I. ANALYSIS.—FIRST YEAR.
DIFFERENTIAL CALCULUS.
Lessons 1–9.Derivatives and Differentials of Functions of a Single Variable.
Indicationof the original problems which led geometers to the discovery of the infinitesimal calculus.
Use of infinitesimals; condition, subject to which, two infinitely small quantities may be substituted for one another. Indication in simple cases of the advantage of such substitution.
On the different orders of infinitely small quantities. Infinitely small quantities of a certain order may be neglected in respect of those of an inferior order. The infinitely small increment of a function is in general of the same order as the corresponding increment of the variable, that is to say, their ratio has a finite limit.
Definitions of the derivative and differential of a function of a single variable. Tangents and normals to plane curves, whose equation in linear or polar coordinates is given.
A function is increasing or decreasing, according as its derivative is positive or negative. If the derivative is zero for all values of the variable, the function is constant. Concavity and convexity of curves; points of inflection.
Principle of function of functions. Differentiation of inverse functions.
Differentials of the sums, products, quotients, and powers of functions, whose differentials are known. General theorem for the differentiation of functions composed of several functions.
Differentials of exponential and logarithmic functions.
Differentials of direct and inverse circular functions.
Differentiation of implicit functions.
Tangents to curves of double curvature. Normal plane.
Differential of the area and arc of a plane curve, in terms of rectilinear and polar co-ordinates.
Differential of the arc of a curve of double curvature.
Applications to the cycloid, the spiral of Archimedes, the logarithmic spiral, the curve whose normal, sub-normal, or tangent, is constant; the curve whose normal passes through a fixed point; the curve whose arc is proportional to the angle which it subtends at a given point.
Derivatives and differentials of different orders of functions of one variable. Notation adopted.
Remarks upon the singular points of plane curves.
Lessons 10–13.Derivatives and Differentials of Functions of Several Variables.
Partial derivatives and differentials of functions of several variables. The order in which two or any number of differentiations is effected does not influence the result.
Total differentials. Symbolical formula for representing the total differential of thenthorder of a function of several independent variables.
Total differentials of different orders of a function; several dependent variables.Case where these variables are linear functions of the independent variables.
The infinitesimal increment of a function of several variables may in general be regarded as a linear function of the increments assigned to the variables. Exceptional cases.
Tangent and normal planes to curved surfaces.
Lessons 14–18.Analytical Applications of the Differential Calculus.
Development of F(x + h,) according to ascending powers ofh. Limits within which the remainder is confined on stopping at any assigned power ofh.
Development of F(x,) according to powers ofxorx - a;abeing a quantity arbitrarily assumed. Application to the functions sin(x,) cosx,ax, (1 +xm) and log.(1 +x.) Numerical applications. Representation of cosxand sinxby imaginary exponential quantities.
Developments of cosmxand sinmxin terms of sines and curves of multiples ofx.
Development of F(x + h, y + k,) according to powers ofhandk. Development of F(x, y) according to powers ofxandy. Expression for the remainder. Theorem on homogeneous functions.
Maxima and minima of functions of a single variable; of functions of several variables, whether independent or connected by given equations. How to discriminate between maxima and minima values in the case of one and two independent variables.
True values of functions, which upon a particular supposition assume one or another of the forms
00,∞∞, ∞ + 0, 00, 4∞
Lessons 19–23.Geometrical Applications. Curvature of Plane Curves.
Definition of the curvature of a plane curve at any point. Circle of curvature. Center of curvature. This center is the point where two infinitely near normals meet.
Radius of curvature with rectilinear and polar co-ordinates. Change of the independent variable.
Contacts of different orders of plane curves. Osculating curves of a given kind. Osculating straight line. Osculating circle. It is identical with the circle of curvature.
Application of the method of infinitesimals to the determination of the radius of curvature of certain curves geometrically defined. Ellipse, cycloid, epicycloid, &c.
Evolutes of plane curves. Value of the arc of the evolute. Equation to the involute of a curve. Application to the circle. Evolutes considered as envelops. On envelops in general. Application to caustics.
