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bols as ( + )*, ( —)*, &c, which indicate that branches of the curve exist in planes inclined at 120°, &c. to the plane of the paper; but the full development would occupy more space than can be given to it in an elementary treatise.

Section 3.—On the generation of some plane curves of higher orders, and 011 their equations.

193.] The reader is supposed to be familiar with the principal properties of the straight line, the circle, and the three conic sections as exhibited in their algebraical equations; yet, as more copious illustration will be required in the succeeding Chapters than they afford, it is necessary to insert an account of the modes of description, and the equations of some curves of a higher order; most of which possess historical interest from the labour bestowed on them by ancient mathematicians.

In the first place let it be observed, that the equation to a curve may sometimes be expressed by means of a subsidiary angle: the elimination of which from the given equations will produce the better-known equation to the curve.

Thus the equation to the ellipse may be put in the forms,

x - a cos 01 m x* y*

y = bsmo\' "" aJ + 6» ~ lm W

The hyperbola may be expressed by

x = asec0) . *» _ y_2 _

y = bt&n6S' "a* b> ~ K'

Other examples of the same kind will be found in the sequel.

194.] The Cissoid of Diodes. Fig. 34.

Definition.—If at equal distances from o and A, the two extremities of a diameter of a circle, two ordinates M Q and N s are drawn, and if os is drawn cutting Mq in p, the locus of the point F is the Cissoid of Diocles.

Let oc = Cb = Ca = a = the radius of the circle; Om = x, Mp = y.

Then, by the geometry, Om : Mp ;; On : Ns; but On = OA —An = Oa Om = 2a—x,

Ns = Mq = {om x Ma}* = (2ax—x2)b; substituting which values in the above proportion, we have y _ {2az-x*}*m

x ~ 2a-x' 1 l)

y^-u-x- ■■■ *-±<i^r*-. (12>

The equation represents the curve described in fig. 34: the dotted part being that out of the plane of the paper, and when y is affected with + V ; and as the equation to the fundamental curve, viz. the circle, also expresses a rectangular hyperbola out of the plane of reference, that part arises from the hyperbola having been operated upon in a manner analogous to the circle in the above generation of the curve.

The equation will be subsequently completely analysed; but certain salient points of it are at once evident from the geometrical description. Thus the curve lies equally above and below the axis of x; it passes through o and B, and has for an asymptote the line drawn through A and perpendicular to Oca; it lies out of the plane of the paper to the left of o and to the right of A; in the fig. Oe = a, Od = 3a.

If (12) is expressed in terms of polar coordinates, the equa

tl0nis r = 2asin0tan0. (13)

195.] The Witch of Agnesi. Fig. 35.

Def.—In the ordinate Mq of a circle a poiut P is taken, so that Mp : Mq :; Oa : Oh; the locus of the point P is the Witch of Agnesi.

Let oc = Ca = a, Om = x, Mp = y.

Then, by the definition, Mp : Mq ;; oA : Dm; but

Mq = {2ax-x2}*; .-. y : {2ax-x2}i ;; 2a : x; (14)

.-. y* = 4a*—-, y = ±2«J—J . (15)

The dotted parts of the curve are out of the plane of reference, and arise from an analogous operation being performed on that rectangular hyperbola, out of the plane of the paper, which the equation to the fundamental curve also represents.

Although we are obliged to reserve the complete discussion of equation (15) until the next Chapter, yet it appears that the curve cuts the axis of x at A, and that the axis of y is an asymptote; and that the ordinate is affected with 4- whenever x is negative, and whenever x is greater than 2a; in the fig. Ob = Ob'= 2a.

196.] The Conchoid of Nicomedes. Fig. 36.

Dep.—A point A and a straight line Eoe' being given, from A a straight line Aqp is drawn cutting Oe' in q, and p is such that qp is always equal to a given straight line Ob; the locus of the point P, in the different positions of Aqp, is the Conchoid of Nicomedes.

Prom A draw Ao at right angles to Eoe', and let Oa = a; let the straight line qp = Ob = 6; Om = x, M p = y.

