free hand, or with the help of a bent spring, draw a curve, A E, so as to touch all those circular arcs; this will be very nearly the rolled curve required.

The curve A E is called the "Envelope" of the series of arce that it touches.

II. To find a series of points in a rolled curve.-Draw a series of tangent circular arcs as in the preceding rule; then draw the several normals, 11, 22, 33, 44, &c., as radii of those arcs; the direction of each normal being determined by the principle, that at the point where it meets the fixed curve A B, it makes an angle with a tangent to that curve equal to the angle which the correnormal of the epicycloid, T A p; and let the required radius of curvature,

AFP.

=

Let the angular velocity of the rolling cylinder, relatively to the rotating plane O C, be denoted by b, and that of the plane O C by a, so that the resultant angular velocity of the rolling cylinder is a + b. Then, because the angle CTA is the complement of one-half of the angle T C A, it is evident that the angular velocity of T A is a + But according to Article 76, a R = br; therefore

a+b= b 1 +
3 (1

b

In any indefinitely short time, dt, the normal sweeps through an angle whose value in circular measure is

d s = (a + b) p d t = b (1 + i ) pdt:

therefore the radius of curvature of the epicycloid at the point A is

R =

This formula is made to comprehend the case of a cycloid by making = ∞, when it becomes = Ρ 2 p; and that of the involute of a circle by making, when we have pp. When the epicycloid is internal, and R and r denote arithmetical values of those radii, the sign -- is to be substituted for both in the numerator and in the denominator of the formula. The symbolical expression for equation 2 of the text is

with the same understanding as to the sign in the denominator. In the case already referred to at the end of Article 77, when a cylinder rolls inside a cylinder of twice its diameter, we have R - 2 r, and the denominator of the expression for p becomes 0; showing that the radius of curvature is infinite; or, in other words, that the epicycloid traced is a straight line, as stated in the text. When the rolling cylinder is concave, r is negative.

ρ

=

sponding chord on the rolling curve A D, makes with a tangent to that curve at the corresponding point. Thus are found a series of points, 1, 2, 3, 4, &c., on the rolled curve A E, at the ends of the normals from the corresponding points on the fixed curve A B. The two preceding Rules are applicable to fixed and rolling curves of all figures whatsoever. When both curves are circles, the finding of a series of points is facilitated by drawing the circle CC', which contains the successive positions of the centre of the rolling circle; then marking those successive positions, l', 2′, 3′, 4', &c., on the circle CC, by drawing radii through the corresponding points 1, 2, 3, 4, &c., on the circle A B; then drawing the rolling circle in its several successive positions (marked with dots in the figure), and laying off the chords 11, 22, 33, 44, &c., of their proper lengths upon those positions of the rolling circle, which chords will be a series of normals to the rolled curve A E.

III. To approximate to the figure of an epicycloidal arc by means of one circular arc. By the method of the preceding Rule draw the normal to the epicycloidal arc in question at a point near its middle. For example, if A 3 is the arc of the epicycloid A E, whose figure is to be approximated to by means of one circular arc, draw the normal 22 by Rule II. Then conceive that normal to be represented by AT in fig. 46, page 57; and by the method of Article 78 find the corresponding radius of curvature AF and centre of curvature F. A circular arc described about F, with the radius FA (fig. 46), will be an approximation to the epicycloidal arc.

This is the approximation used in Mr. Willis's method of designing teeth for wheels, to be described farther on. It ensures that the circular arc shall have, at or about the middle of its length, the same position, direction, and curvature with the epicycloidal are for which it is substituted. Towards the ends of the arcs they gradually deviate from each other.

IV. To approximate to the figure of an epicycloidal arc by means of two circular arcs. This method of approximation is closer than the preceding, but more laborious. It substitutes for an epicycloidal arc a curve made up of two circular arcs; and the approximate curve coincides exactly with the true curve at the two ends and at one intermediate point, and has also the same tangents at its two ends

Suppose that A and B (fig. 48) are the two ends of the epicycloidal arc to which an approximation is required, and that A. Cand BC are normals to the arc at those points: the positions of the ends of the arc and directions of its normals having been determined by Rule II. of this Article. Let C be the point of intersection of the normals. Draw the tangents A D perpendicular to A C, and BD perpendicular to BC, meeting each other in D. Draw the straight line D C, and bisect it in E. About E, with the radius ED EC, describe a circle, which will pass through the four

=

points, A, D, B, C.

Draw the diameter FE G, bisecting the arc A B in F and the arc B C A in G.

Draw the straight line G D, in which take G H = GA = GB.

