When a solid and heavy body of any form and description is suspended from an axis fixed horizontally, and round which it can turn with freedom, or at least with a very slight degree of friction; if we withdraw it, however little, from the position of equilibrium at which it naturally places itself when at rest, and then abandon it to itself, the force of gravity, which is now no more destroyed by the resistance of the axis of suspension, brings back the body towards its primitive position of equilibrium with a velocity continually accelerated. When it reaches this position, the accelerating force ceases for a moment to act on it, but the body, continuing to move in consequence of the velocity already acquired, rises on the other side of the vertical line, and continues rising until the constantly increasing force of gravity destroys its velocity—then it stops for an instant, and again yielding to the continued action of gravity, to which there is now no more opposition, it again falls with a motion exactly similar to that which it had when it began to descend from the opposite side of the vertical. It returns then, in the same manner, to its primitive position of equilibrium, passes it, and re-ascends on the opposite side of the vertical, to the point where its velocity is destroyed anew; after which it again begins to descend, and again to remount; and the oscillatory motion which results from these alternations only ceases in consequence of the resistance of the air and the friction of the axis, which gradually reduce it to nothing.

An apparatus of this kind is termed a pendulum. The oscillations of pendulums can be calculated completely, and with perfect rigour, by the principles of mechanics, when they are supposed to take place in a vacuum, and round an axis, which presents no friction. The results in regard to this imaginary case are so much the more important to be known, as in the real experiments we always endeavour to approach it as near as possible, by combining to the utmost every circumstance which can tend to prolong the duration of the pendulum's motion. These are the results, then, which it is proper to present first in order, as they exhibit a first approximation to every motion of this kind which can be realized.

In this simple case, whatever be the form of the body which constitutes the pendulum, provided that it remains invariably constant, all the successive oscillations have equal amplitudes, and are also of equal duration among themselves, so that the motion, once begun, never ceases to go on. In the case of oscillations with different amplitudes, the duration is in general unequal; but this inequality diminishes in proportion as the amplitudes become less, and it ceases altogether at the limit where they become infinitely small; so that all the oscillations made with amplitudes, which, in a physical sense, may be reckoned infinitely small, are sensibly of equal duration.

In regard to the nature of the motion in each oscillation, it is absolutely the same for every body, whatever be its form; and we may always consider it as identical with that of a pendulum formed by a material gravitating point suspended at the extremity of a thread, supposed to be inflexible and without weight. Let \( l \) denote the ideal length of such a pendulum, which is called a simple pendulum. Let \( m \) be the mass of an oscillating body (fig. 1) with which we wish to compare it, and which we shall call, in opposition to the other, a compound pendulum. From the centre of gravity of this body, denoted by \( G \), conceive a perpendicular \( SG \) drawn to the axis of suspension, and call \( h \) the length of this line. If we multiply each element of the mass \( m \) by the square of its distance from the same axis, and denote the sum of the whole by \( c \), the product \( c \) thus formed will be what is called in mechanics the momentum of inertia of the body \( m \) relatively to the axis in question. In order that the motion of the simple pendulum \( SP \) be exactly isochronous with that of the body \( m \), it is sufficient that we have the equation

\[ l = \frac{c}{mh} \]

and, besides this, that the lines \( SP \) and \( SG \) have at any one instant equal angular velocities at the same distance from the vertical. This last condition will be fulfilled if, for example, at the beginning of the motion the lines \( SP \), \( SG \) are equally distant from the vertical, and that the simple and compound pendulum be then abandoned together to the action of gravity, or be driven with equal velocities in the plane of their oscillations. The simple pendulum will then accompany the compound one in all its successive excursions, and its direction will always coincide with the line \( SG \) drawn from the centre of gravity of the body \( m \), perpendicular to the axis of suspension. The length \( l \) being ascertained by this formula, we can lay it off on the line \( SG \), setting out from the axis of suspension \( S \); and the point \( P \), where it terminates, is called the centre of oscillation of the body \( m \).

The initial conditions above stated can always be established, and the analytical value of \( l \) is also always real. For every given compound pendulum, then, we can always assign a simple pendulum, which is isochronous with it, and of which the motion is absolutely similar to that of the line \( SG \). By means of this substitution, we have nothing more to consider or to compare, but the different lengths of the simple pendulums, and it then only remains to ascertain the mode in which such pendulums perform their oscillations.

To do this in the simplest manner possible, let us conceive that the arc \( ZP \) (fig. 2) is half the extent of the oscillations through the vertical \( SZ \), and suppose that the pendulum arrived or placed in this position is there abandoned to the sole action of gravity without any initial velocity of impulsion. Call \( \alpha \) the angle \( PSZ \), and denote by \( g \) the intensity of gravity measured by the double of the space which heavy bodies describe at the place where the experiment is made, when they fall freely in a right line during the unity of time. Then denoting always by \( l \) the length of a simple pendulum \( SP \) or \( SZ \), the time \( T \) of its whole oscillation in the arc \( PZP' \) or \( PP' \) will be expressed by the following series:

\[ (1.) \quad T = \sqrt{\frac{l}{g}} \left[ 1 + \left( \frac{1}{2} \right) \sin^2 \frac{1}{2} \alpha + \left( \frac{1}{24} \right) \sin^4 \frac{1}{2} \alpha + \left( \frac{1}{35} \right) \sin^6 \frac{1}{2} \alpha + \ldots \right] \]

\( \sigma \) being the ratio of the circumference of the circle to its diameter, or \( 3:14159 \).

If, besides, we denote the velocity of the pendulum in any point of its oscillation by \( V \), \( \delta \) being its angular distance from the vertical, we have

\[ (2.) \quad V^2 = 2gl (\cos \delta - \cos \alpha), \]

or, what comes to the same thing,

\[ V^2 = 4gl \sin \frac{1}{2} (\alpha + \delta) \sin \frac{1}{2} (\alpha - \delta). \] These formulae will still serve if the pendulum, instead of falling freely from the extremity of the arc, receives there an initial velocity expressed by \( V \); provided always this velocity is within the limits which permit the oscillatory motion to take place. If this be the case, indeed, it will be sufficient to consider the pendulum as setting out with an initial impulse from another angular distance, \( \alpha' \). Then it will be necessary that this unknown distance \( \alpha' \) satisfies, instead of \( \alpha \), the general equation of the velocities, and that \( \alpha \) becomes in it \( \delta \), which gives

\[ (3) \quad V^2 = 2gl \{ \cos \alpha - \cos \alpha' \}. \]

The half amplitude \( \alpha' \) of the oscillations being the only unknown quantity in this equation, will be thus determined, and their duration will be then obtained by the equation (1), but putting in it \( \alpha' \) instead of \( \alpha \). This same substitution made in the equation (2) will give

\[ V^2 = 2gl (\cos \delta - \cos \alpha') \]

for the velocity in any point whatever of the oscillation.

But these transformations are only possible when the equation (3) gives for \( \alpha' \) a real arc, and consequently for \( \cos \alpha' \) a value comprehended between +1 and −1. We may easily conceive, that, when the \( \cos \alpha' \) exceeds these limits, it is because the velocity of impulse \( V \) exceeds the greatest velocity of the fall which the pendulum can acquire in a circle of a radius \( l \), even supposing it to fall from the very summit. It is evident indeed that the oscillatory motion can then no more produce such a velocity, and we know also that, in that case, it will change into a continued motion of rotation. If we exclude this circumstance, the formulae (1) and (2) will determine generally every particular regarding oscillatory motions. When the amplitudes of the oscillations become so small that we can, in the series (1), neglect all the powers of \( \sin^2 \frac{1}{2} \alpha \), compared with the unity which precedes them, we will have simply

\[ T = \pi \sqrt{\frac{l}{g}}. \]

Now, the angle \( \alpha \) entering no more into the value of \( T \), it appears that its value will have no influence on it; that is to say, that for the same pendulum moved in a vacuum, all the oscillations which are performed with amplitudes infinitely small, are of equal durations.

In actual experiment, the oscillations can never be altogether infinitely small, but we may take care, at least, to confine them within amplitudes so limited that the angle \( \alpha \) has a very small value. We have then an approximation perfectly sufficient in limiting the series to the term which contains the square of the \( \sin \frac{1}{2} \alpha \). We may then, in the same order of approximation, substitute \( \frac{1}{2} \sin^2 \alpha \) for \( \sin^2 \frac{1}{2} \alpha \); and the series (1), being thus limited to its two first terms, gives

\[ T = \pi \sqrt{\frac{l}{g}} \left\{ 1 + \frac{1}{16} \sin^2 \alpha \right\}, \]

\( \alpha \) being always the half amplitude of the oscillation.

We have seen above, that, by supposing the simple pendulum \( l \) isochronous with the compound pendulum of the mass \( m \), we have

\[ l = \frac{c}{mh}, \]

\( c \) being the momentum of inertia of the mass \( m \), relatively to the axis of suspension. But if we call \( c' \) the momentum of the same mass, relatively to an axis parallel to the preceding, and passing through the centre of gravity \( G \), we find, by mechanics, that the quantities \( c, c' \) have between them the following relation:

\[ c = mh^2 + c'. \]

This value of \( c \), being substituted in the expression of \( l \), gives evidently

\[ l = h + \frac{c'}{mh}. \]

Now, when \( h \) is given, this expression only furnishes one value of \( l \); that is to say, a single length for a simple pendulum isochronous with the mass \( m \). But if \( l \) be given, then there are two values of \( h \) which give the same value to \( l \); and these are deducible from the preceding equation, by taking \( h \) in it as the unknown quantity. If we denote these two values of \( h \) by \( h' \) and \( h'' \), it is easy to see that their sum is \( l \), and that thus the first being SG (fig. 1), the second will be PG. If, then, after having placed the axis of suspension in S, we place it in P, that is, in the centre of oscillation itself, preserving it always parallel to its first direction, the oscillations performed round the axis P will be of the same duration as those performed round the axis S, provided always that in both cases the amplitudes of the oscillation, as well as the initial velocities, are equal. This remarkable theorem we owe to Huygens.

It is easy to extend it to one much more general. In all solid bodies, whatever be their figure, we may draw through the centre of gravity three rectangular axes, termed in mechanics principal axes, and which possess several properties extremely remarkable. Let the momentum of inertia of the mass \( m \) relatively to these axes be denoted by \( A, B, C \). Then, if we consider any axis of suspension of which the distance from the centre of gravity is expressed as above by \( h \), and which forms with the preceding certain angles, \( X, Y, Z \); it is shown, in mechanics, that the momentum of inertia \( c'' \), relative to this axis, can be expressed in the following manner:

\[ c'' = mh^2 + A \cos^2 X + B \cos^2 Y + C \cos^2 Z; \]

and this value being substituted in \( l \), instead of the letter \( c \), gives

\[ l = h + \frac{A \cos^2 X + B \cos^2 Y + C \cos^2 Z}{mh}. \]

If, now, the axis of suspension be given along with the distance \( h \), this expression gives but a single value for \( l \); but if we regard \( l \) as given, and constant, then there arises between the angles \( X, Y, Z \), and the distance \( h \), a simple relation, which we can satisfy in an infinity of different ways, so that there result as many different axes of suspension, which are all isochronous with each other. To be sensible of the extensive application of such solutions, let us transform the preceding relation into one with rectilinear co-ordinates. Let \( x, y, z \) be such co-ordinates directed rectangularly, according to the three principal axes of the mass \( m \), and having their common origin at the centre of gravity of this mass, the axis of suspension relatively to these co-ordinates will have its equations of the form

\[ (4) \quad x = ax + a, \quad y = bx + b, \]

\( a, b, a, b \) being four constant indeterminate quantities, depending on their position in space. We have, besides, by the well-known theorems of analytical geometry,

\[ \cos X = \frac{a}{\sqrt{1 + a^2 + b^2}}, \quad \cos Y = \frac{b}{\sqrt{1 + a^2 + b^2}}, \]

\[ \cos Z = \frac{1}{\sqrt{1 + a^2 + b^2}}, \quad \text{and, lastly, } h = \frac{\sqrt{a^2 + b^2}}{\sqrt{1 + a^2 + b^2}}. \]

By substituting these values in the general expression of \( l \), it becomes

\[ (5) \quad l = \frac{\sqrt{a^2 + b^2}}{\sqrt{1 + a^2 + b^2}} + \frac{Aa^2 + Bb^2 + C}{m\sqrt{a^2 + b^2}\sqrt{1 + a^2 + b^2}}. \]

By supposing \( l \) constant, this relation, combined with the equation (4), will characterize the isochronous axes; but as this combination only furnishes three equations, while there are four constant indeterminate quantities \( a, b, a, b \); in the position of the axis, it hence appears that we may still assume at pleasure an additional condition among the quan- tities themselves, after which, by eliminating them, we shall have, in \(xyz\), the equation of a surface on which will be found the isochronous axes fitted to satisfy the condition prescribed.

Having thus made known the laws of oscillatory motion in a vacuum and round an axis altogether free from friction, let us now consider them in a feebly resisting medium like air, and supposing a slight degree of friction round the axis, such as is invariably the case in the experiments.

In the first place, whatever be the nature of the physical process by which the two causes operate, their definite effect will always be to retard the pendulum, according to a certain function of the velocity. But whatever be the form of this function, provided it be such as to become nothing when the velocity is nothing, which is an essential condition of the kind of obstacles it is designed to express, we may always assign a simple pendulum which, moving with the same laws of friction and resistance, will be exactly isochronous with the compound pendulum we are considering; and, what is very remarkable, the length of this simple pendulum is exactly the same as it would be if the oscillations were performed in a vacuum, and consequently the same as that of which we have given the expression above. Thus the centre of oscillation of solid bodies, such as we have defined it, has in each of them a situation independent of the medium in which they move, and of the resistances of every kind which their motions may suffer. This important proposition was first demonstrated by Clairault.

Now, for the simple pendulum, as well as for the compound one which accompanies it, the resistance of the air and the friction of the axis diminish continually the extent of the arcs in which the successive oscillations are performed; but it happens, from a circumstance well worthy of remark, that when this retarding force is very slight, and acts with continued and equal effect on both sides of the vertical, the durations of the oscillations are not altered on this account. For, although the resistance which the pendulum suffers must retard, no doubt, its fall, and consequently prolong its duration in each half oscillation in descending; yet in each half oscillation in ascending, this same cause accelerates the extinction of the velocity, and rather brings on the instant when this half oscillation is terminated. And whatever be the mathematical law of the motion thus performed, if the amplitudes of the successive oscillations diminish very slowly, which always takes place when the body put into oscillation has a very considerable density relative to that of the air, and if we make it perform vibrations only of very small extent, and round an axis of suspension so worked as to present but a slight degree of friction, then the motion of the pendulum presents a succession of velocities almost exactly similar in each descending half oscillation, and in the ascending half which follows it. The alterations produced in these velocities by the friction and the resistance of the air are then almost equal, so that their effects are almost exactly compensated in the actual observations. Hence it follows that the isochronism of small oscillations, though altered in each particular half oscillation, is still found to subsist in the total oscillations, notwithstanding the friction of the axis and the resistance of the air, provided always that these two forces are rendered so feeble as to have but a very gradual influence on the motion. This is at least proved by experiment; for, when a compound pendulum of any form whatever oscillates in the air round a suspension, so free that the decrease of its vibrations goes on with great slowness, if we observe the amplitudes of these vibrations at intervals so near each other that their absolute diminution is inconsiderable, and apply to the number of oscillations performed during this interval the reduction of amplitude calculated according to the mean value of the arcs thus observed; the number of oscillations corrected and reduced in this manner to the case of amplitudes infinitely small, is also found invariably the same for the same pendulum, at least with all the degree of exactness admitted by physical experiments; which shows that the correction of the amplitude is the only one which the oscillations require in order to reduce the motion of the pendulum to a uniformity quite mathematical. This spontaneous compensation, which is produced in the effects of the resistance of the air on the two descending and ascending half oscillations, had first been remarked and pointed out by Newton in his Principia, lib. ii. prop. xxvii., theor. xxii. coroll. 2. He even gives a rigorous demonstration of it in prop. xxvi. and xxvii., for the case of a resistance proportional to the two first powers of the velocity—the motion being then in the cycloid. M. Poisson has given the analogous demonstration for a circular motion in the seventh volume of the Journal de l'École Polytechnique. These demonstrations, however, only apply to that part of the resistance which arises from the direct impulse with which the moving body strikes the aerial particles, supposing these particles quite removed after the stroke, and consequently without regard to the peculiar agitation which their displacement produces in the medium itself. But, as Newton remarks in the corollary above cited, the descending half oscillation, which is performed with a motion continually accelerated, must, on this account, excite a resistance in a slight degree stronger than the half oscillation ascending, which goes on with a motion continually retarded; because, in this second case, the aerial particles struck by the pendulum may fly from it, and withdraw themselves from its action more easily than in the former. This diminution of the resistance in the second half of the oscillation must cause it to last a little longer than it would have done without this circumstance, and thus the time of the whole oscillation must be a little augmented. Fortunately this cause, it appears, becomes insensible in the most important experiments to which the pendulum is applied; for in these the observations are never made but with very small amplitudes, which produce very small velocities, and these cannot excite any sensible resistance except by the direct impulse communicated to the ambient medium.

But, independently of its resistance, the air, by its mere presence, floating round the oscillating body, produces on the motion another effect, which may be called statical, and which must be attended to, in order to compare the observations made in different states of this fluid. As a gravitating medium, in fact, it deprives the oscillating body of a part of its weight equal to that of the volume of air which the body displaces, so that the latter, in reality, only gravitates in consequence of the difference between these two quantities. To calculate the resulting effect on the oscillations, call \(P\) the absolute weight of the body in vacuo, \(\Delta\) its density, compared with that of the air in the circumstances under which we are operating. The weights of bodies of equal volume being proportional to their densities, the weight of the air displaced by the body will be \(P - \frac{1}{\Delta}\); thus the apparent weight of this same body, during its oscillations, will be \(P - \frac{P}{\Delta}\), so that it will be to its absolute weight as \(1 - \frac{1}{\Delta}\) to 1. The effect, then, will be the same as if the absolute weight \(P\) were acted on, not by the actual gravity itself, but by a force diminished in this ratio. We have only, therefore, to reduce the elements of this correction to terms that we can compare together. For this purpose, suppose that at the temperature of freezing, and under an atmospheric pressure measured by a column of mercury of 0·76 metre in height, \(D\) represents the density of the substance of the pendulum, that of the air being If we denote the cubic dilatation of this substance for a change of temperature equal to a centesimal degree, by \( c \), its density at \( t \) degrees will become very nearly \( D (1 - ct) \); and if \( p \) is the atmospheric pressure at this temperature, the corresponding density of the air, according to the known law of the dilatation of this fluid, will be

\[ \frac{p}{0.76(1 + ct \cdot 0.00375)} \]

Then denoting the absolute intensity of gravity, as it is exerted on the body in vacuo, by \( g \), and the apparent force with which it really moves the body in the air by \( g' \), we shall have

\[ g' = g \left[ 1 - \frac{p}{0.76(1 + ct \cdot 0.00375)(1 - ct)D} \right] \]

To illustrate the use of this correction, let \( l \) represent the length of a simple pendulum, which performs its oscillations in the time \( T \), under the influence of the apparent gravity \( g' \), and with the amplitude \( 2a \), we shall have

\[ T = \sqrt{\frac{l}{g'}} \left[ 1 + \left(\frac{a}{l}\right)^2 \sin^2 \frac{a}{l} + \ldots \right] \]

In the same manner, if we call \( l \) the length of a simple pendulum, which makes its oscillations in the time \( T \), under the influence of gravity \( g \), and with the amplitude \( 2a \), we shall have

\[ T = \sqrt{\frac{l}{g}} \left[ 1 + \left(\frac{a}{l}\right)^2 \sin^2 \frac{a}{l} + \ldots \right] \]

If now we wish the two pendulums to oscillate with equal amplitudes, we have only to make \( a = a' \); if we wish also, to have their times of oscillation equal, we have only further to suppose \( T' = T \), then the two preceding expressions being equal to each other, we obtain \( \frac{l}{g} = \frac{l}{g'} \) and \( l = \frac{g}{g'} \), from which we can calculate \( l \), when we know from observation \( l \) and \( g' \).

As the density of the solid mass of the pendulum is usually very great, compared with that of the air, \( D \) is a very considerable number, so that this correction is always very small. Bouguer appears to have been the first philosopher who made use of it, as appears by his work on the figure of the earth. But before him Newton was well aware of the necessity of paying attention to it, as we may conclude even from the enunciation which he gives to the propositions regarding the resistance of the air above alluded to. For he there compares the motion of the pendulum, affected by this resistance, to that which would take place in a medium of the same specific gravity, and which would present no resistance.

Having thus explained in general the mathematical laws of the motion of the pendulum, whether in the air or in vacuo, we shall now describe the principal applications which have been made of them in physical science. These are, 1st, the measurement of time; 2d, the estimating of the resistance of fluid media; 3d, the comparison of the intensities of gravity on different parts of the surface of the terrestrial spheroid, from which certain positive conclusions have been drawn regarding the figure of a spheroid, as well as the arrangement and the density of the strata of which it is formed.

The first idea of employing the pendulum as a measure of time is due to Galileo; and it occurred to him when he was observing the apparent isochronism of the small oscillations of suspended bodies. But the variation in the length of these oscillations, in proportion as the resistance of the air diminishes their amplitude, the necessity of frequently renewing, by a new impulse, the motion which this resistance was destroying; and, lastly, the tedious necessity of following, and counting directly, the oscillations, one by one, during the whole interval that is to be measured; these proved serious obstacles to a practical and certain use of the instrument. Huygens had the merit of surmounting all these difficulties, by employing the pendulum in clocks to regulate the motions of a system of wheels, acted on by a constant power which tends continually to make them revolve; the pendulum determining the rate of their gradual rotation, by acting on them at equidistant intervals. The pendulum carries at its upper extremity a piece in the form of an anchor, which is termed the escapement, and of which the two ends, carried successively from right to left, and from left to right, by the oscillatory motion, are alternately engaged and disengaged with the teeth of a principal wheel, whose rotation they thus serve to check, and which, in its turn, serves as a similar alternate check to the other wheels. These now turn more or less slowly, according to the relation of the number of their teeth to that of the principal wheel. By applying, then, to their axes one or more indices, which turn on a dial-plate divided on the outside, we obtain by their indications so many unities of different kinds, the amount of which shows the number of oscillations that have been made. These unities of time are hours, minutes, and seconds. Great care is taken in the construction as well as in the application of the wheels, so that their motion may be as easy as possible, and that they may always obey, with equal facility, the intermitting impressions of the pendulum. The body of the pendulum itself is constructed with particular precautions. It is formed of a rod, or system of rods, of metal, terminated below by a mass also of metal, and very heavy; generally of a lenticular form, which, as the edge lies in the direction of the plane of oscillation, possesses the advantage of diminishing the effect of the air's resistance. Besides this, as the dilatations and contractions of the metal, by the changes of temperature, would lengthen or contract the pendulum, and thus cause it to alter the duration of its oscillations, the stalk of the pendulum is composed of a number of slips of different metals, which are so combined that the centre of oscillation of these slips and of the lenticular weight remains constantly at the same height.

Such is, in general, the mode of applying the pendulum to clocks, which we owe to Huygens, and which, by the exactness it has introduced in the measurement of time, is one of the finest and most valuable presents which the sciences have ever received from the hands of genius.

The second application of the pendulum, namely, its use in determining the resistance of fluid media, we owe to Newton, who has explained it with much detail in the sixth section of the first book of the Principia. The intensity, and the law of the resistance, are estimated from the progressive diminution of the amplitudes, determined by observation. We may see in that part of the work above referred to, the profound nature of the theory on which this deduction is founded, as well as the experiments themselves to which Newton applies it. The pendulums which Newton made use of were, in general, spheres of wood, or of metal suspended by threads. Besides the law, also, of resistances, several important points in physics depend on this sort of observation. Newton, for example, made use of it to establish the fact, that the action of terrestrial gravity upon all bodies is proportional to their mass; and also to inquire if these bodies, when in motion, suffer any sensible resistance by the presence within them of subtle media, which have been supposed to spread throughout the whole universe.

