On the basis of this assumption our first problem is to estimate the frequency with which a star, visible or dark, is likely to approach near enough to the sun to produce a _tidal_ effect four times that of Jupiter. The number of visible stars is known or at least well estimated. As to dark stars, which have grown cool, Arrhenius believed that they are a hundred times as numerous as bright stars; few astronomers believe that there are less than three or four times as many. Dr. Shapley of the Harvard Observatory states that a new investigation of the matter suggests that eight or ten is probably a maximum figure. Let us assume that nine is correct. The average visible star, so far as measured, has a mass about twice that of the sun, or about 2100 times that of Jupiter. The distances of the stars have been measured in hundreds of cases and thus we can estimate how many stars, both visible and invisible, are on an average contained in a given volume of space. On this basis Jeans estimates that there is only one chance in thirty billion years that a visible star will approach within 2.8 times the distance of Neptune from the sun, that is, within about eight billion miles. If we include the invisible stars the chances become one in three billion years. In order to produce four times the tidal effect of Jupiter, however, the average star would have to approach within about four billion miles of the sun, and the chances of that are only one in twelve billion years. The disturbing star would be only 40 per cent farther from the sun than Neptune, and would almost pass within the solar system. Even though Jeans holds that the frequency of the mutual approach of the sun and a star was probably much greater in the distant past than at present, the figures just given lend little support to the tidal hypothesis. In fact, they apparently throw it out of court. It will be remembered that Jeans has made assumptions which give as high a frequency of stellar encounters as is consistent with the astronomical facts. We have assumed nine dark stars for every bright one, which may be a liberal estimate. Also, although we have assumed that a disturbance of the sun's atmosphere sufficient to cause a glacial period would arise from a tidal effect only twenty times as great as the difference in Jupiter's effect when nearest the sun and farthest away, in our computations this has actually been reduced to thirteen. With all these favorable assumptions the chances of a stellar approach of the sort here described are now only one in twelve billion years. Yet within a hundred million years, according to many estimates of geological time, and almost certainly within a billion, there have been at least half a dozen glaciations.