x --- (.0000904 - .0000066) = .0000066 X 39.14, whence x = 6¼ in. 2
It must, however, be remembered that the glass jar has some weight and that it does not rise by anything like the amount of the mercury. This tends to keep the centre of gravity down. So that the height of mercury of 6¼ in. will not be sufficient to effect the compensation, and about 6¾ to 7 in. will be required. Some authors specify 7 in.; this is when the diameter of the jar is small. A certain amount of negative compensation must also be deducted to allow for the changes of temperature in the air, as will presently be seen; this amounts in the case of mercury to about 1/5 in.
In consequence of the complication of all these calculations it is usual to allow about 6¾ to 7 in. of mercury in the glass vessel and to adjust the exact amount of mercury by trial.
Another very good form of mercurial pendulum was proposed by E. J. Dent; it consists of a cast-iron jar into the top of which the steel pendulum rod is screwed, having its end plunged into the mercury contained in the jar. By this means the mercury, jar and rod rapidly acquire the same temperature. This pendulum is less likely to break than the form just described. The depth of mercury required in an iron jar is stated by Lord Grimthorpe to be 8½ to 9 in. The reason why it is greater than it is when a glass jar is employed is that iron has a larger coefficient of expansion than glass, and that it is also heavier. In all cases, however, of mercury pendulums experiment seems to be the only ultimate test of the quantity of mercury required, for the results are so complicated by the behaviour of the oil and the barometric errors that at its best the regulation of a clock can only be ultimately a matter of scientifically guided compromise. A small amount of compensation of a purely experimental character is also allowed to compensate the changes which temperature effects on the suspension spring. This is sometimes made as much as 1/6 of the length correction.
As an alternative to the mercurial pendulum other systems have been employed. The "gridiron" pendulum consists of a group of alternate rods of steel and brass, so arranged that the expansion of the brass acts upwards and counteracts that of the steel downwards. It was invented in 1726 by John Harrison. Assuming that 9 rods are used--5 of steel and 4 of brass--their lengths may be as follows from pin to pin:--Centre steel rod 31.5 in.; 2 steel rods next the centre 24.5 in.; 2 steel rods farthest from centre 29.5 in.; from the lower end of outside steel rods to centre of bob 3 in.; total 89.5 in. Of the 4 brass rods the 2 outside ones are 26.87 in.; and the two inside ones 22.25 in.; total 49.12 in. Thus the expansion of 88½ in. of steel is counteracted by the expansion of 49 1/8 in. of brass. Everything depends, however, on the expansion coefficient of the steel and brass employed, the requirement in every case being that of total lengths of the brass and iron should be in proportion to the linear coefficients of expansion of those metals. The above figures are for a very soft brass and steel. Thos. Reid, with more ordinary steel and brass, prescribed a ratio of 112 to 71, Lord Grimthorpe a ratio of 100 to 61. It is absolutely necessary to put the actual rods to be used for making the pendulum in a hot water bath, and measure their expansions with a microscope.