Lastly, it now remains to consider the use of the pendulum in measuring the intensity of gravity on different parts of the terrestrial spheroid; and we have kept this application for the last, on account of the delicate nature of the experiments which it requires, and which are now really performed. It would be of no use to enumerate here all the methods which have been successively employed, and successively abandoned as experiments of greater exactness came to be required. Even the results of these first attempts, though they may have been at the time very useful, cannot now be any more employed, so much do the limits of the errors which they admitted exceed those which are allowed by our actual processes. These can be reduced to three principal methods; two of them give the absolute measure of the pendulum; the one is due to Borda, the other to Captain H. Kater; the third gives merely the relations of the lengths of pendulums in different places, and deduces these by comparing the number of oscillations performed in the same interval of time by the same compound pendulum, supposed to be of an invariable form, and which is carried successively to the different places of observation.

I.—BORDA'S METHOD.

The method used by Borda was originally described in a memoir inserted in the third volume of the work which Delambre has published under the name of Base du Système Métrique Decimal. The same memoir includes a detailed account of a very great number of experiments performed in this manner by Messrs Borda and Cassini, to determine the length of the seconds pendulum at the observatory at Paris. The method of Borda has since been simplified by the French astronomers, so that, without losing any of its original exactness, it has been rendered more easy of execution in travelling, and in places where the observer can only reckon upon the resources he carries with him. Under this new form this method has been employed on a great number of points of the terrestrial arc, comprehended between the Pithouse Islands and the Shetland Islands. The description of these modifications, and of the results thus obtained, will be found in a volume which forms a sequel to that of Delambre, and which has been published by Biot and Arago. It is from thence that we shall take our general account of this method, the description of which will serve also for the explanation of the others, these having many points in common with it.

The fundamental principle of this method consists in employing for a pendulum a system of bodies which approaches the nearest possible in its properties to the simple pendulum, and which we can reduce to this ideal case by corrections equally simple to calculate, and exact in their application. The pendulum is formed by a ball of platinum, suspended to a metal wire (fig. 3). The under extremity of the wire is screwed into the bottom of a spherical cap of copper, of the same radius as the ball, and which being applied on its surface with a little tallow, adheres to it in consequence of the pressure of the atmosphere, and of the perfect contact resulting from its sphericity. The other end of the wire is attached to a suspended knife (fig. 4), which oscillates on a plane of agate (fig. 5), furnished with adjusting screws, by which it can be brought perfectly horizontal; a circumstance which is ascertained by placing on this plane a glass spirit-level without its frame. The mass of the knife is previously adjusted, so that its oscillations may be very nearly isochronous with those of the clock, by which the whole pendulum must be regulated. This is done by the motion of a small ring of metal A, which screws round a metal rod T fixed to the knife, and which, by screwing and unscrewing, approaches to or recedes from the plane of suspension, giving to the momentum of inertia of its mass a greater or less influence on the motion of the system of the knife and its rod. When the isochronism of the oscillations of the knife and of the clock is as perfect as can be obtained by this method, we suspend from the knife the wire and ball, giving to the wire such a length that the oscillations of the whole system may differ but little from those of the clock, consequently from those of the knife itself. It can then be shown, as well by calculation as by experiment, that the mass of the knife exerts no sensible influence on the length of the pendulum, which arises from its centre of gravity being then excessively near the plane of suspension. The whole system of the knife, the ball, and the wire, has only now to exert an effort infinitely small to complete the exact regulation of the oscillations of the knife, and to make them agree with those of the whole system.

The pendulum is enclosed, with the clock, in a glass case, where it is exempt from the agitation of the air. Behind the wire, at a very small distance, is fixed horizontally a scale of equal parts, which serves to measure the amplitudes of the oscillations. Two sensible thermometers, carefully adjusted, are fixed near the wire, the one at the height of the plane of suspension, the other at the height of the ball, in order to indicate, at every instant, the temperature of the air around the wire. But as the wire, on account of the smallness of its mass, receives the impressions of temperature much more rapidly than the most sensible thermometer, the experiment is made in a room so large and sheltered, that the temperature of the air in it may change very slowly. The state of the thermometers is observed through the glasses of the case without ever opening it during the period of the oscillations.

Every thing being thus disposed, we place, at the distance of seven or eight metres, a telescope fixed horizontally, and the eye-glass of which has a wire fixed vertically before it. We direct this wire upon that of the pendulum when in a state of rest, and we then place in the same direction, on the ball of the clock, also at rest, a small circle of paper to serve for an index. These preparations being made, the clock is set to oscillate, and is no more stopped. When its rate of going has become very steady, we cause the pendulum also to oscillate, and shutting the door of the glass case, we proceed to observe it from without with the telescope. If it should move exactly at the same rate with the clock, it would always be found in the same position in relation to the index in all its consecutive oscillations. But this never happens, and the pendulum goes always quicker or slower than the clock. If it goes quicker, it only coincides for an instant with the index, after which it passes it, recedes from it, returns to it in the opposite direction, passes it anew, and, after having receded from it again, returns to coincide with it a second time, and follows its motion of oscillation for an instant. The telescope which serves to observe these separations and these coincidences magnifies the are described by the pendulum and by the clock, augments their apparent velocity, and thus enables us to judge of the instants of coincidence with singular precision. Between two consecutive coincidences the pendulum gains or loses two oscillations upon the clock, and a simple proportion determines how much it must gain or lose in twenty- Pendulum four hours of the clock if it be sexagesimal, or in ten hours if it be decimal. If we suppose $N$ to denote the interval between two coincidences, in clock time, it follows, that while the clock makes $N$ oscillations, the pendulum makes $N \pm 2$; the sign $+$ being employed if the pendulum goes quicker than the clock, and the sign $-$ if it goes slower. Thus, during any number of oscillations of the clock, denoted by $J$, the number of oscillations of the pendulum will be proportionally $\frac{J}{N} = \frac{2J}{N}$, or $J = \frac{2J}{N}$.

If the clock be sexagesimal, the number $J$ of its beats in twenty-four hours is 86,400. If it be decimal, this number is equal to 100,000. Both these systems have been employed by the French observers. Whatever may be the one which we adopt, we regulate the length of the wire of suspension in such a manner that the coincidences of the pendulum with the clock may not be very near to each other, which would multiply unnecessarily the trouble of the observer. But neither must they be made too distant, because in that case the pendulum and the clock, detaching themselves too slowly from each other, the precise instants of each coincidence become more difficult to observe. A few trials will soon point out a convenient medium between these extremes. Then the difference in the diurnal rate $\frac{2J}{N}$, or $n$, between the pendulum and the clock, always forms a very small number of oscillations. But the extent of the arcs described by the pendulum, diminishing always by the effect of the resistance of the air, while the clock, having its motion restored by the action of its weight, preserves always the same amplitude, it hence always happens that the intervals between the successive coincidences of the same pendulum vary with the time, which alters the value of the number $n$. During this inevitable change, the period when the coincidences are observed with the greatest precision is that where the amplitudes of the oscillations of the pendulum and of the clock are equal to each other; so that if we are obliged, by any consideration, only to observe a small number of coincidences, we must regulate the primitive range of the pendulum, so as to approach as near as possible this condition of equality.

The difference in the rate $\frac{2J}{N}$ or $n$ corresponds with those oscillations of the pendulum which are performed between the coincidences which we compare together, that is to say, with an amplitude of arc varying from $2\alpha$ at the beginning of the interval to $2\alpha'$ at the end of it. The duration of these oscillations is larger than if the oscillations had been performed with the same pendulum, but with amplitudes infinitely small; and, therefore, to render the results comparable with each other, they must be reduced to this latter case. For this purpose, at the moment of each coincidence, we observe, through the fixed telescope, the point of the horizontal scale at which the wire stops in its excursions on each side of the vertical. This furnishes sufficient data to calculate the angular deviation of the pendulum from the vertical, at the instant of the coincidence, since we know the distance of the scale from the plane of suspension at which the centre of rotation lies. We mark also the state of the interior thermometers, and that of the barometer, at the same instant. If the arcs $\alpha$ and $\alpha'$ are both very small, as it is usual to make them, we may, without sensible error, suppose all the oscillations made with the mean amplitude $\alpha + \alpha'$. Then, after what has been shown above, each of them, expressed in oscillations infinitely small, will be equal to $1 + \frac{1}{16} \sin^2 (\alpha + \alpha')$, which we may express in an abridged form by $1 + \mu$, and consequently the $J + n$ oscillations of the pendulum supposed to be made in this arc will be equal to a number of oscillations infinitely small, expressed by

$$ (J + n)(1 + \mu), $$

or $J + n + \mu(J + n)$,

a result which we may represent in an abridged form by $J + n'$; the number $n'$, according to what has been above established, never being any way considerable.

If the arcs $2\alpha, 2\alpha'$ differ more than in a slight degree from each other, as, for example, when the interval between the coincidences which we compare is large enough for permitting the resistance of the air to have a considerable effect in modifying the first of these arcs, it will then be no more sufficiently exact to suppose all the oscillations made with the mean amplitude $\alpha + \alpha'$. But this inconvenience may be remedied, by observing experimentally the law of the gradual decrease of the amplitudes. This law is in geometrical progression when the number of oscillations increases in arithmetical progression; that is, if we begin with the instant when the half amplitude was $\alpha$, and represent by $\alpha_n$ the amplitude which takes place after $n$ oscillations, we find $\alpha_n = \frac{\alpha}{K^n}$, or because $\alpha$ and $\alpha_n$ are supposed very small, $\sin \alpha_n = \frac{\sin \alpha}{K^n}$; $K$ being a coefficient, which, in the same state of the air, is constant for the same pendulum, and depends on its length, its shape, and its other physical qualities. This law, first remarked by Borda, and since confirmed by the other French observers, is a necessary consequence of the smallness of the amplitudes, and of the feebleness of the resistance, which alters each amplitude in succession, proportionally to its extent. But however this may be, it is enough that it really subsists, to enable us to calculate by it the exact sum of the squares of the half amplitudes in the successive oscillations; a problem which is reduced to the summing of a geometrical progression of $n$ terms, of which the ratio is $\frac{1}{K}$. This sum is simplified when we consider the extreme minuteness of the arcs which we compare, and by then pushing the approximation to their second power inclusively, which is the limit of the correction necessary for each individual amplitude. We thus find

$$ \mu = \frac{\sin(\alpha + \alpha') \sin(\alpha - \alpha')}{S^2 M \left\{ \log \sin \alpha - \log \sin \alpha' \right\}} $$

$M$ being the modulus of the tables of common logarithms, or $230258599$. If we suppose the arcs $\alpha$ and $\alpha'$ so small, that in the development of their series and of their logarithms we may limit ourselves to their first power, this expression of $\mu$ becomes what we have already obtained by our first approximation.

By these calculations we ascertain the rate of the relative going of the pendulum on the clock, which serves to measure the intervals between the coincidences. We know that it performs $J + n'$ infinitely small oscillations, while the clock makes $J$ of them. Suppose now that the latter advances, during the mean solar day, a number of oscillations equal to $h$, that is, that it performs $J + h$ oscillations during the same time that a clock, exactly regulated by mean time, performs the exact number $J$, we shall evidently obtain this proportion; $J$ oscillations of the clock are to $J + n'$ infinitely small oscillations of the pendulum, as $J + h$ oscillations of the clock, or a mean solar day, are to the number of oscillations of the pendulum during a mean solar day. The latter number is thus found equal to

$$ \frac{(J + n')(J + h)}{J}, $$

or $J + n' + h + \frac{hn'}{J}$, the quantity which, for simplicity, we shall represent by $J + n'$. With the apparatus so disposed as we have described, if the clock is not very far from mean time, so that \( h \) denotes a small number of oscillations, the correction expressed by the last term, the only one which demands a calculation, will be of an extreme minuteness, and easily obtained with great precision.

It now remains to measure the length of the pendulum from the plane of suspension to the bottom of the ball of platinum. For this purpose we place beforehand, under this ball, a small plate of metal, well polished, perfectly horizontal, and which can be made to sink or rise vertically by means of a screw, of which the threads being very fine, permit the smallest motion. When the coincidences are finished, we open the glass case, and we raise gently this plane until it comes in contact with the ball of platinum. We must be equally careful to avoid raising it too much, which would raise the ball, and make the pendulum too short, or not raising it so high as the contact, which would give a pendulum too long; but if we take for an index the disappearance of a thread of light between the plane and the ball, at their common point of contact, we may then succeed, by a little skill, in fixing this contact with the utmost degree of rigour. This, however, is never done at the first attempt, for the entry of the observer into the glass case, however short, always elevates in a slight degree the temperature of the air contained in it, and consequently that of the wire, which acquires this temperature in the same instant, from whence arises a small increase of its length, which we ought to be aware of. On this account, instead of establishing a perfect contact between the plane and the ball, in that accidental state of the wire which the interior thermometers, less sensible than it, do not perhaps indicate with sufficient exactness, it is better to confine ourselves at first to the mere preparing for the operation, by making the little plane approach extremely near the ball, without, at this time, actually touching it. We observe now the point where the index of the screw that moves the ball stops, and then coming out of the glass case, we shut it until the temperature within, and the thermometers which measure it, have had time to return to a state of rest. We then open the case anew, and finish in an instant the operation of contact, which is easily done, as we have only to give to the plane of contact a very slight degree of motion, and such as we are previously quite prepared for. At this moment, or rather before entering the glass case, we mark the temperature of the thermometers within, and consider this as the temperature of the wire at the instant of contact.

The distance of the plane of suspension from the bottom of the ball is now fixed, and in such a manner that it is henceforth invariable, or at least we may suppose it such during a long interval of time. For, the supports of the plane of suspension being fixed in the wall itself; and those of the plane of contact being cemented to a large stone resting on this wall, or sunk into the ground, the accidental variations of temperature cannot alter the distance which separates them, excepting in a very slow degree. It remains then to measure this interval by means of a divided rod of metal; but to determine the length of such a Pendulum rule, its extremities must be quite free; and how can we, in that case, apply its summit exactly on a level with the plane of suspension? Borda has very happily resolved this difficulty, by adapting to one of the extremities of the rule a knife of suspension, which is fixed to it, so as to touch it on its edge. Suppose we wish first to measure the length of the rule, we take off the knife, and apply the rule itself to the apparatus intended for that object. If we wish then to measure the length of the pendulum, we replace the knife, and suspend the rule, thus armed, on the plane of suspension, in place of the pendulum itself. In our experiments, the knife is adapted to the rule by means of a metal case; the rule is inserted into this case until it touches the knife, when it is fixed in this position by means of a strong pinching screw, denoted by V, and which is screwed by an iron key. It only remains to alter the length of this rule in such a manner as to render it exactly equal to the actual distance which is found between the bottom of the ball and the plane of suspension. For this purpose, in the experiments of Borda, the rule carried, on its under part, a divided tongue, having a free motion. When the observations of the coincidences were finished, the pendulum was removed, and for it was substituted this rule, of which the tongue was let down until it fell upon the little plane which had touched the ball of the pendulum. Then, by reading, with a magnifying glass, the divisions of the tongue, it is easy to know the distance between the bottom of the ball and the axis of suspension.

In Borda's experiments, the pendulum was twelve feet long. A rule of such a length could not have been carried in travelling, or even in stations of difficult access, without the risk of serious errors, resulting from the bending which it must have received. On this account the French observers, who were intrusted with such experiments, thought it necessary to modify in this point the apparatus of Borda, and they confined themselves to pendulums much shorter; as are those which swing mean sexagesimal or mean decimal seconds. This enabled them to make use of rules much shorter, more portable, and which they could also make larger and more solid without increasing too much their weight; but then it became indispensable to introduce a still greater degree of precision than before into the determination of the length of the rules; into that of the divisions traced upon the tongues which were fixed to them; and, lastly, into the measurement of the variable parts of these tongues, which were used in each experiment, in order to adapt them to the different lengths of the pendulum which they were intended to measure. All these elements were obtained with unexpected exactness, by employing for their determination the apparatus already used with such success for the comparison of metrical scales, under the name of comparateur. All the details of this application may be seen in the work of Biot and Arago, above referred to.

By means of the operations above described, we find the total length of the pendulum, from the plane of suspension to the bottom of the ball of platinum; such at least as it is at the instant of contact of the latter with the little plane. But this length may not be, and is not in general, the same which the pendulum has when it began to oscillate; because

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1 Considering the indispensable necessity which there is of preserving rigorously this quantity invariable, it appears to be of extreme importance that the ground may not yield when the observer approaches to complete the measurement. The only means of avoiding, beyond suspicion, this possibility, is to construct round the pendulum a platform, supported on certain points of the ground, at a distance from the stone which carries the little plane with which we effect the contacts of the ball, so that this apparatus may become quite independent of the motions of the observer. Pendulum: the temperature, which modifies almost instantaneously the length of the wire, cannot have been the same at the time of the contact of the plane, and during the observation of the coincidences. But it is to this state that we must evidently reduce the length that we have attained. For this purpose, let it be denoted by \( A \), and suppose \( t \) the temperature of the wire in degrees of the thermometer at the instant of contact, \( t \) being its mean value during the coincidences; then if \( R' \) represent the radius of the ball of platinum at the temperature of \( t \) when the contact was produced, the length of the wire at that instant was \( A - 2R' \); so that, calling \( K \) the lineal dilatation of the matter of the wire, for a difference of one degree in the temperature, the length of the wire at the time of the oscillations must have been \( (A - 2R') \{1 + K(t - t)\} \). In the same manner, if \( K' \) be the lineal dilatation of the substance of the ball, its diameter, at the time of the oscillations, will be \( 2R' \{1 + K'(t - t)\} \); and adding this quantity to the length of the wire, we obtain \( A + \Delta K(t - t) + 2R'(K' - K)(t - t) \) for the distance of the plane of suspension from the bottom of the platinum ball during the actual time of the oscillations. By deducting from this length the radius of the ball, such as it was at the same instant, that is, \( R' \{1 + K'(t - t)\} \), we shall have the distance of the centre of the ball from the plane of suspension, a distance which we shall call \( h \). This being determined, if the wire which sustains the ball, and the cap which fits upon its surface, were both without weight, or if their weight could be altogether neglected in comparison with that of the ball, the length \( l \) of the simple pendulum isochronous with the compound one thus formed would be obtained by the above formula, and would be \( l = h + \frac{c}{m} \), \( m \) being the mass of the ball, and \( c \) its momentum of inertia relative to an axis drawn through its centre. But calling \( \rho \) the density of the mass of the ball, and \( R \) its radius, at the temperature at which the pendulum oscillates, its mass \( m \) is equal to \( \frac{4}{3} \pi R^3 \rho \), and the value of \( c \) is \( \frac{8}{15} \pi R^5 \rho \). Substituting these values, we have \( l = h + \frac{2R^2}{5h} \); hence it appears, that it would be easy to calculate \( l \), since \( R \) and \( h \) are known. But, in truth, the weight of the wire and that of the cap can never be absolutely nothing. They are only very small, relatively to the weight of the ball; so that the preceding value of \( l \) is but an approximation, which, to become quite exact, requires a small correction, depending on the relation of these masses. This correction being rather complicated in its expression, we shall not repeat it here, but refer to the memoir of Borda, or the work of Biot and Arago, already mentioned, and represent it by \( Q \); as it is always negative, the length \( l \) of the simple pendulum isochronous with the pendulum observed will become \( l = h + \frac{2R^2}{5h} - Q \).

Now we have seen, that, in these experiments, the apparent gravity which impels the pendulum is less than the real gravity which operates in vacuo, on account of the statical effect of the ambient medium; but for a simple pendulum of the length \( l \) moved by the force of gravity \( g \), the time \( T \) of its infinitely small oscillations is expressed by \( \sqrt{\frac{l}{g}} \); and if we wish to obtain oscillations of equal duration with different forces of gravity, we must vary the lengths in proportion to these forces, so that the relation \( \frac{l}{g} \) may remain constant. Now, after what we have before seen, if we denote by \( D \) the density of the substance of the pendulum at the temperature of freezing, and under the atmospheric pressure of \( 0^\circ76 \), that of the air being \( 1 \), if, besides, we denote by \( e \) the cubic dilatation of this same substance, the relation of the apparent gravity in air to the gravity in vacuo, under the pressure \( p \), and at the temperature \( t \), will be expressed by \( 1 - \frac{p}{0^\circ76(1 + t - 0^\circ00375)(1 - et)}D \), which, for simplicity, may be represented by \( 1 - \gamma \). Then, to obtain the length \( l' \) of the simple pendulum, which, making its oscillations in vacuo under the influence of the gravity \( g \), would be isochronous with the actual pendulum \( l \), going in the open air, we must take \( l' = \frac{l}{1 - \gamma} \), which, on account of the smallness of \( \gamma \), may be reduced to \( l' = l + \gamma l \). We have denoted above by \( J + n^2 \) the number of infinitely small oscillations performed by the actual pendulum in a mean solar day. Such, then, is also the rate of the pendulum \( l' \). If we wish, in fine, to obtain the length \( l'' \) of a pendulum which would move exactly to mean time in vacuo, under the influence of the same power of gravity as \( l' \), we have only to consider that, according to the preceding expression of \( T \), the lengths \( l'' \) must be directly proportional to the squares of the times of their oscillations, and therefore reciprocally as the squares of the number of oscillations made in equal times. We must take, then, \( \frac{l''}{l'} = \frac{(J + n^2)^2}{J^2} \), whence we obtain \( l'' = l' \frac{2n^2}{J} + \frac{n^4}{J^2} \).

The length \( l'' \) thus obtained is now free from all the variable elements, which depend on particular circumstances of their observations. This constitutes what we should properly call the absolute length of the simple pendulum in the place of observation.

The experimental method which we have described, when it is employed with all due care, gives results which, in the same place, are perfectly comparable with each other. For, with various lengths, such as the sexagesimal pendulum, for example, or the decimal pendulum; the deductions from particular experiments do not differ generally from each other more than in the thousandth parts of a millimetre. To establish, however, completely the theoretical certainty of this method, it is necessary to examine more particularly some of the circumstances which form a part of it.

Our first remark relates to the extensibility of the wire to which the platinum ball is suspended. It is clear, that, during the period of each oscillation, the wire is impelled in the direction of its length by two forces of different kinds, and of different intensities; of which the one is the varying traction, acting on it every instant by the weight of the ball decomposed into its direction; and the other is the centrifugal force, which the motion of oscillation generates. It may evidently be a question, whether this double action has not on the oscillations a sensible influence, which, disappearing in the measurement of the length taken when the pendulum is at rest, would alter the result which we have obtained. M. Poisson has submitted this question to calculation in the eighth volume of the Journal de l'École Polytechnique, and he has found, first, that the symmetry of the oscillations on each side of the vertical is not altered by the extensibility of the wire; at least if we suppose them to be performed in vacuo, a circumstance which it was easy to anticipate from the symmetry itself of the mode of action of this force. But he has found that their duration is affected with a periodical inequality, in consequence of which the successive oscillations are not isochronous among themselves. When the total extension, however, suffered by the wire is very small, which is generally the case in experiments where the wires are formed of metallic substances, the effect of this inequality neutralizes itself in the mean duration of a great number of oscillations; only this mean period is a little longer than if the wire had been altogether inextensible. Let \( l \), for example, be the linear length which the wire would naturally have if left to the sole attractive action of its particles on each other; and suppose that, by suspending the platinum ball at its lower extremity, it lengthens by a small quantity \( \lambda \). This being the case, if the half amplitude of the oscillation is denoted by \( a \), M. Poisson finds that, limiting the results to the square of \( a \), the mean duration \( T \) of the whole oscillations will be

\[ T = \frac{1}{g} \left\{ 1 + \frac{1}{16} \left( 1 + \frac{11\lambda}{l} \right) \sin^2 a \right\}. \]

If we wish to suppose the wire inextensible, we have only to make \( \lambda \) equal to nothing, and it hence appears that the extensibility only modifies the correction of amplitude, already in itself so small, and alters it by a quantity which, from the small extensibility of the metals, cannot in general produce any effect that could be detected by observation.