Lessons24–27.Geometrical Applications continued. Curvature of Lines of Double Curvature and of Surfaces.
Osculating plane of a curve of double curvature. It may be considered as passing through three points infinitely near to one another, or as drawn through a tangent parallel to the tangent infinitely near to the former. Center and radius of curvature of a curve of double curvature. Osculating circle. Application to the helix.
Radii of curvature of normal s of a surface. Maximum and minimum radii. Relations between these and that of any , normal or oblique.
Use of the indicatrix for the demonstration of the preceding results. Conjugate tangents. Definition of the lines of curvature. Lines of curvature of certain simple surfaces. Surface of revolution. Developable surfaces. Differential equation of lines of curvature in general.
Lesson 28.Cylindrical, Conical, Conoidal surfaces, and Surfaces of Revolution.
Equations of these surfaces in finite terms. Differential equations of the same deduced from their characteristic geometrical properties.
INTEGRAL CALCULUS.
Lessons 29–34.Integration of Functions of a Single Variable.
Object of the integral calculus. There always exists a function which has a given function for its derivative.
Indefinite integrals. Definite integrals. Notation. Integration by separation, by substitution, by parts.
Integration of rational differentials, integer or fractional, in the several cases which may present themselves. Integration of the algebraical differentials, which contain a radical of the second degree of the form √c+bx+ax2. Different transformations which render the differential rational. Reduction of the radical to one of the forms
√x2+x2, √a2-x2, √x2-a2.
Integration of the algebraical differentials which contain two radicals of the form
√a+x, √b+x,
or any number of monomials affected with fractional indices. Application to the expressions
Integration of the differentials
F(logx)dxx, F sin-1xdx√1-x2,x(logxn)dx,xmeaxdx, (sin-1xm)dx.
Integration of the differentialseaxsinbxdxandeaxcosbxdx.
Integration of (sinxm.)(cosxn)dx.
Integration by series. Application to the expression
Application of integration by series to the development of functions, the development of whose derivatives is given: tan-1x, sin-1x, log(1 +x.)
Lessons 35–38.Geometrical Applications.
Quadrature of certain curves. Circle, hyperbola, cycloid, logarithmic spiral, &c.
Rectification of curves by rectilinear or polar co-ordinates. Examples. Numerical applications.
Cubic content of solids of revolution. Quadrature of their surfaces.
Cubic content of solids in general, with rectilinear or polar co-ordinates. Numerical applications.
Quadrature of any curved surfaces expressed by rectangular co-ordinates. Application to the sphere.
Lessons 39–42.Mechanical Applications.
General formula for the determination of the center of gravity of solids, curved or plane surfaces, and arcs of curves. Various applications.
Guldin’s theorem.
Volume of the truncated cylinder.
General formula which represent the components of the attraction of a body upon a material point, upon the supposition that the action upon each element varies inversely as the square of the distance. Attraction of a spherical shell on an external or internal point.
Definition of moments of inertia. How to calculate the moment of inertia of a body in relation to a straight line, when the moment in relation to a parallel straight line is known. How to represent the moments of inertia of a body relative to the straight lines which pass through a given point by means of the radii vectores of an ellipsoid. What is meant by theprincipal axes of inertia.
Determination of the principal moments of inertia of certain homogeneous bodies, sphere, ellipsoid, prism, &c.
Lessons 43–45.Calculus of Differences.
Calculation of differences of different orders of a function of one variable by means of values of the function corresponding to equidistant values of the variable.
Expression for any one of the values of the function by means of the first, and its differences. Numerical applications; construction of tables representing a function whose differences beyond a certain order may be neglected. Application to the theory of interpolation. Formulæ for approximation by quadratures. Numerical exercises relative to the area of equilateral hyperbola or the calculation of a logarithm.
Lessons 46–48.Revision.
General reflections on the subjects contained in the preceding course.
ANALYSIS.—SECOND YEAR.
CONTINUATION OF THE INTEGRAL CALCULUS.
Lessons 1–2.Definite Integrals.
