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J. O'KINEALY,
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A TREATISE
OF
MECHANICS,
BY
S. D. POISSON,
MEMBER OF THE INSTITUTE, ETC.
TRANSLATED FROM THE FRENCH, AND ELUCIDATED WITH
EXPLANATORY NOTES,
BY
THE REV. HENRY H. HARTE,
LATE FELLOW OF TRINITY COLLEGE, Dublin.
IN TWO VOLUMES.
VOL. I.
LONDON: LONGMAN AND CO.
DUBLIN: A. MILLIKEN; AND HODGES AND SMITH.
MDCCCXLII.

J
DUBLIN:
PRINTED AT THE UNIVERSITY PRESS,
BY GRAISBERRY AND GILL.
©12-7-08&.7.J.
PREFACE.
2
THOUGH there are several works in our language that
treat of detached parts of mechanics in a clear and
comprehensive manner, there is no single treatise in
which the entire subject of rational mechanics, in all
its bearings and generality, is brought before the reader.
The consequence of this has been, that the Mechanique
of Poisson is the work which is now in the hands of
all who propose to acquire such a knowledge of the
principles of mechanics as may qualify and enable
them to extend their reading to the more abstruse
works of Laplace, La Grange, and also of Poisson him-
self. The object of the present translation is to render
this treatise of Poisson more accessible to the English
reader.
As several analytic operations and integrations are
taken for granted by the author, it has been suggested
that the work would be still more easily understood,
if the most difficult of these operations were given in
detail in the form of notes appended to the end of each
volume. It is likely the experienced reader may con-
sider many of them to be superfluous, but as the object
*
187329
F
A
iv
PREFACE.
of the translator was to render this work accessible to
the English student, whose acquaintance with the higher
branches of the calculus was not so extensive as the
author takes for granted, he trusts he will not be
deemed unnecessarily diffuse if he has insisted in the
notes more at length on several points, than the ac-
complished reader may consider to be necessary.
The translator did not venture to make any change
in the text, though in some few cases the author ap-
pears to have fallen into mistake; see Nos. 390 and
607. He also retained the numerical coefficient which
is given by the author for the dilatation of gas, which it
appears from the experiments of Rudberg, ought to be
00,364, instead of 00,375; see Scientific Memoirs, No. 7.
The reader, however, will find all these points adverted
to in the corresponding notes.
1
1
CONTENTS OF VOLUME I.
Nos.
INTRODUCTION.-page 1.
Definitions of matter, of bodies, of the mass, of a material point, and of
force,
•
•
•
1, 2
Object of mechanics; division of this science into two parts-Statics
and Dynamics,
3
4
•
The point of application of a material point is determined by means
of three coordinates, either rectangular or polar,
What is meant by equal forces; numerical expression of the intensity
of a force,
LO
5
6, 7, 8
The direction of a force is determined by means of three angles
(which may be either acute or obtuse), that are connected together
by an equation, or by two angles that are independent, the one of
the other; conversion of an arc expressed in degrees, into parts of
the radius,
Expression of the cosine of the angle contained by two right lines;
the equation which has place when they are perpendicular the one
to the other; transformation of rectangular into polar coordinates,
Projections of one right line on another, and of a plane area on another
plane,
9
10
How the two opposite directions of different parallel forces can be de-
termined,
11
In this treatise the infinitesimal method is exclusively employed, . 12
Definitions of the differential of a variable, of that of a function; defi-
nitions and notation of a definite integral; this integral is, in gene-
ral, the sum of the values of the differential,
Differentiation of an integral, with respect to a quantity that is consi-
dered as constant in the integration,
Formula of quadratures,
•
•
13
14
15
vi
CONTENTS.
Nos.
In the case of infinitely small quantities, the ratio of the arc of a curve
to the chord is unity; this enables us to regard a curve as a polygon
consisting of an infinite number of infinitely small sides,
Definition of the tangent to a curve; formulæ which determine its
direction; differential element of the curve; equation of the normal
plane; cosines of the angles which the perpendicular to any plane
makes with parallels to the axes of the coordinates,
Expressions for the angle of contact and for the radius of curvature,
Equation of the osculating plane; formulæ relative to the direction of
the perpendicular to this plane,
Coordinates of the centre of curvature,
Equation of the plane which touches a curved surface; differential
element of the surface; formulæ relative to the direction of the
normal. The reader is referred to a memoir inserted in the 21st
number of the Journal of the Polytechnic School for all that respects
the curvature of surfaces,
Rule for deducing the one from the other, the formulæ relative to
three rectangular axes, when every thing is similar in a problem,
with respect to each of them,
•
16
17
18
19
20
21
22
General conditions, which equations that contain quantities of diffe-
rent natures ought to satisfy,
. 23
BOOK THE FIRST.
STATICS.
FIRST PART.
CHAPTER I. —OF THE COMPOSITION AND EQUILIBRIUM OF FORCES
Applied to thE SAME POINT.-p. 25.
What is meant by the resultant of any number whatever of forces ap-
plied to the same point; its value, when all these forces act along
the same right line,
The resultant of two equal forces, which contain an angle of 120º, is
equal to each of these forces, and divides the angle into two equal
parts,
24
25
26, 27, 28
29
Value and direction of the resultant of two forces which make any
angle whatever; rule of the parallellogram of forces,
Immediate consequences of this theorem,
"
CONTENTS.
vii
Nos.
30
31
Geometric construction for determining, in magnitude and direction,
the resultant of any number whatever of forces,
Composition of three rectangular forces into one, and resolution or
decomposition of this last into three rectangular forces, .
Determination of the resultant of any number whatever of given
forces; values of the angles that determine its direction; expres-
sion of this resultant in a function of the components, and of the
angles contained between their directions,
Particular property of this same resultant,
Equation of the equilibrium of a material point entirely free; it is
shown that in virtue of these equations, each of the forces that act
on this point is equal and contrary to the resultant of all the others, 35
Equation of the equilibrium of a material point constrained to remain
on a given surface; pressure that the surface sustains; direction in
which it acts,
32, 33
34
36, 37
Equation of equilibrium of a material point constrained to exist on a
given curve,
38
Equation of virtual velocities, containing the equations of equilibrium
relative to the three preceding cases,
39
CHAPTER II.—Of the Equilibrium of the LeVER.-p. 58.
Definition of the lever, object of this chapter,
40
Displacement of the point of application of a force applied to a system
of an invariable form,
41
Definition of the moment of a force with respect to a point; equili-
brium of two forces applied to a lever; this equation is independent
of the angle contained between the two arms of the lever; case in
which the two given forces are parallel,
42, 43
Two parallel forces which act in opposite directions, and are not di-
rectly opposed to each other, are not reducible to one sole force;
this couple of forces may be transformed, in an infinite variety of
ways, into another couple of forces that are irreducible to one sole
force,
Condition of the equilibrium of any number whatever of forces applied
to a lever,
44
45
Theorem relative to the moment of the resultant of two forces; ex-
tension of this theorem to the case of any number whatever of forces
acting in the same plane; quantity which continues invariable in
all the transformations of this system of forces; equation of equili-
brium of these forces about a fixed point situated in their plane, 46, 47, 48
viii
CONTENTS.
It is shown that the equation of virtual velocities has place in the
equilibrium of the lever,
CHAPTER III. — OF THE COMPOSITION AND EQUILIBRIUM OF PA-
RALLEL FORCES.-P. 72.
•
Nos.
49
50, 51
Direct demonstration of the composition of two parallel forces, which
was already deduced in No. 43, from that of forces converging to
the same point; the magnitude and point of application of the re-
sultant of any number whatever of forces is deduced from this,
When parallel forces turn about their respective points of application,
the parallellism still continuing, their resultant in like manner
turns about its point of application; definition of the centre of pa-
rallel forces; definition of the moment of a force with respect to a
plane,.
52, 53
The moment of the resultant of any number whatever of parallel
forces, with respect to a plane, is equal to the sum of the moments
of these forces, with respect to this same plane; coordinates of the
centre of parallel forces,
54, 55, 56
Equation of the equilibrium of a system of parallel forces, applied to a
solid body, whether this body is entirely free, or retained by a point
or fixed axis,
57, 58
CHAPTER IV.-General ConSIDERATIONS About Solid Bodies and
CENTRES OF GRAVITY.-p. 84.
The weight may be regarded as a force that is constant, both in in-
tensity and direction, throughout the entire extent of the same
body,
59
·
Definition of the weight and of the density; equations which exist be-
tween the weight, the mass, the volume of a body, and the intensity
of the gravity,
60
Definition of the gramme, ratio of its weight to that of a volume of wa-
ter at the temperature of melting ice; densities of air and mercury, 61
Weights may be made use of to compare all other forces; they fur-
nish the most convenient measure of the mass,
Definition of the centre of gravity; practical rule for determining the
position of it, in the interior of a solid body,
62
Equations by means of which the coordinates of the centre of gravity
of a system of bodies, whose centres of gravity are already known,
may be determined; case in which the masses of the bodies are in-
finitely small; what is meant by the centres of gravity of a volume,
of a surface, of a line,
63
64, 65
CONTENTS.
ix
•
Equation which exists between the mutual distances of the centres of
gravity of different bodies, and their distances from the centre of
gravity of the entire system,
Remarkable property of the equilibrium of a material point entirely
free,
Enumeration of the different cases in which the centre of gravity is
immediately known, .
CHAPTER V.-DETERMINATION OF Centres of GRAVITY.—p. 96.
I. Centres of Gravity of Curved Lines.—Ibid.
Coordinates of the centres of gravity of any line whatever; application
to the right line,
Case of a plane curve; applications to the circle and to the three conic
sections,
Nos.
66
67
. 68
69
70, 71
Equation of the cycloid; statement of its different properties; coor-
dinates of the centre of gravity of any arch whatever of this curve, 72, 73
Rule for determining the area of a surface of revolution, when the
centre of gravity of its generating curve is known, without the aid
of the calculus,
74
75
II. Centres of Gravity of Surfaces. p. 104.
Coordinates of the centre of gravity of any surface whatever; case
in which the surface is plane,
Application to the centre of gravity of a triangle; determination of
this point without the aid of the integral calculus; how the centres
of gravity of a sector and segment of a circle may be obtained from
76, 77, 78
it,
The determination of the centres of gravity of the three conic sections
is suggested as an example for practice; the two coordinates of the
centre of gravity of any portion of the area of the cycloid are com-
pletely determined,
79, 80
Centre of gravity of the zone of a surface of revolution; application
to the convex and concave surfaces generated by the cycloid,
Rule for determining the volume of a solid of revolution when the
centre of gravity of the generating area is known, without the aid
of the calculus; extension of this rule to other kinds of bodies,
Volume of a prism or truncated cone,
•
81, 82
83, 84
85
III. Centres of Gravity of Volumes and of Bodies.—p. 121.
Centre of gravity of any pyramid or cone whatever,
Determination of the centre of gravity of a triangular pyramid, with-
b
86
X
CONTENTS.
Nos.
out the aid of the integral calculus; how the centres of gravity of a
spherical sector or segment may be inferred from it,
Centre of gravity of a body that is symmetrical about an axis; and, in
particular, of a portion of the ellipsoid,
87, 88
89
Centre of gravity of a solid of revolution, and, in particular, of the
concave and convex solids generated by the cycloid,
Different expressions, in triple integrals, of the coordinates of the
centre of gravity of any body whatever; application to a portion of
a heterogeneous sphere,
90
91, 92
Differential element of a volume expressed in terms of the differen-
tials of the polar coordinates,
CHAPTER VI.-CALCULATION OF THE ATTRACTION OF BODIES.-p. 135.
I. Formulæ relative to any Body whatever, and to a Sphere in par-
ticular.-Ibid.
93
General expressions in triple integrals, of the three rectangular com-
ponents of the attraction exerted by a body on a material point, 94, 95
Reduction of these three triple integrals to partial differences of one
sole integral,
In consequence of a difficulty that has been already remarked in
determining the coordinates of the centre of gravity of any body
whatever (No. 91), it is necessary to consider more particularly the
intimate constitution of natural bodies. Definitions of atoms and of
molecules; what is to be understood by the density of a body in any
point whatever; definition of the mean interval of the molecules at
this same point; it is shown how the formulæ relative to the masses
of bodies, to the coordinates of the centres of gravity, and to at-
tractions in the inverse ratio of the square of the distances, may,
without sensible error, be applied to natural bodies,
The attraction of a body which is very distant, is very nearly the
same as if the entire mass of this body was condensed in its centre
of gravity; mutual attraction of two homogeneous spheres, .
Theorems relative to the attractions of spherical bodies, on two mate-
rial points, either external or internal,
Direct demonstration of the equilibrium of a material point situated
within the space bounded by a spherical stratum,
II. Formulæ relative to the Ellipsoid.-p. 149.
96
97, 98
99
100, 101
Transformation of the general formulæ of No. 95, which will be prin-
cipally useful in the case in which the attracted point is a part of
the attracting body,
. 102
. 103
CONTENTS.
xi
Application to the homogeneous ellipsoid; the formulæ relative to its
attraction on an interior point, may be reduced to simple integrals,
that can be calculated by the means of tables of elliptic functions;
extension of the theorem of No. 102 to an elliptic stratum,
The integrations may be effected under a finite form, in the case of the
ellipsoid of revolution; particular case of an ellipse very little com-
pressed,
Nos.
104, 105
Remarkable theorem, by means of which the attraction of an ellipsoid
on an external point is made to depend on the attraction of another
ellipsoid on an internal point; this theorem is independent of the
law of the attraction in a function of the distance: application to
the particular case of two concentrical spheres,
•
106
. 107, 108, 109
BOOK THE SECOND.
DYNAMICS.
FIRST PART.
CHAPTER I.-OF Rectilinear MOTION, AND OF THE MEASURE of
FORCES.-P. 163.
I. Formulæ of Rectilinear Motion.—Ibid.
Definition and equation of uniform motion,
110
Remark on the measure of time; invariability of the sidereal day; its
duration compared with that of the mean day,
. Ill
112
. 113
Definition of the velocity in uniform motion, and afterwards in va-
riable motion,
In what the inertia of matter consists,
•
Expression of the velocity in any motion whatever; expression of the
space described in an infinitely short time, abstracting from the
consideration of the acquired velocity,
Definition and equation of uniformly accelerated and retarded motion;
the force which produces it is a constant force; this motion is that
of heavy bodies in a vacuum; in the same place, the acceleration is
the same for all these bodies; its magnitude at the Observatory of
Paris,
·
•
It is demonstrated that the magnitudes of forces which act successively
on the same material point, are to each other as the infinitely
small velocities impressed on them in the same infinitely small
time,
114
115
. 116
xii
CONTENTS.
A
1
Nos.
In the case of constant forces, their intensities are as the velocities
which they produce in the unit of time; example of the ratio of
forces inferred from that of the observed velocities; converse ex-
ample of the ratio of the velocities inferred from that of the forces, . 117
Measure of force in any variable motion whatever, either by means of
the velocity that it produces, or by means of the space which it
causes to be described in an infinitely short time,
General formulæ of variable motion,
•
. 118
. 119
•
II. Measures of Forces, regard being had to the Masses.-p. 177.
Inaccuracy of the expression, force of inertia,
What is meant by material points being equal in mass; two forces
which act on two different points are to each other as their masses
multiplied by the velocities produced by these forces in the same
instant,
•
Definition of motive force; its value in any motion whatever; it be-
comes a pressure when the motion is destroyed,
From the identity of the motion of heavy bodies in all places on the
earth's surface, the proportionality of the weight to the mass is in-
ferred;
•
When the motive force is given, the accelerating force is deduced
from it by dividing it by the mass of the moveable; the resistance
of a medium, and a given weight, applied successively to different
masses, are given as examples of this,
120
121
. 122
123
124, 125
Definitions of the quantity of motion, and of the percussion or impul
sion; decomposition of a percussion into two others; application to
the wedge,
•
Condition of the equivalency of two percussions; principle of equili-
brium in the case of the impact of two bodies destitute of elasticity;
when they move in opposite directions and meet, they are reduced
to a state of rest, when the velocities are in the inverse ratio of the
masses,
How weights and percussions may be compared together,
CHAPTER II.-EXAMPLES OF RECTILINEAR MOTION.-P. 190.
•
Differential equations of rectilinear motion; the integration is not
possible under a finite form, except when the accelerating force is
constant or given in a function of only one of the three following
variables, namely, the time, the velocity, or the space described,
Vertical motion of a heavy body in a vacuum,
Motion of this body on an inclined plane,
126
127
. 128
•
•
129
130
. 131
CONTENTS.
xiii
Vertical motion of a heavy body in a resisting medium; when it falls
from a great height its velocity approaches more and more to uni-
formity according as the height increases; method of determining
the coefficient of the resistance, by observing the entire time of the
successive ascent and descent of the moveable,
Nos.
132, 133, 134, 135
Example of the application of particular solutions, in the problems of
dynamics,
•
Motion of a body attracted towards a fixed centre, either in the direct
ratio of the distance, or in the inverse ratio of the square of the dis-
tance,
136
137, 138
Motion of a body attracted towards two fixed centres; case in which
these two centres are those of the moon and the earth; diminu-
tion of the velocity of a projectile produced by its gravity towards
the body from which it was projected, when it has attained to a
great distance from this body,
139, 140, 141, 142, 143
CHAPTER III.-OF CURVILINEAR MOTION.-P. 212.
I. General Formulæ of this Motion.—Ibid.
The determination of the curvilinear motion of a material point is
reduced to that of the rectilinear motions of its three projections on
the axes of the coordinates,
. 144
Expression of the velocity of the moveable; its direction is that of a
tangent to the trajectory; the velocities of its three projections are
what are termed the components of the velocity of the moveable;
the composition and decomposition of the velocities are effected ac-
cording to the same rules as the composition and decomposition of
forces,
145
Whatever be the variation of the velocity of a material point in mag-
nitude and direction, during an infinitely short portion of time,
there is always a certain direction, along which the increase of the
velocity is a maximum, and perpendicular to which the components
of the velocity are neither increased nor diminished,
This determinate direction is what is meant by the direction of the
force which acts on a material point in motion; and setting out
from this definition, it is demonstrated that the increment of the
velocity estimated in any direction whatever, during an instant, is
due solely to the force that acts in this direction, and is the same
as if the others had no existence,
•
146
147
Construction of the trajectory by points, which results from the pre-
ceding principle, and determination of the velocity and position of
the moveable at each instant on this curve,
•
148
xiv
CONTENTS.
Nos.
Differential equations of curvilinear motion, whether the origin of the
coordinates be fixed or in motion,
149, 150
Differential equations of the motion of a material point on a given
curve or surface; expression of the accelerating force along the tan-
gent to the trajectory,
151, 152
154, 155
•
156
II. Principal Consequences of the preceding Formulæ.—p. 227.
First integrals of the differential equations of curvilinear motion,
which obtain when the force is constantly directed to a fixed point, 153
Principle of areas contained in these integrals,
Differential elements of the area and of the length of the curve ex-
pressed in polar coordinates; definition of angular velocity,
First integral of the equations of motion, which gives in a very gene-
ral case, the square of the velocity of the moveable, independently
of the curve described; this velocity is constant when the move.
able, being either entirely free, or constrained to exist on a given
curve or surface, is not solicited by any accelerating force what-
ever;
the integral obtains as often as the moveable is subjected to
the action of forces directed towards fixed centres, whose intensities
are functions of the distances from these points,
157, 158
Expression of the velocity of a heavy body on any curve whatever, in
a function of the height through which the moveable has fallen;
immediate consequences which result from this,
Property of the motion of a material point, which has been denomi-
nated the principle of least action,
•
. 159
160
In virtue of this principle, a material point constrained to move on a
given surface, and not solicited by any accelerating force, describes
in general the shortest line from one point to another; in forming
the differential equation of the trajectory; it is shown that this
shortest line has its osculating plane every where normal to the given
surface,
. 161
III. Digression on the Motion of Light.—p. 242.
In the system of emission, the general laws of refraction and of re-
flexion may be easily deduced from the principle of least action, 162,
163, 164
•
Differential equations of the motion of a ray of light when it passes
from one medium to another; consequences of these equations re-
latively to the two different cases of reflexion and refraction; di-
rection of a ray that traverses two parallel surfaces; phenomenon
of dispersion,
165, 166, 167
•
+
CONTENTS.
XV
The composition of the velocity of light with that of the earth, which
produces the phenomenon of aberration, has, notwithstanding, no
appreciable influence on the magnitude of the refraction; in a va-
cuum, the velocity of reflected and direct light is the same, whether
it reaches us from the sun, the fixed stars, or the planets; magni-
tude of this velocity; diminution, which it ought to experience, in
virtue of the gravitation of the luminous rays towards the sun,
CHAPTER IV. OF THE CENTRIFUGAL FORCE.-P. 256.
Definition of the centrifugal force; determination of this motive
force, from the consideration of the normal velocity destroyed in
the transit of the moveable from one element of its trajectory to the
following; the angle of contact being infinitely small, this passage
does not produce any diminution of the velocity along the tangent;
complete determination in magnitude and direction of the pressure
exerted on the trajectory in virtue of the centrifugal force, and of
the given forces which act on the moveable,
Determination of the three components of this same pressure, by
means of the differential equations of the motion,
Consequences that result from the value of this pressure, and from
its direction, when the moveable is constrained to move on a given
surface, and when it is entirely free,
Determination of the centrifugal force, from the consideration of cir-
cular motion,
Nos.
. 168
169, 170
. 171
172, 173
174
175
Comparison of the centrifugal force in a circle with the weight; ten-
sion of a thread to which a weight is attached when it turns about
a fixed point,
Diminution of gravity at the equator and on different parallels, pro-
duced by the centrifugal force that results from the rotation of the
earth; total variation of gravity due to this cause, and to the com-
pression of the terrestrial spheroid,
176, 177, 178
CHAPTER V.—Examples of the Motion of a Material Point on
a given Curve OR SURFACE.-p. 272.
I. Oscillations of the Simple Pendulum.—Ibid.
Definition of the simple pendulum; it will be demonstrated in the se-
quel that there is always a simple pendulum, the motion of which
in a vacuum or in air is the same as that of a given pendulum,
Differential formula of the motion of the simple pendulum in a va-
. 179
cuum,
. 180
Case in which this formula may be integrated under a finite form,
Case of very small oscillations,
. 181
182
xvi
CONTENTS.
