HISTORY OF PHYSICAL ASTROXOMY.

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Clairaut ’s theorem enables us to ascertain the value of this element when' once we know the relative intensities of gravity in two different latitudes.The pendulum which Huygens had already applied so admirably to themeasurement of time derived from this circumstance a vast accession ofimportance in the estimation of astronomers. In order to understand howthe oscillations of a pendulum lead to a knowledge of the figure of theearth, it is necessary to obtain a clear view of the various circumstanceswhich affect the rate of oscillation. A brief notice of them here maynot perhaps be unacceptable to the reader, as they are inseparablyassociated with the history of the application of the pendulum to scientificpurposes.

When a pendulum oscillates in vacuo through very small arcs, the rateof oscillation will depend on the length of the pendulum and the intensityof the moving force. If the oscillations be supposed indefinitely small, asimple relation connects these three elements together, so that by meansof it we can ascertain the value of any one of them when the other two arealready known. Hence the intensity of the moving force may be readilyderived from the length of the pendulum and the rate of oscillation ; andif the latter element remain constant, or, in other words, if the pendulumcontinue to perform the same number of oscillations in the same time, thevariation of the moving force will be indicated by the different lengthswhich it will be found necessary to assign to the pendulum. On the otherhand, if the length of the pendulum be assumed to be constant, thevariation of the moving force will he indicated by the greater or lessquickness with which the oscillations are performed. Hence arise twodistinct methods of comparing the different intensities of a force by meansof experiments with the pendulum. In the one case, the variation of forceis thrown upon the length of the pendulum; in the other, it is thrownupon the rate of oscillation. Both these methods have been employed irj.determining the variation of gravity at the earth’s surface.

The preceding remarks have reference to the purely mathematicaltheory of the pendulum, and therefore do not take cognizance of thosedisturbing causes which in all [eases complicate physical inquiries. It isnecessary then to investigate the effects produced by these disturbingcauses, so that by subducting them from the phenomenon we may arriveat the abstract conditions which form the basis of our reasoning. , Hencearise various corrections which it is necessary to take into account beforethe experiments with the pendulum can be made available for the purposeof determining the ellipticity of the earth.

In the first place, the theory of oscillation above stated supposes that, inall experiments with the same pendulum, its length remains constant.This, however, is a condition which cannot obtain in nature, for thefluctuations of temperature will cause the pendulum continually to vary inlength, and the effect of this variation will manifestly be to disturb therate of oscillation. In order then to render the theory applicable to theactual results of experiment, a certain standard of temperature is assumed,and the effect due to the deviation from this standard as indicated by thethermometer is then computed and applied to the actual rate of oscillation.This is termed the correction for temperature.

Again, the oscillations of the pendulum are supposed to be indefinitelysmall. In reality, however, they all possess a finite magnitude, and alsovary continually from one oscillation to another. Both these circumstanceswill cause the number of oscillations actually performed in a given time to