should employ such, we see not on what grounds he could lay claim to any invention or considerable improvement. La Grange in fact employs them. In cases simple as those of the catenary, it would not be worth while to recur to the formulas of Euler and La Grange, since the business can be effected more expeditiously. M. AMPÈRE solves the problem which we have considered, but we see nothing remarkable nor new in his solation. The properties of the catenary, if we understand by such the values of its subnormal, evolute, &c. if not formally put down in any treatise, may easily be deduced; and the facility and obvious method of deducing them render their insertion in the National Memoirs of France unsuitable and unnecessary. We have no great opinion of the mathematical abilities of M. AMPÈRE.

A general and complete Integration of two important Equations that occur in the Motion of Fluids. By MARK ANTONY PARSEVAL. In the second part of his Mécaniq. Analytique, (p. 501.) M. La Grange gave an equation for the propagation of sound; and it is the object of the present memoir to solve that equation, under the reduced form which it assumes when the fluid is not supposed to be agitated by an accelerating force: in which case, the equation is this:

If we suppose the oscillations to be very small, then the equation becomes of this form:

on which equation, it is well known, the solution of the problem of vibrating chords depends. This latter equation was first solved by D'Alembert, and afterward by Euler; and the problem occasioned a long controversy between those mathematicians.

M. PARSEVAL transforms the equation (1) by a method analogous to that which Euler employed in his last solution (for he gave more than one) of the problem of vibrating chords. The method consists in supposing to be a function of and v, and then and to be functions of two other variable quantities p and q the peculiar form of the function for and is stated by the author. Having made the necessary substitutions, &c. M. PARSEVAL transforms his equation (1) into one of this form:

V

μ

V

of which, by the combination of his own method with that of La Place, (Mem. Acad. 1779) he gives the integral.

The second part of the memoir relates to the integration of a differential equation given at p. 489 of the Mécanique Analy tique, (M. PARSEVAL has inserted a wrong reference,) which expresses the conditions of the motion of a fluid contained in a canal of small depth, and nearly horizontal. M. La Grange solved the equation only under a certain restriction, viz. that the fluid in its motion is only elevated to a very small height above the level: but M. PARSEVAL integrates the equation ge nerally, or with this condition alone, that the horizontal canal is composed of fluid lamine always moving according to the same law.

A Method of Summing, by Definite Integrals, the Series given by the Theorem of M. La Grange, by means of which he finds a Value that satisfies an algebraic or transcendental Equation. By the Same. If a be the least root of the equation,

being any number positive or negative. This theorem is due to M. La Grange, and we have expressed it in the language employed by that mathematician in his Fonctions Analytiques.

If r, then (ur)' = (u)' = 1 since u in English notation

; and in this case the series assumes a more simple form. It is the intention of the present author to sum the series under such simple form; which he effects in about 20 pages, and then applies his result to examples.

"Memoir on Series, and on the complete Integration of an Equation of partial linear Differences of the second Order, with constant Coefficients. By the Same.-De Moivre has given (and, if our recollection be right, he was the original author,) an expression for the sum of a series a + bx + ca+ &c. of which the ntb dif ferences of the coefficients are equal.

Euler has extended the expression; and he has shewn that if S be the sum of a + bx + cx2+ &c. and if each term be multiplied respectively by the terms A, B, C, D, &c. quantities such that their ultimate mth differences (Am A) are equal o, then the sum of

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

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 # 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 ʼn 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 42 sin

dz du

=

Consequently

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

Equating the possible and the impossible parts, expressions for the chords of,, &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. Van. quelin.

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,

sofar R.W.

11

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 Phy sique, 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. Ey 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. Diz.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. DETEUX.

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