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TREATISE ON DYNAMICS.
BY W. P. WILSON, M.A.,
FELLOW OF SAINT JOHN'S COLLEGE, CAMBRIDGE;
AND PROFESSOR OF MATHEMATICS IN QUEEN'S COLLEGE, BELFAST.
?<-' -— ^^^..
juiormy
ia^ambrOrge : MACMILLAN AM
ilonl>on; GEORGE BEIX. Oubltn • TIODGEh
¥
The following pages form the first portion of a Treatise
on Dynamics, and contain the fundamental principles of
the science, with their application to the motion of parti-
cles, and to the simpler cases of the motion of bodies of
finite magnitude. The remainder, which is in course of
preparation, will contain the higher investigations, and
the application of the principles to physical Astronomy,
together with a history of the discovery of the Laws of
Motion, and an account of the principal experiments that
have been made for the pm'pose of testing their accuracy,
and determining the attraction and mean density of the
earth.
Saint John's College,
Oct. 12, 1850.
3 CO. ,
)
VI) s.
CONTENTS.
CHAPTER I.
PAGE
Elementary Notions — Laws of Motion — Equations of Motion . 1 — 35
CHAPTER n.
Methods of Integrating the Equations of Motion when the
Motion is in one line — and in one plane . . . 36 — 64
CHAPTER HI.
Methods of Integrating the Equations of Motion when the
Force is Central — Elliptic Motion .... 65 — 86
CHAPTER IV.
Properties of Motion not in one plane — Motion in a Resisting
Medium — Constrained Motion — Principle of Least Ac-
tion 87 — 112
CHAPTER V.
Motion of a System of Particles under their Mutual Attractions
— General Properties ...... 113 — 128
CHAPTER VI.
Motion of Bodies of Finite Magnitude — D'Alembert's Principle
— Motions of Translation and Rotation — 'Motion about a
Fixed Axis 129—148
CHAPTER VII.
Motion of a System of Bodies — Conservation of the Motion
of the Center of Gravity — and of Areas — and of Vis Viva
— Examples — Centrifugal Force .... 149—107
CHAPTER VIII.
Nature of Impact — Determination of the Motion after Impact
— ^Examples ........ 168—176
TREATISE ON DYNAMICS.
CHAPTER I.
1. Mechanics is the science which treats of the effects produced
by force acting on material bodies. When the different forces acting
on any body counteract each other so that the body remains at rest,
the forces are said to be in equiHbrium. The consideration of the
conditions to which the forces must be subject that this may be the
case, is the object of the science of Statics.
When these conditions are not fulfilled by the forces, the body
will not remain at rest. The investigation of the laws which regulate
the motion which takes place, and of the nature of the motion itself,
will form the subject of the present treatise.
Before we proceed, however, there are certain ideas of which it
is necessary to form an accurate conception ; and, in order to be
definite, we will at present confine our attention to bodies so small
that they may without sensible error be treated as geometrical points,
but we must at the same time consider them to be possessed of all the
properties of matter. This is what must be understood when par-
ticles, or material points, or molecules are spoken of.
2. It would be useless to attempt to define space and time.
No explanation could in any way render the ideas clearer. The
measures of them, on the contrary, require the greatest degree of
attention. In Dynamics we are only concerned with linear space, or
length, or distance, and this is always to be understood when the
term space is used. Now every concrete magnitude must be mea-
sured by some definite magnitude of the same kind; thus, space must
be measured by space, time by time.
1
2 A TREATISE ON DYNAMICS. [cHAP.
With respect to space nothing can be easier. We can fix on some
definite length, and can take a rule or a piece of string of that length,
and apply it over and over again to the space which we wish to
measure to find how often it is contained in it ; and we consider that
we know the magnitude of that space w^hen we know the number of
times that our measure is contained in it. Thus, if it be the distance
between two points A and B that we wish to determine, we consider
that we know it when we have found that our measure will go in it
a times, a being whole or fractional ; and we say that
AB = a times the measure,
or more commonly, suppressing the measure, we say that AB = a.
In this case our measure is called the " unit of length," and the space
AB is said to be numerically represented by a. It is quite obvious
that the greater or less our measure or unit is chosen, the less or
greater will be the numerical representation of any given space AB.
The choice of this unit is perfectly arbitrary ; but we must always
bear in mind that it is a space.
3. We cannot apply the same method of proceeding to " time :"
for we are not able to take an interval of time and apply it to other
intervals to determine how often it is contained in them. An instant
of time, corresponding to a point in space, can only be marked by
the occurrence of some event. Now we can easily conceive a number
of events happening one after the other in such a manner that the
interval between any two shall be of the same duration as the interval
between any other two. It is possible and easy to conceive, a priori,
this equality of intervals, but it is impossible, a priori, to fix on any
series of events which we can be sure will so occur at equal intervals.
We shall see, when we have advanced somewhat further, that such
a series of events can be procured, but till then we must rest satisfied
with being able to conceive such a series. The possibility or impos-
sibility of such a series existing will in no wise affect the accuracy of
the notion of equal times. If Ave take one of these intervals as our
measure, we shall have any interval which lasts during n of those
intervals expressed by ?^ times the interval, where « may be whole or
fractional. And after this interval is agreed on, we consider the
length of any other interval determined when n is known, and w^e
express it simply by n: in other words, we take the agreed-on
interval as our unit. The duration of this " unit of time" is perfectly
arbitrary, and as it is greater or less, h the numerical representation
of any proposed interval will be less or greater.
I.j A TREATISE ON DYNAMICS. 3
4. Having now explained the method of measuring time and
space, and of representing them numerically, we come to the motion
of a material point. The simplest kind of motion that we can con-
ceive is what is called " uniform rectilinear motion." When a point
moves so that at every instant during the whole duration of the
motion it is found in the same straight line, the motion is said to be
rectilinear. If, moreover, it moves so as always to pass over equal
spaces in equal intervals of time, at what part soever of the motion
those intervals be taken, or however long or short their duration,
then the motion is said to be uniform. Hence if it pass over a space
s in one interval, it would pass over a space 2* in two such intervals,
and a space ns in n such intervals.
5. Like space and time ''velocity" is a tei'ra which cannot be
defined. We may say it is the ''rate" of motion or the "pace" or
" speed," but we are only giving other words of the same meaning,
not explaining the idea. We can however, as in those cases, explain
accurately how it is measured.
Suppose two points to be moving uniformly as described above,
and suppose that the space which one describes in any interval of
time is equal to the space which the other describes in the same
interval, then the velocities of these two points are the same.
Now before we can express velocity numerically, it is necessary
to know what is meant when one velocity is said to be twice another.
We saw in the case of space, that if we took any given length, and at
the end of that took another length equal to the former, the whole
length so taken was twice the original length.
Similarly, if we supposed an interval of any definite duration,
and then at the end of it another interval of the same duration, the
whole interval would be twice one of the first intervals.
We cannot however adopt the same method with velocity, we
cannot form any distinct conception of what is meant by one velocity
being the sum of two other velocities without bringing to our aid
ideas which tacitly assume the point in question. The common
notion, however, is very simple and sufficiently exact.
Generally one thing is said to move twice as fast as another, or
to have twice the velocity of another when it passes over twice the
space in the same time. This is the foundation of the method of
representing velocity numerically. And this definition of double
velocity, or of "twice as fast," is at once applicable to any multiple
of the velocity. Thus, if a particle moving with a given velocity
passes over in a given time a space s (that is, a space whose nume-
1—2
4 A TREATISE ON DYNAMICS. [cHAP.
rical representation is s), a particle moving with twice that velocity
would pass over in the same interval of time a space 2*; and a
particle moving with n times the given velocity would in the same
interval pass over a space 7is. If the first-mentioned velocity be
that chosen as a unit, the latter velocity would be represented nume-
rically by n.
The definition just given, of one velocity being ?? times another,
is also expressed by saying that the velocity varies as the space
described in a given interval of time.
Now suppose a point moving with a velocity represented as
above explained by v to pass over the space * in a given interval of
time t, and suppose another point to move so as to pass over an
equal space s in an n^^ part of that interval of time, then this latter
point would pass over a space 7is in this interval of time t, and
therefore its velocity would be represented by )iv.
This is expressed by saying that when two points describe equal
spaces their velocities are inversely as the times of describing them.
If a point describes uniformly a known space in a known time,
the velocity of that point is known as much as we can conceive a
velocity to be. Suppose then that the velocity by which we measure
other velocities, or which we take as our unit of velocity, is that
with which a point would move through a space s^ in a time /,.
And suppose that a body moving with v' times this velocity would
describe a space s in the same time i^, then we have seen that
And suppose that a body moving with v" times this last velocity
would move over the space s in time /, then we have seen that
V
Also this last velocity is v'v" times the velocity which we have
taken as our measure. Let v'v"=v, then the last velocity is v times
our measure, and with it our point moves over the space s in the
time /• And referring to our equations we see that
, „ s t^ s /i
Si t t Si
If we choose as our measure such a velocity that the ratio — is
unity, then we have ^ = j'
This will be the case if we have /i = 1 and .Jj = 1, which is the
same thing as taking for our measure the velocity with which a
body would uniformly describe the unit of space in the unit of time.
I.] A TREATISE ON DYNAMICS. 5
This for the future we shall always do. In that case the equation
v = -, which we shall consider as our fundamental equation, expresses
that the numerical representation of the velocity of a point is the
fraction whose numerator and denominator are respectively the
numerical representations of the space described and of the time
in which it is described.
6. We may observe that the particular length which we fix on
as our unit or measure of space, and the particular interval whose
duration we fix on as the unit or measure of time, are both perfectly
arbitrary and perfectly independent, but that when we have fixed
on them, we shall have fixed on the particular velocity which we
take as our unit of velocity. In different investigations different
units of time and space are chosen, but it is obviously essential that
throughout the same investigation the same units should be ad-
hered to.
7. We have hitherto considered the case of a material point
moving uniformly, and have seen how its velocity is measured and
numerically represented, and have obtained our fundamental equa-
tion V = T > which gives the same numerical value to v however large
or small t be taken. We will now consider the case of a body whose
velocity is changing from one instant to another.
The first question that arises is, " What do we mean when we
speak of the velocity of the point at any instant?" A little con-
sideration however will shew that our idea of velocity is indepen-
dent of the supposition of its being uniform. We can suppose a
body moving uniformly to be moving just as fast as another body,
whose motion is not uniform, was at some particular instant: in
other words, that at some particular instant their velocities were
equal.
If then at any instant we suppose the motion of a body to
become uniform, the velocity at that instant remaining unaltered,
the numerical representation of this uniform velocity will be the
numerical representation of the variable velocity at that instant.
Suppose now for distinctness that the velocity of a body at a
certain instant is increasing, and that its numerical representation
estimated as we have explained is v; and suppose that after an
interval 0/ has elapsed the velocity is v + lv, and that the space
which has been described in that time is ?.v.
6 A TREATISE ON DYNAMICS. [cHAP.
(The expressions It, 6v, hs must be understood as single symbols
expressing no more than we have hitherto done by /, v, and j, that
is, they represent numbers, and are used as single symbols or single
letters.)
Also let S,?! be the space that would have been described, in the
time ht if the body had moved uniformly during that time with the
velocity v, and CS2 the space that would have been described in the
same time S t had the body moved uniformly with the velocity v + hv.
Then, from what has been shewn before, we have
V = -^ , and V + cv = -^ ' .
ct ct
Now the space hs which is actually described will be greater
than the space Isi and less than the space Is^. Therefore we shall
have
the fraction ^ greater tlian-j^,
and less than ^ f ,
dt '
that is, we shall have the fraction
ht
and this is true however small S^ be taken ; but by diminishing li we
may make hv less than any assignable quantity, and S* will also
be diminished at the same time ; if, therefore, in accordance with
the notation of the diiferential calculus, we represent by -j- the
di
it
limit to which the fraction -^ approaches when Zt is diminished
indefinitely, we shall have
ds
Tt = '-
This is another equation of the utmost importance. Let us stop
for a moment to consider its meaning.
Suppose that the distance of the body from some fixed point in
the line of its motion at the instant under consideration is s, and
that the whole time that has elapsed since some fixed epoch is t.
Had we considered the body at some different time the position
would have been different, that is, any change in the value of t is
accompanied by some corresponding change in the value of s, or
in analytical language, s is a function of f.
Now this being the case, we see that when the particle moves
forward through a space hs, its distance from the fixed point be-
I.] A TREATISE ON DYNAMICS. 7
comes s + ls, or is increased by S*; similarly, the time that has
elapsed since the fixed epoch is t + cf, or t has been increased by It.
Hence ds and S< are the corresponding increments of s and t, and
g „
their ratio is — and the limit of this ratio, which we have shewn
ci
to be equal to v, is the differential coefficient of s with respect to
t, consistently with the notation we have used for it. Hence then,
when it is known that a body moves so that the distance from a
fixed point is always represented by a certain function of the time
which has elapsed since a definite instant, the differential coeffi-
cient of this function, with respect to t, will express the velocity
of the particle at any time. And, conversely, if we know the velocity
of a particle at any time as a function of the time, and can find
a quantity of which this function is the differential coefficient, this
quantity will express the distance of the particle at any time from
some fixed point in its line of motion.
In the case of motions arising from natural causes these functions
always exist ; it would, however, be possible by artificial means to
cause motions which should follow no law whatever, or be perfectly
irregular ; in these cases we should still have v equal to the limit of
7^ , but since in this case v, and therefore the limit of -7- which is
ct i>t
equal to it, would vary from one instant to another arbitrarily, it
would be impossible that it should be represented by any function,
and consequently could not be the differential coefficient of any
function.
Motions of this irregular description are, however, never the
subject of mathematical investigation, we may therefore, in all cases
that will come under our consideration, adopt the equation ^--jft
considering — to be, as the notation indicates, the differential coeffi-
cient of s with respect to /.
It must be carefully borne in mind that, though in all these
investigations we have spoken of s, v, and /, as space, velocity, and
time, we only mean the numerical' representations of these quantities
expressed on the principles before laid down, and such is always to
be understood whenever such language is used, unless the contrary
be expressly stated.
8. We will now proceed to describe another kind of motion.
We have already considered the case of uniform rectilinear motion,
and have seen that the characteristic of this kind of motion is, that
at every instant the velocity remains the same ; and from the consi-
8 A TREATISE ON DYNAMICS. [cHAP
deration of this kind of motion, we have obtained our measures
and numerical representations of velocity. We have also obtained
an expression for the velocity in the case where it changes from one
instant to another, but we have not at all considered the case of that
change taking place according to any law.
Now, conceive a body moving in such a manner that the differ-
ence of its velocities at the beginning and end of any interval is
equal to the difference of its velocities at the beginning and end of
any other equal interval, whatever the duration of the intervals, or
at whatever parts of the motion they may be taken. Such motion is
called uniformly accelerated motion.
Suppose, for instance, that the velocity at any instant is ?>„, that
after a time t it is ^i, after another time t it is v^y and so on.
Then v^, fj, v,. . .are the velocities at instants separated by equal
intervals t.
Hence then, by the definition of this kind of motion,
V-^ — Vq = V2—Vi = V3 — Vs, &c.,
if the velocity be increasing, or,
z;„ — ^1 = ^1-^2 = ^2 — ^3j &c.,
if it be diminishing. We may, however, consider the second case
as included in the first by the ordinary conventions of negative
quantities.
If we add n of these together, we have
and v„, % are the velocities at instants separated by an interval nt.
nt nt t
and this is true whatever the magnitudes of n and /. We see, then,
that in this kind of motion, the ratio of the increments or decrements
of the velocity in any intervals of time, is equal to the ratio of
those times.
Or, if we consider the numerical representations of the incre-
ments of the velocity and of the times in which these increments
take place, the ratio of the increment of the velocity to the time in
which it takes place is independent of the time.
Let then v be the velocity at a time t from some fixed epoch.
And let i; + St) be the velocity at a time t + ll from the same epoch,
then we have ^- constant for all values of It, large or small. This
o t
is the characteristic property of this kind of motion. Since this ratio
I.] A TREATISE ON DYNAMICS. 9
is constant for all values of ^Mt will be the same when we pass to
the limit, and suppose it indefinitely diminished, in Avhich case the
ratio — becomes -j-, the differential coefficient of v with respect
to /. This differential coefficient is, however, in this kind of
motion, constant. Now conceive two bodies moving each with
uniformly accelerated motion and so that the increase of the velocity
of one of them in any time is equal to the increase of the velocity of
the other in the same time, and consequently so that the ratio ^^
6 t
and its limit -7- is the same for both, then the motions of these
at
two bodies are said to be equally accelerated, or the accelerations of
the motions to be equal. Of course it does not follow from this that
the velocities are equal at the same time, any more than that two
bodies moving with equal velocities are equidistant from the same
fixed point or from the fixed points from which their positions are
estimated. But whatever difference there is between the velocities
at any one time, there will always be the same difference at any
other time, because the increase of velocity in the interval is the same
for each body. This is the idea we must bear in mind when we
say that the motions of two bodies are equally accelerated.
We will now proceed to find a measure for the acceleration of
any motion : and first we will explain what is meant when the
acceleration of the motion of one body is said to be twice as great
as the acceleration of the motion of another.
Here the manner of estimating velocity will supply us with an
analogy.
The velocity of one body is twice that of another when the one
body passes over in any given time twice the space which the other
passes over in the same time.
In a similar manner the acceleration of the motion of one body is
said to be twice the acceleration of the motion of another body when
the increase of the velocity of the one body in any time is twice the
increase of the velocity of the other in the same time.
Thus at any given instant let the velocity of one body be ii^ and
of another Vq, and after a time / let the velocity of the one be v^ and
of the other be v/, then the increments of the velocities of the two
bodies in the time t are respectively Vy — Vo and v,' — 1;„'. If u,'— «„'
be twice v^ — v^ the motion of the second body is twice as much
accelerated as that of the first. This explanation of what is meant,
when one motion is said to be twice as much accelerated as another.
10 A TREATISE ON DYNAMICS. [cHAP.
must be considered as a definition. It follows at once from the
nature of the motion that if this relation holds for any value of t it
will hold for all values of t.
"Let then o v and 6 v be the increments of velocity in the same
time It of two bodies moving with uniformly accelerated motion, if
the acceleration of the latter is twice that of the former Iv = ^lv
and vice versa. Hence we have the equations
^v _ Iv
Jt^'^lt'
- dv dv
^''^-dt-^-Tt'
And the same reasoning is immediately applicable to any multiple.
So that if the acceleration of the motion of one body is n times the
acceleration of the motion of another, we have
Iv _ Bu
, dv dv
''''^ dt^''-Tr
or, since the ratio remains constant, whatever the magnitude of It,
we need not have it the same on both sides, so that
Iv _ Iv
Jl'^'-zt'
dv' dv
li'^^^'Tt'
It now remains to find a numerical representation for the degree
of acceleration of the motion of any body. If the acceleration of the
motion of any body be considered to be known when the increment
of the velocity in any specified time is known, and some particular
degree of acceleration be fixed on, then the magnitude of any other
accelei'ation will be determined, when the ratio it bears to that par-
ticular degree of acceleration which has been chosen as our standard
is determined. Let this particular degree of acceleration be repre-
sented by the letter a^ , and let any other acceleration be represented
by a, and let ^ and t^- be the corresponding ratios of the incre-
^ ' oil dt r »
ments of the velocity and time; then if we have
S« CVi
we have also a = n. a,.
I.] A TREATISE ON DYNAMICS. 11
From which we have « = -—.«),
It,
Or, suppressing a which amounts to taking our standard acceleration
as our unit, we have the numerical representation of the other
acceleration
If then we take as our unit that particular degree of acceleration
for which ^= 1, we have the equation
_6v _dv
This unit is always adopted.
The letter / will in future be used to denote the numerical
lepresentation of the acceleration, so that our equation is
_ Iv _dv
9. All we have said respecting uniformly accelerated motion is
equally applicable to uniformly retarded motion, if throughout the
whole reasoning we use the decrement of the velocity instead of the
increment. In this case we have, if/ be the retardation, and Zv the
decrement of the velocity in time ht,
J'= —. as before.
Or our reasoning will hold if we still take Sv to represent the
increment of the velocity in time ol, and consider it negative when
the velocity is diminishing or the motion is retarded : that is, essen-
tially negative, not, having a negative sign prefixed. This is merely
the ordinary algebraical generalization.
In this case our equation
6v _dv
gives for f a negative value when the motion is retarded, agreeably
to the opposition expressed by the signs + and — .
12 A TREATISE ON DYNAMICS. [cHAP.
If then in the result of any investigation the expression for the
acceleration should be negative, we must conclude from that cir-
cumstance that the motion is retarded, and the magnitude of the
expression for the acceleration, independently of its sign, will give
the magnitude of the retardation.
We have dwelt at some length on this particular kind of motion,
because, as will be seen afterwards, it is of great importance.
10. In considering the motion of a body moving with variable
velocity, we saw at once that at every instant the body was moving
with some definite velocity, and it only remained to express that
velocity in terms of the space described and the time of describing
it : which was easily done .
It is not, however, so easy to see that when a body moves in any
manner whatever (excepting that sort of motion which we have
termed irregular) there is, at every instant, some definite degree of
acceleration of the motion ; that is, to conceive some uniformly acce-
lerated motion in which the velocity is being increased at the same
rate as in the proposed motion at the instant under consideration.
That this is the case will nevertheless be seen by a little careful
consideration of the nature of acceleration. It may perhaps be ren-
dered clearer by the following analogy.
Velocity is the rate at which the space described by the body
increases ; the acceleration is the rate at Avhich the velocity acquired
by the body increases. When the space described increases uniformly,
the motion is said to be uniform; when the velocity increases uni-
formly, the motion is said to be uniformly accelerated. Carrying
the analogy one step farther, when the motion is variable, the velo-
city at any instant is the rate at which the space described is in-
creasing at that instant ; when the motion is not uniformly accele-
rated, the acceleration at any instant is the rate at which the velocity
is increasing at that instant.
This must not be considered as a proof, since, in fact, there is
nothing to be proved, but merely as an explanation tending to facili-
tate the formation of the idea.
Let then jT be the acceleration at some instant at which the velo-
city is V, and after an interval o^, letj'+ If be the acceleration, and
V + hv the velocity.
Then j- will lie between / andf+ If.
%
I-] A TREATISIiJ ON DYNAMICS. 13
But when it is indefinitely diminished, Sy is also indefinitely
diminished, and ultimately vanishes, and we have therefore,
,. . ^ Su dv „
limit of —=-—=:/.
ht (It ^
When the motion is uniformly accelerated, it is indifferent whe-
ther we take ^ , or the limit of that quantity -j- , since the quan-
tity being constant is equal to its limit. When, however, the accele-
ration is not uniform, the ratio i^- is no longer independent of the
magnitude of It, and we must take as our numerical representation
he acceleration -7- , the lim
at
Hence we have generally,
of the acceleration -7- , the limit of the ratio -^--
at ot
-^ ~Tt'
Also, in uniformly accelerated motion,
•^ ot
11. We have then in uniform motion,
also we have, whether the motion be uniform or not,
ds \
' = Tt'
and whether the motion be uniformly accelerated or not,
-^ ~Tt-
From this last equation we obtain at once,
d^
•^ df
When the motion is uniformly accelerated, f, and therefore -p and
d^s
-7-5 will be constant. When the motion is not uniformly accelerated,
f will vary from one instant to another, and will therefore be a
function of /, the time which has elapsed since some fixed epoch.
It does not always happen that -j- and -j-, are expressed explicitly
as functions of ^ ; but they must be either functions of I, or of some
quantity which is itself a function of t.
14 A TKEATISK ON DYNAMICS. [cHAP.
When we know the expressions for these quantities, and are
able to integrate them, we can find the values of the velocity and
of the space described at every instant throughout the whole dura-
tion of the motion.
12. Hitherto we have only considered rectilinear motion ; we will
now proceed to discuss the method of representing analytically motion
which is not rectilinear. When the term Velocity was first used,
no definition was given of it, and it will appear from a little con-
sideration, that the idea employed is equally applicable whether the
motion is in a straight line or a curved one. We can conceive
a point moving along a curved line Avith different degrees of quick-
ness, and our idea of its pace or velocity is as definite as if its path
were a straight line. And if s is its distance at time t from a fixed
ds
point in its path, measured along that path, -j- is its velocity at that
d's
time, and j-^ is the measure of the acceleration of its motion. In this
case, however, there is something besides the velocity required to
complete our knowledge of the state of motion of the body : we must
know the direction of its motion at every instant. Now to determine
the position in space of any point we use three co-ordinates, that is,
the distances from some fixed point of its projections on three
fixed lines passing through 0. Call the three lines Ox, Oij, Oz, and
let L, M, N be the projections of the point P on them made by planes
parallel respectively to yOz, zOx, xOy. Then when OL, OM, ON
are known, the position of P is completely determined ; and when the
velocities of L, M, iV along Ox, Oy, Os, that is, the rates of increase
of Oh, OM, ON are known, the velocity and direction of P's motion
can be found from the geometry.
In the simple case where Ox, Oy, Oz are mutually at right
angles to each other this is easily expressed.
Calling the coordinates of P, x, y, z, the velocities of L, M, and
N will be
dx dy J dz
Tt' Tt ^^"^Tf
and if v be the velocity of P, a, /3, 7 the angles which its direction
makes with Ox, Oy, Os,
ds /7d^' AW 7^
\ dx ^1 dy 1 dz
-cos a = -.-77, cos/i = -. jTT-, cos 7 =-.-7-;
V dt V dl ' V di
l.J A TUEATISE ON DYNAMICS. 15
-J , -^ , and -J- , are called the resolved parts of the velocity of P
in the directions Ox, Oij, Oz.
dv
When Ox, Oj/, Oz are not at right angles to each other, -^ ^
~L, -j^ are called the components in those directions. Hence it
appears that the component of a velocity in any direction depends on
the directions in which the other components are taken ; when, how-
ever, the resolved part in any direction is spoken of, the other com-
ponents are understood to be at right angles to it.
13. The propriety of the term components will be better seen
from the following proposition called the parallelogram of velocities.
This proposition may be stated as follows :
If a point be moving in any direction with any velocity, and
if to this motion another motion be superadded in any direction, then
if the two simple velocities be represented in magnitude and direction
by lines drawn from any point, the actual velocity will be represented
in magnitude and direction by the diagonal of the parallelogram
described on these two lines.
What is meant by superadding a motion will be best understood
by an illustration. Suppose a point to be moving uniformly across
a table in a straight line. This will be the first motion. Now while
this motion relatively to the table continues, suppose the table itself
to be moved uniformly in some other direction, this will be the second
motion superimposed on the particle : neither the one nor the other
will be the motion of the particle relatively to the floor of the room.
It must not be supposed that we here assert that if a ball were
rolling or sliding across a table, and the table were to be pushed
forward, the motion of the ball relatively to the table would be
unaffected ; whether this would or would not be the case, is a subject
for after-investigation : all we suppose is, that a point, a merely
geometrical point if you please, is constrained artificially to advance
uniformly in a straight line relatively to the table.
Let AB be a line chosen
arbitrarily to represent in
magnitude and direction the
first velocity of the point, and
let / be the time in which the
particle would describe JB
with this velocity.
16 A TREATISE ON DYNAMICS. [cHAP.
Now, in order to impress on it the second velocity, suppose it
inclosed in a tube which originally coincides with AB. Then from
the first motion the point would move uniformly from the end A of
the tube to the end B in the time t, and any velocity in any direction
would be superimposed on the former velocity by moving the tube
parallel to itself in that direction and with that velocity, the other
motion relatively to the tube being supposed to continue unaffected
by the motion of the tube.
Let the velocity of the tube be such as would carry it uniformly
from the position AB to the position CD in the time t. Then A C
will represent in magnitude and direction the second velocity super-
imposed on the former: it is to be shewn that the diagonal AD will
represent the actual velocity of the particle resulting from these two.
At the end of the time t, the point will be at D : at the end of any
shorter time t^, let a 6 be the position of the tube: then, since its
motion is uniform, we have
t^ : t = Aa : AC
^aP : CD by similar triangles,
= aP : ab.
therefore, since the motion of the point in the tube is uniform, P is
the position of the point, that is, the point at every instant is in the
diagonal and therefore moves along it. Also
AP : AD = Aa : AC,
-^t, : t,
which shews that the point moves uniformly along the diagonal in
the time t. Therefore AD represents the resultant velocity in mag-
nitude and direction. Which was to be shewn.
From this it follows that if a particle be moving in any direction,
we may suppose the velocity to be the resultant of two velocities in
any two directions in the same plane with the actual velocity, or, if
we extend the parallelogram of velocities to a parallelepiped, of three
velocities in any three directions.
Since the sum of the squares of three adjacent sides of a right
solid is the square of the diagonal, if w^, Vy^ v^ represent three
velocities in directions at right angles to each other coexisting in a
body, V the actual velocity,
v^ = v^ + v/ + v^,
and if a, /3, 7 be the angles between the direction of v and those of
v^, Vy, and v^,
v^ = V cos a, Vy = V cos /3, v^ = V cos 7.
I.] A TREATISE ON DYNAMICS. 17
It must not be supposed that the parallelogram of velocities and
what follows are in reality different from the method in Art. 12. It
has, howevei*, afforded an opportunity for explaining the sense in
which we shall afterwards speak of adding one velocity to another,
and of any number of velocities coexisting.
14. If any number of velocities be superadded at the same time
to a body (in the sense which we have just explained) the re-
sultant velocity may be easily found.
We may take the resultant of any two, and by combining this
resultant with another, obtain the resultant of three, and so on to
any number.
Or we may proceed as follows :
Fixing arbitrarily on three directions, we may resolve each
velocity into its components in these three directions, and add
together the components in each of the three directions. We shall
then have the resultant velocity expressed in terms of three compo-
nents, and can find its magnitude and direction at once by combining
these three.
15. Having explained certain terms which will constantly
occur, and described certain kinds of motion which will continu-
ally come under our notice, we will now proceed to investigate
the causes which would produce those motions, and to establish
rules for determining the nature of the motions which would result
from the action of specified causes.
We have hitherto used the terms point, particle, body, merely to
denote a position in space. A geometrical point supposed capa-
ble of motion would have answered our purpose equally well. We
will now, however, confine our attention to such bodies or points as
we have described in Art. 1. And this is to be understood what-
ever term is used to express it.
Force is defined to be " any cause which produces, or tends to
produce, any change in a body's state of rest or motion."
In the science of Statics, however, we are only concerned with
forces which tend to produce a change in a body's state of rest. In
the present subject the whole of the definition is required.
The first inquiry that suggests itself is. What will become of
a body if left to itself, that is, if not influenced by any bodies
external to itself?
Suppose a body in motion ; we are unable to say, d priori,
whether if left to itself the motion will gradually die away and the
body ultimately stop ; whether it will move in a straight line or be
18 A TREATISE ON DYNAMICS. [cHAP.
deflected from it in any direction ; whether or not it will have a
tendency to move in a circle; in fact, we cannot say any thing
about it. It is found, however, that it will move in obedience to the
following law, which is called the First Law of Motion.
