Search

In the second volume of the Memoirs of the National Institute, M. Bossut announces a theorem relative to the solidity of a sphere similar to Viviani’s, by which quadrable portions of a hemispherical vault are assigned. M. Bossut withholds the analysis that led him to his result, but mentions that it involves an integration much more complicated than that which occurs in Viviani’s problem. In the present paper, Mr. Woodhouse furnishes the analysis that leads to the result announced in Bossut’s theorem, and, by a transformation of the co-ordinates of the sphere, arrives at a differential expression, the integration of which does not appear more complicated than that employed in the solution of Viviani’s problem.

The author, in the prefatory part of this paper, points out the dif­ference between the two methods of solving problems,—the one using lines and diagrams as the signs of quantity, and making an individual to represent a genus; and the other employing generic terms and signs, which bear no resemblance to the things signified: and insists that, in order to make the process of deduction distinct, exact, and luminous, only one of the two methods ought to be adhered to. This, he says, has not been sufficiently attended to, expressions and formu­las of the two methods having often been blended together, the con­sequence of which has been much ambiguity and paradox; since the true method of combining algebraical formulas cannot be well under­stood, unless we duly attend to their true analytical source and com­bination. To show that the language of algebra need not be infected with the mode of expression adopted by geometricians, and that it is of itself an adequate instrument of argumentation, is the principal object of Mr. Woodhouse’s paper. And he declares that he has entered on this inquiry, not merely for the sake of gratifying specu­lative curiosity, being firmly of opinion that the process of calculation will be much more direct, sure, and expeditious, if it be duly freed from all foreign encumbrances. In order to illustrate and confirm this opinion, he has selected a few cases from those expressions and formulas which are supposed to require for their solution the aid of geometrical theorems, and of the properties of curves.

The importance of the subject handled in this paper, which, as it particularly relates to the process of tanning leather, will be allowed to be of sufficient magnitude, has of late excited the attention of several able philosophers, among whom Mr. Seguin was the first who ascertained the peculiar vegetable matter which is essential to this process, and which is possessed of the characteristic property of precipitating gelatine from its solutions. Mr. Proust has since investigated many other properties of this substance ; but neither these, nor any other chemists, have as yet carried their investigations so far as to determine the various affinities of tannin, and especially how its action upon animal matters is modified by combination with other substances. This task was reserved for our author, who during the two last years bestowed most of his leisure hours on a course of experiments on this subject ; and he here lays before the Society an account of their general results. His chief design was to elucidate the practical part of the process ; but in pursuing it he found himself necessarily led to general chemical inquiries concerning the analysis of the different vegetable substances containing tannin, and their peculiar properties. The paper consists of five parts, the titles of which are as follows :—1. Observations on the analysis of astringent vegetable infusions. 2. Experiments on the infusions of galls. 3. Experiments and observations on the extracts of Catechu, or Terra Japonica. 4. Experiments and observations on the astringent infusions of barks, and other vegetable productions ; and 5. General observations.

After some general observations on the art of embalming, as it was practised by the ancient Egyptians, and on the various kinds of animals embalmed by them, Mr. Pearson proceeds to- give a particular description of the very perfect mummy of an Ibis, which forms the chief subject of the present paper. This mummy was taken out of the catacombs at Thebes, by the late Major Hayes, in the year 1802 or 1803. It was enveloped in cloth, and contained in an earthen jar, similar to those which are found at Saccara. Upon unrolling the bandage with which the mummy was covered, it was found to consist of strips of cloth, about three inches broad, which were strong and firm. The first circumvolutions of this cloth separated easily; but as the work proceeded, they were found to adhere more firmly, and at last were so closely united, that it was necessary to divide them by means of a strong knife. Each layer of cloth seemed to have been imbued with some bituminous substance in a liquid state; and the bandages were further secured by means of thread, in such a manner that the whole mass was rendered firm and coherent; when the bandage was removed, the bird appeared to be covered with the same kind of bituminous substance that had cemented the strips of cloth. As much of this substance as could be removed without injuring the bird was now carefully taken off; and after the labour of several hours, Mr. Pearson succeeded in displaying the whole bird as it had been originally deposited by the embalmer.

Mr. Knight, in the papers formerly communicated by him to the Royal Society, endeavoured to prove that the fluid by which the va­rious parts added to trees, &c. are generated, has previously circu­lated through their leaves, either in the same or in the preceding season, and has subsequently descended through their barks. There is, however, a circumstance stated by Hales and by Du Hamel, which appears to militate against the above hypothesis, namely, that when two circular incisions are made, at a small distance from each other, through the hark, round the stem of a tree, and the bark between these incisions is wholly taken away; that portion of the stem which is below the incisions continues to live, and to increase in size, though much more slowly than the parts above the incisions. The above-mentioned naturalists have also observed, that a small elevated ridge is formed round the lower lip of the wound, which makes some slight advances to meet the hark and wood, projected in larger quantities from the upper lip of the wound. Our author, in a former paper, attempted to explain the above cir­cumstance, by supposing that a small part of the true sap, descending from the leaves, escapes downwards, through the porous substance of the alburnum: in another paper he has shown, from the growth of inverted cuttings, the existence of a power in the alburnum to carry the sap in different directions; and he now describes some ex­periments made in order to show that the conclusions drawn by him are not inconsistent with the facts stated by Hales and Du Hamel; and that although the ascending sap usually rises through the al­burnum and central vessels, yet the alburnous vessels appear to be also capable of an inverted action, when such action becomes neces­sary to preserve the existence of the plant.

This paper is merely an extension of one formerly communicated to the Society by Mr. Robertson, and printed in the Philosophical Transactions for the year 1795. It is, the author says, so far as re­lates to the raising of integral powers, the same as that paper, and is confessedly new only to the extent mentioned in the title, namely, that the present demonstration is applicable when the exponent is a positive or a negative fraction. The nature of the paper is obviously such, as to render it unsusceptible of abridgement.

