wheel, A. This wheel has no shaft, but is carried and kept in position by three pairs of rollers, as shown, so that its axis of rotation is the same as that of the shaft, S; and it is toothed externally as well as internally. The strands pass through the hollow axes of the pinions, and thence each to its own opening through the laying-top, T, fixed upon S, which completes the operation of twisting them into a rope. The annular wheel, A, it will be perceived, may be driven by a pinion, E, engaging with its external teeth, at a rate of speed different from that of the central shaft; and by varying the speed of that pinion, the velocity of the wheel, A, may be changed without affecting the velocity of S.

It is true that in making a certain kind of rope, the velocity ratio of A and S must remain constant, in order that the strands may be equally twisted throughout; but if for another kind of rope a different degree of twist is wanted, the velocity of the pinion, E, may be altered by means of change-wheels, and thus the same machine may be used for manufacturing many different sorts.

The second combination of this kind was devised by the writer as a "tell-tale" for showing whether the engines driving a pair of twin screw-propellers were going at the same rate. In Fig. 33, an index, P, is carried by the wheel, F: the wheel, A, is loose upon the shaft of the train-arm, which latter is driven by the wheel, E. The wheels, F and f, are of the same size, but a is twice as large as A; if then A be driven by one engine, and E by the other, at the same rate but in the opposite direction, the index will remain stationary, whatever the absolute velocities. But if either engine go faster than the other, the index will turn to the right or the left accordingly. The same object may also be accomplished as shown in Fig. 34, the index being carried by the train-arm. It makes no difference what the actual value of the ratio A/a may be, but it must be equal to F/f: under which condition it is evident that if A and F be driven contrary ways at equal speeds, small or great, the train-arm will remain at rest; but any inequality will cause the index to turn.

In some cases, particularly when annular wheels are used, the train-arm may become very short, so that it may be impossible to mount the planet-wheel in the manner thus far represented, upon a pin carried by a crank. This difficulty may be surmounted as shown in Fig. 35, which illustrates an arrangement originally forming a part of Nelson's steam steering gear. The Internal pinions, a, f, are but little smaller than the annular wheels, A, F, and are hung upon an eccentric E formed in one solid piece with the driving shaft, D.

The action of a complete epicyclic train involves virtually and always the action of two suns and two planets; but it has already been shown that the two planets may merge into one piece, as in Fig. 10, where the planet-wheel gears externally with one sun-wheel, and internally with the other.

But the train may be reduced still further, and yet retain the essential character of completeness in the same sense, though composed actually of but two toothed wheels. An instance of this is shown in Fig. 36, the annular planet being hung upon and carried by the pins of three cranks, c, c, c, which are all equal and parallel to the virtual train-arm, T. These cranks turning about fixed axes, communicate to f a motion of circular translation, which is the resultant of a revolution, v', about the axis of F in one direction, and a rotation, v, at the same rate in the opposite direction about its own axis, as has been already explained. The cranks then supply the place of a fixed sun-wheel and a planet of equal size, with an intermediate idler for reversing the, direction of the rotation of the planet; and the velocity of F is

V'= v'(1 - f/F).

A modification of this train better suited for practical use is shown in Fig. 37, in which the sun-wheel, instead of the planet, is annular, and the latter is carried by the two eccentrics, E, E, whose throw is equal to the difference between the diameters of the two pitch circles; these eccentrics must, of course, be driven in the same direction and at equal speeds, like the cranks in Fig. 36.

PLANETARY WHEEL TRAINS.

A curious arrangement of pin-gearing is shown in Fig. 38: in this case the diameter of the pinion is half that of the annular wheel, and the latter being the driver, the elementary hypocycloidal faces of its teeth are diameters of its pitch circle; the derived working tooth-outlines for pins of sensible diameter are parallels to these diameters, of which fact advantage is taken to make the pins turn in blocks which slide in straight slots as shown. The formula is the same as that for Fig. 36, viz.:

V' = v'(1 - f/F),

which, since f = 2F, reduces to V' = -v'.

Of the same general nature is the combination known as the "Epicycloidal Multiplying Gear" of Elihu Galloway, represented in Fig. 39. Upon examination it will be seen, although we are not aware that attention has previously been called to the fact, that this differs from the ordinary forms of "pin gearing" only in this particular, viz., that the elementary tooth of the driver consists of a complete branch, instead of a comparatively small part of the hypocycloid traced by rolling the smaller pitch-circle within the larger. It is self-evident that the hypocycloid must return into itself at the point of beginning, without crossing: each branch, then, must subtend an aliquot part of the circumference, and can be traced also by another and a smaller describing circle, whose diameter therefore must be an aliquot part of the diameter of the outer pitch-circle; and since this last must be equal to the sum of the diameters of the two describing circles, it follows that the radii of the pitch circles must be to each other in the ratio of two successive integers; and this is also the ratio of the number of pins to that of the epicycloidal branches.

Thus in Fig. 39, the diameters of the two pitch circles are to each other as 4 to 5; the hypocycloid has 5 branches, and 4 pins are used. These pins must in practice have a sensible diameter, and in order to reduce the friction this diameter is made large, and the pins themselves are in the form of rollers. The original hypocycloid is shown in dotted line, the working curve being at a constant normal distance from it equal to the radius of the roller; this forms a sort of frame or yoke, which is hung upon cranks as in Figs. 36 and 38. The expression for the velocity ratio is the same as in the preceding case:

V¹ = v'(1 - f/F); which in Fig. 39 gives

V¹ = v'(1 - 5/4)= -¼v':

the planet wheel, or epicycloidal yoke, then, has the higher speed, so that if it be desired to "gear up," and drive the propeller faster than the engine goes (and this, we believe, was the purpose of the inventor), the pin-wheel must be made the driver; which is the reverse of advantageous in respect to the relative amounts of approaching and receding action.

In Figs. 40 and 41 are given the skeletons of Galloway's device for ratios of 3:4 and 2:3 respectively, the former having four branches and three pins, the latter three branches and two pins. Following the analogy, it would seem that the next step should be to employ two branches with only one pin; but the rectilinear hypocycloid of Fig. 38 is a complete diameter, and the second branch is identical with the first; the straight tooth, then, could theoretically drive the pin half way round, but upon its reaching the center of the outer wheel, the driving action would cease: this renders it necessary to employ two pins and two