PENDULUM

P E N D U L U M.

Pendulum. 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 any 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 reascends 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.

Pendulum. 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 $l$ (Plate CIX. fig. 1) denote the ideal length of such a pendulum, which is called a simple pendulum. Let $m$ be the mass of an oscillating body 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 $l\; =\; \backslash frac\{C\}\{mh^2\}$, and be-

sides this, that the lines $SP$ and $SG$ have at any one instant an angular velocity equal 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 be always 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 $ZP$ (Plate CIX. fig. 2) is half the extent of the oscillations round 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 $\backslash alpha$

Pendulum. 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 $PZP\text{'}$ or $PP\text{'}$ will be expressed by the following series:

$\backslash pi$ 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 $V$, $\backslash theta$ being its angular distance from the vertical, we have

$$(2.)\; V^2\; =\; 2gl\; (\backslash cos\; \backslash theta\; -\; \backslash cos\; \backslash alpha)$$

or, what comes to the same thing,

$$V^2\; =\; 4gl\; \backslash sin\; \backslash frac\{1\}\{2\}\; (\backslash alpha\; +\; \backslash theta)\; \backslash sin\; \backslash frac\{1\}\{2\}\; (\backslash alpha\; -\; \backslash theta).$$

These formulæ will still serve if the pendulum, instead of falling freely from the extremity of the arc, receives there an initial velocity expressed by $V\text{'}$, 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, $\backslash alpha\text{'}$. Then it will be necessary that this unknown distance $\backslash alpha\text{'}$ satisfy instead of $\backslash alpha$, the general equation of the velocities, and that $\backslash alpha$ becomes in it $\backslash theta$, which gives

$$(3.)\; V\text{'}^2\; =\; 2gl\; \backslash left\backslash \{\; \backslash cos\; \backslash alpha\; -\; \backslash cos\; \backslash alpha\text{'}\; \backslash right\backslash \}$$

The half amplitude $\backslash alpha\text{'}$ 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 $\backslash alpha\text{'}$ instead of $\backslash alpha$. This same substitution made in the equation (2) will give

$$V^2\; =\; 2gl\; (\backslash cos\; \backslash theta\; -\; \backslash cos\; \backslash alpha\text{'})$$

for the velocity in any point whatever of the oscillation. But these transformations are only possible when the equation (3) gives for $\backslash alpha\text{'}$ a real arc, and consequently for $\backslash cos\; \backslash alpha\text{'}$ a value comprehended between +1 and -1. We may easily conceive, that, when the $\backslash cos\; \backslash alpha\text{'}$ exceeds these limits, it is because the velocity of impulse $V\text{'}$ 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 formulæ (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 $\backslash sin^2\; \backslash frac\{1\}\{2\}\; \backslash alpha$, compared with the unity which precedes

them, we will have simply $T\; =\; \backslash pi\; \backslash sqrt\{\backslash frac\{l\}\{g\}\}$.

Pendulum. Now, the angle $\backslash alpha$ 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 duration.

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 $\backslash alpha$ 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 $\backslash sin\; \backslash frac\{1\}\{2\}\; \backslash alpha$. We may then, in the same order of approximation, substitute $\backslash frac\{1\}\{2\}\; \backslash sin^2\; \backslash alpha$ for $\backslash sin^2\; \backslash frac\{1\}\{2\}\; \backslash alpha$; and the series (1) being thus limited to its two first terms, gives

$$T\; =\; \backslash pi\; \backslash sqrt\{\backslash frac\{l\}\{g\}\}\; \backslash left\backslash \{\; 1\; +\; \backslash frac\{1\}\{16\}\; \backslash sin^2\; \backslash alpha\; \backslash right\backslash \}$$

$\backslash alpha$ 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

$$l\; =\; \backslash frac\{c\}\{mh\},$$

$c$ being the momentum of inertia of the mass $m$, relatively to the axis of suspension. But if we call $c\text{'}$ 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 $c,\; c\text{'}$ have between them the following relation:

$$c\; =\; mh^2\; +\; c\text{'}.$$

This value of $c$ being substituted in the expression of $l$, gives evidently

$$l\; =\; h\; +\; \backslash frac\{c\text{'}\}\{mh\}.$$

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 $h\text{'}$ and $h\text{'}\text{'}$, 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 suspen-

Pendulum. sion 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 $c^2$, relatively to this axis, can be expressed in the following manner:

