traordinary refraction; from which the author deduces the relations that necessarily have place between the direction and the velocity of light. On this principle, it follows that light passes from a point without to a point within a crystal; in such a manner that, if we add the product of the right line which it: describes without the crystal by its primitive velocity, to the product of the right line that it describes in the interior by its. corresponding velocity, the sum will be a minimum. This principle gives, therefore, the velocity of light in a diaphanous. medium, when the law of refraction is known, and, recipro¬ cally, it gives this law when we know the velocity:- but, in the case of extraordinary refraction, another condition must be. fulfilled, viz. that the velocity of the luminous ray in the crystal may be independent of the manner in which it enters, depending only on its position with regard to the axis of the crystal, that is, with the angle which the ray forms with a line parallel to the axis, and which therefore is as some function of that angle.

The author pursues these ideas with his usual acuteness, and deduces from them two differential equations which give the principle of least action; concluding his memoir by shewing the perfect identity of the laws of Huygens with this principle, which leaves no doubt of its being due to attractive and repulsive forces that are effective only at insensible distances. Thus, as M. Malus has observed on this subject, after a century of research and discussion relative to this phænomenon, we must admit as incontestible the remarkable law of Huygens, which the authority of Newton had rendered doubtful, and must replace one of the finest discoveries of the former celebrated philosopher in the rank which it is intitled to hold in the system of scientific truths.

Second Memoir on the Theory of the Variation of the arbitrary constant Quantities in all Problems of Mechanics. By J. L. LAGRANGE. - In our account of the Memoirs of the Institute, vol. ix., we entered at some length on the new calculus with which this learned author has enriched the theory of mechanics; and we saw in what manner, from a partial problem, he was led to a general calculus, and afterward simplified it, by drawing his principal formulæ from his primitive equations. Still, however, he had not yet fully completed his purpose; because the application of these general formula to particular problems required a tedious calculation, on account of the eliminations which were necessary to obtain separately the expression of the variation of each of the constant quantities when they were become variable. This the author proposes to make the subject of the present memoir; and a very happy idea,

which had before escaped him, is now introduced, to simplify this part of the operation, which (as the author himself affirms) ⚫ leaves nothing more to be desired in the analytical theory of the variation of constant quantities, as connected with the theory of mechanics.'

Memoir on the Approximations of those Formula which are the Functions of very great Numbers, and their Application to the Doctrine of Probabilities. By M. LA PLACE. With a Supplement. — In a variety of analytical investigations, and particularly in those relating to the doctrine of Probabilities, we are frequently led to formulæ in which, from the nature of the problem under consideration, we have to substitute very large numbers; the computation of which, therefore, in those cases, becomes nearly or entirely impracticable. Suppose, for example, we had found the probability of an event to be

the change of signs taking place by pairs: it is obvious that, if t and n are very great numbers, though the probability may be still expressible either absolutely or approximatively in small numbers, yet it would be impossible to arrive at it unless we could convert the above formula into one of another sort. The first transformation of this kind we owe to Stirling; who reduced the mean term of a very high power of a binomial into a rapidly converging series; a transformation which, M. LA PLACE says, may be considered as one of the finest inventions in analysis.' The means, however, employed by Stirling were indirect, and in some measure limited in their application; which made it desirable that we should possess a more general and direct method in those cases in which such transformations

become necessary. This is the problem which the present author has now proposed to himself; and he here displays that depth of thought and profound analytical knowlege for which he is so justly celebrated. The idea is simple and the transformation easy but the length of the investigation prevents us from attempting any abstract of it.

Investigations relating to various definitive Integrals. By M. LEGENDRE. Euier, in many parts of his works, has bestowed considerable attention on the subject of definitive integrals; a species of investigation to which he seems to have been peculiarly attached, but which has scarcely attracted the attention of any other mathematician since his time. Little, therefore, has been added to what he has left us on this subject; and M. LEGENDRE appears to have been the first who made any ad

vances relative to it, in his memoir published in 1794 on "Elliptic Transcendentals;" of which, we believe, a translation has been given in Leybourn's "Mathematical Repository." As these theorems, however, were not the principal objects which the author had then in view, they were very slightly treated in that memoir: but, having since found that the methods there indicated might be connected with others of the same kind, from the union of which some new theorems and easy approximations would be the result, he has been induced to reconsider the sub-. ject, and to bring the whole into one connected memoir. This is divided into four parts. The first treats of integrals of the form

for all values of x between 0 and 1.

In the second, the author proves that the ratio of the definite integrals

is always expressible by a function which contains no other transcendentals than circular arcs and logarithms; and he thus completely generalizes the theorem of Euler relative to these forms of integrals.

The third part treats of the successive integrals,

from which are drawn several curious and interesting results.

I

The fourth treats of the integral dx (log. I taken

between the same limits, viz. x and x=1, at the conclusion of which is given a table for the several values between *=1 and *= {, which very considerably abridges the calculation in many integrals, both of this form and those belonging to the form given in the first part of this paper.

Fourth Memoir on the Measurements of Altitudes by means of the Barometer. By M. RAMOND..-This memoir, which occupies nearly 100 pages, exhibits the details of a great variety of barometrical observations. M. RAMOND had deduced from his previous computation a certain modification of the coefficients of La Place's barometrical formula, which Prony thought were defective for small heights, and that in such cases the original formula of La Place was more correct; and this doubt, which is

still not completely removed, induced the author to undertake another extensive series of observations, which are here detailed, but of which it is impossible to give the reader any intelligible abstract within our limits.

Examination of the different Methods employed for determining the Azimuths of the Sides of Triangles in geodetic Operations. By F. C. BURCKHARD.It is obvious that this problem requires nothing more than the determination of the angular distance of an object from the meridian; and to persons who are unacquainted with the great degree of accuracy which is requisite ingeodetic operations, nothing can appear more easy. Those, however, who are or have been engaged in similar undertakings know well the defects of all the methods commonly employed on such occasions. In operations which have in view the measurement of an arc of the meridian, the azimuths of the sides of the triangles are not of the highest importance: but, in measuring a perpendicular to a meridian, the utmost accuracy is required; and, as M. BURCKHARD had projected a great undertaking of this kind, it became interesting to him to ascertain the best method of what the French call orienting a chain of triangles, which is immediately reduced to that of finding the azimuth of given terrestrial objects.

The method commonly adopted for this determination is to place, by the aid of a transit instrument, an object at a sufficient distance in the plane of the meridian, by means of the superior and inferior passages of the circumpolar stars:-but, as this requires an excellent clock, and considerable time, it can only be advantageously adopted in fixed observatories. We may likewise observe the passage of two stars, the one of which is very high and the other very low, and of which the difference in right ascension is known. This method was introduced by a missionary some years ago: but it was attended with so little success that it was abandoned, in consequence of those errors, which were attributable only to the inaccuracy of observation, being improperly referred to the principles employed. DELAMBRE again introduced it in his operations in 1780; and, though other methods have been since proposed, this appears to be best adapted to those cases in which the observer is constantly moving from place to place. Accordingly, after having examined all others, M. BURCKHARD has preferred this mode, and has entered at some length into a consideration of the amount of the probable errors to which it is subject, with the means of obviating some of them, and of appreciating the amount of those which are by their nature unavoidable. The memoir is concluded with several useful remarks on the best method of taking observations with Borda's repeating circle.

INDEX

To the REMARKABLE PASSAGES in this Volume.

N. B. To find any particular Book, er Pamphlet, see the
Table of Contents, prefixed to the Volume.