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the generating circle in its first position, to meet the corresponding generating circle drawn from the centres C1, 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 CD 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, I T, 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.

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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

angle PA Q

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and as the arc PQ equals the circumference of the generating circle, this becomes

angle PAQ =

360 rad. of generating circle

rad. of directing circle

Set off this angle. Draw the arc B C which is the locus of the centre of the generating circle, and, as before, divide it and the circle into the same number of equal parts, and then proceed as with the cycloid. The construction is clearly shown in the figure.

Normals and Tangents are drawn exactly as to a cycloid. Thus in the figure, N is the centre of the generating circle corresponding to the position O in the curve of the generating point, and M is the contact point of the rolling and directing circles. Then MO is a normal, and TR at right angles a tangent.

The evolutes are the curves PR, RQ, drawn tangent to the normals of the curve as before. They are similar curves to the original curve, PS Q, and are, therefore, epicycloids, but are not equal to the original curve. The hypocycloid and its evolutes are drawn in precisely the same way as the epicycloid, and do not, therefore, need separate explanation. In Fig. 55 the curve P'S'Q' is the hypocycloid, the evolutes not being shown for want of space. They are, however, drawn touching the normals to the curve as before. EF is a tangent, and H K a normal. The generating circle rolls in the direction from P' towards Q'.

Notice that both the epi- and hypocycloids are traced by the end of a thread unwound from the evolutes, as with a cycloid. No difficulty should be experienced in drawing the epi- and hypotrochoids, as the construction is exactly similar.

EXAMPLES.

EX. 9.-Draw an epicycloid and its evolutes when the diameters of the directing and generating circles are 10" and 4" respectively, and draw a tangent and normal at any point. in the curve not found in the construction. Show that the evolute is an epicycloid traced by a point on a circle of diameter equal to R G (Fig. 55) rolling on a circle of radius, A R.

EX. 10.-Draw a hypocycloid and its evolutes when the diameters of the directing and generating circles are 10" and 3′′ respectively, and draw a tangent and normal at any point in the curve not found in the construction.

EX. 11.-Show that when the diameter of the directing circle is twice the diameter of the generating circle, the hypocycloid is a straight line.

EX. 12.-Draw a hypotrochoid when the diameter of the directing circle is twice the diameter of the generating circle, and show that half the curve is a quadrant of an ellipse.

Involute of a Circle.-The involute of a circle is the curve traced out by the end of a piece of thread unwound from the circle, the thread being kept tight. The circle is then the evolute to this curve.

PROBLEM XL. (Fig. 56).-To draw the involute of a

circle.

Let the circle have the centre C, and let P be the starting point of the curve or end of the supposed thread.

Let the thread be partly unwound, so that it assumes the position P3 3.

It is evident P3 3 must be a tangent to the circle, and be, therefore, at right angles to the radius C 3. Also P33 must equal the length of the arc P 3. Then P3 is a point in the involute.

If the arc P3 be divided into a number of small parts, and the

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same number of parts be marked off from 3 to P3, then the length P3 3 may be assumed equal to the chord P 3 and P be a point in the curve. But it is better to divide the circumference of the circle into, say, twelve equal parts, in which case the length of the tangent P3 3 would be one-quarter of the circumference (which

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