VI v
ill!
A TREATISE
ON
DYNAMICS OF A PAKTICLE
*
A TREATISE ON
DYNAMICS OF A PARTICLE
WITH NUMEROUS EXAMPLES
BY
PETER GUTHRIE TAIT, M.A.
HONORARY FELLOW OF ST PETER'S COLLEGE, CAMBRIDGE,
PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH,
AND THE LATE
WILLIAM JOHN STEELE, B.A.
FELLOW OF ST PETER'S COLLEGE, CAMBRIDGE.
SEVENTH EDITION
Carefully revised
MACMILLAN AND CO., LIMITED
NEW YORK: THE MACMILLAN COMPANY
1900
[All Rights reserved]
First Edition, 1856. Second Edition, 1865.
Third Edition, 1871. Fourth Edition, 1878.
Fifth Edition, 1882. Sixth Edition, 1889.
Seventh Edition, 1900.
PREFACE.
THE fact that this work has already reached a seventh
edition has not merely delighted me but has, in no small
degree, perplexed and puzzled me. Comparing its prospects
with those of its more recent rivals, which have had the
unquestionable advantage of being written by authors in
close touch with the present phase of the ever-changing
atmosphere of Cambridge, they appeared to be by no means
hopeful : but in this, it seems, I was unduly despondent.
I must, therefore, attribute its continued success to the
main features in which it differs from its concurrents:
viz. its plain, almost blunt, language ; and its absolute free-
dom from pedantry : especially that particularly dangerous
species which exaggerates molehills into mountains, and
leaves the unfortunate reader, who has toiled through some
laborious investigation or explanation, utterly mystified as
to the necessity for it : perhaps even ignorant of its object.
[It is scarcely possible to overestimate the mischief which
may thus be done to really intelligent, though (for once !)
sufficiently incautious, beginners.]
Vlll PREFACE.
I may be wholly mistaken in this opinion, but it furnishes
the only satisfactory solution of the problem which has yet
occurred to me, and it is based on facts plain to everyone.
Meanwhile, I once more despatch the Veteran on a
campaign, with a few necessary patches on his battered
harness.
P. G. TAIT.
COLLEGE, EDINBURGH.
Nov. 7, 1900.
PEEFACE TO THE SIXTH EDITION.
THIS work, commenced by Mr STEELE and myself towards
the end of 1852, first appeared in 1856. At Mr STEELE'S
early death his allotted share of the work was uncompleted,
and I had to undertake the final arrangement of the whole.
In the subsequent editions it has derived much benefit from
revision: first by Mr STIRLING of Trinity in 1865, then by
Mr W. D. NIVEN of Trinity in 1871, and by Prof. GREENHILL
of Emmanuel in 1878. It last appeared after a general
revision by myself, with the assistance of Dr C. G. KNOTT
and of my colleague Prof. CHRYSTAL. The present edition
has been prepared by me, with the assistance of Dr W.
PEDDIE.
Under such circumstances it could not fail to be a patch-
work of a somewhat complicated kind ; but the comparatively
rapid exhaustion of the later editions shows that, with all its
many faults, it meets not very inadequately a real want.
I have no doubt that, with a few months' leisure, I could
immensely improve it ; if merely by giving it more unity of
plan. But the time I am able to devote to such things has
to be snatched at irregular intervals from other engrossing
work ; and I am led, therefore, very naturally rather to the
making of hastily improvised insertions than to carrying
X PREFACE TO THE SIXTH EDITION.
out any well-considered scheme of compression or co-ordi-
nation.
The book's most important fault is its bulk ; yet I do not
think it can be honestly accused of prolixity. I have always
considered undue prolixity to be, next of course to inaccuracy,
the greatest fault that a scientific work could exhibit. The
number of Examples is perhaps unduly large, but experience
has shown me that there are many readers who do not con-
sider this a defect. For their quality, their respective authors
(mainly Cambridge Examiners) are alone responsible.
My attention has been called to the fact that several
sections of this book, in which some novelties appear, have
been translated almost letter for letter and transferred, with-
out the slightest allusion to their source, to the pages of a
German work. Several other books have obviously been
similarly treated by the same compiler. It is well that this
should be generally known, as the British authors might
otherwise come to be supposed to have adopted these
passages simpliciter from the German.
P. G. TAIT.
COLLEGE, EDINBURGH,
May, 1889.
CONTENTS.
PAGES
PREFACE vii viii
CHAPTER IX KINEMATICS 1 34
Division of the subject, i 3.
Velocity, 47.
Composition and Resolution of Velocities, 8 n.
Acceleration, 12 19.
Hodograph, 20.
Moment of Velocity, 21 24.
Motion of a point deduced from the given acceleration, 25.
Relative Velocity and Acceleration, 16 36.
Angular Velocity and Acceleration, 37 40.
Velocity and Acceleration relative to Moving Axes, 41 43.
EXAMPLES 3441
CHAPTER Ily LAWS OP MOTION 42 58
\ Definitions of Mass, Density, Particle, Force, Momentum,
Vis Viva, Kinetic Energy, Measure of Force, Compo-
nent of Force, &c. &c., 44 57.
Definition, and Properties, of Centre of Inertia, -;8.
Definition of Moment of Momentum, 59.
Definition of Work done by a force, and consequences of
the definition, 60, 61.
Definition of Potential Energy, 62.
Newton's Laws of Motion, with their consequences as
Measure of Time, Parallelogram of Forces, Conserva-
tion of Momentum and of Moment of Momentum, t&c.,
572.
Xll CONTENTS.
PAGES
Scholium to the Third Law, with its interpretation.
D'Alembert's principle, Horse-Power, Conservation of
Energy in Ordinary Mechanics, 73 75.
Conservation of Energy, Impossibility of Perpetual Motion,
Joule's experimental results, 7678*.
CHAPTER III. RECTILINEAR MOTION 59 79
Constant Force, 79 87.
Force varying according to different powers of the distance,
* 10-v
\
EXAMPLES ... ... ... ... ... ... 79 85
CHAPTER IV. PARABOLIC MOTION 86 108
Projectile in vacuo, 106 119.
Projectile in vacuo when the changes in the direction and
magnitude of gravity are considered, 120, 121.
Force constant in direction, but not in magnitude, 122
129.
Newton's investigation of the motion of a luminous cor-
puscle, 130.
EXAMPLES ... ... ... ... ... ... 108 112
CHAPTER V. CENTRAL ORBITS ... 113 144
A General Equations, 131, 132.
Attraction proportional to the distance, 133.
Polar Form of General Equations, and consequences,
134144-
Properties of Apses, 145 148.
Orbits under the Law of Gravitation, 149 158.
Elliptic motion : definitions and immediate deductions,
159162.
Kepler's Problem, 163167.
Lambert's Theorem, 168.
EXAMPLES 145 166
CHAPTER VI. CONSTRAINED MOTION 167 222
Preliminary remarks on Constraint, 169.
Motion on Smooth Plane Curve, Cycloidal and Common
Pendulum, &c., Direct Problem, 170 179.
CONTENTS. Xlll
PAGES
Inverse Problems Brachistochrone, &c., 180 186.
Motion on Smooth Surface, 187189.
Particular Case Spherical Pendulum, 190, 191.
Double Pendulum, 192.
Effect of the Earth's rotation on simple Pendulum,
I93I95-
Constraint by String attached to a moving Point, 196
198.
Constraint by Smooth Tube in motion, 199 203.
Constraint by Rough Curve, 204, 205.
Constraint by Eough Surface, 206.
EXAMPLES ... ... ... ... ... ... 222 237
CHAPTER VII. MOTION IN A RESISTING MEDIUM ... 238 251
General Statement of the Problem, 207.
%l Rectilinear Motion with various applied forces and various
laws of Resistance Terminal Velocity, &c., 208
212.
Curvilinear Motion, under various laws of resistance and
various forces. Approximate determination of path of
projectile with low trajectory, 213 217.
Equation of Central Orbit in resisting medium, 218, 219.
EXAMPLES ... ... ... ... ... ... 252 259
CHAPTER VIII. GENERAL THEOREMS 260309
Constraint perpendicular to direction of motion, 220,
221.
J All central forces have a potential, 222.
Conservation of Energy, and Equipotential Surfaces,
323, 224.
Inverse Problem as to conservative forces, 225.
Deductions from Conservation of Energy, 226 229.
Least, or Stationary, Action, 230 237.
Varying Action, 238243.
The principle applied to the investigation of a planetary
orbit, 244-248.
Application to Cotes' spirals, 249.
Lagrange's Equations in Generalized Co-ordinates, 250,
251-
XIV CONTENTS.
Application of Varying Action to Brachistochrones, 252.
The brachistoclirone when the force is central, 253.
The brachistochrone normal to a series of isochronous sur-
faces, 254.
Connection between the forces under which curves may
be described as free paths or brachistochrones,
255260.
Motion about moving centre, 261, 262.
Hodographs, 263270.
Case of resisted motion in an equiangular spiral, 271.
EXAMPLES ... ... ... ... ... ... 310 319
CHAPTER IX. IMPACT 320337
Preliminary Kemarks, Coefficient of Kestitution, 272.
Direct Impact of Spheres, 273, 274.
Impact of Sphere on Fixed Surface, 275.
Impact of Smooth Spheres generally, Apparent Loss of
Energy, 276, 277.
Impulsive Tension in Chain, 278 283.
Continuous Series of indefinitely Small Impacts, 284
286.
Disturbed Planet, 287.
EXAMPLES ... 337 352
CHAPTER X. MOTION OF Two OR MORE PARTICLES 353 374
I. Free Motion. General Equations, 289, 290.
Conservation of Momentum, 291 ; of Moment of Momen-
tum, 292; of Energy, 293.
The Virial Equation, 294.
Particular Case of Two Particles, only, 295 297.
II. Constrained Motion. Conditions of Constraint, 298.
Two particles, in space, connected by inextensible string,
299.
String constrained by pulley, 300.
Chain slipping over pulley, 301*
Complex pendulum, 302, 303.
Limits of the treatise, 304.
EXAMPLES 374 381
GENKKAL EXAMPLES ... ... ... .. ... 382 399
CONTENTS. XV
PAGES
APPENDIX 400412
A. On the integration of the equations of motion about
a centre of attraction ... ... ... ... ... 400
B. Motion on a cycloid ... ... ... ... ... 405
C. Brachistochrone, for gravity ... ... ... ... 406
Cj. , for any forces ... ... ... ... 408
C.,. General property of free path and brachistochrone for
any force whose direction is constant ... ... ... 409
D. Of two curves, convex upwards, joining two points in
a vertical plane, the inner is described in less time
than the outer ... ... ... ... ... ... ibid.
E. Inverse problem To find the equation of the con-
straining curve when the time of descent, to the
lowest point, through any arc, is given as a
function of the vertical height fallen through ... 411
DYNAMICS OF A PAKTICLE.
CHAPTER I.
KINEMATICS.
1. DYNAMICS (consistently with its derivation) is the
Science which investigates the action of Force ; and naturally
divides itself into two parts as follows.
2. Force is recognized as acting in two ways : in Statics
so as to compel rest or to prevent change of motion, and in
Kinetics so as to produce or to change motion.
3. In Kinetics it is not mere motion which is investi-
gated, but the relation of forces to motion. The circumstances
of mere motion, considered without reference to the bodies
moved, or to the forces producing the motion, or to the forces
called into action by the motion, constitute the subject of a
branch of Pure Mathematics, which is called Kinematics.
To this, as a necessary introduction, we devote the present
chapter.
4. The rate of motion (or the rate of change of position)
of a point is called its Velocity. [But see 8.] It is greater
or less as the space passed over in a given time is greater or
less : and it may be constant, i.e. the same at every instant ;
or it may be variable.
Constant velocity is measured by the space passed over in
unit of time, and is, in general, expressed in feet per second ;
if very great, as in the case of light, it may be measured in
miles per second. It is to be observed, that Time is here
used in the abstract sense of a uniformly-increasing quantity
what in the differential calculus is called an independent
variable. Its physical definition is given in Chap. II.
CJ T. D, 1
2 KINEMATICS.
5. Thus, a point moving uniformly with the velocity v
describes a space of v feet each second, and therefore vt feet
in t seconds, t being any number whatever, whole or fractional.
Putting s for the space described in t seconds, we have
s = vt.
Hence with unit velocity a point describes unit of space in
unit of time. The path may be straight or curved.
6. It is well to observe that since, by our formula, we
have generally, for constant or uniform velocity
and since nothing has been said as to the magnitudes of s
and t, we may take these as small as we choose. Thus we
get the same result whether we derive v from the space described
in a million seconds, or from that described in a millionth of a
second. This idea is very useful, as it will give confidence
hi results when a variable velocity has to be measured, and
we find ourselves obliged to approximate to its value by
considering the space described in an interval so short, that
during its lapse the velocity does not sensibly alter in value.
7. Velocity is said to be variable when the moving point
does not describe equal spaces in equal times. The velocity
at any instant is then measured by the space which would
have been described in a unit of time, if the point had moved
on uniformly for that interval with the velocity which it had
at the instant contemplated. This is a most important, and
in fact a fundamental, conception, which the student must
thoroughly realize before he can usefully proceed farther. It
lies at the root of all the correct methods ever devised for
the purpose of measuring the rate at which change, of any
(essentially continuous) kind, is going on.
Let v be the velocity of the point at the time t, measured
from a fixed epoch, s the space described by it during that
time, and s + &s the space described during a greater interval
1 4- &t. Suppose V-L to be the greatest, and v 2 the least, velo-
city with which the point moves during the time St ; then
vt, vt would be the spaces which a point would describe
in that interval, moving uniformly with these velocities
KINEMATICS. 3
respectively. But the actual velocity of the point is not
greater than v l , and not less than # 2 > therefore as regards the
actual space described,
&s is not greater than v^t, and not less than v t 8t,
Ss
or si Vl v *>
however small 8t may be. But, as St continually diminishes,
v l and v 2 tend continually to, and ultimately become each
equal to, v. Therefore, proceeding to the limit,
ds
dt = V '
If v be negative in this expression, it indicates that s
diminishes as t increases ; the positive case, which we have
taken as the standard one, referring to that in which s and t
increase together. It follows that, if a velocity in one direc-
tion be considered positive, in the opposite direction it must
be considered negative; and consequently the sign of the
velocity indicates the direction of motion, when the path is
given.
This investigation rests on the supposition that the velocity
alters CONTINUOUSLY and not by jerks. It would require an
infinite force to produce in an infinitely short time such a
change of velocity in a material particle. Hence as we are
preparing for physical applications only, such cases may be
entirely excluded. [The action of great forces for short periods
of time will be treated in the chapter on Impact."]
In all cases, however, we may define as the Average Speed
the quotient of the space described, by the time employed.
This is applicable even to the motion of a postman or a
lamplighter. But its interest for our present purpose, where
continuous change of speed, alone, is considered, consists in
the fact that the average speed approaches without limit the
actual speed at any instant the shorter the interval of time
is. Now ~- is the average speed, and becomes in the limit
IWf ,
37 as above.
at
12
4 KINEMATICS.
8. So far as we have yet spoken of it, velocity has been
regarded merely as Speed, and all that is said above is equally
applicable whether the point be considered as moving in a
straight, or in a curved, line. In the latter case, however,
the direction of motion continually changes ; and it is neces-
sary to know at every instant the direction, as well as the
magnitude, of the point's velocity. This is usually, and in
general most conveniently, done by considering the speeds
of the point parallel to the three co-ordinate axes respec-
tively. [In fact velocity is properly a directed magnitude (or
VECTOR, as it is now called) involving at once the direction and
the speed of the motion.] If the co-ordinates of the moving
point be represented by x, y, z, the rates of increase of these,
or the speeds parallel to the corresponding axes, will by
reasoning analogous to that in 7 be
dx dy dz
~dt' di' di'
Denoting by v the speed of the motion, we have
V = j~ t = t/U } +(-77l (
and, if a, /3, 7 be the angles which the direction of the
motion makes with the axes,
dx
_dx _ dt
dt
dx
or ~7f =v cos a ~ Vx > su PP se -
Similarly, -j- v cos /3 = v y ,
^= VC os
dt
Hence, - - are to be found from the whole
at at at
speed v, by resolving as it is called; i.e. by multiplying by
the direction-cosines of the direction of motion. They may
KINEMATICS. 5
be looked on as the Component Velocities of the point : and,
with reference to them, v is called the Resultant Velocity.
9. It follows from the above, that, if a point be moving
in any direction, we may look on its velocity as the result-
ant of three coexistent velocities in any three directions at
right angles to each other ; or, more generally, in any three
directions not coplanar. But the rectangular resolution is
the simplest and best except in some very special applications.
Let v x , v y , v z be the rectangular components of the velo-
city v of a moving point, then the resolved part of v along
a line inclined at angles X, //,, v to the axes will be
v x cos X -f v y cos yu- 4- v z cos v.
For, let a, 0, 7 be the angles which the direction of the
point's motion makes with the axes, the angle between
this direction and the given line. Then since
cos 6 = cos a. cos X + cos /3 cos /JL -f cos y cos v
the resolved part of v along tnat line is
v cos 6 = v (cos a cos X + cos /3 cos /JL + cos 7 cos i;}
= V x COS X + V y COS fl + V z COS V.
10. These propositions are virtually equivalent to the
following obvious geometrical construction, which is the Law
of Composition of Vectors :
To compound any two velocities as OA, OB in the figure;
where 0-4, for instance, represents in magnitude and direc-
tion the space which would be described in one second by
a point moving with the first of the given velocities and
o A
similarly OB for the second ; from A draw A C parallel and
6 KINEMATICS.
equal to OB. Join OC: then OC is the resultant velocity
in magnitude and direction. For the motions parallel to OA
and OB are independent.
OC is evidently the diagonal of the parallelogram two of
whose sides are OA, OB.
Hence the resultant of any two velocities as OA, AC, in
the figure is a velocity represented by the third side, OC, of
the triangle OAC.
Hence if a point have, simultaneously, velocities repre-
sented by OA, AC, and CO, the sides of a triangle taken in
the same order, it is at rest.
Hence the resultant of velocities represented by the sides
of any closed polygon whatever, whether in one plane or not,
taken all in the same order, is zero.
Hence also the resultant of velocities represented by all
the sides of a polygon but one, taken in order, is represented
by that one taken in the opposite direction.
When there are two velocities or three velocities in two
or in three rectangular directions, the resultant speed is the
square root of the sum of their squares and the cosines of
the inclination of its direction to the given directions are the
ratios of the components to the resultant.
[Newton's Method of Fluxions was devised simply to ex-
press this and other fundamental conceptions in Kinematics.
To him s, x, y, z, or (as we now somewhat less conveniently
,. x ds dx dy dz , , . .
write them) , , -r , - t -77 , are simply the velocity of the
moving point and its components parallel to the axes. It
may be convenient, or even necessary, to use the idea of
Limits or of Infinitesimals to calculate their values ; but the
Fluxions themselves do not involve any such idea.]
11. When a point moves in a plane curve, to express its
component velocities at any instant along, and perpendicular
to, the radius vector drawn from a fixed point in the plane of
the carve.
KINEMATICS.
Let x, y be the rectangular, r t 6 the polar, co-ordinates of
the moving point ; so that
x = r cos 0, y = r sin 0.
We have at once, by differentiation,
and
dx dr - adO
-j- = -j- cos 6 r sm 6 -=-
dt dt dt
dy dr . Q A dO
- = -77 sin 6 -f r cos 6 -TT
(1),
which are the velocities parallel to x and y. But by 9 the
velocity along the radius vector is
dO , ,,.
= r ,-, by (1).
and the velocity perpendicular to it is
dy - dx
dt C S ~~ dt S
dt
12. The velocity of a point (in the sense of its speed) is
popularly said to be accelerated or retarded according as it
increases or diminishes, but the word Acceleration is scien-
tifically used in both senses ; and may be defined as the rate
of change of the velocity per unit of time.
Acceleration may be either constant or variable. It is
said to be constant when the point receives equal increments
of velocity in equal times, and is then measured by the actual
increase of velocity generated in unit of time. Let the unit
8 KINEMATICS.
of acceleration be so taken that a point under its action would
receive an increment of a unit of velocity in a unit of time ;
then a point under the influence of a units of acceleration
would receive an increment of a units of velocity in a unit of
time, and consequently at units of acceleration in t units of
time. If the point starts from rest we have
where v denotes the velocity at the end of the interval t, and
a the acceleration.
13. Acceleration is variable when the point does not re-
ceive equal increments of velocity in equal increments of time.
The acceleration at any instant is then measured by the in-
crement of velocity which would have been generated in a
unit of time had the acceleration remained constant during
that interval and equal to the value at its commencement.
Let v be the velocity of the point at the end of the time
t, a. the acceleration at that instant, v 4- Bv the velocity at the
end of the time t + Bt ; and let a l , a^ be the greatest and least
values of the acceleration during the interval Bt, then a^t,
o^Bt would be the increments of velocity in that interval, of a
point under those accelerations respectively. But the actual
acceleration is not greater Jjian a a and not less than 02, there-
fore the actual increment of velocity
Bv is not greater than a^t and not less than c^Bt,
Bv
or & ........................ ttl ........................ * 2 '
however small Bt may be. But, as Bt continually diminishes,
! and 2 tend continually to and ultimately become each
equal to a. Therefore, proceeding to the limit,
dv
The positive sign given to a shews that v increases with t,
while a negative sign would shew that v decreases as t in-
creases, in other words a negative acceleration is a retardation.
As in 7 we have, for the Average change of speed during
KINEMATICS. 9
St, the expression Sv/Bt. Its limit dv/dt is therefore the
acceleration.
Combining the above equation with
ds
we have = a
at*
considering t as the independent variable.
[Here, again, Newton employs the symbol s to represent
the rate of increase of s, a quantity whose conception is
altogether independent of the methods (infinitesimal or not)
which may be employed to calculate its value.]
14. Thus far we have been dealing with a point's
motion in some definite path, which may be either straight
or curved, but in which there is only one degree of freedom
to move, and in which therefore the position at any time
is determined by one variable, s. But when we consider
velocity as a directed magnitude ( 8) we are led to generalize
the definition of Acceleration (see 20 below).
If the path be curved, the accelerations of the rates of
increase of the co-ordinates of the moving point are called
the Component Accelerations parallel to the axes. If these
be denoted by a x , a y , a. z , we shall have
With reference to these, Vc^ 2 + Oy a -f a/ is called the
Resultant Acceleration.
15. The acceleration -=- is not the complete resultant
at
^ fit* ' fit* ' ^7/2 ' as ma y eas % ke seen : f r i* s s q uare
does not in general equal the sum of the squares of those
three accelerations. It is, however, the only part of their
resultant which has any effect on the magnitude of the
velocity ; in short , 2 is the sum of the resolved parts of
10 KINEMATICS.
//2/Yi fj^^l r/~^
-j-- , -T^ , -j-- in the direction of motion, as the following
Ott ft* Cut
identical equation shews :
d?s _ dx d' 2 x dy d 2 y dz d*z
df z ~"dsd^dsdt 2 ~ds dt z '
This follows immediately from the equation of 8
.
dt \dt) \dt
by differentiation. And it shews that acceleration is to be
resolved according to the same law as velocity. For to find
c ^ (fa r r (i T
-r- , the acceleration along s, -^- has to be multiplied by -r- ,
&c. &c. which is the vector law.
The other part, of the resultant is at right angles to this,
and its sole effect is to change the direction of the motion
of the point. And this leads us to another form of accelera-
tion, viz. when the magnitude of the velocity is unaltered,
but the direction of motion changes. Its value in terms of
the speed and the curvature will be given later.
mi . dx dy dz
Ine above equation also shews, since -7-, -- , -j- are
ds ds ds
the direction-cosines of the small arc ds, which may have any
direction whatever, that to obtain the acceleration along any
line inclined at given angles to the axes, we must resolve
the component accelerations parallel to the axes along it,
and take the sum of the resolved parts. Thus the accelera-
tion along a line inclined at angles X, /JL, v to the axes is
OB cos X 4- a y cos IJL + a z cos v.
16. A point moves in a plane curve, to express its com-
ponent accelerations at any instant along, and perpendicular
to, the radius vector.
Let x, y be the rectangular, r, 6 the polar, co-ordinates ;
so that
x r cos 0,
y = r sin 6 ;
KINEMATICS.
11
we have
dx dr . Q d0
-j- = -T7 cos 6 - r sin 6 -j-
dt dt dt
I
, d?x \*r fd\* a f-drd0 d*0\ . a
and -j- = l-j- - r (-j- H cos 6 - 2 -j- -^ +r -j- sin ^.
eft 2 (rf^ 2 \/ J \ dt dt dtfJ
Similarly,
f,drd0 d*0\
2 -3;^ + r ^^ cos #
\ dtdt dtfJ
- - r sm
These are the accelerations parallel to x and y. And
since, by 15, the acceleration along the radius vector is
the above expressions give it in the form
d?r _ (d0\*
dt*~ r (di) '
The acceleration perpendicular to the radius vector is
d? c( "^ !
. dr dd cfr
that is, 2 -r -y- + r -r
1 rJ / fl b\
which may be written - -y- (r 2 -^ ) .
r rf^ \ a^/
12 KINEMATICS.
17. When a point is in motion in any curve, to find its
accelerations along, and perpendicular to, the tangent, at any
instant.
Let x, y, z be the co-ordinates of the point at the end
of the time t, s the length of the arc described during that
interval. Then, since by the equations of the curve x, y and
z are functions of s,
dx dx ds
di = ~dsdi'
-, a x a x t as \ ax a s
n~r\r\ ^ _____ I I I _
/V/2 y7o2 \ /// / fl & /V/2
UiO Uo \Uil// (Jib UiV
r*' 'i i cL if a ii f as\ aif a s
Similarly, ~ = -j-^ I L H - ,
ds\ 2 dz d*s
(l 1" I fl <? flf^
Remembering the law of resolution of acceleration, the
form of these equations shews that in them are resolved
along x, y, z, 1st an acceleration -^ , whose direction-cosines
dx dy dz , , 1 fds\* ,
are -=- , -f- . -=- , and 2nd an acceleration - [ -=- , whose
ds ds ds p \dt)
direction- cosines are p -^ , p -^ , p -=-$ ;' where p is a linear
cts as as
quantity, which will be presently recognized as the radius of
curvature of the path. This process might have been
employed with advantage in some previous sections. But,
for the beginner, we must take a more laborious method.
18. To find the acceleration along the tangent, we must
multiply these component accelerations by -j-', -j- , -7- ,
cts ds ds
respectively, and add. Thus the tangential acceleration is
dx d*x dy (fry dz d*z _ d 2 s _ dv
KINEMATICS. 13
as we have already seen. Also in the normal, towards the
centre of curvature, we have the acceleration
dt
We assume here the following equations from Analytical
Geometry,
where p is the radius of curvature, whose direction-cosines
are
d?x d? d?z
and
whence
*
* The accelerations of the moving point may be found in the following
manner. There is obviously no acceleration perpendicular to the osculating
plane, as that plane contains two consecutive directions of the point's motion.
Of the two consecutive directions let the first make an angle 6 with any fixed
line in the osculating plane, then v cos 6 and v sin $ are the velocities of the
point parallel and perpendicular to the fixed line respectively. Consequently
(v cos 6) and (v sin 6) are the accelerations in the same directions. These
(It at
expressions, when expanded, become -^cos - v sin 6 , and ~ sin + v cos 6 -=-
dt dt dt dt
Therefore the accelerations along the tangent and the normal are and
j/j
v j- , the last being positive in the direction of the centre of curvature. Since
d0 1 dd ds v 2
= - , the normal acceleration, being = v -^ . , may be expressed as .
14
KINEMATICS.
thus
19. We might have treated the component accelerations
d*Z\ *
r ( resultant acceleration) 2
by adding the squares of their values as given in 17.
d*s
Now ^ is the acceleration along the tangent, and the
1 / ds\ * v*
other part (-7: 1 , or , acts at right angles to it as the
p \dtj p
form of the equation shews, and consequently is the accelera-
tion perpendicular to the tangent.
From the expressions for -j- , -yf , -j- , we also obtain
ctt ctt at
2 x fdyd^z dz d z y
tf \ds~ds* ~ ds d&
*y (dz d?x _ dx d 2 z
^ \dsds* dsd*
d 2 z idx d 2 y dy d 2 x\ _ _
+ dt 2 (dsd* dsd*)** '
_
which may be written in the form of a determinant
d*x d*y d*z
~dt* ~dt* dt*
dx dy dz
ds ds ds
~ds* ds* ds*
This signifies that the Resultant Acceleration lies in the
KINEMATICS. 15
plane containing the tangent and the radius of absolute
curvature, or that there is no acceleration perpendicular to
the osculating plane. The acceleration must therefore be
along a normal to the path drawn in the osculating plane ;
that is, along the radius of absolute curvature.
20. We are therefore led to expand the definition given
in 12 thus: Acceleration is the rate of change of velocity
whether that change take place in the direction of motion or not.
What is meant by change of velocity is evident from 10.
For if a velocity OA (in the figure of that section) become
0(7, its change is AC, or OB.
Hence, just as the direction of motion of a point is the
tangent to its path so the direction of acceleration of a
moving point is to be found by the following construction.
From any point draw lines OP, OQ, etc., representing
in magnitude and direction the velocity of the moving point
at every instant. The points, P, Q, etc., form in all cases of
motion of a material particle a continuous curve, for an infi-
nitely great force is requisite to change the velocity of a par-
ticle abruptly either in direction or magnitude. Now if Q be
a point near to P, OP and OQ represent two successive values
of the velocity. Hence PQ is the whole change of velocity
during the interval. As the interval becomes smaller, the
direction PQ more and more nearly (without limit) becomes
16 KINEMATICS.
the tangent at P. Hence the direction of acceleration is
that of the tangent to the curve thus described, called by
its inventor, Sir W. R. Hamilton, the Hodograph.
The amount of acceleration is the rate of change of
velocity, and is therefore measured by the velocity of P in
the curve PQ.
21. The Moment of a velocity about any point is the
rectangle under its magnitude and the perpendicular from the
point upon its direction. The moment of the resultant velo-
city of a point about any point in the plane of the components
is equal to the algebraic sum of the moments of the components,
the proper sign of each moment depending on the direction
of motion about the point. The same is true of moments of
acceleration, and of moments of momentum as denned later.
Consider two component velocities, AB and AC, and let
AD be their resultant ( 10). Their half moments round
the point are respectively the areas OAB, OAC. The half
moment of the resultant, AD, is the area OAD. But all
three areas are on the same base, A ; and the sum of the
distances of C and B from that line is equal to the distance
of D from it. Hence the moment of the resultant is equal
to the sum of the moments of the two components. By
attending to the signs of the moments, we see that the
proposition holds when is within the angle CAB.
22. Now if the direction of one of the components always
passes through the point 0, its moment vanishes. This is the
case of a motion in which the acceleration is directed to a
KINEMATICS. 17
fixed point, and we thus prove the theorem that in the case of
acceleration always directed to a fixed point the path is plane
and the areas described by the radius-vector are proportional
to the times ; for the moment of velocity, which in this case
is constant, is evidently double the rate at which the area
is traced out by the radius-vector.
23. Hence in this case the velocity at any point is
inversely as the perpendicular from the fixed point upon the
tangent to the path, the momentary direction of motion.
For evidently the product of this perpendicular and the
velocity at any instant gives double the area described in one
second about the fixed point, which has ju'st been shewn to
be a constant quantity.
24. The results of the last three sections may be easily
obtained analytically, thus. Let the plane of motion be
taken as that of x, y ; and let the origin be the point about
which moments are taken. Then if a), y be the position of
the moving point at time t, the perpendicular from the origin
on the tangent to its path is
dy dx d6 . ,.
p = a;-^ 7/-=- = r 2 -7-, in polar co-ordinates.
ds y ds ds
From this we have at once
ds dy dx n d6
or with the notation of 8,
which is the theorem of 21.
d , . d z y d-x /ox
Also (pv} = X - y .................. (2)>
T. D 2
18 KINEMATICS.
Now, if the acceleration be directed to or from 0, its
moment about 0, which is evidently
d 2 y d?x
*-;
must vanish. Hence (2) gives
pv = constant, which is 23.
By means of (1) this gives
r 2 -%- = constant, which is 22 ;
since, if A be the area traced out by the radius-vector,
dA_r*
dd ~2'
25. To determine the motion of a point when the accelera-
tion of its velocity is given.
This is one of the most general of the Problems suggested
by the Kinematics of a point, for it includes, as will be seen,
the determination of the motion when the component velo-
cities are given.
Let a, ft, 7 be the components of the given acceleration ;
we have
.(1).
_
eft 2 " 7 '
AT . . dx dy dz
Now a, p, 7 may be functions of x, y y z, t } -7- , -j- , or -r- ,
at at at
or of two or more of these quantities. Equations (1) must be
integrated as simultaneous differential equations if possible.
KINEMATICS. 19
Thus by one integration we have the values of -^- , -^ , -^- ,
in terms of one or more of the quantities x, y, z and t ; that
is, the component velocities are known.
Another integration, if it can be performed, gives a, y,
and z, in terms of t ; and, it' the latter variable be eliminated
from the three integrated equations, we have the two equa-
tions of the path in space : and thus, theoretically at least,
the motion is completely determined.
It is unnecessary to give examples of the integration of
such equations here, as the major part of the following
chapters will be devoted to them.
26. So far for a single point. Whe*n more points than
one are considered, Kinematics enables us to determine, from
the given motions of all, their relative motions with respect
to any one of them ; or conversely, from the actual motion
of one, and the motions relative to it of the others, to de-
termine the actual motions of the latter in space. This
depends on the following self-evident proposition.
If the velocity of any point of a system be reversed in
direction, and be communicated to each point of the system in
composition with that which it already possesses, the relative
motions of all about the first, thus reduced to rest, will be
the same as their relative motions about it when all were in
motion.
For the proof it is sufficient to notice that if at every
instant the distance of two points, and the direction of the
line joining them be the same, as for two other points, the
relative motions of one of each pair about the other will be
the same. The simplest illustrations of this proposition are
furnished by the relative motions of objects in a vessel or
carriage, which are independent of the common velocity of
the whole or, on a grander scale, of terrestrial objects,
whose relative motions are unaffected by the earth's rotation,
or by its motion in space.
Since accelerations are compounded according to the same
law as velocities, the above theorem is true of them also.
22
20 KINEMATICS.
27. Two points describe similar orbits about each other
and about any point dividing in a given ratio the line which
joins them.
Let A and B be the points, G a point in AB such that
AG
j^ = a constant.
(JT-D
The .path of B about A will evidently be the same as
that of A about B, since the length and direction of the
line AB are the same whichever end be supposed fixed.
Also if be fixed, the path of B about it will evidently
differ from that of B about A by having corresponding
radii-vectores diminished in the ratio -- . But this is the
definition of similar curves. The same of course would hold
with respect to the relative path of A with respect to G.
This proposition will be found of considerable use afterwards,
as it enables us materially to simplify the equations of
motion of two mutually attracting tree particles.
28. As an instance of relative motion, consider two
points, one of which moves uniformly in a straight line,
while the other moves uniformly in a circle about the first as
centre; determine the path of the second point, the motion
being in one plane.
Take the line of motion of the first as the axis of #, v its
velocity, the plane of the circle as that of xy, a the radius of
the relative circular orbit, w the angular velocity in it, 37.
Suppose the revolving point to be initially in the axis. Also
at time t suppose the line joining the points to be inclined
at an angle 6 to the axis of x. Then for the co-ordinates of
the revolving point we have
y = a sin 6,
x = vt + a cos 6.
KINEMATICS. 21
But = 0)f,
I
^i fit
hence x = - sin" 1 - + V(& 2 y*)
<u ft
is the equation of the absolute path required. This belongs
to ihe class of cycloids ; it is prolate or curtate according as
v is greater or less than a&>, or the absolute motion of the
first point greater or less than that of the other in its circular
orbit. If the two are equal, or v = aco, we have the equation
of the common cycloid, as is indeed evident, for the circular
path may be supposed the generating circle, and the velocity
of the centre in its rectilinear path is equal to that of the
tracing point about that centre.
29. It is evident that, whatever be the relative path, if
the first point be still supposed to move in the axis of x, and
if r, denote the relative co-ordinates of the second point
with respect to the first at time t, x, y, and x the absolute
co-ordinates at the same time,
,..,
Now in the first case, when the motion of the first point,
and that in the relative orbit are given, x, r, and 6 are known
functions of t ; if therefore these values be substituted in (1),
and t be eliminated, we shall have the equation between x
and y, which is required.
Again, if the absolute orbits of both are given, x, y, and
x are given in terms of t, and thus equations (1) serve to
give r and 6 in terms of t, which furnishes the complete
determination of the relative path, and the circumstances of
its description.
30. The following is a most useful case, having many
important applications in Physical Optics, &c.
A point A is fixed. B describes uniformly a circle about
A, and C describes uniformly (in the same plane) a circle about
B. Find the motion of C relative to A.
22 KINEMATICS.
Let a be the length of AB, b that of BC, r that of AC]
and at time t let theifo make angles <f>, %, with some fixed
line in the plane of motion. Then
r cos 6 = a cos < + b cos %,
r sin 6 = a sin $ + 6 sin ^.
But $ and % increase uniformly. Hence
<j) = mt + a,
X = nt + A
where m, w, a, /3, are constants. Thus
r cos = a cos (w + a) + b cos (w -f ft),
r sin = a sin (ra + a) + b sin (n + /3).
These are the general equations of Epicycloids and Hypo-
cycloids ; and from them all their properties may be derived.
We confine ourselves to one or two very simple cases.
(1) Let m = n, a = b. (This is the composition of two
equal circular motions, in the same direction and of equal
period.) We have
r cos 6 = 2a cos =-*- cos ( mt +
(->+-)
rsind 2a cos sin (mt H
whence
r 2a cos
2
This also denotes uniform circular motion, and of the same
period, and in the same direction, as the components.
(2) Let m = n, a = b. (Here we compound equal
circular motions, of equal period, but in opposite directions.)
As before we have
r cos 6 = 2a cos --^ cos [mt + -
2 \ Lt
r sin = 2a sin - ^- cos f mt H ^ J .
KINEMATICS. 23
Therefore
2a cos (mt + ~ j ,
=
2 '
and this denotes vibratory motion in a definite straight line.
31. In any system of moving points, to determine the rela-
tive from the absolute motions ; and vice versa.
Let a?,, y lf z 1 , # 2 , 3/2, z^ be the co-ordinates of two of the
points, x, y, z the relative co-ordinates of the second with
regard to the first, u lt v ly w l} u 2 , v 2 , w 2 the velocities of each
parallel to the axes, u, v, w the velocities of the second
relatively to the first.
Then x = x. 2 x l , u = u 2 u l ,
y = y*-y^ v = v 2 -v 1 ,
z = 2 z l , w = w, 2 w l .
The second group may be derived from the first by differ-
entiation with respect to t.
Now, when the actual motions of the two are given, all
the subscribed quantities are known. Hence the above
equations give the circumstances of the relative motion.
Or if the actual motion of the first, and the relative
motion about it of the second, be known, we have xyz, uvw,
x \y\z\> ^ifiWi, to find the other six quantities for the actual
motion of the second in space.
A second differentiation proves the statement in 26
regarding relative acceleration.
32. Some of the best illustrations of this part of our
subject are to be found in what are called Curves of Pursuit.
These questions arose from the consideration of the path
taken by a dog, who in following his master always directs
his course towards him.
In order to simplify the question the rates of motion of
both master and dog are supposed to continue constant ; or
at least to have a constant ratio.
KINEMATICS.
33. As an instance of the curve of pursuit, suppose it be
required to determine^ the path of a point P which continually,
with constant velocity M, moves towards another point Q which
is describing a straight line with constant velocity v.
The curve of course is plane. Take the line of motion
of the second point Q as the axis of #, and let x denote
its position at the instant when the co-ordinates of the first,
M
P, are x, y. The axis of y is chosen as that tangent to the
curve of pursuit which is perpendicular to the axis of x, and
the distance between the points in that position is called a.
e, then by the conditions of the problem we have
Let -
and PQ a tangent at P.
Expressed analytically these lead to the following equa-
tions ;
dx
@S ~~ vCt 3u ~~ Y/ ~~^ .
y dy
The mode of solution is precisely the same whether x or y
be taken as independent variable : but y is to be preferred as
it leads to less cumbrous expressions.
Differentiating therefore with respect to y, we have
ds d*x
e ~T~ V J~T-
dy J dy*
KINEMATICS.
But s increases as y diminishes,
e dy*
Hence
y
Integrating, and noting that y = a, -j- 0, together,
ay
Hence, ()' = A / ^1 + ( ^] J. + ^ ,
\**/
and therefore, taking reciprocals,
\y) 'y (_ \dy) ) dy '
Subtracting, we have finally
2 = (yy_(?y.. ..(i),
dy \aJ \yl
But a; = 0, y = a, together ; which gives (7= - =- .
G ~~ A
Hence 2(^+- n )= C H (2).
x>2 1 / ^.e /^, i 1 \ /..e l / si 1 \ V y
This is the correct integral for all values of e except unity,
when it ceases to have any meaning. To this case we will
presently recur.
There are two cases of curves represented by equation (2),
1st, e> 1, 2nd, e< 1.
26 KINEMATICS.
In the first case Q moves the faster, and P can never
overtake it ; the curve therefore never meets the axis of x,
which indeed will be seen by (2) to be an asymptote.
In the second case equation (2) becomes
2 L ae }- y l+ L ae y'~ e
*\ JU ' l-eV~a e (l + e) \- e '
and for x - '- we have 2/ = 0, and also by (1) -7- infinite.
1 e dy
Hence the curve touches the axis at this point. The re-
mainder of the curve satisfies an obvious modification of the
question, whence it is called the Curve of Flight, jit is to be
ae . /I + 0\stel
observed, however, that x = -- - gives also ?/= + , I - k
1 e \1 ej }
The distance between P and Q, being
V(# - x) z + ?/ 2 ,
is easily seen by the fundamental equations to be
ds
or, by (1),
+!<
where the sign is to be chosen so as to make the expression
positive.
When e > 1, this expression is infinite both for y= oo and
for y = 0. The minimum value is easily found to be
_^
ae (e -
Ve 2 -l \e + lj
When e< 1, the distance vanishes, as we have seen it
must, when y = 0.
34. When e = 1, the corrected integral of (1) is
KINEMATICS.
27
This is the only case in which we do not obtain an alge-
braic curve. Here again the axis of x is an asymptote, and
we easily find
which shews that the limit to which the distance tends is ^ .
2i
The same result may at once be obtained by putting e 1 in
the expression for the minimum distance found above in the
case of e > 1.
35. As an instance of relative motion let us consider the
path of P with regard to Q. It will be easy to see that this
corresponds exactly to the following question.
A boat, propelled (relatively to the water) with constant
velocity u, starts from a point A in the bank of a river which
runs with velocity v parallel to Qx, and tends continually to
the point Q, on the other bank, directly opposite o A ; to find
its path.
The constant velocity of the stream in this case com-
municated to P corresponds to the constant velocity of Q in
the last example, but is in the opposite direction. In fact,
if the earth were to be supposed moving in the direction xQ
with constant velocity v, the river would be at rest in space,
28 KINEMATICS.
and the actual motions of P and Q would be the same as inl
the last example. (See 26.)
To investigate the path, take Q as origin, Qx, QA as the I
axes. Then the component velocities of P are v parallel to
Qx and it along PQ, and the tangent to its path is in the
direction of the resultant of these two. Putting 6 for PQx,
we have ' T = v u cos 0, and ~ = u sin 6,
at at
dy u sin 6 sin 6
whence -/- = - =
ax v u cos 6 e cos 6
This, being a homogeneous equation, is easily integrated
and we have, taking x = 0, y = a, as co-ordinates of A,
yi+e
a e
or 2# = a e y l ~ e
- cos 6
or
\ a ) sin
in polar co-ordinates. This evidently gives a parabola about
Q as focus, if e = 1.
[Note. The student is not unlikely to be led into a curious
error in looking at this problem from a geometrical point of
view. Thus, the velocity along PQ is always in a definite
ratio to that in MP produced ; why is not the path always
a conic section of which Q is a focus ? The idea is com-
pletely erroneous (as in fact the above investigation shews),
but it forms the very best training in a science like Kinematics
to seek to explain such difficulties without any aid from
analysis.]
KINEMATICS. 29
36. To find the time of crossing the stream.
This may easily be effected by considering the actual
velocity parallel to the axis of y :
dt
u
Now taking quotients of y* by both sides of (1),
a e y l ~ e = V(# 2 + 2/ 2 ) + #
Hence 2 \7(# 2 + 2/ 2 ) = a^y 1 " 6 4- a~ e y l+e ;
and therefore (a e y l ~ e + a~ e y l+e ) = - 2udt.
3
Taking the integral from a to 0, and putting ^ for the
time of crossing,
But, if there had been no current, we should have had
for the time of crossing,
a T, ' u*
T = -- whence ^ = - .
In the integration we have, of course, e < 1, else the boat
could not reach Q.
If e = I, the boat will (after an infinite time) reach the
farther bank, but not at Q. The solution of this case
presents no special difficulty.
37. If the motion of a point in a plane be considered
with reference to a fixed point in that plane, the rate of
increase of the angle made by the line joining the two points,
with some fixed line in the plane, is called the Angular
Velocity of the former point about the latter. Unit of
angular velocity corresponds to the description of unit angle
(arc equal to radius) in unit of time.
30 KINEMATICS.
Suppose the above-mentioned angle to be represented
by 6 at time t ; then at time t + &t it has the value 6 + BO,
and it may be shewn as before ( 7), that if &> represent the
angular velocity required, then
dd
Ex. A point moves with constant velocity v in a straight
line ; to find at any instant its angular velocity about a fixed
point whose distance from the straight line is a.
Taking as initial line the perpendicular from the fixed
point on the line of motion, the polar equation of the path is
r = a sec 6.
Also, if 6 = 0, when t = 0, we have
r sin = vt.
Hence, a tan 6 = vt,
dd va va
and a> = -rr = - - = .
dt a? + vH 2 r 2
7/1
Or thus, by 24 ; r 2 -j- = av, whence a> = av/r 2 .
38. A point describes a circle with constant velocity ; it
is required to find the actual velocity, and the angular velocity
(about the centre) in any orthographic projection.
Let ApA' be an ellipse and APA' the auxiliary circle.
Then the former will be the orthographic projection of the
latter if its axes be made in the ratio of the cosine of the
angle (a) between the planes of projection. Also if PpM be
perpendicular to A A', P and p will be corresponding points
in the two. Draw the tangents pT, PT\ then
actual velocity at = T : ^ . f
velocity at p _ V(PT 2 sin 2 6 + PT* cos 2 6 cos 2 a)
~ ~~
= V(sin- 6 + cos 2 9 cos 2 a)
sin 2 a cos 2 6).
KINEMATICS.
31
= -fa tan" 1 (cos a tan 6)
au
cos a
cos 2 4- cos 2 a sin 2
_ cos a
1 sin 2 a sin 2 6'
This is a maximum if 6 = , when its value is sec a,
z
......... minimum ... =0 ........................ cos a.
Hence, if a> x and o> 2 be the greatest and least angular
velocities in the projection,
is the angular velocity in the original path.
39. Evidently, the product of the radius-vector into the
angular velocity is the velocity perpendicular to the radius-
vector. ( 11.) This is to the whole velocity as the perpen-
dicular on the tangent is to the radius-vector : and therefore
the product of the square of the radius-vector by the angular
velocity is equal to the product of the whole velocity by the
perpendicular on the tangent, i.e. to the moment of velocity
about the pole, 24, (1).
40. When the angular velocity is variable, its rate of
change per unit of time is called the Angular Acceleration,
and is measured with reference to the same unit angle.
32 KINEMATICS.
Thus, in the Ex. 37, the angular acceleration is
da) 2vadr
_ _ __ _ _ / K* rt
dt" r 3 dt ~ ' r* ^ T
41. The motion of a point in a plane being given with
respect to fixed axes, to investigate expressions for its velocity
and acceleration relative to axes in the same plane, which
revolve about the same origin with constant angular velocity.
Let co be this angular velocity ; then, if at time t = the
fixed and revolving axes coincide, at time t they will be
inclined to one another at an angle cot. Hence, if x, y, f, t)
be the co-ordinates of the point at time t, referred to the
fixed and to the revolving axes respectively, we have by
the ordinary formulae for transformation of co-ordinates
% = x cos cot + y sin cot} ,-^
)
rj = y cos cot x sin cot
These give, by differentiation,
d dx dy . ,
-^ = -r- cos cot + ~ sm cot co (x sin cot y cos cot)
dt dt ' dt
iscot + -^ sin cot + 0)77. I
(2),
. ., T dij dy dx .
Similarly, -TL -T: cos cot -7- sin cot col;,
at dt at
which determine the velocities relative to the revolving axes.
Again,
d 2 d 2 x d*y . (dx . dy
d 2 y d*x . _ /dy . dx \
~-2a) ^sma)^4--j A cosw^ }-co z rj
\at at )
~^7o
ar dr at-
or
(3'),
*# ey a 2 I
the relative accelerations.
KINEMATICS. 33
Now the component accelerations along fixed axes, with
which at the time t the moving axes coincide, are evidently
represented by the first two terms of the right-hand sides of
these equations ; or, in terms of the co-ordinates with respect
to the moving axes, by
*|_ 2 *?-, and *? + 2tt $-, (4).
dt 2 dt dt 2 dt
Ex. If the point be at rest, # and y are constant, and
df dr,
These expressions are obvious, as in this case the relative
motion of the point with respect to the moving axes is a
uniform circular motion about the origin, in the negative
direction, i.e. from the axis of 77 to that of f.
42. Suppose the new axes not to revolve uniformly.
In this case the investigation is precisely the same as the
above, with the exception that 6, a given function of t, must
be substituted for at If &>, now no longer constant, be put
for -=- , the student will have no difficulty in verifying the
following expressions, which take the place of (2), (3') and (4)
of the preceding section.
dg dx * dy . ~
IT = -77 cos V + -?- sm 6 + a
drj dy * dx . ~
~ji = jl cos - -ji sm -
dt dt dt
da
**
'" "
T. D.
34 KINEMATICS.
These expressions might have been deduced at once from
the expressions in 16, by the consideration of relative
accelerations as in 26. Let OM = , MP = 77, be the
co-ordinates of the point referred to the moving axes. Then,
by 16, the acceleration of M along OM is
Also, as MP revolves with angular velocity a), the ac-
celeration of P relative to M, in the direction perpendicular
to MP, is
This is in the direction of the negative part of the axis of
f. Hence the resolved part parallel to Of, of the accelera-
tion of P with respect to 0, is
d* 1 d
43. The principles already enunciated, and the examples
given of their application, will suffice for the solution of
problems on this part of the subject.
Other examples of the application of these principles,
such as the kinematical part of the investigations of the
Hodograph, &c., will be more appropriately introduced in
future chapters.
EXAMPLES.
(1) A point moves from rest in a given path, and its
velocity at any instant is proportional to the time elapsed
since its motion commenced; find the space described in a
given time.
- (2) If a point begin to move with velocity v, and at
equal intervals of time a velocity u be communicated to it
in the same direction; find the space described in n such
intervals.
KINEMATICS. 35
(3) A man six feet high walks in a straight line at the
rate of four miles an hour away from a street lamp, the height
of which is 10 feet; supposing the man to start from the
lamp-post, find the rate at which the end of his shadow
travels, and also the rate at which the end of his shadow
separates from himself.
(4) If the position of a point moving in a plane be
determined by the co-ordinates p and </>, p being measured
from a fixed circle (radius a) along a tangent which has
revolved through an angle <f) from a fixed tangent ; investi-
gate the following expressions for the accelerations along and
perpendicular to p respectively,
/ (5) Prove that it is not possible for a point to move so
that its velocity in any position may be proportional to the
length of the path which it has described from rest : also that
if its velocity be proportional to the space it has to describe,
however small, it will never accomplish it.
(6) The velocity of a point parallel to each of three
rectangular axes is proportional to the product of the other
two co-ordinates; what are the equations of the path, and
what is the time of describing a given portion when the
curve passes through the origin ?
(7) A point moves in a plane, so that its velocities
parallel to the axes of x and y are
u -f ey and v + ex respectively,
shew that it moves in a conic section.
(8) Two points are moving with constant velocity in two
straight lines, 1st in a plane, 2nd in space; given the initial
circumstances, find when they are nearest to each other.
Shew also that in both cases the relative path is a straight
line, described with constant velocity.
32
36 KINEMATICS.
(9) A number of points are moving, each with constant
velocity, in straight lines in space ; determine the motion of
their common centre of inertia. ( 58.)
(10) A cannon-ball is moving in a direction making an
acute angle 6 with a line drawn from the ball to an observer ;
if F be the velocity of sound, and nV that of the ball, prove
that the whizzing of the ball at different points of its course
will be heard in the order in which it is produced, or in the
reverse order, according as n < > sec 6.
(11) A particle, projected with a velocity u, is acted on
by a force, which produces a constant acceleration f, in the
plane of motion, inclined at a constant angle a to the
direction of motion. Obtain the intrinsic equation of the
curve described, and shew that the particle will be moving in
the opposite direction to that of projection at the time
u f
' J air COt a
/cos a V
(12) Shew that any infinitely small motion given to a
plane figure in its own plane is equivalent to a rotation
through an infinitely small angle about some point in the
figure.
Hence shew that the relative motion of two figures in a
plane may be produced by rolling a curve fixed to one figure
on a curve fixed to the other figure. (These curves are
called Centroids.)
(13) The highest point of the wheel of a carriage rolling
on a road moves twice as fast as each of two points in the
rim whose distance from the ground is half the radius of the
wheel.
(14) A rod of given length moves with its ends in two
given lines which intersect ; shew how to draw a tangent to
the path described by any point of the rod.
(15) Investigate the position of the instantaneous centre
about which the rod is turning, and apply this also to solve
the preceding question.
(16) One circle rolls on another whose centre is fixed.
From the initial and final positions of a diameter in each
KINEMATICS. 37
determine what portions of their circumferences have been
in contact.
(17) One point describes the diameter AB of a circle
with constant velocity, and another the semi-circumference
AB from rest with constant tangential acceleration; they
start together from A and arrive together at B ; shew that
the velocities at B are as TT : 1.
(18) In the example of 33 find in the case of e > 1 the
length of time occupied in the pursuit.
(19) In the example of 35 find the greatest distance
the boat is carried down the stream, and shew that when
it is in that position its velocity is \/(u 2 v 2 ).
When u = v, shew directly that the curve described is a
parabola.
(20) Shew that if p be the radius of curvature of the
curve of pursuit, we have in the figure of 33,
(21) In the case of a boat propelled with velocity u
relatively to the water in a stream running with velocity v,
shew that the boat passes from one given point to another in
the least possible time when its actual path is a straight
line.
(22) The velocity of a stream varies as the distance from
the nearest bank ; shew that a man attempting to swim
directly across will describe two semiparabolas. (Shew that
the sub-normal is constant.) Find by how much the mean
velocity is increased.
(23) A point moves with constant velocity in a circle ;
find an expression for its angular velocity about any point
in the plane of the circle.
(24) If the velocity of a point moving in a plane curve
vary as the radius of curvature, shew that the direction of
motion revolves with constant angular velocity.
