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HISTOBY OF PHYSICAL ASTRONOMY.

differ in all eases from the number of indefinitely small oscillations whichwould be performed in the same time. In every case of actual experimentthe time of a single oscillation will exceed a certain finite quantitydepending on the length of the pendulum and the intensity of the movingforce; but, the smaller the arc is, the more nearly will the time ofoscillation approach this quantity; which may, therefore, be considered asthe time‘corresponding to indefinitely small oscillations. Hence, in orderto reduce the experiments to an accordance with theory, the effect due tothe magnitude of the arc is computed, and then applied to the actual rateof oscillation. This is termed the correction for the amplitude of the arc.

The third correction which it is necessary to take into account, inconducting experiments with the pendulum, depends on the fact that theforce of gravity varies at different distances from the centre of the earth.Clairaut ’s theorem has reference to the variation of gravity at the surfaceonly of the terrestrial spheroid. It is necessary, therefore, to makeallowance for the altitude of the place where the experiments are made, bydiminishing the force indicated by the oscillations, in the ratio of thesquare of the distance from the earth’s centre.

Lastly the oscillations are supposed to take place in a vacuum. Inreality, however, they are performed in a medium of air. Newton firstconsidered, in accordance with mathematical principles, the effect producedon the motion of a pendulum by the resistance of the air. He hasdemonstrated, on the two hypotheses of the resistance being proportionalto the first and second powers of the velocity, that, when a pendulummakes very small oscillations in a cycloid, it does not experience anysensible retardation from this cause*. Poisson subsequently showed thatthis was also true for a pendulum oscillating in a circle when the arcs aresupposed very small f; in fact, although the resistance of the air opposes theaction of gravity when the pendulum is descending to a vertical position,and in consequence tends to prolong the time of oscillation, yet, when thependulum is ascending on the opposite side, the same cause conspires withgravity in destroying the velocity, and on this account tends to shortenthe time of oscillation. A compensation thus takes place between every twohalf oscillations, which restores the times of the whole oscillations to nearlya uniform state, rendering necessary only the correction due to theamplitude of the arc. It must be remarked, however, that this conclusionis founded on an imperfect conception of the nature of the resistanceoffered by the air; for it is assumed that the particles of the latter, as soonas impinged on, immediately afterwards cease to exercise any influenceupon the motion of the pendulum, either directly or by means of theagitation they excite among the surrounding particles of the medium.Newton, indeed, did not fail to perceive that this view of the question didnot accord with the real condition of nature, for he remarked that the pen-dulum while descending with accelerated velocity would continue to operateupon the particles of the medium after the first impact, and that the timeof oscillation would in consequence be prolonged, while, on the other hand,when the pendulum was ascending on the opposite side, the continualdiminution of the velocity would allow the particles to escape from a similarsuccession of impacts; and hence the effect of the resistance in the onehalf of the arc of oscillation would not be exactly compensated by the

* Princip., lib. ii. prop. 26 et 27.

f Journal de l’Ecole Polytechnique, tome vii.