Notwithstanding the rapid progress of zoology in other departments, the Heteropoda still remain imperfectly known, if one may form a judgment from the scantiness of definite information respecting them to be found in systematic works. Nearly all the available space is usually occupied with an exposition of the errors and doubts of the great men who gave us the first outlines of the order, while comparatively little is done to improve the subject, or make it intelligible to the student.

1. It is a well-known property of simple harmonic functions, that the sum of any two or more of them having the same period, is itself a simple harmonic function having the same period as its components. The same thing must be true of their mean, since this is equal to the sum divided by a constant; and it will still be true when the number of components is indefinitely great, and the mean becomes an integral.

Nearly a hundred years ago, Dr Heberden of London made the following experiment:—Having prepared three exactly similar rain-gauges, he placed one of them on the roof of Westminster Abbey, another on the roof of a neighbouring house, but at a much lower level, and the third in the garden of the same house. At the end of twelve months he found that the gauge on the roof of the Abbey had received 12·099 inches of rain, the gauge on the neighbouring house-top 18·139 inches, and the gauge on the ground 22·608 inches.

This paradoxical result required, of course, to be confirmed by other observers, and in other localities; and the similar results obtained by Dobson, Dalton, Howard, and especially by Arago at the Paris Observatory, and by Phillips at York, have amply supplied all that was wanting in this respect.

In the years 1823 and 1826 I communicated to this Society two papers “On the Existence of Two New Fluids in the Cavities of Precious Stones and other Minerals.” These two fluids were generally found together in the same cavity, though sometimes the cavities were occupied only by one of them. They were perfectly transparent and immiscible. The denser of the two occupied the angles of the cavities, or the necks, or narrow passages, or canals which united two or more larger cavities; while the rarer fluid floated, as it were, on the other in deep cavities, or filled the body of shallower ones, with the exception of a circular vacuity, which diminished and disappeared with the slightest increase of temperature, or enlarged itself and disappeared in consequence of the fluid being converted into vapour.

The object of this paper will be, to give a connected view of some theorems of importance, which are often found in books rather obscurely demonstrated, and in some cases are inaccurately given, or are liable to exceptions which are not mentioned.

Before entering on the main subject of this paper, the composition of rainwater, I shall give a brief account of the rain that has fallen in the Lake-District during the year just past; and this I shall do chiefly by means of tables. The quantity of rain registered has been so greatly in excess of any former year, that of itself it is deserving of record, and the more so, comparing the weather which has prevailed here with the weather in the south, for the most part, happily for the harvest, as remarkable for an opposite state, an excess of dryness.

The very curious Parasitic Crustacean, the structure and larval form of which we propose to consider in the following communication, we obtained in the month of August last, in considerable numbers, in the branchial pouches of a Lophius piscatorius, which we captured at Whitby in Yorkshire.

The first example of this, apparently little known, species of Parasitic Crustacean appears to have been noted by Professor Otto. The celebrated Swedish naturalist Retzius was, however, the first to give, in 1829, an anatomical description of it. He named it Lernœa Dalmanni. His description was accompanied by several figures, which, though in many respects imperfect, enable one to recognise the chief external characters of the animal. He found three specimens at Christian Sound, in the nasal cavity of Raia Batis. Von Nordmann obtained from Rudolphi the specimen discovered by Otto, but it was so injured, that he adopted, in his account of the anatomy of the animal, the description of Retzius. Some years afterwards, in 1836, Kröyer added it to the Danish fauna. He states that he obtained two specimens from the nasal chamber of a skate brought to him by a fisherman from Aal-back, and that specimens from Iceland had been for several years in the possession of the Natural History Society. As naturalists had now begun to subdivide the old Linnean genus Lernæa into various genera, Kröyer added this animal to the genus Lerneopoda of De Blainville, and, continuing its specific name, termed it Lerneopoda Dalmanni.

As the means for making observations on the heavenly bodies become more and more exact, astronomers are compelled to introduce new refinements into their calculations; new inequalities are discovered, and the computation of those whose sources are already known has to be carried to a greater number of approximative steps.

Discussions on the amount of solar parallax, of aberration and of nutation, are now carried on to the third and fourth decimal fraction of a second; with such a refinement of computation, it seems almost impossible to proceed too far in the refinement of theoretical deductions, and on that account, it may not be inopportune to discuss the influence which the sun's and moon's attraction exert upon the direction of the plumb-line.

In searching for crystals of titanium between films of mica, I was surprised to observe something resembling the remains of insects. Upon submitting the specimen which contained them to a more careful examination, I found, in different parts of it, five completely formed Acari, all of which were much smaller than those found in cheese, brown sugar, and damp bran, specimens of which I obtained for the purpose of comparison.

When occupied several years ago in studying the fluid and gaseous cavities in minerals, my attention was drawn to certain remarkable formations in perfectly transparent specimens of calcareous spar. These formations were of two kinds—the one apparently of vegetable origin, and the other obviously mineral.

It does not always happen that the memory of inquirers into nature, who have the merit or the fortune to strike first into a right path, is cherished as it deserves. This remark applies forcibly to the eminent person, whether regarded as a physiologist or as a physician, of whose life aud labours a brief memoir is now laid before the Society. The name of Robert Whytt was familiar to his contemporaries both at home and abroad. Increase of distance should hardly yet have dimmed its lustre. Yet, in proportion as the views which he initiated have expanded more and more in growing to maturity, the less and less is heard of their author. Biography—which never did Whytt great justice—begins already to put him aside. A few particulars of his life, with a catalogue of his works, have hitherto been common in books of that description, principally in those of Germany and France. In some newer French biographies his name has dropped out. But of a late Edinburgh Biographical Dictionary, extending to not a few volumes, while restricted to the lives of eminent Scotsmen, it will hardly obtain credit that an early luminary of the rising University, conspicuous among the European leaders of medical science during a busy period of the eighteenth century, should, amidst a cloud of mediocrity, be there sought for in vain.

1. The experiments to which I shall have to refer in this paper were all made above ten years ago. A brief report of the general results, so far as they were then obtained, was made to the British Association in 1852, at whose expense the instrumental apparatus had been provided. The causes of the delay which has occurred in the publication of the numerical results will be presently explained.

1. The object of the present paper is to draw a comparison between the results of the mechanical theory of heat, and those of the recent experiments of Messrs Fairbairn and Tate on the density of steam, published in the “Philosophical Transactions” for 1860.

1. For eighteen years it has pressed on my mind, that essential principles of Thermo-dynamics have been overlooked by those geologists who uncompromisingly oppose all paroxysmal hypotheses, and maintain not only that we have examples now before us, on the earth, of all the different actions by which its crust has been modified in geological history, but that these actions have never, or have not on the whole, been more violent in past time than they are at present.