Nos.
On any curve whatever, the duration of the infinitely small oscilla-
tions of a heavy material point, is of a finite magnitude, and inde-
pendent of the magnitude of their amplitude,
Correction which should be made to the duration of the very small
oscillations of a simple pendulum, in order to obtain from it the du-
ration of the infinitely small oscillations,
Reduction into a series, of the time of an oscillation of any magnitude
whatever,
•
•
183
184
. 185
186, 187
Motion of the simple pendulum in the air when the resistance is sup-
posed to be proportional to the velocity; the successive amplitudes
of very small oscillations decrease in geometrical progression; their
duration is not sensibly influenced by the resistance of the me-
dium,
Motion of the simple pendulum in the air when the resistance is sup-
posed to be proportional to the square of the velocity; law of the
decrease of successive amplitudes; in the case of small oscillations,
it is demonstrated that the duration of an ascending semi-oscillation
is as much diminished, as that of the preceding descending semi-
oscillation is increased,
188, 189, 190
Correction in the length of the pendulum and in the duration of small
oscillations, which is termed the reduction to a vacuum; increase
which ought to be made to this correction, in consequence of the
air being in a state of motion,
•
191
On the surface of the earth the weight is proportional to the length of
the pendulum that vibrates seconds; values of those quantities at
the Observatory at Paris; experiments made with the pendulum
demonstrate that in all places of its surface, the attraction of the
earth is the same on different descriptions of matter,
Value of the weight and of the length of the pendulum that vibrates
seconds in functions of the latitude; a time-piece regulated at Pa-
ris, and then transferred to the equator, loses,
II. Motion on the Cycloid.-p. 297.
The time of descent of a heavy material point on a cycloid, is inde-
pendent of the elevation of the point of departure above the lowest
point, whether the motion is performed in a vacuum or in air, the
resistance in this last case being supposed to be proportional to the
square of the velocity,
Cycloidal pendulum,
In a vacuum the cycloid is the only tutochronous curve,
•
192
193
•
194, 195
· 196
197
Investigation of the brachystochrone in a vacuum; formulæ relative
.
B
CONTENTS.
xvii
Nos.
to the case in which the line of swiftest descent is traced on a given
surface; the formulæ in the case in which its length is given, will
enable us, in the sequel, to resolve another problem of the same na-
ture,
198, 199, 200, 201
The equation of the brachystochrone properly so called, is that of a
cycloid situated in a vertical plane; case in which the point of de-
parture and the point of arrival exist in the same vertical,
111. Motion on a given Surface.—p. 312.
Differential equations of the motion of the simple pendulum, which
does not move in the same fixed plane,
Differential formulæ relative to the conical oscillations of a simple
pendulum in a vacuum,
202
203
204, 205
Case of small oscillations; case in which the pendulum describes uni-
formly the surface of a right cone with a circular base; the curve
described by the horizontal projection of the moveable, is always an
ellipse whose centre is the point of suspension,
206, 207
CHAPTER VI.—EXAMPLES OF THE MOTION OF A MOVEABLE ENTIRELY
FREE. p. 321.
I. Motion of Projectiles.-Ibid.
The trajectory of a material point in a vacuum is a parabola; ampli-
tude of the projection; velocity in any point whatever,
. 208
The initial velocity being given, to find its direction, in order that the
projectile may strike a given mark; the curve beyond which the
projectile cannot fall,
. 209
Equations of the motion of the projectile in air; construction of the
trajectory by points; computation of the time; expression for the
velocity at any point whatever,
210, 211, 212
If the moveable ascends to a great height, its motion, after it begins
to descend, becomes continually more uniform, and its direction
approaches more and more to the vertical; determination of the
vertical asymptote of the descending branch,
The other branch of the trajectory has also an asymptote; direction
of this line, and its distance to the point from which the moveable
commences its motion,
Equation of the trajectory when the angle of projection is very small;
determination of the horizontal range, and of the time of flight,
by means of the magnitude of the initial velocity; different values
of the range and velocity that are furnished by observation; uncer-
C
•
. 213
214
xviii
CONTENTS.
tainty as to the magnitude of the coefficient of the resistance;
means of determining it by experiment,
Laws of Kepler,
II. Motion of the Planets.—p. 337.
Nos.
•
215, 216
Equations furnished by the two first of these laws,
Definition of some terms employed in astronomy; durations of the
sidereal and equinoctial years; magnitude of the annual precession
of the equinoxes,
. 217
218
219
Expressions of the two polar coordinates of the planet, and of the
time, in functions of the eccentric anomaly,
Method of reducing the radius vector and equation of the centre into
series arranged according to the sines and cosines of the multiples
of the mean motion,
220
•
Formula which determine, at any point of the ellipse described by a
planet, the magnitude and direction of its velocity,
Position of a planet with respect to any plane whatever; its longi-
tude and its latitude, its right ascension and its declension; obliquity
of the ecliptic; its annual diminution; magnitude and period of
the nutation,
It follows from the laws of Kepler, that the force which retains the
planets in their orbits is constantly directed towards the centre of
the sun; that it varies for each planet, in the inverse ratio of the
square of the distance from this point; that at the unit of distance,
the accelerating force is the same for all the planets; these laws are
also applicable to the comets and to the motions of the satellites
about their respective primaries, and to the relative motions of the
double stars,
221
222
223
227, 228
224, 225, 226
Differential equations of the motion of a planet in a resisting medium;
in the case in which the resistance is neglected, the complete num-
ber of arbitrary constants which their integrals ought to contain
is given,
Method of the variations of arbitrary constants, for the integration of
differential equations, .
Application of this method to the equations of the motion of a planet
or comet in a resisting medium; why the resistance of the ether
may be appreciable in the motion of a comet, and insensible in the
motion of a planet,
231, 232, 233
•
229, 230
CONTENTS.
xix
Nos.
III. Motion of a material Point acted on by a Central Force.-p. 362.
Equations of the motion of a material point attracted towards a fixed
centre, by a force given in a function of the distance from this
centre,
Case in which the force is proportional to the distance,
•
Case in which the force is in the inverse cube of the distance,
Case in which the force is in the inverse ratio of the square of the dis-
tance: the trajectory may be then one of the three conic sections;
circumstances which determine each of the three curves,
Particular discussion of the parabolical motion; in what the astro-
nomical problem of the complete determination of the orbit of a
comet consists,
. 234
235
236
. 237, 238
239, 240
CHAPTER VII.-DIGRESSION ON UNIVERSAL ATTRACTION.-P. 376.
Law of universal attraction,
·
Motive force resulting from the mutual attraction of the sun and of a
planet; invariability of the attractive power,
Accelerating force of a planet in its motion about the sun; correction
which should be made to the third law of Kepler; smallness of the
masses of the planets relatively to the mass of the sun,
Enunciation of the different kinds of perturbations of the elliptic mo-
tion of the planets produced by their mutual attraction; these ob-
served effects make known the masses of the disturbing planets,
that of the sun being assumed for unity; invariability of the greater
axes; the motion of the moon is accelerated from one century to
another,
•
•
•
Another means of determining the masses of the planets that are ac-
companied with satellites,
Determination of the forces arising from the action of the sun and
moon, to raise the waters of the ocean; mass of the moon inferred
from the lunar tide compared with the solar tide; diminution of
the gravity at the surface of the earth, produced by the action
of the moon,
241
242
243
244
245
248
246, 247
At the distance of the moon from the earth, the terrestrial gravity is
very nearly equal to the force which retains this satellite in its
orbit,
Determination of the mass of the earth; parallax of the sun, its den-
sity, its distance from the earth; exact determination of the greater
axis of the orbit of a planet whose mass is known,
Deviation of the plumb line produced by local attractions,
•
249, 250
251
XX
CONTENTS.
Nos.
252, 253
Balance of torsion, peculiarly fitted for measuring very small forces;
experiment of Cavendish; mean density of the earth,
Stability of the equilibrium of the ocean, it results from the circum-
stance of this density being greater than that of water; increase of
the density of the strata of the earth from its surface to the centre;
inequality of the motion of the moon caused by the non-sphericity
of the earth; influence of local attractions on the length of the pen-
dulum which vibrates seconds,
254
Reduction to the level of the sea, of the length of the pendulum, ob-
served at a given elevation,
255
BOOK THE THIRD.
STATICS.
SECOND PART.
Chapter I.—OF THE EQUILIBRIUM of a solid Body.—p. 403.
Remarks on the compressibility and change of form of the body that is
considered, .
256
Transformation of a system of any forces whatever, applied to a solid
body, into three groups of forces; the first composed of forces per-
pendicular to a given plane, the second of parallel forces, and com-
prised in this plane, and the third of forces acting in the direction
of a line perpendicular to the preceding, and traced in this same
plane, .
. 257,258,259
Equations that are necessary and sufficient for the equilibrium of a
solid body entirely free,
These equations are also necessary for the equilibrium of every other
system which does not contain any fixed obstacle,
. 260
•
Particular cases of parallel forces, and of forces which are all comprised
in the same plane,
Condition required in order that the given forces may have an unique
resultant; equations of this resultant; its magnitude and direction;
in all cases, the given forces may be reduced to two, in an infinite
variety of different ways,
•
Equation of equilibrium of two solid bodies, which press the one
against the other,
261
262
263, 264
265
Equations of equilibrium of a solid body retained by fixed obstacles,
in the principal cases which present themselves,
266
CONTENTS.
xxi
•
Nos.
. 268
269
Transformation of the equation of equilibrium relative to a fixed axis, 267
Equilibrium of a heavy body on an inclined plane,
Measure of the friction at the instant the equilibrium gives way,
Pressure on the different legs of a horizontal table which supports a
given weight; whence arises the apparent indetermination of the
problem,
CHAPTER II. THEORY OF MOMENTS.-P. 427.
Forces being represented by right lines, their moments are repre-
sented by plane areas; the theorem of No. 46, relative to the mo-
ment of the resultant of two forces, is then a geometrical propo-
sition the demonstration of which is given,
The moment of the projection of a force on a plane is the projection
of the moment of this force on this same plane,
What is meant by the moment of a system of forces with respect to an
axis; the moments of the same system with respect to two axes, of
which one is the production of the other, are equal, and affected
with contrary signs; it is the case also with respect to the moments
relatively to the same axis, of two systems of equal and opposite
forces,
. 270
271
272
274
. 275
. 273
Expressions of the moments of a system of forces with respect to the
three axes of the positive coordinates of their points of application;
how the signs of the terms of these formulæ are determined,
Values of the cosines of the angles relative to the direction of the
normal to the plane, in which a line and given point exist,
Formulæ relative to the projections of a system of plane areas on dif-
ferent planes; identity of these formula with those which respect
the projections of right lines on other right lines,
Plane and magnitude of the minimum area; characteristic property
of this plane,
Properties of moments deduced from those of plane areas; identity of
the composition of moments and of the composition of forces; re-
sultant of that of the projections of plane areas and of the projec-
tions of right lines,
. 276, 277
278, 279, 280
Principal moment of a system of forces; new enunciation of the con-
ditions of the equilibrium of this system; conditions necessary in
order that two systems of forces may be equivalent,
•
Variation of the principal moment produced by the displacement of
the centre of moments; principal minima moments; how the con-
dition, which is necessary and sufficient for the existence of an
unique resultant, is obtained,
•
281
282
283, 284
xxii
CONTENTS.
CHAPTER III.—EXAMPLES OF the EquilibrIUM OF A FLEXIBLE
BODY.-P. 445.
1. Equilibrium of the Funicular Polygon.-Ibid.
In the state of equilibrium of the polygon, each side must be drawn
in the directions of its productions, by equal and contrary forces;
equations necessary for the equilibrium of forces applied to the
polygon,
Construction of the figure of the polygon in equilibrio; determina-
tions of the tensions of its sides; case in which its extreme points
are not supposed to be fixed,
Nos.
285
286, 287
The extensions of the sides of the polygon are proportional to the ten-
sions to which they are subjected,
When one of the nodes of the polygon is replaced by a ring, the force
applied at this point should divide into two equal parts the angle
between the two adjacent sides,
Condition relative to the direction of the forces which should obtain
in all systems of material points in equilibrio, and of which the pre-
ceding is a particular case,
Equilibrium of a polygon loaded with weights; pressures experienced
by the fixed points to which it is attached,
Remark analogous to that of No. 270, on the tensions of cords which
sustain a given weight; whatever may be the number of these cords,
their tensions and the pressures of the fixed points may be obtained
by measuring the quantity by which their length is increased,
II. Equilibrium of a Flexible Thread.-p. 458.
•
288
289
. 290
291
292
The three general equations of the equilibrium of a thread acted on
by gravity, may be reduced to two,
293
The integrals of these equations in a finite form; equation of the ca-
tenary; expression of the tension at any point whatever,
Determination of the tension at the lowest point, and of the pressures
on the two points of suspension,
Among all isoperimetrical curves, the catenary is that whose centre of
gravity is the lowest,
294
295
296
•
Case in which the vertical forces that act on the elements of the thread
are proportional to their horizontal projections; the curve which
the thread then assumes when in equilibrio is a parabola; compu-
tation of the tension at the lowest point, and of the pressures on
the extreme points, which may be useful in the construction of sus-
pension bridges,
•
297
CONTENTS.
xxiii
Nos.
Equations of equilibrium of a thread solicited by any forces whatever, 298
Case of a heavy thread suspended vertically at a fixed point, and
loaded with a weight at its lower extremity; calculation of the en-
tire increase of length,
Expression of the tension in the general case; the curve is deter- -
mined by two differential equations of the second order; value of
the radius of curvature from knowing the direction of the tangent
at each point,
Application of the preceding formula to the case of a thread stretched
on the surface of a solid body, by forces applied at its extremities,
and which are the only ones that solicit it; the tension is the same
throughout its entire length, in the state of stable equilibrium the
curve traces on the surface the shortest line that can be drawn
from one point to another; the pressure exerted on each of the
points of the surface is in the inverse ratio of the radius of curvature
of this line, and proportional to the tension,
These results are modified by the friction of the thread against the
surface of the solid body; determination of the friction of a thread
on the throat of a fixed pulley,
299
300
301, 302
303
The six general equations of equilibrium of No. 261, are verified in
the case of the equilibrium of a flexible thread; use of these equa-
tions in determining the coordinates of the extreme points, where
they are free, or the pressures which they experience, when they
are fixed and given in position,
304, 305
Equilibrium of an Elastic Rod.-p. 485.
Condition necessary in order that the rod may be elastic when it is
bent; different effects which these parts experience when it has
been made to deviate from its position of equilibrium; definition
of an elastic plate,
·
.306
307
Hypothesis relative to the forces which result from the extension or
contraction of the longitudinal filaments, and from the magnitude
of their curvature; value of the entire force of contraction of an
element of the plate; value of the moment of elasticity,
In the elastic curve, properly so called, the tension is constant, and
has no influence on the curvature of the plate; differential equa-
tion of this curve; conditions relative to its extremities, 308, 309
Case in which the plate is horizontal, firmly fixed at one end, and
loaded with a given weight at its other extremity; determination
of the entire flexion; comparison of the extension and flexion of a
plate, which may be produced by the same weight,
. 310
•
xxiv
CONTENTS.
Nos.
Case in which the plate is a flexible spring placed on a horizontal
plane, and loaded with a weight at its upper extremity; detailed
examination of the different forms which the spring may assume, 311, 312
What is meant by the force of a spring; determination of the force
when either the extension or flexion of the spring, produced by a
given weight, is known,
313
•
Extension of the preceding results to the case of an elastic rod either
straight or curved, which is not twisted on itself; what is meant by
the mean filament; value of the moment of elasticity,
Formula which gives the flexion of a straight rod, by means of the
force of this spring; determination of this force in the different hy-
potheses that may be made respecting the contour of the normal
section; comparison of the force of a cylindrical rod which is hol-
low in its interior with one which is not,
Value of the differential of the tension at any point of an elastic rod,
all whose points are solicited by any forces whatever; the volume
of a rod stretched at its extremity, becomes greater, according as
its length is augmented,
•
314
315
. 316
General equations of the equilibrium of an elastic rod, when the tor-
sion is taken into account,
. 317
•
The moment of the torsion is constant throughout its entire length;
its value by means of the forces which act at one of its extremities, 318
Reduction, to one single equation, of the three general equations,
when the mean filament is a plain curve; equations relative to the
particular forces which act at the two extremities of the rod, . 319
Case of a heavy rod; determination of its figure; determination of its
flexion, and of the pressures on the points of support, 320, 321, 322
Case in which the entire weight of the rod is unequally distributed be-
tween its points of support; formula of La Grange, for expressing,
in a series of periodical quantities, the values of a given function,
within given limits of the values of the variable,
Determination of the figure of a rod loaded with a weight suspended
at its middle point; determination of the flexion and of the pres-
sures on its points of support,
323
326
Demonstration of the formula cited in No. 323; other formulæ of the
same nature,
325, 326
Use of this description of formulæ in the summation of series,
Formula of Fourier, deduced from the preceding,
327
328
CONTENTS.
XXV
CHAPTER IV.-Principle of Virtual VelocitiES.—p. 531.
•
Nos.
329, '330
Verification of this principle, in the case of two forces, applied to a
system of pulleys, to the axle in the wheel, to the screw, to the
lever,
General equation of the equilibrium of any system whatever of mate-
rial points, which results from the principle of virtual velocities;
this equation only obtains in the case of the infinitely small motions
that are compatible with the conditions of the system, the motions
directly opposed to which are equally possible; it has been already
demonstrated (No. 39) in the case of an isolated material point,
Notions relative to the tensions and contractions which have place be-
tween the points of a system in equilibrio, that are connected to-
gether in any manner whatever; mode in which these interior
forces are connected; mode of representing the variations of dis-
tance of the points of the system; equations which obtain between
the total variation and the partial variation of each distance,
A very general demonstration of the principle of virtual veloci-
ties,
•
331
332
:
•
334, 335
It is shown that the direct proposition being proved, the inverse pro-
position is an immediate consequence of it,
This principle obtains also in the equilibrium of fluids, as will be
shown hereafter; another demonstration of this founded on the
consideration of systems of pulleys,
336
340
337, 338, 339
The rules relative to the parallellogram of forces and to the compo-
sition of parallel forces, may be deduced from the principle of vir-
tual velocities; how the equations of equilibrium of a solid body,
which have been already found, may be deduced from it,
Transformation of the general equation of virtual velocities; rules for
deducing from it all the equations of equilibrium of a system of ma-
terial points, whose mutual connexions with each other and with
fixed points are expressed by equations between their coordi-
nates,
341, 342
The internal forces which result from these connexions are at the same
time determined, in magnitude and direction; the principle of
virtual velocities is necessary to make known, relatively to two or
more points connected together by the same equation, the relations
between the magnitudes of these forces, for what has been stated
in No. 290 only determines the directions,
Application of the preceding formula to the example of the funicular
polygon,
•
d
•
343, 344
•
345
1
xxvi
CONTENTS.
2
Property of maximum or minimum that obtains in the equilibrium
of a system of material points subjected to the actions of their mu-
tual attractions or repulsions, which are expressed in functions of
the distances, and to other similar forces, directed towards fixed
centres,
Distinction between stable and instantaneous equilibrium,
Nos.
346
347
Property of the centre of gravity of a system of heavy bodies in these
two states of equilibrium,
Example of the stability and instability of this system,
348
349
ERRATA.
Page 52, line 9, for (e) read (b), and for (c) read (d).
2, for AME read ANE.
α
101,
113,
11, for (#
read
line for a³ read aª.
2
155, last line, for e³ read e².
24
2
(
a
&c.; and for (ax-x)* read ‡ (ax-xª); and last
‡
180, line 1, for i'n'h' read i'n'h.
186, 21, for after displaced read in a sensible manner.
203, last line, for 2gh read √2gh.
244, last line, for n'r read n'z.
do⁹
dtª
286, line 14, for + read
d0*
dta
5, for de read de; and line 11, after equations read (h).
360,
370,
10, for a read D.
373,
3 from bottom, for 3nt read 3ny.
383,
9, for ↓ read p.
401,
13, for 2πof'h read 2 πp'fh.
402,
5, for " read r.
466,
2 from bottom, for eV(+) read c +√(x²+c)".
475,
475,
496,
1, put the bracket [ before (dødªy, and ]³ after dzd³y)ª.
11, for (dad³y read [(død³y ; and for dzd*y)" read dzdªy)*]$
18, for bread 7.
514, last line, for g=gywa read q=gywa.
522, line 3 from bottom, for 1—h³ +4h sin” § (0
—h³ +4h sin³ † (0—a) read (1 − h)* 1-4 h sin*}(0—a).
(1—h)*
nx (x'—x)
nr(x'—a)
dx = px.
525,
10, for cos
=px read cos
a
a
529,
3 from bottom, for
nx
2a
=ne, read
NT
2a
ne.
2
600,
Sede Br
-12, after Jr.dr read VBxx.
INTRODUCTION.
THAT which can affect our senses in any manner whatever,
is termed matter.
Bodies are such portions of matter as are bounded in
every direction; they must consequently have a determinate
form and volume. By the mass of a body, is understood the
quantity of matter of which it is composed.
A material point is a body infinitely small in all its di-
mensions; so that the length of every line comprised in its
interior, is infinitely small, that is to say, less than any as-
signable length. A body of finite dimensions may be con-
sidered as an aggregate of an infinite number of material
points, and its mass as the sum of all their infinitely small
masses.
2. A body is in motion, when this body, or its parts,
occupy successively different places in space.
But space
being indefinite and every where the same, we can only judge
of the state of rest or motion of a body, by comparing it,
either with other bodies, or with ourselves; and on this
account, all the motions which we observe are necessarily
relative motions.
All bodies are moveable; but matter never moves sponta-
neously, for there is no reason why a material point should move
in one direction rather than in another; and in fact, if we con-
sider a body at the instant it passes from a state of rest to
a state of motion, we may always observe, that this change
B
2
INTRODUCTION.
is owing to the action of an extraneous cause, i. e. of one
without which we may conceive that this body may exist.
Any cause which excites motion in a body, or which only
tends to excite it, when its effect is interrupted or prevented
by any other cause, is called force.