(25) Two bevelled wheels roll together; having given
the inclinations of the axes of the cones, find their vertical
38 KINEMATICS.
angles that the wheels may revolve with angular velocities
in a given ratio.
(26) Supposing the Earth and Venus to describe in the
same plane circles about the Sun as centre ; investigate an
expression for the angular velocity of the Earth about Venus
in any position, the actual velocities being inversely as the
square roots of their distances from the Sun.
(27) A particle moving uniformly round the circular
base of an oblique cone is projected by generating lines on
a sub-contrary section ; find its angular velocity about the
centre of the latter.
(28) If f , 77 denote the co-ordinates of a moving point re-
ferred to two axes, one of which is fixed and the other rotates
with constant angular velocity co, prove that its component
accelerations parallel to these axes are
4 2&) cosec cot -37 ,
dP dt
fl - ^V + 2&) cot cot -^ .
at 2 dt
(29) Two lines are moving in their own plane about
their point of intersection with constant angular velocities
&), ft/ ; if the co-ordinates of a moving point referred to them
be x, y at a time t, prove that its accelerations parallel to the
axes are
-y- tfx - 2ft> cot (ft)' ) -y7 ~ 2ft)' cosec (w w} t -~ ,
- <>yy - 2ft) cosec (a/ &>) t -r. 2o>' cot (&)' w)t.
-r.
(30) Employ the formulae of 30 to trace approximately
the form of the path of C about A, when ra is nearly, but not
exactly, equal to + n or to n.
(31) If an odd number n of rods OA l} A^A^, A 2 A 3) ...
whose lengths are a, ^ , ,...-, are hinged together at
2 o n
KINEMATICS. 39
A lt A z ,... and revolve with constant angular accelerations
a, 2a, 3a, ...no., about their extremities O l A l A z ,.,.A n _i, shew
that the direction of motion of the point A n at any time is
perpendicular to the direction of the middle rod ; the motion
commencing from rest with the rods in a straight line.
(32) A man is in a boat, on a river, at a distance a
from the shore, and 6 from a fall of water ahead. If the velo-
city of the stream be F, prove that he cannot escape the fall
unless he can row with a velocity -= U^ F; and that in case
he can just row at this pace, the direction in which he must
row is at right angles to the line joining his position with the
point of the bank opposite the fall. Find also the direction
in which he will have the least distance to row to reach the
bank, supposing his velocity greater than this minimum.
(33) If a point is moving in a hypocycloid with velo-
city u ; and v, V represent the velocities of the centre of
curvature and the centre of the generating circle corre-
sponding to the position of the point, prove that
u- v* 4F 2
(c-6) 2 + (c-f 6) 2 m (c-6) 2 '
c being the distance between the centres of the generating
circles, and 6 the radius of the moving circle.
(34) N particles are arranged equably along the circum-
ference of a circle of radius a ; each continually moves towards
the next in order with a constant velocity v ; shew that they
will all arrive together at the centre of the circle in the time
a TT
- cosec -^ .
v N
(35) A point P moves with constant velocity in a circle ;
Q is a point in the same radius at double the distance from the
centre, PR is a tangent at P equal to the arc described by P
from the beginning of the motion : shew that the acceleration
of the point R is represented in direction and magnitude
by RQ.
(36) If a point move in an orbit so that the area de-
scribed in any time by the radius of curvature is proportional
40 KINEMATICS.
to that time, prove that the direction of the acceleration of the
point is perpendicular to the line joining the point to the
corresponding centre of curvature of the evolute, and its
magnitude (F) is given by the equation
where u is the index of curvature at the point, and c is twice
the area described in a unit of time.
(37) A body P is describing an ellipse in any manner:
Q is a fixed point on the major-axis and PG the normal at
P. Shew that at the moment when G coincides with Q, the
angular velocity of P about Q is to its angular velocity about
G as CD 2 to CB 2 .
(38) A plane is moving about an axis perpendicular to it,
and a point is moving in a given curve traced on the plane ;
in any position ro is the angular velocity of the plane, v the
velocity of the particle relative to the plane, r its distance
from the axis, p the perpendicular on the tangent, s the arc
described along the plane ; prove that the acceleration along
the tangent to the curve is
fdv du>\ , dr
v -j- + P-J- ) - tfr-j-.
\ds ^ dsj ds
(39) A particle moves on a surface : v, v' are the com-
ponents of its velocity along the lines of curvature, p, p' the
principal radii of curvature; prove that the acceleration along
v- v 2
the normal to the surface = I , .
P P
(40) The intrinsic equation of a curve being s =/(<),
the curve is described by a point with accelerations XY
parallel to the tangent and normal at the point for which
(f> == ; prove that
cos <> - 3X - sin <> + 3F
d(f>
KINEMATICS. 41
(41) Obtain expressions for the accelerations of a moving
point whose co-ordinates are r, 0, <f>, (1) in the direction of r,
(2) in the direction perpendicular to the radius vector and in
the plane of 0, (3) in the direction perpendicular to the plane
of 0.
A point describes a rhumb line on a sphere in such a way
that its longitude increases uniformly; prove that the re-
sultant acceleration varies as the cosine of the latitude, and
that its direction makes with the normal an angle equal
to the latitude.
(42) A rigid plane sheet is deprived by guide-pieces of
all freedom of motion save parallel to a fixed line in its plane.
If it be set in motion by the end of a crank, describing a
given path in a given manner and working in a slot of given
form cut in the sheet, form the equation of rectilinear motion
of the sheet.
(43) Investigate completely the cases of Example (42)
when
(a) the slot is straight,
(b) the slot is a circular arc,
the motion of the crank being circular and uniform.
(42)
CHAPTER II.
LAWS OF MOTION.
44. HAVING, in the preceding chapter, very briefly
considered the purely geometrical properties of the motion of
a point, we must now treat of the causes which produce
various circumstanc.es of motion of a Particle ; and of the
experimental laws, on the assumed truth of which all our
succeeding investigations are founded. And it is obvious
that we now introduce for the first time the ideas of Matter,
and of Force.
We commence with a few definitions and explanations,
necessary to the full enunciation of Newton's Laws and their
consequences.
45. The Quantity of Matter in a body, or the Mass of
a body, is proportional to the Volume and the Density
conjointly. The Density may therefore be defined as the
quantity of matter in unit volume.
If M be the mass, p the density, and Fthe volume, of a
homogeneous body, we have at once
M=V P ;
if we so take our units that unit of mass is the mass of unit
volume of a body of unit density.
As will be presently explained, the most convenient unit
mass is an Imperial Pound of matter.
46. A Particle of matter is supposed to be so small that,
though retaining its material properties, it may be treated, so
LAWS OF MOTION. 43
far as its co-ordinates, &c. are concerned, as a geometrical
point.
47. The Quantity of Motion, or the Momentum, of a
moving body is proportional to its mass and velocity con-
jointly.
Hence, if we take as unit of momentum the momentum
of a unit of mass moving with unit velocity, the momentum
of a mass M moving with velocity v is Mv.
48. Change of Quantity of Motion, or Change of Momen-
tum, is proportional to the mass moving and the change of
its velocity conjointly.
Change of velocity is to be understood in the general
sense of | 10. Thus, with the notation of that section, if a
velocity represented by OA be changed to another represented
by OC, the change of velocity is represented in magnitude
and direction by AC.
49. Rate of Change of Momentum, or Acceleration of
Momentum, is proportional to the mass moving and the
acceleration of its velocity conjointly. Thus ( 17) the
acceleration of momentum of a particle moving in a curve
is M j- along the tangent, and M- in the radius of absolute
curvature.
50. The Kinetic Energy of a moving body is proportional
to the mass and the square of the velocity, conjointly. If
we adopt the same units of mass and velocity as before, there
is particular advantage in defining kinetic energy as half the
product of the mass into the square of its velocity.
51. Rate of Change of Kinetic Energy (when defined as
above) is the product of the velocity into the component of
acceleration of momentum in the direction of motion.
d fMv-\ r dv
44 LAWS OF MOTION.
52. Matter has the innate property of resisting external
influences, so that every body, as far as it can, remains at rest,
or moves with constant velocity in a straight line.
This, the Inertia of matter, is proportional to the quan-
tity of matter in the body. And it follows that some cause
is requisite to disturb a body's uniformity of motion, or to
change its direction from the natural rectilinear path.
53. Impressed Force, or Force simply, is any cause which
tends to alter a body's natural state of rest, or of uniform
motion in a straight line.
The three elements specifying a force, or the three ele-
ments which must be known, before a clear notion of the force
under consideration can be formed, are, its place of application,
its direction, and its magnitude.
54. The Measure of a Force is the quantity of motion
which it produces in unit of time. According to this method
of measurement, the standard or unit force is that force
which, acting on the unit of matter during the unit of time,
generates the unit of velocity.
Hence the British absolute unit force is the force which,
acting on one pound of matter for one second, generates a
velocity of one foot per second.
to the common system followed till lately in
mathematical treatises on dynamics, the unit of mass is g
times the mass of the standard or unit weight ; g being the
numerical value of the acceleration produced (in some par-
ticular locality) by the earth's attraction on felling bodies.
This definition, giving a varying and a very unnatural unit
of mass, is exceedingly inconvenient. In reality, standards of
weight are masses, not forces. They are employed primarily
in commerce for the purpose of measuring out a definite
quantity of matter ; not an amount of matter which shall
be attracted by the earth with a given force.]
55. To render this standard intelligible, all that has to
be done is to find how many absolute units will produce", in
any particular locality, the same effect as gravity. The way
LAWS OF MOTION. 45
to do this is to measure the effect of gravity in producing
acceleration on a body unresisted in any way. The most
accurate method is indirect, by means of the pendulum.
The result of pendulum experiments made at Leith Fort, by
Captain Kater, is, that the velocity acquired by a body falling
unresisted for one second is at that place 32'207 feet per
second. The variation in gravity for one degree of difference
of latitude about the latitude of Leith is only '0000832
of its own amount. The average value for the whole of
Great Britain differs but little from 32*2; that is, the
attraction of gravity on a pound of matter in this country is
32'2 times the force which, acting on a pound for a second,
would generate a velocity of one foot per second ; in other
words, 32'2 is the number of absolute units which measures
the weight of a pound. Thus, speaking very roughly, the
British absolute unit of force is equal to the weight of about
half an ounce.
56. Forces (since they involve only direction and mag-
nitude) may be represented, as velocities are, by vectors,
that is, by straight lines drawn in their directions, and of
lengths proportional to their magnitudes, respectively.
Also the laws of composition and resolution of any number
of forces acting at the same point, are, as we shall presently
shew, 67, the same as those which we have already proved
to hold for velocities ; so that, with the substitution of force
for velocity, 10 is still true.
57. The Component of a force in any direction, sometimes
called the Effective Component in that direction, is therefore
found by multiplying the magnitude of the force by the cosine
of the angle between the directions of the force and the
component. The remaining component in this case is per-
pendicular to the other.
It is very generally convenient to resolve forces into
components parallel to three lines at right angles to each
other ; each such resolution being effected by multiplying by
the cosine of the angle concerned.
The magnitude of the resultant of two, or of three, forces
46 LAWS OF MOTION.
in directions at right angles to each other, is the square root
of the sum of their squares.
58. The Centre of Inertia or Mass of any system of
material points whatever (whether rigidly connected with
one another, or connected in any way, or quite detached),
is the point whose distance from any plane is equal to the
sum of the products of each mass into its distance from the
same plane divided by the sum of the masses.
The distance from the plane of yz, of the centre of inertia
of masses ra^ m^, etc., whose distances from the plane are
a?!, # 2 , etc., is therefore
_ _ m^j + m&s + etc. _ 2 (mx)
l + m z + etc.
And, similarly, for the other co-ordinates.
Hence its distance from the plane
8 = \x + py + vz a = 6,
is Z) = X^ + fjuy + vz a,
S \m (\x + py + vz - a)} 2 (mS)
as stated above. And its velocity perpendicular to that
plane is
s / dS\
<# = JLvf f dx dy d?\\ = n dt)
dt 2m \ V dt dt dt)} 2m
from which, by multiplying by S??t, and noting that 8 is the
distance of x, y y z from 8 = 0, we see that the sum of the
momenta of the parts of the system in any direction is
equal to the momentum in that direction of the whole mass
collected at the centre of mass.
59. By introducing, in the definition of moment of
LAWS OF MOTION. 47
velocity ( 21), the mass of the moving body as a factor, we
have an important element of dynamical science, the Moment
of Momentum. The laws of composition and resolution are
the same as those already explained.
60. A force is said to do Work if it moves the body to
which it is applied, and the work done is measured by the
resistance overcome, and the space through which it is over-
come, conjointly.
Thus, in lifting coals from a pit, the amount of work done
is proportional to the weight of the coals lifted ; that is, to
the force overcome in raising them ; and also to the height
through which they are raised. The unit for the measure-
ment of work, adopted in practice by British engineers, is
that required to overcome the weight of a pound through
the height of a foot, and is called a foot-pound.
In purely scientific measurements, the unit of work is not
the foot-pound, but the absolute unit force ( 54) acting
through unit of length.
If the weight be raised obliquely, as, for instance, along
a smooth inclined plane, the distance through which the force
has to be overcome is increased in the ratio of the length to
the height of the plane ; but the force to be overcome is not
the whole weight, but only the resolved part of the weight
parallel to the plane ; and this is less than the weight in the
ratio of the height of the plane to its length. By multiplying
these two expressions together, we find, as we might expect,
that the amount of work required is unchanged by the
substitution of the oblique for the vertical path.
61. Generally, if s be an arc of the path of a particle, S
the tangential component of the applied forces, the work
done on the particle between any two points of its path is
Sds,
taken between limits corresponding to the initial and final
positions.
48 LAWS OF MOTION.
Referred to rectangular co-ordinates, it is easy to see, by
the law of resolution of forces, 67, that this becomes
as as ds
Thus it appears that, for any force, the work done during
an indefinitely small displacement of the point of application
is the product of the resolved part of the force in the direction
of the displacement into the displacement.
From this it follows that, if the motion of a body be
always perpendicular to the direction in which a force acts,
such a force does no work. Thus the mutual normal pressure
between a fixed and a moving body, the tension of the cord
to which a pendulum bob is attached, the attraction of the
sun on a planet if the planet describe a circle with the sun
in the centre, are all cases in which no work is done by the
force.
In fact the geometrical condition that the resultant of
X, Y, Z shall be perpendicular to ds is
and this makes the above expression for the work vanish.
62. Work done on a body by a force is always shewn
by a corresponding increase of kinetic energy, if no other
forces act on the body which can do work or have work
done against them. If work be done against any forces,
the increase of kinetic energy is less than in the former case
by the amount of work so done. In virtue of this, however,
the body possesses an equivalent in the form of Potential
Energy, if its physical conditions are such that these forces
will act equally, and in the same directions, when the motion
of the system is reversed. Thus there may be no change of
kinetic energy produced, and the work done may be wholly
stored up as potential energy.
Thus a weight requires work to raise it to a height, a
spring requires work to bend it, air requires work to com-
LAWS OF MOTION. 49
press it, etc. ; but a raised weight, a bent spring, compressed
air, etc., are stores of energy which can be made use of at
pleasure.
These definitions being premised, we give Newton's Laws
of Motion.
63. LAW I. Every body continues in its state of rest or of
uniform motion in a straight line, except in so far as it is
compelled by forces to change that state.
We may logically convert the assertion of the first law
of motion as to velocity into the following statements :
The times during which any particular body, not com-
pelled by force to alter the speed of its motion, passes through
equal distances, are equal And, again Every other body
in the universe, not compelled by force to alter the speed of
its motion, moves over equal distances in successive intervals,
during which the particular chosen body moves over equal
distances.
64. The first part merely expresses the convention
universally adopted for the measurement of Time. The
earth, in its rotation about its axis, presents us with a case
of motion in which the condition of not being compelled by
force to alter its speed, is more nearly fulfilled than in any
other which we can easily or accurately observe. Hence the
numerical measurement of time practically rests on defining
equal intervals of time, as times during which the earth turns
through equal angles. This is, of course, a mere convention,
and not a law of nature ; and, as we now see it, is a part of
Newton's first law.
The remainder of the law is not a convention, but a great
truth of nature, which we may illustrate by referring to small
and trivial cases as well as to the grandest phenomena we
can conceive.
65. LAW II. Change of motion is proportional to the
force, and takes place in the direction of the straight line in
which the force acts.
We have considered change of velocity, or acceleration, as
T. D. 4
50 LAWS OF MOTION.
a purely geometrical quantity, and have seen how it may be
at once inferred from the given initial and final velocities of
a body. By the definition of motion, or quantity of motion
( 47), we see that, if we multiply the change of velocity,
thus geometrically determined, by the mass of the body, we
have the change of motion ( 48) referred to in Newton's
law as the measure of the force which produces it.
It is to be particularly noticed, that in this statement there
is nothing said about the actual motion of the body before it
was acted on by the force : it is only the change of motion
that concerns us. Thus the same force will produce precisely
the same change of motion in a body, whether the body be
at rest, or in motion with any velocity whatever.
66. Again, it is to be noticed that nothing is said as to
the body being under the action of one force only ; so that we
may logically put part of the second law in the following
(apparently) amplified form :
When any forces whatever act on a body, then, whether
the body be originally at rest or moving with any velocity
and in any direction, each force produces in the body the
exact change of motion which it would have produced if it,
had acted singly on the body originally at rest.
67. A remarkable consequence follows immediately from
this view of the second law. Since forces are measured by
the changes of motion they produce, and their directions
assigned by the directions in which these changes are pro-
duced ; and since the changes of motion of one and the same
body are in the directions of, and proportional to, the changes
of velocity a single force, measured by the resultant change
of velocity, and in its direction, will be the equivalent of any
number of simultaneously acting forces. Hence
The resultant of any number of forces (applied at one
point) is to be found by the same geometrical process as the
resultant of any number of simultaneous velocities.
From this follows at once ( 10) the construction of
the Parallelogram of Forces for finding the resultant of two
forces acting at the same point, and the Polygon of Forces for
LAWS OF MOTION. 51
the resultant ofjany number of forces acting at a point. And,
so far as a single particle is concerned, we have at once the
whole subject of Statics.
68. The second law gives us the means of measuring
force, and also of measuring the mass of a body.
For, if we consider the actions of various forces upon
the same body for equal times, we evidently have changes
of velocity produced, which are proportional to the forces.
The changes of velocity, then, give us in this case the means
of comparing the magnitudes of different forces. Thus the
velocities acquired in one second by the same mass (falling
freely) at different parts of the earth's surface, give us the
relative amounts of the earth's attraction at these places.
Again, if equal forces be exerted on different bodies, the
changes of velocity produced in equal times must be inversely
as the masses of the various bodies. This is approximately
the case, for instance, with trains of various lengths drawn by
the same locomotive.
Again, if we find a case in which different bodies, each
acted on by a force, acquire in the same time the same
changes of velocity, the forces must be proportional to the
masses of the bodies. This, when the resistance of the air
is removed, is the case of falling bodies ; and from it we
conclude that the weight of a body in any given locality,
or the force with which the earth attracts it, is proportional to
its mass. The student must be careful to observe that this
is no mere truism, but is an important part of the grand Law
of Gravitation. Gravity is not, like magnetism for instance,
a force depending on the quality as well as the quantity of
matter in a particle.
69. It appears, lastly, from this law, that every theorem
of Kinematics connected with acceleration has its counter-
part in Kinetics. Thus, for instance ( 18), we see that
the force, under which a particle describes any curve, may
be resolved into two components, one in the tangent to the
curve, the other towards the centre of curvature ; their
magnitudes being the acceleration of momentum, and the
product of the momentum into the angular velocity about
52 LAWS OF MOTION.
the centre of curvature, respectively. In the case of uni-
form motion, the first of these vanishes, or, the whole force
is perpendicular to the direction of motion. When there is
no force perpendicular to the direction of motion, there is no
curvature, or the path is a straight line.
Hence, if we resolve the forces, acting on a particle of
mass m whose co-ordinates are x, y, z, into the three rect-
angular components X, Y, Z] we have
In many of the future chapters these equations will be
somewhat simplified by assuming unity as the mass of the
moving particle. When this cannot be done, it is sometimes
convenient to assume X, Y, Z as the component forces on
unit mass, and the previous equations become
d*x v
m dj?^ ' '
from which m may of course be omitted.
[Some confusion is often introduced by the division of
forces into "accelerating" and "moving" forces; and it is
even stated occasionally that the former are of one, and the
latter of four linear dimensions. The fact, however, is that
an equation such as
may be interpreted either as dynamical, or as merely kine-
matical. If kinematical, the meanings of the terms are
obvious ; if dynamical, the unit of mass must be understood
as a factor on the left-hand side, and in that case X is the
^-component, per unit of mass, of the whole force exerted on
the moving body.]
If there be no acceleration, we have of course equilibrium
among the forces. Hence the equations of motion of a particle
are changed into those of equilibrium by putting
LAWS OF MOTION. 53
70. We have, by means of the first two laws, arrived
at a definition and a measure of force ; and have found how
to compound, and therefore how to resolve, forces ; and also
how to investigate the conditions of equilibrium or motion
of a single particle subjected to given forces. But more
is required before we can completely understand the more
complex cases of motion, especially those in which we have
mutual actions between or amongst two or more bodies ; such
as, for instance, tensions or pressures or transference of energy
in any form. This is perfectly supplied by
71. LAW III. To every action there is always an equal and
contrary reaction: or, the mutual actions of any two bodies are
always equal and oppositely directed in the same straight line.
If one body presses or draws another, it is pressed or
drawn by this other with an equal force in the opposite
direction. If any one presses a stone with his finger, his
finger is pressed with an equal force in the opposite direction
by the stone. A horse, towing a boat on a canal, is dragged
backwards by a force equal to that which he impresses on the
towing-rope forwards. By whatever amount, and in what-
ever direction, one body has its motion changed by impact
upon another, this other body has its motion changed by the
same amount in the opposite direction ; for at each instant
during the impact they exerted on each other equal and
opposite pressures. When neither of the two bodies has
any rotation, whether before or after impact, the changes of
velocity which they experience are inversely as their masses.
When one body attracts another from a distance, this other
attracts it with an equal and opposite force.
72. We shall for the present take for granted, that the
mutual action between two particles may in every case be
imagined as composed of equal and opposite forces in the
straight line joining them, two such equal and opposite forces
constituting a " stress " between the particles. From this it
follows that the sum of the quantities of motion, parallel to
any fixed direction, of the particles of any system influencing
one another in any possible way, remains unchanged by their
mutual action ; also that the sum of the moments of momen-
tum of all the particles round any line in a fixed direction in
54 LAWS OF MOTION.
space, and passing through any point moving uniformly in
a straight line in any direction, remains constant. From
the first of these propositions we infer that the centre of
mass of any system of mutually influencing particles, if in
motion, continues moving uniformly in a straight line, unless
in so far as the direction or velocity of its motion is changed
by stresses between the particles and some other matter not
belonging to the system ; also that the centre of mass of
any system of particles moves just as all their matter, if con-
centrated in a point, would move under the influence of forces
equal and parallel to the forces really acting on its different
parts. From the second we infer that the axis of resultant
rotation through the centre of mass of any system of par-
ticles, or through any point either at rest or moving uniformly
in a straight line, remains unchanged in direction, and the
sum of moments of momenta round it remains constant if the
system experiences no force from without. [This principle is
sometimes called Conservation of Areas, a very misleading
designation.] These results will be deduced analytically in
Chap. XII.
73. What precedes is founded upon Newton's own
comments on the third law, and the actions and reactions
contemplated are the pairs of forces, of which each pair
constitutes a " stress." In the scholium appended, he makes
the following remarkable statement, introducing another
specification of actions and reactions subject to his third
law :
Si cestimetur agentis actio ex ejus vi et velocitate conjunc-
tim; et similiter resistentis reactio cestimetur conjunctim ex
ejus partium sinyularum velocitatibas et viribus resistendi ab
earum attritione, cohcesione, ponder e, et acceleration oriundis;
erunt actio et reactio, in omni instrumentorum usu, sibi invicem
semper cequales.
In a previous discussion Newton has shewn what is to
be understood by the velocity of a force or resistance ; i.e.,
that it is the velocity of the point of application of the force
resolved in the direction of the force. Bearing this in mind,
we may read the above statement as follows :
If the Activity of an agent be measured by its amount and
LAWS OF MOTION. 55
its velocity conjointly ; and if, similarly, the Counter activity
of the resistance be measured by the velocities of its several
parts and their several amounts conjointly, whether these arise
from friction, cohesion, weight, or acceleration ; Activity and
Counter activity y in all combinations of machines, will be
equal and opposite.
74. Newton here points out that resistances against
acceleration are to be reckoned as reactions equal and oppo-
site to the actions by which the acceleration is produced.
Thus, if we consider any one material point of a system, its
reaction against acceleration must be equal and opposite to the
resultant of the forces which that point experiences, whether
by the actions of other parts of the system upon it, or by the
influence of matter not belonging to the system. In other
words, it must be in equilibrium with these forces. Hence
Newton's view amounts to this, that all the forces of the
system, with the reactions against acceleration of the material
points composing it, form groups of equilibrating systems for
these points considered individually. Hence, by the prin-
ciple of superposition of forces in equilibrium, all the forces
acting on points of the system form, with the reactions
against acceleration, an equilibrating set of forces on the
whole system. This is the celebrated principle first explicitly
stated and very usefully applied by D'Alembert in 1742 and
still known by his name.
Newton in the sentence just quoted lays, in an admirably
distinct and compact manner, the foundations of the abstract
theory of Energy, which recent experimental discovery has
raised to the position of the grandest of known physical laws.
He points out, however, only its application to mechanics.
The actio agentis, as he defines it, which is evidently equiva-
lent to the product of the effective component of the force, into
the velocity of the point at which it acts, is simply, in modern
English phraseology, the rate at which the agent works, called
the Power of the agent. The subject for measurement here
is precisely the same as that for which Watt, a hundred years
later, introduced the practical unit of a " Horse-power" or the
rate at which an agent works when overcoming 33,000 times
the weight of a pound through the distance of a foot in a
minute; that is, producing 550 foot-pounds of work per
56 LAWS OF MOTION.
second. The unit, however, which is most generally conve-
nient is that which Newton's definition implies, namely, the
rate of doing work in which the unit of work or energy is
produced in the unit of time.
75. Looking at Newton's words in this light, we see by
51 that they may be logically converted into the following
form :
" Work done on any system of bodies (in Newton's state-
ment, the parts of any machine) has its equivalent in work
done against friction, molecular forces, or gravity, if there be
no acceleration ; but if there be acceleration, part of the work
is expended in overcoming the resistance to acceleration, and
the additional kinetic energy developed is equivalent to the
work so spent."
When part of the work is done against molecular forces,
as in bending a spring; or against gravity, as in raising a
weight ; the recoil of the spring, and the fall of the weight,
are capable, at any future time, of reproducing the work
originally expended ( 62). But in Newton's day, and long
afterwards, it was supposed that work was absolutely lost by
friction.
76. If a system of bodies, given either at rest or in
motion, be influenced by no forces from without, the sum of
the kinetic energies of all its parts is augmented in any time
by an amount equal to the whole work done in that time by
the mutual actions, which we may imagine as acting between
its points. When the lines in which these actions act remain
all unchanged in length, the forces do no work, and the sum
of the kinetic energies of the whole system remains constant.
If, on the other hand, one of these lines varies in length during
the motion, the mutual actions in it will do work, or will
consume work, according as the distance varies with or
against them.
77. Experiment has shewn that the mutual actions be-
tween the parts of any system of natural bodies always per-
form, or always consume, the same amount of work during
any motion whatever, by which the system can pass from one
particular configuration to another : so that each configuration
LAWS OF MOTION. 57
corresponds to a definite amount of kinetic energy. [For the
apparent violation of this by friction, impact, &c., see 78*.]
Hence no arrangement is possible, in which a gain of kinetic
energy can be obtained when the system is restored to its
initial configuration. In other words, " The Perpetual Motion
is impossible"
78. The potential energy ( 62) of such a system, in the
configuration which it has at any instant, is the amount of
work that its mutual forces perform during the passage of the
system from any one chosen configuration to the configura-
tion at the time referred to. It is generally convenient so to
fix the particular configuration, chosen for the zero of reckon-
ing of potential energy, that the potential energy in every
other configuration practically considered shall be positive.
To put this in an analytical form, we have merely to
notice that by what has just been said, the value of
-
as as as
is independent of the paths pursued from the initial to the
final positions, and therefore that
2 (Xdx + Ydy + Zdz)
is a. complete differential. If, in accordance with what has
just been said, this be called dV, V is the potential energy,
and
--
dx,"
Also, by the second law of motion, if m be the mass of
a particle of the system whose co-ordinates are #, y, z, we
have
-*., &c.=&c.
The integral is
V=H,
58 LAWS OF MOTION.
that is, the sum of the kinetic and potential energies is con-
stant. This is called the Conservation of Energy.
In abstract dynamics, with which alone this treatise
is concerned, there is loss of energy by friction, impact, &c.
This we simply leave as loss, to be afterwards accounted for
in physics.
78*. [The theory of energy cannot be completed until we
are able to examine the physical influences which accompany
loss of energy. We then see that in every case in which
energy is lost by resistance, heat is generated ; and we learn
from Joule's investigations that the quantity of heat so gene-
rated is a perfectly definite equivalent for the energy lost.
Also that in no natural action is there ever a development of
energy which cannot be accounted for by the disappearance
of an equal amount elsewhere by means of some known phy-
sical agency. Thus we conclude that, if any limited portion
of the material universe could be perfectly isolated, so as to
be prevented from either giving energy to, or taking energy
from, matter external to it, the sum of its potential and kinetic
energies would be the same at all times. But it is only when
the inscrutably minute motions among small parts, possibly
the ultimate molecules of matter, which constitute light, heat,
and magnetism ; and the intermolecular forces of chemical
affinity ; are taken into account, along with the palpable
motions and measurable forces of which we become cognizant
by direct observation, that we can recognise the universally
conservative character of all natural dynamic action, and per-
ceive the bearing of the principle of reversibility on the whole
class of natural actions involving resistance, which seem to
violate it. It is not consistent with the object of the present
work to enter into details regarding transformations of energy.
But it has been considered advisable to introduce the very
brief sketch given above, not only in order that the student
may be aware, from the beginning of his reading, what an
intimate Connection exists between Dynamics and the modern
theories of Heat, Light, Electricity, &c. ; but also that we may
be enabled to use such terms as ''potential energy" &c. in-
stead of the unnatural "Force-functions" &c. which disfigure
some of the modern analytical treatises on our subject.]
(59)
CHAPTER III.
RECTILINEAR MOTION.
79. THE simplest case of motion of a particle which we
have to consider is that in a straight line. This may be caused
by the applied force acting at every instant in the direction
of motion; or the particle may be supposed to be constrained
to move in a straight line by being enclosed in a straight
tube of indefinitely small bore. As already mentioned, 69,
we shall in every case suppose the mass of the particle to be
unity.
80. A particle moves in a straight line, under the action
of any forces, whose resultant is in that line ; to determine
the motion.
Let P be the position of the particle at any time t,fihe
resultant acceleration along OP, being a fixed point in the
line of motion.
p
Let OP = x, then the equation of motion is (by 69)
CM *7*
In this equation f may be given as a function of a?, of -7- ,
at
or of t, or of any two or all three combined ; but in any case
the first and second integrals of the equation (if they can be
dx
obtained) -will give -7- and x in terms of t\ that is, the position
at
and velocity of the particle at any instant will be known.
60 RECTILINEAR MOTION.
The only one of these cases which we will now consider
is that in which f is given as a function of a?; those in which
CM (JT CM IT
f is a function of -^- , or of -^- and x, being reserved for the
Chapter on Motion in a Resisting Medium : while those in
which f involves t explicitly possess little interest, as they
cannot be procured except by special adaptations ; and can
even then appear only in an incomplete statement of the
circumstances of the particular arrangement.
The simplest supposition we can make is that /is constant.
81. A particle, projected from a given point with a given
velocity, is acted on by a constant force in the line of its motion;
to determine the position and velocity of the particle at any time.
Let A be the initial position of the particle, P its position
at any time t, v its velocity at that time, and / the constant
acceleration of its velocity. Take any fixed point in the
line of motion as origin, and let OA = a, OP = x. The
equation of motion is
= / (1)
dP J '
Integrating once, we have
dx
C being a constant to be determined by the initial circum-
stances of the motion. Suppose the particle projected from
A in the positive direction with velocity V, then when t = 0,
v=V\ hence C = F, and
Integrating again,
RECTILINEAR MOTION. 61
But when t = 0, x a ; hence C' = a, and
/* 2 ..................... (3).
Equations (2) and (3) give the velocity and position of the
particle in terms of t\ and the velocity may be determined in
terms of x by eliminating t between them : but the same
result will be obtained more directly by multiplying (1) by
-j- and integrating. This gives the equation of -energy
V-
But when x = a, t> = F; hence C" ' = -^ fa, and
2S
1 2-ll
99
82. The most important case of the motion of a particle
under the action of a constant acceleration in its line of
motion is that of gravity. For the weights of bodies at a
given latitude may be considered constant at small distances
above the Earth's surface, and therefore if we denote the
acceleration due to the Earth's attraction by g, and consider
the particle to be projected verticallv downwards, equations
(2), (3), (4) of 81 become
v=V+gt
x being measured as before from a fixed point
in the line of motion. As a particular instance
suppose the particle to be dropped from rest at 0.
At that instant A coincides with 0, and a = 0.
F=0.
62 RECTILINEAR MOTION.
Hence v=gt .................... .......... (1),
* = i<7< 2 ........................... (2),
v*=g* .............................. (3).
The last of these equations may also be obtained from
d?x _dv _dv dx dv
y dt* dt dx dt ~ dx
by a single integration.
83. As another particular instance, suppose the particle
to be projected vertically upwards. Here it must be re-
membered that if we measure x upwards from the point of
projection, the acceleration tends to diminish x and the
equation of motion is
In other respects the solution is the same. Taking,
therefore, a = in equations (A) and changing the sign of g,
we obtain
v= V-gt ........................... (4),
-F*-** ......................... (5),
(6).
From equation (4) we see that the velocity continually
diminishes, and becomes zero when t = ; and from (6) that
the height corresponding to v = 0, or the greatest height to
which the particle will ascend, is -^-. After this the velocity
becomes negative, or the particle begins to descend, and
(5) shews that it will return to the point of projection when
2 V
t= , as x then becomes ; and the velocity with which
u
RECTILINEAR MOTION.
63
it returns to that point is, by (6), equal to the velocity of
projection.
84. A particle descends a smooth inclined plane under
the acceleration of gravity, the motion taking place in a vertical
plane perpendicular to the intersection of the inclined with any
horizontal plane ; to determine the motion.
Let P be the position of the particle at any time t on the
inclined plane OA, OP x its distance from a fixed point
iri the line of motion, and let a be the inclination of OA to
the horizontal line AB. The only impressed force on the
particle is its weight g which acts vertically downwards, and
this may be resolved into two, gsina along, and gcosa
perpendicular to OA. Besides these there is the unknown
force R, the pressure on the plane, which is perpendicular
to OA : but neither this nor the component g cos a can affect
the motion along the plane. The equation of motion is
therefore
d'ae
the solution of which, as gsina. is constant, is included in
that of the proposition of 82, and all the results for particles
moving vertically under the action of gravity will be made
to apply to it by writing g sin a for g. Thus, if the particle
64 RECTILINEAR MOTION.
start from rest at 0, we get from equations (1), (2), (3) of
82 by this means,
x ^g sin a . t 2
...(2),
1
v o sin a . x. . .
...(3)
85. Equation (3) proves an important property with
regard to the velocity acquired at any point of the descent.
For, draw PN parallel to AB, and let it meet the vertical line
through in N, then if v be the velocity at P, we have
2 v*=gsma. OP
=9. ON.
Comparing this with equation (3) of 82, we see that
the velocity at P is the same as that which a particle would
acquire by falling freely from rest through the vertical dis-
tance ON\ in other words the velocity at any point, of a
particle sliding down a smooth inclined plane, is that due to
the vertical height through which it has descended; a par-
ticular case of the conservation of energy.
86. Again from (2) we derive immediately the following
curious and useful result.
The times of descent down all chords drawn through the
highest or lowest point of a vertical circle are equal.
Let AB be the vertical diameter of the circle, AC any
chord through A\ join BC ; then if T be the time of descent
down A C, we have by equation (2) of 84,
But AC = AB cos BAG ; whence
RECTILINEAR MOTION.
65
which, being independent of the position of the chord, gives
the same time of descent for all.
It may similarly be shewn that the time of descent down
all chords through B is the same. In fact parallel chords,
drawn through A and B respectively, are of equal length.
To find the straight line of swiftest descent to a given curve
from any point in the same vertical plane, all that is required
is to draw a circle having the given point as the upper ex-
tremity of its vertical diameter, and the smallest which can
meet the curve. Hence if BC be the curve, A the point,
draw AD vertical ; and, with centre in AD, describe a circle
passing through A and touching BC. Let P be the point of
contact, then AP is the required line. For, if we take any
other point, p, in BC, Ap cuts the circle in some point q, and
time down Ap> time down Aq, i.e. > time down AP.
T. D. 5
66 RECTILINEAR MOTION.
If the given curve be not plane, or if it be required
to find the straight line of swiftest descent to a surface,
a sphere must be described passing through A, with centre
in AD, so as to touch the curve or surface; and the proof
is precisely as before.
87. In 84 we have supposed the inclined plane to be
smooth, but the motion will still be constantly accelerated
when the plane is rough. For since there is no motion,
and therefore no acceleration, perpendicular to OA (see fig.
84), we must have
Q = R-gcosa. ( 69).
If p then be the coefficient of kinetic friction, which is
known by experiment to be independent of the velocity of
the particle, the retardation due to friction will be pit or
fig cos a, and the equation of motion will become
the second member still being constant, and the solution there-
fore similar to those we have already considered.
88. When a particle moves under an attraction in its line
of motion, varying directly as the distance of the particle from
a fixed point in that line, to determine the motion.
Let be the fixed point, P the position of the particle at
any time t, v its velocity at that time, and let OP = x. Then
if IJL be the acceleration of a particle due to the attraction at
a unit of distance from 0, which is supposed known, and is
called the strength of the attraction, the acceleration at P
will be fix, and if it be directed towards will tend to
diminish x. Therefore the equation of motion is
RECTILINEAR MOTION. 67
or
dx
Multiplying this equation by -- , and integrating, we
obtain
the equation of energy. This may be written
the negative sign being employed if we suppose the motion
to be towards 0, and a being the distance from the centre at
which the velocity is zero. Integrating again,
^/ fJL (t T) = COS" 1 ~ 5
LL
or x = a cos V/u, (t r) (3),
the complete integral of (1); involving two arbitrary constants
a and T, the values of which are to be determined from the
initial distance, and the velocity of projection. Thus from (3),
T) ............... (4).
89. Suppose the particle to be projected from A in the
positive direction with the velocity F, and let OA = b ; then
when = 0, we have x =b, v=V; and therefore from (3)
and (4)
6 = a cos V/rr ;
V = a ^fj, sin Vyu,r,
which determine a and T, and then (3) and (4) give the
position and velocity of the particle at any instant. The
velocity in terms of x is obtained directly from (2), for when
x = b, we have v= F; whence F 2 = /i,(a 2 6 2 ), and
52
68 RECTILINEAR MOTION.
90. Equations (3) and (4) give periodical values of x and
v, such that all the circumstances of motion are the same at
the time + -- as at the time t. They also shew that the
Vy^
velocity becomes zero when t = r, and that the correspond-
ing value of x is the greatest possible. Hence the par-
ticle will move in the positive direction to a distance a from
0, and then begin to return. Also, since when \V (t T) = TT,
we have v = again, and x = a, it will pass through 0,
move to an equal distance on the other side, and so on : the
time of a complete oscillation, that is, the time from its
leaving any point until it passes through it again in the
same direction, being -j- - This result is remarkable, as
it shews that the time of oscillation is independent of the
velocity and distance of projection, and depends solely on
the strength of the centre of attraction.
The above proposition includes the motion of a particle
within a homogeneous sphere of ordinary matter, in a straight
bore to the centre. For the attraction of such a sphere on a
particle within it is proportional to the distance from the
centre, and the equation of motion is therefore the same as
that which we have just considered.
91. Suppose itself to be in motion in the line OA, and
let f denote its position at time t. The equation of motion
and is integrable when is given in terms of t. This may
be at once changed into the equation of relative motion
d* dt^
which is the same as when the point is at rest if -= = 0,
CLv
i.e. if the velocity of be constant. If move with constant
acceleration, a, the oscillatory motion will be the same as
before, but the mean position will be - behind 0.
RECTILINEAR MOTION. 69
92. If we have a repulsion from the centre, the equation
of motion becomes
the integral of which is known to be
a; = Ac?*" + Be-**" ',
and the motion is not oscillatory. If, when t = 0, # = a,
v = a V/*, the particle constantly approaches the centre but
never reaches it.
93. It is to be remarked that we cannot always apply
the same equation of motion to the negative and positive sides
of the origin as we have done in the case of 88. Our being
able to do so arises from the fact that the expression, //,#, for
acceleration changes sign with x\ for by looking at the figure
it will be seen that when x is negative the acceleration tends
to increase x algebraically, and the equation ought properly
to be written
In general, when the acceleration is proportional to the n th
power of the distance, the equations of motion for the posi-
tive and negative sides of the origin are respectively
and dF""^"^
The only cases, therefore, in which the same equation of
motion will apply to both sides of the origin, occur when n
is of the form - , - , where ra, m' are any whole numbers
including zero, since it is only in these cases that we have
94. In other cases the investigation of the motion will
generally consist of two parts, one for each side of the origin ;
70 RECTILINEAR MOTION.
and in one case even when n is of the form -' - it is
2m + 1
necessary to consider these parts separately, because the form
of the integral is not sufficiently general to include both.
This is when ra = and m! = 1, for in that case the equation
of motion becomes
dt* = ~x'
dx
Multiplying this by -j- and integrating we have
at
1 dx
which becomes impossible when x is negative. But it is
evident that we may then write the integral
1 dx 2
which is, of course, the proper form for the negative side of
the origin. These equations cannot generally be integrated
farther, but we will shew towards the end of the Chapter
how the time of reaching the origin may be determined.
95. A particle, constrained to move in a straight line, is
acted on by an attraction always directed to a point outside
the line, and varying directly as the distance of the particle
from that point ; to determine the motion.
The constraint here contemplated may be conceived by
considering the particle either as an indefinitely small ring
sliding on a thin smooth wire, or as a material particle sliding
in a smooth tube of indefinitely small bore.
Let AB be the straight line, P the position of the particle
at any time, the point to which the attraction on P is
always directed. Draw ON perpendicular to AS, and let
NP x ; then if OP = r, and if /-i as formerly be the attraction
at a unit of distance, the attraction on P along PO is pr. This
may be resolved into two, one along and the other perpen-
dicular to AB, of which the latter has no effect on the motion
RECTILINEAR MOTION.
71
of the particle. The equation of motion is, therefore, since
the acceleration is /j,r cos OPN or p
the same as in 88. The motion of the particle will there-
fore be oscillatory about N, the time of a complete oscillation
being - , and all the circumstances of motion the same as
vV
for a free particle moving in AS under the action of an equal
centre placed at N.
96. A particle moves in a straight line under the action
of an attraction always directed to a point in the line and
varying inversely as the square of the distance from that point;
to determine the motion.
Let be the fixed point, P the position of the particle at
Jr P A
time t, OP x ; the equation of motion is
IJL being, as before, the acceleration at unit distance from 0,
or the strength of the centre.
dx
Multiplying by - and integrating, we get
72 RECTILINEAR MOTION.
1 dx\*
the equation of energy, supposing the particle to start from
rest at a point A distant a from 0,
1/cfo v* I 1
which gives the velocity of the particle at any distance x
from the origin. Again from (1)
=-VfyI
dt
the negative sign being taken, since in the motion towards 0,
x diminishes as t increases. This gives
dt _ /a x
dx V 2/A fj(ax x z )
ya ( 1 a 2x _ a 1
2^ * (2 *J(ax-x*) ~ 2 V(#-
Integrating, we have
TT / -^ . / / o\ a . 2a? . TT&
Hence A /-f^ = V(^-^ 2 )-^
which is the relation between x and t.
97. Putting 57=0, we find that the time of arriving at
Ois
7T /fl^
2V 2i*'
RECTILINEAR MOTION. 73
and (1) shews that the velocity at is infinite. On this
account we are precluded from applying our formulae to
determine the motion after arriving at ; but it is to be
observed that, although at any point very near to there
is a very great attraction tending towards 0, at the point
itself there is no attraction at all : and therefore the particle,
approaching the centre with an indefinitely great velocity,
must pass through it. Also, everything being the same at
equal distances on either side of the centre, we see that the
motion must be checked as rapidly as it was generated, and
therefore the particle will proceed to a distance on the other
side of 8 equal to that from which it started. The motion
will then continue oscillatory.
98. The above case of motion includes that of a body
falling from a great height above the Earth's surface. For a
sphere attracts an external particle with an intensity varying
inversely as the square of the distance of the particle from
its centre, and therefore if x be the distance of a body from
the Earth's centre, R the Earth's radius, and g the kinetic
measure of gravity on unit of mass at the Earth's surface,
the equation of motion will be
the same equation as before, if we write //, for gR 2 . The
results just obtained will therefore apply to this case. Thus
if we wish to find the velocity which a body would acquire
in falling to the Earth's surface from a height h above it, we
have from (1), putting p =
and therefore if V be the velocity when x = R, i.e. the
required velocity,
If A, be small compared with R, this may be written
74
RECTILINEAR MOTION.
from which we see the amount of error introduced by the
ordinary formula, 82, in which the acceleration due to
gravity was treated as constant,
\V-gh.
If the fall be from an infinite distance, a = oo , and we
have
Expressed in terms of the radius and the mean density of
the Earth, this becomes
which is the kinetic energy acquired by unit of mass falling
from rest in infinite space to the Earth's surface.
99. A particle is constrained to move in a straight line,
and is acted on by an attraction, always directed to a point
outside that line, and varying inversely as the square of the
distance from that point ; to determine the motion.
Let AB be the straight Hue, P the position of the particle
at any time, the point to which the attraction is always
N
directed, p, the strength of the centre. Draw ON perpendicular
to AB and let ON=b, NP = x; then the attraction on P
along PO is - , and, as in 95, the only part of this
RECTILINEAR MOTION. 75
which produces motion is the resolved part along PN.
Therefore the equation of motion is
m
-j-
Multiplying by -j- and integrating, we have
where a is the distance from N to the point where the
velocity is zero.
100. This equation cannot generally be integrated farther,
but in this and every similar case the integration can be per-
formed if we suppose x always very small. Suppose the
particle to have been at rest at N, and to have been slightly
displaced from this position of equilibrium, the displacement
being so small that throughout the motion j- may be neglected
Of*
in comparison with y. We have from (1),
,
nearly;
LLX
the same form of equation of motion as that of 88. The
motion will therefore be oscillatory, the time of each small
yb 3
.
76 RECTILINEAR MOTION.
101. A particle moves in a straight line under the action
of attraction varying inversely as the n th power of the distance
of the particle from a fixed point in that line ; to determine
the motion.
Measuring x as before, the equation of motion will be
dx
Multiplying by -j- and integrating, we have
Cut
supposing the particle to start from rest at a distance a from
the fixed point.
102. This equation cannot generally be integrated farther,
but if we suppose the particle to have started from a point
at an infinite distance, we have a = oo , and
~ n 1 ^- 1 '
where v is the velocity from infinity, at the distance x.
We have therefore in this particular case
dx _ l 2/4 \i 1
di ~ (n^l) ~lFi
x
-i
/n-l\* ""-
= -o - ^
V 2yLC /
dt
Integrating this between the limits = a, x = yS, we have
for the time of moving from x = a to x ft,
RECTILINEAR MOTION. 77
103. To find r, the time of oscillation, when the ampli-
tude of oscillation is 2a,
dt In -I I a n - l x n - 1
dx~ V 2a V a n - l -x n - 1 '
-l r a I a n - l x n ~ l
Put
n
2 r 1 _JL
rc-lW Q 2
and T =_^== *-i'(l-
n+1
*?_! te Wj.l 1\
~V2u(n-l) U-l + 2' 2^
.
n+1
n-l
r
2 ^ r(-
\W 1
104. The above solution fails when w = l, but the time
of falling to the centre may be found as follows. The equa-
tion for this case, as given in 94, is
1 AM 2 a
(i *y
since when x = a, -=- = 0. Hence
at
78 RECTILINEAR MOTION.
the negative sign being taken since x diminishes as t in-
creases. Put T for the required time, then
To transform the integral, put A /log - = ;?/. Then we
have
dx
and T - = -
and the limits of y are and oo . Hence
o
= 2a.
Hence ^
and is therefore directly as the distance traversed.
105. A particle is constrained to move in a straight line,
and is acted on by an attraction directed to a point not in
that line, and expressed by a function (f>(r) of the distance; to
determine the time of a small oscillation.
Employing the same notation as in 99, the acceleration
05
along PO being < (r), its component along PN is (/> (r) - ,
therefore the equation of motion is
d z x . x
But r
= b approximately.
RECTILINEAR MOTION. 79
&X 0(6)
Hence -=- + ~ -'# = (),
d& b
and therefore by 90, the time of a small oscillation is
EXAMPLES.
(1) A body is projected vertically upwards with a velocity
which will carry it to a height 2g ; shew that after three
seconds it will be descending with a velocity g.
(2) Find the position of a point on the circumference of
a vertical circle, in order that the time of rectilinear descent
from it to the centre may be the same as the time of descent
to the lowest point.
(3) The straight line down which a particle will slide in
the shortest time from a given point to a given circle in the
same vertical plane, is the line joining the point to the upper
or lower extremity of the vertical diameter, according as the
point is within or without the circle.
(4) Find the locus of all points from which the time of
rectilinear descent to each of two given points is the same.
Shew also that in the particular case in which the given
points are in the same vertical, the locus is formed by the
revolution of a rectangular hyperbola.
(5) Find the line of quickest descent from the focus to
a parabola whose axis is vertical and vertex upwards, and
shew that its length is equal to that of the latus rectum.
(6) Find the straight line of quickest descent from the
focus of a parabola to the curve when the axis is horizontal.
(7) The locus of all points in the same vertical plane as
a given circle, for which the least time of sliding down an
inclined plane to the circle is constant is another circle.
80 RECTILINEAR MOTION.
(8) Two bodies fall in the same time from two given
points in space in the same vertical down two straight lines
drawn to any point of a surface ; shew that the surface is an
equilateral hyperboloid of revolution, having the given points
as vertices.
(9) Find the form of a curve in a vertical plane, such
that if heavy particles be simultaneously let fall from each
point of it so as to slide freely along the normal at that point,
they may all reach a given horizontal straight line at the
same instant.
(10) A semicycloid is placed with its axis vertical and
vertex downwards, and from different points in it a number of
particles are let fall at the same instant, each moving down
the tangent at the point from which it sets out ; prove that
they will reach the involute (which passes through the vertex)
all at the same instant.
(11) A particle moves in a straight line under the action
/3\ th
of an attraction varying inversely as the ( ~ ) power of the
distance ; shew that the velocity acquired by falling from an
infinite distance to a distance a from the centre is equal to the
velocity which would be acquired in moving from rest at a
distance a to a distance T .
4
(12) A particle moves in a straight line from a distance a
towards a centre of attraction varying inversely as the cube
of the distance ; shew that the whole time of descent
(13) A particle is placed at a given point between two
centres of equal intensity attracting directly as the distance ;
to determine the motion and the time of an oscillation.
Let 2a be the distance between the centres, x the distance
of the particle at any time from the middle point between
them, then the equation of motion is
RECTILINEAR MOTION. 81
Hence, the time of an oscillation =
(14) If a particle begin to move directly towards a fixed
centre which repels with an intensity = //, (distance), and
with an initial velocity = $ (initial distance), prove that it
will continually approach the fixed centre, but never attain
to it.
(15) A particle acted upon by two centres of attraction,
each attracting with an intensity varying inversely as the
square of the distance, is projected from a given point be-
tween them, to find the velocity of projection that the particle
may just arrive at the neutral point of attraction and remain
at rest there.
If //,, p be the strength of the centres ; a 1} a 2 the distances
of the point of projection from them ; and F the initial velo-
city; we have
(16) Supposing the earth a homogeneous spheroid of
equilibrium, the time of descent of a body let fall from any
point P on the surface down a hole bored to the centre (7,
varies as (7P, and the velocity at the centre is constant.
(17) A material particle placed at a centre of attraction
varying as the distance, is urged from rest by a constant force
which acts for one-sixth of the time of a complete oscillation
about the centre, ceases for the same period, and then acts as
before, shew that the particle will then be retained at rest,
and that the distances moved through in the two periods are
equal.
(18) A body moves from rest at a distance a towards
a centre of attraction varying inversely as the distance, shew
that the time of describing the space between /3a and j3 n a
will be a maximum if y3 = - .
T. D.
82 RECTILINEAR MOTION.
(19) If the time of a body's descent in a straight line
towards a given centre of attraction vary inversely as the
square of the distance fallen through, determine the law of
the attraction.
(20) Assuming the velocity of a body falling to a centre
/
, where a is the initial and
x
x the variable distance from the centre, find the law of the
attraction.
(21) Find the time of falling to the centre when the
attraction x (dist.) ? .
(22) Shew that the time of descent, to a centre of
attraction x (dist.)~ 2 , through the first half of the initial
distance, is to that through the last half as TT + 2 : TT 2.
(23) A particle descends to a centre of attraction, inten-
sity x (dist.) n . Find n so that the velocity acquired from
infinity to distance a, shall be equal to that acquired from
distance a to distance ^a, from the centre.
(24) A particle is placed at the extremity of the axis of
a thin attracting cylinder of infinite length and of radius a,
shew that its velocity after describing a space x is propor-
tional to
/
log
(25) A particle falls to an infinite homogeneous solid
bounded by parallel plane faces, find the time of descent.
(26) Every point of a fine uniform ring repels with an
intensity x (dist.)~ 2 , find the time of a small oscillation in its
plane, about the centre.
(27) Shew that a body cannot move so that the velocity
shall vary as the distance from the beginning of the motion.
And if the velocity vary as the cube root of that distance,
determine the acceleration, and the time of describing a
given distance.
RECTILINEAR MOTION. 83
(28) Shew that the time of quickest descent down a
focal chord of a parabola whose axis is vertical is
/3*l
V7
where I is the latus rectum.
(29) An ellipse is suspended with its major axis vertical,
find the diameter down which a particle will fall in the least
time, and the limiting value of the excentricity that this may
not be the axis major itself.
(30) Particles slide down chords from a point to a
curved surface, under the attraction of a plane whose attrac-
tion is as the distance, and they reach the surface in the same
time ; shew that the surface is generated by the revolution
(about a line whose length is a through perpendicular to
the plane) of the curve whose polar equation about is
p cos 6 = a (1 cos (k cos 8)}.
(31) If the particles commence their motion at the
surface, and reach after a given time, the equation of the
generating curve is
p cos 6 = a (sec (k cos 0) 1).
