When the vibrations of the balance are at the lowest point, the resistance of the pendulum spring is at the least ; but the more it is bent or unbent, the greater is the resistance ; consequently, when at the height of the wedge or tooth, it is greater than wheo the tooth first begins to act. Two or three different curves for this purpose have been imagined ; one approaching nearly to a right line, which is supposed to give the wheel time to acquire a velocity during the passing of two-thirds of the curve, and the least resistance of the spring, by which the other third more readily overcomes, when the resistance to it is at the greatest. This has been thought to give a greater extent to the arc of vibration, and has been adopted by the French artists. Another curve, where equal spaces make the balance describe equal portions of a circle, is thought to give the least wearing to the edges of the cylinder, and is that which is practised by our 'scapement makers. Arguments equally good for either, it appears, might be given.
The weight and diameter of the balance, are circum stances very materially connected with the wearing on the cylinder edges. Whatever will prevent this wearing, should be carefully attended to. When the diameter is large, the balance must of consequence he less heavy ; a sort of sluggishness in its motion takes place, the pendu lum spring making great resistance to the teeth passing the cylinder edges, and causing wearing to go rapidly on. On the contrary, when the diameter is small, and the weight at a proper medium, there is an alertness in the vibration ; the momentum of the balance has such force over the pendulum spring, that it allows the teeth to pass the edges quickly ; and hence there is a less tendency to wear them. The diameter of the balance should be less than that in a verge watch of equal size, nor should it be heavier than just not to allow setting, unless where a going in time of winding is used. The cylinder 'scape ment, on the whole, must be allowed to be a very excel lent one ; and where care is taken to have it made as it ought to be, such watches will give very good perfor mance. Provision for oil on the cylinder should be made as ample as can be admitted ; that is, the part where the tooth acts, should be as distant from the notch where the wheel bottom passes as possible, and at the same time more distant from the upper copper plug ; the lower notch should not be longer than to give freedom to the wheel bottom to pass easily. When they are made long, which they frequently are, the cylinder will break there, if the watch receive a slight shock from falling. The hcting
part of the tooth, as has already been noticed, should not be too thin, nor the stems too short. If the diameter of the balance is too great, any addition of motive force will make the watch go slow ; if too little, the watch will go fast ; and if of a proper weight and diameter, any addi tion of motive force will make no change on the time keeping. We have made the motive force more than double, and no change took place ; the pendulum spring no doubt had its share in keeping up this uniformity. Balances whose diameters are rather small, will have a natural tendency to cross farther, that is, the arcs of vibra tion will be greater than where the diameters are great. Their weight will be in the ratio of the squares of their diameters ; from which it follows, that if the balance is taken away from a watch which has been regulated, and another put in its place, having the diameter only one half of the former, before the watch could be regulated with the same pendulum spring, the balance would require to be four times heavier than the first. One way of estimat ing the force of a body in motion, is to multiply the mass by the velocity. Let us then calculate the respective forces of two balances, whose diameters are to one another as two to four. The radii in this case express the velo city. According to this principle, we shall have for the small balance two for the radius, multiplied by eight of the mass, equal to sixteen ; and for the great one, four of the radius by two of the mass, equal to eight; sixteen and eight are then the products of the mass by the velocities ; consequently they express the force from the centre of percussion of each balance ; and as it is double in the small one, it is evident that the arcs of vibration will be greater, having the faculty of overcoming easily any re sistance opposed to it by the pendulum spring, without requiring any additional motive lorce.
Let us take an example done in another way, which is the square of the product of the diameter multiplied by the velocity or number of degrees in the vibration, and this again multiplied by the mass or weight, so as to com pare the relative momentum of two balances of different diameters, Stc. Suppose one balance to be .8 of an inch in diameter, the degrees of vibration 240, and the weight eight grains ; the other .7 of an inch in diameter, the arc of vibration 280°, and the weight 10 grains 540 x .8 = 192 x 192 = 36764 X 8 = 294112.