A body in motion not acted on by any external force will move
in a straight line with uniform velocity.
This law assures us of two facts.
First, that matter has no property inherent in itself which
enables it to change the direction of its motion, or to alter or
destroy its velocity ; and
Secondly, that the molecular forces which keep together the
ultimate atoms of which our particle is composed have no such
tendency.
It follows from the definition of force, that if the body be not
acted on by any force, no change can take place in its motion, and
therefore it must move on uniformly in a straight line. For any
change must have some cause, and by the definition, the name
" force " is given to that cause, whatever it be.
The First Law of Motion, then, assures us that when there is no
external force acting on the body, there is no force acting at all
which can avail to change the motion.
We will, at present, say nothing about the proof of this law.
The true nature of the proof, and the true bearing of the experi-
ments which suggested the law, will be better appreciated when
farther progress has been made in the subject.
16. Before proceeding farthei", we will make a remark which
must never be lost sight of. It is this. Force, by whatever means
exerted, or from whatever cause arising, is always necessarily essen-
tially of the same nature.
Forces may differ from one another in their intensity, and in
their directions, and in the points at which they are exerted, but
they are always forces, and as such are always magnitudes of the
same kind.
Thus, a force may be exerted by means of a string, in which
case it is called tension. It may be exerted by pushing one body
against another, in which case it is called pressure or reaction. We
may have the force of friction, pressure exerted by a spring, by a
body's weight which arises from the attraction of the earth, by any
other attraction or repulsion whatever, but still all are the same.
Forces may produce equilibrium, or they may produce motion,
but still the force itself is the same.
I.] A TREATISE ON DYNAMICS. 19
When, therefore, we use tlie term " force," we always mean such
a force as a pressure or tension.
Two forces are equal to one another when a point on which they
act in opposite directions continues at rest under their action.
Two equal forces acting on the same point in the same direction,
constitute together a foi'ce double of either of them.
From these two definitions we are able to represent a force
numerically, by referring it to some standard force as a unit.
17. We will now consider the effect of a force when acting on a
material point.
Suppose the point at rest originally ; when the force acts it will
generate a velocity in the particle, and if the force acts for a certain
time and then ceases, we know from the first law of motion that the
particle will continue to move in a straight line with the velocity
which it had when the force "ceased acting.
Now suppose a force exactly equal to this one to act for exactly
the same time on another particle originally at rest, exactly similar
and equal to the other one ; by the end of the time this particle will
liave acquired, the same velocity as the other, and will continue to
move on with this same velocity when the force ceases acting.
Or, in other words, when a given force acts on a given particle,
originally at rest, for a given time, there is always some definite
velocity which it will generate in the time.
Now suppose that a force acts for a certain time on a particle and
produces a certain velocity, if it goes on acting for another equal
time in the same direction, it will increase the velocity, but we
cannot, a priori, say by how much. Here then we are stopped again.
Also, suppose that together with our force another equal force
acts on the body for the time specified, in the same direction with
it, thus amounting to a double force, we cannot say, (i priori, what
proportion the velocity generated in any given time, by these two
forces, will bear to that which would have been generated in the
same time by one only.
To meet these two difficulties we have another rule to guide us,
called The Second Law of Motion, which may be thus stated.
When any number of forces act upon a particle in motion, the
effect of each force on the velocity of the particle is the same in
magnitude and direction as if it acted singly on the particle at rest.
The best commentary on this law will be the manner in which
it is applied. When a force is designated by a letter (as P) that
letter must be considered as representing numerically the intensity
20 A TREATISE ON DYNAMICS. [cHAP.
of the force, estimated by reference to some standard unit. All
forces whose intensities are equal and which act in the same direction
will be considered as the same force, from whatever different causes
they may arise. For instance, a body pulled by a string, and an-
other attracted by a magnet in the same direction and with the
same intensity, would be considered as acted on by the same force.
Forces of different intensities, even though they arise from the same
cause (as from the attraction of the same body at different distances),
are considered different.
18. Suppose then a force P to act on a particle for a time t in
which it generates a velocity v; then if it acts during another
interval t it will, by our law, produce the same effect as during the
first interval, and therefore during this second interval t it will
generate an additional velocity v, so that after a time 9,1 the particle
will be moving with a velocity 'Zv, and generally after a time nt \t
will be moving with a velocity 7iv, which may be expressed in
words by saying that when a constant force acts upon a particle, the
velocities generated from rest in different times are to one another in
the same ratio as the times in which they are generated.
From the manner in which the velocity of the particle increases,
we see that its motion will be of that kind which we have called
uniformly-accelerated motion.
Hence then we have the conclusion, that a constant force acting
on a particle will produce uniformly-accelerated motion. And this
result will of course be true for every constant force, and for every
pai'ticle. We will however at present confine our attention to the
same particle or to particles which are alike in every respect.
19. Now suppose two forces each, P, to act together in the same
direction on the same particle for the time t. Each would in this
time, if it acted separately, generate from rest the velocity v. But
by our law the effect of each force will be produced as if the other
did not act. Each therefore will generate its velocity v; so that
the whole velocity generated in the time t will be 9.v: and these two
forces P constitute a force 2 P. Hence then a force 2 P generates
in any time twice the velocity which a force P would generate : and
by similar reasoning it appears that a force nP generates n times
the velocity which a force P would generate in the same time. And
from what has preceded we know that the motion of the particle
under the action of this force n P will be uniformly accelerated : and
the increment of the velocity in any time will be n times the in-
I.] A TREATISE ON DYNAMICS. 21
crement in the same time under the action of the force P: or, in
other words, the acceleration of the motion under the action of the
force M P is h times the acceleration under the action of the force P.
Hence then, if P and P' be any two forces, andy and/' the acce-
lerations of the motions which they produce in the same particle,
we shall have
Here then we have come to the following conclusion. When a
constant force acts on a particle it produces uniformly-accelerated
motion during its action. And the accelerations of the motions
which different forces produce in the same or equal particles are to
each other in the same ratio as the forces which produce them.
20. We have hitherto considered the action of forces on particles
exactly similar and equal in all respects, and, consequently, with
exactly the same properties. It remains to consider the action of
forces on different particles, and to do this we must first endeavour
to convey an idea of what is meant by the term " mass."
The definition generally given is that the " mass " of a body is
the quantity of matter in it. Unless, however, it be accurately stated
how this quantity of matter is to be estimated, we are still as far off
as ever from knowing what is meant by the term "mass."
We will take the above definition, and to render it definite will
join it with the following.
The masses of two particles are said to be equal when the
accelerations of the motions generated in them by the action of
equal forces are equal. This may or may not be the same thing as
saying that their volumes are equal or that their weights are equal.
That must be determined by after considerations. We here give
the definition that will be of most service to us, being perfectly at
liberty to give whatever arbitrary definitions we please, on the sole
condition of adhering to them throughout.
If we join together two particles whose masses are equal, we
shall have a particle whose mass is double that of either of the
constituent particles. This is our definition of a double mass, and
so on for any multiple. And exactly as was the case with respect to
space and time we shall have the numerical representation M of
any mass, that number which expresses the ratio which its mass
bears to that particular mass which we have chosen as our unit.
21. Now, suppose a force P acting on a mass M to produce a
motion whose acceleration is J\ and suppose n such particles to be
arranged side by side close together, and each to be acted on by a
22 A TREATISE ON DYNAMICS. [cHAP.
force P in the same direction. Then it follows from our definition
of equal masses that the acceleration of the motion of each will be^^
and therefore since they start together they will move on together
as one body.
It will not affect this motion if we suppose all these particles
united so as to form one particle, in which case we have a motion
Avhose acceleration is / produced by the action of a force 71 P on a
mass nM.
Now a motion whose acceleration is / is produced by the action
of a force P on a mass M, and therefore from what has been proved
before, a motion whose acceleration is 7if would be produced by the
action of a force n P on a mass M.
And this is true whatever be the magnitude of n.
Hence the accelerations of the motions produced in different
particles by the action of the same force nP are inversely propor-
tional to the masses of those particles.
Or, generally, if_/andy' be the accelerations of the motions
produced in masses M and M' by forces P and P', and/j the accele-
ration produced in the mass 31' by the force P, we have
And it was shewn in Art. 19 that
f P"
f P M'
whence it follows that "iy = irj • "p7 »
P P'
22. It has been shewn in the last few articles that when a con-
stant force P acts on a particle whose mass is M, it produces uni-
formly-accelerated motion, and that the amount of the acceleration
P
varies as the fx-action .^y. If then two forces P and P' act on two
P P'
particles whose masses M and M'are such that -=r>= ^s-r, , the motions
M M
of these two particles will be equally accelerated, or the power
which the force P has of accelerating the motion of M is equal to
the power which the force P' has of accelerating the motion of M'.
. P P'
And if ^ = ?z — , , the power which P has of accelerating the motion
of M is n times as great as the power which P' has of accelerating
the motion of M' This power which a force has of accelerating
I.] A TREATISE ON DYNAMICS. 23
the motion of a particle is called the accelerating-force of that force
on the particle ; and in the first of the two cases above we say that
the accelerating-force of P on M is equal to the accelerating-force
of P' on M', and in the second the accelerating-force of P on M is
71 times the accelerating-force of P' on M'. From this definition it
appears that the accelerating-force of any force on any particle is
proportional to the acceleration of the motion which that force, acting
alone, would produce in the particle, and for convenience the iniit
of accelerating-force is so chosen that the same number represents
the two.
It may seem at first sight to have been unnecessary to make this
distinction between the accelerating-force of a force and the accele-
ration of the motion produced by the force; but a little consideration
will shew that the idea of acceleration can be formed without any
reference to force and mass, while the accelerating-force on a definite
mass is a property of the force which exists even though the effect
of the force may be counteracted or modified by other causes.
The term "accelerating-force" will be constantly used as an
abbreviation of the longer expi'ession "^the accelerating-force of a
force on a particle ;" it must not however be concluded from this
independent use of the expression, that accelerating-force is itself
a force of the kind referred to in Art. l6. The term is a bad one,
inasmuch as it favours this misconstruction, but will lead to no
error if the preceding explanation is borne in mind, and the term
considered as one word, equivalent to some such term as accele-
rativeness.
23. It has been shewn that the acceleration /of the motion pro-
P
duced in a mass M by a force P is proportional to -p.. To simplify
our expression, such a relation is assigned between the units offeree
and of mass that
•^ M
If we consider the unit of mass as the arbitrary quantity in this
case, it amounts to defining the unit of force as that force which
will produce the unit of acceleration in the unit of mass, or the force
whose accelerating-force on the unit of mass is unity.
Combining the preceding equation with those obtained before,
we have
ds
' = dl'
^~ M~ dt~ de'
24 A TREATISE ON DYNAMICS. [cHAP.
24. We have hitherto confined our attention to the action of a
constant force. The force P which we have supposed to act did not
vary from one instant to another, and we have shewn from the
second law of motion, that the motion produced in a particle by such
a force was uniformly accelerated. We will now consider the action
of a variable force, that is, a force whose intensity is continually
changing.
Suppose, for the sake of definiteness, that the force is increasing;
and at a time t from some fixed epoch let its intensity be P, and at
a time t + ct suppose it to be P + 2 P.
Let M be the mass of the particle on which it is acting, and v the
velocity with which it is moving at the time /, v + ^v the velocity
with which it is moving at the time t + dt, and w + d v, and v + ^Vg the
velocities with which it would have been moving at the time t + ht,
had the force remained constant during the time hi, with the re-
spective intensities P and P+cP. Then the velocity v + hv will he
between v + Zvi and v+hv2, or hv will lie between Bi'i and hvz and
therefore t— will lie between 1=-^ and ~ .
bt Of ct
Now from our investigations with reference to a constant "force
we know that
h^ P
ht ^ M'
, aUa P + cP
and -^
It M
Hence then ^r- always lies between -^ and -=-= + — - .
ht ^ MM M
But as dt decreases oP decreases, since the intensity of the force
is supposed to vary continuously : and in the limit ^— will become
dv
■J- , the two quantities between which its magnitude always lies will
coincide, and it must therefore be equal to either of them ; we have
therefore the equation
p
M"
dv
P
M'-
d'i
= 17'
and, consequently, -Trf =
This equation is true at every instant during the motion, P re-
presenting the magnitude of the force at that instant. If the letter
P be considered as standing for such a function of /, or quantities
depending on /, as will express the magnitude of the force at every
proposed instant, the equations
1.] A TREATISE ON DYNAMICS. 25
P d^s , ds
M rf^' dt
when solved Avill give the velocity of the body and the space it has
moved over, in terms of the time elapsed since a fixed epoch.
When forces arising from different causes act on a particle in the
direction of its motion, P in the preceding equation must be con-
sidered to represent the resultant of all such forces ; or the equation
might be written
Pi, P2, Sec. being essentially positive or negative as they tend
respectively to increase or diminish s.
25. Suppose a body whose mass is M to be moving with a
velocity v, then the product of the numbers M and v, viz. Mv, is
called the momentum of the body.
The assignment of this name to this particular product may be
considered perfectly arbitrary. As, however, the product presents
itself in a very extensive class of problems, as will be seen hereafter,
it is found convenient to have a name expressing it. If we put the
equation
M J
under the form Mf = P,
we see that the product of the numerical representatives of the mass
and the accelerating-force of the force P on it, is the numerical
representative of the force P. The name of moving-force is given
to this product. The terms, force, moving-force, and pressure, are
however used indiscriminately.
Since f=--r, and M is constant,
■^ dt
•KT r -KIT dv d .Mv
Hence, the moving-force Mf, bears exactly the same relation to
the momentum Mv, that the accelerating-force of the force P, viz./,
does to the velocity v.
This, however, is not of much importance.
26. When a pressure P acts on a body whose mass is M, and
generates motion in it, if v be the velocity at time /,
p_d.Mv
dt '
26 A TREATISE ON DYNAMICS. [cHAP.
where P may be either constant or variable, and, when the latter is
the case, must be either explicitly or implicitly a function of t.
Integrating this equation, we have
Mv=fPdt:
at the times /j and t.^, let i\ and Vo be the velocities of the body,
then
or the definite integral L ^Pdt expresses the change in the momen-
tum of the particle which takes place between the times /„ and ^,
from the action of the force P.
When P is known as a function of t, this integration can be
effected, and the change of momentuna deduced from it.
It sometimes happens that, although P is not known as a func-
tion of t, and although the interval t^— ti is not known, the value of
the definite integral is known. Or, in other words, we sometimes
know the whole change in velocity or momentum which is caused by
a certain force, when we do not know the law of the force, or the
time during which it acts. When this is the case, although from the
velocity of the particle before the action of the force commenced, we
can determine the velocity when it ceased to act, or the converse, we
cannot determine the position of the body, because we are ignorant
of the nature of its motion during the time tz — tv
In the cases, however, of this kind which come under our notice,
the interval <2 - h is extremely short, so that we may neglect without
sensible error the change of position of the body which takes place
during that interval, or consider it in the same position at the time
^2 as it was at the time ^i.
The force P, on the contrary, is generally so great as to produce
a considerable change in the velocity v, or the momentum Mv.
In these cases the change of velocity from v to V2 is considered
to take place instantaneously : the action of the force is said to be
impulsive, and the definite integral J 'Pdt is called the "impulse,"
or " blow," perhaps an abbreviation for " effect of the impulse," or
" measure of the impulse."
When we consider the definite integral L 'Pdt, or the change of
momentum M (v-j-v^), as expressing the magnitude of the impulse, we
fix on the particular magnitude of the impulse which we take as our
1.] A TREATISE ON DYNAMICS. 27
unit, namely, that impulse which if applied to a unit of mass at rest
would cause it to move with the unit of velocity. Thus, if we
denote the impulse by B, and suppose the body originally at rest, we
have
and R will equal unity when
iV/=l, and f= 1.
The term "impulse" must not be confounded with the term
'• impulsive- force."
The impulsive-force is the force P, which is generally very large
and not known, whicif acting through the finite but extremely short
time fi — /j, produces a finite effect.
The impulse is the whole action of the impulsive force during
the interval t.^ - ti, and is, as we have seen, generally known.
The impulsive force P is of the same nature as all other forces,
and is, of course, comparable with them in magnitude.
The impulse R is entirely different in its nature from forces, and
can only be compared in magnitude with other impulses.
The magnitude of R depends jointly on P and ^2 — U) and though
the latter is very small, yet its value in one case may bear any
ratio to its value in another, without violating this condition.
Since an impulsive-force is not different from any other force,
all the results that we shall arrive at for forces generally will be
true of impulsive forces, unless there is some particular consideration
which excludes them.
It would be premature at present to describe the nature of the
action which takes place between bodies when an impulse is pro-
duced, or, as it is said, when there is "impact" or "collision." All
this will be fully discussed hereafter. The class of problems to which
it leads are particularly beautiful, and have this advantage, that all
the differential equations to which they lead are of the first order
instead of the second.
27. From the identity of our ideas of force, whether considered
as in equilibrium or producing motion, it follows that when any
number of forces arising from different causes act at the same time
on a particle in motion, Ave may replace these forces by their re-
sultant calculated on the principles of statics, and if we suppose this
force to change from instant to instant, so as always to be the
resultant of the variable forces which act on the particle, the effect
of this single force will be the same as the effects of those forces of
Avhich it is the resultant. Or, which is more convenient, Ave may
28 A TREATISE ON DYNAMICS. [c'HAP.
replace the forces by the three components of that resultant in three
constant directions.
Let u, V, w be the components at time t of the velocity of the
particle in the direction of three axes ; X, Y^ Z the sums of the
components in directions of the axes of all the forces which act on
the particle at time t.
X+tX, Y+IY, Z + hZ the same quantities at time t + ct. cX,
3F, SZ may depend on 2/ explicitly, and also on the change of
position and velocity of the particle, but in all cases they will vanish
with ht.
Now we are told by the second law of motion, that each of these
forces will produce the same effect on the velocity of the particle
during the time 0/ as if the particle were at rest at the commence-
ment of ht, and that force were the only one acting.
Hence the variable force X will produce in that time a velocity
cu which lies between -^ and ^^ ^jr^-^ — , and the other forces
will produce effects in their own directions, namely,
, , ^ YU .(Y + hY)U
d V between -^y and ^r? >
M M
, . , Zot ^ (Z + lZ)ht
and 6 7V between ^rj- and ^r~ — .
so that the components of the velocity at the time t + ht will be
ti + ^u, v + dv, fv + Zw, where 2?<, hv, Ziv are properly the increments
of u, V, and w.
Hence, passing to the limit, we have
M -r = X, M -7-= F, and M -i- = Z.
dt dt ' dt
If X, y, z are the coordinates of the particle at time t,
dx dy dz
'' = Tr'=dt''' = Tt'
d^r d^7i d^z
These equations must be considered as the fundamental equations of
dynamics where the motion of a particle only is considered.
28. It may be objected to what has preceded, that we have
used terms which have in themselves a meaning in ordinary lan-
guage, and that we have assigned to these terms a sense differing
I,] A TREATISE ON DYNAMICS. 29
from, or only partially agreeing with, their ordinary meaning, and
that this sense has been assigned arbitrarily.
To this we reply that we are perfectly at liberty to give any
arbitrary definitions that we please, provided only that we adhere to
them throughout the subject.
The Science of Mechanics, however, is particularly unfortunate
in this respect, that the terms used in it have, in addition to the
meaning given them by definition, a meaning in common conversa-
tion. It arises from the fact, that in the early ages of the science
erroneous, very erroneous notions Avere entertained of the principles
of the science, and the phraseology adopted expressed those erro-
neous notions. When the true principles of the science were
gradually developed, it was found convenient to retain the old
names expressing the same analytical combinations and numerical
quantities as before, though all idea of the name expressing any
principle was given up.
In popular language, however, the old notions still cling to the
words.
Such terms are " momentum," " accelerating-force," " vis viva,"
&c.
We caution the student, once for all, to take the words in the
sense assigned them by definition and in no other.
29. It is a matter of every-day experience, that bodies of any
material if left unsupported fall to the ground, and that some amount
of force is requisite to countei-act this tendency. This is caused by
the attraction of the earth on the body. This force, with which the
earth pulls the body, is called the weight of that body, and it is
found by experiment that the weight of a given volume of any
particular substance (as lead) is very nearly the same at all points on
the surface of the earth, and for small distances above it. Assuming,
for the present, that it is accurately so, a heavy particle when aban-
doned to itself in vacuo is acted on only by this constant force, and
will consequently, as has been shewn in the preceding articles, move
with uniformly-accelerated motion. It must not be imagined that we
assert here that we have proved that a particle abandoned to itself m
vacuo will move with uniformly-accelerated motion, but we have
proved it on the assumption of the two laws of motion and of the
fact that the weight of a body at the same place is constant.
If we take two similar and equal particles, these will have
the same weight and the same mass, and consequently the accele-
rating forces of their weights will be the same, and if the particles
30 A TREATISE ON DYNAMICS. [CHAP.
are let loose together they will move on together: and this will not
be altered if we suppose them joined, and so for any number. That
is, for the same substance the accelerating force of the attraction of
the earth is constant.
This might have been deduced at once from the consideration
that both the masses and weights of different portions of the same
substance would be proportional to the volume and therefore to each
other.
It is found by experiment that, for bodies of all substances, and
not for bodies of the same substance only, the accelerating-force of
the earth's attraction is the same, and such as to generate in any body
whatever, i?i vacuo, in a second, a velocity with which it would move
over 32-2 feet in a second. If then we take a second as the unit of
time and a foot as the unit of space, the numerical representation of
the accelerating-force of the earth's attraction will be 32-2. This
is written g, and is called the accelerating-force of gravity.
If then W be the weight of a body and M its mass, we have the
equation W - M.g
By the weight we mean the force with which it presses down-
wards or which is requisite to support it.
Since the quantity g is the same for all bodies, we see that the
weight and mass of a body are proportional. The quantity g, though
very nearly constant, is not accurately so. Its magnitude is different
in different latitudes, and at different altitudes above the earth's
surface. The variations, however, for positions within our reach are
so small that they are generally omitted, and g is considered constant.
When we are concerned with very great heights on the earth, or
with bodies at distances from the earth, we are obliged to take into
account the variations in its magnitude. The law of its variation, and
the great principle of which it forms a part, will all be fully explained
in their places. They have however more to do with the facts than
with the principles of dynamics.
30. The fact that the accelerating-force of the attraction of the
earth on any body is the same whatever the weight of the body, has
led in the first class of problems that we are concerned with, viz.
falling bodies, to the neglect of the weight and mass of the bodies
and the consideration only of the accelerating force of that weight,
viz. g. And this has extended itself very'much to other problems,
so that we constantly meet with the term accelerating-force applied
absolutely, without any reference to the force or mass on which
it acts. If in all such cases we understand by the term "accelerating-
I.] A TREATISE ON DYNAMICS. 31
force" such an expression as "a force whose accelerating-force" it can
create no ambiguity, and is a very admissible abbreviation. For
instance, -sve should find it said, "A body descends under the action
of a constant accelerating-force g " Instead of "A body
descends under the action of a force, whose accelerating-force is
constant and equal g " The term "accelerating-force" is in
constant use in this sense, and, provided this meaning is always
attached to the word, there is no objection to its use. In Dynamics
the weight of a body is scarcely ever used, we employ instead M.g,
because it very often happens that the factor M may be divided out
from both sides of our equations.
31. The unit of force has been chosen so that the equation
F — MJ" may hold. In common affairs the unit of force is chosen
arbitrarily, and this unit is generally used in the science of Statics.
It is necessary, therefore, to explain how the relation between the
two may be found. For this purpose, we must be able to express
the weight of the unit of mass in terms of the common standard; in
pounds, for instance.
Let then the unit of mass weigh ?i lbs. and let TV be its weight
in terms of the dynamical unit; from the equation P — Mfwe have
W=Mg = g,
and therefore g and n express the same weight referred to different
units. Hence, to reduce any dynamical force P to lbs. we must
multiply it by the factor - .
g
32. We will now proceed to explain the theory of gravitation
which was mentioned above, and the nature of the reasoning on
which the truth of it, and of the laws of motion, depends.
When one particle of matter is said to attract another, it is
meant that if these particles be placed at a distance from each other
and then abandoned to themselves, they will commence moving
towards each other, or, that if they be not so abandoned to them-
selves, a certain degree of force must be applied to them to prevent
their approaching each other. The intensity of this force for
different particles at different distances depends on the law of at-
traction and its intensity.
For instance, every substance is attracted to the earth, and we
see that if any substance is left to itself it immediately commences
moving towards the earth, or if not, a certain degree of force is
required to support it.
The law of gravitation is this.
32 A TREATISE ON DYNAMICS. [cHAP.
Every particle of matter attracts every other particle of matter
with a force which vai-ies directly as the product of the masses of
the two particles, and inversely as the square of the distance between
them.
Thus if m and in! be the masses of two particles, and r the
distance between them, the force with which they attract each
other, or whicli it would be necessary to apply to each of them to
keep them asunder, will be C . — \ — , where C is some constant
quantity which is independent of the masses of the particles and of
the distance between them.
^ • , <. ■ , . . . ^ m.m'
Smce the force actmg on m to move it towards ot is C . — ^— ,
the accelerating-force of this force on m' is C .-^, which is indepen-
dent of the mass of the attracted particle.
That mass is frequently chosen for the unit of mass whose
attraction at the unit of distance has the unit of accelerating-force ;
in this case C^l, and the attraction between two particles is
- , m . m'
expressed by — 5— ,
The determination of this mass, that is, the expression of it in
terms of a given volume of some specified substance, requires most
difficult and delicate experiments. It has, however, been effected
with great accuracy. An account of the experiments will be given
hereafter.
33. Since, then, every particle of matter in the universe attracts
every other particle of matter according to the law just explained,
we are able by this law and the equations of motion which we
have obtained, to calculate to any degree of accuracy the motions
of any one of the heavenly bodies, and to predict the exact direc-
tion in which it will be seen from the earth at any proposed
future instant, however remote.
These calculations are very intricate and laborious, and involve
many complicated considerations. Astronomical Instruments have
also arrived at extreme perfection, so that we are able also to
observe at any instant the directions in which these heavenly bodies
are seen from the earth. And the more exact our calculations are
made, and the more perfect our instruments and methods of
observing, the more minute are found to be the agreements be-
tween these results of calculation and observation.
I.] A TREATISE ON DYNAMICS. 33
Nothing can afford a more complete proof of the truth of our
assumptions than this. In order, however, that the proof may be
fully appreciated, the whole of the processes of calculation and obser-
vation should be familiarly known. It is impossible, without some
acquaintance with an observatory, to realize the degree of accuracy
to which observations can be carried, and it is this accuracy which
conveys the most perfect conviction to the mind.
Perhaps the recent discovery of the planet Neptune may afford
a more striki?ig proof of the truth of these laws. Some minute
disagreements between the results of calculation and observation had
long been noticed in the positions of the planet Uranus ; and it had
been suspected that there was some other body in our system, to the
neglect of whose action these differences might be due. But it was
long deemed past the powers of calculation to discover where a body
must be to produce exactly such differences. Till at length the
labour was undertaken and completed by Mr. Adams, in England,
and shortly after by M. Leverrier, in France ; and their efforts
were crowned with success by the consequent discovery of the
planet Neptune by Dr. Galle, at Berlin, and, almost, on the same day
by Professor Challis, at our own Observatory.
34. When the term " mass" was first introduced, it was spoken
of as the " quantity of matter" of a body. The propriety of this
expression appears from the following consideration.
The mass of a body remains unaltered whatever changes of state
the body undergoes, provided nothing is added to or taken from it.
If a lump of iron be heated so that its volume is increased, its
mass remains unaltered. If it were removed to the Moon, where its
weight would be diminished to one- sixth, or to Jupiter, where it
would be increased to two and a half times its amount, its mass would
remain unaltered.
If a quantity of water be converted into ice or steam, or decom-
posed into its component gases, the mass still remains unaltered. We
see then that mass, as it has been defined, has two important pro-
perties besides that which is taken as the basis of the definition, and
which are by no means consequences of the definition. The one,
that which has been just described, which entitles it to the name
" quantity of matter," and the other, that in the law of gravitation
the attraction of two particles varies as their masses jointly.
35. We will conclude this chapter by explaining a condition
which any equation expressing a relation between concrete quanti-
34 A TREATISE ON DYNAMICS. [cHAP.
ties must satisfy. It must be homogeneous with respect to each of
the independent concrete units that enter into it, provided no parti-
cular unit has been assumed in obtaining the equation.
Since every letter represents a number, any combination of letters
will also represent a number, and any equation is algebraically correct
when the number on one side of it is the same as that on the other.
When, however, a letter represents a concrete magnitude expressed
numerically, and the unit to which the magnitude is referred
changes, the number representing the magnitude must change in
the inverse ratio. Now in this subject certain of the units are
defined with reference to others, so that three only are independent,
and this must be considered in estimating the dimensions of any
term of an equation. Ihe independent units are those of duration,
length, and mass. If the unit of length be increased ?^-fold the units
of velocity and acceleration will also be increased w-fold. If the
unit of duration be increased w-fold, the unit of velocity will be
diminished to one m* of its magnitude, and the unit of acceleration
to one («^)'^. And similarly the other units will vary. This is what
is meant, when velocity is said to be of one dimension in space and
minus one dimension in time, and when accelerating-force is said to
be of one dimension in space and minus two dimensions in time :
when letters referring to these quantities occur in an equation they
must be estimated in this manner; and when so estimated the
dimensions of every term of the equation in each of the independent
units must be the same. If this were not so, by changing one of
the units for which it did not hold, we should change the several
terms of the equation in different ratios, and the equality would be
destroyed. But the relation expressed by the equation is true in-
dependently of any units.
Let us examine, for example, the equation
Mk'^ = -Wh sine.
M is a mass, and W which is a force is of one dimension in mass,
so that the condition is satisfied with reference to the unit of mass.
Again, A; is a line and it enters to the second power, W is of one
dimension in space, and h is also a line thus making two dimensions,
and the condition is satisfied with reference to the unit of length.
Again, dt enters to the second power in the denominator, and
W which is a force is of minus two dimensions in time, and the
condition is satisfied in this case also : sin d is a ratio and of no
dimensions. This equation also furnishes an exception : on the left
^•]
A TREATISE ON DYNAMICS.
35
hand side there is in the numerator d^d, or the angle enters to the
first power, on the right hand side it does not enter at all: and
what is the cause of this? The equation is only true when the
angle subtended by an arc equal to the radius is used as the angular
unit. Any equation may be deficient in this respect, if it be re-
stricted to one particular unit.
Library, ))
Of
Oajlft.ri)^?'
3— a
36 A TREATISE ON DYNAMICS. [CHAP.
CHAPTER 11.
36. Having found in the last chapter the equations of motion of
a particle, the object of the present one is to explain the manner of
proceeding so as to determine from them the nature of the motion
which will take place.
The simplest case is when the motion is in one line and all the
forces act in that line.