A second circumstance, which deserves equally to be examined, is the probable influence of the motion of rotation of the platinum ball round the direction of the wire, and any twisting which the wire may suffer during the oscillations. M. Poisson has examined, by a calculation in the *Connaissances des Temps* for 1815, the effect of such a motion; and he has found that, in the ordinary disposition of the pendulum of Borda, it is rendered in a manner insensible by this circumstance, that the momentum of inertia of the ball, and of the whole pendulum, relatively to an axis drawn through the direction of the wire, is a very small quantity. Let \( R \) be the radius of the platinum ball, \( h \) the distance of its centre from the axis of suspension, \( \alpha \) the angle which the rotation of the ball makes each of the points of its surface describe during the period of an oscillation, this angle being measured in a circle of a radius equal to 1. \( T \) the time of an infinitely small oscillation of the same pendulum, in the case when the rotation is nothing; M. Poisson finds that the real duration of the oscillations will be

\[ T \left\{ 1 - \frac{\alpha^2 R^4}{50h^2 \pi^2} \right\}, \]

\( \pi \) being always, as before, the circumference to the diameter, which is unit, or, what is the same thing, the semi-circumference, of which the radius is 1. In regard to the effect of the torsion communicated to the wire by the rotation of the ball, M. Poisson proves that it can have no influence on the duration of the oscillations. In the shortest lengths of pendulum which have been observed by the above process, \( R \) was less than 0°02, and \( l \) nearly equal to 0°74. Adopting these numbers, and supposing, besides, that the ball describes two whole circumferences for each oscillation, which would be a very rapid motion of rotation, we shall then have

\[ \frac{R}{h} = \frac{1}{37}, \quad \frac{\alpha}{\pi} = 4, \]

and consequently

\[ \frac{\alpha^2 R^4}{50h^2 \pi^2} = \frac{1}{5856569}; \]

whence it appears that the time of the oscillation would be diminished only by a quantity altogether insensible, even on the above suppositions. But it is far from being the case that a rotation so rapid as we have supposed really takes place in the experiments; on the contrary, when we set the pendulum in motion, we take great care to avoid every movement of this kind; we also pay particular attention to let fall the ball without any lateral impulse, so that its oscillation may be performed as exactly as possible in a vertical plane, which we also take care to verify by observation, when the pendulum is in motion. It would be useless to attempt to obtain this condi-

tion in a manner more exact; for it is known by the calculation of conical oscillations, that when these take place in an orbit much flattened, their duration is almost exactly the same as if they were quite plain.

The last object to be considered, and of which the discussion is as important as it is delicate, is the influence which the form of the suspending knife may have upon the oscillations of the same pendulum. Comparing, indeed, the motion of the pendulum to that which would take place round an axis of suspension perfectly rectilineal and mathematically straight, we suppose, or at least seem tacitly to suppose, that the edge of the knife forms such an axis, which is physically impossible, since the most perfect art cannot give it any other form than that of a round surface, the breadth of which is sensible to the microscope, and which, even there, appears always like a saw indented with teeth more or less deep. Now, if this surface were a circular cylinder, a simple calculation, which was first made by Euler, shows that the durations of the oscillations will be the same as if they were performed round a rectilineal axis placed under the surface of the cylinder, and at a distance equal to the radius of its curvature; and in the case of very small oscillations, this result may be extended to a knife of any form, if we take for its curvature that of its osculating circle. Hence it follows, that in order to have the true length of the simple pendulum in this circumstance, we must subtract the radius of this circle from the length calculated on the hypothesis of a rectilineal axis, according to the oscillations observed. But such a correction would throw great uncertainty upon the results; for the osculatory curvature of the knife cannot be measured, or even appreciated, by any process, and it must vary considerably, either by the difference of workmanship in different knives, or by the inevitable wearing which the edge of the knife undergoes when the weight suspended from the wire presses it against the plane of suspension. Fortunately, the extent, and the variability itself, of the effects which this cause should produce, serve to prove that it has no action whatever in experiments; for, in the first place, by loading successively the head of the same knife with several weights very different among themselves, in order to observe if these different systems, previously according with the same clock, would have an influence on the length of the pendulum, Borda has found that this influence was absolutely insensible, although the curvature of the edge, to which he did not pay attention, was then undergoing very different modifications under the unequal compressions to which they were subject. Secondly, the length of the simple pendulum, beating seconds at the Observatory of Paris, which Borda had deduced from a pendulum of twelve feet long, has since been found as exactly the same as the difficulties of the operation would permit, by employing, with the same knife and the same ball, wires four times shorter, which gave a much greater influence to the alterations of length which the curvature of the knife could produce. Lastly, by observing successively, at the same place, with the same ball, and the same length of wire, but with knives whose edges presented an extreme diversity, from the highest possible finish to the greatest coarseness in the execution, M. Biot has obtained, at Leith Fort, in Scotland, such lengths for the simple pendulum, between which no sensible difference could be observed, although no correction whatever was made for the curvature of the knife. These proofs of different kinds, but all agreeing in their consequences, seem to show evidently that in the process of Borda the shape of the edge of the knife has no sensible influence on the results, and that it is unnecessary, therefore, to pay any attention to it; and yet, as the theory of oscillations round cylindrical axes cannot be questioned, we must either conclude that in this circumstance the oscillation is really not performed on a cylinder of sensible dimensions, but upon Pendulum: the ideal axis of insensible dimensions, formed by the asperities which still exist in the grain of metal of which the knife is composed; or that the agreement previously established between the proper motion of the knife and the total motion of the pendulum, compensates physically the effect which the curvature of the knife would have upon the oscillations, if it consisted really of a simple cylinder without mass attached to the wire.

II.—METHOD OF KATER.

The method employed by Captain Kater to measure the length of the pendulum is founded upon this theorem of Huygens, that whatever be the form of the oscillating body, the centre of oscillation and the centre of suspension are reciprocals to each other, a theorem of which we have already given the demonstration. To realise this disposition, Mr Kater chose a body of such a form that it was easy to determine by calculation the approximate position of its centre of oscillation for a given position of the axis of suspension. These two points being thus known, he fixes there immovably two knives parallel to each other. In the space which separates them he then adapts to the body a moveable weight, and having first placed it at random, he makes the system oscillate successively upon the one knife and upon the other. If, as it almost always happens in the first trial, the oscillations performed in the two cases are of unequal duration, he moves the intermediate weight, so as to bring them nearer to an equality; then, comparing these anew by observation, he finds necessarily a less disparity between them, which he again reduces, until at last, after a few trials, the duration of the oscillations performed round the two axes become exactly equal. The justness as well as the rapidity of these reductions are favoured by the form which Mr Kater has chosen for his oscillating body. This form is a simple rectilineal bar of brass, towards the two extremities of which are placed two known weights; the one of which is immovably fixed, and the other, being moveable, but at the same time capable of being fixed in a similar manner, serves, first, by its motion, to establish between the two knives, not exactly, but approximately, that reciprocity between the oscillations to which it is desired to bring up the system. This reciprocity is then rendered rigorously exact, by the much more delicate motion of a third smaller weight placed between the knives, in that part of the rod where we know, by calculation, the effect of its displacement will have the least sensible influence upon the oscillations, which is found to be towards the middle of the rod, in the division of the weights adopted by Mr Kater. A divided scale, engraved upon the bar of the pendulum itself, serves to measure the displacement of this latter moveable weight. In the experiments of Mr Kater, this scale was divided into twelfths of an inch, and a displacement of twelve parts produced a difference of about four seconds in the diurnal rate of the system, reckoning sexagesimally; whence we may be able to appreciate the extreme delicateness of this mode of regulation. The bar, with its weights, is represented in fig. 1, Plate CCCCHIII. Fig. 2 shows its disposition during the observations.

It is then, as appears, placed before a clock, with which it is compared, by means of a fixed telescope, after the method of coincidences of Borda. But the mode of experiment employed by Kater, requiring the oscillations round the two axes to be observed in a state of rigorous equality, it becomes necessary to avoid, in their comparison, every change of temperature, and thus it is necessary to make the results independent of the variations of this kind which inevitably arise in the atmosphere. That could only be obtained by rendering the series of coincidences very short, and multiplying the successive inversions of the apparatus. But then, to obtain the same exactness, it is necessary to fix the coincidences with much greater precision than in the method of Borda, where the little influence which they have is one of the principal advantages. Mr Kater has attained this object, by fixing upon the lentil of his clock a white disc, traced upon a black ground, and of such a size that it is exactly covered, and no more, by the interposition of the bar of the pendulum, when this is at rest in the situation of the vertical. This same occultation, being then observed during the motion of the clock and of the pendulum, serves to fix the instant of the coincidence; and Mr Kater finds, that in this manner there cannot be any error greater than a second on the instant to which each coincidence belongs.

The rest of the details of the observation are the same as in Borda's process. The amplitude of the arc of oscillation is observed at each coincidence, as well as the state of the barometer and thermometer; and these are employed in the same manner to reduce the oscillations to what they would have been if they had taken place in vacuo, and with amplitudes infinitely small, at the observed mean temperature of the oscillating body.

It still remains to obtain the length of the simple pendulum corresponding to this rate of going. After an equality has been obtained in the oscillations round the two knives, this length is equal to the distance between the edges of the knives at the moment of the oscillation of the pendulum, at least if we consider these edges as lineal axes. It would be evidently impossible to observe the distance in question during the actual motion of the pendulum; but this defect may be supplied by determining it first for some known temperature, and reducing it by calculation to the value which it ought to have during the coincidences, according to the temperature at the time these took place, and the dilatation of the substance of the pendulum, which is also known. It is thus that Mr Kater operated, and he has obtained the true distance between the knives, in comparing it by a microscopic process, with the metal rule which he employed as a standard of measure. In order that this operation may be put in practice, the two edges of the knives must be very exactly parallel. Mr Kater accordingly disposed them in this manner before the experiments, employing the measure itself of their distance to determine and to prove the accuracy of their position, to which they were gradually brought by means of adjusting screws, which allowed each knife to move by very small displacements. He also took advantage of this method to render their direction quite perpendicular to the length of the bar. Lastly, as the distance between the knives, which we are seeking to determine, is that which took place when the pendulum was in a vertical situation, Mr Kater, during the measurement, applied to the bar, now horizontal, a force of longitudinal traction equal to what it exerted on itself by its own weight, and in a state of oscillation. In calculating the influence which the curvature of the knives, supposing it to be circular, can have upon the length of the simple pendulum, deduced from the oscillations of a similar apparatus, M. Laplace has found it to be equal to nothing, and that this length was always rigorously equal to the distance between the edges of the knives. This theorem is only true on the supposition that the two edges are of the same curvature; but whatever precaution may be taken to render them identical, even making them together, and with a single piece of steel, it will be impossible to be assured that there may not be found differences, not merely very small, but very considerable, in the radii of curvature of their osculatory circles; since these circles are the result, not of any measured and geometrical operation, but of a work of reducing and polishing necessarily vague and irregular. It would appear then by this, that the results of this method would still be subject to the same uncertainty in theory as the results of the others are; but these uncertainties are dispelled in both cases, by the experimental proofs already described, that, in the process of Borda, the figure of the edge of the knives has no influence upon the length of the simple pendulum, deduced from their oscillations. In short, what completes the proof, that these two methods do not include in themselves any source of inaccuracy, is the surprising and almost ideal coincidence of the results which they afford, notwithstanding the diversity of the two processes. We shall have occasion to give a striking proof of it at the conclusion of the ensuing paragraph.

III.—PENDULUMS OF COMPARISON.

The two methods which we have been explaining make known the absolute lengths of the simple pendulum in every place where it is observed. Both of them, therefore, require to be absolutely determined, on the spot, in lineal measures; and this cannot be done with sufficient exactness but by a process of extreme delicacy, the practice of which implies numerous preparations. But when we wish merely to determine the ratios of the lengths of the simple pendulum to each other, for different places on the earth, we may obtain this without any absolute measurement, and by the mere comparison of the oscillations made in these places in equal times by a compound pendulum of any form. To demonstrate this, suppose, first, that the figure of the mass of this pendulum is quite invariable, and that it suffers neither dilatation nor contraction by the changes of temperature; or, what comes to the same thing, suppose the observations are always made during temperatures exactly equal; in this case, according to the formulae laid down in the beginning of this article, the length $l$ of the simple pendulum isochronous with that compound pendulum, may be expressed by $\frac{c}{mh}$, $m$ being the mass of the oscillating body,

& the distance of its centre of gravity from the axis of suspension, and, lastly, $c$ its momentum of inertia; that is to say, the sum of all the elements of its mass, multiplied by the squares of their respective distances from the axis of suspension. This length $l$, then, will be the same in whatever place we observe it, since its analytical expression depends only on the figure of the oscillating body and the density of its parts, but in no respect on the intensity of the gravitating force which impels it; so that if we take any compound pendulum, of any form whatever, but having its mass and figure constantly the same, and make it oscillate successively in different parts of the earth, it is the same thing as to cause a simple pendulum to oscillate successively in the same places. But, supposing the oscillations performed in vacuo, and with infinitely small amplitudes, or, what is the same thing, supposing them reduced to these conditions by calculation, the durations $T'$, $T''$ of the oscillations of the same simple pendulum, whose length is $l$, are connected with the intensities $g'$, $g''$ of the gravitating force, which impels them by the following relations,

$$T'^2 = \frac{\pi^2 l}{g'}, \quad T''^2 = \frac{\pi^2 l}{g''};$$

whence we obtain $\frac{g'}{g''} = \frac{T'^2}{T''^2}$,

that is to say, that the intensities of the gravitating forces are reciprocally as the squares of the times of the oscillations; or, what is the same thing, they are directly proportional to the squares of the numbers of oscillations made in equal times. For let $N'$, $N''$ be these numbers, and $T$ the total time which corresponds with them, then $T'$ will be equal to $\frac{T}{N'}$, and $T''$ equal to $\frac{T}{N''}$; so that, by substituting their values, the preceding relation will become $\frac{g'}{g''} = \frac{N'^2}{N''^2}$.

By such experiments, then, made with the same compound pendulum, we may be able to determine the relative forces of gravity in the different places of observation. But we may, with equal facility, deduce from them the ratios of the absolute lengths which it would be necessary to give to two simple pendulums in the same places, for making them bear an equal number of oscillations in a given time; for example, to beat the mean second. For let $\lambda'$, $\lambda''$ be these unknown lengths, since the corresponding times of oscillation are each one second, we shall have by our general formula $1' = \frac{\pi^2 \lambda'}{g'}, \quad 1'' = \frac{\pi^2 \lambda''}{g''}$; whence we obtain $\frac{\lambda'}{\lambda''} = \frac{g''}{g'}$;

that is, that the lengths required are proportional to the intensities of gravity in the two places. But we have seen that the ratio of the gravitating forces may be deduced from the observations made with the same compound pendulum; and introducing this determination into the preceding expression, we obtain $\frac{\lambda'}{\lambda''} = \frac{N'^2}{N''^2}$. Hence, when we have observed the numbers of infinitely small oscillations made in two different parts of the earth, by the same compound pendulum, of a constant form, the ratio of the squares of these numbers will be equal to the ratio of the lengths of the simple pendulums, which swing seconds in the same places. All these results suppose, as we have seen, that the mass and form of the compound pendulum are rigorously the same at the two stations. To obtain a degree of permanency in the mass, we form the pendulum of metal, cast in one piece, to which we adapt for suspension a knife edge made by a process which insures the firmness of the connection, and we take every precaution possible to prevent any physical or chemical alteration from modifying it during the carriage. But the permanency in its dimensions and figure is much more difficult to be obtained, because the inequalities of temperature in the different places of observation, and the accidental variations of natural heat, even in the same place, tend perpetually to disturb it. It is physically impossible to prevent the effects of these alterations in any other way, than by preserving constantly round the pendulum the same artificial temperature; a method which has been really employed, but which requires very great precautions to render the temperature round the pendulum uniform, and a constant attention to manage the sources of cold and of heat, in order to maintain it at the same fixed degree. It is on this account more simple, and perhaps more accurate, when the thing is possible, to dispose the experiment so as to have only very slow changes of temperature; then to allow the pendulum to partake of these changes, and to correct this effect on its form by calculation, from the observation of the temperature, and the knowledge of the proper dilatation of the substance of which it is composed. This correction is extremely easy; for if we resume the expression $l = \frac{c}{mh}$, which expresses the corresponding length of the simple pendulum, the momentum of inertia $c$ is of the same order as the mass, multiplied by the square of the dimensions of the oscillating body, and the denominator $mh$ is the product of this same mass by a single dimension; whence it is easy to conclude, that if the dimensions should vary in the same proportion in every direction, which really happens in changes of temperature, the quantity $\frac{c}{mh}$ will vary according to this simple proportion. Hence, if we name $l$ the length of this simple pendulum, isochronous with the compound pendulum, when the latter is at the standard temperature $t$, and denote by $l'$ the analogous length when the temperature is $t'$, representing also by $K$ the lineal dilatation of the mass of the pendulum, for a change of one degree of the temperature, and for a length equal to unity, we shall evidently have \( \frac{t}{l} = \left\{ 1 + K(t - t') \right\} \). Suppose now the pendulum \( l \) has made a number \( N' \) of oscillations in a given interval of time, for example, a mean solar day, it will be easy to calculate how many the standard pendulum \( l \) would have made in the same time, if it had been acted on by the same force of gravity; for the squares of the numbers of oscillations made in equal times being reciprocally as the lengths of the pendulums which perform them, the square of the number sought will then be

\[ N'^2 \frac{l}{p}, \text{ or } N'^2 \left\{ 1 + K(t - t') \right\}. \]

Hence we obtain for this number itself \( N'^2 \sqrt{1 + K(t - t')} \); or, reducing the radical into a series, and limiting it to the first power of \( K \), \( N'^2 + \frac{1}{2} N'^2 K(t - t') \). This approximation is always sufficient, because the co-efficient \( K \) of the lineal dilatation is always very small in solid bodies, and the difference \( t - t' \) of the natural temperatures in the places of observation can never exceed a small number of degrees. Captain Kater, for example, has operated with a similar pendulum, made of brass, the observed lineal dilatation of which was 0.00000982 for one degree of Fahrenheit. In an experiment made in London, this pendulum was found to perform a number of oscillations equal to 86051-32, the temperature being 71°6. If we wish to reduce this experiment to the standard temperature of 62 degrees, which was adopted by Mr Kater, we shall have \( N' = 86051-32, K = 0.00000982, t - t' = + 9°6 \), whence we obtain, for the correction of the temperature, \( \frac{1}{2} N'^2 K(t - t') = + 4°04 \), that is to say, four oscillations, and four hundred parts, added to the number of oscillations observed.

By operating in the same manner in every other case, we can reduce all the observations to the constant temperature once fixed as a standard.

But one of the indispensable elements in this reduction is the actual temperature of the pendulum during the experiments, and some precautions are necessary to obtain it with exactness. For the pendulum being always very large compared with the thermometers, which we can place by the side of it, it partakes much more slowly than them of the variations of temperature, so that it ought always to be a little colder than the thermometers when the temperature of the air is rising, and a little warmer when it descends. It would be impossible to estimate these differences of state; but we can render their effect insensible, by operating in a room, so large and sheltered from the sun that the temperature remains in it nearly constant, or at least suffers such slow variations, that the mass of the pendulum has time to partake of them. For in that case the thermometers will point out the state of this mass in indicating that of the ambient air; or if there remains some difference between both, the effect of this will disappear by compensation in a series of experiments sufficiently repeated.

We have mentioned above, that the corrections relative to the amplitude of the arcs and the density of the air, are made in every place for the compound pendulums, the same as in the experiments with the absolute pendulum. The duration of the oscillations may also be determined in the same manner by the method of coincidences, comparing the experimental pendulum with a clock that is actually regulated by astronomical observations. Thus, in applying this process, and these corrections, we shall obtain the numbers of oscillations which a compound pendulum would have made at the different stations, if it had oscillated in vacuo, and at a temperature always constant. Whence we may then deduce the relation of the intensity of gravity at these stations, or the ratio of the lengths of the simple pendulum, swinging the same fixed number of seconds in a given time.

To give an example of this deduction, we shall relate the following result, obtained in 1818, by Captain Kater, with the same compound pendulum the dilatation of which is stated above. The numbers of oscillations expressed in the last column are reduced by calculation to the ease of amplitudes infinitely small, the pendulum in vacuo, and the temperature being the standard of 62° Fahrenheit.

| Names of the Places | Latitudes of the Stations | Number of Oscillations of the Compound Pendulum in 24 mean Solar Hours | |---------------------|--------------------------|---------------------------------------------------------------| | London | 51° 31' 8" | 86061°30 | | Leith Fort | 55° 58' 37" | 86079°22 | | Unst | 60° 45' 25" | 86096°84 |

From other experiments previously made by means of the method of inversion, Captain Kater had determined the length of the absolute seconds pendulum at London, precisely at the same place, and in the same room, where he since made his compound pendulums oscillate. This length, expressed in English inches, on Sir George Shuckburgh's scale, was found to be 39-13908 inches. If then we call this length \( \lambda_s \) and \( \lambda_u \) the analogous lengths for the two other stations of Leith and of Unst; also \( N_s, N_u \) the number of oscillations of the portable pendulum in these three stations, we shall have, according to the formulae above laid down, \( \lambda_s = \lambda_u \cdot \frac{N_s^2}{N_u^2} \), which will give for the length of the simple pendulum, at the station of Leith, 39-15538 inches, at the station of Unst 39-17141 inches. Now, by comparing, by methods of extreme precision, the scale of Sir George Shuckburgh, with a metre of platinum, executed under the directions of the Board of Longitude of France, and verified by a commission of several members of this body, Mr Kater has found that the metre, taken at its own standard temperature, which is that of melting ice, is equal to 39-37079 inches of the scale of Sir George Shuckburgh, taken also at its own standard temperature, which is 62° degrees Fahrenheit. Hence it follows, that any length \( l \) expressed in inches of this scale, taken at its standard temperature, is equal in millimetres to 1000\( l \).

The preceding lengths of the simple pendulum, both at Leith and Unst, being already reduced to this standard state, we may apply to them directly this formula, and deduce in millimetres the following values, which are set down, compared with those of Biot, obtained by the method of Borda, from observations made with great care, the preceding year, in the same stations of Leith and Unst.

| Names of the Stations | Lengths of the Simple Pendulum according to Kater | Lengths of the Simple Pendulum according to Biot | Differences of Kater's Measurement | |-----------------------|---------------------------------------------------|--------------------------------------------------|----------------------------------| | Leith Fort | 994-528685 | 994-524453 | + 0-004232 | | Unst | 994-935840 | 994-943083 | - 0-007243 |

The differences of the results, it will be seen, are excessively minute, for they consist only in some thousandth parts of a millimetre, which is equal to \( \frac{39}{1,000,000} \) of an English inch; and they are, besides, affected with contrary signs at Unst and at Leith. We may reasonably conclude, then, that they fall within the limits of that uncertainty to which all physical results are subject; and it may therefore be inferred, that the method of Borda and that of Captain Kater are equally precise, and both give, with exactness, the absolute measure of the pendulum.