Differentiation of a definite integral with respect to a parameter in it, which is made to vary. Geometricaldemonstrationof the formula. Integration under the sign of integration. Application to the determination of certain definite integrals.
Determination of the integrals ∫sinaxxdx, and ∫cosbxsinaxxdx, between the limits0andx. Remarkable discontinuity which these integrals present.
Determination of ∫e-x²dxand ∫e-x²cosmx dxbetween the limits 0 and ∞.
Lesson3.Integration of Differentials containing several Variables.
Condition that an expression of the form Mdx+ Ndyin which M and N are given functions ofxandymay be an exact differential of two independent variablesxandy. When this condition is satisfied, to find the function.
Extension of this theory to the case of three variables.
Lessons 4–6.Integration of Differential Equations of the First Order.
Differential equations of the first order with two variables. Problem in geometry to which these equations correspond. What is meant by their integral. This integral always exists, and its expression contains an arbitrary constant.
Integration of the equation Mdx+ Ndy= 0 when its first member is an exact differential. Whatever the functions M and N may be there always exists a factorµ, such thatµ(Mdx+ Ndy) is an exact differential.
Integration of homogeneous equations. Their general integral represents a system of similar curves. The equation (a+b x+c y)dx+ (a’+b’ x+c’ y)dy=c, may be rendered homogeneous. Particular case where the method fails. How the integration may be effected in such case.
Integration of the linear equation of the first orderdydx+ Py= Q, where P and Q denote functions ofx. Examples.
Remarks on the integration of equations of the first order which contain a higher power than the first ofdydx. Case in which it may be resolved in respect ofdydx. Case in which it may be resolved in respect ofxory.
Integrations of the equationy=xdydx+ φ (dydx). Its general integral represents a system of straight lines. A particular solution represents the envelop of this system.
Solution of various problems in geometry which lead to differential equations of the first order.
Lessons 7–8.Integration of Differential Equations of Orders superior to the First.
The general integral of an equation of themorder containsmarbitrary constants.
(The demonstration is made to depend on the consideration of infinitely small quantities.)
Integration of the equationdmydxm= φ (x.)
Integration of the equationd2ydx2= φ (y,dydx).
How this is reduced to an equation of the first order. Solution of various problems in geometry which conduct to differential equations of the second order.
Lessons 9–10.On Linear Equations.
When a linear equation of themthorder contains no term independent of the unknown function and its derivatives, the sum of any number whatever ofparticular integrals multiplied by arbitrary constants is also an integral. From this the conclusion is drawn that the general integral of this equation is deducible from the knowledge ofmparticular integrals.
Application to linear equations with constant co-efficients. Their integration is made to depend on the resolution of an algebraical equation. Case where this equation has imaginary roots. Case where it has equal roots. The general integral of a linear equation of any order, which contains a term independent of the function, may be reduced by the aid of quadratures to the integration of the same equation with this term omitted.
Lesson 11.Simultaneous Equations.
General considerations on the integration of simultaneous equations. It may be made to depend on the integrations of a single differential equation. Integration of a system of two simultaneous linear equations of the first order.
Lesson 12.Integrations of Equations by Series.
Development of the unknown function of the variablexaccording to the powers ofx-a. In certain cases only a particular integral is obtained. If the equation is linear, the general integral may be deduced from it by the variation of constants.
Lessons 13–16.Partial Differential Equations.
Elimination of the arbitrary functions which enter into an equation by means of partial derivatives. Integration of an equation of partial differences with two independent variables, in the case where it is linear in respect to the derivatives of the unknown function. The general integral contains an arbitrary function.
Indication of the geometrical problem, of which the partial differential equation expresses analytically the enunciation. Integration of the partial differential equations to cylindrical, conical, conoidal surfaces of revolution. Determination of the arbitrary functions.
Integration of the equationd2udy2= a2d2udx2.The general integral contains two arbitrary functions. Determination of these functions.
Lessons 17–23.Applications to Mechanics.
Equation to the catenary.