$c^2\; =\; mh^2\; +\; A\; \backslash cos^2\; X\; +\; B\; \backslash cos^2\; Y\; +\; C\; \backslash cos^2\; Z$; and this value being substituted in $l$, instead of the letter $c$, gives

$$l\; =\; h\; +\; \backslash frac\{A\; \backslash cos^2\; X\; +\; B\; \backslash cos^2\; Y\; +\; C\; \backslash cos^2\; Z\}\{mh\}$$

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 $x,\; y$, and $z$, 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

$$(4.)\; \backslash quad\; x\; =\; az\; +\; \backslash alpha,\; \backslash quad\; y\; =\; bz\; +\; \backslash beta,$$

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

$$\backslash cos\; X\; =\; \backslash frac\{a\}\{\backslash sqrt\{1+a^2+b^2\}\},\; \backslash quad\; \backslash cos\; Y\; =\; \backslash frac\{b\}\{\backslash sqrt\{1+a^2+b^2\}\}$$

$$\backslash cos\; Z\; =\; \backslash frac\{1\}\{\backslash sqrt\{1+a^2+b^2\}\}\; \backslash text\{\; and,\; lastly,\; \}\; h\; =\; \backslash frac\{\backslash sqrt\{a^2+b^2\}\}\{\backslash sqrt\{1+a^2+b^2\}\}.$$

By substituting these values in the general expression of $l$, it becomes

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

By supposing $l$ constant, this relation, combined with the equation (4), will characterise the isochronous axes: but as this combination only furnishes three equations, while there are four constant indeterminate quantities $a,\; b,\; \backslash alpha,\; \backslash beta$, in the position of the axis, it hence appears that we may still assume at pleasure an additional condition among the quantities themselves, after which, by eliminating them, we shall have in $x,\; y,\; z$, 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 Clair-ault.

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 duration 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 of 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

Pendulum. 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'Ecole 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, independent 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, $\backslash Delta$ 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 Pendulum. weight of the air displaced by the body will be

$P\; \backslash frac\{1\}\{\backslash Delta\}$, thus the apparent weight of this same body,

during its oscillations, will be $P\; -\; \backslash frac\{P\}\{\backslash Delta\}$, so that it

will be to its absolute weight as $1\; -\; \backslash frac\{1\}\{\backslash Delta\}$ 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 metres in height, $D$ represents the density of the substance of the pendulum, that of the air being taken as unity. 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

$$\backslash frac\{p\}\{0.76(1+t.0.00375)\}$$

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 $g\text{'}$, we shall have

$$g\text{'}\; =\; g\; \backslash left\backslash \{\; 1\; -\; \backslash frac\{p\}\{0.76(1+t.0.00375)(1-ct)D\}\; \backslash right\backslash \}$$

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

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 $2\backslash alpha$, we shall have,

If now we wish the two pendulums to oscillate with equal amplitudes, we have only to make $\backslash alpha\; =\; \backslash alpha\text{'}$; if we wish, also, to have their times of oscillation equal, we have only further to suppose $T\text{'}\; =\; T$, then the two preceding expressions being equal to each

other, we obtain $\backslash frac\{l\}\{g\}\; =\; \backslash frac\{l\text{'}\}\{g\text{'}\}$, and $l\; =\; l\text{'}\; \backslash frac\{g\}\{g\text{'}\}$, from which we

can calculate $l$, when we know from observation $l\text{'}$ and $\backslash frac\{g\}\{g\text{'}\}$.

Pendulum. 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 possi-

ble, and that they may always obey, with equal facility, to the intermitting impressions of the pendulum. 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 dilations 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, is 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 of 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

Pendulum. 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 Systeme Metrique 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 been since 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 platina, suspended to a metal wire. (Plate CIX. 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 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 inclosed 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 arc 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 24 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,

Pendulum. the pendulum makes $N\; \backslash pm\; 2$. 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 $J\; \backslash left\backslash \{\; \backslash frac\{N\; \backslash pm\; 2\}\{N\}\; \backslash right\backslash \}$ or $J\; \backslash pm\; \backslash frac\{2J\}\{N\}$, a result which we may represent in an abridged form by $J\; \backslash pm\; n$.

