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VE 23 A and 2B 2 d are complements about the diagonal O 2', and are, therefore, equal (Euclid i., 43), while EOB 22 is common-that is, the figure VOBA figure EO 2 d, and similarly for each of the remaining points.

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The curve can evidently be produced backwards by a similar construction, so as to approach the line O V. But it will be evident that however much the curve be produced in either direction it can never touch the lines O V and O P, hence the lines may be said to be continually approaching the hyperbola, yet never touching it. Such lines are called "asymptotes," and when the asymptotes, as in this example, are at right angles to each other the curve is distinguished as a rectangular hyperbola.

EXAMPLES.

EX. 5.-Draw two lines, OV, OP, at right angles, intersecting at O, as in Fig. 536, and mark a point A 1" from OV and 21" from O P. Find at least seven points in the rectangular hyperbola drawn from A.

EX. 6.-Draw two lines, OV, OP, as in Ex. 5, and mark a point A " away from O P and 3" from O V. Draw the curve of a rectangular hyperbola from A towards OV to within " of O V.

Cycloidal or Rolling Curves.-There are three principal curves of this class, each being generated by a fixed point on the circumference of a circle, rolling in contact with a fixed line or circle in the same plane. These curves are all used in the construction of wheel teeth.

Cycloids. A cycloid is the curve traced out by a fixed point on the circumference of a circle, rolling along a fixed straight line.

Epicycloids.—An epicycloid is the curve traced out by a fixed point on the circumference of a circle rolling round another circle, and outside it.

Hypocycloids.-A hypocycloid is the curve traced out by a fixed point on the circumference of a circle, rolling round another circle, and inside it.

Trochoids.-A trochoid is the curve traced out by a point rigidly fixed to a circle, within or without its circumference, as the circle rolls along a fixed straight line. When the fixed point is without the circumference, the curve is termed superior, and when inside the circumference, inferior. When the circle rolls round another circle, either outside or inside it, the curves are known as Epitrochoids and Hypotrochoids respectively.

The rolling circle is called the generating circle.

The fixed point is called the generating point.

The fixed line or circle is called the directing line or circle. Evolutes and Involutes.-An evolute is the curve formed by the intersection of normals to a curve. An involute is the curve formed by drawing tangents to a curve, the length of each tangent being equal to the arc of the original curve from its point of contact to its intersection with the curve. It is the curve traced out by the end of a flexible thread unwrapped from the original curve. Tangents to an evolute are normals to an involute. Thus, in Fig. 54, a number of normals are drawn to the cycloid P SQ and the curves P R, R Q are drawn tangent to the normals; these curves are evolutes. Also, the cycloid PS Q is an involute, for, as will be seen later, it passes through the ends of tangents to the curves PR, R Q, each of which fulfils the condition that, if P be the intersection of the involute with the original curve, P' a point in the involute, and E' the contact point of the tangent with the original curve, then P' E' = length of arc P E'.

The vertex of the cycloid is at the point S; the points P and Q are called cusps.

PROBLEM XXXVIII. (Fig. 54).-To construct a cycloid when the size of the generating circle is given.

Let A B be the directing line, and P the generating point. In one revolution of the circle P will reach a point, Q, on A B, so that the distance P Q equals the circumference of the circle where d diameter).

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CD is the locus or path of the centre of the circle for this revolution.

Divide C D into any number of equal parts, and mark as shown C, C1, C2, . . . C7, &c., draw lines from these points to A B.

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While C is moving to D, the point P moves round the circumference of the circle, therefore divide the circumference into the same number of equal parts, and mark the points 1, 2, 3, . . . 7.

To find the position of the generating point, at any position of the generating circle, proceed as follows:-If the generating circle move to the position C1, the point P will have moved in the same time through the arc P 1. Draw the generating circle with centre C1, and mark its contact point with the directing line E; from E mark off the distance P 1 along the circle to the point P1, then P1 is a point in the cycloid. Proceed in the same way for the other points until the curve is completed. Notice that only parts of the circles need be drawn.