If, Mr. Manning observes, there existed as full and extensive lo­garithmic tables as ever will be wanted, and of whose accuracy we were absolutely certain, and if the evidence for that accuracy could remain unimpaired through all ages, then any new method of com­puting logarithms would be totally superfluous, so far as concerns the formation of tables, and could only be valuable indirectly, and inasmuch as it might show some curious and new views of mathe­matical truth. But the above kind of evidence is necessarily im­paired by the lapse of time, even while the original record remains, and still more when the record must from time to time be renewed by copies; nor is the uncertainty of copies being accurately taken greater in any case than in that of copied numbers. It is conse­quently useful to contrive new and easy methods for computing new tables, or for examining those we already have; and it is particularly useful to contrive methods by which any part of a table may be ve­rified, independently of the rest; for by examining parts taken at random, we may, in some cases, acquire a moral certainty respecting the accuracy of the whole. Among the various methods of computing logarithms, none, our author says, possesses the advantage of forming them with tolerable ease, independently of each other, by means of a few easy bases. This desideratum, he trusts, the method described by him will supply; being very easy of application, as it requires no division, multiplica­tion, or extraction of roots, and has its relative advantages highly in­creased by increasing the number of decimal places to which the computation is carried.

The same property which at the last meeting was stated by Mr. Garrard to belong to the tangents of any three parts of a semicircle, was in this paper extended to all cases of trisection of the whole circle; but the demonstration of course could not be read to the Society.

Since the degree and manner in which the circulation of the blood depends upon the muscular and elastic powers of the heart and arteries are questions belonging to the most refined departments of hydraulics, the author has already submitted to the Society those general principles upon which he designs, in the present lecture, 1st, to inquire what would be the nature of the circulation if the vessels were as inelastic as glass or bone; 2ndly, in what manner the pulse would be transmitted if the tubes were merely elastic; 3rdly, what actions may be ascribed to their muscular coats; and, lastly, what disturbances are occasioned in different kinds of fevers and inflammations. In order to determine the velocity of the blood in different parts, it is necessary to estimate the pressure by which it is urged forward, and the resistance opposed to its motion. From the experiments of Hales, the pressure may be considered as equivalent to a column of seven feet. In order to calculate the resistance , the author employs the theorems contained in his former communication, and adopts the measurements of Keill and others, for the diameters of the aorta, and of its successive subdivisions. The quantity of blood in the arteries is estimated at nine or ten pounds; its velocity in the aorta about eight inches and a half in a second; while that in the capillary arteries is about 1/93rd of an inch in a second (the diameter of these vessels being about 1/1100th of an inch). The resistance that would be opposed to water circulating under the same circumstances is calculated to be equivalent to a pressure of a column of twenty inches; but the resistance to the motion of the blood is supposed in consequence of its viscidity to be about four times as great, and is consequently stated as eighty inches.

This calculus, which is considered as the largest on record, weighs 44 ounces avoirdupois. It was taken after death from the bladder of Sir James Ogilvie, who had submitted to a fruitless attempt to remove it by the usual operation of lithotomy, rather than prolong an existence extremely miserable from this among other consequences of a blow on his back thirty years before. The stone so completely filled the cavity of the bladder, that it was with difficulty taken out, although there was no real adhesion. Its texture was less compact than that of calculi in general, but agreed in appearance with that species which has been called the Fusible Calculus; and it was ascertained by Dr. Powel to contain the same ingredients as it was composed of, the ammoniacal phosphate of magnesia, with phosphate of lime.

An opinion having been entertained by many persons, that alcohol which has been distilled from wine does not exist ready formed in the liquor, but is generated during the process of distillation, Mr. Brande undertook a repetition of Fabroni’s experiment, on which this opinion is principally founded; but when he added four ounces of dry subcarbonate of potash to eight fluid ounces of port, no alcohol was separated, although some of the same wine had previously been ascertained to yield one fifth part of alcohol by distillation. When the same experiment was repeated on the same wine, to which one seventh part of alcohol had been previously added, still none was separated by subcarbonate of potash: but when so much as one third part had been added, then a very small proportion was found to float upon the surface after it had stood twenty-four hours. When madeira or sherry were employed instead of port, the results were nearly the same. Since the method of Fabroni failed of detecting the presence of alcohol, unless the quantity was very considerable, it became necessary to have recourse to some other method of proving or disproving the presence of it as a product of fermentation; and Mr. Brande conceived, that if it were formed by the heat applied in distillation, the quantity should in that case be different when the same liquor was distilled at different temperatures.

Sir Isaac Newton, who first considered the figure of the earth and planets, confined his view to the supposition of their having been originally in a fluid state; and he conceived them to retain the same figure which they assumed in their primitive condition; and those mathematicians who succeeded him in the same path of inquiry have seldom ventured beyond this limited hypothesis, and have shown, that when a body composed of one uniform fluid revolves about its axis, or even if it consists of several fluids of different densities, its parts will be in equilibrium, and it will preserve its figure when it has the form of an elliptic spheroid of revolution oblate at the poles. But though the supposition of original fluidity of the mass simplifies the investigation, it does not seem to be warranted by what we see of the surface; for in that case, Mr. Ivory observes, the arrangement of all the heterogeneous matters would have been according to their densities; those least dense occupying the surface with gradual increase of density to the centre; whereas, on the contrary, nothing can be more irregular than the density of such solid parts of the earth as come under our observation, and the elevation of continents above the level of the sea, as well as the depths of the different channels which contain the waters of the ocean.

