tbn are too complicated for ordinary use, except in bodies of certain simple and regular forms ; but they are fortunately not requisite in practice, because in all clock pendulums the centre of oscillation is only a short distance below the centre of gravity of the whole pendulum, and generally so near to the centre of gravity of the bob in fact a little above it that there is no difficulty in making a pendulum for any given time of vibration near enough to the proper length at once, and then adjusting it by screwing the bob up or down until it is found to vibrate in the proper time.
Revolving or Conical Pendulum.
Thus far we have been speaking of vibrating pendulums but now useu universally, in all but some interior foreign clocks, which have strings instead, is a thin and short spring, with one end let into the top of the pendulum, and the other screwed between two luiu me lujj vi uic peuuiuum, iinu uie inner screwed ueiwecn two chops ot metal with a pin through them, which rests firmly in n nick in the cock which carries the pendulum as shown in fig. 2 a little farther on ; and the steadiness of this cock, and its firm Qxing to a wall, are essential to the accurate performance of the rlnc V. Tlio thinner flip Qnrinrr flip 1^o<fnT. -rn^t-i.!*./! ff isMii*cr> if lilUS I UT We nave uccll suettmiig ui iuianii^ ^/CUILIUIUUO , L>uii the notice of pendulums would be incomplete without some allusion to revolving or conical pendulums, as they are called, because they describe a cone in revolving. Such pendulums are used where a continuous instead of an intermittent motion of the clock train is required, as in the clocks for keeping an equatorial telescope directed to a star, by driving it the opposite way to the motion of the earth, to whose axis the axis on which the telescope turns is made parallel. Clocks with such pendulums may also be used in bedrooms by persons who cannot bear the ticking of a common clock. The pendulum, instead of being hung by a flat spring, is hung by a thin piece of piano-forte wire ; and it should be under stood that it has no tendency to twist on its own axis, and so to twist off the wire, as may be apprehended ; in fact, it would require some extra force to make it twist, if it were wanted to do so. The time of revolution of such a pendulum may be easily ascertained as follows : Let Z be its length ; a the angle which it makes with the vertical axis of the cone which it describes ; w the angular velocity ; then the centrifugal force ? u> 2 ? sin. a ; and as this is the force which keeps the pendulum away from the vertical, it must balance the force which draws it to the vertical, which is g tan. a : and therefore / 2 ? the angular velocity, or the angle de- v/ i cos. 6 scribed in a second of time ; and the time of complete revolution through the angle 360 or 2w is^?2ir^/L2L ; that is to say, the time of revolution of a pendulum of any given length is less than the time of a double oscillation of the same pendulum, in the proportion of the cosine of the angle which it makes with the axis of revolution to unity.
A rotary pendulum is kept in motion by the train of the clock ending in a horizontal wheel with a vertical axis, from which pro jects an arm pressing against a spike at the bottom of the pendu lum ; and it has this disadvantage that any inequality in the force of the train, arising from variations of friction or any other cause, is immediately transmitted to the pendulum ; whereas it will be seen that in several kinds of escapements which can be applied to a vibrating pendulum, the variations of force can be rendered nearly or quiteinsensible. And it is a mistake to imagine that there is any self-correcting power in a conical pendulum analogous to that of the governor of a steam-engine ; for that apparatus, though it is a couple of conical pendulums, has also a communication by a system of levers with the valve which supplies the steam. _ The governor apparatus has itself been applied to telescope-driving clocks, with a lever ending in a spring which acts by friction on some revolving plate in the clock, increasing the friction, and so diminishing the force as the balls of the governor fly out farther under any increase in the force . And with the addition of some connection with the hand of the observer, by which the action can be farther moderated, the motion can be made sufficiently uniform for that purpose.
Various other contrivances have been invented for producing a continuous clock-motion. The great equatorial telescope at Green wich is kept in motion by a kind of water clock called in books on hydrostatics Barker s Mill, in which two horizontal pipes branching out from a vertical tubular axis have each a hole near their ends on opposite sides, from which water flows, being poured constantly into the tubular axis, which revolves on a pivot, resistance of the air to the water issuing from the holes drives the mill round, and there are means of regulating it. Another plan is to connect a clock train having a vibrating pendulum with another clock havin" a conical pendulum by one of the lower wheels in the train, with a spring connection ; the telescope is driven by the revolving clock train, and the other pendulum keeps it sufficiently in order, though allowing it to expatiate enough for each beat ot the pendulum The more complicated plan of Wagner of I ans described in Sir E. Beckett s Rudimentary Treatise on Clocks and JFatches and Bdls does not appear to have ever come i and therefore it is now omitted.
