The shaking of heavy cylinders, grindstones, millstones, etc., etc., when out of balance, depends upon and is caused by the irregularity of centrifugal force on the opposite sides of the wheel. That is, if the centrifugal force, or the sum of the tendencies to fly off in tangents to their arcs of revolution, possessed by all the partielesf * tte lighter side, be represented by a, and that on the heavier side by 6, the power to which the shaking is due will be represented by the expression b — a. Suppose, in a given case, b equals a pressure of 4,000 pounds.and a equals 3,750, the force with which the cylinder would be shaken would be 250 pounds moving from one side of the axis to the other, and a vibration would be produced upon each interchange of place between the heavier and lighter sides, having for one of its elements of measure the ratio existing between the difference of the centrifugal forces of the opposite sides, and the weight of the cylinder. If all the supports of such unbalanced wheels or cylinders were perfectly rigid and inelastic, no vibration would be felt, but the strain upon the axle journals and boxes would remain undiminished, so that greater strength of parts would be necessary in order to avoid breakages.and loss of power would accrue. In order that a cylinder may be perfectly balanced, when in motion, it is necessary that the sum of the moments of the particles on one side (that is the sum of their several weights multiplied into their several velocities), should exactly equal the sum of the moments of the particles on the other side, when the cylinder is running at any speed. This can never be more than approximately attained in practice. The writer, who has had considerable experience in balancing heavy cylinders, designed to run at high speeds, has found the following method the best : The cylinder being keyed upon its axle, as it is intended to run, is lifted by a tackle or crane, and lowered, so that each of its journals rests upon a stout steel straight-edge placed so that its upper surface is exactly level, and parallel with its fellow. These straight-edges should not only be so rigid as to suffer no sensible deflection from the weight of the cylinder to be balanced, but they should be very hard and as smooth as it is possible to make them; and great care should be taken to keep them free from indentations. The journals of the cylinder must also be round and polished in order to secure delicate action. All the friction is thus converted into rolling friction, and this is reduced to a minimum. The cylinder can now be loaded on its lighter side, or vine versa, until it will remain perfectly motionless when stopped in any part of its revolution. We have balanced heavy cylinders in this way until they would revolve by placing upon either side one twenty-thousandth of their weight. The method sometimes practiced of suspending a cylinder by the centers of the journals is not sufficiently delicate. Either the lathe centers will be so forced in as to greatly increase friction, or there will be some play, so that the center of suspension will be outside the center of the axle. The latter makes no difference where the cylinder can roll, as on the steel straight-edges, but when suspended from a point, it will certainly defeat the attainment of any great degree of accuracy.