In his second paper, Mr. Ivory investigates the attractions of that particular class of spheroids mentioned in the former; for though it is to these that the theorems of Laplace may strictly be applied, it is liable to the important objection, that the terms of his series near the beginning cannot be found without previously computing all the rest. The analysis of Mr. Ivory, on the contrary, is direct; and every term of his series is deduced directly from the radius of the spheroid. In an appendix to these papers, Mr. Ivory adds some remarks upon a memoir of Lagrange, upon the same subject, published at Paris in December 1809, but which had not till lately been received in this country.

The substance here examined by the author, we are told, was first made known by the celebrated Klaproth. It has been ranked as a distinct principle, soluble in water, but insoluble in alcohol or ether, and convertible, by the action of nitric or oxymuriatic acids, into resinous matter no longer soluble in water, but now rendered soluble in alcohol, by a supposed union with oxygen derived from these acids. Mr. Smithson being in possession of ulmin, sent to him from Pa­lermo by the same person who had furnished M. Klaproth with the subject of his researches, has made various experiments, which lead to a different opinion of its nature from that which has been enter­tained.

In a paper communicated to the Royal Irish Academy by Dr. Mooney, the method of exterminating any number of quadratic surds is pointed out by successively squaring them when brought alone to one side of the equation; and the present is an extension of the same method: first, to all surds whose indices are any integral power of 2, as the fourth, eighth, sixteenth, thirty-second power, &c.; and next to cubic surds, and to any number of surds whose common indices are in any manner compounded of the factors 2 and 3; next to any combinations of surds whose indices do not exceed the number 6, and to as many as three surds, neither of whose indices exceed 12, as well as to various others which cannot be concisely specified.

That the perspective representation of an entire circle is an ellipse, is simply a part of the doctrine of conic sections, and is not in need of demonstration. The principal inquiry of the author has been with regard to the positions of the axes of such ellipses. Selecting the case of any number of circles in the same plane, and having their centres in the same line, he examines what law is to be observed in the directions of the axes of the representing ellipses.

The present instrument depends upon a new extension of the principle of the common sliding-rule; for as in that numbers themselves are multiplied or divided by the mechanical addition of their logarithms, so in this their logarithms are multiplied or divided by mechanical application of corresponding logometric spaces. In the common tables of logarithms, that of 10 is 1, and those of its simple powers are 2, 3, 4, &c.; so also the logarithm of the square root of 10 is ½ or ⋅5; the fourth root is ¼, or ⋅25, being a decimal index expressing a power of 10 less than unity. In the same manner all other numbers are considered as powers of 10, and their logarithms are integral or decimal indices of those powers.

The subject here considered by Mr. Herschel relates to the celebrated theorems of Lagrange, expressing the connection between sim­ple exponential indices and those of differentiation and of integration. Since the theorems have been demonstrated by various subsequent analysts, as by Laplace, by Arbogast, and by Dr. Brinkley, the author takes them for granted; but observes that in their original form they are but abridged expressions of their meaning; and that in order to become practically useful, their exponential functions require further development.

The circulation of the blood in these animals is different from that of any other animal known to the author. There is one point to which all the blood is brought, and from which it is again emitted to all parts of the body. To this the author gives the name of heart, although it be not, in fact, the principal agent in carrying on the circulation, and although so small as hardly to deserve the name of ventricle. It is situated directly in the middle line of the belly, and is that point in a vessel which comes from the head, where the rest of the blood is received from two auricles that lie above it at the back of the animal, one on each side. Sir Everard, after describing the unsuccessful attempts which he had made to discover the arrangement of the vessels by transparency of the body of the animal in full sunshine, acknowledges that he is indebted to Mr. Clift for the means of detecting the course of its circulation, by steeping it in vinegar, which presently coagulates the blood in the vessels, and gives it a deep black colour.

At the commencement of this paper the author states the advan­tages which may be derived from the employment of analogical rea­soning in mathematics, and recommends it as a very useful guide to new discoveries: he then proceeds to point out the striking resem­blance which subsists between several parts of common algebra and the integral calculus, and similar parts of the calculus of functions. Mr. Babbage then notices certain fractions which, by peculiar re­lations among the functions of which they consist, become evanescent. The true values of these fractions are ascertained, and they are ap­plied to the solution of a class of functional equations which the author had solved in a former paper, from which the following result is obtained :—“ Whenever the mode of solution there adopted seems to fail, the failure is apparent only, and the general solution may al­ways be deduced from it.”

In this note the author expresses his regret at finding that the demonstration of the binomial theorem; and the first proposition of his paper on the construction of logarithms, formerly presented to the Royal Society, had been previously given by Mr. Spence in his Essay on Logarithmic Transcendents. This author, however, says Mr. Knight, is not particularly happy in the manner of developing the kind of functions treated of in his preface, and therefore in the present note gives a solution of a class of equations of which Mr. Spence has considered a particular case, without however resolving.

Most of the attempts which have been made by writers on vege­table physiology, to account for the force with which the sap of trees ascends during the spring, having proved unsatisfactory and inade­quate, Mr. Knight was induced some years ago to suggest the ex­pansion and contraction of the cellular processes proceeding from the bark to the medulla, and which he called the true or silver grain of the wood, as concerned in this process. The present paper contains further experiments, showing this power to be active in living trees, and were made on many kinds of timber with nearly similar results. Some boards of ash and beech wood were cut in opposite directions relative to their medulla, so that the convergent cellular processes crossed the surfaces of some of them at right angles, and were parallel with the surfaces of others. These were placed, under similar circumstances, in a warm room, and the former warped about ten times more than the latter, contracting nearly 14 per cent, in breadth, while the others only contracted 3½ per cent. During his experiment Mr. Knight was led to infer that the medullary canal must be liable to considerable changes of dia­meter, as the moisture of wood increases or diminishes. To ascertain this, parts of the stems of young trees were carefully dried, the me­dulla was removed, and metal cylinders driven with force into the empty space. The pieces of wood were then suffered to absorb moisture, and the medullary canal became so much enlarged as to suffer the cylinders to fall out.

