sign at such a point, there is a maximum or minimum ordinate, du such as is drawn in figs. 12 and 13; but if — does not change sign, then the form of the curve will be such as in figs. 14 and 15, according as ^ is positive or negative. dv If = oo , the tangent, and therefore the curve at the point of contact, is perpendicular to the axis of x, and may be such as oy in fig. 39, or as is drawn in one or other of the diagrams dy of fig. 48; that is, if ~, in passing through oo, changes sign from + to —, the point of the curve may be such as is represented in (a): if it changes sign from — to +, it may be that represented in (/3); but if ~ does not change sign and is positive throughout, the curve is that indicated in (y), and if it is negative throughout, it is that indicated in (8). If at any point of a curve ~jt = 2; that is, if = 0 and {^r-) = 0, the direction of the tangent at the point will be dy indeterminate as far as the form of ~ defines it; but it may be evaluated, and the means of doing so will be discussed in Section 4 of the present Chapter. 221.] Illustrative examples on the preceding Articles. Ex.1. Properties of the ellipse. idr\ _ 2x idr\ _ 2y_ dy _ b2x Hence the equation to the tangent is the equation to the normal is a1 The intercept of the axis of x by the tangent = £0 = —, b1 y = "° = j; dec a ^ v ^ The subtangent = y_ = The subnormal = w-r^ = 3 dx a2 The perpendicular from origin on tangent = {b*x% + a*y2}* Ex. 2. Properties of the Cissoid of Diocles. xs The equation is yl =; lid X dy x* (3 a — x) _ 5* *~ - (2a-x)i' therefore at the origin, ^ = 0, and the curve touches the axis of x; when x = 3 a, -y- = 0, and changes sign, and therefore the ordinates are severally a maximum and a minimum; when x = 2a, ^ = °o, and therefore the curve is perpendicular to the axis of x; when x = a, y — a, = 2, and therefore the curve cuts its fundamental circle at tan-12. These several values of the tangent are exhibited in fig. 34. Ex. 3. Let the curve be the hypocycloid whose equation is, Art. 206, 3 3 , x* + yt — a*. The equation to the tangent is x* y* and therefore f0 = o* and t/o = a*; • •• V + & = ai (** + yi) = a2; and therefore the length of the tangent line intercepted between the coordinate axes is constant. Ex. 4. To find the differential equation to the equitangential curve; see Art. 200. According to the definition, the line Pt of fig. 38 is to be of constant length; let the length be a; _ \ dx2)* dy _ y the negative sign being taken because, according to the form y = a'; .•. the sub tangent = y — = j— - = a constant. du Also when x = 0, y = 1; in which case = log a, which is the tangent of the angle at which the curve is inclined to the axis of x, at the point where it cuts the axis of y. Ex. 6. Properties of the cycloid. (a) Firstly, let the starting point be the origin; the equation to the curve is, x = a versin-1 - — {2ay — y2}^. dx v which is the equation to the tangent in the reduced form; and is of n — 1 dimensions in terms of x and y. If the equation to the curve is expressed in terms of three variables x, y, z, this result is evident immediately: the equation to the tangent in this case is given by (18), and (^~) > (^) > are plainly of « —1 dimensions in terms of x, y and z. If z — C — 1, (18) and (52) are identical; in which case C (^) = «r>-l +2«n_2+ ... + (n— \)UX + »!i0. Let us return to (52). If the origin is on the curve, «o = 0; and if the tangent is in this case drawn at the origin, then omitting the terms which vanish, and replacing and by their values from (49), we have and if we replace f and rj by x and y, remembering that x and y are in this case the current coordinates of the tangenti (53) becomes, by reason of Euler's Theorem, Ml = 0; (54) so that if % + Ut + ... + un = 0 is the equation to a curve, «i = 0 is the equation to the tangent at the origin. Thus if the equation to a conic is Ax* + Bxy + cy* + vx+ By — 0, the equation to the tangent at the origin is ox + Ey = 0. Since a straight line may cut a curve of the »th degree in n points, it follows that if two of these points become coincident, that is, if a line touches a curve at a given point, it can cut it in only n —2 other points. Thus if a straight line touches a conic, it cannot meet the curve again. If a straight line touches a curve of the third order, it must cut it at some other point. Generally, if a line touches a curve of an even order, it will either meet it again in an even number of points or not at all. And if a line touches a curve of an odd order it will meet it again in an odd number of points, and in one point at least. Since a tangent passes through two points of a curve, it is PRICE, Vol. i. Y y |