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scientific prophecy were founded upon this knowledge, and if at present we cannot help wondering at the precise anticipations of the nautical almanack, we may imagine the wonder excited by such predictions in early times.

Combined Periodic Changes.

We shall seldom find a body subject to a single periodic variation, and free from other disturbances. We may expect the periodic variation itself to undergo variation, which may possibly be secular, but is more likely to prove periodic; nor is there any limit to the complication of periods beyond periods, or periods within periods, which may ultimately be disclosed. In studying a phenomenon of rhythmical character we have a succession of questions to ask. Is the periodic variation uniform? If not, is the change uniform? If not, is the change itself periodic? Is that new period uniform, or subject to any other change, or not? and so on ad infinitum.

In some cases there may be many distinct causes of periodic variations, and according to the principle of the superposition of small effects, to be afterwards considered, these periodic effects will be simply added together, or at least approximately so, and the joint result may present a very complicated subject of investigation. The tides of the ocean consist of a series of superimposed undulations. Not only are there the ordinary semi-diurnal tides caused by sun and moon, but a series of minor tides, such as the lunar diurnal, the solar diurnal, the lunar monthly, the lunar fortnightly, the solar annual and solar semi-annual are gradually being disentangled by the labours of Sir W. Thomson, Professor Haughton and others.

Variable stars present interesting periodic phenomena; while some stars, & Cephei for instance, are subject to very regular variations, others, like Mira Ceti, are less constant in the degrees of brilliancy which they attain or the rapidity of the changes, possibly on account of some longer periodic variation.1 The star B Lyræ presents a double maximum and minimum in each of its periods of nearly 13 days, and since the discovery of this variation the period

1 Herschel's Outlines of Astronomy, 4th edit. pp. 555—557.

"At first

in a period has probably been on the increase. the variability was more rapid, then it became gradually slower; and this decrease in the length of time reached its limit between the years 1840 and 1844. During that time its period was nearly invariable; at present it is again decidedly on the decrease.1 The tracing out of such complicated variations presents an unlimited field for interesting investigation. The number of such variable stars. already known is considerable, and there is no reason to suppose that any appreciable fraction of the whole. number has yet been detected.

Principle of Forced Vibrations.

Investigations of the connection of periodic causes and effects rest upon a principle, which has been demonstrated by Sir John Herschel for some special cases, and clearly explained by him in several of his works.2 The principle may be formally stated in the following manner: "If one part of any system connected together either by material ties, or by the mutual attractions of its members, be continually maintained by any cause, whether inherent in the constitution of the system or external to it, in a state of regular periodic motion, that motion will be propagated throughout the whole system, and will give rise, in every member of it, and in every part of each member, to periodic movements executed in equal periods, with that to which they owe their origin, though not necessarily synchronous with them in their maxima and minima." The meaning of the proposition is that the effect of a periodic cause will be periodic, and will recur at intervals equal to those of the cause. Accordingly when we find two phenomena which do proceed, time after time, through changes of the same period, there is much probability that they are connected. In this manner, doubtless, Pliny correctly inferred that the cause of the tides lies in the sun and the moon, the intervals between successive high tides being equal to the intervals between the moon's

1 Humboldt's Cosmos (Bohn), vol. iii. p. 229.

2 Encyclopædia Metropolitana, art. Sound, § 323; Outlines of Astronomy, 4th edit., § 650. pp. 410, 487-88; Meteorology, Encyclopædia Britannica, Reprint, p. 197.

passage across the meridian. Kepler and Descartes too admitted the connection previous to Newton's demonstration of its precise nature. When Bradley discovered the apparent motion of the stars arising from the aberration of light, he was soon able to attribute it to the earth's annual motion, because it went through its phases in a

year.

The most beautiful instance of induction concerning periodic changes which can be cited, is the discovery of an eleven-year period in various meteorological phenomena. It would be difficult to mention any two things apparently more disconnected than the spots upon the sun and auroras. As long ago as 1826, Schwabe commenced a regular series of observations of the spots upon the sun, which has been continued to the present time, and he was able to show that at intervals of about eleven years the spots increased much in size and number. Hardly was this discovery made known, when Lamont pointed out a nearly equal period of variation in the declination of the magnetic needle. Magnetic storms or sudden disturbances of the needle were next shown to take place most frequently at the times when sun-spots were prevalent, and as auroras are generally coincident with magnetic storms, these phenomena were brought into the cycle. It has since been shown by Professor Piazzi Smyth and Mr. E. J. Stone, that the temperature of the earth's surface as indicated by sunken thermometers gives some evidence of a like period. The existence of a periodic cause having once been established, it is quite to be expected, according to the principle of forced vibrations, that its influence will be detected in all meteorological phenomena.