The principle of applying diagonal frame-work to ships of war was first partially and successfully adopted in the Kent, of 74 guns, in the year 1805, and since that period has been successfully employed in the construction of thirty-eight sail of the line and thirty frigates. These circumstances might be deemed conclusive as to the advantages of the new system; but as the Royal Society have already published this author’s account of it at a very early period of its adoption, he is induced to offer the result of a new experiment in proof of the correctness of the principles before laid down, which, as far as his knowledge extends, has never been previously applied, nor ever suggested by any continental writer, though, says the author, it has been pretty broadly insinuated that the hint was borrowed from the French. In the early part of the present year, the Justitia, an old 74, was ordered to be broken up; when Mr. Seppings, notwithstanding her shattered condition, determined to apply the trussing principle. Prior to her being taken into dock, sights were placed in the lower and upper gun-deck, to ascertain, when she had grounded on the blocks, how much she deviated from her state afloat. She was then partially trussed, as described by reference to an annexed drawing, and floated out into the basin. After lying one hour, it was found, by the sights placed on the gun-deck, that she had come down in the mid-ship 1 foot; and by those on the upper-deck, 1 foot 2⅝ inches. In twenty-four hours she further hogged 2⅝ inches, and then appeared stationary. The trusses in the hold were then removed, and she further hogged 6 inches, and 3½ inches in removing those in the ports.

In the first section of this letter, Dr. Young proceeds to examine in what manner the apparent constancy of many general results, subject to numerous causes of diversity, may be best explained; and shows that the combination of many independent causes of error, each liable to incessant fluctuation, has a natural tendency, dependent on their multiplicity and independence, to diminish the aggregate variation of their joint effect; a position illustrated by the simple case of supposing an equal large number of black and white balls to be thrown into a box, and 100 of them to be drawn out at once or in succession; when it is demonstrated that there is 1 chance in 12 1/2; that exactly 50 of each kind will be drawn, and an even chance that there will not be more than 53 of either; and that it is barely possible that 100 black, or 100 white, should be drawn in succession. From calculations contained in this paper, Dr. Young infers that the original conditions of the probability of different errors do not considerably modify the conclusions respecting the accuracy of the mean result, because their effect is comprehended in the magnitude of the mean error from which these conclusions are deduced. The author also shows that the error of the mean, on account of this limitation is never likely to be greater than six sevenths of the mean of all the errors divided by the square root of the number of observations.

The object of this paper is to explain two methods of finding the sums of a variety of infinite series. One of these the author discovered several years ago; but finding that some of the results to which it led were erroneous, he then declined publishing it. In inquiring into the causes of these errors, he was led to the second method, which employs the process of integration relative to finite differences. The cause of the fallacies in the former method was afterwards discovered, and in this paper a criterion is proposed for judging of the truth of the results, and a mode of correcting them where found to be erroneous. The sums of a variety of series are found by these methods; and the author concludes by observing, that he has since been informed by M. Poisson, that that gentleman had arrived at some nearly similar results in investigating a problem in physical astronomy, and also that some investigations of a similar nature were found amongst the papers of Lagrange, but that neither of these mathematicians had explained the cause of the errors, or given a method of correcting them.

In this communication Captain Kater, having noticed the circumstances to which his researches owe their origin, proceeds to detail his investigations, and to describe the implements and apparatus employed in his various inquiries; the construction of the pendulum and its appendages is minutely explained, as also the rate of its expansion for each thermometric degree, whence is deduced the corresponding correction to be applied to the number of its vibrations. The operations at each station, with their results, are enumerated at length, and illustrated by numerous tables. The length of the seconds pendulum for the latitude of London is 39·13722 inches in parts of the scale which forms the basis of the trigonometrical survey; for the latitude of Unst 39·16939 inches, of Portsay 39·15952, of Leith Fort 39·15347, of Clifton 39·14393, of Arbury Hill 39·14043, and of Shanklin Farm 39·13407 inches. The calculation of the latitude of each of these stations is given at length, to afford the opportunity of any further examination desirable on that subject; but these and the other details relating to calculation do not admit of abridgement. Captain Kater concludes this paper with some observations respecting the figure of the earth. It having been shown by Clairaut that the sum of the two fractions, expressing the ellipticity and the diminution of gravity, from the pole to the equator, is always a constant quantity, and equal to 5/2 of the fraction, expressing the ratio of centrifugal force, find that of gravity at the equator, it follows that if the decrease of gravity from the pole to the equator be subtracted from this constant quantity, the remaining fraction will express the ellipticity of the spheroid. The diminution of gravity may be known by finding the difference of the length of two pendulums, vibrating in equal times at the equator and pole, which are to each other directly as gravitation; but as such experiments cannot be made at the pole, Captain Kater proceeds to describe the means of obtaining the desired result by observations at intermediate stations; whence it appears that the length of the seconds pendulum at the equator, deduced from the observations at Unst and Dunnose, is 39·00527 inches, and gravitation at the equator 16·040 feet; hence the centrifugal force at the equator is 1/288 of gravitation, or 1/282 of gravity, which last being multiplied by 5/2 gives ·0086505 for the sum of the fractions, expressing the ellipticity of the earth and diminution of gravity from the pole to the equator.

In this communication Sir Everard announces the existence in certain animal structures of globules Smaller than and independent of those ordinarily belonging to the blood; they were first remarked by Mr. Bauer, during the microscopic examination of the coats of an aneurismal tumour; their number was in the proportion of one to four of the larger globules, and their diameter was 1/28000 of an inch in the larger in contact with the circulating blood; in the other layers they became more numerous, and in that longest coagulated were in the proportion of four to one. In the section of an aneurismal tumour, the author notices the uncommon appearance of crystals of sulphate of lime with muriate and phosphate of soda, which, as well as the globules, he supposes to have been originally dissolved in the serum, since they are only brought to view by the act of coagulation.