In attempting to carry compound pendulums on distant journeys, or when we are obliged to observe them in places where the apparatus cannot be fixed to solid buildings, this must necessarily be modified, so as to be complete in itself. To do this, we may prepare for the pendulum a support of metal, made from a single casting, the feet of which spreading out, can be firmly fixed in the ground, while they allow the pendulum, at the same time, to oscillate at freedom between them. The upper part of this support must consist of a plate having a longitudinal opening in it, to allow the stem and knife of the pendulum to pass through. On this is fixed, with long screws, a polished plate pierced with a similar opening, and which can be set horizontally with a spirit-level before fixing it; and it is on this plane that we place the knife of the pendulum. A divided scale, unconnected with the pendulum, is placed horizontally, immediately under the lower extremity of its stem, which being furnished with a point, indicates, by its excursions upon this scale, the amplitudes of the oscillations. In order, now, to determine the rate of the pendulum's going, it is not always possible to procure the necessary facilities for employing the method of coincidences. In that case, we may substitute for it the comparison of the pendulum with an adjoining clock, or else with a chronometer, counting, as Bouguer did, the whole oscillations which the pendulum performs during a given time, and determining the fractions of oscillations, by observing the part of the amplitude with which the point of the stalk corresponds at the commencement and termination of the interval of the time observed. But in making use of this last process, which is indispensable for fixing the extreme terms of each compared interval, we can dispense with the counting of the oscillations one by one; for it will be sufficient to follow them with a counter, the rate of which is adjusted very nearly to that of the experimental pendulum, and which we take care from time to time to regulate according to it; accelerating or retarding its motion by an impulse given to its lens before it has lost or gained a whole oscillation. This last part of the proceeding has been suggested by M. Arago, and employed by Captain Freycinet in his voyage round the world. Then it only remains to fix, by observation, the position of the stem upon the arc of amplitudes at the periods of comparison with the chronometer or the clock, and from thence to deduce the fractions of oscillations which the counter could not indicate. These fractions may be obtained from the mathematical law which regulates the motion of the pendulum in each oscillation. If we call $2a$ the whole amplitude with which the pendulum oscillates, and $T$ the total time which it takes to describe it, also $\theta$ the arc which it describes, during the time $t$, in falling from the extremity of this amplitude, the law of the descent, limited to small amplitudes, gives:

$$\theta = 2a \sin^2 \left( \frac{90^\circ}{T} \right) t$$

so that, by representing the half amplitude $a$ by 1000 parts, and supposing $t$ successively equal to $\frac{1}{10}$th, $\frac{2}{10}$ths, $\frac{3}{10}$ths of $T$, or of the duration of a whole oscillation, we obtain for $\theta$ the following values:

| Values of $t$ in 10ths of the whole Oscillation | Portions of the Half Amplitude described | |-----------------------------------------------|----------------------------------------| | 1 | 48·9 | | 2 | 191·0 | | 3 | 412·2 | | 4 | 691·0 | | 5 | 1000·0 |

It would evidently be of no use to push the calculation of these numbers beyond a half amplitude, since they must be symmetrical on each side of the vertical, when the values of the time $t$ are reckoned, as they always can be, from the extremity of the half oscillation in which the pendulum actually is. This being understood, the use of the table is easily explained; for the immediate observation gives the demi-amplitude $\alpha$ at the period of the comparison with the chronometer; it gives also, at this instant, the value of the arc $\theta$, according to the division on the scale of amplitudes to which the stem of the pendulum corresponds. Dividing $\theta$ by $\alpha$, the decimal fraction which will hence result, being multiplied by the number 1000, may be compared with the numbers contained in the second column of the above table; and the first column will immediately give, either directly or by interpolation, the fraction of time corresponding to this position of the stem, a fraction which must be added to the whole number $N$ of the preceding oscillations if the pendulum is on its descent towards the vertical, and subtracted from $N + 1$ if the pendulum is on its ascent towards the end of the oscillation.

The experiments on the variation of gravity at different places on the earth were not at first made with a free pendulum such as we have now described, but with a pendulum adapted to a clock. It was in this manner that Richer discovered the existence of this phenomenon in 1672, in a voyage which he made to Cayenne by order of the Academy of Sciences, for the prosecution of various researches in physics and astronomy, among the number of which was the measurement of the pendulum. On his arrival at Cayenne, Richer remarked that his clock, the weights of which had not been altered since his departure, had a diagonal rate of going of $2' 28''$ slower than at Paris; and not only did this observation prove the fact of the diminution of gravity, in going from the pole towards the equator, but, if we had known the details, particularly in regard to the relative differences of temperature, we might then probably deduce a more certain and exact measure of this diminution than what can be drawn from the absolute length of the equatorial pendulum, determined by the same Richer at Cayenne, with the imperfect methods which were then in use. We shall not dissemble that this assertion requires some proof; for the mode of observing by clocks appears necessarily subject to great uncertainty, the pendulum's own motion being constrained or modified by the motion of the wheels. But this influence is not, perhaps, in reality so great as one would be led to suppose; in fact, it is not the pendulum, but the weight applied to the clock, which makes the wheels move; the pendulum merely regulates the intermitances in the fall of this weight, by its oscillations, which stop it and set it free by turns; and this alternation is performed by means of the escapement, which now disengages itself from the teeth, and then lays hold of them again. When it is disengaged, the action of the weight which turns the wheel excites it, and accelerates its fall; its descent. But in the ascending half oscillation which follows, the same action confines the pendulum and retards it; so that these two contrary efforts, which both operate with very slight degrees of friction, appear, like the resistance of the air, and every other constant friction, to balance their mutual influences on the motion in each whole oscillation, and merely to limit the amplitude of the arcs in which this oscillation takes place; a limitation which we can easily take into account, by observing the amplitudes, and reducing all the oscillations, by calculation, to the case of their being infinitely small. It would be curious to make experiments on this subject, and it could be easily done; for it would be sufficient to vary the weight applied to a clock, and to see if the variations of amplitude which would result are such that, in paying attention to them, the clock may be brought back to its original rate. Some observations already made indicate the exactness of this restitution, or at least the very near approach to it.

In the travels in Lapland, for example, undertaken by the French academicians in 1736, an excellent clock by Graham was carried out to be employed in determining the variation of gravity. This celebrated artist had constructed it for this purpose, and had done so with very particular care. In order to render it more steadily comparable with itself, he had adapted to it a pendulum, formed of a simple rod of brass, to the bottom of which was affixed a lenticular mass of a constant weight; and he had provided pieces which raised the rod up during the voyage, and kept its summit free from all contact, so that the knife edge could not be altered by any friction against the plane of suspension, although the rod was always at liberty to follow the dilatations and contractions produced by the changes of temperature. Now, in the account of the labours connected with this operation, which Maupertuis has published, under the title of Figure de la Terre Determinée, we find, that with the action of the weight usually applied to this clock, it made, at Paris, $86394^2\cdot4$, during a revolution of the fixed stars, in describing arcs of $2^\circ 10'$ on each side of the vertical, while, with a weight twice as small, it made $4'$ more, that is, $86398^2\cdot4$ in the same interval, describing arcs of $1^\circ 15'$. Now, if we apply here the correction relative to the amplitude of the arcs, which is

$$\frac{1}{16} N \sin^2 \alpha,$$

calling $N$ the number of oscillations, and $\alpha$ the demi-amplitude, we shall then find, that in the first case it is necessary to add $7'\cdot711$ oscillations, and in the second $2'\cdot563$, to reduce each of them to the case of amplitudes infinitely small, which gives $86402\cdot1$ and $86401\cdot0$ for the total number of oscillations infinitely small in the two cases. These quantities only differ by $1'\cdot1$, and as the observations at this period were not carried to a greater exactness than this difference, it would be of no use to look for a more perfect agreement between them. We have still the example of a similar proof made by Graham himself, upon another clock, which he had constructed to determine the variation of gravity between London and Jamaica; an object for which it was really employed, its rate having been observed for this purpose by Graham in London, and at Jamaica by C. Campbell, a skilful observer, and the friend of Bradley. In the account of this operation, which has been given by Bradley himself, in No. 431 of the Philosophical Transactions, it appears that Graham having taken away the weight from this clock, which was $12lb.\ 10\frac{1}{2}oz.$, and having replaced it by another of $6lb.\ 3oz.$, the amplitudes of the oscillations, which were at first $3^\circ 30'$, were reduced to $3^\circ 2'$, and the diurnal rate of the clock slackened by $1'\cdot4$. Now, if we reduce each of these rates to the case of amplitudes infinitely small, in taking successively for $\alpha$, $1^\circ 45'$ and $1^\circ 15'$, we find for the reduction in the first case $5'\cdot03$, in the second $2'\cdot6$, of which the difference is $2'\cdot4$, instead of $1'\cdot4$, which Graham had observed; and as this able artist had not had any other end in view than to prove the small alteration in the diurnal motion by a change of weight so considerable, it is possible that he may not have taken the same pains in determining the temperature and other details of the observation, which he would have done if he had been seeking to determine an element of correction with a perfect accuracy. It appears very probable, then, by these examples, that in clocks constructed in this manner, the action of the weight, transmitted by the wheels, accelerates the proper motion of the pendulum, during each descending half oscillation, as much nearly as it retards it in the ascending half oscillation which follows. So that these opposite modifications seem to compensate each other, at least with a sensible equality, in each complete oscillation.

Whence we may conclude, with equal probability, that the greater or less facility in the motions, and the various energies of friction produced by the unequal tenacity of the oil, at different periods and at different temperatures, can have but a very small influence on the proper motion of a pendulum, and which must become quite insensible by employing an oil of tried permanence of constitution; and, above all, by producing artificially, at all the stations, the same fixed temperature as was done by the French academicians in 1736, in their journeys in Lapland, and also by the intrepid English mariners in 1820 in their memorable voyage to the North Pole. But admitting the constancy of the results obtained at the same place by this mode of observation, which, we repeat, still wants to be completely proved by new experiments, it is clear that no other method could be more convenient. For it would be sufficient in every place to prepare the clock; to set it up, with every precaution in levelling which can place it in a state and situation similar to itself; then to compare its rate of going with the diurnal motion of the heavens, either with a small transit instrument, or even by means of a simple telescope, firmly fixed to some immoveable mass, and directed towards a star, the diurnal return of which could be observed with fixed wires stretched in the focus. MM. Breguet instituted a series of experiments on this subject; but as the processes for determining results of this nature cannot be too severely scrutinized, we sincerely wish that other observers would make similar attempts, and publish the results deducible from them.

USE OF THE PENDULUM IN DETERMINING THE OBLATENESS OF THE EARTH, AND THE INTENSITY OF GRAVITY AT DIFFERENT LATITUDES.

According to the theory of universal attraction, if we consider the earth and the planets as having been originally masses in a fluid state, endowed with a motion of rotation round themselves, they must have taken the form of a spheroid, flattened at its poles; and the force of gravity, which is observed at their surfaces, would then be the result of two distinct forces, of which the one is the general attraction, exerted upon each point of the surface by all the particles of matter in the spheroid, according to their masses and distances; and the other is the centrifugal force, excited at the same point by the motion of rotation. But the intensity of the attraction, exerted upon different points of the surface by the whole mass, must be in general variable, as well as the centrifugal force. The union of these two causes, then, must produce in the force of gravity inequalities, which observation may discover. But we may easily rid these inequalities of the effect of the centrifugal force; for this can be calculated for each point, when we know the dimensions of the spheroid, its rotation, and the axis round which it turns. The observations thus reduced present results which are only dependent on the attraction of the spheroid upon which they are made; and they may consequently serve to determine its exterior configuration, as well as the laws of density, by which the attractive matter is distributed through its interior. The remarkable discovery of these relations, between the force of gravity at the surface of the heavenly bodies, and their form, as well as their internal constitution, we owe to Newton; and this great man, in following them out, determined even the value of the oblateness which the terrestrial spheroid ought to have, supposing it elliptical and homogeneous, in order to be in equilibrio with the actual velocity of its rotation. He thus found, that denoting by $p$ the observed ratio of the centrifugal force to the force of gravity at the equator, the oblateness of the spheroid must be $\frac{5}{4} \phi$; and as $\phi$, from observation, may be estimated for the earth at $\frac{1}{289}$ or at $0.00346031$, there results the oblateness $\frac{5}{4} \phi = \frac{1}{231}$, or $0.004325$, a quantity much superior to the observed value, $0.00326$, which shows that the terrestrial spheroid is not homogeneous. But, as this element of measure had not as yet been determined in the time of Newton, he could not draw this consequence. He confined himself, therefore, to the determination of the variations of gravity in the case of the supposed homogeneity, and he found it, as it is in fact, proportional to the square of the sine of the latitude. But he erred in endeavouring to extend these determinations to the case of any ellipsoid, composed of concentric strata of unequal density. For, finding that the observations of the pendulum gave the actual variation of gravity in proceeding from the equator to the pole, greater than the calculation established for the case of homogeneity, he thought that the oblateness ought to increase at the same time with this variation, although the real measures of degrees have since pointed out the inverse of this result; for they agreed in giving a slighter oblateness than $0.004325$, with a more considerable variation of gravity. Clairaut, in his admirable work on the figure of the earth, was the first to point out this error, which had escaped Newton; and he demonstrated at the same time this remarkable theorem, that, in all the hypotheses, the most probable that can be formed regarding the density of the interior parts of the earth, which must always be supposed most dense towards the centre, there is always such a connection between the fraction which expresses the difference of the axes, and that which expresses the diminution of gravity from the pole to the equator, that, if the one of these two fractions exceeds $0.004325$, the second will fall short of it by the same quantity; so that their sum must be always equal to the double of $0.004325$, or to $0.00865$. In this case, also, the length of the seconds pendulum varies from the equator to the pole, in proportion to the square of the sine of the latitude. Thus, calling $\lambda$ this length at any latitude $L$, and $A$ the length at the equator itself, we have in general $\lambda = A + B \sin^2 L$, $B$ being a constant co-efficient, to be determined by observation. It must be remarked, however, that this result supposes the lengths of the pendulum to be observed at the very surface of the terrestrial spheroid; for, in receding from this surface, although at the same latitude, the intensity of gravity diminishes nearly in proportion to the square of the distance from the centre, and consequently the length of the pendulum must diminish according to the same law. Reciprocally, then, if we have observed this length at any height $h$ above the terrestrial surface, and that we have found for it a value expressed by $l$, $a$ being the radius of the earth at this latitude, the length reduced to the level of the sea will be $\frac{l(a + h)^2}{a^2}$, or $l + \frac{2hl}{a} + \frac{hl^2}{a^2}$. But this reduction may be simplified, by considering that, on account of the small height to which we can rise above the earth's surface, $\frac{h}{a}$ is always a fraction so excessively small, that the first power of it is sufficient to be used. So that, limiting ourselves to this order of approximation, the reduced length will become $l + \frac{2hl}{a}$. The term $\frac{2hl}{a}$ forms, then, the correction which the experiments require that they may be reduced to the level of the sea, and thus rendered comparable with each other. Such is Pendulum, in fact, the mode of reduction generally employed; but we must remark, that it is itself subject to uncertainty. For the mountains on which we ascend attract the pendulum by virtue of their own mass; in consequence of which, it becomes necessary to pay attention to this attraction, that the reductions may be made rigorously exact, instead of applying the bare formula, which supposes the observations to have been made in the open atmosphere. But this is an inconvenience which is unavoidable; for it is impossible to calculate exactly the peculiar attraction of the masses on which we operate, since this would require the knowledge of their relative density, and even of the arrangement of the materials which enter into their composition. But, as we cannot avoid this uncertainty, we must endeavour to render it as small as possible, by making our observations as near the level of the sea as we are able. We must then recollect, that, by ascribing at the highest station the whole of the force of gravity to the sole and distant action of the earth, we suppose it to be more powerful than it really is. So that, by reducing it, on this hypothesis, to what it would really be if it had been observed at the level of the sea itself, we commit a double error; the mountain's own attraction tending, in this second case, by its contrary direction, to weaken the effect of gravity, which it had before augmented. Fortunately, the excessive smallness of the highest mountains, compared with the mass of the globe, must diminish extremely their relative influence, and render equally minute the errors which may arise from neglecting it.

To determine, now, the co-efficients $A$ and $B$ of the general formula, we shall employ the oblateness $0.00326$, or $\frac{1}{30675}$, which M. Laplace has obtained by submitting to a general and profound discussion the measures of the terrestrial degrees, and the lunar inequalities depending on the oblateness of the earth. We shall join to it the length of the simple pendulum of sexagesimal seconds, found by Biot at the station of Unst, a length which, we think, may be considered as one of the most certain that has been observed; first, because having been the last of the observations made by Borda's method, it must have been taken with all the precautions suggested by preceding experiments; secondly, on account of the great number of series from which it results, these being fifty-six in number, and made with different rules, and pendulums of unequal lengths, which all agreed in assigning for the definitive result values differing excessively little from each other; and, lastly, from the perfect agreement which is found between it and the results of the observations of Captain Kater. This single absolute length, together with the oblateness $0.00326$, will suffice for determining the two constant quantities $A$ and $B$ of the general formula, which expresses the length of the pendulum at any latitude. Now, according to this formula, the length of the pendulum at the equator, where $L$ is nothing, is equal to $A$, and at the pole, where $L = 90^\circ$, it is $A + B$. So that $\frac{B}{A}$ is the relation of the total variation of the pendulum to its absolute length at the equator; a ratio which is the same as that of the increments of gravity to the absolute force of gravity itself. Adding, then, $\frac{B}{A}$ to the oblateness $0.00326$, we shall have, by the theorem of Clairaut, the following condition, $\frac{B}{A} + 0.00326 = 0.00865$, whence we obtain $B = A \cdot 0.00539$, and consequently $\lambda = A (1 + 0.00539 \sin^2 L)$. Now we have seen above, that at the station of Unst, in latitude $60^\circ 45' 25''$, the length of the sexagesimal seconds pendulum determined by the ob- The height of this station was only \(9^\circ\) above the level of the sea, which gives for the reduction \(+0^\circ = 0.02818\). Whence there results, at the level of the sea, the height \(994.45923\). Putting this value, then, and that of \(L\) in the formula, the coefficient \(A\) is determined, and we find \(A = 990.879660\), consequently \(B = 5.340843\); which gives for any latitude \(L\),

\[ \lambda = 990.879660 + 5.340843 \sin^2 L. \]

If we wish to reduce this formula to English inches, all the terms must be multiplied by \(\frac{39-87079}{1000}\), and then

\[ \lambda = 39.0117150 + 0.2102732 \sin^2 L. \]

Finally, if we wish to reduce it to the decimal pendulum employed by the French observers in their calculations, we must multiply the terms by \(\left(\frac{864}{1000}\right)^5\), the ratio of the decimal to the sexagesimal pendulum. We then have

\[ \lambda = 739.687686 + 3.986917 \sin^2 L. \]

If we calculate from this last formula the lengths of the decimal pendulum for the stations where the French observers have operated, from Formentera to Unst, and compare them with their results, we obtain the following table.

| Names of the Places | Names of the Observers | North Latitudes | Length of the Decimal Pendulum at the Level of the Sea | Excess of Calculation | |---------------------|------------------------|----------------|------------------------------------------------------|----------------------| | Unst | Biot | 60° 45' 25" | 742.723136 | 0.000000 | | Leith Fort | Biot, Mudge | 55° 58' 37" | 742.426416 | +0.012981 | | Dunkirk | Biot, Mathieu | 51° 2' 10" | 742.098066 | +0.021036 | | Paris | Biot, Mathieu, Bouvard | 48° 50' 14" | 741.947360 | +0.029870 | | Clermont | Biot, Mathieu | 45° 46' 48" | 741.735412 | +0.030232 | | Bordeaux | Biot, Mathieu | 44° 50' 26" | 741.670048 | +0.061328 | | Figec | Biot, Mathieu | 44° 36' 45" | 741.654181 | +0.041901 | | Formentera | Biot, Arago, Chaix | 38° 39' 56" | 741.243950 | -0.008050 |

The progression of the deviations contained in the last column of this table shows, in proceeding from the north to the south, a progressive decrease of gravity, greater in a slight degree than the elliptical figure requires; a result which had already been remarked in regard to Scotland and England by Captain Kater. It may be observed here, that the absolute value of this variation for Unst, Leith, and Dunkirk, agrees exactly with that which Captain Kater has found, or what could be deduced from his experiments. But the same effect is observed to continue throughout France, being most sensible at the station of Bordeaux. It becomes less even at Figec, situated more inland, and on a more solid base. It again becomes nothing at Formentera, where the deviation of the formula compared with observation is \(1000\) in a contrary direction, which would seem rather to indicate a slight local excess in the intensity of gravity. This singular anomaly, which is so stated, in regard to the force of gravity, throughout the terrestrial arc which extends over all this part of Europe, is, without doubt, owing to peculiarities in the geological constitution of the countries which are situate on it; and it appears by this example, how well the observations of the pendulum are adapted for pointing out the irregularities of this constitution. But, for this purpose, the observations must possess so great a degree of exactness that the peculiar uncertainties to which they are liable may be, as we may suppose they were in those which we have employed, much smaller than the variations of constitution which they are intended to indicate. Here it may be remarked, that the part of France where these variations are the most sensible, are precisely the same where there were found, by Delambre's observations, the greatest anomalies in the lengths of the degrees.

From the preceding formulae may be deduced the variation in the diurnal rate which a compound pendulum, of an invariable form, must present when carried to different latitudes. If we denote by \(N_1, N_2\) the number of oscillations of this pendulum at two different stations, where the lengths of the simple seconds pendulum are \(\lambda, \lambda'\), we have shown

\[ N_2 = \frac{A + B \sin^2 \lambda'}{A + B \sin^2 \lambda} \cdot N_1. \]

Now, calling \(L, L'\) the latitudes of the two stations, the above formulæ give the values of \(\lambda\) and of \(\lambda'\); as well as their relations; substituting, then, these values in the preceding equation, we obtain

\[ N_2 = \frac{A + B \sin^2 \lambda'}{A + B \sin^2 \lambda} \cdot N_1; \]

an expression by means of which we can calculate \(N'\) when we know \(N\).

The total variation of gravity from the equator to the pole is so inconsiderable that the difference between the numbers \(N, N'\) is always very small compared with these numbers themselves. This difference, then, is the element which we must try to put in evidence in the formula. But nothing is easier; for, if we denote it by \(n\), so that \(N'\) is represented by \(N + n\), the preceding equation will become

\[ N + 2n + n^2 = \frac{A + B \sin^2 \lambda'}{A + B \sin^2 \lambda} \cdot N. \]

Whence we deduce

\[ 2n + n^2 = \frac{B \sin (L' + L) \sin (L' - L)}{A + B \sin^2 L} \cdot N; \]

and resolving the value of \(n\) into a series,

\[ n = \frac{B \sin (L' + L) \sin (L' - L)}{2(A + B \sin^2 L)} \cdot N, \]

\[ n = \frac{-B^2 \sin^2 (L' + L) \sin^2 (L' - L)}{8(A + B \sin^2 L)^2} \cdot N, \text{ &c.} \]

But from the value of the oblateness which we have adopted, we have seen that \(B\) is equal to \(0.00539 A\). Substituting this value in our series, it becomes

\[ n = \frac{0.00539 N \sin (L' + L) \sin (L' - L)}{2(1 + 0.00539 \sin^2 L)} \cdot N, \]

\[ n = \frac{-0.000029 N \sin^2 (L' + L) \sin^2 (L' - L)}{8(1 + 0.00539 \sin^2 L)^2}, \text{ &c.} \]

The second term will be almost always insensible, and it will be quite needless to take in any of the following ones.