3. When several forces are applied at once to the same
body, they modify each other's effects, in virtue of the con-
nexion that exists between its parts, and which hinders them
from assuming the motion, that the force to which it is sub-
jected, tends to impress on each of them. It may happen,
that these forces completely destroy each other's effects, so
that the body does not move at all; this particular state of
the moveable body is termed equilibrium, in which the body
remains at rest, though solicited by several forces, or in other
words, the forces are said to constitute an equilibrium.
Mechanics is the science which treats of the equilibrium
and motion of bodies. The part of which the object is, in
general, to discover the conditions of equilibrium, is called
statics. The part of which the object is to determine the
motion which a body assumes, when the forces which are
applied to it do not constitute an equilibrium, is called
dynamics.
As geometers have succeeded, as will be hereafter shewn,
in reducing all questions relating to motion to mere problems
of equilibrium, the natural mode of proceeding would be to
treat first of statics, and then to enter on the consideration of
dynamics; but in order to facilitate the understanding of the
subject, it seems preferable, in a treatise designed for instruc-
tion, to direct our attention to the elementary and simple
parts of dynamics, before we enter on the consideration of the
general questions of equilibrium. This is the order which
will be pursued in the following treatise.
4. In the case of a force acting on a material point, there
are always three things to be considered, namely, the position
of this point, the intensity of the force, and its direction, that
INTRODUCTION.
3
is, the rectilineal space which it tends to cause to be described
at its point of application. Nevertheless, we must not sup-
pose that a material point is the same with what is called a
point in geometry, where this term denotes the extremity of
a line, or the intersection of two lines which cut each other;
neither is the space which a material point describes a line
having only one dimension; but as this body is infinitely
small in every direction, and as the breadth and thickness of
the space, which the force tends to make it describe, are also
infinitely small, the position and direction of this force can be
determined in the same manner, as the position and direction
of a right line are determined in geometry.
Thus, the position of the point of application of a force in
space, will be determined, in general, by means of its three
coordinates parallel to the intersections of three rectangular
planes; and this will not leave any ambiguity with respect to
the direction, when we take into account both the sign and
magnitude of each coordinate. Sometimes also polar coordi-
nates are employed; these are the radius vector of the given
point, or its distance from the origin, the angle this radius
makes with a fixed line drawn through the origin, and the
angle comprised between a plane passing through these lines,
and a fixed plane passing through the second.
5. In order to measure forces, it is necessary to take some
known force as unit, and then to express in numbers the rela-
tions of the other forces to this unit; this requires that we
should precisely define what is meant when one force is said to
be equal to another, double, triple, or quadruple ....of another,
independently altogether of the particular nature of these differ-
ent causes of motion.
Two forces are said to be equal, when being applied in
opposite directions to the same material point, or to two
points connected by a right line, which is of an invariable
length, they constitute an equilibrium.
If, after having recognized that two forces are equal, we
4
INTRODUCTION.
apply them in the same direction to the same point, we shall
have a double force; if we combine in this manner three equal
forces, we shall have a triple force; if we combine four, we
shall have a quadruple force; and so on.
When, therefore, we say that a force applied to a material
point is a certain multiple of another force, we understand
that the first may be considered as made up of a certain
number of forces respectively equal to the second, and acting
in the same direction. It is by considering them in this
manner, that forces can be regarded, whatever their particular
nature may be, as quantities susceptible of being measured,
which may, therefore, be expressed in numbers, like every
other description of quantity, by referring them to the unit of
their species. We may likewise represent their intensities by
lines proportional to these numbers, which lines we lay off in
their several directions, commencing at their respective points
of application; one advantage effected by this is, that me-
chanical theorems may be stated with great simplicity.
6. The points of application, and the intensities of forces,
being thus determined, it only remains to shew how their
directions are ascertained.
Let M (fig. 1) be the point of application of a force, and
let its direction be represented by the line MD, in which case
this force tends to cause the point м to move from M to D ;
through the point м, conceive three rectangular axes MA,
MB, MC, to be drawn, these will be, in general, parallel to the
axes of the coordinates, and in the direction of the positive
coordinates; let a, ß, y, be the obtuse or acute angles which
the direction MD makes with these axes, so that
AMD α, BMD - B,
CMD=Y;
this direction will be completely determined, when these three
angles are given.
In fact, if we only consider the two angles a and ẞ, the
line mò must exist at the same time on two right cones, of
I
INTRODUCTION.
5
which the common summit is м, and of which the respective
axes are ma and мB. Therefore it is necessary that a and ß
should be such that these two cones may intersect; this
will have place in the direction of two lines, drawn from M,
situated in the same plane perpendicular to the plane AMB,
and which will make with the axis MC, two angles that are
supplements the one of the other. Therefore the line MD
may have also two different positions, but as the angle y is
also given, we know whether it is acute or obtuse, and we
can select between these two positions that which answers to
the direction of the force. This construction also shews, that
the three angles a, ß, y, cannot be arbitrarily taken. In fact,
there exists between the cosines of the angles which the same
right line MD makes with three rectangular axes, the equation
cos² a + cos² ß + cos² y = 1,
:
(1)
which can be demonstrated by taking on MD, reckoning from
the point м, a line equal to unity, and then forming a rec-
tangular parallelopiped, of which this line is the diagonal, and
of which the three adjacent sides lie on the three axes MA, MB,
MC. These three sides will be the cosines of the angles a, ß,y;
and as by a known theorem, the sum of their squares must
be equal to the square of the diagonal, there results the equa-
tion which has been stated above.
7. In this treatise, the division of the circumference into
360°, of the degree into 60 minutes, and of the minute into
60 seconds, will be adopted. The letter π will be constantly
employed to represent the semicircumference of which the
radius is equal to unity, so that we shall have
π = 3,1415926....
T
The fourth part of the circumference corresponds to a
right angle, or to the angle 324000"(a), hence it follows, that
the length of an arc corresponding to an angle, containing
any number such as n of seconds, will be the fourth term of a
6
INTRODUCTION.
proportion, of which the three first terms are r n and 32400",
and denoting this length by w there will result(b)
n
@= 206264,8...
The logarithm, in the common system, of this constant divi-
sor is
5,3144251.
In numerical calculations, the arcs computed in this man-
ner ought to be employed in place of angles, which will not be
comprised under the trigonometric signs sin, cos, tang.
In order that we may be able, by means of the angles
a, ß, y, to represent the direction of a force in all possible
positions about its point of application, it is only necessary
that they be reckoned from zero up to 180° inclusively. If, for
example, the axis Mc lies above the plane of the two other
axes MA and MB, the angle y will be less or greater than 90°,
according as the right line MD exists above or below this
plane: it will be zero when the direction MD coincides with MC,
and equal to 180°, when MD coincides with Mc' the production
of MC. The cosines of a, ß, and y, may therefore be positive
or negative; but their sines will be always positive, because
these angles never exceed 180°.
In general, if we consider MD' the production of any line
MD, in the opposite direction, it is evident that the angles
which it makes with the three axes are the supplements of
a, ß, y. Therefore if we make
we shall have
AMD'a', BMD'B', CMD'y',
cos a, cos B'= cos ß, cos y' =
cos y ;
cos a' =
hence it follows that the directions of two forces which act in
opposite directions on the same point м, the one along MD and
the other along MD', are distinguished from each other, by the
INTRODUCTION.
7
signs of the cosines of the angles which respectively correspond
to them.
8. In place of three angles a, ß, y, connected together by
equation (1), we may only employ two angles, perfectly in-
dependent of each other, to determine the direction of a force.
In fact, let me be the projection of MD on the plane AMB,
and let & be the angle which this projection makes with the
axis MA, so that we have AMES. When the angle 8 is
given, it will make known the position of the plane CME, and
then the angle y will completely determine the position of
the line MD comprised in this plane. It is necessary that the
angle 8 should be reckoned from MA in a definite direction,
which may extend from zero to 360°; the angle y, as was
before stated, is comprised between the limits zero and 180°.
The projection of the diagonal of the parallelopiped, ad-
verted to in No. 6, on the plane AMB, will be the cosine of
the angle DME, or equal to sin y. If we project a second
time this projection on the axis MA, this second projection is
equal to first multiplied by cos &; moreover it coincides with
the projection of the diagonal of the parallelopiped on this
same axis ma, and consequently will be equal to cosa; there-
fore we shall have
cos asin y. cos d.
In like manner we shall have
cos ẞsin y. sin S.
4.8.2.8.
equa-
These two formulæ will enable us to transform the
tions in which the angles a, ß, y, are made use of, into others
in which we only employ y and S. It is easy to perceive
that they satisfy equation (1).
9. There exists another equation that comprises equa-
tion (1) as a particular case, and which will be frequently
useful.
In order to obtain it, let x, y, z, be the three coordinates
of any point м (fig. 2) referred to the three rectangular axes
8
INTRODUCTION.
Qx, oy, oz. Let r denote its radius vector om, and a, ß, y, the
acute or obtuse angles which this radius makes with the three
axes, so that, for example, we may have
ZOMY.
If from the point м a perpendicular MN be let fall on the
axis ox, the right line on will be the ordinate z, and in the
right angled triangle MON we shall have
z=r.cos y ;
in like manner there results
y=r.cos ß, x =r.cos a.
Let M' be another point and let x', y', x', r', a', B', y', be
respectively its coordinates, its radius vector, and the angles,
relatively to this line; we shall have
x' = r' cos a', y' = r'. cos ẞ', z' = r'. cos y'.
Let u denote the distance MM', then we have the known
relation
u² — (x' — x)²+(y' — y)² + (2' — 2)²;
and if e denote the angle MOM', we shall have at the same
time
u² = p² + p⁄2 — 2rr'. cos ε,
-
in the triangle of which r, r', and u are the three sides.
Because
x²+y²+z² = r², x²² + y²² + 2/2 = p²²,
2
the first value of u² is the same thing as
u² = p² + p/2 — 2 (xx′ + yy' + zz′) ;
2(xx′+yy'
by comparing it with the second, we obtain
rr'. cosɛ = xx′ +yy' + zx';
and if in this equation, we substitute the preceding values of
x, y, z, x'y', z', there will result
cos ε = cos a. cos a' + cos B. cos B'+ cosy.cos y'; (2)
which it was proposed to find out.
INTRODUCTION.
9
When the two lines oм and oм' coincide, the angles
a',ẞ',y', are the same as a, ß,y, and this formula is reduced to
equation (1). When these two right lines are at right angles
to each other, we have & 90°, and consequently
ɛ =
cosa.cos a' + cos ß.cos B' + cos y.cos y' = 0.
By substituting in the values of x, y, z, those of cos a,
cos B, which have been found in the preceding number, we
shall have
x=r.sin y.cos d, y = r. sin y. sin d, ≈=r.cos y;
in which formulæ, the three variables r, y, d, are the three
polar coordinates of the point м, such as they have been de-
fined in No. 4, and which will consequently enable us to
transform rectangular into polar coordinates.
10. The consideration of projections which were made use
of in No. 8, will be frequently adverted to in this treatise, it
will therefore be useful here to explain their first principles.
The projection of a right line on another right line, is the
part of the latter which is comprised between two perpendi-
culars let fall from the two extremities of the projected line
on the other. Thus, the differences xx, y' — y, z' — Z,
of the extreme coordinates are the projections of the same
right line MM' on the axes of x, y, z; and, from the first ex-
pression which is given above for u², it follows, that the sum
of the squares of the projections of the same right line, on
three rectangular axes, is equal to the square of this right
line. If the projected line, and the line on which it is pro-
jected, exist in the same plane, the projection is equal and
parallel to the base of a right angled triangle, of which the
projected line is the hypothenuse, so that if I represents the
length of this line, A that of its projection, and i the angle con-
tained by these two right lines, we shall have
λ=l.cos i.
l
The projection of a plane surface on another plane, is the
C
10
INTRODUCTION.
part of this plane which is terminated by the projection of
the outline of the projected surface, that is to say, by the
curve formed by perpendiculars let fall from all the points of
this outline. Now the preceding equation likewise obtains,
if in place of 1, the area of the projected surface be substituted,
and in place of λA, the area of its projection; i being then the
inclination of one plane upon the other, which is evidently
equal to the angle contained by the two perpendiculars to
these planes.
In fact, if the area of the projected surface be divided into
a number of indefinitely small elements perpendicular to the
intersection of its plane with that on which the projection
is made; the projection of each element will be equal to this
element multiplied by the cosine of their mutual inclination ;
consequently as this inclination is the same and equal to ¿ for
all the elements, the sum of their projections or λ, will be
equal to their sum, or to the entire area 1, multiplied by
cos i. Hence it follows, that the square of the area of a
plane surface, is equal to the sum of the squares of its pro-
jections on three rectangular planes; this follows from equa-
tion (1), and from considering that the inclination on each
plane is the angle which a normal to the given surface makes
with the perpendiculars to this plane(c).
11. When in a question, we consider a system of parallel
forces, we may suppose that one of the three rectangular axes
Ma, MB, MC, (fig. 1) is also parallel to them. Then two of
the three angles a, ß, y, the two last, for instance, will be
right angles for all these forces; and the equation (1) will be
reduced to
cos² a = 1;
hence it follows that a=0 or 180°.
In this manner, the direction of each force will be deter-
mined by stating, that it makes with the axis MA an angle,
which either vanishes or is equal to 180°; but in this parti-
cular case, it will be simpler to determine this direction by the
INTRODUCTION.
11
sign of the force, those forces which act in one direction being
considered as positive, while those which act in an opposite
direction are regarded as negative.
In fine, the case of parallel forces will be the only one in
which forces are considered as positive and negative ; in all
other cases, the quantities which represent the magnitudes of
forces are considered in computations as positive; and the
variation of the sign will fall on the cosines of the angles
which their directions make with fixed axes.
12. What precedes respects the preliminary definitions and
details that are required for determining the magnitudes and
directions of forces; but as the method of infinitely small
quantities will be exclusively adopted in the following treatise,
it is necessary to advert to in this introduction, the principles
of the infinitesimal analysis, and among the formulæ which are
immediately deducible from them, to enumerate those which
will be useful in the sequel.
An infinitely small quantity is a magnitude less than any
other given magnitude of the same nature.
When we consider the successive variations of a magnitude
subject to the law of continuity, the notion of infinitely small
quantities is necessarily suggested. Thus, time increases by
degrees less than any assignable interval, however small.
The spaces traversed by the different points of a body, like-
wise increase by infinitely small quantities; for each point
cannot go from one position to another, without traversing
all the intermediate positions, and we cannot assign a distance,
as small as we please, between two consecutive positions. There-
fore, infinitely small quantities have a real existence, and are
not to be considered as merely a means of investigation de-
vised by geometers.
An infinitely small quantity may be double, triple, qua-
druple.... of another, and between such quantities any relations
may exist, the determination of which is an essential object
of the infinitesimal analysis.
12
INTRODUCTION.
If a and b be infinitely small quantities, and if the ratio
of a to b be also infinitely small, b is what is termed an infi-
nitely small quantity of the second order. For example, the
chord of an arc of a circle being supposed infinitely small, the
versed sine of the same arc is an infinitely small quantity of
the second order, because the ratio of the versed sine to the
chord is always the same as that of the chord to the diameter,
and consequently becomes infinitely small at the same time as
the second ratio. In like manner, if b be an infinitely small
quantity of the second order, and if we suppose the ratio of
c to b to be an infinitely small quantity of the first order,
c will be an infinitely small quantity of the third order, and
Hence it follows, that a product composed of a num-
ber n of infinitely small factors of the first order, must be
ranged in the class of infinitely small quantities of the order
N. The area of a surface infinitely small in all its dimensions,
is at least an infinitely small quantity of the second order;
for it is less than the square of the longest line which can be
drawn from one point to another of its outline, which is by
hypothesis an infinitely small quantity. Likewise it is evi-
dent, that a volume of which all the dimensions are infinitely
small, is at least an infinitely small quantity of the third
order, for it is less than the cube of the longest line drawn
from one point to another of its superficies.
so on.
This being premised, the fundamental principle of the infi-
nitesimal calculus consists in this, that two finite quantities,
which only differ from each other by an infinitely small
quantity, must be considered as equal, since we cannot assign
any inequality between them, however small. This is also
true with respect to two infinitely small quantities of the first
order, of which the difference is an infinitely small quantity of
the second order, and in general two infinitely small quantities
of any order whatever, which only differ from each other by
an infinitely small quantity of a higher order, may be con-
sidered as quantities rigorously equal, and their ratio as equal
to unity.
INTRODUCTION.
13
These principles may be expressed in a different manner,
by stating that in a calculation we are permitted to neglect,
without any apprehension of altering the results, either infi-
nitely small quantities added to finite quantities, or infinitely
small quantities of any order, which are added to those of an
inferior order.
13. dx, the differential of an independent variable x, is an
infinitely small increment ascribed to this variable; dy the
differential of y a function of x, is the corresponding increment
of this function, reduced to the same order of magnitude as
that of the independent variable, by the suppression of infi-
nitely small quantities of a higher order; hence it follows,
that this differential dy is always of the form xdx; x being
another function of x. It may happen, that for some parti-
cular values of x, the differential coefficient x becomes infi-
nite, which will render the differential xdx indeterminate;
but this case will not occur in mechanics. Let fx be
a given function of x, c a constant arbitrary, and Fx + c
the complete or indefinite integral of fxdx. Let also a and b
be two given constants. If the constant c be determined in
such a manner, that this integral may be nothing, or com-
mence when x = a, and if then x be assumed equal to b, the
result Fb - Fa will be what is termed the definite integral,
taken between the limits x = ɑ and x = b. We shall denote
it by Sofada, agreeably to the very commodious notation
proposed by Fourier; consequently, we shall have
Fb Fα=
So fxdx.
If we assign to x an infinite number of values, increasing
successively from a to b by infinitely small differences, and if
these differences (whether equal or unequal) be assumed as the
values of dx, it is easy to shew that the sum of all the values
of the differential fada will be equal to the definite integral
}
}
14
INTRODUCTION.
Fb ་ Fa. In fact, if infinitely small quantities of an order
higher than the first, be neglected, we have by the definition of
the differential
F(x+dx) - Fxfxdx.
Therefore if d₁, 82, 83, ... En, denote an infinite number of
infinitely small quantities, such that
81 +82 +83+.... +dn=ba;
and if we take successively for x and dx, the pairs of values
a and S₁, a+ Si and S₂, a + S₁ + S2 and S3,
and Sn, there will result
F (a + d₁) — Fa= fadı,
....b-dn-1
F (a + d₁ + 82) — F (a + d₁) = ƒ (a + d₁) 82,
d1
F (a + d +82 +83) − F (a + &; +d₂) = ƒ (a + &₁ +82) 83,
Fb — F (b — dn) = ƒ (b − dn) dn ;
the sum of these equations is
Fb
Fa=fadı +ƒ(a+81) 8½ +ƒ (a + 81 +82).83
+ƒ (b − Sn) En ;
which expresses the theorem, that was proposed to be proved.
When the function fr becomes infinite, between the two
limits a and b, this demonstration does not obtain, and the
theorem fails. In this case of exception, which will not occur
in this treatise, the definite integral has no connexion with
the sum of the values of the differential, and it may be ne-
gative when all these values are positive, or positive when
they are all negative. In order then to make this theorem
obtain, we should take care, that fr does not become infinite
between x = a and xb, by causing the variable x to pass
from the first to the second of these limits, through a series of
imaginary values(₫).
The preceding theorem may be extended without difficulty
to multiple integrals. Thus for example, if f(x, y) be a given
INTRODUCTION.
15
function of two independent variables x and y, and if we
assign to these variables successively, series of values in-
creasing by infinitely small differences, and if at the same
time we assume the differences between the consecutive values
of a to be equal to da, and those of the consecutive values of
y to be equal to dy, the sum of all the values of ƒ (x, y) dxdy,
will be the integral SSƒ(x, y) dady, taken between suitable
limits.
14. When the function fr contains a quantity a that has
been considered as constant during the course of the integration,
the value of the integral Sofada will be itself a function of a.
Questions occur, in which this integral not being known in a
finite form, it will be nevertheless necessary to determine its dif-
ferential with respect to a. Now, in this operation, two dif-
ferent cases present themselves, according as the limits a and
b are independent of a, or depend on it in some way. In the
first case, it will be sufficient to difference fr with respect to a,
under the sign §, so that we shall have
d.
So fxdx
da
-
Sb dfc dx.
a da
In fact, it follows from the theorem of the preceding num-
ber, that the first member of this equation is the differential
coefficient with respect to a of the sum of the values of fxdx
comprised between xa and xb; while the second member
is the sum of the values between the same limits, of the dif-
ferential coefficient of fxdx taken relatively to a, and it is
evident, that these two sums are identically the same. In
the second case, when a becomes a + da, the limit b becomes
b+
db
da
da, and on this account, the sum of the values of fxdx,
or the integral Sofrdr is increased by the value of fædx,
16
INTRODUCTION.
db
da
which answers to x = b, and dx = da, that is to say, of
da
da; at the same time, the limit a is changed into
db
fb
da
a+
da
corresponding to xa and dx =
da
da
da, this diminishes this integral by the value of fxdx,
da
da, or of fa da; there-
Ta
fore, on account of the simultaneous variation of the two limits
a and b, produced by that of a, the integral will be increased
by the differential
db
da r
fb-da fa) da,
and its differential coefficient with respect to a, by this coeffi-
cient of da. Consequently, by adding it to the second mem-
ber of the preceding equation, we shall have
b
1.So fx dx
d
α
=
da
dx +
a da
b dfx db
dafb
da
da fa,
for the complete value of the differential coefficient of fxdx.
α
When a does not occur in fr, if this quantity be one of the
two limits b or a, and if these two limits do not depend the
one on the other, this expression will be reduced to
d. So fxdx
db
d
a Sofxdx
=fb or
=-fa;
da
which is, otherwise, evident of itself.
Similar remarks are applicable to multiple integrals, of
which the differential coefficients with respect to a quantity
which in the first instance is considered as constant, may like-
wise be obtained, by differentiating under the signs of inte-
gration, and by adding to the result the terms which depend
on the variations of the limits, when they are functions of this
quantity considered as variable.
INTRODUCTION.