AO MP Ao + Mp a + y Then, bv the geometry, — = •— = =;



.-. oq = .

a + y

Also pq2 = M p2 + MQ2,

b2 = y* + (x;

y T \ a + y''

.-. *V = (6*-y2) (a+y)2, (16) x - + °L±1 (17)


From which equation it appears that x = 00 , when y = 0, and therefore that Eoe' is an asymptote.

The line Eoe' is called the rule of the conchoid, and Pq or o B the modulus. If the line b is measured from q towards A instead of along Aq produced, then another curve is generated which is called the Inferior Conchoid, and is represented in the figure.

(1) If b is less than a, the upper and lower conchoids, as shewn in the figure, are somewhat similar in form.

(2) If b = a, the lower conchoid passes through A, and is somewhat like the lower conchoid drawn in the figure, but without the loop.

(3) If b is greater than a, the lower conchoid has an oval or loop, a double point of which is at A, and is that drawn in the figure.

The equation may be easily expressed in polar coordinates. Let A be the pole, and Pao = 0, Ap = r;

.;. r = a sec 6 + b, (18)


the upper and lower signs referring respectively to the upper and lower conchoids.

197.] The Lemniscata of James Bernoulli. Pig. 37.

Dep.—If from the centre of an equilateral hyperbola perpendiculars are drawn to the tangents, the locus of the points of intersection is the Lemniscata.

Let x and y be the current coordinates of the lines Pq and Op; and let x', y' be the coordinates to Q, the point on the hyperbola at which the tangent is drawn; the equations to the hyperbola and the tangent are respectively

x'2-y'2 = a2, (19)

xaf-yy' = a2; (20)


whence the equation to Op is y = — x;

... * = m

and multiplying each term of (20) by one or other of these equalities, we have

x2 + y2

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x2 + y2' y x2 + y2'

and therefore, by means of (19),

(x2 + y2)2 = a2 (x2 - y2). (22)

The curve, as is manifest from the generation of it, consists of two ovals, meeting in a double point at o; the tangents to which are coincident with the asymptotes of the hyperbola, and form angles of 45° on each side of Oa.

The polar equation is

r2 = a2 cos 20. (23)

198.] The Logarithmic Curve. Pig. 32. No better definition of the curve can be given than that expressed by its equation,

y = «*; (24) which means that the abscissa is the logarithm of the ordinate to the base a.

Hence, when x = 0, y = = l; when x \y y = a; when x = oo, y oo; when x = — e©, y = 0.

Therefore Oa = 1; and as the ordinate recedes further from oy, it increases and ultimately becomes infinite; and as x decreases, that is, increases negatively, y decreases, and the axis of x is an asymptote to the curve.

199.] The Catenary. Fig. 33.

The catenary is the curve in which a perfectly flexible and uniform, though heavy, string hangs, when suspended in vacuo from two points.

Let Om = x, Mp = y, oc = c; its equation is

where e is the Napierian logarithmic base; but as a knowledge of mechanics is requisite for a determination of the equation, the discussion of it will be found in Chapter V of the third volume of our work. It is manifest however that when x = 0, y = c; and as the equation is not altered when — x is written for + x, that the curve is symmetrical with respect to the axis of y.

200.] The Tractory, or Equitangential Curve. Fig. 38.

Dep.—If Ap is a curve, such that Pt, the length of the tangent intercepted between the point of contact and the axis of x, is always equal to Oa; then the locus of p is the equitangential curve.

Let Om = x, Mp = y, Oa = Pt = a; then the definition of the curve above given leads, as will be seen in the next Chapter, to an equation of the form,

and the equation to the curve is that of which (26) is the derivedfunction, and is therefore, as will be seen in Vol. II, Chap. VI,

This curve is sometimes considered as generated by attaching one end of a string of constant length a to a weight at A, and by moving the other end along ox; the weight is supposed to trace out the curve: hence arises the name Tractory or Tractrix. But the mode of generation is incorrect, unless we also consider the friction produced by traction to be infinitely great, so that the weight's momentum, which is caused by its motion, may be instantly destroyed.

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