CHAPTER XX.

METHOD OF VARIATIONS.

Experiments may be of two kinds, experiments of simple fact, and experiments of quantity. In the first class of experiments we combine certain conditions, and wish to ascertain whether or not a certain effect of any quantity exists. Hooke wished to ascertain whether or not there was any difference in the force of gravity at the top and bottom of St. Paul’s Cathedral. The chemist continually performs analyses for the purpose of ascertaining whether or not a given element exists in a particular mineral or mixture; all such experiments and analyses are qualitative rather than quantitative, because though the result may be more or less, the particular amount of the result is not the object of the inquiry.

So soon, however, as a result is known to be discoverable, the scientific man ought to proceed to the quantitative inquiry, how great a result follows from a certain amount of the conditions which are supposed to constitute the cause? The possible numbers of experiments are now infinitely great, for every variation in a quantitative condition will usually produce a variation in the amount of the effect. The method of variation which thus arises is no narrow or special method, but it is the general application of experiment to phenomena capable of continuous variation. As Mr. Fowler has well remarked,[356] the observation of variations is really an integration of a supposed infinite number of applications of the so-called method of difference, that is of experiment in its perfect form.

[356] *Elements of Inductive Logic*, 1st edit. p. 175.

In induction we aim at establishing a general law, and if we deal with quantities that law must really be expressed more or less obviously in the form of an equation, or equations. We treat as before of conditions, and of what happens under those conditions. But the conditions will now vary, not in quality, but quantity, and the effect will also vary in quantity, so that the result of quantitative induction is always to arrive at some mathematical expression involving the quantity of each condition, and expressing the quantity of the result. In other words, we wish to know what function the effect is of its conditions. We shall find that it is one thing to obtain the numerical results, and quite another thing to detect the law obeyed by those results, the latter being an operation of an inverse and tentative character.

*The Variable and the Variant.*

Almost every series of quantitative experiments is directed to obtain the relation between the different values of one quantity which is varied at will, and another quantity which is caused thereby to vary. We may conveniently distinguish these as respectively the *variable* and the *variant*. When we are examining the effect of heat in expanding bodies, heat, or one of its dimensions, temperature, is the variable, length the variant. If we compress a body to observe how much it is thereby heated, pressure, or it may be the dimensions of the body, forms the variable, heat the variant. In the thermo-electric pile we make heat the variable and measure electricity as the variant. That one of the two measured quantities which is an antecedent condition of the other will be the variable.

It is always convenient to have the variable entirely under our command. Experiments may indeed be made with accuracy, provided we can exactly measure the variable at the moment when the quantity of the effect is determined. But if we have to trust to the action of some capricious force, there may be great difficulty in making exact measurements, and those results may not be disposed over the whole range of quantity in a convenient manner. It is one prime object of the experimenter, therefore, to obtain a regular and governable supply of the force which he is investigating. To determine correctly the efficiency of windmills, when the natural winds were constantly varying in force, would be exceedingly difficult. Smeaton, therefore, in his experiments on the subject, created a uniform wind of the required force by moving his models against the air on the extremity of a revolving arm.[357] The velocity of the wind could thus be rendered greater or less, it could be maintained uniform for any length of time, and its amount could be exactly ascertained. In determining the laws of the chemical action of light it would be out of the question to employ the rays of the sun, which vary in intensity with the clearness of the atmosphere, and with every passing cloud. One great difficulty in photometry and the investigation of the chemical action of light consists in obtaining a uniform and governable source of light rays.[358]

[357] *Philosophical Transactions*, vol. li. p. 138; abridgment, vol. xi. p. 355.

[358] See Bunsen and Roscoe’s researches, in *Philosophical Transactions* (1859), vol. cxlix. p. 880, &c., where they describe a constant flame of carbon monoxide gas.

Fizeau’s method of measuring the velocity of light enabled him to appreciate the time occupied by light in travelling through a distance of eight or nine thousand metres. But the revolving mirror of Wheatstone subsequently enabled Foucault and Fizeau to measure the velocity in a space of four metres. In this latter method there was the advantage that various media could be substituted for air, and the temperature, density, and other conditions of the experiment could be accurately governed and measured.