17
15. The integral calculus furnishes rules which enable us
to determine either exactly, or by approximation, the nume-
rical values of definite integrals, whether simple or multiple ;
so that a problem is considered to be resolved, when we are
enabled to express the unknown quantities by integrals of this
nature. The problem is then said to be reduced to quadra-
tures, because on the one hand, a multiple integral is nothing
else than a simple integral several times repeated, and also
because on the other, an integral such as Sofada may always
X
fxdx
be represented by a square equal to the area of the plane curve,
in which x and for are the coordinates of any point whatever,
and a and b the abscissæ of the extreme points.
Among the different formula which are in use for deter-
mining the approximate values of this integral So fade, we
will cite the following, in which it is assumed that the functions
fr and
dfx
dx
do not become infinite between the limits a and b.
Retaining the preceding notation, and moreover making
dfx
d²fx
=fx',
=fix", &c.
dx
dx2
If the differences 81, 82, 83, &c. are not infinitely small,
but only very small, when they are all equal and represented
respectively by d, we shall have by Taylor's theorem
F (a + 8) = Fa + Sƒa + §8²ƒ'a + &c.
F (a + 28) = F (a + d) + dƒ (a + 8) + § 8²ƒ'(a+8)+&c.
F (a +38) = F (a+28) +8ƒ(a+28) + 18²ƒ'(a + 28) + &c.
F (a + nd) = F (a + (nd − d)) + Sƒ(a + nd—8)
+38²ƒ¼(a + nd→ 8) + &c.
Therefore, if we suppose ndb-a, we shall have, by
taking the sum of these equations,
D
F
18
INTRODUCTION.
Fb - Fa= 8. Σf (a + id) + § 8². Σf' (a +id)
+ } 8³ Σƒ". (a+id) +&c.
83
ï being an integer number or cypher, and the characteristics
Σ denoting the sums which extend to the n values of i, taken
between i = 0 and in — 1.
1. If we assume fx and fx', fx'
and fx", &c. successively, in place of Fx and fx, we shall
likewise have
fb — fa = dΣƒ” (a + i8) + } 8² Σƒ" (a + id) + &c.,
ƒ'b− ƒ'a = 8Σƒ''(a + id) + &c.
This being established, if we wish to neglect powers of
8 higher than the square, in the value of Fb - Fa, we can, by
means of the preceding equations, assume
/ d² Σƒ' (a + id) = § 8 (fb — fa) — 18² (ƒ'b — ƒ'a),
} 8³ Σƒ¹¹ (a + id) = † 8² (ƒ'b — ƒ'a),
for the values of its two last terms; its entire value will con-
sequently become
Fb — Fa = 8Σƒ(a+id) +18 (ƒb− fa) — 1½d² (ƒ'b—ƒ'a),
-
or which is the same thing(e),
So fxdx = 8 [§ƒa +ƒ (a + d) + ƒ(a + 28) .
+ƒ(a+nd − 8) + {ƒb] − 1½ 8² (ƒ'b — ƒ'a).
This formula will be more exact according as the differ-
ence d, or — (b − a), is less, and as the values of fæ vary less
N
rapidly between the limits a and b. For the most part we
can neglect the term depending on 82; the formula will then
only contain the values of fx, which may be given in num-
bers, although the form of this function is not known.
16. In the theory of infinitely small quantities, curves are
treated as polygons, which consist of an infinite number of
infinitely small sides. This implies that the chord of an in-
finitely small arc is equal to this arc, or that we can assume
INTRODUCTION.
19
the ratio of their respective lengths to be equal to unity;
which can be plainly demonstrated in the following manner.
Let мmm'м' (fig. 3) be an infinitely small are of a curve, let
the chords mm, mm', m'm', be drawn, and let the third be
produced to meet мT the production of the first, in a point K,
The arc mm' is greater than the chord mm', and less than the
line mK+Km'; therefore, if we prove that this line and this
chord, when infinitely small, differ only by an infinitely small
quantity of a higher order, and that consequently we can
assume their ratio to be equal to unity, it will follow à fortiori,
that the ratio of the arc mm' to its chord, is that of equality.
Now if in the arc мmm'м', there does not exist any singular
point in which the direction of the curve changes abruptly,
chords drawn from one of its points to two other will contain
an angle differing from two right angles by an infinitely small
quantity. Consequently the angle ткм' the supplement of
MKM' will be an infinitely small quantity, denoting it by d,
and moreover, making,
mk = a, m'k = b, mm'≈ c,
we shall have in the triangle mкm', the equation
c² = a²+b²+ 2 ab cos d,
which can be converted into
(because
c² = (a + b)² — 4ab.sin² 8,
cos 81-2 sin² 8).
Consequently we shall have
C²
4ab
=1
1-
(a+b) 2
sin28,
(a + b)² ·
which expresses the square of the ratio of the chord mm' to the
line mK+ Km'. Moreover as
4ab
b\2
(a+b)²
=1
;
ab.
+
it follows that the coefficient of sin² 8, can never become in-
20
INTRODUCTION.
finite, because it is always less than unity. Therefore, if we
neglect infinitely small quantities of the second order, the
ratio of c to a + b will be always expressed by unity.
17. Considering a curve as an infinitesimal polygon, the
tangents will be the productions of the infinitely small sides;
at the point м, where the side is мm, the tangent will be the
indefinite line T'мmт. If x, y, z, denote the three rectangu-
lar coordinates of the point м, those of the point m will be
x + dx, y+dy, z + dz. If ds denotes the element of the
curve, i. e. its side мm, the differentials dr, dy, dz, will be its
projections on the axes of x, y, z; consequently, if a, ß, y,
be the three angles which the direction of the line мт makes
with lines parallel to these axes respectively, drawn through
the point м, we have
dx
dy
dz
COS α =
ds
cos B =
ds'
cos y =
ds'
(1)
and likewise at the same time,
dx²+ dy²+ dz²= ds².
Assuming on the curve cмmm'м' a fixed point c, and
supposing that s denotes the arc Cм reckoned from this origin,
this arc can be considered as the independent variable; con-
sequently x, y, ≈, will be given functions of s, depending on
the equations of the curve. In this case, ds will be positive,
but dx, dy, dz, and consequently cos a, cos ẞ, cos y, may be
either positive or negative. The angles a, ß, y, always refer to
mr the production of the side мm, or to the part мmт of the
tangent, the angles relatively to the other part MT' will be the
supplements of a, ß, y, (No. 7.) As the direction of the
tangent at the point м is determined by equations (1), we can
deduce from them also the equation of the normal plane at
this same point; but this equation may be directly obtained
in the following manner. Let k denote the radius of a sphere,
the centre of which is at the point м, its equation will be
(x' — x)² + (y' — y)² + (z' — z)² — k²;
INTRODUCTION.
21
x', y', z', denoting the coordinates at the other extremity of
the radius k. The equation of the sphere having the same
radius, and of which the centre is at the point m, may be
deduced from this, by substituting x + dx, y + dy, z+dz,
in place of x, y, z; and if we subtract one of these equations
from the other, we shall have, by neglecting infinitely small
quantities of the second order(ƒ),
(x' — x) dx + (y' — y) dy + (≈' — z) dz = 0;
which is the equation for the intersection of two spherical
surfaces. As it is the equation of a plane of which x', y', z',
are the coordinates, it will be that of the plane of this curve,
and consequently the required equation of the normal plane,
because the intersection of these two spheres is a circle per-
pendicular to the line TT' which passes through their centres M
and m.
If we divide this equation by ds, and then substitute for
dx dy dz
ds' ds' ds'
their values given by equation (1), it will become
(x' — x) cos a + (y' − y) cos ß + (≈′ — z) cos y = 0.
Therefore if
a (x' — x)+b (y' − y) + c (z' — z) = 0,
represents the equation of a plane drawn through a point of
which the coordinates are x, y, z, and perpendicular to the
line of which the direction is determined by the angles a, ß, y,
it must coincide with the preceding; this requires that we
should have
a=h.cos a, b = h cos ß, ch cos y,
h being an indeterminate factor. Moreover from equation (1)
of No. 6, we can infer
a² + b²+c²= h²;
by means of which we are enabled to determine the value of
22
INTRODUCTION.
!
h in every particular, with the exception of the sign. Hence
then there results
α
COS α =
b
= cos B==/ cosy ==/;
B:
h³
(2)
which coincides with those known formulæ by means of which
the direction of a perpendicular to a given plane can be de-
termined. The reason why the sign of h is undetermined, is
because the plane has two sides, and the angles a, ß, y, may
refer to this line considered indifferently as existing on one
side or the other.
18. The angle of contact is the indefinitely small angle
contained between two consecutive tangents. Thus if мm
and mm' (fig. 4) be consecutive sides of the curve, this angle
at the point m is the supplement of мmm', or the angle
Tmt, made by the tangent Tmt and the consecutive one mm't.
If we denote it by 8, and suppose that the angles a, ß, y,
refer always to the direction of мT, and if a', ẞ', y', denote
what they become with respect to the direction of mt, we
shall have in virtue of equation (2) of No. 9,
sin² 8 = 1 − (cos a cos a' + cos ß cos ẞ' + cos y cos y')².
Likewise by Taylor's theorem we have
cos a' = cos a + d.cos a + .d² cos a +&c.,
cos B'= cos ß + d. cosß + 1.d² cos ß + &c.,
cos y' = cos y + d.cos y + {.d² cos y + &c.
Now, if we substitute these values in those of sin² 8, and if
we take into account the equation
cos² a + cos² ß + cos² y = 1,
and its differential
cosa.d.cosa + cos B.d.cos ẞ+ cos y.d.cos y.=0,
it is apparent that the finite quantities, and also the infinitely
small quantities of the first order mutually destroy each other,
INTRODUCTION.
23
L
hence, therefore, if we neglect infinitely small quantities of
a higher order than the second, we shall have
sin28 (cos a d². cos a + cos ẞd.cos ß+ cos yd². cos y);
=
(g)
Moreover, if the preceding equation be differentiated, we
shall obtain
cos ad² cos a + cos Bd2. cos ẞ+ cos yd². cos y
+(d. cos a)² + (d.cos ß)² + (d.cos y)² = 0,
hence the preceding value of sin² & will become
sin² 8 (d.cos a)² + (d.cos ẞ)² + (d.cos y),
=
and this will be also the value of 82, because the infinitely
small arc is equal to its sine.
The differentials of cos a, cos B, cos y, may be deduced
from formulæ (1) of the preceding number. For if the in-
dependent variable be not specified, we shall have
d cos a
and as we have
ds2.d²x-dx.ds d's
ds3
ds²- dx² + dy²+ dz²,
dsd's = dx d²x + dy d²y + dz ď²z,
consequently there results(h)
dy
ds³•
do
ds
d.cosa = (dyd²x — dxd³y) +33 (dzd³x — dxd³z);
and in like manner
dx
dz
ds3
d.cosẞ= (dxd³y — dyd²x) +
ds³
(dzď³y — dyd²x),
dy
ds³ (dyd²z — dzd³y);
d.cos y = det(dxd²z—dzd³x) +
now, if we take the sums of the squares of these three vari-
ables, after some reductions, we shall find that the expression
for sin² 8 or 82 may be exhibited under the form(¿)
*
24
INTRODUCTION.
82= —[(dxd³y — dyd²x)² + (dzd³x — dxd²z)²
ds
+(dyd²z — dzd²y)²].
The osculating circle is that which has two consecutive
sides common with the curve, consequently at the point м, this
circle is that which passes through the three points м,m,m',
therefore the centre is at o the intersection of two perpendi-
culars to мm and mm' in the plane of these two consecutive
elements raised at their points of bisection, and its radius is the
right line мo. If these two elements be supposed to be
equal, this line will bisect the angle мmm'; and we may
assume this to be the case, without any apprehension that the
value of Mo will be altered; for it is easy to be assured that
the numerical ratio of the two infinitely small sides мm and mm'
can only affect the magnitude of this radius by an infinitely
small quantity, and it is consequently the same, whether the
two consecutive sides be assumed to be equal or unequal.
Denoting the length of the sides мm by ds, and that of the
radius mo by p, the projection of p on мm will be ds; so
that we shall have
¿ds = p.cos мmo;
and because this angle мmo is half the supplement of d, or
equal to ½π — §8, there will result
dsp sin 18 = pd,
the arc 18 being substituted in place of its sine.
This being established, if the value of the radius of cur-
vature was known in any other manner, we would have
ds
d=
P
for the value of the angle of contact; and conversely, from the
preceding value of 82, which has been given above, there would
result that of p
p =
ds³
[(dxd³y—dyd²x)²+(dzd²x—dxd²z)²+(dyd²z—dzd²y)²]* ·
INTRODUCTION.
25
19. In order to have a complete knowledge of the nature of
the curve at any point м, its osculating plane should be also
determined, that is, the plane of the two consecutive sides мm
and mm'. If this plane passes through the point м, its equa-
tion may be represented by
A (x' — x) + B (y' — y) + c (z' — 2) = 0;
x', y', z' being any coordinates whatever, and because it must
pass through the points m and m', the first and second differ-
entials of this equation, namely,
Adx' +вdy' + cdz' = 0,
Ad²x² + Bd²y' + cd²z' = 0,
must be satisfied like the equation itself, by making xx,
y' = y, z=z; so that we shall have
=y, z'
Adx + вdy + cdz = 0,
Ad²x + Bd²y + cd²%= 0.
The values of A, B, C, which satisfy these two conditions,
are, as may be easily verified (k),
c =D (dxd²y — dyd²x),
B =D (dzd²x — dxd²z),
AD (dyd²z — dzd²y);
D being an indeterminate factor. By substituting them in the
equation of the osculating plane, and then suppressing the
factor which is common to all its terms, it becomes
(z' — z) (dxd³y — dyd²x) + (y' — y) (dzd²x — dxd²z)
+ (2 – x) (dydz – dzd^y) = 0.
Naming λ,u,v, the angles that the normal to the osculating
plane makes with the parallels to the axes of x, y, z, drawn
through the point м, we shall have by means of equations (2)
of No. 17,
E
26
INTRODUCTION.
1
cos λ = {} (dyd²z — dzd³y),
h
1
cos μ == (dzd²x - dxd²z),
-
h
1
COS v =
—
(dxd³y — dyd²x),
(3)
h² denoting the sum of the squares of the three numerators.
The infinitely small angle contained between two conse-
cutive normals, which is the angle between two consecutive
osculating planes, can be determined in the same manner as
the angle between two tangents has been already determined.
If we denote it by ε, by a method similar to that employed in
the preceding number, we shall have
&² = (d. cos λ)² + (d. cos µ)² + (d. cos v)².
20. As the centre of curvature exists at the same time on
the osculating plane, and on the intersection of two conse-
cutive normal planes, we are enabled to determine its coordi-
nates by means of the equations of these three planes, which
now we can consider as known.
The equation of the normal plane at м being (Ng. 17)
(x' — x) dx + (y' — y) dy + (z' — z) dz=0,
that of the consecutive plane may be deduced from it, by sub-
stituting x + dx, y+dy, z+dz, in place of x, y, z; conse-
quently, the differential of the equation of the first of these
two planes taken with respect to x, y, z, namely,
(x' — x) d²x + (y' — y) d²y + (z' —) d²z = ds²,
will be that of their intersection.
From these two equations, we infer(1)
(x' — x) (dxd³y — dyd²x) = (2'- z) (dyd²z — dzd³y) — ds²dy,
(y' — y) (dyd²x — dxd²y) = (z' — z)(dxd²z - dzd²x) — ds²dx;
and by means of the equation of the osculating plane, there
results
INTRODUCTION.
27
2
z' — z = — [dy (dyd³z — dzd³y) — dx (dzď³x — dxd²z)],
ds4
p denoting, in order to abridge, the same expression as in
No. 18. We can obtain in the same manner
y' — y =
L
ds¹
4
[dx (dxd³y — dyd²x) — dz (dyd²z — dzď²y)],
=
2
x' — x' — få [dz (dzd³x — dxd³z) — dy (dxd³y — dyd²x)];
ds4
hence the three coordinates x', y', z', of o the centre of the
curve may be known, and, consequently, the direction of the
curvature of which the radius of the osculating circle only de-
termines the magnitude. If the squares of these values of
x' — x, y' — y, z'-z, be added together, we shall obtain,
after all reductions(m),
(x' — x)² + (y' — y)² + (z' — 2)² = p²;
hence it follows that the quantity p is the distance of the point
o from the point м, or the radius of curvature Mo, as we know
from other considerations.
As to
21. The formulæ given in the five preceding numbers,
contain every thing which respects the direction and curvature
of any line, either of single or double curvature. With
respect to any surface, we have also to consider its cur-
vature, and the direction of the tangential plane.
the curvature, the reader will find it fully discussed in the
twenty-first volume of the Journal of the Polytechnic School,
so that here we shall only concern ourselves with what res-
pects the tangential plane and normal. At the point м, of
which the coordinates are x, y, z, the equation of the tangen-
tial plane may be represented by
A (x' — x) + B (y' — y) + c (z' — z) = 0 ;
where x', y', z', are indeterminate coordinates. This plane
must pass through м' another point of the surface infinitely
1
28
INTRODUCTION.
near to M, consequently this equation must be satisfied by
making x'= x + dx, y' = y + dy, z' = z + dz, or, its differ-
ential taken relatively to x', y', z', must be true, when x, y, z,
are substituted in place of these variables. Therefore we shall
have
Adx+вdy+cdz = 0.
The equation of the surface being
dz = pdx + qdy ;
in which p and q are known functions of x, y, z, the preceding
becomes
(A + pc) dx + (B + qc) dy = 0 ;
and as it must obtain for every direction which the line MM
can assume, and consequently for every relation which can
subsist between dx and dy, the coefficients of each of these
differentials must be separately equal to cypher; hence results
A+pc = 0, B+qc = 0.
If the values of A and B be deduced from these equations,
and if they be then substituted in the equation of the tangen-
tial plane, we shall have
z' — z — p (x' —x) — q (y' — y) = 0.
If a, b, c, denote the angles, which the normal at the point
м, makes with the productions of the coordinates x, y, z, we
shall have by means of the equations (2) of No. 17,
cos α =
p
√1 + p² + q
q
cos b =
(4)
√1 + p² + q²
1
COS C =
√ I + p² + q²
The radical will be positive or negative in these three for-
mulæ, according as the part of the normal in question makes
i
.
INTRODUCTION.
29
an acute or obtuse angle c with the line drawn through the
the point м, in the positive direction of z. If w represents the
element of the surface, of which the projection on the plane
of x and y is dxdy, we shall have
dxdy = ±w cos c,
according as c is acute or obtuse; for this element, which is
infinitely small in every direction, lies in the tangential plane,
of which the angle c, or its supplement, is the inclination to
the plane passing through x and y; and the theorem of No.
10 is true also in the case of the projection of an infinitely
small plane surface. Consequently from what has been just
established, we have
w = dxdy √1+p²+q²,
the sign of the radical being always considered as positive. If
L be a given function of x, y, z, and if
L=0,
represents the equation of the surface, which we have con-
sidered, we shall obtain, by differentiating it successively
with respect to x and y
dL
dL
dL
di
dx
+P dz
0,
dy
+9az=0.
If the values of p and q be determined by means of these
equations, and then substituted in the equation of the tan-
gential plane, it will assume the form
dL
di
— +(≈' — z)
(x-x) +(y' − y) dy
dx
At the same time, formulæ (4) become
dL
= 0.
dz
dL
cos α = V
cos b = v
dx
>
dL
dy'
dL
COS CV
(5)
dz
in which equations, for the sake of abridging, we put
30
INTRODUCTION.
dx
2
dy
2
2
V.
dz
22. We shall here make a remark that will be useful in
verifying and in deducing from one another, formula which
refer to different axes.
Suppose that in any question all things were alike with
respect to the three axes of the coordinates x, y, z. If an equa-
tion x = 0, obtains with respect to the axis of x, a similar
one y = 0 will have place for the axis of y, and likewise a
third z = 0, for the axis of z, and these two last equations
y=0, z=0, may be inferred from x=0, by simple changes
of the letters x, y, z. The following is the manner in which
these permutations are effected.
In x let all quantities relative to the axis of x, be substi-
tuted in place of the corresponding quantities which respect
the axis of y, then substitute these quantities instead of those
which respect the axis of z; and finally, these last in place of
the first, which respect the axis of x. By this mutual per-
mutation z is deduced from x; by a second permutation of
the same nature effected on z, we shall obtain y, and by a
third permutation effected on y, we shall light on the first
equation x. For example, if these operations were instituted
on equations (3) of No. 19, of which the first respects the
axis of x, the second the axis of y, and the third the axis
of z, let the coordinates x, y, z, and the angles λ, u, v, which
correspond to them respectively, be written down in the same
line, but distributed into two classes; then in a second line
let these six quantities be also arranged into two classes, but
in a different order, so that we may have
x, y, z,
Z, X, Y,
λ, μ, ν ;
ν, λ, μ.
This being done, in the first equation (3), let each of the
quantities of the superior line be replaced by a corresponding
quantity in the inferior line; it is evident that by this permu-
INTRODUCTION.
31
tation no change is produced in h, and the third equation (3)
will be obtained. If now in this last, the quantities of the
inferior line be substituted in place of those which correspond
to them in the superior line, the second equation (3) will be
given; and by operating in the same manner on this equation,
we shall arrive at the first equation (3), from which we set out.
Each of these operations implies a change in the axes of
the coordinates, in which we first make the axes of x and of y
revolve in their plane, in such a manner that the axis of the
positive as may fall on the axis of the positive ys, then this
last on the axis of the negative xs, and afterwards this axis
of the positive ys thus displaced, and the axis of the positive
zs are made to revolve in such a manner, that the first may
fall on the axis of the positive zs, and this last on the primi-
tive axis of the positive xs; so that, finally, each axis of the
positive coordinates takes the place of another axis of the
positive coordinates. It is on this account, that the equations
relative to the three axes of the coordinates may be inferred
the one from the other, by simple permutations of letters, and
without any change of sign, which would not be the case
unless the three coordinates, and the quantities which relate
to them, were changed simultaneously, in the manner which
has been pointed out.
23. As the following general observation is of frequent oc-
currence, it will be useful to advert to it here, at the conclusion
of this introduction. The equations which will come under
our observation will frequently contain abstract numbers,
such as the number 7, logarithms, trigonometrical lines, &c.;
they will also contain other quantities of different natures,
which will likewise be represented by numbers expressing
their ratio to different units arbitrarily selected; however each
unit must be the same for all quantities of the same species.