The experiments above adverted to are detailed in the Philoso­phical Transactions for 1798; and though the greatest attention was bestowed on those parts of the inquiry relating to the weight of the solids, the method of measuring them is not so fully detailed; Cap­tain Kater, therefore, was desirous of re-investigating the latter sub­ject before the Commissioners of Weights and Measures should make their final report. The author then proceeds to describe the state of the apparatus; and the means which he adopted in effecting this measurement of three sides of the cube gives for its content 124.1969 inches.

Dr. Ure commences this paper by adverting to the fallacies to which the modes of analysing organic substances hitherto practised are subject; and in detailing the peculiar methods adopted in his own researches, he shows the means of obviating them, and of diminishing the various sources of inaccuracy to which these complicated processes of analytical chemistry are necessarily more or less liable. Where oxide of copper is used, its hygrometric quality has generally been overlooked, or not duly allowed for; and the animal and vegetable substances have not in general been exposed to any process of desiccation sufficiently exact or uniform; the author therefore always used the oxide of copper in some known or ascertained degree of humidity; and he dried the organic bodies in the air.pump vacuum, aided by the absorbent powers of a surface of sulphuric acid in the apparatus, and with precautions which he fully describes. He then details the best means of applying heat for the decomposition of organic substances, and describes a drawing representing the construction of his furnace, and other implements. Lastly, he points out the method of examining the results and products, and gives in detail the analysis of sulphuric ether, as illustrating the mode of computing the relations of the constituents, while the results of the other analyses are, for the sake of brevity, thrown into a tabular form. Dr. Ure concludes his paper with some general remarks on the analytical details. In respect to sugar, he observes, that on comparing pure crystalline sugar with diabetic sugar, the latter exhibits a notable excess of oxygen; and he considers weak sugars (as the refiners call them), in general, to exhibit the same peculiarity. In applying the atomic theory to his experimental results, the author enlarges on the different views which may be taken of the ultimate constitution of a variety of organic products, and enters at considerable length into details relating to the vegetable acids, with a view of determining with exactness their prime equivalents, and the relative proportions of combined water which they contain in their crystalline states.

The bars of steel used in these experiments were supported at the ends by two blocks of cast iron, resting upon a wooden frame, and a scale for weights was suspended from the middle of the length of the bar, by a cylindrical steel pin, three eighths of an inch in diameter. To measure the flexure a quadrantal piece of mahogany was attached to the frame, with a vertical bar sliding in two guides at its edge, and moving an index. The bar and index were so balanced, that one end of the bar bore with constant pressure upon the specimen, and the graduated arc was divided into inches, tenths, and hundredths. The thousandths were measured by a vernier. A bar of blistered steel of file hardness, 13 inches long between the supports, underwent no permanent alteration of form when loaded with 110lbs. The temper of the bar was then successively lowered, and it was ultimately again hardened; but in these different states its flexure and resistance to permanent change of form remained the same. These experiments were repeated with bars of other dimensions, which were loaded till they broke; and from them the author also infers that the elastic force of steel is not altered by temper, and that the force which produces permanent alteration is to that which causes fracture in hard steel, as 1:1·66; and in the same steel of a straw yellow temper, as 1:2·56. From comparisons of the strain required to cause permanent alteration in different kinds of steel, the author concludes, that in the process of hardening, the particles are put into a state of tension among themselves, which lessens their power to resist extraneous force; and the phenomena of hardening may be referred to the more rapid abstraction of heat from the surface of the metal than can be supplied from the internal parts, whence a contraction of the superficial parts round the expanded central ones, and a subsequent shrinking of the latter, by which the state of tension is produced.

The experiments of M. Arago having excited much interest, the authors of this communication were induced to erect an apparatus for their verification; and after a few trials they succeeded in causing a compass to deviate from the magnetic meridian, by setting in rotation under it plates of copper, zinc, lead, &c. To obtain more visible and regular effects, however, they found it necessary to reverse the experiment, by setting in rotation a powerful horse-shoe magnet, and suspending over it the various metals and other substances to be examined, which were found to follow with various degrees of readiness the motion of the magnet. The substances in which they succeeded in developing signs of magnetism were, copper, zinc, silver, tin, lead, antimony, mercury, gold, bismuth, and carbon, in that peculiar metalloidal state in which it is precipitated from carburetted hydrogen in gas-works. In the case of mercury the rigorous absence of iron was secured. In other bodies, such as sulphuric acid, resin, glass, and other non-conductors, or imperfect conductors of electricity, no positive evidence of magnetism was obtained.

This communication consists of a table stating the annual variations of 23 of the principal fixed stars, as deduced from Dr. Brinkley’s observations, and those of the Astronomer Royal. On. these Mr. Pond remarks, that out of 16 stars observed at Dublin, 13 either indicate a southern deviation, or at least are not inconsistent with it, and that of these 13, about half indicate a greater deviation than that assigned by Mr. Pond himself. The other half a less, while the three remaining stars deviate northwards. Mr. Pond further remarks, that the examination of this table is calculated rather to increase than to diminish scepticism on the subject of the determination of such very small quantities by astronomical observations. He concludes by disclaiming all intention of placing he subject in a controversial point of view, and by expressing a hope that the difficulty will in a very few years be satisfactorily cleared up.

In this paper the author states that his attention was first directed to a consideration of suspension bridges, when the plan for the Menai Bridge was submitted to the Commissioners of Roads and Bridges. It then appeared to him that the proposed depth of curvature was insufficient for insuring a due degree of strength; and this opinion was confirmed by some investigations, which are printed in the Quarterly Journal of Science. In consequence of this, the interval between the road-way and the points of support has been augmented to 50 feet, and its strength now appears sufficient. The object of this paper is the expansion of the formulae, from which the above-mentioned approximation was derived, into tables adapted to general use; and the derivation of other formulae and tables for the catenary of equal strength; a curve not merely of speculative curiosity, but of practical use when bridges of very wide span are to be constructed. The author first remarks, that as all catenaries, like circles, parabolas, &c., are similar curves; tables constructed for one value of the parameter apply to all by simple proportion.

