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that there are three planets having mean motions between 550″ and 560″. There are also four planets between 560″ and 570″, and one between 570″ and 580″. Then there are no more till we pass 610″, when we find six planets between 610″ and 620″, followed by a multitude of others.

Examining the diagram we are able to distinguish five or six groups. The outermost one is between 400″ and 460″, and is nearest to Jupiter. The times of revolution are not far from eight years. Then there is a wide gap extending to 560″, when we have a group of ten planets between 540″ and 580″. From this point downwards the planets are more numerous, but we find very sparse or empty points at 700″, 750″, and 900″. Now the most singular feature of the case is that these empty spaces are those in which the motion of a planet would have a simple relation to that of Jupiter. A planet with a mean motion of 900″ would make its circuit round the sun in one third the time that Jupiter does; one of 600″ in half the time; one of 750″ in two fifths of the time. It is a law of celestial mechanics that the orbits of planets having these simple relations to another undergo great changes in the course of time from their action on each other. It was therefore supposed by Kirkwood, who first pointed out these gaps in the series, that they arose because a planet within them could not keep its orbit permanently. But it is curious that there is no gap, but on the contrary, a group of planets whose mean motion is nearly two thirds of that of Jupiter. Hence the view is doubtful.