Pendulum Suspension.
The suspension of the pendulum on what are called like those of a scale-beam, has often been advocated. But it may do well enough for short experiments, in which th iiii.iv in me LUIIV nuivii lituTieH uiu pciKimuiii as snown in rig. z a little farther on ; and the steadiness of this cock, and its firm Gxing to a wall, are essential to the accurate performance of the clock. The thinner the spring the better ; provided, of course, it is strong enough to carry the pendulum without being bent beyond its elasticity, or bent short ; not that there is much risk of that in practice. Pendulum springs are much oftener too thick than too thin ; and it is worth notice that, independently of their greater effect on the natural time of vibration of the pendulum, thick and narrow springs are more liable to break than thin nnd broad ones of the same strength. It is of great importance that the spring should be of uniform thickness throughout its breadth ; and the bottom of the chops which carry it should be exactly horizontal ; otherwise the pendulum will swing with a twist, as they may he often seen to do in ill-made clocks. If the bottom of the chops is left sharp, where they clip the spring, it is very likely to break there ; and therefore the sharp edges should be taken off.
The bob of the pendulum used to be generally made in the shape of a lens, with a view to its passing through the air with the least resistance. But after the importance of making the bob heavy was discovered, it became almost necessary to adopt a form of more solid content in proportion to its surface. A sphere has beOn occa sionally used, but it is not a good shape, because a slight error in the place of the hole for the rod may make a serious difference in the amount of weight on each side, and give the pendulum a ten dency to twist in motion. The mercurial jar pendulum suggested the cylindrical form, which is now generally adopted for astronomical clocks, and in the best turret clocks, with a round top to prevent any bits of mortar or dirt falling and resting upon it, which would alter the time ; it also looks better than a flat-topped cylinder. There is no rule to be given for the weight of pendulums. It will be shown hereafter that, whatever escapement may be used, the errors due to any variation of force are expressed in fractions which inva riably have the weight and the length of the pendulum in the denominator, though some kind of escapements require a heavy pendulum to correct their errors much less than others. And as a heavy pendulum requires very little more force to keep it in motion than a light one, being less affected by the resistance of the air, we may almost say that the heavier and longer a pendulum can be made the better ; at any rate, the only limit is one of convenience ; for instance, it would obviously be inconvenient to put a large pen dulum of 100 lb weight in the case of an astronomical or common house clock. It may perhaps be laid down as a rule, that no astronomical clock or regulator (as they are also called) will go as well as is now expected of such clocks with a pendulum of less than 28 lb weight, and no turret clock with less than 1 cwt. Long pendulums are generally made with heavier bobs than short ones ; and such a clock as that of the Houses of Par liament, with a two-seconds pendulum of 6 cwt., ought to go 44 times as well as a small turret clock with a one-second pendulum of 60 lb. Pendulums longer than 14 feet (2 seconds) are incon venient, liable to be disturbed by wind, and expensive to compen sate, and they are now quite disused, and most or all of the old ones removed, with their clocks, for better ones.
Pendulum Regulation.
The regulation of pendulums, or their exact adjustment to the proper length, is primarily effected by a nut on the end of the rod by which the bob can be screwed up or down. In the best clocks the rim of this nut is divided, with an index over it ; so the exact quantity of rise or fall, or the exact acceleration or retardation, may be known, the amount due to one turn ot tlic nut being previously ascertained. By the calculation used below for compensation of pendulums, it may be seen that if the J gtfl of the pendulum rod is I, and the breadth of one thread of the screw is called dl, then one turn of the nut will alte rate of the clock by 43200 y seconds a day ; which would be iust 80 seconds, if the pendulum rod is 45 inches long and the screw has 32 threads in the inch. To accelerate the clock the nut has always to be turned to the right, as it is called, and t But in astronomical and in large turret clocks, it i to avoid stopping, or in any way disturbing the pendulum ; and f the finer adjustments other methods of regu ation are adopted The best is that of fixing a collar, as shown in fig 2, capable o having very small weights laid upon it, half-way down th dulunf, this being the place where the addition of any smal 1 eig ht produces the greatest effect, and where, it may be added aiij moving of that weight up or down on the rod produces
