considerations: in (99) let x = y = ∞, then all the terms must be neglected except the first; and thus the direction of the curve at infinity is given by means of the first term of (99). Now this breaks up into two factors a1 = 0 and Un-1 = 0, of which the former represents a straight line, and the latter a curve of the (n-1)th degree: to each of these therefore is the curve infinitesimally near at infinity; and thus a1 = 0, and Un-1 = 0, are the equations to asymptotic lines. Hereby therefore we have not only rectilinear but also curvilinear asymptotes. Thus, for instance, in Art. 235, Ex. 2, the equation y3+x3-ax2 = 0 may be expressed in the form of which the first is a straight line and the second is a hyperbola. Similarly, in Ex. 4, the equation may be expressed in the form (x+y) (x2 — xy + y2)+3a2x+3b2y + c3 = 0, and x+y=0 is the equation to a rectilinear asymptote. Suppose also a1 and a2 to be linear functions of the coordinates; and suppose Un-2 to be a function of x and y of n-2 dimensions; and un+Un-1+ Un-2 to be capable of expression in the form a1 a U-2; so that the equation to the curve becomes (100) а1 а2 Un ... then if we consider the points on the curve where x = y = ∞, only the first term of (100) is to be taken account of, and we have a1a2 Un-2 = which is satisfied by a1 = 0, a2 = 0, Un-2 = 0; of which the first two equations are those of straight lines, and the last is that of a curve of the (n-2)th degree: the curve therefore at infinity breaks up into two straight lines which are asymptotes to it, and into another curve of the (n-2)th degree which is also asymptotic. Thus, if the equation to a hyperbola is Again, if the first four terms of the general equation are such that Un + Un-1 + Un-2 + Un-3 = а1 A2 A3 Un-3, = (101) then the curve has three rectilinear asymptotes, and a curvilinear one whose equation is U-30: it is unnecessary to enter farther into the investigation of special cases. It is however to be observed that all the terms in the equation of lower dimensions may be altered when the equation is expressed in the form (101). Such is the case in the first example given above. It is also to be observed that, generally, a curve of the nth degree has n rectilinear asymptotes. For if x = y=x, all the terms of lower dimensions must be neglected, and we have Un = ... which is a homogeneous equation of n dimensions in terms of a and y; and which is therefore capable of resolution into n factors of the form y-mix, Y—M2X, y-mx; each of which gives the direction in which the curve is going at infinity; and will become the equations to rectilinear asymptotes, or of lines parallel to them, if m1, m2, ... are real numbers; if the m's are, some or all, imaginary, the asymptotes refer to branches of the curves in other planes. m2, SECTION 3.-The direction of curvature, and points of inflexion. dy dx' 240.] The value of which we have discussed in the preceding Sections, enables us to determine the inclination to either of the coordinate axes of a curve at any point, but tells us nothing as to the direction of curvature, that is, as to whether the curve is concave or convex towards a certain line or in a given direction; we use these words in their common meaning; and we proceed to discover criteria which will determine such direction. To simplify the formulæ we shall take a to be an equicrescent variable. d2y = (sec)2, which is always positive; it follows that if is positive, and ≈ are dx2 increasing and decreasing simultaneously; and if tive, as x increases, 7 decreases, and vice versa. T and a simul Now from the geometry it is plain, that if taneously increase and decrease, the form of the curve must be such as that of fig. 53, that is, the curve must be convex downwards; and if as x increases 7 decreases, or vice versa, the form must be such as that of fig. 54, that is, the curve must be concave downwards. Hence we have the following criteria of the direction of curvature. day If is positive, the curve is convex downwards; and if dx2 is negative, the curve is concave downwards. day d2y dx2 Suppose that at a certain point changes its sign, which in algebraical curves it can do only by passing through 0 or ∞, the direction of curvature changes; if the change of sign is from + to, then the curvature, having been convex downwards, becomes concave; if the change is from to +, the reverse is the case. A point where such a change of curvature takes place is called a point of inflexion. To determine d2y its position we must equate to 0, and to ∞o; and if at the dx2 corresponding critical value there is a change of sign, then such a point is a point of inflexion; fig. 55, (y) and (8) of fig. 48, fig. 14, and fig. 15 illustrate such points of inflexion. This is also evident from the following considerations. 241.] Let y = f(x) be the equation to a curve; and suppose that we consider it not only at the point (x, y), but also at another point (x+h, y+k), so that y + k = f(x+h); (102) then, expanding f(x+h) to three terms by Taylor's Theorem, If a tangent is drawn to the curve at the point (x, y), its and therefore its ordinate, when έ is x+h, is (104) (105) from (103), we obtain the difference between the ordinates to the curve and to the tangent corresponding to the point x+h, and we have h2 y + k − n = - f'(x+0h); 1.2 (106) and taking to be infinitesimal, that is, considering only the point in the curve which is next to (x+dx, y+dy), whereby greater than the ordinate to the tangent, and the curve lies above the tangent, whatever is the sign of h; that is, the curve d2y is convex downwards, as in fig. 53; but if is negative, con trary results follow, and the curve is concave downwards, as in fig. 54. If therefore at any point the curve passes through the tangent so as to be above it on one side of the point of contact and section, and below it on the other, then y+k-n must change sign as h changes sign, which can only be the case when d2y d3y = 0 and is a finite quantity; or in general, when the dx2 dx3 term in the expansion of f(x+h), which gives sign to the whole, is one involving an odd power of h, in which case the curve will pass from below the tangent to above it, or vice versâ, in one or other of the ways indicated in fig. 55. It is plain PRICE, VOL. I. 3 B dy that, at a point of inflexion, and therefore the angle T, attains to a maximum or minimum, and is in fact at the point stationary; that is, is the same for three consecutive points on the curve: the corresponding tangent is called a stationary tangent. The curvature of a curve towards the right or the left may be determined in a similar manner by the value and sign of d2x dy2* 242.] Examples illustrative of the foregoing theory. Ex. 1. To determine the direction of curvature of the curve whose equation is y = (x-1)(x-3) x-2 that is, d2y dx2 is positive or negative according as a is less or greater than 2; and therefore the curve is convex downwards for all values of a less than 2, and concave downwards for all values of a greater than 2; and when a = 2, there is a point of inflexion. x Ex. 2. To determine the direction of curvature of the Witch of Agnesi. ence is convex downwards for all values of x between 0 and at the point therefore, the curve is concave downwards; x= there is a point of inflexion. This investigation ex 3 a plains the form of the curve as drawn in fig. 35. Ex. 3. To prove that the equitangential curve, see Art. 200, is always convex downwards. |