To show the use of this formula, we shall apply it to the following observations, which belong to the most distant countries on the earth. | Names of the Observers | Names of the Stations | Longitudes, reckoned from Greenwich | Latitudes | Number of Oscillations performed by the Compound Pendulum in a Sidereal Day, or in a mean Solar Day, at the same Temperature | Variation of the Diurnal Rate, by Observation | Variation of the Diurnal Rate, by Calculation | Difference | |------------------------|----------------------|-----------------------------------|----------|-------------------------------------------------|---------------------------------|---------------------------------|-----------| | G. Graham & C. Campbell, in 1731 and 1732 | Jamaica | 76°45'15" W. | 18°0'0" N. | 86283·0 | 118·2 | 119·96 | -1·76 | | Maupertuis, Clairaut, Lemonnier, 1738 | London | 0°0'0" | 51°31'0" N. | 86401·2 | | | | | | Paris | 2°20'15" E. | 48°50'14" N. | 86394·4 | | | | | Graham, 1738 | London | 0°0'0" | 51°31'0" N. | 86453·5 | 59·1 | 64·70 | -5·60 | | Freycinet | Paris | 2°20'15" E. | 48°50'14" N. | 86402·1 | 51·4 | 53·85 | -2·45 | | | Rio Janeiro | 43°18'37" W. | 22°55'2" S. | 89048·8 | 95·0 | 99·43 | -4·43 | | | Cape of Good Hope | 18°24'0" E. | 33°55'15" S. | 89086·4 | 57·4 | 61·10 | -3·70 | | Sabine, 1818 | London | 0°0'0" | 51°31'8" N. | 86497·40 | 33·11 | 31·82 | +1·29 | | | Brassy | 60°9'42" N. | 86530·51 | 65·24 | 62·46 | +2·78 | | | Hare Island | 70°26'17" N. | 86562·64 | | | | | Sabine, 1820 | London | 0°0'0" | 51°31'8" N. | 86455·65 | 74·73 | 73·93 | +0·80 | | | Melville Island | 110°49'0" W. | 74°47'14" N. | 86530·38 | | | |

The experiments of Captain Sabine were made with two pendulums applied to two different clocks, the results of which have agreed very well in their relations. We have only stated here the mean of these results. The observations of Maupertuis, Clairaut, and Lemonnier, were made by a process of the same kind, but with a single clock by Graham. In these two expeditions the observers produced artificially at the second station the same temperature as at the first. In the operation by Campbell, the same pains were not taken; but in Bradley's computations, an allowance was made according to the indications of the thermometers. The experiments of Captain Freycinet were made on detached pendulums, the rate of which, first determined by Arago, Mathieu, and himself, at the Royal Observatory of Paris, was compared in the voyage with well-regulated chronometers, making allowance, by calculation, for the changes of temperature. The smallness and the irregularity of the differences which are found between the results of these different experiments, and the numbers given by the formula, show that the latter is the general expression of them, modified only by the accidental variations which may be occasioned in each place by small differences of density in the neighbouring strata of the surface of the earth. The formula being grounded upon the oblateness 0·00326, or $\frac{1}{306·75}$, its agreement with the facts proves that this value of the oblateness, if not rigorously exact, is at least a very near approximation, and is, besides, common to the two hemispheres of the globe; since the observations of Captain Freycinet in the southern hemisphere, at the Cape of Good Hope, are as correctly represented by it as the observations made in the northern hemisphere. This puts an end, then, to the notion entertained after the measurement of the degree by Lacaille in this part of the globe, that the southern hemisphere was more oblate than the northern; a notion, however, already much weakened by the agreement of the oblateness observed in this latter hemisphere, with that which was deduced from the inequalities of the moon; since the motion of this satellite must be influenced by the mean of the two ellipticities, if they were different; but it was nevertheless of consequence to see this suspicion wholly extinguished, as it is now by Freycinet's observations.

The general experiments on the length of the pendulum which we have above described, being verified by the different observations by which we have compared them, will serve to determine the intensity of gravity, whether absolute or relative, on any of the places of the terrestrial globe. For, calling $\lambda$ the length of the simple pendulum, which makes its oscillations in a second of time in a given place, and denoting by $g$ the double of the space which gravity makes bodies describe in their fall in the same place, and during the same interval of a second, the fundamental formula of oscillations, infinitely small, gives $1^r = \pi \sqrt{\frac{\lambda}{g}}$, consequently $g = \sigma^2 \lambda$, $\sigma$ being the ratio of the circumference to the diameter, or 3·14159. But we have already given for any latitude the value of $\lambda$ expressed in millimetres and in English inches, taking for the unity of time either the decimal or the sexagesimal second. Multiplying these expressions by the square of $\pi$, we shall have the value of $g$ for the same latitude, and the same kind of unity of time which may be chosen.

It may be objected, that we have not made use of the lengths of the simple pendulum observed under the equator by Bouguer, and detailed in his work on the figure of the earth. The reason is, that, notwithstanding the ability of Bouguer as a philosopher and an observer, and the infinite pains which he took in his measurements of the pendulum, it appears to us, on account of the nature of the processes he made use of, that they are too inexact to be employed with advantage. The method of Bouguer consisted in forming a sort of simple pendulum, with a very small weight suspended to a stem, the other extremity of which was attached to a pincer fixed into a solid wall. He made this little pendulum always of the same length, by comparing it with an iron rule, which served him for a standard; after which he determined the value of its oscillations by comparing its rate with that of a clock regulated by the heavens. But M. Laplace has justly remarked, that the bending of the stem at the point of suspension, where it is inserted into the pincer, must produce the same effect on the oscillations as a contraction in the wire; so that the length, measured in a state of repose, must be too great, and would appear to give the pendulum too long. This effect, indeed, must have been produced on all the lengths given by Bouguer, since they were all observed in the same manner. From that it would seem that these ob- Pendulum. servations might at least be employed in comparison with each other, and in that case give exact ratios. But the process by which Bouguer judged of the length of his little pendulum, and compared it with his standard rule, appears to us not accurate enough for giving a sufficient certainty in his results. For it consisted in laying this rule close to the pendulum, placing its upper end in contact with the point, and judging of its equality by the eye, in comparing it with the pendulum at its lower extremity. But no one, by such an operation, can answer for an exactness greater than $\frac{1}{100}$th of a line. Now $\frac{1}{100}$th of a line being equal to $\frac{1}{100}$th of a millimetre, such an error, with the methods now actually employed, would be accounted gross, and such indeed as, with the least attention, it is quite impossible to commit. These results cannot, then, be compared with the observations which are made now; and, unfortunately, the same remark applies with equal justice to the measurements of the absolute lengths of the pendulum which were made about the same period, as well in France as in various other parts of the globe. We think it extremely probable, that to the want of exactness in the methods employed at that time may be ascribed, at least in a great measure, the strange anomalies observed by Grischow in the lengths of the pendulum, in the neighbourhood of St Petersburg, between stations very little distant from each other; anomalies so much more justly suspected, since the different instruments employed by Grischow to establish them are far from agreeing with each other. Nevertheless, for removing entirely all suspicion with regard to a point so important, it would be a useful undertaking to repeat these experiments in the same places where Grischow's observations were made; employing for this purpose our present much more accurate methods.

Much has been done in this department of science since the preceding part of this article, which was written by M. Biot for the Supplement to the sixth edition of this work, was published. This took place not long after a most important addition had been made both to the instruments of research and to the methods of experimenting, by the invention, or at least the construction and application, of the convertible pendulum.

IV.—DETACHED PENDULUMS.

Although Professor Bohnenberger, in a treatise on astronomy, published at Tübingen in the year 1811, had suggested that the length of the seconds pendulum might be obtained by means of Huygens's theorem, yet he seems to have done nothing towards putting the scheme in practice; nor had any notice of such a suggestion reached Captain Kater till fully eight years after he had devised and executed his valuable experiments, as described in the Philosophical Transactions for 1818. These form a new era in the history of the pendulum, and have greatly increased the interest and importance which, only a few years before, had begun to be attached to researches on this subject, and which have ever since continued to increase. This is evinced, not only by the repeated and valuable labours of several of the most distinguished mathematicians and experimentalists of the present age, but also by the numerous scientific voyages that have been undertaken by various governments, with the view of ascertaining and comparing the results of different pendulum experiments made in various parts of the world, and thence to determine the true figure of the earth. The most valuable and extensive experiments of this sort are those made by Major Sabine and those by the late Captain Henry Foster, in almost every practicable latitude, and embracing a greater variety and range of temperature than those of any other experimen- talist. As is the usual method among the English experi- menters, invariable pendulums, which had been first ob- served in London to ascertain the number of vibrations made there per day, were afterwards observed in the same manner at all the stations, and similarly again on returning to London. In this manner, without regarding the abso- lute force of gravity at any one place, the proportion of it at different places is found probably with greater accuracy than by any other method. With the French philosophers, however, it is more usual to observe the absolute length of the seconds pendulum at every station. In this manner they have experimented in a great variety of places; but in some cases they have also employed invariable pendu- lums, which are the same with those called "pendulums of comparison" in the preceding article; for the term irra- variable does not refer to any compensation for temperature, &c., but merely implies that the effective length is never interfered with, by shifting or altering the positions or mag- nitudes of any of the parts. In this way the results at dif- ferent stations are more capable of being compared.

It is not many years since such results, especially the number of vibrations which are made in a mean solar day, whether by the same or by different pendulums, were con- sidered as strictly comparable with each other, provided only they were reduced, by means of certain hypothetical rules, to what it was believed they should have been under the following circumstances: 1st, at the mean level of the sea; 2d, in a vacuum; 3d, in indefinitely small arcs; and, 4th, at a common standard of temperature.

The late Dr Thomas Young, however, at length demon- strated that the formerly-received formula for the reduc- tion to the level of the sea is in most cases too great. M. Bessel next questioned the old formula for the reduction to a vacuum, as having the contrary fault of being too small. This he had inferred from some very far-fetched experi- ments; but that the old correction was really too small, has since been amply confirmed by the direct experiments of Major Sabine and Mr Baily. In this reduction to a va- cuum they had all to a certain extent been anticipated by the Chevalier Du Buat, who has treated the question ex- perimentally, and at great length, though in a much less satisfactory manner, in his Principes d'Hydraulique, second edition, 1786, and third edition, 1816. But his researches having appeared before any great interest had been taken in this question, they were till lately almost totally over- looked and unknown. The like may be said of some hints given by Newton, and well deserving of notice, in his Prin- cipia, lib. ii. prop. 27, cor. 2. Fortunately, this defect in the old formula for the reduction to a vacuum has not been productive of any considerable errors in the determinations of the proportions of the force of gravity at different places. In the Philosophical Transactions for 1831, Major Sabine has described a variety of experiments, which have led him to question the accuracy of the usual formula for the reduc- tion to indefinitely small arcs. He had previously, in his work on the figure of the earth, pointed out the discordant results arising from the use of different agate planes with the same knife-edge; and had also stated his decided opinion on the powerful effects of certain geological strata, or even of an increase of buildings, in the immediate neighbourhood of the pendulum. Mr Baily, again, in the Philosophical Transac- tions for 1832, has gone much farther, in having pointed out great discordances between experiments executed with the same knife-edge of the same pendulum, and on the same plane. He likewise acknowledges, that to whatever cause the observed anomalies may be owing, he has, during a long course of experiments on various pendulums, at different seasons of the year, and under a variety of circumstances, frequently met with discordances that have baffled every attempt at explanation by any of the known laws applica- ble to the subject; and other persons, also, who have had much practice in such experiments, have occasionally met with anomalies for which they have been as unable to account. As it is desirable, however, that these difficulties should be cleared up, if possible, and as every information connected with so important a subject, founded on such delicate experiments, must add to our means of removing them, we shall by and by give a summary account of the results of a variety of experiments made by Major Sabine and Mr Baily, with pendulums of various forms and constructions, immediately bearing on the discordances in question.

It is evident that till two pendulums can be constructed which will always give precisely the same results, cleared of all these discordances, the important problem of determining the length of the simple pendulum cannot be considered as fully solved; neither can the observations made by different experimentalists, in different parts of the globe, with different pendulums, be strictly and directly comparable with each other. We have, it is true, two pendulums, differing widely in form and construction, and yet agreeing surprisingly in their results, viz. Borda's and Kater's. But although this evinces the talent and skill of the distinguished persons who made the experiments, it should not be forgotten that the reductions to a vacuum were, in both cases, made agreeably to the old formula; and that since the date of M. Bessel's important researches on this subject, which indicate the necessity of revising the computations of all preceding experiments, no rigid comparison of the results has yet been repeated. The amount of the additional corrections for Borda's pendulum and for that of Kater differ materially, as will be shown in the sequel, so that we are as yet ignorant whether the results of any two pendulums that have ever been differently constructed are in strict accordance with each other; nor, until this is practically accomplished and repeated, can the true length of the seconds pendulum be considered as satisfactorily determined.

It had long been the practice to make allowance for the height of any station above the level of the sea, without taking into account the attraction of the elevated parts themselves. Dr Young showed that this was far from being correct; but no one has yet been able to estimate the exact amount of the error, nor is it likely ever to be attained. It is, however, obvious, that if we were raised on a sphere of earth a mile in diameter, its attraction would be about \( \frac{1}{200} \) of that of the whole globe; and instead of a reduction of \( \frac{1}{200} \) in the force of gravity, we should obtain only \( \frac{3}{400} \), or three fourths as much. Nor is it at all probable that the attraction of any hill a mile in height would be so small as this, even supposing it to have been only two thirds the mean density of the earth. That of a hemispherical hill of the same height would be fully one half more than the sphere, or would exceed its attraction in the ratio of 1:586 to 1. And it may be easily shown that the attraction of a large tract of table-land, considered as an extensive flat stratum a mile in thickness, would be three times that of a sphere a mile in diameter, or about twice as great as that of such a sphere having the mean density of the earth; so that for a place so situated the allowance for elevation would be reduced to one half; and, in almost any country that could be chosen for the experiment, it must remain less than three fourths of the whole correction deduced immediately from the reciprocals of the squares of the distances from the earth's centre. From Dr Young's view of the subject, it also appears that the correction for the elevation above the sea will vary, according to the nature of the eminence, and also its density, from one half to three fourths of the quantity which it was customary to deduce from the inverse ratio of the squares of the distances from the earth's centre. Thus, if the mean density of the Pendulum earth be taken at 5:5, and that of the matter surrounding the station at 2:5, the quantity deduced from the duplicate distances should be multiplied by \( \frac{16}{15} \) to obtain the correction for a table-land, and by \( \frac{7}{10} \) for that of an eminence of moderate declivity. (Philosophical Transactions for 1819.)

Perhaps some estimate might be formed of the local effects of a cavity, or of a circumscribed mass of any peculiar density, and hid under the surface, by swinging the pendulum at several places around the principal station, and not far from it; and it might even throw some light on the subject to swing the pendulum both in the plane of the meridian and at right angles to it.

In 1824 a series of pendulum experiments was made by Carlini at the summit of Mont Cenis, to ascertain the diminution of gravity at the height of a geographical mile, as detailed in the Milan Ephemeris. He obtained for the mean density of the earth 4:39; but the changes which experiment has shown to be necessary on the elements of reduction throw some doubt on its value. Mont Cenis is peculiarly favourable for such experiments, as it forms nearly a segment of a sphere, of which the chord is about eleven geographical miles, and the versed sine one mile. It consists of calcareous rocks whose mean density is 2:66. The mountain Schallien, on the attraction of which Maskelyne and Hutton made experiments, has since then undergone a mineralogical survey; and from the more correct data so obtained some alteration has been made on the original numerical results. The calculations of Cavendish's experiments on attraction were also corrected by the late Dr C. Hutton, as detailed in the Philosophical Magazine for July 1821. Professor Reich of Freiberg has recently repeated Cavendish's experiment, though in some respects on a smaller scale; for the larger leaden sphere was only 7:75 inches in diameter, and he used only one such sphere at a time. The result gave 5:44 for the mean density of the earth, and an iron ball of the same size gave 5:48. Cavendish's experiment is also about to be repeated in this country, and a new proposal will be noticed afterwards.

Newton and Du Buat had, as already mentioned, made Reduction it plain that the air affects in several ways the motions of pendulums; and more recently M. Bessel has called attention to the subject by his very interesting work on the pendulum, in which he has shown that the usual formula for the reduction to a vacuum, as far as the specific gravity of the moving body is concerned, is very defective, and by no means expresses the whole of the correction which ought to be applied; in fact, that a quantity of air is also set in motion by and adheres to the pendulum (varying according to its form and construction), and thus a compound pendulum is in all cases produced, the specific gravity of which will be much less than that of the metal itself. He observes, that if we denote by \( m \) the mass of a body moving through a fluid, and by \( m' \) the mass of the fluid displaced thereby, the accelerating force acting on the body has, since the time of Newton, been considered equal to \( \frac{m - m'}{m} \).

This formula is founded on the assumption that the moving force which the body undergoes, and which is denoted by \( m - m' \), is confined to the mass \( m \). But it must be distributed not only over the moving body, but on all the particles of the fluid set in motion by that body, and consequently the denominator of the expression denoting the accelerating force must necessarily be greater than \( m \). M. Bessel then enters into a mathematical investigation of the principles from which the results of his experiments are deduced, and at length concludes, that a fluid of very small density surrounding a pendulum has no other influence on

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1 Untersuchungen über die Länge des einfachen Sekundenpendels, 4to, Berlin, 1828. Pendulum, the duration of the vibrations than that it diminishes its gravity and increases the momentum of inertia. When the increase in the motion of the fluid is proportional to the arc of vibration of the pendulum, this increase of the momentum of inertia is very nearly constant; in all other cases it will depend on the magnitude of that arc.

In the Mémoires de l'Académie, tome xi. 1832, and in the Connaissance des Tems for 1834, Baron Poisson has endeavoured to investigate the relations between the simultaneous movements of a pendulum and the surrounding air, and has deduced that the loss of weight of a sphere exceeds by one half the weight of air it displaces. Such investigations are, however, unavoidably mixed up with assumptions. The resistance of the air to a ball pendulum has also been considered by M. Plana, in a memoir published at Turin in 1835, in which the case of an incompressible fluid is first discussed, and then that of an elastic fluid; and from both the author concludes, as Poisson had done, that the loss of weight of the sphere exceeds by just one half the weight of the fluid it displaces. The question, however, has not yet received a satisfactory solution, since theory has hitherto failed to account for one of the leading circumstances of the case, viz. that the co-efficient of resistance is different for spheres of different diameters. This difference, it appears, would equally exist whether the balls vibrated in a confined apparatus or in free air. The vibrations of a pendulum in fluid media have likewise been investigated by Mr Green, in the Transactions of the Royal Society of Edinburgh, vol. xiii.

In the theoretical researches on this question, it does not seem to be considered, that since each vibration of a pendulum in air tends to create a current of that fluid, against which it is to return in the succeeding vibration, the alternating motion of the pendulum must subject it to a very different and much greater resistance from the air than if its motion were continuous and not reciprocating. But on what principles a formula could be framed to express the velocity and mass of such currents, or their resistance, we cannot pretend to say. No doubt, if the space in which the pendulum vibrates be very confined, that will farther affect the result. Such considerations seem to set at defiance every attempt at framing any thing beyond an empirical formula for these reductions.

But those experiments and researches, though very imperfect, are sufficient to show that the amount of the correction will not only vary according to the size, figure, weight, and density of the moving body, but also, that in the case of the convertible pendulum (except, perhaps, when it makes the fewest vibrations possible) the correction will not be the same for the two knife-edges, and consequently that a pendulum which has been made convertible in air, will not be so when tried in a vacuum. It becomes, therefore, of importance to know how far the differently constructed pendulums used by various experimentalists are affected by this newly-discovered principle, in order that their results may be strictly comparable with each other. The amount of the required correction, however, cannot in the present state of the question, and most probably never will, be determined by calculation founded on any satisfactory theory, but must for every pendulum be ascertained by actual experiment. The most direct, and perhaps the only certain method of effecting this, appears to be by swinging the pendulum in a vacuum; although M. Bessel himself, on account of some undefinable doubts which he entertained regarding it, adopted two other and very different methods of experimenting for this purpose. The first was by swinging in air one sphere of brass and another of ivory, each being about 2½ inches in diameter, and suspended by a fine steel wire; the other method consisted in swinging the same brass sphere, first in air and then in water. The results persuaded him that the usual reduction to a vacuum was much too small. The first method, which he considered the preferable one, gave 1·956, and the second 1·625, as the factor by which the old formula should be multiplied, in order to obtain the true correction. To such experiments we can attach very little importance, because these numbers are still reached nearly as much through the medium of a hypothesis as the old formula. What the swinging of a ball in water has to do with the rate of a pendulum in a vacuum, we cannot pretend to see. The experiments of Du Buat were very similar to those of M. Bessel, but if possible still more objectionable.

The old correction for reducing the number of vibrations to a vacuum had been deduced from the relative weights of the air and of the pendulum by means of the formula

\[ N \times \frac{1}{2(S - 1)} \times \frac{\beta}{\beta'} \times \frac{1}{1 + (a + \mu)(t - t')} \]

where \( N \) is the number of vibrations in a mean solar day, \( S \) the specific gravity of the pendulum, \( \sigma \) the specific gravity of air, \( \mu \) the expansion of mercury, and \( a \) the expansion of air for one degree, \( \beta' \) the height of the barometer, and \( t \) the temperature during the experiments, \( \beta \) and \( t \) the standard pressure and temperature for computing the specific gravities. But in this, \( S \) ought not, as hitherto, to denote the mean specific gravity, as determined in the usual manner, for the pendulum, when at rest, if it consists of several parts, whose specific gravities are different; for in all such cases we should compute the vibrating specific gravity of the mass in the following manner. Let \( d, d', d'' \), &c. denote the distance of the centre of gravity of each body respectively from the axis of suspension (these being positive if below, and negative if above it); \( w, w', w'' \), &c. the weight in the air of each body; \( s, s', s'' \), &c. their specific gravities determined in the usual manner. Then will the required vibrating specific gravity of the pendulum be

\[ S = \frac{w'd + w'd' + w'd'' + \ldots}{s' + s'' + \ldots} \]

which is Professor Airy's formula.

The other parts of the above formula forming the old correction are so far erroneous, that no account is taken of the effect of the air set in motion by and accompanying the pendulum, as if adhering thereto; and which is now found to influence the result very materially. Mr Baily thinks, that in the case of the gridiron pendulum, it may be a matter of doubt whether the air between the vertical rods may not diminish their specific gravity, when considered as a vibrating body.

Early in the year 1828, a proposal was made to the late Board of Longitude, that, on account of the uncertainty in the extent of the defects which M. Bessel had discovered in the old formula for reducing the vibrations of a pendulum in air to what they should be in a vacuum, it would be of great importance to submit the question to the most direct experiment, by the construction of an apparatus in which a pendulum might be alternately vibrated in air of full atmospheric pressure, and in air rarefied nearly to a vacuum. This proposal having been favourably entertained by the Board, orders were given for fitting up an apparatus for this purpose at the public expense, and which was ably executed by Mr Newman, under the direction of Major Sabine, by whom it was to be used; but, that it might also be useful to others who might wish to avail themselves of it for similar purposes, a place was allotted for it in the Royal Observatory at Greenwich. This apparatus, or chamber for forming the vacuum, consists principally of six pieces, exclusive of the iron frame by which the suspension is fixed securely to the wall of the apartment. The The third column is the pressure within the apparatus by the gauge; the fourth the difference between that of the barometer and the third; the fifth the observed retardation, or the difference between the number of vibrations made per day under the full pressure and that in column third; the sixth the retardation by calculation; and the seventh their difference.

In the seventh and eighth experiments, a near approach had been made to a vacuum by repeatedly filling the apparatus with hydrogen, and then exhausting it as far as practicable. At the same time experiments were made on the rate of the pendulum in hydrogen having the full atmospheric pressure, for the purpose of comparing it with the rate in air. From one experiment, it appeared that at the same height of the barometer, 30-193 inches, the pendulum made 8515 vibrations per day more in hydrogen than in air, the temperature of the gas being 39°-32, and of the air 38°-1. From another experiment it appeared, first, that at nearly equal heights of the barometer (30-120 for hydrogen, and 30-118 for air), the pendulum made 856 vibrations more in hydrogen at 39°-75, than in air at 41°-25; second, that it made 1-95 vibration per day more in hydrogen of 0-872 inch pressure, than when the pressure of the gas was 30-120. Hence 1-95 vibration corresponds to a difference of 29-248 inches pressure of hydrogen, and, therefore, two vibrations per day will be the reduction to a vacuum for hydrogen of thirty inches pressure at 40°. The retardation of the pendulum in air is to that in hydrogen as 10-41 to two, and as 10-55 to two, when both are reduced to the same pressure and temperature; the mean being as 5-24 to one nearly. Now, the densities of air and hydrogen being nearly as thirteen to one, shows that the retardation in different elastic fluids is not in the simple ratio of their densities. Major Sabine supposes it to indicate an inherent property in the elastic fluids, analogous to that of viscosity in liquids, of resistance to the motion of bodies passing through them independently of their density.

But perhaps a less hypothetical reason might be given why hydrogen gas should occasion a greater retardation than one thirteenth that of air of full pressure. The retardation of air in these experiments appears to be as its density; thus, for example, air rarefied thirteen times should only offer one thirteenth of the retardation of air of full pressure. Now, it has been shown in the article Hydrometry, vol. xii. page 118, that the specific heat of a given volume of air is as its density; so that air which is thirteen times rarer will emit only one thirteenth as much heat on being compressed. But, from Dr Haycraft's experiments, it appears that, under equal volumes, pressures, and temperatures, hydrogen and air have equal specific heats. Consequently hydrogen of full pressure, on being compressed by the motion of the pendulum, will emit thirteen times more heat than air equally rare would do on being so compressed; and the like may be said of any cold due to the rarefaction occasioned by the motion of the pendulum. Here, it is true, the changes of temperature in the two gases themselves would still be equal, were it not that the materials of the apparatus will have far more power to check any fluctuations or undulations of temperature in the rarefied air than in the hydrogen, which is thirteen times better supplied with sensible heat, and which, therefore, having the equilibrium between the temperatures of its different parts more disturbed by the motions of the pendulum, will be thereby put into a state of greater commotion than the rarefied air, and consequently will so much the more disturb the free motion of the pendulum.

