LIBRARY
I
A TREATISE
DYNAMICS OF A PARTICLE.
A TREATISE ON
DYNAMICS OF A PARTICLE,
WITH NUMEROUS EXAMPLES.
BY
PETER GUTHRIE TAIT, M.A.,
F0KJIE1U.Y FELLOW OF ST PETEK'S COLLEGE, CAMBRIDGE,
FEOFESSOB OF KATUBAL PHILOSOPHY IK THE CNITEBSITT OF EDINBUKGH,
AND THE LATE
WILLIAM JOHN STEELE, B.A.,
FELLOW OF ST PETER'S COLLEGE, CtMBRIPGE.
or THr V
UNIVEnsiTY J
FIFTH EDITION.
Carefully revised.
Hontfon :
MACMILLAN AND CO.
1882
[All Rights reserved.]
( c
Camtrilige :
PRINTED BY C. J. CLAY, M.A. & SON,
AT THE UNIVERSITY PRESS.
PEEFACE.
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 Kas derived much benefit from
revision : first by Mr Stirling of Trinity in 1865, then by
Mr W. D. NiVEN of Trinity in 1871, and recently by Prof.
Greenhill of Emmanuel in 1878.
It now appears after a general revision by myself, with
the assistance of Dr C. G. Knott and of my colleague
Prof. Chrystal.
Under such circumstances it could not fail to be a patch-
work of a somewhat complicated kind; but the comparatively
rapid exhaustion of the latest edition 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 en-
grossing work ; and I am led, therefore, very naturally rather
to the making of hastily improvised insertions than to
carrying out any well-considered scheme of compression or
co-ordination.
t. d. b
VI PREFACE.
The book's most important fault is its bulk ; yet I do not
think it can be honestly accused of prolixity. And 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 will not consider this a defect.
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 simpUciter from the German.
P. GUTHRIE TAIT.
College, Edinbubgii,
July, 1882.
CONTENTS.
PAGES
Preface v — vi
Chapter I. Kinematics ... ... ... ... 1 — 34
Dmsion of the subject, §§ i — 3.
Velocity, §§ 4^7.
Composition and Kesolution of Velocities, §§ 8— 11.
Acceleration, §§ 12 — 19.
Hodograpli, § 20.
Moment of Velocity, §§ 21 — 24.
Motion of a point deduced from the given acceleration, § 25.
Kelative Velocity and Acceleration, §§ 26 — 36.
Angular Velocity and Acceleration, §§ 37 — 40.
Velocity and Acceleration relative to Moving Axes, §§41—43.
Examples 34 — 41
Chapter II. Laws of 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 Center of Inertia, § 58.
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, &c.
§§ 63—72.
CONTENTS.
Scholium to the Third Law, with its interpretation.
D'Alembert's principle, Horse-power, Conservation of
Energj- in Ordinary Mechanics, §§ 73—75.
Conservation of Energy, Impossibility of Perpetual Motion,
Joale's experimental results, §§ 76 — 78*.
Chapter III. Rectilinear Motion ... ... 59 — 79
Constant Force, §§ 79 — 87.
Force varying according to different powers of the distance,
§§ 88—105.
Examples 79 — S5
Chapter IV. Parabolic Motion ... ... ... 8G — 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 — 113
Chapter V. Central Orbits ... ... ... 113 — 144
General Equations, §§ 131, 132.
Attraction proportional to the distance, § 133.
Polar Form of General Equations, and consequences, §g 134
— 144-
Properties of Apses, §§ 145 — 148.
Orbits under the Law of Gravitation, §§ 149— 1 58.
Elliptic motion ; definitions and immediate deductions,
§§ 159-162.
Kepler's Problem, §§ 163 — 167.
Lambert's Theorem, § 168.
Ex.vMPLEs 145—166
C'hapter VI. Constrained Motion ... ... 1G7 — 222
Preliminary remarks on Constraint, § 169.
Motion on Smooth Piano Curve, Cycloidal and Common
Pendulum, Ac., Direct Problem, §§ 170 — 179.
CONTENTS.
Inverse Problems— Braehistoehrone, &c., §§ 180—186.
Motion on Smooth Surface, §§ 187 — 189.
Particular Case — Spherical Pendulum, §§ 190, 191.
Double Pendulum, § 192.
Effect of the Earth's rotation on simple pendulum, §§ 193 —
195-
Constraint by String attached to a moving Point, §§ 196—
198.
Constraint by Smooth Tube in motion, §§ 199 — 203.
Constraint by Eough 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.
Bectilinear Motion with various apphed 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 260—309
Constraint perpendicular to direction of motion, §§ 220,
221.
All central forces have a potential, § 222.
Conser^'ation of Energy, and Equipotential Surfaces, §§
223, 224.
Inverse Problem as to conservative forces, § 225.
Deductions from Conservation of Energy, §§ 226 — 229.
Least, or Stationary, Action, §§ 230 — 237.
Varying Action, §§ 238 — 243.
The principle applied to the investigation of a planetary
orbit, §§ 244 — 248.
AppUcation to Cotes' spirals, § 249.
Lagrange's Equations in Generalized Co-ordinates, §§ 250,
251-
CONTENTS.
A 1- 1- PAGEa
Application of Varying Action to Brachistochroncs, § 2^2.
The brachistochrone when the force is central, § 2^3.
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, §§ 255—
260.
Motion about moving centre, §§ 261, 262.
Hodographs, §§ 263—270.
Case of resisted motion in an equiangular spiral, § 271.
Examples
310—319
Chapter IX. Impact 3-^0—337
Preliminary Remarks, Coefficient of Restitution, § 272.
Direct Impact of Spheres, §§ 273, 274.
Impact of Sphere on Fixed Surface, § 275.
Impact of Smooth Spheres generaUy, 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
CiiAPTEE X. Motion of Two or More Particles 353—373
I. Free Motion. General Equations, §§ 289, 290.
Conservation of Momentum, § 291 ; of Moment of Momen-
tum, § 292 ; of Energy, § 293.
Particular Case of Two Particles, only, §§ 294_2()-.
II. Comtrained Motion. Conditions of Constraint, § 298.
Two particles, in space, connected by incxtcusible st'rinc
_ § ^99-
String constrained by pulley, § 300.
Chain slipping over pulley, § 301.
Complex pendulum, §§ 302, 303.
Limits of the treatise, § 304.
Examples
GlNEUAL EXAIIPJ
CONTENTS. XI
PAGES
Appendix 399—411
A. On the integration of the equations of motion about
a centre of attraction 399
B. Motion on a cycloid 404
C. Brachistochrone, for gravity 405
Cj. , for any forces 407
Cj. General property of free path and brachistochrone for
any force whose direction is constant 408
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 410
DYNAMICS OF A PARTICLE.
CHAPTER I.
KINEMATICS.
1. Dynamios 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 /orces 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 opposition)
of a point is called its Velocity. 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
T. D. 1
2 KINEMATICS.
— what in the differential calculus is called an independent
Miriable. Its physical definition is given in Chap. ii.
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. 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
liave generally
s
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
in 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 antj instant is tlien measured bi/ the space which would
have been described in a unit of time, if the point had moved
on imiformly for tliat interval ivith the velocity which it had
at the instant contemplated. This is a most important, and
in fact a fuudamentaL conception, which the student must
thoroughly realize befTre he can usefully proceed fartlier. It
lies at tlie root of all the correct metliods ever devised for the
])urpose of measuring the rate at which change, of any kind,
is going on.
Let V be the velocity of the point at the tinu^ t, measured
from a fixeil epoch, s the space (k\soribed by it iluring that
time, antl s + hs the space described during a greater interval
KINEMATICS. 3
t + U. Suppose t\ to be the greatest, and v^ the least, velo-
city with which the point moves during the time ht; then
v^U, v.M would he the spaces which a point would describe
in that interval, moving uniformly with these velocities
respectively. But the actual velocity of the point is not
greater than v^, and not less than v.^, therefore as regards the
actual space described,
hs is not greater than v^t, and not less than v^ht,
"' u "' "-
however small Zt may be. But, as ht continually diminishes,
v^ and v^ tend continually to, and ultimately become each
equal to, v. Therefore, proceeding to the limit,
ds
dr'-
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
excluded for the present. The action of great force for short
periods of time will be treated in the chapter on Impact.
8. So far as we have yet spoken of it, velocity has been
regarded merely as sjjeed, 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
1—2
4 KINEMATICS.
general most conveniently, done by considering the velocities
of the point parallel to the three co-ordinate axes respec-
tively. In fact velocity is properly a directed ii]agnitude (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 velocities parallel to the corresponding axes, will by
reasoning analogous to that in § 7 be
dx dy dz
di' di' dt'
Denoting by v the speed of the motion, we have
and, if a, j3, 7 be the angles which the direction of the mo-
tion makes with the axes,
d_x
dx dt
cos a = — = -7- ;
ds as
dt
or -^ = t; cos a = i\, suppose.
Similarly, -^ = v cos /3 = i'^ ,
dz
77 = V cos 7 = v..
dt
Hence, -r. , , , -r- are to be found from the whole velo-
dt dt dt
city V, by resolving as it is called; i.e. by multiplying by
the direction-cosines of the direction of motion. Thoy are
called the Component Velocities of the point : and, witli refer-
ence to them, V is called the licstdtant Velocity.
9. It follows from the above, that, if a point be moving
in any direction, we may suppose its velocity to be the result-
ant of three coexistent velocities in any three directions at
KINEMATICS. 5
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^, Vy, V, be the rectangular components of the velo-
city I) of a moving point, then the resolved part of v along
a line inclined at angles \, /ju, v to the axes will be
V^ cos X + Vj, COS /J'+V^ COS V.
For, let a, /S, 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 ^ = cos a cos A, + cos /3 cos /a + cos 7 cos v
the resolved part of v along that line is
V cos d = v {cos a cos A, + cos /3 cos fj, + cos 7 cos v]
— v^ cos X + v^coa /ii + V, cos v.
10. These propositions are virtually . equivalent to the
folloAving obvious geometrical construction, which is the Law
of Composition of Vectors : —
To comjDound any two velocities as OA, OB in the figure;
Avhere OA, 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
similarly 05 for the second; from A draw AC parallel and
equal to OB. Join OG: — then 00 is the resultant velocity
in magnitude and direction. For the motions parallel to OA
and OB are independent.
0(7 is evidently the diagonal of the parallelogram two of
whose sides are OA, OB.
6 KINEMATICS.
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, AG, 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 is the square
root of the sum of their squares — and the cosines of the in-
clination of its direction to the given directions are the ratios
of the components to the resultant.
[NeAvton's Method of Fliixtoiis was devised simply to
express this and other fundamental conceptions in Kinematics.
To him X, d; y, i, or (as we now somewhat less conveniently write
them) ' , ,- , ; , T, are simply the velocity of the moving
' dt dt' dt (It ^ '' '' °
point and its components parallel to the axes. It may be
convenient, or even necessary, to use the idea of Limits or of
Iitjinitesiin(ds to calculate their values ; but the Fluxions
themselves do not involve any such idea.]
11. When a point moves in a plane curve, to e.rpress its
component velocities at any instant alomj, and perj)cndicular
to, the radius vector drawn from a fixed point in the plane of
the curve.
Let X, y be the rectangular, r, the polar, co-ordinates of
the moving point; so that
x = r cos d, y=r sin 6.
KINEMATICS.
We have at once, by differentiation,
and
dx dr n • /I ^Q'
= y; cos 6 — r sm d -y-
dt dt dt
dy dr . ^ ^ ^ dd
•^ — sm ^ + r cos ^
dt dt
dt
.(1),
•which are the velocities parallel to x and ?/. But by § 9 the
velocity along the radius vector is
dy . -. dx ^ dr , ,^.
and the velocity perpendicular to it is
d^l ^ dx . -
-f' cos ^ — J- sm 6 ■■
dt dt
cie
'dt'-
by (1).
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
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 ;
8 KINEMATICS.
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 b}' 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 + 8v the velocity at the
end of the time t + Bt ; and let a,, a„ be the greatest and least
values of the acceleration during the interval Bt, then a^Bt,
a,^at would be the increments of velocity in that interval, of a
point under those accelerations respectively. But the actual
acceleration is not greater than a^ and not less than a^, there-
fore the actual increment of velocity
Bv is not greater than a^Bt and not less than aJBt,
Bv
^•^ Bt ^' ^^'
however small Bt may be. But, as Bt continually diminishes,
Gj and Ojj tend continually to and ultimately become each
equal to a. Therefore, proceeding to the limit,
dv
dt=^-
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.
KINEMATICS. 9
Combining the above equation with
ds
we have
d's
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 alto-
gether independent of the methods (infinitesimal or not)
which may be employed to calculate its value.]
14. Thus far w^e 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 we are led to generalize
the definition of Acceleration (see § 20 below).
If the path be curved, the accelerations of the rates of in-
crease of the co-ordinates of the moving point are called the
Component Accelerations parallel to the axes. If these be
denoted by a^, ay, a,, Ave shall have
d'^x _ d-y d'z
With reference to these, Ja^ -\- ciy -\- a/ is called the Re-
sultant Acceleration.
d~s .
15. The acceleration -jy is not the complete resultant
„ d^x d^y d^z ., , r -x
01 ^ttt > 'ifi > j^ > ^s may easily be seen : lor its square
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
d^s
velocity ; in short -^ is the sum of the resolved parts of
10 KINEMATICS.
-r^ , 7/ . ,r in the direction of motion, as the followincj
dt^ ' de ' df ' o
identical equation shews :
d^s _dx d'x dxj d^y dz d^z
df ~dsde'^ds de ^dsdf'
This follows immediately from the equation of § (8)
(dsV _ fdxV fdyV fdzV
[dtj ~[dt) "^U/ '^[dtJ
by diflferentiation. And it shews that acceleration is to be
resolved according to the same law as velocity. For to find
-v^ , the acceleration along s, -t-t, has to be multiplied by -j- ,
dt' ° dt" ^ -^ ds
&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 velocity and the curvature will be given later.
The above equation also shews, since _. ^ ~t~ t j' are
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 \, yb, v to the axes is
a^ cos X + 5r„ cos yu, + a, cos v.
16. A point moves in a plane curve, to express its com-
ponent accelerations at any imitant 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 ^ . ^dO
, tPx fdV Idey) „ (,-,drde d'e\ .
Similarly,
d^y
df-idf-'^[dt)r'''^^[^dtdt+''df^'''^-
These are the accelerations parallel to x and y. And
since, by § 15, the acceleration along the radius vector is
d'l/ . ^ , d-x
^,sm^ + ^,cos^,
the above expressions give it in the form
df '^[dtj '
The acceleration perpendicular to the radius vector is
df
sin 6,
that
^^drdd d^
^ di dt'^'^ df-
Id f ,de\
which may be written - 37 (^"^ jt) •
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
dt~dsdt'
, d'x _ d^x /dsV dx d^s
df~ d? [dij '^ d^de'
similarly, -^^ = <& f-^^Y + ^^ g ,
•^ dt' ds'\dtj ds df
d'^z _ d^z fds\^ dz d's
d¥ ~ ds" [dtj '^chdi''
Remembering the law of resolution of acceleration, the
form of these equations shews that in them are resolved
d's
along X, 11, z, 1st an acceleration , , , whose direction-cosines
dx dy dz 1^.1 i x- 1 i<^^\^ 1
are T- , -r- , -j- , and 2nd an acceleration - { -r 1 > whose
ds ds ds p \dt/
,. . . d^x d'u d^z , • 1-
direction-cosmes are p -^-7^, p -rj , p -y-^; wiiere /3 is a linear
quantity, which will be presently recognized as the radius of
curvature of the path. This process might have been em-
ployed 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
niultijily these component accelerations by -^ , / . ' T *
respectively, and add. Thus the tangential acceleration is
dx d-x dy dy dz d'z _ d^s ^ dv
ds de '*' ds dt' "^ dsdi' ~ dtr ~ dt '
KINEMATICS. 13
as we have already seen. Also in tlie normal, towards the
centre of curvature, we have the acceleration
fdsV
im
-m
p\dt)
p'
We assume here the following equations from Analytical
Geometry,
p' W) [dsO ■*" W
where p is the radius of curvature, whose direction-cosines
are
d'^x d'^y d^2
Pd?' Pis'' Pd?
and
fdxV /dy\^ fdzV ,
whence d_x (Tx dy d^ dz d^
^^^^^ ds ds' ^ ds ds' ^ ds ds' ^ '
* The accelerations of the moving point may be found in the following
manner. There is obviously no acceleration i:)erpendicular to the osculating
plane, as that plane contains two consecutive directions of the point's motion.
Of the two consecutive dii-ections let the first make an angle d with any fixed
line in the osculating plane, then v cos 6 and v sin 6 are the velocities of the
point parallel and perpendicular to the fixed line respectively. Consequently
— [v cos 6) and — (r sin 6) are the accelerations in the same directions. These
dv dd dv '"'
expressions, whenexpanded, become -.cos 6*- r sin ^— , and — siu^+rcos^
dt dt' dt at
dv
dt
Therefore the accelerations along the tangent and the normal are -- and
V - , the last being positive in the direction of the centre of curvature. Since
— = , the normal acceleration, being =v t- . t: > J^iay be expressed as — .
ds p ° as dt "^ p
14
KINE5IATICS.
thus
19. We might have treated the component accelerations
(Yi) + ( ~r^ ) or (resultant acceleration)'
~ pAclt) [de,
by adding the squares of their values as given in § 17
d's
!
1 /d,<
Now ^ is the acceleration along the tanfrent, and the
other part
O'
or - , acts at right angles to it as the
p \ai/ p o o
form of the equation shews, and consequently is the accelera-
tion perpendicular to the tangent.
„ ,, • r. d^^^ dSi (fz
From the expressions lor -j-^ , -~ , -j-^ ,
we also obtain
d^x fdy d\
'dt'\d.
dzd^\
\^ds ds^ ds ds'J
d'y /dz d'x dx d^z
~df \ds ds
dx^
dJ^z /dx d^y _ dy d^x \ _
"^ dfKd's d? ~ d^ ds^J ~ '
which may be written in the form of a determinant
d'x
df
d'y
df
d'z
df
=
dx
ds
dy
ds
dz
ds
d'x
dy
ds'
d'z
ds'
This signifies that the Resultant Acceleration lies in the
plane containing the tangent and the radius of absolute cur-
vature, or that there is no acceleration perpendicular to the
KINEMATICS. 15
osculating plane. The acceleration - must tlierefore be along
a normal to the path drawn in the osculating jjlane ; 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
00, 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 constniction.
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 ahruptly either in direction or magnitude. Now if Q be
a point near to P, OP and Q 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 becomes 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.
16 KINEMATICS.
The amount of acceleration is the rate of change of velo-
city, 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 defined later.
Consider two component velocities, AB and AC, and let
AD be their resultant (§ 10). Their half moments round
the point O^are respectively the areas OAB, OCA. Now
OCA, together with half the area of the parallelogram CABD,
is equal to OBD.. Hence the sum of the two half moments
together with half the area of the parallelogram is equal to
A OB together with BOD, that is to say, to the area of the
whole figure OABD. But ABD, a part of this figure, is
equal to half the area of the parallelogram ; and therefore the
remainder, OAD, is equal to the sum of the two half mo-
ments. And OAD is half the moment of the resultant velo-
city round the point 0. Hence the moment of the resultant
is equal to the sum of the moments of the two com]»<»nents.
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 thiongh 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 i^oint the path is plane
and the areas described by the radius-vector are proportional
to the times ; for tlie 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 just 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 oi x, y; and let the origin be the point about
which moments are taken. Then if x, 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 ^dd .
polar CO
■ordinates.
From this we have at once
ds du dx
' df
•(1)
or with the notation of § 8,
pv = xVy-yv^,
which is the theorem of § 19.
Also |^(j„) = x^f-
d'x
..(2).
T. D.
2
18 KINEMATICS.
Now, if the acceleration be dii-ected to or from 0, its
moment about 0, Avhicli is evidently
^ de y df '
must vanish. Hence (2) gives
pv = constant, which is § 23.
By means of (1) this gives
r^ -,- = constant, which is § 22 ;
since, if A be the area traced out by the radius-vector,
dA^r^
de 2 •
25. To determine the motion of a point luhen 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, y8, 7 be the components of the given acceleration ;
we have
d'x
df '
df ^'
d'z
•d)
Now a, 13, 7 may be functions of oc, y, z, t, ^ , -,- , or -j- ,
dt dt dt
or of two or more of these quantities. ■ Equations (1) must
be integrated as simultaneous differential equations if possible.
KINEMATICS. 19
^, , . . , ,1 1 ,. dx dy dz
Thus by one integration we nave the values oi ~j1'~ji^~j±i
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 x, y, and
z, in terms of t\ and, if the latter variable be eliminated from
the three integrated equations, we have the two equations 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. When 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 de-
pends on the following self-evident proposition.
If the velocity of any j^oint of a system he reversed in
direction, and be communicated to each point of the system in
composition zuith that which it already possesses, the relative
motions^ of all about the first, thus reduced to rest, will he
the same as their relative motions about it luhen all tuere in
motion.
For the proof it is sufScieut to notice that if at every
instant the distance of two points, and the direction of thf-
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.
2—2
20 KINEMATICS.
27. Two 2>oints describe similar orbits about each other
and about any jJoint dividing in a given ratio the line luhich
joins them.
Let A and B be the points, G a point in AB sucli that
—y-7T = a constant.
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 G be fixed, the path of B about it will evidently
differ from that of B about A by having corresponding radii-
BC
vectores diminished in the ratio r^ . But this is the defi-
AH
nition 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 free 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 x, v its
velocity, the plane of the circle as tliat of av/, a the radius of
the relative circular orbit, to tlic 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 to the axis of .r. Then for the co-ordinates of
the revolving point we have
y = a sin 6,
.X = vt + a cos 0.
KINEMATICS. 21
But = cot;
V 11
hence x = — sin"^ - + \J {a^ — if)
CO a J ' .
is the equation of the absolute path required. This belongs
to the class of cycloids ; it is prolate or curtate according as
V is greater or less than aw, 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 w = aw, 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
r, Q denote the relative co-ordinates of the second point with
respect to the first at time t, x, y, and a; the absolute co-ordi-
nates at the same time,
x = x + r cos d\ , .
y = rsme ] ^ ^•
Now in the first case, when the motion of the first point,
and that in the relative orbit are given, x, r, and ^ are known
functions of ^ ; 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, h that of BC, r that of ^C;
and at time t let them make angles >, ;j^, 6 with some fixed
line in the plane of motion. Then
r cos 6 = a cos > + 6 cos ;!^,
r sin ^ = o sin ^ + i sin x-
But ^ and ^ increase uniformly. Hence
nt + -
/)
Therefore
KINEMATICS. 23
r = 2a cos ( mt -\ — ~
and this denotes vibratory motion in a definite straight line.
31. In any system of moving points, to determine the i^ela-
tivefrom the absolute motions; and vice versa.
Let x^, 2/j, z^ , x^, 3/2, ^2 t>e the co-ordinates of two of the points,
./", y, z the relative co-ordinates of the second with regard to
the first, u^, v^,w^, u^,v^, lu^ the velocities of each parallel to the
axes, u, V, lu the velocities of the second relatively to the first.
Then x = x^ — x^, u = u^ — u^ ,
y = y',-y,> v = v,-v^,
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 xy z,uvw, x^ y^ z^,
u^ v^ lu^, 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 sub-
ject 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.
24
KINEMATICS.
33. As an instance of the curve of pursuit, suppose it be
required to determine tlie i^ath of a point P which continually,
with constant velocity u, moves towards another point Q which
is describing a straiyht line ivith constant velocity \.
The curve of course is plane. Take the line of motion
of the second point Q as the axis of x, and let x denote
its position at the instant when the co-ordinates of the first.
P, are cc, y. The axis of y is chosen as that tangent to the
curve of pursuit which is pei'pendicular to the axis of x, and
the distance between the points in that position is a.
Let - = e, then by the conditions of the problem we have
eAP=OQ,
and PQ & tangent at P.
Expressed analytically thes6' lead to the following equa-
tions ;
_ dx
-^ dy
The mode of solution is precisely the same whether x or y
be taken as independent variable : but y is to be prefeired as
it leads to less cumbrous expressions.
Differentiating therefore with respect to y, we have
ds
dra
'dy- Uy^
KINEMATICS. 25
But s increases, as y diminishes,
whence | = _yjl + g)'
Hence ' ^ ^
' ^/h(i)}
Integrating, and noting that y= a, -^ = 0, together,
-- (!yv{^-(i)i4;^
and therefore, taking reciprocals,
dy
(^y=v/h(i)i "
Subtracting, we have finally
2$=f^Y-(^! (1),
or 2 (« + C)
But x = 0, y = a, together ; which gives C =
1
Hence 2 x+^]= J ,, + ^, f ,, (2).
V e'-iy ^"(6 + 1) 2/(^-1). ^ '
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<\.
26 KINEilATICS.
In the first case Q moves the faster, and P can never over-
take it ; the curve therefore never meets the axis of oc, which
indeed will be seen by (2) to be an asymptote.
In the second case equation (2) becomes
ae \ y""' a'y"
l-eV o'(l + e) 1-e
and for a; = ^ we have y = 0, and also by (1) ;7- infinite.
Hence the cui-ve touches the axis at this point. The re-
mainder of the cui-ve satisfies an obvious modification of the
question, whence it is called the Curve of Fligld. -lit is to be
observed, however, that x = ^ gives also y = ±a (- jY.
The distance between P and Q, being
J{x-xf+y\
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>l, this expression is infinite both for 7/ = oo and
for 2/ = 0. The minimum value is easily found to be
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
2f. + «) = :5''-a!og».
V 4/ 2a ° a
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
^ 2a^2'
which shews that the limit to which the distance tends is ^.
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
velocitij u, starts from a point A in the hank of a river which
runs with velocity v parallel to Qx, and tends continually to
the point Q, on the other bank, directly opposite to 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 iu
the last example. (See § 26.)
To investigate the path, take Q as origin, Qx, QA as the
axes. Then the component velocities of P are v parallel to
Qx and u along PQ, and the tangent to its jmth is in the
direction of the resultant of these two. Putting 6 for PQx,
we have -v: = ^ — w cos 6, and -n = — u sin 6,
at dt
whence
dy _ u sin d _ sin ^
dx V — u cos 6 e — cos 6
y
e \l [p? ■\- y") -x'
This, being a homogeneous equation, is easily integrated
and we have, taking a; = 0, y = a, as co-ordinates of A,
^^ = ^{x'^ + f)-^ (1).
y -a y
/r sin ^y _ 1 — cos 9
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 iu a definite
ratio to that in MP produced ; why is not the path alivays
a conic section of which Q is a focus ? Tlie 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.]
KINE>IATICS. 29
36. To find the time of crossing tJie stream.
This may easily be effected by considering the actual
velocity parallel to the axis of y :
du . ^
-^ = — w sm ^
dt
Now taking quotients of y^ by both sides of (1),
a^y^~" =\/{x' + if) + X.
Hence 2 V («' + 2/') = a^y"-' + a'f^' ;
and therefore ^ {i^f-" + dTf^'^) = - 2udt.
