same time. Now consider the resistance of the air; this cannot depend in any way on the nature of the inside of the balls, and so must be the same on two balls of the same size, shape, and texture of surface, if they move with the same velocity. But by Art. 132 this force would produce less velocity in the solid ball than in the hollow ball; and so in the actual case we may readily suppose that it will take away much less of the downward velocity of the solid ball than of the hollow ball.

134. An example will illustrate the difference of the influence of the resistance of the air on bodies differing only in size. Suppose, for example, two cannon balls, one 4 inches in diameter and the other 5, but formed of the same material. The masses of the balls are in the same proportion as the cubes of the diameters, that is in the proportion of 64 to 125. It appears by experiment and theory that the resistances of the air are in the same proportion as the squares of the diameters, that is in the proportion of 16 to 25, that is in the proportion of 80 to 125. Hence we see that the resistance on the smaller ball bears to that on the larger ball a greater proportion than the mass of the smaller ball bears to that of the larger; and so the resistance exercises more influence on the smaller than on the larger ball, supposing the velocities equal.

135. We have not given a very full account of the influence of the resistance of the air because we have not attended to the way in which the resistance depends on the velocity of the moving body. In reality the resistance increases very rapidly as the velocity of the moving body increases. For an example, it has been found that under certain circumstances the range of a cannon ball would be 23000 feet if there were no such resistance, while it was in fact about 6400 feet. Another example is furnished by an experiment with a railway engine. The engine was started down an inclined plane with a velocity of 45 miles an hour; the velocity gradually diminished until it became 32 miles an hour and remained at that. Thus the resistance of the air together with that caused by the want of perfect smoothness in the wheels and iron rails just balanced the influence of the force of gravity in urging the engine down the plane and maintained uniform velocity.

T. P.

VIII. THIRD LAW OF MOTION.

136. Third Law of Motion. To every action there is always an equal and contrary reaction: or the mutual actions of any two bodies are always equal and oppositely directed in the same straight line.

Newton gives three illustrations of this Law:

If any one presses a stone with his finger, his finger is also pressed by the stone.

If a horse draws a stone fastened to a rope, the horse is drawn backwards, so to speak, equally towards the stone. If one body impinges on another body and changes the motion of the other body, its own motion experiences an equal change in the opposite direction.

In the third illustration motion is to be measured by momentum as in all cases. We shall return to the discus

sion of this illustration hereafter.

137. One of the most important examples of this Law is furnished by the attraction of bodies. The earth, for instance, attracts a body, and that body attracts the earth again with equal power. Thus when the earth produces velocity in a falling body that falling body also produces velocity in the earth, although the latter velocity is so small as to be imperceptible. For, according to the third Law of Motion, the stone gives to the earth as much momentum as the earth gives to the stone, and as the mass of the earth is incomparably greater than that of the stone the velocity given to the earth is incomparably less than that given to the stone. In the science of Astronomy the mutual attraction of bodies is a principle of supreme importance; the earth, for instance, attracts the moon, and the moon attracts the earth again with equal power.

138. The fact that not merely the earth as a whole attracts, but that distinct portions of the earth also do so, has been made obvious by noticing the action of moun

tains on plumb-lines hanging at places near them. It is thus discovered that the weight at the end of a plumb-line is drawn a little towards a neighbouring mountain; so that the plumb-line does not hang quite in the direction in which it would hang if there were no mountain. In very accurate surveys of the earth made for the purpose of determining its exact size and shape, it is necessary to pay great attention to the deviation which the action of mountains produces in the direction of the plumb-line.

어

139. A very interesting example of motion is furnished by a contrivance of which the essential part is indicated by the diagram. Two heavy bodies are connected by a string which passes over a smooth peg. Here the force of gravity tends to draw each body down, while the force exerted by the string tends to draw each body up. The force exerted by the string is the same on the two bodies in agreement with the third Law of Motion, which makes the action of one body on the other equal to the reaction of the latter on the former. Experiment will shew that if the two bodies are of unequal weight and are left to themselves the heavier will descend; so that the force exerted by the string is less than the weight of the heavier body, but greater than the weight of the lighter body. When the case is examined by the aid of a little mathematics it is found that the motion is just like what would take place if a force equal to the difference of the two weights were employed to move a mass equal to the sum of the two masses. Thus if one body weighs 13 pounds, and the other weighs 12 pounds, the motion will be just like that of body which weighs 25 pounds acted on by a force of 1 pound. Therefore the motion will be like that of a falling body but much slower, namely, at the rate of 1 foot for every 25 feet of the body falling freely.

140. The preceding example is one of those which justify our confidence in the truth of the laws of falling bodies: see Art. 89. We have here a case of motion which by the aid of sound theory we can shew to be of the same kind as that of falling bodies; while the motion is so much

less rapid that it can be easily observed. A machine is made, named after its inventor, Atwood, which is furnished with appliances for performing the experiment easily, but which in principle is the contrivance of the preceding Article. The results are very satisfactory, and the student will be pleased when he has the opportunity of seeing them exhibited in a lecture-room.

141. It is usual to call the force exerted by a string, as in Art. 139, the tension of the string. There is nothing special in the nature of the force exerted in this way, but it is convenient to give it a name.

142. The solution of the problem of motion noticed in Art. 139 involves more mathematics than we assume in the reader; but it may be instructive to verify by an example the result which is asserted to hold, at least so far as to shew that it is reasonable and consistent with itself. It will be seen that both bodies move, and that by the nature of the contrivance the weights of the two bodies are set in opposition as it were; so that the motion may naturally be that which would be produced in the sum of the masses by the difference of the weights. Now in the example we say that the heavier body will descend at of the rate of a body falling freely; thus in fact of the weight of the body is taken away by the tension of the string. Again, the lighter body rises at of the rate of a body falling freely; thus in fact the weight of the body is taken away by the tension of the string and besides a force equal to of the weight exerted upwards. Thus the tension of the string must be of the weight of the heavier body, and must also be g of the weight of the lighter body; so that our statement will not be consistent unless these two results are equal: it is easily found by trial that they are equal, each of them being 12 pounds.

IX. COMPOSITION OF FORCES AT A POINT.

143. In Chapters IV. to VIII. we have discussed the motion of falling bodies, and also the Laws which relate to the connexion between force and the motion produced by it; we must now devote some Chapters to the consideration of forces not producing motion but checking the action of other forces. It is a matter of observation that forces may act on a body without putting it in motion. A man may try to lift a body and find it too heavy for him: in this case the body is acted on by the force of gravity downwards, by the resistance of the ground on which it is placed which acts upwards, and by the effort of the man which also acts upwards; and the body remains at rest. When a body remains at rest though acted on by forces, it is said to be in equilibrium; and the forces are said to counteract each other or to balance each other.

144. There are three things to consider with respect to a force acting on a body; the point of application, that is the point of the body at which the force is applied; the direction of the force; and the magnitude of the force. It is necessary for simplicity to confine ourselves for some time to the case of a very small body, which we call a particle. In this case the forces which we have to consider all act at one point, namely that at which the particle is situated. The direction of any force is the straight line along which it tends to move the particle. We have seen in Art. 123 that a similar restriction as to the size of the bodies we consider is advantageous in treating the subject of motion.

145. The magnitudes of forces are conveniently measured by the weights which they will support. Thus we speak of a force of 5 pounds; by this we mean a force which will just support a weight of 5 pounds, that is a force which will just counteract the force exerted by gravity on a body weighing 5 pounds.

146. Forces may be conveniently represented by straight lines. For we may take a point to denote the