have been included. "We thus come to the important conclusion that the effect of gravity upon a body of any shape is to produce one, force which acts vertically downwards. It remains to be shown that the direction of this force passes through the centre of gravity of the body. Suspend a body of any shape by a cord, then when the body is at rest the centre of gravity must lie vertically beneath the point of suspension. If the direction of that one force which constitutes the effect of gravity does not pass through the centre of gravity, then its line of action cannot coincide with the direction of the cord of sus pension ; but it is impossible that two forces should equilibrate unless their lines of action are coincident, whence we are led to the import ant conclusion that the effect of the attraction of the earth upon a rigid body is to produce a single force which passes through the centre of gravity of the body and acts vertically downwards.
III. Universal Gravitation.
§ 9. Law of Gravitation between Two Masses.—Investigations of the motions of the planets have conducted us to the conclusion that each planet is attracted towards the sun by a force which varies according to the inverse square of the distance. We likewise see that each of the satellites appears to be attracted towards the correspond ing primary planet by a force which obeys the same law. We are therefore tempted to generalize these results into the proposition that any two masses in the universe attract each other with a force u-hich varies according to the inverse square of the distance. Obser vations of the most widely different character have combined to show us that this law, which was discovered by Sir Isaac Newton, is true. It is called the law of Gravitation. We suppose that the distance between the two bodies is so exceed ingly great compared with their dimensions that we may practically consider each of the bodies as a particle. Let m, m be the masses of the two bodies, and let r be the dis tance. The force with which m attracts m is equal in magnitude though opposite in direction to the force with which m attracts m. The reader may perhaps feel some difficulty at first in admitting the truth of this statement. We speak so often of the effects which the attraction of the sun produces on the planets that it may seem strange to hear that each planet reacts on the sun with a force pre cisely equal and opposite to the force with which the sun acts upon the planets. We can, however, by the aid of a simple illustration, show that such is really the case. Siippose the sun and the earth to be placed at rest in space, and abandoned to the influence of their mutual gravitation. It is evident that the two bodies would begin moving towards each other, and would after a certain time come into collision. If, however, the earth and the sun had been separated by a rigid rod, it would then be impossible for the two bodies to move closer to each other, so we must consider whether any other motion could be produced by their mutual attraction. As the two bodies were initially at rest, it is clear that there will be no tendency of the rod to move out of the line in which it was originally placed", and consequently if the rod begin to move at all it must move in that line, and carry with it the earth at one end, and the sun at the other. As, however, the earth and the sun would remain separated at a constant distance, the statical energy due to their separation would remain constant, and therefore if they began to move we should have the kinetic energy of their motion created out of no thing, which is now well known to be impossible. It therefore follows that, under the circumstances we have assumed, the rod would remain for ever at rest. But what are the forces which act iipon the rod ? At one end of the rod the earth presses upon it with a force which is equal to the attraction of the sun upon the earth. At the other end the sun presses upon the rod with a force which is equal to the attraction of the earth upon the sun. But since the rod remains at rest, these two forces must be equal and opposite, and hence the force with whicli the sun attracts the earth must be equal and opposite to the force with which the earth attracts the sun. If we express the gravitation of two masses in the form f(m, m )-=-r 2 , then m and m must enter symmetrically into the expression, for if the two masses were interchanged the gravitation must not be altered. We should here also advert to a circumstance connected with gravitation which is of the very highest importance. Suppose we take two definite masses (for simplicity, two pounds) separated at a definite distance (for simplicity, one foot), then the gravitation of these two masses to each other is a certain definite force (which we shall subsequently calculate). What we want here to lay special stress upon is that, so far as we know at present, this force appears to be the same whatever be the material of which the two masses are composed. Thus two pounds of iron at a distance of one foot attract each other with the same force as a pound of iron would attract a pound of lead at the same distance. The signifi cance, or perhaps it should be said the vast importance, of this statement is apt to be lost sight of from a