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.
In either instrument, the semi-major axis C X is equal to S R, and the semi-minor axis to S P.
The ellipse, then, is described by these arrangements because it is a special form of the epitrochoid; and various other epitrochoids may be traced with Suardi's pen by substituting other wheels, with different numbers of teeth, for a in Fig. 43.
Another disguised planetary arrangement is found in Oldham's coupling, Fig. 44. The two sections of shafting, A and B, have each a flange or collar forged or keyed upon them; and in each flange is planed a transverse groove. A third piece, C, equal in diameter to the flanges, is provided on each side with a tongue, fitted to slide in one of the grooves, and these tongues are at right angles to each other. The axes of A and B must be parallel, but need not coincide; and the result of this connection is that the two shafts will turn in the same direction at the same rate.
The fact that C in this arrangement is in reality a planetary wheel, will be perceived by the aid of the diagram, Fig. 45. Let C D be two pieces rotating about fixed parallel axes, each having a groove in which slides freely one of the arms, A C, A D, which are rigidly secured to each other at right angles.
The point C of the upper arm can at the instant move only in the direction C A; and the point D of the lower arm only in the direction A D, at the same instant; the instantaneous axis is therefore at the intersection, K, of perpendiculars to A C and A D, at the points C and D. C A D K being then a rectangle, A K and C D will be two diameters of a circle whose center, O, bisects C D; and K will also be the point of contact between this circle and another whose center is A, and radius A K = C D. If then we extend the arms so as to form the cross, P K, M N, and suppose this to be carried by the outer circle, f, rolling upon the inner one, F, its motion will be the same as that determined by the pieces, C D; and such a cross is identical with that formed by the tongues on the coupling-piece, C, of Fig. 44.
A O is the virtual train-arm; let the center, A, of the cross move to the position B, then since the angles A O B at the center, and A C B in the circumference, stand on the same arc, A B, the former is double the latter, showing that the cross revolves twice round the center O during each rotation of C; and since A C B = A D B, C and D rotate with equal velocities, and these rotations and the revolution about O have the same direction. While revolving, the cross rotates about its traveling center, A, in the opposite direction, the contact between the two circles being internal, and at a rate equal to that of the rotations of C and D, because the velocities of the axial and the orbital motion are to each other as f is to F, that is to say, as 1 is to 2. Since in the course of the revolution the points P and K must each coincide with C, and the points M and N with D, it follows that each tongue in Fig. 44 must slide in its groove a distance equal to twice that between the axes of the shafts.
Another example of a disguised planetary train is shown in Fig. 46. Let C be the center about which the train arm, T, revolves, and suppose it required that the distant shaft, B, carried by T, shall turn once backward for each forward revolution of the arm. E is a fixed eccentric of any convenient diameter, in the upper side of which is a pin, D. On the shaft, B, is keyed a crank, B G, equal in length to C D; and at any convenient point, H, on B C, or its prolongation, another crank, H F, equal also to C D, is provided with a bearing in the train-arm. The three crank pins, F, D, G, are connected by a rod, like the parallel rod of a locomotive; F D, D G, being respectively equal to H C, C B. Then, as the train-arm revolves, the three cranks must remain parallel to each other; but C D being fixed, the cranks, H F and B G, will remain always parallel to their original positions, thus receiving the required motion of circular translation.