To prove that zinc is not an efficient metal for compensation pendulums when employed for the exact measurement of time, a short calculation may be made—using the above conclusions—that a zinc rod one meter in length, after being subjected to a difference of temperature of 44° C. will alter its length 0.018 mm. after having been brought back to its initial degree. For a seconds pendulum with zinc compensation each of the zinc rods would require a length of 64.9 cm. With the above computations we get a difference in length of 0.0117 mm. at the same degree of temperature. Since a lengthening of the zinc rods without a suitable and contemporaneous expansion of the steel rods is synonymous with a shortening of the effectual pendulum length, we have, notwithstanding the compensation, a shortening of the pendulum length of 0.017 mm., which corresponds to a change in the daily rate of about 0.5 seconds.
This will sufficiently prove that zinc is unquestionably not suitable for extremely accurate compensation pendulums, and as neither is permanent under extremes of temperature the advantages of first cost and of correction of error appear to lie with the mercurial form.
The average mercurial compensation pendulums, on sale in the trade are often only partially compensated, as the mercury is nearly always deficient in quantity relatively, and not high enough in the jar to neutralize the action of the rigid metallic elements, composing the structure. The trouble generally is that the mercury forms too small a proportion of the total weight of the pendulum bob. There is a fundamental principle governing these compensating pendulums that has to be kept in mind, and that is that one of the compensating elements is expected to just undo what the other does and so establish through the medium of physical things the condition of the ideal pendulum, without weight or elements outside of the bob. As iron and mercury, for instance, have a pretty fixed relative expansive ratio, then whatever these ratios are after being found, must be maintained in the construction of the pendulum, or the results cannot be satisfactory.
First, there are 39.2 inches of rod of steel to hold the bob between the point of suspension and the center of oscillation, and it has been found that, constructively, in all the ordinary forms of these pendulums, the height of mercury in the bob cannot usually be less than 7.5 inches. Second, that in all seconds pendulums the length of the metal is fixed substantially, while the height of the mercury is a varying one, due to the differing weights of the jars, straps, etc.
Third, the mercury, at its minimum, cannot with jars of ordinary weight be less in height in the jar than 7.5 inches, to effectually counteract what the 39.2 inches of iron does in the way of expanding and contracting under the same exposure.
Whoever observes the great mass of pendulums of this description on sale and in use will find that the height of the mercury in the jar is not up to the amount given above for the least quantity that will serve under the most favorable circumstances of construction. The less weight there is in the rod, jar and frame, the less is the height of mercury which is required; but with most of the pendulums made in the present day for the market, the height given cannot be cut short without impairing the quality and efficiency of the compensation. Any amount less will have the effect of leaving the rigid metal in the ascendancy; or, in other words, the pendulum will be under compensated and leave the pendulum to feel heat and cold by raising and lowering the center of oscillation of the pendulum and hence only partly compensating. A jar with only six inches in height of mercury will in round numbers only correct the temperature error about six-sevenths.
Calculations of Weights.—As to how to calculate the amount of mercury required to compensate a seconds pendulum, the following explanation should make the matter clear to anyone having a fair knowledge of arithmetic only, though there are several points to be considered which render it a rather more complicated process than would appear at first sight.
1st. The expansion in length of steel and cast iron, as given in the tables (these tables differ somewhat in the various books), is respectively .0064 and .0066, while mercury expands .1 in bulk for the same increase of temperature. If the mercury were contained in a jar which itself had no expansion in diameter, then all its expansion would take place in height, and in round numbers it would expand sixteen times more than steel, and we should only require (neglecting at present the allowance to be explained under head 3) to make the height of the mercury—reckoned from the bottom of the jar (inside) to the middle of the column of mercury contained therein—one-sixteenth of the total length of the pendulum measured from the point of suspension to the bottom of the jar, assuming that the rod and the jar are both of steel, and that the center of oscillation is coincident with the center of the column of mercury. Practically in these pendulums, the center of oscillation is almost identical with the center of the bob.
2d. As we cannot obtain a jar having no expansion in diameter, we must allow for such expansion as follows, and as cast iron or steel jars of cylindrical shape are undoubtedly the best, we will consider that material and form only.
As above stated, cast iron expands .0066, so that if the original diameter of the jar be represented by 1, its expanded diameter will be 1.0066. Now the area of any circle varies as the square of its diameter, so that before and after its expansion the areas of the jar will be in the ratio of 1² to 1.0066²; that is, in the proportion of 1 to 1.013243; or in round numbers it will be one-seventy-sixth larger in area after expansion than before. It is evident that the mercury will then expand sideways, and that its vertical rise will be diminished to the same extent. Deduct, therefore, the one-seventy-sixth from its expansion in bulk (one-tenth) and we get one-eleventh (or more exactly .086757) remaining. This, then, is the actual vertical rise in the jar, and when compared with the expansion of steel in length it will be found to be about thirteen and a half times greater (more exactly 13.556).