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 slots, but it is not essential that the latter should be perpendicular to each other.

In these last arrangements, the forms of the parts are so different from those of ordinary wheels, that the true nature of the combinations is at least partially disguised. But it may be still more completely hidden, as for instance in the common elliptic trammel, Fig. 42. The slotted cross is here fixed, and the pins, R and P, sliding respectively in the vertical and horizontal lines, control the motion of the bar which carries the pencil, S. At first glance there would seem to be nothing here resembling wheel works. But if we describe a circle upon R P as a diameter, its circumference will always pass through C, because R C P is a right angle, and the instantaneous axis of the bar being at the intersection O of a vertical line through P, with a horizontal line through R, will also lie upon this circumference. Again, since O is diametrically opposite to C, we have C O = R P, whence a circle about center C with radius R P will also pass through O, which therefore is the point of contact of these two circles.

It will now be seen that the motion of the bar is the same as though carried by the inner circle while rolling within the outer one, the latter being fixed; the points P and R describing the diameters L M and K N, the point D a circle, and S an ellipse; C D being the train-arm. The distance R P being always the diameter of one circle and the radius of the other, the sizes of the wheels can be in effect varied by altering that distance.

Thus we see that this combination is virtually the same in its action as the one shown in Fig. 43, known as Suardi's Geometrical Pen. In this particular case the diameter of a is half of that of A; these wheels are connected by the idler, E, which merely reverses the direction without affecting the velocity of a's rotation. The working train arm is jointed so as to pivot about the axis of E, and may be clamped at any angle within its range, thus changing the length of the virtual train arm, C D. The bar being fixed to a, then, moves as though carried by the wheel, a¹, rolling within A¹; the radius of a¹ being C D, and that of A¹ twice as great.