If the clock be sexagesimal, the number $J$ of its beats in 24 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 $\backslash pm\; \backslash frac\{2J\}\{N\}$, 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 $\backslash frac\{2J\}\{N\}$ 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 $2\backslash alpha$ at the beginning of the interval to $2\backslash alpha\text{'}$ 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

Pendulum. 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 $\backslash alpha$ and $\backslash alpha\text{'}$ 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 $\backslash alpha\; +\; \backslash alpha\text{'}$. Then, after what has been shown above, each of them expressed in oscillations infinitely small, will be

equal to $1\; +\; \backslash frac\{1\}\{16\}\; \backslash sin^2(\backslash alpha\; +\; \backslash alpha\text{'})$, which we may express in

an abridged form by $1\; +\; \backslash mu$, 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

$$(J\; +\; n)(1\; +\; \backslash mu)\; \backslash text\{\; or\; \}\; J\; +\; n\; +\; \backslash mu(J\; +\; n)$$

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

If the arcs $2\backslash alpha,\; 2\backslash alpha\text{'}$ 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 $\backslash alpha\; +\; \backslash alpha\text{'}$. 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 $\backslash alpha$, and represent by $\backslash alpha\_n$ the amplitude, which

takes place after $n$ oscillations, we find $\backslash alpha\_n\; =\; \backslash frac\{\backslash alpha\}\{K^n\}$ or because $\backslash alpha$ and $\backslash alpha\_n$ are supposed very small $\backslash sin\; \backslash alpha\_n\; =\; \backslash frac\{\backslash sin\; \backslash alpha\}\{K^n\}$. $K$ 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 $\backslash frac\{1\}\{K\}$. 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

$$\backslash mu\; =\; \backslash frac\{\backslash sin(\backslash alpha\; +\; \backslash alpha\text{'})\; \backslash sin(\backslash alpha\; -\; \backslash alpha\text{'})\}\{32M\; \backslash left\backslash \{\; \backslash log\; \backslash sin\; \backslash alpha\; -\; \backslash log\; \backslash sin\; \backslash alpha\text{'}\; \backslash right\backslash \}\}$$

M being the modulus of the tables of common logarithms, or 2.30258509. If we suppose the arcs $\backslash alpha$ and $\backslash alpha\text{'}$ so small, that in the development of their series and of their logarithms, we may limit ourselves to their first power, this expression of $\backslash mu$ 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 $J+n\text{'}$ 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 $J+n\text{'}$ 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 $\backslash frac\{(J+n\text{'})(J+h)\}\{J\}$ or $J+n\text{'}+h+\backslash frac\{hn\text{'}\}\{J\}$,

the quantity which, for simplicity, we shall represent by $J+n\text{'}$. 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 platina. 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. (See Plate GIX. fig. 6.) 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 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. (See fig. 7.) 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

* 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. 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 entrusted 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 platina; 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 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 $\backslash Lambda$, and suppose $t\text{'}$ 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 $R\text{'}$ represent the radius of the ball of platina at the temperature of $t\text{'}$ when the contact was produced, the length of the wire at that instant was $\backslash Lambda\; -\; 2R\text{'}$; so that, calling $K$

the lineal dilatation of the matter of the wire, for a Pendulum. difference of one degree in the temperature, the length of the wire at the time of the oscillations must have been $(\backslash Lambda\; -\; 2R\text{'})\; \backslash \{\; 1\; +\; K(t\; -\; t\text{'})\; \backslash \}$. In the same manner, if $K\text{'}$ be the lineal dilatation of the substance of the ball, its diameter, at the time of the oscillations, will be $2R\text{'}\; \backslash \{\; 1\; +\; K\text{'}(t\; -\; t\text{'})\; \backslash \}$, and adding this quantity to the length of the wire, we obtain $\backslash Lambda\; +\; \backslash Lambda\; K(t\; -\; t\text{'})\; +\; 2R\text{'}(K\text{'}\; -\; K)(t\; -\; t\text{'})$ for the distance of the plane of suspension from the bottom of the platina 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, $R\text{'}\; \backslash \{\; 1\; +\; K\text{'}(t\; -\; t\text{'})\; \backslash \}$, 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 $l\; =\; h\; +\; \backslash frac\{c\text{'}\}\{m\; h\}\; m$

being the mass of the ball, and $c\text{'}$ its momentum of inertia relative to an axis drawn through its centre. But calling $\backslash rho$ 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 $\backslash frac\{4\}\{3\}\; \backslash pi\; \backslash rho\; R^3$, and

the value of $c\text{'}$ is $\backslash frac\{8\}\{15\}\; \backslash pi\; \backslash rho\; R^5$. Substituting these values, we have $l\; =\; h\; +\; \backslash frac\{2R^2\}\{5h\}$; 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 $l\text{'}\; =\; h\; +\; \backslash frac\{2R^2\}\{5h\}\; -\; Q$. 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 $\backslash pi\; \backslash sqrt\{\backslash frac\{l\}\{g\}\}$; and if we wish to obtain oscillations of equal duration with different forces of gravity, we must vary the lengths in proportion