Vertical motion of a heavy particle, taking into account the variation of gravity according to the distance from the center of the earth. Vertical motion of a heavy point in a resisting medium, the resistance being supposed proportional to the square of the velocity.
Motion of a heavy point compelled to remain in a circle or cycloid. Simple pendulum. Indication of the analytical problem to which we are led in investigating the motion of a free point.
Motion of projectiles in a vacuum. Calculation of the longitudinal and transversal vibrations of cords. Longitudinal vibrations of elastic rods. Vibration of gases in cylindical tubes.
Lessons 24–26.Applications to Astronomy.
Calculation of the force which attracts the planets, deduced from Kepler’s laws. Numerical data of the question.
Calculation of the relative motion of two points attracting one another, according to the inverse square of the distance.
Determination of the masses of the earth and of the planets accompanied by satellites. Numerical applications.
Lessons 27–30.
Elements of the calculus of probabilities and social arithmetic.
General principles of the calculus of chances. Simple probability, compound probability, partial probability, total probability. Repeated trials. Enunciation of Bernouilli’s theorem (without proof.)
Mathematical expectation. Applications to various cases, and especially to lotteries.
Tables of population and mortality. Mean life annuities, life interests, assurances, &c.
Lessons 31–32.Revision.
General reflections on the subjects comprised in the course.
II. DESCRIPTIVE GEOMETRY AND STEREOTOMY.
General Arrangements.
The pupils take in the lecture-room notes and sketches upon sheets, which are presented to the professor and the “répétiteurs” at each interrogation. The care with which these notes are taken is determined by “marks,” of which account is taken in arranging the pupils in order of merit.
The plans are made according to programmes, of which the conditions are different for different pupils. The drawings are in general accompanied with decimal scales, expressing a simple ratio to the meter. They carry inscriptions written conformably to the admitted models, and are, when necessary, accompanied with verbal descriptions.
In the graphic exercises of the first part of the course, the principal object is to familiarize the pupils with the different kinds of geometrical drawing, such as elevations and shaded s, oblique projections and various kinds of perspective. The pupils are also accustomed to different constructions useful in stereotomy.
The subjects for graphic exercises in stereotomy are taken from roofs, vaults, and staircases. Skew and oblique arches are the subject of detailed plans.
FIRST YEAR.
DESCRIPTIVE GEOMETRY.—GEOMETRICAL DRAWING.
Lessons1–3.Revision and Completion of the Subjects of Descriptive Geometry comprised in the Programme for Admission into the School.
Object of geometrical drawing. Methods of projection. Representation of points, lines, planes, cones, cylinders, and surfaces of revolution. Construction of tangent planes to surfaces, of curves, of inter of surfaces, of their tangents and their assymplotes.
Osculating plane of a curve of double curvature. A curve in general cuts its osculating plane.
When the generating line of a cylinder or a cone becomes a tangent to the directrix, the cylinder or cone in general has an edge of regression along thisgenerating line. The osculating plane of the directrix at the point of contact touches the surface along this edge.
Projections of curves of double curvature; infinite branches and their assymplotes, inflections, nodes, cusps, &c.
Change of planes of projection.
Reduction of scale; transposition.
Advantage and employment of curves of error; their irrelevant solutions.
Lessons 4–6.Modes of Representation for the Complete Definition of Objects.
Representation by plans, s, and elevation.
Projection by the method of contours. Representation of a point, a line, and a plane; questions relative to the straight line and plane. Representation of cones and cylinders; tangent planes to these surfaces.
Lessons 7–11.Modes of Representation which are not enough in themselves to define objectscompletely.
Isometrical and other kinds of perspective.
Oblique projections.
Conical perspective: vanishing points; scales of perspective; method of squares; perspective of curved lines; diverse applications. Choice of the point of sight. Rules for putting an elevation in perspective. Rule for determining the point of sight of a given picture, and for passing from the perspective to the plan as far as that is possible. Perspective of reflected images. Notions on panoramas.
Lessons 12–13.Representations with Shadows.
General observations on envelops and characteristics.
A developable surface is the envelop of the position of a movable plane; it is composed of two sheets which meet. It may be considered as generated by a straight line, which moves so as to remain always a tangent to a fixed curve.