Taking the integi'al from a to 0, and putting T^^ for the
time of crossing,
^ s = uT, ; or r = -, ^r .
But, if there had been no current, we should have had for
the time of crossing,
To = - ; whence -^ = — ^ .
In the integTation we have, of course, e < 1, else the boat
could not reach Q.
If e = 1, the boat will 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 in-
crease of the angle made by the line joining the two points,
with some fixed line in the plane, is called the Angular Velo-
city of the former point about the latter. Unit of angular
velocity corresponds to the description of an arc equal to
radius in unit of time.
80 KINEMATICS.
Suppose the above-mentioned angle to be represented
by 6 at time t ; then at time t-\-ht it has the value 6 + ^6,
and it may be shewn as before (§ 7), that if g> 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 angidar 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 9.
and
Also, if ^ = 0, Avhen ^ = 0, we have
r sin 6 = vt.
Hence, a tan 6 = vt,
_d6 _ va _ va
'^~"dt~ a^+^'~7
38. A point describes a circle luitlt consta7it velocity; it
is required to find the actual velocity, and the angular velocity
[about the centre) in any oithographic projection.
Let ApA' be an ellipse and A PA' the auxiliaiy circle.
Then the former mil 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 AA', P and p will be corresponding j)oints
in the two. Draw the tangents pT, PT ; then
actual velocity at » pT ^ -r rr,,-n /i
p = prn • f^U J if ^OP = 6,
velocity at j? _ ^/{PT^ sin'^ + PT^ cos'^ cos'a)
P~ PT
= V (sin" 6 + COS" 6 cos'^ a)
= V(l — sin'"acos"^).
KINEMATICS.
31
Now, if TOp = i>, " ^"^;^^"^,"^3 = fe
J-: tan ^ (cos a tan i
do
cos"'' 6 + cos^ a sin'' 6
cos a
This is a maximum if 6 ■■
when its value is sec
minimum ... =0 cos a.
Hence, if w^ and m^ be the greatest and least angular
velocities in the projection,
\J{w^oi^ is the angular velocity in the original path.
39. Evidently, the product
angular velocity is the velocity
vector. (§ 11.) This is to the
dicular on the tangent is to the
the product of the square of the
velocity is equal to the product
perpendicular on the tangent, i
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.
of the radius-vector into the
perpendicular to the radius-
whole velocity as the perpen-
radius-vector ; and therefore
radius-vector by the angular
of the whole velocity by the
.e. to the moment of velocity
32 KINE^LA-TICS.
Thus, in the Ex. § 37, the angular acceleration is
rfct) _ 2va dr _ '2.v^a , ^ ^
dt ~~~r^ dt ^n/''-«-
41. The motion of a point in a jilane being given with
respect to fixed axes, to investigate expressions for its velocity
and acceleration relative to axes in the same plane, which re-
volve about a common origin luith constant angidar velocity.
Let ft) 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 (ot. Hence, if x, y, ^, rjhe
the co-ordinates of the jjoint at time t, refeiTed to the fixed and
to the revolving axes resiDectively, we have by the ordinary
formulae for transformation of co-ordinates
^ = X cos cot + y s'm cot] . .
?7 = y cos cot — x sin cot}
These give, by differentiation,
d^ dx J. dy . ^ , . -
'T. — j7 cos ci)t + -j- sm cot — co{x sin oot—y cos cot)
dx ^ dy .
= J- cos cot + ,7 sm cot + COT],
at at , -.^v
Similarly, -^ — ~- cos wt — ,- sin cot — w?.
^' dt dt dt ^
which determine the velocities relative to the revolving axes.
Again,
d^^ d^x ^ d^y . _ [dx . dy \ -.^1
-J- = ~fYCOScot+-~ smcot — 2co\ -,j sm cot — -^^ cos cot j-co't
df dv dt"- \dt dt J * I ,„.
d'T] dSi d'x . ^ . (dy . ^dx \ ' ^ ^'
-jT = -nr COS cot --r^, smcot — 'Zcol ,- sm&)^+-,- cosco^ —
dt' dt' dt' \dt dt
,C0t]—C0''7}
d'^ d'x d'y . „ dn .,-
dt' = If '°' ""^ -^ d? '''' "' + ^"^ dt + ""^
d'r} d'y ^ d'x . ^ ., d^ .,
-y^ = 7-T- cos cot - , ., sm cot — Jco -y: + CO y
dt df dt' dt
the relative accelerations.
(3'),
KINEMATICS. 83
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
f-2.^J-„',..d§+2.f-.V (4,
Ex. If the point be at rest, x and y are constant, and
Also J = _«f, - = _.,.
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 ^.
42. Suppose the new axes not to revolve uniformly.
In this case the investigation is precisely the same as the
above, with the excejDtion that 6, a given function of t, must
be substituted for wt. If to, 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.
.(2J.
dP dx r, dy . ^ 1
dr) dij a dx . I
-y: = ir cos ^ — ^ sm d — cotl
dt dt dt ^J
d^^ d-x a , d^y . ^ „^ - dr] dco ■)
dv dt' df ' dt dt ^
W-'^^—rjJ^''^^' df-'^'^^^dt^''^^ (^*^^-
T. D. 3
.(3\).
34 KINEMATICS.
These expressions might have been deduced at once from
the expressions in § 16, by the consideration of rehitive accele-
rations as in § 26. Let 0M=^, MP = tj, be the co-ordinates
of the point referred to tlie moving axes. Then, by § 16, the
acceleration of M along OM is
Also, as il/P revolves Avith angidar velocity w, the ac-
celeration of F relative to M, in the direction perpendicular
to MP, is
This is in the direction of the negative part of the axis of
^. Hence the resolved part parallel to 0^, of the accele-
ration of P with respect to 0, is
43. The principles already enunciated, and the examples
given of their application, will suffice for the solution of pro-
blems 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, ami its
velocity at any instant is proportional to tiio time elapsed
since its motion commenced ; liud 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 sec 6.
(11) A particle, projected with a velocity u, is acted on by
a force, which produces a constant acceleration y^ 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
_ ^ ( g'rCOta_ Y
/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 o( & 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 § 34 find the greatest distance
the boat is carried down the stream, and shew that when
it is in that position its velocity is \/{ii^—v').
When u = v, shew directly that the curve described is a
parabola.
^ (20) Shew that if p be the radius of curvature of the
curv^e of pursuit, we have in the figure of § 33,
_^:
P~ePM'
(21) In the case of a boat propelled with velocity w
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. t
(22) The velocity of a stream varies as the distance from
the nearest bank ; shew that a man attempting to swim di-
rectly across will describe two semiparabolas. (Shew that the
sub-normal is constant.) Find by how much the mean velo-
city 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.
(2.5) 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.
(2G) 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 a^ 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, 7] 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
d'^ , dv
—r^ — zco cosec cot ,, ,
dt dt
d'^V 2 , o + /^^
-T ., — 03 71 + la) cot (ot ,, .
dt dt
(29) Two lines are moving in their own plane about
their point of intersection with constant angular velocities
ft), w ; if the co-ordinates of a moving point referred to them
be iT, 3/ at a time t, prove that its accelerations parallel to the
axes are
d^x , „ ^ , , X ^dx ^ , , , ^ dxj
,,, — (ji'x — 1(£) cot {w — (o) t , — 2u> cosec [oi —CO) t -,'- ,
dt- ^ ^ dt dt
-~—co"y — 2£u cosec (&>'— co) t ,- - 2a)' cot {co' — co) t ■ •
(30) Employ the formuhc of § (30) to trace approximately
the form of the path of G about A, when m is nearly, but not
exactly, equal to + n or to — ??.
(31) If an odd number n of rods OA^,A^A,^, A„A ^,. . .whose
lengths are a,-, ,...-, are hinged together at ^1 ,, ^1 ,,, . . .and
KINEMATICS. 39
revolve with constant angular accelerations a, 2^, 3a,...?ia,
about their extremities O^^A^,...A^_^, shew that the direction
of motion of the point A^ 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 b from a fall of water ahead. If the velo-
city of the stream be V, prove that he cannot escape the fall
unless he can row with a velocity - V; 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 rej^resent the velocities of the centre of cur-
vature and the centre of the generating circle corresponding
to the position of the point, prove that
{c-bf "^ {c + bf ~ {c-by '
c being the distance between the centres of the generating
circles, and b the radius of the moving circle.
(3-i) iV 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 IT
- cosec ^, .
(8.5) A point P moves with constant velocity in a circle;
^ 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
byPQ.
(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
nuignitude {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 :
() is a fixed point on the major-axis and FG the normal at
P. Shew that at the moment when G coincides with Q, the
angular velocity of P about Q is to its angidar velocity about
G as CD' to CB\
(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 w 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
dv dQ)\ , dr
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'-
the normal to the surface = — h ~, •
P P
(40) The intrinsic equation of a curve being s = f((f>),
the curve is described by a point with accelerations Xl^ pa-
rallel to the tangent and normal at the point for which ^ = ;
prove that
KINEMATICS. 41
(41) Obtain expressions for the accelerations of a moving
point whose co-ordinates are r, 0, ^, (1) in the direction of r,
(2) in the direction perpendicular to the radius vector and in
the plane of 6, (3) in the direction perpendicular to the plane
of 6*.
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 beincr 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 circumstances of motion of a Particle ; and of the
experimental laws, on the assimied truth of which all our
succeeding investigations are founded. And it is 'obvious
that Ave 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
conse({uences.
45. The Quantity of Matter in a body, or tlie Mass of
a body, is proportional to the Volume and the Density con-
jointly. The Density may therefore be defined as the quan-
tity of matter in unit volume.
If M be the mass, p the density, and V the volume, of a
homogeneous body, Ave have at once
M=Vp;
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 Iviperial Found of matter.
46. A Particle of matter is sup])osod to bo so small that,
though retaining its material properties, it may be treated, so
LAWS OF MOTIOX. 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 DC, the change of velocity is represented in magnitude
and direction by A G.
49. Rate of Change of Momentum, or Acceleration of Mo-
mentum, is proportional to the mass moving and the accelera-
tion of its velocity conjointly. Thus r§ 17) the acceleration
cPs
of momentum of a particle moving in a curve is i/yg along
the tano-ent, and M - in the radius of absolute curvature.
P
50. The Vis Viva, or 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 velo-
city 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 /Mv*\ -.^ dv f ^rd's'
^- i(f)=^4:-K
44< LAWS OF MOTION.
52. Matter has the innate property of resi.sting 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 mo-
tion 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 tJie 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,
[According 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 falling bodies.
This definition, giving a varying and a very unnatural unit
of mass, is exceedingly inconvenient. In reality, standards of
weight are fiiasses, not forces. They are employed primarily
in commerce for the purpose of measuring out a definite quan-
iity of matter ; not an amount of matter which shall be at-
tracted 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 etfect 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
322 times the force which, acting on a pound for a second,
would generate a velocity of one foot per second ; in other
words, 82'2 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 directicns, 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 sliall 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 com-
ponent. The remaining component in this case is perpen-
dicular to the other.
It is very generally convenient to resolve forces into com-
ponents 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
4G LAWS OF MOTION.
in directions at right angles to each other, is the square root
of the sum of their squares.
58. The Centime of Inertia or Mass of any system of
material points M'hatever (whether rigidly connected \dt\i
one another, or connected in any way, or quite detached),
is a point Avhose 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 m^, m^, etc., whose distances from the plane are
x^, x^y etc., is therefore
And, similarly, for the other co-ordinates.
Hence its distance from the plane
S = Xa; + /LIT/ + j;^ — a = 0,
is D = \x -\- fiy + vz — a,
_ S {m (Xx + fiy + vz — a)] _'S (mS)
as . stated above. And its velocity peri:)endicular to that
plane is
^( d8\
dD 1 ^ f frlv (hi dz\\ -^K'^'dt.
-dt^x^^X'T'dt-^^tt-^'dtJi — ^;;7-'
from which, l)y multiplying by Xm, and noting that 8 is the
distance of x, y, z from 8 = 0, we see that the sum of the mo-
menta of the parts of the system in any direction is equal to
the momentum in that direction of the wdiole mass collected
at the centre of mass.
59. By introducing, in the definition of moment of velo-
LAWS OF MOTION. 47
city (§ 21), the mass of tlie 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 sub-
stitution 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
\scls.
taken between limits corresponding to the initial and final
positions.
48 LAWS OF MOTION.
E-eferrcd to rectangular co-ordinates, it is easy to see, by
the law of resolution of forces, § 67, that this becomes
/(
„ dx -^ dii „ f?2\ ,
Z ,-+ F~i + Z ,-]ds.
as ds 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 whicli 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 wdiich 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
ds ds ds
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 tlie former case
by the amount of work so done.. In virtue of this, however,
the body possesses an equivalent in the form of Potential
Energj/, 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 re(|uiros work to com-
LAWS OF MOTIOIT. 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 uni-
versally 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 nu-
merical measurement of time practically rests on defining
equal intervals of time, as times during ivhich 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,
T. D. 4
50 LAWS OF MOTION.
as a piirely geometrical quantity, and have seen how it
may be at once inferred from the given initial and final velo-
cities of a body. By the definition of motion, or quantity of
motion (§ 47), we see that, if we multiply the change of velo-
city, 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 : —
WJien any forces luhatever act on a hoch/, then, ^vhether
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 tlie 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 resultaid 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
LAWS OF MOTION. 51
forces acting at the same point, and the Polygon of Forces for
the resultant of any number of foixes 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 iwoportional 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 ichich 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 jDarticle.
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 toivarcls the centre of curvature ; their
magnitudes being the acceleration of momentum, and the
product of the momentum into the angular velocity about
4—2
52 LAWS OF MOTION.
the centre of cui-\-ature, respectively. In the case of uni-
Ibrm 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
d'x ,. d'y 1^ cVz „
7?z , „ = A, m , f = Y, m -,-, = Z.
df ' dt' dt'
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
m-^ = mX, &c.;
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 oi four linear dimensions. The fact, however, is that
an equation such as
d^x ^
5? = ^
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 A' is the
a;-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
d'x . „
df ='"'''''•
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
alivays 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 op-
posite pressures. When neither of the two bodies has any
rotation, whether before or after impact, the changes of velo-
city 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 tlirough 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
beh)nging 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 fVijjiL withojut. [This principle is
sometimes called Conservation of Areas, a very misleading
designation.] These results will bo deduced analytically in
Chap. XII.
73. Wliat precedes is founded upon Newton's own com-
ments on the third law, and the actions and reactions con-
templated are the pairs of forces, of which each pair consti-
tutes a " stress." In the scholium appended, he makes the
following remarkable statement, introducing another speci-
fication of actions and reactions subject to liis third law : —
Si oistimetur agentis actio ex ejus vi et velocitate conjunc-
tim ; et similiter resistentis reactio lestimetur conjunctini ex ejus
partium singularum velocitatibus et viribus resistendi ah earum
attritione, cohaisione, 'pondere, et acceleratione oHundis ; erunt
actio et reactio, in omni instrumentorum usu,sihi invicem sem-
per 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 Action of an agent be measured hj/ its amount and its
velocity conjointlj ; and if, siniilarli/, the licaction of the resist-
LAWS OF MOTION. 55
ance he measured hy the velocities of its several parts and their
several amounts conjointly, tohether these arise from friction,
cohesion, tueight, or acceleration ; — Action and Reaction, 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 sys-
tem. 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-jwiuer," or the
rate at which an agent works when overcoming 33,000 times
the weight of a pound throvigh the distance of a foot in a
minute ; that is, producing 550 foot-pounds of work per
second. The unit, however, which is most generally conve-
nient is that which Newton's definition implies, namely, the
56 LAWS OF MOTION.
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 Avork
originally expended (§ 02). But in Newton's day, and long
afterwards, it was supposed that work was ahsolidebj lost by
friction.
76. If a system of bodies, given either at rest or in mo-
tion, 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 sura 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 Avill do work, or will con-
sume 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 botlios always ])er-
form, or always consume, the same amount of work during
any motion whatever, by which the system can p:iss from one
particular configuration to another: so tliat each configuration
corresponds to a dcfiniti> ainotnit of kinetic energy. [For the
apparent violation of this by friction, impact, vS:c., see § Tii*.]
LAWS OF MOTION. 57
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 snch a system, in the
configuration which it has at any instant, is the amount of
Avork 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
^/G-
as as dsj
is independent of the paths pursued from the initial to the
final positions, and therefore that
t {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
X =- —
^ dx^ '
Also, by the second law of motion, if m be the mass of
a particle of the system whose co-ordinates are x, y, z, we
have
m, -r.,^ = X, , &c. = &c.
^ dt ^
= -dV.
The intecrral
^l.{nvf)+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 theoiy 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 agenc3^ 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 e)ier(j>/," Sec. in-
stead of the unnatural "Force-functions,'' Sec. which ilisfigure
some of the modern analytical treatises on our subject.]
( 59 )
CHAPTER III.
EECTILIXEAR ilOTIOX.
79. The simplest case of motion of a particle whicli 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 OMy forces, vjliose resultant is in that line; to determine
the motion.
Let P be the position of the particle at any time t, f the
resultant acceleration along OP, being a fixed point in the
line of motion.
Let OP=x, then the equation of motion is (by § G9)
d-x _ .
df ~^'
dx
In this equation / may be given as a function of x, of ~r. >
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 -y- and x in terms of i; that is, the position
and velocity of the particle at any instant will be known.
The only one of these cases which we will now consider
is that in which f is given as a function of x ; those in which
/' is a function of ^ , or of -,- and x, being reserved for the
•' dt dt ^
GO RECTILINEAR MOTION.
Chapter on Motion in a Resisting Medium : while those in
which / 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 cir-
cumstances 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 hy a constant force in the line of its motion;
to determine the jJosition 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
*=/ •
Integrating again,
1
= C'^-Vt^%ft\
RECTILIXEAR MOTION. 61
But -when t= 0, x = a; hence C = a, and
x = a+Vt + ^ft\.... (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
IfdxV 1 , „,, .
But when x = a, v = V ; hence C" = -^ — fa, and
^v' = ^V'+f(x-a) (4).
82. The most important case of the motion of a particle
under the action of a constant acceleration in its hne 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 vertically downwards, equations
(2), (3), (4) of §81 become
v = V+gt 1
^=^a+Vt + ye I ^^^^
-v^ = ^V'+g{x-a)
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.
02 RECTILINEAR MOTION.
Hence r =gt (1),
^0 = 10^" (2),
\v'' = 0^ (^)-
The last of these equations may also be obtained from
d'^x dv dv dx dv
^ df dt dxdt ^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
d'x
df^-^'
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),
^^Vt-lfff (5),
lv^ = lv^-gx (6).
From equation (4) we see that the velocity continually
V
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 -y . After this the velocity
becomes negative, or the particle begins to descend, and
(5) sliews that it will return to the point of projection when
2V
t=--,a.sx then becomes 0; ami the velocity with which
RECTILINEAR MOTION.
63
it returns to that point is, by (G), equal to the velocity of
projection.
84. A particle descends a smooth inclined plane under
the acceleration of gravity , the motion taking j)lace in a vertical
plane peyyendicidar 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
in 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, g sin a along, and g cos a
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'x
77^5 =5' sm a,
the solution of which, as g sin a 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 got from equations (1), (2), (3) of
§ 82 by this means,
v=g ^ma .t (1),
X = Ig ?AXi a . f (2),
^v''=g?>ma.x (3).
85. Equation (3) proves an important property with
regard to the velocity acquired at any point of the descent.
For, draw PiV parallel to AB, and let it meet the vertical line
through in N, then if v be the velocity at P, we have
-^v" = g sin a . OP
= g.OX.
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 tlie conservation of energy.
86. Again from (2) we derive immediately the following
curious and useful result,
TJie times of descent doivn all chords d)'aiun through the
highest or lowest point of a vertical circle are equal.
Let AB hQ the vertical diameter of the circle, A G any
chord through A ; join JBC; then if T be the time of descent
down A C, we have by equation (2) of § 84,
AC =hgr cos BAG.
Bwt AG = AB cos BAG; whence
AB=yr\
,_ /2AB,
EECTILIXEAR MOTIOX. 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 BG. Let P be the point of
contact, then AP is the required line. For, if we take any
T. D. 5
66 EECTILINEAR MOTION.
other point, p, in BC, Ap cuts the circle in some point q, and
time down A}^ > time down Aq, i.e. > time down AP.
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
= R-gcos'x. (§G9).
If /x then be the coefficient of kinetic friction, -which is
known by experiment to be independent of the velocity of the
particle, the retardation of friction will be fjuR or /x^ cos o,
and the equation of motion will become
^=g&ma-fMffcosa,
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 ofthejiarticlefroin
a fixed p)oint 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
/JiJC
if /i, 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 /xa', and if it be directed towards will tend to
diminish x. Therefore the equation of motion is
d'x
RECTILINEAR MOTION. 67
or ^ + ixx = (1).
Multiplying this equation by -^ , and integrating, we
obtain
i(sy=2^(»'-') (^).
the equation of energy. This may be written
^-__1 1 .
dx ~ ^JjM sj{a^ — x^) '
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,
VyU, it — T) = cos~^ - :
or X = acosWfj,(t — t) (3),
the complete integral of (1) ; involving two arbitrary constants
a and r, the values of which are to be determined from the
initial distance, and the velocity of projection. Thus from (3),
-r: =v = — ^/xa sin ^/ fj,{t — T) (4).
89. Suppose the particle to be projected from A in the
positive direction with the velocity V, and let OA = b ; then
when ^ = 0, we have x = b, v = V ; and therefore from (3)
and (4)
b = a cos VytiT,
V — a^/ [M sin J fiTy
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
£c = a, we have v=V; whence V^ = /x {a^ — 6^), and
v-^^V' + f^ia'-x')
5—2
68 RECTILINEAR MOTION.
90. Equations (3) and (4) give periodical values of cc and
V, such that all the circumstances of motion are the same at
27r
the time t + ^ as at tlie time t. They also shew that the
velocity becomes zero when i = 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 VyLt(^ — r) = 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 leav-
ing any point until it passes through it again in the same
27J-
direction, beinsj - - . 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 Oxi, and
let ^ 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
df ^^'^ ^> df '
which is the same as when the point is at rest if — = 0,
i.e. if the velocity of be constant. If move with constant
acceleration, ot, the osciHatory motion will be the same as be-
fore, 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
d'x
the integral of which is knowai to be
and the motion is not oscillatory. If, when ^ = 0, x = a,
V = — a \Jfjb, the particle constantly approaches the centre but
never reaches it.
93. It is to be remarked that we cannot always apply
the same ecpiation 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, fxx, 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^^
power of the distance, the equations of motion for the posi-
tive and negative sides of the origin are respectively
d'x
and -^ = -/j,{-x).
The only cases, therefore, in which the same equation of
motion will apply to both sides of the origin, occur when n is
2m 4- 1
of the form ~— , — ^ , where m, m' are any whole numbers in-
eluding zero, since it is only in these cases that we have
- (- xT = x'\
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 ^^^-— ^ — z^ it is
2771 + 1
necessary to consider these parts separately, because the form
of the integral is not sufficiently general to include both.
This is when m = and m'= — 1, for in that case the equation
of motion becomes
d'x _ fi
de~~x'
dx
Multiplying this by -,- and integrating we have
which becomes impossible when x is negative. But it is evi-
dent that we may then write the integral
i(iy=c-Miog(_.).
which is, of course, the proper form for the negative side of the
origin. These equations cannot generally be integrated far-
ther, but we will shew towards the end of the Chapter how
the time of reaching the origin may be determined.
95. A particle, comtrained to move in a strairjht 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.
^LiQiAB 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 AB, and let
NP = x; then if 0P= r, and if fx as formerly be the attraction
at a unit of distance, the attraction on i* along PO is ^r. This
may be resolved into two, one along and the other perpen-
dicular to AB, of which the latter has no eftect on the motion
EECTILIXEAE MOTIOX. 71
i of the particle. The equation of motion is, therefore, since
, the acceleration is jxr cos OPN or fMPJS^,
df ~ ''^'
the same as in § 88. The motion of the particle will there-
fore be oscillatory about JS^, the time of a complete oscillation
2
being -r- , and all the circumstances of motion the same as
° V/^ . ,
for a free particle moving in AB under the action of an equal
centre placed at N.
96. A imrticle moves in a straight line under the action
of an attraction alivays 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
time t, OP=x; the equation of motion is
de'
fj, being, as before, the acceleration at unit distance from 0,
or the strength of the centre.
dx
Multiplying by ~J- and integrating, we get
etc
72 RECTILINEAR MOTION.
the equation of energy, supposing the particle to start from
rest at a point A distant a from 0,
Avhieh gives the velocity of the particle at any distance x
from the origin. Again from (1)
'x I , la — X
the negative sign being taken, since in the motion towards 0,
X diminishes as t increases. This gives
dt _ / ft X
dx~~y Yfji' ^{ax-x^)
_ / a {\ a — 2x a 1 |
Integrating, we have
vs^f
,, .,, a ' _,2j-|^
V {ax — x) — -^ vers - -Y .
^^ /2ll^ ,, ,. a _i2.r 7ra
Hence . / -^<= v(aa; — a?) — ^ vers •" 9 >
which is the relation bctw^een x and t.
97. Putting x= 0, we find that the time of arriving at
Ois
TT /a'
2 V 2/A
RECTILINEAR MOTION. 73
and (1) shews that the velocity at is infinite. On this
account we are ^ precluded from applying our formulse 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, ap-
proaching 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 a; be the distance of a body from
the Earth's centre, H 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 /x for gW. The re-
sults 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 [ji = gR^,
2 -^ \x E + k
and therefore if F be the velocity ■whenx=B, i.e. the re-
quired velocity,
If h be small compared with R, this may be written
lv"-=gh(l-^^ + &c),
74f RECTILINEAR MOTION.
from which we see the amount of error introduced by the
ordinary formula, § 82,
If the fall be from an infinite distance, a = oo , and we
have
V'=gR.
Expressed in terms of the radius and the mean density of
the Earth, this becomes
1 „2 _ 47rp 2
2 3'
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 hy an attraction, always directed to a point
outside that line, and varying inversely as the square of tJie
distance from that point ; to determine the motion.
Let ABhe the straight line, P the position of the particle
at any time, the point to which the attraction is always
directed, /u,the strength of the centre. Draw OA'perpendicular
to AB and let ON=h,NP = x\ then the attraction on P
along PO is -Jy^ji > ^i^^> as in § 95, the only part of this
which produces motion is the resolved part along PN. There-
RECTILINEAR MOTION,
fore the equation of motion is
Multiplying by -^ and integrating, we have
.(1).
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
x^
being so small that throughout the motion ^ may be neglected
oc
in comparison with j . We have from (1),
cT'x _ IJ'X f x"\~^
df~~T'[^b\
nearly ;
the same form of equation of motion as that of § 88. The
motion -svill therefore be oscillatory, the time of each small
oscillation being 27r
-f(l
2 6'^
&c.
fix
drx
lix
V a'
76 EFX'TILIXEAE MOTION.
101. A particle moves in a straight line under the action
of attraction varying inversely as the n'^^;o?<-e?- of the distance of
the particle from a fixed point in that line ; to determine the
1 notion.
