Our attention, naturally, is most drawn to the difficult parts of an inquiry; the easier steps are apt to be overlooked. Thus, while much attention and deep thought have been bestowed on the solution of equations, on the computation of logarithms, on the construction of astronomical canons; such a simple process as the repeated addition of some known quantity, has not been considered worthy of the notice of arithmeticians. Yet the tables constructed by help of this process are much more numerous, and are as important as the others, often indeed serving as their foundation.

The case when the quantity to be added is expressed accurately in numbers, needs no remark; but when that quantity is represented only approximately, we have to consider the accumulation of that part which is unavoidably left off, and to see that this does not lead to an error of so much as the half of unit in the last place that is to be retained.

At the end of my previous paper on the Algebra of Relationship (Proc. Roy. Soc. Edinb., vol. x. p. 224), I promised to return to the investigation of the subject, as it seemed capable of further extension. This anticipation has proved correct, and I have now the honour of bringing before the Society the further developments I have made.

In this investigation we consider a particular class of objects, and that class is in its widest extent mankind, by which term I mean the entire number of men who have existed, exist, or will exist. The universal properties of the symbols are deduced from the universal properties of mankind. In our consideration of this universe of mankind, we restrict our attention to the classes into which it is divided by qualities depending on ties of anterior or of posterior relationship, that is, of consanguinity or of affinity.

Dust, fogs, and clouds seem to have but little connection with each other, and we might think they could be better treated of under two separate and distinct heads. Yet I think we shall presently see that they are more closely related than might at first sight appear, and that dust is the germ of which fogs and clouds are the developed phenomena.

This was illustrated by an experiment in which steam was mixed with air in two large glass receivers; the one receiver was filled with common air, the other with air which had been carefully passed through a cotton-wool filter and all dust removed from it. In the unfiltered air the steam gave the usual and well-known cloudy form of condensation, while in the filtered air no cloudiness whatever appeared. The air remained supersaturated and perfectly transparent.

The elements for the compution of eclipses are given in the “Nautical Almanac” with precision sufficient for all ordinary purposes; but, when we wish to compare the lunar ephemeris with actual observation for the purpose of verifying or of improving our data, we must go somewhat more minutely into the investigation.

Thus, in the List of Elements, p. 403, the changes in the right-ascension and declination of the sun and moon are supposed to be proportional to the times, while the moon's geocentric semidiameter, as well as the horizontal parallax, is supposed to be constant during the eclipse. In this way some exceedingly small errors are introduced into the calculation.

The preparation of adamantine carbon or diamond has exercised the genius of philosophers from the very earliest times; but it was not until the middle of the last century (1772) that Lavoisier established the diamond's true nature—notwithstanding the simplicity of the experiments required to demonstrate the fact—and showed it to consist of pure carbon in a crystallised state. Since that time very many attempts have been made to prepare it artificially, but until the recent and now famous experiments of Mr J. B. Hannay there has not been the slightest approach towards the solution of this problem. Great obstacles stood in the way of success, the chief being the complete insolubility of carbon in all known liquids, coupled with its non-volatility and infusibility; while the subject was rendered even more difficult and obscure, by ignorance of the conditions under which the diamond is produced in nature, its peculiar crystalline form, together with extreme rarity, indicating a probable very slow formation, and rare natural existence of the conditions necessary for its formation.

The apparatus consists of an ordinary sonometer with five feet clear space between the two end bridges, and having a wire stretched from one end to the other. A current from eight or ten Grove's cells, interrupted by a tuning fork which vibrates 128 times per second, is sent through the wire. At a distance about a fifth of its length from the end of the wire a large electro-magnet, with pointed poles, is placed so that the line joining the poles is at right angles to the wires. The poles are also put close to the wire, but leaving it freedom to vibrate. When a current from eight Grove's cells is sent through the coils of the electro-magnet, the wire begins to sound, and by altering its tension the fundamental note of the wire comes out loud and clear. The wire is also seen to be vibrating as a whole; and the vibrations are also seen to be in the plane perpendicular to the line joining the poles. By shifting the electro-magnet a little, and regulating the tension of the wire, it is seen to divide into nodes and loops with one, with two, with three nodes in its length, thus giving the harmonics of the fundamental note.

A large number of unicellular plants and animals, and many of the cellular units which make up the complex tissues of man and more evolved animals, exhibit certain movements. These are termed “Amœboid,” as they are well seen, and were at an early period studied in the Amœba.

If an amœba or a white blood corpuscle of the newt be examined with a good lens, the following facts, among others, may be made out. The corpuscle looks like a granular lump of jelly containing two or three nuclei, and it is, we will suppose, spherical to begin with. Soon the shape changes, for a little process is seen to protrude at one side, which may become retracted, or go on elongating. In the substance of the cell and in the processes, movements may be observed, consisting evidently of a flowing of the protoplasm, as indicated by the embedded granules which are carried along. The little processes are termed pseudopodia, and vary much in shape, some being thick and either pointed or club-shaped, others are filamentous, as in the perforated Foraminiferæ, where large numbers of these are seen radiating from the test. These movements may be automatic, and they are modified by various external forces; for example, heat and electricity.

