274.] Examples illustrative of the preceding theory. Ex. 1. To find the positions of the asymptotes of the hyperbola whose polar equation is Therefore the asymptotes are inclined to the prime radius at which is equal to 0, at the critical angles; therefore both asymptotes pass through the pole. Ex. 2. To determine the position of the asymptote to the Conchoid of Nicomedes; see Art. 196, equation (18). The asymptote therefore cuts the prime radius at right angles, and at a distance a from the pole in the positive direction; see fig. 36. Ex. 3. To determine the asymptotes to the lituus. The prime radius therefore is an asymptote, as delineated in fig. 85. Ex. 4. To determine the asymptotes of the reciprocal spiral. The asymptote therefore is a line parallel to the prime radius, at a distance a from it, to be measured in antecedentia, since Suppose the equation to a polar curve to be such that r approaches to a finite limit, say a, as is infinitely increased; then the curve approaches more and more nearly to a circle whose radius is a, which circle is said to be asymptotic to the curve; and if the curve approaches to it from the outside, the circle is called an interior asymptotic circle, and if from the inside, an exterior asymptotic circle. First, let be positive; then r is always greater than a; and line parallel to the prime radius at a distance a above it is an asymptote to the curve; and when = x ra; whence we have an interior asymptotic circle such as is drawn in fig. 91. Secondly, let be negative; then T = a 0, r = - and therefore when = -∞, and is negative as increases until = 1, in which case r = 0, and thence is always less than a until 0 = ∞, when r = a. Thus we have the curve dotted in the figure, and with an exterior asymptotic circle of radius SA = a; the continuity of the two branches of the curve is worth remarking. Ex. 2. To determine the asymptotic circle to the curve, 0 = ∞, 0 (2ar-r2) = 1; when r = 2a, and when r = 0. Therefore the circle whose radius is 2a is asymptotic to the curve; and as r must be always less than 2 a, otherwise would be affected with √, the asymptotic circle is exterior to the curve. SECTION 4.-Direction of curvature, and points of inflexion. 276.] On an inspection of the figures numbered 92 and 93 it is manifest that, if a curve referred to polar coordinates is concave towards the pole, as r increases, p increases also, and theredr fore is positive; and if the curve is convex towards the pole, dp dr as r increases, p decreases, and vice versa, and therefore is dp negative. If therefore the equation to the curve is given in the form rƒ(0), in order to determine whether the curve is concave or convex towards the pole, we must transform the equation into its equivalent between r and p, by means of the relations given in (19) or (21) of Art. 270, and thence find dr dp is positive, the curve is concave dr towards the pole; and for all values for which is negative, dp ; the curve is convex towards the pole; and therefore if at any dr point changes sign by passing through 0 or ∞, at such a dp point the direction of curvature changes and there is a point of inflexion; hence, to determine such points, we must equate dr dr dp to 0 and to ∞, and examine whether changes sign; if it does, there is a point of inflexion. dp Ex. 1. To determine the point of inflexion of the lituus. r = = 0, if ra√2, and changes sign from+to; the curve therefore having been concave towards the pole for values of r less than a √2, changes its direction of curvature at that point, and becomes convex towards the pole; see fig. 85, SB = a = a√2. Ex. 2. To prove that the equiangular spiral is always concave towards the pole. r = ao; dp and by (25), Art. 272, p = mr; therefore =m, which is always positive, and therefore the curve is always concave towards the pole. SECTION 5. On tracing polar curves by means of their equations. 277.] Having discussed all the peculiarities which curves referred to polar coordinates generally admit of, we are now in a condition to analyse the equations, and to give general rules for tracing the curves of which they are the mathematical expressions and definitions. 1) If the equation is of the form r = f(0) + (0), so that r = f(0) is diametral to the curve to be traced, we had better trace separately the two curves r = f(0) and r = (0), and then by addition and subtraction of the radii vectores trace the required curve. Thus if it is required to draw the curve whose equation is r = a (2 + sin 0), the circle whose radius is 2 a is diametral to the required curve, and its radii are to be increased and diminished by a sin @ corresponding to the several values of 0; see fig. 94. 2) Investigate the several values of 0 which make r = 0, and = ∞ ; and in the latter case, if the value of 0 is finite, determine whether the polar subtangent is finite or not, as this is the criterion whether the rectilinear asymptote can be constructed or not. Give such particular values to 0 as the equa tion suggests; as e. g. if the equation involves a function of 30, put = 15°, 30°, 45°, &c.; or if the equation involves a function to values such that r may be constructed; and, by giving to the values 0 and nπ, we find the values of r when the curve cuts the prime radius, or the prime radius produced backwards; and make r revolve in both directions. dr 3) It is convenient to find as it is the ratio of the cor responding increments of r and 0; and therefore, if it is positive, as increases, r increases; and, if it is negative, r decreases as O increases, and vice versa. And if dr do = O, we have no increase of r corresponding to an increase of 0; that is, the curve is at right angles to the radius vector; which is also manifest from equation (14), Art. 269, because at such a point tan SPT = ∞. And if = dr do O, and changes its sign, we have a maximum or minimum value of r, the point corresponding to which is called an apse; of which there are instances in the ellipse, if the focus is the pole, at the extremities of the major axis: and of the circle, if the centre is the pole, every point is an apse. 4) Nothing more need be said on the subject of rectilinear asymptotes and asymptotic circles; or 5) On the direction of curvature and points of inflexion. 278.] Hence then to trace a curve referred to polar coordinates, I. Investigate, arrange, and tabulate with their proper signs, all the particular values of which render r = 0, and = ∞ ; and equal to a value that may be constructed without difficulty. dr do II. Find examine its sign, and the values of 0 at which it is equal to 0, and to ∞, and whether it changes its sign; if it does, at such points there are maximum and minimum radii vectores. do III. Determine whether any finite values of 0 render r = ∞ ; if so, find the value of r2 corresponding to this value of 0, PRICE, VOL. I. 3 I |