*Measurement of the Variable.*

There is little use in obtaining exact measurements of an effect unless we can also exactly measure its conditions.

It is absurd to measure the electrical resistance of a piece of metal, its elasticity, tenacity, density, or other physical qualities, if these vary, not only with the minute impurities of the metal, but also with its physical condition. If the same bar changes its properties by being heated and cooled, and we cannot exactly define the state in which it is at any moment, our care in measuring will be wasted, because it can lead to no law. It is of little use to determine very exactly the electric conductibility of carbon, which as graphite or gas carbon conducts like a metal, as diamond is almost a non-conductor, and in several other forms possesses variable and intermediate powers of conduction. It will be of use only for immediate practical applications. Before measuring these we ought to have something to measure of which the conditions are capable of exact definition, and to which at a future time we can recur. Similarly the accuracy of our measurement need not much surpass the accuracy with which we can define the conditions of the object treated.

The speed of electricity in passing through a conductor mainly depends upon the inductive capacity of the surrounding substances, and, except for technical or special purposes, there is little use in measuring velocities which in some cases are one hundred times as great as in other cases. But the maximum speed of electric conduction is probably a constant quantity of great scientific importance, and according to Prof. Clerk Maxwell’s determination in 1868 is 174,800 miles per second, or little less than that of light. The true boiling point of water is a point on which practical thermometry depends, and it is highly important to determine that point in relation to the absolute thermometric scale. But when water free from air and impurity is heated there seems to be no definite limit to the temperature it may reach, a temperature of 180° Cent. having been actually observed. Such temperatures, therefore, do not require accurate measurement. All meteorological measurements depending on the accidental condition of the sky are of far less importance than physical measurements in which such accidental conditions do not intervene. Many profound investigations depend upon our knowledge of the radiant energy continually poured upon the earth by the sun; but this must be measured when the sky is perfectly clear, and the absorption of the atmosphere at its minimum. The slightest interference of cloud destroys the value of such a measurement, except for meteorological purposes, which are of vastly less generality and importance. It is seldom useful, again, to measure the height of a snow-covered mountain within a foot, when the thickness of the snow alone may cause it to vary 25 feet or more, when in short the height itself is indefinite to that extent.[359]

[359] Humboldt’s *Cosmos* (Bohn), vol. i. p. 7.

*Maintenance of Similar Conditions.*

Our ultimate object in induction must be to obtain the complete relation between the conditions and the effect, but this relation will generally be so complex that we can only attack it in detail. We must, as far as possible, confine the variation to one condition at a time, and establish a separate relation between each condition and the effect. This is at any rate the first step in approximating to the complete law, and it will be a subsequent question how far the simultaneous variation of several conditions modifies their separate actions. In many experiments, indeed, it is only one condition which we wish to study, and the others are interfering forces which we would avoid if possible. One of the conditions of the motion of a pendulum is the resistance of the air, or other medium in which it swings; but when Newton was desirous of proving the equal gravitation of all substances, he had no interest in the air. His object was to observe a single force only, and so it is in a great many other experiments. Accordingly, one of the most important precautions in investigation consists in maintaining all conditions constant except that which is to be studied. As that admirable experimental philosopher, Gilbert, expressed it,[360] “There is always need of similar preparation, of similar figure, and of equal magnitude, for in dissimilar and unequal circumstances the experiment is doubtful.”

[360] Gilbert, *De Magnete*, p. 109.

In Newton’s decisive experiment similar conditions were provided for, with the simplicity which characterises the highest art. The pendulums of which the oscillations were compared consisted of equal boxes of wood, hanging by equal threads, and filled with different substances, so that the total weights should be equal and the centres of oscillation at the same distance from the points of suspension. Hence the resistance of the air became approximately a matter of indifference; for the outward size and shape of the pendulums being the same, the absolute force of resistance would be the same, so long as the pendulums vibrated with equal velocity; and the weights being equal the resistance would diminish the velocity equally. Hence if any inequality were observed in the vibrations of the two pendulums, it must arise from the only circumstance which was different, namely the chemical nature of the matter within the boxes. No inequality being observed, the chemical nature of substances can have no appreciable influence upon the force of gravitation.[361]

[361] *Principia*, bk. iii. Prop. vi.

