Let us now examine how far these conclusions will be altered if the body from time to time is forced towards S. We will suppose it moving uniformly from A to B as before, no matter for the present how it got to A, or into the direction AB. If left to itself it would, in an equal time (say 1´´) go through BC´ in the same straight line with and equal to AB. But just as it reaches B, and is beginning to move along BC´, let it be suddenly pulled towards S with a motion which, had it been at rest, would have carried it in the same time, 1´´ through any other space B_c_. According to the second law of motion, its direction during this 1´´, in consequence of the two motions combined, will be along BC, the diagonal of the parallelogram of which BC´, B_c_, are sides. In this case, as this figure is drawn, BC, though passed in the same time, is longer than AB; that is to say, the body is moving quicker than at first. How is it with the triangular areas, supposed as before to be swept by a string constantly stretched between S and the body? It will soon be seen that these still remain equal, notwithstanding the change of direction, and increased swiftness. For since CC´ is parallel to B_c_, the triangles SCB, SC´B are equal, being on the same base SB, and between the same parallels SB, CC´, and SC´B is equal to SBA as before, therefore SCB, SBA are equal. The body is now moving uniformly (though quicker than along AB) along BC. As before, it would in a time equal to the time of passing along BC, go through an equal space CD´ in the same straight line. But if at C it has a second pull towards S, strong enough to carry it to _d_ in the same time, its direction will change a second time to CD, the diagonal of the parallelogram, whose sides are CD´, C_d_; and the circumstances being exactly similar to those at the first pull, it is shewn in the same manner that the triangular area SDC = SCB = SBA.
Thus it appears, that in consequence of these intermitting pulls towards S, the body may be moving round, sometimes faster, sometimes slower, but that the triangles formed by any of the straight portions of its path (which are all described in equal times), and the lines joining S to the ends of that portion, are all equal. The path it will take depends of course, in other respects, upon the frequency and strength of the different pulls, and it might happen, if they were duly proportionate, that when at H, and moving off in the direction HA´, the pull H_a_ might be such as just to carry the body back to A, the point from which it started, and with such a motion, that after one pull more, A_b_, at A, it might move along AB as it did at first. If this were so, the body would continue to move round in the same polygonal path, alternately approaching and receding from S, as long as the same pulls were repeated in the same order, and at the same intervals.