Let M be the mass of the particle, x its distance at the time t
from some fixed point in its line of motion, v its velocity, P the
resultant of all the forces which act on it : then the equations of
motion are
^■^-^''
dx
Since, however, the mass of the particle is constant, we will divide
p
both sides of the first equation by iVf, and write y instead of -^.
f is the accelerating-force of P on M. We have then
dt' "-''
dx
'' = dt-
Now the motion of the particle will be completely determined
when we are able to assign at every instant the position of the body,
and the velocity with which it is moving ; that is, when v and x
are expressed as functions of t and known quantities. When this is
done the velocity in any position may be found by eliminating t
between the two equations giving v and x.
37. As a particular case suppose the body to be acted on by a
constant force: in this case /is constant, and we have
W-r: 0)
•■-4r/'+C' (^>
I and x = ifl'+CH-C' (S)
II.] A TREATISE ON DYNAMICS. 37
We have here obtained the expressions for v and x in terms of t,
but these expressions contain unknown constant quantities C and C,
which have been introduced by integration : to determine these it is
necessary to know the circumstances of the motion at some specified
time. Suppose, for instance, it is given that at a time /j the body is
at a distance x^ from the origin, and moving with a velocity i\ ;
making these substitutions in the above equation, we have
v,=/.t,+ C
from which to determine C ancf C. When these are determined
and substituted in equations (2) and (3) the values of u and x will be
completely expressed in terms of t and known quantities.
The constants C and C might be equally well determined from
the position at one time and the velocity at another ; or from the
positions at two different times ; or from other conditions under
different forms. Variations of this sort constitute the difference
between one problem and another.
Eliminating t between the equations (2) and (3) we have an
equation between v and x. This might also be found as follows :
de ~^ '
dx d'x _ c p dx
''' ^Tt dt^^^-^'Tt'
d-fy-/-
c
or v^ = 2f.x + C"; (4)
and C" must be determined from conditions analogous to those
already alluded to. Such as that the velocity is y, when x = x^,
from which vi' = 2fxi + C"
and.-. v^= «,'+ 2/. (x-x,).
When a body moves in a vertical line acted on only by the
attraction of the earth, within moderate distances, it may, without
sensible error, be considered as acted on by a constant force. In
this case, if x be measured upwards, and the accelerating force of
the earth's attraction be written as usual ^, we shall have J'- — g
since the force acts in the negative direction, or tends to diminish x.
The preceding equations become
v = -gt + C, (2)
x = -lgt'+Ct + C', (3)
and v' = -2gx+C" (4).
38 A TREATISE ON DYNAMICS. [cHAP.
Suppose now that the body is projected upwards from the
origin with the velocity V, then, if t be measured from the time of
projection, we have v=V when t = and x = 0, and therefore our
equations give
o = c\
and therefore v = V — gt (5)
^=Ft-^ge (6)
v'=V'--2^x (7).
From equation (5) we see that as t increases v diminishes, that
J/
when t = —, V is zero, and that afterwards v becomes negative and
increases in actual magnitude ; also from equation (6) we see that
X commencing at zero increases till t = — when it equals — , that
2 V
it then diminishes, becomes zero when t = - — , and after this
g
becomes negative and continues to increase in actual magnitude.
From this it follows that the body will move upwards at first with
V-
a continually decreasing velocity till it attains a height , when
''to
it comes to rest and immediately commences moving down-
wards with an increasing velocity; that the time of attaining its
V .
greatest height is — , and the time of returning to the point from
which it started is twice this. Equation (7) shews that the velocity
at each point in its descending course is the same as at the corres-
ponding point in its ascending course. It is unnecessary to dwell
longer on this case ; the reader will be able to frame for himself
endless variations of it.
38. If the force be not constant, f will have different values
at different times, and consequently will be a function of the time.
Suppose f expressed explicitly as a function of t, in this case the
proceeding is as follows :
(Px _ r
He' J'
x=fJfdi' + Ct + C',
and the determination of the constants is the same as in the preced-
ing case.
II.] A TREATISE ON DYNAMICS. 39
30. If/ does not depend directly on t it must depend on the
position oi- velocity of the body, or on some circumstances external
to the body, or on some combination of them. The first of these,
where / depends on the position of the body, is the only case to
which any general principles of solution are applicable : in the
others, except in some simple cases, the equations can only be solved
approximately, though it must be borne in mind that they are
themselves strictly accurate.
When the force depends only on the position of the particle,
it can only vary with x, and therefore y will be a function of x;
let this bey(.i*); so that the equation of motion is
cFx „
^ dx drx ^ rf ^ dx
and integrating this, we have
This equation gives the velocity of the particle in any position.
To determine C we must know the velocity is some particular
position, or have some equivalent condition.
Suppose the integration on the right hand side of the equation
effected, as it generally can be ; v^ is then given as a function of
X and constant quantities ;
^dxV .
Fix);
Q
and to determine the position at any time, we have
di__^ 1
where the upper or lower sign must be used according as x is in-
creasing or diminishing with t.
Integrating this we get
C:
= i.
or, supposing the integration on the right hand side effected, and
the constant included
l^iPix),
and consequently x = \//(/),
and the problem is completely solved.
4^ A TREATISE ON DYNAMICS. [cHAP.
The different forms that may be given to f{x) and the different
conditions that may be assigned for the determination of the con-
stants, are the points in which one problem differs from another.
Readiness in performing the operations, in expressing analytically
the conditions given for the determination of the constants, and in
interpreting the results obtained is only to be acquired by the solu-
tion of a great number of problems.
drx
The equation -rp=fi^)
can sometimes be solved at once as a differential equation ; in this
case it is generally the best method of solving the problem. We
thus get at once
and hence v = -j- = -4^' (t),
by differentiation: in this case \/^(0 will contain two unknown con-
stants, which have to be determined as before explained.
40. We will take the case where
f{x)=-nx,
fx being constant, as affording an example of both methods of solu-
tion, and being itself most important. It is the case in which the
force varies directly as the distance of the body from some fixed
point from which x is measured, and acts towards that point.
d'x . .
-d? — ^^' (')
dx d^x ,, dx
^dxV
Tf dt'~ '^^''■'dt
, fdx\^
fXX
Suppose now that when the particle is at a distance x^ from the
origin it is moving from it with a velocity Vi ; then we have
v,' = -iJiX,'+C;
and .-. v' = v,'-fx(x'-x,') (2)
This equation gives the velocity in any position- From it we
have
^^=^Jv,^-^{x^-xO,
we must take the upper sign because we suppose the body to be
dx .
moving from the origin, and therefore -j- is positive.
"•] A TREATISE ON DYNAMICS. 41
we have therefore -r- = , ■ ,
■h
or altering the form of the equation.
from which t = - _ sin~^ ^^'^
J/J- Jvr+fx.
^s/f
x,^ .smj~n{t-c'); (3)
suppose that when / = 4 the value of x is x.,, we then have
1 . , Xzjui
t, = -j^. sin-' - j^^^ + c',
which gives a value of c': we will not substitute this value because
it is complicated, and the letter c' may stand as its representative.
Equation (3) shews that the value of x can never exceed
J'
— +Xi- : this is a known quantity ; call it a ; then equations
(2) and (3) may be written
v^ = ix(a-- X-), (4)
x = a.&inj^(t-c') (5).
Equation (5) shews that as t increases x will increase till
J^{i -c') first becomes an odd multiple of -; x will then be equal
to a ; and equation (4) shews that v will then be zero or the body
will come to rest. As t goes on increasing x will diminish and the
body will move in the opposite direction ; when Jfx (i - c') becomes
a multiple of tt, x will be zero and the body will be at the origin
moving with a velocity a Jjx in the negative direction : as x in-
creases negative the velocity will decrease, and when x = -a or
Jix{t-c') is again an odd multiple of ^j ^ will again be zero, and
th body will be at rest at a distance a on the negative side of the
origin. After this it will again move in the positive direction, and
will continue moving backwards and forwards between the limits
X = a and x = — a. Any motion of this kind is called oscillatory.
We saw that the body came to rest whenever Jfx {t - c') was an
odd multiple of ^ ; hence if t', t", &c. be the successive values of/
for which this is the case, we must have
42 A TREATISE ON DYNAMICS. [cHAP.
JJi(t"'-c')^(2 7i + 5)~,
.: t"-t' = t"'-l" = 8zc.= ^.
This result is curious, inasmuch as it shews that the time of
moving from rest to rest is independent of the distance between the
points of rest.
The time from rest to rest is called the time of an oscillation, the
distance between the points of rest is called the amplitude of the
oscillation. The peculiarity of this particular law of force is that the
time of oscillation is independent of the amplitude.
41. We will now apply the other method of solving the equation
rfF-"/^^ = ^ (^)-
The integral of this is known to be
X = A sin{J]x . t + B), (2)
where A and B are unknown constants.
Differentiating (2) we have
V = -£ = A Jil. cos(Jii J + B), (3)
Equations (2) and (.3) are exactly of the same form as those
obtained before for a: and v. The constants may be determined in a
similar manner and the interpretation is exactly the same.
Whenever an equation appears under the preceding form, this
latter method should always be adopted in integrating it. The
equation is one which presents itself continually, and should be
familiar to every one.
If the equation is of the form
which corresponds to a repulsive force varying according to the same
law, the integral is
X = C, e VJ^- « + Ca eV^r. <,
and the motion will not be oscillatory at all.
11. J A TREATISE OX DYNAMICS.
43
42. Let us now consider the case where the force varies as the
«"' power of the distance ; in this case
d'x
-11^ = -^'=''
•0)
dx d^x „ „ dx
„ (dxV ^ ^fxx"^'
^;
If I' = 111 when x - a, we have
w + 1
O ,1
r-)..
....(2)
cy
which gives the velocity in any position.
The next integration which would give the relation between x
and t cannot generally be effected.
If t>i and a are both positive the body is then moving from the
origin, and equation (2) shews that as x increases v diminishes; let
c be the value of x that makes v = 0, then c will be given by the
equation
and equation (2) may be replaced by
^,^^ _
There is a peculiarity about this expression for the velocity when n
is even which is worthy of attention, as it illustrates a point re-
quiring great care when physical conditions are to be expressed
analytically.
When x = c, i; = 0, and the particle is at rest at a distance c from
the origin, acted on by a force tending towards the origin ; it will
therefore commence moving in that direction, and its velocity at any
distance will be given by equation (3).
First let ?« + 1 be positive : then if V be the velocity of the body
when it reaches the origin,
«+ 1
So far the force has been increasing the velocity of the particle, its
direction now changes, since it is supposed to act towards tlie point
44 A TREATISE ON DYNAMICS. [CHAP.
chosen as origin, and it will therefore tend to diminish the velocity ;
and since the intensity of the force is the same at equal distances
on opposite sides of the origin, it is to be expected that the velocity
will be destroyed by exactly the same steps by which it was gene-
rated. This conclusion follows from the similarity of the action of
forces when they accelerate and retard the motion of a particle. It
also follows from the equation
^,^J±_ (c«+._j,n+n /gX
w+ 1 ^ ^ ^ '
when 71 + \ is even, since in this case the equation gives the same
value to V for values of x equal in magnitude and of opposite signs,
and therefore as x increases negative the velocity will gradually
diminish, being always the same as it was at an equal distance on
the opposite side, and will ultimately equal zero when x = — c.
When, however, w + 1 is odd, negative values of a; substituted in
equation (3) give for v values greater than V. Now it is impossible
that a force acting in a direction opposite to the motion of a particle
can increase its velocity : therefore equation (3) cannot correctly
represent the velocity on the negative side of the origin. Referring
to the equation
-di^^-'^' ^')
from which we started, we find that - fxx"^ the analytical expression
for the force, represents correctly a force tending towards the origin,
both for positive and negative values of x, when n is odd ; but that
when n is even it represents a force tending towards the origin for
positive values of x only, and therefore conclusions derived from it
will not be true for negative values of x. For negative values of x
the correct equation is
d'x
from which v^ = 1- C,
?i+ 1
and making v ^- . / ~ ,
when x = 0, which amounts to supposing the particle to pass through
the origin in the negative direction with the velocity V, we have
as the equation to be used in this case instead of equation (3).
II.] A TREATISE ON DYNAMICS. 45
This equation shews that the velocity gradually diminishes as x
changes from to - c when it equals zero, consistently with what
takes place when ti is odd.
Now let 71 be negative, and ^ - 7/1. Equation (3) becomes
„.=^(_i,_A^) (5)
m-1 \x" ^ c"^V '
and we see that the velocity increases without limit as the body
approaches the origin. This was to be expected since the force
which produces the motion itself increases without limit. As in the
previous case, when ?« - 1 is even, this equation shews that the
velocity decreases, as the body recedes from the origin in the nega-
tive direction, till it comes to rest at a distance c. When 7n - 1 is
odd, negative values of x substituted in (5) give impossible values to
V. The explanation in this case is the same as before: when m is
odd — ^ represents the force correctly on both sides of the origin,
when m is even - ^ must be used for positive values of x, and -^for
negative values, and the velocity on the negative side of the origin
will be given by the equation
2m / 1 J_\
^ ~ m-l \x'"-' "^ c"-7 ■
In all these cases then the body will pass through the origin and
move in the negative direction, losing its velocity at the same rate
at which it acquired it, and will come to rest when x=-c. It will
then move towards the origin and pass through it as before, con-
tinuing to oscillate between the limits x ^c and x = - c.
43. In the cases of rectilinear motion which have hitherto been
considered, the forces which acted on the particle were either
constant, or some functions of the position of the particle.
The force may also be some function of the velocity with which
the particle is moving; that is, the equation of motion which has
hitherto always been of the form
d'x .. .
d'x Jdx\
may now become 'Trr=J\jj)>
d'x „{ dx\
""' di'
46 A TREATISE ON DYNAMICS. [cHAP.
These equations when they occur must be solved by the ordinary
methods applied to differential equations.
We will solve two or three of them to exhibit the methods.
Let the equation of motion be
Ti
^-a)" (■)
that is, suppose the only force which acts on the body to be a
retarding force varying as the w* power of the velocity. The above
equation may be written
1 dv
The integral of which is
Let V = Di when t = 0, then
1
C,
and — -r ;rT=(«-l)M^-
This equation shews that as / increases the velocity will con-
tinually diminish, but will never be actually zero. From this
equation
1+V"~^. (}l — l)fxt'
and therefore -tt = v
at
from which C + .-^^l+^'"~'-("-^>'t
1 + Vi"-^.(n- l)fjii]"-^
t),"-'.(«-l )M
which gives a; as a function of t.
The above solution becomes nugatory when 7i=l ; in this case
the solution of the differential equation takes a different form ;
we have
d'x _ dv
"de^'^Tt'
dx ^
let Ui be the velocity when x^a, then
(1)
II.] A TREATISE ON DYNAMICS. 47
dx
dt
-v,=(x{a-x); (2)
1 iI^^ .
fxa + Vi — nx ' dt
.-. log {n{a-x) + v,} = -fxt + C,
let t = when x = a, then
C = log v„
and /i (a - x) + ?;,=??, e"**', (3)
which gives x as a function of t.
Equation (1) might have been integrated as follows:
1 dv
.: log v= — nt + C,
log Vi = C;
.-. t' = v,6-'^' (4).
44. Cases of motion of this sort are more curious than useful.
The only case known to exist in nature of a force depending on the
velocity of the moving body, is when the body moves in a fluid, or,
as it is commonly called, a resisting medium. In this case it is
found that the fluid exerts a pressure on the particle in a direction
opposite to that in which it is moving, and varying directly as the
square of the velocity. If the body is acted on by no other force the
solution of this is included in the general case just discussed. Let
us consider the case where the body is acted on by a constant force
as well.
The equation of motion is
£^=/-*"'. 0)
dv
or -y- = f-kv^ ;
dt ^
_J_ ^-
•*• f-k^-^ ' dt~ '
l=.log^^C±H?=^+C.
2 Jfl^ J/- V J/c
Let v = Vi when t = 0, then
48 A TREATISE ON DYNAMICS. [CHAP.
or \//+ ^ V^ ^ J/+VjJ^ ^2 ViS . '
Jf-vJk-Jf-vJk' '
=y
where m = ^'^^4 !-^ :
we may also solve equation (l) as follows :
dv
dt
dv dx
^Tx -Ji^
dv
'■Tx-
So that
(1) becomes
dv , ,
V ^f- + kv' =
dx
/•
Let 2 w = ^;^ then
dm
dx
dv
-Tx'
and
.■.p-,2k
dx
or
»' = |.2C
. e-'*-.
Let v =
Vj when ar
— a,
then
f-
kv' = (f-kv
')e-^'^-'
(3).
Equations (2) and (3) comprise the complete solution of the
problem ; they give two relations between v, t and x.
It appears from an examination of either of these equations that
the velocity continually approaches kJ r, as its limit; if the original
velocity Vi is greater than this it will continue diminishing, but will
never become so small as a/t; if the original velocity is less than
Kj T it will continually increase, but never become so great as ^/ r ;
the name "terminal velocity" has been given to ,/ j .
II.} A TREATISE ON DYNAMICS. 49
In motion of this kind considered as motion in a resisting medium,
the resistance is always in the direction opposite to that in which the
body is moving, so that if the forces whicli act on the body are such
as cause it to come to rest and then commence moving in the opposite
direction, the direction of the resistance will change. In any case of
this sort the problem must be separated into two, the one commenc-
ing where the other left off, at the point where the discontinuity-
takes place. The following is an instance.
45. Let a constant force act on the body in a direction opposite
to that in which the body is moving. The equation of motion is
de- J ^"^
dv J. J 2 - ^ /
The integral of this is
1
Let i\ be the initial velocity, then
• tan-y^.^^C-^
t-^n-' ^ J. v-tan-\^ J. v, = -Jfk.t (1).
As t increases v diminishes, and becomes zero when
Adopting the other method of integration, we have
dv
v^^kv' = -/; -r— - ^""^ '. ^ L^\^^
dx
.H.X.
f-.w^-Ir^C.r^
2 k
.: f+kv^=9.kC.e-"'\
Let X be measured from the starting point, then
f+kv;' = 2kc,
and/ + /.V=(/4-^06-*- (2),
which gives the velocity of the body in any position.
By elimhiating v between (l) and (2) we may obtain a relation
between x and /. If in equation (2) we put u = 0, we have
e"^ = y^+l (3),
50 A TREATISE ON DYNAMICS. [cHAP.
which gives the distance to which the body will move before it
comes to rest. After this it will commence moving in the opposite
direction, and the equations which determine the velocity and po-
sition in terms of the time, will be deduced from the case previously
discussed, by putting »>, = ; we thus get
and v-' = ^{l-e-"'% (5)
in equation (5) x is measured in the new direction of motion from
the point where the body came to rest.
If a be the value of x which satisfies equation (3), equation (5)
may be written
i;2 = /{l_e-^^C«-)}^ (6)
in which x is measured from the original starting point in the
direction of projection.
When a; = in this equation we have
or, substituting for a its value given by equation (3)
which gives the velocity with which the body returns to the point
from which it started.
This is the case of a body projected upwards in the air : writing
g for / the results may be stated as follows.
A body projected upwards with the velocity »j, will rise to a
height a given by the equation
g.^'-^g + kv,^
in the time /, given by the equation
Ujgk = t&n-' ^-.v,,
it will then return in the time t^ given by the equation
Jg + kv^'-v.Jk'
and reach the starting point with the velocity v, */ — ~—^ .
n.] A TREATISE ON DYNAMICS. ol
These examples will be sufficient to shew how cases of motion of
this kind are to be treated ; it will be seen at once that the difficulties
they present are entirely analytical.
Before leaving the subject of resistances, however, we will make
one remark which will be understood at once by any one familiar
with the nature of fluid pressure.
The actual force of resistance exercised by the fluid will be the
same for all bodies of the same size and shape. Consequently the
accelerating, or rather retarding, force of the resistance will vary
inversely as the mass of the body in motion. For bodies of the same
shape and size may have very different masses if composed of diffe-
rent materials. Now it is with the retarding force that we are
concerned in dynamics. We draw attention to it in this case because
it is apt to be overlooked, from the fact that most of the forces with
which we ai'e concerned in dynamics vary directly as the masses of
the bodies on which they act, and consequently their accelerating
forces are independent of those masses.
46. The preceding examples will be sufficient to shew how any
problem on the rectilinear motion of a particle is to be treated. The
first step is to express analytically the forces Avhich act on the
particle, and to form the equation of motion; the next step is to
integrate this equation ; the third is to express analytically the con-
ditions given for determining the constants introduced by integration ;
and the last and by no means least important is to interpret the
equations thus obtained, that is, to examine them for different values
of the variable quantities which enter into them, and so to trace the
motion of the particle through any peculiarities it may present.
47. Let us now proceed to the case of curvilinear motion, and
first, for the sake of simplicity, we will consider motion in one
plane.
The equations we have obtained are
If X = and F=0, the particle will move on in a straight line Ijy
the first law of motion.
Let X= and F = - Mg. This will be the case of a body pro-
jected from the surface of the earth and acted on by gravity, the
4—2
52 A TREATISE ON DYNAMICS. [cHAP.
axis of X being measured horizontally and the axis of j/ vertically
upwards. The equations of motion are
(1)
d'x
df
= 0,
_J.........
Integrating
these
we
hav
dx
dt
e
dt
-c-
•«'•
Suppose the body to be projected from the origin with a velocity
V, in a direction making an angle a with the axis of x, and let t be
measured from that instant ; then we have
dx
(2)
, - FCOS a,
at
-~ = ^sin a — £•/.
dt
And integrating again, we have
X — Fcos a.t + C",
y =V s,\n a. t-\gf + C",
or since x and 3/ are zero when ^ = 0, we have
w= Fcosa J ) /g-\
^ = Fsin a .^- ^gt^ J
Equations (2) give the resolved parts of the velocity, and equa-
tions (3) the position at any time.
From (2) -/ = tana-77-^
^ ^ dx v cos a
which gives the direction of motion,
= JV'-2gtV^\na-^gH\
which gives the velocity at any time.
Eliminating t between equations (3) we have
^■*'
y = tan a .x- —rj^ j- ,
^ 2 F^ cos'- a
a relation between the co-ordinates of the particle independent of the
time, and therefore the equation to its path.
"•] A TREATISE ON DYNAMICS. 53
This equation shews that the curve in which the particle moves
is a parabola whose axis is vertical, and concavity downwards;
putting it in the form
/ _ ^-sina cos a Y_ 2F-cos'« / V'sm' a
we see that the co-ordinates of the vertex are
F* sin a cos a , F^sin^a
and — ,
g ^g
, ,, , . • 2 F^ cos- a
and the latus rectum is .
g
The height of the directrix above the origin is
r^sin^a F^cos^a - . , V'
-i which
2g 2g 2g
If the body fell from rest vertically through this height it would
acquire the velocity T; and since any point of the body's course
may be considered the starting point, it follows that the velocity at
any point in the path is that which the body would acquire in falling
verticall}'^ from the directrix to that point.
We see from equations (2) that the resolved part of the velocity
in a horizontal direction is constant throughout the motion, and that
the vertical component varies as it would do if the body were
moving in a vertical line. This consideration is useful in discussing
many points in the motion of projectiles.
From these equations there will be no difficulty in obtaining the
" range" of the projectile, that is, the distance of the point at which
it will strike a horizontal or inclined plane passing through the point
of projection ; and the time that will be occupied in it, which is
called " the time of flight."
Sometimes it will happen that V and a are not given, and have
to be determined by the conditions of the problem, as for instance,
to find V and a, that the path may pass through two given points.
By varying the conditions of this sort, an endless number of
problems may be formed, all of which may be solved by the appli-
cation of the above equations, either in their present or in a modified
form.
If instead of X=0 we had X constant, the case would not be
altered, for the two constant forces X and 1' would be equivalent to
some constant force in some constant direction ; and no force would
act in a direction at right angles to this.
54 A TREATISE ON DYNAMICS. [ciIAP.
48. It is unnecessary to consider the case of X and Y being
functions of t explicitly as this never occurs in nature.
Let then X and Y depend on the position of the particle, and
first let
X be a function of (x) alone
and Y a function of (y) alone.
In this case we have
~ dt • dl' ^y^ dt '
0)
(2).
These two equations give the resolved parts of the velocity, and
therefore the actual velocity and direction in any position. C and C
must be determined from the velocity and direction in some known
position.
From equations (2) we have
\A'xl ~ 2jf{x)dx+C' ^-"^
which gives the direction of motion at any instant, or the position of
the tangent to the path.
If we can integrate equations (2), we shall obtain two relations,
between x and t, and between 7/ and t ; each of these will contain
a new constant which must be determined from the position of the
particle at some instant : if between these two equations we eliminate
t, we shall obtain a relation between x and ?/ which will be the
equation to the path of the particle.
The equation to the path may also be found at once by inte-
grating equation (3). The constant in this case will be determined
from any corresponding values of x and y, or, in other words, any
point through which the curve passes. We may remark that equa-
tion (3) will always be integrable when equations (2) are, and never
when they are not.
II.] A TREATISE ON DYNAMICS. 55
dx , dy
pressed as functions of x and y ; we also have x and y as functions
of <; we may therefore by elimination find -7- and —- as functions
of ^; we shall then know the position, velocity, and direction of
motion of the particle at every instant, and the problem will be
completely solved.
49. As an example,
let X = -ixx, Y = -iiy,
so that our equations are
d^x \
d^y
d¥=-^y
'dx
(1)
m
., =G-«..
whenx = /«, and^ = ^, let Vhe the velocity of the particle and a
the angle which the direction of motion makes with the axis of x,
then we have F^cos^a = d —fxh^,
V'sin-a = C2-fxk',
from which (-jjj = F'cos^a + fxk'-fxx'',
{^^'=V'-shva+,.k^-j,y^;
d^ =fcl
" d^~ jF'cos'a + fxh'-^x''
dl ±1
dy ~ J V'sin^ a + fxk'-ixy' '
Taking the upper sign in each case and integrating, we have
- '~
J^ . < + C3 = sin"
JJ^.t + C.=
JV^ cos- a + [xh '
JV'Jm^a + fxk'
and therefore J/x . x = JV'^ cos" a + fx h' sin {J~ix . / + C3),
Jf^-y = JV-s>\n^a + (xk- . sin {JJi . t + CJ.
56 A TREATISE ON DYNAMICS, [cHAP.
If in these equations we substitute corresponding values oi x, y,
and t we sh^ll determine C3 and d, and x and 1/ will then be com-
pletely known as functions of t, and also -^ and --^ . Also by elimi-
nating t we should obtain an equation between x and y, which would
be the equation to the curve described by the body. It will be
found on elimination that this curve is an ellipse whose centre is the
origin.
We might have integrated equations (1) at once, as linear equa-
tions, and that would in this case have been the easier method of
solution though not so generally applicable. We will illustrate this
method with particular values of the constants h, k and a.
Suppose the particle to be projected from a point in the axis of .r
at a distance a from the origin in a direction parallel to the axis of 3^,
with a velocity F. The equations are
d'x
77?-^'^^ = ^)
The integrals of which are
x = A^ cos {Jn . 1 4 Bi),
y = A., cos {J'fi . t + Z?.),
and therefore -jj = - A, Jn sin {Jn . t + B,),
-j- = - A, Jn sin (Jn .t + B.):
when / = 0, we have
dx_ ihj_
making these substitutions, we have the equations
a = 7^1 cos B„ (2)
0=^ocos5„ (3)
= -A,J~n&mB„ (4)
V=-A,/llsmB, (5)
From (2) and (4), we have
B,=0, A^^a.
From (3) and (5), we have
II.J A TREATISE ON DYNAMICS. 57
Substituting these values we have
X = a cos (sfi^ . t)
y = -^sin(VM.0|' ^^^
-j-:=-aJnsm{J^.t)
"^^ [; (7)
^=Fcos(JJ..t)
^■^•'^""-.1 (8).
and eliminating i between equations (6) we have
a' ' F'
The equation to an elhpse whose semiaxes are a and ( 'nj; oi*
putting (b) for the last, we have
x= a cos ^t,
ij=:b sin Jfx t,
■£ = bJ',IcosJfxt,
a' b' '
In these equations / is the time from the extremity of that axis
which coincides Avith the axis of x.
These equations are very often useful and are worth remem-
bering.
Ill the preceding example* the resultant of the forces X and Y
always passes through the origin, and is equal to n times the
distance from the origin. This is a particular case of a class of
forces called central forces which will shortly occupy a large share
of our attention.
The example we have given will be sufficient to illustrate the
method of proceeding in problems of this class. In particular cases,
artifices will frequently suggest themselves which will render the
solution much easier; these however are best acquired by practice.
50. We will now enter upon the consideration of the case where
X and Y are each functions of both x and i/.
The above method of solution evidently will not apply, nor can
we offer any that will be generally applicable. The following how-
ever will often be found useful.
58
A
TREATISE
ON
The
equations
are
df
df
= F
[chap.
•(1)
" ^\Tt' dt'^dfdej~~\^dt ^^ dtj'
or d.v- = ^{Xdx+Ydy) (2)
When the right hand side of this equation is a perfect differential
of X and y considered as independent variables, we have by inte-
gration
If F be the velocity when x = a, and y=-h,
V'=c+f{a,b);
.-. v^-V'=fix,y)-fia,h)
which shews that the change in the velocity in passing from any one
point to any other depends only on the co-ordinates of those points,
and is independent of the path by which the body has passed from
one to the other ; that is, if from any point (a, h') we can project the
body with a velocity V in any number of different ways so as to
pass through the point (x, y), it will in each case pass through it
with the velocity v.
Again, referring to equations (l), we have
d^y d'x
''d¥--ydw='=^--y^-
If the right-hand side of this equation is constant and equal to
c we have by integration
dy dx ^ ,
dt -^ dt
The only case in which this method of integration is useful is
when c = 0, in this case the resultant of the forces X, Y passes
through the origin, and we have
dy dx J
dt ^ dt
If A be the area swept out by the line joining the origin and the
particle, we have
^ dA dy dx J
.-. 2A = ht+ C,
II.] A TREATISE ON DYNAMICS. 59
or if A be reckoned from the time when t = 0,
2A= hi,
that is, the area described in any time is proportional to the time of
describing it.
This is true whenever the resultant force which acts on the par-
ticle passes through the origin ; that is, through any fixed point
which may be taken as the origin.
These equations when obtained are only the first integrals of the
equations of motion ; the second must be obtained by artifices sug-
gested by the forms of the equations in each particular case.
51. There is a class of problems the inverse of those we have
been considering, viz. where it is required to find what must be the
forces under whose action a particle would move in a certain speci-
fied manner.
If the co-ordinates x, y of the particle were given as functions
of t we should obtain the resolved parts of the force at once by two
differentiations, this would be the simplest form of the pi'oblem. A
more common form of it is to find the force under the action of
which a particle would move in a certain given curve.
