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degree of heat, the expansion of several kinds of metal will be in the following proportion; viz. Iron 80; steel 85; copper 89; brass 110; tin 153; lead 155.

Of all metals iron, being least expansible by heat, is properest for the rods of pendulums. Wood does not lengthen much by heat; and therefore, when well dried, it will answer the purpose better than metal. But glass will answer still better, if kept of the same length:*

Various methods have been suggested to remedy this inconvenience in pendulums. If a pendulum be made of a glass tube partly filled with mercury, while the tube expands by the heat and sinks the center of oscillation, the mercury will also expand, and rising in the tube will raise the center of oscillation as much as it was depressed when they are properly adjusted to each other; and thereby the length of the pendulum and consequently the times of vibration continue invariable.

Pendulums have also been compounded of different bars of iron and brass, in such a manner, that while the center of oscillation is depressed by the expansion of some, it is equally raised by the expansion of others. In this manner the pendulums for astronomical clocks are generally made.

The loss of time occasioned by the vibrations being performed in larger arches at some times than at others, by an unequal application of force to the wheels, has been in some measure prevented, by a flat board fixed on the pendulum, to give it a retardation from the re

Glass will expand regularly and equably by heat; whereas metals expand irregularly, when they have imbibed a certain quantity; and the increase of length in a metallic pendulum is not exactly proportional to the degree of heat.

sistance of the air, to moderate the velocity given by an unequal application of the weight to the wheels.*

* If a pendulum be made to vibrate in a cycloid, all its vibrations, whether through smaller or larger arches, will be perform ed in equal times.

If a circle perform a revolution on a right line, any point in the circumference will describe on an opposite plane, a curve called the cycloid; whose axis is equal to the diameter, and whose base is equal to the periphery of the generating circle.

If the generating circle (see plate 2, fig. 1.) DBI move along the line AEF, any point as A will describe the curve ABVGF, called the cycloid. From the generation of the curve, it is evident, that the arch of the circle BD=AD, and the arch BI-DE=KL =BN, that the cords BD,NE are parallel, as also BI,NV, that BD is perpendicular to the curve, and that BI is a tangent to the cycloid at the point B, because DBI is a right angle.

The arch of the cycloid BV, cut off by a line BNG parallel to the base, is double of the corresponding chord BI or NB of the generating circle.

Let POR be drawn infinitely near to BNG, cutting VN produced in S, and the generating circle VNE in O, and OT be drawn perpendicular to VS; draw also the tangents QV and QN of the circle at the points V and N, then the triangles QNV,ONS will be similar as QV is parallel to SO, and because QN and QV are equal, SO, and ON will be equal, and NT the increment of the chord VN, will be one half of NS or BP the simultaneous increment of the arch VB, as VT=VO. Therefore as the arch of the cycloid increases twice as fast as the corresponding chord of the generating circle it must, in every situation, be double of the chord Q.E.D.

If then a pendulum be suspended at the point H by a flexible thread applying to the semicycloidal cheeks AH and FH, so that it may vibrate in the cycloid AVF, all its vibrations through larger or smaller arches will be performed in the same time.

If the pendulum descend from R or G, the last acquired velocity will be as OV and NV respectively, or as 20V=RV and 2NV=GV. Therefore, when the velocities of any kind of motion are as the spaces moved through, the times of description must be the same. Hence whether the pendulum descend from R or G, it will arrive at V in the same time.

MOTION OF PROJECTILES.

A PROJECTILE is a heavy body thrown in any direction. The force with which it is thrown is called the impetus and as its velocity is measured by the height, from which a body must fall to acquire this velocity, this height is also called the impetus. The random or amplitude of the projection is the horizontal distance to which the body is thrown. The angle which the piece of ordnance makes with the horizon is called the elevation: and the perpendicular height to which the ball rises above the horizon is called its altitude.

A projectile is acted upon by two forces in different directions, viz. the force of gravity, which acts in right lines directed to the center of the earth, by which alone, it would pass over spaces proportional to the squares of the times; and by the projectile force, which acts by a single exertion, and therefore produces an equable

To make a pendulum vibrate in a given cycloid DKG;* let two semicycloids ABD, and AG, of equal magnitude with the given cycloid be described, whose vertices shall be at D and G the ends of the base, meeting together in the axis of the given cycloid produced in A, making AK double of the axis FK. If a pendulum be suspended at the point A, and of such a length as to reach to K, and in its vibration apply to the semicycloidal cheeks it will still be in the cycloid, in every part of its vibration.