Pendulum. to these forces, so that the relation $\backslash frac\{l\}\{g\}$ 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 $0^\{\backslash text\{th\}\}.76$, that of the air being 1, if, besides, we denote by $c$ 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

$$\backslash text\{expressed\; by\; \}\; 1\; -\; \backslash frac\{p\}\{0^\{\backslash text\{th\}\}.76(1+t.000375)(1-t)D\},$$

which, for simplicity, may be represented by $1-\backslash gamma$. Then, to obtain the length $l\text{'}$ 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 $l\text{'}\; =\; \backslash frac\{l\}\{1-\backslash gamma\}$, which, on account of the

smallness of $\backslash gamma$, may be reduced to $l\text{'}\; =\; l\; +\; \backslash gamma\; l$. 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 $l\text{'}$. If we wish, in fine, to obtain the length $l\text{'}\text{'}$ of a pendulum which would move exactly to mean time in vacuo, under the influence of the same power of gravity as $l\text{'}$, we have only to consider that, according to the preceding expression of $T$, the lengths $l\text{'}\text{'}$ 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,

$$l\text{'}\text{'}\; =\; \backslash frac\{(J+n^2)^2\}\{J^2\},\; \backslash text\{\; whence\; we\; obtain\; \}\; l\text{'}\text{'}\; =\; l\text{'}\; +\; \backslash frac\{2n^2l\text{'}\}\{J\}\; +\; \backslash frac\{n^4l\text{'}^2\}\{J^2\}.$$

The length $l\text{'}\text{'}$ 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 platina 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 de-

composed 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'Ecole 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 neutralises 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 lineal 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 platina ball at its lower extremity, it lengthens by a small quantity $\backslash lambda$. This being the case, if the half-amplitude of the oscillation is denoted by $\backslash alpha$, M. Poisson finds that limiting the results to the square of $\backslash alpha$, the mean duration $T$ of the whole oscillations will be,

$$T\; =\; \backslash pi\; \backslash sqrt\{\backslash frac\{l\}\{g\}\}\; \backslash left\backslash \{\; 1\; +\; \backslash frac\{1\}\{16\}\; \backslash left(\; 1\; +\; \backslash frac\{11\backslash lambda\}\{l\}\; \backslash right)\; \backslash sin^2\; \backslash alpha\; \backslash right\backslash \}.$$

If we wish to suppose the wire inextensible, we have only to make $\backslash lambda$ 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 platina 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 Connaissance 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 platina ball, $h$ the distance of its centre from the axis of suspension, $\backslash alpha$ 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;

Pendulum. M. Poisson finds that the real duration of the oscillations will be $T\; \backslash left\backslash \{\; 1\; -\; \backslash frac\{\backslash alpha^2\; R^4\}\{50h^4\; \backslash pi^2\}\; \backslash right\backslash \}$, $\backslash pi$ being always as before the relation of the circumference to the diameter, 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 $0^m.02$, and $l$ nearly equal to $0^m.74$. 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 $\backslash frac\{R\}\{h\}\; =\; \backslash frac\{1\}\{37\}$, $\backslash frac\{\backslash alpha\}\{\backslash pi\}\; =\; 4$, and consequently $\backslash frac\{\backslash alpha^2\; R^4\}\{50h^4\; \backslash pi^2\}\; =\; \backslash frac\{1\}\{5856569\}$, 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 condition 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 rectilinear 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 rectilinear 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

rectilinear axis, according to the oscillations observed. Pendulum. 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 variableness 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; now 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 12 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 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

Pendulum. 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 rectilinear bar of copper, 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 12ths of an inch, and a displacement of 12 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 fig. 8, Plate CIX. Fig. 9 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. La Place 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 sup-

