acceleration vertically downwards, and equal to the acceleration due to its weight (Art. 42), and therefore the path of such a body ought to be a parabola whose axis is vertical, and concavity downwards. The resistance of the air will of course cause a deviation from this motion, but the deviation is so small as to lead us to conclude that this statement is true. This might also be tested on an inclined plane, as in the two former cases. From all this it appears that the effect of a force is independent of any velocity the body may have.

(4) The accelerations produced by constant forces are proportional to their intensities.

If a body of weight W be placed successively on two planes inclined to the horizon at angles, ', then the forces which cause its motion are in the respective cases Wsin ɩ, Wsin '; therefore if this statement be true, the accelerations of the motion in these cases are in the ratio of Wsin: Wsin, i. e. sin sin', or if the planes be of the same length, in the ratio of their heights. This can easily be tested from observations of the motions; and it may be observed that the friction may be made very small, and if the inclinations of the planes be small, the velocity of the body will never be very great, so that the resistance of the air will have much less effect than if the body were moving with a great velocity, and the observations on the motion can be more conveniently taken.

(5) If any number of constant forces act in the same direction, the acceleration produced will be equal to the sum of the accelerations which they would separately produce.

Let the forces be F, F, &c. and the accelerations they would separately produce be a,, a2, &c.; also let F be the resulting force, and a the acceleration produced by it.

Then Fis in the same direction as F1, F2, &c., and therefore a is in the same direction as a1, ɑ2ı

&c.

= &c.,

But F=F+F12 + ... ; · · α = α1 + α2+...

From this it appears that we can correctly obtain the motion. by considering the component forces, taking their effects separately, and then combining them.

C

A

b

B

(6) The effect of any constant force acting on a particle is independent of any other constant force that may be acting. Let AB, AC represent two constant forces in direction and magnitude, then completing the parallelogram BC, their resultant is represented by AD.

C

D

The acceleration produced will consequently be in the direction of AD, and let it be represented by Ad.

Draw db, de parallel to DB, DC.

:

appears

Then Ab Ac: Ad: AB: AC: AD; and therefore Ab, Ac represent the accelerations due to AB, AC. Whence it that we may take the separate effects of the component forces and combine them by the parallelogram of accelerations, and thus correctly investigate the motion. This can be extended to more than two forces, after the manner of Art. 9.

(7) The foregoing statements will also be true for any forces.

For the effect of a variable force at the instant in question will be measured by the effect of an equal force continued constant for a certain time, and therefore the measures of the accelerations will be the same (for the instant) as if the forces were continued constant, and consequently will be subject to the above laws. On this plan the truth of the Second Law of Motion is established.

106. We have yet to determine the coefficient of the proportionality between the measures of a force and the acceleration

produced by it: this evidently depends on the nature of the particle acted upon, for the same force acting upon different particles is found to produce different accelerations.

It is evident that if any number of particles be taken with equal quantities of matter in them, the same force must be exerted on each to make them move in the same way: and if they be connected together, the effect of the above system of forces will not be altered: i. e. if F be the force acting on each particle to produce an acceleration a in its motion, there being n particles; when they are connected together, or formed into one particle, all the forces, i. e. nF, must act in order to produce the same acceleration a as before. Now the quantity of matter in this last particle is n times as great as the quantity of matter in one of the original particles: wherefore, in order that the accelerations on different particles may be the same, the intensity of the forces acting on them must be proportional to the quantity of matter in the particles. This is the only plan on which we can proceed to estimate the quantity of matter in bodies. The measure of the quantity of matter in a body is called its mass.

107. The force then acting on a particle varies as the acceleration produced as long as the mass of the particle is the same, and as the mass when the acceleration is the same; therefore generally the force varies as the mass of the particle and the acceleration produced jointly; i.e. if forces F, F" acting on particles whose masses are M, M', produce accelerations a, a', F Ma FM'a

then

=

We must assume the unit of mass: let the

mass of the second particle M' be the unit of mass, then

F", a' are at present undetermined: let them both = 1,

This assumption fixes the unit of mass to be that quantity of

matter in which the unit of force produces the unit of accele

ration, and then we have the numerical measure of the intensity of a force equal to the product of the numerical measures of the mass moved and the acceleration produced.

108. The mass of a body is proportional to its weight, for the acceleration of gravity is the same on all bodies, as is established by the experiment of letting fall at the same instant two bodies of very different weights, such as a sovereign and a feather from the same height within the exhausted receiver of an air-pump, when they are found to reach the bottom at the same instant. Therefore, as the accelerations on these are equal, the force causing motion is proportional to the mass moved (Art. 106), i. e. the mass of a body is proportional to its weight.

109. The expression Ma is called the moving or motional effect of a force, and Mx (the measure of the velocity) is called the momentum or quantity of motion of a body. Also M× (velocity) is called the vis viva of the body.

110. It now remains to determine what effects particles produce on the motion of each other, and we must be guided by the analogous case in Statics, which is that of bodies pressing against each other, or exerting forces by means of strings or rods: in this case the forces exerted by any two particles are equal in magnitude and opposite in direction, and we should therefore expect the same to hold good in Dynamics. Since under the conventions we have adopted the intensity of the force is measured by the product of the measures of the mass and the acceleration, i. e. by the motional effect, we state in the following manner

THE THIRD LAW OF MOTION.

"If one particle act on another particle, the motional effect produced by the first on the second is equal in magnitude and opposite in direction to that produced by the second on the first." Or concisely thus: "Action and reaction are equal and opposite."

111. We may test this by observing the motion of two bodies of different weights hanging by a fine inextensible string over a pulley.

Let W, W' be the weights of the bodies, (W> W'), and g the acceleration due to gravity, i. e. the acceleration of motion in a body falling freely under its own weight. Also let T be the tension of the string; then the force downwards on the heavier particle is W-T, and that upwards on the other is T— W', if the law be true. Now on the motion of the heavier particle the force W, and on the other the force W', produces an acceleration g, therefore (by Art. 105, (4)) the accelerations on these are respectively

But the string being always stretched, the downward motion of the heavier particle is identical with the upward motion of the other, and consequently the above accelerations are equal,

This is the acceleration on the motion of each particle; and as it is a constant acceleration, the motion possesses the properties investigated in Arts. 39, 40, 41. In making experiments we can see whether these properties are possessed in any case, and as it is found that they are, we conclude that the law is true.

* If M, M' be the masses of the particles, these accelerations can be represented