Let F(x,j/) = (1)
be the equation to the given curve : differentiating this equation
we have
If we differentiate this again, we have
Now our object is to find -^- and -~^ and we see that from
equation (1) we are able to obtain only a single relation (3) between
these quantities and others : some other condition therefore is neces-
sary to render the problem definite. We might, for instance, assign
one of the forces arbitrarily as a function of x and y, and equation
(3) would then enable us to determine the other : thus let
60 A TREATISE ON DYNAMICS. [ciIAP.
eliminating y between this and (1) we obtain
from which we may find -7- .
•^ at
dn d 2J
Substituting this value in equation (2) we obtain -jj , and ■— may
then be found by differentiation, or from equation (3).
In this case we had to integrate -r-^ to determine -7- ; and gene-
rally, whatever condition is given in addition to the equation to the
curve, it will be necessary to express it in the form
/(-^'§''J)-^ w
that is, in the form of an equation involving one or both of the
first differential coefficients and neither of the second differential
coefficients.
From (1), (2) and (4) we can always deduce equations of the
form
from which we obtain
The resolved parts of the force are then known.
Problems of this kind seldom present any difficulty, as the
operations consist for the most part of differentiation, and not of
integration.
52. For example, let it be required to find under the action of
what force a body would describe an ellipse with uniform velocity.
I-' ^^l'=» (■)
be the equation to the ellipse, v the constant velocity, then
(syns)'-' (^)
, . X dx V dy ^ fo\
from equation (1) -.-+^.^ = (3)
II.] A TREATISE ON DYNAMICS. 61
Eliminating y and ~ from these three equations we have
\dt) a*-(fl»-6^)a:' ^^
Differentiating (4) Ave have
Similarly we may obtain
Y^_ d^y _ c^h^v'y
M~dP~~ \v^{p~ra:ijf\^'
X and Y are the resolved parts of the force parallel to the axes.
Since l-^ i a*-(a^-b^)x' Y_a^y
we]|see that the resultant of X and Y coincides with the normal to the
ellipse at every point.
Let a = b and the ellipse be a circle ; then
and the resultant, which acts in the normal, and therefore always
passes through the centre
= JX'+Y-' = M-;
that is, a body will describe a circle under the action of a constant
force M— tending to the centre.
We may shew, generally, that when a body moves with uniform
velocity, the resultant force on it acts in direction of the normal : for
since the velocity is uniform
i^hm
a constant quantity ;
dx d^x dy
•'• df'd^^'di'
■g=-
Y de
•''X = d^"
dx
dt^
which proves the proposition.
62 A TREATISE ON DYNAMICS. [CHAP.
53. It is sometimes convenient to employ polar co-ordinates in
the solution of problems.
In this case the force is resolved in direction of, and perpen-
dicular to the radius vector.
We will conclude this Chapter by making the requisite trans-
formations.
Let P be the part of the force tending to the pole, T the part
perpendicular to that direction,
X = r cos d, y -f sin Q,
then we have P = - X cos - F sin 0,
r = - X sin + F cos Q.
The equations of motion are -r-y = X,
cosaJ|-sina^! = r.
dt^ dt^
-^ dx ^dr . ^dd
NoWj -r- = cos t^ J- - r sm o -j-r,
' dl dt dt
dy . .dr .de
-yf = sm -r- + r cos -T- ;
dt dt dt
fdev . .d'd
d^x .d'r ^ . .dr de
-^,=cosa^-2sm0^.^-.
d'v . „d'r ^dr dd . . {ddV . d'd
^ = sm0^4-2cos0^.^-rsmay ...cosa-.
Hence substituting
'■r fdey
7-'\dt)=-
de
dr de d'e_
'~~d'f'dt^''-dt'~-^'
These are the polar equations of motion.
The preceding transformation is the most direct, and leaves the
problem still in the form of two differential equations of the second
order between r, 6 and t.
The following transformation is more useful though more indi-
rect : we have
II.]
A TREATISE ON DYNAMICS.
63
~ = -Pcos0-rsme
^,=- P sine +T COS
From these
dy dx ,
Let X -f- -y -rT = "i
dt ^ dt
J . odd dy dx ,
and since K-r-=x~r--y-r- = h,
dt dt ^ dt
dh_dh dd_h dh
dt~ dd' dt ~r'' dd
and integrating
where - is written for r.
dd.
Again,
dr
dt
df
(1).
(2)
(3)
1 du do du
u'-Jd 'Tt^~''Td'
dh du .„ J d^u
"di'TQ'''''' 'W
Substituting this value of -j-5 in the equation
, dh dii d-ic ,„ 3 _ ^
we have -rr . -rz+ h^u^ . -r^ + «' ?r - P = ;
dt ' dd
dd'
, ^, . d^u P T du ^
and therefore, _ + „-_+ ^^,-^^3- • _ = O;
and substituting for hr the vahie given by (3) we obtain
P T du
d'u h.'ii'' h.-u""' de
1 +2
(4),
de
de
Tt'-
h
A TREATISE ON
\ DYNAMICS.
and
= Kn-Jl^2^
L^l.a
64 A TREATISE ON DYNAMICS. [cilAP.
(5).
Equations (4) and (5) though very complicated in appearance
will not be so in practice, since in the cases which occur the forms
of P and T will be such as to introduce considerable simplifications.
When the whole force which acts on the body is central, we have
T =0, and these equations become much simpler. We have in
this case
d'u P
. dt 1
These equations, however, and the class of problems depending
on them deserve an independent consideration.
III.] A TREATISE ON DYNAMICS. 65
CHAPTER III.
54. When the direction of the force that acts on a particle
always passes through a fixed point the force is said to be central,
and that point is called the center of force. In the cases that will
present themselves the intensity of the force will generally be a
function of the distance of the particle from the center and of
nothing else. Many of the properties of motion round centers of
force are true, whether this is or is not the case. The student will,
however, see from the proof whether this is requisite or not.
We will first shew, that however a body be projected, if it is
subject to the action of a central force only, it will always move in
one plane: and we shall afterwards confine our attention to that
plane. Let P be the accelerating-force to the center, x, y, z the
co-ordinates of the particle at time t, r its distance from the origin
which is the center of force. Then the equations of motion are
d^x _ Px
de~ r '
de r '
Combining these equations two and two, we obtain ^^^ O/ p.,.- y^
d'z d'y „ "* *"" '^
d^x d'2
^ df^'^'W'-^'
d'y d'x
Integrating these, we have
dz dy ,
dx dz .
'irr'-dt = ^'''
dy dx ,
""dt-yTt^^'^^
66 A TREATISE ON DYNAMICS, [cHAP.
where h^, h^, h^ are constants introduced in integration. Multiply-
ing them by x, y, and z respectively and adding, we get
= ^1 X + h-^y + h^z,
a relation to which or, y, and z are always subject.
But this is the equation to a plane passing through the origin,
therefore the body is at every instant in the same plane, and this is
its equation.
The curved line in which a body moves round a center of force
is called its orbit.
It may be seen from the above equations that the areas described
on the co-ordinate planes by the projections of the radius vector are
proportional to the times of describing them.
We will for the future consider the plane of motion as that of xy,
and retain the rest of the notation.
55. The equations of motion are
^ = -P- = - Pcos^l
r
g--f-— r ''
X = r cos 6, 7/ = rsmd, (2)
2 , „2 _ ,.2
X +
f = ^,..-..^ (3)
before, a: --^--1/-^ = h.
dt ^ dt
■ivT dy dx .dd
■-'fr" w-
It is found convenient to use the reciprocal of r instead of r, and
to denote it by u, so that
rr"" (=)•
Now since x=^- cos Q ;
u
dx __sme de cos d du d^
•*• It' u dt u' dd dt
— — hu sin 6 — h cos B -r^ ;
av
III.] A TREATISE ON DYNAMICS. 67
(Px J add J ad'u dB
.-. -,=-/„,cos0^-/.COS0^^ ^
= - Jl^U^ COS Q - h' U^ cos Q -TTi .
av
Substituting this in the first of equations (1), and dividing by
cos 0, we have
Equations (5) and (6) are the polar equations of motion : they are
equivalent to the two equations (1).
Since however equations (l) are both of the second order, their
integrals will contain four arbitrary constants. Equation (5) is of
the first order, consequently only three constants will be introduced
in the solution of equations (5) and (6). The remaining one h has
been already introduced in the process of transformation.
These two may be considered as fundamental equations by which
all questions relating to central forces may be solved. The constants
which are introduced in integration are determined by various con-
ditions which form part of the data of the problem. The quantity
h which occurs in these equations is twice the sectorial area swept
out by the radius vector in a unit of time. It is constant throughout
the motion, and must be known in order that the problem may be
completely solved. It will therefore be found useful to express it in
terms of some of the other quantities that enter our equations.
We have /e = r*-r-
at
dt
Now V-
i_ /ddv ^ 1 /dev
^ p'u* \dt) ~ tt« sin> \dt) '
5—2
6S A TREATISE ON DYNAMICS. [cHAP.
Where p and (p axe, as usual, the perpendicular on the tangent,
and the angle between the tangent and radius vector.
From these equations we have the following relations :
t;^sm-0 o , . 2
m
.(7).
These relations are useful in solving the equations
^"=rf^' ^5)
rfa^ + ^'-AV = ^ (^)-
Though equations (5) and (6) are generally preferable, as being
the most direct, and giving the solutions under the form of an
equation between r and Q, there are some other relations which in
particular cases are advantageous. We have
^JB _ -p X
df ~~ r'
dC r'
" dt df dt de~ 'r\ dt^ dtj'
dv y^dr ,„,
'''''Tr-^-dt («^-
From this equation the velocity can at once be determined as a
function of the distance. We have by integrating it
v"-=f{r) + a
If V be the velocity at distance R we have
... v^-F^=f{r)-f(R).
This equation is a particular case of the more general one
obtained in Art. 50.
Equation (8) may be written
dr
and is easily remembered under that form.
III.] A TREATISE ON DYNAMICS. 69
Again, from (7) we have
h
V =- ;
P
.: log V = log k - log p,
and difFerentiatine - . -j- = . -^ •
^ V dt p dt'
and from (8) 4^ = -P^(;
••-•'•p-"! (»
If q be the chord of curvature of the orbit through the center of
force
therefore equation (9) may be written
«^ = 2P.| (10)
which shews that the velocity at any point is that which would be
acquired in falling through a distance equal to a quarter of the chord
of curvature of the orbit at that point, under the action of a constant
force equal in intensity to the force at that point.
Again, combining the equation
p-
with equation (9) we have
pA^> (11)
p^ dr ^ '
another relation which is sometimes found useful.
The results here obtained agree with those arrived at by inde-
pendent considerations in The Pri?icipia, Book I. Section 2, with
which they should be carefully compared.
56. We will now apply these equations to a very important
example ; namely, where the force varies inversely as the square of
the distance.
This, it will be remembered, is the law of attraction in nature, and
it is from this fact that it derives its importance.
We have in this case P = fxu',
and substituting this in the equation
d'u P
70 A TREATISE ON DYNAMICS. [cHAP.
we get the equation
The integral of this equation is
u = p{l + ecos{e-a)} (2).
This is well known as the polar equation to a conic section, the
focus being the pole, and — a the angular distance from the nearer
vertex.
Hence then a body moving about a center of force which varies
inversely as the square of the distance will move in some conic
section.
The constant quantities e and a which have been introduced in
integration must be determined from the circumstances of the motion
at some known epoch, or other equivalent data; h also must be
determined from similar considerations. For example, suppose it
known that, when 6 — O, u = c^, the velocity = V and the direction of
motion makes an angle (3 with the radius vector : differentiating
equation (2) we get
and .-. cot = - 1 . ^ /^r /^ ^ - ^£^
^ u dd ■ , ' ' -
= l.^.esin(0-«).
Hence we get Cj = ^ . (1 + e cos a), . ,- /jr \ '■ ''^^ '^'Ct '-r/l,^
J. r> 1 M
cot /i = , y5 . c sm a, , , - ,
The last equation determines h, the other equations give — -^
e cos a = — 1,
IXC,
e sin a = . sin ft cos ft ;
fxc.
„ V'sWft Vsln'ft
M c/ nci
which gives the magnitude of e, and it was assumed to be positive
HI.] A TREATISE ON DYNAMICS. 71
when 6 — a was considered as measured from the nearer vertex : a
may then be found without ambiguity from the two preceding
equations.
The constants which enter into equation (2) are then determined
and the equation to the orbit is completely known.
If we had not substituted for h' we should then have had
e cos a = 1,
e sm a = — - — . cot S ;
c ^k* c h'
and .-. e^ = -^ .cosec^/3-2.-^— + 1,
which is convenient when k and not V is known. V, Cj, /3 are
called the circumstances of projection.
The nature of the conic section will of course depend on the
magnitude of e. If e = 0, the orbit is a circle. That this may be
the case we must have
5 o -~ 2 t 1- 1 — 0.
To satisfy this equation we must have ft = -^ s (a condition which
is also evident independently) and V'^ = jxc^.
This condition agrees with that deduced previously for motion in
a circle.
The velocity of a body moving in a circle is often taken as a
standard by comparison with which the velocities of bodies in their
orbits are estimated. Thus the velocity of a body is sometimes
expressed by saying that it is q times that in a circle at the same
distance. The substitution of q^jxci for V^ will simplify the equa-
tions. It gives
e' = 5^sin^y^-2g*sin'/3+l.
The most important case is where e is less than unity, or the
orbit is elliptical.
Now e^ ^ (q- - 1 y sin- f3 + cos*^.
Hence that e may be less than unity q' must be less than 2,
or the velocity at every point is less than ^2 x that in a circle at the
same distance.
72 A TREATISE ON DYNAMICS. [cHAP.
Let a be the semi-axis major of the ellipse ; then from equation (2)
>?
a h'
" V'sin'(3 \ IXC, ix'c,' )
an important equation, -which shews that the axis-major of the orbit
is independent of the direction of projection, and depends only on
the distance and velocity.
If we write v for V, and - for c„ we have.
\r a)'
which gives the velocity at any distance in a simple and useful form.
a determines the position of the axis-major of the orbit: when
the body is at the extremity of the axis-major its motion is perpen-
dicular to the radius- vector : all places in any orbit where the
direction of the motion of the body is perpendicular to the radius-
vector are called " apsides," or, more commonly, " apses." a is the
angular distance of the apse from the line corresponding to 6 = 0, or,
as it is commonly called, the '^ longitude " of the apse.
6 is the "longitude " of the body ; e, the eccentricity, determines
the form of the orbit, and a, the semi-axis major, determines its
magnitude.
a, e and a are called the " elements " of the orbit ; when they are
known, the form, magnitude, and position of the orbit are completely
determined.
The first object of the direct problem, therefore, is to determine
the elements of the orbit from the circumstances of projection, or
other equivalent conditions. Sometimes it is required to determine
the circumstances of projection from certain given conditions which
the motion must satisfy.
We have seen that for the orbit to be an ellipse we must have e
less than unity, which requires that q be less than J2, or V^ less
than SjuCp
When q = ^2,
or r^=2)UCi,
in.] A TREATISE ON DYNAMICS. 73
we have e = 1, and the orbit is a parabola. The equation then
becomes
u = ^,{l + cos(d-a)},
h^ _ F"sin'/3 ^ 2 sin' /3
is the semi-latus-rectura of the parabola, a, as in the previous case,
is the longitude of the apse. When these are known the orbit is
completely determined.
The velocity at any distance r in a parabolic orbit is given by the
very simple equation
„ 2m
v = — .
r
If 7 be greater than ^^2, or F greater than J^fxc^^ we shall have
e greater than unity, and the orbit will be an hyperbola. The
elements in this case will be determined by a process similar to that
used in the elhpse. This case, however, is of comparatively small
importance.
We have at present only found the equation to the orbit, and its
nature, form, magnitude and position; and these, we have seen,
remain constant. In order that the problem may be completely
solved, we must moreover determine the position in the oi'bit which
the body occupies at any instant: that is, we must find u and Q, or
r and d as functions of t.
To do this we must integrate the equation
having given u = j^{l + e cos (6 — a)},
J a 2
from Avhich -j- = j-^ {I + e cos (6 - a)}^
This could be integrated accurately in finite terms, the solution,
however, which is thus obtained is in a form which is practically
useless ; as, however, in the cases in which it is generally required
e is a small quantity, the solution is obtained in the form of a series
ascending by powers of e, and may be carried to any degree of
accuracy.
We are able, without integrating the equation at all, to determine
the time of a complete revolution. For since h is twice the area
74 A TREATISE ON DYNAMICS. [cHAP.
swept out in the unit of time^ if we divide twice the whole area of
the elHpse by h the quotient will be the time of a complete revolu-
tion. Calling this T, we have
h
h" a{\-e')'
T is called the ''period" or "periodic time" of the body.
In the time T the longitude of the body increases by 2 ir, but
this increase does not take place uniformly. If it did take place
uniformly, the increase of longitude in time t would be
y.STT,
which
•yi^'
n is generally used for the quantity */ — , and therefore the increase
of longitude in time t on this false supposition is n/; n\% called the
" mean motion" of the body ; it is useful in simplifying some of the
equations.
It is connected with the other constants that have been used by
the equations
We will now proceed to determine as a function of t. To
facilitate this certain auxiliary quantities are used which we will
now define.
In Astronomy (on which this part of the subject more particu-
larly bears) the name "anomaly" is given to certain angles that are
frequently used.
Let AVA' be the ellipse which the body describes; AQA' a circle
on the axis-major; S the focus coinciding with the center of force;
P the position of the particle at the time t.
III.]
A TREATISE ON DYNAMICS.
75
Then A is the nearer apse and
A' is the farther apse. The angle
ASP which is what has been
written d-a is the angular distance
from the apse. It is called the
" true anomaly," We will denote
it by the letter v.
If the ordinate iV^Pbe produced
to meet the circle in Q, the angle
^CQis called the "eccentric ano-
maly," and is denoted by the letter u. Also the angle which a body
moving with the mean motion 7i would describe while the real body
is moving from A to P, is called the ''mean anomaly/' and is denoted
by the letter J7i.
„- , , m area ASP
We have then — - = ^~^j^.
Stt area ot elapse
area ASQ
Now area ASQ, = Sivea ACQ- area SCO,
= I a^u -^ae . a sin u;
.'. m = u-e sin u ;
also, r = a- e . CN
= a (1 - e cos u) ;
_ SN a (cos u — e)
cos V -
SP a(l-ecosM) l-ecosw
1 - cos V 1 + e 1 - cos M
1 + cos u ~ 1 - e ' 1 + cos 7< '
■'
V /l+e ^ u
Now V = Q - a;
dv dd h
'*' dt~ dt~ r'
h
nO
rt^(l —e cos tif
= «(l-e7'(l-ecos?<)-'
. .)(1 +2e cos u + 3e' cos'' « + 4 e' cos' u + kc.)
(6cos'?<-l) + &c.}.
76 A TREATISE ON DYNAMICS. [CHAP.
From the equation m= u-e sin u
we must express cosm, cos*?^, &c. as functions of m by means of
Lagrange's theorem, and transform them so as to involve only simple
powers of cosines of multiples of ?n.
XT ^"^
Now -y- = n ;
dt
.-. 7)1 = nt + C.
We now have -7- expressed in a series of simple powers of cosines
of nt + C and its multiples, and can therefore find v at once by
integration.
If we only proceed as far as e^ it is not necessary to employ
Lagrange's theorem. We have
u=m + e sin u.
To a first approximation, u = 7«, where the error committed is of
the same order of magnitude as e.
Therefore to a second approximation
u = tn+ e sin tn ;
.'. cos u = COS (m + e sin ?«)
= COS m . COS (e sin 7n) — sin m . sin (e sin m)
= cosm-e sin^m
= cos m - -(1 - cos 2 m),
the error being of the order e^.
1 + cos 2m
Also, cos^ u =
2
1 + cos 2 m
with an error of the order e.
Therefore, substituting, we have
-7- = w {1 + 2 e cos 7« + - e^ cos 2 7«}
dt '^ 2 '
= ??{! +2ecos(«i + C) + -e-cos2(M/ + C)},
where the error is of the order e' ;
.-. v = nt + C'+ 2esin(w^+ C) + -e'sin2(w/4 C).
Now u = when wz = 0, that is, when nt + C = ;
.-. C'=C;
III.] A TREATISE ON DYNAMICS. 77
.-. e-a = v =nt + C + 2e sin (nf + C) + |e' sin 2(n< + C).
It is customary to use tii instead of a, and instead of C to write
c-w, so that
0=«/ + 6 + 2esin (nt + e-'CT) + — sin 2 (nt + e- m);
6 is the true longitude of the body ; we observe that it consists of a
part nt + € which increases uniformly, together with a number of
terms which are sometimes positive and sometimes negative, but
which always lie between certain limits ; the periods in which they
go through all their values being different.
nt + e is called the "mean longitude."
e is the mean longitude when « = 0, it is called the " epoch," it
is the new constant that has been introduced in the integration ; it
is of course known when the longitude of the body at any one instant
is known.
w is the longitude of the apse, «/ + e — «t is the mean anomaly.
V - m or 6 -(nt + e) is sometimes called the "equation to the
center," and sometimes the "elliptic inequality."
It still remains to express r the distance of the body from the
center as a function of ^.
This is easily done ; we have
r = a(l — e cos u).
Substituting for cos w its approximate value, we have
r =a {1 -e COS TO + — (1- cos 2 7tt)}
= a {1 - e COS (72 / + e - w) + - [1 - COS 2 («/ + e - w)]},
where the error is of the order e'.
We have thus obtained an approximation to the complete solution
of the problem.
By using Lagrange's Theorem, we might have carried this ap-
proximation to any degree of accuracy.
The four constants introduced in integration which constitute
the four elements of the orbit are cr, e, a, and e.
The first determines the position of the orbit, the second its form,
the third its magnitude, and the fourth the position of the body in it.
78 A TREATISE ON DYNAMICS. [cHAP.
When a and e are known^ n and h are also known.
In solving the problem, we have considered ju to be a known
quantity, when this is not the case it is usual to consider n as the
quantity to be determined, since when n and a are both known, fx is
known from the equation /x - n^a^.
The conditions for determining these elements will appear under
very different forms. One very important case is, when we know
the values of Q corresponding to four different values of t, that is, the
longitude at four different times.
These will give us four equations,
01 = w/i + 6 + 2e sin {nti + e - w) + &c.,
Q., = nt2 + e + &c.,
03 = w/3 + e + &c,
04 = 71^4+6+ &c.
These equations enable us to determine e, n, e, and w ; and if fx
is known, we then know a. If (x is not known we may remark that
more values of d will not assist in determining it, since the above four
quantities are the only ones which enter into the equations.
If we had the four longitudes, and the distance or value of r at
any given instant we could also find a and thence ft.
Or we might have three longitudes and two distances, or these
conditions might be varied in any way.
It is however necessary to have one distance at least, and one
longitude at least in order to determine a and e.
If fx is given the distance is not necessary.
The method of approximation which we adopted in obtaining
the above equations is one with which the student will do well to
make himself familiar as early as possible. It will be continually
occurring in the higher parts of physical and plane Astronomy, the
same in principle but with variations in the form of applying it.
57. There is also another method which is largely used in the
Planetary Theory, which we will illustrate by the problem which
is now before us. Let us first, however, collect the principal
equations which connect the elements of the orbit and the circum-
stances of projection.
III.] A TREATISE ON DYNAMICS. 79
^^Fw/j_^rw^
= £!!l'cosec^/3-2^+l (2)
h-^^s\nl3 (3)
r*sin/3cos/3
•""" ^c,-F-sin-/3 W
0(1-^) = ^ (S)
«^ = ^ (8)
The question which we propose to consider is the nature and
magnitude of the change Avhich would be produced in the elements
of the orbit by a small change in the circumstances of projection,
or of the intensity of the force, or the converse.
The manner of doing this will be best shewn by taking two or
three particular cases.
Suppose that the velocity of projection is slightly increased, the
distance and direction remaining unaltered. We see, by inspecting
the preceding equations, that any change in V will be accompanied
by a change in most or all of the elements, and consequently that
these elements are functions of V.
If therefore we suppose V to be increased to V+IV, Ave must
substitute V -^ cV for F in the above equations, in order to obtain
the new values of the elements. We may expand these new values
by Taylor's Theorem in a series ascending by powers of 2 F : and
if, as we suppose, S F is small, we shall obtain these new values very
approximately by neglecting all the terms of the series which con-
tain powers of D r higher than the first; so that the new value of a
will be very approximately
From equation (l), we have
IXCy VC /
80 A TREATISE ON DYNAMICS. [cHAP.
This shews that an increase of the velocity at any point would
increase or decrease the eccentricity according as V^ is greater or
less than fxc ; that is, according as the velocity is greater or less than
that in a circle at the same distance. From equation (6) we see
that V^ is greater or less than fxc^, as Cj is greater or less than -;
that is, as the distance is less or greater than a. Now the distance
is a when the body is at the extremity of the axis-minor. Hence
then an increase of the velocity will be accompanied by an increase
or decrease of the eccentricity according as the body is at the time
on the nearer or farther side of the axis-minor. Exactly opposite
results would follow a diminution of the velocity.
Again, from equation (6)
1 a ^"
a /JL
a ju
•which shews that an increase of the velocity is always accompanied
by an increase of the axis-major.
From equation (4) we have
sec- a.da= -f-l — „/. ^^.7 . S r.
Since cos /3 is the only factor of this expression which can change
its sign, we see that an increase of the velocity will cause an increase
or decrease of the longitude of the apse according as /3 is less or
greater than - ; that is, according as the body is moving from the
nearer apse to the farther, or from the farther to the nearer.
As another example suppose V to remain unaltered, and the
angle /3 to be slightly increased.
We have from equation (l)
e.le = — A 2 y sin ^. cos /3. 2/3.
Here, since V" is less than S^c,, e will be increased or diminished
according as cos ft is negative or positive, that is, as ft is greater or
less than - . An exactly opposite result would follow a diminution
of /3 ; so that we see in all cases e is increased or diminished as ft
is made farther from, or nearer to, a right angle.
III.] A TREATISE ON DYNAMICS. 81
From equation (6) we see that a is unaffected by a change in /3 ;
a result which we have already remarked.
From equation (4) we get
(/xc, - F^ sm^fty '
which shews that the longitude of the apse will be increased or
decreased by an increase of /3, according as
H c, cos"* (3 + (V — fiCt) sin^ /3,
or, fx(aci- sin-/3) is positive or negative.
When the body is on the nearer side of the axis-minor ac, is
greater than unity, and therefore this is positive ; when the body is
very near the farther apse a c, - sin* /3 is negative ; the determination
of the point Avhere (ac, — sin* ft) changes sign is a geometrical problem
of little importance. We see, therefore, that by an increase of ft
which takes place near the nearer apse the longitude of the apse is
increased, and diminished by an increase of ft which takes place near
the farther apse.
These will be sufficient to illustrate the method, which goes by
the name of the ''Variation of Parameters."
We have selected these examples as themselves of importance.
The results should be carefully compared with the Corollaries to the
66th Proposition of the eleventh Section of the Principia.
We have dwelt at considerable length on the motion in an ellipse
because it forms the basis of the calculations of the motions of the
heavenly bodies.
58. These methods of expressing d and r as functions of t
entirely fail when the motion is in a parabola or hyperbola, since
the convergence of the series depends on e being less than unity. In
the case of a parabola, however, the equation can be easily in-
tegrated.
iL-'l^^ 1
de~ h " / ' {1+ cos (0 - a)P
h' J- a
Let 2/ be the latus-rectum of the parabola, then h' = fxl,
and TTi = T A / - • ( 1 + tan'^ ^ i —
fZ^ 4 V M \ 2 7 2
„ 1 //'A e-a 1 30- aN
C = -./-.(tan +- tan'— -— 1 .
2V n \ 2 3 2 J
From this equation 6 may be expressed in terms of /.
'4'
82 A TREATISE ON DYNAMICS. [cHAP.
59. The method of integrating the equation
dhi P
dd'
-^"-7?^-
= 0,
which has been
adopted in the case of
is not generally applicable.
P=HU%
Suppose P =
= M w" ; we must proceed
as follows
d'u
^•■"--
...(1)
Multiplying both sides by ^^-r?, > and integrating, we have
Let V be the velocity of the particle when u = c, then
It will be convenient to express V in terms of the velocity which
a body would acquire in moving from rest at an infinite distance to
a distance - from the center of force : to find this we have
c
d'-x _ p.
d¥^~l^"
0=C;
w - 1
Let V^qV,, then
(2)
This equation can always be integrated when ^=1. In this
case it becomes
de
du u J^nu"-^-{n-\)ie
Let
{n--[)h'
III.] A TREATISE ON DYNAMICS. 83
dO 2 1 .
then -J- = — I "; , >
dy n-3 r/ Jf-l
2
.-. 6 - a = — ^ sec"' y ;
71-3 -^
the general equation to the orbit described.
This method of solution fails when n = 3, and when n = 1 .
When n = 3 equation (2) becomes
d^_ 1
The integral of this equation will be logarithmic, algebraical, or
trigonometrical as j-s is greater than, equal to, or less than unity.
In the first case.
In this case the value of q can be either greater or less than unity.
In the second case we have j2=l, and the integral is
' — ej^l'
or u = c Jq^-l . (d - a),
in this case q^ can never be less than unity, for
'^ ,jic'~ fx sin' /3 sin-/3*
Lastly, when ^ is less than unity, we have
dl_ 1
dti '
h . , / h--ix u
. 6 -a= — - ^;= sin ^ ^ / —1-1, — -T . -
J/i'-fx V m(9--1) c
6—2
84 A TREATISE ON DYNAMICS. [cHAP.
h" . .
in this case also q^ = — ^~2~1d is necessarily greater than unity.
We might have obtained these equations at once from the general
equation
d'u P
ad- h ti
which in this case becomes a linear equation
ru ( u \
When n = \, we have
•'• ^''{(^y+«'}=2Mlog« + C,
F^=2)ulogc+ C;
We cannot generally perform the next integration. These cases
however are of little importance. In all these cases after deter-
mining the equation to the orbit we must find the position of the
rl ft
body in the orbit from the equation ~ =hu^.
60. It was stated in Article 56 that a point in an orbit where the
motion of the body is at right angles to the radius-vector is called an
apse, and the corresponding radius-vector is called the apsidal line,
or apsidal distance. From the nature of an apse we must have at
such points
dr ^ . ^ . dti
TH" = : and therefore ^^ = 0.
dd ' do
Let P = ixu", then the equation to the orbit is
d^u mm"~* „
and at an apse we must have
-G
III.] A TREATISE ON DYNAMICS. 85
This equation cannot have more than two changes of sign, and
therefore cannot have more than two positive roots : so that in an
orbit round a center of force varying as any power of the distance
there cannot be more than two different apsidal distances.
