cases proceed as follows:-Divide the circumference exactly, by plane geometry, into such a number of equal arcs as may be required in order to give sufficient precision to the approximative part of the process. Let the number of equal arcs in that preliminary division be called n. Divide one of them, by means of Rule V., into the required number of equal parts; n times one of those parts will be one of the required equal arcs into which the whole circumference is to be divided.

Rules I., III., and V., are applicable to arcs of other curves besides the circle, provided the changes of curvature in such arcs are small and gradual.

52. Relative Translation of a Pair of Points in a Rotating Piece.— In fig. 19, page 26 (where O, as already explained, is at once the projection and the trace of a fixed axis of rotation on a plane perpendicular to it, and A the projection of a point in the rotating piece), let B be the projection of another point in the rotating piece, and A B the projection of the straight line connecting those two points. The point B describes a circle of the radius O B about the fixed axis; and the radii OA and O B sweep round with the angular velocity common to all parts of the rotating piece, so that by the time that A has moved to the position A', B has moved to the position B', such that the angles A O A' and BO B' are equal. In order to determine the motion of one of those moving points (as A) relatively to the other (as B), it is to be considered that, owing to the rigidity of the body, the length of A B is invariable, and that the change of direction of that line (as projected on the plane of rotation), consists in turning in a given time through an angle equal to that through which the whole piece turns. In fig. 20, take B to represent at once the trace and the projection, on a plane of rotation, of an axis parallel to the fixed axis, and traversing the point B. Draw BA in fig. 20 parallel and equal to B A in fig. 19; and B A' in fig. 20 parallel and equal to B' A' in fig. 19. Then A and A' in fig. 20 represent two successive positions of A with the summit pointing away from the centre of the arc; a straight line from the centre of the arc to that summit will bisect the arc. (2.) To mark the sixth part of the circumference of a circle. Lay off C a chord equal to the radius. (3.) To mark the tenth part of the circumference of a circle. In fig. 24 A, draw the straight line A B the radius of the circle; and perpendicular to A B, draw BCA B. Join A C, and from it cut off CD = C B. AD will be the B chord of one-tenth part of the circumference of the circle. (4.) For the fifteenth part, take the difference between one-sixth and one-tenth. It may be added, that Gauss discovered a method of dividing the circumference of a circle by geometry exactly, when the number of equal parts is any prime number that is equal to 1+ a power of 2; such as 1 + 2 = 17; 1 + 2 3 = 257, &c; but the method is too laborious for use in designing mechanism.

D

Fig. 24 A.

relatively to the axis traversing B, at the beginning and end respectively of the interval of time in which the rotating piece turns through the angle A O A′ (fig. 19) = A B A' (fig. 20). The translation of A relatively to this new axis consists in revolution in a circle of the radius B A, in the same direction with the rotation (that is, in the present example, right-handed); and the velocity of that relative translation is B A x the angular velocity of rotation. Fig. 21 shows how, by a similar construction, the motion of B relatively to an axis traversing A is represented. Take A in fig. 21 to represent at once the trace and the projection, on a plane of rotation, of an axis parallel to the fixed axis, and traversing A. Draw A B and A B' in fig. 21 parallel and equal respectively to A B and A' B′ in fig. 19. Then B and B' in fig. 21 represent two successive positions of B relatively to the axis traversing A, at the beginning and end respectively of the interval of time in which the rotating piece turns through the angle AOA' (fig. 19) = BA B' (fig. 21); the translation of B relatively to this new axis consists in reyolution in a circle of the radius A B, in the same direction with the rotation (that is, in the present example, right-handed); and the velocity of that relative translation is A B x the angular velocity, and is at each instant equal, parallel, and contrary to the velocity of translation of A relatively to B, agreeably to the general principle stated at the end of Article 42, page 21.

53. Comparative Motion of Points in a Rotating Piece.-In fig. 19, page 26, as before, let A and B be the projections at a given instant, on a plane of rotation, of two points whose motions are to be compared. The directions of motion of those points at that instant are represented by the straight lines A a, Bb, tangents to the circles in which the points revolve about the axis O; and the directional relation of the points is expressed by the fact, that the angle between those directions of motion is equal to the angle A O B, between the perpendiculars let fall from the two points on the axis O; or, in other words, the angle between the planes traversing that axis and the two points respectively; of which planes O A and O B are the traces upon the plane of rotation; for the directions of motion, A a, B b, are respectively perpendicular to those two planes.

The velocity-ratio of the two points is equal to the ratio O B : 0 A borne to each other by the radii of their circular paths. In other words, if A a = A A' be taken, as before, to represent the velocity of A, and B b B B' to represent the velocity of B, then OA:OB:: Aa: Bb;

=

and if the velocities of any number of points in a rotating piece are compared together, they are all proportional respectively to

the perpendicular distances of those points from the axis of rotation.

It is obvious that all points in a circular cylindrical surface described about the axis of rotation have equal velocities. The dotted circles in fig. 19, page 26, represent the traces of two such surfaces.

