must also involve the obliquity of the ecliptic, the inclination of the moon's orbit, and the direction of the moon's axis. I have been perforce compelled to omit the discussion of these attributes of the earth-moon system, and in doing so I have inflicted what is really an injustice on the tidal theory. For it is the chief claim of the theory of tidal evolution, as expounded by Professor Darwin, that it links together all these various features of the earth-moon system. It affords a connected explanation, not only of the fact that the moon always turns the same face to the earth, but also of the eccentricity of the moon's path around the earth, and the still more difficult points about the inclinations of the various axes and orbits of the planets. It is the consideration of these points that forms the stronghold of the doctrine of tidal evolution. For when we find that a theory depending upon influences that undoubtedly exist, and are in ceaseless action around us, can at the same time bring into connection and offer a common explanation of a number of phenomena which would otherwise have no common bond of union, it is impossible to refuse to believe that such a theory does actually correspond to nature. The greatest of mathematicians have ever found in astronomy problems which tax, and problems which greatly surpass, the utmost efforts of which they are capable. The usual way in which the powers of the mathematician have been awakened into action is by the effort to remove some glaring discrepancy between an imperfect theory and the facts of observation. The genius of a Laplace or a Lagrange was expended, and worthily expended, in efforts to show how one planet acted on another planet, and produced irregularities in its orbit; the genius of an Adams and a Leverrier was nobly applied to explain the irregularities in the motion of Uranus, and to discover a cause of those irregularities in the unseen Neptune. In all these cases, and in many others which might be mentioned, the mathematician has been stimulated by the laudable anxiety to clear away some blemish from the theory of gravitation throughout the system. The blemish was seen to exist before its removal was suggested. In that application of mathematics with which we have been concerned in these lectures the call for the mathematician has been of quite a different kind. A certain familiar phenomenon on our sea-coasts has invited attention. The tidal ripples murmur a secret, but not for every ear. To interpret that secret fully, the hearer must be a mathematician. Even then the interpretation can only be won after the profoundest efforts of thought and attention, but at last the language has been made intelligible. The labour has been gloriously rewarded, and an interesting chapter of our earth's history has for the first time been written.