the telescope is sought to within a small fraction, as the twentieth or fiftieth part of a second. We must be able, therefore, to depend on the going of the clock from one observation of a standard star till another within this degree of precision, otherwise the care bestowed on the collimation of the instrument is thrown away. Now a rise of half an inch in the barometer will, in twelve hours, produce a retardation of \( \frac{1}{4} \)th of a second; so that, if an Pendulum: astronomers aim at precision in his observations, he must be able to make allowance for the effect of the air's buoyancy on the going of his clock.

By carefully collating the observed variations in the clock with the indications of the thermometer and barometer, an empirical formula may be constructed, by means of which the error may be computed up to any given time.

When steel or iron rods enter into the composition of the pendulum, an error of another kind is induced. The rod, particularly in latitudes where the needle dips much, becomes magnetic, and is then influenced by the earth's magnetism. The amount of this influence increases until the maximum magnetism is induced; and thus newly-made clocks which have steel and mercury compensation pendulums are liable to show an acceleration of rate for a considerable time after they are set up; they are also liable to changes of rate depending on variations in the intensity of the earth's magnetism.

CONICAL PENDULUM.

When a pendulum, instead of being hung on an axis which restricts its motion to one plane, is hung on a point merely, its movements are much more complex. Laplace has partially investigated the case of a single material point suspended by means of an imponderable thread; but the general subject has hardly yet been inquired into.

If the motion of a simple pendulum be restricted to a very minute distance from the point of rest, it describes almost exactly an ellipse; but whenever the amplitudes become perceptible, the major axis of the curve turns in space so as to imitate in some respects the motion of the line of apsides of a planet. When, however, the pendulum is compound, a much more intricate movement takes place.

To illustrate the nature of this complex motion, let us suppose that a pendulum of irregular shape is swung from a Hooke's universal joint, the two axes of which are parallel to the horizontal axes of greatest and least rotation. Then if the pendulum be drawn aside in the direction of one axis, and let go, it will continue to oscillate on the other axis; but the times of the oscillation on these axes will not be the same: let us represent them by $T$ and $t$. If now the pendulum be drawn aside in a direction not parallel to either axis, both oscillations will go on conjointly. The determination of the complex motion thus induced is too difficult for the calculus in its present state; it, however, we restrict the amplitudes to be exceedingly minute, the form of the curve may be obtained by the following process:

Assume O (fig. 13) as the central position, AOa, BOb as the extents of the oscillations parallel to the two axes, then the rectangle CDed circumscribes the curve. On eD and DC describe two semicircles, and divide each of them into equal parts, the numbers being proportional to the times $T$ and $t$ (the ratio assumed in the adjoining fig. is 5:3), and through the points of section draw lines parallel to the sides of the rectangle. Then, beginning at any of the crossings, let a curve be traced diagonally through the adjoining meshes; this line represents the path of the pendulum. It is of the same nature with the curves described by the free extremity of a straight wire held firmly by the other end.

If the periodic times of the oscillations be nearly alike, the curve assumes the appearance of an ellipse, of which the major axis oscillates between the diagonals Ce and Dd, and which collapses into a straight line in each of them.

When, however, the amplitudes are considerable, this method fails to give a correct representation of the movement.

If the suspension could be by means of a thread offering no resistance to torsion, the motions of an irregular mass would be still more complex; and when the cord resists torsion, an oscillatory-rotatory motion is superadded, and the complexity becomes enormous.

One species of conical pendulum deserves attention as being used for regulating the speed of steam-engines and water-wheels. In it the pendulum is hinged on a horizontal axis, which is caused to revolve round a vertical one, and the effect of the motion is to throw the pendulum aside from the vertical position. If there were no friction on the hinge, the rotatory motion would be combined with an oscillatory motion on the horizontal axis; but in the actual apparatus, the friction and other resistances destroy the oscillation, while the rotation is kept up by the machinery; and, on this account, the only object of interest is to determine the degree of inclination which belongs to a given speed.

Let AB (fig 14) be the vertical axis of a governor, C and C' the hinges, G and G' the centre of gravity of the two pendulums; then having drawn GD perpendicular to AB, GD is the radius of the circle described by G, and therefore, if $T$ be the time of a rotation, $m \frac{4\pi^2}{g} \frac{DG}{T^2}$ is the centrifugal tendency in the direction DG; and the resultant obtained by combining this with $m$ in a vertical direction must lie along CG; wherefore, having produced GC to meet AB in E—

$$\tan DAC = \frac{4\pi^2}{g} \frac{DG}{ED} = \frac{ED}{ED};$$

whence $ED = \frac{g}{4\pi^2} T^2$.

If we notice that in reckoning the oscillations of a pendulum we counted from one side to another, whereas a complete oscillation requires a return to the same place, we shall see that the height DE of the conical pendulum is equal to the length of a simple pendulum oscillating along with it.

FOUCAULT'S EXPERIMENT.

When a round body, suspended by means of a flexible thread, is once set to oscillate in a plane, it continues to move in that plane. M. Foucault has taken advantage of this property in order to demonstrate the diurnal rotation of the earth. If the earth were at rest, the direction of the vibrations would remain, and would appear to remain, fixed; but as the earth turns while the plane of oscillation preserves its parallelism, that plane appears, in reference to surrounding objects, to turn in the direction of the apparent motion of the stars.

In repeating this beautiful experiment, care must be taken that the weight be symmetric in shape, and that no bias be given at the outset, lest some of the complex movements above mentioned be induced.