Although meconic acid is constantly taken, even in the most recent handbooks of chemistry, as an instance of a tribasic acid, it is by no means certain that it possesses that nature. Some years ago, Dittmar and Dewar investigated the matter, and came to the conclusion that meconic acid is dibasic though triatomic; but their experiments are not supposed to completely elucidate the subject. All published statements regarding this acid are consistent with it being only dibasic, if we except one or two analyses of its metallic salts. Only two ethyl ethers are known, while hydromeconic acid, which is formed from meconic acid by the action of sodium-amalgam, forms dibasic salts alone. With morphia and with aniline tribasic compounds are not known, though the dibasic salts are easily prepared.

The crystallization of silica from fused metals, although at first sight appearing to be of little importance, nevertheless presents some features of peculiar interest. It also constitutes a field almost entirely new to the investigator, though the subject is one which, from a technical point of view, may prove to be of very considerable importance.

I have therefore undertaken the examination of some of the facts relating to this subject—at first more particularly inquiring into the nature of the change which occurs when silica itself is kept at a high temperature for a number of hours and subsequently submitted to a process of very slow cooling.

The experiments of Professor Crum Brown and the author on the “Thetines” and their derivatives have clearly shown that very striking analogies exist between certain compounds of nitrogen and sulphur. Thus sulphide of methyl closely resembles trimethylamine (and ammonia), in many of its reactions, and in the products which it gives rise to. Like trimethylamine it combines with a molecule of bromacetic acid, and the resulting product, which was named hydrobromate of dimethyl-thetine, behaves in certain respects like the compound of bromacetic acid and trimethylamine (hydrobromate of betaine).

Analogous phosphorus compounds have been obtained: in the ethyl series by Hofmann, in the methyl series by A. H. Meyer. The latter compound is simply the betaine salt in which the nitrogen is replaced by phosphorus, and may be called a salt either of phosphorus-betaine or of trimethyl-phosphorus-betaine.

The difficulty of preparing tertiary phosphines either by heating alcohols with phosphonium iodide or by acting on zinc ethers with terchloride of phosphorus, of which we have had much experience, induced us to seek for other and less troublesome methods for obtaining these bodies in considerable quantity.

The ease with which phosphorus combines with certain metals, and the readiness with which the resulting compounds react on haloid ethers of hydrocarbon radicals led us to think that, if only comparatively pure metallic phosphides could be obtained by a simple process, we should have no difficulty in preparing the tertiary phosphines and phosphonium compounds.

Just as the past progress of astronomy has, in a great measure, been due to the improved construction of our instruments, so its future progress must depend much on the accurate confection of our telescopes, and most of all on the excellence of the object-glasses.

In the actual formation of a compound lens, after the thicknesses and curvatures have been fixed, we have to make the tools to the proper shapes and with those to grind and polish the glass. It is impossible, even with the greatest care, to bring the workmanship so close to the computations as that appreciable differences may not be found, and hence the finished lens hardly ever comes so near to our expectations as that it may not be necessary to test it by actual trial. One important matter to be so settled is the exact position of the principal focus, since that is needed for determining the length of the tube.

The author described the results of a number of experiments, made during the examination of the “Challenger” Deep-Sea Thermometers, with the view of testing, at pressures of 3 tons weight per square inch and upwards, Thomson's formula for the heat developed by compression (Proc. Roy. Soc., 1857, p. 568).

When, for instance, the bulb of one of the thermometers was surrounded by a shell of lard upwards of half an inch thick, the total effect produced by a pressure of 3½ tons weight was 5° F.; while for the same pressure, without the lard, the effect was only l°·8 F. The temperature of the water in the compression apparatus was 43° F., so that the temperature effect due to the compression of water was less than 0°·2 F. In obtaining this number it was assumed from Kopp's experiments that the coefficient of expansion of water at a temperature t° C., near its maximum density point (roughly, 4° C.), . Hence the effect due to the compression of the lard was 3°·4 F., or about 1° F. per ton weight. This is subject to corrections (which will increase its value) depending on the heat developed by friction in the pump and in the narrow connecting tubes, and on another cause not yet fully ascertained.

This family includes two of Savigny's genera—Boltenia and Cynthia, and may be defined by their common characters if we omit or modify one point on which considerable stress is usually laid, namely, the number of lobes round the branchial and atrial apertures.

The amended definition of the family is as follows:—

Body attached, sessile or pedunculated; apertures four-lobed or having less than four lobes.

Test coriaceous; rarely cartilaginous or gelatinous.

This paper contains a detailed account of the structure of the soft parts of the four varieties of Pedicellariæ in this species describes particularly certain new pseudo-skeletal structures, and concludes with a reconciliation of the hitherto contradictory views as to the structure of Echinoderm muscles.

The author had noticed that a jet of mercury from a funnel, falling horizontally and nearly tangentially on a slightly inclined glass plate, seemed to roll upwards along the plate. Attributing this to a rotation of the jet, he endeavoured, with success, to re-produce the phenomenon with water jets escaping from a rapidly rotating tube, and falling on a slightly inclined glass surface covered with a thin layer of paraffin.