But, if the magnitude of one or more of these units be changed,
the numbers which express corresponding quantities, will be
changed in the inverse ratio of this magnitude; notwithstand-
!
32
INTRODUCTION.
犨
​ing, however, this change, which is entirely arbitrary, the
equations in which they occur must still subsist. For this
purpose, it is necessary that their form should satisfy certain
conditions, which may be easily verified in each particular
case, and which, in the most extended acceptation, may be
termed the conditions of the homogeneity of quantities. Every
equation that does not satisfy these, will, on this sole account,
be inaccurate, and ought to be rejected.
Thus if F denotes a given function, and if
F (f, ƒ' . . . . l, l',
m, m,
t, t' . . . . ) = 0; (a)
in which equation f,f', .... denote forces, l,l',.... lines, m, m',
masses, t,t',.... times, then if n,n',n',n',
times, then if n,n',n',n"", .... represent
abstract numbers, and if, at the same time, we diminish the
unit of force in the ratio of one to n, the linear unit in the
ratio of one to n', the unit of mass in the ratio of one to n",
the unit of time in the ratio of one to n"", the numbers ff',....
l, l',.... m, m', .. t, t'
t, t',.... will become nf, nf", n'l,
n'l', .... n'm, n'm .... n'"'t, n'"'t ...., and the equation (a)
must still obtain, that is to say, we must still have
F (nƒ, n'f', ………. n'l, n'l', ………. n'm, n''m', ....n'''t, n'"'t ……..) = 0,
whatever may the magnitudes of n, n', n", n'". If the equation
(a) contains the surfaces s, s',.... and the volumes v, v', ..
...and
. .
their dimensions must be referred to the same unit as the lines
l, l',...., and these quantities will consequently become nºs,
n's', .. n'³v, n'³v′,………. by the change of this unit.
The equation of No. 18, which gives the value of p, evi-
dently satisfies this condition; for it only contains finite lines,
or infinitely small quantities, p,ds, dx, dy, dz, d²x, d³y,ď²z; and
when the linear unit is changed, and each of the lines is mul-
tiplied, in the manner pointed out, by the same number n',
this number disappears, and so the equation will not be changed.
The equation in the same number which expresses the value
of 8², also satisfies the condition of homogeneity, since 82 is
INTRODUCTION.
33
an abstract number which like this value does not change
with the magnitude of the linear unit. It will be impossible
for equation (a) to contain only one quantity of the same
species; when it contains but two, for example two forces
fand f', if it be resolved with respect to one of them, we
we shall have
ƒ' = F (ƒ, l, l, . ... m, m', . . . . t, t', . . .
...
.),
it is necessary in order to secure the homogeneity of these quan-
tities, that ƒ should be a common factor to all the terms of the
new function F, or in other words, we must have
f' = Nf';
N being a factor which does not contain any quantity of the
same nature as ƒ and ƒ, nor vary with the unit of force.
Sometimes the principle of the homogeneity of quantities will
appear not to have place, when there is taken for the unit of
force, one of the forces which are considered in the question,
or for the linear unit, the distance between two of the mate-
rial points to which the question refers; or for the unit of
mass that of one of the bodies of the problem, &c. But, then,
if these units be arbitrarily changed, and if the force, the line,
the mass, and time, which have been in the first place taken
for units, be now expressed by p, λ, µ, 0, the other quantities
of these different species which occur in the equation (a) will
become
ƒ ƒ¹
φ' φ
therefore have
1 ľ
m m'
μμ
t ť
we must
0' 0
(ƒ ƒ'
ιν
m m'
t t'
F
Τ' Τ'
…..)=0;
μ μ
which equation may be written
F1 (p, ƒ,ƒ', … … … … λ, l, l', . . . . µ, m, m',.... 0, t, t', ....) = 0,
and this should now satisfy the condition of homogeneity:
F₁ indicates here a function which, in each case, may be de-
duced from the given function F.
F
过
​4
1
,
?
BOOK THE FIRST.
STATICS.
FIRST PART.
CHAPTER I.
OF THE COMPOSITION AND EQUILIBRIUM OF FORCES
APPLIED TO THE SAME POINT.
24. WHEN a material point is subjected to the simultaneous
action of several forces which do not constitute an equilibrium,
it moves in some determinate direction, and this motion with
which the point is actuated, may be ascribed to an unique
force acting in this direction. This force is termed the re-
sultant of the forces which produce motion in the moveable,
and these last are called the components of the first. If the re-
sultant be applied to the point in a direction directly opposite
to that in which it acts, it will constitute an equilibrium with
the components, because it tends to impress on the moveable a
motion equal and opposite to that which it would receive from
the simultaneous action of the components, and there is no
reason consequently why it should move in one direction(a)
rather than in the other. If all the components act in the
same direction and along the same right line, it follows from
what has been stated in No. 5, about the measure of forces,
36
OF THE COMPOSITION AND EQUILIBRIUM OF FORCES
!
that the resultant is equal to their sum. If the point is so-
licited by two forces directly opposed to each other, and if
the greater be resolved into two others, of which one is equal
to the less, this last will be destroyed, and what remains,
namely, the excess of the greater above the less, will be the
resultant. From these two propositions it follows, that if
there be any number of components, of which some act in the
direction of the same right line, and others in the opposite
direction, the resultant will be equal to the sum of those
which act in one direction, minus the sum of those which act
in the opposite direction, and it will act in the direction of
the greater sum. When the two sums are equal, the resultant
is cypher, and the given forces constitute an equilibrium.
24. There is also another case in which the magnitude and
direction of the resultant may be very easily determined.
Let MA, MB, MC, (fig. 5), be the directions of three equal
forces applied to the point м; if these forces exist in the same
plane, and if the three angles AMB, BMC, CMA, are respec-
tively equal to each other, or each to 120°; the point м will
remain in equilibrio; for there is no reason why it should de-
viate from the plane in which the three forces act, nor why it
should move in one rather than in another of these three
angles. Consequently, each of the three forces will be equal
and opposite to the resultant of the other two. Now, if on
the directions MA and MB of two of the forces, the equal lines
MG and мH be taken to represent their intensities, and if the
rhombus GMHK be completed, the diagonal мK will fall on
MD the production of MC; the angle мGK will be 60º, as will
also be each of the other angles of the same triangle, which
will be equilateral; hence we shall have MK = MG; conse-
quently мк, the diagonal of the rhombus constructed on the
two forces MG and MH, represents the resultant of these two
*forces in magnitude and direction.
This proposition is contained in another, which we now
proceed to demonstrate in the case of two equal forces, the
APPLIED TO THE SAME POINT.
37
directions of which make any angle whatever with each other, and
which we shall afterwards extend to the case of unequal forces.
26. The resultant of two equal forces always divides the
angle comprised between their directions into two equal parts;
for there is no reason why it should be nearer to one of these
two forces than to the other, nor why its direction should de-
viate from their plane on one side, rather than on the other ;
therefore, as its direction is known, it is only necessary for us
to determine its magnitude.
For this purpose, let MA and MB (fig. 6) be the directions
of the components, and let P denote their common value.
Likewise let the angle AMB be represented by 2x, and let MD
be the direction of the resultant, so that we may have AMD
BMD=x. Its intensity depends solely on the quantities P
and x; therefore, if it be denoted by R, we shall have
R =ƒ(P, x).
In this equation, R and P are the only quantities of which
the numerical expression varies with the unit of force; by the
principle of the homogeneity of quantities (No. 23), the func-
tion ƒ(P,x) must therefore be of the form Ppx. Hence we
have
R = Ppx;
and the question is reduced to the determination of the form
of the function px.
For this purpose, let the four lines MA', MA", MB′, MB",
be drawn arbitrarily through the point м, so that the four
angles A'MA, AʼMA, B'MB, B"MB, may be equal to each other,
and respectively represented by z. Let the force P acting in
the direction MA, be decomposed into two equal forces, acting
in the directions MA', MA", that is to say, let the force P be re-
garded as the resultant of two equal forces of which the value
is unknown, and which act in the directions MA', MA"; if this
value be denoted by a we shall have
PQ02
38
OF THE COMPOSITION AND EQUILIBRIUM OF FORCES
for the same relation must exist between the quantities P, Q,2,
as between the quantities R, P, x. Likewise the force P acting
in the direction MB, may be decomposed into two forces
respectively equal to Q, and acting in the directions MB' and
MB"; in this manner the two forces P will be replaced by the
four forces ; consequently, the resultant of these last ought
to coincide in magnitude and direction with the force R,
which is the resultant of the two forces P. Now, if q' de-
notes the resultant of the two forces Q, which act in the
respective directions MA' and MB', this force will act in the
direction MD, and since A'MDB'MD-8, we shall have
=
Q' = Qp (x − x).
In like manner, the resultant of the two other forces a
will act in the direction MD, since this line also divides the
angle A"MB" into two equal parts; and because A″MD = B″MD
=x+≈, we shall have
Q″ = q p (x + 2) ;
Q" denoting this second resultant. As the two forces q' and
Q" act in the direction of the same right line MD, their re-
sultant, which is likewise that of the four forces a, must be
equal to their sum; consequently we must have
But we have already
R = Q'+Q".
R = Ppx = Qpspx ;
and by substituting this value of R and those of Q' and o″ in
the preceding equation, there results, by suppressing the factor
Q, which is common to all the terms,
pxpz = p(x + x) + 4 (x − ≈).
(1)
This is the equation which must be resolved in order to obtain
the expression of px.
27. It is evident that it may be satisfied by assuming
px=2 cos ax;
APPLIED TO THE SAME POINT.
39
a being a constant arbitrary, so that we may have at the
same time,
ps = 2 cos az,
4(x+x) = 2 cos a (x + ≈),
4(x-2)=2 cos a (x ≈) ;
and, in fact, if these values be substituted in equation (1),
there results the known equation,
2 cos ax cos az = cos a (x + z) + cos a (x — x).
Now it is to be remarked, that this expression of the func-
tion or is the only one which satisfies equation (1), and more-
over, that in the present question, the constant quantity a is
unity; so that we have
px=2 cos x.
(2)
This is evident when x = 0; for then the directions of the
two forces coincide, and the resultant R is equal to 2P, which
implies that px = 2.
If we admit that there is another value such as a of x, for
which we have also pa = 2 cosa; then equation (2) will like-
wise subsist for all values 2a, 3a, 4a, .... Ja, ja, ja, .... of x,
and generally for
X =
=
ma
2n
(3)
m and n being any whole numbers whatever. In fact, if
equation (2) is true for the three angles x, x, x - x, so that we
have
px=2 cos x, px. 2 cos z, p(x
=
2)=2 cos(x2),
it will be also true for a fourth angle x+; for, in virtue of
equation (1), we shall then have
4 (x + z) = 4 cos x cos z − 2 cos (x − z) ;
which equation may be reduced to
4 (x + z) = 2 cos (x + z).
#
1
40
OF THE COMPOSITION AND EQUILIBRIUM OF FORCES
Thus, as equation (2) obtains for x = 0 and a = a, it fol-
lows that it subsists for a 2a, and as it obtains for x = a
and x2a, it will also subsist for a 3a; by continuing in
this manner, it appears that it will obtain generally for xma.
Now if we make ma = ß, we shall have
+B=2 cos ß;
from which it follows that equation (2) will likewise obtain for
x=ẞ, for assuming x = ≈ = 1ß, equation (1) will become
Ꮳ
(418)² = 2 cos ẞ+2;
consequently we shall have(b)
-
+1ẞ= 2 cos ẞ.
2
If in the next place, we make x = z = 1ß, we shall have
by equation (1) and this last,
(41ß)² = 2 cos 1ß +2,
16 = 2 cos 1ß;
and by continuing in this manner, equation (2) will be de-
monstrated for x=
prised in formula (3).
β
>
2n
that is to say, for all values of x com-
Now, as the numbers m and n may be rendered as great
as we please, and may even become infinite, these values of
x may be made to increase by infinitely small quantities.
Therefore formula (3) embraces all possible values of the
angle x, and equation (2) is completely demonstrated, pro-
vided it is true for a particular value x = a, different from
zero. Now by the theorem of No. 25, the resultant R is equal
to P, in the case of x = 60°; therefore in this case we have
px12 cos 60°;
hence equation (2) obtains for x = 60°, and consequently for
all values of x.
28. By means of this equation we shall have
R2P cos x.
APPLIED TO THE SAME POINT.
41
Hence if the resultant R and the two components P are
represented, as in No. 25, by lines taken in their respective
directions, reckoning from their point of application, the force
R will be equal to twice the projection of P on its direction, or
equal to the diagonal of the rhombus constructed on the two
forces P.
Now let two unequal forces P and Q be applied to the
point м (fig. 7) in the directions MA and MB, also let their in-
tensities be represented by the lines MG and мí, taken in
their respective directions, and let the parallellogram MGHK
be completed; there are two cases to be considered, first,
when the angle AMB is a right angle, and, secondly, when it
is acute or obtuse. In the first case, let the two diagonals
MK and GH be drawn intersecting in the point L; and through
the points G and H, let GN and нo be drawn parallel to ML,
meeting in N and o, the line drawn through the point M
parallel to GH. MK and GH bisect each other at the point L,
and since in a rectangle the two diagonals are equal, we have
GL = LH = LM.
Hence each of the parallellograms GLMN, HLMO is a rhombus,
consequently it follows from the preceding proposition that
the force MG may be considered as the resultant of the two
forces MN and ML, and the force мH, as the resultant of Mo
and ML. Therefore, if instead of the two given forces there
be substituted their components, we shall have for мH and
MG the two forces MN and мo (which since they are equal
and opposite, they mutually destroy each other) and the two
forces ML, which added together give a resultant represented
in magnitude and direction by the diagonal MK. In the
second case, let GE and HF be drawn through the points G
and н, perpendicular to the diagonal мк (fig. 8,) and the lines
GN and нo parallel to this diagonal; likewise let NMO be drawn
through the point м perpendicular to the same line. The two
parallellograms GEMN and HFMO will be rectangles, and their
G
42
OF THE COMPOSITION AND EQUILIBRIUM OF FORCES
kes.
sides MN and MO will be equal, as being the altitudes of the
two equal triangles GMK and HMK. By the first case, we can
replace the forces мG and мH by their rectangular components
Me and MN, MF and мo; therefore, in place of the two given
forces we shall have the two forces MN and MO, which destroy
each other, as being equal and opposite, and the two forces
ME and MF acting in the same direction, which being added
together will give (because ME = FK) a resultant expressed
in magnitude and direction by the diagonal мK. Hence we
may conclude generally, that the resultant of any two forces
whatever, applied to the same point and represented by lines
taken on their directions, reckoning from this point, is repre-
sented in magnitude and direction by the diagonal of the
parallellogram constructed on the two given forces.
29. The following consequences may be immediately de-
duced from this theorem.
T
In the first place, it appears that all questions which can
be proposed relatively to the composition of two forces into
one or relatively to the decomposition of one force into two
others, are reduced to the resolution of a triangle. In fact,
the magnitudes of the resultant and of the two components
are represented by the three sides MK, MG, GK, of the triangle
MGK; and the three angles of this triangle are those which
the resultant makes with each of the components and the
supplement of the angle comprised between the components.
It follows therefore, that any three of these six quantities,
namely, the three forces and the three angles comprised
between their directions, being given, the three remaining
quantities may be found by the resolution of the triangle
MGK; this, however, supposes that in the number of data,
there is one force at least. For example, let P and a be the
values of the two components, and m the angle contained
between their directions; it is required to determine their re-
sultant R and a the angle which it makes with the force P.
In the first place, we have the equation
APPLIED TO THE SAME POINT.
43
2
2
R²= p²+Q²+ 2PQ cos m,
by means of which the value of R can be determined; and that
of a can be deduced from the proportion
sin x: sin m:: Q: R.
If the three forces P, Q, s, applied to the same point м,
(fig. 9,) in the directions MA, MB, MC, are in equilibrio, each
of these forces must be equal and directly opposite to the re-
sultant of the two others; and as this resultant exists in the
plane of these two forces, it follows that these three forces
must also exist in the same plane. Let MD be the production
of MC; the resultant of P and Q will act in the direction of MD,
and if it be represented by R, we shall have Rs. More-
over, if the force R be compared with each of its components,
we have, agreeably to what is stated above,
R:Q::.
I sin AMB: sin amp,
R:P:: sin AMB; sin BMD;
and since
sin AMD = sin AMC, sin BMD sin BMC,
there results
S:Q:P::sin AMB: sin AMC: sin BMC ;
this shows that when three forces are in equilibrio about the
same point, the magnitude of each of them may be represented
by the sine of the angle comprised between the directions of
the other two.
If from o, a point assumed on the direction of the resultant
R, or on its production, the perpendiculars OE and or be let
fall on the directions of the components P and Q; we shall
have
OEMO sin AMD, OF MO sin BMD.
If, therefore, the two last terms of the proportion
P: Q: sin BMD: sin AMD,
20
44
OF THE COMPOSITION AND EQUILIBRIUM OF FORCES
:
be multiplied by Mo, there will result
P: Q:: OF: OE;
Som which it appears that the components are in the inverse
ratio of perpendiculars let fall on their directions, from any
point in the direction of the resultant. Conversely, if the
components P and Q are in the inverse ratio of OE and OF, per-
pendiculars let fall on their directions, from any point o taken
in their plane, this point will exist on the direction of the re-
sultant; for if the two last terms of the last proportion be
divided by mo, the preceding will be obtained, which deter-
mines this direction.
30. The resultant of two forces being known, it is easy to
deduce that of any number of forces applied to the same point
and situated, or not, in the same plane. First, the resultant
of two of these forces is taken, then this resultant is com-
pounded with a third force, this will give a second resultant,
which is compounded in the same manner with a fourth force,
and so on, until all the given forces are exhausted. In this
construction it is easy to perceive, that if the magnitudes of
all the forces are represented by the sides of a portion of a
polygon parallel to their respective directions, and traced in
the direction of their actions, the resultant will be represented
in magnitude and direction by the line which connects its two
extreme points, and consequently closes the polygon. It is
indifferent in what order the sides parallel to the forces succeed
each other. When the polygon is closed of itself, the resultant
vanishes, and the given forces constitute an equilibrium.
Hence it follows, that when there are but three forces
which do not exist in the same plane, their resultant is repre-
sented in magnitude and direction by the diagonal of a paral-
lelopiped, of which the three forces constitute the adjacent
sides.
31. This reduction of any number of forces to one, may
be effected in a simpler manner, by considering, first, the par-
APPLIED TO THE SAME POINT.
45
ticular case of three rectangular forces, to which the general
case may then be reduced.
Let x, y, z be the three components, R their resultant,
a, b, c the angles which it makes with x, y, z. From what
has been already observed, R is evidently the diagonal of a
parallelopiped of which x, y, z are the three adjacent sides,
now as this parallelopiped is rectangular, it follows that
R²= x²+ y²+ z².
(a)
It likewise follows that if the extremity of the diagonal R be
joined to those of the three sides X, Y, Z, three right angled
triangles will be formed, of which R will be the common
hypothenuse; hence we have
X= R coś α, Y= R cos b, Z = R cos c;
(b)
which equations agree with the preceding, for the three
angles a, b, c are connected by the equation (No. 6.)
cos² a + cos² b + cos² c = 1.
When the components X, Y, Z are given, equation (a) deter-
mines the value of the resultant, and equations (b) determine
the direction, by means of the three angles a, b, c; if, on the
other hand, the force R is given, and it is required to decom-
pose it, into three rectangular forces x, y, z, which make with
it the given angles a, b, c, the values of the required forces
will be immediately determined by means of equation (b).
If one of the components, the force z for example, vanishes,
R is then only the resultant of two forces x and y; it exists in
their plane, and its direction depends solely on the two
angles a and b. These angles and the value of R are then
determined by the equations
2
R²= x²+Y², X = R cos α, Y = R cos b.
32. Let us now suppose that м (fig. 1) is the point of
application of any number of given forces. Let these forces
be represented by P, P', p", &c., and for greater clearness, let
the line MD be the direction of the force P.
It is unnecessary
46 OF THE COMPOSITION AND EQUILIBRIUm of forces
+
to indicate the directions of the other forces in the figure.
Let a, ẞ, y be the angles which the direction MD makes with
the three rectangular axes MA, MB, MC, drawn arbitrarily
through the point м. Likewise let a', B', y' be the angles
which the force p' makes with the same axes, a", ẞ", y",
those which correspond to the force p", &c. All these angles
are given, and are supposed to include every angle from zero
to 180° (No. 7), in order that the forces P, P', P", &c., may
have all possible positions about the point м. If each of these
forces be resolved into three others in the direction of the
axes MA, MB, MC; the components of the force P will be
P cos a, P cos ß, P cos y; those of the force p' will be p′ cos a',
P′ cos B', P' cos y', &c.; and these components will act in the
direction of the axes or of their productions, according as they
are positive or negative. For example, as the direction MD
falls like the axis Mc above AMB, the plane of the two other
axes, the component P cos y of the force P tends to elevate
the point м, that is to say, it acts in the direction мc; and
in this case P cos y is a positive quantity, because y is less
than 90º. On the contrary, if this direction MD falls below
the plane AMB, we would have y> 90°; and the component
P cos y will be negative, and, at the same time, it will tend
to depress the point м, that is to say, it will act along the
production of Mc. Therefore, taking into account the signs
of the components, it appears from what has been stated in
No. 24, that all forces acting in the direction of the same
axis and its production, are reduced to one sole force, equal to
their difference.
In this manner the given forces P, P', P", &c., may be
replaced by three rectangular forces; and if these last be de-
noted by x, y, z we shall have
(c)
X = P cos a + P′ cos a' + P' cos a' +, &c.
Y = P COSẞ+P' cos B'+P" cos ẞ"+, &c.
z = P cos y +P' cos y'+p" cos y"+, &c.
The values of x, y, z may be either positive or negative,
APPLIED TO THE SAME POINT.
47
and their signs make known the direction of their action. If
the force x is positive, it follows that it acts in the direction
of the axis MA, or in the direction of the components P cos a,
P' cos a', &c., which are positive; if it is negative, we must
infer that it acts along the production of MA or in the direction
of the negative components, and the same is true for the
forces y and z.
