which it would describe if the comet and sun were the only two bodies in existence. The earth, Jupiter, all the planets are, in fact, pulling and hauling at the unfortunate body: first one drags it a little one way, then another pulls it in a different direction. The real path of the comet about the sun is, then, a very complicated, wavy sort of a curve, which, as a rule, does not depart very much from the ellipse above figured.
Now, while mathematicians have succeeded in completely solving the problem of two bodies, yet, up to the present day, that of three or more bodies is still unsolved. If the sun and comet were the only two bodies in the universe, then could a mathematician, after a few moments' calculation, predict exactly where each would be a thousand years hence; could tell where they were ten thousand years ago. But as soon as there is introduced into such a simple system the earth, Jupiter, and the other planets, our mathematics fails to give a complete solution. All that can be done is to trace the course of the comet step by step, day by day, almost. We know its position to-day, and we can accurately calculate the direction and the amount of the pull of each planet; hence, we can find where it will be to-morrow, and, by repeating the process, where it will be the next day, and the next. Of course, this is a very laborious process; the calculation of the pull of a single planet requires the writing and the combination of one hundred and fifty numbers of six figures each. But, fortunately, the sun is over a thousand times as strong as the great planet Jupiter, and over three hundred thousand times as strong as the earth; so that, unless the comet approach very near to one of the larger planets, it will never deviate much from its simple orbit around the sun. The steps in our computation may be, therefore, lengthened. The ordinary length of step in such work is forty days; and, in a first computation, the pulls of the smaller politicians—as the earth, Venus, and Mars—can be neglected beside the very strong ones of Jupiter and Saturn.
As we wished to trace the history of comet V, we started with the earliest observed position, that of July 6, 1889, and we began by taking steps of forty days each. Thus the path the comet had traveled was slowly traced backward, and it was found to approach nearer and nearer to Jupiter. Proceeding backward thus over a period of two years, we find that in March, 1887, the pull of Jupiter was so strong that, in order to keep the work at all accurate, we were obliged to shorten the steps to ten days. Continuing thus, the pull of Jupiter grew stronger and stronger, until, in October, 1886, it was actually greater than that of the sun, and a change of method had to be used in order to trace the path beyond that point, and with this change in methods appears the interesting mathematical part of the problem.