Page images

the equation of the parabola is y2 = 4 ax, the equation of its normal is y


and if we eliminate x from these two equations we have

y3-4a(f-2a)y-8a2>7 = 0; (82) which is a cubic; so that if the roots of this are all real, three normals may be drawn to a parabola; if two of the roots are equal, only two normals; and if two roots are imaginary, only one normal can be drawn: and as one root of a cubic is always real, one normal can be drawn, whatever is the position of (f, Tj); if two roots of (82) are equal, we have

3y3-4a(f-2a) = 0; (83) and if we eliminate y from this equation and (82), we have

27ar,2 = 4(f-2a)3; (84) from all points on which curve two normals can be drawn. Thus (84) divides the plane of reference into two districts, from all points in one of which three normals, and from all points in the other of which only one normal can be drawn to the parabola. (84) is called the Evolute of the Parabola.

In a memoir by Professor Steiner of Berlin, and contained in Crelle's Journal, Vol. XLIX. 1855, three geometrical proofs are given of the theorem of this Article. One of them I will insert here. Let P be the point from which the normals are to be drawn. Let the curve be of the nth degree, and let it be moved round in its plane about the point p; then the curve in its new position will intersect that in the old position, in n3 points, which are either real or imaginary: and when the displacement is infinitesimal, the radii vectores drawn from p to the same points on the curves, when the second is displaced, will at the points of intersection be equal; and thus the line joining these two points will be perpendicular to the radii vectores, and the radius vector will be a normal; and as the number of points at which these circumstances may occur will be n2, so from r may n2 normals be drawn to a curve of the nth degree.

230.] The normal is the longest or the shortest line that can be drawn to a plane curve from a given point in its plane.

Let jjo and f0 be the coordinates to the given point, and eland y the current coordinates to the curve; then, if r is the distance between (r)0, £0) and (x, y),

[ocr errors]

Section 2.—Asymptotes to j)lane curves referred to rectangular


231.] A line is said to be an asymptote to a curve, when the curve approaches continually nearer and nearer to it, but is not coincident with it within a finite distance.

From this definition it is plain that there are two classes of asymptotes, rectilinear and curvilinear, which it is convenient to discuss separately.

If the curve has asymptotes which are either the coordinate axes themselves or straight lines parallel to them, they may be determined in the following manner. If y = oo, when x = 0, the axis of y is an asymptote; and if y = 0, when x = oo , the axis of x is an asymptote: such are the axes of coordinates to the curve, xy = k2.

Again, if y = ao, when x = a, a line parallel to the axis of y, at a distance a from it, is an asymptote; and if x oo, when y = b, a line parallel to the axis of x, at a distance b from it, is an asymptote.

Thus suppose the equation to a curve to be xy ay bx = 0;


then, as it may be put under either of the forms,

bx ay • — or x =

[ocr errors]

that is, two lines parallel to the coordinate axes are asymptotes; the curve is represented in fig. 51; wherein Oa = S, Ob = b.

So of the logarithmic curve, see fig. 32, the axis of x is an asymptote; its equation is

y = a";
.-. y = 0, when x = — to;

therefore Ob is an asymptote to the branch Ac.

So in the cissoid, equation (12), Art. 194, and fig. 34, y = oc , when x = 2a; therefore the ordinate through A is an asymptote to the curve.

And in the tractory, equation (27), Art. 200, and fig. 38, y = 0, when x = 00, and therefore the axis of x is an asymptote.

If however a curve has rectilinear asymptotes not parallel to the axes of coordinates, they are to be determined by one or other of the following methods.

232.] Method of determining rectilinear asymptotes by expansion in descending powers of x.

If by any artifice, as by the Binomial Theorem, or by Maclaurin's Theorem, the equation to a curve can be expanded in a series of the form

y = aix + a0 + + ^|+; (88)

x x

then, as all and every term after the first two, that is, every term which involves a negative power of x, diminishes without limit, and ultimately becomes infinitesimal, when x becomes infinity, the difference between the ordinate to the curve represented in equation (88), and that to the straight line whose

equation is , ,on.

M y = axx + ao, (89)

is infinitesimal, and the straight line represented by equation (89) is an asymptote to the curve.

And according as the first term after a<>, be it — or

x x*

is positive or negative, so will the ordinate to the curve be greater or less than the ordinate to the asymptote, and the curve will be above or below the asymptote.

The equation (89) is to be constructed in the ordinary way.

And if, finally, the equation to the asymptote is affected with + -%/—, it indicates that the asymptote lies out of the plane of reference, and is asymptotic therefore to a branch of a curve similarly placed, and to be drawn according to the methods of Section 2 of the last Chapter.

233.] Ex. 1. To find the equations to the asymptotes of the Cissoid of Diocles.


[ocr errors]


••• y = ± (*+a),

are the equations to the asymptotes, and represent two straight lines out of the plane of reference inclined to the axis of x at + 45°, and cutting the axis of x at a distance — a from the origin, which are delineated by the dotted straight lines of fig. 34.

Ex. 2. To determine the asymptotes of the Witch of Agnesi. From the equation (15) of Art. 195, we have la x

[ocr errors]
[ocr errors]

.-. y = ± -J —2a, when x = oc;

which equations are those to the asymptotes, and express two straight lines out of the plane of reference, and parallel to the axis of x, at distances ±2a from it; see fig.35.

Ex. 3. To determine the asymptotes of

y - * x^Tl' r-±.(i-if(i + ir*. = ±*(1-Ki-C» + -)(i-Hi + B» + -). = ±*(i-p + ...)'

therefore neglecting terms involving negative powers of x, the equations to the asymptotes are y = + x; and as the next term of the series is negative, it follows that the curve is, in the first quadrant, below the asymptote.

234.] The preceding method does not indicate any general property of equations of curves which have asymptotic straight lines; and frequently the equation does not admit of development in the form (88) with the requisite facility. These defects are supplied by the following process, which is easy of application in all cases.

If a curve approaches nearer and nearer to a definite straight line and meets that line at an infinite distance, it is evident that this line is a rectilinear asymptote and that it is the tangent to the curve at infinity; thus the equation to this particular tangent is the equation to the asymptote; and the equation to it may be deduced from the general equation to the tangent, if the necessary alterations are made. This is the theory of these asymptotes in the general case. If the asymptote is parallel to one of the coordinate axes, say to the axis of x, that we may fix our thoughts; then, if it is at a distance b from it, y = b, when x = oo : and if it is parallel to the axis of y at a distance a from it, then x = a, when y = oo. If however it is not parallel to either of these axes, the solution of the problem requires the determination of a tangent when x = y = oo .

Let us take the equation to a curve in the general form given in (48), Art. 207; then the equation to the tangent is, see (52), Art. 222,

^ (^) + 17 (jjk) +m»-i + 2m'-2+ ...+(»-l)«i + n«o = 0. (90)

When x = y = oo , the terms of the highest dimensions alone may be retained; in which case,

(dr\ _ /<fa„\ /dj\ _ /dM„\.
W ~ \dx>' \dy> ~ \dy>'

and (90) becomes

this is evidently an equation of n — 1 dimensions in all its terms; and it is the equation to the asymptote when the several terms in it are evaluated for x = y = oo .

As the asymptote is a tangent to the curve, so the tangent of


the angle contained between it and the axis of # is ^-. In this case then

« PreviousContinue »