(32) Prove that the times of falling through a given dis-
tance AC towards a centre S, under the action of two attrac-
tions, one of which varies as the distance, and the other is
constant and equal to the original value of the first, are as the
arc (whose versed sine is AC) to the chord, in a circle whose
radius is AS.
(33) The earth being supposed a thin uniform spherical
shell, in the surface of which a circular aperture of given radius
is made, if a particle be dropped from the centre of the aper-
ture, determine its velocity at any point of the descent.
(34) If a particle fall down a radius of a circle under
the action of an attraction oc (D) 3 in the centre, and ascend
the opposite radius under the action of a repulsion of equal
intensity at equal distances from the centre, shew that it will
acquire a velocity which is a geometric mean between the
radius and the intensity at the circumference.
62
84 RECTILINEAR MOTION.
(35) If a particle fall to a centre of attraction of inten-
sity oc (D) ; determine the constant attraction which would
produce the effect in the same time, and compare the final
velocities.
(36) Find the equation of the curve down each of whose
tangents a particle will slide to the horizontal axis in a given
time.
(37) A sphere is composed of an infinite number of free
particles, equally distributed, which gravitate to each other
without interfering ; supposing the particles to have no initial
velocity, prove that the mean density about a given particle
will vary inversely as the cube of its distance from the centre.
(38) Prove that if PQ be a chord of quickest descent
from one curve in a vertical plane to another, the tangents
at P and Q are parallel and PQ bisects the angles between
the normals and the vertical.
(39) A rough horizontal plane has the coefficients of fric-
tion at any point proportional to the distance from a fixed
point S to which an attraction tends whose intensity is
/j, (dist.)~ 3 , prove that if a particle be placed at a distance
a tan a from S it will arrive at S in time
log (sec 2),
a being the distance at which the particle must be placed so
as to be on the point of moving.
(40) If a particle P move from rest under the action of
an attraction tending to a point S measured by the accelera-
tion ri*SP, determine the time from rest to rest ; and shew
that, if a small constant retardation y act through a portion of
he path extending equally on each side of S the time will be
unaltered, and the diminution of the amplitude of one oscilla-
2/*
tion will be -~ cos nr, r being the time when the disturbance
begins.
RECTILINEAR MOTION. 85
(41) A fine thread having two masses each equal to P
suspended at its extremities is hung over two smooth pegs in
the same horizontal line ; a mass Q is then attached to the
middle point of the portion of the string between the pegs,
and allowed to descend under gravity; shew that the velocity
of Q at any depth x below the horizontal line is
Q (of +<*)
(42) An elastic string has its ends fastened to the ends
of a rod of equal length. The middle point of the string is
fastened, and at that point is placed a centre of repulsion,
which repels every particle of the rod with an intensity T^r-
The rod is then moved parallel to itself through a distance
equal to half its length. If in this position the elasticity of
the string is such that the rod is in equilibrium, shew that if
slightly displaced perpendicular to its length, the time of a
small oscillation
47T
(43) A particle moves in a straight line under an
a?
attraction to a centre in the straight line px + 2// , and
starts from rest at a distance a from the centre ; shew that
after a time t the distance from the centre will be
a en [ Vft ,
where K Z
(86)
CHAPTER IV.
PARABOLIC MOTION.
106. IN this chapter we intend to treat principally of the
motion of a free particle which, is subject to the action of
forces whose resultant is parallel to a given fixed line.
The simplest case of course will be when that resultant is
constant. The problem then becomes the determination of
the motion of a projectile in vacuo and unresisted, since the
attraction of the earth may be considered within moderate
limits as constant and parallel to a fixed line. This we will
now consider.
107. A free particle moves under the action of a vertical
attraction whose intensity is constant ; to determine the form
of the path, and the circumstances of its description.
Taking the axis of a; horizontal and in the vertical plane
and sense of projection, and that of y vertically upwards, it
is evident that the particle will continue to move in the plane
of xy, as it is projected in it, and is subject to no force which
would tend to witnd^w it from that plane.
The equations of motion then are
d**_ dy_
dt z dt* ~ 9>
if g be the kinetic measure of the attraction per unit of mass.
Suppose that the point from which the particle is projected
is taken as origin, that the velocity of projection is K, and
that the direction of projection makes an angle a with the
axis of x.
The first and second integrals of the above equations will
then be
PARABOLIC MOTION. 87
-* ............ (1).
x V cos a . t, y = V sin'a . t ^gt 2 ...... (2).
These equations give the co-ordinates of the particle and
its velocity parallel to either axis for any assumed value of
the time.
Eliminating t between equations (2) we obtain the equa-
tion of the trajectory, viz.
(3),
which shews that the particle will move in a parabola whose
axis is vertical, and vertex upwards.
108. Equation (3) may be written
2 F 2 sin a cos a 2 F 2 cos 2 a
a? -- -# = -- -y,
9 9
( F 2 sin a cos aV 2 2F 2 cos 2 a / F 2 sin 2 a
or \x -- = -- [11 --
V g ) g V 2g
By comparing this with the equation of a parabola re-
ferred to its vertex as origin, we find for the co-ordinates # , y Q
of the vertex
F 2 sin a cos a F 2 sin 2 a
v- - - g - >^ = ^
Hence we obtain the equation of the directrix
F 2 sin 2 a F 2 cos 2 a ' F 2
2/ = yo + i (parameter) = %jj + ^~ = ^-
Now if v be the velocity of the particle at any point of
its path,
= ( F 2 cos 2 a) + ( F 2 sin 2 a - 2 Vg sin a.t + g 2 ?)
by (2).
88 PVRABOLIC MOTION.
To acquire this velocity in falling from rest, the particle
y 2 F 2
must have fallen ( 82) through a height , or y, i.e.
through the distance from the directrix.
109. To find the time of flight along a horizontal plane.
Put 2/ = in equation (3). The corresponding values of
2V 2
x are and sin a cos a. But the horizontal velocity is
9
Fcos'a. Hence the time of flight is - - ; and, ceteris
paribus, varies as the sine of the elevation (inclination to the
horizon) of the direction of projection.
110. To find the time of flight along an inclined plane
passing through the point of projection.
Let its intersection with the plane of projection make an
angle ft with the horizon ; it is evident that we have only to
eliminate y between (3) and y = x tan ft.
This gives for the abscissa of the point where the pro-
jectile meets the plane,
2F 2
x l = (sin a cos a tan ft cos 2 a)
&
_2F 2 cosasin(a-/9)
y cos ft
Hence time of flight
a?! _2Fsin(a-/3)
F cos a g cos ft
111. To find the direction of projection which gives the
greatest range on a given plane.
F 2
The range on the horizontal plane is sin 2. For a
c/
given value of F this will be greatest when
7T 7T
PARABOLIC MOTION.
89
That on the inclined plane is
2F 2
.77,
, or
g cos 2
cos a sin (a - j3).
That this may be a maximum for a given value of F we
must equate to zero its differential coefficient with respect to
a, which gives the equation
cos a. cos (a ft) sin a sin (a @) = 0,
or cos (2a ft) = ;
whence a =
Hence the" direction of projection required for the greatest
range makes with the vertical an angle
7T _ 1 /7T
2 " ~ 2 V2 ~
that is, it bisects the angle between the vertical and the
plane on which the range is measured.
112. To find the elevation necessary to the particles
passing through a given point.
Suppose the point in the axis of x and distant a from the
origin. Then we must have
F 2
sin 2a = a,
9
so that a must not be greater than .
9
Let a be the smallest positive angle whose sine is ^ .
The admissible values of a are -= and = ; so that we
see there are two directions in which a particle may be pro-
jected so as to reach the given point, and that these are
(7T\
a = - j which
gives the greatest range.
90 PARABOLIC MOTION.
Suppose the given point to lie in the plane which makes
an angle with the horizon. Then if its abscissa be a t we
must have
2F 2 . ,
-= cos a sin (a p) = a.
gcos/3
If a', a" be the two values of a which satisfy this equa-
tion, we must have
cos a' sin (of - 0) = cos a" sin (a" 0) ;
and therefore a" - ft - a',
'-
Hence, as before, the two directions of projection, which
enable the particle to strike a point in a given plane through
the point of projection, are equally inclined to the direction of
projection required for the greatest range along that plane.
113. To find the envelop of all the trajectories correspond-
ing to different values of a..
Differentiating equation (3) with respect to a, we get
qx sin a
sec 2 a-%, =- = 0,
V 2 cos 3 a
V 2
or tan a= (4).
gx
The elimination of a. between (3) and (4) gives us as the
equation of the required envelop
2F 2 / 7 2
or a?-
This represents a parabola, whose axis is vertical, whose
focus is the point of projection, and whose vertex is in the
common directrix of the trajectories.
PARABOLIC MOTION. 91
It will easily be seen from what has gone before that there
are two directions of projection, so that the particle may pass
through any given point within this parabola, only one for a
point on it ; and of course there is no possibility of its reaching
(with the given velocity V) any point without this parabola.
114. By a somewhat simpler method of considering the
problem we might easily have arrived at some of the more
obvious properties of the trajectory, thus
Take the direction of projection as the axis of cc, and the
vertical downwards from the point of projection as that of y.
By the second law of motion we may consider the velocity
due to projection to be maintained constant (= F) parallel to
the axis of x, while we have in addition parallel to the axis
of y the portion due to gravity as investigated in 82.
Hence x = Vt
at any time,
2F 2
and therefore a? = y,
J
the equation of a parabola referred to a diameter and the
tangent at its vertex. The distance of the origin from the
directrix, being th of the coefficient of y, is ^- , and the
velocity due to a fall through that height is as before
'.%F.
115. Many properties of parabolic motion are more
easily obtained by geometry than by analysis. We give a
few examples.
Thus suppose in the figure to be the point of projection,
MN the directrix common to the trajectories of all particles
projected from in the plane of the figure with a given velo-
city, and suppose it be required to determine the direction of
projection for the greatest range along the plane OS. Since
is a point in each trajectory and MN the common directrix,
92
PARABOLIC MOTION.
the foci of all possible trajectories lie in the circle MF' FF"
described with centre and touching MN in M.
Take any point F' in this circle, then the path whose
focus is F' will intersect OS again in a point P' such that if
P'N' be drawn perpendicular to MN, F'P' = P'N'. Now in
order that P' may be as far as possible from 0, at P suppose,
it is evident (ex absurdo) that the focus must be taken at the
point F where OS meets the circle. But the tangent at
bisects the angle between the diameter MO and the focal
distance OF. Hence the direction of projection for the
greatest range on an inclined plane bisects the angle between
the plane and the vertical.
Again, if with centre P' and radius P'F' an arc be de-
scribed cutting F'FF" in F", it is evident that the trajectories
whose foci are F', F", will intersect OS in the same point P'.
Hence, since the directions of projection for these cases will
bisect the angles MOF', MOF" respectively, we see that to
strike a given object there are in general two directions of
PARABOLIC MOTION.
93
projection, and that these are equally inclined to the direc-
tion which gives the greatest range on the plane passing
through the object and the point of projection.
Again, for the envelop of all the trajectories. It is evi-
dent that P must be a point in the envelop ; since it is the
ultimate position of P', when the two parabolas which inter-
sect in that point have become indefinitely nearly coincident.
Draw PN perpendicular to MN, and produce it till NQ = FO.
Draw QR parallel to NM, and cutting OM in R. RQ is a
fixed line since RM = MO, and as OP = PQ we see that the
envelop is a parabola whose focus is and directrix RQ.
It may be seen at once that it touches in P the only tra-
jectory which can pass through that point. For the tangent
to either curve at P bisects the angle OPQ or FPN.
116. Ex. It is required to throw a shell with given
velocity so as to strike at right angles an inclined plane through
the point of projection.
The letters being the same as before, join ST cutting
MF'F" in F". Draw F" P' N f perpendicular to MS cutting
OS in P'. Find F' so that P F' = P F" = P N f . P' is a
point in the trajectory whose focus is F'. Hence the tangent
94
PARABOLIC MOTION.
at P bisects F'PN'. But OP' bisects F'P'F". Hence the
trajectory at P is perpendicular to OS.
Also as F" is the focus of the other path by which the
point P' might be reached, P' will be the vertex of that path,
and therefore the particle will be moving horizontally when
it reaches P'.
117. Even if the plane along which the range is measured
do not pass through the point of projection, a somewhat similar
construction will enable us to find the direction of projection
for the maximum range. Thus,
Let it be required to find the direction of projection from
A P' P
M Q' N
with velocity due to AO in order that the range on a hori-
zontal line MN may be a maximum.
Suppose Q the point where the projectile falls. Join
Q'F', F'O, F f being the focus of the path. Then if Q'P be
vertical and meet the horizontal line through A in P', we
have F'Q = Q'P'. This is true of each of the paths, and
Q'P is constant. The farthest point Q which can be reached
will therefore be determined by inflecting OQ to MN, where
OQ= OA + PQ, and therefore if A0 = a, AM = b the cosine
of double the requisite angle of elevation will be f T J .
Should M N be an inclined plane, we must evidently draw
a line QO, and the corresponding vertical QP ; such that if
QO meet the circle in F, FQ = QP.
PARABOLIC MOTION.
95
This resolves itself into the well-known geometrical
problem of describing a circle whose centre is in a given line,
and which touches a given circle, and a given straight line.
Of the two solutions, which this problem admits of, one
belongs to MN, the other to MN produced to the other side
of the point of projection.
118. Perhaps, however, the most satisfactory method of
solving all such problems about the maximum range, is to
describe the parabola which envelops all the trajectories.
The point where this cuts the plane, &c. on which the range
is estimated, gives the maximum value of the range, and it is
then easy from known properties of the envelop to construct
for the required path.
119. Let P be any point in the trajectory, S its focus,
BN, AL, the directrix, and the tangent at the vertex.
96 PARABOLIC MOTION.
Then (velocity at P) 2 = 2g PN = 2g SP
= (by a property of the parabola) -^ SL 2 = ^-.
Hence velocity at P oc SN ; and, since by the figure
SL = LN, PL is the tangent at P and is perpendicular to
8N.
Hence as SN is perpendicular to the direction of motion
at P, proportional to the velocity at P, and drawn from a
fixed point S, the locus of N is the Hodograph ( 20) turned
through a right angle about S. As this is a horizontal
straight line, the Hodograph is a vertical line.
This result will be found of considerable utility in solving
various problems in the common vacuum theory of projectiles.
It is evident that SB, BN represent the horizontal and vertical
velocities at P, on the same scale on which SN represents the
entire velocity at that point.
120. It may be interesting to anticipate a little here, by
introducing matter properly belonging to the next chapter.
We wish to shew that the above geometrical constructions
can easily be extended to paths of projectiles when they are
so large as to require us to take account of the variations in
the direction and amount of gravity. The following sections
are taken from the Proc. R. S. E. 1865-6.
121. When, instead of supposing gravity to be of con-
stant amount, and to act in parallel lines, we take the more
accurate assumption that it tends to the centre of the Earth,
and varies inversely as the square of the distance from that
point, Chapter V. shews us that in general the path of the
projectile is an Ellipse, one of whose foci is at the Earth's
centre, and the length of whose major axis depends only
on the velocity of projection. The following propositions
(among many others analogous to those just given) may then
be enuntiated.
1. The locus of the second foci of the paths of all pro-
jectiles leaving a given point, with a given velocity, in a
vertical plane, is a circle.
PARABOLIC MOTION.
97
2. The direction of projection, for the greatest range on
a given line, passing through the point of projection, bisects
the angle between the vertical and the line.
3. Any other point on the line, which can be reached at
all, can be reached by two different paths, and the directions
of projection for these are equally inclined to the direction
which gives the maximum range.
4. If a projectile meet the line at right angles, the point
which it strikes is the vertex of the other path by which it
may be reached.
5. The envelop of all possible paths in a vertical plane
is an ellipse, one of whose foci is the centre of the earth, and
the other the point of projection.
The proofs of these propositions are extremely simple.
Thus, let E be the earth's centre, P the point of projection,
A the point which the projectile would reach if fired verti-
cally upwards. With centre E, and radius EA, describe a
T. D.
98 PARABOLIC MOTION.
circle in the common plane of projection. This, the circle of
zero velocity, corresponds to the common directrix of the
parabolic paths in the ordinary theory. If F be the second
focus of any path, we must have EP + PF constant, be-
cause the axis major depends on the velocity, not the
direction, of projection. Hence (1) the locus of F is the
circle AFO. Again, since, if F be the focus of the path
which meets PR in Q, we must have FQ = QS, it is obvious
that the greatest range Pq is to be found by the condition
Oq = qs. is therefore the second focus of this trajectory,
and therefore (2) the direction of projection for the greatest
range on PR bisects the angle APR. If QF=QF' = QS, F
and F' are the second foci of the two paths by which Q may be
reached ; and, as Z FPO = Z F'PO, we see the truth of (3).
If Q be a point . reached by the projectile when moving in
a direction perpendicular to PR, we must evidently have
PQF' = Z PQF = Z SQR = Z EQP ; i.e. EQ passes through F'.
Thi-s case is represented on the other side of the diagram
where f'g = gh fg. The ellipse whose second focus is/
evidently meets Pr at right angles : and that whose second
focus is/' has (4) its vertex at g. The locus of q is evidently
the envelop of all the trajectories. Now
Eq = Es-sq = EA- Oq.
Hence
PA+AE,
or (5) the envelop is an ellipse, whose foci are E and P, and
which passes through A.
122. When a particle moves subject to the action of two
centres, one attractive and the other repulsive, where the law is
the direct distance and the strengths the same, its motion will
be the same as that of a projectile in vacuo.
For the whole force on the particle resolved perpendicular
to the line joining the centres is evidently zero, and that
parallel to this line is equal to that which would be exerted
by either of the centres on a particle placed at the other ; and
PARABOLIC MOTION. 99
always tends in the direction parallel to that from the repel-
ling, to the attracting, centre. It corresponds therefore
exactly to gravity, within moderate elevations above the
earth's surface.
123. Again, if a particle moves on a plane inclined to the
horizon at an angle 6, the acceleration is, by 84, g sin 6
parallel to the line of greatest slope on the plane, and there-
fore the trajectory will still be a parabola, whose dimensions
will depend upon 6.
Ex. A particle is projected from a given point with a
given velocity, and moves on an inclined plane; find the locus
of the directrices of its path for different inclinations of the
plane.
It will be easily seen that when a particle moves on an
inclined plane, the velocity at any point is equal to that
which would have been acquired by sliding from the direc-
trix ; that is ( 85) equal to the velocity due to the fall from
a horizontal plane through the directrix. Now the velocity
is given constant, hence the locus of the directrices is a hori-
zontal plane.
124. A particle moves subject to an attraction always
perpendicular to a given plane, its intensity being a function
of the distance of the particle from the plane : to determine
the motion.
It is evident that the motion will be confined entirely to
a plane through the direction of projection perpendicular to
the attracting plane. Let us take the plane of motion as
that of xy t the axis of x lying in the attracting plane. Let
<j>' (D) be the acceleration at distance D, where (/>' is the
derived function of </>. Then the equations of motion are
- S-*
Suppose the particle projected from a point (a, b), in a
direction making an angle a with the axis of x, and with a
velocity F.
72
100 PARABOLIC MOTION.
Multiplying by -j- , -jj , and integrating we get
= const-
Hence
or
a particular case of conservation of energy.
To find the differential equation of the path, we have
dy
<K = dy = V[F 2 sin 2 a +
dx dx V cos a
an equation integrable for particular forms only of the func-
tion (f).
An interesting case is that in which the attraction of the
plane is inversely as the cube of the distance,
or </>' (y) = ^ , and therefore </> (y) = - % ^ .
The differential equation becomes
or
dx V cos a
dx _ ydy
PARABOLIC MOTION. 101
and integrating
x a
Fcosa
the equation of a conic ; an ellipse, parabola, or hyperbola,
according as
is negative, zero, or positive.
We might have obtained the above results by integrating
separately the two equations of motion, and then eliminating
t between them.
For a repulsion, instead of an attraction, it is easy to see
that we must simply change the sign of //,, and thus that the
curve described is a hyperbola whose conjugate axis lies in
the intersection of the plane of projection and the attracting
plane.
From this we see that the conic sections are the only
curves which can be described by a free particle moving in
a plane with acceleration in the direction, and inversely as
the cube, of the perpendicular distance from a given line in
that plane.
The converse of either of the above propositions is easily
investigated ; thus, taking the first, our problem becomes
125. To find the law of force perpendicular to an axis
that a free particle may describe a conic section.
Take the axis as that of x, and the vertex as origin, then
the equation
\f = 2mx + no? ..................... (1)
102 PARABOLIC MOTION.
will represent, by properly taking m and n, any parabola,
any hyperbola referred to its transverse axis, or any ellipse
referred to either axis.
Since the attraction is perpendicular to the axis, we have
dx_
~dt~
Hence y -j, = me + nxc ;
_, ' d 2 y 1 f / dy^ 2
From these -^ = - ( we 2 ( Hr
at y
lf (m-
y\
f
C 2
= (ny* m 2
t/
C 2 m 2 , . .
= -- by equation (1).
u
This indicates an attractive force, inversely as the cube of
the distance from the axis.
For a hyperbola with its conjugate axis in the axis of
x, the equation is
Hence
dy . dx
from which we have immediately
v^
y dt* '
PARABOLIC MOTION. 103
That is, ^ =
dt
l-^} =
y
which indicates a repulsion inversely as the cube of the
distance from the conjugate axis.
126. To find the force which must act perpendicularly to
a plane, in terms of the distance from that plane, that a given
(plane) path may be described.
Take the axes as before ; then, F being the acceleration
due to the repulsion (a function of y only), we have
d 2 x dx
= 0, or -77 = const. = a, suppose ;
at at
Let y = f(x) be the equation of the given curve, then
orby(l),
by the equation of the curve. Hence; as / is a given function,
the acceleration and the repulsion are found.
127. It is necessary to observe that, in the case of 124,
when the particle actually reaches the axis, it will not proceed
to describe the portion of the same curve which lies on the
104 PARABOLIC MOTION.
other side of the axis, as this would involve a change in sign
of the constant horizontal velocity. It is, in fact, evident that
in such cases the particle having described ABC will, instead
of pursuing the course Cba, actually describe CDE similar
and equal to Cba, but turned in the opposite direction.
And a similar remark applies to the general problem in
126.
Although, in the case of AEG being a conic, one of whose
axes is CG, and therefore cutting it at right angles in G, it
might seem that at C the horizontal velocity vanishes, yet it
is to be recollected that the velocity at C is infinitely great ;
and it may easily be shewn by independent methods, such as
the method of limits, if the foregoing analysis do not appear
satisfactory, that the velocity parallel to CG is really constant
throughout the motion.
128. It may be useful to notice that cases of this kind
are reduced at once to investigations similar to those of the
last Chapter, by considering, separately, the equations of
motion parallel and perpendicular to the attracting plane.
Whenever, then, we can completely determine the motion
of a particle in a straight line towards a centre, we can also
completely solve the problem of the motion of a particle
anyhow projected, and attracted by an infinite plane ; the
intensity in terms of the distance being the same in the two
cases.
PARABOLIC MOTION. 105
129. Generally, when a particle is anyhow projected and
subject only to an acceleration whose direction is perpen-
dicular to a given plane, and whose magnitude depends solely
on the distance from the plane ; the velocity parallel to that
plane is constant; and, in passing from any point to another,
the square of the velocity is altered by a quantity depending
only upon the distances of those two points from the given
plane.
Take the axis of y perpendicular to the given plane, and
the axis of x in it, so that the direction of projection lies
in xy. This will evidently be the plane of motion ; and the
equations are
dx
Hence -
or
V being the velocity of projection, and y the co-ordinate of
the point of projection ; which proves the proposition.
This is, of course, merely a particular case of the general
principle of Conservation of Energy ( 78) ; <f> (y) being the
Potential, and = v* the Kinetic, Energy of unit mass.
130. As another example of the motion of a particle
under the action of forces whose direction is constant, let us
consider the motion of a particle of light in the corpuscular
106 PARABOLIC MOTION.
theory, at the confines of two homogeneous isotropic media
whose bounding surface is plane.
In this theory the hypothesis is that the attractions or
repulsions, exerted by the particles of any medium on a
particle of light passing through it, are insensible at sensible
distances but enormously great at infinitely small distances.
Hence of course the path of such a particle in a homogeneous
medium will be a straight line, and will be described with
constant velocity, until the particle is infinitely near to the
bounding surface of the medium.
Thus, suppose A B to be the common plane surface of two
such media. Draw CD at a distance from AB equal to that
at which the intensity of the attractions of the particles of
the medium begins to be sensible ; and draw EF parallel to
CD and equidistant from it with AB. By what we have just
noticed, a particle of light moving along PQ will arrive at Q
without any change of velocity or direction. Also from the
symmetry of the figure, the resultant of all the sensible
attractions or repulsions on it will always be perpendicular
to AB. This shews, 129, that the velocity resolved parallel
to AB is constant throughout the motion, and also that
whatever be the direction of PQ, the change in the square
of the velocity in passing from Q to any point of the path
will depend only on the distance of that point from AB.
Let PQR represent a portion of the path.
We have no means of determining its actual form, since
the extent through which the attraction is sensible, the law
of its variation, and whether it change from attraction to
repulsion with the distance, are unknown.
Through any point R draw KRL parallel to AB, and let
GH be equidistant from KL with AB.
Then at R the particle is subject only to the actions of the
upper medium beyond GH, and of the lower medium.
If the resultant effects of these two should, at a point S
in the superior medium, destroy the velocity perpendicular
to AB, the particle will evidently pursue a course SR'Q'P'
PARABOLIC MOTION.
107
similar and equal to SRQP, and the angles P'Q'G and PQD
will be equal, as also the velocities in PQ and P'Qf. ( 129.)
Here we have the case of a ray reflected at a plane surface.
If, however, the attraction of the lower medium should
so prevail that the particle actually enters it, then we may
consider its motion, while it is still within the range of
action of both media, precisely as before ; but there will be
two cases.
I. At some point as S whose distance from AB (the
bounding surface) is less than that of AB from CD, the
velocity perpendicular to AB may be destroyed ; then, as
before, the particle will pursue the path ST'Q'P', similar and
equal to STQP, and will be reflected at an angle equal to
that of incidence and with its original velocity.
II. The particle may pass into the lower medium so far
as to be independent of the action of the upper medium.
After this it will move in a straight line as before, arid the
change of the square of its velocity will be, 129, independent
108 PARABOLIC MOTION.
of the path pursued. Hence, if V be the velocity, and a the
angle, of incidence ; V, a those of refraction, we have, by the
condition that the velocity parallel to the surface is unaltered,
V sin a = V sin a'.
Also by the fixed amount of change in the square of the
whole velocity,
F 2 = F' 2 a 2 ,
where a is a constant depending on the nature of the two
media.
V
sin a V //- _ a 2 \
Hence, , = -= = / 1+ ,
sin a V V V J
and, therefore, for particles of light which have the same
velocity the ratio of the sines of the angles of incidence and
refraction is constant. This is the known law of ordinary
refraction. Unfortunately, however, in order that a ray may
be bent, at refraction, towards the normal to the refracting
surface (i.e. so that a' < a) we must have V > V\ a result
lately shewn to be inconsistent with experiment.
We have introduced this example, although belonging
to a theory now completely exploded, as it forms a good
illustration of the application of the results of this Chapter,
and afforded the first instance of the solution of a problem
connected with molecular actions. It is due to Newton.
EXAMPLES.
(1) The time of describing any portion PQ of the para-
bolic path of a particle under gravity, is proportional to the
difference of the tangents of the angles which the tangents at
P and Q make with the horizon. ( 119.)
(2) If a shell burst, all the fragments receiving equal
velocities from the explosion ; shew that the locus of the foci
of the paths of the fragments is a sphere, of the vertices an
oblate spheroid, and that the particles themselves at any
instant will lie on a sphere.
PARABOLIC MOTION. 109
(3) Two bodies, projected from the same point A, in
directions making angles a, a' with the vertical, pass through
the point B in the horizontal plane through A ; prove that
if t, t' be the times of flight from A to B,
sin(a-a')_t'' 2 -t 2
(4) If u and v be the velocities at the ends of a focal
chord of a projectile's path, V x the horizontal velocity, shew
that
(5) From a point in an inclined plane two bodies are
projected with the same velocity in the same vertical plane
in (directions at right angles to each other; prove that the
difference of their ranges is constant.
(6) If v, v', v", be the velocities at three points P, Q, R,
of the path of a projectile where the inclinations to the hori-
zon are a /3, a, a + ft ; and if t, t' be the times of describing
PQ, QR respectively, shew that
* *> j 1 1 2 . cos /3 /ff i , ft \
v t = vt , and - + , = - 7-^ . ( 119.)
(7) If two particles be projected from the same point at
the same instant in the same vertical plane, with velocities v
and v l in directions making angles a and ^ with the horizon;
shew that the interval between their transits through the
other point which is common to their paths is
2 Wi sin (a - o^)
g v l cos ot! + v cos a
(8) Particles slide from rest at the highest point of a
vertical circle down chords, and are then allowed to move
freely; shew that the locus of the foci of their paths is a
circle of half the radius, and that all the paths bisect the
vertical radius.
110 PARABOLIC MOTION.
(9) If the particles slide down chords to the lowest
point, and be then suffered to move freely, the locus of the
foci is a cardioid.
(10) Down what chord from the vertex of a vertical
circle must a particle slide so as to have when falling freely
the greatest range on a given horizontal plane ?
(11) Find the locus of the foci of all trajectories which
pass through two given points.
(12) Particles fall down diameters of a vertical circle;
the locus of the foci of their subsequent paths is the circle.
(13) If a body describe a cycloid under an attraction
to the axis, shew that the attraction varies inversely as
2 sin 6 sin 20, 6 being the corresponding arc of the gene-
rating circle measured from the vertex.
(14) If the acceleration be perpendicular to a plane and
vary as the distance, shew that the curves described have
equations of the form
y = Aa x + Bar~ x , ] for a repulsion or attraction
or y = A cos (mx + B)} respectively.
Find the circumstances of projection in the two cases that
the curves may be the catenary, and the companion to the
cycloid, respectively.
(15) Particles are projected in the same plane and from
the same point, in such a manner that the parabolas de-
scribed are equal ; prove that the locus of the vertices of these
parabolas will be a parabola.
(16) Find the direction of projection, with a given velo-
city, from a given point, so that a given plane, not passing
through the point, may be reached in the least possible time.
(17) Particles slide down radii vectores of the curve
whose equation is r=f(0), the plane of the curve being
PARABOLIC MOTION. Ill
vertical and 6 being measured from a horizontal line, prove
that the locus of the foci of their future paths is the curve
6
(18) Through a point an inclined plane is drawn, and
from that point a particle is projected with a given velocity so
that its direction of motion when it meets the plane again cuts
it at right angles ; shew that the locus of the point of meeting
for different positions of the inclined plane is an ellipse.
(19) The attraction between two particles is ^-, where
m is the mass of each particle, and r the distance between
them, and they are projected with equal velocities on the
same side of the line (c) joining them in directions not
parallel but equally inclined to that line ; prove that the
path of each will be an ellipse, parabola, or hyperbola, ac-
cording as the initial component of each velocity in direction
of the line (c) is less than, equal to, or greater than */ '^ .
(20) A perfectly elastic particle is projected 'so as to
strike on the inside a surface of revolution of which the axis
is vertical and given in position. Shew that the vertices of
all the parabolic orbits described after successive rebounds
lie on a surface which is independent of the surface of
revolution.
(21) If a be the angle of elevation required in order
that a bullet may have a certain range on a horizontal plane,
the additional elevation required above a plane inclined to
the horizon at an angle /3,
tan 0..E|JE?!.
(22) A particle is projected from a given point with a
given velocity u so that the range on a given inclined plane
may be the greatest possible: prove that, if v be its final
112 PARABOLIC MOTION.
velocity, and a perpendicular be let fall on the given plane
from the point of intersection of the initial and final directions
of motion, the length of the perpendicular is ~- .
(23) A cycloidal arc is placed with its axis vertical and
vertex upwards, and a particle is projected so as, after moving
in contact with the arc for a finite distance, to describe a
parabola freely ; prove that the focus of the parabola lies on
a cycloid of half the dimensions having the same base.
(24) Shew that the whole area commanded by a gun
on a hill-side is an ellipse whose focus is at the gun, whose
excentricity is the sine of the inclination of the hill to the
horizon, and whose semi-latus-rectum is twice the greatest
height to which the gun could send a ball.
(113)
CHAPTER V.
CENTRAL ORBITS.
131. IN this part of the subject we consider the motion
of a particle under the action of an attraction or repulsion
whose direction always passes through, and whose intensity
is some function of the distance from, a fixed point. The
fixed point is called the Centre. The case of attraction, as
including the most important applications of the subject, we
will take as our standard case ; but it will be seen that a
simple change of sign will adapt our general formulae to
repulsion. If the centre of attraction be itself in motion,
the methods of 26, 31, enable us easily to treat it as
fixed; but in this case the relative acceleration is not in
general directed to the centre, so that the problem no longer
belongs to Central Orbits strictly so called. It will be con-
sidered later. If the centre be moving with constant velocity
in a straight line, the results of this chapter are at once
applicable to the relative motion.
132. A particle is projected in a plane, and is acted on
by an attraction P directed to the fixed point in that plane ;
to determine the motion.
The whole motion will clearly take place in the plane, as
there is nothing to withdraw the particle from it. Let Ox,
Oy, any two lines through at right angles to each other, be
taken as the axes of co-ordinates. Let M be the position of
the particle at the time t ; and draw MN perpendicular to
Ox, and join MO. Let ON = x, NM=y, OM = r, and the
T ?/
angle NOM = 0. Then, since cos 6 - , sin 6 = - , the cora-
r r
T. D. 8
114
CENTRAL ORBITS.
ponents of P, parallel to the axes, are P - , P - . But by
the second law of motion we may consider the accelerations
N
in the directions of x and y separately, and we have therefore
d*x
In these, since P is a function of r, and therefore of x
and ?/, the second members will generally contain both these
variables, and the equations must be treated as simultaneous
differential equations. Their integrals will give x, y, -nt-jz*
in terms of t ; from which the position and velocity of the
particle at any instant will be known, and the problem com-
pletely solved. In one case, however, viz. when P is pro-
portional to r, the first equation will involve x and t, and the
second y and t, only, and each equation may be integrated
by itself. As it is the simplest example of its class, and of
great importance in its applications, especially to Acoustics
and to Physical Optics, we will begin by considering it.
133. A particle moves about a centre of attraction vary-
ing directly as the distance : to determine the motion.
Let //, be the acceleration at unit of distance, called the
CENTRAL ORBITS. 115
strength of the centre, then P = /xr, and equations (A)
become
^-^
d*y_
dP~
the integrals of which, see 88, are
B\ ' (1),
(2),
A, B, A', B' being the constants introduced in the integration,
to be determined by the initial circumstances of motion.
Consider the particle projected from a point on the axis of x,
at distance a from the centre, with velocity F, and in a
direction making an angle a with Ox. When t 0, we have
<c = a,y = Q, -TT = F cos a, -j- = V sin a. Hence,
at at
a = A cos B,
Q = A' cos B',
Fcos a = A V/"- sin J5,
Fsin a = A' *Jju sin '.
Expanding the cosines in (1) and (2), and substituting
these expressions for the constants, we obtain
Fcos a . . .
x -. - sm tJ/jLt + a cos \lpt ............... (3),
VA*
Fsin a .
^
which contain the complete solution of the problem. Elimi-
nating t, we have
LLQ?
(x sm a y cos a) 2 -I- ^ y 2 = a 2 sin 2 a,
82
116 CENTRAL ORBITS.
the equation of the path of the particle; which is therefore an
ellipse whose centre is 0. Equations (3) and (4) give periodic
values for x t y, -=- , -^ , such that all the circumstances of
2_
motion will be the same at the time t + r- as at the time t.
*/p
The period of revolution is therefore : a most remarkable
V/i
result, as it is independent of the dimensions of the ellipse,
and depends solely on the intensity of the force.
By taking fj, negative in equations (J9), we may apply
them to the case of a repulsion varying as the distance from
0. In the integration for this supposition the sines and
cosines would be replaced by exponentials, and the curve
described would be a hyperbola having as centre ; but
the motion would not be one of revolution, as the particle
would necessarily always remain on the same branch of the
hyperbola.
134. Recurring to equations (A), it will in all cases but
the one we have just considered be more convenient to trans-
form them to polar co-ordinates, especially as the general
polar differential equation of the orbit described by a particle
under the action of a central force can be easily formed, as
follows.
135. A particle being acted on by a central attraction ;
it is required to determine the polar equation of the path.
Multiplying the second of equations (A), 132, by x t and
the first by y, and subtracting, we obtain
Integrating,
dy dx ,
x~ y -T- = constant = h suppose.
Changing the variables from x, y, to r, 6, where x = r cos
y = r sin 6, we get, as in 24,
CENTRAL ORBITS.
r *T t = k
or, substituting - for r,
Again,
dt
x = r cos 6 =
COS0
u
1 U Sin (j T v/v^o v j-r, j,]
cfo? a# dv
which gives ~ji~~ """"
,
and therefore -7 = h ( u
dt
117
(1)>
.(2).
d z u\ d6
= - h?u 2 \u cos 6 + cos -^ J , by (2).
But, by the first of equations (^1),
Equating these values of -^ , and dividing by cos 6 y we
have
or
(4).
This is the differential equation of the orbit described;
and as, in any particular instance, P will be given in terms
of r, and therefore in terms of u, its integral will be the polar
equation of the required path.
118 CENTRAL ORBITS.
136. It may easily be obtained by the formulae of 16,
and, as this method is instructive as well as useful, we give
it for the case, when in addition to the central acceleration
due to the attraction P there is a transverse acceleration T
impressed on the particle.
Instead of equations (A) we may evidently write (by
$ 16, 69),
dV_ (
dt
r dt V dt
Putting r 2 -n=h, and u - , then -=- =hu z , and the second
equation becomes
dh T
dfr 2T
~ '
dr dr dO , du
= = - k
d?~ d&>
//7/3\2
and
-.
Therefore
d?u _P^ T_du
+ ~
137. The general integrals of (A), ( 132), which are
differential equations of the second order, ought to contain
four constants. One of these has been already introduced in
(1), and two more will be introduced by the integration of (4).
If the value of r in terms of deduced from the integral of
CENTRAL ORBITS. 119
(4) be substituted in (1), and that equation be then integrated,
the remaining constant will be introduced; and the path of
the particle and its position at any time will be obtained.
The four constants involved in the resulting equations must
be determined from the initial circumstances of motion;
namely, the initial position of the particle (depending on
two independent co-ordinates), its initial velocity, and its
direction of projection.
138. Equation (3) may be used to ascertain the law of
central attraction which must act upon a particle to cause it
to describe a given curve. To effect this we must determine
the relation between u and 6 from the polar equation of the
proposed orbit referred to the required centre as pole: we
must then differentiate u twice with respect to 0, and substi-
tute the result in the expression for P ; eliminating 6, if it
be involved, by means of the relation between u and 6. In
this way we shall obtain P in terms of u alone, and therefore
of r alone.
When we know the relation between r and 0, from (4), we
make use of equation (1) to determine the time of describing
a given portion of the orbit ; or, conversely, to find the posi-
tion of the particle in its orbit at any time.
139. The equation of the orbit between r and p, the
radius vector and the perpendicular on the tangent at any
point, may be easily obtained from (4). For by Diff. Calc.
we have
dht, 1 dp
-rjr -f- U ~T 1 J j
and therefore P = -f- .
p 3 ar
140. The sectorial area swept out by the radius vector of
the particle in any time is proportional to the time ( 24).
If A denote this area, we have, by Diff. Calc.,
dA _\ dO^
dt ~**dt'
120 CENTRAL ORBITS.
and therefore, by equation (1) of 135,
dA 1 ,
W = 2 k >
whence A = ^ht,
if A and t be supposed to vanish together.
Therefore the areas described in different intervals are
proportional to these intervals.
We also see, by taking = 1, that the value of h is twice
the area described in a unit of time.
141. The velocity of the particle at each point of its path
is inversely proportional to the perpendicular from the centre
on the tangent at that point. ( 23.)
ds
For Velocity = v = ji
= ds d6
~ dB dt
(p being the perpendicular on the tangent from the centre)
= - , by equation (1) of 135.
Hence, as above, v oc - .
P
142. This equation enables us to express h in terms of
the initial circumstances of the motion. For, let R be the
distance of the point of projection from the centre, V the
velocity, and ft the angle which the direction of projection
makes with that of R. Then evidently the perpendicular on
tangent at point of projection = R sin /9 ;
CENTRAL ORBITS. 121
h
whence h = VR sin fi.
Again, since by Diff. Gale.,
we have
another important expression for the velocity.
143. ^e velocity at any point of a central orbit is
independent of the path described, and depends solely on the
intensity of the attraction, the distance of the point from
the centre, and the velocity and distance of projection.
(i */* ft '11
Multiply equations (A) 132, by -j- , -^ respectively, and
add, then
dx d*x dy d*y _ P ( dx dy
~~ + ~~~ " + 2/
dt'
Also, since P is a function of r alone, let P = <f> (r), then
^2_7 2= _ P + Mdr
J R
if at the point of projection v = V, r = R.
122 CENTRAL ORBITS.
If the velocity vanishes at a distance a from the centre,
Jt?-fc(a)-f (r)
and a is called the radius of the circle of zero velocity.
(Compare 78.)
144. The velocity of a particle at any point of a central
orbit is the same as that which would be acquired by a par-
ticle moving freely from rest along one-fourth of the chord of
curvature at the point, drawn through the centre, under the
action of a constant force whose intensity is equal to that of
the central attraction at the point.
By 143,
1 d(v*)_ _ dr
2 ~di dt
dv n
or v -r = - P.
dr
And by 141,
h
v = .
P
Taking the logarithmic differential, we obtain
1 dv _ _ 1 dp
v dr p dr '
and, dividing the former equation by this,
* >**:
dp r 4 dp
where q is the chord of curvature through the centre. Hence
the proposition, 82.
From this it follows that the velocity, V, of a particle
CENTRAL ORBITS. 123
moving in a circle of radius R, under the action of an attrac-
tion P to the centre, is given by the equation
a simple, and most useful expression*.
145. DEF. An Apse is a point in a central orbit at
which the radius vector is a maximum or minimum, and the
corresponding value of the radius vector is called an Apsidal
Distance.
The analytical conditions for such a point are that
^777
-j0 should vanish, and that the first succeeding differential
coefficient which does Dot vanish should be of an even order.
The first condition ensures that the tangent at an apse is
perpendicular to the radius vector.
* The results of the last few Articles may be obtained in the following
manner.
By 49 and 65
-na= Resolved part of P along the tangent to the orbit = --P-r- ......... (1)>
-= Resolved part of P along the normal =P- .............................. (2).
Multiply (1) by -? and integrate, then
From (2) v 2 =Px
P q
4*
Also if in (2) we put - for v, 141, and r -j- for />, we obtain
r
pr
dp
P- h * dp
-pd^'
which is the result contained in Art. 139.
124 CENTRAL ORBITS.
Every apsidal line divides the orbit into two parts which
are equal and similar.
For the acceleration at any point being a function of the
distance from the centre of attraction, when the particle has
reached an apse it must proceed to describe on the other
side of the apse a path equal, similar and symmetrical with
the path it has already described, and hence an apsidal line
divides the orbit into two parts which are equal and similar.
(Compare, however, Ex. 80 at end of Chapter.)
146. In a central orbit there cannot be more than two
apsidal distances.
For, since the parts of the orbit on opposite sides of an
apse are similar, the particle after passing two apses must
come next to one at an equal distance with that of the first,
then to one at an equal distance with that of the second, and
so on. Hence there can be but two apsidal distances.
147. When the central attraction varies as a power of
the distance, we may obtain the above result, as well as the
equation for determining the apsidal distances, directly from
equation (4) of 135. Suppose P = /j,u n , then we have
d z u u,
- 4- tl ti n ~ 2
+ u
Multiplying by h* and integrating, we have
= = ""-'
Suppose the particle projected with a velocity equal to
q times the velocity from infinity at the same distance, and
let c be the initial value of u, then when u = c,
CENTRAL ORBITS. 125
whence ' C = (q* - 1) --= c n ~ l ;
and therefore M 2 1 -y^ + u?\ = - . \u n ~ l -t- (o 2
(We'/ J n 1
To determine the apsidal distances we must put -^ = 0,
which gives
S) ' \1 /
The form of this equation shews that it can have at most
two positive roots, which are therefore the two apsidal dis-
tances.
Although there can be but two apsidal distances, there
may be any number of apses, and the angle between two
consecutive apsidal distances is called the apsidal angle.
Generally, to determine this angle, the equation of the orbit
must first be found for the particular case considered ; but
the apsidal angle may be determined approximately for any
law of attraction, without first finding the form of the orbit,
if we assume that it does not differ much from a circle.
148. A particle revolves in an orbit which is very nearly
circular, and is acted on by an attraction varying as any
function of the distance and directed towards the centre of
the circle: to determine the apsidal angle.
If we put P in the form fiu 2 ^ (u) the differential equation
of the orbit is
If the orbit were circular, we should have
an d -7^ = 0,
in which case
126 CENTRAL ORBITS.
When the orbit is very jiearly circular we may put
u = c + x, where x is always very small. Hence
or ~ + c + x - fa {(/> (c) + x# (c)} = 0, nearly ;
and (a) enables us to reduce this to
or, by a second application of (a),
the integral of which is ( 88)
(
Hence the general value of which renders -^ = 0, is
given by the equation
n being any integer ; and consequently the difference between
any two such successive values of 6 is
4
the approximate apsidal angle.
CENTRAL ORBITS. 127
Thus if the attraction vary directly as the n th power of
the distance, we have
/jLU 2 <j) (u) = fjuu~ n ; and </> (u) = u~ n ~ 2 ,
whence <' (u) = (n + 2) ur"-*
and the apsidal angle is
7T
This shews that n cannot be less than 3, or that ttfe
attraction must vary according to a lower inverse power
of the distance than the third, if the circle with the centre
of attraction at its centre is to be an approximation to the
path of the particle : and the investigation furnishes a simple
example of the determination of the conditions of Kinetic
Stability, which we cannot discuss in this elementary treatise.
To find the law of attraction that the apsidal angle in
the nearly circular orbit, whatever be its radius, may be
equal to a given angle, a suppose, we have
7T
from which ' _ n _ )
4>(c) c\ a 2 /'
or, by integration,
i < ( c ) /-. T 8 \ i
l 8 -g- = \ tf J g '
whence ^> (c) = (7c a8 ;
3--
and therefore the law of attraction, yu,M 2 < (u), is pu * 2 .
Thus for a = TT we have the law of the inverse square of
the distance, for a = the law of the direct distance, while
a = -jz corresponds to a constant central attraction.
If 1 -- be zero or negative, the form of the integral of
[6) above shews that x does not remain infinitely small ; i.e^
128 CENTRAL ORBITS.
that a circle is not a kinetically stable path under the con-
ditions. In this case all that (6) can furnish is an account of
the way in which the orbit begins to differ from a circle in
consequence of a slight disturbance.
149. A particle is projected from a given point in a given
direction and with a given velocity, and moves under the action
of a central attraction varying inversely as the square of the
distance ; to determine the orbit.
We have P pu 2 , and therefore
the integral of which is
or, as it is usually written,
' ( 1 i //3 \1 /1\
n?
This is the polar equation of a conic section, the focus
(the centre of force) being the pole.
It gives by differentiation
-^ = ^ e sin (6 a) (2).
Let R be the distance of the point of projection from the
centre ; j3 the angle, and V the velocity, of projection; then
when 6 = 0,
du\
h*
Hence, by (1), ^ l = e cos a,
h 2
and by (2), -= cot ft = e sin a.
fJLld
CENTRAL ORBITS. 120
From these, tana = ^ ^ ^ .............................. (3),
and e 2 = -^ cosec 2 @ -- ^ + 1 ......... (4).
jjfJy pM
But/^FlR'sin 2 /?, 142;
FIR sin cos
wherefore tan a = ---- T721 T . a - b ............... (3 )
/i - F'72 sin'' ft
F 2 jR 2 sin 2 /5/2 F 2 \
and 1-02= - -(^- i4 - ............ (4).
Now (1) is the general polar equation of a conic section
focus the pole ; and, as its nature depends on the value of
the excentricity e given by (4'), we see that
if F 2 > - , e > 1, and the orbit is a hyperbola,
F 2 = -~ , e = 1, ..................... a parabola,
F 2 < ~ , e < 1, ..................... an ellipse.
150. By 102, the square of the velocity from infinity at
distance E, for the law of attraction we are considering, is
~, and the above conditions may therefore be expressed
Jtt>
more concisely by saying that the orbit will be a hyperbola,
a parabola, or an ellipse, according as the velocity of pro-
jection is greater than, equal to, or less than, the velocity
from infinity. Illustrations of this proposition are found in
the cases of comets and meteor swarms.
The velocity of a particle moving in a circle is also often
taken as the standard of comparison for estimating the velo-
cities of bodies in their orbits. For the gravitation law of
attraction the square of the velocity in a circle of radius R
is ^ ; and the above conditions may be expressed in another
T. D. 9
130 CENTRAL ORBITS.
form by saying that the orbit will be a hyperbola, a parabola,
or an ellipse, according as the velocity of projection is greater
than, equal to, or less than, V2 times the velocity in a circle
at the same distance.
151. Supposing the orbit to be an ellipse, we shall obtain
its major axis and latus rectum most easily by a different pro-
cess of integrating the differential equation. Multiplying it
by h 2 -= and integrating, we obtain
But when u = ~p, v= V', which gives
n-\ v*.
hence h*{(jY 4 u-l = L- = J F' - + ^ (5).
z Vote// ^t
Now to determine the apsidal distances, we must put
--0-
d6~
and this gives us the condition
which is a quadratic equation whose roots are the reciprocals
of the two apsidal distances. But if a be the semi-axis
major, and e the excentricity, these distances are
a (1 e) and a (1 + e).
Hence, as the coefficient of the second term of (6) is the
sum of the roots with their signs changed, we have
1 1 2u
or a(l-O = - ....................... (7).
CENTRAL ORBITS. 131
And, as the third term is the product of the roots,
or
i - 1
a~R
.(8),
or
and therefore
-
R 2o '
.
2a'
.(9).
Equations (7) and (8) give the latus rectum and major axis
of the orbit, and shew that the major axis is independent of
the direction of projection.
Equation (9) gives a useful expression for the velocity at
any point, and shews that the radius of the circle of zero
velocity is 2a.
152. The time of describing any given angle is to be
obtained from the formula,
&2 )}> by equation (7).
From this, combined with the polar equation of a conic
section about the focus, we have
dt r 2
/K
VI
(l+ecos0)'-"
measuring the angle from the nearest apse. To integrate
this, let
sin 9
I + e cos 6 '
92
132
CENTRAL ORBITS.
Then
cos
1 1 e 2
- (1 + e cos 0) --
-f e g v y e
dd (I + e cos Of (l+e cos 0) 2
_1 1 _\^& _ l
~e 1 +ecos0 e (1
e sn
1 e 2 l+0cos0 1
sec 2
e sinfl 2 _ :
1 - e 2 1 + e-cos0 + (F- #)t
(if c<l);
l-e\ 6}
T.J tan 2} '
6
(if e>l).
Hence the time of describing, about the focus, an angle 6
measured from the nearer apse is, in the ellipse,
2
that is, y of the sectorial area ASP (figure to 160); and,
n>
in the hyperbola,
v ^
sn
~5
sm-
2
that is, - of the sectorial area ASP of the hyperbola.
CENTRAL ORBITS.
133
Hence these expressions for the time through any area
of an elliptic or hyperbolic orbit about a focus might have
been written down from the known expressions for the area
of an elliptic or hyperbolic sector.
153. In the parabola, if d be the apsidal distance, the
integral becomes
[since e = l, a (I e) = d, a (I e*) = 'Id],
= \/ 8 ^!l
/2d?
~v /*
tan 8
e i
d tan -
(ton j+.| ton*
154. From the result for the ellipse we see that the
IQ*
periodic time is 2?r . / . This might also have been found
from the consideration of equable description of areas by the
radius vector.
Thu y = 2 area of ellipse
h
In the notation commonly employed we write
where n, which is called the Mean Motion, is
1.34 CENTRAL ORBITS.
155. By laborious calculation from an immense series of
observations of the planets, and of Mars in particular, Kepler
was led to enuntiate the following as the laws of the planetary
motions about the Sun.
I. The planets describe, relatively to the Sun, Ellipses
of which the Sun occupies a focus.
II. The radius vector of each planet traces out equal
areas in equal times.
III. The squares of the periodic times of any two planets
are as the cubes of the major axes of their orbits.
156. From the second of these laws we conclude that
the planets are retained in their orbits by an attraction
tending to the Sun. For,
If the radius vector of a particle moving in a plane de-
scribe equal areas in equal times about a point in that plane,
the resultant attraction on the particle tends to that point.
Take the point as origin, and let x, y be the co-ordinates
of the particle at time t ; X, Y the component accelerations
due to the attraction acting on it, resolved parallel to the
axes; the equations of motion are
fcx _ d 2 y_
- x > ~
But by hypothesis, if A be the area traced out by the
,. dA .
radius vector, -3- is constant.
at
dA dy dx n
Hence, 2.C.
Differentiating, x -^ - y ^ = ;
or,by(l),
CENTRAL ORBITS. 135
Hence,
and by the parallelogram of forces (67) the resultant of X
and F passes through the origin.
157. From the first of these laws it follows that the law
of the intensity of the attraction is that of the inverse square
of the distance.
The polar equation of an Ellipse referred to its focus is
2
u = y(l + ecos0),
I
where I is the latus rectum.
TJ dhi 2e
Hence, = ..^^0,
and therefore the attraction to the focus requisite for the
description of the ellipse is ( 135)
Hence, if the orbit be an ellipse, described about a centre
of attraction at the focus, the law of intensity is that of the
inverse square of the distance.
158. From the third it follows that the attraction of the
Sun (supposed fixed) which acts on unit of mass of each of the
planets is the same for each planet at the same distance.
For, in the formula in 154, T* will not vary as a 3 unless
fju be constant, i.e. unless the strength of attraction of the
Sun be the same for all the planets.
We shall find afterwards that for more reasons than one
Kepler's laws are only approximate, but their emmtiation
was sufficient to enable Newton to propound the doctrine of
Universal Gravitation ; viz. that every particle of matter in
131) CENTRAL ORBITS.
the universe attracts every other with an attraction whose
direction is that of the line joining them and whose magnitude
is as the product of the masses directly, and as the square of
the distance inversely; or according to Maxwell's "Matter
and Motion," between every pair of particles there is a stress
of the nature of a tension, proportional to the product of the
masses of the particles divided by the square of their distance.
On this hypothesis, neglecting the mutual attractions of
the planets, Kepler's third law should be stated (Chap. XL) :
The cubes of the major axes of the orbits are as the squares of
the periodic times and the sums of the masses of the Sun and
the planet.
159. Suppose APA' to be an elliptic orbit described
about a centre of attraction in the focus S. Also suppose P
to be the position of the particle at any time t. Draw PM
perpendicular to the major axis AC A', and produce it to cut
the auxiliary circle in the point Q. Let C be the common
centre of the curves. Join CQ.
