Now, if the acceleration be directed to or from O, its mowhich is evidently ment about = ро constant; which is § 21. By means of (1) this gives since, if A be the area traced out by the radius-vector, dA = de 2 23. To determine the motion of a point when the accelerations to which it is subjected are given. This includes also, as will be seen, the determination of the motion when the component velocities are given. Let a, ẞ, y be the given accelerations parallel to the axes, we have Now a, B, y may be functions of x, y, z, t, or of two or more of these quantities. Equations (1) must be integrated as simultaneous differential equations if possible. Thus we have the values of de dy dz in terms of one or more of the quantities x, y, z and t; that is the component velocities are known. Another integration, if it can be performed, gives x, y, and zin terms of t; and, if the latter be eliminated from the three integrated equations, we have the two equations to the path in space, and thus theoretically the motion is completely determined. It is unnecessary to give examples of the integration of such equations, as the major part of the following chapters will be devoted to them. 24. So far for a single point. When more points than one are considered, Kinematics enables us to determine, from the given motions of all, their relative motions with respect to any one of them; or conversely, from the actual motion of one, and the motions relative to it of the others, to determine the actual motions of the latter in space. This depends on the following self-evident proposition. If the velocity of any point of a system be reversed in direction, and be communicated to each point of the system in composition with that which it already possesses, the relative motions of all about the first, thus reduced to rest, will be the same as their relative motions about it when all were in motion. For the proof it is sufficient to notice that if at every instant the distance of two points, and the direction of the line joining them be the same as for two other points, the relative motions of one of each pair about the other will be the same. The simplest illustrations of this proposition are furnished by the relative motions of objects in a vessel or carriage, which are independent of the common velocity of the whole-or, on a grander scale, of terrestrial objects, whose relative motions are unaffected by the earth's rotation, or by its motion in space. Since accelerations are compounded according to the same law as velocities, the above theorem is true of them also. 25. Two points describe similar orbits about each other and about any point dividing in a given ratio the line which joins them. AG GB Let A and B be the points, G a point in AB such that = a constant. The path of B about A will evidently be the same as that of A about B, since the length and direction of the line AB are the same whichever end be supposed fixed. Also if G be fixed the path of B about it will evidently differ from that of B about A by having corresponding radii vectores diminished in the ratio BG But this is the defi nition of similar curves. The same of course would hold with respect to the relative path of A with respect to G. This proposition will be found of considerable use afterwards, as it enables us materially to simplify the equations of motion of two mutually attracting free particles. 26. As an instance of relative motion, consider two points, one of which moves uniformly in a straight line, while the other moves uniformly in a circle about the first as center; to determine the path of the second point, the motion being in one plane. Take the line of motion of the first as the axis of x, v its velocity, the plane of the circle as xy, a the radius of the relative circular orbit, w the angular velocity in it, § 32. Suppose the revolving point to be initially in the axis. Also at time t suppose the line joining the points to be inclined at an angle to the axis of x. Then for the co-ordinates of the revolving point we have is the equation to the absolute path required. This belongs to the class of cycloids; it is prolate or curtate according as v is greater or less than aw, or the absolute motion of the first point greater or less than that of the other in its circular orbit. If the two are equal, or v=aw, we have the equation to the common cycloid, as is indeed evident, for the circular path may be supposed the generating circle, and the velocity of the center in its rectilinear path is equal to that of the tracing point about that center. 27. It is evident that, whatever be the relative path, if r, denote the relative co-ordinates of the second point with respect to the first at time t, x, y, and a the absolute co-ordinates at the same time, Now in the first case, when the motion of the first point, and that in the relative orbit are given, , r and are known functions of t, if therefore these values be substituted in (1), and t be eliminated, we shall have the equation between x and y, which is required. Again, if the absolute orbits of both are given, x, y, and are known in terms of t, and thus equations (1) serve to giver and in terms of t, which furnishes the complete determination of the relative path, and the circumstances of its description. 28. In any system of moving points, to determine the relative, from the absolute, motions; and vice versa. Let x, y, z,, x, y, z, be the co-ordinates of two of the points, x, y, z the relative co-ordinates of the second with regard to the first, u, v, w1, u, v, w, the velocities of each parallel to the axes, u, v, w the velocities of the second relatively to the first. The second group may be derived from the first by differentiation with respect to t. Now, when the actual motions of the two are given, all the subscribed quantities are known. Hence the above equations give the circumstances of the relative motion. Or if the actual motion of the first, and the relative motion about it of the second, be known, we have xyz, u v w x1 y117 u1v, w1, to find the other six quantities for the actual motion of the second in space. A second differentiation proves the statement in § 24 regarding relative acceleration." 29. Some of the best illustrations of this part of our subject are to be found in what are called Curves of Pursuit. These questions arose from the consideration of the path taken by a dog who in following his master always directs. his course towards him. In order to simplify the question the rates of motion of both master and dog are supposed to continue uniform; or at least to have a constant ratio. 30. As an instance of the curve of pursuit, suppose it be required to determine the path of a point which continually with uniform velocity u moves towards another which is describing a straight line with uniform velocity v. The curve of course is plane. Take the line of motion अ A P M |