In the first place, the candidates for admission are required to have studied the subjects set forth in the following pro

gramme:

"1st. The whole of arithmetic, comprising the theory of proportion, progressions, logarithms, and the use of the tables, and exposition of the metrical system.

"2nd. Elementary geometry, comprehending the properties of spherical triangles; the method of limits will be required exclusively in the demonstrations relating to the measurement of the circle and of all circular bodies.

"3rd. Algebra, comprising the resolution of equations of the first and second degrees; indeterminate equations of the first degree, fractional exponents and exponentials, the demonstration of the Binomial Theorem of Newton, only in the case of whole and positive exponents; general composition of equations; rules for the signs of Descartes; determination of commensurable roots; those of equal roots; resolution of binomial and trinomial equations by means of trigonometrical lines; decomposition of rational fractions into simple fractions; resolution of numerical equations by approximation; the elimination of one unknown quantity between two equations of any degree by two unknown quantities, without exposition of any procedure to clear the final equation by the different solutions that it may include.

"4th. Plane trigonometry and use of the table of signs; the four principal formulæ of spherical trigonometry without the application of logarithms, or the resolution of triangles.

"5th. Statics demonstrated in a synthetical manner; composition and decomposition of forces and parallelograms; reduction of a system of forces to a parallelogram and to one force; equation of the equilibrium of a solid, free to move or fixed at a point, or axis, centre of parallel forces and co-ordinates of this centre; determination of the centre of gravity; of the triangle and pyramid; equilibrium of simple machines, the lever, pulley, inclined plane, wedge, wheel, and axle, screw and tackle of pulleys.

"6. Analytical geometry, comprehending the complete discussion of lines represented by equations of the first and second degree by two unknown quantities; and the principal properties of conic sections; equations to the straight-line in space; the equation of the plane, the solution of problems derived from it, and the transformation of co-ordinates.

7th. The first elements of descriptive geometry, relating to the straight-line and plane."

In addition to these subjects, the candidates are all required to stand the following trials :

"1st. They must write a mathematical paper on some subject contained in the programme already cited.

"2nd. An example on the solution of the plane triangle is to be proposed to them in order to prove that they know how to use the tables of logarithms. The tables used in the calculations must extend to seven decimals.

"3rd. They must translate, under the inspection of one of the examiners, passages from a Latin author, and express the force of these by rhetorical paraphrase. They are also expected to write, in French, upon a given subject as an exercise in composition.

"4. They must copy a human figure shaded by crayons, after a model which will be placed before them by the examiner, or by those whom he appoints, and they must draw a plan of a building by descriptive geometry, relating to a question contained in the programme.

"5th. The candidates must write a composition on a subject of Natural Philosophy, comprised in the following pro

gramme:

"General properties of bodies; laws of weight deduced from experience; first principles of hydrostatics; Mariotte's law; barometer; pneumatical machine; weights and specific weights; proper instruments to determine them; thermometer; exposition of the laws of radiant heat; of specific heat; tension of vapours; hygrometer with a fibre, and hygrometer on the principle of condensation; electric attraction and repulsion; electric machine condenser; Leyden-jar; electroscope; lightning conducter; description of the voltaic pile; magnetic attraction and repulsion; declination and inclination of the magnetic needle; production and propagation of sound; reflection of sound; musical intervals; propagation of light; shade, and half or imperfect shade; variation of the intensity of light by reason of the distance and inclination of surfaces; laws of reflection and simple refraction; mirrors and spherical lenses; formulæ relating to the determination of their foci; dispersion of light; properties of simple bodies, not metallic."

The candidates are only examined on the knowledge required by the programme; they are nevertheless recommended to study chemistry, as well as the German and English languages.

The candidates admitted have to undergo new trials, in order to prove that they are the real authors of the literary compositions, drawings, plans and colourings which they have presented. In the case of fraud the pupil is at once expelled. Those candidates whose knowledge of natural philosophy, literature, and drawing is not sufficient are at once declared inadmissible.

The general scheme of instruction will be best understood from the following programme of the courses of analysis, probabilities, and mechanics :

66 ANALYSIS.-DIFFERENTIAL AND INTEGRAL CALCULUS.

"First Year.

"Of functions in general; geometrical representation of functions with one variable; of the ratio between the increment of a function and the increment of its variable; value of this ratio when the increment becomes infinitely small; definition of a derived function, and differential of a function of one principal variable; differentials of simple functions, and also the differentials of functions of other functions. They are required to consider successively, algebraical, logarithmic, exponential, and circular functions.

"Differentials of functions of several independent variables, and of those which depend on a single variable.

"Differentials of implicit functions.

"Differentials of different kinds of functions of only one

variable.