Measuring x as before, the equation of motion ^Yill be
d'x _ fi
df x" '
Multiplying by -', and integrating, we have
1 /dx\' 1 „ /ti / 1 1
2\dtJ ~2'''~n-lW-' a"-V ^^^'
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, Ave have a = oo , and
n — 1 x"'^'
where v is the velocity from infinity, at the distance x.
We have therefore in this particular case
dx
dt~
m
1
X
dt _
dx
(>-;y
x
Integrating this between the limits x = a, x = yS, we havi
for the time of moving from a; = a to a; = /S,
RECTILINEAE MOTION. 77
103. To find T, the time of oscillation, when the ampli-
tude of .oscillation is 2a,
1 /dxV ^ _j^ /_1 1^ >
2[dt) n-iW'' a""'
dt _ /n — 1 / a" ^ x'' ^
dx~ V 2/^ V ct"~^ — x"-~^ '
n — 1 f^ / a" ^ x'^ ^ ,
Put
T=4
- =«; -r= -.z'
aj dz n — 1
d T = ■ ^" ' (1 - ^) ^
dz.
4a^ 5f JL +1 1
72/^(^.-1) ^^-1 2' 2
^ 4a^ U-1 2^ V2
\n — 1
, '-f / 7 /V^ (,7:^+2
= 4a J"^-^\Jy
\n — 1
104. The above solution fails when n = l, but the time
of falling to the centre may be found as follows. The equation
for this case, as given in § 94, is
1 /dxV , a
since when x = a, ,7 = 0. Hence
■ dt
J<'
78 RECTILINEAR MOTION.
the negative sign being taken since x diminishes as t increases.
Put T for the required time, then
To transform the integral, put /log"- =7/. Then we
have
o clx
X = a€~y', and -,- = — 2ae"2'' y,
and the Hmits of y are and co . Hence
Jo
= 2a . A Vtt.
Hence ^^^'^V^il'
and is therefore directly as the distance traversed.
105. A imrtkle is comtrained 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 (p (r) of the distance ; to
determine the time of a small oscillation.
Employing the same notation as in § 99, the acceleration
along FO being ^ (?•), its component along PX is ^ (r) - ,
therefore the equation of motion is
But r = ^{b' + x')^h^(l + '^^
= b approximately.
EECTILIXEAR MOTION. 79
Hence ^-;r + — y — x= 0,
and therefore by § 90, the time of a small oscillation is
b
>(b)-
-y,
EXAMPLES.
(1) A body is projected vertically upwards with a velocity
which Avill 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 for
which the least time of sliding down an inclined plane to
a 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 tlie 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 ditferent 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
of an attraction varying inversely as the f ^ j 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 -. •
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 ilistance ;
to determine the motion and the time of an oscillation.
Let 2a be the distance between the centres, .r 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 = ,,^ , .
V(2/t)
(14) If a particle begin to move directly towards a fixed
centre wliich repels with an intensity = /u, (distance), and
with an initial velocity = /i^ (initial distance), prove that it
Avill continually approach the fixed centre, but never attain
to it.
(15) A particle acted upon by two centres of attraction,
each attracting Avith 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 jDoint of attraction and remain
at rest there.
If /Lt, fM be the strength of the centres ; a^, a^ the distances
of the point of projection from them ; and V 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 C,
varies as CF, 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 fi"a will
be a maximum if /3 = — ^ — .
T. D. 6
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
of attraction to be as . / , where a is the initial and x
V ^
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.)"^. .f^*... ..■■.. c<- = ^v,,
(22) Shew that the time of descent, to a centre of at-
traction cc (dist.)"^ through the first half of the initial dis-
tance, is to that through the last half as tt -1- 2 : tt — 2.
(23) A particle descends to a centre of attraction, inten-
sity cc (dist.)". Find n so that the velocity acquired from
infinity to distance a, shall be equal to that acquired frorri
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 proper^
tional to
^/■■
iQ„f^±vV±_^
(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 oc (dist.)"^, find the time of a small oscillation in its
plane, about the centre.
(27) Shew that a body cannot move so that the ve-
locity 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 describ-
ing a given distance.
EECTILINEAR MOTION. 83 ■
(28) Shew that the time of quickest descent down a focal
chord of a parabola whose axis is vertical is
/^
where I is the latus rectum.
Y X (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 Otoa curved
surface, under the attraction of a plane whose attraction 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 ^ = a {1 — cos {k cos 0)].
(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 6) — 1}.
(32) Prove that the times of falling through a given dis-
tance AC towards a centre 8, under the action of two attrac-
tions, one of which varies as the distance, and the other is con-
stant and equal to the original value of the first, are as the
arc (whose versed sine is A G) 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 x {Dy in the centre, and ascend the
opposite radius under the action of a repulsion of equal in-
tensity 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.
6—2
84- RECTILINEAR MOTION.
(35) If a particle fall to a centre of attraction of inten-
sity oc (D) ; determine tlie 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 nurnher of free
particles, equally distributed, which gi-avitate 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 iu a vertical plane to another, the tangents at P and
Q are parallel and FQ 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
/^(dist.)~^ prove that if a particle be placed at a distance
a tan a from S it will arrive at S in time
"_ log (sec 2a),
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 it'SF, determine the time from rest to rest; and shew
that, if a small constant retardation / act through a portion of
the 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
n
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
/-T—~., /2glQx + TPa^2PjxU^^)
^^ +a Y Q{x' + a;)-\-2Px'
(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 , . „ .
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
47r
y;
(43) A particle moves in a straight line under an attraction
,x^
to a centre in the straight line fxx + 2/i' ^ , and starts from
0.
rest at a distance a from the centre ; shew that after a time t
the distance from the centre will be
a en ( \^fM — , K
\ UK
where k^ = — ., ^ , ,
fj,u. + z/i-a
( 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 vioves under the action of a vertical
attraction ivhose 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 jorojected in it, and is subject to no force which
would tend to withdraw it from that plane.
The equations of motion then are
d'x ^ d^y
df=^'df=-^'
if (jrbe 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 V, and
that the direction of projection makes an angle a with the
axis of cc.
The first and second integrals of the above equations will
then be
. =Fcosa, jy = Fsin a— at (1).
at dt ^ ^ '
x = V^cosa. t, y = Fsiu a. t — lyf (2).
PARABOLIC MOTION. 87
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.
y = ic tan a — ^ ,,„ — :^—x^ (3),
-^ 2 k 'cos- a ^ ^
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^ sin Oleosa 2F^cos'^a
[ F'sinacosa\' 2F^cos'^a/' F^sin"ct
"' (" -9 — )= ^(^--2,-
By comparing this with the equation of a parabola re-
ferred to its vertex as origin, we find for the co-ordinates x^,y^
of the vertex
_ V^ sin a cos a _V^ sin^ ct
Hence we obtain the equation of the directrix
, , F'^sin^a F^cos^a V^
y = 3/0 + i (parameter) = — g^-- + -^ = ^ .
Now if V be the velocity of the particle at any point of
its path,
^" = '-J+U7j' ''^^^^^
= (F' cos' a) + (F'sin' a - 2Fr/ sin a . < ^g^^)
= V'-2g {V sin a. t-hgf)
= T'-2gy, by (2).
S8 PARABOLIC MOTION.
To acquire this velocity in falling from rest, the par-
tide 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 i-)Jane.
Put w = in equation (3). The correspondiug values
2r'
of X are and - — sin a cos a. But the horizontal velocity
. 2Fsina
is Fcosa. Hence the time of flight is ; and, ceteris
paribus, varies as the sine of the elevation (inclination to the
liorizon) of the direction of projection.
110. To find the time of flight along an inclined plane
passing through the j^oint of projection.
Let its intersection with the plane of projection make an
angle ^ with the horizon ; it is evident that Ave have only to
eliminate y between (3) and y = x tan /3.
This gives for the abscissa of the point where the pro-
jectile meets the plane,
x^ = - — (sin a cos a — tan ^ cos' a)
_ 2F''cosgsin(QC-/3)
~ g cos^
Hence time of flight
_ a\ _2Fsin(a-y3)
y cos a g cos y8
111. To find the direction of projection which gives the
greatest range on a given plane.
V-
The ranGfe on the horizontal plane is — -sin 2a. For a
given value of V this will be greatest when
TT TT
22 = ^ , or a = ^ .
PARABOLIC MOTION. 89
That on the Inclined plane is — V^ , or
cos p
cos a sin (a — p).
g cos''/?
That this may be a maximum for a given value of V we
must equate to zero its differential coefficient with respect to
a, which gives the equation
cos a cos (a — /3) — sin a sin (a — ^) = 0,
or cos(2a — yS) = 0;
whence
-i(| + /3)
Hence the direction of projection required for the greatest
range makes with the vertical an angle
77
2-"
lih^y
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 j^article's pass-
ing through a given point.
Suppose the point in the axis of x and distant a from the
origin. Then we must have
F' .
— sm 2a = a,
9
F*
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
equally inclined to the direction of projection (« = ^) 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, we
must have
-^ — ^ cos "x sin (oL — B) = a.
^cos/3
If a, a" be the two values of a which satisfy tliis equa-
tion, we must have
cos a' sin (a — /3) = cos a" sin (a" — /3);
and therefore a" — /3 = ;y — r/ ,
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
2 {y)=\v"-,\n^r. + 4>{h)-{y).\^
Hence
or \v-'+{y) = \v"- + 4>{h),
a particular case of conservation of energy.
To find the differential equation of the path, -we have
dy
di^dj/^ V[F^sin^a + 2 \±{b)-Jij)]]
dx dx V cos a '
It
an equation integrable for particular forms only of the func-
tion (j).
An interesting case is that in Avhich the attraction of the
plane is inversely as the cube of the distance,
^1' ^' iy) — :i ' ^^^ therefore cj) (y) = — ^ ^^ .
J J
The differential equation becomes
x/C^
dy _ \l \f f^
dx V cus a
dx yd If
PARABOLIC MOTION. 101
and inteofratincf
X — a
Tcos a
sin^a-f; U'
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,
by a slight modification of the above process, that there is
only one case, and that the curve described is a hyperbola
Avhose 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 attraction perpendicular to an axis that
a free paiiicle may describe a conic section.
Take the axis as that of x, and the vertex as origin, then
the equation
2/^ = 2.mx + nx" (1)
102 PARABOLIC MOTION.
will represent, by proiDcrly taking m and n, any parabola, any
hyperbola referred to its transverse axis, or any ellipse re-
ferred to either axis.
Since the attraction is perpendicular to the axis, we have
dx
-01 = ''
Hence
drj
and
d'y fdyV
From these
df yX \dt),
y\ y
3 {ny — m^ — Imnx - n'x-)
= — ^—j- by equation (1).
For a hyperbola with its conjugate axis in the axis of
X, the equation is
TT dy „ dx
Hence ytt=i''''dt
— p ex,
from which we have immediately
d"y (d}/\\
,/1^ + f^^/) =„v
PARABOLIC MOTIOX. 103
That is ^'^-■^.W-Z'^Vl
y \ f
which indicates a repulsion inversely as the cube of the
distance from the conjugate axis.
" 126. To find the repulsion luhich must act perpendicular
from a plane, in terms of the distance from that plane, that
a given path may he described.
Take the axes as before ; then, Y being the acceleration
due to the repulsion (a function of y only), we have
-7^ = 0, or -^ = const. = a, suppose ;
S=^^ <^)-
Let y —f{j:) be the equation of the given curve, then
dy r,f s
orby(l), Y^a'f'ix)
= af"[f-\y)],
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.
otlier side of tlie 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 .^5 C being a conic, one of whose
axes is CG. and therefore cutting it at right angles in C, 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.
PARAEOLIC MOTION. 105
129. Generally, luhen a particle is anyhow projected and
subject only to an acceleration whose direction is pierpen-
dicular to a given plane, and ivhose magnitude depends solely
on the distance from the plane; the velocity p)o.i'cdlel to that
plane is constant ; and, in jmssing from any 2^oint to another,
the square of the velocity is altered by a quantity depending
only xipon the distances of those two points from the given
p)lane.
Take the axis of y pei-pendicular to tlie 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
^'{y) suppose.
and
dc "• de
dx
dl = '-
1 .,
1'-
-im^m^
-b^' + M-^OA
lv' + 4>{y)^lv' + (y,),
V being the
the point
;he velocity of projection, and y^ the co-ordinate of
of projection ; which proves the proposition.
This is, of course, merely a particular case of the general
principle of Conservation of Energy (§ 78); S' whose distance from AB (the
bounding surface) is less than that of AB from CD, the velo-
city perpendicular to AB may be destroyed ; then, as before,
the particle will pursue the path STQ'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, and 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^
Fsin a = F'sin a'.
Also by the fixed amount of change in the square of the
whole velocity,
where a is a constant depending on the nature of the two
media.
„ sin a V I (^ _ a^\
Hence, —-, = -^ = ^(^1+ ^j
sin a
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 hioiun laiu 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 illus-
tration of the application of the results of this Chapter, and
afforded the first instance of the solution of a problem con-
nected 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
Pand 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 vortices an
oblate spheroid, and that the particles themselves at any
instant will lie on a sphere.
PAEABOLIC MOTION. 100
(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 (2 - g ) _ r^- f
sin (a + a) t"' + f '
^ (4) If u and v be the velocities at the ends of a focal
chord of a projectile's path, V^ the horizontal velocity, shew
that
14- h- ^§"'->
(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 ditfer-
ence of their ranges is constant.
(6) If V, v, v", be the velocities at three points P, Q, B,
of the path of a projectile where the inclinations to the horizon
are a — /3, ex, a + /3; and if t, t' be the times of describing
PQ, QB respectively, shew that
v"t = vt\ and - + \ = ^ • T-^ . (§ 119.)
V V V
(7) If two particles be projected from the same point at
the same instant in the same vertical plane, with velocities v
and v^ in directions making angles a and a^ with the horizon ;
shew that the interval between their transits thi'ough the
other point which is common to their paths is
2 vv, sin (a ~ a)
g v^ cos a^ + 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 tlie particles slide down chords to the lowest
])oint, and be then sutfercd 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
l^ass 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
2sin^ — sin2^, 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" + Ba~''y ] 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 described
are equal; prove that the locus of the vertices of these para-
bolas will be a parabola.
(10) Find the direction of projection, Avith 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 vcctorcs of the curve
whose e(|uation is r=f{0), the plane of the curve being
PARABOLIC MOTION. Ill
vertical and 9 being measured from a horizontal line, prove
that the locus of the foci of their future paths is the curve
r = cos-/(^^
(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 ^— y- , where
7)1 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 pa-
rallel but equally inclined to that line; prove that the path
of each will be an ellipse, parabola, or hyperbola, according 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 revo-
lution.
(21) If a be the angle of elevation required in order that
a bullet may have a certain range on a horizontal plane, 6 the
additional elevation required above a plane inclined to the
horizon at an angle 13,
^ sin/3sin^a
sin (2a +
(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 ^^ .
"J %
(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.
(2-i) 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 the greatest height
to Avhich the sun 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 som6 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 O 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, 0M = r, and the
ancfle N0M=6. Then, since cos ^ = -, sin ^ = — , the com-
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
in the directions of x and y separately, and we have therefore
df
>.y
.(A).
In
and
these, since P is a function of r, and therefore of x
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, -^' , -^ ,
(it do
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 wall involve x and t, and the
second y and t, only, and each equation may be integi-ated
by itself. As it is the simplest example of its class, and of
great importance in its applications, especially to Acoustics
and to Pliysical Optics, we will begin by considering it.
133. A particle moves about a centre of attraction
varyinrj directly as the distance: to determine the motion.
Let jjL be the acceleration at unit of distance, called the
CENTRAL ORBITS. 115
Strength of the centre, then P = fir, and equations {A)
become
% (^''
the integrals of which, see § 83, are
x=Aco^yiit + B]., (1),
y = A'co&yfit + B'] (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 V, and in a
direction making an angle a with Ox. When t=0, we have
a; = a, 2/ = 0,^= Fcosa, -^= Fsina. Hence,
a= A cos B,
= A'cosB',
Fcos a^ — AsJfi sin B,
Fsina = — ^' V/* sin B'.
Expanding the cosines in (1) and (2), and substituting
these expressions for the constants, we obtain
FcOS IX
x = — ~ — sin \/jxt + a cos \fjlt (3),
Fsin a . ^
y= r^ sm^fjlt (4),
which contain the complete solution of the problem. Elimi-
nating t, we have
{x sin a — 2/ cos 7)^ + ^r^y^ = ct^ sin^ a.
J— 2
116 CENTRAL ORBITS.
the equcation of the path of the particle ; which is therefore an
ellipse whose centre is 0. Equations (3) and (4) give periodic
values for x, v, -,- , -^ , such that all the circumstances of
motion will be the same at the time f + -t- as at the time t.
The period of revolution is therefore ~- : a most remarkable
result, as it is independent of the dimensions of the ellipse,
and depends solely on the intensity of the force.
By taking fi negative in equations (B), 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 hi/ a central attraction;
it is required to determine the polar equation of the path.
Multiplying the second of equations {A), § 132, by x, and
the first by y, and subtracting, we obtain
d-y d^x
d
Integrating,
dy dx
~dt ~y dt
Changing the variables from x, y, to r, 6, where x = r cos 6,
1/ = r sin 6, we get, as in § 24,
(
^/*=-^'df' = "•
CENTRAL ORBITS. Il7
or, substituting - for r,
M- W'
(2).
dd . ,
Again,
^ cos^
lich gives
, u sin ^ + cos ^ -y7. , .
dx dd dd
dt u' dt
= -ALsin6^ + cos^^),by (2);
d therefore
d^x ^ ( . ^ d'u\ dd
- /iV fu cos ^ + cos ^ ^ J , by (2).
But, by the first of equations (A),
d^x „ ^
^^ = — P cos ^.
dt
d^x
Equating these values of -7^ , and dividing by cos Q, we
have
^=«@+'') (3),
•
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 m, its integral will be the polar
equation of the required path.
118 CENTRAL ORBITS.
136. It may easily be obtained by the formulae of § l(j,
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 im-
on the particle.
Instead
§§ 16, 69),
of equations {A) we may evidently
write (by
df ' [dt) ^'
(A-
ld( dj^
rdtV dt) ^•
^
Putting
,d0 , . 1 ,, de
r -n = h, and u = -, then ,^ =
dt r dt
?^u^and the second
equation becomes
or
,-/^ ^ dt u'
dk' 2T V'
de u' '
Also
dr drde du
dt dedt "cw
df "'' de^'udd
and
Kf)^--
Therefo]
•e
de' u de
P,
or
d\ P T du
de''^''~hhi' h'u'de-
137. The general integrals of (A), 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 (4) be
CENTEAL ORBITS. 119
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 Uvo
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 6, and substitute
the result in the expression for P; eliminating 6, if it be in-
volved, by means of the relation between u and ^. 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 6, 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, tlie
radius vector and the perpendicular on the tangent at any
point, may be easily obtained from (4). For by Diff. Calc.
we have
d\ _ 1 dp •:•;.
and therefore
K^ dp
p^ dr'
140. The sectorial area siuept out hy 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^l ^d£
dt~2'' dt'
120 CENTllAL ORBITS,
and therefore, by equation (1) of § 135,
dA 1 -
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.)
For Velocity = y = -r:
-^ dt
^dsde
~ d6 dt
{p being the perpendicular on the tangent from the centre)
= - , by equation (1) of § 135.
Hence, as above, v x -.
P
142. This equation enables us to express h in terms of
the initial circumstances of the motion. For, let li be the dis-
tance of the point of projection from the centre, Kthe velocity,
and /3 the angle which the direction of projection makes with
that of R. Then evidently the perpendicular on tangent at
point of projection = li sin /3 ;
CENTRAL ORBITS.
or
i^ sin /3 '
whence
h = VR sin /3.
Again,
we have
since by Diff.
1
V
Calc,
another important expression for the velocity.
121
143. . The velocity at any point of a central orbit is inde-
pendent of the path described, and depends solely on the inten-
sity of the attraction, the distance of the point from the centre,
and the velocity and distance of projection.
Multiply equations {A) § 132, ^J~n>-K respectively, and
add, then
dx (^x dy d?y _ P / dx dy
didf'^didf~~rVdi'^^dt
" dt'
(since ^^^-y^ = r^ we have a;^ + ^g = r|).
Also, since P is a function of r alone, let P = (?'), then
.' 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,
-y = c/), (a) -(/),(;•)
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 woidd be acquired by a par-
ticle moving freely from rest along onefourth of the chord of
curvature at the point, drawn through the centre, under the
action of a constant force tvhose intensity is equal to that of
the central attraction at the p)oi'nt.
By § 143,
ld{v'')
2 dt
__ dr
^dt
dv
'dr-
-P.
And by § 141,
V
_h
Differentiating the logari
thm of t
ldv_
=-'t.
vdr
pdr'
and, dividing- the former equation by this,
" ^ dp r ^ dp
where q is the chord of curvature through the centre. Hence
the proposition, § 82.
From this it follows that the velocity, I'', of a particle
CENTRAL ORBITS. 123
moving in a circle of radius B, under the action of an attrac-
tion F to the centre, is given by the equation
V = PR,
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
-j-z should vanish, and that the first succeeding differential
coefficient which does not vanish should be of an even order.
The first condition ensures that the tangent at an apse is per-
pendicular to the radius vector.
* The results of the last few Articles
may be obtained
in the foUowinj
manner.
By §§ 49 and 65
j^ = Eesolved part of P along the tangent of the orbit - -
4:-
(1),
11 2
— = Resolved part of P along the normal :
P
=^f
(2).
Multiply (1) by ^^ and mtegrate, then
m-
- jPdr.
From (2) hi-=Jr
-K^'?)
Also if in (2) we put - for v, § 141, and r -z- for p, we obtain
„ dr r'
P = ^'^
p^ dr '
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 tiuo
apsidal distances.
For, since the joarts 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 =1^11"', then we have
d^"^ /* n-2 ri
Multiplying by h^ -7^ 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 = (o^ - 1 ) -^ - c""' ;
^^ 71 — 1
and therefore i A= If^'y + uA = ^_^ [ir' + (^^ - 1) c""^}.
To determine the apsidal distances we must put -Tn = 0,
which gives
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 ivhich is very nearly
circular, and is acted on by an attraction varying as any func-
tion of the distance and directed toiuards the centre of the circle:
to determine the apsidal angle.
If we put P in the form fiit^ (j) (u) the differential equation
of the orbit is
d^^^ ;"■ . / ^ /^
If the orbit were circular, we should have
u = c,
and ^ = 0,
in which case
c-^(/>(c)=0 (a).
126 CENTRAL ORBITS.
When the orbit is very nearly circular we may put
u = c + x, where x is always very small. Hence
+ c + x-^,{(f>{c)+ x(f>' (c)} = 0, nearly ;
and (a) enables us to reduce this to
or, by a second application of (a),
^4-|l_^-^^)U = (6)
the integral of which is (§ 88)
dx
Hence the general vahie of 6 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
the approximate apsidal angle.
CENTRAL ORBITS. 127
Thus if the attraction vary directly as the ?i* power of
the distance, we have
fjiu^ (]) (w) = fin'"'; and (f) (u) = ii'"'^,
whence j) (u) = — {n + 2) u'"*'^
and the apsidal angle is
This shews that n cannot be less than — 3, or that the
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 coDditions 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, luhatever be its radius, may be
equal to a given angle, a suppose, we have
•JT
vi^ >(c)j
from which
or, by integration,
4>{c) c
log^) = (l-^^)loge,
whence j>{c) = C& a^;
_7r2
and therefore the law of attraction, fj,u^
c o ' o
(6) above shows 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 (b) 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 jjoint 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 = fiu^, and therefore
d'u
dO'
+ u-
0,
or
d' (
1)
K«-
-^l-
the integral of which is
u-^
= A
cos ((9
+ B)
or,
as it is usually written
u = l^{l + ecos(0-o^)] (1).
This is the polar equation of a conic section, the focus
(the centre of force) being the pole.
It gives by differentiation
5g = -,-,«sm(«-a) ; (2),
Let R be the distance of the point of projection from the
centre; /3 the angle, and F the velocity, of projection; then
when ^ = 0,
1
cot/3 = -(l^
Hence, by (1),
h' ,
„ — 1 = e cos a ,
fiR
d by (2),
-^ cot /3 = — e sin a.
CENTRAL ORBITS. 129
From these, tan a = —p — ,3 (3)>
and > e* = -^^2 cosec^ yS h+1 (4).
fill /jbH
1 f , F^i? sin y 3 cos /3 ,„,.
wherefore tan a = fr 2 7^ • 2/^ (3 )
fjb—VBsmp ^ '
i_,.=^'«L-i^f|_n ,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
2/u,
if V^> ^ , e>\, and the orbit is a hyperbola,
2it
F^ = ^,e = l, a parabola,
JX
F^<-^, e'
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{l —e) and a{l + e).
Hence, as the coefficient of the second term of (G) is the
sum of the roots with their signs changed, we have
1 ^ 1 2/..
a(l-e) a{l + e) W
«(l-0 = ^ (7).
CENTRAL OEBITS.
And,
as the third term is
the product
of the roots,
1
2/. V
WR 1' •
a'(l-e')
or
1
— =
a
2 y ■
R f. ■■'■
or
and therefore
2 ^ -
R 2a'
h-
r 2a
131
(8),
(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,
= Vj/cia (1 —e^)], by equation (7).
From this, combined with the polar equation of a conic
section about the focus, we have
dt r"
dd ^J\^la{l-e^)]
vf
(1 + e cos df
measuring the angle from the nearest apse. To integrate
this, let
sin^
=
1 + e cos ^ '
132 CENTRAL ORBITS.
1 l-e^
(l + ecos^)
dS _ ^os^^e _e__ e
^^""^ le" {l + e cosdf " {1+e cos d)'
1 1 l-e' 1
e 1 + e cos ^ e (1+e cos df
dd
sec"^ do f
•'•J (1+ecos^j' 1 - e'' ^ 1 - e' j 1 + e cos ^
e sm6> 1 r —2"'^ i
J{l+e)+{l-e)i&ii'^ I
e sin 6* 2 ^ _if //1-eN, ^|
(if el).
Hence the time of describing, about the focus, an angle
measured from the nearer apse is, in the ellipse,
2
that is, y of the sectorial area ASP (figure to § IGO) ; and,
in the hyperbola,
[ v(.+ l)eos|-V(.-l)sinf ] ^._^^
*- U/(e+l)cos- + \/((^-l)sin- -*
2
that is, T 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 = 1, a(l—e) = d, a (1 — e'^) = 2c?},
dd
(1 + cos ey
sec* K dd
1 + tan^ ^ j d tan ^
'2cZV, 1^ 6'^
tan- + 3 tan 2^
154. From the result for the ellipse we see that the
periodic time is Stt . /— , This might also have been found
from the consideration of equable description of areas by the
radius vector.
2 area of ellipse
Thus T =
h
27ra' V(l - e")
V H'
In the notation commonly employed we write
27r
n '
where n, which is called the Mean Motion, is
^/
134 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 describe
equal areas in equal times about a point in that plane, the re-
sultant 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
d?x_ d-'y_
df-^' df-^ ^^^-
But by hypothesis, if A be the area traced out by the
radius vector, -^- is constant.
dt
Hence,
^^-^
dy dx_
dt~y dt~
G.