From the mean of the results in the above table we have 9-042 vibrations per day as the reduction to a vacuum of the invariable pendulum vibrating in air of 45° under a pressure of 26-2 inches, and consequently in the same proportion we have 10-36 vibrations under a pressure of thirty inches; but which, if reduced to 32°, by multiplying it by \( \frac{448 + 45}{448 + 32} \) as explained below, will be 10-64 vibrations.

As we have no information regarding the humidity of the air when Major Sabine made his experiments, we have taken it to be perfectly dry, which it never is. The old "correction for buoyancy," as formerly computed, would have been 6-26 vibrations, which is less than 10-36 in the ratio of one to 1-655, and less than 10-64 in the ratio of one to 1-77.

Now, from the principles developed in the article Hy- Pendulum. Geometry, vol. xii. p. 131, it appears that if the specific gravity of dry air under the pressure \( P \) and at the Fahrenheit temperature \( T \) be reckoned unit, the specific gravity of a mixture of air and aqueous vapour whose joint pressure is \( p \) and temperature \( t \) will be

\[ \frac{p - 375f}{P} \times \frac{T + 448}{t + 448} \]

where \( f \) is the force of the vapour. If, therefore, the reduction to a vacuum, which amounts to 10-36 vibrations when \( P = 30 \) and \( T = 45^\circ \), vary as the density of the medium, it will in all cases be

\[ \frac{p - 375f}{30} \times \frac{45 + 448}{t + 448} \times 10-36 = \frac{16p - 6f}{t + 448} \times 10-64. \]

Hence, when \( p = 30 \) inches and \( t = 80^\circ \), the reduction will be diminished about an eightieth part by the presence of aqueous vapour in a state of saturation, for then \( f = 1 \) inch. However, from the circumstance that hydrogen, compared with air, requires a greater reduction than in the ratio of the density, something of the sort may in a slight degree belong to aqueous vapour. It is farther to be observed, that the constant factor of the above correction applies to the invariable pendulum used by Major Sabine; and that almost every pendulum of a different form or construction will require a different number. A table of these is given on next page.

That part of the difference between the old and new corrections which is occasioned by barometric variations, must be extremely small in all cases of comparison between stations which differ little in their heights above the level of the sea. The specific gravity of an invariable brass pendulum being about 8-4, an inch in the barometer will correspond in buoyancy to about 0-21 of a vibration a day, which, multiplied by 0-65, is about 0-14 vibration. In the comparison of tropical and extratropical stations, the barometer, in the middle and high latitudes, is liable to fluctuate an inch and more from the mean height, which is uniform, or nearly so, within the tropics; but as the observations generally include several days at each station, and as, in proportion to their continuance, the barometer will approximate to its mean height, it will be found that a difference of half an inch in the barometric height at two stations is a rare occurrence. The correction for this is not more than 0-07 of a vibration to be added to the daily number at the station where the barometer was highest. The liability to error from variations of temperature at different stations is, however, far more considerable than from variations of pressure; sufficiently so, indeed, to become in some cases influential on the ellipticity deduced. A difference of 40° of Fahrenheit between the temperatures in the tropics and the high latitudes is by no means unfrequent; and as 16° Fahrenheit are nearly equivalent in their influence on the density of the air to one inch of the barometer, the error in such a case may amount to 0-52 × 0-65 = 0-34 of a vibration per day. Moreover, as the difference of temperature is always in favour of the tropical stations, the error will be of a constant nature, unlike the greater part of the small irregularities to which pendulum experiments are liable, and which may compensate themselves by the multiplication of the experiments.

The correction for the expansion or contraction of an invariable pendulum is sometimes derived from pyrometric experiments; but as these are apt to give rather uncertain results, it is more accurate to ascertain, by direct experiments, the difference in the number of vibrations which the pendulum makes in extreme temperatures. This was the method followed by Major Sabine, who found the correction for a brass pendulum to be about 0-44 of a vibration per day for each degree of Fahrenheit; but this depends on the particular materials of which the pendulum is formed.

The experiments of Captain Kater were made, as is well known, in the free air, and its influence in retarding the vibrations, and thereby interfering with the effect of gravity, was only computed and allowed for on the old principle then generally received, but now known to be erroneous. To clear up this point, Major Sabine, at Captain Kater's request, undertook to ascertain, by direct experiment, with the apparatus above described, the actual acceleration which his pendulum would exhibit in a vacuum; and especially to determine what influence the wooden tail-pieces might have had on the original results. For, in the account of Captain Kater's experiments, it is recorded, that on the air suddenly becoming drier, the number of vibrations, which had previously been the same in both positions of the pendulum, ceased to be so; an effect which he attributed to the wooden tail-pieces losing weight by becoming drier. Major Sabine having removed the pendulum to the Royal Observatory, commenced by ascertaining its rate upon both knife-edges, in its original form unaltered, first in air of full pressure, and then in the vacuum apparatus nearly exhausted. The like experiments were repeated with the wooden tail-pieces reduced from their original length of seventeen inches to 6-4 inches, the moveable weights being altered to restore the equality of vibration. The wooden tail-pieces were next entirely removed, and brass ones seven inches long substituted for them, the weights being again altered to restore the convertibility. Lastly, the brass tail-pieces, together with the larger moveable weight, were entirely removed; and a part was filed from the heavy end, to enable the small slider to restore the convertibility.

The following are the results in the four different forms, and for both positions of the pendulum. With the wooden tail-pieces of their original length, and with the heavy end above, the daily acceleration over that in air having a pressure of thirty inches and temperature 49°, was 16-1 vibrations; and with the heavy end below, 15-7 vibrations. With wooden tail-pieces of 6-4 inches, and heavy end above, at 53°5, the acceleration was 14-9 vibrations; below, 12-4. With brass tail-pieces, and heavy end above, at 60°, it was 12-8; below, 11-8. Lastly, without any tail-pieces, and heavy end above, at 57°4, it was 13-7; below, 12-1. But in all of these the actual difference of pressure having been rather less than thirty inches, an allowance was made for it by simple proportion. The details are given at great length in the Philosophical Transactions for 1829 and 1831.

Major Sabine considers it a fortunate circumstance that Captain Kater had used the wooden tail-pieces just of the very sort and size as they happened to be in his original experiments, as by that means his pendulum in its original state was almost as nearly convertible in vacuo as in air. But in making this remark, Major Sabine seems to have forgotten what he had just before mentioned regarding the great effect which a change of dryness was found to have in deranging the convertibility. For if, as there is every reason to believe, the wood became much drier in vacuo from evaporating its moisture, we should be very apt to doubt the accuracy of any experiments upon, or of any comparison between, the rates of the pendulum in air and in vacuo, while so changeable a material as wood entered into its composition.

Mr Baily being also desirous of ascertaining, by a more direct process than that of M. Bessel, the true value of the correction for the numerous and various pendulums in his possession, and considering the subject to be otherwise of importance to science, resolved to devote some time to examine it more minutely, and for this purpose had a vacuum apparatus fitted up, but which is of a much simpler construction than that erected in the Royal Observatory at Greenwich, as already described. It consists of a brass cylindrical tube about five feet long and six and a half inches in diameter, rounded at the bottom, and soldered at the | Kind of Pendulum | No. | Old Correction | Factor n. | New Correction | Grains of Adhesive Air | |------------------|-----|---------------|-----------|---------------|-----------------------| | Sphere of platina 1·44 inch diameter, like Borda's | 1 | 2·709 | 1·861 | 5·104 | 0·496 | | Sphere of lead 1·46 inch diameter | 2 | 5·003 | 1·871 | 9·362 | 0·468 | | Sphere of brass 1·46 inch diameter | 3 | 7·343 | 1·834 | 13·467 | 0·457 | | Sphere of ivory 1·46 inch diameter | 4 | 30·080 | 1·872 | 56·310 | 0·472 | | Spheres 2·06 inches diameter | 5 | 4·988 | 1·739 | 8·668 | 1·115 | | on knife-edge | Lead | 6 | 7·032 | 1·751 | 12·317 | 1·140 | | Brass | 7 | 32·143 | 1·755 | 56·420 | 1·164 | | Ivory | 8 | 4·988 | 1·746 | doubtful | doubtful | | on cylinder | Lead | 9 | 32·143 | 1·741 | doubtful | doubtful | | Brass cylinder 2·06 inches diameter, filled with wire, flat ends horizontal | 10 | 6·882 | 1·860 | 12·800 | 1·945 | | Flat ends vertical | 11 | 6·859 | 1·920 | 13·169 | 2·378 | | Flat ends horizontal | 12 | 6·859 | 1·950 | 13·377 | 2·451 | | Flat ends horizontal | 13 | 6·859 | 1·922 | 13·188 | 2·382 | | Brass tube 2·06 inches diameter, filled with lead, ends horizontal | 14 | 5·448 | 2·032 | 11·070 | 4·558 | | Ditto, hollow, both ends open | 15 | 22·172 | 1·925 | 42·686 | 4·045 | | Ditto, top open, bottom closed | 16 | 21·437 | 1·940 | 41·582 | 4·165 | | Ditto, top closed, bottom open | 17 | 21·955 | 1·975 | 43·378 | 4·283 | | Ditto, both ends closed | 18 | 21·227 | 2·000 | 42·468 | 4·454 | | Ditto, hermetically sealed | 19 | 25·191 | 2·070 | 52·150 | 4·532 | | Lens of lead one inch thick, 2·06 diameter, and horizontal | 20 | 5·000 | 1·580 | 7·900 | 0·438 | | Round copper rod 5·88 inches long and 0·41 diameter | 21 | 6·519 | 2·932 | 19·117 | 4·904 | | Kater's invariable brass pendulum | 22 | 6·697 | 1·590 | 10·649 | 8·339 | | Kater's convertible with the wooden tail-pieces | 23 | 7·630 | 2·144 | 16·356 | | | Great weight below | 24 | 7·630 | 2·204 | 16·815 | | | Plain bars two inches broad and convertible | 25 | 7·002 | 1·848 | 12·938 | 16·705 | | ½ inch thick | Copper | 26 | 7·002 | 1·968 | 13·780 | 19·049 | | Iron | 27 | 6·519 | 1·891 | 12·330 | 20·986 | | ¼ inch thick | Brass | 28 | 6·519 | 1·991 | 12·980 | 23·276 | | ¼ inch thick | Brass | 29 | 7·319 | 1·945 | 14·237 | 22·455 | | ¼ inch thick | Brass | 30 | 7·319 | 2·064 | 15·107 | 25·435 | | ¼ inch thick | Brass | 31 | 6·980 | 2·071 | 14·460 | 40·594 | | ¼ inch thick | Brass | 32 | 6·980 | 2·078 | 14·569 | 40·594 | | ¼ inch thick | Brass | 33 | 6·980 | 2·099 | 14·506 | 40·594 | | ¼ inch thick | Brass | 34 | 6·980 | 2·087 | 14·612 | 40·594 | | Long brass tube | Plane A | 35 | 18·546 | 2·318 | 42·990 | 45·937 | | Plane C | 36 | 18·546 | 2·258 | 41·874 | 43·563 | | Plane a | 37 | 18·546 | 2·267 | 42·048 | 44·195 | | Plane e | 38 | 18·546 | 2·317 | 42·974 | 45·900 | | Mercurial clock pendulum, spring suspension | 39 | 5·312 | 2·343 | 12·448 | 17·003 | | Leaden cylinder upon round wooden rod | 40 | 5·190 | 2·589 | 13·104 | 17·462 | | Ditto, flat rod | 41 | 5·190 | 2·627 | 14·312 | 20·120 | There are several things in the table which will require some farther explanation to make them quite intelligible.

No. 1 is similar to Borda's pendulum, which is particularly described in the preceding article. Nos. 2, 3, 4 were also suspended as Borda's, and so were Nos. 5, 6, 7. Nos. 8, 9 are the same as 5, 7, but they were merely suspended by a fine wire, which was slightly held aside by the round surface of a steel wire one-fifteenth inch in diameter, a contrivance of M. Bessel, the relation of which to any practical application of the pendulum we cannot pretend to see. No. 10 is a brass cylinder with its axis vertical, and suspended by a fine wire and knife-edge. No. 11 is the same, with its axis across the plane of motion, but suspended by a knife-edge and brass rod. No. 12 is the same as the last, but with flat ends across the plane of motion. No. 13 the same, with flat ends horizontal. No. 14 is a brass tube 2-06 inches diameter and four inches long, filled with lead. It was suspended with its axis vertical by a fine wire and knife-edge. Nos. 15, 16, 17, 18, 19 are the same tube empty. No. 20 has a flat circumference about a quarter of an inch wide. It is suspended horizontally by a knife-edge and wire. No. 21 has a similar suspension. No. 22 is Kater's invariable pendulum, with which Major Sabine made the experiments described on page 205. It has a brass tail-piece about sixteen and a half inches long, for which there was not room in Mr Baily's apparatus; so that the results are those of Major Sabine, and so are the results with Nos. 23 and 24. But Major Sabine having removed the tail-pieces altogether, found the value of $n = 1.875$ for knife-edge A, or with the heavy end below, and $= 2.135$ for knife-edge B, or with heavy end above. Mr Baily, it is true, has slightly altered Major Sabine's numbers, by assuming a different specific gravity for the pendulum, and also for the air at a different temperature. Nos. 25, 26 are the results on the knife-edges A and B of a convertible pendulum formed of a plain brass bar 62-2 inches long; Nos. 27, 28 are those of a copper bar 62-5 inches long; and Nos. 29, 30 those of an iron bar 62-1 inches long.

The last two pendulums were taken out by the late Captain Foster, in his scientific expedition to the south, which we shall notice farther on. Nos. 31, 32, 33, 34, are the results with the four knife-edges, named A, B, C, D, of a doubly-convertible pendulum, described by Mr Baily in the Philosophical Magazine for February 1829, along with the account of some experiments which differ to an extent for which he acknowledges himself unable to give any satisfactory reason, unless it be owing to irregularities in the knife-edges and planes. It consists of a plain brass rod sixty-two inches long, without any moveable weights or sliding-pieces, and has its knife-edges convertible in pairs taken alternately; thus forming, in effect, two convertible pendulums. The distance between the knife-edges A and C was 39-3038 inches, and the number of synchronous vibrations 862183. The distance between B and D was 39-3084 inches, and made 862046 vibrations. Consequently, from these data, the lengths of the seconds pendulum will be respectively

$$\frac{862183}{86400} \times 39-3038 = 39-1386,$$

$$\frac{862046}{86400} \times 39-3084 = 39-1307.$$

The first agrees nearly with that of Captain Kater; but it is not easy to account for the discrepancy between these two results. Mr Baily does not inform us whether he had ascertained that this pendulum was perfectly free from magnetic influence, which is far from being always the case with articles of brass or copper. Nos. 35, 36, 37, 38 are the results with the four planes named A, C, a, c, which are used in place of knife-edges on a doubly-convertible pendulum formed of seven different brass tubes drawn closely one within the other, their joint thickness being 0-13 inch. The diameter of the outside is 1-5 inch, the ends are open, and the length is fifty-six inches. This pendulum is of a very singular construction; for, instead of being fitted up with steel knife-edges vibrating on agate planes, it is furnished with circular steel planes, which vibrate on a pair of fixed agate knife-edges, which are common to all the planes. The mode of suspension, therefore, is in this case reversed. The planes A and a are convertible, and so are C and c; the length between each of these pairs is very nearly a standard yard. We should, however, consider this to be rather a vague kind of pendulum, because the axis of suspension is so indeterminate that it is not restricted to such straight lines in the surface of the plane as pass through the axis of the tubes, and therefore the results must be rather indeterminate. No. 39 does not materially differ from the mercurial pendulum described farther on. No. 40 is a pendulum with a spring suspension, and deal-rod three eighths of an inch in diameter, the lower part of which is encased by a leaden tube 13-5 inches long and 1-8 inch in diameter. This tube serves as the bob, and is also meant to act, by its upward expansion, as a compensation for temperature. No. 41 only differs from the preceding in having a flat deal-rod one inch broad and 0-14 inch thick, which was bevilled to a thin edge, and moved edgewise. The factor $n$, it will be seen, is greater with the flat than with the round rod. This was also found to be the case when rods of the same form as the wooden ones, but of other materials, were substituted for them.

The preceding table does not contain the half of Mr Baily's experiments. The results generally accord pretty well with the theory that a quantity of air adheres to every pendulum when in motion, and, by thus forming a portion of the moving body, diminishes its specific gravity, or rather adds to its inertia. This adhesive air seems to be confined chiefly to the two opposite portions of the pendulum which lie in the line of its motion, and very little of it adheres to or is dragged by the sides of the moving body. The shape of this coating of air will consequently partake in some measure of the form of the pendulum, subject probably to some slight modifications, of the nature of which nothing is yet known. The quantity of air dragged by the pendulum seems to depend on the extent and form of surface opposed to its action, and is not affected by the density of the body. In the case of a sphere one inch in diameter suspended by a fine wire, the weight of air dragged by the sphere alone appears to be about 0-123 grain; and for different spheres it is nearly as the cubes of their diameters. The weight of air dragged by the wire of the length of the seconds pendulum may amount to, but probably does not exceed, 0-1 grain, and perhaps is nearly the same for all fine wires of that length; so that with spheres less than an inch in diameter the weight of air dragged by the wire is nearly the same as that dragged by the sphere.

With respect to cylinders suspended by rods and swung with their flat sides opposed to the line of motion, the law of the variation is not so manifest, owing to the precise effect of the edge of the cylinder being as yet unknown. Neither are there yet sufficient data to develope the effect.

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1 In the Philosophical Transactions for 1830, p. 207, Mr Lubbock professes to prove, that if the knife-edge is rounded or cylindrical, the distance between the planes is still the length of the equivalent simple pendulum. But in this he has deceived himself, and only proved the converse of the proposition; for it is easily shown (as we have done farther on in the precisely similar case of Laplace's theorem), that whenever the centre of gravity is within a certain distance of the middle between the planes, their distance is greater than the length of the equivalent simple pendulum. Besides, Mr Lubbock entirely overlooks the uncertainty in the position of the axis of suspension. of the air on cylinders suspended by rods or wires, and swung with their flat sides in a horizontal position, similar to the pendulums Nos. 10 and 14. In these cases the cylinder four inches long drags much more than double the quantity of air adhering to the cylinder of two inches, although they have precisely the same diameter. With respect to very thin cylinders or discs swung with their flat sides opposed to the line of motion, the weight of air dragged by a disc of one inch in diameter appears to be about 0.149 grain, and for different discs is nearly in the ratio of the cube of the diameter; whence it appears that a thin disc drags more air than a sphere of the same diameter.

The last column of the table contains the weight of air which is supposed to adhere to and be dragged by the pendulum, in consequence of the air put in motion thereby, when vibrating in the mean state of the atmosphere; or rather the quantity of air which, if applied to the centre of gyration of the pendulum, would produce the retardation shown by the experiment. This view of the subject was suggested by Professor Airy, the present distinguished astronomer royal, who also gives the following investigation and formula for computing the weight of adhesive air. Let $N$ denote the number of vibrations made by a pendulum in a mean solar day, when swung in air; and let $\nu$ be the additional number which it makes when swung in vacuo. Also, let $w$ be the weight of the pendulum in grains troy, $S$ its vibrating specific gravity, and $\varepsilon$ the specific gravity of the air. Now, since the force of gravity diminishes in the ratio of $(N + \nu)^2$ to $N^2$, or in the ratio nearly of $\left(1 + \frac{2\nu}{N}\right)$ to 1, it follows, that when the pendulum vibrates in air, it is as if, retaining the inertia of its weight $w$, it had the gravity of only $w \times \frac{N^2}{(N + \nu)^2} = w \left(1 - \frac{2\nu}{N}\right)$ nearly; or as if it had lost the weight $w \times \frac{2\nu}{N}$. But the weight which it has really lost from the displacement of a quantity of air is $w \times \frac{\varepsilon}{S}$. Consequently, the portion which is not accounted for by the mere displacement of the air is $w \left(\frac{2\nu}{N} - \frac{\varepsilon}{S}\right)$, and which may be considered as the additional weight gained by the pendulum (or rather the addition to its inertia) when moving in air supposed to be applied to the centre of gyration. The inertia of the whole pendulum in resisting angular motion is the same as if it were collected at the centre of gyration. The immediate result of the experiment and formula above given is, that the inertia of the whole pendulum ought to be increased in the proportion of $\left(1 + \frac{2\nu}{N} - \frac{\varepsilon}{S}\right)$ to 1; or that, instead of supposing the inertia $w$ to be applied at the centre of gyration, the inertia $w \left(1 + \frac{2\nu}{N} - \frac{\varepsilon}{S}\right)$ ought to be applied there. The addition to the inertia is therefore $w \left(\frac{2\nu}{N} - \frac{\varepsilon}{S}\right)$ applied where that of the whole pendulum may be supposed to be applied, that is, at the centre of gyration.

As already mentioned, Major Sabine has alleged, that the usual formula for the reduction of the vibrations of a pendulum to indefinitely small arcs is erroneous, as it does not agree with the result of a series of experiments which he had undertaken for the purpose of testing it, and which, in the case of the convertible pendulum tried by him, would require the hitherto assumed corrections to be multiplied by 1.13 when the great weight is below, and by 1.4 when above. As this was somewhat at variance with Mr Baily's view of the matter, he determined on making a few trials, in order to ascertain more minutely, by experiment, the difference which really arises from the use of larger and smaller arcs; and for this purpose he took a brass bar convertible pendulum, placing it in the vacuum apparatus, under about one inch pressure of the atmosphere. Two series were made on the knife-edge which was nearest the end of the bar, and which he calls A, and two on the other, named B; and each of these series was divided into three portions, in the first of which the width of the arc was taken from about 1° to 0°6, in the second from 0°6 to 0°88, and in the third from 0°38 to 0°2 and 0°1. The first series on the knife-edge A required the usual correction to be increased about a tenth, which agrees nearly with Major Sabine's experiments; but the second series on the same knife-edge required the correction to be diminished nearly as much. Mr Baily, therefore, considers these two series as neutralizing each other, and that the difference observed must lie within the errors of observation. On the knife-edge B, both series required the correction to be increased one fifth, which is only half the amount indicated by Major Sabine's experiments. Further inquiries, therefore, are necessary to clear up this point; not only as to the cause of the anomaly, whether it arises, as Major Sabine supposes, from a sliding of the knife-edges on the agate planes, in which case it may differ in different pendulums, but also as to the accuracy of the generally received formula, which several mathematicians allege to become inaccurate when the arc is very large. But whether the difference really arises from a defect in the formula, or from a sliding of the knife-edges, or from the variable effect of the air on the pendulum, or from all three, remains still to be decided. Perhaps magnetism is not always sufficiently guarded against. There are instances of articles of brass having become highly magnetic from the manner in which they have been worked; so much so, indeed, in parts of chronometers, as not only to derange their performance, but even to arrest their motions altogether. According to M. Bessel, the knife-edges slide on the planes. The sliding is proportional to the extent of the arc of vibration, and is always in the direction in which the pendulum is moving. Now, if the sliding can really be so well observed and appreciated, it must be very considerable; probably far exceeding the sliding and friction of a fine cylindric surface turning, or rather sliding round, in a nicely jewelled groove.

