have DK = lcose; we shall also have DE = lcosa; and

since h is equal to DK

DE, we shall have,

Which, being substituted in the preceding formula, gives,

Equating these two values of v, we have,

If we develop cose and cosa into series, by McLAURIN'S

being

When a is very small, say one or two degrees, still smaller, we may neglect all the terms after the second as inappreciable, giving

Substituting in Equation (93), it becomes,

a, and

A = +a, t will denote the time of one vibration, and we shall have,

Hence, the time of vibration of a simple pendulum is equal to the number 3.1416, multiplied into the square root of the quotient obtained by dividing the length of the pendulum by the force of gravity.

For a pendulum, whose length is l', we shall have,

From Equations (95) and (96), we have, by division,

t ť'

=

V or, t:t':: √T: √ (97.)

That is, the times of vibration of two simple pendulums, are to each other as the square roots of their lengths.

If we suppose the lengths of two pendulums to be the same, but the force of gravity to vary, as it does slightly in different latitudes, and at different elevations, we shall have,

That is, the times of vibration of the same simple pendulum, at two different places, are to each other inversely as the square roots of the forces of gravity at the two places. If we suppose the times of vibration to be the same, and the force of gravity to vary, the lengths will vary also, and we shall have,

That is, the lengths of simple pendulums which vibrate in equal times at different places, are to each other as the forces of gravity at those places.

Vibrations of equal duration are called isochronal.

The Compound Pendulum.

124. A COMPOUND PENDULUM is a heavy body free to oscillate about a horizontal axis. This axis is called the axis of suspension. The straight line drawn from the centre of gravity of the pendulum perpendicular to the axis of suspension is called the axis of the pendulum.

In all practical applications, the pendulum is so taken that the plane through the axis of suspension and the centre of gravity divides it symmetrically.

Were the elementary particles of the pendulum entirely disconnected, but constrained to remain at invariable distances from the axis of suspension, we should have a collection of simple pendulums. Those at equal distances from

the axis would vibrate in equal times; those unequally distant from it would vibrate in unequal times.

Those particles which are at the same distance from the axis of suspension lie upon the surface of a cylinder, whose axis coincides with the axis of suspension, and we may, without at all affecting the time of vibration, suppose them all to be concentrated at the point in which the cylinder cuts the axis of the pendulum. If we suppose the same to be done for each of the concentric cylinders, we may regard the pendulum as made up of a succession of heavy points, a, b, . . . p, k, lying on the axis, firmly connected with each

Fig. 105.

a

b

G

10

k

other and with the point of suspension C. The particles a, b, &c., nearest to C will tend to accelerate the motion of the entire pendulum, whilst those most remote, as p, k, &c., will tend to retard it. There must, therefore, be some intermediate point, as O, which will vibrate precisely as though it were not connected with the system; were the entire mass of the pendulum concentrated at this point it would vibrate in the same time as the given pendulum. This point is called the centre of oscillation. Hence, the centre of oscillation of a pendulum is that point of its axis, at which, if the entire mass of the pendulum were concentrated, its time of vibration would be unchanged. A line drawn through this point, parallel to the axis of suspension, is called the axis of oscillation. The distance from the axis of oscillation to the axis of suspension is the length of an equivalent simple pendulum, that is, of a simple pendulum, whose time of vibration is the same as that of the compound pendulum.

To find an expression for CO, C being the axis of suspension, and O the axis of oscillation. Denote CO by l; let G be the centre of gravity, and denote the distance CG by k; denote the masses concentrated at a, b, p, k, by m, m'...m", m", and their distances from C by ", "'... ''', yo!!!. r, r'

...

Whatever may be the position of CO, the effective com

ponent of gravity is the same for each particle, and were they free to move, each would have impressed upon it the same velocity that is actually impressed upon 0. Denote the angular velocity at any instant, by ; then will the actual velocity of the mass m, be equal to rw, and the effective moving force will be equal to mrw (Art. 24). Had the mass m been at O, instead of at a, the entire moving force impressed would have been effective, and its measure would have been mlw. The difference between these forces, or m(lr), is that portion of the force applied at a which goes to accelerate the motion of the system. The moment of this force with respect to C, is m(lr)rw. In like manner, for the force acting at b, which also tends to accelerate the system, we have m'(l — r')r'w, and so on, for all of the particles between 0 and C. By a similar course of reasoning, we get, for the moments of the force tending to retard the system, and which are applied at the points p, k, &c., m''(r'' — l)r''w, m''' (r'''' — 1)r'''w, &c. But since there is neither acceleration nor retardation, in consequence of the action of these forces, they must be in equilibrium, and, consequently, the sum of the moments of the forces which tend to accelerate the system, must be equal to the sum of the moments, which tend to retard the system. Hence, we have,

m(l− r)rw + m2(l — r')r'w + &c.

= m''(r'' — l)r''w + m''' (p''' — 1)p'''w + &c.

Striking out the factor w, and reducing, we have,

(mr + m'r'+m''r'' + &c.) l = mr2 + m'r'2 + m''r''2+ &c.,