Pendulum. position 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 $\backslash frac\{C\}\{mh\}$, $m$ being the mass of the oscillating body, $h$ 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 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, $T\text{'}$, $T\text{'}\text{'}$, of the oscillations of the same simple pendulum, whose length is $l$, are connected with the intensities, $g\text{'}$, $g\text{'}\text{'}$, of the gravitating force, which impels them by the following relations,

Pendulum.

$$T\text{'}^2\; =\; \backslash pi^2\; \backslash frac\{l\}\{g\text{'}\};\; \backslash quad\; T\text{'}\text{'}^2\; =\; \backslash pi^2\; \backslash frac\{l\}\{g\text{'}\text{'}\};\; \backslash quad\; \backslash text\{whence\; we\; obtain,\; \}\; \backslash frac\{g\text{'}\text{'}\}\{g\text{'}\}\; =\; \backslash frac\{T\text{'}^2\}\{T\text{'}\text{'}^2\},$$

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 $N\text{'}$, $N\text{'}\text{'}$, be these numbers, and $T$ the total time which corresponds with them, then

$$T\text{'}\; \backslash text\{\; will\; be\; equal\; to\; \}\; \backslash frac\{T\}\{N\text{'}\},\; \backslash text\{\; and\; \}\; T\text{'}\text{'}\; \backslash text\{\; equal\; to\; \}\; \backslash frac\{T\}\{N\text{'}\text{'}\};\; \backslash text\{\; so\; that,\}$$

by substituting their values, the preceding relation

$$\backslash text\{will\; become\; \}\; \backslash frac\{g\text{'}\text{'}\}\{g\text{'}\}\; =\; \backslash frac\{N\text{'}^2\}\{N\text{'}\text{'}^2\}.\; \backslash text\{\; By\; such\; experiments,\; then,\}$$

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 $\backslash lambda\text{'}$, $\backslash lambda\text{'}\text{'}$, be these unknown lengths, since the corresponding times of oscillation are each one second, we shall have by our general

$$\backslash text\{formulae\; \}\; 1\text{'}^2\; =\; \backslash frac\{\backslash pi^2\; \backslash lambda\text{'}\}\{g\text{'}\},\; \backslash quad\; 1\text{'}\text{'}^2\; =\; \backslash frac\{\backslash pi^2\; \backslash lambda\text{'}\text{'}\}\{g\text{'}\text{'}\},\; \backslash quad\; \backslash text\{whence\; we\; obtain\}$$

$$\backslash frac\{\backslash lambda\text{'}\text{'}\}\{\backslash lambda\text{'}\}\; =\; \backslash frac\{g\text{'}\text{'}\}\{g\text{'}\},\; \backslash quad\; \backslash text\{that\; is,\; that\; the\; lengths\; required\; are\; pro-$$

portional 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 $\backslash frac\{\backslash lambda\text{'}\text{'}\}\{\backslash lambda\text{'}\}\; =\; \backslash frac\{N\text{'}^2\}\{N\text{'}\text{'}^2\}$. Hence, when

$$\backslash frac\{\backslash lambda\text{'}\text{'}\}\{\backslash lambda\text{'}\}\; =\; \backslash frac\{N\text{'}^2\}\{N\text{'}\text{'}^2\}.$$

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 ensures the firmness of the connection, and we take

Pendulum. 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 $l\; =\; \backslash frac\{C\}\{mh\}$, which expresses the corre-

sponding 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 tempera-

ture, the quantity $\backslash frac\{C\}\{mh\}$ will vary according to this sim-

ple proportion. Hence, if we name $l$ the length of the simple pendulum, isochronous with the compound pendulum, when the latter is at the standard temperature $t$, and denote by $l\text{'}$ the analogous length when the temperature is $t\text{'}$, 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

$l\text{'}\; =\; l\; \backslash \{\; 1\; +\; K(t\text{'}\; -\; t)\; \backslash \}$. Suppose now the pendulum

$l\text{'}$ has made a number $N\text{'}$ 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 length of the pendulums which perform them, the square of the number sought will then be

$$\backslash frac\{N\text{'}^2\; l\text{'}\}\{l\},\; \backslash text\{\; or\; \}\; N\text{'}^2\; \backslash \{\; 1\; +\; K(t\text{'}\; -\; t)\; \backslash \}.$$

