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of a piano or violin, a tuning-fork, or the membrane of
a drum, but in those minute excursions of particles of
air which carry sound from one place to another, in the
waves and tides of the sea, and in the amazingly rapid
tremor of the luminiferous ether which, in its varying
action on different bodies, makes itself known as light
or radiant heat or chemical action. Simple harmonic
motions differ from one another in three respects ; in
the extent or amplitude of the swing, which is measured
by the distance from the middle point to either extreme ;
in the period or interval of time between two successive
passages through an extreme position ; and in the time
of starting, or epoch, as it is called, which is named
by saying what particular stage of the vibration was
being executed at a certain instant of time. One of the
most astonishing and fruitful theorems of mathematical
science is this ; that every periodic motion whatever,
that is to say, every motion which exactly repeats itself
again and again at definite intervals of time, is a com-
pound of simple harmonic motions, whose periods are
successively smaller and smaller ah quo t parts of the
original period, and whose amplitudes (after a certain
number of them) are less and less as their periods are
more rapid. The ' harmonic ' tones of a string, which
VOL. n. c


are always heard along with the fundamental tone, are
a particular case of these constituents. The theorem
was given by Fourier in connexion with the flow of
heat, but its applications are innumerable, and extend
over the whole range of physical science.

The laws of combination of harmonic motions have
been illustrated by some ingenious apparatus of Messrs.
Tisley and Spiller, and by a machine invented by Mr.
Donkin ; but the most important practical application
of these laws is to be found in Sir W. Thomson's Tidal
Clock, and in a more elaborate machine which draws
curves predicting the height of the tide at a given port
for all times of the day and night with as much
accuracy as can be obtained by direct observation.
One special combination is worthy of notice. The
union of a vertical vibration with a horizontal one of
half the period gives rise to that figure of 8 which M.
Marey has observed by his beautiful methods in the
motion of the tip of a bird's or insect's wing.

Elliptic Motion.

The motion of the sun and moon relative to the
earth was at first described by a combination of circular
motions ; and this was the immortal achievement of the
Greek astronomers Hipparchus and Ptolemy. Indeed,
in so far as these motions are periodic, it follows from
Fourier's theorem mentioned above that this mode
of description is mathematically sufficient to represent
them ; and astronomical tables are to this day calcu-
lated by a method which practically comes to the
same thing. But this representation is not the simplest
that can be found ; it requires theoretically an infinite


number of component motions, and gives no informa-
tion about the way in which these are connected with
one another. We owe to Kepler the accurate and com-
plete description of planetary or elliptic motion. Hi a
investigation applied in the first instance to the orbit
of the planet Mars about the sun, but it was found true
of the orbits of all planets about the sun, and of the
moon about the earth. The path of the moving body in
each of these motions is an ellipse, or oval shadow of a
circle, a curve having various properties in relation to
two internal points or foci, which replace as it were
the one centre of a circle. In the case of the ellipse
described by a planet, the sun is in one of these foci ;
in the case of the moon, the earth is in one focus. So
much for the geometrical description of the motion.
Kepler further observed that a line drawn from the sun
to a planet, or from the earth to the moon, and sup-
posed to move round with the moving body, would
sweep out equal areas in equal times. These two laws,
called Kepler's first and second laws, complete the
kinematic description of elliptic motion ; but to obtain
formulae fit for computation, it was necessary to cal-
culate from these laws the various harmonic compo-
nents of the motion to and from the sun, and round it ;
this calculation has much occupied the attention of

The laws of rotatory motion of rigid bodies are
somewhat difficult to describe without mathematical
symbols, but they are thoroughly known. Examples
of them are given by the apparatus called a gyroscope,
aad the motion of the earth ; and an application of the
former to prove the nature of the latter, made by

c 2


Foucault, is one of the most beautiful experiments
belonging entirely to dynamics.


Next in simplicity after the translation of a rigid
body, come two kinds of motion which are at first sight
very different, but between which a closer observation
discovers very striking analogies. These are the motion
of rotation about a fixed point, and the motion of slid-
ing on a fixed plane. The first of these is most easily
produced in practice by what is well known as a ball-
and-socket joint ; that is to say, a body ending in a
portion of a spherical surface which can move about in
a spherical cavity of the same size. The centre of the
spherical surface is then a fixed point, and the motion
is reduced to the sliding of one sphere inside another.
In the same way, if we consider, for instance, the
motion of a flat-iron on an ironing-board, we may see
that this is not a pure translation, for the iron is
frequently turned round as well as carried about ; but
the motion may be described as the sliding of one plane
upon another. Thus in each case the matter to be
studied is the sliding of one surface on another which it
exactly fits. For two surfaces to fit one another exactly,
in all positions, they must be either both spheres of the
same size, or both planes ; and the latter case is really in-
cluded under the former, for a plane may be regarded
as a sphere whose radius has increased without limit.
Thus, if a piece of ice be made to slide about on the
frozen surface of a perfectly smooth pond, it is really
rotating about a fixed point at the centre of the earth ;
for the frozen surface may be regarded as part of an


enormous sphere, having that point for centre. And
yet the motion cannot be practically distinguished from
that of sliding on a plane.

