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sidered to be only of one dimension, the differential equation which resulted was

nad v d ,

. V dx?

di2 which equation, it is known, D'Alembert solved. The air, however, being considered to be of more than of one dimension, the differential equations that arose in consequence of such consideration became of more difficult solution. To this subject, M. Parseval has repeatedly turned his attention ; and, after many trials, he flatters himself that in this memoir he has overcome the difficulty, and has resolved the problem. He confesses, however, that it would have been fortunate if his solution had possessed the advantage of that of D'Alembert ; in which, by the simple inspection of its form, he finds the laws of the propagation of sound, which are known independently of the nature of arbitrary functions; while in M. PARSEVAL'S solution this advantage cannot take place, since the variable quantities relative to the definite integrals are envcloped under the sign of the functions, and since there are no means of performing the integration unaided by a previous knowlege of the nature of those functions.

Without many symbols and processes, we cannot present to our readers the method pursued by this mathematician: but, in a concise way, we can represent it to consist in reducing the total integration to a single definite integration for the case of two dimensions, and to two definite integrals for the case of three dimensions. We must add, however, that M. PARSEVAL's method appears to us not remarkably elegant and general; and that many parts of it are indirect.

Memoir on Curves of double Curvature. By M. LANCRET.This paper contains several properties relating to the curves of double curvature ; some of which (viz. those due to the author of the memoir) are demonstrated, and other properties discovered by Clairaut, Euler, &c. as having connection with those which are demonstrated, ale premised. As the nature of the paper and of the discussion precludes us from any particular comment, we must content ourselves with recommending it to the notice of mathematical readers.

The French mathematicians have attended particularly to the subject of curves of double curvature, and especially M101:. Many curious researches, indeed, are connected with the subject. It is rather strange, but we believe the fact 10 be, that no English mathematician has created these curves; if there be any, it must be Waring, in his Proprietates Curvarum: but we have not the power of immediate reference.


Researches on the Application of the general Formulas of the Calculus of Variations to Problems in Mechanics. By A. M. AMPÈRE.-If P, Q, R, be the forces acting on a body of which the element, according to the differential notation, be dm, then the equation of equilibrium is

(P&pt 289+ Rör) dm and to this equation must be added the equation of condition ; for instance, Lda.

This is the method adopted by M. La Grange in his . caniq. Analytique ; and in page 55 he says that “the equation will have a form analogous to equations which the Calculus of Variations furnishes for the determination of the maxima and minima of integral formulas ; and consequently the rules in the Calculus of Variations may be applied to the Equation of Equilibrium.” As a precept this is sufficiently clear, and of its application he gives several instances. One of his problems regards the catenary: the equation of condition must be derived from the circumstance of its inextensibility : thus, if ds be the element of the arc, ds=v{dx? + dy}

dx. dx + dye dy and ods=o=

ds Hence integrating saods by parts according to the known method in the Calculus of Variations, the terms to be added to the equation of condition are

adx d.

ds Then if X be the forces parallel to the axis *, and y the forces parallel to the axis y, we have

X dm-d

ds rdmad

2. dy

ds adx and hence * = A +/Xdm

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the equation to the curve.'

We have perused the memoir of M. AMPÈRE with some attention, but cannot discover its precise object. He wishes to employ the formulas invented by Euler and La Grange in the Calculus of Variations for the solution of problems like the preceding problem ; and, no doubt, in complicated cases it would be very commodious to use them: but if a mathematician



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 solution. 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: do do

dap dx

(1) d x2 dx dx.di dt? If we suppose the oscillations to be very small, then the equation becomes of this form :



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dx2 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 mathe. maticians.

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 u ands, and then and v to be functions of two other variable quantities p and q: the peculiar form of the function for M and v is stated by the author.

Having made the necessary substitutions, &c. M. PARSEVAL transforms his equation (1) into one of this form:

s do

du do 4k ? du dy



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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 Analytique, (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 generally, or with this condition alone, that the horizontal canal is composed of fluid laminæ 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,


(ur)' x f*,

then w = v +(wy"X fu+ (YXA)'+ &c.

r 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=1, then (ur)' = (u)' = I 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+cm2 + &c. of which the nibdifferences of the coefficients are equal.

Euler has extended the expression; and he has shewn that if S be the sum of a + bx + 0*+ &c, and if each term be multiplied respectively by the terms A, B, C, D, &c. quantities such that their ultimate mil differences (Am A) are equal o, then the sum of

Aa + Bb.x + &c.


+ A'A..


+ &c.


Mr. Parseval has invented an analogous method for sum. ming such a series as

Aa + Bb -+ Cct &c. formed by the multiplication of two series

A + Bf + cft &c.
a + b

+ c.

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 of 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 po then the chord of nA can be expressed in terms of p; and from such expression, putting n A 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 n A 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 2 the arc, then since u'= 2 sin?



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