Hence we obtain for this number itself $N\text{'}\; \backslash sqrt\{1\; +\; K(t\text{'}\; -\; t)\}$; or reducing the radical into a series, and limiting it to the first power of $K$; $N\text{'}\; +\; \backslash frac\{1\}\{2\}\; N\text{'}\; K(t\text{'}\; -\; t)$. This approximation is always sufficient, because the co-

efficient $K$ of the lineal dilatation is always very small in solid bodies, and the difference $t\text{'}\; -\; t$ 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 $71^\{\backslash circ\}.6$. If we wish to reduce this experiment to the standard temperature of $62^\{\backslash circ\}$ degrees, which was adopted by Mr Kater, we shall have $N\text{'}\; =\; 86051.32$; $K\; =\; 0.00000982$; $t\text{'}\; -\; t\; =\; +9^\{\backslash circ\}.6$, whence we obtain, for the correction of the temperature, $\backslash frac\{1\}\{2\}\; N\text{'}\; K(t\text{'}\; -\; t)\; =\; +4^\{\backslash circ\}.04$, 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

Pendulum. oscillations expressed in the last column are reduced by calculation to the case 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 second pendulum at London, precisely at the same place, and in the same room, where he since made his compound pendulum 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 $\backslash lambda$, and $\backslash lambda\text{'}$, $\backslash lambda\text{'}\text{'}$ the analogous lengths for the two other stations of Leith and of Unst; also $N$, $N\text{'}$, $N\text{'}\text{'}$ the number of oscillations of the portable pendulum in these three stations, we shall have, according to the formulae

above laid down, $\backslash lambda\text{'}\; =\; \backslash lambda\; \backslash frac\{N^2\}\{N\text{'}^2\}$; $\backslash lambda\text{'}\text{'}\; =\; \backslash lambda\; \backslash frac\{N^2\}\{N\text{'}\text{'}^2\}$, 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 platina, 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 $\backslash frac\{1000\; \backslash cdot\; l\}\{39.37079\}$. 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, where Captain Kater has since gone.

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 thou-

sand parts of a millimetre, which is equal to $\backslash frac\{39\}\{1,000,000\}$ Pendulum. 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 give both, 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 horizontal 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,

Pendulum. and $T$ the total time which it takes to describe it, also $s$ 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

$$s\; =\; 2a\; \backslash sin^2\; \backslash left\backslash \{\; 90^\backslash circ\; \backslash frac\{t\}\{T\}\; \backslash right\backslash \}$$

so that, by representing the half-amplitude $a$ by 1000 parts, and supposing $t$ successively equal to $\backslash frac\{1\}\{10\}$, $\backslash frac\{2\}\{10\}$, $\backslash frac\{3\}\{10\}$, $\backslash frac\{4\}\{10\}$, $\backslash frac\{5\}\{10\}$ of $T$, or of the duration of a whole oscillation, we obtain for $s$ 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 $a$ at the period of the comparison with the chronometer; it gives also, at this instant, the value of the arc $s$, according to the division on the scale of amplitudes to which the stem of the pendulum corresponds. Dividing $s$ by $a$, 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 diurnal rate of going of $2^\{\backslash circ\}\; 28\text{'}\text{'}$ 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 di-

minution 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 intermittances 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 in 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 would 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 for 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 verge of copper, 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, $86394^\{\backslash circ\}.4$, during a revolution of the fixed stars, in describing arcs of $2^\{\backslash circ\}\; 10\text{'}$ on each side of the vertical, while, with a weight twice as small, it made $4\text{'}\text{'}$ more, that is,

Pendulum. 86398".4 in the same interval, describing arcs of $1^\{\backslash circ\}\; 15\text{'}$. Now, if we apply here the correction relative

to the amplitude of the arcs, which is $\backslash frac\{1\}\{16\}\; N\; \backslash sin^2\; \backslash alpha$, calling $N$ the number of oscillations, and $\backslash alpha$ the demi-amplitude, we shall then find, that in the first case it is necessary to add 7" 711 oscillations, and in the second, 2" 563, to reduce each of them to the case of amplitudes infinitely small, which gives 86402.1 and 86401.0 for the total number of oscillations infinitely small in the two cases. These quantities only differ by 1.1, 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. 101oz. and having replaced it by another of 6lb. 30$\backslash frac\{1\}\{2\}$oz., the amplitudes of the oscillations, which were at first $3^\{\backslash circ\}\; 30\text{'}$, were reduced to $3^\{\backslash circ\}$, and the diurnal rate of the clock slackened by $1\backslash frac\{1\}\{4\}$". Now, if we reduce each of these rates to the case of amplitudes infinitely small, in taking successively for $\backslash alpha$, $1^\{\backslash circ\}\; 45\text{'}$ and $1^\{\backslash circ\}\; 15\text{'}$, we find for the reduction in the first case 5".03, in the second 2".6, of which the difference is 2".4, instead of 1", 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 admit-

