We have not, however, inserted here all the properties of surfaces of the second order, but have retained only those which it is indispensable to know and to retain. The transformation of rectilinear co-ordinates, for example, must be executed with simplicity, and the teacher must restrict himself to giving his pupils a succinct explanation of the course to be pursued; this will suffice to them for the very rare cases in which they may happen to have need of them. No questions will be asked relating to the general considerations, which require very complicated theoretical discussions, and especially that of the general reduction of the equation of the second degree with three variables. We have omitted from the problems relating to the right line and to the plane, the determination of the shortest distance of two right lines. The properties of surfaces of the second order will be deduced from the equations of those surfaces, taken directly in the simplest forms. Among these properties, we place in the first rank, for their valuable applications, those of the surfaces which can be generated by the movement of a right line. PROGRAMME OF ANALYTICAL GEOMETRY. 1. GEOMETRY OF TWO DIMENSIONS. Rectilinear co-ordinates.--Position of a point on a plane. Representation of geometric loci by equations. Homogeneity of equations and of formulas.--Construction of algebraic expressions. Transformation of rectilinear co-ordinates. Construction of equations of the first degree.--Problems on the right line. Construction of equations of the second degree.--Division of the curves which they represent into three classes.--Reduction of the equation to its simplest form by the change of co-ordinates.[5] [Footnote 5: The students will apply these reductions to a numerical equation of the second degree, and will determine the situation of the new axes with respect to the original axes, by means of trigonometrical tables. They will show to the examiner the complete calculations of this reduction and the trace of the two systems of axes and of the curves.] Problem of tangents.--The coefficient of inclination of the tangent to the curve, to the axis of the abscissas, is equal to the derivative of the ordinate with respect to the abscissa. _Of the ellipse._ Centre and axes.--The squares of the ordinates perpendicular to one of the axes are to each other as the products of the corresponding segments formed on that axis. The ordinates perpendicular to the major axis are to the corresponding ordinates of the circle described on that axis as a diameter, in the constant ratio of the minor axis to the major.--Construction of the curve by points, by means of this property. Foci; eccentricity of the ellipse.--The sum of the radii vectors drawn to any point of the ellipse is constant and equal to the major axis.--Description of the ellipse by means of this property.