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value of dt must be zero, except the first; therefore b must be zero, and i + 1, or i; thus ds= az dz; the integral of which is s2azt, the equation to a cycloid DzE, fig. 30, with a horizontal base, the only curve in vacuo having the property required. Hence the oscillations of a pendulum moving in a cycloid are rigorously isochronous in vacuo. If r 2BC, by the properties of the cycloid r = 2a2, and if the preceding value of ds be put in

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It is unnecessary to add a constant quantity if z = h when t=0. If T be the time that the particle takes to descend to the lowest point in the curve where z = 0, then

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Thus the time of descent through the cycloidal arc is equal to a semioscillation of the pendulum whose length is r, and whose oscillations are very small, because at the lowest point of the curve the cycloidal arc ds coincides with the indefinitely small arc of the osculating circle whose vertical diameter is 2r.

110. The cycloid in question is formed by supposing a circle ABC, fig. 30, to roll along a straight line ED. The curve EAD traced by a point A in its circumference is a cycloid. In the same manner the cycloidal arcs SD, SE, may be traced by a point in a circle, rolling on the other side of DE. These arcs are such, that if we imagine a thread fixed at S to be applied to SD, and then unrolled so that it may always be tangent to SD, its extremity D will trace the cycloid DzE; and the tangent zS is equal to the corresponding arc DS. It is evident also, that the line DE is equal to the circumference of the circle ABC. The curve SD is called the

D

fig. 30.

E

involute, and the curve Dz the evolute. In applying this principle to the construction of clocks, it is so difficult to make the cycloidal arcs SE, SD, round which the thread of the pendulum winds at each vibration, that the motion in small circular arcs

is preferred. The properties of the isochronous curve were discovered by Huygens, who first applied the pendulum to clocks.

111. The time of the very small oscillation of

fig. 31.

a circular pendulum is expressed by TT,

g

r being the length of the pendulum, and consequently the radius of the circle AmB, fig. 31. Also t= 22 is the time employed by a heavy

g

A

m

B

body to fall by the force of gravitation through a height equal to ≈. Now the time employed by a heavy body to fall through a space

equal to twice the length of the pendulum will be t =

4r

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E

fig. 32.

that is, the time employed to move through the arc Am, which is half an oscillation, is to the time of falling through twice the length of the pendulum, as a fourth of the circumference of the circle AmB to its diameter. But the times of falling through all chords drawn to the lowest point A, fig. 32, of a circle are equal: for the accelerating force F in any chord AB, is to that of gravitation as AC: AB, or as AB to AD, since the triangles are similar. But the forces being as the spaces, the times are equal: for as

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A

C

B

it follows that T = t.

112. Hence the time of falling through the chord AB, is the same with that of falling through the diameter; and thus the time of falling through the arc AB is to the time of falling through the chord 2, that is, as one-fourth of the circumference to the dia

AB as

π

2

meter, or as 1.57079 to 2. Thus the straight line AB, though the shortest that can be drawn between the points B and A, is not the line of quickest descent.

Curve of quickest Descent.

113. In order to find the curve in which a heavy body will descend from one given point to another in the shortest time possible, let CP, PM y, and CM=s, fig. 33. The velocity of a body moving in the

C

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must be a minimum, or, by the method of variations,

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The values of z and z' are the same for any curves that can be drawn between the points M and m': hence ddz= 0 ddz' = 0. Besides, whatever the curves may be, the ordinate om' is the same for all; hence dydy' is constant, therefore (dy + dy') = 0: whence ds'

ddyddy'; and ♪.

ds

√2

+d- = 0, from these considerations,

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second term of this equation is only the first term in which each variable quantity is augmented by its increment, so that

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dy

But is the sine of the angle that the tangent to the curve

ds

makes with the line of the abscissæ, and at the point where the tan

dy

gent is horizontal this angle is a right angle, so that

=1: hence

ds

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the equation to the cycloid, which is the curve of quickest descent.

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CHAPTER III.

ON THE EQUILIBRIUM OF A SYSTEM OF BODIES.

Definitions and Axioms.

114. ANY number of bodies which can in any way mutually affect each other's motion or rest, is a system of bodies.

115. Momentum is the product of the mass and the velocity of a body.

116. Force is proportional to velocity, and momentum is proportional to the product of the velocity and the mass; hence the only difference between the equilibrium of a particle and that of a solid body is, that a particle is balanced by equal and contrary forces, whereas a body is balanced by equal and contrary momenta.

117. For the same reason, the motion of a solid body differs from the motion of a particle by the mass alone, and thus the equation of the equilibrium or motion of a particle will determine the equilibrium or motion of a solid body, if they be multiplied by its mass.

118. A moving force is proportional to the quantity of momentum generated by it.

Reaction equal and contrary to Action.

119. The law of reaction being equal and contrary to action, is a general induction from observations made on the motions of bodies when placed within certain distances of one another; the law is, that the sum of the momenta generated and estimated in a given direction is zero. It is found by experiment, that if two spheres A and B of the same dimensions and of homogeneous matter, as of gold, be suspended by two threads so as to touch one another when at rest, then if they be drawn aside from the perpendicular to equal heights and let fall at the same instant, they will strike one another centrically, and will destroy each other's motion, so as to remain at rest in the perpendicular. The experiment being repeated with spheres of homogeneous matter, but of different dimensions, if the velocities be inversely as the quantities of matter, the bodies

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