A beautiful experiment was devised by Dr. Joule for the purpose of showing that the gain or loss of heat by a gas is connected, not with the mere change of its volume and density, but with the energy received or given out by the gas. Two strong vessels, connected by a tube and stopcock, were placed in water after the air had been exhausted from one vessel and condensed in the other to the extent of twenty atmospheres. The whole apparatus having been brought to a uniform temperature by agitating the water, and the temperature having been exactly observed, the stopcock was opened, so that the air at once expanded and filled the two vessels uniformly. The temperature of the water being again noted was found to be almost unchanged. The experiment was then repeated in an exactly similar manner, except that the strong vessels were placed in separate portions of the water. Now cold was produced in the vessel from which the air rushed, and an almost exactly equal quantity of heat appeared in that to which it was conducted. Thus Dr. Joule clearly proved that rarefaction produces as much heat as cold, and that only when there is disappearance of mechanical energy will there be production of heat.[362] What we have to notice, however, is not so much the result of the experiment, as the simple manner in which a single change in the apparatus, the separation of the portions of water surrounding the air vessels, is made to give indications of the utmost significance.

[362] *Philosophical Magazine*, 3rd Series, vol. xxvi. p. 375.

*Collective Experiments.*

There is an interesting class of experiments which enable us to observe a number of quantitative results in one act. Generally speaking, each experiment yields us but one number, and before we can approach the real processes of reasoning we must laboriously repeat measurement after measurement, until we can lay out a curve of the variation of one quantity as depending on another. We can sometimes abbreviate this labour, by making a quantity vary in different parts of the same apparatus through every required amount. In observing the height to which water rises by the capillary attraction of a glass vessel, we may take a series of glass tubes of different bore, and measure the height through which it rises in each. But if we take two glass plates, and place them vertically in water, so as to be in contact at one vertical side, and slightly separated at the other side, the interval between the plates varies through every intermediate width, and the water rises to a corresponding height, producing at its upper surface a hyperbolic curve.

The absorption of light in passing through a coloured liquid may be beautifully shown by enclosing the liquid in a wedge-shaped glass, so that we have at a single glance an infinite variety of thicknesses in view. As Newton himself remarked, a red liquid viewed in this manner is found to have a pale yellow colour at the thinnest part, and it passes through orange into red, which gradually becomes of a deeper and darker tint.[363] The effect may be noticed in a conical wine-glass. The prismatic analysis of light from such a wedge-shaped vessel discloses the reason, by exhibiting the progressive absorption of different rays of the spectrum as investigated by Dr. J. H. Gladstone.[364]

[363] *Opticks*, 3rd edit. p. 159.

[364] Watts, *Dictionary of Chemistry*, vol. iii. p. 637.

A moving body may sometimes be made to mark out its own course, like a shooting star which leaves a tail behind it. Thus an inclined jet of water exhibits in the clearest manner the parabolic path of a projectile. In Wheatstone’s Kaleidophone the curves produced by the combination of vibrations of different ratios are shown by placing bright reflective buttons on the tops of wires of various forms. The motions are performed so quickly that the eye receives the impression of the path as a complete whole, just as a burning stick whirled round produces a continuous circle. The laws of electric induction are beautifully shown when iron filings are brought under the influence of a magnet, and fall into curves corresponding to what Faraday called the Lines of Magnetic Force. When Faraday tried to define what he meant by his lines of force, he was obliged to refer to the filings. “By magnetic curves,” he says,[365] “I mean lines of magnetic forces which would be depicted by iron filings.” Robison had previously produced similar curves by the action of frictional electricity, and from a mathematical investigation of the forms of such curves we may infer that magnetic and electric attractions obey the general law of emanation, that of the inverse square of the distance. In the electric brush we have a similar exhibition of the laws of electric attraction.

