in which the minute-hand points to true time, and its motion therefore consists of the equable motion of an ordinary minutehand, plus or minus the equation, or difference between true and mean time. The same principle has been applied with the greatest success to the bobbin and fly-frame. 406. The train which is carried on the arm, and the arm itself, receive various forms; the train should be as light as possible, and consist of few wheels, especially when it revolves in a vertical plane; because being excentric its weight interferes with the equable rotation of the arm or wheel which carries it, unless it be balanced very carefully. When the excentric train is necessarily heavy, this difficulty is in some degree got over by making the train-bearing axis vertical, as in planetary machinery and in rope-laying machinery. EXAMPLES OF THE FIRST USE OF EPICYCLIC TRAINS. 407. Ex. 1. Ferguson's Mechanical Paradox.-This was contrived to show the properties of a simple epicyclic train, of which the first wheel is fixed to the frame of the machine. It consists of a wheel A, fig. 290, of 20 teeth, fixed to the top of a stud which is planted in a stand that serves to support the apparatus. An arm CD can be made to revolve round this stud, and has two pins m and n fixed into it, upon one of which is a thick idle wheel B of any number of teeth, which wheel geers with A and also with three loose wheels E, F, and G, which lie one on the other about the pin n. When the arm CD is turned round, motion is given to these three wheels which form respectively with the intermediate wheel B and the wheel A three epicyclic trains. Now in this machine the extreme wheels of each epicyclic train revolve in the same direction, and therefore & is positive, and the formula applicable to this case is n a ε = 1 E, where n and a are the absolute synchronal rotations of the last wheel and of the arm. But the object of this machine is only to show the directions of rotation. =0, and the last wheel of the train will have no absolute rotation. If be less than unity will be positive, n and the last wheel will revolve absolutely in the same direction as the arm. But if & be greater than unity will be negative, ε n a and the absolute rotations of the arm and wheel will be in opposite directions. Let E, F, G have respectively 21, 20, and 19 teeth, then in is less than unity, and E will revolve the same way as the arm: equals unity, =0 and F will have no absolute revolution : n a is greater than unity, and G will revolve backwards. A The principle of the middle train, is employed in the me F' chanism of an elaborate rotary book-desk by Ramelli, fig. 188, published in 1588. It follows from this that when the arm is turned round, E will revolve one way, G the other, and F will stand still, or rather continually point in the same direction. Which being an apparent paradox, gave rise to the name of the apparatus, which is well adapted to show the more obvious properties of trains of this kind. But Ferguson was not the first who studied the motions of epicyclic trains; Graham's orrery in 1715, appears to be the original of this curious class of machinery, but for which no general formula appears to have been hitherto given.* 408. Ex. 2. The contrivance termed sun and planet-wheels was invented by Watt as a substitute for the common crank in * In Rees' Cyclopædia, Art. 'Planetary Numbers,' are a few arithmetical rules for the calculation of planetary trains, given without demonstration. Fig. 291. converting the reciprocating motion of the beam of the steam engine into the circular motion of the fly-wheel. The rod DB, fig. 291, has a toothed wheel B fixed to it, and the fly-wheel has a toothed wheel A also attached to it, a link BA serves to keep these wheels in geer. Now when the beam is in action the link or arm BA will be made to revolve round the center A, just as a common crank would, but as the wheel B is attached to the rod DB so as to prevent it from revolving absolutely on its own center B, every part of its circumference is in turn presented to the wheel A, which thus receives a rotatory motion, the proportionate value of which is easily ascertained by the formula already given. A B The wheels AB with the arm constitute an epicyclic train =, in which a is negative, since the wheels revolve in opposite directions considered with respect to the arm, and in which the last wheel B has no absolute rotation, being pinned to the rod DB; the formula In Watt's engine the wheels were equal and therefore m=2a, and the fly-wheel revolved twice as fast as the crank-arm. 409. Ex. 3. Planetary Mechanism. mn is a fixed central axis, upon which a train-bearing arm fg turns, carrying two separate epicyclic trains &, and ɛ. ει One of these, &,, has a first wheel D, and a last wheel F, connected by any train of wheel-work, and the axis of this last wheel passes through the end of the arm fg, and carries a second arm pq. The other train ɛ, has a first wheel A connected to its last wheel B, by any train of wheel-work, but this last wheel is united to the first wheel of an epicyclic train & borne by the arm p q, of which train the last wheel is C. The question is, to find the absolute rotations of this last axis. The arrangement is one that occurs in some shape or other in most orreries, for the purpose of representing the diurnal rotation of the Earth's axis, in which case f g is the annual bar, and E a ball representing the Earth. Let the absolute synchronal rotations of the bar fg=a, those of D=m1; of F (and therefore of the arm p q)=n1; of A=m,; of B (and therefore of the first wheel of the train &,)=n; and of C (and therefore of the Earth)=n.. In an orrery by Mr. Pearson for equated motions, described in Rees' Cyclopædia,' the arm or annual bar fg, is carried round by hand, and the wheels A and D are fixed to the central axis. In this case m, and m, vanish, and we obtain the formula a = But the arm p q which carries the Earth's axis must preserve its parallelism, and therefore having no absolute rotation n1 =0. The train &, will therefore = +1; which must be positive, since the Earth performs its daily and annual revolutions in the same direction. The train in ε3 Mr. Pearson's orrery consists of three wheels of 40 each en suite; .' . 83 = +1, in which the extreme wheels revolve in opposite directions, therefore, is negative; In making these calculations it must be remembered that the absolute period of E is a sidereal day and its period relative to the arm f g is a solar day, also the period of f g is a year. Now from Art. 398 it appears that the absolute revolutions of any wheel or piece of an epicyclic train are equal to the sum of its relative revolutions and the revolutions of the arm when they revolve in the same direction, and the same reasoning shows that the number of sidereal days in a year is equal to the number of solar days +1. Also n3 and a are the synchronal absolute rotations of the arm or annual bar ƒ g, and Earth's axis CE; therefore 23= number a of sidereal days in a year; but the fractions in Art. 346 represent the number of solar days in a year, and we may therefore employ them for by adding unity as above. We may thus n3 a obtain other and simpler trains than that already giyen. The train & being carried by a small arm should be as simple and light as possible. But it may be reduced to only two wheels by making & negative, and at the same time & positive, since 94963 П3 a must be positive. For example, employing the fraction (vide p. 280) and remembering that the rotations n, are sidereal days, we have |