If they were equal in weight, they would meet mid-way. Now let us suppose the second body to have an onward motion given to it. The moment it felt the attraction of the first body, it would attract that in return, share its motion with it, and force it to move round their common centre of gravity. If the two bodies were of equal weight, they would revolve at equal distances from the centre of gravity; if not, the heaviest would move in the smallest orbit; but both would revolve in the same time. CHAPTER XV. LAWS OF MOTION-(CONTINUED.) Path of a Pro- Three general Laws of Motion. Composition of Forces. § 343. There are three general laws which a body obeys in its motion, whatever be the kind of body or the kind of force that impels it; whether it be a particle of dust driven by the wind, or a planet revolving in consequence of an original impulse through the celestial spaces. A body does not change its state, either of rest or motion, unless in consequence of some external cause. The effect is always proportional to the force impressed, and takes place in the direction in which the force acts. Action and reaction are equal and contrary. This law holds whether the bodies attract or repel one another, and whether they act at a distance or in apparent contact. A body is often acted on by several forces at once, and the effect of their joint action is an exact compound of their several effects, or the same as if each had acted successively. The body may be acted on by two forces applied at the same point of the body. If they act in the same direction, the resultant will be in that direction, and equal to their sum. If in opposite directions, the resultant will be in the direction of the greater, and equal to their difference. If at an angle, the resultant will be in the same plane, and represented by the diagonal of a parallelogram of which the two sides represent the simple forces. In this case it is less than the sum of the forces, and greater than their difference. Thus a body may move in a certain direction in consequence of one, two, or a dozen impulses. It is often desirable to resolve a single force into others to which it is equivalent, in order to find its effect in a given direction. We need only resolve the force into two, one of which is parallel to the given direction, and the other at right angles to it. The latter can have no effect in the given direction, and therefore the other will express the whole effect. Thus when we wish to know how much one planet draws another from the plane of its orbit, the line representing the influence of the planet and in the direction of their centres is considered the diagonal of a parallelogram of which one side shows how much the planet is moved from its orbit. Celestial motions are caused, not by two impulses, but by an impulse and a pressure. The impulse was imparted by a force of whose nature we can only form a vague guess. The pressure is the constant attraction of gravitation. If the heavenly bodies had received only the impulse, they would have moved on in straight lines forever. If other impulses had interfered successively, they would have moved on in broken lines, making angles with each other. Since the second force is a pressure, and acts incessantly, the straight lines will become infinitely short, and the path consequently will become curvilineal. Since a body, if left to itself, moves in a straight line, we may conclude, when it moves in a curve, without being compelled to it by a fixed obstacle, that there is a force of pressure constantly deflecting it from the direction of the tangent. We are now therefore ready to consider the effect of attraction on the motion of bodies. § 344. If a body is projected in the direction in which gravity draws it, its velocity is increased. If gravity acts directly contrary to the projectile force, it gradually weakens and at length overcomes it, as when an arrow shot vertically is brought to the ground. A more important case for astronomy than either of these is when the body is projected transverse to the direction in which the force draws it. The simplest instance of this motion that we can imagine is the motion of a stone when it is thrown from the hand in a horizontal direction. It does not move in a straight line. It begins to move in the direction in which it is thrown; but this direction is speedily changed. It continues to change gradually and constantly, and the stone strikes the ground moving at that time in a direction much inclined to the original direction. The most powerful effort that we can make is not sufficient to prevent the body from falling at last. This experiment therefore will not enable us to judge immediately what will become of a body (as a planet) which is put in motion at a great distance from another body which attracts it (as the sun). But it will assist us much in judging generally what is the nature of the motion when a body is projected in a direction transverse to the direction in which the force acts upon it. § 345. The general nature of the motion is this. The body describes a curved path, of which the first part has the same direction as the line in which it is projected. If A (Fig. 8, Plate I.) is the point from which the stone was thrown, and A H the direction in which it was thrown; and if we wish to know where the stone will be at the end of any particular time, (suppose three seconds,) and if the velocity with which it is thrown would, in three seconds, have carried it from P to F, supposing gravity not to have acted upon it; and if gravity would have made it fall from A to P, supposing it to have been merely dropped from the hand; then, at the end of three seconds, stone really will be at the point F. And it will have reached it by a curved path  F, of which different points the can be determined in the same way for different instants of time. The calculation of the stone's course is easy, because during the whole motion of the stone gravity is acting upon it, with the same force and in the same direction. The motion of a body attracted by a planet or the sun, where the force varies as the distance alters, and is not the same either in amount or direction at the point F as it is at the point A, cannot be computed by the same simple method. But the same method will apply, provided we restrict the intervals for which the calculations are made to times so short that the alterations in the amount of the force and in its direction, during each of those times will be very small. Thus, in the motion of the earth, as affected by the attraction of the sun, if we used the process that we have described to find where the earth will be at the end of a month from the present time, the place that we should find would be very far wrong. If we calculated for the end of a week, since the direction of the force and its magnitude would have been less altered, the error would be much less than before. Fig. 8, Plate I., shows the paths described in obedience to several different combinations of the projectile force with gravity. § 346. Every body which is under the influence of a constant attractive force, and of an impulse originally given, moves in a curved path round the centre of attraction. And on the other hand, when a body moves in a curve, there must be one impulsive and one constantly restraining force acting upon it. If the projectile force acted at perceptible intervals, the body would describe the perimeter of a polygon, having as many sides as the number of impulses given. Since the projectile force acts at intervals infinitely small, the polygon has an infinite number of sides, is a circle. The projectile force urging the body on at a tangent, diminishes its tendency to the centre or the centripetal force, and creates a centrifugal force in the opposite direction from the centripetal force. The centrifugal force arises from and is inseparable from a curvilinear track. It is not a tendency which the body originally has to fly from the centre, but arises from its constrained continuance in a curved orbit, when, if unattracted to the centre, it would proceed in a tangent. The projectile and centripetal forces cannot be directly compared, for they are different in kind, one is impulsive and the other is incessant; but the centrifugal force, generated by a given projectile force, acts incessantly like the centripetal, and may be compared with it. We may also calculate through how long a space gravity must act to balance a given projectile force. § 347. Every body moving under the influence of gravity describes one of the conic sections. If a cone is cut parallel to its base the section is a circle ; if cut obliquely to its base, but not in such a way as to intersect it, the section will be an ellipse more or less elongated. If it is cut parallel to the curved surface of the cone, the section will be a parabola; if perpendicular to the base and not through the axis of the cone, it will be a hyperbola. A cone divided by these sections best explains this. The curve described cannot be a circle unless the line of projection is perpendicular to the line of attraction, and unless the velocity with which the planet is projected is neither greater nor less than one particular velocity determined by the distance and mass of the attracting body. If it exceeds this velocity a little, or falls a little short of it, the body will move in an ellipse. If the projectile force gives a rapidity equal to that acquired by a body in falling through a height equal to one third the radius of the circle, and if also the projectile act at right angles to the centripetal force, the body will describe a circle. If the projectile force is to that required for a circle as 2 to 1, or equal to that acquired by a body falling through one half of the radius, the body will describe a parabola. Any ratio of the central forces between these two will cause the body to describe an ellipse. If the projectile force is stronger than that required for a parabola, or such as a body would gain by falling through |