at P' will balance G at 1, the weight at P" will balance G at n, ♣ the weight at P"" will balance G at n. It is in this manner that the steel-yard is graduated, any weight in equilibrium with G is determined for knowing the number of divisions from the fulcrum at which the weight is placed.

4. When a body in equilibrium having been slightly disturbed, has a tendency to arrive at its original position, then the equilibrium is called stable.

When after having been slightly disturbed it removes from its original position, and has no tendency whatever to arrive at it, then the equilibrium is called unstable.

5. Accelerating force is measured by the velocity produced in a given time as one second.

Momentum is measured by the product of the numbers which represent the weight and the velocity.

Force in statics is measured by the number of pounds which it would support.

Force in dynamics is measured by the space through which it would cause a body to move in a given time as one second.

=

the length of a second's pendulum

G the force of gravity, then from the expression for the time of an oscillation, we get

G Gn2

..g

=

=

t2

N2

Now the variations of gravity being very small,

and if n be nearly equal to N, we may obtain approximations for the difference; If g be greater than G, then n will be greater than N, and the pendulum will gain, and vice-versa. Let now g G (1 + h), and let q be the number of oscillations gained, then

=

nearly, N being put for n in the

numerator of the second number q' being neglected

Now this formula will hold when the

intensity of gravity is diminished and the pendulum loses.

Suppose the loss in the number of oscillations of a second's pendulum at the top of a mountain is q, we wish to find its height.

Then since the force of gravity varies inversely as the square of the distance from the centre of the earth, we have

G (1-h): G :: r2 : (r + x)2

— G (1 − 2) by neglecting all the terms of the

=

r

expansion after the second, for x being very small compared with r,

—is a small fraction, and its powers may therefore be safely neglected.

r

7.

A

H

Let a body be projected from A in the direction AL; then by the first law of motion, the body would move in that direction with a uniform velocity, if no other force impeded its motion. But from the moment that the body is projected, the force of gravity begins to act on it, and deflects it from the straight line AL, and the real path of the body becomes a curve, for

by the second law of motion, the motion which the force of gravity would produce on the body at rest is compounded with the projectile motion of the body; and if the path of the body be the curve AHQ, LQ a vertical line will be exactly equal to AV through which the body would fall at the same time by the force of gravity. Join VQ.

Now AL=V. T, if V be the velocity of projection, and T be the time of flight.

Now AV, LQ being equal and parallel, AL is equal and parallel to VQ, and

VQ
AV

=

4 h, or QV24 h AV which is the equation to a parabola referred to oblique axes AL, and AV, AL being a tangent at A, and AV any diameter.

9. The pressure of the atmosphere is acurately measured by the height of the column of mercury in the barometer; but the height of the mercurial column is subject to fluctuations, hence it is plain that the atmospheric pressure is subject to variations. But the variations of the mercurial column are confined within certain limits, 30 inches being the mean height. Hence the mean pressure of the atmosphere on any surface is equal to the weight of a column of mercury, whose height is 30 inches, and base equal to the given surface. Therefore the pressure

of the atmosphere on every square inch is the weight of 30 cubic inches of mercury. But the weight of a cublic inch of mercury is Therefore atmospheric pressure on every square

7.85 ounces.

inch

H =

=

64

115

Since the specific gravities are as the densities of the bodies, the density of mercury is to the density of acid so is 13.6 to 1.84. Now suppose the sections of the two barometers are each equal to unity, and the height of the mercury barometer, and h height of acid barometer. Then the weights of columns of mercury and acid which are each equal to the atmospheric column are as 13.6 × g × H: 1.84 x gx h; and these are in a ratio of equality.

=

10.

The syphon is a bent tube MLQ, having both ends open, and one of them is capable of being placed in a vessel of fluid, the other end being lower than the surface of the fluid in the vessel. The vertical height MT corresponding to the part ML of the bent tube must be less than the height of the column of fluid in the vessel, which equals the atmospheric pressure. Then if the tube be filled with the fluid

by suction or other means, the fluid will continue to flow through the syphon tube as a natural fountain, until the surface of the fluid falls below G. Suppose water is the fluid used.

=

Let HM be the height of the water barometer 34 feet nearly. Draw LT and PQ horizontal. Now the pressure of the atmosphere at M is the weight of a column of water whose height is HM, and the downward pressure at M on account of the weight of water in LM is proportional to MT, therefore the remaining pressure or the force with which water is impelled into the syphon is proportional to HT, and this is the force with which the water moves. If now a finger be placed at Q, the whole pressure at Q, will be proportional to HT together with TP the vertical height corresponding to the column of water in the tube which is rushing out. That is the downward at Q is the weight of a column of water whose height is HP. But the upward pressure at Q of the atmosphere is proportional to HM, therefore the pressure which the finger at Q really sustains is the weight of a column of water whose height is MP.