Another way of finding the points is to draw lines parallel to the directing line through the division points 1, 2, 3, . . . of

the generating circle in its first position, to meet the corresponding generating circle drawn from the centres C', C2, C3, ... This is shown in the figure.

Tangents and Normals. In all rolling curves the normal at any point passes through the corresponding point of contact of the generating circle with the directing line or circle. The tangent is at right angles to the normal.

To draw a tangent and normal at any point, O, in the curve (Fig. 54). With the point O as centre and the radius of the generating circle as distance, describe an arc cutting the line OD in N, and draw N M perpendicular to the directing line, meeting it in M. Then N is the centre of the generating circle

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corresponding to the position O of the generating point, and M is its point of contact with the directing line; therefore the line through MO is a normal, and the line TR at right angles a tangent.

Evolutes of Cycloid.-Draw normals through each of the points found in constructing the curve, and produce them below the directing line. Then draw the curves QR, R P tangent to the normals as shown. These curves are the evolutes.

The evolutes of a cycloid together make an equal cycloid. Thus the curves R Q and SQ are identical. This can be proved by cutting out the curve R Q in paper or card and applying to the curve SQ.

If a piece of thread be fixed at R, and wound round the curve of one of the evolutes as R P, so that the other end of the thread

reaches to P, and then be slowly unwound from the curve, the end P, if the thread be kept tight, will trace out the cycloid PS Q. Hence the reason for this curve being called an "involute." This arrangement forms what is known as an isochronous or equal timed pendulum, the pendulum bob being at one end of the thread P, the other end being fixed at R, curved guides being fixed in place of the evolutes. The time taken by the pendulum to swing through different arcs is then always the same, whatever be the length of the arc.

The cycloid has an important property in mechanics in that the evolutes R P or R Q are the curves of quickest descent from R to P or Q.

Trochoids. The method of constructing the inferior and superior trochoids differs but little from the above, and should present no difficulty. Having found the new position of point P on the generating circles, having centres C1, C2, C3, draw the radius through P in each case.

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Then for the superior trochoid these radii must be produced the given distance, and their ends then represent points in the

curve.

For the inferior trochoid points in the curve are obtained by marking along each radius from the centre the given distance.

These two curves are shown on the right hand of Fig. 54, the full-looped curve, S T, being the "superior trochoid," and the dotted curve, IT, the "inferior trochoid."

Tangents and normals to trochoids are drawn in a similar way as to cycloids, and the necessary construction will present no difficulty.

EXAMPLES.

EX. 7.-Describe a cycloid and its evolutes when the diameter of the generating circle is 5", and draw a normal and tangent at any point in the cycloid, not being one of the points found in constructing it. Then work the following: (a) Show by cutting out a paper pattern that the curve of the evolute is a similar and equal cycloid; (b) show that the length of the normals from the directing line to the cycloid is equal to the length from the directing line to the evolute (note how this suggests an accurate way of finding points where the evolutes touch the normals); (c) measure the length of the cycloid, and show that it is eight times the radius of the generating circle; (d) find area between cycloid and directing line, and show that it is three times the area of generating circle; (e) find area between evolutes and directing line, and show that it is equal to the area of the generating circle.

EX. 8.-Draw the superior and inferior trochoids, when the diameter of the generating circle is 4", the point for the superior curve being beyond the circumference and for the inferior curve 3" within. Draw a normal and tangent to each curve at points not found in the construction.

Epicycloids. - PROBLEM XXXIX. (Fig. 55). — To draw an epicycloid and its evolutes, given the directing and generating circles.

Let the directing and generating circles have centres A and B respectively, P being the generating point. The construction is identical in principle with that of the cycloid, allowing only for the change from a directing line to a directing circle. But

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it is necessary to first find the position Q of the point P after one revolution of the generating circle. This is done by knowing that

arc P Q
circ. of directing circle

angle PA Q

360°

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