This being agreed on, let R be the resultant of the given
forces P, P', P", &c., or of the three forces X, Y, Z; let also
a, b, c be the angles which its unknown direction makes with
the axes MA, MB, MC. The values of R, a, b, c will be given
by equations (a) and (b), in which the formulæ (c) are sub-
stituted in place of x, y, z. The angles a, b, c may be acute
Y,
or obtuse; but because the force R must be always a positive
quantity, the signs of their cosines must be always the same
as those of the quantities x, y, z, in virtue of equations (b).
In this manner, the force R will be completely determined in
magnitude and direction.
33. The magnitude of the resultant R does not depend on
the arbitrary direction of the axes MA, MB, MC, it depends
solely on the magnitude of the given forces and on the angles
comprised between their directions, and in fact, we may find
an expression for it, which contains these quantities only.
For this purpose, let PMP, PMP", P'MP"", &c., denote the
angles contained between the directions of the forces P and P',
P and P", P' and p"", &c. By equation (2) of No. 9, we shall have
COS PMP' — cos a cos a' + cos ß cos ß' + cos y cos y',
COS PMP"= cos a cos a"+cos ẞ cos ẞ"+cos y cos y“,
COS P'MP”— cos a' cos a"+cos ẞ' cos ẞ" + cos y'cos y",
&c.
We shall likewise have
2
cos² a + cos² ß+cos² y = 1,
cos² a' + cos² ß'+cos² y' = 1,
cos² a" + cos² ẞ"+cos² y" 1,
&c.
$
48 OF THE COmposition and EQUILIBRIUM of forces
and, this being the case, if the squares of formulæ (c) be
added, we shall obtain, by taking into account equation (a),
2
12
R²= p²+P²²+ p/2+, &c.
+ 2PP' COS PMP'+2PP" COS PMP"
+2P'p" COS P'MP"+, &c.
for the value of the square of R which was required to be de-
termined.
34. We may also deduce from a consideration of equations
(b) and (c), a property of the resultant which will be useful in
one of the following numbers.
Let there be drawn through the point м, in any direction
whatever, a right line, of which let the point o be the other
extremity, and let g, h, k denote the angles AMO, BмO, CMO,
which this line makes with the three axes MA, MB, MC, if the
angles comprised between this same right line мo and the
directions of the forces R, P, P, &c., be denoted by RMO, PMO,
P'MO, &c., we shall have, as has been just stated, by equation
(2) of No. 9,
COS RMO = cos g cos a + cos h cos b + cos k cos c,
cos PMO = cos g cos a + cos h cos ß + cos k cos y,
COS P'MO = cos g cos a' + cos h cos ẞ'+ cos k cos y', &c.
From the first of these formula and equations (b), we
obtain
R COS RMO = X cos g + Y cos h + z cos k;
and in virtue of the subsequent formulæ, if, after having mul-
tiplied the first of equations (c) by cos g, the second by
cos h, the third by cos k, we add them together, there will
result (c)
R.COS RMO = P. COS PMO + P' COS P'MO + &c. ;
which shows that the resultant R, resolved in any direction
Mo, is equal to the sum of the components P, P', P", &c., re-
solved in the same direction.
APPLIED TO THE SAME POINT.
49
This being established, let the line мo be projected on the
directions of the forces R, P, P', P", &c., and let these pro-
jections be denoted respectively by r, p, p', p", &c., so that
r MO COS RMO,
p➡ MO COS PMO, p' MO COS P'MO, &c.,
each of the quantities r, p, p', p", &c., being considered as
positive or negative, according as the projection which it repre-
sents, falls on the direction of the force, or on its production.
Then, if the preceding equation be multiplied by мo, we shall
have
Rr = Pp + Þp + P"p" + &c. ;
(d)
which expresses the property of the resultant that was pro-
posed to be demonstrated.
35. In order that the forces P, P', p", &c., may be in equi-
librio, it is sufficient that the resultant R should vanish, and
this condition is necessary, if their point of application м is
entirely free; but the equation R=0, or
x²+ y²+z²= 0,
cannot have place, unless we have separately
x = 0,
y = 0,
Y = 0,
z=0,
that is to say, in virtue of equation (c),
P cos a + P'. cos a' + P" cos a' + &c. = 0,
P cos ẞ + P'cos ß' + P" cos ẞ”+ &c. = 0,
P cos y + P'cos y′+ p″ cos y″ + &c. = 0.
(e)
Therefore these are the equations of equilibrium of a ma-
terial point, which is supposed to be entirely free. In this
state, each of the forces that solicit it, must be equal and
directly contrary to the resultant of all the others, which may
be easily verified in the following manner.
Let R' be the resultant of the forces P', p", &c. Denoting
H
50 forces
ON THE COMPOSITION AND EQUILIBRIUM OF FORCES.
the angles which it makes with the axes MA, MB, мC, by
a', b', c', &c., and in order to abridge, making
X'P'cos a' + P" cos a" + &c.,
Y' — P' cos B'+p" cos ß“ + &c.,
z'P'cos y′+P" cos y" + &c. ;
we shall have, by No. 32,
X'R' cos a', 'R' cos b', z'R' cos c',
and consequently, in virtue of the equations of equilibrium,
P COS α =
a
P cos B =
P COS Y =
R'. cos a',
R'. cos b',
R'. cos c'.
If the squares of each of the members of these equations
be taken, and if they be then added together, we shall have
because by (No. 6)
p² - R2,
cos² a + cos²ß + cos² y = 1,
cos² a'+cos² b'+ cos² c² 1;
therefore we shall have p±R'; but as these forces must be
respectively positive, we must take PR'. The preceding
equations will then become
COS α = -
cos b',
cos y =
cos a', coś ß=
cos c';
consequently, the angles a, ß, y, are the supplements of a', b', c',
and belong to a force, of which the direction is the production
of the force R' (No. 7). It follows, therefore, that the force P
is equal and directly opposed to R' the resultant of all the
other forces p', p", &c. ; which it was proposed to establish.
36. If the point м, to which the forces P, P', P", &c., are
applied, be subjected to exist on a given surface, it is no
longer necessary, in order to the equilibrium, that their re-
APPLIED TO THE SAME POINT.
51
sultant should vanish; it is sufficient if it be normal to the sur-
face, because then the point м cannot slide in any direction on
this surface; and moreover, this condition will be necessary;
for if it was not satisfied, the resultant might be decomposed
into two forces, the one normal to the surface, which will be
destroyed, the other a tangent to the surface which there is
nothing to prevent from causing the moveable to slide. It is
only therefore necessary to determine in each case, the direc-
tion of the resultant of the forces P, P', P", &c., and then ex-
amine if it is perpendicular to the given surface, in order to
know whether the equilibrium exists; but it is more conve-
nient in practice to express, as has been done in the case of a
free point, the conditions of the equilibrium by means of the
equations which exist between the data of the question. Now
the normal component of each of the forces which acts on the
point м, is destroyed by the resistance of the surface; conse-
quently, this resistance is equivalent to a force equal and con-
trary to all the forces destroyed. Hence it appears, that we
may abstract from the consideration of the given surface, and
regard the material point as entirely free, provided that to the
given forces P, P', P", &c., a new force of an unknown mag-
nitude and perpendicular to this surface be added.
Therefore, if N represents this force, and if X, u, v, denote
the angles which its direction makes with the axes MA, MB, MC;
each of the equations of equilibrium already given, will be in-
creased by a new term, so that instead of equations (e), we
shall have
N.cosλ+P.cos a + P'. cos a' + P". cos a" + &c.—0,
N.cos μ+P.cos ẞ+P'.cos ẞ'+p". cos ẞ" + &c. =0,} (f)
N.cos v+P.cos y + P'.cos y'+P". cos y" + &c. = 0,]
Let x, y, z, denote the three coordinates of м referred to
the axes parallel to MA, MB, MC, and let Lo, denote the
equation of the given surface; the direction of the force being,
52 ON THE COMPOSITION AND EQUILIBRIUM OF FORCES
by hypothesis, that of the normal at the point м, we shall
have, by means of equations (5) of No. 21,
cosλ = v.
dL
dx,
cos μ = V
•
dL
dy
dL
,
cos vv.
dz'
in which, in order to abridge, we suppose
dx
2
2
The sign is undetermined, because we do not know before-
hand in what direction of the normal, the force v acts; but v
disappears in the elimination of N between equations (f); and
by means of the formula (c), we obtain (c)
dr
Y.
dz
dL
X 0, z
dy
dL
dL
X
0,
(g)
.
dx dz
for the two equations which are necessary and sufficient for
the equilibrium of a material point subjected to exist on a
given surface.
37. If the position of this point on this surface, is not
known, equations (g) combined with the given equation L=0,
will enable us to determine the coordinates of the different
points of this surface, in which the material point may be in
equilibrio. When this position is given, we shall only have
to verify whether x, y, z, the coordinates of the points of ap-
plication of the given forces, satisfy equations (g). But, in
this case, these equations may be simplified by making one of
the axes MA, MB, MC, as for instance the first, to coincide with
one of the two parts of the normal; hence there will result
cosλ1, cos μ= 0, cos y = 0;
and this changes equations (f) into the following
+N+P. cosa + P' cos a' + P". cos a" + &c. = 0,
P. cosẞ+P'cos ẞ' + p" cos ẞ" + &c. = 0,
P.cos y + P. cos y' + p" cos y" + &c. = 0.
APPLIED TO THE SAME POINT.
53
1
From the two last of these equations it appears, as is in-
deed evident of itself, that in the plane which touches the
given surface at the point where the normal meets it, the
components of the forces applied to the material point, con-
stitute an equilibrium, in the same manner as if this surface
did not exist at all.
The resistance N which the surface opposes to the forces
P, P', P", &c., is equal and contrary to the pressure that it
experiences from them. From equations (f) it appears, that
in the case of equilibrium, this pressure is the resultant itself
of these forces. In practice, the magnitude of this force
ought to be calculated by means of equation (a), in order to
know whether the surface is capable of supporting it. If the
moveable is merely laid on this surface, which is that of a
solid body, it is moreover necessary that the direction of this
pressure should be such as to press the moveable against this
surface; which condition, as it cannot be expressed by any
equation, must be verified in each case by determining the
direction of this force, by means of equations (b). This veri-
fication can be effected in a simpler manner, by means of the
first of the three preceding equations.
In fact, let us suppose for greater clearness, that the part
of the normal with which the axis MA coincides, is situated in
the concave part of the surface. As it is known whether the
given angles a, a', a", &c., are acute or obtuse; the sign of
x, the sum of the components acting in the direction of
this line will be known, and as the quantity N should be
positive, it is necessary that in the equation in question, that
is to say,
±N+X=0,
we should take the sign -or the sign + before N, according
as the sum x is positive or negative. In the first case, we
shall have cosλ= 1, and the pressure contrary to N will
act in the direction MA; in the second case, we shall have
54
ON THE COMPOSITION AND EQUILIBRIUM OF FORCES
cos X1, and the pressure will act along the production of
this determinate part of the normal.
38. When the material point м, on which the forces P, P′, P",
&c., act, is constrained to exist on two given surfaces, or on
their curve of intersection, it is sufficient, in order to insure
the equilibrium, that the resultant of all these forces should
be decomposable into two forces perpendicular to the given
surfaces, and which will be destroyed by their resistances.
Consequently, if there be added to the forces P, P', p", &c., two
forces perpendicular to these surfaces, but of an unknown
magnitude, we may then altogether abstract from the con-
sideration of these surfaces, and consider the material point as
entirely free.
N, N', being therefore these new forces; λ, u, v, the angles
which determine the direction of N with respect to the axes
MA, MB, MC, and X', u', v', those which determine the direction
of N' with respect to the same axes, equations (c) will become
N.cosλ+N'.cosλ'+P.cosa+P'.cos a'+P".cosa"+&c.=0,"
α
N.cosu+N'.cosμ'+P.cosẞ+P'.cosß'+P".cosß"+&c.=0,} (h)
N.COs v+N'.cos v'+P.cosy+P'.cosy'+p".cosy" +&c.=0,]
Moreover, if x, y, z, denote the coordinates of the point м
referred to axes parallel to MA, MB, MC, and if L = 0, L'= 0,
represent the equations of the two given surfaces, the values
of cos λ, cos µ, cosv, will be the same as in the last case, and
those of cos X', cos μ', cos v', will be obtained from them by
changing L into L'. If these values be substituted in the
three equations (h), and if N and N' be then eliminated be-
tween them, the equation of equilibrium which the given
forces P, P', P", &c., ought to satisfy, will be obtained; also,
if the position of the moveable on the intersection of the two
surfaces is not given, this equation of equilibrium combined
with equations L 0, L'o, will determine its three coor-
dinates x, y, z.
APPLIED TO THE SAME POINT.
55
When the position of the moveable is given on the curve
on which it should remain, the equation of equilibrium of
the forces P, P', p", &c., may be obtained at once, by taking
the axes MB and мc, to which the angles µ, ß, ß', &c. v,y,y',
&c., refer, in the plane of the normals to the two given
surfaces. The third axis MA will then fall on the tangent to
their curve of intersection, it is, therefore, perpendicular to
the normal forces N and N', so that we have λ= 90°, A'≈ 90°,
and likewise, in virtue of the first equation (h),
P. cos a + P'.cos a' + p". cos a" + &c. = 0,
for the required equation.
This equation indicates, that the sum of the components
of P, P′, P", &c., acting tangentially to the intersection of the
given surfaces, is equal to cypher; this is in fact the con-
dition which must be satisfied, in order that the point м shall
not slide on this curve. After having assured ourselves that
it is satisfied, the values of the forces N and N', and the di-
rection in which they act, can be determined by means of the
two last equations (h). If then two forces equal and contrary
to N and N' be taken, and if these are reduced to one by the
rule for the composition of forces, this will be the resultant of
the forces P, P', P", &c., and will make known the pressure
exerted on the given curve, to which it will be perpendicular.
39. It appears from what precedes, that when the move-
able is constrained to exist on a given curve, there is only one
equation of equilibrium, that there are two when it is restricted
to move on a given surface, and three when the point is alto-
gether free; so that the number of these equations increases,
as they evidently ought, according as the possible movements
of the moveable are less restricted. These different equa-
tions may be all embraced in one formula, which will con-
sequently be the general equation of equilibrium applicable
to any system whatever of material points. In order to obtain
this formula, let us suppose that the moveable is transferred
56
ON THE COMPOSITION AND EQUILIBRIUM OF forces
from the point м, which it occupied in its position of equili-
brium, to o another point infinitely near to M, and such that
this displacement may be compatible with the condition to
which the moveable is subjected, if it is not altogether free.
Letr,p,p′,p",&c., denote the projections of the infinitely small
line мo on the directions of the forces R, P, P', P", &c., in the
first position of the material point, and let each of these pro-
jections be considered as positive or negative, according as it
falls on the direction itself of the force to which it refers, or
on its production. If the force R be supposed to be the re-
sultant of P, P', P", &c., the product Rr will always vanish in
the case of equilibrium; it will vanish for a material point
entirely free, because then the resultant R should be equal to
cypher; it will also vanish, for a point subjected to exist on
a given surface or curve, because the force R must act in
the direction of the normal, hence as the infinitely small line
Mo exists in the tangent plane, or on the tangent, r its pro-
jection on the direction of R will be equal to cypher. Therefore
by equation (d), which has been already demonstrated, and
which obtains equally when the line mo is infinitely small, we
have
Pp + P'p' + P'p" + &c. = 0,
(i)
as often as the forces P, P', P", &c., constitute an equilibrium.
Conversely, the equilibrium will subsist, when this equation
obtains for all possible displacements of a material point en-
tirely free, or constrained to exist on a given surface or curve.
Every infinitely small line, such as Mo, which the material
point can be made to describe, consistently with the conditions
to which it is subjected, is called its virtual velocity; and the
principle of equilibrium contained in the equation which has
been just expressed, is termed the principle of virtual veloci-
ties. Now, if it be successively applied to the case of a
material point entirely free, to one subjected to exist on a
surface, or constrained to remain on a given curve, we can,
without any difficulty, arrive at the equations of equilibrium
APPLIED TO THE SAME POINT.
57
which have been already obtained. Each of the equations
(e) may be deduced from formula (i), by taking for мo, the
displacement of M on one of the axes MA, MB, MC; the equa-
tions of equilibrium, which have place in the case of a point
subjected to exist on a given surface, will be obtained by con-
sidering its displacements in the direction of two axes traced
in the tangential plane; and formula (i) furnishes immediately
the equation of equilibrium of a point constrained to exist on
a given curve, by assuming for мo the element of this curve,
and for p, p', p", &c., the projections of this element on the
directions of the forces P, P', P", &c. The angles which
these directions make with the tangent to the curve being
a, a', a", &c., we shall then have
p = MO cos a, p' MO cos a', p"
MO cos a", &c. ;
and if мo, which is a common factor to all the terms of equa-
tion (i), be suppressed, there will result
as before.
P cos a + P'cos a' + P" cos a”+ &c. — 0,
I
CHAPTER II.
OF THE EQUILIBRIUM OF THE LEVER.
40. In this chapter the lever is supposed to be an inex-
tensible right line or curve, of an invariable form, which can
only turn in a plane about c, (fig. 10,) one of its points,
that is supposed to be fixed, and which is termed the prop of
the lever. Most commonly there are only two forces applied
to this machine, of which one is employed in keeping the
other in equilibrio; the first is termed the power, and the
second the resistance. But, in order that our conclusions may
be as general as possible, we shall suppose that any number
of forces whatever directed in the plane of the lever, act on
different points of this line; and the object of this chapter is
to find the conditions of their equilibrium.
It is not proposed, in the present treatise, to apply the
laws of equilibrium, which will be explained here, to the dif-
ferent mechanic powers or machines. The student is referred
to the various elementary treatises on statics for information
on this subject; but as the law of the equilibrium of the lever
is a fundamental principle of mechanics, it will be necessary
for us to dwell on it here; and we now proceed to show how
this principle is connected with that of the composition of
forces which act on a detached point.
41. When several forces are applied to a body which is
supposed to be of an invariable form, the point of application
of each of these forces may be transferred to another point of
the body, assumed either on its direction or on its production.
For example, if a given force P acts at E, the extremity of the
lever, in the direction of the line AE, and if м be another point
OF THE EQUILIBRIUM OF THE LEVER.
59
belonging to this direction, which is supposed to be connected
with the lever in an invariable manner, then we may replace
the force P by another force of the same intensity, acting at
the point м in the direction of the line MA. In fact, we can,
in the first place, apply to the point м two equal forces, and
acting in opposite directions, the one along MA, the other
along its production MA'; moreover, if each of these forces be
supposed equal to P, that which acts in the direction MA' will
destroy the force P applied to the point E in the direction ea,
because these two equal forces act in opposite directions at the
extremities of the line ME, which by hypothesis is of an in-
variable length; therefore, the only force which remains is P
acting at the point м in the direction MA, by which, therefore,
the given force P, which acts at the point E, is replaced.
Forces act frequently on those bodies, in which they excite,
or have a tendency to excite, motion, either by pulling them by
means of a string which is attached to them, or by pushing them
by means of a bar that presses against their surface. As this string
or this bar may be more or less extended or contracted, it is only
when they cease to extend or contract that they are considered
as invariable lines, which represent the direction of each force,
the action of which is then the same as if it was exerted im-
mediately at the points of the surface of the body at which
these lines terminate. Strictly speaking, a lever is not, as
has been supposed above, a line of an invariable form, it is a
bar which is susceptible of flexure, however small it may be,
and which also contracts or extends by a small quantity in
consequence of the forces which are applied to it. It would
be extremely difficult to determine before hand, the form
which it should assume; but it is only when it has attained
this form, that it is considered as invariable; and it is to this
figure, which differs very little from its natural form, that the
conditions of equilibrium which we propose to investigate, are
referred.
42. Let a second force o act at F, the other extremity of
60
OF THE EQUILIBRIUM OF THE LEVER.
丰
​the lever, in the direction FB, and let the two directions EA
and FB be comprised in the plane in which the lever can
turn; these two right lines, or their productions, must meet
in a certain point м, which, from what has been already es-
tablished, may be assumed as the common point of applica-
tion of P and Q. This being agreed on, we can determine by
the rule of the parallellogram of forces, the resultant of these
two forces, of which resultant M will also be the point of ap-
plication. Now, in order that it may be destroyed, and that
the lever may remain in equilibrio, it will be necessary that
its direction should pass through the prop c; and this will be
sufficient, because by transferring this resultant to this point,
which is fixed, it will be destroyed by its resistance. From
what has been stated in No. 29, it appears that if from the
point c, the perpendiculars CG and CH be let fall, on the di-
rections of the forces P and Q, we shall have in the case of an
equilibrium
P:Q::CH: CG;
and conversely, the equilibrium will subsist, when this pro-
portion obtains. Consequently, if we denote the perpendi-
culars CG and CH by p and q, the equation of equilibrium will
be
Pp Qq.
By the moment of a force with respect to a point, is meant
the product of this force into the perpendicular let fall from
this point on its direction. Hence the condition of equili-
brium in the case of the lever consists in this, that the mo-
ments of the power and resistance, taken relatively to the
prop or point on which the lever turns, are equal; these two
forces tending besides to make the lever turn in opposite
directions.
If the lines CG and CH be supposed to be invariably at-
tached to the lever, G and H may be taken for the points of
application of the forces P and Q, and whatever be the figure
OF THE EQUILIBRIUM OF THE Lever.
61
ECF of the lever, it may be replaced by the bent lever GCH
(fig. 11.) The perpendiculars ce and CH, are termed the arms
of the lever. The condition of equilibrium does not depend
at all on the magnitude of the angle GCH; this indeed is evi-
dent à priori.
In fact, if from the point c as centre, and with a radius
equal to cн, we describe an arc of the circle нн', which is
supposed to be invariably attached to the lever, and if at the
point H' two forces equal to o be applied, acting in opposite
directions along the parts H'B' and H'B" of the tangent at this
point, it is evident that the force a acting in the direction.
H'B", will be destroyed by the force a acting in the direction
HB'; for as the two forces tend to make the system revolve in
opposite directions, there is no reason why it should obey one
in preference to the other. Hence the second of these two
forces will be replaced by the force Q acting in the direction
H'B', and the angle GCH will be changed into the angle GCH',
which may be greater or less, without the equilibrium being
deranged.