In this latter case it is found that, excepting in the
case of a pure translation, there is at every instant a
certain point which is at rest, and about which as a
centre the body is turning. This point is called the in-
stantaneous centre of rotation ; it travels about as the
motion goes on, but at any instant its position is per-
fectly definite. From this fact follows a very important
consequence ; namely that every possible motion of a
plane sliding on a plane may be produced by the rolling
of a curve in one plane upon a curve in the other. The
point of contact of the two curves at any instant is the
instantaneous centre at that instant. The problems to
be considered in this subject are thus of two kinds:
Given the curves of rolling to find the path described
by any point of the moving plane ; and, Given the
paths described by two points of the moving plane
(enough to determine the motion) to find the curves of
rolling and the paths of all other points. An important
case of the first problem is that in which one circle rolls
on another, either inside or outside ; the curves de-
scribed by points in the moving plane are used for the
teeth of wheels. To the second problem belongs the
valuable and now rapidly increasing theory of link-work,
which, starting from the wonderful discovery of an
exact parallel motion by M. Peaucellier, has received an
immense and most unexpected development at the
hands of Professor Sylvester, Mr. Hart, and- Mr. A. B.

Passing now to the spherical form of this motion,


we find that the instantaneous centre of rotation (which
is clearly equivalent to an instantaneous axis perpen-
dicular to the plane) is replaced by an instantaneous
axis passing through the common centre of the moving
spheres. In the same way the rolling of one curve on
another in the plane is replaced by the rolling of one
cone upon another, the two cones having a common
vertex at the same centre.

Analogous theorems have been proved for the most
general motion of a rigid body. It was shown by M.
Chasles that this is always similar to the motion of a
corkscrew desending into a cork ; that is to say, there is
always a rotation about a certain instantaneous axis,
combined with translation along this axis. The amount
of translation per unit of rotation is called the pitch of
the screw. The instantaneous screw moves about as
the motion goes on, but at any given instant it is per-
fectly definite in position and pitch. And any motion
whatever of a rigid body may be produced by the
rolling and sliding of one surface on another, both
surfaces being produced by the motion of straight
lines. This crowning theorem in the geometry 01
motion is due to Professor Cayley. The laws of combina-
tion of screw motions have been investigated by Dr. Ball.

Thus, proceeding gradually from the more simple to
the more complex, we have been able to describe every
change in the position of a body. It remains only to
describe changes of size and shape. Of these there are
three kinds, but they are all included under the same
name strains. We may have, first, a change 01 size
without any change of shape, a uniform dilatation or
contraction of the whole body in all directions, such as


happens to a sphere of metal when it is heated or cooled.
Next, we may have an elongation or contraction in one
direction only, all lines of this body pointing in this
direction being increased or diminished in the same
ratio ; such as would happen to a rod six feet long and
an inch square, if it were stretched to seven feet long,
still remaining an inch square. Thirdly, we may have a
change of shape produced by the sliding of layers over
one another, a mode of deformation which is easily pro-
duced in a pack of cards ; this is called a shear. By
appropriate combinations of these three, every change
of size and shape may be produced ; or we may even
leave out the second element, and produce any strain
whatever by a dilatation or contraction, and two


We have already said- that the change of motion
of a body depends upon the position and state of sur-
rounding bodies. To make this intelligible it will be
necessary to notice a certain property of the three kinds
of motion of a point which we described.

The combination of velocities may be understood
from the case of a body carried in any sort of cart or
vehicle in which it moves about. The whole velocity
of the body is then compounded of the velocity of the
vehicle and of its velocity relative to the vehicle.
Thus, if a man walks across a railway carriage his
whole velocity is compounded of the velocity of the
railway carriage and of the velocity with which he
walks across.

When the velocity of a body is changed by adding


to it a velocity in the same direction or in the opposite
direction, it is only altered in amount ; but when a
transverse velocity is compounded with it, a change of
direction is produced. Thus, if a man walks fore and
aft on a steamboat, he only travels a little faster or
slower ; but if he walks across from one side to the
other, he slightly changes the direction in which he is

Now, in the parabolic motion of a projectile, we
found that while the horizontal velocity continues
unchanged, the vertical velocity increases at a uniform
rate. Such a body is having a downwards velocity
continually poured into it, as it were. This gradual
change of the velocity is called acceleration : we may
say that the acceleration of a projectile is always the
same, and is directed vertically downwards.

