The parabola may be considered as an ellipse, in which the major axis is indefinitely increased while the distance from one focus to the adjacent summit remains constant. Equations of the tangent and of the normal.--Sub-tangent and sub-normal. They furnish means of drawing a tangent at any point of the curve. The tangent makes equal angles with the axis and with the radius vector drawn to the point of contact.--To draw, by means of this property, a tangent to the parabola, 1º through a point on the curve; 2º through an exterior point. All the diameters of the parabola are right lines parallel to the axis, and reciprocally.--The chords which a diameter bisects are parallel to the tangent drawn at the extremity of that diameter. Expression of the area of a parabolic segment. * * * Polar co-ordinates.--To pass from a system of rectilinear and rectangular co-ordinates to a system of polar co-ordinates, and reciprocally. Polar equations of the three curves of the second order, the pole being situated at a focus, and the angles being reckoned from the axis which passes through that focus. Summary discussion of some transcendental curves.--Determination of the tangent at one of their points. Construction of the real roots of equations of any form with one unknown quantity.--Investigation of the intersections of two curves of the second degree.--Numerical applications of these formulas. 2. GEOMETRY OF THREE DIMENSIONS. The sum of the projections of several consecutive right lines upon an axis is equal to the projection of the resulting line.--The sum of the projections of a right line on three rectangular axes is equal to the square of the right line.--The sum of the squares of the cosines of the angles which a right line makes with three rectangular right lines is equal to unity. The projection of a plane area on a plane is equal to the product of that area by the cosine of the angle of the two planes. Representation of a point by its co-ordinates.--Equations of lines and of surfaces. Transformation of rectilinear co-ordinates. _Of the right line and of the plane._ Equations of the right line.--Equation of the plane. To find the equations of a right line, 1º which passes through two given points, 2º which passes through a given point and which is parallel to a given line. To determine the point of intersection of two right lines whose equations are known. To pass a plane, 1º through three given points; 2º through a given point and parallel to a given plane; 3º through a point and through a given right line. Knowing the equations of two planes, to find the projections of their intersection. To find the intersection of a right line and of a plane, their equations being known. Knowing the co-ordinates of two points, to find their distance. From a given point to let fall a perpendicular on a plane; to find the foot and the length of that perpendicular (rectangular co-ordinates).