Change of the independent variable.

"Differentials of various orders of functions of several variables: they shall prove that the order of differentiations does not influence the results.

"Taylor's series for functions of one variable. Determination of the limit in which the rest of the series is comprised.

"Value of quantities which are presented under the form g. "Application of Taylor's theorem to the development of any power whatever of a binomial, exponential, logarithm, sine and cosine.

"Representation of the series which express the sine and cosine, by imaginary exponentials.

"Extension of Taylor's theorem to functions of several variables.

"Theorems on the derived parts of homogeneous functions. "Maxima and minima of functions of one or more variables. Proper criterion for distinguishing the values of the maxima and minima.

"Expression for the differential of the arc of a plane curve. Equation of the tangent and normal to plane curves. "Asymptotes.

"Expression for the tangent and subtangent, normal and subnormal to plane curves.

66

Tracing the concavity and convexity of a curve.

"Points of inflexion or contrary flexure; and points of reflexion or cusps, &c.

"On the oscillation of the circle, angle of inclination and radius of curvature of a plane curve.

"On the contact of plane curves.

Theory of developments. Application to conic sections, to the cycloid, logarithmic spiral, &c.

"Tangent and normal plane to a curve of double curvature. Expression for the differential of its arc.

"On the osculating plane, the angle of inclination, and the radius of curvature to curves of double curvature: they shall take the spiral for example.

"Tangent and normal planes to a curved surface.

"Of the curve of contact of a cylinder or circumscribed cone to a surface.

"Definition of the integral of a differential function. An integral taken between given limits is the sum of the infinite number of small values of the differential comprised within these limits.

"Method of integrating rational functions, adfected differentials of a radical of the second degree, and those which include logarithmic, exponential and circular functions. Indication of the methods of integration by reduction for binomial differentials. "Integration by parts. Application to the devolopment of log. (1+a), arc. tang. x, and arc. sin. x.

Applications of the integral calculus for the rectification of plane curves and double curvature, also for the quadrature of plane curves, transformed to rectangular co-ordinates or to polar co-ordinates, quadrature of curved surfaces, cubature of solids. They must be separately occupied with the solids of revolution, and solids terminated by any surfaces whatever. The formulæ ought to be deduced from the consideration of infinitesimals, and applied to several examples.

"They shall insist on the determination of the limits of simple and double integrals.

"Differentiation and integration under the sign г. Determination of some definite integrals. Conditions of integration for differential functions of the first order to several independent variables. Integrations of the same functions when they satisfy these conditions.

"ANALYSIS.
"Second Year.

"Integration of differential equations of the first order to two variables. General integral of those equations containing an arbitrary constant. Of the factor proper to render the equation integrable.

66

Integration of the linear equation of the first order, and all homogeneous equations.

"Particular solutions of the differential equations of the first

order deduced from the general integral. General integral and particular solution of equations where the two variables only enter in the first degree.

"Number of arbitrary constants which ought to enter into the complete integral of a differential equation of any order whatever.

"Theorems relating to the integration of linear equations of any order. Application to the case of equations of constant coefficients with a last term, constant or variable.

"Elimination of the variables between simultaneous differential equations. Integration of simultaneous linear equations. "Integrations by parts of some differential equations.

"Elimination of arbitrary functions by means of equations to partial differences.

"Integration of linear equations to partial differences of the first order. Determination of the arbitrary function.

Integration of the linear equation of the second order which is found in the vibrating cord; determination of the arbitrary functions.

"Elements of the method of variations. Variation of a simple integral which only contains differentials of the first and second order.

66

Equations which determine the maxima and minima of these integrals; 1st, when the variables are independent; 2nd, when they are joined with finite equations; 3rd, when an integral where they enter ought to have a constant value. Applications to several examples.

"Elements of the calculation of finite differences direct and inverse. Application to the summation of series. Examples. "Formulæ of interpolations. Use of these formulæ for the approximation of quadratures, cubatures and rectifications.

"Theory of the curvature of surfaces. Directrix. Conjugate tangents. Radius of curvature made by normal planes. Radius of greater and less curvature. Relations which exist between these radii of curvature and that of the normal or inclined section. "Equation of the lines of curvature. These two lines are tangents to the sections of greatest and least curvature.

Elements of the calculation of probabilities and social arithmetic.

"General principles of the calculation of chances; simple, compound, partial, and total probability. Repeated proofs; theorems of Bernouilli. Probability of events to come deduced from the observation of former events of the same nature.

"Mathematical trust; application to various cases, and particularly to lotteries.

"Tables of population and mortality. Average length of life. "Annuities. Rents during life. Tontines, assurances, &c." (To be continued.)

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