In a simple harmonic motion it is found that the
acceleration is directed towards the centre, and is
always proportional to the distance from it. In the
case of elliptic motion it was proved by Newton that
the acceleration is directed towards the focus, and is
inversely proportional to the square of the distance
from it.

Let us now consider the circumstances under
which these motions take place. To produce a simple
harmonic motion we may take a piece of elastic string,
whose length is equal to the height of a smooth table ;
then fasten one end of the string to a bullet and the
other end to the floor, having passed it through a hole
in the table, so that the bullet just rests on the top of
the hole when the string is unstretched. If the bullet
be now pulled away from the hole so that the string is


stretched, and then let it go, it will oscillate to and fro
on either side of the hole with a simple harmonic
motion. The acceleration (or rate of change of velo-
city) is here proportional to the distance from the hole;
that is, to the amount of elongation of the string. It is
directed towards the hole ; that is, in the direction of
this elongation. In the case of the moon moving round
the earth, the acceleration is directed towards the
earth, and is inversely proportional to the square of
the distance from the earth.

In both these cases, then, the change of velocity
depends upon surrounding circumstances ; but in the
case of the bullet, this circumstance is the strained con-
dition of an adjoining body, namely, the elastic string ;
while in the case of the moon the circumstance is the
position of a distant body, namely, the earth. The
motion of a projectile turns out to be only a special
case of the motion of the moon ; for the parabola
which it describes may be regarded as one end of a
very long ellipse, whose other end goes round the
earth's centre.

There is a remarkable difference between the two
cases. The swing of the bullet depends upon its size ;
a large bullet will oscillate more slowly than a small
one. .This leads us to modify the rule. If a large
bullet is equivalent to two small ones, then when it is
going at the same rate it must contain twice as much
motion as one of the small ones ; or, as we now say,
with the same velocity it has twice the momentum.
Now the change of momentum is found to be the same
for all bullets, when the momentum is reckoned as pro-
portional to the quantity of matter in the bullet as well


as to the velocity. The quantity of matter in a body is
called its mass : for bodies of the same substance it is, of
course, simply the quantity of that substance ; but for
bodies of different substances it is so reckoned as to
make the rule hold good. The rule for this case may
then be stated thus ; the change of momentum of a
body (that is, the change of velocity multiplied by the
mass), depends on the state of strain of adjoining bodies.
Eegarded as so depending, this change of momentum is
called the pressure or tension of the adjoining body,
according to the nature of the strain ; both of these
are included in the name stress, introduced by Eankine.

But in the case of projectiles, the acceleration is
found to be the same for all bodies at the same place ;
and this rule holds good in all cases of planetary
motion. So that it seems as if the change of velocity,
and not the change of momentum, depended upon the
position of distant bodies. But this case is brought
under the same rule as the other by supposing that the
mass of the moving body is to be reckoned among the
' circumstances.' The change of momentum is in this
case called the attraction of gravitation, and we say
that the attraction is proportional to the mass of the
attracted body. And this way of representing the facts
is borne out by the electrical and magnetic attractions
and repulsions, where the change of momentum depends
on the position and state of the attracting thing, and
upon the electric charge or the induced magnetism of
the attracted thing.

Force, then, is of two kinds ; the stress of a strained
adjoining body, and the attraction or repulsion of a
distant body. Attempts have been made with more or


less success to explain each of these by means of the
other. In common discourse the word ' force ' means
muscular effort exerted by the human frame. In this
case the part of the human body which is in contact
with the object to be moved is in a state of strain, and
the force, dynamically considered, is of the first kind.
But this state of strain is preceded and followed by
nervous discharges, which are accompanied by the
sensations of effort and of muscular strain ; a complica-
tion of circumstances which does not occur in the
action of inanimate bodies. What is common to the
two cases is, that the change of momentum depends on
the strain.

Having thus explained the law of Force, which is
the foundation of Dynamics, we may consider the
remaining laws of motion. It is convenient to state
them first for particles, or bodies so small that we need
take account only of their position. Every particle,
then, has a rate of change of momentum due to the
position or state of every other particle, whether
adjoining it or distant from it. These are compounded
together by the law of composition of velocities, and
the result of the whole is the actual change of momen-
tum of the particle. This statement, and the law of
Force stated above, amount together to Newton's first
and second laws of motion. His third law is, that the
change of momentum in one particle, due to the posi-
tion or state of another, is equal and opposite to the
change of momentum in the other, due to the position
or state of the first.