When the moving particle is at A, the nearest point of
the orbit to S, it is said to be in Perihelion.
The angle ASP, or the excess of the particle's longitude
over that of the perihelion, is called the True Anomaly. Let
us denote it by 8.
The angle ACQ is called the Excentric Anomaly, and is
2-7T
generally denoted by u. And if - - be the time of a complete
revolution, nt is the circular measure of an imaginary angle
called the Mean Anomaly; it would evidently be the true
anomaly if the particle's angular velocity about $ were
constant.
160. It is easy from known properties of the ellipse to
deduce relations between the mean and excentric, and also
between the true and excentric, anomalies ; this we proceed
to do.
ELLIPTIC MOTION. 137
To find the relation between the inean and eccentric
nomalies.
In the figure QCA is the exceritric anomaly, and the
mean anomaly is evidently to 2?r as the area PSA is to the
whole area of the elliptic orbit ( 154 159), or as area
QSA to area of auxiliary circle.
Now area QSA = area QCA - area QCS
= \a z u ^a.ae. sin u
(a being the major semi-axis of the orbit and e the excentricity)
Hence
or
/ 'X
-~ (u e sin u)
til
2-7T 7ra' J
nt = u e sin ?.
138
ELLIPTIC MOTION.
161. To find the relation between the true and excentric
anomalies.
We have (by Conies)
SP= "
But
Hence
\-\-e cos 6 '
= a- eCM = a (1 - e cos u).
1 4- e cos 6
= e cos u.
Hence
cos u e
cos 6 =
1 e cos t
and
tan|= v j
- cos d
1 e cos u - cos u + e
j\ e cos u
v 1 e cos u
-cosw)
-e)(l + COSM)
therefore
tan
u_ /(l-_
2 v u +
tan 2'
' J l+TcoT0'
substituted in nt = u e sin u give the expressions obtained
in 152.
162. By far the most important problem is to find the
values of 6 and r as functions of t, so that the direction and
length of a planet's radius vector may be determined for any
given time. This generally goes by the name of Kepler's
Problem.
Before entering on the systematic development of u, r
and 6 in terms of t from our equations, it may be useful to
ELLIPTIC MOTION. 139
remark, that if e be so small that higher terms than its
square may be neglected, we may easily obtain developments
correct to the first three terms.
Thus u = nt 4- e sin u
= nt 4 e sin (nt 4- e sin nt) nearly,
e 2
= nt + e sin nt 4 ~ sin 2nt.
Also - = 1 e cos u
a
\e cos (nt 4- e sin nt)
= 1 e cos w 4- ~ (1 ~~ cos 2w).
And r 2 ^
which may be written ( 154)
or (1 e-)- (l + e cos 0)~ 2 -^- = n.
Keeping powers of e lower than the third
( 1 - 2e cos 4 o e 2 cos 20 ) -y- = w,
\ JL / d/v
3
or wi = 2e sin + T e 2 sin 20 ;
4
whence = ?i + 2e sin - j e 2 sin 20
3
2e sin (nt 4 2e sin w^) ^ e 2 sin 2nt
o
2e sin nt 4 4e 2 cos ?i^ sin ?i^ -^e 2 sin
4
5
2e sin ?i^ 4 T ^' 2 sin 2t
14-0 ELLIPTIC MOTION.
163. KEPLER'S PROBLEM. To find r and 6 as functions
of t from the equations
r a (1 e cos u) .................. (1) ;
nt = u e sin u ..................... (3).
These equations evidently give r, 6, and t directly for any
assigned value of u, but this is of little value in practice.
The method of solution which we proceed to give is that
of Lagrange, and the general principle of it is this
We can develop from equation (2) in a series ascending
by powers of a small quantity, a function of e, the coefficients
of these powers involving u and the sines of multiples of u.
Now by Lagrange's Theorem we may from equation (3)
express u t \e cos u t sin u, sin 2u, &c. in series ascending
by powers of e, whose coefficients are sines or cosines of
multiples of nt Hence by substituting these values in
equation (1) and in the development of (2), we have r and
6 expressed in series whose terms rapidly decrease, and
whose coefficients are sines or cosines of multiples of nt.
This is the complete practical solution of the problem.
164. To express the true, as a function of the excentric,
anomaly.
Substituting in (2) the exponential expressions for the
tangents, and writing i for "/I, we have
whence
- e)} + iV(l - e) -
.;,/(! - e) - V(l + e)\ + ;V(1 ~e) + v '(l + e)}
V(l + --
or, putting X = -
ELLIPTIC MOTION. 141
Taking the logarithm of each side and dividing by i,
6 = u + * [e iu - -*} + g (e 2 - e- 2 -} + . . .
= u 4- 2 f X sin u + -~ sin 2u + ^ sin 3w + &c. J ......... (4).
165. To develop u in terms oft.
If we have
y = z + axf)(y) ..................... (5),
we obtain, by Lagrange's Theorem, the development
=/(*) + * (*)/' () + '
Now equation (3) may be put in the form
u = nt + e sin u,
which is identical with (5) if
y = u, z = nt, x = e, and <j> (y) = sin y.
Also, as it is the development of u that we require, we must
put
f(u) = ?/, and f'(u) = 1. Hence, by (6)
y = z + x sin z + ^ S ( sin2 *) + iT?. 3 (s) < sin3 ^ + &(X ;
and, substituting for the powers of sin z their corresponding
expressions in sines and cosines of multiples of z t
a? d fl-cos2z\ a? f d\ 2 fZsmz -sin3z\
,+*sin*+ os ^-- )+ r: ^y ( -
a* l d Y /3 - 4 cos 2^ + cos 4^
= z -I- a? sin 2r + sin 2z + ~ (3 sin 3^ sin z) +
142 ELLIPTIC MOTION.
or, substituting for x, y, z their values as above,
e' 2 e*
u = n t + e sin n t + sin 2nt + (3 sin 3nt sin nt)
Z o
e 4
+ -^ (2 sin 4?tf - sin 2nt) + &c ................ (7).
To develop sin ?/, we recur to equation (3), which gives,
after the elimination of u by means of (7),
sin M = sin nt + ^ s ^ n 2 ^ + g ( 3 s ^ n 3/ ^ ~ s i n W + &c.. . .(8).
By the application of Lagrange's Theorem to equation (3),
it is easy to deduce the following expressions :
sin 2u = sin 2nt + e (sin 3nt - sin nt) + e 2 (sin bnt sin 2n)
e 3
+ (4 sin n - 27 sin 3nt + 25 sin 5nt) -f &c.
sin 3w = sin 3/i^ + ^ (sin 4n^ sin 2nt)
g2
+ - (15 sin 5nt 18 sin 3nt + 3 sin nt) + &c.
o
&C. = &C.
Substituting these values in (4), we obtain the value
of 0, containing however the quantity X. If we take as its
e e 3
approximate value ^ + Q , and make the requisite substitu-
o
tions, we obtain
= nt + l2e- T & } sin nt + 7 e 2 sin 2nt + jg e 3 sin Snt + ......
which is correct as far as e 3 .
[The development of u in terms of t is
7=00 |
ft ss n -f- 22 / m (me) sin m nt,
m = I Wl
1 f ""
where / m (me) = - I cos m (t e sin t) dt
vrj o
is Bessel's function of the ?? th order.]
ELLIPTIC MOTION. 143
For the development of r and 6 in terms of t, the co-
efficients being Bessel's functions, see Todhunter's Treatise
on Legendre's, Laplace's, and BesseVs Functions.
166. In proceeding farther with the development, it
becomes necessary to expand X and its powers in series
ascending by powers of e. This is readily done as follows.
We have
e _ e_
- ^ E supp01
Hence E = 2 - ^ ,
from which, by Lagrange's Theorem,
and thus the value of \ p , being e p E~ p , is known.
The correct value of 6 to the fifth power of e is thus
found to be
502 03
nt -f 20 sin nt + sin 2nt + ^ ^ (13 sin 3ra 3 sin nt)
+ s^- (103 sin 4snt - 44 sin Znt)
2 . .3
05
(1097 sin 5nt - 645 sin 3nt + 50 sin nt).
1 2 6 .3.5
167. To develop r m terras o/ 1.
From (1) it is evident that all we have to do is to
develop by Lagrange's Theorem, 1 e cos u as a function
of t, from nt =. u e sin u.
To develop (1 e cos u) m terras of t.
#ere /(y) = 1 ~ e cos 2/>
/ ' (y) = e sin y ;
144 ELLIPTIC MOTION.
and the form of (j> is the same as before ; hence
1 e cos y = (1 e cos z) + x sin z (e sin z)
Hence, as before, substituting for the powers of sines their
equivalent expressions in sines and cosines of multiple arcs,
differentiating, and substituting u for y, nt for z, and e for as,
we have
fY* fft
I e cos u = -=le cos nt + H (1 cos
a 2 v
+ - (3 cos n 3 cos
o
+ Q (cos 2/i^ cos 4<nt) + &c.
o
which gives the radius vector in terms of the time.
168. Lambert's Theorem. The area of a focal elliptic
sector and therefore the time through any arc of the ellipse,
described about the focus, can be expressed in terms of the
chord and the focal distances of the ends of the arc.
If Ti t r. 2 be the focal distances of the ends and c the chord
of the arc, it is proved in Williamson's Integral Calculus,
137, that the sectorial area is
J ab [0! - </> 2 - (sin fa - sin fa)},
where fa and fa are given by the equations
+ n>-c\
-
and therefore if t denote the time in the arc,
nt = fa fa (sin fa sin fa).
CENTRAL ORBITS. 145
EXAMPLES.
(1) A particle describes an ellipse under an attraction i
always directed to the centre, to determine the law of the
attraction.
From the polar equation of the ellipse, centre pole,
cos 2 sin 2 du (I \\
if = 2 I ,- 2 - ; we have a -^ = ( ^- 2 J cos sin 0]
,,' ,
= i
and therefore the law is that of the direct distance.
(2) A particle describes a conic section under an at-
traction always directed to one of the foci, to find the law of
attraction.
In this case
and therefore P = h*tf + tt
T. D. 10
146 CENTRAL ORBITS.
(3) Find the attraction to the pole under which a
particle may describe an equiangular spiral.
(4) Find the attraction by which a particle may describe
the lemniscate of Bernouilli, the centre being the node.
(5) Find the attraction by which a particle may describe
a circle, the centre of attraction being in the circumference
of the circle.
(6) Find the attraction to the pole under which a
particle will describe the curve
r n = a n cos nd,
and interpret the result when ?i = 1. Deduce the law of
attraction for (1) a rectangular hyperbola, (2) a lemniscate,
(3) a circle about a point in the circumference, (4) a cardioid,
(5) a parabola.
(7) Prove that the attraction to the pole under which a
particle will describe the n th pedal of a cardioid varies as
_3n+8
r n+2 . Deduce the law of attraction for a circle about a
point on the circumference.
(8) A particle is projected from a given point in a given
direction with the velocity from an infinite distance, and is
under an attraction varying inversely as the n th power of the
distance, to determine the orbit.
Here P = fj,u n , and therefore
CENTRAL ORBITS.
147
Multiplying by /t 2 -^ and integrating,
_
" ~F('
Now if a be the apsidal distance,
therefore
n-l
n-l ''
du\*
d0)
+ if = a 1
integrating
n-3
= sec" 1 (au)
or
the polar equation of the required orbit.
(9) A particle, under an attraction varying inversely as
the cube of the distance, is projected from a given point with
any velocity in any direction ; to classify the paths described
according to the circumstances of projection. The curves in
question are called Cotes Spirals.
The equation of motion is
The integral of this equation involves exponential or
circular functions according as ~ is greater or less than
HI
102
148 CENTRAL ORBITS.
unity, that is, according as the velocity at an apse is less or
greater than the velocity from infinity.
I. Let be > 1, and let ^ - 1 = k 2 ; then
rr fi 2
the integral of which is
u = A<P + B<r M ..................... (2).
SPECIES 1. Let A and B have the same sign ; then
and ~
The values of A and B may in these equations be ex-
pressed in terms of the initial distance, and angle of projection ;
but we may put the equation of the curve in a simpler form
as follows. Let a be the value of 6 corresponding to an apse,
then when 6 = a, -7^ = ;
or = AJ* - Be-**,
which always gives a possible value of a ; and therefore
= Be~ ka = ^- , suppose.
CENTRAL ORBITS.
149
Substituting, au = -
+ -
Hence when = a, au=l, or a is the apsidal distance.
As 6 increases, u increases, or r diminishes ; and when 6 = oo ,
u = oo , or r = 0. Hence the curve forms an infinite number
of convolutions about the pole ; and, as it is symmetrical on
both sides of the apse, it must be as represented in the figure,
where A is the apse and the centre of attraction.
SPECIES 2. Let p >1, 5 = 0, then the equation (2)
becomes
au = e ke ,
the equation of the logarithmic spiral. The nature of the
curve will be the same if A t instead of B, vanish.
SPECIES 3. Let ^ > 1, and B negative, then by equa-
tion (2),
Putting it = 0, when 6 a, we obtain as for Species 1,
Hence, when 6 = a, u = or r = x . As 6 increases r
decreases, and when 6 is infinite r = ; so that there is an
150
CENTRAL ORBITS.
infinite number of convolutions round the pole. The curve
has an asymptote parallel to OA, at a distance -.
II. SPECIES 4. Let ^ = 1, then equation (1) becomes
the integral of which is
a,
the equation of the reciprocal spiral.
III. SPECIES 5. Let , < 1, and let 1 ~ = &*, then by
equation (1),
7 it + &U = 0.
the integral of which is
j~.
whence
du T 7 / /i \
a -vn = ^ sin K (6 a),
du
Then a is the value of 6 corresponding to an apse, and a
is the apsidal distance. The asymptotes to this curve are
easily found for any assigned value of k. One case is re-
presented in the annexed figure.
CENTRAL ORBITS. 151
(10) A particle of mass m under a central repulsion
~ is projected from an apse at a distance a with velocity
575- Find the orbit, and prove that the time from the
apse to the distance a V2 is ~ \/2 ~j- .
6 v/*
(11) A particle under an attraction inversely propor-
tional to the fourth power of the distance from a centre is
projected in any manner; for instance, from an apse with
velocity n times the velocity from infinity : determine the
orbit.
(12) A particle under a central attraction varying in-
versely as the fifth power of the distance is projected in any
manner, determine the orbit.
Here P = ^u 5 , and we have
whence
If the particle be projected from an apse at a distance a
with velocity ?i times the velocity from infinity, then
IX/ tl' -
4 a 4
and therefore (7 = ^ (/?-' 1) 4 ;
T? tt
and /t 2
2 a 2 '
Therefore ( -^ ) + u- = a *-- + - - ,
Vrf^/ n- n-a-
152 CENTRAL ORBITS.
?i 2 2 w 2 - 1
" "
-l
or
and therefore r is an elliptic function of 0.
For instance, suppose n< 1, we have
r = a en md,
2 11* I - n"
where ra 2 = and & 2 = ^~ - 2 .
(13) A body moves under a central attraction
- 2 j(a 2 + b z + c 2 ) u s - 2a 8 6V},
being projected from an apse whose distance is a (> b) with a
velocity , shew
whose equation is
velocity , shew that it will proceed to describe the orbit
r- a 2 en 2 \-b 2 sn 2
c c
the modulus of the elliptic functions being the excentricity of
an ellipse whose semi-axes are a and b.
This may be written r = a dn .
(14) If the central attraction be
and the body be projected as in the last example, prove that
the orbit will be the pedal of the ellipse with respect to the
centre.
CENTRAL ORBITS.
153
(15) A particle under a central attraction varying in-
versely as the fifth power of the distance is projected from
a given point with a velocity which is to the velocity from
f\ /n
infinity as 5 to 3, in a direction making an angle sin" 1 ~~-
5
with the radius vector; find the orbit.
Here we have
d?u u,
+ u<
But if V be the velocity of projection, c the initial value
of u,
and when
-
"92'
u = c> v= V, :. = ;
2 '
18c 2 25
' A 2 4c 2 '
Substituting and integrating we find, after the necessary
reductions,
where Ji is the initial distance, and a a constant to be deter-
mined by the position of the initial line.
154 CENTRAL ORBITS.
(16) If P = 2/i + yrf, and a particle be projected at an
C"
angle of JTT with the initial distance (R =) -, with a velocity
c
which is to the velocity in a circle at the same distance as
V2 to \/3> find the curve described.
r = R(\-0).
(17) A particle under a central attraction, varying partly
as the inverse third, and partly as the inverse fifth, power of
the distance, is projected with the velocity from infinity at
an angle with the distance, the tangent of which is \/2, the
intensities being equal at the point of projection ; determine
the orbit.
&-r = W
(18) If P = ^ (or 2 - 8c 2 ), and a particle be projected
from an apse at a distance c with the velocity from infinity ;
prove that the equation of the orbit is
r = c(e' 2e e~ 2e ).
(19) If P = 2/ji I - -J, and the particle be projected
from an apse at a distance a with velocity , prove that it
CL
will be at a distance r after a time
27 /4 ( aM '- -^-^ +rV '"- 0> )- ]
(20) The attraction tending to the centre of a circle
(2ft :i \
r H - J , find the velocity with
which a particle will describe the circle ; and shew that if
the velocity be suddenly doubled the particle will come to
an apse at the distance 3a.
CENTRAL ORBITS. 155
(21) If P = /u,r-f-, prove that the equation of the
orbit is of the form
J. = cos 2 &0 sin'^0
r 2 ~ a a b*
If the particle be projected from an apse at a distance
a = / - , with velocity ^ /JLV, prove that the equation of the
orbit is
a 2
1 + P '
and that the time of describing the angle 6 from the apse is
- tan- 1 6.
V>
(22) If a particle move under a central attraction
fiu~ + vu 3 , shew that the equation of the orbit is generally
of the form
a
\e cos (kd) '
In the case when the projection takes place at an apse,
the apsidal distance being ^ , and v being equal to /t- 2 , shew
that the equation of the path is
~
and that the time of describing an angle a is
i tan 0(0 + | sin 20) where tan0= -*-.
Determine generally the relation between the orbits
when P = fjiU 2 <f) (u) and when P = fiu*<f>(u) + vit.
(23) A particle is projected in any direction from one
end of a uniform straight line each particle of which attracts
it with an intensity proportional to the distance, prove that
the particle will pass through the other end.
156 CENTRAL ORBITS.
(24) A particle moves in an ellipse under an attraction
tending to a fixed point ; prove that the acceleration due
DD' 6
to the attraction at any point P varies as ^ p. 2 ~ppt* , where
PP' is the chord of the ellipse passing through 0, and DU
the diameter parallel to PP'.
(25) A particle describes an equilateral hyperbola about
a centre of attraction in the centre, shew that an angle
from the apsidal line is connected with the time t of its
description by the formula
* (26) If v be the velocity of a particle moving in an
ellipse about the centre, v' its velocity when the direction of
its motion is at right angles to the former direction, the time
of describing the intercepted arc = -j- sin" 1 , .
(27) A particle moves under a central repulsion which
varies as the distance from a fixed point ; shew that the
equation of the path described is
x ?/ 2 -b 2 -y x> -a? = c,
where a, b, c are constants, and determine the curve which
this equation represents.
(28) Find the time in which a particle would move from
the vertex to the end of the latus rectum of a parabola, the
centre of attraction being at the focus ; and shew that if
the velocity be there suddenly altered in the ratio m to 1
(m being < 1) the body will proceed to describe an ellipse,
the excentricity of which is (1 2ra 2 + 2m 4 )*.
(29) If the Earth's orbit be taken an exact circle, and
a comet be supposed to describe round the Sun a parabolic
orbit in the same plane ; shew that the comet cannot possibly
/ 2 \ th
continue within the Earth's orbit longer than the ( ) part
\O7T/
of a year.
CENTRAL ORBITS. 157
(30) If a particle, under a central attraction varying
inversely as the square of the distance, be projected with
a velocity equal to n times the velocity in a circle at the
same distance ; the angle a between the major axis and this
distance may be determined from the equation
tan (a - 0) = (1 - ft 2 ) tan ft,
being the angle between the radius vector and the direction
of projection.
(31) A particle describes a parabola about a centre of
attraction (oc D~ 2 ) residing in a point in the circumference of
a given ellipse, the foci of which are in the circumference of
the parabola ; shew that the time of moving from one focus
to the other is the same, at whatever point in the circum-
ference of the ellipse the centre of attraction is placed.
(32) A particle is projected from a given point with a
given velocity and is under a central attraction varying
inversely as the square of the distance ; shew that whatever
be the direction of projection the centre of the orbit described
will lie on the surface of a certain sphere.
(33) A particle revolves in a circle about a centre of
attraction in the centre, the intensity x =^ ; the strength is
suddenly increased in the ratio of m : 1 when the particle
is at any assigned point of its path, and when the particle
arrives again at the same point the strength is again in-
creased in the same ratio ; shew that the path which the
particle will describe is an ellipse whose excentricity
(34) A particle is moving in an ellipse about a centre of
attraction in the focus ; supposing that every time the particle
arrives at the nearer apse the strength is diminished in the
158 CENTRAL ORBITS.
ratio of 1 to 1 n, find the excentricity of the elliptic orbit
after p revolutions, the original excentricity being e.
*+ r i.
(35) If the attraction vary inversely as the square of
the distance, prove that there are two initial directions in
which a particle can move so that its apse line may coincide
with a given line. If j , 2 be the angles which these direc-
tions make with the initial distance c, and 2a be the length
of the apse line, prove that
/
cot a l . cot a, = 1.
a
(36) If the perihelion distance of a comet's orbit be J of
the radius of the Earth's orbit supposed circular, find the
number of days the comet will remain within the Earth's
orbit.
(37) If a comet describe 90 from perihelion in 100 days,
compare its perihelion distance with the distance of a planet
which describes its circular orbit in 942 days.
(38) In the case of planets and comets prove the follow-
ing formulae, the letters being the same as in the text,
d6
a
2 (X cos u + J\ a cos 2u + %\ 3 cos 3u + &c.).
(39) A body describes an ellipse about the focu^ : prove
that the times of describing the two parts, into which the
orbit is divided by the minor axis, are to one another as
TT + 2e to TT 2e, where e is the excentricity of the ellipse.
CENTRAL ORBITS. 159
(40) If Pp, Qq be chords parallel to the major axis of an
elliptic orbit, shew that the difference of the times through
the arcs PQ, pq varies as the distance between the chords.
(41) If a comet whose orbit is inclined to the plane of
the ecliptic were observed to pass over the Sun's disc, and
three months after to strike the planet Mars, determine its
distance from the Earth at the first observation, the Earth
and Mars describing about the Sun circles in the same plane
whose radii are as 2 : 3.
(42) Shew that the arithmetic mean of the distances of
a planet from the Sun, at equal indefinitely small intervals
of time, is
d + l J
(43) The time through an arc of a parabolic orbit
bounded by a focal chord oc (chord)*.
(44) If a circle be described passing through the focus
and vertex of a parabolic orbit, and also through the position
of the moving particle at each instant, shew that its centre
describes with constant velocity a straight line bisecting at
right angles the perihelion distance.
(45) Shew that the velocity of a comet perpendicular to
the major axis varies inversely as its radius vector.
(46) DI, D. 2 being two distances of a comet, on opposite
sides of perihelion, including a known angle, shew that the
position of perihelion may be found from the equation
TTv- /yx 2 = tan J (sum of true anomalies), tan J (difference).
V -Is i + \JJ 9
(47) In an elliptic orbit find the relation between the
mean angular velocity about the centre of attraction and the
angular velocity about the other focus, and thence shew that
when e is small the latter is nearly constant.
160 CENTRAL ORBITS.
(48) If a, ft be the greatest and least angular velocities
in an ellipse about the focus, the mean angular velocity is
(49) Find the maximum value of 6 - nt in an elliptic
orbit, and develop it in powers of e, shewing that it cannot
contain even powers.
If be this quantity,
lie 3 599e- 5
(50) If P = iJLiC 1 (1 4- k 2 sin 2 #)~ ? , find the orbit, and in-
terpret the result geometrically.
Find the equation of the orbit generally when P = fiit?f(0).
(51) Shew that if the central repulsion be constant
(=/, suppose) we have the following relation between the
radius vector and the time,
t = r rdr
and from this, with the help of the equation of constant mo-
ment of momentum, deduce the differential equation of the
orbit. Shew also how the apsidal angle may be determined.
If a particle, under a constant central repulsion, be pro-
jected from an apse with the velocity acquired from the
centre, find the orbit.
(52) A particle moves about a centre of attraction, arid
its velocity at any point is inversely proportional to the dis-
tance from the centre of attraction ; shew that its path will be
a logarithmic spiral.
(53) Shew that the only law of central attraction for
which the velocity at each point of the orbit can be equal to
that in a circle at the same distance is that of the inverse
third power, and that the orbit is the logarithmic spiral.
CENTRAL ORBITS. 161
(54) If a number of particles, describing different circles
in the same plane about a centre of attraction x D~ s , start
together from the same radius, find the curve in which they
all lie when that which moves in the circle whose radius is a
has completed a revolution.
(55) If v be the velocity, and P the attraction at distance
r in a central orbit, and if v', P', r be similar quantities for
the corresponding point of the locus of the foot of the perpen-
dicular on the tangent, shew that
Pr i/ 2 "
(56) A particle attached to one end of an elastic string
moves on a smooth horizontal plane, the other end of the
string being fixed to a point in the plane. If the path of the
particle be a circle, shew that the periodic time oc [ - - ) ,
\r - aj
a and r being the natural and stretched lengths of the string.
If the orbit be nearly circular, find the apsidal angle.
(57) A particle is describing a curve about a centre of
attraction, and its velocity x - n , find the law of attraction and
the equation of the path.
(58) A particle projected in a given direction with a
given velocity and attracted towards a given centre has its
velocity at every point to the velocity in a circle at the same
distance as 1 to ^2 ; find the orbit described, the position of
the apse, and the law of attraction.
(59) If a particle move in a circle of radius ?, about a
centre of attraction distant a from the centre of the circle,
T. D. 11
162 CENTRAL ORBITS.
shew that the time from distance r to the nearer apse is
2*r*
where is the initial attraction ; and that the periodic time is
2?rr
where </> is the attraction at the nearer apse.
(60) If the m th power of the periodic time be proportional
to the n fch power of the velocity in a circle, find the law of
attraction in terms of the radius.
(61) A particle is projected at a distance c from a fixed
centre of attraction with a velocity */~- 4 , and in a direction
s*
making an angle sin" 1 with the distance ; the intensity of
Gb
Ll/*
the attraction at the distance r being ----- ^ . Shew that
the orbit described will be a circle, of radius a.
(62) A point describes a parabola, latus rectum 4a, with
an acceleration tending to a point in the axis distant c from
the vertex : prove that the time of moving from the vertex to
i/ s
a point distant y from the axis is proportional to ^~ -- 1- y.
(63) If a body describes a parabola under an attraction
tending to a point on the axis, prove that the acceleration
(1 1 \~~ 2
^rp + 7p OP~ 2 , p being the point of
(Jr Up/
intersection of PO produced with the curve.
Also prove that the time of passing from one end of the
8 /2
ordinate through to the other = -./-.
CENTRAL ORBITS. 163
(64) A particle P describes a cycloid ABC under an
attraction tending to the middle point of the base. If
PM be drawn perpendicular to the axis OB, and PT the
tangent meet OB in T : the angular velocity of the tangent
will vary as OM . OT inversely.
(65) If r, p be the radius vector and perpendicular on the
tangent at any point of the curve described by a particle under
an attraction P towards the pole, and a force T along the
tangent, shew that
ggr = d L ( 3p dr\
_* drv d)'
dp,
For an attraction P to the pole, and a force N in the
normal, prove that
d ( ^ dr\ d
(66) A particle describes the ?zth pedal freely under
an attraction tending to a pole : find the law of at-
traction. If the curve be a rectangular hyperbola, and
the pedals be formed with respect to its centre, prove that
the nth pedal will be the orbit of a particle moving under
_
an attraction varying as r 2n-1 , where r is the distance from
the centre of attraction.
(67) A particle describes an orbit round a centre of at-
traction in a periodic time P. Straight lines are drawn from
a point to represent the accelerations of the particle at equal
intervals of time r, during a complete revolution. If P = HT,
when n is an indefinitely great whole number, shew that
these straight lines will represent a system of forces in equi-
librium. Shew also that if the attraction vary directly as the
distance, the result is true if n be not great.
(68) A particle describes an orbit about a centre of at-
traction. If the centre of attraction be replaced by the
particle, and the orbit for any complete number of revolutions
by a fine wire whose section varies inversely as the velocity
in the corresponding orbit, and every point of which attracts
11-2
164 CENTRAL ORBITS.
by the same law as the centre of attraction did, shew that
the particle will be in equilibrium : determine also the
nature of this equilibrium (1) when the attraction varies as
the distance, (2) when it varies inversely as the square of the
distance.
Shew that if the orbit be an ellipse, described about a
centre of attraction in the focus, the centre of mass of the
wire is midway between the centre and the other focus.
(69) If a uniform string under a central repulsion P per
unit of length assume the form of a certain curve, prove that
the same curve will be described by a particle of unit mass
under a central attraction PT, the velocity at any point being
numerically equal to the tension T of the string.
(70) If P = -j ----- 2 \2> an d if the particle be projected
(r c )"
from an apse at a distance nc(n > 1) with velocity which is
to that in a circle as vV "1 : n, prove that it will describe
a branch of an epicycloid, and find the time to a cusp.
(71) Shew that if an ellipse be described under an
attraction / to the focus 8, and an attraction /' to the focus
H, and 8P = r, HP = r',
df df _ (f f'\
dr dr ~~ \r r'j '
(72) Prove that if f=,f' = > the
ellipse can be described freely, and that the velocity at any
r '2 _|_ rr ' _J_ r '-2
point will be n , n being the mean motion m
2v?r'
the ellipse under an attraction ~ to a focus.
(73) A particle describes an ellipse under two attrac-
tions tending to the foci which are to one another at any
point inversely as the focal distances: prove that the velocity
CENTRAL -ORBITS. 165
varies as the perpendicular from the centre on the tangent,
and that the periodic time = -7 (-=- + -), ka, kb being the
K \ Cl/
velocities at the ends of the axes.
(74) Prove that a particle can describe a parabola under
a repulsion in the focus varying as the distance, and another
force parallel to the axis always of three times the magnitude
of the repulsion ; and that if two equal particles describe the
same parabola under these forces, their directions of motion
will always intersect in a fixed confocal parabola.
(75) Prove that a lemniscate can be described freely by
a particle under two central attractions of equal strength to
the foci each varying inversely as the distance ; and that the
velocity will be always equal to . / -- , p being the strength
of each attraction.
(76) If a particle move under an attraction /nr to the
point S, and a repulsion fifr' from the point $', prove that
dO , . dO'
a constant, where 0, & are the angles r, r' make with SS'.
(77) The velocity of a point is the resultant of the
velocities v and v' along radii-vectores r and r' measured
from two fixed points at a distance a apart. Prove that the
corresponding accelerations are
(78) A particle describes a circular orbit about a centre
of attraction situated in the centre of the circle ; prove that
the form of the orbit will be stable or unstable according as
the value of -7-, , for u = - , is less or not less than 3, P
a log u a
16G CENTRAL ORBITS.
being the central attraction, u the reciprocal of the radius
vector, and a the radius of the circle.
(79) If the equation for determining the apsidal distances
in a central orbit contain the factor (u a)?, shew that u = a
cannot correspond to an apse unless p be of one of the forms
A t <rn I O
4m 4- 2 or - s =- . If the fa,ctor u a occur twice, then a
will be a root of the equation
where < (u) is the central attraction.
(80) Examine carefully the case of an apse where the
centre of attraction coincides with the centre of curvature.
Shew that the particle will, after passing such an apse, de-
scribe a circle about the centre of attraction, but that the
motion will be unstable.
(81) A particle is projected from an apse under the
attraction '~ with a velocity - J_\_l ^ n b e i n g very
small and a the initial distance, determine the apsidal angle
and the other apsidal distance.
(82) A particle moving in an ellipse about the focus
is under a central disturbance which varies as cos k0,
where 6 is the longitude measured from the nearer apse,
and k is nearly unity. Prove that in one revolution the
apse line turns through an angle a, given by
(2?r + a) cot a = constant.
( 167 )
CHAPTER VI.
CONSTRAINED MOTION.
169. WE come now to the case of the motion of a
particle subject not only to given forces, but to undetermined
reactions. This occurs when the particle is attached to a
fixed, or moving, point by means of a rod or string, and when
it is forced to move on a curve or surface.
In applying to a problem of this kind the general equations
of motion of a free particle, we must assume directions and
intensities for the unknown reactions, treating them then as
known, and it will always be found that the geometrical
circumstances of the motion will furnish the requisite number
of additional equations for the determination of all the
unknown quantities. Thus, if the particle be attached to a
string, there is no tension of the string unless it be straight ;
and then its length furnishes an additional datum.
One case of this kind has been already treated of ( 84),
namely, that of a particle moving on an inclined plane under
gravity. There the undetermined reaction is the pressure
on the plane, which however is evidently constant, and equal
to the resolved part of the particle's weight perpendicular to
the plane.
The laws of kinetic friction are but imperfectly known,
and the few investigations which will be given of motion on
a rough curve or surface are of very slight importance.
170. The simplest case is
A particle is constrained to move on a given smooth plane
curve, under given forces in the plane of the curve, to determine
the motion.
Taking rectangular axes in this plane, the given forces
may be resolved into two, X, Y, parallel respectively to the
168
CONSTRAINED MOTION.
axes of x and y y the mass of the particle being taken as unity.
In addition there will be R, the pressure between the curve
and particle, which acts in the normal to the curve, since the
curve is smooth and there is therefore no friction.
Let P be the position of the particle at the time t ; and
T
let the forces X, Y, R, act on the particle as in the figure,
R being estimated positive towards the centre of curvature.
Draw TP, a tangent to the constraining curve at P. Then
the direction cosines of TP are
dx
dy
~ds'
and those of PR are
The equations of motion are
(1),
cffi
ds*
(2).
These two equations, together with the equation of the
given curve, are sufficient to determine the motion completely.
CONSTRAINED MOTION. 169
To eliminate R, multiply (J) by ^, (2) by | , arid add.
We thus obtain,
dx d?x dy d-y
since -j- -j + -/- -j^ = 0,
<fo ds 2 ds cfe 2
fadfo dydty dsdte Y dac ydy .-,,
d< <ft 2 + dt dt* ~dtdt*~ dt^ "eft 1 "
or, as we may write it,
d*s dx dy
~TT = ** ^T + * ~T~ 5
dtf ds as
which might at once have been obtained by resolving along
the tangent.
Now, it has been shewn in Chap. II. that if the forces
resolved into X and F are such as occur in nature,
Xdx + Ydy
is the complete differential of some function cf> (x, y).
Integrating (3) on this hypothesis, we have
supposing v to represent the velocity of the particle at the
point xy.
Suppose the particle to start at the time t = 0, from a
point whose co-ordinates are a, 6, with a velocity F.
We have, from (4),
and therefore = V * + < (a, b) - (#, y) ...... (5).
This shews that a particle, constrained to move under the
forces X, Y, along any path whatever from the point a, b to
the point x, y, has, on arriving at the latter point, the kinetic
170 CONSTRAINED MOTION.
energy increased by a quantity entirely independent of the
path pursued : another simple case of the conservation of
energy.
171. To find the reaction of the constraining curve.
Resolving along the normal PR, towards the centre of
curvature,
v Tr
B = p- z ^- r i > '
which may also be written
R^+X^-T^.
p as as
This might, of course, have been obtained from (1) and
d?x
(2) above, by multiplying them respectively by p -^ and
d*y , , ,.
p ~ , and adding.
172. To find the point where the particle will leave the
constraining curve.
For this it is evident that we have only to put R = 0, as
then the motion will be free.
This condition gives
= FcosFPR,
if F be the resultant of X and Y.
Hence
where Q is the chord of curvature in the direction PF.
CONSTRAINED MOTION. 171
Comparing this with the formula ^v* =fs ( 82), we see
that the particle will leave the curve at a point where its
velocity is such as would be produced by the resultant force
then acting on it, if continued constant during its fall from rest
through a space equal to J of the chord of curvature parallel
to that resultant. (Compare 144.)
This result is, from the analytical point of view, of little
importance ; but it is of great interest in connection with
Newton's mode of treating such questions.
173. The formulae just given are much simplified when
we consider gravity only to be acting. Taking in this case
the axis of y vertically upwards, our forces become
X = and Y=-g;
and the velocity, and the pressure on the curve, are given by
if v = V when y = k\
v 2 dx
and - = H q -7- .
p y ds
Suppose we change the origin to the point from which the
particle's motion is supposed to commence ; and take the axis
of y vertically downwards ; we shall evidently have
and if the particle starts from rest
This shews that the velocity depends merely on the
distance beneath a horizontal plane through the original
position of rest. Hence, whatever be the nature of the curve
on which a particle slides under gravity, its motion will
always be in the same direction along the curve till it rises
to the same level as that to the fall from which its velocity
is due. If it cannot do so, its motion will be constantly in
the same direction ; if it can, its velocity will become zero,
and the particle will then either come permanently to rest, or
return to the point from which it started.
172
CONSTRAINED MOTION.
174. To find the time of a particles sliding down any
arc of a curve under gravity, from rest at the upper extremity
of the arc.
Taking the upper extremity as origin and the axis of y
vertically downwards ; we have
ds _
di~
ds
and
.(1)
if ?/! be the vertical co-ordinate of the lower extremity of the
given arc.
Or, taking the lower point as origin, and axis of y upwards,
we have, since in this case v tends to decrease s,
ds , P> ds ,
! =
7 ./ff
.(2).
175. To find the time of descending from rest at any
point of an inverted cycloid to the vertex.
Taking formula (2) ; since in this case the vertex is the
CONSTRAINED MOTION. 173
origin, and the axis is the axis of y, we have from the figure
s = OP = 2 chord OP' = 2J(AO. ON) =
if a be the radius of the generating circle.
ds /2a
-7- = / ;
cfy V y
Hence
and
dy
m / ? r
V gio
/ /a 1 2yV
= . / - vers- 1 - 1 ,
\v g 2/1/0
/
'v* ;
which is independent of y lt that is, of the point from which
the particle begins its descent.
The reason of this remarkable property will be more
easily seen if we take the formula for the acceleration in the
direction of the arc. We have thus
(since OP' is parallel to the tangent to the cycloid at P)
= -gsm(OAP')
OP'
-9
OA
or the acceleration is proportional to the distance -from the
vertex measured along the cycloid.
176. A particle, under gravity, moves in a vertical circle,
to determine the motion.
Taking the vertical diameter as axis of y, and its lower
extremity as origin, the equation of the circle is
x =
174 CONSTRAINED MOTION.
But dt
if we suppose the motion to be due to the level y^ above the
lowest point ; and therefore
dt _ a I ^ ,
^ ~~ " ~//o~\ /(/. ,,.\ /o^,,, 7]2\1 \ /*
I. Suppose yi_ less than 2a, the particle will then oscil-
late, and we must put y = y\ sin 2 <f>, and then
* =
an elliptic integral of the first kind, of which </> is the ampli-
tude and k the modulus.
Instead of considering t as a function of <, we must con-
sider (j) as a function of given by this equation, and then
with Jacobi's notation put
and therefore y y\ sn2 fu t>
and the time of vibrating from rest to rest is therefore
2J5TA/ - , where K is the complete elliptic integral
If the oscillations are indefinitely small, k = and K =
4
therefore the time of a complete oscillation is 2?r A/ - .
and the time of vibration from rest to rest is TT A / - , and
9
CONSTRAINED MOTION. 175
II. Suppose ^ greater than 2a, the particle will then
perform complete revolutions, and we must put
y 2a sin a $,
which gives
te i A
V
and therefore </> = am / ,
and y = 2a sn 2 A / - T ;
V CL rv
and the time of a complete revolution is
9
When the particle is supposed to be suspended by a
thread without weight, it becomes what is termed a simple
pendulum. Such a machine can exist only in theory, but
Dynamics furnishes us with the means of reducing the calcu-
lation of the motion of such a pendulum as we can construct,
to that of the simple pendulum. It is evident that by its
means we may determine the value of g, if the length of the
pendulum, its arc of oscillation, and the number of vibrations
it makes in a given time, be known. Since gravity decreases
(according to a known law) as we ascend above the Earth's
surface, the comparison of the times of vibration of the same
pendulum on the top of a mountain and at its base would
give approximately the height. Similarly, the comparison of
the times of vibration above ground, and at the bottom of a
coal-pit, gives information as to the Mean Density of the
Earth. One of the most important applications of the pen-
dulum is that made by Newton. It is evident that if the
weight of a body be not proportional to its mass, the value of
g will be different for different materials. Hence the fact
that pendulums of the same length vibrate in equal times at
the same place whatever be the matter of which the bob is
made, proves, by means of the above formula, the truth of
176
CONSTRAINED MOTION.
one
uuc part of the Law of Gravitation : viz. that, ceteris paribus,
the attraction exerted by one body on another is proportional
to the quantity of matter it contains, and independent of its
quality.
177. We may determine the motion of the simple circular
pendulum by resolving along the arc. The details of the
process will shew the nature of the Elliptic Function trans-
formations.
Let be the centre of the circle, OA the vertical radius,
P the position of the particle at the time t, and let AOP = 0.
Suppose the motion to be due to the level EG : then we
must distinguish the two cases in which BG does and does
not cut the circle.
I. Suppose BG to cut the circle in B and C, and let
= a.\ then the pendulum will oscillate through an
angle 2 a.
The equation of motion will be
But
therefore
x = (1.0,
CONSTRAINED MOTION. 177
7/J
Multiplying by -r and integrating
Let BC cut OA in D\ on AD as diameter describe a
circle, and let PM drawn perpendicular to OA cut this circle
in Q, and let
n
Then since AM= a (1 cos 0) = 2a sin 2 ^
and AM AD sin 2 </> = 2a sin 2 ^ sin 2 </> ;
a
therefore sin = sin sin <f>.
Substituting in equation (1), we obtain
/a f* dd> . OL
and t = A / - / -77- -- .^ . . x , k = sm s .
V #J V(l - A; 2 sm 2 <#>) ' 2
Therefore as before
sin ^ = &sn
2
cos ^ =
therefore ^1P = ^4 5 sn A - 1,
V a
T. D. 12
178
CONSTRAINED MOTION.
and
as before.
dt
II. Suppose BC not to cut the circle, then the pendulum
will perform complete revolutions; let
Then, as before, resolving along the tangent
and
^ COS0
a
therefore
= - d -cos
CONSTRAINED MOTION. 179
Let AEP = <I>, therefore 6 = 2$,
JL i _ fr sin 2 <f>),
Therefore c/> = am
and
as before.
In the separating case 5(7 touches the circle at its highest
point E, and y l = 2a. A; = 1 ; therefore
o cos</>
1 -f sin
1 - sin
which determines the motion completely.
Since < = TT when t = <x> , the pendulum will continually
approach the highest position, but never reach it.
122
180
CONSTRAINED MOTION.
178. Let DM be a horizontal line, and let DA be taken
equal to the tangent from D to the circle BPC\ whose centre
C is vertically under D. Also let PAQ be any line through A,
cutting in Q the semicircle on AC. Also make E the image
of A in DM. Then if P move with velocity due to the level
of DM, Q moves with velocity due to the level of E ; so that
we have the means of comparing, arc for arc, two different
continuous forms of pendulum motion, in one of which the
rotation takes place in half the time of that in the other.
Let a) be a small increment of the circular measure of
AP PC
BAP, then arc at Q = CA.a), arc at P =
Also,
velocity at P =
Hence,
velocity at Q
PQ
CA.PQ
AP.P(
I /T
IV AC'
AP
PC
.PQ.
CONSTRAINED MOTION.
181
But
\/CP'-CR.CA (where QR is horizontal)
Hence,
^ /CP'-CA*
CA
velocity at Q = - *Jg . ER.
Thus Q moves with velocity due to the level of E, and
constant acceleration
AC*
We have at once the means of comparing continuous
rotation with oscillation, as follows :
Let two circles touch one another at their lowest points ;
compare the arcual motions of points P and p, which are
always in the same horizontal line. Draw the horizontal
tangent AB. Then, if the line MPp be slightly displaced,
arc at P __AO
arc at p
Mp_AO /aM.MO _AO /
' aO ~ aOV ~AM'7MO ~ aO V
aM
AM '
182 CONSTRAINED MOTION.
Thus, if P move, with velocity due to g and level a,
continuously in its circle, p oscillates with velocity due to
g . . na and level AB.
Combining the two propositions, we are enabled to
compare the times of oscillation in two different arcs of the
same or of different circles.
It is obvious that the squares of the sines of the quarter
arcs of vibration which the combination of the above pro-
cesses leads us to compare are (in the first figure),
CA , C'B
CE WD res P ectlvel y-
Calling them and 7^, and putting DA = a, AC = e, we
have
a + e +
Hence
4
1 k
or
k, i+k'
Lagrange's transformation is equivalent to
sin cp = ,
1 + k sin- d
which, for limits and sin" 1 j for 6, gives and sin" 1 j- for </> ;
CONSTRAINED MOTION. 183
and we thus have
"- 1 d(f> 2&i f""- 1 d0
o
whose application to the pendulum problem is obvious.
Proc. R.S.E., 18712.
179. To find the pressure between the circle and the
particle, or the tension of the string.
The reaction R being measured positively as a tension
between the particle and the centre,
^2
R = h g cos 6.
In the figure of 177, let AD = y l} EC being the level
to which the motion is due ; then
1
and therefore
R = g j^l _ 2 (1 - cos 0) + cos 01
--
This expression for R admits of the value zero, if
^2
It may happen however that when the particle oscillates,
the points thus found may not lie within the arc which the
particle passes over.
The particle will oscillate if y l < 2. Now in order that
the points where R vanishes may lie within the limits of
184 CONSTRAINED MOTION.
oscillation, the value of cos 6 for the former must not be less
than that for the latter, and therefore
or y, <t a.
Hence, if a< y l <-^a, there will be a point at which R
vanishes ; and if the particle be moving on the concave side
of a smooth circle, or be attached by a string to a fixed point,
the circular motion will cease at this point ; the particle will
fall off the circle in the one case, and the string will cease to
be stretched in the other.
If, however, the particle be confined in a circular tube, or
attached to the centre by a light rigid rod, the pressure on
the particle will act outwards from the centre, or the stress
in the rod will change from a tension to a pressure.
Beyond these limits it is evident we shall have complete
revolutions if yi>~ a > an d oscillations if y l < a, without
ft
change of sign of R, or without the string becoming slack.
Also, by what we have before shown, if the particle be
constrained by a circular tube or light rigid rod, it will
oscillate if y^ < 2a ; if y^ = 2a, the particle will reach the
highest point after the lapse of an infinite time, and if
2/j > 2a, the particle will revolve continuously.
180. Two points being given, which are neither in a
vertical nor in a horizontal line, to find the curve joining them,
down which a particle sliding under gravity, and starting
from rest at the higher, will reach the other in the least possible
time.
The curve must evidently lie in the vertical plane passing
through the points. For suppose it not to lie in that plane,
project it orthogonally on the plane, and call corresponding
elements of the curve and its projection o- and a-'. Then if a
particle slide down the projected curve its velocity at <r' will
CONSTRAINED MOTION. 185
be the same as the velocity in the other at a. But cr is never
less than a', and is generally greater. Hence the time through
cr' is generally less than that through cr, and never greater.
That is, the whole time of falling through the projected curve
is less than that through the curve itself. Or the required
curve lies in the vertical plane through the points.
Taking the axes of x and y, horizontal, and vertically
downwards, respectively, from the starting point; if x be the
abscissa of the other point, the time of descent will be
Applying the rules of the Calculus of Variations, we have,
since V or k - is a function of y and p, the condition
. vjr
tor a minimum,
the differential coefficient being partial.
This gives ^!+>=- _
or )/y>J(l+p*) = = ^ a suppose.
Hence
+p*) / a
-^ ' = A / - - ,
p V -y
the differential equation of a cycloid, the origin being a cusp
and the base the axis of x.
This is a problem celebrated in the history of Dynamics.
The cycloid has received on account of this property the name
186 CONSTRAINED MOTION.
of Brachistochrone. Farther on we propose to investigate
the nature and some of the properties of Brachistochrones
for other forces besides gravity. For an investigation not
directly involving the Calculus of Variations see Appendix.
181. A particle moves on a smooth plane curve under
an attraction to a fixed centre in the plane of the curve ; to
determine the motion.
Let r=f(6) be the polar equation of the constraining
curve about the centre of force as, pole, and let P = <j>(r) be
the attraction on a particle whose distance from the centre
is r.
Resolving along the tangent at any point,
---p- r m
dP ds"
Hence, (' = &'- = C -ft(r)dr ............... (2).
Equation (2) contains the complete solution of the problem
so far as the motion is concerned ; since, by means of the
equation of the curve, either r or s may be eliminated from
it, and if the resulting differential equation be integrable, it
will give s or r in terms of t.
For the pressure on the curve. Resolving along the
normal at any point, p being the radius of curvature, we
have
an expression which by means of the foregoing equations will
give R in terms of t or r.
Hence the solution is complete.
182. When the constraining curve is tortuous.
All we know directly about R is that it is perpendicular
to the tangent line at any point.
CONSTRAINED MOTION. 187
Resolve then the given forces acting upon the particle into
three, one, T, along the tangent, which in all cases in nature
will be a function of x, y, z and therefore of s ; another, N,
in the line of intersection of the normal and osculating planes
(or radius of absolute curvature) ; and the third, P, perpen-
dicular to the osculating plane.
Let the resolved parts of R in the directions of N and
P be RH R z . Now the acceleration of a point moving in any
manner is compounded of two accelerations, one -y or v -r-
along the tangent to the path, and the other - towards the
centre of absolute curvature, the acceleration perpendicular
to the osculating plane being zero ; and therefore
-*-
This equation together with the two of the curve is suffi
cient to determine the motion completely.
Also -=R, + N... ...(2),
P
R l and T being considered positive when acting towards the
centre of absolute curvature; this equation determines R^
Now R% is the reaction which prevents P's withdrawing
the particle from the osculating plane ; and therefore
R^-P ........................... (3).
(2) and (3) give the resolved parts of the pressure on the
curve.
Also R <J(Ri 2 + RJ), and its direction makes an angle
7? \
-p- 2 1 with the osculating plane.
183. In Art. 17") we arrived at the remarkable property
of the inverted cycloid, that a particle falling under gravity
from rest at any point of the curve reaches the lowest point
188 CONSTRAINED MOTION.
in the same time, whatever be the point of the curve from
which it starts. Let us find for what forces an analogous
property is possessed by any other given curve.
Let the forces resolved along the curve have a component
= (f)'(s), where s is the distance from the point to which
the time of fall is constant : then,
If the particle starts at a distance k from the fixed point,
the velocity = when s = k. Hence the corrected integral
of (1) is
ds
fl
and we have \/2 T = I
if T be the time of fall to the fixed point, which is by
hypothesis to be independent of k.
Put s = kz, the limits of z are 1 and 0, and
kdz
and, that this may be independent of k, we must obviously
have
This may be put in the form
4>(k)
from which, by inspection, we obtain
CONSTRAINED MOTION. 189
Or we might have proceeded as follows.
Put ^ = ^fe, then ^ (k) - z^ (kz} =fz.
By differentiation with regard to lc,
This shows that kPty' (k) is an absolute constant.
Hence, or by (2),
<$>' (s) = Cs.
Thus, by (1), 2 = -^ ........................ ( 3 )'
that is, the resolved force along the curve must be propor-
tional to the arcual distance from the fixed point.
Hence, if X, Y, Z be the impressed forces,
is the condition they must satisfy at every point x, y, z of the
given curve. For such forces the given curve is said to be a
Tautochrone.
By 90, the time of descent is
184. To find the Brachistochrone for a particle subjected
to any forces which make Xdx 4- Ydy + Zdz a complete dif-
ferential of three independent variables.
Generally
*=P,
between proper limits, is to be a minimum ; and therefore,
taking its variation,
r^^J n J.
o (i).
190 CONSTRAINED MOTION.
But the equation of energy is
i v 2 = j(Xdx + Ydy + Zdz) :
and gives vBv = X$x + YSy + ZBz,
or rfs 6v = (A ox + J oy + z/o^) a^ (2).
Again ds 2 = d^ 2 + df + dz 2 ,
ds ~ , ^ , da?
and
Hence (1) becomes, by (2) and (3), and since d and
follow the commutative law,
dz
fl (dx* dy ^ dz . \"|
= - -Ji ^ + ^ % + ^7 ^
l_v 2 V^ dt ' dt J ]
-. dy ^ d^ . \)
OP + 7. ow H- ji o-er ) >
d* l dt 1}
by integrating the first term by parts. The integrated terms
in [] belong to the superior, those in {} to the inferior, limit.
But, if the terminal points are given, we have at both limits
&e = 0, &/ = 0, &z = 0,
and therefore the terms independent of the integral sign
vanish. In order that the integral may be identically zero,
we must have, since $x, Sy, Sz are independent,
CONSTRAINED MOTION. 191
with similar expressions in y and z. The elimination of t,
ds
and v or -7- , from these equations will give us the two diffe-
at
rential equations of the curve required, the forces X, F, Z
being by hypothesis functions of x, y, z only.
185. But without getting rid of v we may prove two
properties common to all such Brachistochrones.
Eliminating t from (4) we have
__/_, | _i Q
ds \v ds) v- ~
^x dvdx v
or v z -r-, V-JT- + X = (o),
ds* dsds
with similar expressions in y and z.
e in order by '>
it
X *? + M *& + *'
Multiplying these in order by X, /A, v and adding ; if we
take X, fj,, v such that
.(6),
= o'
we shall have also
i^ = .................. (7).
Now (6) shows that the line whose direction cosines are
as A,, p, v, is perpendicular to the radius of absolute curvature
of the path, and also to the tangent ; that is, it is normal to
the osculating plane. Also by (7) the same line is perpen-
dicular to the resultant of X, F, Z.
Hence, the osculating plane at any point contains the
resultant of the impressed forces.
Again, if p be the radius of absolute curvature,
192 CONSTRAINED MOTION.
and its direction cosines are
dte dfy d*z
p <te' p di 2 ' p <& :
therefore, multiplying equations (5) by
and adding ; noting that, since
dx\*
)
we have
da; d z x dy d*y dz d*z __
we obtain the equation
p
or the normal component of the impressed forces in a
Brachistochrone is equal and opposite to the normal com-
ponent of the forces which with the same velocity would
cause the Brachistochrone to be described freely.
The velocity in the curve supposed to be a Brachis-
tochrone or a free path being the same, the tangential
component of the impressed forces must be the same, and
therefore if we reflect the impressed force in the tangent at
every point, the Brachistochrone becomes a free path, and
vice versa; in this way from the known properties of free
paths we can find for what forces they are Brachistochrones
and conversely.
Thus from the properties of free parabolic or elliptic
motion we obtain that, a parabola for a constant repulsion
from the focus, or an ellipse for a repulsion from one
focus inversely as the square of the distance from the other
focus is a Brachistochrone, the circle of zero velocity being
evanescent.