It has been shown by Major Sabine in his work on the Defects of figure of the earth, page 195, that in a pendulum with knife-edges and planes, a considerable difference may arise in the results, if they be used with different planes; but it does not seem to have occurred to any one versed in these experiments, till discovered by Mr Baily, that a much greater difference than that just referred to may arise while using the same knife-edge with the same plane. This fact had probably long escaped detection, from the peculiar manner in which pendulum experiments are usually conducted; for, on examining the detail of most of them, it will be found that, after the pendulum, at any one station, has been placed in its V's, it has never been removed from them, but merely raised and lowered again as occasion may require, till it has been ultimately dismounted and packed up for another station. In this way any inconsistency that might otherwise have occurred is avoided, and consequently escapes detection. Mr Baily considers the pendulum furnished with a

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1 Mr Baily advises, that any future experiments on this question should be made in the free air, and not, as formerly, in vacuo. We should, however, have great doubts of the correctness of this opinion; for how then could the very different effects of the air on wider and narrower vibrations be distinguished from the effects which belong to the differences of the arcs themselves? Pendulum knife-edge and agate planes, as at present constructed, to be a very inadequate instrument for the delicate purposes for which it was originally intended; and that a more rigid examination and adjustment of those parts of the instrument are requisite, before we can depend on it, either for the determination of the length of the seconds pendulum, or even for the comparison of results obtained in different parts of the world. The knife-edge is seldom or never perfectly straight; the planes, whether from being carelessly ground, or being so thin as to be flexible, are seldom or never perfectly true; so that as there is generally a little play in the Y's, the knife-edge is not always let down on the same parts of the agate plane. This may be detected by holding a candle behind the knife-edge while it is resting on the plane; for, by means of its rays, the smallest inequalities in the points of contact are readily discernible.

But the fact is rendered still more evident, by reversing the ends of the same knife-edge in the Y's, when a sensible difference in the result generally takes place. Among the numerous pendulums in Mr Baily's possession, he has only met with one whose results were not altered by an appreciable quantity when the knife-edge is reversed in the Y's, or turned half round in azimuth. If the knife-edge and plane were perfectly correct and true, there ought to be no difference in the results, whichever side of the pendulum is next the observer; whereas a difference of upwards of two vibrations in a day actually occurs in one of the pendulums above alluded to.

The theory of pendulums suspended on rolling cylinders under various forms and circumstances, is treated at considerable length, and with such ability, by Euler, in the *Acta Petropolitana* (tom. iv. part ii. for 1780), and again in the *Nova Acta* (tom. vi. for 1788), that in place of here entering into any elaborate investigation on this head, we shall now, for brevity's sake, take advantage of Euler's results, especially in examining the merits of that celebrated theorem which the late illustrious Laplace discovered about twenty years ago, and which, when it appeared, tended greatly to allay the very considerable fears which had previously existed regarding the accuracy of the experiments with convertible pendulums. For the knife-edges on which these had been suspended were considered liable to become blunted, or in some measure changed into cylindric surfaces, by their turning, under considerable pressure, on the horizontal planes; and it had been feared that this might affect the time of vibration. The theorem, though sometimes a little differently expressed, amounts to this: when a rolling pendulum is convertible, vibrating alike fast on each of two equal and parallel cylinders rolling on a horizontal plane, the distance between the cylindric surfaces is the length of the equivalent simple pendulum.

Though usually regarded as universally true, this theorem fails whenever the distances of the two cylinders from the centre of gravity do not differ by a quantity which greatly exceeds a diameter of the cylinders. Fortunately, it holds good in the cases which are of most importance for the convertible pendulum; but it is not perhaps much to the credit of science that the true limits of the theorem should not have been ascertained; and probably, too, from mathematicians having so much mistaken the nature of the proposition, that the reasonings hitherto employed by them in proof of it have at best amounted to demonstrations of its converse, or rather of the converse only of its possible cases, namely, that positions may always be found for two rolling suspensions of the sort above described, such that they shall make the pendulum convertible, and have their distance equal to the given length of an equivalent simple pendulum.

The investigation of Laplace himself, while it has the defects just noticed, is so unnecessarily intricate, that it would be tedious to discuss its merits; but in the Philosophical Transactions for 1818, p. 95, the late eminent philosopher, Dr Thomas Young, has given what he considered to be a more elementary demonstration of this theorem, though it is, in fact, of its converse; and he has again, in the Philosophical Transactions for 1819, p. 94, attempted, from some of Euler's results, to deduce the same thing, but which comes to be still its converse, as will be afterwards noticed.

There is likewise an abridgement of Euler's investigations for 1788 given in the Philosophical Magazine for December 1821, and with it is coupled a similar, but not more successful, attempt at deducing the theorem in question; for in all of these the proof is applicable to nothing more than the converse of the possible cases of Laplace's theorem, though that converse itself really forms a theorem which never fails.

If $a$ be the distance of the centre of gravity of the rolling pendulum from the centre of the upper cylinder, $e$ the radius of each cylinder, and $k$ the radius of gyration, with respect to an axis passing through the centre of gravity of the pendulum, and parallel to the axes of the cylinders; then it has been demonstrated by Euler, in the *Nova Acta Petropolitana* for 1788 (p. 149), that this rolling pendulum will perform a small vibration in the same time as a simple pendulum whose length is

$$ l = \frac{k^2 + (a - e)^2}{a} $$

But it is easily shown that this must vibrate in a shorter time, especially with the heavier end of the convertible pendulum uppermost, than the same pendulum would do were it suspended by knife-edges substituted for, and put exactly in the places of, the cylindric surfaces. For, supposing the pendulum to continue a convertible one, after being thus modified, we should, from the well-known relation between the centres of suspension, gravity, and oscillation, have $k$ a mean proportional between the segments into which the centre of gravity divides $l$ the distance between the knife-edges, or we should have

$$ k^2 = (l - a + c)(a - c) $$

and hence

$$ l = \frac{k^2 + (a - e)^2}{a - e} $$

which corresponds to a simple pendulum longer than the former in the ratio of $a$ to $a - e$. Hence it appears, that a rolling pendulum always vibrates faster than if it were suspended by a knife-edge substituted exactly in the place of the cylindric surface; so that when the rolling pendulum is a convertible one, the cylindric surfaces always have positions different from those of two knife-edges, which, with an equal interval between them, would also render the same pendulum convertible, and preserve the same time of vibration; and, therefore, to make the two proposed knife-edges keep time with the cylinders, the former would need to be shifted so as to shorten the time of vibration, which, when possible, may be effected by bringing the more distant knife-edge nearer the centre of gravity, and removing the other as much farther off, so as still to preserve an interval $= l$ between them. In that case, putting $x$ for the space to be so shifted, we should, for the reason above given, have

$$ k^2 = (l - a + c - x)(a - c + x) $$

and

$$ l = \frac{k^2 + (a - e + x)^2}{a - e + x} $$

which, according to the theorem, is to equal the efficient length of the rolling pendulum, namely,

$$ \frac{k^2 + (a - e)}{a} $$

Wherefore, substituting the above value for $k^2$ in this last formula, should afford a ready means of finding such an expression for $x$ as shall reconcile, when possible, the times of vibration on the two different sorts of suspensions; for it then becomes

$$ l = \frac{(a - c + x) - 2(a - e)x - x^2}{a} $$

whence $x^2 - (l - 2(a - e))x = -cl$, and $x = \frac{1}{2}l - a$, $+c = \sqrt{\left(\frac{1}{2}l - a + c\right)^2 - cl}$, where both values of $x$, with a deviation in altitude; for one degree on that position Pendulum of the knife-edge produces an acceleration of three seconds in twenty-four hours. Still more considerable is the effect of a deviation from horizontality in the agate planes; for ten minutes of a degree produce a daily acceleration of six seconds. Both the two latter deviations render the distance between the knife-edges greater than the length of the equivalent simple pendulum.

During the last twenty years, a vast number of experiments have been made by means of the pendulum, on the relative force of gravity in different parts of the world, with the view of contributing towards the determination of the true figure of the earth. Accounts of these have been published in the scientific periodicals, and in separate works; but our limits will only permit us to notice briefly a few which are more particularly interesting.

The length of the seconds pendulum having been determined in London by the method and apparatus of Kater, and in Paris by those of Borda and Biot, and the standard linear measures of the two countries having been respectively referred to these pendulums for future verification, an attempt was made about twenty years ago to ascertain the difference between the actual number of vibrations which invariable pendulums conveyed between London and Paris would make at these places. A summary account of the operations is given in the *Buse du Système Métrique*, tom. iii.; but, from certain accidental causes, some doubts have always attached to the result. To clear up this point by solving the question anew, Major Sabine, early in the year 1827, had two invariable pendulums conveyed to Paris, where an extensive series of experiments were made by himself, and several eminent French philosophers, in the very spot where M. Biot had determined the length of the seconds pendulum. The two pendulums were next brought back to London, and similar operations were made with them in the room of Mr Browne's house, Portland Place, where Captain Kater had originally made his experiments. The comparison of the results showed, that in the latter place the pendulums made twelve vibrations more per day than at Paris, answering to 0·01088 inch in the length. The reductions, it is true, were made according to the old rules, though it is likely that this would occasion nearly equal errors in both quantities, so as not to affect their difference. The details are given at great length in the Philosophical Transactions for 1828, and the new corrections may still be applied.

In order the better to insure that the various pendulum experiments which had been made in Mr Browne's house, Portland Place, London, particularly those of Captain Kater, might be rendered comparable with any similar experiments made at the Royal Observatory at Greenwich, Major Sabine, at the request of the Royal Society, undertook to settle the question by direct experiment, as detailed at great length in the Philosophical Transactions for 1829. For this purpose, he first made a series of experiments on the daily rate of an invariable pendulum in Mr Browne's house, and then did the like with the same instrument taken to Greenwich. The result was, that the pendulum made 0·52 of a vibration more per day at Greenwich than at London. The knife-edge of the pendulum was then fresh ground, and a new series of experiments made at both places, with a similar result of 0·44. The mean is 0·48 of a vibration. Now, as the latitude of the Royal Observatory is 28° south of Mr Browne's house, and as it is about fifty feet above its level, a retardation, from these causes combined, of about 0·27 of a vibration, was to have been expected at Greenwich, instead of an acceleration. The retardation, as computed for the difference in latitude, would be 0·15 vibration; and that for fifty feet of difference in elevation, being about 0·20, would, when multiplied by Dr Young's co-efficient of 0·6, be 0·12 of a Pendulum vibration. These added to 0.48, make 0.75 of a vibration; by which amount the result of actual experiment differs from what might have been anticipated by mere theory.

In the autumn of the same year, 1828, Major Sabine transported the same invariable pendulum to Altona, where he made a series of observations with it. The like was done with the same instrument again in 1829, at Greenwich. The result was, that the pendulum made 833 more vibrations per day at Altona than at Greenwich.

The late lamented Captain Henry Foster, who had been sent out on a scientific expedition towards the south, the principal object of which was to swing the pendulum near the equator, and also at various places in the southern hemisphere, took out with him four pendulums, two of which were of the kind called Kater's invariable pendulum, but the other two were of a new construction, recommended by Mr Baily, the one of iron and the other of copper, each of which was furnished with two knife-edges; so that, in effect, Captain Foster might be considered as having taken out six different and independent pendulums. These were swung at fourteen different places, and, with the exception of the experiments with one of the brass pendulums at South Shetland, the results are very accordant, and show that the pendulum, even with its present imperfections, affords an accurate measure of the relative force of gravity at different places. But the sources of minute errors more recently detected in this instrument will render future experiments more valuable and comparable with each other.

As our limits will not admit of entering into a comparative examination of the results obtained at each place, it must suffice here to state the general result at the several stations, referring, for the whole details, to the Memoirs of the Astronomical Society, vol. vii. For the purpose of deducing these results with the least chance of error, Mr Baily has adopted the usual method of minimum squares, whereby he finds that if \( v \) be the number of vibrations at the equator, \( f \) the co-efficient of the increase of the force of gravity, \( L \) the latitude of any other place where the same pendulum makes \( V \) vibrations, the general formula \( V = v(1 + f \sin^2 L) \) becomes, in the present case, and according to Captain Foster's experiments,

\[ V = (7441507482 + 386866418 \sin^2 L). \]

The following table shows the observed mean results of all the pendulums, deduced from the whole of the experiments, at each station, as well as the value computed from the above formula, together with the error or difference thence arising.

| Stations | Latitude | Observed Vibrations | Computed Vibrations | Difference | |----------------|----------|---------------------|---------------------|------------| | Para | 1° 27' O'S. | 86360-61 | 86264-30 | -9.69 | | Maranhao | 2° 31' 25" | 86258-74 | 86264-60 | -5.86 | | Fernando de Noronha | 3° 49' 59" | 86271-20 | 86265-16 | +6.04 | | Ascension | 7° 55' 23" | 86272-25 | 86268-44 | -3.81 | | Porto Bello | 9° 32' 30"N.| 86272-01 | 86270-32 | +1.69 | | Trinidad | 10° 38' 55" | 86267-24 | 86271-84 | -4.60 | | St Helena | 15° 56' 7"S.| 86288-29 | 86281-06 | +7.23 | | Cape of Good Hope | 33° 54' 37" | 86331-33 | 86333-90 | -2.57 | | Monte Video | 34° 54' 26" | 86334-96 | 86337-52 | -3.16 | | Greenwich | 51° 28' 40"N.| 86398-90 | 86401-24 | -2.34 | | London | 51° 31' 17" | 86400-00 | 86401-40 | -1.40 | | Staten Island | 54° 46' 23"S.| 86415-22 | 86413-58 | +1.64 | | Cape Horn | 55° 51' 20" | 86417-98 | 86417-54 | +0.44 | | South Shetland | 62° 56' 11" | 86444-52 | 86441-72 | +2.80 |

The last column clearly indicates that the pendulum is powerfully affected by local circumstances, since the difference between the observed and computed results in most of the cases far exceeds the probable errors of observation, and all the pendulums agree in their indication of the degree of intensity at the several places. This fact, however, is rendered more striking by combining the experiments hitherto made with the invariable pendulum by the several English, French, and Russian voyagers. In this manner Mr Baily has obtained fifty-one different places where the pendulum has been swung; and at several of these stations this has been done by more than one experimentalist. The results, therefore, being in several instances confirmed either by the experiments of various persons, or by various pendulums swung by the same person, show most decidedly that there is some local influence on the pendulum at such stations, with the exact nature of which we are unacquainted, and which baffles all our efforts to deduce the true figure of the earth from pendulum experiments made at a few places only; for the results deduced from such experiments will vary according to the selection which is made of the stations. And it is a remarkable circumstance, that the force of gravity seems to be greater in islands remote from the mainland, such as St Helena, Ascension, &c., than it is on continents.

The ellipticity of the earth, deduced from Captain Foster's experiments, is \( \frac{1}{289.48} \); Major Sabine made it \( \frac{1}{288.40} \); the mean of the French and Russian experiments is \( \frac{1}{267.23} \); the mean of the whole combined, \( \frac{1}{285.26} \). The total number of coincidences made by Captain Foster was more than 20,000, and occupied upwards of 3180 hours. The total time occupied by Major Sabine was only 598 hours; that by Captain Freycinet only 367 hours; and that by Captain Duperrey only 256 hours. So that Captain Foster's experiments are five times more extensive than Major Sabine's, and fully twice and a half more extensive than those of all the above experimentalists united. They have also the advantage of having been made with a greater variety of pendulums, and cannot fail to be duly appreciated in all inquiries connected with that important subject which they were intended to elucidate, the true figure of the earth.

Among the various sources of discordance between the results of different experimenters, a fertile one seems to be the very different reductions which they use for the expansion of the air, or for reducing its density from one temperature to another. Thus, although few things in natural philosophy are better determined than the expansion of the air, almost every one uses his own favourite reduction for it. According to the experiments of Dalton, Gay-Lussac, &c., the expansion of air for one degree of Fahrenheit, and whether dry or moist, is equal to the \((t + 448)\)th part of its bulk, under a constant pressure, at the temperature \( t \). From the circumstance of this being the 480th part of the bulk at 32°, and from 32° being often used as a standard temperature, seems to have arisen the common-place phrase, that "the expansion of air for one degree is the 480th part," a phrase which is always more or less erroneous, unless it mean the bulk at 32°; for it is the 448th of the bulk at the zero of Fahrenheit, the 528th of it at 80°, and the 660th of the bulk at 212°, &c. This, though an extremely simple affair, seems much too abstruse for the greater part of writers on chemistry, and for not a few on natural philosophy, among whom we find the above phrase means the 480th part of the bulk at any temperature whatever. The error is precisely of the same sort as if, because a shilling is the 480th part of twenty-four pounds, it should be reckoned the 480th part of every other sum. The greatest dunces rarely do anything like this in the ordinary affairs of life; and why should such blunders pervade writings which profess to be scientific? In 1830, Major Sabine having entirely removed the tail-pieces and the larger moveable weight from Captain Carter's convertible pendulum, executed a very extensive series of experiments with it at the Royal Observatory, Greenwich, for the purpose of accurately determining the length of the seconds pendulum there; and the result was, that at the temperature of $62^\circ$, and in a vacuum, it is $39\cdot13734$ inches. But notwithstanding the very great care which seems to have been bestowed on these experiments, doubts regarding their accuracy have been started by Mr Baily, particularly on account of an anomaly which occurs on the face of the observations. For it appears, that when the small sliding weight was shifted about $0\cdot133$ inch, it caused a daily acceleration of $0\cdot10$ vibration on the knife-edge A, or with the heavy end below; whilst it caused a retardation of $1\cdot12$ vibration on the knife-edge B. Now this is quite contrary to the known principles of the pendulum, since the effect of a slider of this sort is to produce an alteration of the same kind in each knife-edge, differing only in degree.

We suspect that with convertible pendulums sufficient care is not always taken to insure that the centre of gravity and both knife-edges shall be accurately in the same plane; without this the results must be very doubtful.

V.—ATTACHED PENDULUMS.

Thus far we have treated principally of what relates to detached pendulums, and of the corrections to be computed and applied to the numerical results. What follows more especially belongs to clock pendulums, and to the contrivances which enable them to correct or compensate themselves for the effects of atmospherical pressure, temperature, &c.

On examining the new correction for the mercurial pendulum, which is the one now generally employed in clocks for astronomical purposes, though, for reasons given farther on, we do not consider it as affording by any means the best compensation for temperature, it will be found that a difference of one inch in the pressure of the atmosphere should alter the daily rate of the clock by $0\cdot414$ second, which is more than double the quantity hitherto assumed as depending on the change of the barometer, and which therefore should no longer be disregarded by the astronomer. To obviate this defect in the clock at the Observatory at Armagh, Dr Robinson has attached two syphon-ba-rometers to the mercurial pendulum, one on each side of the rod, and so placed that the variations in the height of the mercury in their tubes may exactly compensate the effect of a change of atmospheric pressure. In the Quarterly Journal of Science (vol. xv.) Mr Davies Gilbert has endeavoured to show how a compensation for the effects of pressure may be produced, by making the arc of vibration just of such an extent that the effect produced by the difference of density in the atmosphere may be exactly counterbalanced by the effect arising from the difference in the arc of vibration caused by such difference of density. But by proceeding agreeably to the formula which Mr Gilbert has there given for finding the value of such arc, and on the assumption of the accuracy of the new correction above mentioned, Mr Baily finds that the required arc of semi-vibration should be $29^\circ 45'$, whereas in astronomical clocks it seldom exceeds $29^\circ$, which would produce only half the requisite compensation; so that, after following up Mr Gilbert's theory, there still remains a part of the effect, exceeding $0\cdot2$ second, on the daily rate of the mercurial pendulum, as produced by a difference of one inch of pressure. Did the extent of the arc depend on no other variable cause than the density of the air, we should think that an increase in the bulk of the pendulum would readily give the requisite addition to the compensation. For, from Mr Baily's experiments, it appears that the retardation of the air depends principally on the bulk of the pendulum, though it seems to be affected by other circumstances at present not well understood. But since the extent of the arc of vibration depends very much on the variable state of the oil, and the degree of cleanness of the clock, neither of which keeps pace with the density of the air, it seems greatly preferable to have recourse to barometers like those of Dr Robinson, and to employ a spring of suspension, having its length, if possible, so adjusted as to equalize the times of vibration in the wider and narrower arcs, which is more particularly considered farther on.

All bodies expand by heat and contract by cold in a greater or less degree; and consequently every attempt to discover any simple pendulum-rod, whether natural or artificial, which could continue of an invariable length, so as to have its rate of motion independent of variations of temperature, has uniformly failed. To obviate this inconvenience, a variety of devices have been resorted to by ingenious persons, and some of them with success. About the year 1715, the celebrated artist Mr George Graham attempted, by means of the very imperfect pyrometers then in use, to ascertain the relative expansions of several metals, in the expectation that the difference in some two or more of them, when applied to act in opposite directions, would enable him to construct a compound rod of an invariable length. But, owing to the defects of his pyrometer, the difference between their expansions seemed so inconsiderable, that he then relinquished all hope of effecting a compensation in this way. Some time after, it occurred to him that the great expansibility of mercury might be rendered available for this purpose; and, accordingly, in 1721 he had succeeded so far in the construction of the following kind of pendulum, that the usual error in the extremes of temperature was reduced to one eighth, as is described at length in the Philosophical Transactions for 1726.

The bob or weight of the mercurial pendulum, represented in fig. 3, Plate CCCCHI., consists of a cylindrical pendulum glass jar A, containing mercury. The rod BB is of steel, and single till it reaches down thirty-two inches from the suspension, or within ten inches of the bottom, where it screws into the milled-headed nut D, in the middle of the cross bar EE of the stirrup FF, which carries the glass jar resting on the circular base G. This base, shown separately in fig. 4, is of brass, slightly hollowed to fit the jar, and a cover precisely similar to it is used on the top of the jar. Besides this, Mr Browne, with the view of keeping the surface of the mercury perfectly flat, covers it with a float of plate-glass. The jar is generally about two inches wide and eight deep, having about $6\cdot8$ inches of this depth filled with mercury; but the quantity of mercury depends very much on the weight and expansibility of the other materials. If, upon trial, the clock go too slow with an increase of temperature, showing the downward expansion of the rods to be too great for the upward expansion of the mercury, more of this fluid must be poured into the jar; if the reverse take place, some mercury must be withdrawn. In this manner may the expansion and contraction of the mercury be made exactly to compensate those of the other materials, so as to preserve invariable the distance of the centre of oscillation from the axis or other suspension. This pendulum, though in great repute, is by no means free from objection; for, since there is often a sensible difference of temperature in a small difference of height, the mercury, being much lower down than the greater part of the rod, is liable to have a somewhat different temperature, and con-

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See Memoirs of the Astronomical Society, vol. v. Pendulum. sequently to fail in producing at all times an exact compensation. It is also alleged, that such a great mass of mercury, being enclosed in so bad a conductor as glass, does not acquire the changes of temperature so promptly as the metallic rods. To obviate this last, glass is sometimes dispensed with, and an iron cylinder, or one of brass, varnished inside to prevent the action of the mercury, is substituted for the glass. On the other hand, Captain Kater had a cylinder and rod blown all in one piece of glass, similar to a bottle with a very long neck. He had no other rod or frame. A cap of brass was clamped by screws to the top of the glass rod, and to this was pinned the suspension spring. Pendulums of this sort are said to have been at one time used in France. They are much less expensive than those with the metal rods, &c.

In the Philosophical Magazine for August 1819, Dr Firminger has given a description of a mercurial pendulum, with the dimensions of all the parts, apparently to a great degree of exactness, as computed by Mr Gavin Lowe for the Royal Observatory, and which have, therefore, been often copied and referred to as standard authority. But, on examination, they will be found to be very incorrect; for, in computing the position of the centre of oscillation, and even by means of an algebraic formula, no regard is had to any dimensions of the materials but their lengths. Thus, neither the diameter of the cylinder, nor the still greater breadth of the frame or stirrup, is taken into account, although both have a considerable influence on the result.

The late ingenious Mr Troughton contrived a method of constructing a compensating pendulum with much less mercury than is usually employed. Instead of the cylindrical cistern, he put the mercury into a bulb and tube in the form of a large thermometer. In that case, forty-five ounces of mercury sufficed for the compensation, but he also used an ordinary bob. According to Dr Pearson, it is a property of this contrivance, that if the column of mercury be made to reach either above or not quite up to the middle of the pendulum, at a mean temperature, the compensation will be in defect; but when the column reaches exactly to the middle, at a mean temperature, the compensation is exact. But in this there must be some mistake; for nothing can be more evident than that the column may reach exactly to the middle, while the sizes of the bulb and tube may be such as to produce nothing at all approaching to a correct compensation.