Through H, parallel to FEG, draw the straight line HKL, cutting AC in K and BC in L. Then about K, with the radius K A K H, draw the circular arc AH; and about L, with the radius L H = L B, draw the circular arc H B: the curve made up of those two circular arcs will be a close approximation to the epicycloidal arc, having the same position and tangents at its two ends, and being very near to the true arc at all intermediate points.

=

It may be remarked that GH GA= GB=/(HK HL) approximates very closely to the mean radius of curvature of the epicycloidal arc A B; also that the process described is applicable to the approximate drawing of many curves besides epicycloids; and that the ratio of the two radii, HL: HK, deviates less from equality than that of the radii of any other pair of circular arcs which can be drawn so as to touch A D in A and B D in B, and also to touch each other at an intermediate point.*

Fig. 48.

This may be expressed symbolically by stating that

minimum; or that (log. His a minimum.

80. Resolution of Rotation in General.-The following propositions show how the rotation of a rigid body about a given axis, fixed or instantaneous, may be resolved into two component rotations about any two axes in the same plane with the actual axis.

I. PARALLEL AXES.-The rotation of a rigid body about a given axis is equivalent to the resultant of two component rotations about two axes parallel to the given axis and in the same plane, the angular velocity of each of the three rotations being proportional to the distance between the axes of the other two rotations.

In fig. 49, let the plane of the paper be perpendicular to the three parallel axes, and let C be the trace of the axis of the resultant rotation, and A and B the traces of the axes of the component rotations; all three axes being in the same plane, whose trace is A C B. Let the angular velocities

G

E

1

C,

BC, СА, A B.

As the figure is drawn, all three angular velocities are of the same sign, because A B ВС + СА. If C lay beyond A and B, instead of between them, A B would be the difference of BC and CA, instead of their sum; and the lesser of these two distances and of the corresponding angular velocities would have to be considered as negative.

Fig. 49.

Let H be the projection of a particle in the rigid body, which particle is moving in a direction perpendicular to H C, with a velocity proportional to CHA B. Then, first, from H let fall HD perpendicular to A B; then, by the principles of Article 55, page 33, the component velocity of H in the direction H D, whether due to rotation about A, B, or C, is the same with that of a particle at D. Now the velocities of a particle at D due to the rotations about C

A,

are proportional respectively to

B,

+ AD BC; – BD ·CA; + CD · A B;

and CD A BAD BCBD CA; therefore this component of the velocity of the particle H due to the rotation about C is the resultant of the corresponding components due to the rotations about A and B respectively.

Secondly. Through H draw EGHF parallel to AC B, and on it let fall the perpendiculars A E, BF, CG. Then, by the

1

principles of Article 55, page 33, the component velocities of H along E F due to the rotations about the axes A, B, and C are respectively equal to the velocities of E due to rotation about A, of F due to rotation about B, and of G due to rotation about C'; and because AE = BF = CG, these velocities are respectively proportional to

BC, CA, A B;

But A B = BCCA; therefore the component along EF of the velocity of the particle H due to the rotation about C is the resultant of the corresponding component velocities due to the rotations about A and B respectively. Therefore the whole velocity of the particle H due to rotation about C, with an angular velocity proportional to A B, is the resultant of the velocities of the same particle due respectively to rotations about A, with an angular velocity proportional to B C, and about B, with an angular velocity proportional to CA. And this being true for every particle of the rotating body, is true for the whole body: Q. E. D.

II. INTERSECTING AXES.-The rotation of a rigid body about a given axis is equivalent to the resultant of two component rotations about two axes in the same plane with the first axis, and cutting it in one point; the angular velocities of the component and resultant rotations being proportional respectively to the sides and diagonal of a parallelogram, which are parallel respectively to the three axes of

rotation.

In fig. 50 the upper right-hand part of the figure represents a plane perpendicular to the resultant axis of rotation, O". F" is the projection of any particle on that plane; and the direction of motion of any particle whose projection is F" is perpendicular to

O" F".

O" Y" and O" Z" are the traces of two planes perpendicular to the first plane of projection and to each other; and D' and E" are the projections of F" on those planes respectively. According to the principle of Article 55, page 33, the component velocity parallel to O" Y" of the particle whose projection is F is the same with the velocity of a particle at D"; and its component velocity parallel to O"Z" is the same with that of a particle at E".

The upper left-hand part of the figure represents the plane whose trace on the first plane of projection is O" Z'; O' X', on this second plane, is the axis of rotation; O'Z' is the trace of the first plane of projection; and D' is the projection of F", and is the same point that is marked D" on the first plane. The lower part of the figure represents the plane whose trace on the first plane of projection is O" Y", and on the second plane, O X'. On this third plane O X is the axis of rotation, and also the trace of the second plane; OY is the trace of the first plane; E is the projection of