DifFerentiating,
^§
d'x .
-ydi^=^-^
by (1).
xY
-yX=0.
CENTRAL ORBITS. 135
Hence, -f^ = - ,
and by the parallelogram of forces (§ 67) the resultant of X
and Y 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
M = y (1 + e cos 6),
where I is the latus rectum.
XT d% 2e ^
Hence, -rsi = ~ 7~ cos v,
do i
and therefore the attraction to the focus requisite for the
description of the elJipse is (§ 135)
Hence, if the orbit be an ellipse, described about a centre
of attraction at the focus, the laiu 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^ unless
fi 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 enuntiation
was sufiicient to enable Newton to propound the doctrine of
Universal Gravitation ; viz. that evert/ particle of matter in
136 CENTRAL ORBITS.
the universe attracts every other with an attraction whose direc-
tion 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 aud
Motiou," between every imir of particles there is a stress of the
nature of a tension, proportional to the ptroduct 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 pei'iodic times ami the sums of the masses of the Sun and
the ptlanet.
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 per-
pendicular to the major axis ACA', and produce it to cut the
auxiliary circle in the point Q. Let G be the common centre
of the curves. Join GQ.
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 Tru£ Anomaly. Let
us denote it by 6.
The angle ACQ is called the Excentnc Anomaly, and is
27r
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 S 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 mean and excentric ano-
malies.
In the figure QCA is the excentric anomaly, and the
mean anomaly is evidently to 27r as the area PSA is tor the
whole area of the elliptic orbit (§§ 154 — 159), or as area
QSA to area of auxiliary circle.
Now area QSA = area QCA — area QGS
{a being the major semi -axis of the orbit and e^thesexcentricity)
= Q- (w — e sin u).
Hence
a , . .
T7 (u—e sm u)
nt _ 2_
2-77 "~ ira^
nt = u — e sin w.
138 ELLIPTIC MOTION.
161. To find the relation between the true and excentric
anomalies.
We have (by Conies)
SP = ^ (^ ~ ^')
1 + e cos 6 '
But SP = a — eCM = a (1 — e cos u).
Heuce ^ = 1 — e cos m.
1 + e cos ^
Hence cos 6
and tan
2 Vl +
1 — e COS w '
cos^
cos^
\—e cos It — COS u + e
1 — e COS w + cos u — e
(1 + e) (1 — cos u)
(1 — e) (1 + cos ti)
1 + e\ , u
tan 2;
therefore tan ^= . [- — ^ ] tan ^ ,
2 V VI +6/ 2'
sin u = s/l — e
-e\.
sin 6
1 + e cos ^ '
substituted in nt = u — e sinu 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 = lit + e sin u
= nt + e sin {nt + e sin nt) nearly,
= nf + e sin nt + -^ sin 27it.
Also - = 1 — e cos u
a
= 1 — e cos {7it + e sin nt)
And
= 1 — e cos nt + - (1 ~ cos 2nt).
which may be written (§ 154)
(l + ecosdrdt''''' ^^ ^''
or (l-e*)^(l+ecos^r^^ = w.
Keeping powers of e lower than the third
3 2 1/1 \ dd
3
4
(l - 2e cos ^ + I e' cos 26' j ^ = 7i,
nt = 6" - 2e sin 6' + ? e* sin 2^
3
whence 6 = nt ■¥ 2e sin ^ — - e'^ sin 29
4
3
= «^ + 2e sin {nt + 2e sin nt) — ^ e" sin 2?i«
3
= nt + 2e sin ??^ + 4e^ cos ?i^ sin n^ — -: e^ sin 2?ii
4
= nt + 2e sin ?2i + t e*^ siji 2w^.
140 ELLIPTIC MOTION.
163. Kepler's Problem. To Jind r and 6 as functions
of i from the equations
r = a {\ — ecos a) (1) ;
*^°f=y(T^o*'"'5-
(2);
nt = n — e ?>\xi u (3).
These equations evidently give r, d, and t directly for any
assigned value of n, 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 6 from equation (2) in a series ascending
by powers of a small quantity, a function of e, the coefficients
of these powers involving it and the sines of multiples of u.
Now by Lagrange's theorem we may from equation (3) ex-
press u, 1 —e cos u, sin u, sin 2u, &c. in series ascending by
powers of e, whose coefficients are sines or cosines of mul-
tiples of nt. Hence by substituting these values in equa-
tion (1) and in the development of (2), we have r and 6 ex-
pressed 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 J—1, we have
i0 iO iu iu
if-vd-
e^ + e " e' +e '
whence
ie^€^W{l +J) +V(l-e )}+{V (l- e)-V(l-he)|
^ e^« W{l-e) - V(l +e)} + {V(l -e) + J\l+e)]
,,. ^ ^/{l + e)-^/{\-e) e
or, putting A, = -77- : ryi r = :: rrz ,-,
' ^ *= ^/{l+e) + ^/{l-e) 1 + \/(l - e')
.. . l-X6-'«
1 — Xe"*
ELLIPTIC MOTION. 141
Taking the logarithm of each side and dividing by i,
e=u+-. {e^« - e-"^ I + ^. . {e2«' - e'^^"} + . . .
\* V
= M + 2 (X sin ?; + -^ sin 2?* + — sin 3m + &c.) (4).
165. To develop u ^'w. te7'?ws of t.
If we have
y = z + x(\>{y) (5),
we obtain, by Lagrange's Theorem, the development
f{y) =f{z) + x4> {z)f' (z) +^^JJi^V' i^)}
+ l4:^(|y{0(^T(^)} + &c (6).
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 (f) (y) = sin y.
Also, as it is the development of u that we require, we must
put
f^u) = u, and/' (u) = 1. Hence, by (6)
and, substituting for the powers of sin z their corresponding
expressions in sines and cosines of multiples of z,
x^ d /l-cos2s\ x^ /^V/3sin.3-sin35\
.y = ^+^sm.+^-2 dii — 2--; + 1X3^; [ 4 )
+ l-fTsTiidz) [ 8 ) +^-
= 2 + a; sin 3 + ^ sin 2;? + "5^ (3 sin 3s — sin 5;) +
z o
142 ELLIPTIC MOTION.
or, substituting for x, y, z their values as above,
u = nt + e sin nt + -^ sin 2nt + j (3 sin ^nt — sin nt)
+ ^ (2 sin 4??^ - sin %it) + &c (7).
To develop sin xi, we recur to equation (3), which gives,
after the elimination of u by means of (7),
e . e^ .
sin u = ^mnt+ ^ sin 2.nt + 77 (3 sin ^nt — sin nt) + &c. . . .(8).
Z o
By the application of Lagrange's theorem to equation (3),
it is easy to deduce the following expressions :
sin 2it= sin '2nt + e (sin ^nt — sin nt) 4- e^ (sin 4?i^ - sin 2n<)
+ ^ (4 sin nt — 27 sin 8n^ + 25 sin 5n^) + &c.
sin 3t< = sin 8»^ + ^ (sin -^nt — sin 2ni)
+ - (15 sin btit — 18 sin 3??^ + 3 sin nt) + &c.
o
&C. = &C.
Substituting these values in (4), we obtain the value of 6,
containing however the quantity X. If we take as its approxi-
e e^
mate value h + u » ^^^ make the requisite substitutions, we
A o
obtain
= nt + (2e — 7 &^) sin nt + 7 e" sin 2?2f +75 ^^ sin 3»/ 4-
which is correct as far as e^.
[The development of 10 in terms of / is
w = ?ii + 2S,,j^j — /„, {me) sin 7?i /?^,
1 /■"
where / (?«e) = — I cos 7» (< — e sin t) dt
is Bessel's function of the 7j"' order.]
ELLIPTIC MOTION. 143
For the development of r and in terms of t, the co-
efficients being Bessel's functions, see Tod hunter's Treatise on
j's, Laplace's, and Bessel's Functions.
166. In proceeding farther with the development, it be-
comes necessary to expand \ and its powers in series ascending
by powers of e. This is readily done as follows.
We have
Hence E='2-^,
III
from which, by Lagrange's Theorem,
and thus the value of \^, being e^E"^, is known.
The correct value of 6 to the fifth power of e is thus found
to be
nt + 2e sin nt + —7- sin 2nt + ^2—^ (13 sin 3nt — 3 sin nt)
e*
+ H^-5 (103 sin hit - 44 sin 2nt)
(1097 sin 5nt — 645 sin Snt + 50 sin nt).
2\S.
167. To develop r wi ^erms 0/ 1.
From (1) it is evident that all we have to do is to de-
velop by Lagrange's Theorem, 1 — ecos u as a function of t,
from nt = u — e sin u.
To develop (1 — e cosu) in terms oft.
Here/ {y) = l — e cos y,
f{y)=esmy;
144 ELLIPTIC MOTION.
and the form of ^ is the same as before ; hence
1 — e cos 3/ = (1 - eco^z) +x sin z (e sin z)
+ r2i^'^^^'^-^'^^'^+
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 x,
we have
T &^
1 —e cos u = -=1 — e cos nt + ^ (1 — cos 27it)
&
4- Q (3 cos nt — 8 cos 3n0
o
+ ^ (cos Int - cos 4?i«) + &c.
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 r^, r^ be the focal distances of the ends and c the chord ||
of the arc, it is proved in Williamson's Integral Calculus,
§ 1 37, that the sectorial area is
^ ab {(f)^ -(/>,- (sin <^^ - sin <^,)},
where ^j and (j)., are given by the etpiations
and therefore if t denote the time in the arc,
nt = (f)^- (f).^ - (sin 0, - sin )J.
CENTRAL ORBITS. 145
EXAMPLES.
(1) A particle describes an ellipse under an attraction
always directed to the centre, to determine the law of the
attraction.
From the polar equation of the ellipse, centre pole,
2 cos'^ , sin'^ , du (1 1\ . . ^
d\t , (duV (1 1\. ,. . ,.
= ! {"' - {I - ?f "^'^ ''"'' + '' (jf - «-') 'r ^"^ - ''-'')]
_A^ 1 1_ A'_
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
u = — rr rr [1+e cos (6 - a)},
a (1 — e )
and therefore P = h'^u^i jij^ + u
h'u' ' 1
X
ail-e') r''
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.
pA-
(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" = a" cos nO,
and interpret the result when n = — 1. Deduce the law of
attraction for (1) a rectangular hyperbola, (2) a lemniscate,
(3) a circle about a point in a circumference, (4) a cardioid,
(5) a parabola.
(7) Prove that the attraction to the pole under which a
particle will describe the n^^ pedal of a cardioid varies as
3>t-|-8
r~""^. 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 giver
direction with the velocity from an infinite distance, and ii
under an attraction varying inversely as the /i"' power of the
distance, to determine the orbit.
Here F= fxu", and therefore
CENTRAL ORBITS. 147
lu
dd'
Multiplying by K^ -^ and integrating,
^'m-A-\--^iy"^'-'^r
1 ,2 {(dO
2
do^^" 2 _ 2/Aw"
Now if a be the apsidal distance,
?z-l
therefore (j^]\ u' = a"-'ii"-S
(|)=.M(-)"--ll.
de^ 1
du uJ[auY-^-l '
integrating — ^ 6 = sec~^ (aw) * ,
or - r ^ = a - cos — ^ — ^,
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
d^u ifc - ...
55^ + "-^" = " «■
The integral of this equation involves exponential or
circular functions according as is greater or less than
10—2
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 = A'; then
the integral of which is
u = Ae^^^-Be-^^ (2).
Species 1. Let A and B have the same sign ; then
u = Ae^^+ Be-^«;
and ^^=k{Ae^'-Be-^^).
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, -fA = 0;
or = Ae'"' - Be-'"',
Avhich always gives a possible value of a ; and therefore
j^^ka — B^-ka. = - - , suppose.
Substituting,
CENTRAL ORBITS.
, = l|e^(9-a)+^-i(e-a)|
149
Hence when 6 = a, au = l, or a is the apsidal distance.
As increases, it increases, or r diminishes; and when 6 =cc ,
u = cc , OT 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>l, B = 0, then the equation (2)
becomes
the equation of the logarithmic spiral. The nature of the curve
will be the same if A, instead of B, vanish.
Species S. Let p > 1, and B negative, then by equa-
tion (2),
ti = ^e*^-^e-^«.
Putting w = 0, when 6 =^ a, we obtain as for Species 1,
Hence, when 6 = a, u = OoTr=co. 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 r .
II. Species 4. Let
1, then equation (1) becomes
= 0,
the integral of which is
au= 6 — a,
the equation of the reciprocal spiral.
III. Species 5. Let p < 1, and let 1 - t^^ = ^*, then by
equation (1),
d\
+ khi = 0,
the integral of which is
aw = cos A; {6 — a
du
whence
a-Tp. = — k sin k {6 — a).
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 repre-
sented in the annexed figure.
CEXTRAL ORBITS. 151
(10) A particle of mass m under a central repulsion
'^ is projected from an apse at a distance a with velocity
->/^, Find the orbit, and prove that the time from the
2a 72
apse to the distance aj2 is ^ J^ -,- •
(11) A particle under an attraction inversely propor-
tional to the fourth j)ower 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 = fj,u^, and we have
d^u a
whence 2 ^'' I ( cJ) ^ ^ 1 ^ ^ i ^^^^ "*" ^'
If the particle be projected from an apse at a distance a
with velocity n times the velocity from infinity, then
1 2 2 /" " 3 7 1 n'^fM
and therefore ^ ^ Z ^'^^ ~^) i'
2 i i In /J,
and h = v a = tz —^ .
2 a
rr,, „ fdu\^ _ 2 a^u* . n^
Therefore ij^] +u' = -^ +
ddj ' n' ' n"
162 CENTRAL ORBITS.
ddj n'y"" a' '^ a'
d7-Y a-/. r-\ f, . , _r'
dej=.^{^-a^)\'-^'-'^J'
and therefore r is an elliptic function of 6.
For instance, suppose n b) wath a
velocity — , shew
whose equation is
velocity — , shew that it will proceed to describe the orbit
r^ = a" cn^ - + h^ sn"* — ,
c c
the modulus of the elliptic functions being the excentricity of
an ellipse whose semi-axes are a and h.
This may be written ?• = 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 tlu
centre.
CENTRAL DEBITS. 153
(15) A particle under a central attraction varj'ing in-
versely as the fifth power of the distance is projected from
a given point with a velocity which is to the velocity from
infinity as 5 to 3, in a direction making an angle sin~^ ''^~-
o
with the radius vector ; find the orbit.
Here we have
^'^ , At 3 A
•.^^gy..|=.=a
But if V be the velocity of projection, c the initial value
of w,
"55 /xc^
and when u = c, v=V, .-. C +
9 2 '
8fic*
But
V sin' /3 _ 25 fic'24<
c' ~18c^25'
/jl 3
Substituting and integrating we find, after the necessary
reductions,
V3 l-eV^(^-) .
^-2 ^1+6V2(«-)'
where i2 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/x -^ + /Lt^/^ and a particle be projected at an
angle of ^^tt with the initial distance {R=)- , with a velocity
which is to the velocity in a circle at the same distance as
\/2 to \/3, find the curve described.
r = R{l- 6).
(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 /v/2, the
intensities being equal at the point of projection ; determine
the orbit.
(18) If P = -^ (5?'* — 8c^), 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
(19) If P= 2/u,(-3 — 5 j , and the particle be projected
from an apse at a distance a with velocity — - , prove that it
will be at a distance r after a time
(20) The attraction tending to the centre of a circle
whose radius is a being /u. ( r + .^ 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 CEBITS. 155
(21) If P=ixr + -^, prove that the equation of the
orbit is of the form
1 _ cos** kO sin" kd
If the particle be projected frora an apse at a distance
a = a/ - , with velocity ^/xv, prove that the equation of the
orbit is
1 + 6^^'
and that the time of describing the angle 6 from the apse is
^ tan"' d.
Jim
(22) If a particle move under a central attraction
/Lt?r + vic^, shew that the equation of the orbit is generally of
the form
a
r =
1 — e cos (kO)
In the case when the projection takes place at an apse,
the apsidal distance being j , and v being equal to ^^ shew
that the equation of the path is
2M'
and that the time of describing an angle a is
- tan e(e+h sin 29) where tan = f^^,, .
a ' - J{2k')
Determine generally the relation between the orbits
when P = /xii^cf) (m) and when F = fiu^(f) (u) + vv?.
(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 CKNTRAL ORBITS.
(24) A particle moves in an ellipse under an attraction
tending to a fixed point ; prove that the acceleration due
to the attraction at any point F varies as ^^ ^.^ , where
PP' is the chord of the ellipse passing through 0, and DB'
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
sin 26 = -rn — r •
(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 = — — sin"* — . .
(27) A particle moves under a central reimlsion which
varies as the distance from a fixed point; shew that the
equation of the path described is
ccji/'^ — 6* — y Jx^ — 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 in to 1
(m being < 1) the body will proceed to describe an ellipse, the
excentricity of which is (1 — 2ni^ + 2iu*)-.
(29) If tlie Earth's orbit bo 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
continue within the Earth's orbit longer than the [.^-j part
of a year.
CENTRAL ORBITS. 157
(30) If a particle, under a central attraction varying
inversely as the square of the distance, be j^rojected 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 - /3) = (1 - n"") tan l3,
/3 being the angle between the radius vector and the direction
of projection.
(31) A particle describes a parabola about a centre of
attraction (x i)~^) 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 Avith 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 oc j-^ ; the strength is
suddenly increased in the ratio of w : 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
168 CENTRAL ORBITS.
ratio of 1 to 1 — n, find the excentricit}- of the elliptic orbit
after ^j revolutions, the original excentricity being e.
(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 a^, a^ be the angles which these direc-
tions make with the initial distance c, and 2^/ be the length
of the apse line, prove that
cot a, . cot a„ — — 1.
(36) If the perihelion distance of a comet's orbit be ^ 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 fullow-
ing formula;, the letters being the same as in the text,
,-| = aV(i-.');
- sm ^ = — ^- - ^ (v - nt) :
a e
log^ = -log(l+\^)
— 2 (X cos M + IX^ cos 2u + ^V cos Su + &c.).
(39) A body describes an ellipse about the focus : prove
that the times of describing the two parts, into which the
orbit is divided by the minor axis, are to one another as
7r+ 2e to TT- 2^', 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 FQ, 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 Sim'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
(43) The time through an arc of a parabolic orbit
bounded by a focal chord x (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) -Dj, Dg 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
^^TYT — ^7T^ ~ ^^^ 4 (sum of true anomalies) . tan \ (difference).
(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, /3 be the greatest and least angular velocities
iu an ellipse about the focus, the mean angular velocity is
Va + \//3 ■
(49) Find the maximum value of ^ — nt in an elHptic
orbit, and develop it in powers of e, shewing that it cannot
contain even powers.
If be this quantity,
^ ^ lie' 599e' ,
® = ^^ + 372^ + 5.2- + ^^-
(50) If P=/jLii'{l+k'sm''d)'^-, find the orbit,, and in-
terpret the result geometrically.
Find the equation of the orbit generally when P = fjLU^f{d).
(51) Shew that if the central repulsion be constant
(=/ suppose) we have the following relation between the
radius vector and the time,
f-.{ ^^^
Jj2f7-\r + a)-h''
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 pai'ticle moves about a centre of attraction, and
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 cc D"', 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 w 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 v''
(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
/ TCb ^ i
particle be a circle, shew that the periodic time oc f 1",
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 oc — , find the law of attraction and.
the equation of the path.
^^^- g)""=cos(ri-l)(^-a).
(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 r, 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
where is the initial attraction ; and that the periodic time is
27rr^
(r-a)V<^'
Avhere ^ is the attraction at the nearer apse.
(GO) If the m**" power of the periodic time be proportional
to the 11^^ 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 / ~^^ , and in a direction
making an angle sin"' - Avith the distance ; the intensity of
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
a point distant ?/ from the axis is proportional to ' — h y.
(63) If a body describes a parabola under an attraction
tending to a point on the axis, prove that the acceleration
/I 1 x"'^
at any point F is /^f -7yp+ Tp ) 0P~', p being the point of
intersection of PO produced with the curve.
Also prove that the time of passing from one end of the
ordinate through to the other = o a/ -•
CENTRAL ORBITS. 1G3
(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 DM . 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
2TpS;^ _ d / ,pdr\
Jr — p- dr V <^^/ '
For an attraction P to the pole, and a force JV in the
normal, prove that
2 d /,^ dr\ d / ^ o dr^
^dri^-^dj^^ZV^dTp.
0.
(66) A particle describes the ?ith 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
_6n+\
an attraction varying as r ^"-i, 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 = nr,
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 laAv as tlie centre of attraction did, shew that
the particle Avill 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.
LLV
(70) If P=— ^ ^3, and if the particle be projected
from an apse at a distance nc {n > 1) with velocity which is
to that in a circle as Jn'^ — 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 S, and an attraction/' to the focus
iT, and SP = r,HF=r',
dr' dr \r r
(72) Prove that if f=l^-^^i- , / =/^ "sTiV^' *^^
ellipse can be described freely, and that the velocity at any
T -T TV -\- T
point will be n ,^= — , n being the mean motion in
2jrr'
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 tlie perpendicular from the centre on the tangent,
and that the periodic time = r ( r + -) > ^«> kh being the
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 a/-^, /* being the strength
of each attraction.
(70) If a particle move under an attraction fir to the
point S, and a repulsion /*'?•' from the point S', prove that
odO , ,„ dO' „
a constant, where 6, 6' 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 w = - , is less or not less than 3, P
d losf u a
166 CENTRAL ORBITS.
being the central attraction, w tlie 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 (ii — ay, shew that ii = a
cannot correspond to an apse unless p be of one of the forms
^in + 2
Aim + 2 or -7T , . If the factor ii — a occur twice, then a
2?i+ 1
will be a root of the equation
> {u) - h\^ = 0,
where cf) {u) is the central attraction.
(80) Examine carefully the case of an apse where the
centre of attraction coincicles 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 — ^g"^^ with a vclocit}'- \ n being 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 kO,
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
(277 + 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 un-
known quantities in terms of the time, and the position of
the particle.
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 forces may be
resolved into two, X, Y, parallel respectively to the axes of x
and y, the mass of the particle being taken as unity. In
168
CONSTRAINED MOTION.
addition there will be B, 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 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
dec dy
and those of PR are
d'x d"
ds'
ds'
The equations of motion are
drx
d\v ,, „
de=^-^^PcU
•(1).
.(2).
dtr-^-^^Pd^-
These two equations, together with the equation of the
given curve, arc sufticient to determine the motion completely.
CONSTRAINED MOTION. 169
ff T flit
To eliminate R, multiply (1) by-^, (2) by — , and add.
We thus obtain,
dx ^x dy d^y _ _
ds ds' ds ds^ '
dx d?x dy d'y _ ds d^s _ ^.dx ^^dy , .
'dtdf'^dide~dtdf~Jt'^ ~dt ^ '''
or, as we may write it,
cZf ds ds '
which might at once have been obtained by resolving along
the tangent.
Now, it has been shewn in Chap. II. that if the forces re-
solved into X and Y are such as occur in nature,
Xdx + Ydy
is the complete differential of some function — ^ {x,y).
Integrating (8) on this hvpothesis, we have
M(sy-ci)]=^--^(-) (*)■
supposing V to represent the velocity of the particle at the
point xy.
Suppose the particle to start at the time ^ = 0, from a point
whose co-ordinates are a, h, with a velocity V.
We have, from (4),
and therefore -xV- = ^V'^ + (^{a,h) — <^{x,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 ot" the conservation of
energy.
171. To find the reaction of the constraining curve.
Resolving along the normal PR, towards the centre of
curvature,
V- „ ,, d'x _^ d''y
^ v"^ -,, d'^x _^ d^y
or It = Ap-r-r^— 1 p .': ,
p '^ ds' ^ ds-
which may also be written
p ^ ds ds '
This might, of course, have been obtained from (1) and
dj^x
(2) above, by multiplying them respectively by p -j^ and
p -r^ , and adding.
172. To find the point where the x>(i-^'ticle will leave the
constraining curve.
For this it is evident that we have only to put i? = 0, as
then the motion will be free.
This condition gives
v^ ,. d"x ,^ d-u
p ^ ds ^ ds
= F cos FPB,
if F be the resultant of X and Y.
Hence
lv' = FlpcosFrE
f\q,
where Q is the chord of curvature in the direction FF.
CONSTRAINED MOTION. 171
Comparing tins with the formula ^v^=fs (§ 82), we see
that the paii.icle will leave the curve at a point where its
velocity is such as luould he produced hy the resultant force
then acting on it, if continued constant during its fall from rest
through a space equal to \ 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 formulaB 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=Oand Y=-g\
and the velocity, and the pressure on the curve, are given by
if u = F when y = Jc;
v^ „ clx
P
Suppose we change the origin to the point from which the
particle's motion is supposed to commence ; and take the axis
of?/ vertically downwards; we shall evidently have
K-ir = 5r^;
and if the particle starts from rest
iv'=gy.
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 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
panicle will then either come permanently to rest, or return
to 'J le point from which it started.
and -=^'-ffds
1/2 CONSTRAINED MOTION.
174. To find the time of a particle's 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
j^ = v=^{2gy);
and t=\ -^^ (1)
if y, 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.
.(2).
175. To find the time of descending from rest at any
point of an inverted cycloid to the vertex.
O T 3C
Taking formula (2) ; since in this case the vertex i.> the
CONSTRAINED MOTION. 173
origin, and the axis is the axis of y, we have from the figure
s = 0P= 2 chord OF = 2^/{A0 . ON) = 2>^{2mj),
if a be the radius of the generating circle.
ds
dy
TT ds /2a
Hence, -^ = a / — ;
and .. .
^ /gf' dy
' Vyh'^iyy.-f
a _. 2yy'
- vers -^ '
5^ y.
Vg'
9
which is independent of y^ , 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
^ = -^sm(P Ox)
(since OP' is parallel to the tangent to the cycloid at P)
= -gsin{OAP)
OP'
^-^OA
^4a'
or the acceleration is proportional to the distance from the
vertex measured along the cycloid.
176. A particle, ujider 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
= V(2ay -?/').
174 CONSTRAINED MOTION.
ds a
Hence
dy ^{2ay-f)
But § = -V{2^(y.-2/)].
if we suppose the motion to be due to the level y, above the
lowest point ; and therefore
I. Suppose ?/, less than 2a, the particle will then oscil-
late, and we must put y = 2/i sin'^ (f>, and then
an elliptic integral of the first kind, of which
V(l-/.-'sin*(/))"
If the oscillations are indefinitely small, h = and K= W,
and the time of vibration from rest to rest is tta/-, and
therefore the time of a complete oscilhitiou is ^ir \/ - .
CONSTRAINED MOTION. 175
II. Suppose y^ greater than 2a, the particle will then
perform complete revolutions, and we must put
y = 2a sin^ ^,
which gives
*~ V^ioV(l-^^^sin»' "-y,^
and therefore = am * / - y ,
^ y ak
and y = 2a sn^ A/ j, 5
and the time of a complete revolution is
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 oi 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 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.
