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Mr. PARSEVAL has invented an analogous method for summing such a series as

Aa+ Bỏ Cc+ &c.

formed by the multiplication of two series

A + Bƒ + Cƒ2 + &c.

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term with term; and this method he employs, towards the conclusion of his memoir, in integrating a differential partial equation of the second order.-The general method of summing the compounded series is very simple and ingenious: but, applied to easy cases that can be solved otherwise and directly, it is tedious and complicated.

In a short Appendix to this volume are inserted two Mathematical Memoirs, by C. F. DE NIEUPORT. The first contains the solution of a problem proposed by D'Alembert in the 8th volume of his Opuscules, p. 40. relative to the conditions of the equilibrium of a flexible string fastened to its two ends, and passing through a groove cut in a body which is supported by the string. M. NIEUPORT resolves the problem, on the principle that the centre of gravity is always at the lowest or the highest point in the case of equilibrium.

The second memoir is on the general Equation of regular Polygons, and on the Division of any Arc whatever into equal Parts. If the chord of A be expressed by any symbol, as p then the chord of nA can be expressed in terms of p; and from such expression, putting nA equal to the circumference, we should obtain an expression involving p and n. Hence n being given, we should have an equation involving p and the powers of p; and p would be the side of a regular polygon of n sides inscribed in a circle. This method is sufficiently plain when an expression for chord nA is obtained; and that expression is perhaps most easily obtained by the aid of the exponential expression for the sine and cosine of an arc: but it may be obtained otherwise, by Waring's method in his Proprietates Curvarum, or that of La Grange in his Fonctions Analytiques, or that of Arbogast in his Calcul des Dérivations.

The method employed by the author of the present memoir is in fact that which uses imaginary expressions.

If u be the chord and z the arc, then since = 2 sin.

dz du

.

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Consequently,

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and by a simple process, we shall obtain an expression of this kind:

1

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Equating the possible and the impossible parts, expressions for the chords of, 4, &c. of the circumference will result.

We discern little in this memoir that intitles it in our opinion to the honor of insertion. If Euler (Analysis Infinitorum) and La Croix (Introduction, Calcul diff.) have not given the same method, they have suggested those which are very nearly related and similar to it; and the same results may be obtained without the aid of those symbols, on the legitimate use of which all mathematicians are not agreed.

MEDICINE, CHEMISTRY, NATURAL HISTORY, &c.

It appears by the dates prefixed to these memoirs, that a considerable period has elapsed since they were presented to the Institute; and in most cases, either the whole of the papers or an abstract from them has already been given in some of the literary journals: so that, however valuable in themselves, their publication will afford little new information. On this account, we shall be more than usually brief in our notice of

them.

Memoir on the Sap of Plants, and particularly on that of the Vine and the Hornbeam, with an Analysis of this Fluid. By M. DEYEUX-The most important fact ascertained in this paper is that the sap of the vine and hornbeam contains the acetate of lime, united to a quantity of vegeto animal matter, which appears to be similar to the gluten of wheat. The essay is well written, and must have been valuable when it was originally presented to the society, in the year 1796: but it is now, in a great measure, superseded by the experiments of M. Vanquelin.

Memoir on three different Species of Carbonated Hydrogenous Gas, produced by different Processes, from Ether and Alcohol. By M. M. BONDT, DEIMAN, PAATS VAN TROOSTWYK, and LAUWEREN

BURG

BURG, of Amsterdam. This very valuable paper contains an account of the discovery of the olifient gas. An ample abstract of it was afforded to the chemists of France, soon after it was presented to the Institute, in the 45th vol. of the Journal de Physique, and has been laid before the English reader in some of our temporary publications.

Sketch of some Experiments respecting the Division which Cylinders of Camphor experience at the Surface of Water, and Reflections on the Motions which accompany this Division. By J. B. Venturi, Professor of Natural Philosophy at Modena, &c.-An account of these experiments has already appeared in the 21st vol. of the Annales de Chimie.

Memoir on the Blood of Persons affected with Jaundice, considered with respect to its Chemical Relations. By M. DEYEUX.--A principal object of this paper is to ascertain whether the yellow tinge which the blood exhibits in Jaundice be, according to the common opinion, owing to the actual presence of bile in it. Having obtained a quantity of this blood, M. DEYEUX permitted the crassamentum and serum to separate; when, although the latter exhibited the deep yellow color, it had neither the odor nor the taste of bile; nor was alcohol, by being digested on it, impregnated with any degree of bitterness. It was remarkable that the crassamentum was not reddened in the usual manner by exposure to the atmosphere, nor did the serum become solid by the application of heat to it. We think that the author is justified in his conclusion that, in this case, there was no proper bile in the blood. The facts stated in this paper are curious, and deserve the attentive consideration of the physiologist.

Observations on the Concrete Citric Acid. By M. Dizé.— These remarks have already appeared in the Annales de Chimie, and in an English dress.

