and the values of the intercepts of the coordinate axes of r and y by this asymptote are respectively Hence it appears that the asymptote passes through the origin, when un-10; that is, if the equation to the curve is of the nth degree, and has no term of the (n-1)th degree, the asymptotes pass through the origin. If on evaluation (92) becomes zero, the asymptote is parallel to the axis of x; and if it = ∞, the asymptote is parallel to the axis of y: in either case the intercepts of the coordinate axes are to be determined from (93). 235.] Let us apply the process to some examples. Ex. 1. The equation to the hyperbola being is required to find its rectilinear asymptotes. x2 y2 a2 b2 As this equation is of two dimensions and has no term of one dimension, the asymptotes pass through the origin, which is the centre of the curve. The tangents of the angles at which they are inclined to the axis of x are thus found: Let the equation to the hyperbola be considered in the form Uz + Uo= 0, so that Ex. 2. To find the equations to the asymptotes of the curve y3 + x3 — ax2 = 0. PRICE, VOL. I. 3 A 88 dy dx In the next differentiation must be considered constant; because the inclination of the asymptote to the axes is the same for all the points infinitesimally consecutive on the curve at an infinite distance; see also Art. 139; and therefore differentiating Again, from (93) the intercept of the axis of x Ex. 3. To find the asymptotes to the Folium of Descartes, see fig. 63, of which the equation is And from (93) the intercept of the axis of y Ex. 4. If the equation to a curve is x3+y3 +3a2x + 3b2y + c3 = 0, the equation to the asymptote is x + y = 0. I must observe that as the asymptote is a tangent to a curve at an infinite distance, the general properties of tangents are true also, in their degree, of asymptotes. Thus an asymptote must be considered to have two points common with the curve. Now as a straight line can generally cut a curve of the nth degree in n points, so an asymptote cannot cut it in more than n-2 points. Thus an asymptote to a conic cannot cut the conic, at least at a point within a finite distance. An asymptote to a curve of the third order must also cut the curve. Hence also it follows that to a curve of the nth degree not more than n (n-1)-2 tangents can be drawn parallel to an asymptote, for in this aspect the asymptote counts for two tangents. 236.] Sometimes by this method, as well as by the former, we arrive at results affected with ±√, in which case the lines must be drawn in their own planes. And sometimes curves have branches out of the plane of reference, which are asymptotic to straight lines in the plane, as in the following example; and these must be determined in one or other of the methods which have been just explained. Thus if When x is infinitely great, the right-hand member of the equation vanishes; and we have y = x, which is the equation to a line passing through the origin, and inclined at 45° to the axis. of x, and which is asymptotic to two branches of the curve; but, for all values of x not within the limits a, the curve lies out of the plane of reference, and therefore the asymptote is that to which these branches are continually approaching. The form of the curve is given in fig. 52, the dotted lines representing the branches in a plane perpendicular to that of the paper. 237.] Two curves may also be asymptotic to each other; for suppose the equation to a given curve, when expanded in descending powers of x, to be then, if we neglect on the right-hand side of the equation all terms after the first three, which is equivalent to making the curve whose equation is and which is a parabola, is asymptotic to the given curve. = ∞, (96) Also if in equation (88), Art. 232, we omit all terms except the first three, and multiply through by x, we have the equation to a hyperbola, viz. xy = α1x2 +αo x + b1 ; and this curve is asymptotic to the given curve, because the difference between the lengths of their ordinates is a quantity which diminishes without limit as a increases without limit. And so again if we take account of the first four terms of the same equation, and neglect all subsequent ones, we shall obtain the equation to a curve which is nearer to the given curve than either the rectilinear asymptote or the hyperbola; and thus by a similar process we may obtain a series of curves more and more asymptotic to a given curve. Thus also we often find curves which have cubical and semicubical parabolas asymptotic to them as for instance, and therefore, neglecting all terms involving negative powers of x, we have y = ± x3, which is the equation to a semicubical parabola, the form of which is given in fig. 64, and is asymptotic to the curve. 238.] And I must enter further into the relation between the equation of a curve and those of its asymptotes, whether they are curvilineal or rectilineal, because a general theory will be hereby obtained; and because we shall also have the occasion of explaining, algebraically and geometrically, the points of intersection of a curve with a line, straight or curved, when the intersection takes place at an infinite distance. Let the equation to the curve be that given in (48), Art. 207; and let us assume a to be the abridging symbol for a linear function of the form ax+by+c; so that ... and thus a = 0 is the equation to a straight line; and a is proportional to the perpendicular distance from the point (x, y) on that line; it will also be convenient to take a as the type of a1, a2, am, which are other linear functions similar to (97). As heretofore observed, the abscissæ of the points of intersection of a straight line with the curve (48), Art. 207, are the roots of the equation obtained by eliminating y between the equations to the curve and the straight line: and as this equation is generally of the nth degree, it has n roots either real or imaginary. Suppose however that the degree of the equation which determines these abscissæ of the points of intersection is less than n by m units, and yet that no factor involving the variable has been omitted, what explanation does this fact admit of? It is of course due to a relation between the coefficients of the equations, by reason of which the coefficients of the first m terms vanish in the equation resulting from the elimination; and therefore by reason of which m constants become zero. But if a constant = O, see Art. 11, a straight line at an infinite distance is expressed; and therefore if the resulting equation is of n-m dimensions, m of the points of intersection are at an infinite distance. Now as a 0 contains two undetermined constants, we may determine them so that two of its points of intersection with the curve may be at infinity; that is, so that the two highest terms in the resulting equation may vanish; in which case the straight line becomes an asymptote to the curve. Although these results are arrived at in reference to the axis of x yet they are evidently true also with reference to any other line; and therefore we may find the equation to asymptotes in the following way: 239.] Suppose a to be a linear factor of the two highest terms un + un-1 in the equation to the curve: so that if U-1 is a function of x and y of n-1 dimensions, and therefore if a1 = 0 the two highest terms in the equation of the curve disappear, and the line a1 = 0 will intersect the curve in two points at an infinite distance, and thus will be an asymptote to the curve. This result is also evident by the following |