substituted for them which geers at once with the fixed wheel A and the slow-moving wheel D.* Let M, M-1, and K be the numbers of teeth of D, A, and the thick pinion respectively, then where M is the number of teeth of the slow-moving wheel. EXAMPLES OF THE FOURTH USE OF EPICYCLIC TRAINS. 420. The fourth employment of epicyclic trains consists in concentrating the effects of two or more different trains upon one revolving body when these trains move with respect to each other with a variable velocity ratio. I have already shown how this may be effected when the extent of motion is small, as in Arts. 386, 389, but by epicyclic trains an indefinite number of rotations may be produced. As an example of this application I shall take the equation clock, as it is the earliest problem of this class which presents itself for solution in the history of mechanism, and actually occupied the attention of mechanists for a long period.† The object of this machine is to cause the hands of a clock to point on the usual dial, not to mean solar time, but to true solar time. For this purpose we may resolve its motion as astronomers resolve the motion of the sun; namely, into two, one of which is the uniform motion which belongs to the mean time, and the other the difference between mean and true time, or the equation. If, then, two trains of mechanism be provided, one of them an ordinary clock, and the other contrived so as to communicate a slow motion corresponding to the equation of time, and if we then concentrate the effects of these separate trains upon the hands of our equation clock by means of an epicyclic train, we shall obtain the desired result. There are three possible arrangements, as in Art. 397, (1) the equation may be communicated to one end of the train, and the mean motion to the other, the arm receiving the solar motion; (2) the equation may be given to one end of the train, and the mean motion to the arm, the other end of the train will then receive the solar motion; (3) the equation may be commu * In Roberts' self-acting mule. + Vide the Machines Approuvées of the Acad. des Sciences. nicated to the arm, and the mean time to one end of the train, when the other end of the train will receive the solar motion.* I shall describe the mechanism of the latter arrangement. Fig. 296. 421. Fig. 296 is a diagram which will serve to show the wheel-work of that part of an equation clock by which the motion is given to the hands. This wheel-work is commonly called the dial-work. G is the centre of motion of the epicyclic train, G Dethetrainbearing arm. The wheels f and C turn freely upon the axis G, and the axis D carried by the arm has two wheels D and c fixed to it, which geer with f and C respectively. E B A The epicyclic train consists, therefore, of the four wheels C, c, D and f, of which let C be the first wheel. In this arrangement the equation is to be communicated to the train-bearing arm, and the mean motion to the first wheel C of the epicyclic train. Now for this purpose C is driven by the wheel B, dotted in the figure, which derives its motion from a wheel A connected with an ordinary clock, and as the minute-hand M of the clock is fastened to the axis of B, this minute-hand will show mean time upon the dial in the usual manner. The equation is communicated to the train-bearing arm GDe, as follows. E is a cam-plate, which by its connection with the clock is made to revolve in a year (Art. 346). 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 ƒ, 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. * In the clocks of Du Tertre, 1742, and Enderlin. The formula which belongs to this case is, (Art. 402), n=a.1−ɛ+m ɛ, Cc in which is positive and= Now if the synchronal rotations Df of the minute-hand M and of C be M and m respectively, we havem=M., and if those of ƒ and g be n and s, we have g n=s. ; substituting these values in the formula, we obtain f 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 Be M; .. Dg = 1. And for that part of the motion of S which is due to the equation, the expression a. Df- Cc shows the pro Dg portion between the angular motion of the train-bearing 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, then Df- Cc = 1, and this is readily effected by making f=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. 422. 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 movable dial, and communicating to the latter the equation, by which the ordinary minute-hand of the clock will simultaneously show mean time on the fixed, and true time on the movable dial, without the intervention of the epicyclic train.* *This is done in the early equation clocks of Le Bon, 1714, Le Roy, &c. 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. CHAPTER III. ON COMBINATIONS FOR PRODUCING AGGREGATE PATHS. 423. I HAVE already stated in the beginning of this work 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. 424. 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 |