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1. In 1833 the Royal Society did me the honour to publish, in its Transactions, a memoir of mine, entitled “Essay towards a First Approximation to a Map of Cotidal Lines;" and, in subsequent years, a number of further communications on the sub­ject of our knowledge of the tides, as deduced from observations of those phenomena. These later “Researches” have modified my first views,—a result which I from the first contemplated as probable, as I intended to imply by entitling my memoir “An Essay towards a First Approximation,” and as I expressed more fully in the memoir itself. I have also obtained from various persons, since my last communication to the Society, a considerable amount of recent tide observations, made in various quarters of the globe; and I am desirous of pointing out the general bearing of these addi­tional materials of knowledge. I wish especially to bring under the consideration both of mathematicians and of navigators, the problem of the tides of the Pacific Ocean. When I wrote my first memoir on the subject, our knowledge of the tides of that ocean was so imperfect, that I did not even venture upon a first approxima­tion to the cotidal lines. And I have since seen reason to believe that, not only for that ocean but for all large seas, the method of drawing cotidal lines which I formerly adopted, is very precarious. 2. There is another leading feature of the tides, which has been brought clearly into view in the course of these researches, which is of great interest and importance to the navigator, as well as to the mathematician, and of which I have assigned the laws in a general manner, and with an accuracy sufficient for most practical pur­poses; I mean the Diurnal Inequality which makes the common or semidiurnal tides differ alternately in excess and in defect. I have already examined various series of tide observations in which this diurnal inequality prominently appears; but I have now the means of showing it to be much more extensively distributed and larger in amount than has been supposed.

I. The Bakerian Lecture.—Researches on the tides. Thirteenth series. On the tides of the Pacific, and on the diurnal inequality