By the help of these laws D'Alembert showed how
the motion of rigid bodies, or systems of particles, might


be dealt with. It appears from his method that two
stresses, acting on a rigid body, may be equivalent, in
their effect on the body as a whole, to a single stress,
whose direction and position will be totally independent
of the shape and nature of the body considered. The
law of combination of stresses acting on a system of
particles is, in fact, the same as the law of combination
of velocities, so far as regards the motion of the system
as a whole. This beautiful but somewhat complex
result of Dynamics has been used in some text-books as
the independent foundation of Statics, under the name of
the parallelogram of forces ; a singular inversion of the
historical order and of the methods of the great writers.
When the result of all the circumstances surround-
ing a body is that there is no change of momentum, the
body is said to be in equilibrium. In this case, if the
body is at rest, it will remain so ; and on this ^ account
the study of such conditions is called Statics. In deal-
ing with the statics of rigid bodies, we have only to
examine those cases in which the resultant of the
external stresses and attractions acting on the body
amounts to nothing. But the most important part of
statics is that which finds the stresses acting in the
interior of bodies between contiguous parts of them ;
for upon this depends the determination of the requisite
strength of structures which have to bear given loads.
It is found that the way in which the stress due to a
given strain depends on the strain varies according to
the physical nature of the body ; for bodies, however,
which are not crystalline or fibrous, but which have the
same properties in all directions, there are two quanti-
ties which, if known, will enable us always to calculate


the stress due to a given strain. These are, the
elasticity of volume, or resistance to change of size ; and
the rigidity, elasticity of figure, or resistance to change
of shape. Problems relating to the interior state of
bodies are far more difficult than those which regard
them as rigid. Thus, if a beam is supported at its two
ends, it is very easy to find the portion of its weight
which is borne by each support ; but the determination
of the state of stress in the interior is a problem of
great complexity.

There is one theorem of kinetics which must be
mentioned here. If we multiply half the momentum of
every particle of a body by its velocity, and add all the
results together, we shall get what is called the kinetic
energy of the body. When the body is moved from
one position to another, if we multiply each force acting
on it whether attraction or stress by the distance
moved in the direction opposite to the force, and add
the results, we shall get what is called the work done
against the forces during the change of position. It
does not at all depend on the rate at which the change
is made, but only on the two positions. If a body
moves, and loses kinetic energy, it does an amount of
work equal to the kinetic energy lost. If it gains kinetic
energy, an amount of work equal to this gain must be
done to take it back from the new position to the old
one. The amount of work which must be done to take
a body from a certain standard position to the position
which it has at present is called the potential energy of
the body. The theorem may be stated in this form ;
the sum of the potential and kinetic energies is always
the same, provided the surrounding circumstances do


not alter. Hence the theorem is called the Conservation
of Energy. It is one fact out of many that may be
deduced from the equations of motion ; it is not suffi-
cient to determine the motion of a body, but it is ex-
ceedingly useful as giving a general result in cases
where it might be difficult or undesirable to investigate
all the particulars ; and it is especially applicable to
machines, the important question in regard to which is
the amount of work which they can do.

It will have been seen that the science of motion
depends on a few fundamental principles which are
easily verified, and consists almost entirely of mathema-
tical deductions and calculations based on those princi-
ples. It is no longer therefore an experimental science
in the same sense as those are in which the fundamental
facts are still being discovered. The apparatus con-
nected with it may be conveniently classified under
three heads :

(a) Apparatus for illustrating theorems or solving
problems of kinematics, such as those mentioned
above for compounding harmonic motions.
There is reason to hope for great extension of
our powers in this direction.

(b) Apparatus for measuring the dynamical
quantities, such as weight, work, and the
elasticities of different substances. These are
more fully classified under Measurements.

(c) Apparatus designed for purposes belonging to
other sciences, but illustrating by its structure
and functions the results of kinematics or
dynamics. In this class the remainder of the
collection is included.





THE subject of this Lecture is one in regard to which
a great change has recently taken place in the public
mind. Some time ago it was the custom to look with
suspicion upon all questions of a metaphysical nature
as being questions that could not be discussed with any
good result, and which, leading inquirers round and
round in the same circle, never came to an end. But
quite of late years there is an indication that a large
number of people are waking up to the fact that Science
has something to say upon these subjects ; and the
English people have always been very ready to hear
what Science can say understanding by Science what
we shall now understand by it, that is, organized com-
mon sense.

When I say Science, I do not mean what some
people are pleased to call Philosophy. The word ' phi-
losopher,' which meant originally ' lover of wisdom/
has come in some strange way to mean a man who
thinks it his business to explain everything in a certain
number of large books. It will be found, I think, that
in proportion to his colossal ignorance is the perfection
and symmetry of the system which he sets up ; because
it is so much easier to put an empty room tidy than a

1 Sunday Lecture Society, November 1, 1874 ; * Fortnightly Keview/

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Online LibraryWilliam Kingdon CliffordLectures and essays by William Kingdon Clifford (Volume 2) → online text (page 2 of 22)