Theory of shade and shadow, of the penumbra, of the brilliant point, of curves of equal intensity, of bright and dark edges.
Atmospheric light: direction of the principal atmospheric ray. Notions on the degradation of tints; construction of curves of equal tint.
Influence of light reflected by neighboring bodies.
Received convention in geometrical drawing on the direction of the luminous ray, &c.
Perspective of shadows.
Lessons 14–15.Construction of Lines of Shadows and of Perspective of Surfaces.
Use of circumscribed cones and cylinders, and of the normal parallel to a given straight line.
General method of construction of lines of shadow and of perspective of surfaces by plane s and auxiliary cylindrical or conical surfaces.
Construction of lines of shadow and perspective of a surface of revolution.
The curve of contact of a cone circumscribed about a surface of the second degree is a plane curve. Its plane is parallel to the diametral plane, conjugate to the diameter passing through the summit of the cone. The curve of contact of a cylinder circumscribed about a surface of the second degree is a planecurve, and situated in the diametral plane conjugate to the diameter parallel to the axis of the cylinder.
The plane parallel s of a surface of the second degree are similar curves. The locus of their centers is the diameter conjugate to that one of the secant planes which passes through the center of the surface.
General study of surfaces with reference to the geometrical constructions to which their use gives rise.
Lesson 16.Complementary Notions on Developable Surfaces.
Development of a developable surface; construction of transformed curves and their tangents. Developable surface; an envelop of the osculating planes of a curve. The osculating plane of a curve at a given point may be constructed by considering it as the edge of regression of a developable surface; this construction presents some uncertainty in practice. Notions on the helix and the developable helicoid.
Approximate development of a segment of an undevelopable surface.
Lessons 17–18.Hyperbolic Paraboloid.
Double mode of generation of the paraboloid by straight lines; plane-directers; tangent planes, vertex, axis, principal planes; representation of this surface. Construction of the tangent plane parallel to a given plane. Construction of plane s and of curves of contact, of cones, and circumscribed cylinders.
Scalene paraboloid. Isosceles paraboloid.
Identity of the paraboloid with one of the five surfaces of the second degree studied in analytical geometry.
Re-statement without demonstration of the properties of this surface found by analysis, principally as regards its generation by the conic s.
Lessons 19–20.General Properties of Warped or Ruled Surfaces.
Principal modes of generation of warped surfaces. When two warped surfaces touch in three points of a common generatrix, they touch each other in every point of this straight line. Every plane passing through a generatrix touches the surface at one point in this line. The tangent plane at infinity is the plane-directer to all the paraboloids of “raccordement.”
Construction of the tangent planes and curves of contact of circumscribed cones and cylinders. When two infinitely near generatrices of a warped surface are in the same plane, all the curves of contact of the circumscribed cones and cylinders pass through their point of concourse.
The normals to a warped surface along a generatrix form an isosceles paraboloid. The name of central point of a generatrix is given to the point where it is met by the straight line upon which is measured its shortest distance from the adjoining generatrix. The locus of these points forms the line of striction of the surface. The vertex of the normal paraboloid along a generating line is situated at the central point. If the point of contact of a plane touching a warped surface moves along a generatrix, beginning from the central point, the tangent of the angle which the tangent plane makes with its primitive position is proportional to the length described by the point of contact. The tangentplane at the central point is perpendicular to the tangent plane at infinity upon the same generatrix. Construction of the line of striction by aid of this property.
Lessons 21–22.Ruled Surfaces with plane-divecters Conoids.
The plane-directer of the surface is also so to all the paraboloids of “raccordement.” Construction of the tangent planes and curves of contact of the circumscribed cones and cylinders.
The line of striction of the surface is its curve of contact with a circumscribed cylinder perpendicular to the directer-plane. Determination of the nature of the plane s.
The lines of striction of the scalene paraboloid are parabolas; those of the isosceles paraboloid are straight lines.
Construction of the tangent plane parallel to a given plane.