Mr. Bevan took the opportunity of the severe frost of the last winter to determine the modulus of elasticity of ice, which he did by cutting a rectangular plate of that substance from the surface of a pond of 100 inches in length, 10 in width, and about 4 in thickness. The deflection produced by a weight of 25 lbs. was 0·206 inches, from which he concludes the modulus of elasticity to be 2,100,000 feet. The modulus for water he states at 2,178,000 feet, on a certain hypothesis respecting its cubical compression.

The author begins this paper by an enumeration of the various works on the subject extant in our language, and a general mention of the writings of foreign mathematicians, which he considers as leaving room for further inquiry and simplification. He then states the method employed in his experiments for determining the refractive and relative dispersive powers of his glasses, the former of which is that generally known and practised;—of measuring the radii and focal length of a lens, and thence deriving the refractive index; with some refinements in its practical application, consisting chiefly in using the lens as the object-glass of a telescope, and adapting to it a positive eye-piece and cross-wires, which are brought precisely to the true focus by the criterion of the evanescence of parallax arising from a motion of the eye, as is practised in adjusting the stops of astronomical instruments. The only source of error it involves is in the measurement of radii of the tools which it was found could always be performed within 1/500th of their whole values. The dispersive ratio of two glasses was determined by over-correcting the dispersion of a convex lens of the less dispersive glass by a concave of the greater, and then withdrawing the latter from the former till the achromaticity is perfect, or as nearly so as the materials will admit, and measuring the interval between the lenses and their foci, from which data the ratio of their dispersive powers is easily obtained. The refractive indices and dispersive ratio thus determined, the next step is to find the radii of curvature so as to destroy spherical -aberration. In this investigation, the author does not consider it as necessary to limit the indeterminate problem by any further condition, as others before him have done, but regarding it as a matter of great convenience to avoid contact of the interior surfaces in the centre of the glasses, leaves it open to the optician to make a choice within certain limits, thus avoiding what he considers as an intricate equation arising out of the fourth condition. He proceeds, therefore, to express analytically the aberrations of the glasses, and to deduce the equation expressive of its destruction, which of course involves one indeterminate quantity; this may be either of the radii, or any combination of them. The author chooses the ratio of the radii of the interior and exterior surfaces of his flint lens for this indeterminate, which he assumes, as well as may be, to satisfy the condition of the absence of contact and near equi-curvature of the adjacent surfaces; thence deduces, first, the radii of both of the surfaces of the flint lens; next, its aberration to be corrected; and thence, by the solution of a quadratic, or by the use of a table containing its solutions registered in various states of the data, the ratio of the radii of the convex, whence the radii themselves are easily deduced.

The author commences by observing, that the present is the first of a series of communications, which he hopes to have the honour of laying before the Royal Society, on the same subject; and that the object of the whole series is to determine the exact composition of the three great divisions, viz. the saccharine, the oily, and the albuminous, in which the alimentary matters employed by the more perfect animals may be comprehended; and afterwards to inquire into the changes induced in them by the action of the stomach and other organs, during the subsequent stages of assimilation. The present paper includes some preliminary observations on the analysis of organized bodies in general, and the composition of the first of the above classes, viz. the saccharine.

Recent investigations having shown that the method employed by Captain Kater for the reduction of his experiments on the length of the pendulum vibrating seconds in air, to that of the same pendulum in vacuo , was founded on erroneous principles, the author undertook to ascertain, by direct experiment, the actual difference of the num­ber of vibrations of the pendulum employed by Captain Kater, in air of ordinary density, and in highly rarefied air. The alteration of density in the medium in which the pendulum is swung, would, in the first place, if its form were not symmetrical, affect its converti­bility ; that is, the same adjustment of the axes which gave an equa­lity of oscillations in reversed positions, when vibrating in air, would not afford the same equality in a more rarefied medium. It follows also, from the corrected investigation, that the amount of the retar­dation occasioned by the air is considerably greater than what had been originally computed from the simple consideration of buoyancy. These inferences have been fully confirmed by the experiments of Captain Sabine. The increase in the number of vibrations per diem with the convertible pendulum as it was used by Capt. Kater, that is, vibrating with the great weight below, in vacuo , above those in air of the temperature of 49°, under a pressure of 30 inches of mer­cury at 32°, was 15·71: when inverted, the other conditions remain­ing the same, the increase was 16T3 vibrations per diem .

The author, in a former paper, read to the Society in February last, had discussed various objections which had been raised against his mode of geometric representation of the square roots of negative quantities. At that time he had only discovered geometrical repre­sentations for quantities of the form a + b √‒1, of geometrically adding and multiplying such quantities, and also of raising them to powers either whole or fractional, positive or negative; but he was at that time unable to represent geometrically quantities raised to powers, whose indices involve the square roots of negative quantities (such as a + b √‒1 m + n ). His attention has since been drawn to this latter class of quantities by a passage in M. Mourey’s work on this subject, which implied that that gentleman was in posses­sion of methods of representing them geometrically, but that he was at present precluded by circumstances from publishing his discoveries. The author was therefore induced to pursue his own investigations, and arrived at the general result stated by M. Mourey, that all algebraic quantities whatsoever are capable of geometrical representation by lines all situated in the same plane. The object of the present paper is to extend the geometrical representations stated in his former treatise, to the powers of quantities, whose indices involve the square roots of negative quantities. With this view he investigates Various equivalent formulæ suited to the particular cases, and employs a peculiar notation adapted to this express purpose ; but the nature of these investigations is such as renders them incapable of abridgement.

