If the denominator of this expression had been constant, the equation would have belonged to an ellipse, hyperbola, or parabola, according as n was negative, positive or nothing; hence if such constant quantity be replaced by the variable quantity 2 x, the conic section becomes " hyperbolized" by having an infinite branch proceeding to the axis of y as an asymptote.

For the nine figures corresponding to the values of p, see Newton, Enum. Lin. Tert. Ord.

From the last article it appears that all curves of the third order have infinite branches; and this must necessarily be the case, for every equation of an odd degree has at least one real root, so that there is always one real value of y corresponding to any real value of x.

312. The conchoid of Nicomedes.

Let X x (fig. 1) be an indefinite straight line, A a given point, from which draw the straight line A C B perpendicular to Xx, and also any number of straight lines A E P, A E P", &c. ; take EP always equal to CB, then the locus of P is the conchoid.

If in EA we take E P' = EP the locus of P' is called the inferior conchoid; both conchoids form but one curve, that is, both are expressed by the same equation.

C B is called the modulus, and Xr the base or rule;

We have three cases according as b is > a, a, or ◄ a.

B

From (1) X is an asymptote; from (2) the curve passes through B; from (3) and (4) the curve extends from the asymptote upwards to B and no higher; hence the branch B PP". Again from (5) and (6) the curve passes through A and D if CD = b; from (7) there is a branch A x extending from A to the asymptote; and from (8) the curve exists between A and D; the double value of x gives the same results along Cx. Case 2. ba; in the table of values put b = a, and omit (8); thus the figure will be the same as the preceding, with the exception of the oval A P' D, which vanishes by the coincidence of A and D.

Case 3. ba; in the table of values put b for a in (7), and for (8) write "if y is > b, x is impossible;" the upper part of the curve is not altered, but the point D falls between A and C; from (8) no part of the curve is between D and A; but from (5) A is a point not on the curve, but belonging to it, and called a conjugate point. In this case the lower curve is similar to the upper one.

The generation of the conchoid gives a good idea of the nature of an asymptote, for the line E P must always be equal to C B, and this condition manifestly brings the curve continually nearer to C X, as at P", SO that the curve, though never actually coinciding with C X, approaches nearer to it than by any finite distance.

This curve was invented by Nicomedes, a Greek geometrician, who flourished about 200 years B.C. He called it the Conchoid, from a Greek word signifying "a shell;" it was employed by him in solving the problems of the duplication of the cube, and the trisection of an angle.

To show how the curve may be applied to the latter problem, let BCA (fig. 2) be the angle to be trisected; draw A EF meeting the circle in E, and the diameter produced in F, and so that the part E F equal the radius CA, then it is directly seen that the arc D E is one-third of B A.

Now it is not possible by the common geometry, that is, with the straight line and circle alone, to draw the line A EF, so that EF shall be equal to CA (the tentative process, though easy, being never considered geometrically correct), and for a long time the ancient geometricians would not hear of any other mathematical instruments than the ruler and compasses; hence the problem was quite insuperable: finding at last that this was the case, they began to invent some curves to assist in the solution of this and other problems: of these curves, the most celebrated is the conchoid of Nicomedes. It may be thus applied to the present problem. Let A be the pole of the inferior conchoid, B F the asymptote or base, and A C the modulus, the intersection of the curve with the circle evidently gives the required point E. The superior conchoid may also be used for the same purpose.

Ꮖ

Unless the curve could be described by continued motion, the solution would be incomplete. Nicomedes therefore invented the following simple machine for describing it. Let x X be a straight ruler with a groove cut in it; C D is another ruler fixed at right angles to x X; at A there is a fixed pin, which is inserted in the groove of a third ruler A EP; in AP is a fixed pin at E, which is inserted into the groove of a X; PE is any given length; then, by the constrained motion of the ruler PEA, a pencil at P will trace out a conchoid, and another pencil fixed in EA would trace out the inferior curve.

This curve was formerly used by architects; the contour of the shaft of a column being the portion B PP" of a conchoid.

The polar equation to the conchoid is thus found:

Let A (fig. 1, page 156) be the pole, A P = r, PA B = 0;

:.y + a = r cos. 0, and xr sin. 0.

Substituting these values in the equation, and reducing, we arrive at the polar equation r a sec. 0 + b.

The polar equation may, however, be much more easily obtained from the definition of the curve. We have

TAPAE + EP AC sec. C AE + CB a sec. 0 + b. 313. The following method of obtaining the equation to the conchoid will be found applicable to many similar problems.

Let any number of lines, A EP, fig. Î, be drawn cutting C X in different points E, &c.; from each of these points E as centre, and with radius b describe a circle cutting the line A EP in P and P'; the locus of the point P is the conchoid.

Let A be the origin of the rectangular co-ordinates.

A B the axis of y, and A X parallel to C X in the figure.

Let the general equation to the line A EP be ya x, where a is indeterminate;

α

Then y' = a, and a' == are the equations to the point E;

α

The equation to the circle which has the point E for its centre and radius b, is

And eliminating a between this equation, and that to the line A EP, we have the final equation to the curve,

In general if the line CX be a curve whose equation is y = f (x), the co-ordinates of the point E are found by eliminating x and y from the equations ya x, and y = f (x); hence we find x = ƒ' (a), and y= af' (a), and the equation to the circle is

{y - a f' (α)} + {x − ƒ' (α) } * = b2,

314. A perpendicular is drawn from the centre of an hyperbola upon a tangent, find the locus of their intersection.

The equation to the perpendicular on it from the centre is

In order to get the equation to their intersection, we must eliminate and y' from these two equations and that to the hyperbola; from (1) and (2) we find

a2 b3, we have

--

Substituting in the equation a2 y'2 b2x22=

(x2 + y2)2 + b2 y2 — a2 x2 = 0,

which is an equation of the fourth degree.

We shall only investigate the figure in the case when ba, that is, when the hyperbola is equilateral, in which case the equation is (x + y3)a = a2 (x2 — y3),

..y + (2x2 + a3) y3 + x* — a2x2 = 0,

If the sign of the interior root be negative, y is impossible; hence we shall only examine the equation

From (1) the curve passes through C; from (2) it passes through A and A'; from (3) it has two branches from C to A and from C to A'; from (4) it does not extend beyond A and A'.

We may judge yet more nearly of the form of these ovals, for the tangent at the vertex of the hyperbola being perpendicular to the axis, the oval will cut the axis at A at a right angle; and again at C in an angle of 45°, because the tangent nearly coinciding with the asymptote, the perpendicular on it makes an angle of 45° with the axis ultimately.

This curve was invented by James Bernouilli; it is called the Lemniscata, and forms one of a series of curves corresponding to different values of b.

To find the polar equation to the lemniscata,

Let y = r sin. 0, and x = r cos. ;

hence the equation (x2 + y2)2 = a2 (x2 — y2) becomes ra2 cos. 2 0. Any curve that is of the form of this figure is called a lemniscata.

315. In the following example the curve may be easily traced by points. Let a circle be described with centre C and any radius CQ; draw the ordinate QM, and in QC take Q P Q M; the locus of P is a lem- › niscata.

Again, if in MQ we take MR = a third proportional to M Q and C M, the locus of R is another lemniscata whose equation is

x2 - a2 x2 + a2 y2 = 0.