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.