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.