[365] *Faraday’s Life*, by Bence Jones, vol. ii. p. 5.

There are several branches of science in which collective experiments have been used with great advantage. Lichtenberg’s electric figures, produced by scattering electrified powder on an electrified resin cake, so as to show the condition of the latter, suggested to Chladni the notion of discovering the state of vibration of plates by strewing sand upon them. The sand collects at the points where the motion is least, and we gain at a glance a comprehension of the undulations of the plate. To this method of experiment we owe the beautiful observations of Savart. The exquisite coloured figures exhibited by plates of crystal, when examined by polarised light, afford a more complicated example of the same kind of investigation. They led Brewster and Fresnel to an explanation of the properties of the optic axes of crystals. The unequal conduction of heat in crystalline substances has also been shown in a similar manner, by spreading a thin layer of wax over the plate of crystal, and applying heat to a single point. The wax then melts in a circular or elliptic area according as the rate of conduction is uniform or not. Nor should we forget that Newton’s rings were an early and most important instance of investigations of the same kind, showing the effects of interference of light undulations of all magnitudes at a single view. Herschel gave to all such opportunities of observing directly the results of a general law, the name of *Collective Instances*,[366] and I propose to adopt the name *Collective Experiments*.

[366] *Preliminary Discourse*, &c., p. 185.

Such experiments will in many subjects only give the first hint of the nature of the law in question, but will not admit of any exact measurements. The parabolic form of a jet of water may well have suggested to Galileo his views concerning the path of a projectile; but it would not serve now for the exact investigation of the laws of gravity. It is unlikely that capillary attraction could be exactly measured by the use of inclined plates of glass, and tubes would probably be better for precise investigation. As a general rule, these collective experiments would be most useful for popular illustration. But when the curves are of a precise and permanent character, as in the coloured figures produced by crystalline plates, they may admit of exact measurement. Newton’s rings and diffraction fringes allow of very accurate measurements.

Under collective experiments we may perhaps place those in which we render visible the motions of gas or liquid by diffusing some opaque substance in it. The behaviour of a body of air may often be studied in a beautiful way by the use of smoke, as in the production of smoke rings and jets. In the case of liquids lycopodium powder is sometimes employed. To detect the mixture of currents or strata of liquid, I employed very dilute solutions of common salt and silver nitrate, which produce a visible cloud wherever they come into contact.[367] Atmospheric clouds often reveal to us the movements of great volumes of air which would otherwise be quite unapparent.

[367] *Philosophical Magazine*, July, 1857, 4th Series, vol. xiv. p. 24.

*Periodic Variations.*

A large class of investigations is concerned with Periodic Variations. We may define a periodic phenomenon as one which, with the uniform change of the variable, returns time after time to the same value. If we strike a pendulum it presently returns to the point from which we disturbed it, and while time, the variable, progresses uniformly, it goes on making excursions and returning, until stopped by the dissipation of its energy. If one body in space approaches by gravity towards another, they will revolve round each other in elliptic orbits, and return for an indefinite number of times to the same relative positions. On the other hand a single body projected into empty space, free from the action of any extraneous force, would go on moving for ever in a straight line, according to the first law of motion. In the latter case the variation is called *secular*, because it proceeds during ages in a similar manner, and suffers no περίοδος or going round. It may be doubted whether there really is any motion in the universe which is not periodic. Mr. Herbert Spencer long since adopted the doctrine that all motion is ultimately rhythmical,[368] and abundance of evidence may be adduced in favour of his view.

[368] *First Principles*, 3rd edit. chap. x. p. 253.

The so-called secular acceleration of the moon’s motion is certainly periodic, and as, so far as we can tell, no body is beyond the attractive power of other bodies, rectilinear motion becomes purely hypothetical, or at least infinitely improbable. All the motions of all the stars must tend to become periodic. Though certain disturbances in the planetary system seem to be uniformly progressive, Laplace is considered to have proved that they really have their limits, so that after an immense time, all the planetary bodies might return to the same places, and the stability of the system be established. Such a theory of periodic stability is really hypothetical, and does not take into account phenomena resulting in the dissipation of energy, which may be a really secular process. For our present purposes we need not attempt to form an opinion on such questions. Any change which does not present the appearance of a periodic character will be empirically regarded as a secular change, so that there will be plenty of non-periodic variations.

