motion; the resultant of these two will be the required other component motion. For example, in fig. 15, let AD be the given resultant motion, and A B the given component; draw D C equal and parallel to A B, and pointing the opposite way; join A C; this will be the required other component. or otherwise, join BD and draw A C equal and parallel to it.

VI. (Fig. 17.) Given, the vertical projection, A B, and the horizontal projection, A' B', of a straight line representing a motion, to resolve that motion into three rectangular components parallel and perpendicular to the planes of projection. Let OX be the axis of projection (Article 9, page 4). Draw the straight lines A A', BB,

cutting the axis of projection (of course at right angles) in C and D. Then through any convenient point, O, in the axis of projection, draw the straight line ZO Y at right angles to that axis; and take O Y' to represent a transverse horizontal axis, and OZ to represent a vertical axis. (The point O is called the origin.) Then parallel to X O draw A' E' and B' F' to meet O Y', and A G and BH to meet O Z. The three components required will be represented by C D, E' F', and G H.

VII. Given (in fig. 17), the vertical projection, A B, and the horizontal projection, A' B', of a straight line representing a motion, to draw a third projection of the same straight line on a vertical transverse plane of projection perpendicular to the first two planes of projection. Construct fig. 17 as described in the preceding Rule. O Z and O Y' will be the traces of the third plane of projection. Produce X O towards Y"; then O Y" will represent the rabatment

of O Y', and Z O Y" the rabatment of the vertical transverse plane upon the vertical longitudinal plane of projection. In OY" take OE"= 0 E', and O F" = O F; draw E" A" and F"B" parallel to O Z, to meet A G and B H produced in A" and B" respectively; join A B"; this will be the projection required.

According to the rule already stated in Article 19, page 7, the motion of which A B and A' B' are the projections is to be found by making K L = A' B', and joining L B, which line will represent the extent of the resultant motion.

The following are the relations between a resultant motion and its components as expressed by calculation. In fig. 15,

sin C A B sin C A D : sin DAB::AD:AB: A C; also, A D2 A B+ A C2 + 2 A B A C cos CA B.

In fig. 16,

=

A B = AD cos B A D; A C = A D · sin BAD;

In fig. 17,

A D2 A B2+ A C2.

=

L B2 C D2 + E' F'2 + G H2.

=

42. Belative Motion of Two Moving Pieces.-All motion is relative that is to say, every conceivable motion consists in a change of the relative position of two or more points. In speaking of the motions of the moving pieces of machines, motions relatively to the frame are always to be understood, unless it is otherwise specified. It is often requisite, however, to express the motion of a point in a moving piece relatively to a point in the same or in another moving piece.

In the case considered in the present section, where the relative position of two points in the same moving piece remains unaltered, not only as to distance but as to direction, the relative motion of such a pair of points is nothing. The motion of one moving piece relatively to another is determined by the following principle:-Let P, Q, and R denote any three points; then the motion of R relatively to P is the resultant of the motion of R relatively to Q, combined with the motion of Q relatively to P; so that if the motions of Q relatively to P, and of R relatively to P are given, the motion of R relatively to Q is to be found according to Rule V. of the preceding Article, by compounding with the motion of R relatively to P a motion equal and opposite to that of Q relatively to P. For example, let P stand for the frame of a machine, and Q and R for two moving pieces which slide along straight guides;

and in a given interval of time let A B, in fig. 15, page 19, represent the motion of Q relatively to P, and A D the motion of R relatively to P; then A C, found by Rule V. of Article 41, will represent the motion of R relatively to Q.

In all cases whatsoever of relative motion of two bodies, the motion of one relatively to the other is exactly equal and contrary to that of the second relatively to the first. For example, let P and Q be two points; and when P is treated as fixed, let Q move through a given distance in a given direction relatively to P; then if Q is treated as fixed, P moves through the same distance in the contrary direction relatively to Q.

43. Comparative Motion (A. M., 358,) is the relation borne to each other by the simultaneous motions of two points, either in the same body or in different bodies, relatively to one and the same fixed point or body. It consists of two elements: the velocity-ratio, which is the proportion borne to each other by the distances moved through by the two points in the same interval of time; and the directional relation, which is the relation between the directions in which the two points are moving at the same instant.

In the case of two points in a primary piece whose motion is one of translation, the velocity-ratio is that of equality, and the directional relation that of identity; for all points in such a piece are moving with equal speed in parallel directions at the same instant.