Ward's pendulum is a more recent one than the gridiron pendulum; but being essentially the same, and simpler in a very great degree, the description of it will form an easy introduction to that of the other. It is shown in fig. 5, Plate CCCCIII., where aa and bb are two flat rods of iron or steel, about half an inch broad and an eighth thick. Between these is a third rod cc of zinc, nearly the same in breadth, and a quarter of an inch thick. The one steel rod aa is cranked at the upper end, and the other bb, at the lower end, for the purpose of bringing the suspension spring and the centre of the bob to be in the same vertical line with the zinc rod. These three rods are held together by the screws d, d', d", but only so as still to allow them to shift with ease upon each other in a slight degree. These screws, passing through oblong holes in the rods aa and cc, screw into the rod bb, while the rod aa is held firmly to the zinc one cc by the single screw e, which, therefore, forms in effect the lower limit of the zinc rod; but since the proper length of the zinc rod can almost never be hit on at first, several holes are provided for the screw e, in order that on trial it may be put into the one which gives the true compensation. The steel rod bb, which carries the bob of the pendulum, is knoc'd, so as to hook against the upper end of the zinc rod; and the other steel rod aa might in like manner, without the screw e, have hooked against the lower end of the zinc, but in that case it would not have been so easy to adjust the proper length of the zinc. The expansion of zinc exceeds that of steel in the ratio of 27 to one nearly. In some pendulums the zinc rod is about twenty-five inches long, in others fully twenty-seven. It is obvious, that as the two rods of steel aa, bb expand downward, lengthening the pendulum, the zinc rod cc expands upward, drawing up the bob by means of the steel rod bb, so as to raise, or rather preserve the proper position of the centre of oscillation. It is necessary that the expansion of the zinc rod should more than compensate an equal length of each of the steel rods, because they stretch beyond it both ways. But the bob itself, by expanding upward from the nut on which it rests beneath, will a little aid the compensation.

The gridiron pendulum, the invention of the celebrated John Harrison, is represented in fig. 6, Plate CCCCIII., and ordinarily consists of five rods of steel and four of brass, placed in alternate order, the former being shaded darker for distinction; but since eight of these act in pairs for the sake of steadiness, there are in effect only three rods of steel and two of brass. Their mode of action will be readily understood by tracing the expansion of the rods in the order of their connection; 1st, tracing down from the cross bar attached to the suspension, the two outer rods of steel, which expand downward, and act together as one rod only; 2d, tracing up the two brass rods next on the inside of these, which expand upward, and likewise act together; 3d, down the two steel rods inside of the last, which expand downward; 4th, up the two rods of brass next inside of these, which expand upward; and, lastly, down the middle rod of steel, which carries the bob, and expands downward. None of the cross pieces have any hold of the middle rod except the very short one at its upper end. The other cross pieces at some distance from the ends are double, two being on each side of the rods; but they are only fixed to the two outer rods to keep the whole system more evenly. In this arrangement, the expansion of all the steel rods is downward, and that of all the brass upward. It is necessary that the expansion of the latter metal should be much the greater, in order to compensate for there being one rod more of steel than of brass, and for the parts of steel which reach above and below the brass. But the precise compensation required is so uncertain, that the length of the rods can only be previously estimated approximately, and must afterwards be adjusted by trial if necessary. This, when well executed, is a more perfect compensation than the mercurial pendulum; because the average temperatures of the parts whose expansions are opposed to each other must be almost exactly the same. The number, order, length, and thickness of the bars often differ considerably from the above. Thus, it is evident that the whole gridiron part now described might be inverted without altering the effect. Sometimes the gridiron consists of a numerous assemblage of very short rods at the lower end of the pendulum, where they are liable to have a different temperature from that of the principal rod, which they are meant to compensate, and which has its greater part far above them. Berthoud would have liked to have deprived Harrison of the honour of inventing the gridiron pendulum, because he did not publish an account of it till 1767. But it was known to the public almost from the time it was made in 1726; and Short has described it in the Philosophical Transactions for 1751.

Graham had proposed to increase the effect of the difference between the expansions of two metals by means of levers. This was put in practice by Ellicot in a pendulum described by him in the Philosophical Transactions for 1751. Many other pendulums have been made on this principle; but, unless great care be taken to avoid friction, they are liable either not to act at all, or to move by starts in a very uncertain manner. Some pendulums are partly composed of tubes, and these generally differ only in form from those of the gridiron construction. The following is perhaps amongst the best, as well as the simplest, of the tubular sort in use. A rod of steel attached to and descending from the suspension is covered by a tube of zinc, which rests on a shoulder or nut on the bottom of the rod; over this again is put a tube of iron or steel, having a contraction or flat ring at its upper end, to rest or hang upon the top of the zinc; and this outer tube carries at its lower end the bob or weight. This pendulum needs no farther description, being essentially the same as that of Ward, only the parts are not so equally exposed to changes of temperature. Sometimes lead is used in place of zinc, and rods of steel instead of the outer tube.

Smeaton's pendulum has a solid glass rod, covered to the extent of a foot by a tube of zinc, which rests on a nut or shoulder at the bottom of the rod, and expands upward. This is covered by a tube of iron also a foot long, hooking upon the top of the zinc, and expanding downward. The iron again has a shoulder at the bottom, on which rests a massy tube of lead incasing the other two, and expanding upward. These three tubes therefore form both the bob and the compensation. This construction is neither difficult nor expensive, but the parts whose expansions are opposed to each other are, like those of some of the pendulums already described, liable to be of somewhat different temperatures.

Mr Adam Reid's compensation has a very long central rod of steel descending considerably below the bob, and has this lower part covered by a zinc tube. The bottom of the zinc rests on a nut fixed on the lower end of the rod, while the bob, again, rests on the top of the zinc tube. Hence, if the zinc, by expanding upward, raise the centre of oscillation as much as the downward expansion of the long steel rod depresses it, a compensation will be effected. The length of this pendulum exposes it to much resistance from the air; and the zinc being so low down, is apt to differ materially in temperature from the steel rod. Captain Kater's pendulum of 1808, noticed below, is older than this, and only differs from it in having a wooden rod.

Somewhat allied to the last is the very home-made kind of pendulum consisting of a wooden rod partly covered by a leaden tube, and which has often received far more than its due share of commendation. Pendulums with wooden rods are described by Captain Kater, in Nicholson's Journal for 1808, and in the volume on Mechanics in Lardner's Cyclopaedia; by Colonel Beaufort, in the Annals of Philosophy for March 1820; by Mr Baily, in vol. i. Memoirs of Astronomical Society; by Mr Squire, in Philosophical Magazine for January 1825 and July 1827; and by Professor Stevelly, in the Reports of the British Association (vol. v.). This last, though professedly written in refutation of the rules given by Captain Kater and Mr Baily for the construction of compensating pendulums, is itself far from being correct; for Mr Stevelly only takes into account the expansion of the wood in so far as it affects the position of the lead, and quite overlooks it as affecting the position of its own mass; so that he provides no compensation for the wooden part of his pendulum, although he is to have such a mass of wood at the lower end of the rod that the wooden part alone is to vibrate in a second when there is no lead on it at all.

Mr Baily, probably including the suspension spring, makes the rod of deal 45-75 inches long and 0-375 in diameter; the tube of lead 14-3 inches long and from 1-25 to 2-25 inches diameter outside, with a bore just sufficient to hold the rod. Mr Stevelly uses a suspension spring two inches long; a deal-rod 44-995 inches long and 0-6 diameter; a leaden tube 16-965 inches long and 1-5 diameter outside, with a bore to fit the rod. But, without endeavouring to correct either of these rules, the safe way, we presume, is to have the tube at first too long, and then gradually to reduce it by repeated trials, because the expansion of the wood is so very uncertain. Mr Stevelly considers a bracket of wood firmly fixed on the lower end of the rod to be greatly preferable to a nut and screw for the tube to rest upon; and, to regulate the clock, he varies the length of the suspension spring in a very ingenious manner, by means of a micrometer screw, which works a nut and screw at the top of the pendulum. But of this part of the scheme we do not approve, no matter to what sort of pendulum it were applied. For we are much of the opinion which has often been advanced, that the suspension spring ought to be so contrived as to equalize the times of the wider and narrower vibrations, a great desideratum for every nice pendulum, as will be noticed more particularly afterwards. The proper length of such a spring could only be ascertained by actual trials; and if that were done, the length could not afterwards be varied without disturbing the isochronism of the vibrations. The like may be said of all those compensations whose effect depends on their varying the length of the spring, as is the case with those of Deparcieux, Julien Leroy, Fordyce, Wynn, and many others.

Although the effects of moisture in altering the dimensions and weight of the wood may be in a great measure obviated by baking, and by impregnating and coating it with certain substances, yet these again tend to increase its expansion by heat. But another objection, which applies to every pendulum in which the bob acts as the compensation, is, that the middle of the bob being far below the middle of the rod, their temperatures are apt to differ materially; and especially in this case, where the temperature of the wood must change much more slowly than that of the metal.

Bars composed of plates of two different metals, as, for instance, brass and steel soldered or riveted together, have also been used in the construction of various compensations. Thus, a bar of this sort being placed across the pendulum rod, and having the more expansible metal on the under side, carries a weight at each end. The consequence is, that a rise of temperature tends to raise the ends of the bar, together with the weights, and of course to raise the centre of oscillation; and a fall of temperature has just the opposite effect. If the bar is thin, it is apt to be in a state of perpetual tremor; and if so thick as to obviate this, it scarcely yields under the variations of temperature. Other forms of this sort of compensation are generally very complicated.

There are many other compensating pendulums, but the most of them are either the same in principle with those we have described, or they fall under some of the objections stated in the course of this article. We shall, therefore, only add a description of the following one, which we presume to be new, and free from the objections just alluded to.

If an elastic hoop of metal in the figure of an ellipse have a force applied to it, as it were, slightly to alter the length of either axis, the length of the other axis will at the same time undergo a contrary change; and the variations of these axes will be to each other in the inverse ratio of the axes themselves. Thus, if the greater axis AC were nine inches long, and a force were applied to increase it to 9-01, the smaller axis BD, if three inches, would be reduced to 2-97; or if AC were reduced to 8-99, BD would be increased to 3-03. This property we have found by trial upon a large scale to be either accurately or very nearly true, for minute variations of the axes. Whether it admits of demonstration is not of much consequence to our present pur- Indeed, the probability of forming a hoop of an accurately elliptical figure is not very great, and as little is the chance that it would be perfectly uniform in its stiffness. Without these conditions it would be nearly useless to attempt any demonstration.

In the annexed outline of a compensation pendulum on this principle, let the elliptic hoop A B C D be of the dimensions above stated, and of steel, while the horizontal rod A C, occupying the position of the longer axis, is of zinc; since the expansions of these metals are nearly as one to 27, the excess of the expansion of the zinc rod of nine inches over that of an equal length of steel would still compensate $9 \times 1.7 = 15.3$ inches of steel. But, by the property just described, the corresponding excess of the contraction of the shorter axis B D in the vertical direction over its own expansion, being equal to three times the expansion of those 15.3 inches of steel, will compensate 45.9 inches of the steel pendulum rod E F attached in two pieces E B, D F to the opposite sides of the hoop.

This rough computation will, we presume, be sufficient to show the efficiency of such a mode of compensation, in which, if the hoop be only pressed against the rounded ends of the zinc rod by the weight of the pendulum, there need be no friction, and consequently no moving by jerks or starts, whilst the temperature is changing. It is, besides, a great recommendation to this compensation, that the hoop will readily admit of being so placed in the pendulum rod as to have the average temperature of its different parts. The construction will be easy, and attended with little expense. Perhaps, in place of a single hoop, it might in some cases be preferable to use two of a smaller and less eccentric figure, and even to employ other metals. But since the greater the eccentricity of the ellipse, the greater, within proper limits, will be the compensating effect; and therefore, by putting in a longer or shorter rod of zinc, the effect may be easily adjusted. The compensation might be still more readily regulated by forming the zinc rod in two pieces, screwing together in the middle, and thus being capable of lengthening and shortening, like the common contrivance for stretching a hat. The length of the zinc rod, when so adjusted, together with the weight of the pendulum, will readily bring and keep the hoop to the proper eccentricity, unless it has originally been very far from it, in which case it must be altered. It would, however, be an advantage that the hoop should be rather longer than the zinc, in order to lessen the pressure on the ends of the rod, when brought to bear against it by the weight of the bob. But the ends of the rod might be capped with steel or other hard substance, which could both protect the zinc and lessen any chance of friction, by presenting a sharper point to bear against the hoop. It is evident that this principle of compensation is equally applicable to the balances of chronometers. But we cannot here enter into the details of the construction.

The expansion of one material cannot be expected accurately to compensate that of another, unless they either proceed part passu everywhere alike exposed for temperature, or have the average temperatures of their several parts exactly the same; or at least have these parts so arranged that they may with good reason be expected to produce the same effect as if the whole apparatus underwent the same change of temperature. Now, the following considerations render it more than probable that many compensations are liable to this objection, and that they are therefore unfit for the purpose. Generally speaking, the external air is colder as the height is greater, but the reverse is more commonly the case with confined air, or that within doors; so that a thermometer near the floor often indicates a temperature lower by several degrees than another near the ceiling; and this difference is said to be greater as the weather is more inclined to rain. In an upper room the ceiling is liable to great fluctuations of temperature from the vicissitudes of the weather, and, no doubt, may occasionally be much colder than the floor. But at any rate, it is found that, even in confined air, a difference of thirty-nine inches in height, which is far short of the length of many a pendulum rod, is often attended with a sensible difference of temperature. Thus, on examining the extensive tables of Mr Baily's experiments in the Philosophical Transactions for 1832, we find that he had placed two thermometers within his vacuum apparatus, the one being at the axis of suspension of the pendulum, and the other on a level with the centre of oscillation, and that they frequently differ by half a degree and more, though the apartment was one of an uncommonly steady temperature. Generally the upper thermometer was at the higher temperature, though sometimes the reverse. In some cases the air was of the full pressure, in others nearly exhausted. There is likewise reason to fear, that a difference frequently exists between the temperature of a pendulum rod and that of the parts compensating it, when the latter are separate from the pendulum and at rest, and especially when they are at some distance from it, with perhaps some board, stone, or partition intervening. When a tube encloses one or more rods or tubes, it may be expected to partake sooner of any change in the air's temperature than the parts within; and still more will this be the case where several tubes successively incase one another. Besides, different materials differ greatly in the promptitude with which they acquire a change of temperature. Marble, mica, glass, and even earthenware, have sometimes been used for pendulum rods. But such substances, if slender, are extremely liable to be broken; and if thick, they are apt, from their being very bad conductors of heat, to take longer time in changing their temperatures than the other parts of the pendulum which are to form the compensation; hence, none of these materials can be compared with the metals. It is likewise to be feared that some of these will be liable to have their weights continually varying with the changes of humidity. Still more objectionable are vegetable or animal materials, owing to their changeable texture, and the perpetually varying effects of heat and moisture in altering their dimensions and weights in an uncertain manner.

Sudden alternations in the temperature of a pendulum might surely be in a great measure prevented by making the clock-case of double boards, with a small space between them, which might be either left empty, or filled with some bad conductor of heat. But we very much doubt if this, or any other simple method, could insure, that parts which are at a distance and on different levels, as, for instance, that both ends of a pendulum, would have the same temperature, the want of which is a serious objection to many a compensation, not excepting the mercurial pendulum. In all cases in which the parts of the pendulum slide or turn with friction under considerable pressure, the compensation is apt either not to act at all, or to be too long in moving, and then to go too far by a start or jerk.

We have noticed above, and also in the article Clock-Engraving, that the vibrations of a pendulum naturally occupy more time in the wider arcs of the same circle than in the narrower. But since the width of the vibrations is liable to be considerably affected by the state of the oil and the degree of foulness of the clock, it would be of great consequence that the rate of the clock should be rendered independent of the variable extent of the arc, because without this every compensation, whether for temperature or pressure, must at times fail to some extent, though it would only be worth the while to equalize the times in a clock having a dead beat, or detached escapement, or one whose rate is otherwise independent of any inconstant action of the wheels upon the pendulum. Smith, in his Horological Disquisitions, published in 1694, alleged that it is possible to form the spring of suspension such that it would equalize the times of wider and narrower vibrations. The same idea was afterwards advanced by Peter Leroy, in a memoir published in 1770; and again by Berthoud, in his *Supplement au Traité des Montres à Longitude*, 1807, where he says, "la suspension à ressort bien construite tend à rendre isochrones les oscillations du pendule." In the Philosophical Magazine for April 1833, Mr Scrymgeour has given an account of his trying this with success upon the pendulums of two clocks, both of which had dead-beat escapements. The one pendulum had a wooden rod and a common lenticular bob of four pounds, with a brass ball under it of half a pound. The suspending spring was originally about three fourths of an inch long, and of a middling strength. It was then in effect shortened by fixing on each side of it a piece of steel, which was also joined to the rod. Mr Scrymgeour found, that when the length of the acting part of the spring was by this means reduced to about an eighth of an inch, the time in an arc of $3^\circ$ did not differ more than half a second in twenty-four hours from that in an arc of $5^\circ$. The other pendulum was a mercurial one, with a suspending spring nearly half an inch broad and .0083 thick. A moveable clamp was made to fasten near the top of the rod, so as to be shifted up and down at pleasure, and to clasp the spring. The times in different arcs were found to be equalized when the length of the acting part of the spring was reduced to about one twentieth of an inch. After both clocks had been kept going for a considerable time, they had lost in their rate, which Mr Scrymgeour ascribes to the gradual weakening of the spring. In this he was probably right; and if so, it seems also to show that the springs were too soft, otherwise they would rather have broken than shown any symptoms of losing their force. The usual and needless practice of softening the ends of the spring, that it may be more easily pierced for the rivets, was very likely the cause of this. However, the springs may likewise have at length acquired a little play in the slits in which they were held. For, after all, we suspect that this property of isochronism in so short a spring must have been in a great measure owing to its coming to act more closely against the corners of both slits, when the pendulum vibrated in larger arcs than in smaller.

A much longer and stiffer spring, when curved into an arc of $2^\circ$, or of half a vibration, might tend to shorten the pendulum, though by an almost inappreciable quantity; but this seems nearly out of the question with a short slender spring. Neither does Hooke's law of the force of a spring being as the tension at all account for it.

We have sometimes thought, that if two pieces of a slender spring were laid together, and firmly held at both ends, the compound spring so formed would have the property of transgressing Hooke's law, so as to have its force increasing in a higher ratio than the tension. For, whilst bending at first very slightly, the one part of this compound spring would not sensibly interfere with the action of the other, and so the force of both would be only double that of either singly; but as the curvature increased, the one part would, as it were, become too long for the other, and so their joint force would be more than double that of either separately. In this way the force would increase faster than the arc of vibration, which is just the thing wanted. Such a spring would obviously break or be wrinkled useless, if it were forced through a large arc, but would safely bear to be bent through arcs more than sufficiently large for the purposes of the pendulum.

As some clocks, in place of altering the bob, are regulated by having a small weight to shift along the pendulum rod, which was a contrivance of Huygens, it may be useful here to show, that the effect of such regulating weight to accelerate the clock is greatest when it is exactly half-way between the axis of suspension and the centre of oscillation; because, unless this is kept in view, the shifting the weight upward may really make the clock go slower instead of faster, and vice versa. Let $A$ be the sum of the products of all the particles of the pendulum, except those of the moveable weight, into the squares of their distances respectively from the axis of suspension, and let $B$ be the sum of all the products of the same particles into their simple distances from that axis; also, let $C$ be the moveable weight at the variable distance $x$ from the axis of suspension. Since the greatest acceleration will occur when $y$, the effective length of the pendulum, is a minimum, and since, by the well-known property of the pendulum, $y = \frac{A + Cx^2}{B + Cx}$, its fluxion will then be $= 0$; or

$$2(B + Cx)Cx - (A + Cx^2)Cdx = 0.$$ From this we obtain $x = \pm \sqrt{\left(\frac{B}{C}\right)^2 + \frac{A}{C}} - \frac{B}{C}$. Substituting now the positive value of $x$ in the above expression for $y$, and reducing, it will be found that the value of $y$ is just double that of $x$, or that $x = \frac{1}{2}y$. This question is treated in a very different manner, and at great length, in the *Connaissance des Temps* for 1817, by Baron de Prony, who recommends this mode of regulating, because the small weight admits of being shifted over a much more appreciable space than the bob does, and because it may be placed so high on the rod as to be readily shifted without stopping the clock. If placed above the middle of the effective length of the whole pendulum, it must be moved downward to make the clock go faster, and upward to make it go slower.

Mr Browne's mode of regulating consists in having some convenient surface so situated on the pendulum, that crooked bits of lead may either be readily put on it or removed, without disturbing the motion of the clock.

The British government has lately granted a sum of money towards defraying the expenses of repeating Ca-for determinish's experiment for determining the mean density of mining the earth, and the apparatus is in the course of being constructed. But we have often thought that in this case (as well as in that proposed under the article CLOCK-WORK, vol. vi., p. 784, for determining the velocity of sound), the application of clock-work might furnish a more systematic mode of settling the question. If two accurate clocks, perfectly equal in every respect, and provided with very long pendulums, but without any compensations, were adjusted to the same rate; then, by placing one or more great masses of lead near to the ball of the one pendulum, the effect of this on its rate, if at all appreciable, would be very accurately pointed out by the difference in the times shown by the two clocks. The spaces in which the two pendulums moved could still be kept exactly equal, by placing equally near the other pendulum patterns of the lead, but formed of some light substance. In this way, with proper precaution to have both clocks at the same, though perhaps a variable, temperature, no corrections for temperature or pressure would be required, and the two clocks, left to themselves, would repeat the experiment thousands of times, and register the amount. By making the ball of the pendulum vibrate through a massy ring of lead placed vertically, with a slit at the top to make way for the rod, the clock should be accelerated. This acceleration, together with the extent of the vibrations, and the masses and dimensions of the ring and pendulum, would furnish data for computing the attraction and mass of the earth in terms of those of the ring. If the lower end of the pendulum rod were formed into a sufficiently large loop linking into the ring, it might move clear of it without the lead having any slit. But there are various other forms in which the lead might be presented. to the pendulum, either to accelerate or retard it. Nay, masses might be so placed as to accelerate the one clock and retard the other.

Hardy's inverted pendulum, represented in fig. 7, Plate CCCCIII. is similar to an ordinary clock pendulum having a spring suspension, but only inverted. It is also furnished with a weight, which can be shifted along the rod, to render its vibrations of any required duration. This instrument has been much used by Captain Kater and others as a test for the steadiness of the supports on which their pendulums were swung; because its sensibility was believed to be such, that the slightest vibration of any thing on which it stood would set it a nodding. But the following consideration has led us to doubt its extreme sensibility, and therefore also to question whether the discrepancies in some pendulum experiments may not have been owing to the unbounded confidence reposed in this instrument. The spring which supports the rod, and connects it with the pedestal, must be incomparably more stiff in proportion to the mass of the pendulum, than is usual with springs of suspension; otherwise this pendulum would to a certainty lean over to one side, or fall down altogether. Such a stiff spring must therefore render the instrument quite insensible to any very slight vibrations of the article Penisa on which it is set. Another defect is, that the force of gravity acts upon it in the most disadvantageous manner, and makes the durations of the vibrations increase prodigiously with their width. From such considerations, we should be disposed to reject this form of instrument altogether, as not being a trustworthy test for the steadiness of supports. All we would retain of it is the mere frame, from the top of which, if a leaden ball were suspended by a thread or fine wire, it would be possessed of far greater sensibility, would have its wide and narrow vibrations nearly isochronous, and would join in or comply with the vibrations of the support, whatever might be their direction; whereas Hardy's instrument can only act in a single plane. The thread, too, could most readily have its length altered, so as to fit it for commencing its vibrations in accordance with those which might be suspected in any support.

In addition to the various references and authorities which have been mentioned in the course of this article, and in that on Clock-work, we beg to refer to the different scientific periodicals, and to the accounts of the various expeditions which have been more recently undertaken for purposes of scientific investigation.

(P.E.E.)