CONSTRAINED MOTION. 103
186. If the terminal points are not definitely assigned
(if, for instance, it be required to find the line of swiftest
descent from one given curve to another) we have no longer
at the limits ; but, with the requisite modifications, the pro-
cess in 184 enables us to find the proper conditions in any
case. Such questions, however, involve difficulties belonging
rather to Calculus of Variations than to Kinetics.
Thus, suppose that the final point of the path is to lie on
we have
Also that [] may vanish, which is necessary in order that
may be zero, we must have
Now the only relation between 8#, By and $z is (1), to
which (2) must therefore be equivalent : hence
dx dy dz dF dF dF
dt ' dt ' dt" dx ' dy ' dz'
These equations show that the moving particle meets
the terminal surface at right angles. A similar condition is
easily seen to hold if the initial point of the path is also to
lie on a given surface, provided the whole energy be given
and the given surface be an equipotential one. If it be not
equipotential, terms depending on &r , By , &Z Q) will appear in
the integral and must be taken along with {}.
If a terminal point is to lie on a given cume the condition
is to be determined in a similar manner.
T. D. 13
104 CONSTRAINED MOTION.
187. A particle moves under given forces on a given
smooth surface; to determine the motion, and the pressure on
the surface.
Let
F(x t y,z) = Q ..................... (1)
be the equation of the surface, R the reaction, acting in the
normal to the surface, which is the only effect of the con-
straint. Then if X, /JL, v be its direction cosines, we know
that
dF
/i
V \\d
d*
with similar expressions for yu, and v\ the differential coeffi-
cients being partial.
If X, F, Z be the impressed forces on unit of mass, our
equations of motion are, evidently,
fty _
(3).
Multiplying equations (3) respectively by
dx dy dz
di' ~di' ~dt*
and adding, we obtain
dz&z _l
dt dt z dt~dt z dt di? "2 dt
= X d +Y d / f +Z% ..... (4).
dt dt dt
CONSTRAINED MOTION. 195
R disappears from this equation, for its coefficient is
. dx dy dz
and -vanishes, because the line whose direction cosines are
proportional to -7-, &c. being the tangent to the path, is
(it
perpendicular to the normal to the surface.
If we suppose X, Y, Z to be forces such as occur in nature,
(Chap. II.) the integral of (4) will be of the form
i* = 0(a,y,*) + C .................. (5),
and the velocity at any point will depend only on the initial
circumstances of projection, and not on the form of the path
pursued.
To find R, resolve along the normal, then
which gives the reaction of the surface; p being the radius of
curvature of the normal section of the surface through the
tangent to the path.
188. To find the curve which the particle describes on the
surface.
For this purpose we must eliminate R from equations (3).
By this process we obtain
d*x y d?y v ttz
dt*~~ A d?~ Y df*~*
two equations, between which if t be eliminated, the result is
the differential equation of a second surface intersecting the
first in the curve described.
If there be no impressed forces, or if the component
of the impressed force in the tangent plane coincide
with the direction of motion of the particle, then the oscu-
lating plane of the path of the particle, which contains the
132
196 CONSTRAINED MOTION.
resultant of R and the impressed force, will be a normal
plane, and therefore the path will be a geodesic on the
surface.
Thus a particle under no forces on a smooth (or rough)
surface will describe a geodesic.
189. A particle moves on a surface of revolution, under
gravity acting in a direction parallel to the axis of the
surface; to determine the motion.
Take the axis of the surface as that of z, the equation
may be written
This may be put in the form
/(/)-*,= 0,
if p be the distance of any point in the surface from the axis.
Equations (6) become
dfa dfy d?z_
dt* ~dt* ~<&~ g
The first two equal terms give us, for the projection of
the motion on a horizontal plane, the equation
But if 6 be the angle between the plane containing p and
the axis of z, and a fixed plane through that axis ; we see
that this is equivalent to
7/1
p 2 -= = const. = h ..................... (8).
at
If the motion be due to the level k, the second integral of
equations (7) is
CONSTRAINED MOTION. 197
Let u = - , and z = ^>(n) be the equation of the surface ;
then
*
dz , ,. .du
5= *"*<"> 5* ;
therefore
and differentiating with respect to 6, and dividing by ^ ,
the differential equation of the projection of the path on a
horizontal plane.
If we omit the term containing g, we see that the above
equation will represent the projection of a geodesic on the
given surface.
190. Suppose the motion to take place in a spherical
bowl; or let the particle be suspended by a string from a
fixed point.
This is the most general motion of the Simple Pendulum.
Let us take the centre as origin, and the axis of z
vertically downwards.
Then F(x, y, z} = x> + f + * 2 - a =
is the equation of constraint, and the equations of motion are
d?z
198 CONSTRAINED MOTION.
.2 /t?9\^\
w-f -- 1 / \A/w
Hence, j( jfi
if the motion be due to the level Z Q .
T> T1 f , r ** y .. ^ x Q .
dy dx ,
or x ^- y fi
., dx dy _ dz
by the equation of the surface.
Squaring and adding (2) and (3),
(W+\&
and therefore from (1),
and therefore is an elliptic function of t.
The motion will be comprised between two horizontal
circles, and if the depth of these circles below the centre be
b + c and 6 c, the cubic in z in the right-hand side of (4)
must have roots 6 + c and b c, and if d be the third root
If we suppose the particle initially on the lower circle
and put
z = (6 + c) cos 2 <f> + (b c) sin- <
c cos
CONSTRAINED MOTION. 199
then z b c = 2c sin 2 <f>,
z b + c = 2c cos 2 0,
and therefore
72
where & = f : j >
o + c ct
and therefore </> = am ( K , k \ ,
where ?-ft-iJ< +e -*
Therefore the vertical motion of the bob of the pendu-
lum will be the same as that of a point on a simple circular
a 2
pendulum of length -- performing complete revolutions in
c
the same periodic time 2T as the spherical pendulum.
Now (z - b - c) (z b + c) (z d)
h*
= & z^z- o?z + a?z + -5- ;
therefore 26 + d
200 CONSTRAINED MOTION.
Therefore d =
26
and k*
-c 2
If -\/r denote the angle which the vertical plane through
the pendulum has turned through in the time t, then
dy dx .
Now
putting
u= t; and therefore
To reduce these expressions to Jacobi's normal form of
the third elliptic integral, we must put
sn 2
T^ -ti ) - j - , ./x oil WM ;; ,
a+6+c a-b-c
and then a x and c/ 2 will be real.
Therefore sn 2 (i^
CONSTRAINED MOTION.
201
Now
6 + c-d
sn 2 ia = ,
a-b-c
a d
en a
a o c
a _&_j_ c
a 6 c
- (6* - c 2 ) d - a%
d)(b-c +
+ d)(b-c
_
T 2
and therefore
(a + 6 + c) (a + b - c) (a - b + c) (a - b - c) .
c)
- en 2 (tc^ + K) dn 2 (i^ +
T*
and similarly,
1 //- 1
-' cn a wig dn a t
4 a 2 (a-6-c) 2 ^/
202 CONSTRAINED MOTION.
Therefore
en (zaj + K) dn (ia^ + K) en ia. 2 dn m 2
dyjr . sn (ictj 4- K) . sn ia 2
u + l 1 - # 2 sn 2 m 2 sn 2 w '
and therefore, measuring -^ from the lowest position of the
pendulum,
. Jen (ich + K ) dn (ia^ 4- ^0 en ia z dn m 2 )
( sn (zOj 4- .ff") sn ia^ j
^11 (w, mj + K) + til (^, m 2 )
_ [c? log sn (ia! + ^T) rf log sn ia z
( rfd! C^0t 2
d log et
1. -
1 2 ^
(d log ^f (to! + K) d log # m 2 )
~ -~
,
" ~e* 77~ rrt
1 . . O (u - zog - K) e (M - to.)
where u = K -~, } and k' is the modulus complementary to k,
CONSTRAINED MOTION. 203
If we put i|r = ^ + i/r',
j, tf) dlog H(a^
where - = ^4-- --
f , 1 . , %(u - ia, - K) @ (u - t
and -
then M> will be the apsidal angle, and ^ the mean angular
velocity of the vertical plane through the pendulum : also
i|r', the periodic part of o/r, will be such that
and therefore
lS(u-{-ia 1 +K)@(u-{-ia 2 )
L ~
a periodic function of t, of period T.
191. An interesting special case is that of the Conical
Pendulum, as it is called, when the particle moves in a
horizontal plane and therefore in a circular path, the string
describing a right circular cone whose axis is vertical.
Here z is constant and equal to 6, c being zero; and
d^lr h
therefore ^ = a^T"
h* 222
where
26
and therefore -^ = , /| ,
and the time of a complete revolution is
204 CONSTRAINED MOTION.
depending only on the depth of the plane of motion below
the point of suspension.
If the motion be slightly disturbed, the period of a
vibration, putting
k = Q,K = 1 ^, and *? = ~~,
*>n
becomes
and the apsidal angle of the projection of the motion on a
horizontal plane
192. To determine the nature of the small oscillations
executed by a particle, under gravity, about a position of stable
equilibrium on a smooth surface.
The tangent plane at the position of equilibrium must be
horizontal, and the contiguous portion of the surface must
be synclastic and evidently lie above the tangent plane in
order that the equilibrium may be stable.
If p, p l be the radii of curvature of the principal normal
sections, and if the axes of x and y be tangents to these
sections respectively, at the point of contact with the hori-
zontal plane, we know by Analytical Geometry that the
equation of the surface in the immediate neighbourhood of
the origin is of the form
2 *-* 2 _^0 .................. (1).
P Pi
The equations of motion of the particles are, as in 187,
<&y_
.(2).
CONSTRAINED MOTION.
205
If x and y are small, z is of the second order of small
quantities by (1) and may therefore be neglected, as may
x
Hence X = - , /x, = , y = 1, approximately. Elimi-
nating R from equations (2), we have
^--9 X
d?~ p
d?
(),
which show ( 177) that the motion consists of simultaneous
simple pendulum small oscillations in the principal planes,
the lengths of the pendulums being the corresponding radii
of curvature.
The annexed cut shows a very simple arrangement, due to
Prof. Blackburn of Glasgow, by which this species of con-
strnint may easily be produced. Three strings are knotted
together at the point (7, the other ends A and J5 of two of
200 CONSTRAINED MOTION.
them are attached to fixed points, and the third supports the
particle D. Suppose CE to be vertical, then the small oscil-
lations of D will evidently be executed as if on a smooth
surface whose principal planes of curvature at D are in, and
perpendicular to, the plane of the paper. The radii of curva-
ture in these planes are CD and DE respectively.
If we put - =?i 2 , and ^ = nf, the integrals of (3) are
P PI
x = A cos(nt + B) [
? = cos "
The curves corresponding to these equations are very in-
teresting, but we cannot enter at length on the consideration
of them. We may take, as a special case, that in which
DE =4fCD; in which therefore
x = A cos (nt + B)
The circumstances of projection determine in each case the
particular curve described a few of the principal forms are
sketched below, the last of which is a portion of a parabola.
When n^ is nearly, but not exactly, equal to 2n, the curve
described is always for a short time approximately one of the
above figures, but its form slowly passes in succession from
one member of the series to the next, completing the round
when one pendulum has executed one more or less than twice
as many complete oscillations as the other.
193. We must next consider the effect of the earth's
rotation upon the motion of a simple pendulum. Strange
to say it was left for Foucault to point out, in February
1851, that the plane of vibration of a simple pendulum
suspended at either pole would appear to turn through 4
CONSTRAINED MOTION. 207
right angles in 24 hours the plane, in fact, remaining
constant in position while objects beneath the pendulum
were carried round by the diurnal rotation. At the equator,
it was pretty obvious that no such effect would occur, at
least if the original plane of vibration was east and west.
By some process, of which he gives no account, he arrived at
the result that the plane of oscillation must, in any latitude,
appear to make a complete revolution in 24 h x cosec. lat.
This curious result has been amply verified by experiment.
194. The equations of motion of the pendulum, referred
to rectangular axes fixed in space and drawn from the earth's
centre, the polar axis being that of z, are obviously
with similar expressions in y and z\ a, b, c, being the co-
ordinates of the point of suspension, T the tension, I the
length of the string, and X, Y, Z the components of gravity.
The equations of motion referred to a new set of axes,
parallel to the former, but drawn through the point of
suspension, are
m
&c. = &c.
Let us now refer the motion to axes turning with the
earth, but drawn from the point of suspension. If the axis
of f be drawn vertically, and the axes of rj, respectively
southwards and eastwards ; and if cot be the angle at time t
between the planes of xz and 77, X being the co-latitude of
the point of suspension, we have at once (assuming that f
intersects z)
A . . A A
cos x% = sin X cos at, cos xrj = cos X cos cot, cos #f = sin cot,
A A A
cos yt; = sm X sin cot, cos yr) = cos X sin cot, cos yf = cos cot,
A A t A
cos z% cos X, cos zjj = sin X, cos z% = 0.
208 CONSTRAINED MOTION.
By means of these expressions we can at once find the
values of x a, y b, z c in terms of f, 77, f, t, as follows :
x a = % sin X cos cot + 77 cos X cos cot f sin &>,
y b = % sin X sin cot + rj cos X sin &) + f cos &),
# c = f cos X ?? sin X.
Let 7 be the acceleration due to the attraction of gravity
alone, and v the angle (nearly equal to X) which its direction
makes with the polar axis. [We have above in effect as-
sumed that its direction lies in the plane of z%, as we have
assumed that the axis of f intersects the polar axis, while we
know that the centrifugal force lies in their common plane.]
Let r be the distance of the point of suspension from the
earth's centre, p the angle its direction makes with the polar
axis. Then
a = r sin p cos cot, b r sin //, sin cot, c r cos /*.
With these data equations (1) become
sin X ( -i ft) 2 cos cot 2o> ~ sin cot
( -ij[ fft) 2 J
+ cos X -, Tjo) 2 j cos (ot 2co~^ sin o>
f -
2 sin wi - 2o> cos
T
= y- (f sin X cos o> + 17 cos X cos cot f sin <)
677i
7 sin v cos o> + rear sin /z cos o>.
sin X (-jjj -?a> 2 ) sin cot + 2a> ^ cos o>
COS X ( 7^ y w * ) Sm 6>^ + 2ft) TT COS &>
1355 fa) 2 J cos cot 2ft> ~ sin cot
T
= y (f sin X sin cot-{- r) cos X sin a) + fcos ft))
fc?7l "
7 sin v sin cot + ?*&> 2 sin /A sin cot.
CONSTRAINED MOTION. 209
&% d*rj . T , fc
^-cosX- - smX = -y- (f cos X- 77 sin X) 7cosi/.
As we contemplate small vibrations only, we may treat
f as being practically equal to I, and omit its differential
coefficients. We also admit powers and products of rj, f
and terms in o> 2 , except those in which it is multiplied
by a large quantity. For it is known that the centrifugal
force at the equator is about j, Q -^th of gravity, or that
approximately
With these simplifications our equations become
cos \ I -^ cos cot 2co-j{ sin cot } ^ sin cot 2o> -=| cos eat
\dt- dt J dt 2 dt
T
~~j~ (~~ I sm ^ cos cot + rj cos X cos cot fsin cot)
7 sin v cos cot + rco 2 sin /j, cos cot.
cos X (-3 sin o> + 2o> -^ cos oif] + -7^ cos co^ 2o> -^ sin o>^
VcZ^ dt J dt' 2 dt
T
= j ( I sin X sin cot-^-rj cos X sin cot + f cos cot)
7 sin y sin cot + ro> 2 sin JJL sin o>.
j-j yr
-=-^ sin X = j (I cos X + 17 sin X) 7 cos z/.
The two first may be put in the form
?2 J)* rn
,p cos X 2(0 -j- = T ( I sin X + 77 cos X) 7sin v+rco 2 sin (JL.
dy d* T .,
2ft) , . COS X -V7T, = T-
d^ rf^ 2 ^m
But, when rj = 0, f= 0, we have T = nig, so that
</ sin X 7 sin v + rco' 2 sin /A = 0,
g cos X 7 cos i/ = 0,
T. D. H
210 CONSTRAINED MOTION.
and our equations become
-yv cos X - 2o> -jj = ( -- a ) sin X - y rj cos X
dt z dt \m ) lm
d* T
cos X -Tr 2 = r~
C& 2 01*
- y^ sin X = ( -- a } cos X -}- , 77 sin X.
eft 2 \ra / w
The first and last give
-- q = 2o> ^- sin X,
m 9 dt
and therefore, to the degree of approximation above deter-
mined upon,
(it U/C/ V | Xfjx
(*)'
These are the equations of the motion of the bob, referred
to a horizontal plane fixed to the earth. If we omit the
middle terms, which obviously depend upon the earth's
rotation, we fall back upon the equations of 192.
195. To interpret equations (2) it is convenient to
employ a second change of co-ordinates to refer the motion
to axes revolving uniformly in the plane of rj, with angular
velocity O. If 5, ? be the co-ordinates referred to the new
axes, we have by Analytical Geometry
i) = g cos nt $ sin lt, = g sin fit + ? cos fU,
the substitution of which in (2) leads to the equations
if we make the assumption
fl = a) cos \ ........................ (4),
and omit as before terms of the order w 2 .
CONSTRAINED MOTION. 211
(4) shews that the new axes rotate, in the opposite
direction to that of the earth, with the component of the
earth's angular velocity about the vertical at the place.
And in the plane, so revolving, we see by (3) that the bob of
the pendulum describes an approximately elliptic orbit.
A circular path being obviously possible, let us assume
as particular integrals of (2)
77 = a cos (pt + a), ? = a sin (pt 4- a).
The substitution of these values gives the same result
p" + Zap cos \ j =
6
in each of equations (2).
Put j = n 2 , then the values of p are n co cos X, so that the
L t
(apparent) angular velocity of a conical pendulum is increased
or diminished by to cos X according as its direction of rotation
is negative or positive.
196. A particle under any forces, and resting on a
smooth horizontal plane, is attached by an inextensible string
to a point which moves in a given manner in that plane; to
determine the motion of the particle.
Let x, y, x, y be the co-ordinates, at time t, of the particle
and point, a the length of the string, R the tension of the
string, and m the mass of the particle.
For the motion of the particle we have
v ^
m -- = mX R
dt 2 a
(1),
a
with the condition (x ~xf + (y y) 2 = a' 2 .
Now x, y are given functions of t. Take from both sides
CM ~HT (M 'T/
of the equations in (1) the quantities -j- , -^~ , respectively,
H 2
212 CONSTRAINED MOTION.
and we have the equations of relative motion
( v \ rJlTf.
(JU JL) v D X JU U
(2).
These are precisely the equations we should have had if
the point had been fixed, and in addition to the forces X, Y
and R acting on the particle, we had applied, reversed in
direction, the accelerations of the point's motion with the
mass as a factor. It is evident that the same theorem will
hold in three dimensions. The accelerations -= , - are
at" at"
known as functions of t, and therefore the equations of
relative motion are completely determined.
197. Let there be no impressed forces, and suppose first
that the point moves with constant velocity in a straight line.
fl *V f/'?7
Here -r- 7 are constant, and therefore no terms are
at at
introduced in the equations of motion. We have thus the
case of 28.
Again, suppose the point's motion to be rectilinear, but
uniformly accelerated.
The relative motion will evidently be that of a simple
pendulum from side to side of the point's line of motion. In
certain cases, when the angular velocity exceeds a certain
limit, we shall have the string occasionally untended ; and
this will give rise to an impact when it is again tended.
While the string is untended the particle moves, of course,
in a straight line.
198. Suppose the point to move, with constant angular
velocity &), in a circle whose radius is r and centre origin.
Here, supposing the point to start from the axis of x,
x = r cos cot, y r sin cot.
CONSTRAINED MOTION. 213
Hence the equations of motion are, since
Whence ._
or, in polar co-ordinates, for the relative motion,
= _ (o ,
dp a
Now 6 cot is the inclination of the string to the radius
passing through the point ; call it <, and we have
which is the ordinary equation of motion of a simple pendulum
whose length is ~- .
The particle therefore moves, with reference to the uni-
fqrmly revolving radius of the circle described by the point,
just as a simple pendulum with reference to the vertical.
199. To determine the motion of a particle under given
forces, moving in a smooth tube, in the form of a given plane
curve, which revolves in i given manner about an axis in its
plane.
Let the axis of revolution be that of z, and let the position
of the particle at time t be given by its distance r from that
214 CONSTRAINED MOTION.
axis, the plane of the tube at that instant making an angle
with a fixed plane passing through the axis. By the con-
ditions of the problem 6 is a given function of t.
The sole effect of the tube will be to produce a pressure
of constraint, which lies in the normal plane to the tube, and
may therefore be resolved into two parts, one perpendicular
to the plane of the tube, the other in that plane and in the
principal normal to the tube.
Let the impressed forces be resolved into three, P along r,
T perpendicular to the plane of the tube, and S parallel to
the axis of z.
Let R, R' be the two resolved parts of the pressure of
constraint.
The equations of motion will then be (by 16, 69)
dr
where s is the arc of the revolving curve.
In addition to these we have the two equations
e=f(t) ........................ (4),
which gives the position of the tube at any time, and
r = *() ........................ (5),
the equation of the tube.
By means of (4) and (5) we may eliminate 0, r, and s
from (1), (2), (3). Then eliminating R between (1) and (3),
we obtain a differential equation between z and t, whose
integral together with (4) completely determines the position
of the particle at any instant.
R and R may then be found from (1) or (3), and (2).
CONSTRAINED MOTION. 215
In the simplest case when the angular velocity of the tube
7/3
is constant, or -=- = , (4) becomes 6 = cot if the plane from
which 6 is measured be that of the tube at the time t = 0.
We proceed to give an example or two.
200. A particle moves in a smooth straight tube which
revolves with constant angular velocity round a vertical axis to
which it is perpendicular, to determine the motion.
Here z = constant, ,, = constant = o>, P = 0, and we have
at
from (1)
whence r = A e wt + Be~ ut .
Suppose the motion to commence at time = by the
cutting of a string, length a, attaching the particle to the
axis. The velocity of the particle at that instant along the
tube would be zero. Hence at t =
r = a = A + B,
and r= 2^0"' + e- w <).
In the figure, let OM be the initial position of the tube,
A that of the particle ; OX, Q the tube and particle at time t.
Then OA = a, arc AP = ant, OQ = r, and we have
+6
From this we see that OQ and the arc AP are correspond-
ing values of the ordinate and abscissa of a catenary whose
216 CONSTRAINED MOTION.
parameter is OA. (It is not necessary for the tube to meet
the axis of revolution.)
M
Here, by (3), we have evidently R = g.
Also, by (2), R f = - 2 ~ (e*>< - <r') o>
From this equation, combined with the value of r, we
easily deduce
72 / = 2ft>V(r 2 -a 2 ),
and it is therefore proportional at any instant to the tangent
drawn from Q to the circle APN.
201. Suppose the tube to revolve with constant angular
velocity in a vertical plane about a horizontal axis.
We have from equation (1) of 199
, /0 ^&> 2 = 9 cos cot,
if we conceive the tube to be vertical when t = 0. The
integral of this equation is
d\~
CONSTRAINED MOTION.
217
r =
- cos a>t
dr
= a, -rr = 0, when = 0,
at
= A-B;
or
and if
we have
and
or,
which completely determines the motion. R and R' may be
found as before.
202. Let the tube be in the form of a circle turning with
constant angular velocity about a vertical diameter.
Let AO be the axis, P the position of the particle at any
time. Let POA = 6 denote the particle's position. The
accelerations of the particle in the directions ON and NP
being
ffiON
,, - d*cos0 a
therefore a -v - = g R cos 6,
218 CONSTRAINED MOTION.
d?sin0 . ' n A
-TTT w 2 a sin 6 R sin 6.
Eliminating R
72/3
a ~" aft)2 s * n ^ cos ^ = ~~
The position of equilibrium will therefore be given by
sin 6 = 0, or 6 = 7, where cos 7 = -^ .
Integrating (1)
V = 4- 2w 2 cos 7 cos -o> 2 cos 2 (9 ......... (2).
I. Suppose the particle to be making complete revolu-
tions, passing through the lowest point with velocity ao>! ;
therefore
J = coj 2 2o) 2 cos 7 (1 cos 0) + a> 2 sin 2 6
= a> 2 {(1 - cos 7) 2 + Wl 2 - (cos ^ - cos 7> 2 },
and -vr can never vanish if -4- > 4 cos 7, or o^ 2 > , that
WC ft)^ tt
is, if the velocity at the lowest point be greater than that
due to the level of the highest point.
To solve the equation, we must put
where u -cot V(( 5 + 1)(1 r )},
and s, r are the values of cos 6 that make the right-hand
side of the equation vanish, s being >1, and r< 1.
II. If a)j 2 < , the particle will oscillate through the
d
lowest point, and if -=- = 0, when 6 = a, then
CONSTRAINED MOTION. 219
( -=- ) = -^ (cos 6 cos a) - w 2 (cos 2 6 cos 2 a)
\citj a
= a) 2 (cos 6 - cos a) f-~ - cos a - cos 6 J
= o> 2 (cos 6 cos a) (2 cos 7 cos a cos 0),
and therefore if
2 cos 7 cos a > 1,
the particle will oscillate through the lowest point.
W T e must put
tan = = tan ^ en u, where k' = cot = \/2 cos 7 cos a 1,
and then i = o> V(cos 7 cos a).
III. If 1 > 2 cos 7 - cos a > - 1,
then putting 2 cos 7 cos a = cos y&,
( -IT j = a> 2 (cos cos a) (cos /8 cos 0),
and the particle will oscillate on one side of the vertical
diameter.
We must put
6 a, 6 ft I
tan 5 = tan ^ dn u, or tan ^ = tan - -= ,
2> i L z on %
* an ^>
and then w = o> sin ^ cos , k' = .
Z 2S cc
tan g
203. To ywicZ the form of the tube in order that the particle
projected with given velocity may preserve its velocity un-
changed, gravity acting parallel to the axis.
Resolving tangentially, and taking co-ordinates x, y, in
the plane of the curve, the axis of revolution being that of y.
we have
d' 2 s dx dy
'
220 CONSTRAINED MOTION.
Hence, ( J] = x 2 w~ - Zgy + 0.
\atj
ds
But 7- = constant.
Hence, x- = 2 (y + fc),
the equation of a parabola whose axis is vertical and vertex
downwards. This result might easily have been foreseen, as
the velocity can only be constant if the acceleration due to
the impressed forces along the curve be zero at every
point; that is, if the resultant of gravity and the reac-
tion to circular motion (called the centrifugal force) lie in
the normal. That this may be the case, we must have
Centrifugal force : Gravity : : Ordinate : Subnormal. But
the centrifugal force is proportional to the ordinate, hence
the subnormal must be proportional to gravity, i.e. must be
constant : a property peculiar to the parabola. This propo-
sition has a singular application in Hydrostatics.
204. A particle moves on a rough curve, under given
forces; to determine the motion.
If fju be the coefficient of kinetic friction, and
be the normal reaction, friction will cause a resistance
fju V(-Ki 2 + J^2 2 ) acting in the tangent to the curve in the
opposite direction to the particle's motion.
Equation (1) of 182 will therefore become
J = 2- -,WW + *,'),
til/
the other two equations remaining the same.
If from the three we eliminate Jftj and R 2) we may by
means of the equations of the curve eliminate x, y and z, and
the final result, involving only s and t, suffices to determine
the motion completely.
205. Ex. A particle moves in a rough tube in the form
of a plane curve, under no forces ; to determine the motion.
CONSTRAINED MOTION.
221
Here
Now
hence
or
dv d?s
dv v 2
v -j- = /* :
ds ^ p
{ds
~P I -
v = ae p .
But, if A/T be the angle which the tangent at any point
makes with a fixed line,
Hence, v = ae~^, where a is the velocity when
0.
It may be instructive to compare this result with that for
the tension of a string stretched over a rough curve.
ds
If the curve be tortuous, - is the angle between two
successive tangents. If the surface of which the curve is the
cuspidal edge be developed, and if <f> represent the angle
between the tangents corresponding to the initial and final
positions of the particle,
*&"*.
206. A particle under given forces moves on a given
rough surface; to determine the motion.
If R be the normal reaction of the surface, the friction
will cause a resistance pR, and the equations of motion
become
222 CONSTRAINED MOTION.
from which R must be eliminated. The two resulting equa-
tions contain x, y, z and t, and if the latter be eliminated, we
have one equation in #?, y, z, which, with the equation of the
surface, will completely determine the path. In general these
equations are utterly intractable.
EXAMPLES.
(1) If a particle, attached by a string to a point, just
make complete revolutions in a vertical plane, the tension of
the string in the two positions when it is vertical is zero, and
six times the weight of the particle, respectively.
(2) A pendulum which vibrates seconds at a place A
gains n beats in 24 hours at a place B ; compare gravity at
the two places.
(3) Prove that a seconds pendulum when taken to the
top of a mountain h miles high will lose 21'6h beats in a
day nearly.
(4) The times of oscillation of a pendulum are observed
at the earth's surface, and also at a height h above the sur-
face ; from these data find the radius of the earth supposed
spherical.
(5) Shew that a simple circular pendulum under a
central attraction varying as the distance will move as it
does under gravity.
(6) A pendulum oscillates in a small circular arc, and
is acted on in addition to gravity by a small horizontal
attraction as the attraction of a mountain. Shew how to
find this attraction by observing the number of oscillations
gained in a given time. Also find the direction in which
the attraction must act so as not to alter the time of
oscillation.
(7) Prove that a particle moving under gravity on the
convex side of a vertical circle will leave the circle at two-
thirds of the height above the centre of the line to the level
of which the velocity is due.
CONSTRAINED MOTION. 223
(8) A particle is suspended from a fixed point by an
inextensible string : find the level to which the velocity must
be due, so that the particle after the string has ceased to be
stretched may pass through the point of suspension.
(9) Two particles are projected from the same point, in
the same direction, and with the same velocity, but at dif-
ferent instants, in a smooth circular tube of small bore whose
plane is vertical, to shew that the line joining them constantly
touches another circle.
Let the tube be called the circle A, and the horizontal
line, to the level of which the velocity is due, L. Let m, m',
be simultaneous positions of the particles. Suppose that mm'
passes into its next position by turning about 0, these two
lines will intercept two indefinitely small arcs at m and m
which (by a property of the circle) are in the ratio mO : Om'.
Let another circle B be described touching mm' in 0, and
such that L is the radical axis of A and B. Let a be the
distance between their centres, mp, m'p' perpendiculars on L.
Let mp cut A again in q and B in r, s.
Then by Geometry,
mp .qp = rp.sp,
and therefore raO 2 = mr . ms = (mp rp) (mp sp)
= mp (mp + qp rp- sp)
Similarly
= 2a . mp = - (velocity of mf.
Om' 2 = 2a . mp = - (velocity of m'}
i/
Hence the velocities of m and m are as mO : Om', and
therefore by what we have shewn above about elementary
arcs at m and m, the proximate position of mm' is also a
tangent to B, which proves the proposition.
It is easily seen from this, that if one polygon of a given
number of sides can be inscribed in one circle and circum-
224 CONSTRAINED MOTION.
scribed about another, an indefinite number can be drawn.
For this we have only to suppose a number of particles
moving in A with velocities due to a fall from L, and then if
they form at any time the angular points of a polygon whose
sides touch B, they will continue to do so throughout the
motion. This however does not belong to our subject.
(10) Two segments of circles are described on the under
side of the same horizontal line, the one subtending at its
centre double the angle which the other subtends; if a
particle under gravity describes the lower arc, any tangent
to the upper arc will cut off from the lower a portion which
will be described in half the time of a single vibration.
(11) AB is a vertical diameter of a fine circular tube
in which move three equal particles P, Q, Q' of perfect
elasticity ; P starts from A and Q, Q in opposite directions
from B with such velocities that at the first impact all three
have equal velocities ; prove that throughout the motion
the line joining any pair is either horizontal or passes
through one of two fixed points, and that the intervals of
time between successive impacts are all equal.
(12) Two equal smooth circles are fixed so as to touch
the same horizontal plane at their lowest points, their
planes being at different inclinations; two small beads are
projected at the same instant along the circles from their
lowest points, the velocity of each bead being due to the
level of the highest point of the other circle above the
horizontal plane; prove that during the motion the beads
will always be at the same level.
(13) Prove that the time of vibration from rest to rest
of a simple circular pendulum of length a oscillating through
an angle 2a is equal to the time of complete revolution
of the pendulum of length a cosec 2 = a, the velocity being due
to the level 2a cosec 4 -= a, above the lowest point.
(14) A bead can slide on a smooth circular arc AB and
is attracted by it, with intensity f(r) ; if it be displaced
CONSTRAINED MOTION. 225
from its position of equilibrium, the time of oscillation
will be
27T
where G is the middle point of AB, and a the angle AC
subtends at the centre of the circle.
(15) A string passes through a small hole in a smooth
horizontal table, and has equal particles attached to its ends,
one hanging vertically and the other lying on the table at a
distance a from the hole ; the latter is projected with a
velocity \/ga perpendicular to the string; shew that the other
particle will remain at rest, and if it be slightly disturbed the
time of a small oscillation will be 2?r
\ / =- .
V %
(16) A particle, under gravity, is attached to a fixed
point by means of an elastic string of natural length 3a,
the modulus of elasticity being six times the weight of the
particle ; when the string is at its natural length and the
particle vertically above the point of attachment, the particle
is projected horizontally with a velocity 3 A/ -^- ; prove that
the angular velocity of the string will be constant, and that
the particle will describe the Iima9on
r a (4 cos 6).
(17) From a point upon the surface of a smooth vertical
circular hollow cylinder, and inside, a particle is projected
in a direction making an angle a with the generating line
through the point ; find the velocity of projection that the
particle may rise to a given height (h) above the point, and
the condition that the highest point may be vertically above
the point of projection.
Find the condition that after n revolutions the particle
may be again at the point of projection.
(18) A particle slides down a catenary, whose plane is
vertical and vertex upwards, the velocity at any point being
T. D. 15
226 CONSTRAINED MOTION.
due to the level of the directrix ; prove that the pressure
at any point is inversely as the distance of that point from
the directrix.
(19) A particle projected with given velocity, moves
under gravity on a curve in a vertical plane ; find the nature
of the curve that the pressure on it may be constant through-
out the motion.
If the pressure on the curve is always n times the weight,
prove that the vertical distance between the highest and
lowest points of the curve is
2na
and that the interval between the instants at which the
particle is at the same level is
/a n
the length of the curve between two such points being
TTO, 5 .
(rf-ir
Determine the nature of the evolute of this curve, which
is such that the string of a simple pendulum must be
wrapped on it in order that the tension may be constant,
and prove the relation between the length of the arc and the
vertical ordinate from the upper cusp
where I is the length of the string.
(20) The major axis of an ellipse being vertical, shew
that in order that a particle projected along the concave
side of the arc may pass through the centre after leaving the
curve, the velocity must be due to the level
8a 2 + 6 8
above the centre, a and b being the semiaxes of the ellipse.
CONSTRAINED MOTION. 227
(21) A particle is initially at rest at a point of the
equiangular spiral r = ce~ me , distant d from the pole. Shew
that if the pole be a centre of attraction = ~ , the time of
fall to it is
Find the pressure on the curve at any instant.
(22) A particle attached by a string to a point moves on
a horizontal plane. A small ring passing round the string
moves with constant velocity in a straight line from the
point. Shew how to find the equation of the actual path,
and shew that the path relative to the ring is a reciprocal
spiral.
(23) A particle moves in a circular groove radius a
under a central attraction x D~ z situated at a distance b
from the centre of the circle. It is projected from the
nearest point with velocity V, shew that for a complete
revolution
(24) Prove that if a particle move in a smooth tube
under given central attractions, the pressure at any point
of the tube will vary as
P ( pap-
where -s- is the acceleration due to the attraction of any one
dr
of the centres, and p is the radius of curvature ; and hence
that the pressure at any point of the tube will vary as the
curvature whenever the orbit is such as could be described
freely under each of the attractions taken separately.
(25) A particle of mass m moves in a smooth circular
tube of radius a under an attraction nip times the distance to
a point inside the circle at a distance c from the centre. If
the particle be placed very nearly at its greatest distance
152
228 CONSTRAINED MOTION.
from the centre of attraction, prove that it will pass over the
quadrant ending at its least distance in the time
(26) Shew that a particle moving under gravity on a
smooth helix whose axis is vertical, makes the first revolution
from rest in the time
8?ra
J.
g sin 2a
(27) A groove is cut on a right cone of height h,
making an angle ft with the generating lines. Shew that
the time of reaching the base, from a vertical height h^
below the vertex, by a particle sliding in the groove is
2
g cos a cos /
where a is the semivertical angle.
(28) A particle under a central repulsion varying as the
distance moves in a tube of the form of an epicycloid, the
pole being at the centre of repulsion. Shew that the oscilla-
tions are tautochronous.
For an attraction, the curve is a hypocycloid.
(29) Prove that the tautochrone when the attraction is
as the cube root of the distance from and perpendicular
to the axis of x is the hypocycloid
(30) A particle P is attached by strings to two points A
and B in the same horizontal plane, and describes a vertical
circle. When the particle is at the lowest point, the string AP
is cut and the particle proceeds to describe a horizontal circle;
find the ratio of the new tension of BP to the old tension.
(31) A smooth ring slides on a circular wire which
revolves with constant angular velocity about a vertical
diameter. If the ring be attached to the highest point by
CONSTRAINED MOTION. 229
a fine elastic string of natural length equal to the radius
of the wire, and be slightly displaced from the lowest point,
shew that it will just reach the highest point if the modulus
of elasticity is four times the weight of the particle.
(32) A ring slides on a smooth wire bent into the form of
a plane vertical curve, and is attached by an elastic string to
a fixed point in the plane of the curve ; if it start initially
from a position in which the string is just not stretched,
prove that it will descend through a vertical distance which is
a third proportional to the natural length of the string and its
extension at the lowest position, supposing that the modulus
of elasticity is twice the weight of the ring, and the string is
stretched throughout the motion.
(33) Three equal particles are attached to a string of
length 4a, one at its middle point and the others half way
between it and the extremities, which are attached to two
points in a horizontal line at a distance a (\/3 + 1) from each
other ; find the position of equilibrium, and shew that if the
middle particle receive a slight vertical displacement the
time of a small oscillation is the same as that of a pendulum
of length
3-V3
"
(34) A particle, under gravity, is suspended by a light
elastic thread which passes through a ring B above the par-
ticle and is attached to a fixed point A, AB being the natural
length of the string.
If the particle be projected from any point in any direc-
tion, prove that it will describe an ellipse about the position
of equilibrium of the particle as centre.
Prove that the same will hold if the particle be suspended
in a similar way by a number of elastic strings.
(35) A chord AB of a circle is vertical and the inclina-
tion of the tangents at A and B to the horizon is the angle
of friction. Shew that the time down any chord AC or CB
drawn in the smaller of the two segments into which AB
divides the circle is constant.
230 CONSTRAINED MOTION.
(36) A particle, under no forces, is projected with velo-
city V in a rough tube in the form of an equiangular spiral
at a distance a from the pole and towards the pole; shew
that it will arrive at the pole in time
cos a fji, sn a
a being the angle of the spiral and //, (< cot a) the coefficient
of friction.
(37) A bead is projected along a rough plane curved
wire, such that it changes the direction of its motion with
constant angular velocity. Shew that the form of the wire
must be a logarithmic spiral.
(38) A particle attached to a point by a string whose
natural length is a, lies on a rough horizontal plane and is
projected perpendicular to the string with velocity v. If it
comes to rest at a distance a from the point, after describing
a distance 6', -y 2
(39) A particle descends a rough circular tube from the
extremity of the horizontal diameter. If it stops at the
lowest point, shew that
(40) If a particle under no forces be projected with
velocity V along the inner surface of a rough sphere, deter-
mine the motion, and shew that it will return to the point
of projection in the time
where r is the radius of the sphere.
(41) A particle is attached to a smooth string which
passes over a rough circular arc in a vertical plane; the
particle initially at the extremity of a horizontal diameter
is drawn up with constant acceleration - : shew that the
CONSTRAINED MOTION. 231
work expended in drawing it to the vertex of the circle is
where W is the weight of the particle, a the radius of the
circle, and p the coefficient of friction.
(42) A rough wire in the form of an equiangular spiral,
whose angle is cot" 1 2//,, is placed with its plane vertical and
a particle slides down it under gravity, coming to rest at its
lowest point; prove that at the starting-point the tangent
makes with the horizon an angle 2 tan" 1 //,, and that the
velocity is greatest when the angle < which the direction of
motion makes with the horizon is given by the equation
(2yu, 2 - 1) sin </> + 3yLt cos </> = 2/x.
(43) A particle falling under gravity down a rough
cycloidal arc whose axis is vertical comes to rest at the
lowest point: prove that if </> be the angle which the tangent
at the starting-point makes with the horizon, then
pe** = sin <f> fj, cos </>.
(44) Two equal particles attracting each other with
intensity the acceleration of which is p* x distance are placed
in two rough straight tubes at right angles to one another,
arid the friction is equal to the pressure in each tube ; prove
that if they be initially at unequal distances, one moves for
a time ~ before the other begins to move, and that while
they are approaching the point of intersection of the tubes
they move in the same manner as the projections of the two
extremities of a diameter of a circle upon a straight line on
which the circle rolls.
(45) A particle moves on a rough curve under forces T
in the tangent and N in the normal, prove that the velocity
at any point is given by
232 CONSTRAINED MOTION.
(46) A circular tube of small bore revolves with con-
stant angular velocity co about a vertical diameter, and a
particle in it is projected from the lowest point with velocity
due to the level of the highest point. Determine the motion,
and shew that it is at its greatest distance from the axis
after a time
where a is the radius of the tube.
(47) A particle P, attached by a string of given length
a, to a point S in a fixed axis SA, is attracted with constant
intensity g in a direction always parallel to a line SB, which
is inclined at a given angle to the axis SA, and revolves
about it with a given angular velocity w: shew that if V the
velocity of P, &>'=the angular velocity of the plane PSA
about SA, 4>=Z.PSB, = /.P8A,
JF 2 = go, cos < 4- a 2 o>ft/ sin 2 6 + const.
Shew also that the dynamical conditions of this Problem
are the same as those of a ball-pendulum under gravity,
when the Earth's rotation is taken into account.
(48) A smooth circular tube is fixed at one point A and
contains a particle which is initially at rest at the opposite
end of the diameter through A. The tube is then made
to revolve with constant angular velocity o> about an axis
through A perpendicular to the plane of the tube. Prove
that the angle described in the time t by the particle about
the centre of the tube is
p<*t _ i
4 tan- 1
(49) A ring slides on a smooth elliptic wire which moves
with constant angular velocity about an axis through its
centre perpendicular to its plane. Determine the motion ;
and find the time of a small oscillation about the position
of equilibrium where this is possible.
CONSTRAINED MOTION. 233
(50) If a particle slide along a smooth curve, which
turns with constant angular velocity co about an axis per-
pendicular to its plane passing through a fixed point 0,
then the velocity of the particle relatively to the moving
curve is given by the equation
where r is the distance of the particle from the point 0; and
the pressure on the curve will be given by the formula
v 2
R = - + 2cov + tfp,
where p is the perpendicular from on the tangent.
(51) If a curve revolve with constant angular velocity
about a vertical axis in its plane, prove that the time of
a small oscillation of a particle sliding on the curve about
its position of relative equilibrium is
2?r / p sin a
co v r p sin a cos 2 a '
p being the radius of curvature at the point of equilibrium,
a the angle made by the normal at that point with the ver-
tical, r the distance of the point from the axis of revolution,
and co the angular velocity of the curve.
(52) A fine straight tube rotates in a plane with constant
angular velocity co about a point in its length while the plane
rotates with constant angular velocity &>' about a horizontal
axis through that point, prove that the equation of motion of
a particle placed in the tube is
d*r
-j- (co 2 + w' 2 cos 2 cot) r = g sin co't cos cot,
the tube being initially perpendicular to the horizontal axis
and the plane horizontal.
(53) AB is the diameter of a sphere of radius a; a centre
of attraction at A attracts with intensity (/j, x distance); from
234 CONSTRAINED MOTION.
the extremity of a diameter perpendicular to AB a particle
is projected in any direction along the inner surface with a
velocity (2//,)*a : shew that the velocity of the particle at any
point P is 2\/yuasin0, and the pressure is /-ta (3 sin 2 1),
where 6 is the angle PAB, so long as the particle -remains in
contact with the surface.
(54) A particle is attached by a fine string to the apex
of a right vertical cone whose semivertical angle is /:?, and is
projected from a position of rest on the cone with an initial
angular velocity w (about its axis) which is less than H, the
least angular velocity which would make the particle leave
the cone. If the coefficient of friction between the particle
and cone be /x, find the position of the particle and the
tension of the string at a given instant; and shew that it
will come to rest after a time
lo gn IT-
cos
(55) If a particle be projected horizontally inside a
right circular cone of vertical angle 2a whose axis is vertical
and vertex downwards with the velocity which it would ac-
quire by sliding down a generator from the point of projection
to the vertex, shew that :
1. The motion is oscillatory, and that the particle never
descends lower than its initial position.
2. The curve traced out by the particle on the cone if
developed into a plane has its equation
where r is the distance of the particle from the vertex, p its
initial distance.
3. The reaction is given by the equation
f . , cos 2 a/pV)
R = mg \ sin a + 2 - [ - ) > .
*\ sin a W j
CONSTRAINED MOTION. 235
(56) A particle is in equilibrium on the surface of a
smooth thin hemispherical bowl which attracts according to
the law of nature. If it be slightly displaced, shew that the
time of a small oscillation is
47r
A/2tf
V ~s
where a is the radius of the bowl, and M the mass.
(57) If a particle be projected inside a smooth para-
boloid of revolution, axis vertical and vertex downwards, hori-
zontally at the level of the focus with velocity ^2nag, the
initial radius of curvature of the path will be
(58) A particle is projected in a smooth paraboloid
whose axis is vertical so as very nearly to describe a circle
whose diameter is the latus rectum of the generating para-
bola ; prove that the time of a small oscillation is the same
as that of a pendulum of length a, where 4a is the latus
rectum.
(59) A particle moves in the interior of a smooth
paraboloid of revolution whose axis is vertical and vertex
downwards. Shew that the differential equation of its path
on a horizontal plane is
d /I du\ _ aq
where 4a is the latus rectum of the generating parabola.
(60) A particle under an attraction varying inversely
as the cube of the distance from a given plane, is constrained
to move on a smooth spherical surface, having been projected
with the velocity due to an infinite distance ; prove that the
resultant acceleration of the particle always passes through a
fixed point.
236 CONSTRAINED MOTION.
(61) A particle is attached to the highest point of a
smooth fixed sphere of radius a, by an elastic string whose
natural length is , and the weight of the particle is suf-
ficient to stretch the string to double its natural length ; at
first the particle moves with constant velocity in a small
circle, the string being stretched to double its natural length;
prove that if the motion be slightly disturbed the time of a
small oscillation will be
9 ( W2 - 5) TT + 8^2 '
and find the greatest impulse it can receive along the direc-
tion of the string without leaving the sphere.
(62) A rough paraboloid of revolution of latus rectum
4>a and coefficient of friction cot /3 revolves with constant
angular velocity about its axis which is vertical ; prove that
for any given angular velocity greater than
or less than
V 2a 2
A/t-s
a particle can rest anywhere on the surface except within a
certain zone, but that for any intermediate angular velocity
equilibrium is possible at every point of the surface.
(63) An anchor ring is formed by the revolution of a
circle of radius c about an axis in its own plane at a distance
a from the centre. A particle is projected along the smaller
equator with velocity v and is under an attraction to the
centre of the axis /j,r n . If the particle be slightly displaced,
prove that it will cut its original path at equal angular
intervals a, where
(-H'
\/ (
fa c fta (a c) n _ ..
c v 2
CONSTRAINED MOTION. 237
(64) A smooth surface is generated by the revolution of
the curve a?y = c 3 about the axis of y which is directed
vertically downwards, and a particle under gravity is pro-
jected along the surface with velocity due to the level of
the horizontal plane through the origin.
Prove that its path will intersect all the meridians at a
constant angle.
(65) Find the form of the smooth surface on which if a
particle move, under the action of a central force varying,
according to a given law, with the distance, the normal
pressure shall be constant.
(66) Find the Brachistochrone of continuous plane
curvature, when the velocity is always proportional to the
curvature.
(67) Find the Brachistochrone in a medium in which
the velocity depends upon the direction of motion only.
Take particular cases : such as
a) v = A\ + Bp + Cv,
b) v^AW + BfS + Cv*,
c) v = Apv + Bv\ + C\fjL ;
X, IJL } v being direction-cosines, and A, B, C constants.
( 238)
CHAPTER VII.
MOTION IN A RESISTING MEDIUM.
207. WHEN a body moves in a fluid, whether liquid or
gaseous, it must, in displacing the particles of the medium
and in rubbing against them, lose part of its own velocity.
The resistance of a fluid to a body moving in it produces
therefore a retardation ; but, in consequence of the great
difficulty of making accurate experiments on the subject, the
laws of the resistance of fluids have not yet been satisfactorily
ascertained.
For a velocity neither very great nor very small, the
general approximate law seems to be, that the resistance to
a plane surface, moving with its plane at right angles to the
line of motion, varies as the extent of the surface, the density
of the resisting medium, and the square of the velocity taken
conjointly ; and the retardation due to the resistance is
therefore equal to the numerical value of the resistance
divided by the number of units of mass in the body.
It is well that the student should observe that in all cases
of resistance, as in those of friction, the motions are essentially
irreversible. We are no longer dealing with conservative
systems, so long at least as we confine our attention to the
motion of the resisted body alone.
208. A particle under no forces is projected in a resisting
medium of uniform density, of which the resistance varies as
the n th power of the velocity; to determine the motion.
The motion will evidently be rectilinear. Let s be the
distance of the particle from a fixed point in the line of
motion at the time t, v its velocity at that time. The
MOTION IN A RESISTING MEDIUM. 239
retardation due to the resistance may be represented by
kv n , k being a constant, and the equation of motion is
ar-** ......................... ^
or
Therefore
1 dv
1 dv
_ _ _ fa
^ ^^ n/
v n ~ l ds
Integrating, supposing the initial velocity F,
and the elimination of v between (3) and (4) will give s in
terms of t
We see from (3) and (4) that if n > 1, the velocity never
vanishes ; and that if n > 2, the distance gone increases
indefinitely.
209. The Rev. F. Bashforth, Motion of Projectiles, found
that for small variations of velocity we might put n = 3.
If d = diameter of shot in inches, w = number of pounds
in the shot, then the retardation due to the resistance was
put = 10--Kv s , so that k = 10- 9 -K, and K was deter-
w . w
mined by experiment for velocities proceeding by increments
of 10 between 900 and 1700 feet per second, K attaining its
maximum value for a velocity of about 1200.
240 MOTION IN A RESISTING MEDIUM.
The numerical values of K for elongated and spherical
projectiles are given in Tables I. and II. in the " Motion of
Projectiles."
Tables also were calculated by Mr Bashforth from formulae
(3) and (4) (Tables VIII. IX.), giving -s and -t for every
decrement of 10 in the velocity between 1700 and 900, using
the mean value of K between each pair of velocities, and
from these tables we can determine s in terms of v and t in
terms of v for any shot, neglecting gravity, and consequently
s in terms of t.
210. There is one case in which the above solution fails,
namely when n \ t or the resistance varies as the velocity.
In this case k is the reciprocal of a time and may be put
= - , and then
or
Hence
and therefore
dv
V
...(1),
dt
ldv_
T
1
...(2).
v dt
loOf =
T
t
...(3);
8 v
T
_t
Integrating, we have s=Vr(l e r ) ....................... (4).
Equations (3) and (4) determine the velocity and the
position of the particle at any instant. They shew that the
velocity continually diminishes without ever actually be-
coming zero, but that the distance passed over by the particle
has a definite limit, for when
t = oo , s = FT.
MOTION IN A RESISTING MEDIUM. 241
211. A particle, under a constant force in its line of
motion, moves in a resisting medium of uniform density, of
which the resistance varies as the square of the velocity ; to
determine the motion.
Suppose the particle projected from the origin with the
velocity V, and let v be its velocity at any time t, x its
distance from the origin at that time, and / the constant
acceleration due to the force.
Assume K to be the velocity with which the particle {
would have to be animated that the retardation due to thev^
resistance might be equal to/, then the retardation when the
-.5
velocity is v may be represented by/-^.
Let / act so as to diminish x\ then the equation of
motion is / "j 1 7^
dv _ v* ^^
\ I \ -
Integrating, and determining the constants so that when
# = 0, = 0, v=V,
we obtain
ft ,V , ,
-L . tan- y - tan- ^ = tan-
+
g ^ + t^ '
Let T be the time at which the velocity becomes zero,
and h the corresponding value of x, then
JT V Jf 2 / T^ 2 \
T- j tan- , and A = ^ log (l + -p) .
T. D. 16
242 MOTION IN A RESISTING MEDIUM.
After this the particle begins to return, the resistance
therefore tends to increase x, and the equation of motion is
dt
Integrating, and determining the constants so that when
.. r\ 7 j. fji
we obtain
7) 2
Let U be the velocity with which the particle will return
to the point of projection; then, putting x = Q in the latter
equation, we obtain
or, substituting for h its value,
y*
f/ 2 K*
whence
_L _L JL
U 2 F 2 " K* '
This shews, as we might expect, that the particle returns
to the point of projection with diminished velocity.
The integrated equation connecting x and v above, shews
that however far the particle may fall its speed cannot
exceed K. And by taking the constants to indicate a
MOTION IN A RESISTING MEDIUM. 243
projection downwards, with any finite speed, however great,
we see that the speed will tend again towards the terminal
value K.
212. The results of the last Proposition are applicable to
bodies projected in a resisting medium vertically upwards or
downwards under gravity; for the acceleration due to gravity
may still be considered constant, although not the same as
for a particle in vacuo. The effective attraction of gravity is
in fact the difference of the weights of the body and the fluid
displaced, so that if a be the ratio of the density of the fluid
displaced to that of the body, effective gravity
= F(1 -) = %(! -a),
where W and M are the weight and mass of the body, and
therefore the acceleration caused by gravity = #(1 a). By
substituting this for fin the results of 211, we may obtain
formulae for the motion of bodies in a vertical direction
under gravity. Hailstones and raindrops afford a good
illustration of the Terminal Velocity indicated by the result
of 211. So does the arrest of the (planetary) speed of
"shooting stars."
213. To find the equations of motion, in a resisting
medium, of a particle under any forces.
Let x, y, z be the co-ordinates of the particle relative to
an assumed system of rectangular axes, at the time t, and let
X, Y, Z be the component accelerations, parallel to the axes,
due to the forces acting on the particle. Then denoting by
R the retardation due to the resistance, which lies in the
tangent to the path described, and in a direction opposed to
the motion, we have
d 2 x dx
5 = ^1 jK -5 ,
dP ds'
d?z ^dz
d&~ rfs'
162
244 MOTION IN A RESISTING MEDIUM.
These are the general equations of motion. In any
particular case R will be given as a function of the density
of the medium and the velocity of the particle, and par-
ticular methods will be necessary for obtaining the path of
the particle and its position at any time. These equations
will enable us, when X, Y, Z are given, to determine tlie
resistance that a given path may be described.
214. A particle under gravity is projected from a given
point in a given direction with a given velocity, and moves in a
uniform medium whose resistance varies as some power of the
velocity ; to determine the motion.
