lel to the axis of y. curve is in the plane From 1 it appears that the curve passes through the origin, and has two branches, both of which are out of the plane of the paper, and which touch the axis of x; whence, as 2 and 3 shew, the curve recedes from the axis of x, until when a = + a = OA = OA', y = ∞, and there are two asymptotes paralFor values of x beyond these lines, the of reference, and returns towards the axis of x, until the ordinates reach minimum and maximum values when a√2, as is shewn by 4 and 5, whence it recedes again towards the asymptotes whose equations are y=x, and intersects them at infinity in a point of inflexion, as shewn by 6 and 7, the curve lying above the asymptote in the first quadrant, and being symmetrically situated in the others. Its course is traced in fig. 77, where оA = α, OB = √2a, BC = 2a, and where the dotted line represents the curve out of the plane of reference. = If the equation to be discussed had been y2 (a2 — x2) = x4 the branches of the curve which are in the plane of reference would have been out of it, and vice versa. The continuity of curve is remarkable in both cases. Ex. 9. Discuss the curve whose equation is whence it appears that the curve is symmetrical with respect to a = 0, when x = (1 ± √5), a careful inspec dx 2 tion of the above quantities shews that the form of the curve is that drawn in fig. 78, the dotted branches being those out of the plane of the paper; OA = OB = BC = a, Ex. 10. Examine the Folium of Descartes, the equation to which is y3 −3axy + x3 = 0. As shewn above in Ex. 3, Art. 235, the equation to the asymp Also at the origin there is a double point, as shewn in Ex. 3, Art. 249. dy dx = 0, if ay = x2; that is, if x = a (2), and y = a (4)*. Also = dy ∞o, if ax = y2; that is, if x = a(4), and y = a (2) *; and the curve does not extend beyond these limits; it is such as is delineated in fig. 63. In tracing curves of this kind involving circular functions, the arc, of which the trigonometrical function is given, is to be measured along the coordinate axis; in the present case along the axis of x, since sin x is involved in the equation, and the ordinates are to be constructed corresponding to the arcs or abscissæ thus measured; 7, we must remember, is the symbol for the arithmetical number 3.14159; and we must give to a such values as will render y a quantity capable of construction. Thus, in the equations above, let After which the values recur, and the curve is that drawn in fig.79. Ex. 12. Trace the curve whose equation is y = esec, e being the base of the Napierian logarithms. day dx2 sec x = e sec x {(sec x)3 + 2 (sec x)2 — sec x −1}. The curve is that drawn in fig. 80, where OA=AB=BC=CD CHAPTER XI. PROPERTIES OF PLANE CURVES, AS DEFINED BY EQUATIONS REFERRED TO POLAR COORDINATES. SECTION 1.-The mode of interpretation, and the equations, of curves referred to polar coordinates. 261.] In the present Chapter we shall investigate, for polar curves, formulæ somewhat analogous to those discussed in the last Chapter for curves referred to rectangular coordinates; but previously it is necessary to extend the usual mode of interpreting polar equations, so as to accommodate them in a greater degree to the law of continuity. Let rf(0) be the equation to the curve. Then, taking a fixed point s as the origin, which is called the pole, and a fixed line Sx passing through it as the line of origination, which is called the prime radius, see fig. 81, it is manifest that the moveable radius, which is symbolized by r, may revolve about s in either of two directions; and thus, if the only datum is that r makes an angle with the prime radius, it is undetermined whether r is above or below sx: that is, whether r revolves up from sx from right to left, or down from left to right. Hence arises the necessity of some symbol of the direction in which r turns, so that angles formed in one direction may be differently symbolized to those formed in another. This indefiniteness will be avoided if we call angles positive when measured up from sx, as in fig. 81: that is, when the radius vector revolves round s in the direction indicated by the curved arrow; and negative when they are measured down from sx, and the radius vector revolves in the direction indicated by the curved arrow in fig. 82. In this case then, + and, as affecting angles, indicate the two different directions in which r can revolve in the plane of the paper. Again, suppose that for a given value of 0, r is affected with a negative sign, a question arises, in what direction is the negativer to be measured? No doubt, if r is affected with a positive sign, the length of it, determined by the equation to the curve, is to be measured from the pole along the revolving radius vector which is inclined at the given angle to the prime radius; as e. g. if a polar equation between r and is such that, when 0 = π 4' r = a, then a length =a is to be measured from the pole along the revolving radius, which is inclined at 45° to the prime radius. From analogy therefore to what has been said in Art. 189, on the signs and r must be measured along the radius vector produced backwards; i. e. if, when 0 = π 4' a, a line equal to a must be measured from the pole along the revolving radius produced backwards: that is, in a direction making an angle of 225° with the prime radius. That we may the better avoid confusion on this subject, conceive the revolving radius to be an arrow of variable length, such as we have drawn in figs. 81 and 82, the pole being a fixed point in it; then, if is the angle between the prime radius and the part of the arrow towards the barbed end, lines measured from s in the direction sp will be positive, and in the direction sq negative. If therefore r is affected with a positive sign, it is to be measured towards the barbed end, but if with a negative sign, towards the feathered end of the arrow. In the figures different positions of the arrow are drawn to indicate different positive and negative directions of r. In the following Chapter we shall omit those particular values of which are affected with + √√, as no satisfactory interpretation of such symbols in such a relation exists, and we shall consider those only which are affected with; being careful however to make r revolve in both the positive and negative directions, otherwise at certain points the curve will appear to be discontinuous. And for the purpose of illustration in the sequel, we must here insert an account of the mode of description, and the equations of some polar curves, many of which, having been treated of at length by old geometricians, possess no small historical interest. 262.] The Spiral of Archimedes. DEF.-If the length of the radius vector of a spiral is proportional to the angle through which it has moved from its |