. m. 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= 6.
Suppose the motion to be due to the level BC: then we
must distinguish the two cases in which BC does and does
not cut the circle.
T. Suppose BC to cut the circle in B and C, and let
AOB = a; then the pendulum will oscillate through an
angle 2a.
The equation of motion will be
But
therefore
ere g . .
-.-^ = - -- sm 6.
dt a
CONSTEAINED MOTION. 177
Multiplying by -v- and integrating
KSf-iH—^ J ■ OL
and t = \/ - ~/7^i ,2 • 2 , \ > ^' = sm ^ .
V ^Jo\/(l - A;'sin''0)' 2
Therefore as before
, = am(y|U'),
sm ^ = A; snA/ - ^,
cos| = dny/?«;
therefore AP = ABsniJ^t,
'^V!'>
£'P= ^^ di
T. D. 12
178
CONSTRAINED MOTION.
f=V!cn^.
and
AM=AD^n'^U,
as before.
IT. Suppose BG not to cut the circle, then the pendulum
will perform complete revolutions ; let AD = 2/,.
Then, as before, resolving along the tangent
df a
IfdeV ri 9 a
and
and when ^=0,
\dt.J 'a'
1/dd
2
therefore l(^4)' = ^^i^-^il-co.e).
dt
CONSTRAINED MOTION. 179
Let AEP=(j), therefore 6 = 2(f>,
Therefore ^ = am f a/ - r , k\ ,
and ^il/ = 2asn^A/^7,
as before.
In the separating case BG touches the circle at its highest
point E^ and y^ — 2a, k=l ; therefore
V gjQ C0S)
dt
t
^^ log tan (^ + I
, (IT / 2ae -f- e' '
4
1 k
1 _2^/^
kri+k'
Lagrange's transformation is equivalent to
. . 2 Jk sin
^ l-f-A;sin^
which, for limits and sin"' y for 0, gives and sin'^r for (p;
CONSTRAINED MOTION. 183
and we thus have
dd
sin'^
whose application to the pendulum problem is obvious.
Proc. R. S. E., 1871— 2.
179. To find the pressure between the circle and the par-
ticle, or the tension of the string.
The reaction R being measured positively as a tension
between the particle and the centre,
R = — +qco^6.
a
In the figure of § 177, let AJ) = y^, BC being the level
to which the motion is due ; then
^v' = g[y,-(^0~- cos 6)],
and therefore
E = ^ 1?^ - 2 (1 - cos ^) + cos e
= 3.|cos.-^(l-|)}.
This expression for R admits of the value zero, if
2^-1)^-1, or y.:H?a.
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^ < 2a. 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
3 V a/ a
or 3/i -f: a.
,5
Hence, ii a „ a, and oscillations if y^ < a, without
change of sign of R, or without the string becoming slack.
Also by what we have before shewn, 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
y^ > 2a, the particle will revolve continuously.
180. Tvjo points being given, which are neitlier in a ver-
tical 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 a and a'. Then if a
particle slide down the projected curve its velocity at a' will
CONSTRAINED MOTION. 185
be the same as the velocity in the other at a. But a is never
less than a, and is generally greater. Hence the time through
a is generally less than that through a, 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
ds ,
'•^■^ dx .,. d[y
^ Applying the rules of the Calculus of Variations, we have,
since V or "^y --^P-l is a function of y and p, the condition
for a minimum,
dV
dp
the differential coefficient being partial.
V=p . + G,
This gives ^(^= / +C,
sly Vy\/(i+/)
^Jy\/{1 +p^) = z^= \/a suppose.
Hei
ds ^ J{l+f) ^ I a
dy p \/ a — y
the diflferential 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.
ot 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 j)Iane curve under
an attraction to a fixed centre in the plane of the curve ; to
determine the motion.
Let r =f(d) be the polar equation of the constraining
curve about the centre of force as pole, and let P= {s).
dt
'
{{k)-
is)]^
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
V2t
=/:
[j>{k)-{kz)f
and, that this may be independent of k, we must obviously
have 4> {k) - (^ {kz) = kj {z) .
This may be put in the form
<^(/c) ,ik) ^, , C
¥ =^^ k
^^''-C'^Z (2).
CONSTRAINED MOTION. 189
Or we might have proceeded as follows.
Put ^ = yirk, then f {k) - z'f (hz) =fz.
By differentiation with regard to k,
-f' (k) - z^yjr' (kz) = 0.
This shows that k^yjr' (k) is an absolute constant.
Hence, or by (2),
(fi' (s) = Cs.
Thus, by (1), ^ = -^^ 0^)'
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,
as as as
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 + Ydy + Zdz a complete dif-
ferential of three independent variables.
Generally
J V '
between proper Hmits, is to be a minimum; and therefore,
taking its variation,
ht
_ fvhds - dshv _
J v" ~ ^ ''■
190 CONSTRAINED MOTION.
But the equation of energy is
I v' = j{Xdx + Ydy + Zdz) ;
and gives vSv = Xhx + YZy + Zhz,
or dshv = {Xhx+Yhy+Z^z)dt (2).
Again ds^ = dx^ + dy^ + dz"^,
and -,- Ms = vMs = -,- 8o?a; + -^ Sc?y + -y- Mz (3).
dt dt dt ^ dt ^ ^
Hence (1) becomes, by (2) and (3), and since d and 8 fol-
low the commutative law,
\\, {Xhx + FSy + Zhz) dt
-G
dx ^ dy ^ dz ^
1 [dx ^ dy ^ dz
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
ga; = 0, hy=0, hz = Q,
and therefore the terms independent of the integral sign
vanish. In order that the integral may be identically zero,
we must have, since hx, By, Bz are indepeudent,
d fl dx\ . A'
dt\
(?S-^^° (*'.
CONSTRAINED MOTION. 191
with similar expressions in y and z. The elimination of t,
ds
and V or j- , from these equations will give us the two differen-
tial equations of the curve required, the forces X, T, 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
d /I dx\ X -.
ds \v ds/ V
r^d^x dvdx ^ .
''S'-v.s+^^='^ (^>'
with similar expressions in y and z.
Multiplying these in order by \, fi, v and adding ; if we
take X, fi, V such that
^ d^x d^y d^z ^
dx dy dz
ds ^ ds ds
A
.(6),
we shall have also
\X^tJuY+vZ=0 (7).
Now (6) shows that the line whose direction cosines are as
\ fx, 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 perpendicular
to the resultant of X, Y, Z.
Hence, the osculating plane at any point contains the re-
sultant of the impressed forces.
Again, if p be the radius of absolute curvature,
i _ /^V /^V (^\'
192 CONSTRAINED MOTION.
and its direction cosines are
d?x d'y d'z
Pd?' Pds^' Pd?'
therefore, multiplying equations (5) by
d!^x d'y d^z
d^" d?' 5?'
and adding ; noting that, since
we have
dw d^x dy d^y dzdz _
ds ds^ ds ds^ ds ds^
we obtain the equation
p=-[^Pdl^^^Pd?-^^Pd^^
or the normal component of the impressed forces in a Brachis-
tochrone is equal and opposite to the normal component 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 Brachisto-
chrone or a free path being the same, the tangential com-
ponent of the impressed forces must be the same, and there-
fore 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 Bi-achistochrones 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 repulsii)n from one focus
inversely as the square of the tlistance from the other focus
is a Brachistochrone, the circle of zero velocity being evanes-
cent.
= -(^^>rf? + ''''rf# + ^''3?) («)•
CONSTRAINED MOTION, ' 193
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
a^=o, %=o, a^=o
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
F{x,y,z) = 0,
we have
dF^ , clF^ clF
Also that [] may vanish, which is necessary in order that
ht may be zero, we must have
S^-l^^4?^=« <^)-
Now the only relation between hx, hy and hz is (1), to
which (2) must therefore be equivalent : hence
dx dy dz clF 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 equi-
potential, terms depending on ^x^, Sy^, Bz^, will appear in the
integral and must be taken along with { }.
If a terminal point is to lie on a given curve the condition
is to be determined in a similar manner.
T. D. 13
194 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,y,z) = (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, /i, v be its direction cosines, we know that
with similar expressions for /x, and v ; the differential coefiS-
cients being partial.
If X, Y, Z be the impressed forces on unit of mass, our
equations of motion are, evidently,
^, = X+RX
at I
l?=^'+^''[ '^)-
Multiplying equations (3) respectively by
dx dij dz
21' ^t' di'
and adding, wc obtain
dtdf^~dt df^dtdf 2 dt
-^-^>'t w-
CONSTRAINED MOTION. 195
R disappears from this equation, for its coefficient is
^ dx , dy dz
^dt^'^dt^'dV
and vanishes, because the line whose direction cosines are
dx
proportional to -j- , &e. being the tangent to the path, is per-
pendicular 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) wiU be of the form
\v'^^{x,y,z)+G (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 B, resolving along the normal, then
- = X\ + I> + Zi^ + i?,
P
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
df df 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
18—2
196 CONSTRAINED MOTION.
resultant of li 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
F{^,y,z)=fW{sr^' + y')]-z = 0.
This may be put in the foi-m
f{p)-z = Q,
if p be the distance of any point in the surface from the axis.
Equations (6) become
d'x d^y d'z
Jf_ _ df df~^
fipfp f'(p)l "^ ''
The first two equal terms give us, for the projection of
the motion on a horizontal plane, the equation
rfV d^x -
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
p'^-jT — const. = h (8).
If the motion be due to the level k, the second integral of
equations (7) is
ii©^(fy-©]-<-
CONSTRAINED MOTION. 197
Let u = - , and z = ^ (ii) be the equation of the surface ;
then
therefore
and differentiating with respect to 6, and dividing by
dO'
dhi q ,, f \ 1 ,f , \ d { „ ,, , . dii]
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 howl;
or let the particle he suspended hy 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' + y"" + z" - a^ = ()
is the equation of constraint, and the equations of motion are
d'x
df^
---
a
d^y_
df '
= -
a
d'z
de~
■-9
a
198 CONSTRAINED MOTION.
{(t)^(t)"-©]-(-^.) <^)-
„ 1 {ldx\' /dy\' , (dz\'
if the motion be due to the level z.
dy dx
""dt-'nt^^ ^^-
4 , dx dy dz ,.,.
^ ''Tt+ydr-'dt (■'>'
by the equation of the surface.
Squaring and adding (2) and (3),
dA' m' (dz\' '''"""'(a)
°^ [dt> '^[dt) '^{dtj — ?^^
and therefore from (1),
and therefore z 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 b — c, the cubic in z in the right-hand side of (4)
must have roots b + c and b — c, and if d be the third root
If we suppose the particle initially on the lower circle
and put
z = {b + c) cos'*
= 6 + c cos 2(f),
CONSTRAINED MOTION. 199
then z — b — c = — 2c sin^ cos' - c') cZ + {i^ - c' + 2id:) (26 + d)
= 2b{b + c + d){b-c + d),
or ^ = 4^?(6 + c+fZ)(J-c + (Z)
a' a" ^ ^ ^ ^
= 8 — 6 ^ -+^+^ (^^-c + ^)
T' b+c-d
_ K- {a + b + c){a + b-c){a-b + c){a-b- c)
~ r b{b + c-d)
and therefore
1 11 1 ^ K \a + h-c){a-b + c){a-h-c )
4ia:'{a+b+cy~ r b{b + c-d) (a + b + c)
K^ en" (m, + K) dn' (za^ + K) ^
r &n'{ia^ + K) '
and similarly,
Ih^ 1 _ A^ cn^m^dn^g^
4 a'"' (a — 6 — c)"'' ~ T^ su' 10,
202 CONSTRAINED MOTION.
Therefore
en (m, + ^ dn (z'a, + K) en ia^ dn ia^
d-yjr _ . sn (t'g, + K) . sn ia^_
dti~ 1 — K~ mi' (I'ttj + K) sn^u 1 — A'* sn"' ia^ sn* u '
and therefore, measuring -yfr from the lowest position of the
pendulum,
. (en (ia, + K) dn (in. + A")
'^ = '^\' \^ I-. , l^^ +
cnm dn?a„1
u
\ sn (w(j + A) sni'a
+ iTl (u, ia^ + A") + I n {u, t\)
d log sn (m, + A") cZ log sn ta,,
da^ da^
d log (=) (m, + A) ^ losr za,^'
c^a. d>
1 . , (w — taj — K) S {u — la^)
"^ 2 * ° (m + za^TX) (w + ia,)
=1
c?log jH" (/g, + A") d log iija.^
da, da„
10 (u - 1\ - JT ) {u - ia^
+ 2 * ^°« 0"(iM^a^ 4- A") {iL + za,)"
a,+a, cZlog0(a„A:O f^i^^I^'')L
'^tKK''^ da^ "^ da, J
1 ^0 (m - za, - A) (it - rnj
+ 2 * ^^0 (m + ia^ + AT) (« -h wj '
where u = Kjp , and A;' is the modulus complementary to k,
= Ji'k\
CONSTRAINED MOTION. 203
If we put -\/r = '^ — + -v/r',
, ^ a^ + a^ dlo^(t^(a, k') d\ocrH(a„,k')
"^^^^ K--iKK'+ da, +-^-^^-'
then "^ will be the apsidal angle, and ™ the mean angular
velocity of the vertical plane through the pendulum : also
-t/r', the periodic part of i/r, will be such that
and therefore
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 hori-
zontal plane and therefore in a circular path, the string de-
scribing a right circular cone whose axis is vertical.
Here z is constant and equal to h, c being zero ; and
therefore
dyjr h
dt ~d'-b"
where
2(7 26 '
and therefore
d^ /g
dt ~V b'
and the time of i
I complete revolution is
^^Jl
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 vibra-
tion, putting
, _ ,. 1 ,k' 4b
k = 0, A = -, TT, and - = , .^ , ., ,
' 2 ' c a- + W
. 2'Tra Jb
becomes — -=- . — ; ,
and the apsidal angle of the projection of the motion on a
horizontal plane
2vr , ^
Ja' + 3b'
192. To determine the nature of the small oscillations
executed by a particle, under f/ravity, ahoid a position of stable
equilibriuni 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, Pj be the radii of curvature of the principal normal
sections, and if the axes of x and y be tangents to these sec-
tions respectively, at the point of contact -with the horizontal
plane, we know by Analytical Geometry that the equation of
the surface in the immediate neighbourhood of the origin is
of the form
2^_^'_l'=0 (1).
P Pi
The equations of motion of the particle arc, as in § 187,
d'x
df-^'""
.(2)
COXSTEAIXED 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
, (fz
Hence A, = , [jb = —'^, v = l, approximately. Elimi-
nating R from equations (2), we have
d'^x
p
df
-M
.(3),
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-
straint may easily be produced. Three strings are knotted
together at the point C, the other ends A and B of tv.'o of
206 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 1)E respectively.
If we put - = 7^^ and - = ;?,^ the intefrrals of (3) are
P Pi '
x = A cos {ut + 5)| .
y = J^cos{7i^t + B^)\ ^^"
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=^CD; in which therefore
x = A cos {nt + B)) . .
y = A^coz{^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 ??j is nearly, but not exactly, equal to 2«, 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
Avhen 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 Foucavdt to point out, in February
1851, that the plane of vibration of a simple ponduluiu
suspended at either pole would appear to turn through 4
CONSTEAINED MOTION. 207
right angles in 24 hours — the plane, in fact, remaining con-
stant 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 pro-
cess, 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*^ 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, h, c, being the co-ordi-
nates 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 sus-
pension, are
(1).
Let us now refer the motion to axes turning with the
earth, but drawn from the point of suspension. If the axis
of 1^ be drawn vertically, and the axes of t/, f respectively
southwards and eastwards ; and if {z) (5),
the equation of the tube.
By means of (4) and (5) we may eliminate 6, 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 jiosition
of the particle at any instant.
R and R' may then be found from (1) or (3), and (2).
COXSTKAINED MOTION. 21 5
In the simplest case wiien the angular velocity of the tube
is constant, or -^ = o), (4) becomes 6 = o)t 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 imrticle 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.
rlf)
Here z = constant, -r: = constant = w, P = 0, and we have
dt
from (1)
whence r = ^e'"' + Pe~'"'.
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 ^ =
r = a = A + B,
and ^ = 9 « (t"' + e~"0-
In the figure, let OM be the initial position of the tube,
A that of the particle; OL, Q the tube and particle at time t.
Then OA = a, arc AP = atot, OQ = r, and we have
0Q = l0A(e'^ + e~^
From this we see that OQ and the arc AP are correspond-
ing values of the ordinate and abscissa of a catenary whose
21G CONSTRAINED MOTION.
parameter is OA. (It is not necessary for the tube to meet
the axis of revolution.)
Here, by (3), we have evidently R = g.
Also, by (2), R^-^""-^ (e-^ - e-"0 o)
From this equation, combined with the value of r, we
easily deduce
R = 2a)' V(r' - a%
and it is therefore proportional at any instant to the tangent
drawn from Q to the circle APN.
201. Supjyose the tube to revolve with constant amjular
velocity in a vertical plane about a honzontal axis.
We have from equation (1) of § 199
g cos (t>t,
if we conceive the tube to be vertical when t = 0. The inte-
gral of this equation is
r = Ae'-'^ + Be
■"'-©'-4
cos cot,
CONSTRAINED MOTION.
217
r = Ae"^ + Be-"'^ + -— cos wt\
and if
we have
and
or.
a,
0, when t = 0,
a = A + B + £,
= A-S;
which completely determines the motion. B and B' 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 ^ be the axis, P the position of the particle at any
time. Let POA = 9 denote the particle's position. The
accelerations of the particle in the directions ON and NP
being
d'ON , d'NP ,„„
dt^ df
therefore
=g — Bcos 0,
218 CONSTRAINED MOTION.
a — j^ &) a sin ^ = — it sin 6.
atr
Eliminating R
CI-T72 — aw^sin 6 cos 6 = —fj sin 6 (1).
The position of equilibrium will therefore be given by
sin 9 = 0, or 6 = y, where cos y = -^ .
Integrating (1)
(^y=C+2a>'cos7COs6'-a)'cos'^ (2).
I. Suppose the particle to be making complete revolu-
tions, passing through the lowest point with velocity awj ;
therefore
Ji3\ 2
-,- j = Wj^ — 2a)' cos 7 (1 — cos 6) + co^ sin^ 6
2
= co^ {(1 - COS 7)' + ^ - (cos 6 - cos 7)'|,
and -^ can never vanish if * > 4 cos 7, or w^ > -~ , that
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
^ A-1, , ;, /(r + lUs + l)
*^^ 2 = V .TT ^^ ''' ''^''''' ^ = V {r-l){s-l) '
where u=-Q)t\/[{s-\-l)(l-r)],
and s, r are the values of cos^ that make the right-hand
side of the equation vanish, s being > 1, and ?• < — 1.
II. If 0)^ < ' , the particle will oscillate through the
lowest point, and if ,, = 0, when 6 = 2, then
CONSTRAINED MOTION. 219
( .- ) = — (cos 6 — cos a) — (o^ (cos^ 6 — cos^ a)
= ai^ (cos 6 - cos a) ( -^, — cos a - cos 6]
^ ^ \aoi'' J
= o)^ (cos - cos a) (2 cos 7 — cos a — cos 6),
and therefore if
2 cos 7 — cos a > 1,
the particle will oscillate through the lowest point.
We must put
tan - = tan - en u, where k' = cot ^ J 2 cos 7 — cos a - 1,
and then m = w^ \/(cos 7 — cos a).
III. If 1 >2cos7 — cos a> — 1,
then putting
2 cos 7 — cos a = cos /3,
\-T-\ = " (cos Q — cos a) (cos /8 — cos 0),
and the particle will oscillate on one side of the vertical
diameter.
We must put tan - = tan - dn u, or tan - = tan - -5 — ,
^ 2 2 2 2 dn M
and then u = (at sin ^ cos - , ^•' = .
2 2 , a
tan^
203. To find the form of the tube in order that the particle
projected ivith 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^s „ dx dy
dt ds ^ ds
220 CONSTRAINED MOTION.
Hence, (^'^' = xW - 2gy + C.
ds
But -r. = constant.
at
2(7
Hence, x^ = ■-[ (v + k),
to
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 : Sub-normal. 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 iu Hydrostatics.
204. A particle moves on a rough curve, under given
forces; to determine the motion.
If //, be the coefficient of kinetic friction, and
be the normal reaction, friction will cause a resistance
fj, ^/ (R^^ + R^^) acting in the tangent to the curve in the
opposite direction to the particle's motion.
Equation (1) of § 182 will therefore become
the other two equations remaining the same.
If from the three we eliminate R^ and R.,, 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.
Here
Now
dv d's
'" ds~de'
hence
dv v-"
v-r =- a-:
22X
or v = ae -I P.
But, if yjr he the angle which the tangent at any point
makes with a fixed line,
d.s
d^^P-
Hence, v = at' , where a is the velocity when ■>/r = 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
P
successive tangents. If the surface of which the curve is the
cuspidal edge be developed, and if ^ represent the angle
between the tangents corresponding to the initial and final
positions of the particle,
206. A particle tinder 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 [juR, and the equations of motion
become
^, = X + R\-fji'R'^
dt ds
d^y T^ T, i-ndll
^ = Z + R.-^R^,
222 CON'STRAIXED 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 x, 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 Avhich vibrates seconds at a place A
gains n beats in 24 hours at a place B; comjiare gravity at
the two places.
(3) Prove that a seconds pendulum when taken to the
top of a mountain h miles high will lose 2rG/i 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 iKiight 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, mp perpendiculars on L.
Let rtip cut A again in q and B in r, s.
Then by Geometry,
mp .qp = rp. sp,
and therefore mO^ = mr . ms = (mp — rp) (mp — sp)
— mp (mp + qp — rp — sp)
Similarly
2a . mp = - (velocity of m)^.
2a . m'p — - (velocity of m'f.
Hence the velocities of m and m' are as m.0 : Om', and
therefore by what we have shewn above about elementary
arcs at m and m', the proximate position of mm' is also a tan-
gent 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 jDlane ; prove that during the motion the beads
will always be at the same level.
(1.3) 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 acosec'^;^a, the velocity being due
to
the level 2a cosec" ,^ a, above the lowest point.
(14) A bead can slide on a smooth circular arc AB and
is attracted by it, with intensity / (/) ; if it bo displaced
CONSTRAINED MOTION. 225
from its position of equilibrium, the time of oscillation
will be
27r
y
2 cos ^/(^ (7)
where C 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 Jga 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 27r a / ^ .
V Zg
(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/ -j- ; prove that
the angular velocity of the string will be constant, and that
the particle will describe the lima^on
T = a {■ii — cos 9).
(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 (/<) above the j^oint, 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 j)oint 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
"t 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
• f 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
the length of the curve between two such points being
Determine the nature of the evolute of this cur\'e, 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' +6^
6a^/3
above the centre, a and h being the scmiaxcs of the ellipse.
CONSTRAINED MOTION. 227
(21) A particle is initially at rest at a point of the
equiangular spiral r = ce"^, distant d from the pole. Shew
that if the pole be a centre of attraction = y ^ , 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~^ situated at a distance h
from the centre of the circle. It is projected from the
nearest point with velocity V, shew that for a complete
revolution
4/i6
7^
-h'
(2-i) 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
flF
where -,- is the acceleration due to the attraction of any one
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 pai'ticle of mass m moves in a smooth circular
tube of radius a under an attraction m/i 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
15—2
228 CONSTRAINED MOTION.
from tlie centre of attraction, prove that it will pass over the
({uadrant ending at its least distance in the time
y
~c^^Si>J'-^+^)
(2G) Shew that a particle moving under gravity on a
smooth helix whose axis is vertical, makes the first revolution
from rest in the time
V.
g sm za
(27) A groove is cut on a right cone of height h, making
an angle /3 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 J~k^
J\
g cos a cos /i '
where a is the semivertical angle.
(28) A particle under a central j-epulsion 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
a;3 + 7/3 = a^.
(80) A particle P is attached by strings to two points ^1
and B in the same horizontal plane, and elescribes a vertical
circle. When the particle is at the lowestpoint, the string ^4i^
is cut and the jjarticle proceeds to describe a horizontal circle ;
find the ratio of the new tension of BV 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
CO >s STRAINED 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 (J'S + 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 tlie same as that of a pendulum
of length
3-v/3
-3~"-
(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 ^4^ 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 do\\Ti any chord AC ox CB
drasvn in the smaller of the two segments into which AB
divides the circle is constant.
230 CONSTRAINED MOTION.
(3G) 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 tlie pole and towards the pole ; shew-
that it will arrive at the pole in time
V cos a — fM sin 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 s, v^ = 2/j,gs.
(39) A particle descends a rough circular tube from the
extremity of the horizontal diameter. If it stops at the
lowest point, shew that
3/xe-'^" + 2yu,= = 1.
(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 wliioh
passes over a rough circular arc in a vertical piano ; 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 fi the coefficient of friction,
(42) A rough wire in the form of an equiangular spiral,
whose angle is cot~^2/A, 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~V) and that the
velocity is greatest when the angle which the direction of
motion makes with the horizon is given by the equation
(2/jt,^ — 1) sin (}) + 2/ji, cos*ds.
232 CONSTRAINED MOTION.
(4G) A circular tube of small bore revolves with con-
stant angular velocity tu 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 ^ 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 &> : shew that if V= the
velocity of P, w = the angular velocity of the plane PSA
about 'SA, ) = / FSB, 6 = ^ PSA,
^V^ = (ja cos l, 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 ?2. = 3.
If = diameter of shot in inches, ?^= number of pounds
in the shot, then the retardation due to the resistance was
(P d^
put = 10"^ ~ /lv^ so that Jc = 10'^ — K, and K was deter-
^ w to
mined by experiment for velocities proceeding by increments
of 10 between 900 and 1700 feet per second, K attaining its
maK.mum value for a velocitv of about 1200.
240 MOTION IN A RESISTING MEDIUM.
The numerical values of A' 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.— XL), giving ^ s and - t for every
decrement of 10 in the velocity between 1700 and 000, 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
5 in terms of t.
210. There is one case in which the above solution fails,
namely when n=l, or the resistance varies as the velocity.
In this case k is the reciprocal of a time and may be put
= -, and then
dv V
T
Tt--^ W-
»S = -T (-'•
Hence '»S^'^ = ; (S);
ds _?
and therefore v = ,- = Fe r .
at
t
Integrating, we have s = Vr{l — e '') (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 imssed over by the particle
has a defmitc limit, for when
t = x , s = Vt.
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 f 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 the
resistance might be equal to/, then the retardation when the
v^
velocity is v may be represented byy*-™.
Let / act so as to diminish x\ then the equation of
motion is
Integrating, and determining the constants so that when
a; = 0, « = 0, v=F,
we obtain
•'yyr = tan ^— tan -v>= tan -=— — ^ ,
K K K K^ + Vv '
Let T be the time at which the velocity becomes zero,
and h the corresponding value of x, then
T=|tan-|.,and /, = |. log (l + -^.j .