Memoir on the Filaments or Hairs which cover the Plant that produces the Cicer Arietinum; and a Chemical Examination of the Liquor which exudes from these Hairs: By M. DEYEUX.

The hairs with which this plant is so abundantly beset are observed, at particular periods, to be tipped with a minute drop of fluid; of which the author contrived to collect a sufficient quantity to examine its nature, when he found it to consist of pure oxalic acid.

Meteorological Observations made at Cayenne, from the 1st December 1778, to the 30th November 1789. By M. MENTELLE, Geographical Engineer, &c.-The most remarkable circumstance in these observations is the great quantity of rain that

falls

falls at Cayenne, which on the average is nearly 98 inches annually; in the year 1791 it was 116 inches. In the interior part of Guiana, the rain is said to fall in still larger quantity.

History of a Woman who had one of the Ovaries tranformed into Hydatids, and in whose Womb was found a very hard and very compact Bony Substance. By Dr. ÖDIER of Geneva.-An unmarried female, 70 years of age, after having enjoyed a good state of health, was hurt by a fall, and had afterward a number of complaints, which were referred to the stomach: but from some of the symptoms it was suspected that the pancreas was scirrhous. After death this was discovered to be the case; and on continuing the examination, one of the ovaries was found converted into a mass of Hydatids, and the uterus, although of its natural size, was completely filled by a mass of bone.

Collection of different Observations on Chemistry. By M. PROUST. Of this very interesting paper we have already (vol. 48. p. 458.) had an opportunity of giving an ample account, from the historical part of the last volume of the Memoirs of the Institute. We are happy to see it now published in its entire state.

Anatomical Observations on a Patient, on whom the Operation for Popliteal Aneurism had been performed, according to the Method of Hunter. By M. DESCHAMP.-The subject completely recovered from the operation, but, after the space of eight years, was carried off by a disease of the chest, when an opportunity occurred of examining the state of the limb. About 2 inches of the arterial trunk was found to be obliterated.

Experiments on the Influence of Light on Vegetables. By A. P. DECANDOLLE. We have here an account of the comparative effects produced on plants, by being kept in a dark chamber, and in one lighted by lamps, to a certain degree of intensity. It was found that plants, which had been blanched in the dark chamber, were rendered green when removed into the other but the light of the lamps did not appear adequate to detach oxygenous gas from them. M. DECANDOLLE observed the effects of different degrees of light on what has been called the sleep of plants: but the results of his experiments do not seem sufficiently satisfactory to enable us to draw any important conclusion from them.

Memoir on the Pores of the Bark. By the Same.-When the epidermis is removed from a leaf, a net work is discovered, from the fibres of which arise a number of small pores. The use of these bodies has not been accurately ascertained: but

from

from observing on what kinds of plants and on what particular parts of them they existed in the greatest number, the author concludes that their principal office is to carry on the insensible transpiration.

Memoir on the Vegetation of Misletoe. By the Same.-It was found that, by inserting the cut end of a branch which contained misletoe into a red infusion, the coloring matter passed into the vessels of the plant; and that the infusion was also capable of being transmitted from the misletoe into the branch.

Outline of a Plan for establishing Experimental Farms, and for fixing the Principles of Agricultural Improvements. By Sir John Sinclair, Bart. M. P. and founder of the British Board of Agriculture. We learn that this paper was presented to the Institute by M. Otto, in the year 8, (1799) but it is for the most part the same with Essay X. in an Evo. volume, on miscellaneous subjects, which Sir John S. published in London in 1802, and of which we gave some account in M. R. vol. xl. N. S. p. 289. The English essay is intitled "Proposals for establishing by subscription a new Institution to be called the Plough: or Joint Stock Farming and Experimental Society," &c. Though the author seemed seriously to discuss the objections which would probably be urged against his scheme, we did not suppose that he was so sanguine as to cherish the idea of ever seeing it carried into effect. By offering it also to the French, however, it may be concluded that he did not consider the plan as altogether theoretical. He has here stated the outlines, but has deemed it prudent not to anticipate objections. In the Report subjoined, it is admitted by our enemies that their resources for the execution of projects of this nature are very different from those which we enjoy in England; and that though the scheme may appear to involve nothing extra: rdinary in the contemplation of the author's own countrymen, it mus without doubt strike the French as gigantic.

Three plates accompany this communication; the first represents the divisions of an experimental farm, with the crops in each field for one year; the second, the ground plan and elevation of the circular cottages proposed to be erected for the labouring peasants; and the third, a plan and view of a village.

Observations on the Births, Marriages, and Deaths which occurred at Montpellier in the course of twenty-one Years, from 1772 to 1792 inclusive, and the Calculations which result from them respecting the Probabilities of Life. by M. J. A. MOURGUE.-The facts here stated are that during the above mentioned period the number of births at Montpellier was 25,064, viz. 12,919

malt 8,

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