Through a given point to pass a plane perpendicular to a given right line (rectangular co-ordinates). Through a given point, to pass a perpendicular to a given right line; to determine the foot and the length of that perpendicular (rectangular co-ordinates). Knowing the equations of a right line, to determine the angles which that line makes with the axes of the co-ordinates (rectangular co-ordinates). To find the angle of two right lines whose equations are known (rectangular co-ordinates). Knowing the equation of a plane, to find the angles which it makes with the co-ordinate planes (rectangular co-ordinates). To determine the angle of two planes (rectangular co-ordinates). To find the angle of a right line and of a plane (rectangular co-ordinates). _Surfaces of the second degree._ They are divided into two classes; one class having a centre, the other not having any. Co-ordinates of the centre. Of diametric planes. Simplification of the general equation of the second degree by the transformation of co-ordinates. The simplest equations of the ellipsoid, of the hyperboloid of one sheet and of two sheets, of the elliptical and the hyperbolic paraboloid, of cones and of cylinders of the second order. Nature of the plane sections of surfaces of the second order.--Plane sections of the cone, and of the right cylinder with circular base.--Anti-parallel section of the oblique cone with circular base. Cone asymptote to an hyperboloid. Right-lined sections of the hyperboloid of one sheet.--Through each point of a hyperboloid of one sheet two right lines can be drawn, whence result two systems of right-lined generatrices of the hyperboloid.--Two right lines taken in the same system do not meet, and two right lines of different systems always meet.--All the right lines situated on the hyperboloid being transported to the centre, remaining parallel to themselves, coincide with the surface of the asymptote cone.--Three right lines of the same system are never parallel to the same plane.--The hyperboloid of one sheet may be generated by a right line which moves along three fixed right lines, not parallel to the same plane; and, reciprocally, when a right line slides on three fixed lines, not parallel to the same plane, it generates a hyperboloid of one sheet.