170. When a pendulum vibrates in very small circular arcs, if T be the time of an oscillation p the ratio of the circumference of a circle to its diameter 7 the length of the pendulum, and g the measure of the force of gravity, the time of an oscillation may be formed nearly by the following formulæ.

g

POISSON, S

It is by means of the latter formula that the intensity of gravity, or the value of g, may be determined at any point upon the surface of the earth from observations upon the Pendulum. The times of the vibrations of pendulums of unequal lengths are respectively as the square roots of their lengths.

The time of an oscillation is to the time of the fall of a heavy body through half the length of the pendulum as the circumference of a circle is to its diameter.

The lengths of pendulums are inversely as the squares of the number of their respective vibrations in equal times.

The force of gravity in any place whatsoever is proportioned to, and may be measured by, the length of the pendulum vibrating seconds.

171. If a pendulum be placed between two Cycloidal cheeks, and made to vibrate, it will describe in its oscillations arcs of a cycloid identical with that which would be formed by an union of the two cheeks.

172. A Pendulum being thus made to vibrate in a cycloid is at every moment accelerated by a force proportioned to the arc of the cycloid intercepted between it and the lowest point of the

arc.

Hence the vibrations of a pendulum in a cycloidal arc whether great or small are all isochronous. PLAYFAIR, 208. 173. Ifl be the length of the pendulum vibrating in a cycloid, the time of a vibration whether the arc be great or small will be

p=
g

This is the same expression as that deduced for the vibration of a pendulum in a very small circular arc. When, however, the circular arc becomes greater, the value of the time of an oscillation will no longer be the same.

174. The times of vibration in a cycloidal and in a circular arc, and twice the time of descent in the chord of half the circular arc are as

The first and last are constant quantities, but the middle one varies with the extent of the arc of vibration of which a is the versed sine, while ris the length of the pendulum. HUTTON's Tracts, vol. 3. Hence the time in the cycloid is the least, and that in the chord the greatest.

175. There is another singular property of the cycloid, viz, that of being the curve of swiftest descent; for the line in which a heavy body moves by the action of Gravity from one point to another, that is neither in the same vertical nor on the same horizontal plane, in the shortest possible time, is an arc of a cycloid having for its base a horizontal line drawn through one of the given points and intersecting the vertical passing through the other. POISSON, 288.

176. The simple pendulum, such as it has been defined above, and from the consideration of whose properties the foregoing laws have been deduced, cannot exist in nature, inasmuch as we cannot abstract from the figure and extent of the body or bodies of which it is composed. The Pendulums on which experiment may actually be made are called compound pendulums.

177. The Centre of Oscillation in a compound pendulum is the point in which, if all the matter of the pendulum were supposed to be collected and attached by an inflexible line void of weight to the point of suspension, the vibrations would be isochronous with those of the compound pendulum itself

The distance, then, between the point of suspension, and centre of oscillation of a compound pendulum, is equal to the length of an isochronous simple pendulum.

Hence we call the distance between the centre of oscillation and the point of suspension of a compound pendulum, the Length. 178. The distance of the centre of oscillation of a pendulum composed of several bodies from its point of suspension may be found by dividing the sum of the products of each body into the square of its distance from its point of suspension, by the sum of the products of each body into its distance from the point of suspension.

If the bodies be A, B, C, &c. the distances of their centres of gravity from the point of suspension a, b, c, &c. and x the distance between the point of suspension and the centre of oscilla

tion

AaBb2+Cc2 + &c.

AaBb Cc + &c.

a will, of course, be the length of the isochronous simple pendulum. When any of the bodies are situated above the point of suspension their distances from it must be accounted negative quantities. PLAYFAIR, $201. 179. As we may consider that any compound pendulum whatever is composed of an infinite number of small parts, the above theorem may be applied through the intervention of the fluxional calculus to determine the situation of the centre of Oscillation of any regular figure.

In a cylinder suspended from the centre of one of its circular bases, if a be the length of the cylinder, r the radius of the base,

If the thickness of the cylinder be diminished until it become a rod of insensible diameter, there will be but little error in taking

In a cone suspended from its vertex, if a be its altitude, and r the radius of its base,

if r

=

a, which is the case in a right cone, the expression will become x = a

hence, if a right cone be suspended by its vertex, the centre of oscillation is in the middle of its base.

If a homogeneous sphere, whose radius is r, be suspended from a point whose distance from its centre is d,

x=d+

2r2

5d

If the point of suspension be in the surface of the sphere,

GREGORY, 311. PLAYFAIR, § 202, 203, 204.

180. The centres of suspension and oscillation are convertible points; that is to say, if the centre of oscillation of a compound . pendulum be made the point of suspension, the former point of suspension will become the centre of oscillation.

