with the clock is made to revolve in a year (Art. 247). A friction roller e upon the train-bearing arm rests upon the edge of the cam-plate, and is kept in contact with it by means of a spring or weight. The cam-plate is shaped so as to communicate the proper quantity of angular motion to the arm. We have seen how one end of the epicyclic train receives the mean motion, and f, which is the other extremity of the train, geers with a wheel g concentric to the minute-wheel B, and turning freely upon it; the solar hand S is fixed to the tube or cannon of g, and thus receiving the aggregate of the mean motion and the equation, will point upon the dial to the true time which corresponds to the mean time indicated by M.

The formula which belongs to this case is, (Art. 411)

rotations of the minute-hand M and of C be M and m respect

of which the first part belongs to the equation, and the second to the mean motion.

Now the mean motion of S must be the same as that of

= 1. And for that part of the motion of S which

is due to the equation, the expression a.

the proportion between the angular motion of the trainbearing arm and of the hand s, synchronal rotations being

directly proportional to angular velocity (Art. 20.).

If the

arm is to move with the same angular velocity as the hand,

and this is readily effected by making ƒ = c = g and C = 2D; also, since Bc = Dg where c = g, we must have B = D, and these are the actual proportions employed by Enderlin. But if it be required that the arm move through a less angle than the hand, through half the angle, for example, then C = 3D, and so on.

431. In the treatises on Horology, and in the machines of the French Academy, may be found a great number of contrivances for equation clocks, which was a favourite subject with the mechanists of the last century. The machine itself is merely curious, and the desired purpose may be effected in a much more simple manner, if indeed it be worth doing at all, by placing concentrically to the common fixed dial a smaller moveable dial, and communicating to the latter the equation, by which the ordinary minute-hand of the clock will simultaneously shew mean time on the fixed, and true time on the moveable dial, without the intervention of the epicyclic train*.

Nevertheless, I have selected this machine as the best for the purpose of explanation, as being easily intelligible. The most successful machine of this class is undoubtedly the Bobbin and Fly-frame, in which, by means of an epicyclic train, the motions of the spindles are beautifully adjusted to the increasing diameter of the bobbins and consequent varying velocity of the bobbins and flyers. But this machine involves so many other considerations, that the complete explanation of it cannot be given in the present stage of our subject.

This is done in the early equation clocks of Le Bon, 1714, Le Roy, &c.

CHAPTER III.

ON COMBINATIONS FOR PRODUCING

AGGREGATE PATHS.

432. I HAVE already stated in the beginning of this work (Art. 39), that pieces in a train may be required to describe elliptical, epicycloidal, or sinuous lines, and that such motions are produced by combining circular and rectilinear motions by aggregation. The process being, in fact, derived from the well-known geometrical principle by which motion in any curve is resolved into two simultaneous motions in co-ordinate lines or circles.

If the curve in which the piece or point is required to move be referred to rectangular co-ordinates, let the piece be mounted upon a slide attached to a second piece, and let this second piece be again mounted upon a slide attached to the frame of the machine at right angles to the first slide. Then if we assume the direction of one slide for the axis of abscissæ, the direction of the other will be parallel to the ordinates of the required curve. And if we communicate simultaneously such motions to the two sliding pieces as will cause them to describe spaces respectively equal to the corresponding abscissæ and ordinates, the point or piece which is mounted upon the first slide will always be found in the required curve.

This first slide, being itself carried by a transverse slide, falls under the cases described in the first Chapter of this Part, and the motion may be given to it by any contrivance for maintaining the communication of motion between pieces

the position of whose paths is variable, as, for example, by a rack attached to the slide and driven by a long pinion. For the purpose of communicating the velocities to the two slides, any appropriate contrivance from the first part of the work may be chosen.

433. If the curve in which the point is to move be referred to polar co-ordinates, these may be as easily translated into mechanism, by mounting the point upon a slide and causing this slide to revolve round a center, which will be the pole. Then connecting these pieces by mechanism, so that while the slide revolves round its pole the point shall travel along the slide with the proper velocity, this point will always be found in the given curve.

434. Fig. 223 is a very simple arrangement, by which a short curve may be described upon the above principles.

D

a 223

Am B

A C

If

E is the center of motion of an arm Ee which is connected by a link with the describing point s; D is the center of motion of a second arm Dd which is connected by a link ds, with the same describing point s. now Ee be made to move through a small arc, it will communicate to s a motion round d which will be nearly vertical, and if Dd be made to move through a small arc, it will communicate to s a motion round e, which will be nearly horizontal; and as the motion of the describing point s is solely governed by its connexion with these two links, these motions may be separately be separately or simultaneously communicated to it. A is an axis, upon which are fixed two cam-plates, the lower of which, C, is in contact with a roller e at the end of the arm Ee, and the upper, B, in contact with a roller m at the end of an arm Dm, fixed at right angles to the arm Dd.

When the axis A revolves the cams communicate simultaneously motions to the two arms, which motions are given to the describing point, one in a direction nearly perpendicular to the other, the point will thus describe a curve of which the horizontal co-ordinates are determined by the cam B, and the vertical by the cam C.

In practice the shape of the cams may be obtained by trial: the machine must be previously constructed, and plain disks of a sufficient diameter substituted for the cams, then if the required path of s be traced upon paper, and it be placed in succession upon a sufficient number of positions upon this path, the cam-axis being also shifted, the corresponding positions of the rollers e and m may be marked upon the disks, and the shape of the cams thus ascertained.

435. If the object of the machine be merely to trace a few curves upon paper or other material, the principle of relative motion will enable us to dispense with the difficul ties that are introduced by the necessity of maintaining motion with a piece whose path itself travels. For since every complex path is resolvable into two simple paths, let the describing point move in one component path, and the surface upon which it traces the curve move in the other component path with the proper relative velocity, then will the curve be described by the relative motion of the point and

surface. .

Thus to describe polar curves, the surface upon which the curve is to be described may be made to revolve while the describing point travels with the proper velocity along a fixed slide, in a path the direction of which passes through the axis of motion of the surface. And as in this arrangement the axis of motion of the surface and the path of the describing point are both fixed in position, the simultaneous

Already employed in Arts, 256, 404, 405.