By this change, the angle of the two arms may become
180° or zero; the lever will then be strait; the power and re-
sistance will be parallel forces, acting in the same or in con-
trary directions; and in order to an equilibrium, it is always
necessary that their intensities should be in the inverse ratio
of the lengths of the arms of the lever.
43. If R denotes the resultant of the two forces P and Q,
which meet at the point м (fig. 10), and m the angle AMB
comprised between their directions, we shall have (No. 29)
R² = p²+Q²+ 2PQ cos m;
and the value of R will make known the load which the prop
sustains in the case of equilibrium. The direction of the force
R applied to this point, will be along the line CD, the pro-
duction of Mc. In figure 10, the point c is supposed to be
situated between E and F, the points of application of the
62
OF THE EQUILIBRIUM OF THE Lever.
power and of the resistance. In figure 12, the contrary is
the case; but the same reasoning is applicable to both cases;
they differ from each other in this only, that in the first case,
the forces P and Q act on different sides of the lever, and the
angle AMB is acute, while in the second, they act on the same
side, and the angle AMB is obtuse.
If, the three points E, F, C, remaining the same, the point
of concourse of the three forces P, Q, R, be supposed infinitely
distant, these forces will become parallel. In the case
fig. 10, as the angle m becomes then infinitely small, we have
cos m=1, and consequently

R=P+Q.
In the second case, it is the supplement of the angle
which becomes infinitely small; hence we have cos m = -
and
P being supposed
R = Q. - P,
12
Q. Consequently, the resultants of two
parallel forces is equal to their sum, when they act in the
same direction, and to their difference, when they act in op-
posite directions, and in the last case, the resultant acts in the
direction of the greater force. In these two cases, the com-
ponents P and Q are in the inverse ratio of their distances from
the resultant.
This being so, if a common perpendicular to the three forces
be drawn, and if a denotes GH the part of this line (fig. 13
and 14) comprised between the two components P and Q, and
the distance CH of the resultant R from the component a,
which is supposed to be the greater, we shall have
P: Qx: α = x,
A FX's
the superior or inferior sign is to be taken, according as P and
Q act in the same (fig. 13), or in cpposite directions (fig. 14).
Hence we obtain,
P: Q±P::x:α,
OF THE EQUILIBRIUM OF THE LEVER.
63
1
and, consequently,
X
ар
Q ± P
by means of which, the position of the resultant, of which the
value is a ± P, will be determined.
44. When the forces P and Q act in opposite directions,
and differ very little from each other, their resultant, which
always acts in the direction of the greater, will be situated at
a very great distance from the given forces. But when they
are rigorously equal, this distance becomes infinite; which in-
dicates that two equal and parallel forces, acting in opposite
directions, cannot be replaced by a single force; and in point #
of fact, there is no reason why this unique force should act in
one direction in preference to the other.
Two such forces acting at the extremities of the same
line GH, (fig. 15,) will cause this line to revolve about its
middle point K ; which effect cannot evidently be produced by
the action of a single force. They may be replaced in an in-
finite variety of different ways by two other forces, which pre-
sent the same case, for their action is not at all changed, by
applying, for example, to the points & and H along GE and
HF the productions of the line GH, equal forces of any magni-
tude whatever; now the resultant of the forces acting in the di-
rections GA and GE, and that of the forces acting in the directions
HB and HF, will be also equal and parallel forces, acting in
opposite directions along the lines GC and HD, and these re-
sultants will replace the primitive forces, which act along GA
and HB.
If P denotes the common magnitude of these two
forces, and a their mutual distance, each of these two quan-
tities will be changed by the operation that has been indi-
cated; but their product ap will remain constant, as will be
proved immediately.
Ρ
45. In fine, this particular case is the only one in which a
system of any number of forces P, P', p", &c., existing in the
same plane, and acting on material points connected together
64
OF THE EQUILIBRIUM OF THE LEVER.
in an invariable manner, cannot be reduced to a single force.
For whether the two forces P and p' meet in a point, or are
parallel to each other, they may be reduced to a single force
Q, by the rule of the parallelogram of forces, or by that of
the preceding number. In the same manner, this first resultant
Q and p" the third given force, may be reduced to a single force
Q'; then, the second resultant Q', and p"" the fourth force to a
single force o"; and so on until, all the given forces are re-
duced to two only, which may themselves be reduced to a
single force R, unless they fall under the case of exception of
which we have been treating.
In the general case, this force R is the resultant of the
given forces P, P', P", &c., and if to the components a force
R' equal and contrary to R be joined, there will be an equi-
librium in the system. The magnitude of R, and its position
in the plane of the given forces does not depend at all on the
order in which these forces are taken, in the successive re-
ductions which have been indicated; for if, in changing this
order, we should arrive at s a force different from R in magni-
tude and direction, one of these two forces taken in an oppo-
site direction would constitute an equilibrium with the other,
which is impossible. In order to an equilibrium of the forces
P, P', P", &c., when they are applied to a lever situated in
their plane, it is necessary, in the first place, that they should
be reducible to one sole force; for if they were reduced to
s and s', two parallel forces not reducible to a single force, and
if s' was the nearer to the prop, s' may be resolved into two
parallel forces Q and Q', acting in the same direction, of which
the first will be directly opposed to s, and the second will
through the prop; each of these two forces will be less than
s' or s, the force Q' will be destroyed, and there will only
remain the force s Q, which will cause the lever to turn in
the direction of s(a). The given forces being reduced to a
unique force R, it is moreover necessary for the equilibrium of
the lever, that this force should pass through the prop. This
pass
1—
OF THE EQUILIBRIUM OF THE lever.
65
condition will be expressed by an equation, by means of the
following theorem which we proceed to demonstrate.
46. Let us first only consider two forces and their resultant.
The moment of this resultant with respect to a point situated in
the plane of the three forces, is equal to the sum or difference of
the moments of the two components with respect to the same
point; to the difference when the centre of the moments is
situated within the angle of the components, or within its
vertically opposite angle, and to the sum when this point
lies without these two angles. In fact, let P and P' be these
two forces, MA and MA' (fig. 16 and 17) their directions, a
their resultant acting along MB, C the centre of the mo-
ments, p,p′, q, the perpendiculars ca, ca', cb, let fall from
the point c on the directions of P, P', Q. Let each of these
three forces be resolved into others, in the direction of the
line Mc and of KмK' perpendicular to this line; and let the
perpendicular components be considered. We have evidently
COS BMK — sin BMC =
q
C
2
denoting the length of the line MC; hence the component of
Q in the direction мK will be equal to
al .
C
In like manner,
Pp
the components of P and P' perpendicular to мc will be and
C
P'P'. They act in opposite directions when the line Mc tra-
C
verses the angle AMA' (fig. 16), and in the same direction,
when it falls without this angle. Now, the sum of these com-
ponents in the second case, and the excess of the greater over
the less, in the first, ought to reproduce the component of Q,
because a is the resultant of P and P'; consequently, if the
component of P be supposed to be greater than that of p', we
shall obtain, by suppressing the common divisor c,
Qq=pp± P'p';
which was required to be proved.
K
#
32
66
OF THE EQUILIBRIUM OF THE LEVER.
P
If the point c be supposed to be fixed, and if the perpen
diculars ca, ca', cb, constitute an invariable system, the forces
P, P', Q, which may be conceived to act at a, a', b, the ex-
tremities of these lines, can only produce a motion of rotation
about the centre of the moments. Now, it appears from the
inspection of fig. 17, to which the superior sign of the pre-
ceding equation refers, that when the point c falls without
the angle AMB, or its vertically opposite, the three forces
P, P', Q, tend to cause their points of application to turn in
the same direction about the point c; on the contrary, when
this point falls within one of these two angles, figure 16,
which refers to the inferior sign, shows that the forces p and
p' tend to cause the points a and a' to turn in opposite di-
rections; and it is likewise evident, that in this case, the re-
sultant o tends to make its point of application turn in the
same direction as the component which has the greater mo-
ment. It appears from this remark, that the theorem which
which has been now demonstrated implies, that the moment
of the resultant of the two forces is equal to the sum or dif-
ference of the moments of these two forces, according as the
components tend to make their points of application turn in
the same or in opposite directions about the centre of the mo-
ments, and that the resultant tends to make them turn in the
direction of the component, which has the greater moment.
Q
As this theorem obtains for forces of which the directions
make any angle whatever, it must obtain also when they
become parallel; this is, in fact, what it is easy to infer, from
the composition of forces of this kind (No. 43).
47. The advantage of this last mode of stating the problem
is, that it can thus be easily extended to any number of forces
P, P', P", &c., which act in the same plane. The centre of
moments being regarded as a fixed point, about which the
forces tend to make the system, whose points of application
are connected together in an invariable manner, to turn, the
moment of the resultant is equal to the sum of the moments of
1
+
OF THE EQUILIBRIUM OF THE LEVER.
67
the forces which tend to make the system to turn in the same
direction as it turns, minus the sum of the moments of the
forces which tend to make it to turn in a contrary direction.
For greater clearness, let the three first forces P, P', p", be
supposed to make the system turn in one direction, and all the
other forces in a contrary direction. Let the series of reductions
of No. 45 be resumed, and let a be the resultant of P and P',
and qʻthat of Q and P", or of P, P', P". Likewise, let p,p', p",
q, q′, be perpendiculars let fall from the centre of moments on
the directions of P, P', p', q', q; by what has been already
established, we have
Qq=rp + P'p', Q'q' = Qq + P'p",
and, consequently,
Q'q' — Pp + P'p +r'p".
In like manner, if we denote by Q, the resultant of all the
other forces p'"', PIV, &c.; by q1, the perpendicular let fall from
the centre of moments on its direction, by p'", plv, &c., the
perpendiculars let fall from the same point on the directions of
p''', plv, &c., we shall have also
Q191 = p''p""' + pl plv + &c.
Now, R the resultant of all the given forces will be that of
the two forces Q' and Q₁; hence if r denotes the perpendicular
let fall from the centre of moments on the direction of R, and
if we consider that these forces Q' and Q, tend to produce a
revolution in opposite directions, we shall have
Rr = ±(Q'q'— Q191),
according as o'q' will be greater or less than Q191. In the first
case, the force R will tend to cause the line to turn in the
same direction as the force Q', and consequently in the same
direction as the three forces P, P', P". Let the first case be
that which obtains, by substituting for oʻq' and Q₁₁ their
values, we shall then have
68
OF THE EQUILIBRIUM OF THE LEVER.
Rr = Pp+P'p'+r"'p" — r'"'p"" — plV plv &c. ; (1)
which equation contains the theorem we proposed to establish.
If the centre of the moments be supposed to be at the prop
of the lever to which the forces P, P', P'", &c., are applied, it
is necessary in order to insure an equilibrium in this lever,
that we should have
rp + P'p'+ P''p" — p'"'p'"'— plV plv - &c. = 0,
-
—
(2)
since, in this case, these forces must have a resultant which
should pass through the prop (No. 45), and for which we have
therefore r = 0.
48. Equation (1) can be rendered more general, by sup-
posing that by resolutions and recompositions of the forces
P, P', p", &c., they are transformed into other forces s, s', s",
&c., which taken together are equivalent to the given forces.
If s, s', s", &c., denote the perpendiculars let fall from the
centre of moments on the directions of s, s', s", &c., we shall
find by the same mode of reasoning, as in the preceding
number,
$s+s's'+s"s"+&c.=rp+P'p'+P"p"-p""p"-pIpIV-&c., (3)
in which equation the moments of the forces s, s', s", &c.,
which tend to turn in the same direction as P, P', P", should
be affected with the sign+, and the moments of the forces
which tend to turn in the same direction as p", pl, &c., with
the sign
The particular case in which the forces P, P', P",
&c., are irreducible to a single force, is comprised in this last
equation. Let then s and s' be two equal and parallel forces,
not directly opposed to each other; and let h denote their
mutual distance. If the centre of the moments is situated
between their directions, we shall have s + s'h; they will
tend to produce a revolution in the same direction about this
point; therefore, their moments must be affected with the
same sign, and there will result
sss's' sh.
OF THE EQUILIBRIUM OF THE lever.
69
S
If, on the contrary, the centre of moments does not exist
between s and s', and if we suppose s>s', we shall have
sh; these two forces will tend to produce a revolu-
tion in opposite directions; the moment of s must be affected
with the sign + and the moment of s' with the sign
there will result
; and
sss's' sh.
Consequently, equation (3) will always become
sh = Pp + P'p' + p''p" — p'''p'"' — plvp¹— &c.
As the second member of this equation consists entirely of
given quantities, it follows that if the values of s and h undergo
any change, their product must remain constant, as has been
already stated. It appears likewise from this last equation,
that when its second member vanishes, the given forces cannot
fall under the case of exception, in which they are irreducible
to a single force; it follows, therefore, that equation (2) ex-
presses at the same time that the forces P, P', p", &c., have
an unique resultant, and also that this resultant passes through
the centre of moments, consequently this equation is necessary
and sufficient to insure an equilibrium of the lever, of which
this centre is the prop. The resultant R which is obtained by
the series of reductions indicated in No. 45, will express the
pressure or load which the lever will have to support; when
it vanishes, the forces P, P', P", &c., will be in equilibrio in
their plane, without the aid of this fixed point.
49. The condition of equilibrium in the lever may also
be expressed by an equation analogous to formula (i) of
No. 39.
For example, let м, M', M", (fig. 18,) be the points of ap-
plication of three forces P, P´, P", which act on the lever ECF
in the directions MA, M'A', M"A", comprised in its plane. Let
the lever turn by an infinitely small quantity about its prop c,
in such a manner that M, M', M", may assume the positions
m, m', m". By the definition of No. 39, the infinitely small
70
OF THE EQUILIBRIUM OF THE LEVER.
arcs мm, м'm', м"m", which may be assumed to be right
lines, will be the virtual velocities of M, M', M", the points of
application of the three forces, P, P', P". Let the perpendiculars
ma, m'a', m"a", be let fall from m, m', m", on the lines
Ma, m'a', m″a”, or on their productions; ma will be the pro-
jection of мm on the direction of the force P, which tends to
make the lever turn in the direction of rotation that ensues;
M'a' and м"a" will be the projections of м'm' and м"m" on
the productions of the two other forces p' and p”, which tend
to make the lever turn in the opposite direction. On this
account, the first of these projections may be considered as a
positive, and the two others as negative quantities. Let
these three quantities be denoted by π, π',". It is evi-
dent, in virtue of the principle of virtual velocities, that the
sum of the given forces multiplied respectively by the pro-
jections, thus defined, of the virtual velocities of their points
of application, vanishes in the case of equilibrium; and con-
versely, the equilibrium obtains when this sum is cypher; so
that the equation of equilibrium of the lever is
PT + P'T'+P"T"= 0;
(4)
and, in fact, it is easy to show that it coincides with that
which has been deduced from the consideration of moments.
For this purpose, let p, p', p", denote the perpendiculars
CG, CGʻ, CG", let fall from the point c on the directions of the
forces P, P', P"; let c, c', c", be the distances CM, c'M', c"M",
of their points of application from the point c; and y, y', y',
the virtual velocities мm, м'm', м"m". As the infinitely small
arc мm may be considered as coincident with its tangent, the
triangles мma and CMG have their sides perpendicular to each
other and are similar, hence we have
Ma: Mm::CG: CM;
and because
Μα πι
·T,
Mm = Y,
CG = P,
CM = C,
we obtain from it
OF THE EQUILIBRIUM OF THE LEVER.
71
T=
= Py
C
In the same manner we shall have
T'=
p'y'
c'
>
T"=
p"y"
C'
π', π", being by hypothesis negative quantities. Moreover,
the form of the lever being supposed invariable, the three
arcs мm, м'm', м"m", described in the same time, correspond
to the same angle, and if they be divided by their respective
radii CM, C'M', c"m", we shall have three equal ratios. De-
noting by 0, the common magnitude of these ratios, there
will consequently result
and, therefore,
Υ
1=1=1=0,
x=
π=р0, π'=-p'0, π"=- p″0.
Now, if these values be substituted in equation (4), and if
0, the factor common to all its terms, be suppressed, it will
become
Pp — P′p — P″p" = 0;
which is in fact, the equation of equilibrium that we have been
considering. Conversely, if this last equation be multiplied
by 0, it will be changed into equation (4). The reasoning
will be precisely the same, whatever be the number of given
forces P, P', P", &c., and the direction in which they tend to
make the lever to turn.
CHAPTER III.
OF THE COMPOSITION AND EQUILIBRIUM OF PARALLEL
FORCES.
50. It appears from No. 43, that the composition of pa-
rallel forces may be deduced from the rule of the parallelo-
gram of forces, by considering the point of application at an
infinite distance, but by means of this same rule, we may also
obtain the resultant of two parallel forces, by another way
which it will be useful to know.
P
Let p and q be the two components acting at the points
E and F of the inflexible line EF, along the parallel directions
EA and FB, either in the same direction as is represented in
(fig. 19), or in opposite directions as in (fig. 20). No change
will be produced in this system of forces, by applying to the
extremities of this line, equal forces respectively denoted by
s and acting in opposite directions, along its productions,
EC and FD. Let the force P', which is supposed to act in
the direction of EA', that is comprised within the angle AEC,
be the resultant of the forces P and s applied to the point
E; in like manner, let the force o' acting in the direction
of the line FB', which is comprised within the angle BFD, be
the resultant of the forces Q and s, then if the case of No. 44,
in which the given forces P and Q are equal and act in oppo-
site directions, be excepted, the two lines EA' and FB' will not
be parallel. Consequently, if к their point of intersection be
supposed to be invariably attached to the line EF, it may be
assumed as the common point of application of the two forces
p' and qʻ (No. 41). Through this point K, let the lines E'F'
and кH' be drawn parallel to the line EF and to the direction
K
COMPOSITION AND EQUILIBRIUM OF PARALLEL FORCES. 73
+
of the forces P and Q, and if then each of the forces p' and oʻ
be decomposed in the direction of these parallels, it is evident
that we shall find again the components s and P directed along
KE′ and KH, and the components s and a directed along KF
and кH (fig. 19), or along KF' and кH' (fig. 20). Therefore,
we shall have the same four forces as before, but all applied to
the same point K.
And if the two forces s be suppressed,
there will remain the two forces P and Q acting in the direction
of the same line кн in the case of fig. 19, or along this line
and its production кH' in the case of fig. 20, in which it is
supposed that Q is the greater of the two given forces. Hence,
the resultant of these two forces will be parallel to them; and
if it be denoted by R, we shall have
R = Q = P,
±
according as they act in the same or in opposite directions.
In order to determine the point o, where its direction cuts the
line EF or its production, let E' and F' be the intersections of
the lines AE and BF with the line E'F'; then the two quadri-
laterals EE'KO, FF'KO will be parallelograms, and if their
diagonals KE and KF be taken to represent the resultants p
and q', we shall have
S:P::EO: KO,
S:Q::FO: KO,
for the ratios of the components. Hence we infer
P: QFO: EO;
by means of which the position of the point o, which may be
assumed as the point of application of the resultant R, can be
determined. We can likewise infer
P: QP::F0:EF;
Q : Q = P ::EO:EF;
in which the superior signs refer to figure 19, and the inferior
L
74
OF THE COMPOSITION AND
signs to figure 20; therefore, if the preceding value of R be
taken into account, we shall have in the two cases,
P: Q:R::FO:EO:EF;
which shows that each of the three forces is proportional to
the distance comprised between the points of application of
the two others.
This proportion, and consequently the position of the
point 0, are independent of the angle at which the directions
of the given forces are cut by the line EF, which may be any
line whose extremities terminate at these two directions.
51. We can now resolve, without any difficulty, all questions
which may be proposed on the composition of two parallel
forces into one, and on the decomposition of a force into two
others parallel to it. However, we will not enter into any
details on this subject; neither will we hereafter revert to
the particular case of forces which are equal, but not directly
opposed to each other, which has been excluded from the pre-
ceding demonstration, since it has been sufficiently examined
in No. 44. We proceed, therefore, to consider any number
whatever of parallel forces, of which one part acts in one
direction, and the other part in the opposite direction, and
which, while they may or may not exist in the same plane,
are supposed to be applied to points connected together in an
invariable manner, as for example, to different points of the
same solid body.
The magnitude and position in space of the resultant of
all the given forces, will be obtained by compounding two of
these forces into a single one, then this last and a third into a
single one, and so on, until all the forces are exhausted, pro-
vided that the two last forces which are considered, do not fall
under the case of exception of No. 44. This resultant will be
evidently parallel to the common direction of the components;
moreover, it will be equal to the sum of those which act in
one direction, minus the sum of those which act in the oppo-
EQUILIBRIUM OF PARALLEL FORCES.
75
site direction, and it will act in the direction of the greater
sum. If, therefore, the first be considered as positive, and
the other as negative quantities (No. 11), and if they be repre-
sented by P, P', p", &c., and their resultant by R, we shall have
always
RP+ P+P"+ &c.
52. If while the given forces are made to turn about their
points of application, their parallelism continues, the result-
ant of these forces will also turn about one of the points of its
direction; for its point of application, which is found by com-
pounding the given forces one after another, in the manner
already pointed out, does not at all depend on the common
direction of these forces, and consequently, it remains the same,
when this direction changes. Thus, for example, let us sup-
pose that the given forces are three in number, namely P, P', P",
acting in the direction of the lines ma, M'A', M”A”, (fig. 21.)
First, let NB be the direction of the resultant of P and P',
which resultant will be equal to P+P'; let then N'B' be the
direction of the resultant of P + P' and P"; this last force p
being supposed in the figure to act in the opposite direction
from P and P', and to be greater than their sum. If now the
three forces P, P', P", be supposed to turn about the points
M, M', M", retaining their parallelism and the relative direction in
which they act, and if мa, M'a', м"a", be their new directions;
In this new state, the resultant of the forces P and P' will meet
the line мM' at the same point N as before, since the position
of this point depends solely on the ratio of the components,
and not at all on the angle which the line мM' makes with
their directions (No. 50), it will now be directed along the line
No parallel to мa, and м'a', and it will be still equal to P+P'.