The variations which we produce experimentally will often be non-periodic. When we communicate heat to a gas it increases in bulk or pressure, and as far as we can go the higher the temperature the higher the pressure. Our experiments are of course restricted in temperature both above and below, but there is every reason to believe that the bulk being the same, the pressure would never return to the same point at any two different temperatures. We may of course repeatedly raise and lower the temperature at regular or irregular intervals entirely at our will, and the pressure of the gas will vary in like manner and exactly at the same intervals, but such an arbitrary series of changes would not constitute Periodic Variation. It would constitute a succession of distinct experiments, which would place beyond reasonable doubt the connexion of cause and effect.

Whenever a phenomenon recurs at equal or nearly equal intervals, there is, according to the theory of probability, considerable evidence of connexion, because if the recurrences were entirely casual it is unlikely that they would happen at equal intervals. The fact that a brilliant comet had appeared in the years 1301, 1378, 1456, 1531, 1607, and 1682 gave considerable presumption in favour of the identity of the body, apart from similarity of the orbit. There is nothing which so fascinates the attention of men as the recurrence time after time of some unusual event. Things and appearances which remain ever the same, like mountains and valleys, fail to excite the curiosity of a primitive people. It has been remarked by Laplace that even in his day the rising of Venus in its brightest phase never failed to excite surprise and interest. So there is little doubt that the first germ of science arose in the attention given by Eastern people to the changes of the moon and the motions of the planets. Perhaps the earliest astronomical discovery consisted in proving the identity of the morning and evening stars, on the grounds of their similarity of aspect and invariable alternation.[369] Periodical changes of a somewhat complicated kind must have been understood by the Chaldeans, because they were aware of the cycle of 6585 days or 19 years which brings round the new and full moon upon the same days, hours, and even minutes of the year. The earliest efforts of 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.

[369] Laplace, *System of the World*, vol. i. pp. 50, 54, &c.

*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, δ 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.[370] The star β 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 in a period has probably been on the increase. “At first 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.”[371] 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.

[370] Herschel’s *Outlines of Astronomy*, 4th edit. pp. 555–557.

[371] Humboldt’s *Cosmos* (Bohn), vol. iii. p. 229.

*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.[372] 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 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.

[372] *Encyclopædia Metropolitana*, art. *Sound*, § 323; *Outlines of Astronomy*, 4th edit., § 650. pp. 410, 487–88; *Meteorology, Encyclopædia Britannica*, Reprint, p. 197.

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 30-1/2 beats be moving in opposite directions. Hence whatever energy was communicated in the first half minute will be neutralised by the opposite effect of that given in the second half. The effect of the strokes of the second pendulum will therefore be alternately to increase and decrease the vibrations of the first, so that a new kind of vibration will be produced running through its phases in 61 seconds. An effect of this kind was actually observed by Ellicott, a member of the Royal Society, in the case of two clocks.[373] He found that through the wood-work by which the clocks were connected a slight impulse was transmitted, and each pendulum alternately lost and gained momentum. Each clock, in fact, tended to stop the other at regular intervals, and in the intermediate times to be stopped by the other.

[373] *Philosophical Transactions*, (1739), vol. xli. p. 126.

Many disturbances in the planetary system depend upon the same principle; for if one planet happens always to pull another in the same direction in similar parts of their orbits, the effects, however slight, will be accumulated, and a disturbance of large ultimate amount and of long period will be produced. The long inequality in the motions of Jupiter and Saturn is thus due to the fact that five times the mean motion of Saturn is very nearly equal to twice the mean motion of Jupiter, causing a coincidence in their relative positions and disturbing powers. The rolling of ships depends mainly upon the question whether the period of vibration of the ship corresponds or not with the intervals at which the waves strike her. Much which seems at first sight unaccountable in the behaviour of vessels is thus explained, and the loss of the *Captain* is a sad case in point.