Hence the asymptote cuts the axis of y at a distance as it is therefore inclined at an angle of 45° to the axis of (14) Let the equation to the curve be = Therefore y±(x+a) are the equations to two asymptotes at right angles to each other. Another asymptote parallel to the axis of y is given by putting a = a. (15) Let the equation to the curve be The denominator equated to 0 gives a = b, x = 2b; therefore the corresponding ordinates are asymptotes, since for a=b and x = 2b y is infinite. Whence, expanding and rejecting the terms involving negative powers of a, we have y=x -3 (a - b) as the equation to a third asymptote, which is therefore inclined at an angle of 45° to the axis of a. When the equation cannot be solved with respect to y, we are sometimes able to determine the asymptotes by assuming y=xx, and then by means of the equation expressing x and y in terms of x. If the same value of & which renders x and y infinite give a finite value for the intercepts of the tangents, then these determine the position of the asymptotes. be the equation to the curve: then assuming y=xz, we find Now and y are both infinite when x = intercept of the tangent on the axis of y is and consequently becomes (17) If the equation to the curve be y1 − x1 + 2bx2y = 0, we find by the same means the equations to two asymptotes to be the curve has a rectilinear asymptote in the ordinate at a distance b from the origin. It has also a parabolic asymptote, for we have the equation to a common parabola, the latus rectum of which is a, and the axis of which is parallel to that of y. Their common axis is therefore the axis of y, and their latera recta are equal to a, but they are turned in opposite directions. It sometimes happens that we obtain an equation for an asymptote with possible coefficients, though for large values of one variable in the equation to the curve, the other va riable becomes impossible. This apparent anomaly has been explained by Mr Walton*, by availing himself of the general interpretation which may be given to the symbols in analytical geometry. The impossibility of one of the variables, when certain values are assigned to the other, may be interpreted as signifying that the curve for these values leaves the plane to which it is referred. Now when by assigning an indefinitely large value to the one variable, the other tends to become again possible and to assume the form of the equation to a straight line, as is the case when we find a possible rectilinear asymptote, this indicates that the curve tends to return to the plane of reference, and that at an infinite distance it will coincide with it in a line, the equation to which is that of the asymptote. (20) As an example of a curve having a possible asymptote to an impossible branch let us take the equation, x1 (y − c)2 = b1 (a2 − x3). When = 0, y = co and is possible, and therefore the axis of y is an asymptote: this is one of the ordinary kind. But if we put the equation under the form it is easily seen that when a = ∞, y = c. On the other hand, if xa, y is impossible. Hence the line whose equation is y = c is an asymptote to an impossible branch of the curve; that is to say, a branch of the curve leaves the plane of reference when xa, but tends to return to it again when a∞, coinciding then with the line whose equation is y = c. The form of the curve is given in fig. 27, where the dotted curve represents the impossible branches of the curve. lying in a plane at right angles to the plane of the paper. On the subject of asymptotes to curves, the reader may consult in addition to the work of Stirling before referred to, Newton's Enumeratio Linearum Tertii ordinis, Cramer's Analyse des Lignes Courbes, Chap. VIII. and a paper in the Cambridge Mathematical Journal for November, 1843. Cambridge Mathematical Journal, Vol. 11. p. 236. SECT. 2. Polar Co-ordinates. If the curve be expressed by a relation between » and O, then the tangent of the angle (4) between the radius vector and the tangent to the curve is r do The subtangent, which is the portion of a perpendicular to the radius vector at the origin intercepted by the tangent, is perpendicular from the origin on the tangent is d Ꮎ ; and the dr If the curve be expressed by a relation between u and ✪ Asymptotes to spirals are determined by finding what value of makes infinite; and if the same value of 0 make 72 dᎾ dr either finite or equal to zero, a line drawn through the extremity of the subtangent parallel to r is an asymptote to the curve. Spirals may have asymptotic circles: these are found by the condition that an infinite value of gives a finite value for r. Ex. 1. The equation to the spiral of Archimedes is |