ting 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 immovable mass, and directed towards a star, the diurnal return of which could be observed with fixed wires, stretched in the focus. Messrs Breguet have begun on this subject a series of experiments, 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 OB-LATENESS 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 dependant 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 its actual velocity of its rotation. He thus found, that denoting by $\backslash phi$ the observed ratio of the centrifugal force, to the force of gravity at the equator, the oblateness of the spheroid must be $\backslash frac\{5\}\{4\}\backslash phi$; and as $\backslash phi$, from observation, may be

Pendulum. estimated for the earth at $\backslash frac\{1\}\{289\}$, or at 0.00346031, there results the oblateness $\backslash frac\{5\}\{4\}\; =\; \backslash frac\{1\}\{231\}$, 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. Clairault, 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 $\backslash lambda$ this length at any latitude $L$, and $A$ the length at the equator itself, we have in general $\backslash lambda\; =\; A\; +\; B\; \backslash sin^2\; L$, $B$ being a constant coefficient, 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 $\backslash frac\{l(a+h)^2\}\{a^2\}$,

or $l\; +\; \backslash frac\{2hl\}\{a\}\; +\; \backslash frac\{lh^2\}\{a^2\}$. But this reduction may be simpli-

fied, by considering that, on account of the small height Pendulum. to which we can rise above the earth's surface, $\backslash frac\{h\}\{a\}$ 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 $l\; +\; \backslash frac\{2hl\}\{a\}$. The term $\backslash frac\{2hl\}\{a\}$ 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, 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 coefficients $A$ and $B$ of the general formula, we shall employ the oblateness

0.00326, or $\backslash frac\{1\}\{306.75\}$, which M. La Place has obtained

by submitting to a general and profound discussion, the measures of the terrestrial degrees, and the lunar equalities 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 56 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,

Pendulum. 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°, it is A+B. So that $\backslash frac\{B\}\{A\}$ 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, $\backslash frac\{B\}\{A\}$ to the oblateness 0.00326, we shall have, by the theorem of Clairault, the following condition, $\backslash frac\{B\}\{A\}\; +\; 0.00326\; =\; 0.00865$, whence we obtain B=A.00539, and, consequently, $\backslash lambda\; =\; A(1\; +\; 0.00539\; \backslash sin^2\; L)$. Now, we have seen above, that at the station of Unst, in latitude 60° 45' 25", the length of the sexagesimal seconds pendulum determined by the observations of Biot, was 994.943105. The height of this station was only 9^m above the

level of the sea, which gives for the reduction Pendulum +0.^mm002818. Whence there results, at the level of the sea, the height 994.945923. 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, $\backslash lambda\; =\; 990.879660\; +\; 5.340841\; \backslash sin^2\; L$. If we wish to reduce this formula in English inches, all the terms must be multiplied by $\backslash frac\{39.37079\}\{1000\}$, and then $\backslash lambda\; =\; 39.0117150\; +\; 0.2102732\; \backslash sin^2\; L$. 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 $\backslash left(\backslash frac\{864\}\{1000\}\backslash right)^2$, the ratio of the decimal to the sexagesimal pendulum. We then have $\backslash lambda\; =\; 739.687686\; +\; 3.986917\; \backslash sin^2\; L$. 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.
			By Calculation.	By Observation.
			mm	mm
Unst, . . .	Biot	60° 45' 25"	742.723136	742.723136	0.000000
Leith Fort, .	Biot, Mudge	55 58 37	742.426416	742.413435	+0.012981
Dunkirk, . .	Biot, Mathieu	51 2 10	742.098066	742.077030	+0.021036
Paris, . . .	Biot, Mathieu, } Bouvard }	48 50 14	741.947360	741.917490	+0.029870
Clermont, . .	Biot, Mathieu	45 46 48	741.735412	741.705180	+0.030292
Bourdeaux, .	Biot, Mathieu	44 50 26	741.670048	741.608720	+0.061328
Figeac, . .	Biot, Mathieu	44 36 45	741.654181	741.612280	+0.041901
Formentera, .	Biot, Arago, Chaix	38 39 56	741.243950	741.252000	-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 Bourdeaux. It becomes less even at Figeac, 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 $\backslash frac\{8\; \backslash text\{\; mm\}\}\{1000\}$ in a contrary direction, which would seem rather to indicate a slight local excess in the intensity of gra-

vity. 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.