T. D. 16
242 MOTION IN A RESISTING MEDIUM.
After tjiis the particle begins to return, the resistance
therefore tends to increase x, and the equation of motion is
dt K' ^ ^ ^'
Integrating, and determining the constants so that when
^ = 0, x=h, t = T,
we obtain
- (/l-^)=l0gT^^
Let U be the velocity with which the particle will return
to the point of projection ; then, putting x = in the latter
equation, we obtain
IP
%fh
or, substituting
for h its va]
ue,
72
U'
IP
K'
whence
1
1 1
IP-
- Y^ IP
This shews, as we might expect, that the particle returns
to the point of projection with diminished velocity.
212. The results of the last Proposition are applicable to
bodies projected in a resisting medium vertically uj)wards or
downwards under gravity; for the acceleration due to gravity
MOTION IN A RESISTING MEDIUM. 243
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
= Tr(l -«) = %(! -a),
where W and AI 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.
213. To find the equations of motion, in a resisting me-
dium, of a jpai-ticle under any forces.
Let X, y, ^ 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 tan-
gent to the path described, and in a direction opposed to the
motion, we have
d^x _ „ p dx
dt' ~^^~^^Js'
dt' ds '
d'z _ dz
df-'^'-^ds-
These are the general equations of motion. In any par-
ticular case R will be given as a function of the density of
the medium and the velocity of the particle, and particular
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 the resist-
ance that a given path may be described.
IG— 2
244 MOTION IN A RESISTING MEDIUM.
214. A particle under gravity is projected from a given
r)oint 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 x horizontal,
the axis of y vertically upwards, so that the plane of xy may
contain the direction of projection ; let g denote the accele-
ration of gravity, v the velocity of the particle at any point,
u its horizontal component, j> the inclination of the direction
of motion to the horizon, and R = lev" the retardation due to
the resistance.
Then the equations of motion are, resolving horizontally
and vertically,
f=-4^ «-
^-^-^t (^).
or, resolving in the direction of the tangent and normal,
d^s
-^^,=-gsm-R (3),
v"
-=gcos(f) (4).
Since v = --t-, u = vcos^ and p = - , , equations (1) and
(4) may be written
, =- Acos0 (o),
d6
^'^y = -i7cos) (6),
and therefore
dii _ llv
d4>~ 9
U
= ^"""sec-, (7).
MOTION IN A RESISTING MEDIU.^^. 2-i
Integrating this equation, denoting by Uq the velocity at
the vertex of the trajectory,
1,^_1=^>^, (8),
if P^ denote the integral / n sec""*^^ (f)d(f).
Therefore u = — ^ ,
, _ ku* _ retardation at vertex
ff acceleration of gravity
_ resistance at vertex
weight of shot
From equation (6)
dt V , u „ ,
~ri = — sec 6 = — sec d>,
d^ 9 9
therefore, if a be the angle of projection,
^^t.,r<^ sec^ ^^^^
Again, x = \ udt = — sec* ^dcf)
Jo J 9
(10),
(i-7^*r
and ^^Vptan^sec^^ ^^^^^
^^^ a-yp,f
Equations (9), (10), (11) give t, x, y in terms of 0.
For n = 3, P = 3 tan > + tan' <^,
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. VL
in Mr Bashforth's " Motion of Projectiles."
240
MOTION IN A RESISTING MEDIUM,
215. For n = 1, putting R = - , then t is the measure
T
of a time, and
P= sec^ (pdcf) = tan cp,
Jo
7 tan (f)
u^ r* sec^ (i)d(f)
_^o,_l-7tan
ana since
(/7 ° 1 — 7 tan a
, 1 - 7 tan (^
t = T log _— r
1 — 7 tan a
(1);
:^(f)d(f>
" U-7tana 1 -7 tan ^'^V..(:3)-
'tan0 ° l-7tauay ^ ^'
(an
and the elimination of tan (f) between (2) and (3) will give
the relation between x and y.
Or immediately, resolving horizontally and vertically,
(^)
fPy _ 1 di/
d'x _ \ dx
df T dt
de T dt
-9
(•5):
MOTION IN A RESISTING MEDIUM. 247
and integrating, supposing V the velocity and a the angle of
projection,
^^ = Fcosae .,
^ + (jrT = (Fsina + <7T)e"T,
and integrating again,
X = Ft cos a (1 — e '") (6),
y+gtT = {VTsma+gT'){l-e~'') (7).
Eliminating t between (6) and (7),
, , Vt cos a / QT \
y + gr' log -^ = tan a + -^/ x,
•^ '^ ° Ft cos a -a; V Fcos a/
the equation of the trajectory.
Differentiating this equation twice, we obtain
<^+ a^ =0
da? ( Vt cos a — xf
the differential equation of the trajectory.
-I?
216. For 71 = 2, puttinor Ji — — then a is the measure
^ ° a
of a length, and putting -p = tan ^,
P=[ 2sec' = 2Ul+fdp
=2)Jl+p' + log (p + N/r+7/).
The equations of motion are, resolving horizontally and
vertically,
d^x 1 /ds\^ dx .- ,
df a \dt ds
df~ a\dt) ds ^ ^ ^'
248 MOTION IN A RESISTING MEDIUM.
Equation (1) may be written
du _ \ ds
dt adt '
1 du _ 1 ds
u dt~ adt'
and integrating,
M = Fcosae " (3).
From equation (8) of § 214,
01" e - i = /tan a sec a — tan j> sec ^
, tan a + sec a \
+ 'o=ti^^+se^^) W'
the intrinsic equation of the path.
Differentiating this equation with respect to x,
a dx ga dx ga '^ ^ dx '
l+re^^''=o (-^x
the differential equation of the path.
If >Sf, s denote the arcs of the trajectory in a non-resisting
and a resistmg medium, measured from the point of projec-
tion to any two points at which the tangents are parallel ;
then, smce in the non-resisting medium a = oo , therefore
dp q o'
J::~~ JrT — T~ 6 " cos (A,
ds V cos'a ^'
dp _ g .
dS-'V-'co^^'oi''''^'^'
and J =e\
ds
MOTION IN A RESISTING MEDIUM. 249
and integrating,
fy ^ 2'-
^ = ^(«"-l) (6).
217. For a flat trajectory, p being always small, we may
ds
put T- = 1, and then equation (.5) may be written
dx
Integrating,
dx V^ cos^ a dx
ga i'- , qa
P + cnfi — 2~ 6 — tan a + ^pfA — 2-
2V
^ cos^ a
or e « — 1 = (tan a — p).
ga
And substituting again in equation (5)
aa; a k cos a a
Multiplying by e~'" and integrating,
^ 2F^cos^a V2F'-cos''a /
^ dx 2F-cos^a 2 F^ cos a
And integrating again,
y = x (tan a + ^r^ — s- ) - ttt? — ^ (^'" ~ !)•
^ V 2F'cos'a/ ^F^'cos-a^ ^
Expanding e " in ascending powers of - , a being sup-
posed large,
_ ^ra?'' gx^
y-xt^ucL-^y,^^^,^ g-p^^^ ,
of which the first two terms will represent the trajectory in
a non-resisting medium.
250 MOTION IN A KESISTING 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,
...(1),
1 d 1 ,d0\ „rde
rdtV'di)'-^^
...(2).
7/1
Putting r^ -J- = h, equation (2) may be written
dt
dt ds
Idh R
hdi = -^'
and therefore
/,=;v-^'*
...(3).
Or the equation may be written
ldh__R_
h ds v' '
and therefore
h = h,e-^>
(4).
. . . 1 , de J ^
Again, puttmg r = - , we have ^ = «" .
dr 1 du 1 dndO J du
^""^ di~~tL'dt~ u^dddt~ dd'
d'r d\i dO dh du
df^ ''dff^dt'dtdd
, „ „ d'^a , R du
dd V dd
72 2^W , T>^ ^^
MOTION IN A RESISTING MEDIUM. 251
and therefore "
ds
d'u , P ,.,
^^ + ^* = ^v ^'^'
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, ^\e shall obtain
the equation analogous to (5) most simply by resolving in
the normal, and then
- = Psin<^-f- Tcos (/),
where
^ , rdd ud0
tan^ = -^=-^.
Therefore
PH-rcot^=p-T^f;;
p sm (/> Vc/^ '
""^ dd''^''~h'a' fi'u'dd'
an equation of the same form as that obtained in § 136.
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.
it + ki'i -^ n^u = 0,
may be put into the form
, sm n't ( . k .
where n'^ = n^— ^k''-, and i\, u^ are the values of the velocity
and displacement when t = 0.
Deduce the complete solution of
il + ku + ji^u = U,
in the form
M = e-i^-<|(f^ "^ — y +uJcosnt + ^ , sin n'=9
(n.l)-(7, + a)c-'-}
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 t^, t^ 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 /{y), r
beincr the velocity and a/t the inclination to the horizon of the
direction of motion :
vr cos -vlr — ?; sm -vlr = - / iv),
d^ ^ i/
d±
(it
V — ,V = — COS yjr.
Prove also that if -^^ is the initial value of yfr, ami t, x, y
the time and horizontal and vertical distances from the point
of projection,
gt= \ V sec -v/^f/-^,
u
['Po
gx= \ v^ d^lr,
fPo
gy = I V tan •\|rrti/r.
Solve completely the case for which
MOTION IN A EESISTING MEDIUM. 255
(10) If the horizontal distance of a projectile in a re-
sisting medium from the point of projection be connected
with the time by the equation x =f(t), prove that the equa-
tion of the trajectory is
where A and B are constants.
In the case when t = ax + hx^, shew that the equation of
the trajectory is
/I 2 1
y = x tan a—gi^ aV + ^ (^^'^^ + o ^^-^^
(11) A particle moves under gravity in a medium in
which the resistance varies as the «"' power of the velocity,
V^, Fj being its velocities at the two points where its direction
of motion makes an angle (f) with the horizon, and V its
velocity at the highest point ; prove that
J_ 1 _ 2 cos" (f)
(12) If the resistance vary as the w"' 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/' when it is descending at an inclination > in
the trajectory, prove that
1 1 _ 2 cos" (^
/ / I~'
]__1 _2
/ f~~9
cos" ^ I ?i sec""^^ (f)d(f).
J n
(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 — ^ — m, 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 j)ath 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
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 E, is
eV
where S is the sun, H the other focus of the orbit, e the
eccentricity, and V the velocity.
(IC) 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 EESISTING MEDIUM. 257
(19) The retardation due to the resistance of a medium
being kv^, 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^ik.
(20) Shew that the equation
^ + 2^ ^ + 71^^ = n'P (1 + 2Srcos 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
. 27r
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 ^^a={n^ — i^p^) cos ipt + '2.ikp sin ipt
2nk g.
Q I J. — c \syj^ ^' I oijj /&,t/ T' c oAii. COS ll^t
pn^
1 — e ^ cos sm n.t + e ^ sm
eJ ! L
27rfc ^ iTTk
1 — 2e ^ cos i+e
P
9_
where n, = J 71^ — F, 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
pendulum when unresisted beiog - , prove that if the resist-
ance of the air produce retardation 2/i 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
-atano
(■rtana --tana\
e^ —e~ /,
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
9
-: X distance. Prove that the particle will describe a para-
T
bola, tending to come to rest at the origin.
(24) If a particle under a central attraction producing
acceleration /tV move in a medium of which the resistance
produces retardation 2^■ (velocity), prove that it will describe
the curve
(25) A particle moving under gravity in a medium, the
ardation due
plane down the curve
v
retardation due to whose resistance = - , slides in a vortical
b a
where 5 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; .'^hew that the time of
reaching the lowest point is independent of the height from
which it starts.
MOTIOX IX A KESISTING 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 point
from the vertex is c ; prove that the time to the cusp is
a/ — [ — — 1 j , 2a being the length of the axis.
(27) A particle moves in a resisting medium ; state any
reasons, arising from the principle of the conservation of mo-
mentum, 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^^ 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 a small
retardation — , prove that before it next comes to rest energy
has been dissipated which is ^r- of its original value.
17—2
( 260 )
CHAPTER YIIL
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 demonsti-ations,
though these are certainly superfluous.
221. If a particle he suhject to forces, whose resultant is
continually at right angles to the direction of motion; the
velocity of the particle tvill be constant.
Let R be this resultant, X, yit, v its direction cosines, then
if VI be the mass of the particle, the equations of motion
are
dt
™l'=''-^-
(r) = C - V.
Hence, if U be the velocity at a point whose distances from
the centres are i2j, R.^, , and where V= F^,
1 ,^ -^U' = Scjy (E) - 2<^ (r) = F, - V,
or hv''+V=W'+V^,
a result which involves only the co-ordinates of the initial
and final j)ositions. See, again, § 78.
224. Hence if from any point of the surface
V=^(}>{r)=A,
a particle be projected with a given velocity in any direction;
its velocity when it meets the surface
V= l (r).
The above process gives, in this case,
vdv = Xdx + Ydy + Zdz = d(f) (r)
X -, ?/ , z
= <}>'{o-)l-^dx + -^^dy + -dz
r r " r
X Y Z
or - = — = -
X y z
which shew that the force is in the direction of t.
GENERAL THEOREMS. 265
From this again it evidently follows that its magnitude
must he a function of r,
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 their
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 inde-
pendent of the direction of projection. Hence, by considering
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 he 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
m^, m^, (^c. subjected to forces f^, f^, dx. respectively, and
successively projected from the same point in the same direc-
266 GENERAL THEOREMS.
tion with velocities v,, v^, etc. all describe one path; the same
path will also be described by a particle of mass M projected
with velocity \J from the same point in the same direction, and
acted on at once by the same forces f^, f^, dx. provided
Suppose that, in addition to the forces f,fz, &c., a force
R continually acting in a direction at right angles to that of
il/'s motion be required to cause it to move in tlie given
path; i.e. suppose J/ to be constrained by a smooth tube to
move in the required path ; the equations of motion are
il/^ = SmX+i?\ (1),
with similar equations in y and z,
where \, /i, v are the direction cosines of R, and A', Y, Z the
resolved parts of/
Multiplying by ^^ , y ; , -j- in order, and adding, we
eliminate R and have
\ Md{U"-) = tmXdx + SmYdy + tmZdz.
But for the separate particles m^, m„, &c. we have
^ m^d{v^) = m^X^dx + m^Y^dy + m^Z^dz, Szc. ;
therefore, the path being the same for all,
I ^ [md (v')] = tmXdx -y SmYdy + tmZdz.
Hence S [md{v')]=Md{U''),
or t{mv-) = MU-'+G.
But, by hypothesis, 2wy*= MU^,
therefore C = 0.
GENERAL THEOREMS. 267
[Instead of this analj^sis, it is sufficient (by § 78) to notice
that the work done on M is the sum of that done on m^, m^,
«fec. 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^, 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^, m^,
&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 velogity. Now the latter part is as the kinetic
energy, therefore in the case of M it is the sum of the cor-
responding forces in the case of 7??^, m^, &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, he 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, ^yhence the proposition.
230. Least, or Stationary, Action, If v be the velo-
city of a particle whose mass is in, and if s be the arc of the
path described, the value of the quantity
A =mjvds
(taken between proper limits) is called the Action of the
particle.
GENERAL THEOREMS.
If a particle move freely, or on a smooth surface, (under
forces such as occur in nature,) bettveen any two points, the valve
of the integral m/vds for the whole actual path is generally
less than it tuould be if the particle were constrained to pass
from one point to the other by a different 2Mth. This, com-
bined 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, h being the
symbol of the Calculus of Variations, and the mass of the
particle being for simplicity taken as unity,
hA = h^vds = 0.
Now BJvds=JB (vds) = J {vMs + ds Bv)
f ds
I {vBds + vdtSv), since v = ," ,
But generally,
^ v''=J{Xdx + Ydy + Zdz) = y{r {x, y, z),
the constraint, if any, having disappeared ;
hence vhv = Xtx + YBy + Zhz.
But X=^]f-i?X,, &c.
Hence
vhv =(^8x+^,Sy + ^Bz)-R {XSx + ,.By + vBz).
Now if the particle remain on the surface whose equation
isi^=0,
\Bx + fiBy + vBz = kBF= 0,
and if it leave it 72 = 0, so in either case the latter term on,
the riijht vanishes.
GENERAL THEOREMS. 269
Also ds" = dx' + dy" + dz";
which gives dsMs = dxBdx + dyMy + dzSdz,
or vBds = -^ Bdx + ^ Sdy + -rr hdz
dt dt ^ dt
since the order of d and 8 is immaterial.
Hence
=B^^4^^+sH
taken between proper limits. Now at both limits
Bx = 0, By = 0, Bz = 0;
hence we have BA = 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, BA = 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 J.. 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 THEORE.MS.
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 corpiLscular 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 I^Q 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 72 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
hij a jMrticle about a centre of attraction.
GENERAL THEOREMS.
271
Let P be the central attraction at distance r, then
\v'^=C-!Pdr,
= 2^'l>{^)f> suppose, (1)
which gives
^vds = /0 (r) ds.
Hence
= 8J(/) {v) ds
= J[(r)dSs}
=j\^lf^ i^^x + y^y + ^S^) ds
/
■f (r)
(a:;Saj + yZy + 280)
«.^{^(,.)f}-a,^{^(,.)|'}-&j{^(r)|
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
^-^^-s>(4:H«'
with similar equations in y and z. These may be written
X dr dx\ . , , d^x
■#>'('■)
z dr dz
r ds ds
*Mg = o
.(a).
272 GENERAL THEOREMS,
Multiplying by any three constants, A, B, C, and adding,
we have
which is obviously satisfied by
Ax + By + Gz = 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 x and the first by y and sub-
tracting, we obtain
This is immediately integrable, and gives
>(0(^f-y^) = constant.
Since ^ (r) = v, we see by § 24, that this is in polar
co-ordinates
vp=^r^f^=h (6),
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
^^»t-(^)}-*(4^-^S0=« w-
But, since
dr clx du
dn ds ^ ds '
GENERAL THEOREMS. 273
we have by dififerentiation
Substituting in (c), and changing the independent variable
from 5 to ^ by means of the equation
d^' = dj^ + r'de\
we have
'{r) = -P, by (1).
Thus {d) becomes
^ + t.= ^. asm§13o.
236. We might have treated these equations (§ 235 (a))
somewhat differently thus
Hence )•
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
dA dA dA
dx ' dy ' dz
GENERAL THEOREMS.
277
dA d'A . dA d'A
dxd\
d'A
.. __ dA d'A
dx dx^ dy dxdy dz dxdz '
dz d'A
represent the rectangidar components of the velocity of a
jyarticle in a motion jjossible under the forces whose p)otential
is v.
For, by partial differentiation of § 239, (1), we have
di!^ ~ ^ dx
d fdA\ ^ dx d'A ..,
dt\dx) dt dx^ dt dxdy dt dxdz'
Comparing, we see that
dx _ dA dy _ dA dz _ dA
dt dx ' dt dy' dt 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 corresjwndiny path, and the time of its
description, are given hy
where a^, /S^, e are three additional constants.
For these equations give, by differentiation,
di'A dx , d'^A dy . d^A dz
dydoL
d'A dy
t + e,
dxdoL dt
d^A dx
dt dzda dt
d'A dz
dxdfi dt "^ dyd^ dt "^ dzd/S dt
d^A dx d^A dy d^A dz
•(«)•
dxdH dt dydHdt dzdHdt
But, differentiating § 239, (1), we get
d'A_ dA.d'A^dA _(fJ. dA ^ ^
docdx dx dady dy dadz dz
d'A^dJ^^^A^dJ^ d'A dA^^
d/3dx dx djSdy dy d^dz dz
d'A dA ^ d'A dA ^-^±dA^^
difdx dx dHdy dy dHdz dz ~
.{h).
278 GENERAL THEOREMS.
The values of ^ , »S:c. in (a) are evidently equal respec-
tively to those of ( J-] , &c. in [h). Hence the proposition.
243. Equiactional surfaces, i.e. those whose common
eqiiation is
A = const. = C,
are cut at right angles by the trajectories.
For the direction-cosines of the normal are obviously
proportional to (^) , Q , (^) , that is to ^^ , J , ^^ •
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 OT be the normal distance at any point between the
consecutive surfaces
A = C,andA=C+Sa
We have evidently
clt dt ^ at
where hx, Bi/, Bz 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 = vcs = 8C,
since pcoad 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 vhs (where hs, being an element of the path, is the
normal distance between the surfaces) and must therefore be
equal to hG.
244. To deduce, from the principle of Varying Action,
the form and tnode of description of a planet's orbit.
dV
In this case it is obvious that — -jy represents the attrac-
tion of gravity ( — ^)- Hence the right-hand member of
§ 239 (1) may be written 2 f^ + ^V
Let us take the plane of xy as that of the orbit, then the
equation § 239 (1) becomes
H^hQ'-^i^-^) (1)-
It is not difficult to obtain a satisfactory solution o" this
equation ; but the operation is very much simplified by the
use of polar co-ordinates. With this change, (1) becomes
m^m-'i^-^ (^>>
which is obviously satisfied by
^dA^
Hence
constant = a
^=.e+/rfr^2(ir+^)-^: (4).
280 GENERAL THEOREMS.
The final intc^als are therefore, by § 242,
fdA\ . r dr
m^a=e-a\ ^^^' (5),
and
&-—l^;f^^ <«■
These equations contain the complete solution of the
problem, for they involve four constants, a.^, a, U, e. (5)
gives the equation of the orbit, and (G) the time in terms
of the radius vector.
245. To complete the investigation, let us assume
2H_e'-l
a^ ~ I' '
where I and e are two new arbitrary constants introduced in
place of a and H. With these (5) becomes
«. = ^-/-
dr
: A'^-l 2 1
v-ir^ir-T-
1_1
r /
e
I
I
1 + e cos 1^^ — 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
v<
=^d9.
r) r^
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 d^, 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),
=1
'^ + H\dr
'V'
^(^^-^
.(7),
because, by (5),
To integrate (7), remark that (§ 149) -^ < ^ in an ellipti
orbit, and that thus -H"is negative by § 244 (]).
282 GENERAL THEOREMS.
Put -^=-2a
11
fJLCL
and r = a {l—e cos ^),
and (7) becomes, after substitution,
+ e cos ^) (Z^,
which is immediately integrable.
It is obvious from § IGO that <^ represents the excentiic
anomaly. If we measure it from the perihelion we have
evidently
A = J ijua ((/> + e sin ^).
247. By (6) we have
dr
-I
V<^^
By employing the same substitutions as in last section,
<^ being measured from perihelion, it is easy to bring this
expression into the form
t = f./ — \ {1 — e cos (})) (1(f)
= y ^^[<^-esin<^],
the formula of § 160.
248. By the process of § 160 we see that while
—e sin -y^-
Making x and y constant in (1) and (2) we get their
equilibrium values; and measuring x and y from these Ave
get
Thus if
GENERAL THEOREMS. 287
' I 2ma' '
we have x + 2/ = A cos (nt + B),
x — y = A^cos {nj, + BJ.
It depends upon whether the proximate poles of the
magnets attract or rej^el 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
[dxj
dT\ _ 1 dx (dT\ _ l\dx
fdT\__ idt__[ds
[dHj ~ J v'' J v''
corresponding to the group in § (238).
288 GENERAL THEOREMS.
Hence, just as in § (241) it may be shewn that for any
forces, of which V is the potential, a value of t from the
equation
drV /M" . /dry 1 1
ly) "^ U)
M \dy) \dzl r 2{H-Vy
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 T contain, besides H, two arbitrary constants, a, /?,
the equations of the brachistochrone are
(1)=^-
253. To find the Brachistochrone when the attraction is
central, and proportional to a power of the distance; the velo-
city being also proportional to a poiver of the distance, that is,
being the velocity from infinity, for an attraction, from the
centre for a repidsion.
Here v' =2{H -V) = ^„,
and the central attraction at distance r is evidently
dV _ nfi
Thus (2) becomes
or, changing to polar co-ordinates,
\dr) ^ r' \dd) ^ 7'^ sin'^ 6 \dj>) fi '
It is obvious that we must take
©-■
GENERAL THEOREMS. 289
Avhich shews that the path is ia a plane passing through the
centre offeree. The above equation will then be satisfied by
fdr\ _ fdT\ _ /?" a^
[ddj-''' [dr)~\/ ]I~ ?'
r 2
And the equation of the brachistochrone, which is evidently
a plane curve, is
Hence we have
^ = 0+ — r-s J A / — ^ - 1 - cos ' ^^„
r
VS- ^^/^
iia \ r
ixa
=; cos „ , „ .
n + 2 ""-zl
r '^
or
/- w + 2,.
while the equation of the free path is
2
-j 2 =COS^-(^ + ^).
The above integration fails in the case of n = — 2; that is,
for a rejiulsion directly as the distance, the circle of zero
velocity being evanescent. But in this case
T = a^+Y/^-aMog^,
T. D. 19
290 GENERAL THEOREMS.
and the equation of the brachistochrone is
V /x
the logarithmic spiral. Eliminating r between these equa-
tions, "\ve 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 {jji) '
254. It is easily seen that
r=C
is the equation of an Isochronous surface.
Also, since
,dzJ
dx dy dz '
dt dt dt
the brachistochrone cut's all sudh 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,
hs = 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 &\ an undetermined function of x, y, z, we have
dT\ fdT\
dx.) \d^jj
^s^«y-Kr-w^
GENERAL THEOREMS. 291
If we make
e='{T)
and 2{H-V) = ^^^^-^'^^'-
it becomes
Here it is obvious, that (f) (r) is the action in a free path
coinciding with the brachistochrone, and that 2{II^ — Vj) is
the square of the velocity in this path.
Hence the curious result that, if t he the time through any
arc of a given brachistochrone, the same path luUl he described
freely under forces ivhose potential is Vj, where
2(i^.-r,) =
2{H- V)
<^' being any function whatever, and (f) (t) luill 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.
J 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.
19—2
292 GENERAL THEOREMS.
Hence, since the cycloid is the brachistochrone for gravity,
and since in it v' = 'Hgij, it will be a free path \i v^=-^ , that
is for a system of force where the potential is found from
H.
This gives
-^^ = 0,
dy
1
In other words, a cycloid may be described freely under
a reiDulsion 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, § 239, we have
Comparing this wdth that of § 252, we see that t = <1>{A)
is the brachistochronic expression for the time in a path which
is a free path for potential V, provided that (^4) and the
potential for the brachistochrone are connected by the equa-
tion
Hence, if A he the action in a given free path, the same
path will he a hrachistochrone for forces luhose potential is V,,
determined hi/ the condition just given, V heing the potential
in the free patli.
Thus, the parabola
GENERAL THEOREMS. 293
is the free path for v^ = 2gy. And the action is given by
1 2 #
--=A = Xs/a + -{y-ay.
J2g o
Hence this parabola is the brachistochrone for
In the simplest case <^'(J) = 1, and we have
_^Zi = o _^i = --L.
dx ' d^ 4!gy' '
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 ive have
two curves P and Q, of luhich P is a free path, and Q a
hrachistockrone, for a given conservative system of forces ;
P ivill he 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 it is a brachistochrone.
From this property Professor R. Townsend, Quarterly
Journal of Mathematics, Vol. Xlii., has shewn how to deter-
mine the intensity for parallel and concurrent forces for
ivhich given curves are brachistochrones.
For in the brachistochrone the velocity of description v
for parallel forces must be proportional to the sine of the
mgle 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
3very point contains the direction of the force.
