the negative sign being taken, because the motion is in the negative direction, viz. towards the fixed point; this gives the time of reaching the origin and the velocity at that point. 54. The motion now takes place on the negative side of the origin, and will manifestly be the same as if the moving point's velocity (-. a) were suddenly reversed, and the motion were on the positive side. Hence, as the law of the acceleration undergoes no change, the velocity will be destroyed in passing over the same space as before, and in the same time, i. e. the moving point will come to rest at a distance a on the negative side of the origin, after the lapse of a time π 2√λ Now the moving point is in exactly the same circumstances as it initially was, but on the negative side of the origin: the motion is therefore repeated. Whence it is easily seen that the motion is oscillatory between two points whose distances from the origin area and -a, and that the time of a complete oscillation is 4. which is independent of the initial π = 2π position of the moving point. (NEWTON, Prop. 38.) 55. If the point were not initially at rest, its motion may yet be determined from the case now treated of: for if another point were to be initially at rest at a distance a' from the origin, a' may be assumed to be of such magnitude as that, when this point arrives at the distance a from the origin, it may be moving with the same velocity as that which the first point initially has, which we will call v'. Then, by what has been already said, -a' sin (√.7), T a = a' cos (√7), v' = — where is the time required for the second point to move from its initial position (a') to the initial position (a) of the first point. These two equations will determine a and 7; and as both points have the same velocity at the distance a, and also there is the same acceleration on the motion of both, the subsequent motions of the two will be identical, and the motion of the first point will be deduced from the formulæ of the preceding Articles; where it must be borne in mind that the time is reckoned not from the actual beginning of the motion, but from the beginning of the hypothetical motion of the second point, i. e. that the time t in the preceding Articles has to be corrected by the quantity T. 56. If the direction of the acceleration were away from the fixed point, we should have and if the point be initially at rest at a distance a, we should 57. (ij) Let the acceleration vary as the square of the distance from the fixed point inversely, its direction being towards it. Then with the same notation as before taking the negative sign as before (Art. 52); a τα When t=0, 8=a, and .. C'-π=0; 2 * N.B. A is of 3 dimensions in space and -2 in time. 58. After this the motion is on the negative side of the origin, and by reasoning similar to that used in the preceding case, the moving point will come to rest at a distance a from the origin, cumstances as it initially was, but on the negative side of the origin, and therefore the motion is repeated. The whole motion is consequently oscillatory between points whose distances from the origin area and -a, and the time for a complete oscillation 59. If the motion of the point were supposed not to begin from rest, a similar artifice to that employed in Art. 55, may be made use of here also to determine the motion. 60. Any other such case of motion that may arise is to be investigated in a manner similar to the preceding, by integra terms of t, v, or s by the conditions of the problem. 61. (II) Let the initial direction of motion not pass through the fixed point; then it is clear that the path will lie in one plane, which passes through the fixed point and the initial direction of motion. Let a be the acceleration towards the fixed point taken for origin. Then if x, y be rectangular co-ordinates, and r, polar ◊ co-ordinates of the moving point at time t, s the length of path described, v the velocity, and p the radius of curvature; we have for the determination of the motion, as the acceleration is entirely along the radius vector, (See Arts. 32, 34, 35,) Either of these pairs of equations (which are equivalent to each other) will determine the motion. We shall proceed to investigate some general properties of this kind of motion, and then to discuss more fully its nature in the particular cases of the acceleration varying as the distance, or as the inverse square of the distance, from the fixed point. 62. From the second of equations (2) we get Now if A be the sectorial area swept out by the radius vector in time t, A and t being supposed to begin together; i.e. Act, or the area described in any time varies as the time of its description. (NEWTON, Prop. 1.) L. F |