The angle between two consecutive apsidal distances is called the
apsidal angle. When the orbit does not differ much from a circle
this angle can be found approximately, without finding the equation
to the orbit. Let P= n u^(p{u), then the differential equation of the
orbit is
^.+«-^.0(«) = o (1)
Now since the orbit is nearly circular we have
n = c + X,
where x is small compared with u ;
.-. 0(«<) = 0(c) + ^'(c). X nearly,
. d'u d'x
"^^° rf0^ = rf0-
therefore equation (l) becomes
%^{^--i^-¥{o)]x^c-^,.cp{c)=^0 ...(2)
also the velocity cannot differ much from that in a circle at the same
distance, so that
Therefore the integral of equation (2) is
x = Aco%{kd + B) + C,
where k
=y'-^-"^4{^*w-4^
dx
.'. j^ = - A k sin (kd + B).
Let 0j, 6.2, 08, &c. be consecutive values of 6 for which
^=0, then
kd, + Br^n-n^
ke^ + B = {ii + \)'K,
k0s + 5 = (n + 2)7r,
&c. ;
0, _ 0,= 0,-0, = &C.
^k'
which gives the apsidal angle approximately.
86 A TREATISE ON DYNAMICS. [cHAP.
For example^ let P = fx u' + — where fx' is small : here
0(«) =
.'.cp{c) =
-^
<P'ic)-
^j^j_c.0'(c)_^c^ + V
(p{c) fXC' + ^'
.'. the apsidal angle = tt . . / -^ ;
61. The equations
P^;a,2/^'«
enable us at once to solve the inverse problem of finding what must
be the law of force in order that the particle may describe a given
orbit. For example, let it be required to find the law of force that
the orbit may be the equiangular spiral, whose equation is
e cot <
r = ae ,
or « = i.e-^^°t«;
a
, d'u 1 , ~Qf
here -ttt-, = - • cot'a . e
do'- a
= u . cot* a ;
...P = ;,v(S..)
= A' cosec^ a . u^,
or thus,
in the equiangular spiral,
p = r sin a;
dp .
dr
'. P = — .-/- = . „ . -r , as before.
jr dr siTVa r^
IV.] A TREATISE ON DYNAMICS. 87
CHAPTER IV.
62. The integration of the equations
in the cases where the motion is not, as we have hitherto supposed
it, in one plane, is in general attended with great difficulties and can
rarely be effected except approximately. As moreover no principles
are involved in the solution of the equations in this case which have
not been exemplified in the cases of motion in one plane, it is
unnecessary to give any of the few instances in which the equations
can be completely solved. There are however a few general pro-
perties of the motion which can be readily exhibited.
The resultant of the forces X, Y, Z acts in the osculating plane
of the body's path.
This may be seen at once from general considerations.
The body at any instant is moving along the tangent to its path
at that instant, and the deflection from the tangent must take place
in the direction of the resultant force that acts on the particle at that
point ; but the plane passing through the tangent and the direction
of the deflection from the tangent is the osculating plane, which
therefore passes through the direction of the resultant force.
This may also be easily deduced from the equations.
The direction cosines of the normal to the osculating plane are
proportional to
(Py dz d^zdy d'z dx d'xdz . d''x dy d^y dx
d7Tt~'d?~di' Wdi ~ dF dt Jt^ ~di~ dp di '
Or, considering the equations of motion, to
y^^_Z$, Z^^-X^, and X^J'-Y^,
dt dt ' dt dt ' dt dl '
88 A TREATISE ON DYNAMICS. [cHAP.
and the direction cosines of the resultant force at the point (x,
y, z) are
and
JX'-¥ Y^^ Z' JX'+ Y'+Z' JX'+ F'H- Z' '
and if 6 be the angle between the direction of the resultant force and
the normal to the osculating plane, the numerator of cos 6 is zero, or
is a right angle ; that is, the resultant acts in the osculating plane.
Again, from the equations of motion we have
d2
\df dt de dt dedtj~ di dt dt
or since (^^y=(J)\(g)V(^
^dz
^dt
at ds ds ds
The right hand side of this equation is the sum of the resolved
parts of the forces X, F, Z along the tangent, and this is the only
part that affects the velocity : when this is zero, or the whole re-
sultant force acts always in the normal plane, the velocity will be
unaltered thoughout.
When the expressions for the forces X, Y, Z are such as to make
Xdx + Ydy + Zdz a perfect differential of ;r, y, z considered as three
independent variables, that is, when they satisfy the conditions
dXdY dYdZ dX_dZ
dy ~ dx ' dz ~ dy' dz~ dx'
let F{x, y, z) + C=^j{Xdx + Ydy + Zdz),
then Mv^ = F {x, y, z) + C,
from which it appears that
F {x, y, z) = const,
represents a system of surfaces each of which has the property that
the body always crosses it with the same velocity : and if any two of
them be taken, as
F{x,y,z) = C,,
F{x,y,z) = C„
the change of velocity in passing from one to the other is inde-
pendent of the points in the two surfaces, and of the path pursued
by the particle between them ; for if v, and u, be the velocities at the
two surfaces.
IV.]
A TREATISE ON DYNAMICS.
89
Mvi = a+c,
so that M (v,' - v/) = C2 - C„
in which neither x, y^ 2 nor any relation between x, y, and z, appear.
63. The three equations may be transformed to polar co-ordinates
as follows :
Let P be the position of
the particle,
OM = x;
MN = y,
NP = 2.
OP = r,
ON=p,
MON = d.
M
Let P', T' and <S" be the resolved parts of the forces on the par-
ticle in direction iVO, perpendicular to NO, and in PlSl respectively,
so that M^ = -P'--r'-^,
at p p
dr p p
(1)
(2)
(3).
M-
S';
puttin
g^.
T,
' f- Tl
T
' Tl'
S'
TV
d'x
di'
P
-T^--,
P
,
dt'
d'z
dl'~
p
-S
-r^, (
p
(
From
(1) and
(2)
d'x
= Tp.
Lei
' "^ dt
dx
= h;
^ dh
-Tp,
90 A TREATISE ON DYNAMICS. [cHAP.
and h = p^ — ;
putting u for -
P
A^ = V.o|I^, (4)
= — h (cos 6 'j^ + 71 sin 6),
= - COS — -7^ - T Sin - A^?r cos ( -^^ + u).
■ But^ = -Pcos0-rsm0;
^ P_ T du_
•'• dd''^''~h'u''^¥e de~^'
or, substituting for ^^ from (4),
P T^ Jm
^""""TTT^^^ ^^
and -73 = -} — 5 = , (o)
Again, let tan PON= s.
Then 2 = ps = - ,
drz_
dF—^-
Combining these two with equations (5) and (6), we get
S-Ps T du
d's Ku^ ^ h;'ii^ dd ^ . ,
77^» + *+ TTir—'^ (7)
1 +2
fTdl
IV.] A TREATISE ON DYNAMICS. 91
These equations though perfectly general are of little use except
when P, T and S are of particular forms. Their principal applica-
tion is in determining the motion of the moon about the earth when
acted on by the disturbing force of the sun. In this case they are
integrated by a series of approximations.
64. When the forces X, F, Z depend on the velocity of the par-
ticle as well as its position, the integration of the equations of motion
can seldom be completely effected. It frequently happens, however,
that they can be put into a form which will enable us, by means of
laborious calculation, to determine many points connected with the
motion to any required degree of approximation.
Let us take for an example the motion of a body in the air acted
on by gravity. The resistance of the air varies as the square of the
velocity of the body, so that the equations of motion are
de- ^'dS^ ^ .- ^rr^
These may be written
d^x , dx ds ^ ..
j^ + A -J- -J- = 0, (1)
dt^ dt dt ^ ^
g-4ii-^=« (^)
The integral of equation (1) is :^C
dx ^ _k.
-J- = Ce-*'.
dt
Let V be the initial velocity of the body, a the angle its direction
makes with the horizon, then
^=rcosa6-*' (3)
Let p be written for -p, then -^=p^-f; substituting this in
equation (2), and reducing by means of (l), we have
(4) #
, . .i-. '" dx dp
dt dt
d^. ^ " cos* a
kh ' Now, 7rT7=^
^
'^ ^ dx F^cos^a dx'
92 A TREATISE ON DYNAMICS. [CHAP.
The integral of this equation is
pVTT7 + log(p + VTT7)=y-p,,|^^6^^...(5)
where 7 is a constant introduced by integration, and is given by the
equation
tan a sec a + log (tan a + sec a) = 7 - tttb^—t- •
From equations (4) and (5)
<^P y-pJl^'-logip + JT^'Y
.-. //^-y- -P ■ (7)
and Jkg^- = , , _ . ; (8)
these equations cannot be integrated in finite terms ; they give, how-
ever, the values o? x, y and t corresponding to any value of/>, in the
form of a definite integral, whose value may be found to any degree
of approximation by the ordinary methods applicable in such cases.
Thus any number of points in the path of the body, and the respec-
tive times of reaching those points may be determined. The velocity
corresponding to any value of p is given by the equation
.■4- ,,^':^,,.,^^ (9)
T-
pjl+ f- - log (p + ^1 +/)
When p=0 the body is at the highest point of its course, and at
that time
yk
Equation (4-) may be written
dx^ F'cos'a '
which is the differential equation to the path of the body. We may
shew that this curve has a vertical asymptote. Since p becomes
negative and continues to increase in actual magnitude, let pi be a
very great value of p, and ^, the corresponding vahie of x; then in
equation (6), neglecting all powers of p except the highest, we have
approximately for values of p greater than pi
J dx 1
dp p"
IV.] A TREATISE ON DYNAMICS.
KX = KX, -\ :
Pi P
which shews that x is finite when j) = -<x-
dp
'-if
tan a
Hence then - y / . , is finite
P \/l +/- log (p +Jl +/)
let this equal a, then x is always less than a, and approaches a as a
limit when p increases indefinitely ; a line parallel to the axis of ^ at
a distance a from the origin is therefore an asymptote to the curve.
65. Let us take as another example the case of a body describing
a nearly circular orbit round a center of force varying inversely as
the square of the distance, in a resisting medium of very small
density.
The polar equations of motion are
JP^ t du
d'u h;'u^ h.^ii" dd ^
Je^'-'' 7[Tdr=''^ (^)
1 + 2 I ^2-^
^:-."V^ ■■■(^)
In this case if Arw* is the resistance, k is very small, and
--y"H^)'m «
2
Let -A = «^ 7-3 = ^h then
hi' K^
T ka* / , /duy/dds,' , .
p r_ du _
and equation (1) becomes
d^u « ^ /^
'Jhru'
Neglecting small quantities of the first order, we have
dd
2
94 A TREATISE ON DYNAMICS. [cHAP.
Substituting these values and neglecting small quantities of the
second order^
T _ k^
h^u^ ~ a '
fTdd _ kd
•'■ j Vm^ ■" a '
-y-^+u-a-2kd = ;
do
.'. u=a{l + — — + e cos (0 - a)],
where e is a small quantity, the orbit being supposed nearly circular ;
^ = n{l + ^+2ecos(0-a)} (7)
Substituting these values and neglecting small quantities of the
third order,
,03 = U -— ecosfU- a)\ ;
d^u
jj^ + u - a -2kd + 2ke sm (0-a) = O;
f, 2A:0 .. . kd .. ^,
.-. u = a {1 + + e cos (0 - a) + — e cos {Q - a)],
_ = „{!+__ +___ + 2ecosC0-a)+-^ecos(9-a)
k
+ - e sin (0 - a) 4 e^ cos^ (0 - a)}.
By repeating the process the solution might be carried to a
higher degree of approximation; the labour, however, increases
very rapidly. This example has been selected to exhibit the use of
the equations of Art. 53 ; and also as an example of the method of
solution by successive approximation. Since terms multiplied by
occur in the expression for u, this expression will cease to be an
approximation at all when Q becomes large.
To express in terms of/, we must proceed in a similar manner:
<^^ a ,
~r,-n'> •'• V — nt + €.
at
Substituting this value of Q in equation (7)
—=n {I + — .{lit ■\-e) + 2ecos{nt+ e-a)};
IV.] A TREATISE ON DYNAMICS. 95
3k
.'. e = nt + € + -— (fit- + 2€t) + 2e sin (7it + € - a)
correctly to the first order of small quantities : by substituting this
dd
dt
value of d in the next value of -r- and integrating, we might obtain
a more approximate value of 0.
QQ. In the problems which have been hitherto considered the
forces which acted on the particle were known in terms of the
position, velocity, and direction of motion of the particle, which was
perfectly free to move as these forces would cause it to move; in
other words, in the cases in which we have integrated the equations
d^x d^y d^z „
'^l[f=^' "^7?=^' "^1?-=^'
X, Y, and Z have been given functions of x, y, z and their first
differential coefficients. And j, y, and z have been subject to no
other relations than those furnished by the above equations. In
this case we have three equations which are theoretically sufficient
to determine the three unknown quantities x, y, z in terms of ^ ; or
by eliminating t to give two equations between x, y, and z.
There is, however, an extensive class of problems in which X,
Y, and Z contain, in addition to given forces, unknown forces
arising from geometrical conditions to which the motion is subject,
such, for instance, as the unknown tension of a string, the unknown
reaction of some curve, or surface. These forces will enter as
unknown quantities in the three equations of motion, but however
many unknown quantities of this sort may appear, there will always
be just so many additional equations expressing these geometrical
relations.
The methods to be pursued in cases of this sort will of course
depend on the particular geometrical conditions imposed on the
particle, they will, however, be generally very similar to those
pursued in other cases. The principal problems of this class are
those in which a particle is constrained to move on a particular
surface, or some particular curve line. We will first prove some
properties, and deduce some equations which are common to all
problems of this sort, and, at the same time, point out the methods
to be pursued in particular cases.
Let us first consider the case where the motion is in one plane.
96
A TREATISE ON DYNAMICS.
[chap.
67. Let a particle be constrained to move on a smooth plane
curve and be acted on by any given forces in the plane of the
curve.
By the term smooth we understand that the curve is incapable of
exercising any reaction except in the direction of the normal.
Let y=f{x) (1)
be the equation to the curve ; X, Y the resolved parts parallel to
the axes of the forces which act on the particle at the time t. First
let these forces be functions of x,
y and constant quantities only ;
R the reaction of the curve, A
any fixed point in the curve, P
the position of the particle at
the time t,
AP = s, PTN=e.
The equations of motion are
(Fx
d'y
or since sin 6 ■■
X-Rsin
Y+Rcos 0,
cos 6 =
dy
ds
dx
d~s'
These equations may be written
d'x ^
ds
Y+R
dt
y=f{^)
Eliminating R from (£) and (3) we have
( dxd^x dyd^y) ^^dx
"^{Tsdt'^TsTf
dxd'x
•(2)
.(3)
•(1)
ds ds
dtdt' dtdt' f dt dt
(4)
When X and Y are such functions of x and y that Xdx + Ydy is
the differential of some function F(a;, y) of a: and y considered as inde-
pendent, that is, when X and Y satisfy the condition
dXdY
dy dx '
we may integrate (4) without reference to (1).
IV.] A TREATISE ON DYNAMICS. 97
^----m-m^m-
That equation may be written
dv ds d's d.F(x,y)
'"'Tr'^Ttdi^^—dt—'-
The integral of which is
mv'' = 2F{x,y) + C.
Let Vj, Xj,_yj, be corresponding values of v,x,y, then
mv,'' = 2F{x„y,) + C,
and .-. vi{v^ - v^) = 2F{x,y) -2F(x,, y,) (5)
This equation shews that the change of velocity in passing from
the point {x^y^ to the point {xy) is independent of the form of the
curve along which the particle has moved, and by a reference to '
Art. 50 it will be seen that this change is the same as if the body had
been free and had moved from the one point to the other under the
action of the forces X and Y.
AVhen X and Y do not satisfy the above condition, this is not the
case. In that case we may integrate (4) by eliminating x or y from
the right-hand side by means of equation (1). These cases, however,
seldom occur, and are of little importance.
Equation (4) may be put under the form,
d^s .^dx trdy
m-r^= X-T-+Y -f-,
dr ds ds
d^s
or m -T73 = X cos + F sin d.
The right-hand side of this equation is the" resolved part of the
forces X, Y in the direction of the tangent: calling this T the
equation becomes
"'^-'T (6) />7^_ .
This form of the equation is useful in cases where T can be expressed J^
easily as a function of s. // *-» ^'
Again, from equation (2) and (3) we have C
/f^
dx d^y dyd^x\_^dx -ydy j^ds
Ttd?~Tt dp) Tt~ dt'^ df.'
r> V • a V a dtjdxd^y dyd^x)
7?=Asm0-Fcos0 + ..^|^-|--^^.|
A TREATISE ON DYNAMICS.
[chap.
Now if p is the radius of curvature at the point P
\dt)
" dx d'y dy d^x '
TttP~dt~df
.'. R = Xs\ne- rcos0 +
tnfds\'
P\dtj'
The first terms on the right-hand side of this equation are the
resolved parts of the forces X and Y along the normal : if we call
the sum of these N^ the equation becomes
R + N-.
mv
P
(7)
In this equation p is the absolute length of the radius of curva-
ture, and is always positive ; R is the pressure exerted by the curve
on the particle in a direction towards the centre of curvature, and N
is estimated in the same direction.
We have supposed that the curve is capable of exerting a pressure
on the particle in direction of the normal, either outwards or inwards,
as is the case when a particle is enclosed in a small tube ; if, how-
ever, the particle moves on the curve, R is restricted to being only
positive or only negative, according as the particle moves on the
concave or convex side of the curve ; and in either of these cases
when R becomes zero and changes sign, at that point the particle
leaves the curve and the motion becomes that of a free particle.
68. As an example, let us
take the case of a body moving
on a smooth curve under the
action of gravity.
Let the lowest point of
the curve be taken as origin,
and the axis of x be horizontal,
and let OP = s. The equations
of motion will be
(Fx
dt^'
R
d^u „ dx
•(2)
,(3)
IV.] A TREATISE ON DYNAMICS. 90
Proceeding here as in the general case, ^ve have
^ ds d^s ^ dy
^""didT^-'^'^'^dt'*
.: mv^ =C- ^mgy.
Let Vj be the vakie of v when y =y^, then we have
m if -vX) = 2 mg (^, -y),
or v''^v\ = 2g(yi-y).
This equation shews that the change of velocity will be the same as
if the body had moved through the same vertical height freely. It
will be useful to bear this result in mind.
It is sometimes convenient to put the preceding equation under
the form
d^s _ dy
de~~^Ts'
This is useful when -— can be expressed easily as a function of .y.
When this is the case this equation gives at once the motion along
the curve in terms of the time.
To find the pressure exercised by the curve, we have
dx v^
p'
Suppqse the body to fall from rest at a point where y = h,
the equation
v'~v\ = 2g{y,-y)
becomes v'^ = 2g {h -y) ;
and if ?^ be the velocity at the lowest point
u'^2gk,
the body will move on and ascend the other side of the curve, the
velocity at any point being still given by the equation
v^=2g{h~y),
which will become zero when y = k. Or, whatever the form of the
curve, the body will rise to the same height as that from which it
started.
It is easily seen that the body will oscillate, the points of rest
being always at the same height above the horizontal line.
7-2
^ = '"Sds
100 A TREATISE ON DYNAMICS. [cHAP.
69. Let the curve be an inverted cycloid, whose equation is
x = a vers"' - + ./s ay —y^.
From which we have -7- = a. / — '>
dy \ y
ds _ /27i
" Ty~ S J'
from which we have s = zj^ay;
ds 4>a
Substituting this in the equation
d^s dy
'd?^~^'ts'
, d"s £rs
we have -7T2 + 1- = 0.
dr 4-a
The integral of this equation is
and.-4^ = -^y£-.si„(yA,^B).
Suppose the body to fall from rest from a point where s = Si, we
have then
Si = A cos B,
= -Ax/~ sinB
.-. 5 = 0, A = s„
and,., cos yZ
V 4a
when s = 0, we must have
Hence the time of reaching the lowest point is tt x/ ~ >
and the whole time from rest to rest again is 2 tt a/ - . These are
both independent of the length s^.
IV.] A TKEATISE ON DYNAMICS. 101
Hence we see that from whatever point the body falls it will
always reach the lowest point in the same time. On this account
the cycloid is called the " Tautochronous curve."
The equation '^=.-s, ^ f^ sin ^^ t
gives the velocity at any time; and since we know the position at
any time, Ave can find the velocity in any position. It may however
be more readily found from the equation
for we have s^^ = 8a h.
ay.
»'=lifr'-">
The pressure on the curve is given by the equation
R.
dx mv^
Practically it is impossible to obtain a perfectly smooth curve ;
the properties of the evolute, however, enable us to produce the
same dynamical effect.
Suppose AP to be the curve in
which it is required that the particle
should oscillate, and that QB is its
evolute. If a string be unwrapped
from BQ, its extremity will trace out
the curve PA, and since the only
dynamical effect of the curve PA is to
exercise a resistance in the direction
PQ of its normal, this effect will be
equally produced by the string PQ
exercising a tension ; and therefore the
dynamical circumstances of the body will be the same when moving
on the curve, and when attached to the string.
Now, in the case of the cycloid, the evolute is two semi-cycloids
in the position in the figure.
102
A TREATISE ON DYNAMICS.
[chap.
Hence, if a string OP of length 4a is fastened at 0, and the body
P swings backwards and forwards so that the string wraps and
O
unwraps itself on the curves OA^ OB, the body P will describe the
curve AB in the manner we have just investigated. Such an
instrument is called a cycloidal pendulum.
70. Let us now consider the motion of a body in a circular
arc. The equation to the circle is
dx _a—y _
dy~~x~'
ds a a
from which we have
dy
72^
.-^l.
.'. y — a vers - .
dy . s
-7^ = sm - .
ds a
Hence the equation becomes
d's . s
dt^ = -S''''a'
.•.(^jy=2«gcos-%C.
If Si be the value of s when the body is at rest, we have
= 2a^cos- + C ;
(^;J=2a^(cos^-cos5).
IV.] A TREATISE ON DYNAMICS. 103
We might also have obtained the velocity from the equation
^ ds d^s ^ dy
'''"TtdW=-^'"STt'
from which l-j-j =2g{h-i/).
Neither of these equations can be integrated again in finite terms.
The time of reaching the lowest point may, however, be found from
them in a series which converges very rapidly when h is small.
If we require the time to a first approximation only, we can find
it at once from the equation
d's . s
de = -S''''a'
which is to a first approximation
d'j g
dt' a'
'^i-
and the time of a small oscillation will be t . / - , the same as in a
V g'
cycloid where the length of string is a.
When the oscillations are not small though we cannot find the
position at any time, we know the velocity in any position from
the equation
v' = 2g(h~i/).
[£21
be the
velocity
at the lowest point
and v"^ — u^ — 2gy>
If d be the
angular
distance from the lowest point.
t/ = a - a cos ;
.-. w* = ?r - 2 a^ + 2 ag cos Q.
Also, It = m — + mg -j-
= m — h m g cos Q
„2
= m 'Zmg + 3 mg cos 6.
104 A TREATISE ON DYNAMICS. [cHAP.
Hence at the lowest point
K = m — h ms,
at highest point
Jti = ?)i 5 nig,
when the string is horizontal
R = m 2tng.
a
If R becomes zero and changes its sign, the body will leave the
curve. Now if ti^ is greater than 5 ag we shall never have R = 0,
since the least value of R is
(^=^)-
Also, the least value of v'^ is u^—^ag, so that the particle will
never come to rest, but will continue moving round and round.
If u^ = Sag we shall have jR = at the highest point, but it will
not change sign and therefore the body will not leave the curve.
Also V will never become zero, and therefore the body will move
round and round. If however u^ is less than Sag, R will become
zero before the body reaches the highest point, and the correspond-
ing value of 6 will be given by the equation
„ u^-2ag
cos = ^ .
Sag
When u^ is greater than 2ag this value of cos 6 is negative; and
as 6 increases, cos 6 will increase negatively, so that R would change
sign. The particle will leave the curve therefore at this point, and
its motion will be that of a body projected freely.
That it will reach this point is shewn by the equation
•- = u^ -2ag + 2ag cos 6,
which gives a greater value of 6 for the point at which v would
1 n u'^ — 2ag
vanish, namely, cos (^ = ° .
2ag
If u'^ = 2ag the values of 6, for which R = and v = are coin-
cident, 6 therefore will not increase, and R will not become negative.
Hence in this case the body will not leave the curve. If ii^ is less
than 2ag, we have v=0 for a smaller value of 6 than that which
would make R = 0, R therefore will never vanish.
We have dwelt on this case at considerable length, not from any
importance in the question itself; but because it exhibits very clearly
IV.] A TREATISE ON DYNAMICS. 105
the mode of interpreting the equations to be adopted in all cases of
this sort.
We have two quantities R and v^, each a function of the angular
distance Q from the bottom, and each from its nature incapable of
becoming negative; the angle Q will go on increasing till one or
both become zero; and some change will take place in the motion
when either of these happens. If ?;^ = before R = 0, the motion
will be continuous, but Q will have attained its maximum and will
begin to diminish. If iv := first and an increase of would make
R negative there will be a discontinuity in the motion.
If no value of Q makes either R = or v = 0, the angle 6 will go
on increasing.
These remarks as far as the detail is concerned apply only to
this case, but are sufficiently general to exhibit the manner of inter-
preting the equations in other cases.
71. We will now consider the case where the motion is on a
curve of double curvature. The reaction of the curve will in this
case be in the normal plane, but we cannot assign, a priori, in what
direction in that plane.
Let R he the reaction, I, m, n the direction -cosines of the line in
which it acts.
Then the equations of motion are
(1)
M'^, = Y+Rm,
I, m, n are connected by the equation
l-+m!' + ii'=\ (2)
and also by the equation expressing that the direction of R is per-
pendicular to the tangent. This equation is
jdx dy dz ^ ,.
These five equations together with the two equations to the curve
are sufficient to determine x, y, z, /, m, n, and R at any instant.
Eliminating R from equations (1) by means of (3) we have
106 A TREATISE ON DYNAMICS. [CHAP.
If Xdx + Ydy +Zdz\s a. perfect differential of some function of
X, y^ z considered as independent variables, that is, if X, Y, Z satisfy
the three equations
dX^clY dX^^dZ clY^dZ •-
dy ~ dx ' dz dx ' dz ~ dy '
equation (4) can be integrated without making use of the relation
between a;, y and z given by the equations to the curve.
Let this function be M.F(x, y, z) then we shall have
v''^C + 2F(x,y, z),
and v" ~ Vi^ = 2 F (x, y, z)-2F (x^, y^, Si),
or the change of velocity in passing from the point {_X\y^ 2,) to the
point {xyz) will depend only on the positions of those points, and
not on the nature of the curve between them : and this change of
velocity will be the same as would have taken place if the particle
had moved freely from one point to the other.
We also see that the velocity is the same at all points at which
the value of F (x, y, z) is the same, or
F{x,y,z)=C
is the equation to a surface through which the particle will pass with
the same velocity at whatever point of the surface it arrives, and by
whatever path it has travelled, provided only that it started from
the surface F {x, y, z) ^ Ci with the same velocity i\.
Or the change of velocity in passing from any surface
F(x,y,z)=C,
to any surface F {x, y, z) = C^
is the same, whatever the path between those surfaces.
U Xdx + Ydy + Zdz is not a perfect differential of three inde-
pendent variables, we cannot perform the integration without intro-
ducing the relations between x, y and z given by the equations to the
curve ; these equations will therefore modify the form of expression
for the velocity, and the preceding results will not follow.
Equation (4) may also be put into the form
dt^ ds ds ds
where T is the sum of the resolved parts in direction of the tangent
of the forces X, Y, Z.
In cases where this resolved part can be expressed easily in terms
of s this will give the motion along the curve at once. In the case
IV.] A TREATISE ON DYNAMICS. 107
where no forces act we see that the velocity is constant. In this
case we can easily shew that the whole reaction is in the osculating
plane of the curve. The equations of motion in that case are
, dx dy dz ^
I -J- + 711 -f- + n -r = 0.
as ds ds
Now the direction-cosines of the normal to the osculating plane
are proportional to
d^y dz d^z dy d^z dx d'x dz d^x dy d^y dx
~df' di~TfTt^ le- Tt ~ If Tt ' Iftt'll- Jt'
and therefore, considering the equations of motion, proportional to
(dz dy\ f dx , dz\ f,dy dx\
'''Tt-''di)' K'Tt-^Tt)' \!Tt-'''Tt)'
6 be the angle between this line and the direction
the numerator of cos 6 is
,/ dz dy\ ( dx ,dz\ f,dy dx\
\"'Tr''dt)'-'"VTt-^dt)'-''Vtt-"'dt)
which =^ 0,
and .-. = ^,
or R acts in the osculating plane.
72. Let us next consider the motion of a particle which is con-
strained to move on a given smooth surface.
In this case the whole reaction of the surface will be in the
direction of the normal.
Let u = 0, (1)
be the equation to the surface, and let /, 7n, n be the direction-cosines
of the normal at any point, then the equations of motion are
M^=X^Rl, (2) ^
M^=Y+R7n, (3)
M~ = Z+Rn (4)
108 A TREATISE ON DYNAMICS. [cHAP.
I, m, and n are known functions of x, y, and z, determined by equa-
tion (1), and these four equations will be sufficient to determine x, y,
z and Ji as functions of t.
We shaU have, as before,
and the same results may be deduced from this equation as were
deduced in the case of a curve line.
To find the equation which must be joined to (1) to determine
the path of the particle on the surface, we have
du
dx
//diiV fduV {diiV
s/\T.)^\Ty)^{Tz)
du
dy
/fdu\' /duV (duV'
^/[Tx)^[Ty)-'[rJ
dz
//duV (duV fduV '
\/{Tx)-'[dy)^[Tz)
Eliminating R from the equations (2), (3), (4), we shall have two
equations between x, y, z, and t; and these combined with (1) will
give X, y and z as functions of i : or will give another relation
between x, y and z by eliminating t ; R can then be found from any
one of the previous equations.
We can shew that whenever there are no forces acting on the
body, the osculating plane of its path will always pass through the
normal to the surface. The equations are
And, as before, we can shew that the direction-cosines of the
normal to the osculating plane are proportional to
IV.] A TREATISE ON DYNAMICS. 109
. ('"s-40'(4:-'S).(4J-'«frO-
and it is therefore at right angles to the normal.
73. In the particular case where the proposed surface is one
of revolution, some of the preceding equations may be put under
a simpler form.
Let the axis of revolution be the axis of s, and let r =f(z) be the
equation to the generating curve.
X = r cos 0, y = r sin d.
The direction of the reaction R will in this case always pass
through the axis.
From the first two equations of motion we have
^i-ff-^^n-^^-'^-
When X Y -yX=0 the integral of this equation is
This will be the case when X=0, and Y = 0, or when the whole
force is parallel to the axis of revolution ; it will also be the case
when the resultant of X, F and Z passes through the axis of revo-
lution.
Let (p be the angle which the direction of motion makes with the
generating curve through the particle. Then we shall have
. ^ rde h
sm(b = - -r- = — .
^ V at vr
If no forces act, v will be constant, and sin cp will vary inversely
as the distance from the axis.
