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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 planetary body is carried round by a train-arm, and its rotation about its axis is usually given it by a train of gearing, the inner or central wheel of which is stationary, being fastened to the fixed frame supporting the whole.
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.
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 inclination of her axis to the plane of her orbit, and her consequent libration in latitude. Her varying angular velocity, and consequent libration in longitude. The mechanism consists of a train-arm, T, which turns upon the vertical pivot, P, fixed in the stand.
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 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.
Now as the train-arm, T, is carried round, the cranks, B and C, will turn in their bearings; but by their connection with A, they are compelled to remain always parallel to themselves, and thus the axis of the moon receives a motion of translation.
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.
This may be done in two ways; each of the two independent trains may drive one sun-wheel, thus determining the motion of the train-arm; or, the train-arm may be driven by one of them, and the first sun-wheel by the other; then the motion of the second sun-wheel is the resultant.
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