Take the given point as origin, the axis of oc horizontal,
the axis of y vertically upwards, so that the plane of xy
may contain the direction of projection ; let g denote the
acceleration of gravity, v the velocity of the particle at any
point, u its horizontal component, </> the inclination of the
direction of motion to the horizon, and R = kv n the re-
tardation due to the resistance.
Then the equations of motion are, resolving horizontally
and vertically,
-- ........................
$-->-*% ...................
or, resolving in the direction of the tangent and normal,
x-/2o
-R ................... (3),
(4).
v cos 6 and p = -y-r , equations (
d(p
(4) may be written
^--Aco94
= -#cos(/> ....................... (6),
Since v = -j- , u = v cos 6 and p = -y-r , equations (1) and
at d(p
--Aco94 ...................... (5),
MOTION IN A RESISTING MEDIUM. 245
du Rv
and therefore
<) 9
kv n+l
k
= - u n + l sec n+1 </> ............ (7).
c/
Integrating this equation, denoting by u the velocity at
the vertex of the trajectory,
1 1 k
if P denote the interal I n sec n+1
I
Jo
Therefore u = - r ,
ku n retardation at vertex
where 7
g acceleration of gravity
_ resistance at vertex
weight of shot
From equation (6)
dt v u
-j-j sec <p = sec 2 <p,
d<f> g g
therefore, if a be the angle of projection,
f f a u 2
Again, a? = I udt = I -
Jo J $ g
,f
9
and y = *
- sec
246 MOTION IN A RESISTING MEDIUM.
Equations (9), (10), (11) give t, x, y in terms of </>.
For n = 3, P = 3 tan <f> + tan 3 <,
and the integrals in (9), (10), (11) were calculated by quad-
ratures for different values of 7 and for certain ranges of
angle, and the nominal values tabulated, in Tables IV. V. VI.
in Mr Bashforth's " Motion of Projectiles."
215. For n = l, putting R = - , then r is the measure
of a time, and
sec 2 ()d(f) = tan <,
:-/;
u f a se
= I YH
, .
and since
7 tan $
_.. l-yten* .....
#7 5 1 - 7 tan a
1 7 tan <t>
_ *-;
1 - 7 tan a '
sec 2 tan
ts
i 1 - 7 tan ,
~ g -"
and the elimination of tan < between (2) and (3) will give
the relation between x and y.
MOTION IN A KESISTING MEDIUM. 247
Or immediately, resolving horizontally and vertically,
d?x 1 dx
and integrating, supposing F the velocity and a the angle of
projection,
dx - t -
- = Fcosae
at
dv
and integrating again,
_t
x= Frcosa(l-e T ) .................. (6),
t
y + gtr = (VrsmoL + gT 2 )(l-e'~ T ) ......... (7).
Eliminating t between (6) and (7),
FT cos a / ar \
y + gr z log T7 , = tan a + Tr - -- - x,
D Frcosa-# \ Fcosa/
the equation of the trajectory.
Differentiating this equation twice, we obtain
dx 2 (Frcosa-#) 2
the differential equation of the trajectory.
216. For n = 2, putting R = , then a is the measure
CL
of a length, and putting p = tan <f>,
P = I 2 sec 3 4>d$ = 2 j*/I+p 2 dp
248 MOTION IN A RESISTING MEDIUM.
The equations of motion are, resolving horizontally and
vertically,
_ _
dt*~~adi ds
2 ri (c>\
dt 2 a \dtj ds " '
Equation (1) may be written
du __ 1 ds
1 du _ _ 1 ds
u dt adt'
and integrating,
__
From equation (8) of 214,
2 F 2 cos 2 a/
or e a 1 = - tan a sec a tan 6 sec
</a \
tan a. + sec
& tan < + sec (
the intrinsic equation of the path.
Differentiating this equation with respect to x,
2 ^ds__ F 2 cos 2 a^P = 2F 2 cos 2 q^- ^^
a dx ga dx ga ^ dx'
or -^ -f Yr2 o e a = (5),
dx V cos a
the differential equation of the path.
If $, s denote the arcs of the trajectory in a non-resisting
and a resisting medium, measured from the point of pro-
jection to any two points at which the tangents are parallel ;
then, since in the non-resisting medium a = oo , therefore
MOTION IN A RESISTING MEDIUM. 249
dS ^
and -j- = e a ,
as
and integrating,
f-gc 4 - 1 ) w
217. For a flat trajectory, p being always small, we may
x7n
put -7- = 1, and then equation (5) may be written
dp a 2^ ds _ _
c&e F^ a cos 2 a dx
Integrating,
i 2 F 2 cos 2 a
or e 1 = - (tan a p).
ga
And substituting again in equation (5)
rfp_2p_ g _ 2
rfa? a F 2 cos 2 a a
-i x -
Multiplying by e * and integrating,
And integrating again,
250 MOTION IN A RESISTING MEDIUM.
2- or
Expanding e' a in ascending powers of -, a being supposed
a
large,
go? go?
~"
of which the first two terms will represent the trajectory in
a non-resisting medium.
218. A particle moves in a resisting medium under a
central attraction ; to determine the orbit
Let P be the acceleration due to the central attraction,
R the retardation due to the resistance of the medium ; then
resolving along and perpendicular to the radius vector,
f d6Y ^ dr
_
r~dt T -
Putting r 2 -j- = h, equation (2) may be written
dh_ _ Rr ^dO
dt~ ds'
Idh R
h^t = -^
and therefore
h = h e Jv (3).
Or the equation may be written
h ds
and therefore
(4).
Again, putting r = - , we have -^- = hu 2 ,
u at
,
MOTION IN A RESISTING MEDIUM. 251
dr 1 du \dudO ,du
d*r _ , d z u dO^
dffT dfrd* dtlB
, d*u , E du
h*u* -r^r 4- h -j-z
d6* v dO
and therefore
, ^ r> 19
-j-v--j-) = - h*u* -j^ -R-j-- h*u 3
dt* \dt) d& ds
an equation of the same form as that for the motion in
a non-resisting medium, h however being now variable.
219. If in addition to the central attraction, there is
a transversal force producing acceleration T, we shall obtain
the equation analogous to (5) most simply by resolving in
the normal, and then
- = P sin <f> -|- T cos <,
rdd udO
where tan <f> = - = --- = .
dr du
Therefore P
p sm <>
d*u P T du
or f- u = -
dO 2 h?u? h^ii s d6 '
an equation of the same form as that obtained in 136.
252 MOTION IN A RESISTING MEDIUM.
EXAMPLES.
(1) If the time is a quadratic function of the length
traversed, prove that the resistance varies as the cube of the
velocity.
(2) Shew that the solution of the differential equation
for vibrations resisted by friction proportional to the velocity,
but otherwise free, viz.
u + ku + n z u = 0,
may be put into the form
,. f . sin n't ( ,, k . ,\]
u = e -\kt\ UQ - . -- \-u ( cosnt + jT-^smntflk
where n' 2 = ri 2 J& 2 , and u , u are the values of the velocity
and displacement when t = 0.
Deduce the complete solution of
u + ku + n z u = U,
in the form
sin n't
- ;
n
( ,. k . ,\\
cos n t + ^-7 sin n t U
\ *n / )
+ 1, I'e-m-t') sin n ' (t - t') U' dt,
n J
where U' is the same function of t' as U is of t.
MOTION IN A RESISTING MEDIUM. 253
(3) Determine the motion of a body under an attraction
towards a fixed centre proportional to the distance in a
medium whose resistance is proportional to the velocity.
A body performs rectilinear vibrations in this medium in
a period T, and the co-ordinates of the extremities of three
consecutive semi vibrations are a, 6, c; prove that the co-ordi-
nate of the position of equilibrium and the time of vibration
if there were no resistance are respectively
a + c-Zb
(4) If chords be drawn from either extremity of a vertical
diameter of a circle, the time of descent down each of them in
a medium whose resistance varies as the velocity is the same.
(5) One particle begins to fall from the higher extremity
of a vertical line, and at the same instant au other is projected
upwards from the other extremity with a given velocity, the
particles moving in a medium of which the resistance varies
as the velocity ; shew that the time at which they will meet
FT
will be T log y , where a is the length of the line, F the
velocity of projection, and the retardation due to the resist-
ance is - of the velocity.
(6) A light elastic string whose unstretched length is a
is fastened at one end and to the other end is attached a
particle, which hanging freely would stretch the string to a
length 2a. The particle is projected vertically upwards from
the point at which the string is fastened in a medium of
-.2
resistance producing retardation = . If A be the height at-
tained, U the velocity of projection, F the velocity with
which the particle returns to the point of projection,
254 MOTION IN A RESISTING MEDIUM.
(7) Determine the law of attraction that a particle
may always descend to a given centre in the same time from
whatever distance it commences its motion, the medium in
which the particle moves being uniform, and the resistance
varying as the square of the velocity.
(8) If one particle be projected in a medium, the re-
sistance of which varies as the velocity, and another be pro-
jected in vacuo at the same angle, and with the same velocity,
both particles being under gravity, and if ti, t 2 be the times
of describing two arcs in the medium and in vacuo so related
to each other that the tangents at their extremities shall be
parallel to each other, then
(9) Prove the following equations applicable to the
motion of a shot resisted by the air with retardation / (v), v
being the velocity and ^r the inclination to the horizon of the
direction of motion :
dv . v // \
^ cos i/r-vsmA/r =-/(>),
Prove also that if -\|r is the initial value of ifr, and t, x, y
the time and horizontal and vertical distances from the point
of projection,
*.
vsec
*-/;
ffc
gx = I V^dty,
J $
r^o
gy = I v* tan
J ^
Solve completely the case for which
MOTION IN A RESISTING MEDIUM. 255
(10) If the horizontal distance of a projectile in a
resisting medium from the point of projection be connected
with the time by the equation x = f(t), prove that the
equation of the trajectory is
y = -gf (O + g dt + A/ (t) + B,
where A and B are constants.
In the case when t = ax + ba?, shew that the equation of
the trajectory is
y = x tan a g f ~ a 2 # 2 + ^ aba? + - 6 2 # 4 J .
(11) A particle moves under gravity in a medium in
which the resistance varies as the w th power of the velocity,
F 1} F 2 being its velocities at the two points where its direction
of motion makes an angle $ with the horizon, and V its
velocity at the highest point; prove that
(12) If the resistance vary as the n til power of the velocity,
and if /be the retardation due to the resistance when a shot
is ascending at an inclination <,/ when it is moving hori-
zontally, and f when it is descending at an inclination </> in
the trajectory, prove that
I 1 _ 2 cos n <f)
f' + f~ 7T"'
112 f
-^-- = - cos" < I n sec n+1 </>c</>.
/
(13) A body of mass m is describing a parabola under
gravity, and a tangential impulse mu acts on it. Prove
that the focus of the new trajectory moves towards the body
a distance = u, where v was the velocity of the body.
256 MOTION IN A RESISTING MEDIUM.
If the body is acted on by a uniform resistance we may
conceive the path as the envelope of a system of parabolas.
Apply the above to find the relation between corresponding
arcs of the path and the locus of the foci of the enveloping
parabolas.
(14) A particle of weight W moves under gravity in a
medium of which the resistance R is always small and varies
according to a given law ; shew that the velocity of the focus
D
of the instantaneous parabola at any time is ^ x velocity of
the particle.
(15) Prove that the angular velocity of regression of
the apse line of a planet P, moving in a medium producing
retardation R, is
~si
where S is the sun, H the other focus of the orbit, e the
eccentricity, and V the velocity.
(16) Explain how it is that a resisting medium, even
though acting for a short time only, would accelerate the
mean motion of a planet.
(17) A particle is moving amidst rays diverging from a
point, which offer resistance only to motion across them with
resistance proportional to the velocity. Shew that it is
possible for the particle to move with constant angular
velocity about the point, and find the path and the circum-
stances of projection.
(18) A particle describes an equiangular spiral under a
central attraction in a medium of which the resistance varies
as the square of the velocity.
Prove that the distance at which the attraction is a
maximum is half the distance at which the velocity is a
maximum, and that these distances are independent of the
initial distance or initial velocity.
MOTION IN A RESISTING MEDIUM. 257
(19) The retardation due to the resistance of a medium
being ktf, prove that the orbit under a central attraction
^ will be an equiangular spiral if the velocity of projection
be that in a circle at the same distance, and the angle of
projection be cos~ l 2/j,k.
(20) Shew that the equation
+ 2k + n*x = n*P (1 + 2Sr cos ipt),
in which i has all positive integral values, and k is less than
n, represents cycloidal pendulum motion, with viscous resist-
ance, under the action of an infinite series of equal impulses
(in the same direction) succeeding one another at intervals
e 27T
of .
p
Integrate this equation; and, by comparing the result
with that obtained by treating the problem for each impulse
separately from an epoch so distant that the motion has
become independent of the initial circumstances, shew that
1 (n 2 i*p?) cos ipt + 2ikp sin ipt
~2 + A - (n* -%*
_ . _ .
1 e P cos sin nit + e p sin - cos n^t
pni 27HIJ
1 2e P cos - - + 6 P
P
where n a = Jri 2 k*, and t lies between and - .
(21) Prove that the cycloid is still a tautochrone under
gravity when the resistance varies as the velocity.
Prove that the same is true also of any tautochrone.
T. D. 17
258 MOTION IN A RESISTING MEDIUM.
(22) The time of vibration from rest to rest of a cycloidal
7T
pendulum when unresisted being , prove that if the resist-
ance of the air produce retardation 2n sin a x velocity, in
order that the arc of oscillation may be constantly 2c, each
time the bob passes through the lowest point, it must
receive an impulse in the direction of motion
7T , 7T ,
-tana tana
,2 _ e 2
where m is the mass of the bob.
(23) A particle is projected in a medium the resistance
3
of which produces retardation - x velocity, and is under an
attraction to a fixed point which produces acceleration
2
x distance. Prove that the particle will describe a
parabola, tending to come to rest at the origin.
(24) If a particle under a central attraction producing
acceleration fjfr move in a medium of which the resistance
produces retardation 2k (velocity), prove that it will describe
the curve
logW _ - + , tan" 1 ~-^- = 0.
2 (ad ocf \lu? ]& ay ex
(25) A particle moving under gravity in a medium, the
ardation due to wh
plane down the curve
y 2
retardation due to whose resistance = - , slides in a vertical
a
where s is the length of the curve measured from the lowest
point, y the ordinate of the extremity of this arc referred to
a vertical axis, and a a constant ; shew that the time of
reaching the lowest point is independent of the height from
which it starts.
MOTION IN A RESISTING MEDIUM. 259
(26) A particle of mass m falls down a smooth cycloid
whose axis is vertical and vertex upwards, in a medium
whose resistance is -=- , and the distance of the starting
2iC
point from the vertex is c ; prove that the time to the cusp
is A / ( 1 ) , 2a being the length of the axis.
v 9 \ & f
(27) A particle moves in a resisting medium ; state any
reasons, arising from the principle of the conservation of
momentum, which render it probable that the resistance at
any point varies as the density of the medium at that point,
and the square of the velocity of the moving particle.
A particle describes in the medium an ellipse under two
attractions to the foci varying inversely as the n th power of
the distance ; find the density of the medium at any point
of the path ; and shew that if the attractions vary inversely
as the distance, being equal at equal distances, the density
varies as the acceleration with which it would move in a
non-resisting medium, under the same attraction if it were
constrained to move in the ellipse.
(28) A particle is suspended so as to oscillate in a
cycloid whose vertex is at the lowest point : if it begin to
move from a point distant a from the lowest point measured
along the curve, and the medium in which it moves produces
y 2
a small retardation - , prove that before it next comes to
CL
rest energy has been dissipated which is -^- of its original
od
value.
172
( 260 )
CHAPTER VIII.
GENERAL THEOREMS.
220. WE propose now to prove some of the general
theorems connected with the motion of a particle under any
forces, and to investigate the forces requisite for the descrip-
tion of given paths in a given manner. Several of these
results have already occurred as immediate deductions from
the laws of motion ; but to maintain the special character of
the work we give more formal analytical demonstrations,
though these may be considered superfluous.
221. If a particle be subject to forces, whose resultant is
continually at right angles to the direction of motion; the
speed of the particle will be constant.
Let R be this resultant, X, ^ v its direction cosines, then
if m be the mass of the particle, the equations of motion
are
m -7- = \R,
m -=-r = vJK.
dt*
dx
Multiplying by -^- , . . . , adding, and observing that
Cut
dx dy dz
\- r + fjL-^ + v- r = 0,
as as as
since the force R is at right angles to the element of the path,
Id,,, dx d z x dy d z y , dz d*z
we have __(,,). ^ _ + Jf -*+__ = ;
therefore v = const.
GENERAL THEOREMS. 261
Or, we might at once have resolved along the arc ; this
would have given
fl'2 9
=0-
<ft 2
whose integral is
ds
= v = const.
at
The value of R is evidently m ; and therefore R varies
inversely as the radius of absolute curvature of the path.
It is clear that its direction lies in the osculating plane, for
there is no acceleration perpendicular to that plane.
Ex. A particle projected in a plane is under a constant
force R in that plane, continually perpendicular to the direc-
tion of motion ; to find the path described.
v 2
Here R = ; and therefore p is constant, or the path is
a circle.
222. If X, Y, Z be the rectangular components of a force
or forces such as occur in nature, i.e. tending to fixed centres
and being functions of the distances from these centres,
Xdx + Ydy + Zdz = - dV,
i.e. is a complete differential. Compare 78.
Let the points a^, 6 a , c^\ a 2 , 6 2 , c 2 ; &c. be the positions of
the centres of force ; x, y, z the co-ordinates of the attracted
particle; then, if r ly r* 2 , ... be its distances from the centres,
0/(Z)), </> 2 ' (Z)), &c. the laws of attraction to those centres, we
have
"a
262 GENERAL THEOREMS.
But r = *J{(a - xf + (b - y)~ 4- (c - zf] ;
which gives f -y- J = - , &c., for the values of the partial
differential coefficients of r.
Hence
These give
(1),
since every term of the sum is a complete differential. From
78 it is obvious that V is the potential at #, y, z.
223. Under any forces such as occur in nature the
increment of the square of the velocity of a particle in passing
from one point to another is independent of the path pursued,
and depends only on the initial and final positions. This is
true even if the particle be forced to move in any particular
path by a constraint continually perpendicular to its direction
of motion, such as frictionless constraint.
If we choose tangential resolution, the constraint has no
component in that direction, and the equation of motion is
rf 2 5_ dx dy dz
GENERAL THEOREMS. 263
which becomes by (1)
dsd * s dr dV
Therefore ^ 2 = C - 2</> (r) = G - V.
Hence, if U be the velocity at a point whose distances from
the centres are E l} R 2) ...... , and where F=F 1}
or
a result which involves only the co-ordinates of the initial
and final positions. See, again, 78.
224. Hence if from any point of the surface
a particle be projected with a given velocity in any direction;
its velocity when it meets the surface
will be the same, in whatever point it meet that surface;
A and B being any constants.
Now on account of equation (1),
F= 2,<f> (r) = constant
is the equation of a surface on which if smooth a particle will
rest in any position under the action of the given forces.
Hence a particle leaving any point of a surface of equi-
librium with a given velocity, will have on reaching any other
surface of equilibrium a velocity independent of the path
pursued or the point reached. This is evident from 78 if
we notice that a surface of equilibrium is an Equipotential
Surface.
225. To find the condition to which the applied forces
must be subject when the kinetic energy of a particle depends
upon its position only. This is merely the converse of 223.
264 GENERAL THEOREMS.
Here we have
- v* = (x, y, z\
and, therefore,
, fd<j>\ , (d<t>\ , (d$\ ,
vdv = (-f-} dx+(-f)dy + (-f) dz.
\dxj \dy) \dzj
But, in all cases of motion,
vdv = Xdx + Ydy + Zdz.
Hence, in this case we must have
that is,
Zcfo + Fc??/ + Zdz
must be the differential of a function of three independent
variables.
If the seat of the force be in a definite fixed point, which
may be taken as origin, the velocity can evidently depend
only on the distance from that point, not on the direction
of the distance ; hence, if
we have -= v 2 = </> (r).
2
The above process gives, in this case,
vdv = Xdx + Ydy + Zdz = d<f> (r)
= </>' (r) (-dx + y-dy + - dz
X Y Z
or - = = - ,
x y z
which shew that the force is in the direction q/*r.
GENERAL THEOREMS. 265
From this again it evidently follows that its magnitude
must be a function ofr.
226. The proposition of 223 contains the Principle of
the Conservation of Energy for the case of a single particle.
From this principle it follows that if several particles
moving under the influence of the same centre of attraction
have equal velocities at any particular distance from the
centre ; their velocities will always be equal at equal dis-
tances from that centre.
Now we have seen ( 151) that the axis major, 2a, of an
elliptic orbit about a centre of attraction in the focus is
independent of the direction of projection. Hence, by con-
sidering the particular case of a very narrow ellipse, we see
that the velocity at any point is due to a fall, from rest at
a distance 2a, to that point; and that, therefore, in any
elliptic orbit about a focus the velocity at any point is that
due to a fall to the point, through a distance equal to the
distance from the other focus.
227. If the forces acting on a particle, and the square of
its velocity, be increased at any instant in the same ratio, the
path will not be altered.
For the tangent, and the osculating plane, which con-
tains the tangent and the resultant force, are evidently not
altered. And the curvature, being
Normal Component of Forces
Square of velocity
has its numerator and denominator increased in the same
ratio. And the square of the velocity at the end of any arc
is increased in the same ratio as that at the beginning.
Hence each successive elementary arc of the path remains
unchanged.
228. If a number of separate particles whose masses are
nij, m 2 , &c. subjected to forces fj, f 2 , &c. respectively, and
successively projected from the same point in the same direc-
266 GENERAL THEOREMS.
tion with velocities v lf v 2 , &c. all describe one path; the same
path will also_be described by a particle of mass M projected
with velocity U from the same point in the same direction, and
acted on at once by the same forces f l} f 2 , <&c. provided
Suppose that, in addition to the forces f lt f^, &c., a force
R continually acting in a direction at right angles to that of
M's motion be required to cause it to move in the given
path ; i.e. suppose M to be constrained by a smooth tube to
move in the required path ; the equations of motion are
(1),
with similar equations in y and z,
where X, //,, v are the direction cosines of R, and X, F, Z the
resolved parts of y.
, dx dy dz .
Multiplying by -,-, -g-, -=- m order, and adding, we
eliminate R and have
2 Md ( U 2 ) ^mXdx + ^mYdy + ZmZdz.
But for the separate particles m ly m 2 , &c. we have
m l d (^i 2 ) = Wj X l dx + mj Fj c?i/ + ??i x ^ d^, &c. ;
therefore, the path being the same for all,
I 2 {md (v 2 )} = ZmXdx + ^mYdy 4- 2-
Hence S {wid (v 2 )} = Md ( t7 2 ),
But, by hypothesis, 'Zmv* = MU*,
therefore (7=0.
GENERAL THEOREMS. 267
[Instead of this analysis, it is sufficient (by 78) to
notice that the work done on M is the sum of that done
on m lt ra 2 , &c. Hence the increase of kinetic energy must
be the same ; and if, at starting, the kinetic energy of M be
the sum of those of m l} m^, &c. it will remain so throughout
the motion.]
Hence the kinetic energy of M will be at each point of
the orbit equal to the sum of the kinetic energies of m lt m 2 ,
&c., at that point. To find R, notice that in general the
pressure on a constraining curve depends upon two things,
the resolved parts of the impressed forces, and the pressure
due to the velocity. Now the latter part is as the kinetic
energy, therefore in the case of M it is the sum of the
corresponding forces in the case of m l> ra 2 , &c. Also the
same may be said of the resolved parts of the impressed
forces. But in the , case of each particle, these partial
pressures destroyed each other, since the curve was described
freely, hence their sums will destroy each other, or the curve
will be freely described by M.
229. If at any instant the velocity of a particle, moving
under a conservative system of forces, 77, be reversed, the
particle will describe its former path in the reverse direction.
Suppose a smooth tube, in the form of the original path,
requisite to constrain the particle to move backwards along
it. The velocity will be, at each point, of the same magni-
tude as before; the resultant acceleration, and the curvature
of the path, will also be alike ; hence the normal component
of the force will produce the requisite curvature of the path,
and there will be no pressure on the constraining tube. The
tube is, therefore, not required. Whence the proposition.
230. LEAST, OR STATIONARY, ACTION. If v be the velo-
city of a particle whose mass is m, and if s be the arc of the
path described, the value of the quantity
A = mf vds
(taken between proper limits) is called the Action of the
particle.
268 GENERAL THEOREMS.
If a particle move freely, or on a smooth surface, (under
forces such as occur in nature,) between any two points, the
value of the integral m / vds for the whole actual path is
generally less than it would be if the particle were constrained
to pass from one point to the other by a different path. This,
combined with the above definition, is for a single particle
the Principle of Least Action; of which in an elementary
work like the present we can give only a very imperfect
sketch. For further information see Thomson and Tait's
Natural Philosophy, 318.
231. The proposition to be proved is that, 8 being the
symbol of the Calculus of Variations, and the mass of the
particle being for simplicity taken as unity,
Now S fvds=f 8 (vds) =f(v&ds + ds&v)
= I (v&ds + vdtSv), since v = -r .
But generally,
- v* = j(Xdx + Ydy + Zdz) = ^ (x, y, z),
the constraint, if any, having disappeared ;
hence v&v = X$x + Y&y -f ^2;.
But Z-g.
Hence
v ^ v = (7]^+ ^|% + ^ ^)-^(X^ + /A% + ^(
Now if the particle remain on the surface whose equation
18^=0,
+ fift + v^z = kSF = 0,
and if it leave it R = 0, so in either case the latter term on
the right vanishes.
GENERAL THEOREMS. 269
Also ds* = da? + dy* + dz* ;
which gives dsSds = dxbdx + dySdy + dzbdz,
or
dx
since the order of d and 8 is immaterial.
Hence
, * j * j * j dz
+ fed + 8d + 8rf
fc?57
taken between proper limits. Now at both limits
&? = 0, % = 0, S^ = 0;
hence we have 6.A = 0.
232. It is commonly said that as, in general, it is im-
possible to suppose the Action a maximum, this result shews
that it is a minimum. The true interpretation of the ex-
pression, SA = 0, is that the unconstrained path of the particle
is such, that a small deviation from it will produce an infi-
nitely smaller change in the value of A. Hence Hamilton has
suggested the more appropriate title Stationary Action.
233. If no forces act on the particle except the constraint
of the surface, we have v constant, and the above equation
shews that in this case the length of the path is generally a
minimum.
A particle therefore, projected along a surface and subject
to no forces, will trace out between any two points in its path
the shortest line on the surface.
270 GENERAL THEOREMS.
It may happen, in the case of a sphere, for instance, that
the particle will not necessarily trace out the shortest line on
the surface between the two points; but we cannot here enter
into the details necessary to the full elucidation of such cases.
234. We may apply this principle directly to form the
equations of motion in any particular case, or to find the
actual path under the action of any forces.
Ex. I. Let us take again the case of the refraction of
light in the corpuscular theory ( 130), as illustrating the
general principle of Least Action in the case of a particle.
The velocity in the upper medium is supposed to be u, that
in the lower v, AB being an equipotential surface.
In this case the expression for the Action becomes simply
uPQ + vQR,
if PQR be the path of the particle, the mass being unity.
By making this quantity a minimum, as depending on
the position of Q, P and R being given points ; it is easy to
shew that Q must lie in the plane through P and R perpen-
dicular to the surface AB, and also that the resolved parts of
the velocities in the upper and lower medium parallel to the
tangent plane to AB at Q must be equal ; and therefore the
impulse applied to the corpuscle at Q is perpendicular to AB,
while the sines of the angles which PQ and QR make with
the perpendicular to AB are inversely as the velocities in the
two media.
(If we had made the Time from P to R a minimum, we
should have obtained the law of refraction on the undulatory
theory.)
235. Ex. II. To find the equation of the path described
by a particle about a centre of attraction.
GENERAL THEOREMS. 271
Let P be the central attraction at distance r, then
[ 2 , suppose, (1)
which gives
Hence
fvds=f<f)(r)ds.
/{<' (r) Sr ds + < (r
ds
&B + ySy + zSz) ds
The integrated part refers only to the limits, and must
therefore vanish independently of the integral. That the
integral may be identically zero, we must have
with similar equations in y and z. These may be written
'x dr
^r=0
ds ds
dr dz
.(a).
272 GENERAL THEOREMS.
Multiplying by any three constants, A, B, C, and adding,
we have
which is obviously satisfied by
Ax + By + Cz = 0.
This equation shews that the orbit is in a plane passing
through the centre of attraction. Let xy be this plane, then
we may confine ourselves to the first two of equations (a).
Multiplying the second by as and the first by y and sub-
tracting, we obtain
, , , x dr ( dy
This is immediately integrable, and gives
- = constant.
Since <t>(r) = v, we see by 24, that this is in polar
co-ordinates
i/i
which is the equation for the equable description of areas.
Finally, multiplying these two first equations of group (a)
by x and y respectively and adding, we have
But, since
dr dx dy
GENERAL THEOREMS. 273
we have by differentiation
<M 2
ds) ~
Substituting in (c), and changing the independent variable
from s to 6 by means of the equation
ds* = dr z
we have
Putting - for r, this becomes
But, by (6) as developed in 142,
Also < (r) f (r) = - P, by (1).
Thus (d) becomes
d*u P . eio .
5^ + M = AHT- asm13o.
236. We might have treated these equations ( 235 (a))
somewhat differently thus
ds
, , x dx dx
Hence ^ (r ) _ = _ , & c . ;
and we have the equations
T. D. 18
274 GENERAL THEOREMS.
which give, at once,
x y z '
containing the theorems of constant plane and equable de
scription of areas: and since
(r)-<r)*f(r) P,
the ordinary equations in three rectangular directions.
237. We might have simplified the work by using polar
co-ordinates immediately after having proved that the orbit
is plane. For we have
/( /dr\ z ]
A = /</> (r) A / jr 2 + ( - ) \dO, a minimum,
and therefore the calculus of variations gives (by the formula
dr
or reducing, and putting h for (7,
whence r 2 -r- = h,
dt
the equation for the equable description of areas.
GENERAL THEOREMS. 275
Squaring (e) and attending to (1), we have
or, putting
r 41
and differentiating and dividing by 2 - ,
P
+
the general equation of central orbits.
238. VARYING ACTION. If, in 231, we assume
^ = f(Xdx + Ydy + Zdz) + H=H-V,
(with the notation of 78) it is evident that H will depend
on the initial velocity. Supposing that this and the initial
and final co-ordinates vary; then, in addition to the already
considered variation of the form of the path between its
extremities, upon which the unintegrated part of the value of
&A depends, we shall have in BA terms depending on the
variations of initial and final positions and of initial velocity.
The additional term in v$v is SH, and its integral tSH is
at once obtained. Hence in this more general variation of
the conditions we have in the value of 8A the following
additional terms, depending on the limits only, and therefore
to be treated by themselves,
182
0)
276 GENERAL THEOREMS.
Hence, if A could be found in terms of x, y, z, a? 0> y , z (
and H, we should have at once the first and second integrals
of the equations of motion in the form
/dA\ _ dx fdA\ _ _ fdx\
\dx) ~ dt ' \dxjQ
&c. &c.,
with the farther condition
239. A is, of course, a function of ac, y, z, x , y , z , and
H, and we see by the equations above that it must satisfy
the partial differential equations
240. The whole circumstances of the motion are thus
dependent on the function A, called by Hamilton the Cha-
racteristic Function. The above is a brief sketch of the
foundation of his theory of Varying Action, so far as it relates
to the motion of a single free particle. The determination of
the function A is troublesome, even in very simple cases of
motion ; but the fact that such a mode of representation is
possible is extremely remarkable.
241. More generally, omitting all reference to the initial
point, and the equation 239 (2) which belongs to it, let us
consider A simply as a function of x, y, z. Then
Any function, A, which satisfies
possesses the property that
(LA dA dA
dx ' dy ' dz
GENERAL THEOREMS. 277
represent the rectangular components of the velocity of a
particle in a motion possible under the forces whose potential
is V.
For, by partial differentiation of 239, (1), we have
dte_ x _ _d,V^dAd*A dA d?A dA d*A
dt* dx dx dx 2 dy dxdy dz dxdz'
ddA\_dxdA dy d*A dz d* A
~#4*-# dxdy dtdxdz'
Comparing, we see that
dx _ dA dy _dA dz _ dA
~di = ~dx' ~dt~~dy' di~ ~dz '
satisfy this and the other two similar pairs of equations.
242. Also, if a, (3 be constants, which, along with H,
are involved in a complete integral of the above partial differ-
ential equation, the corresponding path, and the time of its
description, are given by
where a a , ft , e are three additional constants.
For these equations give, by differentiation,
d*A dx d*A dy d*A dz _ ,
dxda dt dyda. dt dzdcn dt ~
d*A dx d?A dy d?A dz _
dacd/3 ~dt + dydft 'dt + datp Tt ~
d*A dx d*A dy d*A dz _
dxdH dt + dydH dt + dzdH ~dt "
But, differentiating 239, (1), we get
d*A dA &A dA. d*A dA
I'll 1 '
dctdx dx dady dy dadz dz
d*A dA d z A dA d' 2 A dA [ ,,x
dfidx dx dftdy dy d/3dz dz
dA d*A dA _
? "T 7 TT i 7
dHdx dx T dHdy dy T dHdz dz
278 GENERAL THEOREMS.
The values of -j- , &c. in (a) are evidently equal respec-
Cit
tively to those of ( -?- ) , &c. in (6). Hence the proposition.
V dx j
243. Equiactional surfaces, i.e. those whose common
equation is
A = const. = C,
are cut at right angles by the trajectories.
For the direction-cosines of the normal are obviously
, . (dA\ fdA\ (dA\ ,, ^ . dx dy dz
proportional to , } , - , that to , , .
Thus the determination of equiactional surfaces is re-
solved into the problem of finding the orthogonal trajectories
of a set of given curves in space, whenever the conditions of
the motion are given. We cannot, in the present work, spare
space for much detail on this very curious subject, and there-
fore give but one other singular property of these surfaces
before applying the principle of Varying Action to an im-
portant problem.
Let CT be the normal distance at any point between the
consecutive surfaces
A = C, and A = C+SC.
We have evidently
fdA\ , (dA\ . fdA\ . sn
( 7 )*&B -f ( -j- ) oy + [ -j- ) o-f = oC/,
\dx J \dy } \dz)
dx ~ dy ~ dz ^ _ ~ ~
where Sx, By, &z are the relative co-ordinates of any two
contiguous points on the two surfaces. If p be the length of
the line joining these points, its inclination to the normal
(i.e. the line of motion), this may evidently be written
vp cos 6 = v& = SC,
since p cos 6 is the normal distance between the surfaces.
GENERAL THEOREMS. 279
Thus, the distance between consecutive equiactional surfaces
is, at any point, inversely as the velocity in the corresponding
path.
This may be seen at once as follows ; the element of the
action is v&s (where Ss, being an element of the path, is the
normal distance between the surfaces) and must therefore be
equal to BO.
244. To deduce, from the principle of Varying Action,
the form and mode of description of a planet's orbit.
dV
In this case it is obvious that -=- represents the attrac-
tion of gravity ( . Hence the right-hand member of
239 (1) may be written
Let us take the plane of xy as that of the orbit, then the
equation 239 (1) becomes
It is not difficult to obtain a satisfactory solution of this
equation ; but the operation is very much simplified by the
use of polar co-ordinates. With this change, (1) becomes
which is obviously satisfied by
fdA\
constant = a
Hence
280 GENERAL THEOREMS.
The final integrals are therefore, by 242,
'dA\ dr
A ] = ttl = e - a. i
OL / J
and
Uar'-"-/ /.,_*. v , <">
0^.9-r
r 2
These equations contain the complete solution of the
problem, for they involve four constants, lf a, H, e. (5)
gives the equation of the orbit, and (6) the time in terms
of the radius vector.
245. To complete the investigation, let us assume
a 2 " Z 2
where Z and 6 are two new arbitrary constants introduced in
place of a and H. With these (5) becomes
dr
->-}
Z 2 ' Ir r*
dr
A 2 f 1 l \ 2
V I* ~ (r ~ 1)
_
(9 - cos- 1 r
e
7
I
or
1 -f e cos (0 - aj '
GENERAL THEOREMS. 281
the general polar equation of conic sections referred to the
focus.
Also, by differentiating (5) with respect to r, we have
adr ^ = d6,
from which, by (6), we immediately obtain
This involves, again, the equation of equable description
of areas.
246. To illustrate the subject farther, we will deduce
others of the ordinary results of Chaps. V. and VI. from these
formulae. Thus, let , r denote the polar co-ordinates of
any fixed point in the path, from which the action is to be
reckoned. We have, by (4),
A =
Jr. V \r / r 4
I * ^ fl2 <f).
because, by (5),
6-6, ' adr
/*
Jr*
To integrate (7), remark that ( 149) ^ < - in an elliptic
orbit, and that thus H is negative by 244 (1).
282 GENERAL THEOREMS.
Put |r=-2a,
.
pa
and r = a(le cos
and (7) becomes, after substitution,
/ f*
A=Vfj,a\
'
4
which is immediately integrable.
It is obvious from 160 that </> represents the excentric
anomaly. If we measure it from the perihelion we have
evidently
A = \///,a (<f> + e sin <).
247. By (6) we have
By employing the same substitutions as in last section,
<f> being measured from perihelion, it is easy to bring this
expression into the form
\/~\ (1 ~
ecos
a 3
[0 esin (],
the formula of 160.
248. By the process of 160 we see that while
$ - e sin <f>
is proportional to the area described about the centre of
attraction, and therefore proportional to the time ;
</> + e sin (j>
is proportional to the area described about the other focus,
and [is, by 246, proportional to the Action. Thus in a
GENERAL THEOREMS. 283
planet's elliptic orbit the time is measured by the area described
about one focus, and the Action by that about the other.
An easy verification of this curious result is as follows.
With the usual notation we have
d A = vds
h,
by the result of 141. But in the ellipse or hyperbola, p
being the perpendicular from the second focus,
Hence dA = j- p'ds,
which expresses the result sought. (Proc. R. S. E., March,
1865.)
It is easy to extend this to a parabolic orbit, for which,
indeed, the theorem is even more simple.
249. It may be useful to give another example of
Hamilton's remarkable method. For this purpose we will
again briefly consider Cotes' Spirals. [See Chap. V., Ex. (9).]
Here the central attraction is inversely as the cube of the
distance, and therefore the equation of Action is
Hence, as in 244, we have
fdA\
From these it is easy to find A, but we leave this as an
exercise to the student.
284 GENERAL THEOREMS.
Again,
Substituting exponentials for the logarithm, this takes
the form
This integration fails for certain special values of, or
relations among, the constants, but the reader can have no
difficulty in obtaining the requisite changes in these cases,
and so reproducing all the varieties of possible orbits given
in the Examples to Chap. V.
250. Assuming, for a set of particles, the result of 231,
we may easily obtain the celebrated equations of motion in
generalized co-ordinates due to Lagrange, as well as the
general equations of Varying Action in the form given by
Hamilton. The following is an outline of the process
for the special case in which the geometrical relations are
independent of the time, and in which therefore the con-
servation of energy holds.
Let the co-ordinates of the particles of such a system
be expressed in terms of new co-ordinates 0, </>, ty, ... which
are independent of one another. Then it is easily shewn
(Thomson and Tait's Nat. Phil. 313) that the kinetic
energy, T, is a homogeneous quadratic function of 0, </>, yjr, ...
and also a function of 6, </>, ^, ...
Hence 2* (f) = *T.
where the bracket denotes partial differentiation.
But the equation of energy is
T+V=H.
GENERAL THEOREMS. 285
The variation of the action is
As we have agreed to assume the results of 231, it is
obvious that the unintegrated part of the value of $A must
vanish. Hence we have two sets of equations.
1. From the unintegrated part we have Lagrange's
Equations, equal in number to the generalized co-ordinates,
and of which the following is one :
d_(dT\_(dT\ (dV\
dt\d6> \d) +
2. From the integrated part the Hamiltonian system
along with
(a)-
As a verification, differentiate with regard to t the
equation
and we have the result
which is obviously consistent with the equations of La-
grange.
286 GENERAL THEOREMS.
251. As an example of Lagrange's equations of motion,
consider the case of the small oscillations in the magnetic
meridian of two equal bar-magnets each suspended by two
equal parallel strings from points in a horizontal line.
Let m be the mass of each magnet, 2a the distance
between the adjacent poles when the magnets are in equili-
brium and demagnetized, I the length of each string, and
fju the product of the strengths of the poles.
If a, y be the displacements of the magnets at the time
t ; then, neglecting the vertical velocities,
2a + x y 2
ous poles of
nother.
Hence the equations of motion are
only the contiguous poles of the magnets being supposed
to act on one another.
Adding, -( + y) = - - (x + y).
Subtracting,
d . . a x y
. . a ( x y\ q .
m dt (x - y} = -w( l - a -]-*7<*-y>-
Making x and y constant in (1) and (2) we get their
equilibrium values; and measuring x and y from these we
get
GENERAL THEOREMS. 287
Thus if n 2 =|,
I 2ma*'
we have x + y = A cos (nt + B),
x v = A, cos
It depends upon whether the proximate poles of the
magnets attract or repel one another, whether n or n^ is the
greater.
If the magnets be swung as one piece at their equi-
librium distance from one another, the time of oscillation
will be the same as that of either magnet when left to itself,
since the magnetic attraction does not vary : this is the
character of the first harmonic motion.
Again, if the magnets be swung with equal and opposite
motion, the centre of inertia is fixed, and the time of oscilla-
tion will be the same as if one of the magnets were held
fixed and its magnetic strength doubled; it will therefore
be shorter or longer than the first period, according as the
poles presented to one another attract or repel ; this is the
character of the second harmonic motion.
252. If we treat the investigation of (184), in the way
in which Hamilton treated that of (230), we arrive at a
number of curious theorems connected with Brachistochrones;
of which a few will be given here from the Trans. R. S. E.
1865.
Putting T for the time in the Brachistochrone, we have
'dr'
(*L\ = J/I ^
' \dxj W 2 dt)
(dr\_ [dt_ [ds
\dHJ~ ~J t?~~J v 8 '
corresponding to the group in (238).
288 GENERAL THEOREMS.
Hence, just as in (241) it may be shewn that for any
forces, of which F is the potential, a value of r from the
equation
'dr\* (dr\* 1 1
7 1 ~T I 7
/dry 1
Vefey v>
_ _
2 (#- F) '
is such that its partial differential coefficients represent the
components of velocity in a possible brachistochrone, each
divided by the square of the whole velocity.
Also if r contain, besides H, two arbitrary constants, a, /3,
the equations of the brachistochrone are
253. To find the Brachistochrone when the attraction is
central, and proportional to a power of the distance; the
velocity being also proportional to a power of the distance,
that is, being the velocity from infinity, for an attraction,
from the centre for a repulsion.
Here ^ = 2(#-F) = ^,
and the central attraction at distance r is evidently
dV np
Thus (2) becomes
dr /dr /dr
or, changing to polar co-ordinates,
,drj r v*\d0J ' r 2 sin*0
It is obvious that we must take
GENERAL THEOREMS.
289
which shews that the path is in a plane passing through the
centre of force. The above equation will then be satisfied by
dr
Hence we have
a i 2a
ad 4- - -_
71+2
And the equation of the brachistochrone, which is evidently
a plane curve, is
r n+2
^^^^^Ml I ^^^_ _
/Zwi+2 " '^? / """77/7 2 I
/ ' 1 T 2 /I A^
v^" 1 v 1 -^]
or
= 6>-
n+2
r - =
~r
while the equation of the /ree path is
/r\^ = w--
W 2
The above integration fails in the case of n = 2 ; that
is, for a repulsion directly as the distance, the circle of zero
velocity being evanescent. But in this case
T =
T. D.
19
290 GENERAL THEOREMS.
and the equation of the brachistochrone is
/"*
the logarithmic spiral. Eliminating r between these equa-
tions, we see that the time is proportional to the polar angle.
Since a definite form has been assigned to the expression
for the velocity in this problem, it is obvious that H is given,
and therefore that there is no [-
\an./
254. It is easily seen that
is the equation of an Isochronous surface.
Also, since
dx = \dy = \dz
dx dy dz '
di cfc dt
the brachistochrone cuts all such surfaces at right angles.
And the normal distance between two consecutive iso-
chronous surfaces is proportional to the velocity in the bra-
chistochrone of which it forms an element. For, of course,
8s = v&r.
255. Hamilton's equation for the determination of the
Characteristic Function (A) in the case of the free motion
of a single particle is
The comparison of this with the equation of 252 suggests
a useful transformation. Introducing in that equation a
factor # 2 , au undetermined function of sc t y, z, we have
GENERAL THEOREMS. 291
If we make
tf = f (T)
a " d 2-(^T) = 2(tfl - F ' ) '
it becomes
Here it is obvious, that $ (T) is the action in a free path
coinciding with the brachistochrone, and that 2 (H 1 Fi) is
the square of the velocity in this path.
Hence the curious result that, if T be the time through any
arc of a given brachistochrone, the same path will be described
freely under forces whose potential is V lt where
V)-
l) ~
- V"l ' I/ o / TT -ir\>
- {J3. V )
<j)' being any function whatever; and <f>(r) will represent the
action in the free path.
256. The simplest supposition we can make is that $' (T)
is constant. In this case the velocity in the free path is in-
versely proportional to that in the brachistochrone at the
same point ; and the action in the one is proportional to the
time in the other. In fact, as Sir W. Thomson has pointed
out, in this case the investigation may be made with extreme
simplicity, thus
In the brachistochrone we have
a minimum.
v
Putting v - , and considering v as the velocity in the same
path due to another (easily determinable) potential ; we must
have
jvds a minimum.
This is the ordinary condition of Least Action, and belongs,
therefore, to a free path.
192
292 GENERAL THEOREMS.
Hence, since the cycloid is the brachistochrone for gravity,
and since in it -y 2 = %gy, it will be a free path if v 2 = ~ , that
c/t/
is for a system of force where the potential is found from
This gives
_
dx ~ " dy
In other words, a cycloid may be described freely under
an acceleration inversely as the square of the distance from
the base ; and the velocity at any point will be the reciprocal
of that in the same cycloid when it is the common brachisto-
chrone.
This result is easily verified by a direct process.
257. The converse of the proposition in 255 is also
curious. Taking Hamilton's equation, 2:39, we have
Comparing this with that of 252, we see that r = (A)
is the brachistochronic expression for the time in a path which
is a free path for potential F, provided that <f>(A) and the
potential for the brachistochrone are connected by the equa-
tion
Hence, if A be the action in a given free path, the same
path will be a brachistochrone for forces whose potential is V 1}
determined by the condition just given, V being the potential
in the free path.
Thus, the parabola
O - &) 2 = 4a (y - a)
GENERAL THEOREMS. 293
is the free path for v 2 = 2gy. And the action is given by
Hence this parabola is the brachistochrone for
In the simplest case <f>' (A) = 1, and we have
_ = __
d# dy
Hence, by 256, the parabola is a brachistochrone when a
cycloid is the free path.
258. The examples immediately preceding are but par-
ticular cases of the following general theorem, which is easily
seen to be involved in the results of 255, 257. If we have
two curves P and Q, of which P is a free path, and Q a
brachistochrone, for a given conservative system of forces ;
P will be a brachistochrone for a system of forces for which Q
is a free path and the action and time in any arc of either,
when it is described freely, are functions of the time and action
respectively, in the same arc, when 4t is a brachistochrone.
From this property Professor R. Townsend, Quarterly
Journal of Mathematics, Vol. xin., has shewn how to deter-
mine the intensity for parallel and concurrent forces for
which given curves are brachistochrones.
For in the brachistochrone the velocity of description v
for parallel forces must be proportional to the sine of the
angle i between the directions of force and motion, and for
concurrent forces must be proportional to the length of the
perpendicular p from the centre of force in the direction of
motion; provided that in addition the osculating plane at
every point contains the direction of the force.
Hence
(a) For parallel forces, every curve (necessarily plane
for brachistochronism in that case) for which sin 2 i <f> (z),
294 GENERAL THEOREMS.
where z is the ordinate in the direction of the force, is
brachistochronous, under description with the velocity which
would vanish with i, for the law of force Z = - & 2 <' (z), k
being any constant.
(6) For concurrent forces, every curve (necessarily plane
for brachistochronism in that case also) for which p 2 = < (?-),
where r is the radius vector from the centre of force, is bra-
chistochronous under description with the velocity which
would vanish with p, for the law of force R - k~<f>' (r), k
being any constant.
In the following examples, given by Professor Townsend,
the form of < (z). or < (r) being given, it is left as an exercise
for the student to find the corresponding brachistochronous
curve, the method of description, and the line of zero velocity ;
, . , z z* a a 2 z z*
(a) * = -, - ~, ? , l.-j, i-y,
^
a 4 _
(a 2 - 6 2 ) 2 '
r 3 r 4 t* a 4 r 11
= ar,r-8in-a,--, - 2 , -, -, - 8 ,
tti (Jb (JU I LI
- a 2 , + m 2 (r 2 - a?
^ a 2 a 2 + b 2
(a n r n )
2a r ' a w
These examples will be found to contain most of the ele-
mentary brachistochrones that have been recognized, but
given any curve the process is the same to determine the
forces for which it is a brachistochrone for parallel or concur-
rent forces ; < (z) or (f> (r) being determined from the pro-
perty of the curve and <f)' (z) or <' (r) expressing the required
law of intensity.
GENERAL THEOREMS. . 295
259. To solve the inverse problem, the determination of
the brachistochrone from the law of force ft '(z) or </>' (r)
supposed given, the differential equation between z and x
or r and 6 is immediately obtained from the general relation
(a) or (6), but these differential equations can only be in-
tegrated in particular cases.
Thus if the force vary as the (n l) th power of the
distance, we have
a n sin 2 i= (z n c n \
or a n ~*p z =(r n c n ) ;
leading to the differential equations
dz _ f a n y
dx ~~ \z n c n /
dr /+a n ~ 2 r 2
/-\ i
7/j I ,. n -L
rau \ r n c n
which are not generally integrable in finite terms unless
c = 0; the special case considered in examples 10, 11, and 21
given above.
260. Professor Townsend, Quarterly Jout^nal of Mathe-
matics, Vol. XIV., has also shewn how from the property
( 185) that " if for the same velocity of description any
curve, plane or twisted, be at once a free path for one
system and a brachistochrone for another system of con-
secutive forces, the resultants of the two systems of forces
must, at every point of the curve, be reflexions of each other,
as regards both magnitude and direction, with respect to
the current tangent at the point," cases of the free motion
of a particle may be deduced from familiar cases of bra-
chistochronous motion, and conversely.
Interesting applications are given of the principle to the
comparison of the different methods of description in free
and brachistochronous motion in well-known orbits, such as
the parabola, the bifocal conies, the cycloid, catenary, &c.
Thus every bifocal conic being a free path for any com-
bination of two forces emanating in similar or opposite
296 . GENERAL THEOREMS.
directions from the foci, and varying inversely as the square
of the distance from its own focus, the velocity of description
(real or imaginary) vanishing at each point (real or imagi-
nary) of equal and opposite normal action of the forces; it
follows that every bifocal conic, ellipse or hyperbola, is a
brachistochronous path for any combination of forces ema-
nating in similar or opposite directions from its two foci,
and varying each inversely as the square of the distance
from the other focus ; the velocity of description (real or
imaginary) vanishing at each point (real or imaginary) of
equal and opposite normal action of the two forces.
261. A ^article moves in a plane, under an attraction
directed to a point which moves in a given manner in the
plane : to find the motion.
Let x, y, %, 77 be the co-ordinates of the particle and
point, at time t. f and 77 are given functions of t. Also let
P =f(r) be the acceleration due to the attraction at distance
r. Then
^ = _P _ *-?
are the equations of motion.
The equations of relative motion are, of course,
...(2),
or, putting , 1^, for the relative co-ordinates,
(3).
= P
de
df
GENERAL THEOREMS.
297
These equations illustrate, in a particular case, the general
theorem of 26 ; as they contain, in addition to the terms
due to the attraction of the fixed centre, the two known
d? d?r)
quantities ~ and -, ., , the components of acceleration
at' at"
of the centre revised.
262. Ex. Let the attraction vary directly as th# dis-
tance.
Here P = /*Vf ,* + %', and equations (3) of last section
become
~dt*
dt* I
dt*
which are easily integrated, in the form
Ccos(V/d + l>)-
for particular values of f and 77 in terms of t
Curiously enough, these equations shew that the form and
position of the relative orbit are altered merely by shifting
its centre, which is no longer at the centre of attraction.
As a particular case, suppose the centre of attraction to
move with constant acceleration, a, parallel to a given direc-
tion, which may be taken as the axis of y. The centre of
attraction will in general (Chap. IV.) describe a parabola, and
the relative motion of the particle will be the same as in
133, the centre of the ellipse or hyperbola being not at the
298 GENERAL THEOREMS.
centre of attraction but at a distance - from it in a line
parallel to the axis of y.
Again, suppose the centre to move uniformly in a circle.
We have
= a cos cot, ij = a sin cot,
and f ! = A cos (\fpt + B) ' cos cot,
77 j = C cos (dpt + .Z)) sin ftj,
a) 2 /A
= f+
iia
= ^1 COS (V/-^ + ) COS &),
CO 2 fJb
and y ^ cos (vV^ + -^) s i n ^
w //.
and the absolute path is therefore epitrochoidal.
263. If the radius vector of a curve in space be at each
instant parallel to the direction, and equal to the magnitude,
of the velocity of a particle moving in any path ; the curve is
called the hodograph corresponding to the path ( 20).
The hodograph is evidently a plane curve if the path
is so.
Let x, y, z be the co-ordinates of a point in the path,
f, rj, those of the corresponding point of the hodograph ;
then evidently by the definition,
I-
GENERAL THEOREMS. 299
Hence, if or be the arc of the hodograph,
dt~~\ \dt
and the direction cosines of So- are proportional to
di?' ~dt z ' di?'
Hence we see, as in 20, that
The tangent to the hodograph at any instant is parallel to
the resultant acceleration of the particle at the corresponding
point of its path, and the velocity in it is equal to the ac-
celeration of the particle.
264. The most important case of the hodograph being
that corresponding to an orbit about a single centre of at-
traction, we may deduce the above properties for that case in
a somewhat different manner.
Let P be any point in PA, an arc of an orbit described
about a centre of attraction S. Draw SY perpendicular to
300 GENERAL THEOREMS.
the tangent at P, and take SQ . SYh, then evidently SQ
is equal to the velocity at P, and perpendicular to it in direc-
tion. Hence the locus of Q is the hodograph turned in its
own plane through a right angle.
But we see that it is the polar reciprocal of PA with
regard to a circle whose centre is $ and radius = *Jh. Hence,
by geometry, the tangent at Q is perpendicular to SP. This
evidently corresponds to the first of the two general properties
of the hodograph given in the last section.
Let r, 6, p, s, r', ff, p, s' represent the usual quantities for
corresponding points of the two curves; then if p be the radius
of curvature at Q, we have by the condition that QZ is per-
pendicular to SP,
cM ,d0 ,dr^d0
dt ~~ p dt~ r dp dt
_
dt~ p s dr T
which proves the second property.
