right line perpendicular to the normal MD. Also Mp, Mn are ultimately perpendicular to AM, BM.

Fig. 10.

B

In the small triangle Mpn, the sides Mp, Mn, pn are respectively perpendicular to AM, BM, MD, and consequently make mutually the same angles with each other as these latter lines;

33. From these expressions it appears that in the small triangle pn M, pn can only vanish with respect to nM or pM when sin BMA vanishes; that is, when the radii of contact coincide with the line of centers. But when pn vanishes the sliding vanishes, and the contact becomes rolling contact. Hence it appears that in rolling contact the curves must be so formed, that the point of contact shall always lie on the line of centers. Also the common normal will cut the line of centers at the point T (fig. 7), which will be now the point of contact, and therefore in rolling contact, the angular velocities are inversely as the segments into which the point of contact divides the line of centers.

34. Examples. Let the curves be a pair of involutes of circles, and let BD be a perpendicular from B upon MD. But this perpendicular is constant in the involute;

pn x BM x sin BMA, that is to say as the perpendicular PM upon AM produced.

But if the curves be an epicycloid turning on the center A, in contact with a radial line which turns round B; then DMB is a right angle,

To find the velocity ratio in wrapping connectors (correlation of sliding and wrapping). Let AB be the respective centers of

motion of a pair of curves, SS, $18, in contact at M, and let 5181, S282 Ss be respective points of contact when the curve SS drives s,s, by sliding or rolling contact.

Let WPS, be the evolute of SMS,, described on the plane of that curve, and wQs, the evolute of s,Mw, described on the plane of that curve. The curve SMS, may be, therefore, described

by the point M of the inextensible string WPM, and similarly the curve s, Qw by the inextensible string wQM, and as these strings are always normal to their respective involutes SMS, they together form a common normal at every point of contact of those curves as at M. Consequently, if we suppose an inextensible flexible string WPTQw to be attached at W, w respectively to the evolutes of the contact curves, and the latter move with their edges in contact, this string will wrap upon one evolute and unwrap from the other evolute, always remaining a common normal to the contact curves, and a common tangent to their evolutes, the wrapping curves, and the point M on the string will coincide in every position with the point of contact of the curves. Hence, if the contact curves be removed, the evolutes and the string constitute a pair of curves with a wrapping connector, whose action is equivalent to that of the contact curves, and as the wrapping connector is the common normal of the latter, the proposition (Art. 31) shows that in wrapping connectors the angular motion of the pieces are inversely as the segments into which the connector, or (which is the same thing) the common tangent of the wrapping curves divides the line of centers.*

If any other point m be taken on the wrapping connector, it will trace, during the motion, another pair of involutes, normally

* In the former edition of this work the following demonstration was given :To find the Velocity Ratio in wrapping connections. Let A, B be the centers of motion, PQ the wrapping connector touching the curves at P and Q, and let the point P be moved to p very near to its first position, then will be drawn to q, and the connector will touch the curves in two new points of contact, which may be r and s

respectively. Now, in the action of wrapping or unwrapping, the connector touches the curves in a series of consecutive points between q and S or p and r, and ultimately q coincides with S and p with r. The extremities of the connector may therefore be considered at any given moment as if jointed to the two curves at the points of contact, and turning upon these points in the manner of a link. The relative velocities of the curves are therefore momentarily the same as if AP, BQ were a pair of rods connected by a link PQ. Hence the angular velocities of the pieces are to each other inversely as the segments into which the connector divides the line of centers.

equidistant from the first, on their respective planes. This new pair may be substituted for the first, if more convenient.

It may happen that one or both of the wrapping curves may have salient points, as at P, which is the meeting point of the two tangents P3 and P6, and at Q, which is the meeting point of the tangents Q5 and Q6. Consequently, the lower sliding

curve from 3 to 6 and the upper one from 5 to 6 are arcs of excentric circles, described about P and Q.

The effect of this is that the wrapping connector in the positions between P5, 5Q and P6, 6 Q acts in the manner of a link

* In the points of certain curves changes of curvature take place which are termed points of inflexion, cusps, or salient points.

At a point of inflexion, I, fig. 14, the curvature changes its aspect, and the direction

whose centers are P and Q. But between the positions P3, 3111, and P5, 5Q the connector is jointed as a link at P, but wraps on the curve III IV Q at the other extremity.

This shows that a link is in effect a wrapping connector, of which the wrapping curves are reduced to a point, and that linkwork is a particular case of wrapping connection (F), in which one or both of the wrapping curves are reduced to a point.

35. If the line of direction of the link in link-work, of the common normal to the curves in the rolling and sliding contact motions, and of the connector in wrapping motion, be severally termed the line of action, we can express the separate propositions which relate to the Velocity Ratio, by saying that the angular velocities of two rotating pieces provided with either of the four mechanistic connections, are to each other inversely as the segments into which the line of action divides the line of centers, or inversely as the perpendiculars from the centers of motion upon the line of action (A).

I have confined these investigations, for the present, to motions in the same plane. The cases of motions in different planes are more simply examined as the individual combinations which require them occur.

36. It has been shown that the points of the principal pieces which constitute a train of mechanism are compelled, by their

of the tangents II, It on each side of the point coincide in one straight line so that the curve ab cuts its tangent at that point.

At a point of cuspidation, C, fig. 15, the curve aC is abruptly reflected as at C, so that the tangents of the two branches Ca, Cb at that point or cusp coalesce in one straight line Ct.

At a salient point, S, figs. 16, 17, 18, the curve aSb is abruptly reflected so that two tangents ST, St meet at that point of the curve at an angle TSt. The salient point may be concave, fig. 16, convex, fig. 17, or concavo-convex, fig. 18.