Hence
(a) For parallel forces, every curve (necessarily plane
for brachistochronism in that case) for which sin^ i = {z),
294 GENERAL THEOREMS.
where z is the ordinate in the direction of tlie force, is
brachistochronous, under description with the velocity which
would vanish with i, for the law of force Z = - k' {z) or ^ (r) being determined from the pro-
perty of the curve and j>' [z) or 0' (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 >' (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 (b), but these diiferential equations can only be in-
tegrated in particular cases.
Thus if the force vary as the (w — 1)**^ power of the
distance, we have
tt"sin%'=±(3"-c"),
or a^-^p^ = ± (?•" — c") ;
leading to the differential equations
dz f ^a"
dx V2" — c"
dr /+a"-^?-'' ^\l
rdd
V r" - c /
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 Joiirnal 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 jMrticle moves in a plane, under an attraction
directed to a 'point xuhiclt moves in a given manner in the
2)lane: to find the motion.
Let X, y, ^, t] be the co-ordinates of the particle and
point, at time t. ^ and rj are given functions of t. Also let
P = /■(?•) be the acceleration due to the attraction at distance
r. Then
df
df
= -P
y-f)
•(1).
V(^-^)'+(2/-'?)'j
are the equations of motion.
The equations of relative motion arc, of course,
df Vfa.--^V-^+(//-77)^ df
d'iy-v)
dt:'
^ _ p yjz^. ^^ _ ^'^
.(2),
or, putting ^^, ?;,, fur the relative co-ordinates,
df
vfTlV df
d\
'h = — P _ "^1 _"^V
df vfTT^* df J
.(3).
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
quantities — -rf and — jy, , the components of acceleration
of the centre reversed.
262. Ex. Let the attraction vary directly as the dis-
tance.
Here P = /x 'sf^^ + t]^, and equations (3) of last section
become
df ^^' df
d-v.
- Mi-
d'v
dV "^ '^ dt J
which are easily integrated, in the form
•(4),
^^ = Aco^ytxt + B)-j^ ^
77^= (7 cos {S' is the focus, the locus of Y is
a straight line, and therefore that of ^ is a circle passing
through S.
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 £c, y be the co-ordinates of the j)lanet, f, t] those
of a point in the hodograph, then
'
dx dif
The equations of motion are
d X fix u, ^
at r r
Hence, as usual,
4f-/5='if-'
and therefore
d^x
di'
-i-<=m-
which gives, by integration,
dx
dt
'"- t
(1),
(2),
Similarly f + 5 = ^ + 5 = -f^, !
at hr j
302 GENERAL THEOREMS,
and thence
proving that the hodograph is a circle.
Also, by eliminating -,- , -.y among the three equations
(1), (2), we get for the equation of the orbit
-h-\- A7j-Bx=j r,
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
2u.
orbit is divided by the focus, and its length = -j- •
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 iiave 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. Co7npare 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 cii'cumference; but the diameter of
the latter is ^/2 times the radius of the former (§ 149).
Hence their circumferences are \/2 : 1, or the Enrtii
in its orbit receives in a revolution \/2 times the amount
of light and heat which the comet can receive in its whole
path.
GENEEAL THEOREMS.
803
It is evident that the path apparently described by a
fixed star, in consequence of the Aberration of light, is the
HodograjDh 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 foU'owing proposition :
If tivo circular hodographs, having a common chord, which
jmsses through, or tends to, a common centre of force, he both
cut perpendicularly by a third circle, the times of hodogra-
phically describing the intercepted aixs 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 perpendi-
cularly must have its centre on the common 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, FT'. 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 AP.
304 GENERAL THEOREMS.
Draw OA perpendicular to AP. Let OT=a, AB = h,
OA = c, SP = r, SM= VT, SM' = ^', PO = q, PA = r', and
PT= PT = r. If P be moved through a space hr, the in-
crease of the angle P>S'J/ which is the angle vector in the
orbit, is nearly. But the corresponding radius vector in
the orbit is — (§ 2G4) and therefore the time of hodographi-
cally describing the small arc at T is
8,=i^-ir=?^«r±. (§265.)
h '(s rr VT era ^'' . ^
Hence the whole change produced in the time of hodo-
graphically describing the arc TT' by shifting P is
/xSr / 1 1 \ 2fj,r'87'
a-sT aTjj' j b'rW
[This is easily seen, if we notice that by the figure
ct] . f ._!«,. _iC"| -,
, V = r sm^ sm ' - + sm - K 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 pro-
position.
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 df
e = a}t;
GENERAL THEOREMS.
therefore in the path
dx ^ „ ,
-^ = p cos d = a COS" (ot,
at '^
dy
so;
dt
= p sin.a = a cos cot sm cot.
Integi-ating and properly adapting the constants, as they
affect only the position of the origin,
a; = -— {2cot + sin 2cot),
2/ = -—- (1 — cos 2cot).
Now the equations of a cycloid are
x = A {(f> + &in (f)),
y = A {I -cos 9);
hence the path is a cycloid; and, since 2cot= , 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 proveil.
Geometrically thus, if ^P 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 S. That the motion is due to constant ac-
celeration perpendicular to AS is obvious from the fact that,
\i Fj) be drawn perpendicular to AS, 6'P*oc Pp.
269. If the orbit he central, and be a circle described about
a jyoint in its circumference, the hodograph is a pai'abola de-
scribed about the focus luith angular velocity propoHional to
the radius vector.
For, if S be the centre of attraction, P the particle in its
circular orbit, 7) the corresponding point of the hodograph:
qp, the tangent to the hodograph at p, must bo parallel to
SP ; and, therefore, if SQq be the tangent at S, the triangle
pSq (being similar to PSQ) is isosceles. Thus the locus oi ])
is a parabola, for its tangent, j^q, is equally inclined to the
radiu.s-vector Sp, and to the fixed line Sq. Also the angular
velocity of Sj), being the same as that of PQ, is double that
of SP, and is, therefore, inversely as SP". But the length of
Sj) is inversely as the perpendicular from S uptm PQ, i.e.
inversely as SP\
Or immediately, the pedal of a circle with respect to a
point on the circumference is a cardioid, and the hodogi'aph,
which is the inverse of the pedal, is therefore a parabola.
270. The only central orbits luhose hodoyraphs also are
described as central orbits, are those in which the acceleration
vanes 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, ^j' 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 FSj) is a straight line.
Also, since p belongs (by hypothesis) to a central orbit, the
tangent atp' 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 jp, where Sp is parallel,
20—2
308 GENERAL THEOREMS.
and equal, to FT, is evidently another logarithmic spiral
similar to the former, and described with the same constant
angular velocity. Hence j)t, the acceleration required, is equal
to e.Sp, and makes with *S'p an angle equal to SPT. Hence,
if Pm 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 . (o.
Hence
Pv = Pu =2Jt = -„~y = ^PT= . ,- /SP= 71^ . >ST(suppose)
^ SP sma sm^a ^ ^^ '
and vu = 2Po . cos a = . — PT= 2k . PT (suppose),
sma ^ ^^ ^
s
Thus the central acceleration at unit distance is n' = . , ,
sin a
and the coefficient of resistance is 27.; = "- . "^ - .
GENERAL THEOREMS. 309
The time of oscillation is evidently — ; but, if there had
been no resistance, the properties of simple harmonic motion
27r
shew that it would have been — ; so that it is increased by
the resistance in the ratio cosec a : 1, or n: J if — k"^.
The rate of diminution of SP is evidently
PT.cosa="^^^.5P=yl-^P;
sm 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
at at
See Proc. B. S. E. for farther illustrations.
310 GENERAL THEOREMS.
EXAMPLES.
(1) Investigate the differential equation of the patli of
a particle in a plane
/F- Y^'^
2X= "^ ' "^-^
(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 Jga, where a
is the radius of the generating circle.
(3) If a curve whose equation is i/=f(x) is described
freely by a particle under potential V, and if the same curve
can be described freely under potential
V+{y-fi^c)],
prove that the curve must be a cycloid.
(4) If a particle move on a rough inclined plane, prove
that
Jpp' cos'^ =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
S8ina= (tan'XJ sin'*/3l (cot ^ ) cosec (pd^,
where v = velocity of projection and y9 = 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 equa-
tion 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 AG P. 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
AB in the time—— , /m being the strength of the at-
J/M 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
\[dx) + U^j "^ \dzj \
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 (f> (r) any function of r,
prove that, if j ^ d^\r
s^^ ^dT ^^ .dT .. . dT
dd dcfi ^ dyjr
316 GENERAL THEOREMS.
is independent of f; 6, <^, ^ ... denoting -7-, -^ , ~ ...
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 latus rectum of its orbit.
(26) If P and Q be the accelerations along the tangent
and normal to the path of a particle, and -v/r the angle the
tangent makes with a fixed line, the equation of the hodo-
graph will be
in"*
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 tlie
original orbit being a circle about its centre, when «n^ law
may obtain. § 270.
(28) If P, P' be the central accelerations for an orbit
and its hodograph, prove that
PP'J\l,rr'.
(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 cor-
responding point of the orbit.
(30) Shew that in the hodograph of a central orbit
whose acceleration is f{r), the curvature varies invei"se]y
asry(r).
(31) When the hodograph is a straight line described
with constant velocity, the path is the trajectory of an un-
resisted 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
I ^
e =s,ec T,
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 6 being the angle which the radius
vector makes with the diameter, the angular velocity be
given by
cW^ k
dt ^ji^'-iy
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 accelera-
tion not tending to the centre, the hodograi^h is a lemnis-
cate.
(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
(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 liodograph of a point moving in any
manner ; and find its equation, for a point on the circum-
ference of a wheel, which rolls uniformly within the circum-
ference of a fixed wheel of four times its radius.
(40) A smooth elliptic tube is placed with its major
axis veitical and a particle allowed to slide down it, starting
from rest at the highest point; shew that the liodograph is
given by the equation
2 V^a sin - J cot M ,- cot ^ j [
(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
• J
r sm* ^ = a.
o
(43) A lemniscate whose equation is 7*'=a"cos2^ 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 hodogi'aph is defined by the equation
, , 7r + 2<^ „7r + 2(/>
r = c cos — —- cos- — ,. - ,
6
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 2^ in the opposite direction ; prove
that the equation of the hodograph is
?'''cos3(9 = c',
GENERAL THEOREMS. 319
and the equation of the orbit is
?- cos ^a — a^-
(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 hodograph of a pro-
jectile is
.7;' + Sxy'' = aif + 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 gene-
rating lines at a constant angle ; prove that its hodograph
is a conic section.
( 320 )
CHAPTER IX.
272. We come next to the consideration of the effects of
a class of actions which cannot bo 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, Jinite forces acting along Avith 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 suffi-
cient 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 b}' 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 titking 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 Ave may consider them as
mere particles, since everything is symmetrical about the line
joining tlicir 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 direction with velocity v; and that at the
instant when the mutual compression is completed, the spheres
are moving with a common velocity V. Let F be the pressure
between them at any time t during the compression, and
T the time during which compression takes place, then we
have
3I(v-V)= Pdt = R, suppose,
M'{V-v)=\^Pdt^R-
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-iM'v , „ MM' ,
^='MTW-' and i2 = ^.^^.^-^, (. + .).
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 addi-
tional pressure proportional to R.
Let e be the coefficient of restitution, v^, i\', the velocities
of the balls when finally separated. Then, as before,
M{V-v;) = eR,
M{v;-V) = eR;
T. D. 21
322 IMPACT.
whence
,r ,,Mv + M'v' MM' , .
and
(M- eM') V 4- 31' (1 + e)v M' ,, . ,, ,.
' M-^M M + M^ ^^ '
with a similar expression for v[.
A rather singular result may easily be deduced from the
last formula. Suppose M=M', e= 1, 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 , a'nc^ similarly \\ — 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, fur in rough and non-sphe-
rical 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 impincjing 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 X, and the point at which the impact takes place, as
origin.
The impulsive reaction of the plane will bo perpen-
dicular to it, since it is smooth. Let this be called J{; and
let the velocity of the particle be resolved into two, v^, v^,
respectively parallel to the axes. For the first part of the
impact
^Z0'x-O = 0,
M{i'^-v;) = ii.
IMPACT. 323
But Vy, 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
be the final velocities,
ball. If the ball be elastic, 1
then
et vj', vj' b
- vj') = 0,
M{v;
-v,")=eR.
These equations give
shewing that the velocity parallel to the plane is unaffected;
and
MvJ' = -eR = - eMvy,
or, Vy' = — €v^,
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 in-
clination 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 6, (f) respectively, we have
Fsin 6 = v^, V" sin ^ = v^',
Fcos ^ = Vy, V" cos (^ = Vy';
and the previous results give at once
e cot 6 = cot )
sin (^
Of course these results are applicable to cases of impact
on any smooth surface; by making the legitimate assumption
21—2
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
v' cos a, we have for their final values v^, v^, after restitution,
by § 274, the expressions
Vj = ■?; cos a — -^7 — yp (1 + e) (y cos a — i; cos a ),
■y, = 1/ cos a + -^TT — vr/ (1 + c)(«cosa — y cosa ),
and the value of R is
MM' ,, ,, , ,
ITITM' ^ "^ ^^ ^^ cos a — v cos a ).
Hence, the sphere M has finally a velocity t\ in the line
joining the centres, and a velocity irsina 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
M{v-V) = R,
M'{V-v')=B,
and, eliminating V,
^ = MTW'^'-'^ ^1)-
Hence, if e be the coefficient of restitution, v^, t\ the final
velocities,
^^ i (2).
Hence, Mv^ + M'v^ = Mv + M'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, p/v/ + ii/V=2^^'+i^V'
-R[l+e){v-v) + kR\'^^er^^^
= 1 Mv^ + \ MV^ - 1 (1 - O MVM' ^' - '"^'-
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 ^> vv, \{v — v')'.
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.
126
IMPACT.
Also by (2),
<
— v^ = v'-
--c-)^^-
=: e(v — v), by (1).
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, inas-
much 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 ajipUed in the
direction of the tangent at that end ; it is required to find the
impidsive tension at any other point of the chain.
Let this be T at a point of the chain whose co-ordinates
are x, y ; and let the initial velocities of that point, parallel
to the axes, be v^, Vj,; then, /x being the mass per unit of
length of the chain, we have the following equations :
dx\ 1
ds V dsj ^ ■" I -,,
'(-s)=-|
ds
The geometrical condition is to be determined as follows.
The chain being incxtensible, the length of an element S*^
is invariable, therefore the velocities of its two extremities
resolved along the element must be the same. This gives
evidently
dv.dx_^dv^dy^^ .
ds ds ds ds
IMPACT. 327
Or, if y^, Vp 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
If (f) be the angle the instantaneous direction of motion
at any point makes with the tangent,
*^^^ = tr = -rfr ^'^^•
By the elimination of v^ and Vp we obtain
Js{lTs)-f^^^^ ^^^'
the general differential equation of the impulsive tension at
any point.
This of course cannot be integrated unless the initial form
of the chain is known, i.e. unless /j, 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),
/Kf)'-©]"--.
328 IMPACT.
whence we easily obtain
ds\fi ds / fip^
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 pai't of the
chain is
T and i\ referring to one end, and T', i\ 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 = o,
and s = ad, and (6) becomes
dd'
whose integral is
To determine the arbitrary constants, observe that when
^ = 0, we have T = T„
the original impulse; and when 6 = 7r, or at the free end
of the chain, T=0. Thus we have
T^ = A+B,
= Ae'^ + B€-\
T €'"" Te"
These give A = " , B= — --- ~ ■
and therefore
T=l\- ^ .
The initial velocity at any point can now be e:isily 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^TM-"^
by hypothesis ; | *CJ>11"^'
•'' d? " ^' ^^ ^^ ^'^^ ~p'^^' °^ P = ^^
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 spirttl. Hence the equation
becomes
ds^ e'Y '
or, if we put s = ae^,
d'T _dT_T^^
d(f)" d(ji e'
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 (rpdx\
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
fs,f(> the tangential and normal accelerations at a point, and
330 IMPACT.
S, X the component tangential and normal impressed force::
per unit of mass,
T
and, as before,
(is p'
T now denoting the fiqite 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 n>ove, as if rigid, vertically.
283. When the string is practically inextensible, and if
the tension be great compared with the amount of the
external forces ; the disturbance being 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
df fj, ' dx' ' dt" 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
3/ or z= (l){x — at) + i/r (a- + at),
T
where a' = , (f) and yjr 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 ht, 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 fi be the mass, an,d 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, /u,^ the mass of the particles
picked up in unit of length, obviously
Substitute and integrate, sujjposing a: = 0, i- = V, when i = 0,
and we get
from which x can be easily found.
It is interesting to observe that we have
{M + fMoXf
so that the mass moves as if acted on by an attraction
oc yrj 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
end, it is obvious that we must write
for the mass at time t, where ^ is the increase of length due
to the increase of mass. But ^ is obviously proportional
directly to the accession of matter, i.e. to x-\-^. Hence ^
bears a constant ratio to x; and the only result of this
refinement in the solution of the problem is that n^ (still a
constant) is greater than before : that is, the centre of at-
traction X yrj is at a smaller distance behind the origin.
This problem obviously loads to the same result as the
following:
A cannon-hall 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 iinder gravity,
receives continually by precipitation of vapour an accessio7i of
mass 2>^'oportional to its surface; a being its radius when it
begins to descend, and r its radius after tlie interval t, shew
that its velocity is given by the equation
4 1,'^r + r+rV'
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 = S^v + B.^v be the increase of velocity in
time Bt; the first term due to gravity, the second to the
impacts.
Evidently, S^ii = gBt; and if j]f bo the mass at time t,
B (Mv) = is the condition of the impact.
IMPACT.
This
gives
J/S^v =
- vhM,
B,v-.
= -
SevBt
r
From these
we
have
dv
dt=^-
3ev
a + et
a + et
Multiplying by (a + ety, and transferring the last term to
the left-hand side of the equation, it gives by inspection
Hence
v = T-\(a + et) - .—^ — rS .
Substituting for e from (1),
at ( a'
4 (?' — a)\ r
at /, a a? a^\
4 V r r r J
as required.
To verify this solution, suppose no moisture to be de-
posited, then r = a, and we have v = gt as it ought to be.
Oi', immediately from the dynamical equation
4 4
since M = 7rp?-^ = k tt'P (« + ^^^
^ [(a + etyv] = (a + ef)V or ^ + ^^^^ = 5^.
334 IMPACT.
286. One end, B, of a imiform heavy chain hangs over a
small smooth pulbj A, and the other is coiled up on a table at
C If ^ preponderates, determine the motion.
The moving force due to gravity is the weight of AB
minus that of xi G = fig {x — a) suppose, a being the length
AC, and x the length AB.
Now in an indefinitely small interval ht, this would
generate in the portion BAG of the chain an increment of
velocity
* /j,{x+ a)
But the whole uncoiled chain, being in motion at the com-
mencement of the interval St with velocity v, lifts up a portion
of length vBt from the table during that interval. Hence,
if S^v be the change of velocity arising from this impact, we
have by the condition that no momentum is lost,
ie r= ^'^
_ fi(x+a)v
V + b,v = —. — , — — Y* »
^ fi{x + a) + fivbt
. v'Bt
or bv = - ----- ,
^ X + a
quantities of the second and higher orders being omitted.
_^ Bv B.v B„v
Hence as 8^=8^+1^'
proceeding to the limit we have
dv _ dv _ ff (x — a)- V*
dt dx {x + a) '
which gives {x + a)"v j- -|- v^ (x + a) =g (.c' — a').
Or, immediately, from the equation of momentum,
dx
?<["
(^ + «'«
= IJ.tr{x- a).
IMPACT. 335
dx
Multiplying by {x -\- a) -rr and integrating, supposing
x = }) initially,
= I {x -b) {x' + bx + b' - Sa'),
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 -syheTi b ^ 2a, when the acceleration is
constant and equal to ^^.
(1) If b<2a, then x^ -}- bx + b'^ - Sa"" will split up into
real factors {x + /3} (a? + 7) suppose, and we must put
b + 13 sin' S
C09^
to reduce the solution to elliptic functions.
(2) Ub> '2a, then x' + bx+b''- Za? is of the form
{x-{-\bf + n\
and we must put
x — b + c tan' I (j),
where c' = J6" + n'.
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
336 IMPACT.
At the end of the major axis farthest from the focus this
becomes
a i + e *
Now if at this point V be made 1^+ B V, without change of
direction, we have the condition that in tlienew orbit a(\ + e)
shall have the same value as in the old; since this will still
be the apsidal distance.
Hence
and o{a(H-e)}=0;
al + e
-2y{;'(.-e=)Jar.
And ha = — - — he
1 + e
vei^o^'^-
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
VhV
'
raises the directrix through an equal distance, and increases
the latus rectum by
■iVBV ,
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^ . .
sm a cos a ha.
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 displace-
ment Sa is indefinitely small.
These results may easily be extended by geometrical pro-
cesses, 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 = l impinge directly with
equal velocities, their masses must be as 1 : 3 that one may
be reduced to rest.
(3) Shew that if two equal balls impinge directly
1 +e
with velocities ., V and — F, the former will be reduced
1-e
to 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 il/ may acquire the greatest velocity, is
^J{MM'),
and that that maximum velocity is ■ , . , — 7m^2 •
T. D. 22
338 IMPACT.
(5) Suppose e = 1, and an infinite number of balls to be
interposed, shew that the maxiumm velocity which can thus
be criven to M, is
V i/ •
[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 ?i"i ball after its impact
with the (n - 1)"\
(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 im-
pinge upon that next it ; prove that the whole row will remain
stationary, except the ball at the other end, which will tiy
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 between
the masses and heights that the balance may remain per-
manently 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 C, Avhence it is equally easy
to make the cannon, are circles whose centres lie on a straight
1 4a
line through A, inclined to AB at an angle = ;, vr-f- ^sin"' ,
where a is the radius of the ball, and e = 1.
IMPACT. 839
(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 pro-
jection, is an ellipse.
(11) An impeifectly elastic particle is under the in-
fluence of a smooth gravitating sphere. Shew that (excepting
special circumstances of projection) it will perjjetually 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 npon 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
3e
(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 i, 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 ?; be the initial velocity of the
ball, whose mass is - th of the mass of the cannon, and h
n ' ■
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 chang^ed in the ratio
v"^ + 11- git : n{n+l) gh.
If a cannon be supported in a gunboat in the manner de-
scribed, 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 j
the focus in an ellipse of excentricity f , and impinge at the 1
nearer apse. Find the distances of future impacts, and shew
that if j9 be the original apsidal distance, the particles fall into
the centre of attraction after the time
14V(2/.)'
(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, a;=l,
equally distributed through a hollow sphere, be set in mo-
tion each with any velocity, shew that the resulting con-
tinuous pressure (referred to a unit of area) on the internal
surface is equal to two-tliirds 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 Avhere the shell lay is
M g jv' + Jo' - 7-yf + jv' - J^F^ryf
IMPACT. 841
Avliere M is the mass of the shell, and r is great compared
Avith its radius.
Explain the result when r= — .
9
(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
11 equal compartments by n screw surfaces, the pitch of the
trace of each on the cyhnder being a. It rotates on its axis
with angular velocity cu, 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
X (no. of particles in unit of volume) ;
nV\lv co'i
'^-e;)'1
h cot a r
n
provided co < j-; . 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 J/ 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
initially.
342 IMPACT.
If one year be the unit of time and the sun's radius the
11 n
it 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 jjarticles, 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
2/iiVm {2 (?i -l)av- {n - 2) bu] u,
provided u < —j— <(: . , where a = distance between each pa-
rallelepiped, h, h = breadth and height of each, and N is the
number of particles in a unit of volume.
Can this be used to exj)laiu 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
* / {(>'"" 4*) V(^''/^) sill ^ — h'' ~ *■) cos a],
and ^' being the excentric anomalies, r and '-' the focal
distances at the given points, a the mean distance, 2p the
latus rectum, and /'the atti-action at distiince 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' bo
IMPACT. 343
the mass of the case, &c., shew that it cannot rise at once
eMV
unless e V> g, nor at all unless , , , > g. If it do rise at
once vertically, shew that its greatest velocity is
and the greatest height it reaches
(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 7n 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 7-^^ from the lowest
is
^ — ](2n — 2r — 1) sina + 2 cos a cot ?ia — sin(2?' +1) o.].
2/?icos- a '^
(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 Z7.
the velocity of the system just before the {r+ 1)*^ particle is
set in motion is given by the equation
^,^ga (r + l)(2r + l) ^
3 * r
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
«y
344 IMPACT.
of natural length c ; they receive small longitudinal dis-
turbances, such that each of them proceeds to perform a
liarmonic vibration : prove that there will be two waves of
vibrations, in opposite directions, with the same velocity
sin - , where a is the average distance between
mCTT 11
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
^ /am 2r + 1 TT
when r = 0, 1, 2, ... w — 1, successively. Hence prove that the
periods of vertical oscillation of a heavy elastic string will be
given by the formula T= ^ — \ ^ , where I is the length
of the string, J/ 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 ujion 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 patii 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.
iJiPACT. 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 + 2b of the cable. Prove that if the anchor be let go it
will descend with acceleration ^g.
(32) A chain of given length is at rest on a smooth
horizontal jDlane, with one end fastened to a point on the
plane, under a repulsion from that point varying as the dis-
tance. 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 na,
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
(l-7n)e"' = (l+«)e"".
If w= 1, prove that m = \ approximately.
(3G) 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 A above the ground, the inter-
mediate portion passing round the barrel of a wiutllass placed
above the second coil. An engine, which can do // 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
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 4;lie 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.
(39) A chain is coiled up on a table and is connected
with a weight by a fine thread passing over a smooth pulley :
if the law of density of the chain be ■m(f>[x] ; and the mass
IMPACT. 847
producing motion be ml ; then the velocity when a length x
has been raised is given by
I'^cc— \\\ ^ (oc) dx\ dx
l^^=g iti^ U .
\l+\ <^{x)dx
I -'0
(40) A series of particles m^^, i?^^, ... connected by in-
elastic strings are placed on a smooth horizontal plane, so
that the sti-ings are sides of an unclosed equiangular jDolygon,
each of whose angles is tt — a, and an impulse is applied to
the extreme particle Pj in directiou P^P^ : prove that
T^ - y^_, cos « _ y^^^ cos g - t;
where T^ is the impulsive tension of the ?''^ string.
Deduce the equation , , 4 — ^ -9 = tor the im-
^ ds' fjL ds as p^
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 ppint 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 im-
pulses 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 +sina, 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 givea 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 peri^endicular 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 ^v sin ^, Avhere 20 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 6^, 6^ with the tangent at the vertex of
the catenary. An impulsive tension 1\ 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
cos^j e-d^_
'cos 6 e,-e\'
(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" sin 110 = «", 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 oc ?--t"-i) and
the density of the string must
OC — Sn-X— .
(4G) 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.
(48) If a small velocity ?iy-be impressed on a planet, in
the direction of the radius vector, shew that
he = lie sin {d — -nr),
Sz7 = — n cos (6 — ot).
Calculate also the changes in e and ct 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
— (2 + 6 005 0),
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 6 is the true anomaly of the body.
