periodic likewise; nor is there any limit to the complication of periods beyond periods, or periods within periods, which may ultimately be disclosed. In studying, then, 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 periodie? Is that new period uniform, or subject to any other change, or not? and so on ad infinitum.

In some cases there distinct causes of may be many periodic variations, and according to the principle of the superposition of small effects, to be afterwards more fully 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. Thus the tides of the ocean consist of a series of superimposed undulations, of which the number and character have by no means been determined as yet. Not only are there the ordinary and very obvious 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 and others r.

Variable stars present very interesting periodic phenorena; while some stars, & Cephei for instance are jeet to very regular and equal variations, others, like Ceti, are less constant in the degrees of brilliancy w they attain or the rapidity of the changes, posaccount of some much longer periodic variations.

3 Lyra presents a double maximum and in each of its periods of nearly 13 days, and of this variation the period in a period www.ery ween on the increase. At first the varia

Assertion Report,' 1870, p. 120.

mes of Astronomy,' 4th edit. pp. 555–557.

bility 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 t." It is evident that the tracing out of such complicated variations presents an almost 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.

All investigations of the connection of periodic causes and effects rest upon a most important and general principle, which has been demonstrated by Sir John Herschel for some special cases, and clearly explained by him in several of his works". 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 systems, and will give rise, in every member of it, and in every part of each member, to periodic movements executed in equal period, 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

t Humboldt's 'Cosmos' (Bohn), vol. iii. p. 229.

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

P. 137.

cause. Accordingly whenever we find any two phenomena which do proceed, time after time, through changes of exactly the same period, there is much probability that they are connected. It was in this manner, doubtless, that Pliny correctly conjectured that the cause of the tides lay in the sun and moon, the intervals between successive high tides being equal to the intervals between the moon's 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 all its phases in exactly a year.

The most extensive and beautiful instance of induction concerning periodic changes which can be cited, is that of the discovery of an eleven-year period in various meteorological and astronomical phenomena. It would be difficult mention any two things apparently more disconnected han the spots upon the sun and auroras. As long ago as 1526, Schwabe, of Dessau, commenced a regular series of -xrvations of the spots upon the sun, which has been ...ed to the present time, and he was able to show ...ntervals of about eleven years the spots increased size and number. Hardly was this discovery 1. than Dr. Lamont pointed out a nearly equal ariation in the magnetic needle as regards The occasional magnetic storms or sudden aces of the needle were next shown to --quently at the times when sun spots auroras are generally coincident these strange phenomena were It has since been shown by

Quetelet, 'Sur la Physique du Globe,'

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 more or less considerable in all meteorological phenomena.

Perhaps the most mysterious part of these investigations is that which refers the phenomena to the planetary configurations as an ulterior cause. Professor Balfour Stewart, with Messrs. Warren de la Rue and Loewy, by laborious researches discovered a periodic change of 584 days in the sun spots, coincident with changes in the relative positions of the Earth, Jupiter, and Venus. It has since been rendered probable by the researches of Dr. Kirkwood and others, that Schwabe's eleven-year period is due to the action of Mercury. Several other periods of more or less importance have been supposed to exist, but the subject is yet open to much more inquiry.

Integrated Variations.

In considering the infinite variety of modes in which one effect may depend upon another, we must set apart 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 any given time, we must again integrate with regard to the variable velocity.

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 it does not follow that the temperature of the air is 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 any other way, 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 sun's or 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 the successive tides are exactly equal, in the absence of disturbance, to the intervals between the passage of the sun or moon across the meridian. Thus the principle of forced vibrations holds true of all such

cases.

In periodic phenomena, however, very curious results will sometimes follow from the integration of effects. If we strike a pendulum, and then repeat the stroke time after time when it is in the same part of the vibration, every stroke concurs with every other one 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 successive effects being added together will soon bring it to rest. Now if we