It is to be remarked that in order to show distinctly the principles of the construction of the figures illustrating this article, all those dimensions which give rise to errors—that is, to deviations from the exact law of harmonic motion-are exaggerated; such as the lengths of the link A B, and of the eccentric-arms O E and O F, as compared with that of the eccentric-rods E A and F B. In ordinary practice the link is from one-third to one-fifth, and each of the eccentric-arms about one-twentyfifth of the length of an eccentric-rod; and the effect of these proportions is to make the deviation of the resultant motion of the slide-valve from true harmonic motion practically inappreciable. (See p. 582.) 241. Differential Harmonic Motions.-Two primary pieces, having different motions, may be regarded as constituting an aggregate combination with respect to the motion of one of them relatively to the other; because that motion is the resultant of two components: for example, if A be taken to denote the frame, and B and C the two primary pieces, the motion of C relatively to B is the resultant of the motion of B relatively to A, and of a motion equal and contrary to the motion of B relatively to A. This principle has been already stated in Article 42, page 21 The following is one of its most frequent applications:-In fig. 190, let b and c be two pieces which have approximately harmonic motions in parallel directions and of equal periodic time, but differing in phase, given to them respectively by two cranks or eccentric-arms, A B and A C, which turn as one piece with the same angular velocity about axis A, the angle B A C being the difference of phase. Then the motion of the slide c relatively to the slide b is approximately the same with that which would be produced by a crank or eccentric-arm, B C, turning with the same angular velocity; that is to say, it is an approximately harmonic motion of the same periodic time with the two elementary motions of B and C; its half-travel is equal to B C, and its phase at any instant is that corresponding to the direction of B C at that instant. This is what may be called a differential harmonic motion; and upon such motions depends the action of double slide-valves and moveable slide-valve seats in steam engines. Fig. 190. SECTION IV.-Production of Curved Aggregate Paths. 242. Circular Aggregate Paths.-Some circular aggregate paths are traced by means of mechanical combinations, which are capable also of tracing ellipses, if required; and these will be described further on. The present Article relates to combinations in which circular paths alone are traced. Amongst such combinations may be classed the coupling-rod shown in fig. 32, Article 68, page 44; for every point in or rigidly attached to that rod traces a circle of a radius equal in length to the crank-arms by which the rod is carried; and the same takes place in every case in which a secondary piece has a motion of circular translation without rotation. For example, in fig. 191, A is a centre pin, carrying a fixed spur-wheel-in other words, a spurwheel without rotation. About the axis of that wheel there turns a disc, carrying a set of diverging epicyclic trains. Each epicyclic train consists of a spur-wheel, B, gearing with the fixed wheel A, and another spur-wheel, C, gearing with B. The last spur-wheel, C, is exactly equal in radius and in number of teeth to the fixed wheel A; and the consequence is, that each of the wheels marked C has an angular velocity equal to that of A-that is to say, equal to nothing: in other words, when the disc rotates, the wheels marked C have a motion of circular translation without rotation. Let E be any point in one of the wheels C; and draw A D equal and parallel to CE; then E traces a circle round D, exactly equal to the circle which the centre C of the wheel to which Ě belongs traces round A at the same time. This combination is used in spinning wire ropes. Each of the wheels C carries a bobbin from which a wire or a strand is paid out as the spinning goes on; and the effect of the absence of rotation in the wheels C is, that the wires or strands are spun together without being twisted, which would overstrain the material. The combination shown in fig. 192 serves to guide a point C along an arc, B C A, of a circle of a radius so great that it would be inconvenient to guide the point C by connecting it directly with the centre of the circle. It is based upon this well-known geometrical principle:-Let A and B be any two fixed points in the circular arc to be traced; then the two chords C A, C B make Fig. 192. with each other a constant angle at C-viz., the supplement of one-half of the angle which the arc A B subtends at the centre of the circle. Two rods are fastened together at C, so as to make with each other the proper constant angle; and they are guided by passing through sockets at A and B, which sockets are free to turn about A and B respectively, but not to move otherwise. Then, when the rods are made to slide through the sockets, the point C traces the required circular arc. The angle made by the rods with each other may be made adjustable by means of a screw or otherwise, so as to vary the curvature of the arc when required. 243. Epitrochoidal Paths.—An epitrochoid is the curve traced by a point in or rigidly attached to a circle which rolls either inside or outside of another circle (called the base-circle); also, if two circles (as the pitch-circles of two spur-wheels) turn in rolling contact with each other about fixed axes, a point rigidly attached to one of those circles traces an epitrochoid upon a disc rigidly attached to the other. When the tracing-point is in the circumference of the rolling-circle, the curve traced becomes that particular kind of epitrochoid that is called an epicycloid. The properties of this curve have already been explained in Article 78, page 56, with a view to its adaptation to the figures of the teeth of wheels. If the circumferences of the rolling-circle and of the base-circle are commensurable with each other, the epitrochoid returns into itself, and has a finite number of lobes or coils-viz., the denominator of the fraction which, being in its least terms, expresses the ratio borne by the circumference of the rolling-circle to that of the base-circle. those circumferences are incommensurable, the epitrochoid does not return into itself, so that the number of its lobes or coils is indefinite. If When the rolling-circle rolls outside a base-circle of equal-radius, the epitrochoid is one-lobed, and is called a cardioid. the rolling-circle to the base-circle is 1 so that the epitrochoids are 3' three-lobed. Each figure shows an external and an internal epitrochoid, traced by rolling the rolling-circle outside and inside the base-circle respectively. The centres of the base-circles are marked A; those of the external rolling-circles, B; those of the internal rolling-circles, b; and the tracing-points of the external and internal rolling-circles are marked C and c respectively. In fig. 193 the tracing-points are in the circumferences of the rolling-circles; and the curves traced are epicycloids, distinguished by having cusps at the points where the tracing-point coincides with the base-circle. In fig. 194 the tracing-points are inside the rolling-circles; and the curves traced are prolate epitrochoids, distinguished by their wave-like form. In fig. 195 the tracing B OA Fig. 195. points are outside the rolling-circles; and the curves traced are curtate epitrochoids, distinguished by their looped form. An important property of curves traced by rolling has already been mentioned-viz., that at every instant the straight line joining the tracing-point and the pitch-point, or point of contact of |