The relative motions of any two pairs of points in a rotating piece may be compared together. For example, let it be pro posed to compare the motion of A relatively to B with the motion of B relatively to O. Then, because the velocity of the motion of A relatively to B is proportional to B A, and its direction perpendicular to the plane whose trace is B A, while the velocity of the motion of B relatively to O is proportional to O B, and its direction perpendicular to the plane of which O B is the trace, the directional relation is expressed by the angle made by those planes with each other, and the velocity-ratio by the ratio B A:0 B borne to each other by the projections on the plane of rotation of the two lines of connection of the two pairs of points.

54. Relative and Comparative Translation of a Pair of Rigidly Connected Points.-The following proposition is applicable to all motions whatsoever of a pair of points so connected that the distance between them is invariable. It forms the basis of nearly the whole theory of combinations in mechanism, and many of its consequences will be explained in the ensuing chapters of this Part. At present it is introduced with a view to its application to pairs of points in a rotating piece.

a

A

C

Fig. 25.

Fig. 26.

THEOREM.-If two points are so connected that their distance apart is invariable, the components of their velocities along the straight line which traverses them both must be equal; for if those component velocities are unequal, the distance between the points must necessarily change.

The straight line which traverses the points is called their Line of Connection.

At a

For example, in fig. 25, let A and B represent two points in the plane of the paper, whose distance apart, A B, is invariable. given instant let the velocities of those points be represented by straight lines, which may be in the same plane, or in different planes,

according to circumstances; and let A a and Bb be the projections of those lines. From a and b let fall a c and bc' perpendicular to the line of connection, A B; these will be the traces of two planes perpendicular to the line of connection, and traversing respectively the points of which a and b are the projections; the parts A c and Be', cut off by those planes from the line of connection (produced where necessary), will be the components along that line of the velocities of A and B respectively; and those components must necessarily be equal-that is, B c A c. The component velocities transverse to the line of connection are represented by the lines whose projections are c a and c' b, and may bear to each other any proportion whatsoever.

=

The same principle is illustrated in fig. 19, page 26. In that figure A a and Bb represent the velocities of two points, A and B, whose line of connection is A B, and is of invariable length; ac and be are perpendiculars let fall from a and b upon A B, produced where necessary; and Ac and B c' represent the component velocities of A and B along the line of connection, which are equal to each other.

RULE-Given (in fig. 25), a pair of rigidly connected points, A and B, and the directions of the projections A a and Bb upon a plane traversing A B, of their velocities at a given instant, to find the ratio of those projections or component velocities to each other. In fig. 26, draw Oc of any convenient length parallel to A B, and a cb perpendicular to it; through O draw O a in fig. 26 parallel to A a in fig. 25, and Ob in fig. 26 parallel to Bb in fig. 25; then the required ratio is

U

55. Components of Velocity of a Point in a Rotating Piece— Periodical Motion. (A. M., 380.)-The component parallel to an axis of rotation, of the velocity of a point in a rotating body relatively to that axis, is nothing. That velocity may be resolved into rectangular components parallel to the plane of rotation. Thus let O in fig. 27 represent the projection and trace of the axis of rotation of a body whose plane of rotation is that of the figure; and let A be the projection of a point in the body, the radius of whose circular path is O A. The velocity of that point being OA × angular velocity, let it be represented by the straight line AV perpendicular to O A. Let BA be any direction in the plane of rotation parallel to which it is desired to find the component of the velocity of A. From V

=

D

Fig. 27.

let fall V U perpendicular to B A; then A U represents the component in question. Sometimes the more convenient way of finding that component is the following:

From O let fall O B perpendicular to BA. Then A and B represent a pair of rigidly connected points; therefore, according to Article 24, the component velocities of A and B along A B are equal. But BA, being perpendicular to O B, is the direction of the whole velocity of B; therefore the component, along a given straight line in the plane of rotation, of the velocity of any point whose projection is in that line, is equal to the whole velocity of the point where a perpendicular from the axis meets that line.

=

The whole velocity of B is O B x the angular velocity; and the velocity-ratio of B to A, or, in other words, the ratio of the component velocity of A along BA to the whole velocity of A, is ОВ: О А.

The velocity of a point such as A in a rotating piece may be resolved into components, oblique (see fig. 19) or rectangular (see fig. 27) as the case may

be, by regarding the velocity of A relatively to O as the resultant of the velocity of A relatively to B, and of that of B relatively to O. The directions of that resultant velocity and its two components are respectively perpendicular to OA, BA, and O B, and their ratios to each other are equal to those of the lengths of the same three lines. This is a particular case of a more general

Fig. 28.

LX

proposition, viz,-that the velocities of three points relatively to each other are proportional to the three sides of a triangle which make with each other the same angles that the directions of those three relative velocities do (A. M., 355).

In fig. 28, let O be the trace of the axis on a plane of rotation, and A a point in the rotating piece, revolving in the circle O A, so as to assume successively a series of positions such as 1, 2, 3, 4, 5, 6, 7, 8; and in each position of A, let the component velocity A U, parallel to a fixed plane whose trace is the diameter 8 0 4 be compared with the whole velocity of revolution. A V.