When we say that an asteroid's period is commensurable with that of Jupiter, we mean that a certain whole number of the former is equal to another whole number of the latter. For instance, if a minor planet completes two revolutions to Jupiter's one, or five to Jupiter's two, the periods are commensurable. It must be remarked, however, that Jupiter's effectiveness in disturbing the motion of a minor planet depends on the _order_ of commensurability. Thus, if the ratio of the less to the greater period is expressed by the fraction 1/2, where the difference between the numerator and the denominator is one, the commensurability is of the first order; 1/3 is of the second; 2/5, of the third, etc. The difference between the terms of the ratio indicates the frequency of conjunctions while Jupiter is completing the number of revolutions expressed by the numerator. The distance 3.277, corresponding to the ratio 1/2, is the only case of the first order in the entire ring; those of the second order, answering to 1/3 and 3/5, are 2.50 and 3.70. These orders of commensurability may be thus arranged in a tabular form, the radius of the earth's orbit being the unit of distance: +--------+----------------+-----------+ | Order. | Ratio. | Distance. | +--------+----------------+-----------+ | First | 1/2 | 3.277 | | | | | | Second | 1/3, 3/5 | { 2.50 | | | | { 3.70 | | | | | | | | { 2.82 | | Third | 2/5, 4/7, 5/8 | { 3.58 | | | | { 3.80 | | | | | | | | { 2.95 | | Fourth | 3/7, 5/9, 7/11 | { 3.51 | | | | { 3.85 | +--------+----------------+-----------+ Do these parts of the ring present discontinuities? and, if so, can they be ascribed to a chance distribution? Let us consider them in order. I.--The Distance 3.277. At this distance an asteroid's conjunctions with Jupiter would all occur at the same place, and its perturbations would be there repeated at intervals equal to Jupiter's period (11.86 y.). Now, when the asteroids are arranged in the order of their mean distances (as in Table II.) this part of the zone presents a wide chasm. The space between 3.218 and 3.376 remains, hitherto a perfect blank, while the adjacent portions of equal breadth, interior and exterior, contain fifty-four minor planets. The probability that this distribution is not the result of chance is more than three hundred billions to one. The breadth of this chasm is one-twentieth part of its distance from the sun, or one-eleventh part of the breadth of the entire zone. II.--The Second Order of Commensurability.--The Distances 2.50 and 3.70.