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.

The result then is the same as though the periphery of E were formed into a fixed spurwheel, A, and another, a, of the same size, secured on a shaft, B, the two being connected by the three equal wheels, L, M, N. It need hardly be stated that instead of the eccentric, E, a stationary crank similar and equal to B G may be used, should it be found better suited to the circumstances of the case.

It is possible also to apply the planetary principle to mechanism composed partially of racks; in fact, a rack is merely a wheel of prodigious size - the limiting case, just as a right line is a circle of infinite radius. A very neat application of this principle is found in Villa's Pantograph, of which a full description and illustration was given in SCIENTIFIC AMERICAN SUPPLEMENT, No. 424; the racks, moving side by side, are the sun-wheels, and the planet-wheels are the pinions, carried by the traveling socket, by which the motion of one rack is transmitted to the other.

Thus far attention has been called only to combinations of circular wheels. In these the velocity ratios are constant, if we except the cases in which two independent trains converge, the two sun-wheels, or one of them and the train-arm, being driven separately - and even in those, a variable motion of the ultimate follower is obtained only by varying the speed of one or both drivers. It is not, however, necessary to employ circular wheels exclusively or even at all; wheels of other forms are capable of acting together in the relation of sun and planet, and in this way a varying velocity ratio may be produced even with a fixed sun-wheel and a single driver. We have not found, in the works of any previous writer, any intimation that noncircular wheels have ever been thus combined; and we propose in the following article to illustrate some curious results which may be thus obtained.