Again, suppose the axis of revolution to be vertical and the only
force acting to be gravity : we then have
which gives sin = — , — ■ .
In the cases of constraint that we have considered the curves
and surfaces on which we supposed the particle compelled to move
were fixed in space. We might vary the conditions by supposing
them in motion, either according to some given law, or in some
manner due to the action of the particle. Some of these cases
110 A TREATISE ON DYNAMICS. [cHAP.
will occupy our attention afterwards. We have here given sufficient
to shew how such problems are to be treated when they occur.
(y 74. We will conclude this chapter by proving in the case of the
motion of a particle, a principle of very general application, called
*' The Principle of Least Action,"
When a free particle moves under the action of forces X, Y, Z,
the equations of motion are
^S=^' ^S=^' ^s=^ «
and if ZXdx + 2Ydy + 'ilZdz = Mdf{x, y, z).,.{2')
we have w* = V^ +f{x, y, z) -f{a^ h, c) ... (S)
Let ^vds be taken between limits corresponding to two fixed
points in the path of the body ; then this integral is less than the
corresponding integral would be if the particle were compelled to
describe any other path by forces in addition to X, Y, Z acting at
every instant in a direction at right angles to the direction of the
body's motion. Let X', Y', Z' be these forces, then they satisfy the
condition
X'dx+ Y'dy + Z'dz = 0,
and therefore v is still given by equation (3).
By the principles of the calculus of Variations we have
hfvds = J'h.vds = J{ds .^v + vhds)
ds = vdt ;
.'. ds .t>v = dt .vlv — ^dt .Iv^
= ^{Xlx+ Yly + Zlz) dt by equations (2) and (3)
= dv, .Ix + dvo.ly + dvs.lz-^ {X'lx^ Y'ly + Z'lz)dt,
where f ,, v^, v^ are the resolved parts of the velocity in the directions
of the axes :
also ds^ = dx^ + dy^^dz^;
.'. ds .Ids ^dx.Zdx + dy .Idy + dz .Idz;
.'. vZds = V),ldx-¥v'ldy + V3ldz
= Vidlx -(- Vodly + Vzdlz;
IV.] A TREATISE ON DYNAMICS. Ill
.-, ds,Zv + vhds = dvi.ix + dvo-^i/ + dv3.^z + Vid^x + v,dci/ + V3d6z
«= d (UiBx + Vohi/ + I'aSz) --^{X'lx + Y'hi/+Z'ds)dt ;
.'. fh.vds = v,ha; + v,l7/ + v,Zz-^f{X'lx+ Y'Zy + Z'l2)dt + C,
which is the indefinite integral. Now, since the extreme points are
supposed constant, Zx, ly and Iz are separately zero at each limit,
and therefore the definite integral fZ.vds between those points
= -^f{X'Zx+Y'Zy + Z'Zz)dt
between the same points.
When X' = 0, Y'=0, Z'=0, this definite integral is zero, so
that when the particle is resigned to the action of the forces X, Y, Z,
Z fvds becomes zero, shewing that, in this case, fvds is a maximum
or minimum, and from the nature of the case it cannot be a maxi-
mum, therefore it must be a minimum.
Again, let a particle be restricted to move on a smooth surface ;
the equations of motion are
M~=X+Rcos\,]
M^^=Y+ R cos, X.,
d^z
M -T-^ = Z+ Rcos V.
where cos A, cos n, cos v are the direction-cosines of the normal.
Then, as before, we have
V' = V +f(x, y, z) -f{a, h,c) (2)
where Mdf{x, t/, 2) = 2 {Xdx + Ydy + Zdz).
Let the integral ^vds be taken between limits corresponding to
two fixed points in the path of the body ; then, in this case also, the
integral is less than the corresponding integral would be if the par-
ticle were compelled to describe any other path on the surface by
forces X', Y', Z' in addition to X, Y, Z the direction of whose resul-
tant was always at right angles to the direction of the body's motion.
We have, as before,
6 Jvds = j'h,.vds=: f(ds . cv + vcds),
ds= vdl;
.(1).
112 A TREATISE ON DYNAMICS. [cHAP.
.-. ds.lv = dt.vZv = ^dt.Zv''
= j^(Xtx + Yty + Zcv)dt from equation (2)
Udvi X' + RcosX\^ /dv2 Y'+Rcosn\^
(dvs Z'+Rcosv\^ \ ,^
- dv^.Zx + dvs.ly-^diV . lz-r^{X'lx+ Y'ly + Z'lz)dt
since cosx. Ix + cos fx.ly + cosv .lz = 0, the variation indicated by
Zx, ly, cz being along the surface.
And, as before, vlds = Vidlx + V2dly + ihdlz,
so that l.vds = d{vylx + V2ly ^v^lz)- -^j^iX'lx + Y'ly->rZ'lz)dt,
from which the conclusion follows as in the preceding case.
v.] A TREATISE ON DYNAMICS. 113
CHAPTEE V.
75. Having described the methods of determining the motion
of a particle when acted on by forces either constant or depending
only on the position and motion of the particle under consideration,
we will now proceed to another extensive class of problems, which
can be solved (except so far as our analysis is deficient) by the aid of
the same principles. This class consists of the cases of the motion of
any number of particles under the action of their mutual attractions.
There is one condition to which we shall consider these attractions
always subject ; namely, that when two particles A and B attract
each other, either alone or constituting two out of a system of many,
the actual force of attraction exerted by ^ on B is equal to that
exerted by B on A, and these forces act in the line joining the two
particles, and in opposite directions. This is the case in all attractions
in nature with which we are acquainted, and is always assumed to
hold whenever two particles are said to attract each other. It is in
general assumed without any formal statement, though it is possible
to conceive that it should not be the case. Under the term attraction
is included both attraction and repulsion.
In treating this class of problems we shall not enter into the de-
tails of the solution of any of them, but shall confine our attention to
finding the equations of motion, and indicating the different ways
in which the solutions are usually effected. We shall also prove
certain properties which are common to the motions of all systems
of this class.
We in every case suppose the force of attraction to be a function
of the masses of the particles and of the distance between them,
and of nothing else.
76. First, let us consider the case of two particles moving in a
straight line under their mutual attraction.
Let m, m! be the masses of the particles ; P the force of attraction
between them, estimated so as to be positive when the force is
attractive, and negative when it is repulsive ; x, x' their respective
distances from some fixed point in their line of motion, at a time t
from some fixed epoch. Then P will be a function of m, m', and
X — x'.
8
114 A TREATISE ON DYNAMICS. [cHAP.
Let X be supposed greater than x', then the equations of motion
will be
dt
d'x
f = -P
= P
•(1)
Adding these equations, we have
^-df'-'''-de-=''' (~)
also, dividing by in and m' respectively and subtracting, we have
d"{x-x') _
= -P(l+i,) (3)
df \m m'J ^ '
Since x and x' enter P only in the form x-x' equation (3) is of
the same form as the equation for the motion of one particle only,
and may be integrated by the same methods. The integration of
equation (2) can be easily effected.
The integrals of equations (2) and (3) will be two relations between
Xy x', t and constant quantities, from which x and x' may be found
separately as functions of t, and the absolute motion of each of the
particles determined. The constants introduced in integration may
be determined, as before, by the positions of the particles and the
circumstances of the motion at some known instant.
This is the most direct though not the most convenient method of
obtaining the solution of the problem from equations (1).
Let X be the distance of the center of gravity of the two particles
from the fixed point, at the time t. Then
(??? + m') X = mx + m'x' (4)
Differentiating this equation twice, we have
,. dx dx , dx , ^
(,. + m)^ = ,«^+m^, (5)
,.d^x d'x ,d'x'
and {m + m)-^,=7n^+m-^ (6)
Therefore, from equation (2) we have
d'x ^
(7)
From which -5- = constant.
dt
v.] A TREATISE ON DYNAMICS. 115
Now at some particular instant let the velocities of the two parti-
cles be V and v' respectively, then from equation (5)
[in + m)-j-=mv + mv ,
, ^, « dTt mv + m'v' _
and therefore, -j- = 7- = v suppose.
dt m + m' ^^
We have obtained the following result. If a point move so as to
coincide at every instant with the center of gravity of the two parti-
cles m and m', it will advance with a uniform velocity v ; or, in
other words, the center of gravity moves with a uniform velocity v.
This property, which has been shewn to hold in this particular
case, will be proved to hold in a great many descriptions of mo-
tion. It suggests another method of solving the problem under
consideration.
77- This method is to determine the motions of the particles
relatively to their center of gravity considered as a fixed point.
If the relative position and velocity at any time is known, the
absolute position and velocity can be immediately determined from it.
Let then r and r' be the distances of the particles from their
common center of gravity ; then we have
^_m'{x-a^) ^,_m{x-cc')
/■
m + m' ' 7n+ m' '
or, (x - x') = 7— r = r'.
^ ^ vi' m
We may, therefore, put equation (3) into the
forms :
d'r P
de~ vi'
two
following
rfV_ p
de~~m''
and since P is a function o£ x-x' and therefore of either r or /,
these two equations may be integrated, and the motion relatively
to the center of gravity determined.
In determining the constant after the first integration of the
equation
drr__T
de~ m'
dr
we have to substitute for j- its value at some known time, or in
8—2
116 A TREATISE ON DYNAMICS. [cHAP.
some known position. It must be borne in mind that this value is
not the velocity of the particle m at that time, but its velocity rela-
tively to the center of gravity : that is, it is v~v, not v.
dr
Similarly, vfe should have to substitute for -r- the quantity
v-v', since it is estimated in a direction opposite to that of w'.
This method is sometimes stated as follows. When two particles
move in a straight line under their mutual attractions, the motion
relatively to the center of gravity may be found thus : impress on
each of the particles a velocity equal to that of the center of gravity,
and in an opposite direction ; this will reduce the center of gravity
to rest, and not affect the motion relatively to it; take now the center
of gravity as origin and determine the motion by the ordinary
equations for rectilinear motion.
78. The integration of equation (3) gives the motion of the two
particles relatively to each other. This is also sometimes stated
as an independent method, as follows: to find the motion of a particle
m relatively to a particle m' when they move in a straight line under
their mutual attractions, impress on the particle m' a force equal to
the force which m exerts on it, and in the opposite direction ; there
will then be no force acting on m'; also, in order that the relative
motion may not be affected, impress on m a force whose accelerating
force is equal to, and in the same direction as, that of the force
impressed on m'; also, impress on each particle a velocity equal and
opposite to the velocity of ?«' at some instant : m' will then remain at
rest, and the motion of m relatively to it may be found by the
ordinary equations of rectilinear motion.
It is easily shewn that this method will lead to equation (3) ; thus
let r be the distance of the particles ; the force which m exerts
on m' is P, therefore the force impressed on m' is — P, and its
p
accelerating force is j ; therefore the force which must be im-
mP
pressed on wi to have this accelerating force is ^ , and the force
that does act on it is - P ; therefore the equation for its motion
becomes
d'r „ mP
d'r „/l 1\
d r \m m'J
which is the same as equation (3), with r in the place of x - x',
1
v.] A TREATISE ON DYNAMICS. 117
Though in this case these two last methods are of little import-
ance, from the simplicity of the direct method of solution, they will
be seen to apply in other more general cases, where the direct
solution would be very complicated.
79. There is another method of combining the original equations :
which is sometimes of use ; we have
_ d'x dx
and adding these we have " '"'
^f d'xdx ,d'x' dx'X ^^d{x-x')
from which we have
or, mv^-^ mV» = C - 2jPd(x - x').
Since P is a function of x - x' this integration can generally be
effected.
80. Next, let there be any number of particles moving in a
straight line under their mutual attractions.
Let ?«!, 7/?2, 7ff„ be the masses of the particles.
a*!, Xj, x„ their distances from some fixed point in their
line of motion at time t.
jPo the force of attraction of 7)1., on m^, estimated as positive when
tending to increase x^, with a similar notation for the other forces,
then the equations of motion will be
^.^ = .P. + ^P^+ +^Pn \
^"2-^'= 2^1 +2^3+ +2^.
f;"=„A + «P.+ +,P,_.
0)
118 A TREATISE ON DYNAMICS, [cHAP.
where 1^2 + 2^1 =
,P, + ,P, = \ (2)
and 1P2 is a function of x^-x^, and the other forces are in like
manner functions of the distances of the particles to which they
correspond.
The solution of these n equations would determine the n quan-
tities Xi, Xn,...x„ as functions of t, and thus completely determine
the motion of each of the 7i particles.
If we add together equations (1), and bear in mind equations (2),
we have
'"'-dF^'^'-dF^ + »«„^^,- = o...(3).
Now if X be the distance of the center of gravity of all the
particles from the fixed pointy we have
(rWi + ?«3 + . . . +^«„) 0? = 7«i .Ti + 7W2'^2 + • • • + ^»^n J
- d'^'x d^x, d^x. d^x„ ^
_ dx
.•. t; = — - = const.
at
or the motion of the center of gravity is uniform.
We shall hereafter see that this is the case when particles move
in any way under their mutual attractions.
Again, from equations (1) and (2)
f d^Xi dx-i d'^Xs dxz d^x„ dxA
-r^ ~dF lu-^"^' -d^-dt'"-''-'''''U-dt ]
and integrating
or, as it is generally written,
2 mv' - C + 2 2 /.Pj (/ (a:i - x^).
These integrations can generally be effected.
We have thus obtained two integrals of the n equations (l); the
methods for obtaining others will depend on the forms of the func-
tions P, and must be left to the ingenuity of the analyst.
v.] A TREATISE ON DYNAMICS. 119
81. Let US now take the case of two bodies moving in one plane,
but not in the same straight line. This is possible, for if at any
instant the directions of their motion lie in one plane, they will
always lie in that plane, since no force acts to move them out of it.
Let m, m' be the masses of the two particles x, x' and y, y' their
co-ordinates at time t ; r their distance ; P the force of attraction
between them ; then the equations of motion are
~ = -Jf —
r
dt-
* dt-
r
= -P'
^_
df
= -P
y -y
(1)
with the geometrical relation
r'=(,x-xy + Q^-yy,
where P is a function of r and constant quantities.
From these equations we have
d'x ,d'x'
(2)
' dt'
d'y
"df
de
df
0,
= 0,
,(3)
from which it appears as in the preceding cases that -j- and -j- , the
resolved parts of the velocity of the center of gravity of the two
bodies, are constant ; and therefore the center of gravity moves in a
straight line with uniform velocity.
Again we have
dx d^ dys fclx^ dx' d^y' dy\
Tt^dt' dt)^^ \de 'dJ^'dW Tt)
9.m
(cPx
\dt'
op
_,^ d(x-x')
'^ dt
+ (y-y')
'^^Syjill
dt
= -2P
dt
{^ih(m
= C-2jPdr....
= c
JPdr,
.(4)
120 A TREATISE ON DYNAMICS. [ciIAP.
This is another integral of equations (l). We may obtain another
as follows :
f d'y cPx\ ,( ,d'ij' ,d'x'^
'^{fd^-^-drO'-'''V'd¥-^ d?
From which we have
(4) and (5), together with the equations
dx ,dx'
'"Tt'-'''-dt='
dy ,dy' f ^^^
dt-^'^dt-
which we derive from (3), are the first integrals of equations (1).
If we can solve these the absolute motions of the particles m and m!
will be completely determined.
Generally, however, it is not the absolute motions of the two
particles that are required, but their motions relatively to their
center of gravity, or to each other. First, then, to find the motion
of m relatively to m', let x^y^ be the co-ordinates of wz considering
m' as origin, that is, let
x, = x-x', yi=y-y'.
Then we have, from equations (1),
d^ {x — x')_ -pffi + »«' x-r- x'
df mm' r '
d^Xx ■n'm + m' Xi .
rfr mm r I
and^ = -P^^^J
at Tnm r '
These equations are the same as those for the motion of a particle
round a fixed center of force : all the properties therefore which are
proved to be true in that case, are true also for the motion of one of
two bodies relatively to the other ; and the same methods of solution
apply in this case as in that. Writing P' for P ;- , we have the
two polar equations of motion
v.] A TREATISE ON DYNAMICS. 121
These •will be the equations from which the solution will generally
be best obtained. This method may be enunciated in words thus.
To find the motion of m relatively to m', apply to m' a force equal
and in the opposite direction to that exerted by m, then there will
be no force acting on m' : in order that the relative motion may not
be disturbed apply to m a force such that its accelerating force shall
be equal to that of the force applied to m' : also apply to each of the
bodies a velocity equal and in the opposite direction to that with
which m' is moving, m' will then be reduced to rest and the relative
motion of ?« will be the absolute motion.
A little consideration will shew at once that this, which we have
given as the interpretation of equations (7), might have been assumed
a priori as true. . The motion of in' relatively to ?« will be exactly
similar to this.
To find the motion relatively to their center of gravity.
Let f, »7 be the co-ordinates of m relatively to their center of
gravity, p the distance of ?« from it ; then we have
,(x-x'), t] = (w_y).
?n + m
and equations (7) become
the eijuations of motion relatively to the center of gravity.
P, which is a function of r, will be a function therefore of p.
These are the equations of motion which we should have had if the
body had been moving round the center of gravity fixed, acted on
by the same force. The only points of difference will be in the
determination of the constants introduced in integration. We may
enunciate the method therefore as follows.
Impress on the system a velocity equal and opposite to that with
which the center of gravity is moving ; the center of gravity will
then be reduced to rest, and the motion about it may be determined
by the ordinary methods for determining the motion about a fixed
center, and all the conclusions arrived at in motion of that kind
will be true also in this.
If the absolute motion be required we can compound the relative
motion thus found with the motion of the center of gravity.
122
A TREATISE ON DYNAMICS.
[chap.
82. Let us now consider the case where the motions of the two
particles are not in one plane. Using a notation similar to that we
have already adopted^ the equations of motion are
x'
d^
d?
d^
■ dt'
r
,d'x'
> y -y
r
'-Z
r
...(1),
From these equations we deduce as in the preceding cases, the
equations _ _ _
dx dy dz , .
-dt'='^' Tr'^' Tt='^ (2)
which shew that the center of gravity moves uniformly in a straight
line.
We may also obtain as before the equation
mv'' + m'v'"'=C-2JPdr (3)
bearing in mind that
r^ = ix-xy^{y-yy + {z-zy.
And we shall also have, in place of equation (5) of the preceding
case, the three equations
dy\
f dz
Vdi
( dy dx\ ,( ,dy' , dx'\ ,
V^-^dt)'-'^V-dt-^-dF) = ^'^
dx
Tt
dz\ /f ,dx' ,dz'\
...(4)
Equations (2), (3) and (4), are the first integrals of equations (1),
and when they can be again integrated, give the direct solution of
the problem.
We might deduce the equations giving the motion of w relatively
to m! or of either relatively to their common center of gravity. It is
also easily seen that the principles before stated for reducing the
motion of one relatively to the other, or to their center of gravity
to the case of motion about a fixed center will apply in this case
also. Hence it follows, that in whatever manner two bodies move
under their mutual attractions, the motion of one relatively to the
other, as also the motions of the two relatively to their center of
gravity will be in one plane. This plane however will not coincide
v.] A TREATISE ON DYNAMICS. 123
with the direction of motion of the center of gravity, or of that body
to which the motion of the other is referred.
When we have by one of the principles above stated reduced the
motion to that about a fixed center, it will become only a particular
case of the motion of one particle, which has already been fully con-
sidered.
83. Let us now consider the case of any number of particles
moving under their mutual attractions.
Let nil, 1112, . . . m„ be the masses of the several particles ; a:,, 3/1, 2, ;
^2, ^2) ^a', ••• ar,„ i/„, s„ their co-ordinates at time t; 1P2 the abso-
lute force of attraction between 7M, and m.^ ; Avith similar expressions
for the other forces ; ^r^ the distance between Wj and 7/13 with similar
expressions for the other distances.
The equations of motion are
d^Xi _ Xi — X.,
mi ■j-^ = -iP2 -
dr ir,
m -f7r= - iP2-^ — -
df ii\
d'si _ Z1-Z2
7«i -772- = - 1^2^
dt^ ^ ira
with similar equations for all the particles.
Together with the geometrical conditions
ir^ = {xi - 0:2)- + (j/i -^2)' + (-.
and similar equations for the other distances.
From these equations, we have
1^3
ir„
.p/'-"' ■
.pr--'"
d'x,
'■-dt^-""
d'x.
d\y,
d'vn :
THi ^^, +...
■ "'" dt' "'
m — — - + , .
dl'
+ v?„ ^^, V,
from which, if x, 1/, z are the co-ordinates of the center of gravity.
dx
-dt='^'
dy
dz
or the center of gravity moves in a straight line with uniform
velocity.
K
124 A TREATISE ON DYNAMICS. [CHAP.
This property which we have seen to hold in all the particular
cases before considered, and have now proved to hold in the
general case of the motion of any number of particles under their
mutual attractions, is a particular case of a much more general
principle, called "the principle of the conservation of the motion of
the center of gravity," which will be stated and proved hereafter.
84. Again, multiplying each second differential coefficient by
twice the corresponding first differential coefficient and adding all
together, and bearing in mind the geometrical relations which give
(--)(S-^').
+ 2
we have
And we observe on the right-hand side that jPg is a function of
jTj and constant quantities, and, in like manner, the other forces of
the corresponding distances ; therefore integrating, we have
or, as it may be written,
^{rnv^)=C-^^jPdr.
We have, in particular cases already considered, obtained equations
corresponding to this.
The name " vis viva" has been given to the product mv^, 1.{mv^)
which is the sum of the " vires vivae" of the several particles of the
system is called the " vis viva" of the system.
We will now ascertain what the last formed equation assures us of.
Let 2/,Po<Z,r2 be i/sGr^),
then it becomes '2.{mv^) = C -1. {f{r)].
Now let Fi, F.2"-Vn be the velocities of the several particles at
the time when their mutual distances are lOa, lOg, sOa, &c. then
we have
2 0«n=C-2{/(a)},
from which we have
2 ipuv') - 2 {m n = 2/(«) - 2/(r). _
This equation shews that the change in the vis viva of the
system depends only on the relative positions of the particles at the
two instants when it is estimated, and is independent of the direc-
tions in which they are moving or the paths they have described.
v.] A TREATISE ON DYNAMICS. 125
85. Again, recurring to our equations of motion, it will be seen
on examination that they give the following equations,
/ cf'x, d-z\ f d^x^ d\^^\ „
"'^ (^' rf^-^' rfF j -^ ^"^ (- rfT^ -^^:^ j ■" ^"- = ^^
Integrating these equations, we have
/ dsi dy^\ f dz„^ di/^\
"^^ \y^-dt-'^dt) + ^"^ V'- -di --^) -^ ^^- =_!'^^
( rfx, dz\ ( dxz dzo\ . ,
"^ V- -dt-^^-dt)'- '"^ ( - 77 - ^--77) -*- ^'- = ^'"
(c?Mi dxi\ t dy^ dx\ . ,
or, as we may write them conveniently
^ { dx
dz dy\ y
dx dz
Tt
where h^, kg, h^ are constant.
Let us now consider what these equations express.
If we conceive a line drawn from the origin of co-ordinates to a
particle m to generate an area by its motion, and if A be this area at
time t, and A^, A^, A.^ its projections on the three co-ordinate planes,
or, which amounts to the same thing, if we conceive that Ai, A^, Ag,
are the areas described on the co-ordinate planes by the projections
of this imaginary line, then it is known from geometry, that
dz
y-Tt-
Jy
-'dJ-
-~ dt
dx
dz
-~ dt
dy
dx
-^Ti-
'" dt '
126 A TREATISE ON DYNAMICS. [cHAP.
These three
equations then
give
22(7;
dt)'
-h,
21 fm
dt)-
-h„
21 (m
dJ,\
dt) =
/^3.
And integrating them we have
2'Z(9nAi)^hit,
2l{mA^)=h.2t,
2l.(niAs) = hst,
making the constants which are introduced by integration each
zero, which amounts to supposing that the description of areas com-
mences at the instant from which t is measured.
These equations shew that if the mass of each particle of the
system be multiplied by the area described by the projection on
each of the co-ordinate planes of the line joining it with the origin,
the sum of all these .products will be proportional to the time of
describing them, whatever point be assumed as origin.
This is a case of a general principle which will be proved here-
after, called the " Principle of the Conservation of Areas."
86. Let A, fx, I/, be the direction-cosines of the normal to a
plane, then the projection of the area A on this plane is
XAi+fxAz+vAa,
and a similar expression will give the projection of the area for
every particle.
Therefore, the sum of the products of each mass into the projec-
tion on this plane of the area described by it is
X 2 (?n Ai) + /A 2 (7rt A2) + vl (m A^.
Now, to find the plane on which this sum is a maximum at any
time we must make this expression a maximum by the variation of
A, Mj and V, subject to the single condition
We have then
1(mA,) lX + 1{mA^lfx + 1{mAs)lv = 0,
XlX + fxlix + vlv = ;
v.] A TREATISE ON DYNAMICS. 127
therefore, using the indeterminate multiplier B we must have
2 (»w A^) +B\ = 0,
1.(7)1 As) + Bfx = 0,
2(?tt^3) + Bu =0;
and therefore =7 — -j- = _.^ . . = vT—TT >
2(ot^i) ^(wJig) l.{mAs)^
X fx 1/ _
and therefore, if /r = ^,^ + ^2^ + /^3^ we have
k, h, /«3
That is, the position of the plane is constant during the motion ; it
is called the plane of maximum areas, or the " Invariable plane."
We may remark that the position in space of the plane has
not been proved constant, but merely the direction of its normal ;
all planes parallel to this would have the same property.
If at any instant we know the positions, magnitudes, velocities,
and directions of motion of the several particles of such a system
as we have been considering, we can find the position of this invari-
able plane, with reference to co-ordinate axes arbitrarily chosen;
and if at any future time we take other co-ordinate axes arbitrarily
chosen, we can also determine with reference to them the position of
the invariable plane, and we know that these two positions must
be absolutely the same, as far as regards direction, whatever the
co-ordinates by which they are determined. Thus we see that the
system itself furnishes a plane of reference which is invariable in
direction, and can always be determined from the state of the system
at any instant.
The preceding are the principal general properties of a system
of particles moving under their mutual actions ; the actual determi-
nation of the motion of each particle in anything like the general
case far surpasses our powers of analysis. The solution of a number
of simple cases is the only means of acquiring a familiarity with the
treatment of problems of this sort.
87- We have hitherto supposed the system to be acted on by
no forces but those which arise from the mutual attractions of its
several particles ; in addition to these, all or any of the particles of
the system may be acted on by forces arising from causes extraneous
to the system, as, for instance, the attraction of some body not
128 A TREATISE ON DYNAMICS. [cHAP.
forming a part of the system ; or by forces arising from geometrical
constraints, such as, that some of them move in tubes or on surfaces.
In these cases if the forces are given, they will enter our equa-
tions as known quantities, and we shall still have as many equations
as we have co-ordinates to determine.
If the forces are not given explicitly, but are known to arise from
the particles being compelled to satisfy certain geometrical conditions,
they will then enter the equations as unknown quantities; and,
corresponding to each unknown force, there will be an equation
expressing the geometrical condition from which it arises ; so that
the whole number of equations will in every case be the same as the
whole number of unknown quantities, and will thus be sufficient
to express every such unknown quantity as a function of t.
We shall not, however, enter upon any of these cases here, as our
object is not to obtain solutions of particular problems, but to explain
the principles by which those solutions are to be obtained, and to
investigate certain properties which belong to extensive classes of
such problems.
VI.] A TREATISE ON DYNAMICS. 129
CHAPTER VI.
88. In the preceding pages the motion of a single material
particle, and also of a system of several particles has been consi-
dered, when the particles were acted on by various forces and subject
to different conditions; it remains to investigate the methods of
determining the motion of bodies of finite magnitude.
Now, when a body of finite magnitude is in motion, unknown
forces act on every particle of it arising from the action of every
other particle. The determination of these forces from the equations
expressing the conditions to which the system is subject presents
insuperable difficulties. The consideration of these forces is avoided
by making use of a principle first stated by D'Alembert, which we
will now proceed to explain.
When a material system is in motion under the action of forces
arising from causes external to the system, these external forces are
called "impressed forces;" and those of them which act on any one
particle of the system are called the impressed forces on that particle.
Every particle besides being acted on by the impressed forces, is
also acted on by forces arising from its connexion with the other
parts of the system ; these are called " internal forces."
The whole force acting on any particle is the resultant of the
impressed and internal forces on that particle, and is called the
"effective" force on that particle.
Conceive an equal isolated particle to be moving in the same
direction and with the same velocity as the proposed particle, and to
be acted on by a single force equal to this resultant and in the same
direction; the two particles Avill be in the same state, one being
acted on by a number of forces, the other by a single force which is
their resultant; the motions of the two particles will therefore be
the same. Hence then we sometimes have the following definition
of effective force.
That force which acting alone on any particle of a system sup-
posed isolated would cause it to move as it does when it forms part
of the system; that is, which would generate in the small time 8/
the same change in velocity and direction of motion that takes place
when the particle forms part of the system, is called the " effective"
force on that particle.
9
130 A TREATISE ON DYNAMICS. [cHAP.
Now D'Alembert's principle asserts that if at any instant we
apply to every particle of a system forces equal and opposite to the
eftective forces on that particle, the system, if it were at rest in the
position which it occupies at that instant, would remain in equili-
brium under the action of these together with the impressed forces.
We have enunciated this principle in the form in which it will be
most convenient for obtaining equations of motion for a system. We
will now explain the nature of the assertion contained in it.
Let mhe a particle of the system ; / the
resultant impressed force acting on that
particle ; M the resultant of the internal
forces on it; JE the resultant of M and /,
and therefore the effective force on the par-
ticle ; R the reversed effective force, that is, II
a force equal to E acting in the opposite direction.
Then if the particle m were at rest, and the forces /, M, and R
acted on it, they would keep it in equilibrium : and the same would
be true of every particle.
If therefore the whole system were at rest, and each particle
acted on by the impressed forces, the effective forces reversed, and
the same internal forces that acted on it when the system was in
motion, every particle would be separately in equilibrium, and
therefore the whole system would be in equilibrium.
Instead of this suppose the system to be at rest, and each particle
to be acted on by the impressed forces and reversed effective forces
that correspond to it, and by such internal forces as naturally arise
from this condition ; then D'Alembert's Principle asserts that the
system will in this case also be in equilibrium.
The Principle may also be considered in the following light, in
which it possibly appears less arbitrary.
A system is in motion under the action of impressed forces and
internal forces arising from the connexion of its parts. Let the
impressed forces be replaced by the effective forces: in this case,
since each particle will move under the action of its effective force
alone consistently with the connexion of the parts of the system, the
internal forces arising from this connexion will not be called into
action, and the change of motion in the small time ht will be the
same as under the action of the impressed forces.
Now, forces equal and opposite to the effective forces, acting
together with the effective forces on the system at rest, keep it in
equilibrium ; will they keep it in equilibrium when they act together
with the impressed forces ? D'Alembert's Principle asserts that they
VI.] A TREATISE ON DYNAMICS. 131
will. It states that a set of forces which balances one of two sets that
produce the same change of motion in a system will also balance the
other.