Pendulum. From the preceding formulæ 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 $N,\; N\text{'}$, the number of oscillations of this pendulum at two different stations, where the lengths of the simple seconds pendulum are $\backslash lambda,\; \backslash lambda\text{'}$, we have shown above that $\backslash frac\{\backslash lambda\text{'}\}\{\backslash lambda\}\; =\; \backslash frac\{N^2\}\{N\text{'}^2\}$, whence $N\text{'}^2\; =\; \backslash frac\{\backslash lambda\text{'}\}\{\backslash lambda\}\; N^2$. Now, calling $L,\; L\text{'}$ the latitudes of the two stations, the above formulae give the values of $\backslash lambda$ and of $\backslash lambda\text{'}$, as well as their relations; substituting, then, these values in the preceding equation, we obtain $N\text{'}^2\; =\; \backslash frac\{A+B\; \backslash sin^2\; L\text{'}\}\{A+B\; \backslash sin^2\; L\}\; N^2$.

An expression by means of which we can calculate $N\text{'}$ when we know $N$.

The total variation of gravity from the equator to the pole is so inconsiderable, that the difference between the numbers $N,\; N\text{'}$ 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 formulæ. But nothing is easier; for, if we denote it by $n$, so that $N\text{'}$ is represented by $N+n$, the preceding equation will become $N^2\; +\; 2nN\; +\; n^2\; =\; \backslash frac\{A+B\; \backslash sin^2\; L\text{'}\}\{A+B\; \backslash sin^2\; L\}\; N^2$.

Whence we deduce

$$2nN\; +\; n^2\; =\; \backslash frac\{B\; \backslash sin(L\text{'}+L)\; \backslash sin(L\text{'}-L)\}\{A+B\; \backslash sin^2\; L\}\; N^2,$$

and resolving the value of $N$ into a series,

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

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.	Excess of Observation.
G. Graham & C. Campbell, in 1731 & 1732,	Jamaica,	76° 45' 15" W.	18° 0' 0" N.	86283.0	118° 2'	119° 96'	-1° 76'
	London,	0 0 0	51 31 0 N.	86401.2
Maupertuis, Clairault, Lemonnier, 1735,	Paris,	2 20 15 E.	48 50 14 N.	86394.4	59 1	64 70	-5 60
	Pello,		66 48 0 N.	86453.5
Graham, 1758, Freycinet,	London,	0 0 0	51 31 0 N.	86402.1	51 4	53 85	-2 45
	Paris,	2 20 15 E.	48 50 14 N.	89143.8
	Rio Janeiro,	43 18 37 W.	22 55 2 S.	89048.8
	Cape of Good Hope,	18 24 0 E.	33 55 15 S.	89086.4
Sabine, 1818,	London,	0 0 0	51 31 8 N.	86497.40	33 11	31 82	+1 29
	Brassay,		60 9 42 N.	86530.51
	Hare Island,		70 26 17 N.	86562.64
Sabine, 1820,	London,	0 0 0	51 31 8 N.	86455.65	65 24	62 46	+2 78
	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, Clairault, 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 be-

Pendulum. tween 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 $\backslash frac\{1\}\{326.75\}$, 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 $\backslash lambda$ 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, infinite-

ly small, gives $1\text{'}\text{'}\; =\; \backslash pi\; \backslash sqrt\{\backslash frac\{\backslash lambda\}\{g\}\}$, consequently $g\; =\; \backslash pi^2\; \backslash lambda$, $\backslash pi$ being

the ratio of the circumference to the diameter, or 3.14159. But we have already given for any latitude the value of $\backslash lambda$ 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 $\backslash pi$, 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 of 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 Pendulum. 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. La Place 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 observations 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 $\backslash frac\{1\}\{100\}$ of a line. Now $\backslash frac\{1\}\{100\}$ of a line being equal to $\backslash frac\{1\}\{10\}$ 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 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. (Z. Z.)

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