*While the particle a moved to b, the par ticle b moved to c. In this time the velocity of the particle b, perpendicular to the plane of the paper, must have been diminished by an amount proportional to b E. A small circle containing a dot at its centre, and of a diameter proportional to b, E, indicates the magnitude and direction of the force which must have been applied to the particle as it moved from b to c.
*The force which acted on the particle c as it moved to d, and that which acted on the par ticle d as it moved to e, are represented in a similar manner.
"Owing to the rotation about the line A B, all particles on the right-hand side of the disc are upward through the plane of the paper; thus it follows that the particle e, in moving to the position f, must have acquired a velocity, directed vertically upward through the paper, proportional to G f. It must, therefore, have been acted upon by a force, proportional to G f, directed vertically upward through the paper. The forces acting on the particles f, g, h, can be determined in a similar manner.
*It is obvious that the velocity of the parti cle k, directed upward through the plane of the paper, is diminished as that particle moves to the position previously occupied by the par ticle 1. Consequently, it must have been acted upon by a force, of which the magnitude is de termined in the manner previously explained, acting downward through the plane of the paper. A circle, of which the diameter is pro portional to this force, while the cross at its centre represents the feathered end of an arrow directed downward through the paper, indicates the magnitude and direction of the force acting on the particle k as it moved to 1. The forces acting on the particles, 1, m, n, p, q, r, s, are de termined similarly, and represented by circles containing crosses, to indicate that the forces act downward through the plane of the paper.
"A glance at Fig. 1 shows that all forces acting on the part of the flywheel above the line a k, are directed upward through the plane of the paper; while all forces acting on the part of the flywheel below the line a k, are directed downward through the plane of the paper. All the forces acting above the line o k might be replaced by a single resultant force, acting upward through the paper at some point on the line C e; while all the forces act ing below the line a k might be replaced by a single resultant acting downward through the paper at some point in the line C p. These two
resultant forces, acting parallel to each other, but in opposite directions, constitute a couple, and produce a torque or turning moment about the line a k. Thus, in order to turn the re volving flywheel about the diameter e p, we must apply a torque which, if it acted on the stationary flywheel, would turn it about the per pendicular diameter a k. Conversely, if we apply a torque tending to turn the flywheel about a diameter a k. it will turn, not about a k (as might have been expected), but about the perpendicular diameter e p.
. "The torque necessary to deflect the flywheel might be produced by forces acting directly upon it, as for instance, by blowing air on the upper half of the flywheel from the back, and on the lower half from the front. Gener ally, however, it is more convenient to act on the axle, the end above the plane of the paper being urged in the direction C B, while the end below the plane of the paper is urged by an equal force, in the direction C A.
"Some further points should be noted. Any force acting to the right of the line A B is equal, both in magnitude and direction, to a corresponding force acting to the left of the same line. Consequently as the flywheel turns about the axis A B, no work will be performed by the forces producing this rotation. This follows from the circumstance that whereas one force acts in the direction of motion (so far as relates to rotation about the axis A B) the other equal force is opposed to that motion.
"The actual behavior of a gyroscope can now be easily understood. The flywheel a a (Fig. 2) having been set in rapid rotation in the di rection indicated by the arrow r, the frame carrying it is supported from a projection n at one end, on a pivot o. Instead of falling to the ground, as it would do if it were not ro tating, the gyroscope remains with its axis b horizontal; but the axis turns in a horizontal plane about the point of support o, in the direc tion indicated by the arrow s. The torque pro duced by the pull of gravity is easily seen to be that required to turn the flywheel a a about a vertical diameter in the direction mentioned. The fact that the flywheel, besides rotating about a vertical axis, also revolves in a circle about the point o as centre, is merely due to the circumstance that, under the conditions of the experiment, the rotation cannot occur with out the revolution.