The author proposes the employment of threads of glass in the construction of torsion-balances, in place of the silver wire, used by Coulomb for the measurement of minute electric or magnetic forces. He describes a galvanometer of his invention, acting upon this principle, the intensity of the galvanic current being measured by the torsion of a slender filament of glass, to the lower end of which a magnetized needle is fixed at right angles. He also applies the same power to the improvement of the sensibility of the common balance for weighing minute bodies, by affixing to the beam a long glass thread horizontally in the axis of suspension, by the torsion of which, when the balance has been brought nearly to a level, the more accurate adjustments are to be effected. On the whole he considers that glass, from its perfect elasticity, possesses decided advantages over metallic wires, for the construction of instruments acting on the principle of torsion.

While engaged in the adjustment and verification of the copies of the Imperial standard yard destined for the Exchequer, Guildhall, Dublin, and Edinburgh, the author discovered a source of error arising from the thickness of the bar, upon the surface of which measures of linear dimension are traced. A notice to that effect was published in the Philosophical Transactions for 1826; and the object of the pre­sent paper is to give an account of the experiments the author has since made on this subject, and to describe a scale which he has had constructed, so as almost entirely to obviate the source of error thus introduced. From the experiments detailed in the first part of the paper, the following conclusions are deduced:—First, that in a standard of linear measure, traced upon the surface of a bar, an error arises from the thickness of the bar when it is placed upon a table the surface of which is not plane; Secondly, that this error in bars of the same material, and of unequal thickness, is within certain limits as the thickness of the bar, and depends upon the extension of that surface of the bar which becomes convex, and the compression of the surface which is concave; Thirdly, that the error to which the same scale is liable from this cause, is directly as the versed sine of the curvature of the surface upon which the scale is placed; Fourthly, that the error very far exceeds that which would arise from the difference of length between the arc and its chord under similar circumstances; so much so, that the sum of the errors from this cause, in a bar one inch thick, with a versed sine of not one hundredth of an inch, is nearly one thousandth of an inch, whilst double the difference be­tween the chord and the arc is not one fifty thousandth.

The Prague doctor and naturalist Marcus Marci1 has been frequently cited as a predecessor of Newton in establishing the physical theory of color; he recognized namely that (1) different refractions uniquely correspond to the different colored rays produced by a prism; (2) the monochromatic rays so defined maintain their color unchanged under further refraction: two propositions which in fact form an essential part of the Newtonian theory of color. Now, naturally, this last assertion is correct as far as it goes; however I would not take it as proper to hold Marci up as Newton’s predecessor merely on the basis of this fact. For, completely apart from the question of whether or not Newton was acquainted with Marci’s arguments, the history of physics should not be comprehended as a mere chronological stringing together of isolated discoveries, but rather the main emphasis should be placed on the logical connections between the individual steps which contribute to the development of a physical theory. In order to judge whether, from this standpoint, a given theory can be designated as a predecessor of another, more complete, theory, the following question must above all be decided: Does an extension and modification, in keeping with their meaning, of the essential categories suffice for arriving at the second theory from the first; or is a complete rejection of the basic concept of the first theory required? Only in the first case could one speak of an historical connection. In this sense, for example, Galileian mechanics would be a predecessor of Newtonian mechanics and this latter in turn a predecessor of Einsteinian mechanics; or the older quantum theory would also be a forerunner of modern quantum mechanics.

Based on the Location Quotient (LQ) method, this paper selected datas of Statistical Yearbook from 2001 to 2008, and calculated the LQ of all sectors of industry in Xinjiang, finding three industries has the phenomenon of clustering, namely Extraction of Petroleum and Natural Gas industry, Oil Processing, Coking and Nuclear Fuel Processing industry and Processing of Ferrous Metals Ores industry. The cluster degree of all sectors has also been calculated, sorted and analyzed. In addition, changes of cluster degree from 2001 to 2008 in clustered industrial sectors was described and reasons for, and some suggestions on the industrial cluster development has been put forward of this region.

The table here communicated is supplementary to those accom­panying the paper “On the Mathematical Theory of Suspension Bridges,” which was published in the Philosophical Transactions for 1826, and is deduced from the first of the tables there given; but admits of a far more ready application than the former to all cases of practical investigation. It consists of five columns, exhibiting respectively the deflections or versed sines of the curve; the lengths of the chains; the tension at the middle points, or apices of the curve; the tensions at the extremities; and the angles made by the chains with the horizon at the extremities.

The design of the author in undertaking the experiments of which he gives an account in the present paper, is to carry on the investigation of the phenomena of the inflexion of light from the point at which it was left by Newton. He begins by examining these phenomena in their simplest form, comparing the appearance of the shadow of an opaque body on a screen of white paper at different distances, with the appearance it would exhibit if the rays passed by the edge of the body, without suffering any deviation from a rectilinear course. It is well known that, under these circumstances, the real shadow is broader than the geometrical shadow, indicating a deflexion of the rays from the edge of the intercepting body. By varying the distances at which the observations are taken, it is found that the rays are not bent at a sharp angle, but pursue a curvilinear course, the concavity of which is towards the shadow, the curve itself resembling an hyperbola. A luminous halo also appears beyond the shadow; the breadth of this halo agreeing accurately, at all distances, with the space which the penumbra should occupy, if the rays were not bent. The author thinks it impossible to reconcile the explanation of these phenomena given by Newton, with his own hypothesis concerning the action of solid'bodies on light, as stated in the “Principia:” for, in that hypothesis, the rays passing nearest to the edge of an intercepting body are supposed to be bent towards the edge, as if attracted; whereas the explanation proceeds upon the supposition that they are bent from that body, as if repelled. The actual hyperbolic course of the rays is also inconsistent with that hypothesis, which would assign to them a parabolic path. It also appears that the breadth of the spectrum made by receiving the sun’s rays through an aperture one tenth of an inch, or more, in width, is less than if the rays proceeded in straight lines; but if the aperture is very much diminished, the result is reversed, the real spectrum being broader than the geometrical spectrum.

