360 GYROSCOPE direction in which the bottom of the disk i8 going. But if the axis be prolonged beyond the support, and the disk and ring slightly overpoised by a weight on the other side, then the disk always travels in the direction in which its top is going, and nearly all the phenomena are reversed. Many other curious results may be obtained ; it will here be added further only that the disk below the horizontal is always, and above it usually, slowly falling ; and that the orbital motion invariably takes place toward that side of the disk in which the force of the rotation about its own axis is most resisted or checked. For proof of this latter principle, let any small wheel be rotated, and while turning rub or seize it upon any side ; the rotation in this side being thus checked, and actually or in effect subtracted from, that in the opposite side preponderates, and the wheel is urged to- ward the side in which the checking occurs. Perhaps no completely satisfactory explanation of the phenomena can be given without employ- ing the language and processes of the higher mathematics. This has been done in a very complete manner by Gen. J. G. Barnard in a paper published in the " American Journal of Education " for June, 1857, and also published separately under the title " Analysis of Rotary Motion as applied to the Gyroscope " (New York, 1857). The following explanation pro- posed by Dr. Levi Reuben of New York is perhaps as satisfactory as it is possible to give without the aid of mathematics. There are two facts to be explained : support, and orbital movement, or travelling about the supporting point. For the first, suppose the disk composed of 1,000 equally heavy particles. "When it is set rotating and released, each of these parti- cles is, as a separate ball, acted on by two moving forces, that giving the rotation, and that of gravity ; but the whole is also held together by the constraining action of cohesion. Suppose that, when released, the axis points below the horizontal: gravity acts in vertical lines and equally on all the particles. Its di- rection and amount may be represented by equal short pendent threads dropping down from all the particles. If the particles be also supposed in a single plane, the extremities will all lie in a new plane, slightly without and be- low the plane of the disk, and parallel with it. The forces impressed in giving rotation upon the several particles of the disk will all point in its plane, being represented at any moment by tangents to the circles in which the several particles move, pointing in all directions, and varying in length from the axis, where this is zero, to the periphery, where it is a maximum. But the resultant movements or tendencies of the particles must all terminate in the exact plane in which the gravitative components were seen to terminate. Every particle thus acted upon, then, tends to go outward or forward into the new plane already referred to. The several pressures are to points scattered some- what widely in that plane ; but owing to the cohesion of all the particles, they are con- strained to move or press forward in a body. The effect is as if the whole disk were pulled outward and very slightly downward, while the pivot in the notch reacts or pulls in the oppo- site direction ; and the wheel is supported, in part, as if slung up by strings attached to its two faces and pulled in opposite directions. When the disk is above the horizontal, the new plane is behind or within it; it then pushes against the pivot, and this reacting, there oc- curs support by opposite pressures, instead of tractions. Thus we discover one reason why no material support is needed for the remote end of the axis ; while as a consequence of this view, if the axis be horizontal it must first sink slightly, yet it may be only imperceptibly, be- fore support can occur. This agrees entirely with the results of mathematical analysis. In the second place, why does the disk travel around the supporting point ? When not over- poised, gravitation acting downward, and rota- tion, in the ascending side of the disk, upward, the latter is in effect decomposed into a hori- zontal and a vertical component, the horizontal expressing itself in the pressure already referred to, the vertical being resisted or antagonized by the force of gravity; the result for each particle being the sum which the latter as a negative quantity would form with the former. In the ascending side, therefore, gravity over- balances, equals, or diminishes, according to the place of each particle, the rotative force of ascent acting upon it ; but to the vertical component of the rotative force of all the par- ticles in the descending side it adds alike a quantity of action equal to its own amount. Hence, the whole rotative force in the descend- ing half may be considered as increased, that in the ascending as diminished. There will be some point in the ascending half at which the vertical component of rotation equals gravity ; this will become in effect a point of rest, or of no action. This is then the point pierced by the resultant axis the point about which all the particles under the combined forces will tend to revolve: those in the ascending half starting with less radii to sweep round this point as a centre ; those in the descending start- ing with longer radii, and sweeping in longer curves about the same point. Thus the disk is continually carried to the side in which the action is most checked; and this constitutes the travelling movement. When overpoised on the opposite side, the action of gravity on the disk itself is upward, the axis acting as a lever, the support on which it rests as a ful- crum ; the rotative force of the descending par- ticles is now resisted by it ; and for a like rea- son the disk now moves toward its descend- ing side. When not overpoised, the travelling movement of the disk itself introduces a new element into the case, by resisting the rotating of particles in the upper half backward in the course of movement. This checks and dimin- ishes the action in the upper half of the disk,
