It has long been known that pendulums are affected by variations of density of the air as well as of temperature, though in a much less degreein fact, so little as to be immaterial, except in the bets clocks, where all the other errors are reduced to a maximum. An increase of density of the air is equivalent to a diminution of the specific gravity while the inertia remains the same. And as the velocity of the pendulum varies directly as the force of gravity and inversely as the inertia, an increase of density must diminish the velocity of increase the time. The late Francis Baily, P.R.A.S., also found from some elaborate experiments (See Phil. Trans. of 1832) that swinging pendulums carry so much air with them as to affect their specific gravity much beyond that due to the more difference of stationary weight, and that this also varies with their shapea rod with a flat elliptical section dragging more air with it than a thicker round one (which is not what one would expect), though a lens-shaped bob was less affected than spherical one of the same diameter, which or course is much heavier. The frictional effect of the air is necessarily greater with its increased density, and that diminishes the arc. In the R.A.S Memoirs of 1853 Mr Bloxam remarked also that the current produced in the descent of the pendulum goes along with it in ascending, and therefore does not retard the ascent as much as it did the descent, and therefore the two effects do not counteract each other as Baily assumed that they did. He also found the circular error always less than its theoretical value, and considered that this was due to the resistance of the air. The conclusions which were arrived at by several eminent clockmakers as to the effect of the pendulum spring on the circular error about 40 years ago were evidently erroneous, and the effect due to other causes.
It appears from further investigation of the subject in several papers in the R.A.S. Notices of 1872 and 1873, that the barometrical error also varies with the nature of the escapement, and (as Baily had before concluded from calculation) with the arc of the pendulum, son that it can hardly be determined for any particular clock a priori, except by inference form a similar one. The barometrical error of an ordinary astronomical clock with a dead escapement was said to be a loss of nearly a second a day for an inch rise of barometer, but with a gravity escapement and a very heavy pendulum not more than ·3 second. Dr Robinson of Armagh (see R.A.S· Mem., vol. v.) suggested the addition of a pair of barometer tubes to the sides of the pendulum, with a bulb at the bottom, and such a diameter of tube as would allow a sufficient quantity of mercury to be transposed to the top by the expansion under heat, to balance the direct effect of the heat upon the pendulum, But it is not necessary to have two tubes. In a paper in the R.A.S. Notices of January 1873 Mr. Denison (now Sir E. Beckett) gave the calculations requisite for the barometrical compensation of pendulum of various lengths and weights, the principle of which is just the same as that above given for regulating a pendulum by adding small weights near the middle of its length. The formula is also given at p. 69 of the sixth edition of his Rudimentary Treatise on Clocks. A barometrical correction of a different kind has been applied to the standard clock at Greenwich. An independent barometer is made to raise or lower a magnet so as to bring it into more or less action on the pendulum and so to accelerate or retard it. But we do not see why that should be better than the barometer tube attached to the pendulum. The necessity for this correction seems to be obviated altogether by giving the pendulum a sufficient arc of vibration. Baily calculated that if the arc (reckoned from 0) is about 2º 45' the barometrical error will be self-corrected. And it is remarkable that the Westminster clock pendulum, to which that large arc was given for other reasons, appears to be free from any barometric error, after trying the results of the daily rate as automatically recorded at Greenwich for the whole of the year 1872. We shall see presently that all the escapement errors of clocks are represented by fractions which have the square or the cube of the arc in the denominator and therefore if the arc can be increased and kept constant without any objectionable increase of force and friction, this is an additional reason for preferring a large arc to a small one, though that is contrary to the usual practice in astronomical clocks.
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