This remarkable proposition, which was demonstrated by Huygens along with the other properties of the centre of oscillation, remained without any practical application until the present time, when Kater has applied it with great success to the measurement of the lengths of pendulums.

181. There are three several purposes to which pendulums have been either actually applied, or proposed to be applied, viz. to the measure of time, to the determination of the force of gravity, and as a universal standard of measure.

Galileo was the first who observed the isochronism of the motion of pendulous bodies. Huygens, who investigated the subject more closely, discovered, that as a pendulum was gradually brought to rest by friction round the point of suspension, and by the resistance of the air, the time of its oscillations became less. To obviate this inconvenience he proposed to make the pendulum vibrate in the arc of a cycloid. However good this plan may be in theory it was found impossible to reduce it to practice. The consideration of motion in a cycloid, however, led to the improvement of making the arcs described by pendulums as small as possible.

As a measure of time, pendulums are attached to instruments known by the name of Clocks, that serve the two fold purpose of counting the vibrations of the pendulum, and restoring to it at each oscillation as much force as it loses by the retarding forces that act upon it; this subject however, shall be treated of in another place.

182. The usual form of a pendulum is a lenticular body attached to an inflexible rod of some metallic substance. As it is found that all metallic substances are affected by alternations of heat and cold, the rate of a clock regulated by a pendulum of this sort must be constantly varying. Means have, however, been found of remedying this defect, the general principle of all of which is, to construct the pendulum of two substances combined in such a way that the expansion of the one upwards shall compensate that of the other downwards, and vice versa. The most remarkable of the contrivances for this purpose are the Mercurial Pendulum invented by Graham, and the Gridiron Pendulum invented by Harrison. The invention of the last is also claimed by French artists.

In the Mercurial pendulum the rod is a simple metallic verge, the bob is composed of a glass vessel containing mercury. The quantity of the latter is so adjusted that its expansion upwards in the containing vessel shall exactly equal that of the rod downwards; the centre of oscillation is thus constantly maintained at the same distance from the point of suspension. Graham investigated with great care the relative expansions of the solid metals, but found them so nearly similar as to prevent his succeeding in a compensation by making the bob of one metal and the rod of another, and appears to have abandoned the idea of employing them. Harrison, on the other hand, after observing that the relative expansions of brass and steel were nearly as 5 to 3 thought of applying the two metals in the rod of the pendulum. The rod of his pendulum, instead of being a single metallic verge, is composed of nine bars, alternately of brass and steel,

of such proportionate lengths that by their motion in opposite directions the common length of the rod remains constant. The form of the Mercurial pendulum has been changed, and the instrument improved of late by that eminent maker of philosophical instruments, Troughton. He has also improved the construction of the Gridiron pendulum, and in one of his methods, where the compensating bars are inclosed in a tube, he has made his clocks more fit to be carried from place to place without risk of the adjustment being altered.

Berthoud and Leroy, in France, have also constructed several species of compensation pendulums, the principle of which does not differ from that of Harrison. See Rees' Cyclopedia, Article Pendulum.

183. If a sphere of Platina be suspended by the slenderest wire of iron that is capable of supporting it, we may without any great error abstract from the weight of the latter; if the whole Pendulum be then actually measured, the position of the centre of Gravity may be found by the formula.

x=d+

2r2

5d

184. If a pendulum of this sort, nearly twice the length of a second's pendulum, be set to vibrate, and the number of its vibrations in a given time ascertained, the true length of the second's pendulum may be thence deduced. To ascertain the number of vibrations in a given time, it is by no means necessary to count them; for if the experimental pendulum be suspended in front of the pendulum of a well-regulated clock, if their planes of vibration be parallel, and their axes both in one plane perpendicular to the horizon and to the plane of vibration, and if the pendulum be nearly, but not exactly some known multiple of the length of a second's pendulum, no more will be necessary than to observe the intervals between the times of coincidence, and hence we may infer the number of vibrations without danger of error.

DELAMBRE Astronomie, cap. XXXV. § 125. When the length of the pendulum and its number of vibrations have been thus determined a correction may be applied for the weight of the suspending wire.

DELAMBRE, ubi supra.

It was in this manner that the members of the French Institute determined the length of the pendulum, while measuring an arc of the Meridian. See DELAMBRE, Base du Systeme Metrique. The length of a Pendulum vibrating seconds in the latitude of Paris was found to be 443.296 lines old French measure or 0.993977 inches of the new.

185. If a pendulum be so constructed that it may be suspended at pleasure upon knife-edges either from one extremity, or from a point not far distant from its supposed centre of oscillation; and