When two points in two different pieces are compared, the comparison may give a different result. For example, let P, as before, stand for the frame of a machine, and Q and R for two moving pieces; and while Q performs relatively to P the motion. represented by A B (fig. 15, page 19), let R perform relatively to P the motion represented by A D. Then the comparative motion of R and Q consists of the following elements:

and the directional relation, represented by the angle B A D. In most of the cases which occur in mechanism the motion of each point is limited to two directions--forward or backward—in a fixed path; so that the directional relation of two points may often be sufficiently expressed by prefixing the sign or to their velocityratio, according as their motions are similar or contrary; that is, the sign denotes that those motions are both forward or both backward; and the sign that one is forward and the other backward.

We may compare together the different components of the motion of one point, and the resultant motion. For example, in

figs. 15 and 16, page 19, the velocity-ratios of two component motions, as compared with their resultant, are expressed by

and in fig. 17, page 20, the velocity-ratios of three rectangular component motions, as compared with their resultant, are ex

Strictly speaking, the principles of the geometry of machines, or of pure mechanism, are concerned with comparative motions only, and not with absolute velocities: or, in other words, those principles relate to the motions which different moving points perform in the course of the same interval of time, but not to the length of the interval of time in which such motions are performed. For example, in the case of a direct-acting steam engine, the principles of pure mechanism show that the piston makes one double stroke for each revolution of the crank; that the directional relation of the piston and crank-pin varies periodically, the piston moving to and fro, while the crank-pin moves continuously round in a circle; and that in particular positions of those pieces their velocity-ratio takes particular values; but the question of what interval of time is occupied by a revolution, or of how many revolutions are performed in a minute, belongs not to the geometry, but to the dynamics of machines. Further, in the case of a pair of spur wheels gearing into each other, the principles of pure mechanism. show that in any given interval of time the numbers of revolutions performed by those wheels respectively are inversely as their numbers of teeth, and that the directions in which they turn are contrary; but those principles do not inform us how many revolutions either wheel makes in a minute.

44. Driving Point and Working Point.-The term driving point is used to denote that point, either in a whole machine or in a given moving piece of a machine, where the force is applied that causes the motion; and the term working point is used to denote the point where the useful work is done. These explanations contain references to the dynamics of machines; but it is to be understood that in the geometry of machines, or pure mechanism, it is the comparative motion only of the driving point and working point that is taken into consideration. It is to be observed, too, that the word "point" is here taken in an extended meaning; for the exertion of force or communication of motion at a mathematical point, of no sensible magnitude, is purely ideal; and when the word point is used with reference to the driving or the work of machines,

it is to be held to mean the place where the action that drives or that resists a machine is exerted, of what magnitude soever that place may be, whether a surface or a volume. Thus, the driving point in a steam engine comprehends the whole surface of the piston that is pressed upon by the steam which drives the engine; and the working point, where friction is overcome, comprehends the whole of the rubbing surface, and where a heavy body is lifted, the whole volume of that body. Nevertheless, for the sake of convenience in mathematical investigation, such places of the action of driving or resisting forces are often treated on the supposition that they may be represented by single points; for when such points are properly chosen, no error is incurred by making that supposition.

SECTION III.-Rotation of Primary Pieces.

45. Rotation of a Primary Piece. (A. M., 370-372.)—Rotation or Turning is the motion of a rigid body when lines in it change their directions; and it is the only kind of motion involving change of the relative positions of the particles of a body that is possible consistently with rigidity; that is to say, with the maintenance of the distance between every pair of particles in the body unchanged. An axis of rotation is a line in a rigid body whose direction is unchanged by the rotation; and a fixed axis of rotation is a line whose position, as well as its direction, is unchanged by the rotation. Every line in a rotating body which is parallel to the axis has its direction unchanged by the rotation. The rotation of a primary piece in a machine always takes place about an axis that is fixed relatively to the frame of the machine; that axis being the geometrical axis, or centre line, of a bearing surface (such as that of the journals or gudgeons of a shaft), whose form is that either of a circular cylinder or of some other surface of revolution The plane of rotation is any plane perpendicular to the axis. Every such plane in a rotating body has its position unchanged by the rotation; and straight lines in such a plane-that is, straight lines perpendicular to the axis of rotationchange their directions more rapidly than any other straight lines in the same body.

46. Speed of Rotation. (A. M., 373.)—Although in the case of rotation, as well as in that of translation, the principles of pure mechanism are concerned with comparative velocities only, still it is desirable here to state, that the speed with which a rotating body turns is expressed in two different ways. For most practical purposes it is usually stated in turns and fractions of a turn in some convenient unit of time; such as a second, or (more commonly) a minute. For scientific purposes, and for some practical purposes also, it is expressed in angular velocity; which means, the angle swept through in a second by a line perpendicular to the axis of