Integrated Variations.

In considering the various modes in which one effect may depend upon another, we must set in a distinct class those which arise from the accumulated effects of a constantly acting cause. When water runs out of a cistern, the velocity of motion depends, according to Torricelli's theorem, on the height of the surface of the water above the vent; but the amount of water which

leaves the cistern in a given time depends upon the aggregate result of that velocity, and is only to be ascertained by the mathematical process of integration. When one gravitating body falls towards another, the force of gravity varies according to the inverse square of the distance; to obtain the velocity produced we must integrate or sum the effects of that law; and to obtain the space passed over by the body in a given time, we must integrate again.

In periodic variations the same distinction must be drawn. The heating power of the sun's rays at any place on the earth varies every day with the height attained, and is greatest about noon; but the temperature of the air will not be greatest at the same time. This temperature is an integrated effect of the sun's heating power, and as long as the sun is able to give more heat to the air than the air loses in other ways, the temperature continues to rise, so that the maximum is deferred until about 3 P.M. Similarly the hottest day of the year falls, on an average, about one month later than the summer solstice, and all the seasons lag about a month behind the motions of the sun. In the case of the tides, too, the effect of the moon's attractive power is never greatest when the power is greatest; the effect always lags more or less behind the cause. Yet the intervals between successive tides are equal, in the absence of disturbance, to the intervals between the passages of the moon across the meridian. Thus the principle of forced vibrations holds true.

In periodic phenomena, however, curious results sometimes follow from the integration of effects. If we strike a pendulum, and then repeat the stroke time after time at the same part of the vibration, all the strokes concur in adding to the momentum, and we can thus increase the extent and violence of the vibrations to any degree. We can stop the pendulum again by strokes applied when it is moving in the opposite direction, and the effects being added together will soon bring it to rest. Now if we alter the intervals of the strokes so that each two successive strokes act in opposite manners they will neutralise each other, and the energy expended will be turned into heat or sound at the point of percussion. Similar effects

occur in all cases of rhythmical motion. If a musical note is sounded in a room containing a piano, the string corresponding to it will be thrown into vibration, because every successive stroke of the air-waves upon the string finds it in like position as regards the vibration, and thus adds to its energy of motion. But the other strings being incapable of vibrating with the same rapidity are struck at various points of their vibrations, and one stroke will soon be opposed by one contrary in effect. All phenomena of resonance arise from this coincidence in time of undulation. The air in a pipe closed at one end, and about 12 inches in length, is capable of vibrating 512 times in a second. If, then, the note C is sounded in front of the open end of the pipe, every successive vibration of the air is treasured up as it were in the motion of the air. In a pipe of different length the pulses of air would strike each other, and the mechanical energy being transmuted into heat would become no longer perceptible as sound.

Accumulated vibrations sometimes become so intense as to lead to unexpected results. A glass vessel if touched with a violin bow at a suitable point may be fractured with the violence of vibration. A suspension bridge may be broken down if a company of soldiers walk across it in steps the intervals of which agree with the vibrations of the bridge itself. But if they break the step or march in either quicker or slower pace, they may have no perceptible effect upon the bridge. In fact if the impulses communicated to any vibrating body are synchronous with its vibrations, the energy of those vibrations will be unlimited, and may fracture any body.

Let us now consider what will happen if the strokes be not exactly at the same intervals as the vibrations of the body, but, say, a little slower. Then a succession of strokes will meet the body in nearly but not quite the same position, and their efforts will be accumulated. Afterwards the strokes will begin to fall when the body is in the opposite phase. Imagine that one pendulum moving from one extreme point to another in a second, should be struck by another pendulum which makes 61 beats in a minute; then, if the pendulums commence together, they will at the end of 30 beats be moving in opposite directions. Hence whatever energy was communicated in the first

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