By an analytical process, the author arrives at the following theorem, namely, if three abscissæ of an equilateral hyperbola be materially dependent by reason of two assumed equations, which are symmetrical with respect to these three abscissæ, the sum of the arcs subtended by them is equal to three quarters of the product of the same abscissæ, or only differs therefrom by a constant quantity. In order to satisfy himself of the correctness of this theorem, the author calculated various numerical examples, which entirely confirmed it. This simple result is essentially a relation between three arcs of the equilateral hyperbola, and is by no means reducible to a relation between two; and therefore is not reducible to the celebrated theorem of Fagnani, concerning the difference of two arcs of an ellipse or hyperbola, nor to any other known property of the curve. The reading of Mr. Faraday’s Sixth Series of Experimental Researches in Electricity was commenced.

After some introductory remarks illustrative of the scope and design of this paper, the object of which is sufficiently pointed out in its title, the author considers, 1st, the integration of the equations of motion of a system, the characteristic function of such motion, and the law of varying action; 2nd, the verification of the foregoing integrals; 3rd, the introduction of relative or polar co-ordinates, or other marks of position of a system; 4th, the separation of the relative motion of a system from the motion of its centre of gravity, the characteristic function for such relative motion, and the law of its variation; 5th, the systems of two points in general, and the characteristic function of the motion of any binary system; 6th, the undisturbed motion of a planet or comet about the sun, and the dependence of the characteristic function of elliptic or parabolic motion on the chord and the sum of the radii; 7th, the systems of three points in general, and their characteristic functions; 8th, a general method of improving an approximate expression for the characteristic function of motion of a system, in any dynamical problem; 9th, the application of the foregoing method to the case of a ternary or multiple system, with any laws of attraction or repulsion, and with one predominant mass; 10th, the rigorous transition from the theory of binary to that of multiple systems, by means of the disturbing part of the whole characteristic function, and approximate expressions for the perturbations.

The author, considering it probable that two or more magnetic needles freely suspended at a certain distance in given positions with respect to each other, would develope certain proportionate deflections determined by their position on the earth, made a set of experiments, with a few common compass needles, at Yarmouth, London, and Clifton, which so far confirmed the truth of his conjecture, as to induce him to fix on stations at a greater distance from one another, and to multiply his observations; ascertaining, from time to time, that the magnetic powers of his apparatus had undergone no material change. The results of these observations are given in a tabular form. The author conceives that comparative observations of the amount of deflection produced by one magnet on another, placed in various situations, relative to the meridian, on an horizontal plane, will afford the means of determining the peculiar local influences of the particular situation in which the experiment is made, as distinguished from the general magnetic influence; because the former will act unequally on each magnet, while the latter acts equally on all.

The author observes, that the magnitude of the terms of the fourth order in the disturbing function, relating to the inclinations, in the theory of the secular inequalities of the planets, does not admit of being estimated à priori ; and consequently the amount of error which may arise from neglecting them cannot be appreciated. The object of the present investigation is to ascertain the analytical ex­pressions of these terms; and the method adopted for this purpose is derived from principles already explained by the author in a for­mer paper. He has bestowed great pains in putting these expressions into the simplest form of which they are susceptible; and has finally succeeded, after much labour of reduction, in obtaining ex­pressions of remarkable simplicity. He exemplifies their application by the calculation, on this method, of one of the terms given by Pro­fessor Airy as requisite for the determination of the inequality of Venus; and arrives, by this shorter process, at the same result. The same method, he remarks, is, with certain modifications, applicable to the developement of the disturbing function in terms of the true longitude.

Dr. Davy complains that Mr. Faraday has, in the paper referred to, made certain statements with respect to the opinions of Sir Humphry Davy relative to the conducting powers of dry nitre, and caustic potash and soda, when in fusion by heat, and also with regard to other matters connected with voltaic electricity, wdiich are not correct; and vindicates Sir Humphry Davy from the charge of want of perspicuity in the statement of his views of these subjects. A Note by Mr. Faraday on the preceding Remarks by Dr. Davy was then read, in which he replies to the charges there brought forward, and justifies those statements, the accuracy of which had been impugned by Dr. Davy.

The author premises a brief historical sketch of the progress of discovery in this branch of analytical science. He observes that the first inventors of the integral calculus obtained the exact integration of a certain number of formulæ only ; resolving them into a finite number of terms, involving algebraic, circular, or logarithmic quantities, and developing the integrals of others into in finite series. The first great improvement in this department of analysis was made by Fagnani, about the year 1714, by the discovery of a method of rectifying the differences of two arcs of a given biquadratic parabola, whose equation is x 4 = y . He published, subsequently, a variety of important theorems respecting the division into equal parts of the arcs of the lemniscate, and respecting the ellipse and hyperbola ; in both of which he showed how two arcs may be determined, of which the difference is a known straight line. Further discoveries in the algebraic integration of differential equations of the fourth degree were made by Euler ; and the inquiry was greatly extended by Legendre, who examined and classified the properties of elliptic integrals, and presented the results of his researches in a luminous and well-arranged theory. In the year 1828, Mr. Abel, of Christiana, in Norway, published a remarkable theorem, which gives the sum of a series of integrals of a more general form, and extending to higher powers than those in Euler’s theorem ; and furnishes a multitude of solutions for each particular case of the problem. Legendre, though at an advanced age, devoted a large portion of time to the verification of this important theorem, the truth of which he established upon the basis of the most rigorous demonstration. M. Poisson has, in a recent memoir, considered various forms of integrals which are not comprehended in Abel’s formula.