Discussion of the General Equation of the Second Degree -The Ellipse-The Hyperbola-The Parabola-Abridged Notation. Y 1. In order to fix the position of a point on a plane it is necessary to know its distances from two fixed lines of reference. Thus, for example, if the two lines OX, OY in the plane of the paper be fixed in position, the point P will evidently be determined if we know the length PM drawn parallel to OY and PN drawn parallel to OX: for suppose that PM is three times the unit of measure and PN four times: X N P M then it is only necessary to measure off from OY a length ON equal to three units and from OX, OM equal to four units, and through the points N and M draw the lines NP, MP parallel respectively to OX and OY; their intersection determines the point P. 2. Instead of two lines of reference being given suppose there is only one OX, then if S' it is known that the point P is three times the unit of length PP' S X from this line, it is evident that the position of the point P, though restricted, is yet indeterminate, as it may lie any B where in the line SS', the perpendicular of which from the line OX is three units of length. 3. The fixed lines XX', YY' are called the axes of coordinates, and their intersection is termed the origin; the lengths PM and PN are called the coordinates of the point P. PN or OM being the abscissa and PM or ON the ordinate of P. If the angle between the fixed lines be a right angle, the axes are said to be rectangular, and the distances OM and MP are the rectangular coordinates of the point P. The distances OM and MP measured from O along the axes are usually denoted by x and y respectively; the line OX is called the axis of x or axis of abscissæ, and OY the axis of Y or the ordinate axis. 4. A point is then determined if we have the following equations: x=a and y=b, where a and b are known quantities. It is necessary to know however not only the absolute values of a and b, but also the signs of these quantities. Thus, for example, if positive values of x are measured from O towards X, then negative values must be measured towards X'. Similarly if positive values of y are measured from O towards Y, negative values must be measured towards Y'. X P P P P2 denotes a point P in the angle YOX, |