265. When the central attraction is inversely as the
square of the distance, we have by 264 for the arc of the
hodograph,
ds' _ fju
dt ~r 2 '
, _ ds' _ ds' dt _ fju dt _ fji
p '~d8~dtd0~~r*d0 = h'
Hence for all conic sections described about the focus the
hodograph is a circle, as was first shewn by Hamilton.
This might have been shewn in another way, thus. In
the fig. ( 264) if PA be a portion of an ellipse or hyperbola
of which S is the focus, the locus of Y is the auxiliary circle.
GENERAL THEOREMS. 301
Hence evidently the locus of Q is a circle. If PA be a por-
tion of a parabola of which S is the focus, the locus of Y is
a straight line, and therefore that of Q is a circle passing
through 8.
Hence generally, the hodograph for any orbit about a
.centre of attraction inversely as the square of the distance,
is a circle ; about an internal point for an ellipse, an external
point for a hyperbola, and about a point in the circumference
for a parabola.
A purely analytical proof of the same theorem is easily
given. If x, y be the co-ordinates of the planet, f, 77 those
of a point in the hodograph, then
i._dx dy
*=dt> r)== ~di'
The equations of motion are
d*x juc
Hence, as usual,
d dx
and therefore
d * x -p^Q de ] _p
dt*~h C dt~hdt\r'
which gives, by integration,
- fc _u A
j j~ + A = j .
dt hr
ft,. ..
Similarly ^ + B =17 + B = -^,
302 GENERAL THEOREMS.
and thence
proving that the hodograph is a circle.
Also, by eliminating -,- , -j- amon
(1), (2), we get for the equation of the orbit
Also, by eliminating -,- , -j- among the three equations-
which gives the focus and directrix property at once.
It is evident that that diameter of the circular hodograph
which passes through the centre of force is divided by the
centre of force in the same ratio as the axis major of the
orbit is divided by the focus, and its length = ~ .
266. The law of diffusion of heat and light from a
calorific and luminous body is that of the inverse square of
the distance. Hence an arc of the hodograph of a planet's
orbit, which arc we have already seen to represent the integral
acceleration due to the central attraction, represents also the
entire amount of light or heat derived from the Sun during
the passage through the corresponding arc of its orbit.
Ex. Compare the amounts of light and heat received
throughout their orbits by the Earth moving in a circle,
and a comet moving in a parabola at the same perihelion
distance.
The hodographs are both circles, one about its centre, the
other about a point in its circumference; but the diameter of
the latter is *J'l times the radius of the former ( 149).
Hence their circumferences are V^ : 1, or the Earth
in its orbit receives in a revolution V2 times the amount
of light and heat which the comet can receive in its whole
path.
GENERAL THEOREMS.
303
It is evident that the path apparently described by a
fixed star, in consequence of the Aberration of light, is the
Hodograph of the Earth's orbit, and is therefore a circle in
a plane parallel to the ecliptic, and of the same dimensions
for all stars.
267. Sir W. R. Hamilton enunciates (Lectures on Qua-
ternions, p. 614) the following proposition :
If two circular hodographs, having a common chord, which
passes through, or tends to, a common centre of force, be both
cut perpendicularly by a third circle, the times of hodogra-
phically describing the intercepted arcs will be equal.
It is evident from ( 265), that the two orbits are conic
sections of the same species, and with equal major axes.
Also, every circle which cuts both hodographs perpen-
dicularly must have its centre on the c6mmon chord. Let
the figure represent one of the hodographs, S being the
centre of force, and ABP the common chord. Take any
point P and draw the tangents PT, PT'. We proceed to
investigate the difference of the times of hodographically
describing TT' and the corresponding arc for a position of P
slightly shifted along A P.
304 GENERAL THEOREMS.
Draw OA perpendicular to AP. Let OT = a, AB = b,
OA=c, SP = r, SM=<v, flf' = -Gr', P0 = q, PA=r f , and
PT = PT' =r. If P be moved through a space Sr, the
increase of the angle PSM which is the angle vector in the
orbit, is nearly. But the corresponding radius vector in
the orbit is ( 264) and therefore the time of hodographi-
cally describing the small arc at T is
- . ( 265.)
H r a rr TT vra
Hence the whole change produced in the time of hodo-
graphically describing the arc TT' by shifting P is
( h
TT \aiz- a'
[This is easily seen, if we notice that by the figure
OT] . f . a . c] n
,> = r sin jsm" 1 - sin" 1 -> .]
Now this is the same for both hodographs, and, as the
arc TT' vanishes for each when P is at B, we have the
proposition.
It will readily be seen that this is in substance the same
as Lambert's Theorem ( 168).
268. We now take an instance of the determination,
from the hodograph and the law of its description, of the
curve described and the forces acting.
The hodograph is a circle described with constant angular
velocity about a point in its circumference, find the original
path and the circumstances of its description.
Here we have in the hodograph,
p = a cos 6,
GENERAL THEOREMS. 305
therefore in the path
dx
~n=p cos 6 = a cos 2 cot,
-j- p sin 6 = a cos cot sin cot.
(MI
Integrating and properly adapting the constants, as they
affect only the position of the origin,
x = . (2cot -f sin 2o)t),
y T (1 cos 2cot).
Now the equations of a cycloid are
x A (<f> + sin <),
hence the path is a cycloid ; and, since 2cot (f>, the direction
of motion revolves uniformly. The particle moves under a
constant force perpendicular to the base of the cycloidal con-
straining curve, and the velocity at any point is that due to
the distance from the base, which is the brachistochrone of
180. The converse is easily proved.
Geometrically thus, if AP be the cycloid described by the
point P of the circle SP rolling uniformly on the line AS,
the velocity at P is proportional to SP, and the direction of
motion is perpendicular to SP. Hence the hodograph (turned
through a right angle in its own plane) may be represented
by the circle SP, described with uniform angular velocity
.T. D. 20
306
GENERAL THEOREMS.
about the point 8. That the motion is due to constant ac-
celeration perpendicular to AS is obvious from the fact that,
if Pp be drawn perpendicular to AS, SP 2 oc Pp.
269. If the orbit be central, and be a circle described
about a point in its circumference, the hodograph is a parabola
described about the focus with angular velocity proportional to
the radius vector.
For, if $ be the centre of attraction, P the particle in its
circular orbit, p the corresponding point of the hodograph:
, the tangent to the hodograph at p, must be parallel to
P ; and, therefore, if SQq be the tangent at 8, the triangle
pSq (being similar to PSQ) is isosceles. Thus the locus of p
is a parabola, for its tangent, pq, is equally inclined to the
radius- vector Sp, and to the fixed line Sq. Also the angular
velocity of Sp, being the same as that of PQ, is double that
qp
S
V_y
of $P, and is, therefore, inversely as SP 2 . But the length of
Sp is inversely as the perpendicular from S upon PQ, i.e.
inversely as $P 2 .
Or immediately, the pedal of a circle with respect to a
point on the circumference is a cardioid, and the hodograph,
which is the inverse of the pedal, is therefore a parabola.
270. The only central orbits whose hodographs also are
described as central orbits, are those in which the acceleration
varies directly as the distance from the centre.
Let S be the centre, P any point in the path, p the
corresponding point in the hodograph, p' that in the hodograph
GENERAL THEOREMS.
307
of the hodograph. Then Sp is parallel to the tangent at p,
which again is parallel to SP. Hence PSp' is a straight line.
Also, since p belongs (by hypothesis) to a central orbit, the
tangent at p' is parallel to Sp, i.e. to the tangent at P. Hence
the locus of p' is similar to that of P, and therefore Sp is
proportional to Sp. But Sp' represents the acceleration at P.
Hence the proposition.
271. A point describes a logarithmic spiral with constant
angular velocity about the pole ; find the acceleration.
Since the angular velocity of SP and the inclination of
this line to the tangent are each constant, the linear velocity
of P is as SP. Take a length PT, equal to e . SP, to represent
it. Then the hodograph, the locus of p, where Sp is parallel,
202
308 GENERAL THEOREMS.
and equal, to PT, is evidently another logarithmic spiral
similar to the former, and described with the same constant
angular velocity. Hence pt, the acceleration required, is equal
to e . Sp, and makes with Sp an angle equal to SPT. Hence,
if Pu be drawn parallel and equal to pt, and uv parallel to PT,
the whole acceleration Pu may be resolved into Pv and vu ;
and Pvu is an isosceles triangle, whose base angles are each
equal to the angle of the spiral. Hence Pv and vu bear con-
stant ratios to Pu, and therefore also to SP or PT.
The acceleration, therefore, is composed of a centripetal
acceleration proportional to the distance, and a tangential
retardation proportional to the velocity.
And, if the resolved part of P's motion parallel to any line
in the plane of the spiral be considered, it is obvious that in
it also the acceleration will consist of two parts one directed
towards a point in the line (the projection of the pole of the
spiral), and proportional to the distance from it, the other
proportional to the velocity, but retarding the motion.
Hence a particle which, unresisted, would have a simple
harmonic motion, has, when subject to resistance proportional
to its velocity, a motion represented by the resolved part of
the spiral motion just described.
If a be the angle of the spiral, &> the angular velocity of
SP, we have evidently PT. sin a = SP . w.
Hence
= -~PT = - SP = n*.SP (suppose)
SP sin a sin 2 a
and vu = 2Pv . cos a = 2ft) 7 ^ os _ a PT=2k. PT (suppose).
sma
Thus the central acceleration at unit distance is ?i 2 == -
sm 2 a
and the coefficient of resistance is 2k = r
sma
GENERAL THEOREMS. 309
The time of oscillation is evidently ; but, if there had
been no resistance, the properties of simple harmonic motion
shew that it would have been -- ; so that it is increased by
71
the resistance in the ratio cosec a : 1, or n : Vrz 2 /<?.
The rate of diminution of SP is evidently
sin a
that is, SP diminishes in geometrical progression as time in-
creases, the rate being k per unit of time per unit of length.
By an ordinary result of arithmetic (compound interest pay-
able every instant) the diminution of log . SP in unit of time
is k.
This process of solution is only applicable to resisted har-
monic vibrations when n is greater than k. When n is not
greater than k the auxiliary curve can no longer be a logarith-
mic spiral, for the moving particle never describes more than
a finite angle about the pole. A curve, derived from an equi-
lateral hyperbola, by a process somewhat resembling that by
which the logarithmic spiral is deduced from a circle, must be
introduced; and then the geometrical method ceases to be
simpler than the analytical one, so that it is useless to pursue
the investigation farther, at least from this point of view.
These geometrical results may easily be deduced by the
principles of the preceding chapter, which give at once for the
rectilinear motion the equation
See Proc. R S. E., for farther illustrations.
310 GENERAL THEOREMS.
EXAMPLES.
(1) Investigate the differential equation of the path of
article in a nlanp.
a particle in a plane
(2) A particle slides down an inverted cycloid from rest
at the cusp ; shew that the whole acceleration at any instant
"is g, and that its direction is towards the centre of the gene-
rating circle. Prove also that the motion of the particle will
be produced by rolling the generating circle on the under
side of a horizontal straight line with velocity ^ga, where a
is the radius of the generating circle.
(3) If a curve whose equation is y =/(#) is described
freely by a particle under potential V, and if the same curve
can be described freely under potential
-/()},
prove that the curve must be a cycloid.
(4) If a particle move on a rough inclined plane, prove
that
' cos 3 6 = r,
where p, p are the radii of curvature of the path at the two
points where the tangents are inclined at an angle 6 to the
horizon, and r is the radius of curvature at the highest point.
(5) A particle is projected up a rough inclined plane.
Shew that the intrinsic equation to the curve described is
V 2 f \2,xCOta ^ M , <k\2nCOta
s sin a = ( tan ^ sin 2 ft cot 1 cosec 3 6d6,
g \ *i j p\ */
where v = velocity of projection and ft = angle between direc-
tion of projection and the line of greatest slope.
GENERAL THEOREMS. 311
(6) A particle moves under two constant forces in the
ratio of 9 to 1 whose directions rotate in opposite directions
with constant angular velocities in the ratio of 3 to 1 ; prove
that under certain initial conditions the path of the particle
will be a closed curve of the form represented by the equation
r a cos 20.
(7) A particle is attracted by an infinite straight line
AB with intensity which is inversely proportional to the
cube of the distance of the particle from the line. The
particle is projected with the velocity from infinity from a
point P at a distance a from the nearest point of the line
in a direction perpendicular to OP, and inclined at the angle a
to the plane A OP. Prove that the particle is always on the
sphere of which is the centre; that it meets every meridian
line through AB at the angle a ; and that it reaches the line
a 2
AB in the time = , yu, being the strength of the at-
Vjt cos a
traction.
(8) Shew that if a material particle move under any
conservative system of forces, the projection of the principal
radius of curvature of its path at any point on the direction
of the resultant force at that point is
v denoting the velocity of the particle.
(9) If r be the radius vector of any point on a curve,
p the perpendicular from the origin on the tangent at that
point, s the length of the arc, and </> (r) any function of r,
prove that, if I (r) ds (the integral being taken between finite
limits) be a maximum or minimum, then </> (?*) x - .
(10) Jets of water escape horizontally from orifices along
a generating line of a vertical cylinder kept always full. Shew
312 GENERAL THEOREMS.
that (to axes inclined 45 to the vertical) the equation of the
lines of equal Action for unit mass of water is of the form
Shew also that the line of equal time for particles of water
issuing simultaneously from the orifices is the free path of
the water which leaves the vessel by an orifice at a depth
below the surface due to that time.
(11) A number of particles fall down the arcs of vertical
circles which have their highest points and the tangents at
them in common, from rest at the highest point. Prove that
the equation of the line of equal Action is
_ Ar^sin 3 6
~ ( i"^cos0y '
r being measured from the highest point of the circles.
(12) Of all the different sets of paths along which a
conservative system may be guided to move from one con-
figuration to another, with the sum of its potential and kinetic
energies equal to a given constant, that one for which the
Action is a minimum is such that the system will require
only to be started with the proper velocities to move along it
unguided.
Shew that, if APB be a projectile's path, AB the latus
rectum, AT, TB tangents at A and B, the Action will be the
same for the free path APB as for the constrained path A TB.
(13) A particle attracted towards a fixed centre, with
intensity varying as the distance from that centre, is pro-
jected with a given velocity at right angles to the line join-
ing the point of projection with the centre so as to describe
an ellipse. Prove that its Action in one revolution will be
greater than it would have been if it had been constrained
to describe the circle round the same centre of attraction
having for radius the distance of projection, the velocity of
projection being the same as before.
Is this result inconsistent with the principle of " Least
Action " ?
GENERAL THEOREMS. 313
(14) If a particle move in the brachistochrone between
two given points under gravity on a smooth surface of revo-
lution of which the axis is vertical, prove that the area swept
out by the projection of the ordinate on a horizontal plane is
proportional to the Action.
(15) The velocity of a particle in a central orbit varies
as . Apply the principle of Least Action to find the orbit,
and thence the law of attraction. Deduce the same results
from the Conservation of Energy.
(16) If u = F(x, y, z, a, b, k) + c is a complete solution
of the equation
where U is a given function of x, y, z, and A; is a constant ;
prove that
du\ (du\
U) =/3>
are the equations to any orthogonal trajectory of the system
of surfaces, for points on each of which u has a constant
value, and that, if points move along these trajectories with
velocities, which in any position are equal to the value of
\/2 ( U + k) at that point, their position at any time is de-
termined by the equation
du
t + T,
du\_
dkl~
where T is an arbitrary constant.
(17) Prove that every curve, plane or twisted, for which
s 2 = </>(#, y, z), where s is the length of any arc of it AP
measured from a fixed point A, and x, y, z the rectangular
co-ordinates of its variable extremity P, is tautochronous with
314 GENERAL THEOREMS.
respect to the fixed point for the force, or system of forces,
whose components parallel to the co-ordinate axes are
2 3y' 2 dz'
k being any constant.
(18) Prove that a rhumb line on the surface of a sphere
is tautochronous with respect to either pole, for a force acting
radially from, or perpendicularly from the tangent plane at,
the opposite pole, and varying in either case directly as the
length and inversely as the sine of the spherical distance
from the original pole. (Prof. Toivnsend.}
(19) Prove that for parallel forces, every curve, plane or
twisted, for which s 2 = <f> (z), where z is the ordinate in the
direction of force, is tautochronous with respect to the origin
of s, for the law of force Z = -J^<f) f (z), k being any con-
stant.
Prove that for concurrent forces every curve, plane or
twisted, for which s~ = (f> (r), where r is the vector from the
centre of force, is tautochronous with respect to the origin of
s, for the law of force R^ ^&fi (r), k being any constant.
Zi
Interpret the curves
s 2 = z 2 sec 2 a, s 2 = r 2 sec-ct, s- = ^a(a z), s*4a(a r),
& = z * - a 2 , s 2 = r 2 - a 2 , s 2 = m* (f 2 - a 2 ),
2as = z~ a-, 2as = r 2 a 2 , - = cos"" 1 -
a a
f*
5 = cos" 1 - . (Prof. Townsend.)
C//
(20) A particle under a system of forces describes their
GENERAL THEOREMS. 315
tautochrone in a time T. Shew that the action in a complete
oscillation is
27T 2 C 2
T '
where 2c is the length of the arc described.
(21) Shew that the pressure of a particle of mass m on
a tautochrone under any conservative system of forces is
mF sin <> + -* cos
jsin </> +
where /o is the radius of curvature at the point, the incli-
nation of the resultant force mF to the tangent, and s, s the
distances measured along the curve of the point and starting-
point from the point where the times of fall are equal.
(22) A particle, under a central attraction, the acce-
ULT
leration due to which at a distance r is 2 ^ , a being
(a + r )
a constant, is projected from a given point with the velocity
from infinity ; prove that the form of the groove, in which it
must move in order to arrive at another given point in the
shortest possible time, is an hyperbola whose centre coincides
with the centre of attraction.
(23) A body is such that it is its own level surface.
Shew that the brachistochrone from any point to the body is
the line of force passing through the point.
(24) If 0, 0, >/r... be the generalized co-ordinates of a
conservative system, T its kinetic energy, and if 6, (f>, ^r . . . be
supposed to be expressed explicitly in terms of t and arbitrary
constants, and if A, 8 be the symbols of two independent
variations of the arbitrary constants, prove that
jrti irn
Afl.S^ + AcM
d6 dj>
~ n A dT <, . . dT ~ , A dT
- BO . A . - 60 . A - - S-Jr . A : - ...
316 GENERAL THEOREMS.
is independent of t ; 0, <, ^r . . . denoting -=- , -^ , J~ . . .
respectively. Illustrate this by reference to the motion (1)
of a projectile, (2) of a system of particles attracting each
other with intensities varying as the distance.
(25) Shew that the amounts of heat and light received
by a planet in one revolution are each inversely as the square
root of the iatus rectum of its orbit.
(26) If P and Q be the accelerations along the tangent
and normal to the path of a particle, and ^ the angle the
tangent makes with a fixed line, the equation of the hodo-
graph will be
)>
r = ae ,
where a is a constant.
(27) Find analytically a central orbit whose form and
mode of description correspond with those of the hodograph
of another central orbit.
Shew that there is but one law of central attraction for
which this is possible except, of course, in the case of the
original orbit being a circle about its centre, when any law
may obtain. 270.
(28) If P, P' be the central accelerations for an orbit
and its hodograph, prove that
7/2
PP = " rr f .
h-
(29) Shew that the central acceleration necessary to
make a particle describe the hodograph of a central orbit
is inversely proportional to the normal acceleration at the
corresponding point of the orbit.
(30) Shew that in the hodograph of a central orbit
whose acceleration is f(r), the curvature varies inversely
as r 2 f(r).
(31) When the hodograph is a straight line described
with constant velocity, the path is the trajectory of an
unresisted projectile.
GENERAL THEOREMS. 317
(32) When it is a straight line described with constant
angular velocity about a point, the path is the catenary of
uniform strength
! *
6 =sec P
and the acceleration is parallel to y and varies as the square
of either of these equal quantities.
(33) Prove that the area swept out by the radius vector
of a projectile, drawn from its point of projection, varies as
the cube of the time of describing it.
(34) If the hodograph be a circle about a point in its
circumference, and if being the angle which the radius
vector makes with the diameter, the angular velocity be
given by
d6 k
shew that the path is a cycloid with its vertex upwards, and
that the velocity at any point is that due to a fall from the
tangent at the vertex.
(35) If a circle be described under a constant ac-
celeration not tending to the centre, the hodograph is a
lemniscate.
(36) A particle is moving in a parabolic orbit so that the
velocity of its recession from the focus is constant ; ascertain
the form of the hodograph of the particle.
(37) The hodograph of an orbit is a parabola whose
ordinate increases with constant velocity. Prove that the
orbit is a semi-cubical parabola.
(38) A straight rod, the ends of which are moveable
along two perpendicular straight lines in one plane, revolves
with a constant angular velocity. Prove that the hodographs
of the paths of its points are ellipses enveloped by a hypo-
cycloid.
318 GENERAL THEOREMS.
(39) Define the hodograph of a point moving in any
manner; and find its equation, for a point on the cir-
cumference of a wheel, which rolls uniformly within the
circumference of a fixed wheel of four times its radius.
(40) A smooth elliptic tube is placed with its major
axis vertical and a particle allowed to slide down it, starting
from rest at the highest point ; shew that the hodograph is
given by the equation .
r = 2\/ga sin - \coi~ 1 ( j- cot 6
( \o
(41) Prove that the hodograph of a catenary, described
freely under an acceleration parallel to the axis, is a straight
line described with velocity proportional to that in the
catenary.
(42) Prove that the hodograph of a central orbit is its
reciprocal polar with respect to the centre of attraction.
Prove that the equation of the hodograph of a cardioid
described under an attraction to the cusp may be put in
the form
r sin 3 5 = a.
3
(43) A lemniscate whose equation is r* = a 2 cos20 is
placed with the initial line vertical, and a particle is con-
strained to move on it, moving from rest at the pole ; prove
that the hodograph is defined by the equation
TT + 26 TT + 2<f>
~? cos 2 -
3 b
where c is a constant.
(44) If a particle move under a constant acceleration
which is initially normal, and which, when the direction of
motion of the particle has turned through an angle <, has
turned through an angle 20 in the opposite direction ; prove
that the equation of the hodograph is
GENERAL THEOREMS. 319
and the equation of the orbit is
3 /j
T* cos u = ct*.
(45) Two particles are describing free paths in one plane
which are hodographs to one another; if the particles be
always at corresponding points, prove that the paths must be
conic sections, and find the nature of the forces acting on the
particles.
(46) The resistance of the air being supposed to vary as
the cube of the velocity, shew that the holograph of a pro-
jectile is
p = aip + b,
the axis of x being vertical.
(47) A particle moves freely under a force whose direc-
tion is always parallel to a fixed plane, and describes a curve
which lies on a right circular cone, and crosses the generating
lines at a constant angle ; prove that its hodograph is a
conic section.
( 320 )
CHAPTER IX.
IMPACT.
272. WE come next to the consideration of the effects of
a class of actions which cannot be treated by the methods
employed in the preceding chapters. These are called Impul-
sive actions, and are such as arise in cases of collision ; lasting
(in the case of bodies of moderate dimensions) for an exceed-
ingly short time only, and yet producing finite changes of
momentum. Hence, in dealing with the immediate effects
of such impulses, finite forces acting along with them need
not be considered.
When two balls of glass or ivory impinge on one another,
no doubt there goes on a very complicated operation during
the brief interval of contact. First, the portions of the sur-
faces immediately in contact are disfigured and compressed
until the molecular reactions thus called into action are
sufficient to resist farther distortion and compression. At this
instant it is evident that the points in contact are moving
with the same velocity. But, most solids being endowed with
a certain degree of elasticity of form, the balls tend to recover
their spherical form, and an additional pressure is generated ;
proportional, as Newton found by experiment, to that exerted
during the compression. The coefficient of proportionality is
a quantity determinable by experiment, and may be conveni-
ently termed the Coefficient of Restitution. It is always less
than unity.
The method of treating questions involving actions of
this nature will be best explained by taking as an example
the case of direct impact of one spherical ball on another ;
first, when the balls are inelastic. Again, when their coeffi-
cient of restitution is given.
And it is evident that in the case of direct impact of
smooth or non-rotating spheres we may consider them as
mere particles, since everything is symmetrical about the line
joining their centres.
IMPACT. 321
273. Suppose that a sphere of mass M, moving with a
velocity v, overtakes and impinges on another of mass M',
moving in the same line with velocity v' ; and that at the
instant when the mutual compression is completed, the
spheres are moving with a common velocity F. Let P be
the pressure between them at any time t during the com-
pression, and r the time during which compression takes
place, then we have
M (v - V) = Pdt = R, suppose,
Jo
'(V-v')=\ r Pdt=R\
Jo
Mv + M'v' MM'
Tr
From these results we see that the whole momentum
after impact is the same as before, and that the common
velocity is that of the centre of inertia before impact. Had
the balls been moving in opposite directions, v' would have
been negative, and in that case we should have
Mv-Wv' MM'
From the first of these results it appears that both balls will
be reduced to rest if
Mv = M'v ;
that is, if their momenta were originally equal and opposite.
This is the complete solution of the problem if the balls
be inelastic, or have no tendency to recover their original
form after compression.
274. If the balls be elastic, there will be generated, by
their tendency to recover their original forms, an additional
pressure proportional to R.
Let e be the coefficient of restitution, v lt #/ the velocities
of the balls when finally separated Then, as before,
T. D. 21
322 IMPACT.
whence
^Mv+M'v' MM'
Mv i = M -W~ + M^- e M +
and
l+e)v M f
- = v
with a similar expression for v/.
A rather singular result may easily be deduced from the
last formula. Suppose M=M', e = l, that is, let the balls
be of equal mass, and their coefficient of restitution unity (or,
in the usual, but most misleading phraseology, "Suppose the
balls to be perfectly elastic")] then in this case
V! = v', and similarly v/ = v,
or the balls, whatever be their velocities, interchange them,
and the motion is the same as if they had passed through
one another without exerting any mutual action whatever.
275. The only other case which we can treat in the
present work is that of oblique impact when the balls are
spherical and perfectly smooth, for in rough and non-spherical
balls rotations are generated and the motion of each ball
requires to be treated as that of a rigid body.
The simplest case is that of a particle impinging with
given velocity, and in a given direction, on a smooth fixed
plane.
Suppose the plane of the particle's motion to be taken as
that of reference ; its trace on the given plane as the axis
of a?, and the point at which the impact takes place, as
origin.
The impulsive reaction of the plane will be perpen-
dicular to it, since it is smooth. Let this be called R ; and
let the velocity of the particle be resolved into two, v xt v yj
respectively parallel to the axes. For the first part of the
impact
IMPACT. 323
But v v ', being the common velocity of the plane and ball,
is evidently zero ; hence
or, the velocity parallel to the plane is unchanged, while
that perpendicular to it is destroyed. So far for an inelastic
ball. If the ball be elastic, let v x ", v y " be the final velocities,
then
These equations give
shewing that the velocity parallel to the plane is unaffected ;
and
or, v y = eVy,
that is, the velocity perpendicular to the plane is reversed in
direction, and diminished in the ratio e : 1.
If we designate by the name of angle of incidence the
inclination of the original direction of the ball's motion to
the normal to the plane, and by that of angle of reflexion the
angle made with the same line by the path after impact;
then denoting the total velocities before and after impact by
V and V", and these angles by 0, $ respectively, we have
F sin = v x , V" sin < = v x ",
Vcos0 = v y , V" cos < = v y " ;
and the previous results give at once
e cot 6 = cot <f>\
Of course these results are applicable to cases of impact
on any smooth surface ; by making the legitimate assumption
212
324 IMPACT.
that the impact, and its consequences as regards the motion
of the ball, would be the same if for the surface its tangent
plane at the point of contact were substituted.
276. Two smooth spheres, moving in given directions and
with given velocities, impinge ; to determine the impulse and
the subsequent motion.
Let the masses of the spheres be M, M' ; their velocities
before impact v and v', and let the original directions of
motion make with the line which joins the centres at the
instant of impact, angles a, a'. These angles may easily be
calculated from the data, if the radii of the spheres be given.
It is evident that, since the spheres are smooth, the entire
impulse takes place in the line joining the centres at the
instant of impact, and that therefore the future motion of
each sphere will be in the plane passing through this line
and its original direction of motion.
Let R be the impulse, e the coefficient of restitution;
then since the velocities in the line of impact are v cos a and
y'cosa', we have for their final values v it v/, after restitution,
by 274, the expressions
M'
v l = v cos a .-- -M (1 + e) (v cos a v' cos a'),
Vi = v' cos a' 4- T? TT, (1 + e) (v cos a v f cos a'),
and the value of R is
MM'
-v cosa).
Hence, the sphere M has finally a velocity v l in the line
joining the centres, and a velocity v sin a in a known direc-
tion perpendicular to this, namely in the plane through this
and its original direction of motion. And similarly for the
sphere M '. Thus the impact is completely determined.
IMPACT, 325
277. Recurring to the equations in 273, we have
and, eliminating F,
Hence, if e be the coefficient of restitution, v lt Vi the final
velocities,
Hence, Mv l + M'v-f = Aft; + Af V, whatever e be, or there
is no momentum lost. This is, of course, a direct conse-
quence of the Third Law of Motion.
Again, %Mv? + %M V 2 = i-flfw 8 + \M V 2
The last term of the right-hand side is therefore the
kinetic energy apparently destroyed by the impact. When
e = 0, its magnitude is greatest and equal to jj\ft\ ( v V 'Y-
When e = 1 its magnitude is zero, that is, when the co-
efficient of restitution is unity no kinetic energy is lost.
The kinetic energy which appears to be destroyed in any
of these cases is, as we see from 78*, only transformed
partly it may be into heat, partly into sonorous vibrations, as
in the impact of a hammer on a bell. But, in spite of this,
the elasticity may be perfect. Hence the absurdity of the
common designation alluded to in 274.
326 IMPACT.
Also by (2),
Hence the velocity of separation is e times that of ap-
proach. These results may easily be extended to the more
general case of 276.
The case of a rough sphere cannot be treated here,
inasmuch as it involves the Dynamics of a Rigid Body,
and this is beyond our professed limits.
278. We proceed to some special problems illustrating
the subject of impact.
To one end of a chain, lying in a given curve on a smooth
horizontal plane, a given impulsive tension is applied in the
direction of the tangent at that end ; it is required to find the
impulsive tension at any other point of the chain.
Let this be T at a point of the chain whose co-ordinates
are a?, y ; and let the initial velocities of that point, parallel
to the axes, be v x , v y \ then, //, being the mass per unit of
length of the chain, we have the following equations :
ds \ ds/ ,
d ( T dy\
as\ r Sj"^J
The geometrical condition is to be determined as follows.
The chain being inextensible, the length of an element 8s
is invariable, therefore the velocities of its two extremities
resolved along the element must be the same. This gives
evidently
ds ds ds ds
IMPACT. 327
Or, if v g , v p be the velocities generated at any point, in
the direction of the tangent and normal, we have at once
dT
and the kinematical condition furnished by the inextensi-
bility of the chain
dvs_v,,
~ "
If (j> be the angle the instantaneous direction of motion
at any point makes with the tangent,
By the elimination of v g and v p we obtain
ds
the general differential equation of the impulsive tension at
any point.
This of course cannot be integrated unless the initial form,
and the line-density, of the chain are known, i.e. unless /A and
p are given in terms of s.
Another method of solution is given in Thomson and
Tait's "Natural Philosophy," 310, 311, where it is shewn
that in such a case the chain takes the least possible kinetic
energy ; this gives, by equations (3),
328 IMPACT.
whence we easily obtain
d (ld
ds \yLt
as above.
The work done by an impulse being equal to the impulse
into half the velocity generated [Thomson and Tait, 308],
it follows that the kinetic energy generated in any part of
the chain is
T and v s referring to one end, and T', v s ' to the other end ;
this may also be written
279. Example I. As a particular example, suppose a
uniform chain to form a semicircle of radius a. Then p = a,
and s = a&, and (6) becomes
whose integral is
T
To determine the arbitrary constants, observe that when
(9 = 0, we have T=T 0)
the original impulse ; and when O = TT, or at the free end
of the chain, T=0. Thus we have
= Ae" +
T
These give A=-^
and therefore
The initial velocity at any point can now be easily deter-
mined.
IMPACT. 329
280. Example II. Suppose it be required that the
tension at each point should be proportional to the distance
from the free end of the chain.
Then I being the length, and s denoting the same quantity
as before,
T = T (l - -.) by hypothesis ;
T
.'. = 0, or by (4) - = 0, or p = oo ,
that is, the chain must lie in a straight line, as is otherwise
evident.
281. Example III. Suppose the chain to form a portion
of the logarithmic spiral. In this case p = es where e is the
cotangent of the angle of the spiral. Hence the equation
becomes
d?T_ T _
ds* e& ~ '
or, if we put s = ae*,
<&T_dT_T_
dp d+ #**
This is easily integrated, and thus the problem can be
completely solved ; it is easily shewn that the direction of
motion at any point makes a constant angle with the tangent.
282. The investigation of the motion which takes place
after the impact is not usually considered under Dynamics
of a particle but it is obvious that from what we have
just arrived at we may write down the equations of motion
of a string in the form
d*x d dx
with two similar equations ; the finite forces X, Y, Z, now
coming in as we are no longer dealing with impact.
Or, resolving along the tangent and normal, supposing
/,, /p the tangential and normal accelerations at a point, and
330 IMPACT.
S, N the component tangential and normal impressed forces
per unit of mass,
dT
and, as before, -^ = ,
T now denoting the finite tension at any point.
As a particular case, if finite (or impulsive) tensions be
applied at any two points of a chain of variable density
hanging in a given curve at rest under gravity, the tensions
being proportional to the tensions in the chain when at rest,
the chain will move, as if rigid, vertically.
283. If the string is practically inextensible, and if the
tension be great compared with the amount of the external
forces ; when the disturbance is small we may write x for s if
we take the undisturbed direction of the string as axis
of x.
The equations of transverse vibration become
d*y _T dfy d*z_T d*z
dt 2 " fjL ' da? ' ~dP ~ fi ' dx* '
where T is to be regarded as a constant.
The student is particularly to observe that we have now
been led to partial differential equations ; in fact we have
but two equations to represent, for all time, the motion of
every point of the string, however the motion of one point
may differ from that of another.
The solution is of course of the form
y or z <f> (x at) 4- ^ (x + at),
T
where a 2 = , ^ and ^ denoting arbitrary functions.
IMPACT. 331
284. The only other case we shall consider is that of a
continuous series of indefinitely small impacts, whose effect
is comparable with that of a finite force. The obvious method
of considering such a problem is to estimate separately the
changes in the velocity produced by the finite forces, and
by the impacts, in the same indefinitely small time 8t, and
compound these for the actual effect on the motion in that
period.
Another way is to equate the rate of increase of momen-
tum per unit of time to the impressed force.
A mass, under no forces, moves through a uniform cloud
of little particles which are at rest. Those it meets adhere to
it. Find the motion.
At time t let JJL be the mass, and let x denote its position
in its line of motion. Then, as there is no loss of momentum,
we have
But if M be the original mass, /^ the mass of the particles
picked up in unit of length, obviously
fJb = M + fJLoX.
Substitute and integrate, supposing x 0, x = V, when t = 0,
and we get
from which x can be easily found.
It is interesting to observe that we have
so that the mass moves as if acted on by an attraction
oc -=- towards a point in its line of motion.
If we take account of the increase of length of the mass
in consequence of the deposition of particles on its forward
332 IMPACT.
end, it is obvious that we must write
for the mass at time t, where f is the increase of length due
to the increase of mass. But f is obviously proportional
directly to the accession of matter, i.e. to x + f . Hence f
bears a constant ratio to x\ and the only result of this
refinement in the solution of the problem is that /^ (still a
constant) is greater than before : that is, the centre of at-
traction x -=r is at a smaller distance behind the origin.
This problem obviously leads to the same result as the
following :
A cannon-ball attached to one end of a chain, which is
coiled up on a smooth horizontal plane, is projected along the
plane. Determine its motion.
285. A spherical rain-drop, descending under gravity,
receives continually by precipitation of vapour an accession of
mass proportional to its surface ; a being its radius when it
begins to descend, and r its radius after the interval t, shew
that its velocity is given by the equation
eft / a a 2 a 3 \
*\ (1+- + -S+-. ),
4 V r r 2 r 3 /
the resistance of the air being left out of account. (Challis,
Smith's Prize Examination, 1853.)
Let e be the thickness of the shell of fluid deposited in
unit of time. Then evidently
r = a-\-et .......................... (1).
Also let Sv = ^v + B 2 v be the increase of velocity in time
Bt ; the first term due to gravity, the second to the impacts.
Evidently, S l v = g&t', and if M be the mass at time t,
S(Mv) = is the condition of the impact.
IMPACT. 333
This gives
M&. 2 v = vSM,
. 4>7rr*eSt _ _ SevSt _ _ Sev&t
4 r a + et'
3^
From these we have
dv 3ev
a + et
Multiplying by (a + et) 3 , and transferring the last term to
the left-hand side of the equation, it gives by inspection
= - e -.
Hence v = f \(a + et) - 7 4
4>e (" (a -f et) 3 \
Substituting for e from (1),
as required.
To veri
ited, the
Or, immediately from the dynamical equation
To verify this solution, suppose no moisture to be de
posited, then r = a, and we have v gt as it ought to be.
4 4.
since M=^ Trpr 3 = - ?rp (a 4- etf,
334 IMPACT.
286. One end, B, of a uniform heavy chain hangs over a
small smooth pulley A, and the other is coiled up on a table at
C. If B preponderates, determine the motion.
The moving force due to gravity is the weight of AB
minus that of A C = fjug (x a) suppose, a being the length
AC, and x the length AB.
Now in an indefinitely small interval &t, this would
generate in the portion BAG of the chain an increment of
velocity
a x ~ a
CjV =
But the whole uncoiled chain, being in motion at the
commencement of the interval $t with velocity v, lifts up a
portion of length v$t from the table during that interval.
Hence, if S 2 v be the change of velocity arising from this
impact, we have by the condition that no momentum is lost,
M+M"
~ fi (x + a) v
zV ~
r^ V \SU
or 8 2 v = --- - ,
x + a
quantities of the second and higher orders being omitted.
Sv 80 8 z v
Henceas Wt = Tt + Tt>
proceeding to the limit we have
dv _ dv _a) a v*
~ V ~
dt~dx~ (x + a)
which gives (x + a) 2 v -r + v z (x + a) = g (# 2 a 2 ).
Or, immediately, from the equation of momentum,
d ~ , dx~
IMPACT. 335
Multiplying by (x + a) -^ and integrating, supposing
x = 6 initially,
I<*-M#(J);-*J
-6)( 3 + 6 + 6 2 -3a 2 ),
and this determines for any given initial circumstances the
velocity at any instant. The final integration, for the deter-
mination of t in terms of x, requires the use of Elliptic
Functions ; except when 6 = 2a, when the acceleration is
constant and equal to ^g.
(1) If 6<2a, then # 2 + &# + 6 2 -3a 2 will split up into
real factors (x + )(# + 7) suppose, and we must put
to reduce the solution to elliptic functions.
(2) If b > 2tt, then a? + bx + 6 2 - 3a 2 is of the form
and we must put
x = b + c tan 2 J </>,
where C 2 = j6 2 + w 2 .
287. If we desire the change produced in the form and
position of an orbit by a slight change made in the velocity
or direction of motion, &c. at some particular point, we must
express separately each of the elements of the orbit in terms
of the quantity to be changed ; then taking the differentials
of both sides, we have the required changes of value.
Thus, we have generally in an elliptic orbit
^ = -~' 131 < 9 >-
336 IMPACT.
At the end of the major axis farthest from the focus this
becomes
V 2 = l ~^ e
a 1 +e'
Now if at this point V be made V+ 8F, without change of
direction, we have the condition that in the new orbit a (1 + e)
shall have the same value as in the old ; since this will still
be the apsidal distance.
Hence
a 1 + e
and 3 {a (1 + e)} = ;
or
And Sa = - -- Se
l+e
which determine the increase of the major axis and diminu-
tion of the excentricity ; and the same method is applicable
to more complicated cases.
Again, in the case of a parabolic orbit, as in Chap. IV.,
it is easy to see that a change in the magnitude of the velo-
city shifts the focus in the line joining it with the point of
projection through a distance
FSF
raises the directrix through an equal distance, and increases
the latus rectum by
4FSF
- cos 2 a,
IMPACT. 337
where a is the inclination of the path to the horizon at the
instant of the impact.
If the direction of motion only be changed, the directrix
is unaltered, the focus moves in a direction perpendicular
to the line joining it with the point of projection, and the
latus rectum is diminished by the length
4F 2 .
- sin a cos a oa.
9
In the latter case the new orbit again intersects the old,
and the tangents to either at the two points of intersection
are at right angles to each other; so long as the displacement
Sa is indefinitely small.
These results may easily be extended by geometrical
processes, as in Chap. IV., or deduced by differentiation
from the analytical results there given.
EXAMPLES.
(1) If e=l, one ball cannot be reduced to rest by
direct impact on another equal ball, unless the latter is at
rest.
(2) If two balls for which e = 1 impinge directly with
equal velocities, their masses must be, as X : 3 that one may
be reduced to rest.
(3) Shew that if two equal balls impinge directly with
1 -f 6
velocities - - V and F, the former will be reduced to
1 e
rest.
(4) Shew that the mass of the ball which must be
interposed directly between M at rest, and M' moving with a
given velocity V, so that M may acquire the greatest velocity, is
M'V(\
and that that maximum velocity is -r-rs^r
-r-rsr
\*JM +
T. D. 22
838 IMPACT.
(5) Suppose e = 1, and an infinite number of balls to be
interposed, shew that the maximum velocity which can thus
be given to M, is
v F
V V M'
[Note that, by the result of the preceding question, the
masses must form a geometric series, and the above is easily
deduced.]
(6) A number of balls A, B, C, &c. for which e is given,
are placed in a line ; A is projected with given velocity so as
to impinge on B, B then impinges on C, and so on ; find the
masses of the balls B, C, &c. in order that each of the balls
A, B, C, &c. may be reduced to rest by impinging on the
next ; and find the velocity of the w th ball after its impact
with the(?i-l) th ;
(7) A row of elastic balls hanging by long strings,
so that their centres are all in the same straight line, are
so placed that each ball is almost touching the next ; the
ball at one end of the row is drawn aside, and permitted to
impinge upon that next it ; prove that the whole row will
remain stationary, except the ball at the other end, which
will fly off and rise to a height equal to that from which the
first was allowed to descend ; the coefficient of restitution
being unity.
(8) A given inelastic body is let fall from a given height
on one scale of a balance, and two inelastic bodies are let fall
from different heights on the other scale, so that the three
impacts take place simultaneously ; find the relations be-
tween the masses and heights that the balance may remain
permanently at rest.
(9) Two equal smooth elastic billiard-balls A and B,
are placed at a distance d apart, and a third equal ball C
is hit so that it impinges on B after striking A. Shew that
the loci of all positions of (7, whence it is equally easy to
make the cannon, are circles whose centres lie on a straight
line through A, inclined to AB at an angle = = IT + \ sin" 1 -y
2* Ct
where a is the radius of the ball, and e = 1.
IMPACT. 339
(10) An imperfectly elastic ball is projected from a
given point in a horizontal plane, against a smooth vertical
wall, in a direction making a given angle with the vertical :
find where it strikes the horizontal plane, and prove that
the locus of these points, for different vertical planes of
projection, is an ellipse.
(11) An imperfectly elastic particle is under the in-
fluence of a smooth gravitating sphere. Shew that (excepting
special circumstances of projection) it will perpetually de-
scribe conic sections : determine also the elements of the
orbit described after any number of rebounds.
(12) A particle moving in an ellipse about a focus is
impinged upon directly by an equal particle moving in a
confocal hyperbola about the same centre of attraction. In-
vestigate the nature of the subsequent motion, the coefficient
of restitution being unity.
If the excentricity of the elliptic orbit be e, and that of
the hyperbolic orbit -, shew that the apse-line of the new
&
orbit of the former particle is inclined to the apse-line of its
old orbit at an angle
cosec" 1 jr- V4 + e 2 + 4e*.
06
(13) A boy standing on a bridge lets a ball fall on the
(horizontal) roof of a railway carriage passing under the
bridge at 15 miles an hour. If the modulus of elasticity
between the ball and carriage roof be f , and the coefficient
of friction ^ , find the least height of the boy's hand from the
roof that the ball may again rebound from the same point.
If the boy's hand be at a greater height than this, what will
happen ?
(14) A loaded cannon is suspended from a fixed hori-
zontal axis, and rests with its axis horizontal and perpen-
dicular to the fixed axis, the supporting ropes being equally
inclined to the vertical ; if v be the initial velocity of the
ball, whose mass is - th of the mass of the cannon, and h
n
222
340 IMPACT.
the distance between the axis of the cannon and the fixed
axis of support, shew that when it is fired off, the tension of
each rope is immediately changed in the ratio
v 2 + n*gh : n (n + 1) gh.
If a cannon be supported in a gunboat in the manner
described, with its axis in the direction of the boat's length,
what would be the effect of firing it off ?
(15) Equal particles revolve in opposite directions about
the focus in an ellipse of excentricity f , and impinge at the
nearer apse. Find the distances of future impacts, and shew
that if p be the original apsidal distance, the particles fall
into the centre of attraction after the time
14
(16) A ball is projected in a given direction within a
fixed horizontal hoop, so as to go on rebounding from the
surface of the hoop ; find the limit to which the velocity will
approach, and shew that it attains this limit in a finite time.
(17) If an infinite number of elastic particles, x = 1,
equally distributed through a hollow sphere, be set in
motion each with any velocity, shew that the resulting
continuous pressure (referred to a unit of area) on the
internal surface is equal to two- thirds of the kinetic energy
of the particles divided by the volume of the sphere.
(18) If a spherical bomb-shell resting on the ground
burst into a very large number of fragments, all of which are
projected with the same velocity, v, in directions uniformly
distributed in space, and if the fragments all remain at the
place where they first strike the ground, shew that, when all
have come to rest, the mass of metal sticking in the ground
per unit area at a distance r from where the shell lay is
M g (v* + Vy*^ry) + (v 2 - vV -
IMPACT. 341
where M is the mass of the shell, and r is great compared
with its radius.
.-2
Explain the result when r = - .
(19) A hollow cylinder is filled with a very large number
of perfectly elastic particles moving freely about in all direc-
tions and with all velocities, and impinging on each other
and the walls of the cylinder. The cylinder is placed on one
of the scales of a balance : shew that the weight of the
counterpoise must be equal to the weight of the cylinder and
of all the particles together.
(20) A cylinder, length h and radius r, is divided into
n equal compartments by n screw surfaces, the pitch of the
trace of each on the cylinder being a. It rotates on its axis
with angular velocity o>, and a stream of particles moving
parallel to the axis with velocities evenly distributed between
and V is incident on one end. Shew that the number of
particles which pass through the cylinder in unit of time
without striking the screw surfaces
tan a f . , . .
x (no. of particles in unit of volume);
h cot a r
provided &> < . V.
(21) If at an extremely great distance from the sun
meteorites have been flying about equally in all directions
for an infinite time, shew that the kinetic energy destroyed
per unit of time by meteorites falling into the sun is
where M is the mass of the meteorites in unit of vol. at a
great distance, r = sun's radius, V = velocity from infinity at
the sun's surface, and = the mean velocity of the meteorites
n
initially.
342 IMPACT.
If one year be the unit of time and the sun's radius the
unit of length, shew that this
having given r = 400000 miles, and the earth's mean dis-
tance = 92000000 miles. Also, from the fact that one unit
of heat is equivalent to 772 foot-pounds, find the quantity of
heat received by the sun in one year through the impacts.
(22) A train composed of n smooth parallelepipeds is
travelling with velocity u along a straight line. A stream of
perfectly elastic particles, each of mass m, is moving with
velocity v, perpendicular to the line, and is impinging on the
train. Supposing that the particles do not interfere with
one another, shew that the train experiences a resistance
2hNm {2 (n -l)av-(n- 2) bu] w,
provided u < -r- -^ -j- , where a = distance between each
_)iped, 6, h = breadth and height of each, and N is
the number of particles in a unit of volume.
Can this be used to explain the fact that a train experi-
ences a greater resistance from a cross wind than a head
wind ?
(23) A comet in moving from one given point to another,
throws off at every instant small portions of its mass which
always bear the same ratio n to the mass which remains.
If v be the velocity with which each particle is thrown off,
a the inclination of its direction to the radius vector, prove
that the period t will be diminished by
{(<' - </>) J(ap) sin a - (/ - r) cos a},
<> and <' being the excentric anomalies, r and r the focal
distances at the given points, a the mean distance, 2p the
latus rectum, and f the attraction at distance a.
(24) If a rocket, originally of mass M, throw off every
unit of time a mass eM with relative velocity V and if M ' be
IMPACT. 343
the mass of the case, &c., shew that it cannot rise at once
unless eV>g, nor at all unless -, 7 > g. If it do rise at
once vertically, shew that its greatest velocity is
M g ( M'
W"e\ ~H
and the greatest height it reaches
F 2 /, M\* F/. M' M
(25) Particles (2n 1) in number, connected by inexten-
sible strings, are suspended from two fixed points in a hori-
zontal plane so as to hang symmetrically, their weights being
such that the inclination of each string to the one immediately
below it is a, which is also the inclination of each of the two
lowest strings to the horizon. Find their weights ; and shew
that if the lowest whose mass is m be struck by a vertical
blow P, the horizontal component of the initial velocity of
any particle varies inversely as its weight, and the vertical
component of the initial velocity of the r th from the lowest
is
TT {(2?i 2r 1) sin a + 2 cos a cot no. sin ( 2r + 1) a}.
2m cos 2 a l
(26) A large number of equal particles are attached at
equal intervals to a string, and the whole is heaped up near
the edge of a smooth table ; the particle at one extremity of
the string is just over the edge of the table. Shew that U r
the velocity of the system just before the (r + l) th particle is
set in motion is given by the equation
3' ~7
Calculate the dissipated energy.
(27) A very long row of particles, each of mass m, on
a smooth horizontal table are connected, each with two
adjacent ones, by similar light elastic stretched strings, each
344 IMPACT.
of natural length c ; they receive small longitudinal dis-
turbances, such that each of them proceeds to perform a
harmonic vibration : prove that there will be two waves of
vibrations, in opposite directions, with the same velocity
a A / sin , where a is the average distance between
y men n
two successive particles, n the number of intervals between
two particles in the same phase, and X the modulus of
elasticity.
(28) A light elastic string of length na and coefficient of
elasticity X is loaded with n particles, each of mass m, ranged
at intervals a along it, beginning at one extremity. If it be
hung up by the other extremity, prove that the period of its
vertical oscillations will be given by
when r 0, 1, 2, ... n 1, successively. Hence prove that the
periods of vertical oscillation of a heavy elastic string will be
4 /TM
given by the formula T= ^ 1 A/ ----, where I is the length
of the string, M its mass, and r is zero or any positive
integer.
(29) A uniform chain hangs vertically from its upper end
with the lower end just in contact with an inelastic table ; if
the chain be allowed to fall, prove that the pressure on the
table during the fall of the chain is always equal to three
times the weight of the coil upon the table.
If the chain hang with its lower end just in contact with
a smooth inclined plane, and be let fall, find the pressure on
the plane at any time during the fall.
(30) Snow is spread evenly over a roof. If a mass com-
mences to slide, clearing away a path of uniform breadth as
it goes, prove that its acceleration is constant, and equal
to one-third that of a mass of snow sliding freely down the
roof.
IMPACT. 345
(31) The cable of a ship is led through a hole in the
deck at a height b above the cable-tier and runs along the
deck a distance a, and out at the hawse-hole, immediately
outside of which is the anchor, of mass equal to a length
$a 4- 2b of the cable. Prove that if the anchor be let go it
will descend with acceleration J#.
(32) A chain of given length is at rest on a smooth
horizontal plane, with one end fastened to a point on the
plane, under a repulsion from that point varying as the
distance. If the chain be set free, find the initial change of
tension at any point, and the subsequent motion of the
chain.
If the chain impinge upon a vertical wall perpendicular
to its own direction, find the pressure upon the wall at any
subsequent time.
(33) Two equal weights W are connected by a string
of length 21, whose weight per unit of length is w, which
passes over a small pulley. The system is put in motion by
adding a weight W at one end. Shew that when either
weight has moved through a distance x, the kinetic energy
will be greater than if the string were weightless by
(34) A fine string passing over a smooth pulley supports
two equal scale-pans ; a uniform chain is held by its upper
end above one of the scale-pans, its lower end being just
above the scale-pan ; if the upper end be let go, determine
the motion completely, and find, at any time, the pressure on
the scale-pan.
(35) A pulley is fixed above a horizontal plane. Over
the pulley passes a fine string which has two equal chains
fastened to its two ends. In the position of equilibrium a
length a of each chain is vertical, the remainder of the chains
being coiled up on the table.
If now one chain be drawn down through a distance wo,
346 IMPACT.
find the equation of motion, and prove that the system will
next come to rest when the upper end of the other string is
a distance ma below its mean position where
If n = 1, prove that m = J approximately.
(36) A uniform flexible chain of indefinite length, the
mass of an unit of length of which is m, lies coiled on the
ground, while another portion of the same chain forms a
coil on a platform at height h above the ground, the inter-
mediate portion passing round the barrel of a windlass placed
above the second coil. An engine, which can do H units of
work per unit of time, is employed to wind up the chain
from the ground and let it fall into the upper coil. Shew
that the velocity of the chain can never exceed the value of
v determined from the equation
mghv + ^mv* = H.
(37) A chain whose density varies as the distance from
the end A is coiled up close to the edge of a smooth table
and the end A allowed to hang over. Shew that the motion
is uniformly accelerated and the tension at the edge of the
table varies as the fourth power of the time elapsed since the
commencement of motion.
(38) A string of length I hangs over a smooth peg so as
to be at rest. One end is ignited, and burns away at a
uniform rate v. Shew that the other end will, at the time t,
before the whole slips off the peg, be at a depth x below the
peg, where x is given by the equation
given that the mass of the string per unit of length is
unity.
(.'>!)) A chain is coiled up on a table and is connected
with a weight by a tine thread passing over a smooth pulley :
if the law of density of the chain be m<f> [x\ ; and the mass
IMPACT. 347
ling motion be ml ; then the velocity when a length x
has been raised is given by
r ( rx
- \ 6(of)daf
iJO
dx
,
c x
\l+\ $(x)dx\
(40) A series of particles m 1} m 2 , ... connected by in-
elastic strings are placed on a smooth horizontal plane, so
that the strings are sides of an unclosed equiangular polygon,
each of whose angles is TT a, and an impulse is applied to
the extreme particle P 1 in direction P Z P\ ' prove that
T r -T r _ l cosa = T r+l cosa-T r
m r m r+l
where T r is the impulsive tension of the r th string.
d*T IdfjidT T
Deduce the equation .--= -- -= = for the im-
as 2 fji ds ds p 2
pulsive tension at any point of a chain lying in the form of
any curve on a horizontal plane and set in motion by
tangential impulses, and if the density of the chain vary as
the curvature, deduce from either equation that the impulsive
tension at any point is equal to Ae^ + Be~ <{> , where <f> is the
angle which the tangent at the point makes with a fixed line,
and A, B are constants.
