CHAPTER VIII. Tangents, normals, asymptotes, and tangent planes. 1. Definitions. (1.) A tangent is a right line which meets a curve, but does not cut it in the neighbourhood of the touching point. The touching point is here supposed not to be a singular point. Vid. Ch. 12. (2.) The subtangent is that part of the axis of the abscissæ which is intercepted between the ordinate and the tangent. (3.) A normal is a perpendicular to the curve, intercepted by the axis. (4.) The subnormal is that part of the axis which is intercepted between the ordinate and the normal. Thus in the figure, PT is the tangent; NT the subtangent; PG the normal; NG the subnormal. (5.) Asymptotes are right lines or curves, which, cutting one of the axes at a finite distance from the origin, by increasing the abscissa indefinitely, may be made to approach nearer to the curve than by any assignable distance. 2. Required to find geometrical representations for the fluxions of a curve and of its co-ordinates. Let AN, NP be the co-ordinates; ap the intercepted arc. Draw PQ parallel to the axis, and take it of a finite and determinate magnitude to represent the fluxion of AN; let TPR be a tangent to the curve at P, meeting the axis of the abscissæ in т; draw QR parallel to the axis of the ordinates meeting the tangent in R; then the sides of the triangle PQR shall represent the required fluxions. For draw the ordinate prn near to PN; produce it to meet the tangent in s; join rp, and produce it to meet QR in r: then as pn moves towards PN, there may always be drawn a curve pr similar to Pp, which has the same tangent PR (7. 29.). Now inc. AN: inc. PN: inc. AP :: PT : рÅ : curve pp, :: PQ: Qr: curve pr; and to obtain the limit of these ratios, suppose pn to move towards PN; then ultimately the RPr vanishes (7. 27.), and therefore instead of or and curve pr, we may substitute in the limit QR and PR; whence by the definition, Ch. 1. Art. 7. d.AN: d.PN: d.AP :: PQ : QR : PR. PQR is called the fluxional triangle of the curve at P. Cor. 1. Since Rr ultimately vanishes, the limit of sĩ: pr is a ratio of equality, and consequently the first fluxions of the ordinate of the curve and of the tangent are equal. Cor. 2. Let ON then, when the co-ordinates are NP = y rectangular, PR2 = PQ2 + qr2, or AP S ds2 = dx2 + dy2, or . Cor. If the co-ordinates are not rectangular but inclined at an angle a, ds2 = dx2 + dy2-2cos.adxdy (Trig. p. 24), or 2cos.a dx 3. Given a curve's equation; required the equation of the tangent at any point of the curve. Let x', y' be the co-ordinates of the tangent, x, y those of the curve at P. Since the equation of the tangent is of the form y'=ax'+b (7, 7), and that it passes through the point (x, y); therefore we have, by elimination, y — y = a(x − x); and a tan. 4 PTN tan. RPQ = dy ; wherefore the re quired equation is y' - y = dy (x − x') = p(x' − x). dx — Hence if the equation of the curve is of the form y = Fx, dy by differentiation, and consequently the equation of the dx tangent may be obtained. The equation may also be found if y is an implicit function of x. 4. Required the equation of the normal. Let x', y' be the co-ordinates of the normal PG; then its equation is of the form y=ax'+a, where a=-tan. ≤ PGN ; and it passes through (x, y), dr = = (▲) — tan. ▲ PRQ = dy therefore the required equation is y' - y = If then the curve's equation be given, and we can resolve it, we can find the equations of the tangent and of the normal of the point P in terms of either of its co-ordinates. If the angle at which the axes are inclined be changed, the equations of the tangent and normal remain unaltered. Examples. Ex. 1. To draw a tangent and a normal to any point of a given circle. = x2 + y2 = r2, or y' =— tion of the required line TP. N It gives the following property, ON XOV+NPX vw=op2, a constant quantity. mal passes through o. The position of P or the value of its co-ordinates x and y is supposed to be given. Ex. 2. To draw a tangent and a normal to an ellipse. a2y a2y b2x To find the normal; y' — y = 67 (x' — x'), or y' = b2x y y -(x' — x), or y'= -x+2y. 5. Required the subtangent and the subnormal of a curve. PN X PQ The subtangent is NT; and from A', TN = QR These may be found in terms either of x or of y from the curve's equation. Cor. 1. Hence arises a method of constructing either for the tangent or for the normal. For having calculated NT or NG by the formulæ of this article, join TP or GP, which will be the required lines. Also draw ay perpendicular to PT, and AY = Cor. 3. If two curves have a common abscissa, and their corresponding ordinates be always in a given ratio; they will have the same subtangent. 6. The values of the subtangent and of the subnormal may also be deduced from the equations of Arts. 3 and 4. For in these equations make y y = 0, therefore — y = p(x − x), or x - x', which therefore = TN. In these examples we suppose the origin of the coordinates to coincide either with the vertex or with the centre of the curve. Ex. 1. Required the subtangent and the subnormal of the Apollonian parabola. To construct for the tangent; take NT = 2AN, and join TP, which will be the tangent. the latus rectum, and PG will be perpendicular to the curve |