somewhat peculiar cause. It must never be forgotten that, when it is asserted that two masses are equal to each other, what is really meant is that, if equal forces were to act upon each of the masses for the same time, the masses would receive the same velocity. As this test of the equality of masses is not practically convenient, the weighing scales have come into use for the purpose ; and though it appears to be true that when two masses have equal " sveights," as tested by the scales, then the masses are themselves equal, yet this is so far from being an obvious or necessary truth that it really is the most remarkable phenomenon connected with gravitation. The expression for the gravitation of two attracting masses must therefore depend solely upon their masses, upon their distance, and upon some specific con stant which is characteristic of the intensity of gravitation. Experiment shows that the gravitation" of a body towards the earth is directly proportional to its mass, and hence we see that the expression must be proportional to m, and as/ (m, m ) must be unaltered by the interchange of m and m , it appears finally that the gravitation of the two masses is emm -=-r 2 , whence is a numerical constant which is equal to the gravitation of two units of mass placed at the unit of distance. To form a de finite conception of the intensity of this force, we take some specific instances. It can be shown that two masses A and B, each contain ing 415,000 tons of matter, and situated at a distance of one statute mile apart, will attract each other with a force of one pound. If the masses of A and B remain the same, and if the distance between them be increased to two miles, then the intensity of the force with which the two masses gravitate together is reduced to one quarter of a pound. If either of the masses were doubled, the distance being unaltered, then the force would be doubled. If both the masses were doubled, then the force would be quadrupled.
§ 10. Motion of a Planet round the Sun.—The effect of gravitation when the bodies are in actual motion must next receive our attention. It may so happen that in consequence of the attrac tion of gravitation one of the bodies will actually describe a circle around the other, so that, notwithstanding the etfect of the attrac tion, the distance between the two bodies remains constant. We shall first explain, by elementary considerations, how it is possible for a planet to continue to revolve in a circular or nearly circular orbit about the sun in its centre ; and then we shall proceed to the more exact consideration of the form of the orbit, and the laws according to which that orbit is described.
Let S represent the sun (fig. 6), and let T be the initial position of the planet. If the planet be simply released, it will immediately begin to fall along the line TS into the sun. If, on the other Land, the planet were initially projected along the line TZ perpendicular to TS, the attraction of the sun at S will deflect the planet from the line TZ which it would otherwise have followed, and compel the planet to move in a curved line. The particular form of curve which the planet will describe depends upon the initial velocity. With a small initial velocity the deflecting power of the sun will have a more speedy effect than is possible when the initial velocity is considerable. The rapidly curving path TX will therefore correspond to a small initial velocity, while the flatter curve TV may be the orbit when the initial velocity is considerable. As the movement proceeds, the velocity of the planet will generally alter. If the planet were moving along the curve TY, it is at every instant after leaving T going farther away from the sun. It is manifest that the planet is thus going against the sun s attraction, and therefore its velocity must be diminishing. On the other hand, when the planet is moving along the curve TX, it is constantly getting nearer the sun, and the effect of the sun s attraction is to increase the velocity. It is therefore plain that for a path somewhere between TX and TY the velocity of the planet must be unaltered by the sun s attraction. With centre S and radius ST describe a circle, and take a point P on that circle exceedingly near to T. With a certain initial velocity it is possible to project the planet so that it shidl describe the arc TP. The attraction of the sun always acts along the radius, and hence in describing the arc TP the planet has at every instant been moving perpendicularly to the sun s attraction. It is manifest that under such circumstances the sun s attraction cannot have altered the velocity, for it would be impossible to give a reason for the velocity having been accelerated which could not be rebutted by an equally valid reason for the velocity having been retarded. We thus see that the planet reaches P with an unchanged velocity, and at that point the direction of the motion is perpendicular to the radius. It is therefore clear that after passing P the planet will again desciibe a small portion of the circle, which will again be followed by another, and so on, i.e. , the planet will continue to move in a circular orbit. We have therefore shown that, if a planet were originally projected with a certain specific velocity