Conoid: discussion of the curves of contact of the circumscribed cones and cylinders.
Right conoid. Conoid whose inter with a torus of the same height, whose axis is its rectilinear directrix, has for its projection upon the directer-plane two arcs of Archimedes’ spiral. Construction of the tangents to this curve of inter.
Lessons 23–25.Ruled Surfaces which have not a Directer-Plane. Hyperboloid. Surface of the “biais passe.”
Directer-cone: its advantages for constructing the tangent plane parallel to a given plane, and for determining the nature of the plane s. The tangent planes to the points of the surface, situated at infinity, are respectively parallel to the tangent plane of the directer-cone. Developable surface which is the envelope of these tangent planes at infinity. Construction of a paraboloid ofraccordementto a ruled surface defined by two directrices and a directrix cone.
Hyperboloid; double mode of generation by straight lines; center; assymptotic cone.
Scalene hyperboloid; hyperboloid of revolution. Identity of the hyperboloid with one of the five surfaces of the second degree studied in analytical geometry.
Re-statement without demonstration of the properties of this surface, found by analysis, principally as to what regards the axis, the vertices, the principal planes, and the generation by conic s.
Hyperboloid ofraccordementto a ruled surface along a generatrix; all their centers are in the same plane. Transformation of a hyperboloid ofraccordement.
Surface of thebiais passé. Construction of a hyperboloid ofraccordement; its transformation into a paraboloid.
Construction of the tangent plane at a given point.
Lessons 26–28.Curvature of Surfaces. Lines of Curvature.
Re-statement without proof of the formula of Euler given in the course of analysis.
There exists an infinity of surfaces of the second degree, which at one of their vertices osculate any surface whatever at a given point.
In the tangent plane, at a point of a surface, there exists a conic , whose diameters are proportional to the square roots of the radii of curvature of the normal s to which they are tangents. This curve is called the indicatrix. It is defined in form and position, but not in magnitude. The normal s tangential to the axes of the indicatrix are called the principal s.
The indicatrix an ellipse; convex surfaces; umbilici; line of spherical curvatures.
The indicatrix a hyperbola; surfaces with opposite curvatures.
The assymplotes of the indicatrix have a contact of the second order with the surface, and of the first order with the of the surface by its tangent plane.
A ruled surface has contrary curvatures at every point. The second assymplotes of the indicatrices of all the points of the same generatrix form a hyperboloid, if the surface has not directer-plane,—a paraboloid, if it have one.
Curvature of developable surfaces.
There exists upon every surface two systems of orthogonal lines, such that every straight line subject to move by gliding over either of them, and remaining normal to the surface, will engender a developable surface. These lines are called lines of curvature.
The two lines of curvature which cross at a point, are tangents to the principal s of the surface at that point.
Remarks upon the lines of curvature of developable surfaces, and surfaces of revolution.
Determination of the radii of curvature, and assymplotes of the indicatrix at a point of a surface of revolution.
Lessons 29–30.Division of Curves of Apparent Contour, and of Separation of Light and Shadow into Real and Virtual Parts.
When a cone is circumscribed about a surface, at any point whatever of the curve of contact, the tangent to this curve and the generatrix of the cone are parallel to two conjugate diameters of the indicatrix.
Surfaces, as they are considered in shadows, envelop opaque bodies, and the curve of contact of a circumscribed cone, only forms a separation of light and shadow, for a luminous point at the summit of the cone, when the generatrices of this cone are exterior. This line is thus sometimes real and sometimes virtual.
Upon a convex surface, the curve of separation of light and shade is either all real or all virtual. Upon a surface with contrary curvatures, this curve presents generally a succession of real and virtual parts: the curve of shadow cast from the surface upon itself presents a like succession. These curves meet tangentially, and the transition from the real to the virtual parts upon one and the other, take place at their points of contact in such a way that the real part of the curve of shadow continues the real part of the curve of separation of light and shade. The circumscribed cones have edges of regression along the generatrices, which correspond to the points of transition.
The lines of visible contour present analogous circumstances.
General method of determining the position of the transition points. Special method for a surface of revolution.