(50) If Q be the central disturbing force on a planet,
find by Xewton's method the equations
^ = -— .sm(^-^),
dvT Qr^ ,n ,
where B is the longitude of the planet, -sr the longitude of the
apse, e the excentricity of the instantaneous ellipse, v the
distance of the planet from the sun.
(51) k. 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 aj)ses are determined by the equations
350
IMPACT.
da
'>J
a' \\
dt
"IM
l-e^)|
de
dt
|a(l-
d'^
\ /
/cos (^ — be this change, shew that (in the supposed case)
(59) Shew that, in an' elliptic orbit about the focus, if
the velocity be increased by -th when the true anomaly is
6 — VT, we shall have
^ r sin [9 — ct)
edar = + — ,
~ na
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.
3 i
= ©
a^hl
(62) A particle describes an elliptic orbit about a centre
of attraction of intensity varying as (distance)"". If T be
27r
the periodic time and a small disturbing force X sin ^ . ^
acts in the direction of the radius vector, calculate the varia-
tions in the orbit.
(63) A spherical cloud of small masses, whose mutual
attraction is insensible, and 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 =i^cj cos nt, where p is a large num-
ber, and l)^ is large compared with (j\ shew that the pendu-
lum may oscillate about either of two points distant 2 from
the lowest point with an amplitude ^ where
co».=2'";.. 0=?.
853 )
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 Avhich 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 ather common impressed
forces : and second, when there are in addition constraints ;
such as when two or more of the particles are connected by
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
3xert each on the other equal and opposite forces, in the
lirection of the line joining them.
If then m, m', be the masses of the particles, and the
ittraction between two units of matter at distance D be
p'{B), the intensity is
mm'(j)' (D).
290. A system of free particles is subject only to their
nutual attractio'ns ; to investigate the motion of the system.
Let, at time t, x^, y^,z^ be the co-ordinates of the particle
vhose mass is m^, and let ' {D) be the law of attraction.
Let ^r^ express the distance between the particles m^ and m^;
hen we have for the motion of m^,
T. D. 23
354 MOTION OF TWO Oil MORE PARTICLES.
w,^-/=-|^«I''^.<^ (lO ^^— 'I (1).
«..g-^{»'.»,>'CO^^£-^j ..(^),
-.^■=s{»v«,.f(.0^-;^} (3).
with sirailfir equations for each of the others ; the summations
being taken throughout the system. Before we can make any
attempt at a sohition of these equations, we must know their
number, and the laws of attraction between the several pairs
of particles. But some general theorems, independent of these
data, may easily be obtained : although not nearly so simply
as those in Chap. II.
291. I. Conservation of Momentum. In the ex-
pression for Wp-TTv , we have a term
p' q
cPx
and in Wg -rJ we have
cc„-oc„
mjn
W^V^'
Hence if we add all the equations of the form (1) togcthc
the result will be
"■<;/ + "'=<;/+ =
Similr
MOTION OF TWO OR MORE PARTICLES.
Now if X, y, 0, be at time t the co-ordinates of the centre
of inertia of all the particles, § 58,
X Xm = S {mx),
y'Zm = 'X {my),
z 27/1 = S imz).
And the above equations may be written,
= 0,
Whence
f^»-'
df
f^-«
- or -
cVy
df
fs» = o_
d^
.df
dx
dt
= a
dy
dt
= h
dz
dt
= c
J
0,
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, h, c, may thus be determined,
.^ / dx
_dx _ V dt
dt Xm
Now if the initial velocity of m^ were resolvable into
itj, Vj, w^, parallel to the axes respectively, and similarly
for w„, &c.
2 (mu) T n 1
a = —J, — - , and so for b, &c.
zm
23—2
.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 {p:yz) were
mX, mY, mZ; we should find
tm -tJ = SwZ, Sm -^ = ^mY, 2m ^ = 2?/iZ;
or, which is the same thing,
d^ dr at
proving that the motion of the centre of inertia of the system
is the same as that of a particle of mass '^m, 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 Ave multiply in succession equation
(1) by ;/j, and equation (2) by oc^, 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 cct/ w^e take p, 6, the polar co-
ordinates of the projection of in,
dy _ dx_ ^dO
dt ydi'^di'
therefore 2 imp- ^ ) = 2.4
MOTION OF TWO OR MORE PARTICLES. 357
Now if a^ be the area swept out by the radius vector p
on the plane of xy,
^ ^dO _ da,
^P di~W
and our equation integrated gives
S (ma^) = A^t,
no constant being necessary if we agree to reckon a^ 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 7'emains constant ; or, as it is commonly but inconve-
niently stated, the sum of the products of the mass of each
jyarticle 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^, a^ to represent for the planes yz, xz the same
that a^ represents for xy, then
'% {ma^ = A^t,
2 {ma,) = A J;.
The value of this quantity for a plane, the direction-cosines
of whose normal are \ fi, v, will be
{^KA^ + iiA^^vA^t,
and will be a maximum if
\A^-^ii.A^-\-vA^ is so,
subject to the equation of condition
Tl^is gives X = ^^j^.^\^^.^ = ^ suppose,
with similar values for yu and v ;
and the value of the product for the plane so found is evidently
At
358 MOTION OF TWO OR MORE PARTICLES.
Hence, we see also, that, as indeed is evident from the
simple statement above, the axis ahuut which the moment of
momentum is greatest remains pai'allel to itself, or, as it is
usually put, tlie plane for winch the sum of the products of
the masses of the particles into the sectorial areas described
hy the radii vectores of their jjroject ions is a ma^vimum, is a
fixed 2:)lane or j)arallel 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^t/ d'x\ ^ , „ ,.
-''\''df-yde)=^'''^''^-y^^-
293. III. Conservation of Energy. Multiply
W'y^' ^^''y'ir <«)''y§'
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^m^ .
Written at length it is
mpm
j ^J{A ± B(o)
which may be integrated by putting (o = if. The integral
will be circular or logarithmic according as B is negative or
positive. Thus we have a;' — ^ 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
the formula (3) becomes
2
3 {x -xf-=C±^/[^ {m + m')] t,
?(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
_ 7nx + 7?i V m (x — x)
x — x= ; X = — ^ 7- ;
m + m m+m
and we easily find from (1),
^(Tx' m',
d^x
{m+m')-^ = {m-m')g (1).
This equation completely determines the motion. Also,
if we eliminate x and x\ we have
366 MOTION OF TWO OR MORE PARTICLES.
m -t- m '^
aud 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 veriiication of the laws of motion.
We see, for instance, by equation (1), that we may easily
keep 711 + m constant while m — m 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 i>i — 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.
Let r, 6 be the polar co-ordinates of the vibrating mass —
then, neglecting powers of 6 higher than the second, we have
(§ 250)-
Put |r for r, and J29 fov 0, and we get
(r^d) = - 20.
2
r — H
1 d
rdt
MOTION OF TWO OR MORE PARTICLES. 3()7
Transform to rectangular co-ordinates in the plane of
motion — x being vertically downwards : — then
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 angidar 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 sti'ing, sup-
pose 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 — a; on the other, and the difference or
2{x- a),
is the portion whose weight accelerates the motion. Hence,
/A being the mass of the chain per unit of length, we have
2/^a^=2;i5'(*'-a);
which gives x — a = Ae"' +Be~ " .
If the chain were initially at rest, a portion a + 6 being on
one side of the pulley,
h = A+B,
0=A-B;
368
MOTION OF TWO OR MOKE PARTICLES.
This is true until x = 2a, that is, till the chain leaves the
pulley; the value of t at that instant being t^, we have
2a J'-
■■■"^iJ''
and therefore t
=^/i-i^^/g-)|■
If, for example, h -
were initially as 4 : 1,
3a
.e. if the portions of the chain
*.Vl'°^-=
302. Two jiar'ticles, of masses m (md m', are attached to
different points of an inextensihle string, one of whose ex-
tremities is fixed. If the sijstem he 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, ff
the angles they make with the vertical, ^, <^' the angles which
vertical planes through them at time t make with the plane
of xz. Let X, y, z, x, y', z, be the co-ordinates of the parti-
cles and T, T' the tensions of the strings,
d'x - - ^ - - 1
Then
de
= -Ti
cos ^ + T'sin ^'cos ^'
^.- = -T
de
d\
sin Q sin ^ + T' sin 6' sin >',
- = mg - Tcos6 + T' cos 6',
df
df
T' sin 6' cos >',
^'^' \j'L = — T' sin 6' sin (^', -
m j^^=mg—I cos a.
de
MOTION OF TWO OR MORE PARTICLES. 3G9
Besides these, we have the six equations for oe, y, z,
X , y , z in terms of a, a , 6, ^, 6', '; 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 , (f> and (f)'. A still greater simplification
will be obtained by taking in addition the condition that
6 and 6' are so small, that their squares and higher powers
may be neglected. With these our equations become
at I
And, to a sufficient approximation,
X = ad,
X =ad + a 6',
z = a,
z =a-\- a.
Hence, T = m'g, and T= [m + m')g,
d-'e , d'e'\
Introducing an indeterminate multiplier, and adding,
(m + \m') ^ + \m'- ^,' + ^ {{m + m) 6 + m (K - 1) 6'] = 0.
^ ^ dt a dt a^^
1^ T. D. 24
370 MOTION OF TWO OR MORE PARTICLES.
Let \, \ be the roots of the equation
\ a X — 1
7n + Xni a m + m
Evidently one is positive and the other negative, and
the form of the equation shews that for both vi + Xyyi is
positive.
Put 4> = e+ '^'"l , -ff = e + kff, suppose.
Then the above eqviation gives
(P<^ ^ g m + m'
at a m + \m ^
By the recent remark the coefficient of <^ is positive for
both values of X, ; let its values be n^ and ??/, and we have,
k^, = tan^,
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
dev v' 1
dtJ a' 2 -cos 2^'
where 6 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 V, 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
GENERAL EXAMPLES. 385
the initial lengths of the portions of the string, m, m their
masses, shew that
in m ,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 (f>
l = a — T-^,
9
(•here (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 sether to a planet or comet
moving with the velocity V being assumed to be k -j- and
the sun's attraction being ^^ , obtain the following exact
equations :
T. D. 25
386 GENERAL EXAilPLES.
at
Obtain also the differential equation of the orbit in the
form
\ddj '^ h' (i + kf-' '
(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 progi-esses at a rate
greater by a than that due to tbe attraction of known bodies;
shew that this increment would be accounted for if the law
of force tending to the sun were .,+ . , and if a = -.^ : 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 fi (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""-^ - r"'"') = 6"', shew that the
apsidal angle is -^ nearly.
Jm
(47) A particle of mass m is attached to a fixed point
by an clastic 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* - 2ar' + a* = 0.
GENERAL EXAMPLES. 391
(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', the excentricities of the
old and new orbits, are connected by the relation
e'=' = l^-4e^
(50) A body describes a fixed ellipse under an attraction
to the focus, and a second body describes a similar and equal
ellipse which revolves in its own plane about its focus which
is fixed, while the plane itself moves so as to retain the same
inclination to a fixed plane, the bodies being always at corre-
sponding points in the two ellipses ; if the angular velocity of
the line of intersection of the two planes, and also the angular
velocity of the axis of the ellipse with respect to this line, be
always proportional to the corresponding angular velocity of
the body in the fixed ellipse, find the forces requisite to make
the second body move in the manner thus defined.
Also find the elements of the orbit which would be de-
scribed by the second body, if the forces acting upon it were
at any moment replaced by an attraction tending to the focus
and equal to the attraction in the fixed plane.
(51) A particle moves under a force whose magnitude is
proportional to the distance from the axis of x, and whose
direction is always perpendicular to the path of the particle.
The particle is projected from the point x = — a, y = a,
parallel to the axis of y, with velocity «a/^.
the path described is
V2a' -i/ + x = -j^ log ~ -^ ,-
-^ V2 ° y(l + N/2)
Shew that
392 GENERAL EXAMPLES.
(52) Investigate the equations of motion of a particle
attracted to any number of centres.
A particle can describe a certain orbit under an attraction
P to tlie point S, and it describes the same orbit under an
attraction P' to the point >S". Find the necessary conditions
that it may describe the same path when acted on both by P
and P'.
Two centres attracting inversely as the square of the
distance are distant r, r respectively from a particle moving
under their influence : if 6, & be the angles r, r make with
the line joining the centres of force, then
V r - -77 -77- = a (/i cos ^ + yu, cos ^ + c),
At, fx being the absolute intensities of, and a the distance
between, the centres of force, and c an absolute constant.
(53) If a parabola be described under two forces one
constant and parallel to the axis, and the other a repulsion
from the focus inversely as the square of the distance, find
the time of describing any arc of the parabola.
(54) A particle is under a central attraction
— / U z
e I '
and is projected from an apse at a distance = with velocity
1 4-e
— — /y//x, shew that the orbit described has for equation
(55) A body is placed on a rough inclined piano, whose
inclination is greater than tan"'/^, and is connected with an
elastic string parallel to tlie plane and attached to a fixed
point. If initially the body be at rest and the string of its
natural length, determine the circumstances of the resulting
motion.
GENERAL EXAMPLES. 393
(56) Particles move each in a system of confocal and
co-axial parabolas under a force constant for each particle
and tending to the focus ; at the beginning of the motion
they lie on a straight line passing through the focus : shew
that this will always be true if the forces and velocities of
projection are proportional to the latera recta.
(57) A particle moves under gravity on a smooth curve
in a vertical plane, and after leaving the curve describes a
parabola freely, and whatever be the velocity the vertical
ordinate of the point where it leaves the curve bears to the
vertical ordinate of the highest point attained in the free
path the ratio 2 : m + 1 ; prove that the equation of the
curve is ?/"' = c^"'~\
(58) A particle is placed on the surface of a smooth
fixed sphere, of radius c, at an angular distance a from its
highest point ; prove that the latus rectum of the parabola
which the particle describes after leaving the sphere is
^S-ccos^a; and find the range on the tangent plane at the
Lowest point of the sphere.
(59) A particle is placed very near the vertex of a
smooth cycloid, having its axis vertical and vertex upwards ;
find where the particle runs off the curve, and prove that it
falls upon the base of the cycloid at the distance ( - + Js ) a
from the centre of the base, a being the radius of the gene-
rating circle.
(60) A smooth right circular cylinder is placed with its
axis horizontal, and a particle moving with velocity v along
the lowest generating line receives a horizontal impulse at
right angles to this line and just sufficient to carry it to the
highest point of the cylinder. If the particle be prevented
from leaving the cylinder, shew that its subsequent path is
such that if the cylinder be developed its equation is
y = 7ra — 4 tan~^ e~ . ^a,
and that the highest generating line is an asymj)tote to the
curve.
394 GENERAL EXAMPLES.
(61) A hyperbola is placed in a vertical plane with the
transverse axis horizontal ; prove that when the time of
descent down a diameter is least, the conjugate diameter is
equal to the distance between the foci.
(62) Find a curve such that the time of descent down
all tangents from the point of contact to a given horizontal
line is the same.
(63) Prove in an elementary manner that the line of
quickest descent, from one curve in a vertical plane to
another in the same plane, is such that it bisects the angle
between the normal and the vertical at each extremity.
If the two curves are (i) an ellipse of semi-axes a, h,
having its major axis (2a) vertical, and (ii) a concentric
circle of radius c { V ■ 1
- cos a — sm a
P
(71) A railway train of given mass is travelling due
south at a uniform rate along a line which runs due north
and south : prove that, the earth being supposed perfectly
spherical, the train will exert a pressure on the Western
metals, the magnitude of which varies as the product of the
velocity of the train and the sine of the latitude of its posi-
tion, and a pressure towards the south, the magnitude of
Avhich varies simply as the sine of twice the latitude.
(72) If a very small tangential disturbance, f, act on a
particle oscillating in a cycloid, prove that the increase in the
arc of semi-vibration is equal to
v^.//eo.(y^.,+^..,
the integration extending over the time of a semi-vibration.
Also find an expression for the proportionate increase in the
time of oscillation.
GENERAL EXAMPLES. 397
(73) A particle moves in a resisting medium : shew liow
to find the resistance that a given curve may be described,
the force acting in parallel lines. A particle describes a curve
under a constant acceleration which makes a constant angle
with the tangent to the path : the motion takes place in
a medium resisting as the n^^ power of the velocity. Shew
that the hodograph of the curve described is of the form
^-^'g-nflcota _ ^-n _ ^-n^
(74) A particle is moving under a central attraction
and experiences resistance which varies as the square of
its velocity. Find a differential equation for its orbit.
If the attraction is -^ , and the resistance kv^, k being
j-11, she
given by
r
small, shew that at the beginning of motion the velocity is
^v'=j/M6'''^'asme.dd,
where a ± a are the reciprocals of the maxima and minima
values of the radius vector, a being supposed small, and
the angle described from the beginning of motion.
(75) A rain-drop descending through a damp atmo-
sphere at rest, accumulates moisture so that the radius
increases uniformly. If a sudden gust of wind gives it a
horizontal velocity, shew that it will proceed to describe a
hyperbola one of whose asymptotes is vertical.
(76) A spherical rain-drop descending from rest by
the action of gravity receives continual accessions to its
mass by depositions of vapour proportional to its surface :
the radius of the drop being a at starting, and r after an
interval t, the velocity acquired in the same interval being
V, shew that
y^9l r* - a*
4r='" r-a '
if the resistance of the air be not taken into account.
Solve the same problem, supposing the resistance of the
398 GENERAL EXAMPLES.
air to he in a given ratio to the actual acceleration of the
drop independently of its size.
(77) If a, h, c, d, e, f are the six elements introduced
by the integration of the equations
d^x fix _ dR d^y fiy _ dR d:z fiz _ dR
df'^V'd^' de'^V^d^' de'^'?~d^'
on the hypothesis R=0, and if x', y , z the expressions for
dx dy dz
dt' dt ' ~di'
in terms of the time and these elements have the same form
whether R be zero or finite, prove that
dR r T
h
or
aG ..^ - MC-
III. -z^ — = — a^ Here
O
-^(^ + «) = 2a^°§-r-^
and r' = C (t' - a?),
whence, after reduction,
2a VC^ 1
aC aC aC aC
T. D. 26
402 APPENDIX.
IV. c = o.
These are, of course, the results of the integration of the
usual equation between u and 0. [ComjDare Chap. V. Ex.
(9)-]
As another case, suppose in (1)
-2JPdr-Pr = mr' + ^ (6).
Differentiate, multiply by r^, and integrate, then
P = -hmr-\--..
Hence, in the case of the direct first power, or a combination
of this with the inverse third,
which gives, according as m is positive or negative,
2w Uf cos {t J- 2m + iV) j '
By means of (4), these equations give us 6 in terms of t,
and, the latter being eliminated, Ave have the required orbit,
which becomes the ellipse or hyperbola as usual when n = 0,
it being observed that we have an additional disposable con-
stant introduced by the method employed in obtaining equa-
tion (1). It is evident that results of this kind may be
multiplied indefinitely. To classify the cases in which the
equations for r^ and 6 in terms of t can be completely in-
tegrated would be an interesting, but by no means an easy
problem.
The method here employed is interesting as being that
which is at once suggested by the application of Quaternions
to the problem of Central Orbits, (Tait's (2uatennons, § .S45.)
APPENDIX. 403
As an additional example, take the gravitation case —
then we obtain, as above,
d ( dr\ ,^ //-
dt\;'dt) = ^-^r^
or r~ = JC + Cr^Yfx:r.
But v^ = C + -^ . Hence, in ellipse,
a
Also ^ = for r = a (1 + e). Thus
at
'•S = Va^"'
'e^ — (r — a)".
The form of this suggests the assumption
r — a = — ae cos u,
so that a'^e (l—ecosu)-^ = a / ^ ae,
' dt V a
whence, as usual,
nt = u — e sin u,
as in § (IGO) above.
Another mode of looking at this question is as follows : —
Eliminate between the equations of the central orbit
r-re' = P, r^e = h,
and we have
from which the above results are obtained at once.
The investigation of any central orbit is thus immediately
reduced to a case of rectilinear motion.
26—2
404
APPENDIX.
Another view of the same question, of which the above
is only a special aspect, is Newton's Revolving Orbit. Suppose
r to remain unaltered, as a function of the time, and 6 to
become md — where m is constant. Then
rd''
7mV
The central acceleration thus requires to be altered by a
term depending on -5 alone. This gives, by inspection, many
of the results in Chap. V. above, e.g. Example 22, p. 155.
[See on this subject, Notes on Gentral Forces by A. H. Curtis.
Messenger of Math. April 1882.]
B. To find the time of fall from rest down any arc of an
inverted cycloid.
Let be the point from which the particle commences
its motion. Draw OA' parallel to CA, and on BA' describe
a semicircle. Let P, P', P" be corresponding points of the
curve, the generating circle, and the circle just drawn, and
let us compare the velocities of the particle at P, and the
point P". Let F'T be the tangent at P".
velocity of P" _ element at P"
velocity of P clement at P
APPENDIX. 405
' A'B
AB
BP' BP" V "
A'B / A'B
~2A'P"\/ AB'
But velocity of P= V(2^ • ^'^) = a/]T^ • -"^'P"-
Hence velocity of P" = \ ^ r-n -A'B, a constant.
And, as the length of A'P"B is - .A'B,
time from A' to B in circle = time from to 5 in cycloid
/AB
Cor. It is evident from the proof, that the particle de-
scends half the vertical distance to B in half the time it takes
to reach B.
C. To find the nature of the hrachistochrone tinder
The following is founded on Bernoulli's original solution.
(WooDHOUSE, Isoperimetrical Problems.)
From Art. 180 it is evident that the curve lies in the ver-
tical plane which contains the given points. Also it is easy
to see that if the time of descent through the entire curve is
a minimum, that through any portion of the curve is less
than if that portion were changed into any other curve.
And it is obvious that, between any tivo contiguous equc^l
values of a continuously vai^ying quaiitity, a maximum or
minimum must lie. [This principle, though excessively simple
(witness its application to the barometer or thermometer), is
of very great power, and often enables us to solve problems of
maxima and minima, such as require in analysis not merely
the processes of the Differential Calculus, but those of the
Calculus of Variations. The present is a good example.]
406 APPENDIX.
Let, then, PQ, QR and PQ', Q'R be two pairs of inde-
finitely small sides of polygons such that the time of de-
scending through either pair, starting from P with a given
velocity, may be equal. Let QQ' be horizontal and indefi-
nitely small compared with PQ and QR. The brachisto-
chrone must lie between these paths, and must possess any
property which they possess in common. Hence if v be the
velocity down PQ (supposed uniform) and v' that down QR^
drawing Qm, Q'n perpendicular to RQ', PQ, we must have
Qn _ Q'in
V v
Now if 6 be the inclination of P to the horizon, 0' that of
QR, Qn = QQ' cos 6, Q'm = QQ' cos 6'. Hence the above
equation becomes
This is true for any two consecutive elements of the required
curve ; therefore throughout the curve
V Gc cos 6.
APPENDIX. 407
But V- X vertical distance fallen through. (§ 173.) Hence the
curve required is such that the cosine of the angle it makes
"with the horizontal line through the point of departure varies
as the square root of the distance from that line ; which is
easily seen to be a property of the cycloid, if we remember
that the tangent to that curve is parallel to the corresponding-
chord of its generating circle. For in the fig. p. 172,
cos OFN'= cos OAF ' = ^ = ^ /4^ac V^-^-
AO V AO
The brachistochrone then, under gravity, is an inverted
cycloid whose cusp is at the point from which the particle
descends.
Cj . Were there any number of impressed forces we might
suppose their resultant constant in magnitude and direction
for two successive elements. Then reasoning similar to that
in § 180 would shew that the osculating plane of the brachis-
tochrone always contains the resultant force. Again we
should have as in last Article,
cos 9 _ cos 6'
V v '
where 6 is now the complement of the angle between the
curve and the resultant of the impressed forces.
Let that resultant = F, and let the element PQ = Ss, and
0' = 0-\- 86. Then since F is supposed the same at P and Q,
v" - v' = 2FBs sin d (by Chap. IV.),
or vSv = F8s sin 6.
But V oc cos 6^ ; -^
rt-hich gives
Bv^ sin ^3^
V COS 6
Hence
^ = -^■003 ft
SB
408 APPENDIX.
But in the limit -^ = p, the radius of absolute curvature
OCT
at Q, and i^cos 6 is the normal component of the impressed
force. Hence we obtain the result of § 185 for the general
brachistochrone.
Cg. Now for the unconstrained path from P to R we
have Jvds a minimum. Hence in the same way as before,
<^ being the angle corresponding to 6, v cos cf) =v' cos <^' from
element to element, and therefore throughout the curve, if
the direction of the force be constant.
But in the brachistochrone,
cos 6 _ cos 6'
V V
Now if the velocities in the two paths be equal at any
equipotential surface, they will be equal at every other.
Hence taking the angles for any equipotential surface
cos 6 cos (f) = constant.
As an example, suppose a parabola with its vertex up-
wards to have for directrix the base of an inverted cycloid ;
these curves evidently satisfy the above condition, the one
being the free path, the other the brachistochrone, for gravity,
and the velocities being in each due to the same horizontal
line. And it is seen at once that the product of the cosines
of the angles which they make with any horizontal straight
line which cuts both is a constant whose magnitude depends
on that of the cycloid and pai'abola, its value being ./ -r-
where I is the latus rectum of the parabola, and a the dia-
meter of the generating circle of the cycloid.
D. To sheiv that of two curves both concave in the sense
of gravity, joining the same points in a vertical plane and not
meeting in any other point, a, particle will descend the enveloped
in less time titan it luill the enveloping carve : the initial velocity
being the same in both cases.
APPENDIX. 409
Take the axis of x as the line to the level of which the
initial velocity is due, and the axis of y in the direction of
gravity, then
ds
dt
f'dt
dx
.r-^-^+yh. (1);
(since the limits are constant),
X l\,dx [- 1 ^^^ +^^^ + 1 P' 1—
V3/(i
Ja-1
Now the curve is convex to the axis of x, hence yq is
positive, and by (1) ^/y and V(l +p") have the same sign.
Hence the sign of Bt^ is the opposite of that of Sy, and for an
enveloping curve By is negative. Hence the time of fall will
be longer.
We may thus pass from one curve to any other enveloping
one, even situated at a finite distance, provided the latter be
concave throughout ; else the multiplier of By . dx in the in-
tegral might change sign between the limits. (Bertrand,
Liouvilles Journal, Vol. vil.)
410 APPENDIX.
A simple geometrical proof of this theorem may easily be
obtained by drawing successive normals to the inner curve
and producing them to meet the outer. The velocities in the
pairs of arcs, thus cut out of the two curves, are equal (if the
curves be indefinitely close), but the arcs themselves are
generally longer in the outer curve, since the convexity of
the inner curve is everjnvhere turned to it.
E. To find the curve in which the time of descent to the
lowest point is a given function (f) (a) of a the vertical height
fallen through.
fa fJo
Hence, the problem may be thus stated,
ds
Having given <}> (a) =
J
Ja- X
where (j) is a known function, find s in terms of x. (Abel,
(Euvres, Tom. i.)
Put ds =/' {x) dx, divide by Jz — a and integrate both
sides with regard to a, from a = to a = z.
r^