D'Alembert's Principle, like the laws of motion and the theory
of gravitation, is proved by the agreement between the results of
calculation and observation. We are not, however, able in this case
to avail ourselves of Astronomical observation. The comparison
has been principally made by means of a machine invented by
Atwood, by which the motion of a system of bodies can be very
accurately observed in a great variety of cases.
89. To form the equations of motion by the aid of this principle,
we must express the effective force on each particle in terms of the
co-ordinates of that particle : if m be the mass of a particle, x, y, z
its co-ordinates, the resolved parts of this force at the time t will be
d-x d-y , d'z „,
m-T-j, m -— , and m -^ . We must now suppose the system to be at
rest in the position it occupies at the time t, and apply to every par-
ticle a force equal and opposite to the effective force on it The
system will then be in equilibrium under the action of these and the
impressed forces : and the equations expressing the conditions of
equilibrium of the system will be the equations of motion.
These equations will be different for different systems of bodies.
They will be different according as there is one body or more, as the
bodies are elastic or inelastic, solid or fluid, rigid or flexible.
The equations expressing the conditions of equilibrium assume
different forms in these different cases, and consequently the equa-
tions of motion obtained from them by the aid of D'Alembert's
principle are of different forms.
90. We will now apply this principle in the form in which we
have enunciated it to the solution of a problem ; viz. to calculate the
motion of a uniform cylinder with its axis horizontal, rolling down
an inclined plane so rough as to prevent all sliding.
Let the figure represent a section of the cylinder and inclined
plane; and let a be the radius of the cylinder, h its length, a the
inclination of the plane to the horizon.
Let any point B in the plane be the origin, BP the axis of x, a
normal to the plane atB the axis of ^ ; Q an element of the cylinder,
^7)1 its mass, x, i/, z its co-ordinates ; then BN = x, QN=y.
Let BP^x, QC = r, QCA^tp.
ACP = 6 the angle through which the cylinder has rolled.
9—2
132 ■ A TREATISE ON DYNAMICS. [cHAP,
Since z is constant -7-^ — , and the resolved parts of the effective
at' ^
force on Q are Im -y— and Im ~~, parallel to the axes of a; and y ;
and the impressed force is g^tti vertically downwards. Also on the
line of particles in contact with the plane there will be other im-
pressed forces arising from the reaction of the plane : let the resolved
parts of one of these parallel to PC and PB be R' and F\
Now apply to every particle the reversed effective forces on that
particle, and write down the equations of equilibrium : they ai*e
1.R' - "EZjn-— — ^^me^cosa — 0,
(It
1.F' + 1.^7)1 -j-^ ~ I-hm g sin a = 0.
1.F'a + IhynQc- x)—^^-^lm{y -")ji2 +'^^mg cos a (x - x)
+ 28 mg sin « (^y — a) = 0.
To these equations we must add the geometrical relations
X ^x — r sin (6 + (p), y = a- r cos {Q + (p),
from which we obtain
•i , -.fdey . .d'0
r^-^'-^\dl)^^-'^-''^dF
de di-
d'y . .fdOy . rl^d
dF- ~(^-"Kdi)-^'-''^de'
Let 2/2' = 72, ^F'^F, 1.1m =M, then substituting these values
in the equations we have obtained, and bearing in mind that ^, y,
VI.] A TUEATISE ON DYNAMICS. 133
and d refer to the whole cyhnder, and that the center of gravity of
the cylinder is in the axis, and therefore thSt
1.l7n(x-w) = 0, 1.d7n(i/-a)-0,
we have R - Mg cos a = (1 )
M~ = Mgdna-F (2)
'^^hnr"-=Fa (3)
also X — aO + c (4).
Equation (l) determines the whole pressure on the plane, which
is constant.
Also 'Zcmr^^Ma%
therefore equation (3) becomes
Ma d^_
2 de~
Eliminating F between this equation and (2) and reducing by
means of (4), we have
d^O _ 9.g sin a ^
dQ _2g sxna.t
" dt ~ Sa '
— is called the angular velocity of the cylinder.
C and C may be determined from the values of 6 and -j- corre-
sponding to any value of t ; and the motion is then completely de-
termined.
We see from the preceding example, that even in a simple case
the equations given by the immediate application of the principle
are complicated, and require considerable reduction before they can
be integrated. In the case of rigid bodies, however, this reduction
can be effected generally, as follows:
91. Let there be one rigid body, and let 7/?, be the mass of a
particle, x,, i/i, ri its co-ordinates at time /, X, Yi Zj the resolved
parts of the impressed forces that act on it. The resolved parts of
the effective force on it are
d'x, d'y, d'z,
""^Tt^' "''rfF' "'^di'-
134 A TREATISE ON DYNAMICS. [CHAP.
If we apply these in the opposite directions, the resolved parts
of the whole force on the particle will be
,„,--^andZ.-^,-^
and the body will be in equilibrium under the action of these and
similar forces applied to every particle.
Hence then the six equations of equilibrium give
(I)
^(i
mA--
dt-
)}
(II)
These are the six equations of motion for one rigid body, and
when these equations are solved the motion of the body is com-
pletely determined.
92. The preceding equations contain the co-ordinates of every
particle of the body, the use of them is greatly facilitated by the
following reduction.
Let X, y, J be the co-ordinates of the center of gravity of the
body at the time t ; x' , y', z' the co-ordinates of a particle of the
body referred to axes whose origin is the center of gravity and which
are parallel to the original ones : then
and therefore
d'x d^x (fx'
dt' dt'
df
d^J^ d^
de dt^ de
dt'
di'' "^ dt'
VI.] A TREATISE ON DYNAMICS. 135
Substituting in equations (I), we have
dl^ dt^
but since the origin of x' y' z' is the center of gravity
and therefore
„ d'x' ^ _ d'y' ^ ^ d'z' ^
di' dl' dr
also X, y, z are the same for every "particle of the body, therefore
writing M for 2?h, we have
^^§=-^4 (Ill)
dt'
These three equations are in the form in which they are most
available for determining the motion of the body. They are the
same as those for the motion of a particle whose mass is M acted on
by the forces 1.X, 1.Y, and 2Z,
Hence then, if a body of finite magnitude be acted on by any
forces, its center of gravity will move as a particle of equal mass
Avould if acted on by all the forces impressed on the body.
We also see that the motion of the center of gravity of the body
will not be affected by changing the points of application of any of
the impressed forces, provided their intensities and directions are
not altered.
If all the forces X, Y and Z be given explicitly as functions of
the variables, these equations will give the motion of the center
of gravity at once by integration.
When this is not the case, that is, when unknown forces enter,
arising from geometrical conditions to which the motion is subject,
it will generally be necessary to combine them with the remain-
ing three equations before the integration can be effected.
136
93. Substituting a; + a;', 7/+ y, 2' + s for
(II) we have
X, y, an
2{(i + /)(x-
d'x
-(i + xo(z-
d'l
2|(3 + x0(f-
d'y
-""dP-
d'y'\
-^de)-
.G^+y)(x-
^)}-
d'y'-
dt
Now in these equations x, y, i and their differential coefficients
are common to all the particles of the system, they may therefore
be placed before the symbol, 2.
Also, 2 m x' = 0, 2 my' = 0, 1.mz' = ;
and therefore,
_ d'x' ^ _ n't/ ^ ^ d'2' ^
By means of these relations, and also the equations (III), the
preceding equations may be reduced to
,(IV)
Now we observe that the position of the center of gravity does not
enter into these equations either explicitly or implicitly.
If then the center of gravity of the body were fixed, and the body
were acted on at the same points by the same impressed forces that
act on it when it is free, the equations of motion would be the same,
since the forces introduced by the condition of the center of gravity
being at rest would all act at the center of gravity, the co-ordinates
of their points of application would be zero, and they would therefore
not enter into the equations.
Hence then we conclude that the motion of the body about its
center of gravity is the same as if the center of gravity were fixed,
and the body acted on by the same impressed forces.
These two results are expressed by saying that motions of
translation and rotation are independent. They are of the utmost
importance in the solution of every problem about the motion of one
rigid body.
VI.] A TREATISE ON DYNAMICS. 137
We will for distinctness enunciate them again.
When a body is in motion under the action of any forces, its
center of gravity moves as if the whole mass were collected there and
acted on by iill the forces which act on the body. And the motion
of the body about its center of gravity is the same as if the center of
gravity were fixed, and the body acted on by the same forces that do
act on it
94. The equations (IV) involve the co-ordinates of every par-
ticle in the body. To determine from them the motion of the body,
they must be reduced to others containing only co-ordinates common
to the whole body.
We shall not enter upon this reduction in the general case at
present, as the process is complicated, but shall confine our attention
to the cases where all the particles of the body move in parallel planes.
Let the plane of xi/ be that to which the motion is parallel ; then
the ordinate parallel to z of every particle is constant. Equations
(IV) are therefore reduced to
or, 2,«(^x'— -y^j^2(x'I-yZ).
Let x' = r cos d, y' = r sin 0,
then for each particle r is constant ; so that we have
d-x' . .,rQ aM^\'
-^ = -rs.ne^-rcose[-),
cVy' .(f-d . ./ddV
dt^ dl- \dtj
Making these substitutions, we have
d^O
Now since it is a rigid body — , and therefore j-r is the same
for every particle of the body, we may therefore place it before 2.
So that we have
'^^j ^mr^ = ^{x'Y-y'X).
Now tlie right-hand side of this equation is the resultant moment
of the impressed forces round an axis through the center of gravity
138 A TREATISE ON DYNAMICS. [CHAP.
perpendicular to the plane of motion; calling this L and putting
Mk^ for 2r«r^ we have
at
Combining with this the first two of equations (III), we have
(V)
These three equations are sufficient to determine the motion of a
rigid body which takes place in one plane, whenever the forces
which act on it are given as functions of the variables.
This is seldom the case. Unknown forces generally enter into
these equations, arising from conditions to which the motion of the
body is subject. There will, however, in every case be a geometrical
equation corresponding to each unknown force so introduced, so
that the number of equations will always be equal to the number of
unknown quantities.
In effecting this last transformation all the geometrical conditions
of the rigidity of the body were introduced when we considered r
to be constant for each particle, and -j- to be the same for all the
particles.
95. In the preceding reduction the expression j-^'Lmr^ occurs,
where r is the distance of the particle m from a certain fixed line
in the body, vir^ is called the "Moment of Inertia" of the particle
m round this line, and 'Zmr^ is the moment of inertia of the whole
body round the same line ; it is the sum of the moments of inertia
of every particle. This quantity will frequently occur; it is different
for different bodies and for different lines in the same body. It is
generally written MF, in which case k is called the "Radius of
Gyration" of the body round the particular line. The Moment of
Inertia of any body may be found by the ordinary methods of inte-
j a
gration. Again, — is called the angular velocity of the body ; if we
conceive a plane fixed in the body perpendicular to the plane of
. motion to be inclined at an angle to a fixed plane also perpen-
VI.]
A TREATISE ON DYNAMICS.
139
dicular to the plane of motion, -7- is the rate at which the angle 6 is
increasing, and the propriety of the term angular velocity is manifest.
Sometimes -jt, is called the angular accelerating force.
96. Let us now take an example of the application of these
equations in the case of motion in one plane.
A uniform heavy rod slides down between a smooth vertical and
a smooth horizontal plane, to determine the motion.
Let the figure represent the
rod at the time t.
AB = 2a,
DN=x, NC = y,
ACN=d,
M the mass of the rod,
R, K' the reactions of the
planes, then from equations (V)
we have
>4-^'..
Mk
de
Ra smd-R'a co&Q
(1)
•(2)
.(3)
and corresponding to the two unknown forces R, R' we have the
two geometrical relations
x = a5\nd, y = acosB, (4)
which express that the ends of the rod are in contact with the
planes.
Eliminating R and R' from the equations, we get
^j^^fd^^dx d^ydj^ d^de\_ dy
^^^ W dt'' dt' dt^^deTtJ-'^^^Tf
Integrating this, we have
and using equations (4),
<--')e
C — 2ag COS0.
140 A TREATISE ON DYiSAMICS. [cHAP.
If we suppose the sliding to commence when the rod is vertical,
we have -r-=0 when 6 = 0:
at
0=C~2ag;
dey 2ag . ..
dtj a^ + k'
dd
dl
V a' + k'
(5)
Or, since k^ — -^
dd_ /Si
dt~\ a
H
which gives the angular velocity in any position ; the equations
dx ^dd dt/ . „dd
dJ=''''''^irt^ 57=-^^^"^d7'
give the velocity of the center of gravity.
The integral of equation (5) is
log tan ? = . /5 t + C.
On attempting to determine C by the condition used beforej
we get C = — CO . This might have been expected since the position
from which the motion was assumed to commence is one of equi-
librium, so that the motion would not take place at all. We may,
however, suppose the motion to commence in a position differing
very slightly from that of equilibrium, in which case the velocity
we have obtained will only be affected by a small quantity of the
same order of magnitude, and C will be rendered finite : or we may
suppose a velocity equal to that indicated in the solution to have
been communicated to the rod in the corresponding position.
To find the pressures on the two planes in any position, we have
\-n) •= x^ (1 -cosy);
\dtj 2a ^ ^'
d'0 3g . .
dv 4«
also from (4) a' = a sin (), y — a cos Q ;
d^x So- . ^,
.•.-^=-^sm0(3cos0-2),
i| = -?l(l+2cosa-3 cos^ &),
at 4 '
VI.] A TREATISE ON DYNAMICS. 141
substituting these values in equations (1) and (2), we have
R'=.^-^ sm 6(3 cos 0-2),
4
R=—^(l-6cosd + 9 cos' e),
Mo-
= — ^ {1 + 3 cos e (3 cos 6 - 2)},
which give the pressures on the planes corresponding to any value
of 0.
When S cos 6 — 2 = 0, R' = 0, and as 6 increases R' becomes
negative : this result shews that a force would be required to keep
the point B in contact with the vertical plane : as this force is not
supposed to exist the rod will at that point leave the plane, and the
motion will be different afterwards.
From the expression for R we see that it continues positive up to
this point.
To determine the motion after the rod leaves the plane BD we
have the equations
M2 = R-Mg,
Mk'~=Rasm0,
y — a cos 6.
dx
From the first equation it appears that -j- remains constant Avith
the value it had when the rod left the plane.
Now at that time
dx ,dd
Tt=''''''^dt'
= cos0y^(l_cos0),
and putting for cos 6 its value | we get
This will continue to be the velocity of the center of gravity in a
horizontal direction.
142 A TREATISE ON DYNAMICS. [CHAP.
From the other equations we have
.-. d" (sin^ + 1) (^Y= C-^ag cos Q.
Now when the rod left the vertical plane
... (3sin^0.l)Q^|f(8-9cos0),
which gives the value of -j- in any position, and we may find the
value of R as in the previous case.
97- When the motion of every particle is parallel to one plane,
we have only the three equations (V) instead of the six equations
(III) and (IV).
These six equations however must always be satisfied, let us
therefore consider what they become in this case.
The last of equations (III) becomes
2Z = 0,
which shews that the sum of the resolved parts parallel to the axis of
z of all the forces which act on the body must be zero.
If the body is free this is one of the conditions that must hold
among the forces acting on it that the motion may be of the kind we
have supposed.
If the body is constrained to move in the manner supposed this
will be one of the equations which determine the forces called into
action by this constraint.
The first two of equations (IV) become
2:,«.'g- = 2;(/7-yz),
^mz' -y-^ = ^{z'X-x'Z).
Now from the relations
x' = r cos d, v' = r sin d.
VI.] A TREATISE ON DYNAMICS. 143
the left-hand sides of these equations can be expressed in terms
of -J- and T— , which are known by (V) ; and the equations then
give two more conditions to be satisfied by the impressed forces.
Thus whenever the motion is of this sort, the six equations (III)
and (IV) are reduced by relations between the impressed forces to no
more than three independent equations.
98. It frequently happens that the conditions of equilibrium of
the system under consideration have been found in a form simpler
than the six general equations of equilibrium. Now, as in consi-
dering any particular case of equilibrium we may employ either the
six general equations, or the more simple ones that have been
deduced for the class to which the particular case under considera-
tion belongs, so we may employ either set of equations of equilibrium
to form the equations of motion.
The case of a body moving about a fixed axis affords an illus-
tration of this remark, and deserves consideration from its own
importance.
In the case of a body moveable about a fixed axis, we may
determine the conditions of equilibrium from the six general equa-
tions, in which case we introduce unknown forces for the pressures
exerted by the axis on the body ; or we may take the condition
which has been deduced from these, that the body will be in equi-
librium when the resultant moment round the fixed axis is zero.
We will consider the motion of a body about a fixed axis in both
these points of view.
Let r be the distance from the axis of a particle whose mass is m,
since its motion is at every instant perpendicular to the axis and the
radius vector, the resolved part of the effective force in this direc-
tion is
(Ps , . - cVe
m-, which = ^r^^^,.
If therefore we apply a force equal to this in the opposite direction,
72/1
its moment round the axis will be — mr^ —-^ .
Let L be the whole moment of the impressed forces, then the
condition that the whole moment round the axis is zero gives the
equation
144 A TREATISE ON DYNAMICS. [cHAP.
or, since -y— /and therefore— -^ is common to every particle of the
system
or, putting Mk^ for 2m r'
iV«-^=L.
From this equation the motion of the body may be completely
determined.
We will now see how the same result may be obtained from the
six general equations.
Let the fixed axis be taken as the axis of z, and a plane perpen-
dicular to it through the center of gravity for the plane of xt/; let
X, y be the co-ordinates of the center of gravity, Q the angle which
a line joining the center of gravity and the origin makes with the
axis of .r.
We may suppose the body to be fixed to the axis at two points,
let these be at distances c, and c' from the origin, and let the re-
solved parts parallel to the axes of the forces which these points
exercise on the body be -R^, Ky, jR,, J?/, Hy, -R/.
Then we have from equations (III), since -r-j — 0,
M^ = 2X4-i?. + i?,' (1)
3f^ = 2:F+2?, + i?; (2)
= 2Z + 7i', + i?/ (3)
df
d^z
also from equations (II), since ,72 =0 foi* every particle, we have
-cE,-c'i?/+2(^Z-.y) + 27«.|^ = (4)
cR, + c'RJ + ^{zX-xZ)-l.mz^^ = (5)
^{xY-yX)-^(^nx^,-my^:;) = (6)
equation (6) may be reduced to
VI.] A TREATISE ON DYNAMICS. 145
This is the same as the equation
which was obtained by the other method : it is sufficient to deter-
mine the motion.
The remaining five equations enable us to determine the un-
known forces /?,, Ry, RJ , RJ , and R^ + R/ ; just as in the statical
problem the general method gives the pressures on the axis, while
the particular method tacitly eliminates them.
We will return to these equations, and put them into a more
convenient form for determining the unknown pressures in particular
cases, but for the present will consider the motion of the body.
99. A particular case of motion round a fixed axis worthy
of consideration is where the only impressed force is gravity, and the
axis is horizontal.
Let h be the distance of the center of gravity from the fixed
axis, 9 the angle between a vertical plane through the axis, and a
plane through the axis and the center of gravity.
Then L =-Mgh sin 6, so that the equation is
•••(;
:^:)'=^-f^-^ (^)-
C must be determined from the angular velocity corresponding to
some known value of 6.
Now the equation of motion of a point in a circular arc of radius
I is
comparing this with equation (l) we see that they are identical, if
'4-
Hence then the time of oscillation of the rigid body will be the
same as that of a particle attached to a string whose length is -j ,
for the same amplitude. Tliis is called the isochronous simple pen-
dulum, the body itself being called a compound pendulum.
10
146 A TREATISE ON DYNAMICS. [cHAP.
The equation . -rji+-jj sin =
can only be solved so as to give the i-elation between & and t \n a.
series.
When however Q is always small, so that powers higher than the
first may be neglected, we can find an approximate solution.
Writing for sin d, which amounts to neglecting 0\ we have
l-^4^^- (^)^
Let the body start from rest when 6= a, then
a = A cos B ; . -
.-. jB = and A = a:
a COS . / ~ t ;
■when 6 = we must have
A''=(^->>i=
let ^1, ts, &c. be the successive corresponding values of t, then we
have _
'^-" SI Rh'
2 V gh
_S7r 11^
~ 2" V gh
_57r 111^
~ % SI gh
1^
This shews that the center of gravity of the body will attain the
loAvest position after successive intervals of tt sj — j '
VI.] A TREATISE ON DYNAMICS. ^ 147
We also see that, to the order of approximation, to which we have
proceeded, the time is independent of the value of o, or the ampli-
tude of the oscillation.
We shall return to this subject afterwards when we treat of pen-
dula, and shall then shew how a more approximate value of the
time of oscillation may be obtained; before leaving it, however, there
are a few general properties which deserve notice.
Let the figure represent a section
of the body made by a plane through
its center of gravity perpendicular to the
axis round which it oscillates ; and let C
be the point in which the axis cuts this
plane, G the center of gravity.
The two equations
will be identical if Z = -7- .
h
If therefore we take in CG produced a point such that CO = I,
the position and motion at any time will be the same under the same
initial circuiftstances as if the whole mass were collected at 0.
O is called the center of oscillation.
If k^ be the radius of gyration round an axis through G parallel
to the fixed one through C,
and therefore I = -^ + h.
h
Now if the body were suspended by an axis through parallel
to the same axis, the length of the corresponding isochronous simple
pendulum would be
which is equal to I.
Hence then oscillations about this axis take place in the same
time as those of the same amplitude about the first axis through C.
C is called the center of suspension, and this property is ex-
pressed by saying that the centers of oscillation and suspension ai'e
reciprocal.
10-5
148 A TREATISE ON DYNAMICS. [cHAP.
This property might also have been deduced in the following
manner. Suppose it required to find the distance from G of an
axis parallel to a given direction, about which the oscillations will
take place in the same time as those of a simple pendulum whose
length is /.
Let k be this distance; then h will be determined from the
equation
or h^-lh + ki'' = 0.
Now this being a quadratic equation gives two values of h,
hi and li^ such that hi + hi = /.
So that an oscillation about an axis at distance h^ takes place in
the same time as one about an axis at distance l-h^.
Also, since in this investigation nothing has been introduced to
indicate in which direction from G the required axis lies, we con-
clude that any direction will satisfy the condition. If therefore with
center G we describe circles in the plane of the paper at distances
GO and GC, small oscillations will take place in the same time about
any parallel axis through any point in either of these circles.
The value of h which will make /, and therefore the time of
oscillation a minimum may be easily determined : since
k ^
l^^ + h,
that I may be a minimum, we have
k *
-^ + 1=0,
or h = ki,
and therefore l = 2h or S^i-
In this case therefore the two circles we have mentioned
coincide.
The axes determined by these conditions are not axes about
which the time of oscillation is the least possible for the body, but
only for axes in the body parallel to a given direction. To find those
about which it will be absolutely the least, we must select that
direction for which k is the least. This we know will be one of the
principal axes through the center of gravity of the body.
A TREATISE ON DYNAMICS. 149
CHAPTER VII.
100. To find the conditions of equilibrium of a system of rigid
bodies we have for each body of the system six equations of equi-
librium, as if that particular body were the only one of the system.
And besides these we have equations expressing the geometrical
connexions of the bodies.
The same method must be adopted to determine the motion of a
system of rigid bodies ; we must write down for each body of the
system the six equations (III) and (IV), or the three equations (V),
and then the equations expressing the geometrical conditions which
the bodies satisfy.
When the equations are once written down the difficulties of
solving them are merely analytical, and such artifices must be used
as the forms of the equations suggest.
The constants introduced in integration must be determined in
the same way as in the case of a single body.
When any system is in equilibrium we may either consider each
body separately, and write down for it the six equations of equi-
librium ; in which case the actions of the other bodies of the system
on it will enter as external forces ; or we may consider the whole
system as one body and write down for it the six equations of equi-
librium; in which case the mutual actions of the different parts of
the system will not appear in the equations; or we may consider
any two or more of the bodies as one, and write down for them the
equations of equilibrium.
These different methods may also be adopted in Dynamics. The
dynamical equations, as first written down, are strictly equations of
equilibrium, and it is only from the analytical character of the
quantities that enter them that they enable us to determine the
motion of the system.
Though the methods to be employed in solving the equations
of motion must in each case be suggested by the forms of those
equations, there are certain artifices by which the first integrals may
often be obtained, the applicability of which can be determined at
once from an inspection of the system. The difficulty generally
arises from there being unknown forces which have to be eliminated
150 A TREATISE ON DYNAMICS. [CHAP.
by means of geometrical relations. These artifices enable us in
certain cases to effect that elimination at once.
101. When a body or system of bodies is in motion under the
action of any forces, if x, y, i are the co-ordinates of the center of
gravity at the time t, and M the mass of the system, we have the
equations
(III)
in which no forces appear but such as arise from causes external to
the system ; that is, no pressure of one part of the system against
another enters, no mutual attraction, no tension of a string con-
necting two parts of the system. But all forces do enter which
arise from causes external to the system, as pressures of fixed sur-
faces, or fixed points, tensions of strings passing over fixed pulleys,
&c., together with the given external forces whatever they may be.
Now equations (III) shew that when
2X=0, 2r=0, 2Z = 0,
the center of gravity will move uniformly in a straight line. Thus,
whatever the internal forces of the system, so long as there are no
external forces the center of gravity will move in a straight line
with uniform velocity. This is called the Principle of the Con-
servation of the motion of the center of gravity.
dx
We see from equations (III) that if SX= 0, j- is constant, what-
ever 2 Y and 2Z are. Hence if the resolved parts of all the external
forces actino- on a system parallel to any line have a resultant zero,
the motion of the center of gravity parallel to that line will be
uniform. Similarly, if two of them vanish the resolved part of the
motion parallel to that plane will be uniform and rectilinear. Thus,
for instance, if any number of balls be piled up on a perfectly smooth
horizontal plane and then left to themselves, they will be acted on
by no external horizontal force, they will therefore fall down in
such a manner that their center of gravity will describe a vertical
line.
In the cases that will most frequently occur the center of gravity
of the system will be at rest at the commencement of the motion,
and will therefore continue so throughout.
VII.] A TREATISE ON DYNAMICS. 151
102. The next principle that we shall mention depends on
equations (II), which may be written
Whenever the right-hand sides of these equations are zero the
equations themselves become
_ / d'x d'z\ ^
^ / d'y d'x\ ^
the integrals of which are
^ ( dz dy\ ,
.(1)
Sot
(2)
Now if A^ be the area described in any time by the projection
on the plane of yz of the line joining the origin and the pai'ticle
whose co-ordinates are x, y, z,
^ dt dt dt '
Employing a similar notation for the other co-ordinate planes,
and substituting in equations (2), we have
from which 2 2rn^, = /«,< + C,,
(3)
152 A TREATISE ON DYNAMICS. [CHAP.
or, if we suppose J^, &c., to be the areas described since the instant
from which t is measured,
2'2mA^ = hJ,[ (4)
2l.mA, = kst.)
In these equations or in equations (3), consists the principle
which is called the Principle of the Conservation of Areas.
We may state it in words as follows :
The area described by the projection on the plane o? yz of the
radius vector of any particle, is called the area described by the
particle round the axis of x. Since any fixed line may be taken as
the axis of x, this definition is applicable to any line. In certain
cases the sum of the products of the mass of each particle of the
system, and the area described by that particle round some fixed
line is proportional to the time; whenever this is the case, the
principle of the conservation of areas is said to hold for that system,
round that line.
This may be the case for only one line, or for any line whatever,
or for lines satisfying certain conditions ; let us consider generally
how to determine in what cases it does hold.
We see by the equations from which it is deduced, that it holds
for the axis of x whenever
^{yZ-zY) = 0.
Now in this expression no mutual actions or other internal forces
enter. It is the expression for the resultant moment of the impressed
forces round the axis of x. The principle will therefore hold for any
line round which the moment of the impressed forces is zero.
First, let there be no external forces, or such as would keep the
system in equilibrium in any position, if placed at rest in that
position. In this case whatever line be chosen, the moment of the
forces round that line is zero, and consequently the principle holds
for any line whatever.
Again, let the resultant of the impressed forces always pass
through a fixed point. In this case the moment round any line
through that point is zero, and therefore the principle holds for any
line through that point.
If all the impressed forces which act on the system are parallel,
the principle will hold round any line parallel to their common
direction.
VII.] A TREATISE ON DYNAMICS. 153
If the resultant of the impressed forces always passes through
a fixed line, the principle will hold round that line. It is needless
to enumerate any more particular cases.
103. When the resultant of the impressed forces acting on the
system is either zero, or passes always through a fixed point, we
have
2 2m J, = ^3^
for any directions whatever of the axes ; and in the former case for
any origin, in the latter only when the fixed point is origin.
hi, h.2, hs, are constants introduced by integration, which may
be determined from the circumstances of the motion at some known
instant ; and will be different for different axes of co-ordinates.
If A, fx, V be the direction-cosines of any line passing through the
origin, the sum of the products of the masses and areas described
by them round this line
= \.2'2.mA^ + fx.2l.mAy-i-v.Q'2mA,
= (\hi + fxh2 + vh3)t
where ¥ = hi + /?/ + h^.
Now -T- J -r J -r i^^y he considered as the direction-cosines of
some line, and if be the angle between this line and that round
which the areas are reckoned, the sum in question is h I cos Q.
The greatest value of this is when cos0= 1, which is the case
when
hi K h,
^ = J' "-^ir " = !'
Hence then the line whose direction-cosines are -j^ , ^, -^ is
h h h
such that the sum of the products of the areas round it and the
masses is a maximum.
Since its direction-cosines are independent of t its position is
constant, and from its nature its direction in space is independent of
the axes to which it is referred. We see therefore that in every
such system there exists a line whose direction in space is invariable
154 A TREATISE ON DYNAMICS. [cHAP.
and can be determined at any instant;, from the circumstances of the
system at that instant.
A plane perpendicular to this line is called the " invariable plane/'
and the line itself is called the " invariable axis."
These results agree with those which were determined previously
in a particular case.
The sum of these products round any line inclined at an angle 6
to this is ?it cos ; and therefore becomes zero for any of the lines at
right angles to this one.
Many other curious properties of the motion connected with this
line might be deduced from these equations, these, however, are the
most important.
104. When a system is in equilibrium under the action of forces
X, F, Z, &c. "We have by the principle of virtual velocities
^{Xlx+Yly + Zlz)=0.
We have therefore by D'Alembert's principle for a system in motion
We will make a few remarks on this equation.
No internal forces appear in it which arise from invariable
geometrical connexions of the parts, that is, which arise from con-
nexions of the parts which are geometrically the same before and
after the iiidefinitely small displacement which is supposed to take
place. Such, for instance, as the tensions of flexible and inextensible
strings ; the pressures of smooth surfaces which continue in contact,
or of rough surfaces where the displacement may have been sup-
posed to take place by rolling ; the pressures of fixed points, &c.
