sign at such a point, there is a maximum or minimum ordinate, dy dx 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, dy If =∞ the tangent, and therefore the curve at the point dx 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 of fig. 48; that is, if in passing through ∞, changes sign dy , 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 dy represented in (8); but if does not change sign and is positive throughout, the curve is that indicated in (7), and if it is negative throughout, it is that indicated in (8). dr = 0, the direction of the tangent at the point will be dy dy indeterminate as far as the form of defines it; but it may dx 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. a2 (n-y), or, = a2-b2. The intercept of the axis of a by the tangent Ex. 2. Properties of the Cissoid of Diocles. = a2 b2 {b1x2+a1y2}$ y2 = 2 a -X x3 (3 a−x) ; (2a-x) the ordinates are severally a maximum and a minimum; when dy the axis of x; when x = a, y = a, =2, and therefore the dx curve cuts its fundamental circle at tan-12. values of the tangent are exhibited in fig. 34. These several Ex. 3. Let the curve be the hypocycloid whose equation is, Art. 206, x} + y} = a}. 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; the negative sign being taken because, according to the form drawn in fig. 38, y decreases as x increases. Ex. 5. Properties of the logarithmic curve. Also when x = 0, y = 1 ; in which case dx 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−12 — {2 ay — y2 } ±. a {2ay-y2}} dy {2ay—y2 } $ 2a-y = dx = Therefore in fig. 49, {2ay—y2}} = LP, and 2a-y = LQ, and therefore the tangent at P also passes through the point Q. Hence also the normal at the point P passes through G, the other extremity of the diameter of the generating circle; as is also manifest from the above value of The length of the normal = PG = Y dx dy ... PG2 = QG X GL. (B) Secondly, let the highest point be the origin; see fig.50; therefore PT is parallel to the chord oq. Hence also OT' = PQ= the arc oq; and hence too the normal PG is parallel to the chord QA. 222.] We must return however to the general equations of the tangent and the normal given in Articles 214-217, for they require closer consideration. The general equation (11) to the tangent is Let us suppose the equation to the curve to be expressed in the form (48) of Art. 207; that is, let F(x, y) = Uo + U1 + Uz + + Un = 0, ... (48) where uo, u,... un, are homogeneous functions of 0, 1, 2, ... n dimensions respectively in terms of x and y; then so that the right-hand member of (47) becomes duz dun + x + y (2) + ... + x (du) + y (day) ・ (du2 ) + y dx dy dx but by Euler's Theorems, Art. 82, this expression is equal to u1 + 2 uz + + n un j ... and therefore, by reason of (48), to so that (47) becomes ; (50) - {un-1+2un-2 + ... + (n − 1) u1 + n uo} ; (51) 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 equa tion to the tangent in this case is given by (18), and (dr), (dr), dr dz are plainly of n-1 dimensions in terms of x, y and z. If z == 1, (18) and (52) are identical; in which case dr dz = Un−1 + 2 Un-2 + + (n − 1) u1 + nuo. ... 0; Let us return to (52). If the origin is on the curve, u。 = and if the tangent is in this case drawn at the origin, then omitting the terms which vanish, and replacing (dr) and by their values from (49), we have η dy dx dr (53) and if we replace έ and 7 by x and y, remembering that a and y are in this case the current coordinates of the tangent, (53) becomes, by reason of Euler's Theorem, ... u1 = 0; (54) so that if u1+Uz + +un=0 is the equation to a curve, u1 = 0 is the equation to the tangent at the origin. Thus if the equation to a conic is Ax2+вxу + cy2 + Dx + Ey = the equation to the tangent at the origin is DX + EY = 0. = 0, Since a straight line may cut a curve of the nth 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 |