(41) A uniform chain hangs in equilibrium over two
smooth pegs in the same horizontal line ; if equal vertical
impulses be applied simultaneously to the two free ends, find
the impulsive tension at any point, and prove that the initial
velocity of the vertex of the catenary is to the velocity which
would be imparted to each of the straight pieces of chain,
if disjointed from the catenary, as 1 : 1 + sin a, where a is
the greatest inclination of the catenary to the horizon.
(42) A uniform string is lying in a catenary on a smooth
348 IMPACT.
horizontal plane, and the vertex is suddenly projected towards
the directrix with a given velocity ; shew that the impulsive
tension at any point varies as the ordinate of that point, and
that every point of the string starts in the same direction.
(43) If a chain of mass m be in the form of a portion of
a catenary cut off by a line perpendicular to its axis, and if
tangential impulses each equal to mv be applied simul-
taneously at its two ends, prove that the whole chain will
begin to move with the velocity 2vsiri<, where 2<j> is the
angle between the tangents at the ends.
(44) A chain lies upon a smooth horizontal plane in the
form of a portion of a common catenary, the tangents at the
ends making angles lt @ 2 with the tangent at the vertex of
the catenary. An impulsive tension T^ is applied at the
former extremity; shew that the impulsive tension at a point
of the chain where the tangent makes an angle 6 with the
tangent at the vertex is equal to
r n COS 0j 6 #2
'cosfl 9T-^ g
(45) A string of infinite length is laid on a smooth
table in the form of a portion of one branch of the curve
r n sin nO = a n , so that one extremity of the string is at a finite
distance from the origin of polar co-ordinates ; to this end a
tangential impulse is applied, so that the initial direction of
motion of each point of the string and the radius vector to
the point are equally inclined to the corresponding tangent.
Shew that the impulsive tension at any point x r~ (n ~ l} and
the density of the string must
2 n -l
^ _ a 2n 2 "
00
(46) A string of varying density slides in a smooth
cycloidal tube whose axis is vertical and vertex downwards.
Shew that if the string be let fall from any position in which
its whole length is within the tube, its centre of gravity will
reach the vertex in the same time.
IMPACT. 349
(47) A straight line describes a right circular cone ;
find the acceleration of a point moving along the line. A
string of given length is enclosed in a smooth straight tube,
which is made to revolve uniformly about a vertical axis, so
as to describe a right circular cone ; determine the motion of
the string, and the tension at any point.
iip
(48) If a small velocity n ~ be impressed on a planet,
in the direction of the radius vector, shew that
e = ne sin (6 nr\
g CT = - n cos (6 -cr).
Calculate also the changes in e and tzr produced by a
small transverse impulse.
(49) A body is moving in an ellipse about the focus ;
prove that if the body receive a transversal impulse the apse
line will be unaffected if the impulse is
where m is the mass of the body, I the semi-latus rectum of
its orbit, h is twice the rate of description of area round the
focus, and is the true anomaly of the body.
(50) If Q be the central disturbing force on a planet,
find by Newton's method the equations
where 6 is the longitude of the planet, txr the longitude of the
apse, e the excentricity of the instantaneous ellipse, r the
distance of the planet from the sun.
(51) A particle revolving about a fixed centre to which
it is attracted with intensity inversely as the square of the
distance is acted on by a small disturbing force / in the
direction of the radius vector: prove that the variations of
the major axis, the excentricity and the inclination of the
line of apses are determined by the equations
350 IMPACT.
da ( "
a-* ,
-' /cos (*-
(52) The first term of the central disturbing force on
the moon is m 2 r, where the central force is ; shew that
the apsidal angle (the orbit being nearly circular) is
, 3 m*\
+ ~ near y '
where -- is a mean lunar month.
n
(53) A particle is moving in a circle about a centre of
attraction oc (Dist.)~ 2 . The strength of the centre increases
slowly and uniformly. Determine the approximate elements
of the orbit after a given time.
(54) A particle moves in a focal elliptic orbit in a very
rare medium whose resistance is as the square of the velocity;
determine the effect of the resistance on the periodic time.
(55) A satellite moves about a spherical planet in the
plane of its equator, in a slightly elliptic orbit. Find the
motion of the apse due to an uniform mountain ridge at the
equator.
(56) If when the earth is at an end of the minor axis
of its elliptic orbit, a small meteor were to fall into the sun
of mass J - of the mass of the sun, prove that the year
2
would be diminished by -" of itself.
T m
IMPACT. 351
Prove also that the apse would regrede through the angle
A/ 1, where e is the excentricity of the earth's orbit.
m v e 2
(57) Central attraction varying as the distance, the
velocity of a particle is increased by - th when it is at the
extremity of one of the equal conjugate diameters of its orbit.
Shew that each axis is increased by ?r-th, and that the apse
* 2n
regredes through an angle
1 ab
(58) At what point of an elliptic orbit, described about
the focus, can a small change be made in the direction of
motion without altering the position of the apse ?
If <$</> be this change, shew that (in the supposed case)
Se
= \-<?'
(59) Shew that, in an elliptic orbit about the focus, if
the velocity be increased by -th when the true anomaly is
6 TV, we shall have
JL. rsin(0-*7)
according as the particle is moving to or from the nearer
apse.
(60) A particle moving about a centre of attraction in
the focus, in an ellipse of small excentricity, receives a small
impulse perpendicular to its direction of motion at any
instant. Find the effect on the position of the apse.
(61) Again, if at the extremity of the minor axis the
velocity be increased by -th, and the direction changed so
352 IMPACT.
that h remains the same, find the alteration in the form and
position of the orbit.
(62) A particle describes an elliptic orbit about a centre
of attraction of intensity varying as (distance)" 2 . If T be
the periodic time and a small disturbing force X sin ^- . t
acts in the direction of the radius vector, calculate the
variations in the orbit.
(63) A spherical cloud of small masses, whose mutual
attraction is insensible, an^ whose velocities are very small,
is overtaken by the sun so as to be incorporated into the
solar system. How will the form of the cloud alter as it
pursues its approximately parabolic orbit ?
(64) The bob of a simple pendulum of length I is acted
on by a horizontal force = pg cos nt, where p is a large
number, and In 2 is large compared with g : shew that the
pendulum may oscillate about either of two points distant a
from the lowest point with an amplitude f3 where
,
gp 1 p
( 353 )
CHAPTER X.
MOTION OF TWO OR MORE PARTICLES.
288. HAVING considered in detail the various cases
which occur in the motion of a single particle subject to any
forces, and whose motion is either free, constrained, or re-
sisted, we proceed to the investigation of some very simple
cases in which more particles than one are involved. These
cases will divide themselves naturally into two series ; first,
when the particles are entirely free, and are subject to their
mutual attractions as well as to other common impressed
forces : and second, when there are in addition constraints ;
such as when two or more of the particles are connected by
inextensible strings, &c. Let us take these in order :
I. Free Motion.
289. An immediate application of the third law of
motion shews that if two particles attract each other, they
exert each on the other equal and opposite forces, in the
direction of the line joining them.
If then m, ra', be the masses of the particles, and the
attraction between two units of matter at distance D be
<f>' (D), the intensity is
mm' <
290. A system of free particles is subject only to their
mutual attractions ; to investigate the motion of the system.
Let, at time t, x n , y n) z n be the co-ordinates of the
particle whose mass is ra n , and let $ (D) be the law of
attraction. Let p r q express the distance between the
particles m p and m q \ then we have for the motion of m ly
T. D. '23
354 MOTION OF TWO OR MORE PARTICLES.
(1),
(2),
.-(3),
with similar equations for each of the others; the summations
being taken throughout the system. Before we can make any
attempt at a solution of these equations, we must know their
number, and the laws of attraction between the several pairs
o particles. But some general theorems, independent of
these data, may easily be obtained : although not nearly so
simply as in Chap. II.
291. I. CONSERVATION OF MOMENTUM. In the ex-
ftx 9
pression for m p -~ , we have a term
and in m q --= 9 we have
Hence if we add all the equations of the form (1) to-
gether the result will be
or 2,
Similarly 2 (m ^jj\ = 0,
MOTION OF TWO OR MORE PARTICLES.
355
Now if x, y, 2, be at time t the co-ordinates of the centre
of inertia of all the particles, 58,
2 (mx),
2 (my),
S (mz).
And the above equations may be written,
d*x ^
e^ m =
or
z^
-v- 2m =
dt*
11TU aX \
Whence -yr = a
dt
dt~
dz__
dt~
These equations shew that the velocity of the centre of inertia
parallel to each of the co-ordinate axes remains invariable
during the motion, that is, that the centre of inertia of the
system remains at rest, or moves with constant velocity in a
straight line. See 72.
The values of a, 6, c, may thus be determined,
( dx\
j- * ( m ~ji)
_ dx _ \ dt)
dt 2m
Now if the initial velocity of m^ were resolvable into
u l} v lt Wi, parallel to the axes respectively, and similarly
for m^ &c.
a =
, and so for 6, &c.
232
356 MOTION OF TWO OR MORE PARTICLES.
If forces had acted on the particles, of which the com-
ponents parallel to the axes on the particle m at (xyz) were
mX, mY, mZ\ we should have found
v d*x ^ v ^ d z y
Zm -j7 2 = ZmX, zra ,^
or, which is the same thing,
d z x v v v d?y ^ v
- Zra = SmA , Zra = ^
proving that the motion of the centre of inertia of the system
is the same as that of a particle of mass 2m, acted upon by
the forces moved parallel to themselves, at the centre of
inertia.
292. II. CONSERVATION OF MOMENT OF MOMENTUM.
Again, it is evident that if we multiply in succession equation
(1) by y lt and equation (2) by as lt and subtract, and take the
sum of all such remainders through the system of equations
of the forms (1) and (2), we shall have
Integrating once we have
where the left-hand member is the moment of momentum of
the system about the axis of z.
Now if in the plane of xy we take p, 0, the polar co-
ordinates of the projection of m,
d dx d6
/ rlf)\
therefore 2) f rap 2 -, - ) = 2^1 3 .
\ dtj
MOTION OF TWO OR MORE PARTICLES. 357
Now if a z be the area swept out by the radius vector p
on the plane of xy,
_,
dt ~ dt '
and our equation integrated gives
no constant being necessary if we agree to reckon a z from
the position of p at time t = 0.
This equation shews (since xy is any plane) that generally
in the motion of a free system of particles, subject only to
their mutual attractions, the moment of momentum about
every axis remains constant ; or, as it is commonly but incon-
veniently stated, the sum of the products of the mass of each
particle of the system, into the area swept out by the radius
vector of its projection on any plane, and about any point in
that plane, will be proportional to the time. See 72.
Take a x , a y to represent for the planes yz, xz the same
that a z represents for xy, then
2 (ma x ) = Aj,
2 (ma y ) = A 2 t.
The value of this quantity for a plane, the direction-
cosines of whose normal are X, /*, v, will be
and will be a maximum if
\A l + fj,A z -f vA 3 is so,
subject to the equation of condition
This gives X = vW+2 , + ^. ) = 1 suppose,
with similar values for p and v ;
and the value of the product for the plane so found is
evidently At.
858 MOTION OF TWO OR MORE PARTICLES.
Hence, we see also, that, as indeed is evident from the
simple statement above, the axis about which the moment of
momentum is greatest remains parallel to itself, or, as it is
usually put, the plane for which the sum of the products of
the masses of the particles into the sectorial areas described
by the radii vectores of their projections is a maximum, is a
fixed plane or parallel to a fixed plane during the motion. It
has been called on this account the Invariable Plane.
If external forces had acted on the system, we should
have found
d?x d?z
293. III. CONSERVATION OF ENERGY. Multiply
and, treating similarly all the other equations, add them all
together.
Let us consider the result as regards the term on the
right-hand side involving the product m p m q .
Written at length it is
+ similar terms in y and z\ ;
and the portion in brackets is equal to
- {(x q - x p ) -j- (x q - x p ) + similar terms in y, z\ ;
or, p r q dt
MOTION OF TWO OR MORE PARTICLES. 350
hence
I fdxd*x dyd-y dzd z z\]
\m -77 ~TZ + ~^i j^ + ~ji ~jl f
I \dt dt 2 dt dt 2 dt dtfj)
= ;
therefore, on integration,
We see therefore that the change in the Kinetic Enei^gy
of the system in any time depends only on the relative distances
of the particles at the beginning and end of that time, 78.
294. An extremely remarkable and very simple theorem,
connected with this subject, was obtained by Clausius.
Though specially devised for the purposes of the Kinetic
Theory of Gases, it is capable of very wide application in
Physics. The restrictions to which its application is subject
will appear from the condition assumed in the proof below.
Consider a system of particles, moving in any manner,
under the action of mutual forces (in the direction of the line
joining each pair), but so that the sum of the products of the
mass of each particle into the square of its distance from the
origin shall remain practically constant. If we twice dif-
ferentiate this sum with respect to t we obtain at once
-g 2m (# 2 4- f + * 2 ) = 2m (a? + if* + z*) -f 2m (xx + yy + zz).
Remembering the condition imposed above, and putting
for mx, &c. their equivalents X, &c., we have, as an expres-
sion for the kinetic energy of the system, what is called the
Virial, viz.
i 2m (x 2 + f + z 2 ) = - \ 2 (xX + yT 4- zZ).
By 290, the right-hand side becomes, for the mutual
action between two particles of the system
" \X p (X q - X p ) + &C. + X q (x p - X q ) + &C.)
860 MOTION OF TWO OR MORE PARTICLES.
Hence, if <j> ( p r,j) be written as p R ( j, the part of the virial
depending on stresses between the pairs of particles is
where each pair of particles is to be taken only once into
account.
If, as in the kinetic theory of gases, the particles are sup-
posed to be enclosed in a vessel, from whose walls they rebound
with unit coefficient of restitution, the walls exert external
pressure on the group. If the particles are sufficiently
numerous, they may be regarded as having always the same
uniform distribution, and this pressure may be regarded as
constant over the whole surface. Hence, dS being an
element of the surface of the vessel at x, y, z, and X, /^, v the
direction-cosines of its normal, we have as the corresponding
part of the Virial
- 2pdS (Xa? + p,y + vz).
This sum, taken over the whole (closed) surface of the
vessel, is evidently equal to
where V is the volume of the vessel.
Hence, in this case, the virial equation is
If the particles exert no molecular force, and be so small
as practically never to impinge on one another, the product
of pressure and volume in such a medium is two-thirds of the
whole kinetic energy. This statement includes the two
"gaseous" laws known as those of Boyle and Charles.
295. So far for the case of several particles. The simplest
examples will of course be found in the case of two particles
only, and to such we will confine our attention ; as, when
three or more are involved, the problem does not admit of
exact solution, and in the two most important applications
which have been made of it, namely to the Lunar and
MOTION OF TWO OR MORE PARTICLES. 361
Planetary Theories, it is found that a distinct method of
approximation is required for each.
Since the acceleration of the centre of inertia is zero, it
follows that the motion of each particle with reference to
that point is the same as if the latter were at rest. Also, if
we apply to each particle of the system an acceleration equal
and opposite to that of any one of them, the latter will be
reduced to rest, and the relative motion of the others about
it will be unchanged. Hence, if there are only two, we see
that the relative motion of one about the other will be the
same as if the sum of the masses were substituted for the
latter.
Two particles, moving initially with given velocities in the
same straight line, are subject only to their mutual attraction
which is inversely as the square of the distance ; to determine
the motion.
The motion will evidently be confined to the straight
line. Let m, m f be the masses of the particles estimated on
the hypothesis that one unit of mass exerts unit of force on
another unit at unit of distance ; #, x' their distances at any
time t from a fixed point in the line of motion, then
wn
dt*
Hence, if x be the co-ordinate of the centre of inertia at
time t,
doc ,dx
mV+m'V f ,
if V and V be the initial velocities : hence the momentum
is constant.
362 MOTION OF TWO OR MORE PARTICLES.
Integrating again,
mx -f m'x = (m + m) x = (mV+ m' V) t + C'
= (mV+m'V')t + ma + m'a f (2),
if a, a' denote the initial positions of the particles.
Again, from equations (1),
d? (x' x) _ m + m
from which, by multiplying by m or m, we see that the
relative motion is the same as if the one particle moved to
the sum of the masses collected at the other, the position of
that other being considered fixed.
Integrating once, we have
l{d(x'-x)Y~ _ m + m
2J dt J = h x -x *
At t = 0, this is
and, eliminating (7,
1 (fJ (w f W\\ 2
J. I U/ \iJu (JUJ I
1 dt j 2 V >\x'-x a -a
This is of the form
/do> A
...(3).
therefore t
which may be integrated by putting &> = 7/ 2 . The integral
will be circular or logarithmic according as B is negative or
positive. Thus we have x x in terms of t, and as we also
know mx + m'x' by (2), the motion is completely determined.
If at the instant of projection
l /V F'V- (m + m/ )
2 ( ~cT-a~ '
MOTION OF TWO OR MORE PARTICLES. 363
the formula (3) becomes
'~ X} - V{2 ( + m')),
-
o
|(a'-a)=C,
and the motion is completely determined.
296. There is another method of treating this problem.
Suppose that, instead of determining the relative motion of
the particles, we consider that of each relatively to the com-
mon centre of inertia. The distance of m from the centre of
inertia is
_ mx + m'x _ m' (x f - x)
m+m f m + m'
and we easily find from (1),
m
/d z x d?x\ _ mm'
\dt* dtfj (x-xf (x'-xf'
Hence, for the relative motion of m and the centre of
inertia,
d* (x x) mm'
N /
whence x x may be determined, in finite circular or loga-
rithmic terms, as before.
297. Two particles, anyhow projected, are acted on solely
by their mutual attraction; to shew that the line joining them
is always parallel to a fixed plane. [This is obvious from
26.]
If m and m' be the particles, x, y, z, x', y\ /, their
364 MOTION OF TWO OR MORE PARTICLES.
co-ordinates at time t t r their distance, and P the mutual
attraction, we have the following equations,
d?x ^x' x , d?x' x x'
m -y = P , m j = P - ,
dt 2 r dP r
with similar expressions for the other co-ordinates ; hence
x x y y
and integrating,
^ d(y' y] , , ^ d (x x) ~
v-*--sr } -(y-y^ -4r --
with other two similar equations. Therefore
G s (if - z) + C 2 (y' -y) + C 1 (x 1 - x) = 0.
Hence the line joining the particles is always parallel to
the plane whose direction-cosines are as (7 a , C 2 , C 3 . This
corresponds to 292.
Also it is evident that the motion of the particles with
respect to each other in a plane parallel to this is the same as
if the plane were at rest ( 294).
From the preceding propositions the following are evident
deductions.
The centre of inertia of the two particles is at rest only
when the initial velocities are zero, or when the directions of
projection are the same or parallel, and the momenta equal
and opposite.
The plane of relative motion will be at rest only when the
initial directions lie in one plane.
If the attraction be inversely as the square of the distance,
the relative orbits of the particles about each other, and
therefore ( 27) about their centre of inertia, will be conic
sections about a focus.
It is needless to pursue this any further, as the preceding
results enable us to reduce the problem to cases treated of
in former chapters.
MOTION OF TWO OR MORE PARTICLES. 365
II. Constrained Motion.
298. Of the constrained motion of particles, we can only
take particular examples, but there are some general con-
siderations which deserve attention.
If two particles be connected by an inextensible string,
its only effect is to prevent their relative distance becoming
greater than its own length. If we introduce an unknown
quantity T for the tension of the string, the equations of
motion can be written down, and the condition that the dis-
tance of the particles is equal to a given quantity will give
us an additional equation, enabling us to eliminate, or to find
the value of, this unknown tension. If at any time the value of
T so found becomes equal to zero, the motion of the particles
must be investigated henceforth as if they were free, until the
values of their co-ordinates shew that the string will begin to
be tended again. In such a case, if their velocities resolved
along the line joining them be not equal, an impact will take
place, whose effects mast be investigated by the methods of
Chap. X.
When the particles are connected by a rigid rod without
mass, we have an unknown tension or pressure in the di-
rection of the rod; and, to determine it, we have the
geometrical condition that the distance between the particles
is constant.
If there be more than two particles attached to the rod, it
may exert a transverse force ; but cases of this kind more
properly belong to the Dynamics of a Rigid Body ; and we
therefore omit all consideration of them.
299. Two particles, attached to each other by an inexten-
sible string, are projected with given velocities in space; to
determine the motion.
We may without loss of generality consider the distance
between the particles at the instant of projection, to be equal
to the length of the string. If their velocities are wholly
perpendicular to its direction, or if their resolved parts along
it are equal and in the same direction, there will be no impact.
If not, suppose the masses m and m' to have velocities v and v
366 MOTION OF TWO OK MORE PARTICLES.
parallel to the string at the instant it is stretched. It is evi-
dent that the impact will change each of these into
ra + ra
This then is determinate ; so we may now in addition suppose
the resolved parts of the velocities along the string equal to
each other. Let x, y, z, x, y' y z, be at any time the co-ordi-
nates of the particles, then, if a be the length of the string,
d?x m x'. x , d?M m x x
m = T - m = T ~
and so on.
Also, (x - x? + (y f - 2/) 2 + (z f - zj = a 2 ,
which are seven equations to find T, and the six co-ordinates
of m and m. From the form of the equations, or by treating
them as in 297, we see that the string remains parallel
to a fixed plane, that the centre of inertia moves with con-
stant velocity in a straight line, and that the motion of the
particles about each other and about the centre of in-
ertia is the same as if that point were at rest. Hence,
the particles revolve with uniform angular velocity, and
the tension of the string is constant. From the above
equations
mm' V*
m + m a
where 7- / [f* .-<* -->l'
V [_( dt }
dt dt
is the relative velocity. The same result might have been
easily obtained by using the last formula in 144, when we
consider that the velocity of m relative to the centre of inertia
is , that the radius of the circle it describes about
m -{-m
that point is , , and that T is the tension which main-
m + m
tains it in that circle.
300. Two particles, connected by an inextensible string
which passes over a small smooth pulley, move under gravity ;
to determine the motion.
MOTION OF TWO OR MORE PARTICLES. 367
This was partly anticipated in 298. Let m, m be the
masses, and let x, x denote their distances from the pulley
at time t. Then if T be the tension of the string (the same
throughout since the pulley is smooth), we have
But x + x = length of string = a suppose. Hence sup-
posing m > m,
Ct r t*
') 2 = (ra-ra')# ............... (1).
This equation completely determines the motion. Also,
if we eliminate x and x', we have
T
~mT+m
and it is therefore constant.
This is one of the cases in which theoretical results may
be tested by actual experiment with considerable accuracy.
And it was this combination, with many delicate precautions
against friction, &c. which Atwood made use of for experi-
mental verification of the laws of motion.
We see, for instance, by equation (1), that we may easily
keep m+m' constant while mm' has any value, and thus by
measuring the accelerations produced, find whether they are,
in the same mass, proportional to the forces producing the
motion. Again, keeping m m' constant, m + m may be
varied at will. Hence by this process the second law of mo-
tion may be tested. See 68. Again if, while the masses are
in motion, a portion be suddenly removed from the greater
so that they remain equal, (1) shews us that observation will
enable us to test the first law of motion.
So far for the motion when vertical. When the particles
are equal, but one of them vibrates as a pendulum, the purely
mathematical difficulties of the question become much more
serious. From the following approximation however (Proc.
R. S. E. 1881) we obtain a general idea of the nature of the
motion.
368 MOTION OF TWO OR MORE PARTICLES.
Let r, 6 be the polar co-ordinates of the vibrating mass
then, neglecting powers of higher than the second, we have
( 250)-
2
dt
Put | r for r, and \/2<9 for 0, and we get
Transform to rectangular co-ordinates in the plane of
motion x being vertically downwards: then
V 2 2y
x= ?/ =
a?' J x
This shews that the vertical acceleration of the vibrating
particle is very small but constantly downward. Hence the
energy of the vibratory motion is steadily converted into
energy of translation of the masses. It would be interesting
to pursue this question to higher degrees of approximation.
When both the equal masses vibrate through small arcs,
it is found that the mass whose angular range is the greater
has downward acceleration with diminishing angular range.
Hence it would appear that, if the string be long enough,
the entire motion should be periodic. But the working of
this question also is left to the reader.
301. Instead of two masses, connected by a string,
suppose a uniform chain of length 2a hang over the pulley ;
then if x be the length hanging down on one side at time t,
there will be 2a x on the other, and the difference or 2(# a),
is the portion whose weight accelerates the motion. Hence,
/j, being the mass of the chain per unit of length, we have
, A- r,
which gives x a = Ae + Be
MOTION OF TWO OR MORE PARTICLES.
369
If the chain were initially at rest, a portion a + b being
on one side of the pulley,
This is true until x = 2a, that is, till the chain leaves the
pulley ; the value of t at that instant being t 0) we have
_
,
- l) J .
and therefore t Q = y log +
If, for example, b = - , i.e. if the portions of the chain
o
were initially as 4 : 1,
302. Two particles, of masses m and m', are attached to
different points of an inextensible string, one of whose ex-
tremities is fixed. If the system be displaced, to determine the
motion.
Take the axes of x and y horizontal, and that of z verti-
cally downwards, the extremity of the string being origin.
Let a, a' be the lengths of the portions of the string, 6, 6'
the angles they make with the vertical, <, <f>' the angles
which vertical planes through them at time t make with the
plane of xz. Let x, y, z, x r , y\ z', be the co-ordinates of the
particles and T, T' the tensions of the strings.
Then
in
dt*
Tsin cos $ + T' sin 6' cos
= - Tsin 6 sin <f> + T' sin 6' sin </>',
4- T cos
T. D.
24
3"70 MOTION OF TWO OR MORE PARTICLES.
m' -TT = T' sin & cos </>',
Besides these, we have the six equations for #, i/, #,
#', i/', / in terms of a, a, 6, <f), 0', </>' ; in all, twelve equations
for the determination of the twelve unknown quantities in
terms of t.
303. These equations will be much simplified if we con-
sider the displacement to be in one plane, as the motion will
evidently be confined to that plane. By this means we at
once get rid of -y, y ', $ and <'. A still greater simplification
will be obtained by taking in addition the condition that
6 and & are so small, that their squares and higher powers
may be neglected. With these our equations become
And, to a sufficient approximation,
Hence, T' = m'g, and T=(m + m')g,
7 2 /a
ma = (m + m') gO + mgO f ,
MOTION OF TWO OR MORE PARTICLES. 371
Introducing an indeterminate multiplier, and adding,
(m + \m) + \m' - + {( m + m ') Q + m > (x _ 1) 0'} = 0.
etc CL at a
Let Xj , \2 be the roots of the equation
X of _ X 1
m + Xm' a m + m' '
Evidently one is positive and the other negative, and
the form of the equation shews that for both m + Xm' is
positive.
Put 4> = 6 + Xm ' , -0' = + kff, suppose.
m a
Then the above equation gives
T
at 2 a m + Xm
By the recent remark the coefficient of < is positive for
both values of X ; let its values be n? and n 2 2 , and we have,
h, <h, & 2 , $2, being the corresponding values of & and (/>,
^ = + ^0' = ttl cos
where i, o^, A, A, are arbitrary constants.
Hence,
19 = ^ ttl cos Wl * + A " k ^ cos
T l a i cos ( w ^ + A) ~ 2 cos (nj
Having given the initial values of 0, 0', -=- and |- , we
etc etc
find i, 2 , ft, /3 2 , and thus the solution is complete. It may
be noticed that the values of and & may be found at any
time by taking the algebraic sum of the corresponding values
of the inclinations to the vertical of two pendulums whose
o OTT
times of oscillation are -- and . Also, if 7^, n^, be com-
n a n^
242
372
MOTION OF TWO OR MORE PARTICLES.
mensurable, the system will in time return to its first position,
and the motion will be periodic.
The following discussion of the cases of the simple har-
monic motions of the system when m is much greater than
m and the strings are not approximately equal is taken from
a paper by Sir W. Thomson, " On the rate of a Clock or
Chronometer as influenced by the mode of suspension."
CASE I.
The upper string considerably longer than the lower.
Figure 1 represents the first or graver fundamental mode ;
the period of the upper pendulum CP being made somewhat
(1) (2) (3) (4)
graver by the influence of the lower, which in the course of
the vibration always exerts a force upon it from its middle
position.
Figure 2 represents the second or quicker fundamental
mode ; the vibration of the upper pendulum being in this
case excessively small in comparison with that of the lower,
and forced by the influence of the latter to a period much
smaller than its own would be if undisturbed.
MOTION OF TWO OR MORE PARTICLES. 373
CASE II.
The upper string considerably shorter than the lower.
Figure 3 represents the graver mode; the vibration of
the upper pendulum through but a very small arc in com-
parison with that of the lower, being augmented by the
influence of the lower, which in the course of the vibrations
exerts a force upon it always from its middle position.
Figure 4 represents the quicker mode ; the vibrations of
the upper pendulum being made somewhat faster by the
influence of the lower, and the lower being influenced so as
to vibrate as if it were shortened to the length OA, which is
somewhat less than the length CP.
In each case OA is the length of the simple equivalent
pendulum vibrating in the same period as that of the funda-
mental mode represented.
If P consisted of the frame and work of a spring clock,
and PP' were its pendulum, then in Case I. the vibrations
which would be maintained by the actions of the escapement
would be those represented by figure 2, and the clock would
go faster than if its frame were perfectly fixed.
In Case II. the vibrations maintained by the escapement
would be those represented by figure 3 and the clock would
go somewhat slower than its proper rate.
Case I. could never occur in practice, but may be experi-
mentally illustrated by hanging the works of a clock on a
light stiff frame moveable round a horizontal axis.
Case II., figure 3, with CP much shorter in proportion
to PP' than shewn in the diagram, represents the actual
circumstances of an ordinary pendulum clock, which owing
to want of perfect rigidity of the frame, must experience a
little of the influence of the pendulum there illustrated,
causing the rate of the clock to be somewhat slower than it
would be if the support of the pendulum were absolutely
fixed.
A very slight modification of the process gives us the
result of small displacements not in one plane : but the
student may easily work out these for himself.
374 MOTION OF TWO OR MORE PARTICLES.
We have here a simple example of the principle of the
Coexistence of Small Oscillations; but this principle, for
its satisfactory treatment, requires in general the use of
D'Alembert's Principle ; which, though ( 74) merely a corol-
lary to the Third Law of Motion, and clearly pointed out by
Newton as such, is beyond the professed limits of the present
treatise.
304. The examples, which have just been given, may
suffice to convey an idea of the mode of applying our methods
to any proposed case of motion of two constrained particles.
These methods are applicable to more complicated cases,
when more particles than two are involved ; but nothing
would be gained by such a proceeding, as D'Alembert's
Principle supplies us with a far simpler mode of investi-
gating the motions of any system of free or connected par-
ticles : especially when it is simplified in its application by
the beautiful system of Generalized Co-ordinates introduced
by Lagrange ( 250). See Thomson and Tait's Natural
Philosophy, 318, 327.
EXAMPLES.
(1) Prove that the periodic time of two bodies round
9
each other is j= = , where a is their mean distance and
Vm + m'
m, m their masses expressed in astronomical units.
(2) If the sun were broken up into an indefinite number
of fragments, uniformly filling the sphere of which the earth's
orbit is a great circle, shew that each would revolve in a year.
(3) Supposing the earth's present orbit to be circular,
examine the effect on the earth of a sudden annihilation of
half the sun's mass.
(4) A thin spherical shell of mass M is driven out sym-
metrically by an internal explosion. Shew that if, when the
shell has a radius a, the outward velocity of each particle be
v } the fragments can never be collected by their mutual
attraction unless M
v 2 < .
a
MOTION OF TWO OR MORE PARTICLES. 375
(5) Two equal particles are initially at rest in two
smooth tubes at right angles to each other. Shew that
whatever be their positions, and whatever their law of
attraction, they will reach the intersection of the tubes
together.
(6) In last question suppose the original distances from
the intersection of the tubes to be a, 6, and the attraction as
the square of the distance inversely, find the future paths
if at any instant the constraint is removed.
Solve the same question, supposing the attraction to
vary inversely as the cube of the distance.
(7) A shell is describing an elliptic orbit under an
attraction tending to the centre. Prove that, if it explodes
at any point of its orbit, all the pieces will meet again at
the same moment ; and that, after half the interval between
the explosion and the collision, all the pieces will be moving
with equal velocities in parallel directions.
(8) A number of equal particles, attracting each other
directly as the distance, are constrained to move in parallel
tubes; if the positions of the particles be given at the com-
mencement of the motion, determine the subsequent motion
of each ; and shew that the particles will oscillate sym-
metrically with respect to the plane perpendicular to the
tubes which passed through their centre of inertia at the
commencement of the motion.
(9) Two equal masses M, are connected by a string
which passes through a hole in a smooth horizontal plane.
One of them hanging vertically, shew that the other describes
on the plane a curve whose differential equation is
and that the tension of the string is
376 MOTION OF TWO OR MORE PARTICLES.
(10) Two given monkeys cling to a rope, which hangs
over the common summit of two given inclined planes ; one
monkey remains stationary : find the acceleration of the
other monkey.
(11) Two equal balls repelling each other with intensity
oc _ 2 hang from the same point by strings of length I. Shew
that if when in equilibrium, the strings making an angle 2a
with each other, they be approximated by equal small arcs,
the time of an oscillation is the same as that of a pendulum
whose length is
I cos a
1 + 2 cos 2 a '
(12) One of two equal particles connected by an inelastic
string moves in a straight groove. The other is projected
parallel to the groove, the string being stretched ; determine
the motion, and shew that the greatest tension is four times
the least.
(13) Two particles connected by a rigid rod move on
a vertical circle. Determine the motion, and find the time
of oscillation about the position of stable equilibrium.
(14) Two particles P and Q are connected by a rigid rod.
P is constrained to move in a smooth horizontal groove. If
the particles be initially at rest, PQ making a given angle
with the groove in a vertical plane through it, find the velo-
city of Q when it reaches the groove, and shew that Q's path
in the vertical plane is an ellipse.
(15) A particle of mass m has attached to it two equal
masses in by means of strings passing over pulleys in the
same horizontal plane, and is initially at rest halfway between
them. Determine the motion. Shew that if the distance
between the pulleys be 2a, the velocity of m will be zero
when it has fallen through a distance
4m' 2 m 2 '
MOTION OF TWO OR MORE PARTICLES. 377
(16) Two masses M, M' are connected by a string which
passes over a smooth peg. To M' is attached a string which
supports a mass m such that M' + ra = M , and ra is displaced
through an angle a. Investigate the motion, supposing m so
small that the horizontal motion of M' may be neglected.
Shew that the string M'm will be vertical after the time
where X is the length of M'm. (Steele.)
(17) Two equal masses are attached each at 1 foot from
the ends of a string 3 feet long which are fixed 2 feet apart
in a horizontal line. Compare the times of vibration in the
various degrees of freedom of the system.
(18) A string ABCD, divided into three equal parts
at B and C, has two equal weights attached to it, at B and
C, and the ends A and D are fastened to two fixed points
in the same horizontal plane, the distance AD being two-
thirds of the length of the string.
Find the tension of the different portions of the string
when there is equilibrium, and, if the string CD be cut
through, find the initial changes of tension of the other
portions of the string, and the direction and magnitude of
the initial acceleration of the weight at C.
(19) The point of suspension of a simple pendulum
moves uniformly in a circle in the plane of oscillation of the
pendulum, find the equations of motion of the pendulum,
and solve them in the case where the radius of the circular
arc is very small.
(20) A fine string passes over two smooth pegs in
the same horizontal plane and carries three equal weights,
one at each end and one capable of sliding on the portion of
string between the pegs. If the system be slightly disturbed
vertically from its position of equilibrium, find the time of a
small oscillation.
378 MOTION OF TWO OR MORE PARTICLES.
(21) A particle of mass M is placed near the centre of
a smooth circular horizontal table of radius a; strings are
attached to the particle, and pass over n smooth pullies
which are placed at equal intervals round the circumference
of the circle ; to the other end of each of these strings a
particle of mass M is attached. Shew that the time of a
small oscillation of the system is 2?r ( . - ) .
\ n g)
(22) Two particles are attached together by a fine
thread : the one is oscillating on the lower part of a vertical
circle, the other below the circle and in its plane : if the
motions be small, shew that the motion of each particle is
compounded of two independent oscillations, the sum of the
squares of the periods of which is approximately equal to
, where c is equal to the sum of the lengths of the radius
and the thread.
(23) In a compound pendulum consisting of masses m,
m' attached to strings of length I, l f , in which of course the
most general small motion in one plane consists of two
harmonic vibrations superposed, if the upper mass m be very
large compared with the under mass m, it is clear that one of
the two periodic times (that corresponding to the mode of
vibration in which m is nearly at rest) must be very nearly
the same as in a simple pendulum of length l' t and the other
very nearly the same as in a simple pendulum of length L
By a continuous variation of I', the former may be made to
pass continuously from less to greater than the latter, and
therefore for some value of I' nearly equal to I the two must
be equal. But when a system is in stable equilibrium (as is
clearly the case here), the equation the roots of which give
the times of vibration cannot have equal roots, for that would
imply the transitional condition between stable and unstable.
Point out precisely the fallacy which leads to the above
contradiction.
(24) A string of length na has attached to it at equal
intervals n equal particles, and the whole is suspended so as
to hang vertically from one end : if the system be slightly dis-
MOTION OF TWO OR MORE PARTICLES.
379
1
o
2 i-l rH rH rH
,<">
CO
^^
ng
.2 <N <N <N <N
-p
* ^ ^ ^ ^
1 rH
| 1 1 1
rH
,13 g
CM
O ^~~^
'3 %lx
o
J 1 1 I
C<f rH
F N 1
<M (N
>> 3 ^
i^ ^^ ^_^ j
-O N ^ rH CO
1 1 s ' \
Of r-T
% % s s ^ ^
o *n> i
p, * ^
tL O^ *v
^) CJ 1 H
r? 1 G? &S
'$ 1 vTlx 1 1
Ol rH
^ ~ ^ ^
'1 1 '
*M ^
'a 1 'j ^^^ /^>>
^3 o "3 1 ^ rH C<1 CO
^ ^ 1 1 1 1
~ t
^(N ^
C 3
1 2 -
I ^
^
380 MOTION OF TWO OR MORE PARTICLES.
(25) A spider hangs from the ceiling by an elastic
thread whose modulus of elasticity is equal to his weight.
Shew that it is possible for him to climb to the ceiling up
the thread by the expenditure of f of the amount of work
required to climb to the same height up an inelastic string,
and describe fully the precautions he must take in order
to do so.
If the thread be making very small longitudinal oscilla-
tions while the spider crawls up very slowly, shew that the
time of an oscillation will vary as the square root of the
distance of the spider from the ceiling.
(26) Two given masses are connected by an elastic
string, and projected so as to whirl round ; find the time
of a small oscillation in the length of the string.
Give a numerical result, supposing the masses to be 1 Ib.
and 2 Ibs. respectively, and the natural length of the string
to be one yard, and supposing that it stretches .y^th inch for
a tension of 1 Ib. weight.
(27) Two particles, connected by an elastic string, are
projected in any manner. Shew that in the relative orbit
(28) Two particles connected by an elastic string initially
unstretched, are projected at right angles to it so that their
centre of gravity remains at rest, and their relative velocity
is that of a particle falling under gravity through the length
of the string. The string is of length a, and would be
stretched to a length 2a by the harmonic mean of the weights
of the particles. Shew that the path of one particle relatively
to the other is given by
r (?'= - a?) a 2 - r- (r - a)~
Prove also that the string can never become slack.
MOTION OF TWO OR MORE PARTICLES. 381
(29) Two equal particles can slide on opposite horizontal
generators of a circular cylinder, and are connected by a
stretched elastic string without mass. If the particles are
slightly and equally disturbed from their position of equi-
librium in opposite directions, find the time of an oscillation.
If the unstretched length of string be given, find what must
be the radius of the cylinder in order that the time of
oscillation may be as small as possible.
(30) A small ring is free to slide along the arc of an ellipse
which is placed with its major axis vertical ; the ring is sup-
ported by an elastic string without weight fastened at the
upper focus of the ellipse, and such that its original length
was equal to the semi-latus rectum of the ellipse, and of
such elasticity that the given ring would, by its weight,
stretch it to be equal to the semi-minor axis in length.
The ellipse revolves with an angular velocity such that the
ring just lies at the extremity of the minor axis. Find the
angular velocity, and find the time of a small oscillation of
the ring when slightly disturbed along the arc of the revolv-
ing ellipse.
(31) A circular elastic band is placed round a wheel the
circumference of which is double the natural length of the
band; if the wheel be made to revolve with a constant
angular velocity, find the pressure of the band on the wheel.
(32) A mass M of fluid is running round a circular
channel of radius a with velocity u ; another equal mass of
fluid is running round a channel of radius b with velocity v ;
the radius of the one channel is made to increase and the
other to diminish till each has the original value of the
other : shew that the work required to produce the change
is
Hence shew that the motion of a fluid in a circular whirl-
pool will be stable or unstable according as the areas de-
scribed by particles in equal times increase or diminish from
centre to circumference.
382 GENERAL EXAMPLES.
GENERAL EXAMPLES.
(1) A spiral spring is stretched an inch by each addi-
tional pound appended to its lower end ; find the greatest
velocity which will be acquired by 20 Ibs. appended to the
unstretched spring and allowed to fall.
Also find how far the mass will fall, and the time of a
complete oscillation.
(2) Find the form of the hodograph, and the law of its
description, for any point of one circular disc rolling uni-
formly on another. Hence, find the force under which a free
particle will describe an epitrochoid, as it is described by a
point of the uniformly rolling disc.
(3) The motion of a point P is determined as follows by
its position relative to two fixed points A and B. The
velocity of P is made up of r^ towards A, and
Shew that P describes a circle passing through A and B,
and that its velocity at any point is inversely as AP . PB.
If its velocity in any position be the same in magnitude
as before, but turned through a right angle in the plane
APB, shew that the path is still a circle.
(4) Determine the (unresisted) motion of a body pro-
jected vertically at a given point of the earth's surface with
a velocity of 7 miles per second.
(5) Apply the principle of varying action to the deter-
mination of the (unresisted) motion of a projectile.
(6) Shew that the action and time, in any arc of the
ordinary brachistochrone commencing at the cusp, are repre-
sented by the area and arc of the corresponding segment of
the generating circle.
GENERAL EXAMPLES. 383
(7) In the parabolic motion of a projectile, the action is
represented by the area included between the curve, the
directrix, and the two vertical ordi nates : and the time by
the intercept on the directrix.
(8) Given a central orbit, and the law of its description,
find the differential equation of a curve such that if tangents
be drawn to it from any two points of the orbit, the action
shall be represented by the area included by these tangents
and the two curves.
(9) Find the component harmonic vibrations of two
equal simple pendulums hung up by two points in the same
horizontal plane, the bobs of the pendulums attracting with
intensity as the inverse square of the distance and of magni-
tude a small portion of the weight of either pendulum.
(10) The point of suspension of a simple pendulum has
a horizontal motion expressed by
x = a cos mt.
Find the effect on the motion of the pendulum, especially
when
or nearly so, I being the length of the pendulum.
(11) Determine the most general (small) motion of a
heavy particle attached at a given point to a stretched elastic
string. Shew that it will vibrate with equal rapidity in
all directions of displacement, however much the string be
stretched, provided the particle be placed at a distance from
one end equal to half the length of the unstretched string.
(12) A particle P describes an ellipse freely under the
attraction of a second particle which is constrained to
move along the major axis; G, but not P, is attracted to
the centre ; find the laws of the attractions that PG may
be always the normal at P.
If it were conceivable that P should repel G with the
same intensity that G attracts P, a certain relation between
384 GENERAL EXAMPLES.
the masses of the two particles would render unnecessary
any force to the centre.
(13) A particle P describes an ellipse with constant
velocity under two equal forces, one directed towards the
focus S, and proportional to CD n , and the other towards the
other focus H ; CD being the semidiameter conjugate to CP.
Shew that n = - 2.
(14) A particle P is attached to a point Q by a wire
without weight, and is acted on by a force whose accelerating
effect varies as the distance from a point to which it tends;
prove that, if Q be constrained to move in a circle with the
same velocity as a free particle would describe that circle
under the action of the force, P will in all cases move uni-
formly relatively to Q in a plane parallel to a fixed plane. If
QO be the length of the wire, shew that, if P be ever at rest,
its absolute path will be a straight line.
(15) A number of equal particles are fastened at equal
distances a on an inelastic string, and placed in contact in
a vertical line ; shew that if the lowest be then allowed to
fall freely the velocity with which the n th begins to move is
equal to
>-l)(2n-l)
3n
(16) Two particles, of masses m and m' respectively, are
connected by a light elastic string of length 2a. The system
is then suspended from a smooth pulley and so adjusted that
each particle is at a distance a from the pulley. If the system
be then left to itself and y, y be the distances of the particles
from their original positions at the time t, then
my my' (m m) ~- .
(17) A particle attached to the end of a string rests upon
a smooth horizontal table ; the string passes through a small
hole in the table through which it is pulled with uniform
velocity ; prove that if the particle be acted upon by a force
GENERAL EXAMPLES. 385
inversely proportional to its distance from the hole, and per-
pendicular to the string, it will describe if properly projected
an equiangular spiral.
(18) A centre of attraction which attracts a particle of
mass m with intensity mw x distance, moves on the circum-
ference of a circle of radius a with constant angular velocity
&>, and a particle is placed midway between the centre of
attraction and the centre of the circle ; if r and be its polar
co-ordinates when the centre of attraction has moved through
an arc a<f), prove that
the centre of the circle being the pole, and the initial line
passing through the initial position of the particle.
(19) Three particles each of mass m are lying on a
smooth horizontal table in a straight line joined together by
two strings, each of length a. The two outer particles are
projected simultaneously with the same velocity v in a direc-
tion perpendicular to the strings, prove that
dtj a? 2 -cos 20'
where is the angle the string joining the middle particle
with either of the other two has turned through in any
time.
(20) Three equal particles are joined by two equal strings
and are placed in one straight line on a smooth table ; if the
middle one be projected perpendicular to the string with a
velocity F, the velocity of the other two when they im-
pinge is F.
(21) Two particles are joined by a string which passes
through a small ring, the particles are held in the same
horizontal line, and the string is tightened and then let go ;
if p, p be the radii of curvature of their paths initially, a, a',
T. D. 25
386 GENERAL EXAMPLES.
the initial lengths of the portions of the string, m, m' their
masses, shew that
mm' ,1111
- = and - + -, = - + .
p p a a p p
(22) Investigate the equation of motion of a chain con-
strained to move in a fine tube under given forces.
A uniform chain of length 4a is coiled up on a horizontal
table at the distance a from one edge of the table, and one
end of the chain is then drawn out at right angles to the
edge and just over it ; the height of the table above the floor
being a, investigate completely the motion of the chain.
(23) An elastic string of length a, mass ma, is placed in
a tube in the form of an equiangular spiral with one end
attached to the pole. The plane of the spiral is horizontal,
and the tube is made to revolve with uniform angular velo-
city ft> about a vertical axis through the pole ; prove that its
length, when in relative equilibrium, is given by the equation
, tan d>
/m
where <f> = aco cos a , / .
(24) A particle is suspended from a fixed point by an
elastic string, and performs small oscillations in a vertical
direction ; supposing the string uniform in its natural state
and of small finite mass, shew that the time of oscillation
will be approximately the same as if the string were without
weight and the mass of the particle increased by one-third of
that of the string.
(25) The resistance of the aether to a planet or comet
moving with the velocity V being assumed to be k -=- and
ctt
the sun's attraction being ~, obtain the following exact
equations :
GENERAL EXAMPLES. 387
at
Obtain also the differential equation of the orbit in the
form
(26) A body moves in a plane about a fixed point under
given forces. If the areal velocity and the direction of
motion of the body at a proposed point be known, find the
semi-latus rectum of the elliptic orbit which has a contact
of the second order with the real orbit at that point, its
focus being at the given fixed point.
Also find the changes produced in an indefinitely small
time in the excentricity and in the position of the apse in
this elliptic orbit in terms of the corresponding change of
the semi-latus rectum.
(27) Prove that the apparent path of a comet on the
celestial sphere is concave or convex towards the sun's ap-
parent place according as the comet or the earth is nearer to
the sun.
(28) It has been found by comparing theory with obser-
vation that the perihelion of Mercury progresses at a rate
greater by a than that due to the attraction of known bodies;
shew that this increment would be accounted for if the law
/ .
of force tending to the sun were -, ' + ^ , and if uf = a. A / .
r 2 r* 'V c
the orbit being supposed to be nearly a circle and the mean
distance to be c.
(29) A comet moving in a parabolic orbit makes a near
approach to a planet ; point out from general considerations
the circumstances under which the orbit of the comet is
rendered elliptic or hyperbolic.
252
S88 GENERAL EXAMPLES.
(30) A particle moves under a retardation f(t) which
brings it to rest in a time a ; prove that the distance tra-
versed is
f
Jo
tf(t)dt.
(31) If the velocity of a railway train resisted by friction
differ from its mean velocity by a periodic function of the
time, determine the least horse power of the engine that
will draw the train, and prove that this horse power is
greater than what would be required if the velocity were
constant.
(32) A particle is in motion within a triangle ABC,
and is attracted perpendicularly to the sides with intensities
each equal to //, times the perpendicular distance. Shew
that the motion consists of two periodic terms of the form
P sin [t VV + Q},
where (X - 1) (X - 2) + 2 cos A cos B cos - 0.
Shew distinctly that the roots of this quadratic are real
and positive; examine the case of an equilateral triangle
and in that case verify the above result independently.
(33) A particle is attracted perpendicularly towards
the faces of a tetrahedron with intensities equal respectively
to //, 2 times the perpendicular distances. If the medium
resist with intensity 2kv, then the particle on moving within
the tetrahedron will have its motion stable provided the
equation in X
cos (12), cos (13), cos (14)
cos (21), ^-^-1, cos (23), cos (24)
cos (31), cos (32), ., -1, cos (34)
cos (41), cos (42), cos (43), : -1
= 0,
GENERAL EXAMPLES. 389
has all its roots real, cos (12) being interpreted to mean the
cosine of the angle between two faces which are marked 1, 2
respectively.
(34) A smooth cylinder, whose transverse section is a
cycloid generated by a circle of diameter a, is placed with
its axes horizontal, the axis of the cycloidal section being
vertical and its vertex downwards. A particle is allowed
to fall from rest at any point of the surface and is attracted
by a perfectly elastic plane perpendicular to the axis of
the cylinder, with an intensity varying directly as the dis-
tance from the plane, and strength . Shew that the
QJ
path of the particle will . be such that if the cylinder be
developed it will develop into successive portions of a
parabola.
(35) A vertical wheel rolls on a horizontal plane with
the velocity it would acquire by falling through a height
equal to half its radius ; a particle flies off at the point P ;
shew that the focus of the parabola described by the particle
is the foot of the perpendicular from the lowest point of
the wheel upon the radius through P; and that the focal
distance of P is a mean proportional between the semi-
latus rectum and the radius of the wheel.
(36) From every point of an ellipse particles are pro-
jected in the direction of the tangent with velocities such
that, when under a central attraction - to one of the foci
r 2
of the ellipse, they proceed to describe parabolas. Shew
that the directrices of these parabolas all touch one or other
of two fixed circles whose radii are equal to the major axis
of the given ellipse.
(37) Find the circumstances of projection that a particle
attracted by an infinite straight line with intensity inversely
as the square of the distance may describe a set of complete
cycloids.
(38) A particle is revolving on a smooth plane about
a centre of attraction, of intensity p x distance, and when
390 GENERAL EXAMPLES.
the body arrives at an apse the plane begins to revolve
with an angular velocity \ */3p about the apsidal line ; shew
that the subsequent orbit described on the plane will be a
portion of a parabola ; and that, when the particle leaves the
plane, its velocity will be V3 x velocity at the vertex.
(39) If the component velocities, parallel to two rect-
angular axes, of a particle moving in a plane be proportional
to its distances from two other straight lines in the plane
at right angles to each other, its path will be an equiangular
spiral or a rectangular hyperbola.
(40) Prove that if a point move in a plane curve with
velocity v, and if the direction of its motion make an angle ^r
with a fixed line,, the rate of change per unit time of the
magnitude of v is -=- and of direction v -~ .
dt dt
Deduce the expressions for the accelerations when the
position of the point is given by its distance from a fixed
point and the angle which that distance makes with a fixed
line.
(41) A point is moving on a plane area, which is itself
moving in any way in its own plane. Find the accelerations
of the point with regard to absolutely fixed axes.
A 1 describes an equiangular spiral with uniform angular
velocity about : A% describes an equal spiral with the same
angular velocity about A lt and so on. Prove that A n describes
an equiangular spiral with the same angular velocity relatively
to 0, and find its size.
(42) Assuming that the moon describes areas propor-
tional to the times of describing them by radii drawn to the
centre of the earth, examine the nature of the force which
acts on the moon. On the above assumption and taking the
orbits of the moon and earth to be circular, shew that the
acceleration of the moon in the direction of the tangent to
its orbit on the nth day of the (lunar) month is j^x sin -
GENERAL EXAMPLES. 391
roughly. Given that the distance of the sun from the earth's
centre is 24000 times the earth's radius, and that the mass
of the sun is 320000 times that of the earth.
(43) Two particles revolve about a centre of attraction
(the law of attraction being as the distance), one in an ellipse
2
of excentricity = , and the other in a circle passing through
the foci of the ellipse. Shew that the first particle moves
within the circular path of the second particle during 1 of its
period ; and compare the velocities of the two particles at the
points common to their orbits.
(44) If the resolved part perpendicular to the radius
vector of the velocity of a body revolving in an elliptic orbit
about the focus ever be half its whole velocity, shew that the
/q
excentricity of the ellipse must be > : and that it is im-
possible that at the same time the resolved part of its velo-
city perpendicular to the major axis should be also half its
whole velocity.
(45) If a particle be projected from an apse at a distance
a from a centre of attraction of which the intensity at
distance r is /j, (r a), obtain the equation for determining
the other apsidal distance, and find the velocity of projection
in order that it may be .
(46) If the orbit is p* (a 7 ^ 2 - r m ~ 2 ) = b m , shew that the
apsidal angle is -j nearly.
(47) A particle of mass m is attached to a fixed point
by an elastic string of natural length a, whose coefficient of
elasticity is m. It is projected with the velocity due to half
the length of the string, in a direction perpendicular to the
string which is initially unstretched. Prove that the apsidal
distances of its orbit are given by
r 4 -2ar } + a 4 = 0.
392 GENERAL EXAMPLES.
(48) Particles describe confocal ellipses under the attrac-
tions tending towards the centre. If at any instant they are
all at the ends of the conjugate, or transverse, axes of their
orbits, prove that a hyperbola confocal with the ellipses can
always be drawn through them all.
(49) A particle is moving in an ellipse about a centre of
attraction in the focus, and the centre of attraction is trans-
ferred to one end of the latus rectum as the particle passes
through the other. Prove that e, e f , the excentricities of the
old and new orbits, are connected by the relation
(50) A bod 