Also we may omit any external forces, the displacements of the
points of application of which are such that their virtual velocities
ai*e zero.
Now since in this equation the displacement and therefore the
values of Zx, oi/, &c, are arbitrary, being subject only to the condi-
tion of not violating the geometrical relations referred to above, we
may suppose the displacement to be that which actually takes place
in the indefinitely small time ht, that is, for ex, by, Iz
.^ dx^, dy ^, dz ^
we mav write -7- d ^ , -~-ot, -y.ct,
dt at dt
VII.] A TREATISE ON DYNAMICS. 155
provided the geometrical relations of the system at the beginning
and end of the time ^t are the same. The equation thus becomes
_ fcPx dx d'y dy d^z dz\^^ -f^dx ^dy rydz\^^
the integral of which is
Hit) * (Mj - (s)} - ^ ^ ^ ^^ ('^''^ - ^''^ ^ ""^-
or 27W v»= C + 2 /2 (Xdx + Ydy + Zdz).
Now in this equation no forces appear which are either internal
forces such as we have already mentioned, or external forces such
that their virtual velocities arising from the actual displacement are
zero. Such forces are the tensions of inextensible strings attached to
fixed points ; pressures of smooth fixed surfaces ; pressures of fixed
points against smooth surfaces, &c., where the actual displacement
of the point of application of the force is perpendicular to the direc-
tion of the force ; and also forces where there is no displacement of
the point of application of the force, as when a body moves about a
fixed point or axis ; or rolls without sliding on a rough surface, &c.
We may remark that the internal forces do not enter into the
equation, because if they did for each force there would be an equal
force whose virtual velocity would have a contrary sign, so that they
would enter in pairs which would destroy each other ; and that the
external forces which we have mentioned do not enter because the
virtual velocity of each is zero.
The product of the mass of a particle and the square of the
absolute velocity with which it is moving is called the vis viva of
the particle.
Hence, then, ^mv" is the vis viva of the whole system, and
27«y= = C + 2/2 {Xdx + Ydtj + Zdz)
is an equation which gives the vis viva of the whole system in any
position, whenever the integration on the right-hand side can be
performed. This is easily done in most of the cases that will occur
in practice.
In order that the integration may be possible, the expression
2 {Xdx + Ydy + Zdz)
must be a perfect differential of some function of the co-ordinates of
the different particles.
This will be the case whenever the forces X, F, Z, &c. are
constant, or arise from the attractions of fixed centers, or from the
156 A TREATISE ON DYNAMICS. [cHAP.
mutual attractions of the different particles of the system, with forces
varying as some powers of the distance. It will not be the case
when the forces arise from the attractions of moveable centers not
forming part of the system, or when the forces are functions of the
velocities, as when the motion takes place in a resisting medium, &c.
Let it be a perfect differential, and let its integral be
<P (-^i? ^i> Zi, a^s, t/s, Sg. . .),
then ^mv" = C + 2(p (or,, t/^, 0, . . .)
Let Oj, bi, Cj. . . be the values of .Tj, ?/„ ^,. . . when the velocities
of the different particles are F^, F^.. . then
Stwd^ - Stw F' = 20 (x„ y,...)-2<p («„ i, , . .)
This equation shews that the change in the vis viva of the system
which takes place in any time depends only on the co-ordinates of
the particles at the beginning and end of that time, and not on their
velocities or directions of motion.
This is called the Principle of vis viva.
If there are no forces external to the system but such as do not
enter into the equation of virtual velocites, that is, if
^{Xdx + Ydy + Zdz) = 0, l.mv^= C,
or the vis viva of the system is constant. This is called the Principle
of the Conservation of vis viva.
It is convenient to express the vis viva of the system in terms of
that due to the motion of translation of the center of gravity, and
that due to the motion of rotation about it-
Preserving the notation already used, and writing Ic + x' for x,
&c., we have
dx
dx dx'
Tt
'^Tt'-'dl
dy
_A_l dy'
dt
dt dt
dz
dz dz'
dt
= Tt^-di
and substituting these, we get
.."^^^,4^},
dt dt dt dt
VII.] A TREATISE ON DYNAMICS. 157
as the expression for the vis viva ; or bearing in mind that x, 'y, z are
the same for every particle of the system, and also that
1.mx'=0, 'Lmy' = 0, 2wz'=0,
and therefore 2»j -y- = 0, 2 m -^- = 0, Srn -^- = 0,
at at at
we have
Now in whatever manner a body is moving, its motion at any
instant may be considered as composed of a motion of translation
of the center of gravity, and a motion of rotation romid some line
through the center of gravity, the direction of which will in some
instances be constant, in others will be continually changing. Let v
be the velocity of the center of gravity, -j- the angular velocity of
rotation, and r the distance of a particle from the line through the
center of gravity round which this takes place ; then the expression
for the vis viva becomes
or 3Iv'+Mk'(^J,
and the equation will be
M I F + k' (^J U C + 2/2 (Xrfo; + Ydy + Zdz).
It must be here carefully borne in mind, that in the general case
the axis about which -j- is estimated is continually changing its
position and direction in space.
This method of expressing the vis viva will be of use only when
we can tell, a jmori, the direction of this axis of rotation, as in the
case of motion in parallel planes ; or, as it is commonly called, space
of two dimensions.
The principles just proved will greatly assist us in the solution of
problems. We will now proceed to explain the manner in which
they should be used.
105. Suppose that the system consists of one rigid body, and
that the motion is in parallel planes.
We must first write down the equations (V) with the proper
forces expressed in them. We must then count how many unknown
158 A TREATISE ON DYNAMICS. [cHAP.
quantities appear in thenij including co-ordinates and unknown
forces. Suppose the whole number to be n; then^ to determine
these n unknown quantities, we have the three equations (V) ; we
must therefore have n — 3 additional equations, expressing geo-
metrical relations which exist between the different parts of the
system.
The direct method of proceeding now is to solve these equations
by any analytical artifices that suggest themselves. The unknown
forces however may sometimes be eliminated very easily by means
of some or all of the preceding principles.
We must therefore examine w^hich of the above principles hold ;
or in other words, which of the equations expressing the above
principles will be free from unknown forces. These will give us
so many first integrals of the equations : that is, so many relations
between the positions and velocities of the different pai-ts of the
system. Sometimes we obtain all the first integrals by means of
these principles, sometimes it is necessary to combine particular
artifices wth them.
If we can perform the next integration the problem is completely
solved, that is, the position, velocity, and direction of motion, are
completely determined as functions of the time : this however can
seldom be effected.
No more exact rules than these can be given. A facility in
applying thera to particular cases is only to be acquired by practice.
If the system consists of more than one body, we must write
down the equations of motion for each separately, and then solve
them by the aid of the general principles, as in the case of a single
body.
We may remark that in the cases where the general principles
are applicable we might obtain the first integrals from them at once,
without using the differential equations of motion of the second
order. These however should always be written down.
106. We will now illustrate the application of these principles
by an example of the method of determining the motion of two
bodies.
A rough cylinder is placed with its axis horizontal on a rough
inclined plane, which is itself moveable on a smooth horizontal
plane ; to determine the motion of the system.
Let a be the radius of the cylinder, M its mass; a the inclination
of the plane AB, m the mass of the plane ; a fixed point in the
VII.] A TREATISE ON DYNAMICS. 159
horizontal plane coinciding with A at the beginning of the motion,
when the point JE coincided with D.
O -"i
ON=x,
OA=x',
PCE = d,
AD^L
R and F the reactions of the plane on the cylinder along PC and
PD. Those of the cylinder on the plane will be equal and in the
opposite directions.
The equations of motion for the cylinder are
M^ =Fcosa
dt
M
Mk
d^
df
df'
l^sina (1)
Fsma + Rco&a- Mg.. . ...(2)
F» (3)
and for the plane
m —j-^ = Rsina — F cos a ,
at'
(4).
These equations contain six unknown quantities, we must there-
fore seek for two more equations. These will express that the
cylinder is in contact with the plane, and that it rolls without
sliding.
The first gives
(x — x') tan a + a sec a = i/ (5).
The second gives
(x — x') sec a +a tan a + ad = I (6).
These six equations when integrated will determine the motion
completely. Since we do not know Avhether R and F are constant
or not, it is necessary to eliminate them before we can integrate.
160
A TREATISE ON DYNAMICS.
[chap.
Now in this case the principle of vis viva is applicable, and also
the principle of the conservation of the motion of the -center of
gravity in a horizontal direction.
These principles therefore guide us to the method of elimination.
Adding (l) and (4) we have
M
d'cc
df
dW
dt'
0,
the integral of which is
, -. dx dx'
dt dl
C = 0..
•(7).
This is the expression of the latter of the two principles.
Again, from all the equations we have
^YTt df^^ dtlii'^^'' dtde]^^'"" dt de- ^^^^ df
which when integrated gives
At the beginning of the motion
= C- SiV/g (Z sin a + a cos a) ;
This is the expression of the former principle.
From equations (5) and (6) we have
/dx dx'\ dy
'^"""[Tt-ltj^di'
fdx dx\ dd ^
and sec«(^^--^J4-a^ = 0.
From these equations and (7) we obtain
ma cos a dd
dx
It
d£_
dt
dy
dt'
M + m dt
Ma cos a dd
M + m U'
. dd
(9)
also k^
VII.] A TREATISE ON DYNAMICS. 161
Substituting these values in equation (8), we have
3ma + Ma
M
the integral of which is
3ma + Ma (1+ 2 sm^ a) /ddV , .
=rr^^ -j-J = ^sr sin a . d,
M + m \dtj * '
t)= {M + 7n)gsma ^„
3ma + Ma{l +2sin'a)
The constant is zero, since ^ — when t = 0.
g sin a
JfA =
3m a + Ma (1 + S sin* a) '
d = A{M + m)t^;
.'. ~^=2A{M+m)t',
therefore, from equations (9), we have
-r- = — 2A7na cos a . t,
at
dx'
-7- = 2 AMa cos a . t,
^-^ ==-2A{M + m)asina.t; »
from which x, x' and y may be found in terms of t.
From these equations it appears that the motion both of the plane
and cylinder is uniformly accelerated. Also, if we differentiate these,
and substitute in equations (3) and (1) we shall obtain F and R; and
it will be found that they are constant.
Again, from equations (9), we have
dy M+m^
-f- = tan a ;
dx m
which shews that the center of gravity of the cylinder will descend
in a right line, making with the horizontal plane an angle whose
. M + m
tangent is tan a.
VI
Instead of making use of the principle of vis viva to guide us in
our elimination, we might have proceeded as follows: from (j), (2)
and (3)
<fx . d'y ,^d'0
a cos a -r-j + « sm a j^ — k" ~—- — ag sm a.
(Xt (It it L
Integrating this, we have
dx . dy ,,d6
dl dt dt ^
.-. acosa .X + as\na.y-k^.d=C--agsina.l' ;
11
162
A TREATISE ON DYNAMICS.
[chap.
also, the integral of (7) is
Mx + mx'= C,
from these two equations, with (5) and (6), we can find x, x, y and Q
in terms of t.
There is still another method of proceeding which is always
applicable in cases where the geometrical equations are linear with
respect to all the variable quantities, and where the equations of
motion involve the second differential coefficients only of those co-
ordinates, as is the case in the present example.
From (5) and (6) we have
tan a
'(Px _ rfV\ ^ ^
M'
dfj~dt"
J d'x d'x' d'9 ^
Substituting in these equations the values of the second differ-
ential coefficients given by equations (1), (2), (3) and (4), we
obtain equations which involve only F, R and constant quantities.
F and R therefore are constant and may be found at once
from these equations ; and the equations of motion may be inte-
grated at sight.
This is the shortest and easiest method of proceeding in cases
where it is applicable.
107. As another example we will take the following problem.
In a smooth tube moveable about one extremity in a horizontal
plane, a small ball is placed at a distance a from the fixed end,
and an angular velocity w is
communicated to the tube:
determine the motion.
Let AB be the tube of
length I; P the position of
the ball at time t; AC the
initial position of the tube;
PAN=e, AN=x,
NP=y, AP^r,
M the mass of the tube, m
the mass of the ball, R the ^
reaction between tube and ball.
VII.] A TREATISE ON DYNAMICS. 103
then for the motion of the ball we have
m^^ = -Rsme, (1)
m^ = Rcosd, (2)
Ma
and for the tube Mk--^ = -R.r (3)
We have omitted two of the equations of motion of the tube,
because they would introduce two additional unknown forces arising
from the condition that A is fixed.
Here then we have x, y, r, 6, and R connected by three equations,
we must therefore seek for two more ; they are
x—r cos 6, y = r sin 6 (4)
Now in this case the principles of the Conservation of Areas, and
of vis viva hold.
Conducting our eliminations accordingly we have
or, transforming to polar co-ordinates,
Now at the commencement of the motion
(7na' + Mk')(o'-=C;
'S+(mr'+M¥) (^J= {ma' + Mk^ u>' (5)
Again, "'{^^-y^^.| + ^W^=0.
From which, integrating and changing to polar co-ordinates,
we have
(mr' + Mk')j^ = C,
at the commencement of the motion
(7na^ + Mk')o>=C;
de ma' + Mk"
•■• "".57.
dt mr" + Mk
■2 0)
(6)
11—2
164 A TREATISE ON DYNAMICS. [cHAP.
J a
Eliminating -y- from equations (5) and (6), we have
fdr
\dt
Hence the velocity along the tube and the angular velocity cf
the tube are known for any position of the particle, and depend
only on its distance from A.
We cannot integrate these equations in finite terms so as to
obtain the position in terms of the time.
108. Hitherto the effective force on any particle forming part
of a system has been resolved in the directions of three rectangular
axes fixed in space. When the motion of the particle is in one
plane it is often convenient to employ polar co-ordinates, and the
resolved parts of the effective force in the direction of the radius
vector, and perpendicular to that direction.
The equations of motion of a single particle acted on by a force
P along the radius vector from the pole, and a force T perpendicularly
to that direction are
dr de d-6 „,
^'''Ttdt^''''dt^-^'
Hence then from the definition of effective force if r and 6 are
the polar co-ordinates of a particle m forming part of a system,
the effective force along the radius vector is
and that in a direction perpendicular to it is
^ dr de d'0
"-'"dtdt'-'"'d?-
Another mode of resolving the effective force is also useful,
namely, in the directions of the tangent and normal of the path of
the particle. These resolved parts are readily obtained as follows.
The parts parallel to the axes of x and y are »j -r^ and m -~„ .
^ ^ -^ dt- df
Hence the part in direction of the tangent
^ d^x dx d^y dy
dt^ ds dt- ds
~^d^\dt^'dt'^d? dt
d's
Til.] A TREATISE ON DYNAMICS. 165
And the part along the uoimal
d'x dy J-y dx
dl ds dl ds
dj^ ((£21 dx d'x dy\
~"' ds W dt df dtj'
Let p be the radius of curvature of the path of the particle, v its
velocity
" d^ dx d'x drf^
dJ 'dl~'dJ~ dt
and the expression for the force along the normal becomes
vxv^ m (ds'
T "'■ p U
When the particle moves in a circle 7: = ^ and the two pairs of
expressions coincide Avith each other.
These expressions for the effective force can in any case be used
instead of those referred to rectangular co-ordinates.
109. These modes of resolving the effective force, afford an
explanation of a term, which, originally founded in error, has a ten-
dency to convey an erroneous impression, viz.. Centrifugal Force.
If a smooth straight tube is moved with uniform angular velocity
in a horizontal plane about one extremity, and a particle is placed in
it, the particle will commence moving from the fixed end of the tube,
and will move with accelerated motion till it ultimately flies out of
the tube.
If a particle be fastened by a string to a fixed point on a smooth
horizontal table, and then be projected so as to move round this
point, it will cause a tension of the string, depending on the mass of
the particle and the rapidity with which it moves.
From these and similar cases, it was concluded that bodies moving
round a center have a tendency to fly off from that center, and in
consequence exercise a force to which the name centrifugal force was
given: and its magnitude Avas found to be mro)', where 7* is the
distance from the fixed center, and w the angular velocity with which
the body is moving round that center.
166 A TREATISE ON DYNAMICS. [cHAP.
The true explanation is found in the following method of view-
ing it.
In the former of the preceding examples, let R be the pressure
which the side of the tube exerts on the particle. This is the only
impressed force on it.
Then reversing the effective forces on it, we obtain the equations
of motion
„ ^ dr dO d'0 ^
^-^''TtTt-''''df=^'
From equation (1)
mj^ = mr.\
which shews that the motion along the tube will be the same as if
the tube were at rest, and the particle were acted on by a force mrta-
along it.
Again, in the other case, let T be the tension of the string.
Applying the effective forces reversed we have
0,
: and from the latter
Hence the first equation becomes
T^mru>\
That is, the tension of the string is the same as it would be if the
particle were at rest and acted on by a force mrio^.
A similar result will follow in any other case in which the
equations are applied to the ntiotion of a particle, either alone or
forming part of a system. So that though no such force as centri-
fugal force really exists, yet when a system in motion is supposed at
rest, and the effective forces are applied in the opposite directions to
balance the impressed forces, a portion of one of these effective forces
will correspond in direction and magnitude to the fabulous centri-
fugal force.
-T-
d'r fddV ,
-'''7it^-''"'Kdt) ='
2
dr dd d'd ^
Now
ris
constant, therefore -j- =
equation
dO
is constant.
VII.] A TREATISE ON DYNAMICS. 107
Thus along the radius vectoi*, the term '"''(y:) in the expression
(Pr fdey J . ,, , .1, 1. , . mv'
-7M-7-5 +wirl — j, and m the normal the whole expression
correspond to the centrifugal forces in those directions respectively.
The name " Centrifugal force/' is very objectionable, from the
plausibihty of the reasoning by which the idea is supported. But it
is found convenient in several classes of problems to have a name for
the portion of the reversed effective force represented by inru^, and
with this meaning the term centrifugal force is retained.
168 A TREATISE ON DYNAMICS. [cHAP.
CHAPTEE VIII.
110. In discussing the elementary principles, it was remarked
that there are cases in which the whole effect produced by a force is
known J when the magnitude of the force and the time during which
it acts are not known ; that is, when we know the definite integral
/ ' Pdt without knowing either P or /o— /j. The name "impulse"
was given to this definite integral, and the force P was called an
impulsive force.
Let us now consider under what circumstances forces of this sort
are called into action.
We have used the term " rigid body/' and have investigated the
motions of rigid bodies in several cases, and have given the equations
for determining that motion in all cases. Now the idea which we
have attached to the term rigid is this : a body is said to be rigid
when all the points preserve the same invariable mutual distances,
whatever the forces which act on the different points of the body.
This idea of rigidity is suggested by the first touch of external
objects ; though a slight examination is sufficient to convince us that
it is not strictly applicable to anj' of them.
Different substances require different degrees of force to produce
any sensible change in their form, some require very great force to
do so. This approximate rigidity enables us to form the idea of the
perfect quality, even though there is no substance in existence
endued with that quality ; and a great many substances are so nearly
rigid as to convince us that calculations effected on the hypothesis
of their perfect rigidity cannot be affected with sensible errors on
account of the deviations from it.
In the branch of the subject however, which is now under con-
sideration, this deviation from rigidity forms a principal feature of
the problem. When pressure is exerted on a body by any means,
the body slightly changes its form, or is compressed ; also when the
pressure ceases to act, the body wholly or partially recovers its
original form. When a ball in motion strikes another ball at rest, a
pressure immediately takes place between the two balls, each ball
pressing on the other, and so flattening it at the point of contact.
This pressure is gradually communicated from one particle to
Vm.] A TREATISE ON DYNAMIC'S. 1G9
another of each of the balls^, so that velocity is communicated to the
one which was at rest, and the velocity of that which was in motion
is lessened ; and this action continues as long as the hinder ball
moves faster than the other. When the balls have acquired the
same velocity, the flattening of the two is at its greatest, and if the
balls had no tendency to recover their original forms the pressure
between them would cease, and the two would move on together
with the same velocity. Most substances however have a tendency
to recover their original forms, and in consequence of this the action
between the two continues till they cease to touch each other ; the
one which was originally at rest having the greater velocity of the
two.
Now all this that we have been describing takes place in a very
short time indeed : so short as to be considered instantaneous with-
out any sensible error.
If our knowledge of the structure of matter were perfect, and if
also our mathematical analysis were perfect, we should be able to
investigate, from the principles already laid down, the exact nature
and magnitude of the action which takes place between the two
bodies, and also the degree and manner in which a change of form
takes place. As it is, however, we are obliged to rest on experiment
for the facts which we have stated, and for the laws which we are
going to state.
Let /i be the time when the first contact takes place, /„ the time
when the two bodies are moving with the same velocity, and ^3 the
time when they cease to touch at all. Then the whole action which
takes place during compression, as it is termed, will be represented
by I Pelt and the whole action during restitution (that is, during
the recovery of the original form) by / Pdl.
The experimental law which forms the basis of our calculations
is that
rh (t-2
\ Pdl = e Pdl:
where e is a multiplier less than unity, which is constant for all
values of P, so long as the substances between which the impact
takes place remain unchanged.
The less or the greater the change of form that takes place for a
given value of I Pdt, the harder or softer the body is said to be.
170
A TREATISE ON DYNAMICS.
[chap.
The greater or less e is, the greater or less is said to be the
" elasticity" of the bodies.
If e = 1, the elasticity is said to be perfect.
e is called the coefficient of elasticity, or sometimes the elasticity.
The definite integrals are generally written R and B', so that
i2'= eR.
R and jR' are called the "impulses" during compression and
restitution, and R + R', or {\+e)R is called the whole impulse.
The law has been enunciated and the action between the bodies
explained in a particular case only ; it is however applicable, with a
very slight extension, to all cases.
111. Since an impulsive force is only a force of great intensity,
D'Alembert's principle is applicable to it, as to all other forces.
If we take the case of one rigid body, that is, rigid in the modified
sense in which we have explained the term, the equations of motion
will be the same as before, and may, as in that case, be reduced to
M
dt'
2X,
And 2
M^=2F,
de ^^'
{,(.-.S)-.(r-4|)}=o.|
{.(x-.$)-.(.-.g)}.0.
{.(r-.g)-,(x-,
'5f)}-.|
(III)
(IV)
Now in this case, 2X, 2F and 2Z include all the forces which
act on the body, impulsive or otherwise ; and if we take the definite
integrals of the first three equations between the times /j and tz, the
beginning and end of impact, we have
m(m,-«o = 2/ xd/,
M(v,-v,)=l. j^Ydi,
M{w„-Wi)-
Zdt,
YIII.] A TREATISE ON DYNAMICS. 171
or using A'', Y', Z' to denote the " Impulses,"
M{y,-v,^ = ^Y'\ (VI)
The limits t^ and ^ must be taken so as to include the whole of
every impulse, and consequently may include a longer time than is
occupied by some of them. This however will produce no error;
for suppose one of the impulses as X' to commence at t( and end at
^2' both of Avhich times are between t^ and t^.
Then since
{h ffi' r^i ['2
Xdt = Xdt + Xdt + Xdt,
•'/, hi J ti Jt.j
and since X is zero between /j and //, and also between // and to,
we
have Xdt= Xdt = X'
,•(■>
Also since 1 Xdt has a sensible magnitude in consequence of X
being very large ; this integral will not have a sensible magnitude
for those forces which are not impulsive ; in other words, these forces
will not produce a sensible effect on the velocity during the short
time of impact, and therefore do not appear in equations (VI).
In the same way we may integrate equations (IV) between /, and
t^; and since the interval to — t^ is extremely short x, y, z, &c., may
be considered constant during that interval, so that we have
'^{y\_Z'-m{io'-wY\-z [F'- m (?;'- v)]} = 0, "j
l.{z\X' -in{u' -xi)-\- x{Z' -m{w' -w)-^ = Q, 1 . . . (VII)
2 {x \Y' - m iy'- vY\ ~y \_X' - m (?/ - «)]} = 0. i
In these equations x, y, z are referred to the center of gravity as
dx
origin, ii is the value of-j- before impact, and u' its value after
impact, and similarly for the other letters.
(VI) and (VII) give the state of motion of the body immediately
after impact, without any integration beyond that expressed by 2.
When the motion of every particle before and after impact is
parallel to one plane, equations (VII) are reduced to the one equation
l.m{x{v'-v)-y{u'-u)\ = ^{xY' -yX'),
172 A TREATISE ON DYNAMICS. [cHAP.
Let 0), and w. be the angular velocities of the body before and
after impact, then
11 =- t/w^, u' = — y w.^, v — xm^^ v' = xw.2.
Let L' be the resultant moment of the impulses, and let r^=x°+i/^,
then this equation becomes
1.1117'- (uy^— w,) -L'.
And the three equations in this case are
M(v,-v,) ^1Y', [ (VIII)
112. If the system consists of more than one body, the equations
must be written down for each body separately, and the unknown
forces which enter the equations, must be determined by the relations
existing between the velocities of the points at which impact takes
place. These will be different according as the bodies are elastic or
inelastic, and as their surfaces are rough or smooth.
When the bodies are inelastic and smooth, the whole impulse will
be in the direction of the normal to the surface at the point of con-
tact, and must be determined from the condition that the resolved
parts in the direction of this normal of the velocities of the points in
contact must be equal. If the surfaces are rough, the whole velocities
of the two points will be equal and in the same direction.
When the bodies are elastic, and the coefficient of elasticity is
given, the normal impulse may be found from the consideration that
the normal velocities of the points in contact are the same at the end
of compression, and that the impulse during restitution is in a con-
stant ratio to the impulse during compression : when the surfaces
are rough, the velocities of the points in contact resolved along the
tangent plane Avill be equal at the conclusion of the impact, while
the resolved parts along the normal will be equal only at the end of
compression.
If the surfaces are partially rough, as is always the case in nature,
let /A be the coefficient of friction and R the normal portion of the
impulsive force ; then the tangential portion of the impulsive force
cannot exceed fxR, and therefore the whole tangential impulse cannot
exceed/ ixRdt.
Little is known experimentally of the value of ^x for forces so
great as to be impulsive ; so far as the experiments go, however,
Viri.] A TREATISE ON DYNAMICS. 173
they indicate a diminution of m for large values of the normal pres-
sure. Considering /,t to be approximately constant, we have
rt-2 rh
Rdl = nR',
where R' is the normal impulse. This is the greatest amount of
tangential impulse that can take place for a given amount of normal
impulse ; all will not necessarily be exerted in any particular instance.
The preceding remarks will be better understood by a consideration
of the following examples.
113. Let a ball whose mass is m^ moving with a velocity t\ over-
take a ball Avhose mass is vk moving with a velocity v^, and let e be
the coefficient of elasticity.
Let R be the impulse during compression, u the common velocity
of the two balls after compression, ?/, and j/g their velocities after
impact : then from equations (VIII),
7W, {u -?;,)=— R,
vi^ (m - t'g) = R,
from which Ave have u = =-^ ,
nil + W'a
Vli + 7»2
the whole impulse is (I + e) R, and the velocities after impact are
given by the equations
7«, (?/i -vi) = -(l + e) -^ '^ ,
^ ^ ^ ^ nii + mg ^
W2 (^2 - v'2) = (l+e) -^-^^' ^-'- .
If the balls are equal and the elasticity perfect, these equations
become Uy-Vi, and xi2=Vx.
114. Again, let a sphere of radius a revolving with the angular
velocity w about a horizontal axis fall vertically on a horizontal
plane ; and let /a and e be the coefficients of friction and elasticity
between the sphere and the plane.
Suppose, to fix the ideas, that the axis of x is horizontal, and the
plane of xt/ perpendicular to the axis of rotation.
Let M be the mass of the sphere, V the velocity of its center of
gravity before impact ; v^. , x\ the resolved parts of the velocity, and
174 A TREATISE ON DYNAMICS. [CHAP.
to' the angular velocity after compression: %i^, iiy and w the same
quantities when the impact is concluded ; B. and F the normal and
tangential impulses during compression ; then
Mv, = F (1)
M(vy+F) = R (2)
Mk^{co'-w) = Fa (3)
and the condition that the point of contact is at rest gives
v^ + a)'a = 0, Vy=.0 (4)
we also have the condition that F cannot exceed n R.
From equations (2) and (4), we have
R = MV,
also from (1), (3) and (4), we have
F Fa' ^
Ma Ir a,
~ a^ + k- '
provided this is not greater than MF. Let this be the case, then if
F' is the tangential impulse during restitution
Mu^=F + F' (5)
M(Uy+V) = (l + e)R (6)
iWF(w-(o) = (F + F')a (7)
with the condition Uj. + -aa^O (8)
Equation (6) gives Uy = eV,
also from (5), (7), and (8)
and therefore F' = 0, and from (5) and (7)
and the initial motion after impact is completely determined.
We may remark here that the whole change of angular velocity
took place during compression ; if — — ^ had been greater than V
but less 'than {l+e)V, part of this change would have taken place
during restitution, and the final result would have been the same.
VIII.] A TREATISE ON DYNAMICS. 175
Let J — p be greater than (l+e) F; in this case instead of the
equation u^ + wa = 0, we have
F+F' = -fx{l + e)MF;
.'. n^ = - fx(\ +c) V and w = w - -^ — r— ,
and Uy = eV as before.
115. As a conchiding example we will take the system described
in Art. 106, and preserve the same notation.
Let the cylinder fall from rest so as to strike the plane along a
horizontal line.
Let F be the velocity of the cylinder before impact; v^, Vy the
resolved parts of the velocity of its center of gravity after compres-
sion, 0) its angular velocity, v the velocity of the plane at the same
time; it^, %iy, tn and u the same quantities after restitution.
Then we have
Mv^ = F cosa — R sin a,
M(Vy+ F) = Fsma + R cos a,
Mk''<o = Fa,
mv = Rsina-F cos a.
Since the velocities of the points of the two bodies in contact are the
same, their resolved parts in any two directions will be the same,
therefore
V — v^ + au} cos a,
= t'y + rt £0 sin a,
M+ 3m
from which R = MV cos a
M (l + 2sin'a) + 37w
After restitution the equations are
Mii^ = F' cosa-{l + e) R sin «, ^
M {Uy + V)= F' sin a + {\ + e) R cos a,
Mk''s: = F'a,
mu = (1 + e) 7? sin a — F' cos a,
and the condition in this case is that the velocities along the plane
shall be the same, which gives the equation
(k, — «) cos a + Vy sin a + a ■KT = 0.
R is already known, and from these equations we can determine F'
and the other unknown quantities.
176 A TREATISE ON DYNAMICS. [cHAP.
116, An examination of the proof of the Principles of the
Conservation of Areas and of the Motion of the center of gravity,
will shew that these principles are true when the forces are im-
pulsive, in the same cases that they are true when the forces are
finite. The Principle of the Conservation of vis viva on the con-
trary is not so. In fact the idea of an impulse implies a change in
the geometrical relations of the system.
These principles, however, are of little use in the solution of
problems where the forces are impulsive, since the equations of
motion are all linear, and require no integration.
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