The method developed in Parts I and II are extended to treat all simple types of commensurability of the first order in the theory of the motion of the asteroids, that is, of the types 4/1, 3/2, 3/1, 2/1, 1/1. The motion in the immediate neighbourhood of the commensurability-point was proved to be stable in the first four cases. Thus the motion of the asteroids belonging to the Families: Themis, Flora and Coronis was proved to be stable including the secular perturbation of the third degree, although some of them are situated very near to the point of commensurability of the first order and are provided with small divisors of higher order in the classical treatment of the problem. The consideration of the secular perturbation of higher degrees than the third would affect the nature of the motion in the case of the coinmensurability of the types 4/1, 3/2, 3/1, 2/1. But the labour prohibited me to work out the discussion exhaustively. However the above discussion holds true for an interval of time of the order of magnitude of the reciprocal of the small quantity under consideration, that is, the result is true for the stability of the second degrce. The case 1/1 represents a very interesting example for the existence of the continuous and discrete distributions of the characteristic values for the periodic solutions. For certain values of the difference of the longitudes of an asteroid and of Jupiter there exist doubly infinite periodic solutions, but for the other values the eccentricities are related to the mean motions for the stability of the motion. It is expected that a similar circumstance occurs when we proceed to the discussion of the stability of higher degrees neglected in the above discussions. Thus certain relations between the eccentricities and the longitudes of the perihelia or the mean motions are expected to hold for the solution to be periodic although there were no such relations in the considerations of the stability of the second degree. The probability that an asteroid with these restrictions should exist is far small compared with the probability that an asteroid without such restrictions should exist. An asteroid near the point of commensurability are restricted in the above relations if it is stable. An asteroid far from the commensurability point of a simple type is limited by a quite little or no restriction. The restriction, on the other hand, is of higher order and can be neglected for an ordinary interval of time which we are need to consider, although there is some owing to the dense distribution of the points of commensurability of complicated types. Hence for such an asteroid we may regard that there is no restriction at all. Hence, through the consideration of the two probabilities in the above, the existence of the gaps near the points of commensurabilites of simple types and of the first order may be explained. So far it was proved that the appearance of the so-called small divisors of higher orders in the thcory of the sceular perturbation of higher degrees is only formal and that the apparent instability of the motion caused by the presence of the small dirisors is not of all esscntial for the actual character of the motion itself, when we conrine oursclurs to consider the stabilily of the second degrce

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