Lessons 31–34.Ruled Helicoidal Surfaces.
Surface of the thread of the triangular screw; generation, representation, s by planes and conical cylinders.
Construction of the tangent plane at a given point, or parallel to a given plane. The axis is the line of striction.
Construction of lines of shadow and perspective: their infinite branches, their assymplotes. Determination of the osculating hyperboloid along a generatrix.
Representation and shading of the screw with a triangular thread and its nut.
Surface of the thread of the square screw; generation, s by planes and conical cylinders; tangent planes; curve of contact of a circumscribed cone.
The curve of contact of a circumscribed cylinder is a helix whosestepis half that of the surface. Determination of the osculating paraboloid. At any point whatever of the surface, the absolute lengths of the radii of curvature are equal.
Representation and shading of the screw with a square thread, and of its nut.
Observations on the general ruled helicoidal surface, and on the surface of intrados of the winding staircase.
Lesson 35.Different Helicoidal Surfaces.
Saint-Giles screw, worm-shaped screw and helicoidal surfaces to any generatrix. Every tangent to the meridian generatrix describes a screw surface with triangular thread, which is circumscribed about the surface, along a helix, and may be used to resolve the problems of tangent planes, circumscribed cylinders, &c.
Helicoid of the open screw, its generation, tangent planes.
Lessons 36–37.Topographical Surfaces.
Approximate representation of a surface by the figured horizontal projections of a series of equidistant horizontal s. This method of representation is especially adapted to topographical surfaces, that is to say, surfaces which a vertical line can only meet in one point.
Lines of greatest slope. Trace of a line of equal slope between two given points.
Inter of a plane and a surface, of two surfaces, of a straight line and a surface.
Tangent planes, cones, and cylinders circumscribed about topographical surfaces.
Use of a topographical surface to replace a table of double-entry when the function of two variables, which it represents, is continuous. It is often possible, by a suitable anamorphosis, to make an advantageous transformation in the curves of level.
Lesson 38.Revision.
Review of the different methods of geometrical drawing. Advantages and disadvantages of each.
Comparison of the different kinds of surfaces,résuméof their general properties.
Object, method, and spirit of descriptive geometry.
SECOND YEAR.
STEREOTOMY.—WOOD-WORK.
Lessons 1–4.Generalities.
Notions on the mode of action of forces in carpentry. Resistance of a piece of wood to a longitudinal effort and to a transversal effort. Distinction between resistance to flexure and resistance to rupture. Beams.
Advantages of the triangular system, St. Andrew’s cross.
Lessons 5–8.Roofs.
Ordinary composition of roofs.
Distribution of pressures in the different parts of a girded roof.
Design of the different parts of roofs, &c., &c.
Lessons 9–10.Staircases.
MASONRY.
Lessons 11–12.Generalities.
Notions on the settlement of vaulted roofs. Principal forms of vaults,en berceau, &c., &c.
Distribution of the pressures, &c.
Division of the intrados. Nature of the surfaces at the joints, &c., &c.
Lessons 13–15.Berceaux and descentes.
Lessons 16–22.Skew Arches.
Study of the general problem of skew arches.
First solution. Straight archesen échelon.
Second solution: Orthogonalappareil. True and principal properties of the orthogonal trajectories of the parallel s of an elliptical or circular cylinder. Right conoid, having for directrices the axis of the circular cylinder and an orthogonal trajectory. The inter of this conoid by a cylinder about the same axis is an orthogonal trajectory for a series of parallel s.
Third solution: helicoidal. Determination of the angular elevation at which the surfaces of the beds become normal to the head planes; construction in the orthogonal and helicoidalappareilof the curves of junction upon the heads, and the angles which they form with the curves of intrados. Cutting of the stones in these different constructions. Broken helicoidalappareil, for very long skew arches.
Helicoidaltrompesat the angles of straight arches;voussuresor widenings, which it is necessary to substitute near the heads at the intrados of an arch with a considerable skew; case where the skew is not the same for the two heads. Orthogonal trajectories of the converging s of a cylinder.