Let the pendulum be in the direction ABP; then, because ABD=ABP=AK,BD=2 DC=BP=2 BE; therefore BC and PI will be equally distant from the base DF, and will cut off equal portions DC and IF of the generating circles, and the chords DC and IF will be equal and parallel, the tangent BP being parallel to both: therefore EF=PI. And since BC or DE=the arch DE or the arch IF, PI or EF will be equal to the remainder of the arch IK: and the point P will be in the cycloid DPK.

* See Plate II. Fig. 2.

sistance of the air, to moderate the velocity given by an unequal application of the weight to the wheels.*

* If a pendulum be made to vibrate in a cycloid, all its vibrations, whether through smaller or larger arches, will be performed in equal times.

If a circle perform a revolution on a right line, any point in the circumference will describe on an opposite plane, a curve called the cycloid; whose axis is equal to the diameter, and whose base is equal to the periphery of the generating circle.

If the generating circle (see plate 2, fig. 1.) DBI move along the line AEF, any point as A will describe the curve ABVGF, called the cycloid. From the generation of the curve, it is evident, that the arch of the circle BD=AD, and the arch BI-DE=KL

BN, that the cords BD,NE are parallel, as also BI,NV, that BD is perpendicular to the curve, and that BI is a tangent to the cycloid at the point B, because DBI is a right angle.

The arch of the cycloid BV, cut off by a line BNG parallel to the base, is double of the corresponding chord BI or NB of the generating circle.

Let POR be drawn infinitely near to BNG, cutting VN produced in S, and the generating circle VNE in O, and OT be drawn perpendicular to VS; draw also the tangents QV and QN of the circle at the points V and N, then the triangles QNV,ONS will be similar as QV is parallel to SO, and because QN and QV are equal, SO, and ON will be equal, and NT the increment of the chord VN, will be one half of NS or BP the simultaneous increment of the arch VB, as VT=VO. Therefore as the arch of the cycloid increases twice as fast as the corresponding chord of the generating circle it must, in every situation, be double of the chord Q.E.D.

If then a pendulum be suspended at the point H by a flexible thread applying to the semicycloidal cheeks AH and FH, so that it may vibrate in the cycloid AVF, all its vibrations through larger or smaller arches will be performed in the same time.

If the pendulum descend from R or G, the last acquired velocity will be as OV and NV respectively, or as 20V=RV and 2NV GV. Therefore, when the velocities of any kind of motion are as the spaces moved through, the times of description must be the same. Hence whether the pendulum descend from R or G, it will arrive at V in the same time.

MOTION OF PROJECTILES.

A PROJECTILE is a heavy body thrown in any direction. The force with which it is thrown is called the impetus and as its velocity is measured by the height, from which a body must fall to acquire this velocity, this height is also called the impetus. The random or amplitude of the projection is the horizontal distance to which the body is thrown. The angle which the piece of ordnance makes with the horizon is called the elevation: and the perpendicular height to which the ball rises above the horizon is called its altitude.

A projectile is acted upon by two forces in different directions, viz. the force of gravity, which acts in right lines directed to the center of the earth, by which alone, it would pass over spaces proportional to the squares of the times; and by the projectile force, which acts by a single exertion, and therefore produces an equable

To make a pendulum vibrate in a given cycloid DKG;* let two semicycloids ABD, and AG, of equal magnitude with the given cycloid be described, whose vertices shall be at D and G the ends of the base, meeting together in the axis of the given cycloid produced in A, making AK double of the axis FK. If a pendulum be suspended at the point A, and of such a length as to reach to K, and in its vibration apply to the semicycloidal cheeks it will still be in the cycloid, in every part of its vibration.

Let the pendulum be in the direction ABP; then, because ABD=ABP=AK,BD=2 DC=BP=2 BE; therefore BC and PI will be equally distant from the base DF, and will cut off equal portions DC and IF of the generating circles, and the chords DC and IF will be equal and parallel, the tangent BP being parallel to both: therefore EF-PI. And since BC or DE=the arch DE or the arch IF, PI or EF will be equal to the remainder of the arch IK: and the point P will be in the cycloid DPK.

*See Plate II. Fig. 2.

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