Many attempts have been made to interpret the processes of division and repetition, in terms of mechanics, or at least to refer them to their nearest mechanical analogies, so far with little success. The problem is beset with difficulties as yet insurmountable and of these one must be especially noticed. In the living thing the process by which repetition and patterns come into being consists partly in division but partly also in growth. We have no means of studying the phenomena of pattern-formation except in association with that of growth. Growth soon ceases unless division takes place, and if growth is impossible division soon ceases also. In consequence of this fact that the final pattern is partly a product of growth, it can never be used as unimpeachable evidence of the primary geometrical relations of the members as laid down in the divisions. In the last chapter in referring to the problem of repetition I introduced an analogy, comparing the patterns of the organic world with those produced in unorganised materials by wave-motion. In the preliminary stage of ignorance, having no more trustworthy clue, I do not think it wholly unprofitable to consider the applicability of this analogy somewhat more fully. It possesses, as I hope to show, at least so much validity as to encourage the belief that morphology may safely discard one source of long-standing error and confusion. Those who have studied the structure of parts repeated in series will have encountered the old morphological problem of "Serial Homology," which has absorbed so much of the attention of naturalists and especially of zoologists at various periods. This problem includes two separate questions. The first of these is the origin in evolution of the resemblance between two organs occurring in a repeated series, of which the fore and hind limbs of Vertebrates are the prerogative instance. From the fact that these resemblances can be traced very far, often into minute details of structure, many anatomists have inclined to the opinion that the resemblance must originally have been still more complete, and that the two limbs, for instance, must have acquired their present forms by the differentiation of two identical groups of parts.