For the same reason, the resultant of P + P′ and p” will meet
the production of the line MM' in N' the same point as before,
and it will act in the direction of the line N'b' parallel to Nb;
consequently, while the three forces P, P', P", turn about their
76
OF THE COMPOSITION AND
points of application м, м', м", their resultant will also turn
about the same point N'.
53. The centre of parallel forces is termed the point in
which all the successive directions of the resultant intersect,
when its components revolve about their points of application,
which are supposed to be invariable.
We shall see in the sequel, of what importance the con-
sideration of the centre of parallel forces, is, especially in all
questions respecting the equilibrium and motions of heavy
bodies. We can already perceive, that if a solid body is soli-
cited by any parallel forces whatever, and if the centre of
these forces is determined and supposed fixed, the equi-
librium will obtain in all positions which the body can assume
about this point, provided that the given forces continue
always parallel and applied to the same points of this body;
for then their resultant will constantly pass through the fixed
point, which is sufficient in order that it may be destroyed.
The rectangular coordinates of the centre of parallel forces
depend, as we now proceed to shew, on the products of
these forces multiplied by the coordinates of their points of
application. As these products occur in a great number of
cases, a particular denomination has been given to them; the
product of a force into its distance from a plane, is termed the
moment of the force with respect to this plane. Thus, P
being the intensity of a force applied to a point of which the
coordinates are x, y, z, the products Pz, Py, Px, will be the
moments with respect to the planes of the axes of
the axes of x and x, and of the axes of y and x.
this species of moments has nothing in common with mo-
ments which refer to a point, and which have been defined
in No. 42. These depend upon the direction of the force,
and are independent of its point of application; on the con-
trary, moments with respect to a plane depend on the position
of the point of application of the force, and are independent
of its direction. These last are only made use of in the case
XC and
y, of
In general,
EQUILIBRIUM OF PARALLEL FORces.
77
of parallel forces; so that they may be either positive or ne-
gative, according to the sign of the force and of the coordi-
nates of the point at which it is applied.
54. Let M, M', M", &c. (fig. 22), be the points of applica-
tion of the parallel forces P, P', p", &c., of which it will be
unnecessary to indicate the directions. Let the three rectan-
gular axes ox, oy, os, drawn arbitrarily, be the axes of the
coordinates; let x, y, z, be the coordinates of м, x', y', ',
those of м', x", y", s", those of м", &c.; and let these coor-
dinates and forces be supposed to be given quantities, which
may be either positive or negative. Likewise, let q, q', Q",
&c., be the projections of the points м, м', м", &c., on the
plane of the axes of x and y ; so that we may have
MQ = Z,
M'Q'=z',_м"Q"=x", &c.
Finally, let x₁, y₁, 21, represent the three coordinates of
the centre of parallel forces, of which it is required to find
the values.
P+P', which is equal to the resultant of the two forces
P and P' will meet at the point N, the line мM', or its pro-
duction, according as these two forces have the same or op-
posite signs, but in each case we shall have
P': P+P':: MN: MM'.
Let κ be the projection of N on the plane of the axes of
x and y. Through the point м, let мGH be drawn parallel to
the line QKQ', meeting the lines NK and м'q' at the points G
and н, so that we may have
we shall also have
MG GK HQ';
MN:MM::NG:MH;
and from this proportion, combined with the preceding, we
obtain (a)
(P+P).NG = PMH.
If to this equation, we add the identical equation
[
เ
1
78
there will result
OF THE COMPOSITION and
(P + P′) GK = P MQ + P’HQ';
(P+P) NK = PZ + P′8′.
The resultant of the two forces P+P and p" will meet in
à point n', either the line NM", or its production, according
as these two forces have the same or contrary signs; and if K'
be the projection of N' on the plane of the axes of x and y, we
shall find, as in the preceding case,
(P + P' + P″) N'K′ = (P + P′) NK + P"~";
consequently we shall have
(P + P'+ P") N'K′ = PZ + P'z′+ p"z".
In this manner we can continue, until all the given forces
P, P', P", &c., are exhausted, and if R is their total resultant,
we shall have finally
RZ1 PZP'z' + P"z" + &c.
Figure 22 supposes that all the points м, M', M", &c., n,n',
&c., are situated on the same side of the plane of the axes of
x and
y, or that their ordinates parallel to the axis of z, have
all the same sign; but it is easy to perceive, that if the pre-
ceding equation is true in this case, it will be also true when
these ordinates are partly positive and partly negative. In fact,
if the plane of the axes of x and y be transferred parallel to
itself, to any distance h from its primitive position, and if
z, z', z″, &c., be the coordinates of M, M', M", &c., and z, that
of the centre of parallel forces relatively to this new plane, so
that we may have
-
z₁ = z₁h, z = z — h, z'z' — h, z"-z"- h, &c.;
and if from the preceding equation the identical one
Rh Ph + P'h + ph + &c.,
be subducted, there will result
RZ1 = PZ + P′Z' + P'z″ + &c. ;
EQUILIBRIUM OF PARALLEL FORCES.
79
يد
in which equation the ordinates z, z', z", &c., may be either
positive or negative.
It appears, therefore, that in all cases, the moment of the
resultant of any number of parallel forces with respect to a
plane arbitrarily selected, is equal to the sum of the moments
of these forces, with respect to the same plane.
55. If the moments with respect to the three planes of the
coordinates, be taken successively, we shall have by the pre-
ceding notations,
Rx₁ = Рx+P'x' + p″x" + &c.,
RX1
Ry, Py+P'y' + P"y" + &c.,
RZ₁ = PZ + P'z' + p"z" + &c. ;
R21
}
(1)
and since
R=P+ P+P"+ &c.,
(2)
the three coordinates of the centre of parallel forces will be
completely determined. If through this point a line be drawn
parallel to the given forces, in the direction of the sign indi-
cated by R, the direction of the resultant will be obtained.
These four equations embrace, in the most general manner,
the theory of parallel forces.
The sum of the moments of the forces is zero, with respect
to every plane passing through the centre of parallel forces;
for, if this plane be assumed to be that of the axes of x and
it is necessary that we should have z₁ = 0, and consequently
PZ + P′z' + P'z'' + &c. = 0,
X
Y,
In the particular case in which P, P', p", &c., are reduced
to two equal forces acting in opposite directions, their sum
R0, hence the values of x1, Y1, S1, will be infinite. Con-
sequently, the centre of parallel forces is at an infinite distance,
or rather it does not exist at all, no more than the resultant.
56. When all the points of application м, м', м", &c., of
the given forces are situated in the same plane, it is evident,
from the nature of the centre of parallel forces (No. 52), that
80
OF THE COMPOSITION AND
this centre, if it exists, must also lie in this plane, this is in
fact what may be inferred from equations (1) and (2).
Let a, b, c, be three given constants, in this case we shall
have
z = ax + by + c,
z' — ax' + by' +c,
z"= ax"+by"+c,
&c.
If these values of z, z', z", &c., be substituted in the third
equation (1), it becomes
RZ₁ = (PX + P'x' + "x" + &c.) a
+(PY + P'Y' + p″y" + &c.) b
+(P+P'+P”+ &c.) c.
By means of the two other equations (1), and of equation
(2), the coefficients of a, b, c, may be replaced by R1, RY1, R;
and if then, the common factor R be suppressed, there results
z₁ = ax₁ + by₁+c;
from which it appears that the centre of parallel forces exists
in the same plane as the points M, M′, M", &c.
This centre may be likewise found, when all these points
exist on the same right line, and then the first of equations (1)
suffices to enable us to determine its position, provided that
this line be assumed to be the axis of x. Moreover, if the
forces P, P', P", &c., are perpendicular to this line, the moments
which have been considered at page 76, are the same with
those which refer to a point o, the origin of the abscissæ x,
and the first equation (1) coincides with equation (1) of No.
47. In fact, it is easy to perceive, that among the given
forces P, P', P", &c., those which tend to produce a revolution
about the point o, in the same direction as the resultant R, are
all forces of which the sign is the same as that of their dis-
tances x, x', x", &c., from this point, and that those which
EQUILIBRIUM OF PARALLEL FORces.
81
1
tend to produce a revolution in the opposite direction, are
forces of which the sign is contrary to that of these same dis-
tances; consequently, the moments of the first must be added,
and those of the second subtracted, agreeably to what has
been stated in the number already cited.
57. The equations of equilibrium of the parallel forces
P, P′, P", &c., may be easily deduced from the theorem which
has been just established.
If there is no fixed point in the system, it is necessary, in
order to an equilibrium, that, if one of these forces be ab-
stracted, for example the force P, R' the resultant of all the
others should be equal and directly opposed to P. Since then
the forces P and R' are equal and directly opposed to each
other, they must be equal and affected with contrary signs, or
in other words, we must have P+R' 0. But R' is the sum
of the components P', P", &c., it therefore follows, that
P+ P+P”+ &c. = 0,
(a)
which is the first equation of equilibrium. In order to express
besides, that the forces P and R' are directly opposed to each
other, let a, ẞ, y, be the three coordinates of the centre of
parallel forces p', P", &c., so that we may have
R'a
P'x' + P″x" + &c.,
R'ẞ= P'y' + P"y" + &c.,
R'y = P'z' + P"z" + &c. ;
M
then this centre being the point of application of their result-
ant R', it is necessary that it should lie on the direction of the
force P, in order that R'may be directly opposed to this force,
or, which comes to the same thing, this centre and м the point-
of application of the force P should exist on the same line
parallel to the common direction of the given forces.
therefore, for greater simplicity, the plane of x and y be as-
sumed perpendicular to this direction, it is necessary that these
two points should exist on the same perpendicular to this
M
If
+ In this Care as plan by us L tidirection, this forces may
be ale as if passein from it & Each 2
82
OF THE COMPOSITION AND
plane, they will then have the same projection on this plane;
consequently, their coordinates parallel to the axes of x and y
will be the same, so that we shall have
a=x, B = y;
hence, if x and y be substituted in place of a and ẞ in the
first two of the preceding equations, there results, because
R'P,
Px+P'x'+P"x"+ &c. = 0,
Py+P'y' + P"y" + &c. = 0;
}
(b)
which equations indicate that the sum of the moments of all
the forces P, P', p", &c., is cypher, relatively to the planes of
the axes of x and z, and of the axes of y and z, parallel to their
direction.
Thus it appears, that in order to an equilibrium of these
forces, it is necessary that equations (a) and (b) should ob-
tain simultaneously. Conversely, when these three equations
are satisfied, the equilibrium has place; for we shall have, in
virtue of these equations,
R' — — P,
and, consequently,
R'a = Px, R'ß=
a=x, B = y;
B=y;
Py,
so that this resultant will be equal, and directly contrary to
the force P which has been omitted. It is not necessary, in the
preceding proof, that the two planes relatively to which the sum
of the moments of the given forces is zero, should be perpen-
dicular the one to the other; it is sufficient if they are parallel
to the direction of these forces, and it is easy to perceive, that
if this condition is satisfied with respect to two planes(b)
parallel to this direction, it will be equally so with respect to
all others.
Hence we may infer, that in order to the equilibrium of a
system of parallel forces applied to a solid body entirely free,
1
EQUILIBRIUM OF PARALLEL FORCES.
83
it is necessary and it suffices-first, that the sum of these forces
be equal to zero; secondly, that the sum of their moments be
nothing, with respect to any two planes parallel to their com-
mon direction. When all the forces exist in the same plane,
this second condition will be already satisfied with respect to
this plane, and it suffices that it be so likewise with respect to
some one other plane.
58. If one of the points of this solid body is supposed to
be fixed, it will suffice for the equilibrium of parallel forces,
that the sum of their moments should be cypher with respect
to two planes passing through this point and parallel to their
direction, and it will be no longer necessary that their result-
ant should be equal to cypher; for then the distances of this
resultant from these two parallel planes will be cypher, it will
therefore coincide with their intersection, and be destroyed by
the resistance of the fixed point(c).
When this point is the centre of parallel forces, the sum of
the moments will be zero with respect to all planes passing
through this point; consequently the given forces constitute
an equilibrium, whatever be their common direction, which we
already know from other considerations (No. 53).
If the solid body is retained by a fixed axis, about which
it has solely liberty to turn, it will suffice, in order to the equi-
librium of parallel forces applied to its different points, that
the sum of their moments should be zero, relatively to the
plane drawn through this axis parallel to their direction; for
as then their resultant lies in this plane, it will meet the
fixed axis, and must be destroyed by its resistance. When the
fixed axis is itself parallel to the given forces, the plane in
question is indeterminate; consequently the condition of equi-
librium does not obtain, which indeed, it is evident of itself,
must be the case, since the forces, which are all parallel to a
fixed axis, cannot cause a solid body to turn about this line, so
that in this case the equilibrium obtains independently of their
intensities, and of their distances from this axis.
عيد
1
CHAPTER IV.
GENERAL CONSIDERATIONS RESPECTING HEAVY BODIES AND
CENTRES OF GRAVITY.
59. THE force which precipitates bodies to the surface of
the earth when they are let to descend, is termed indifferently
gravity or weight. It acts on all material points, in directions
perpendicular to this surface, or in the direction of vertical
lines. The directions of gravity in different places of the
earth, must therefore, when produced, converge towards its
centre, because its form is nearly spherical; but considering
the magnitude of the radius of the earth, relatively to the
dimensions of those bodies, which are generally considered,
we may, without sensible error, assume that for all the ma-
terial points of the same body, gravity acts in parallel di-
rections.
Direct experiment proves, that the intensity of this force
at the surface of the earth varies with the latitude; and that
in the same vertical, it likewise varies with the elevation
above this surface. But it is necessary that the changes of
height and latitude should be very considerable, in order that
these variations may become sensible, and they are evidently
not so, in the extent of a body of ordinary dimensions.
60. It follows from this that the resultant of the parallel
forces, that act on all the points of a heavy body, (which
forces are infinite in number,) is independent of its form; this
resultant is what is termed the weight of the body. In homo-
geneous bodies, the weight is evidently proportional to the
volume; but daily experience shows, that bodies of different
natures have not the same weight under the same volume ;
+
With
F
CONSIDERATIONS RELATIVE TO HEAVY BODies, etc. 85
this may either arise from this, that the attraction of the
earth, which is the principal cause of gravity, as will appear
in the sequel, depends on the nature of the material points on
which it acts, or from the circumstance that heterogeneous
bodies contain, under equal volumes, different quantities of
material points which are equally heavy. We will explain in
another chapter, how philosophers have inferred from the ob-
served motion of heavy bodies, that it is the second of these
cases which obtains in nature.
It results from this, that the weight of any body is in a
ratio compounded of its mass and of the intensity of gravity
in the place where it is situated. Hence, if we denote this
weight by P, the mass by M, and the measure of gravity by g,
we have
PgM.
This quantity g, which is independent of the particular
nature of each body, is thus the weight of what has been
arbitrarily assumed to represent the unit of mass. In the sequel
it will appear how its value has been determined in different
points of the earth, by means of the motion of bodies sub-
jected to the sole action of gravity. We may likewise write
P = WV,
w denoting the weight of the body, which represents the unit
of volume, and v its volume. The weight w is what is termed
the specific gravity of the body that is considered, a denomi-
nation which is evidently improper, for gravity is common to
all bodies of different species, and therefore, there ought to be
substituted for it, the denomination specific weight.
Finally, if D represents the mass, under the unit of volume
of the body which is considered, D will be what is termed the
density of this body, and we shall have
M=DV,
PgDV.
The preceding are the equations which obtain between the
!,
86
CONSIDERATIONS RELATIVE TO HEAVY BODIES
five quantities, P, 9, M, D, V, each of which should be expressed
numerically, by referring it to a unit of its species.
61. The gramme, or unit of weight, is that of a cube of
distilled water, of which one side is a centimetre, taken at its
greatest density; which we know has place at about the
temperature of 4º of the centigrade thermometer. This weight
varies with the latitude and elevation of the place, in which it
exists; however as the weights of other bodies, which it is
made use of to ascertain, vary exactly in the same ratio, it
follows that the weight of any body expressed in grammes, is
every where the same, and that there is no occasion to specify
in what latitude, or at what elevation it has been determined.
According to the experiments of M. Hallstrom, the weight of
a cube of distilled water, of which one side is a centimetre, at
the temperature zero, is
0 gram, 9998918.
The density of distilled water at this last temperature, is
most commonly assumed as the unit of density. The densities
of a great number of substances have been determined by
experiment, and expressed in numbers by means of this unit.
Thus, for example, the density of mercury at this same tem-
perature is,
13,5975,
and it increases or diminishes by
1
5550'
for each degree of increase or diminution of the temperature.
The density of air, taken also at that of melting ice, and when
the barometrical pressure is 76 centimetres at the observatory
of Paris, is found to be equal to
1
769,4
and centres of gravity.
87
and for each variation of a degree in the temperature, it varies
in a contrary sense, by
0,00375,
like that of every other gas.
As the weight of a column of mercury which expresses the
barometrical pressure, varies with the latitude and elevation
above the surface of the earth, the density of the air, sub-
jected to a pressure of a given height, varies at the same
time. It is on this account, that it is not sufficient merely to
assign this height, it is likewise necessary to specify in what
place the observation has been made. The ratio of the
density of mercury to that of air, corresponding to the pre-
ceding numbers, is
10462.
Whenever a phenomenon, such as, for example, caloric, can
be ascribed to a material substance, this substance must be
acted on by gravity, and the expression imponderable should
refer solely to matter of which the density is so feeble as to
elude all our means of investigation; so that its presence does
not increase in an appreciable manner either the weight or
mass of the body of which it constitutes a part, however con-
siderable the quantities of it are, that may be supposed to
exist in the body.
62. As weights are forces with which we are most familiar,
and of which the relative values may be determined by means
of the balance, with the greatest precision and facility, it is
natural for us to make use of them, in comparing forces of
different natures. Thus, when the muscular force of an
animal, or any other force, acts on a body by the intervention
of a chord attached to its surface, we can always conceive
that this force is equivalent to a certain determinate weight,
or we can even, without changing its direction, substitute for
its action, that of this weight, by suspending it at the extre-
mity of the chord, this chord having been previously made to
pass over a fixed pulley properly placed.
The weight furnishes us with the most commodious measure
88
CONSIDERATIONS RELATIVE TO HEAVY BODIES
of the mass; in fact, without the aid of gravity it would be
extremely difficult to determine the ratio of the masses of two
bodies. It will appear in the sequel, that this measure may
also be strictly inferred from the mutual collision of two
bodies; but it is much simpler to substitute for the ratio of
the masses, that of the weights, to which it is equal in every
part of the earth, in virtue of the equation P = gm. Never-
*theless, as the weight is only a secondary property of bodies
which may be rendered altogether insensible, by transferring
them to a sufficiently great distance from the earth, without
any change whatever taking place in the masses, we should
be able to obtain a conception of the equality and relation of
the masses, independently of the consideration of the weight;
this point will be adverted to in a subsequent part of this
treatise.
63. Since all the points of a heavy body are solicited by
parallel forces, it follows, that if it be made to assume succes-
sively different positions with respect to the direction of these
forces, their resultant will constantly pass through a given point
of this body; this point, which in general has been designated
the centre of parallel forces (No. 53), is in this particular case
termed the centre of gravity. Its characteristic property in solid
bodies, which are subjected to the sole action of gravity, con-
sists in this, that if it be supposed fixed, the body to which it
appertains remains in equilibrio in all possible positions about
this point, since in all these positions, the resultant of the forces
applied to all the points of the body passes through the fixed point.
It also appears, that when a heavy solid body is retained by
another fixed point, it is necessary, and it suffices in order to
an equilibrium, that the right line which connects this point
with the centre of gravity should be vertical; this centre may
however exist indifferently, above or below the fixed point.
In fact, as the weight of the body is a vertical force applied
at its centre of gravity, its direction will coincide in this hypo-
thesis, with the right line which connects this centre with the
fixed point, or with its production, consequently this force
AND CENTRES OF GRAVITY.
89
will be destroyed by the resistance of the fixed point, just as
if it was immediately applied to it.
For the same reason, if a heavy solid body be sus-
pended to a fixed point, by means of a thread, of which the
inferior extremity is attached to a point of its surface, the di-
rection of this thread will be vertical in the state of equilibrium,
and its production will pass through the centre of gravity of
the body. This will be also the case when the same body is'
suspended from the fixed point by attaching any other point
of its surface to the inferior extremity of the thread, and the
successive productions of the thread drawn in the interior of
the body, will intersect in its centre of gravity; this furnishes
us with a practical method of determining the position of this
centre in a body of any form whatever, whether homogeneous
or heterogeneous.
In all questions relating to the equilibrium of a solid body,
we may abstract from the consideration of the weight of its
different parts, provided that there be added to the given
forces which act on the body, a force equal to its weight, and
applied vertically to its centre of gravity. Thus, for example,
in the case of the equilibrium of the lever, in the number of
given forces of which the sum of the moments is cypher, with
respect to the prop, there should be included the weight of
the lever, acting in the direction of gravity, at its centre of
gravity (No. 47).
64. When G and G', the centres of gravity of the two parts
of a body are known, and also their weights p and p', к the
centre of gravity of this body may be immediately obtained;
for this centre is the point of application on the line GG', of
the resultant of the parallel forces p and p', which act in the
same direction at its extremities G and G'; and we conse-
quently have, in order to determine it,
GK: GG'::p':p+p'.
In like manner, if K and G, the centres of gravity of a body
N
عة
90
CONSIDERATIONS RELATIVE TO HEAVY BODIES
and of one of its parts, be known, and if p and p be the respective
weights of the body and of this part, G' the centre of gravity
of the other part may be deduced from them; for this point will
be situated beyond the point K on the production of the line
GK, and its distance from the point & will be determined by
the proportion
GG':GK::P:P
P:P-P.
If a body is divided into any number of parts of which the
weights and the centres of gravity are known, its centre of
gravity may be deduced by a series of proportions; but it is
more convenient to determine its three coordinates by means
of the theorem which has been established respecting the mo-
ments of parallel forces (No. 54).
For this purpose, let p,p', p", &c., be the weights of the
different parts of the body, and P the total weight, so that
p=p+p'+p" + &c.
Likewise, let x, y, z, be the coordinates of the centre of
gravity of the part of which p is the weight, x', y', z', those of
the centre of gravity of the part of which p' is the weight, &c.
All these quantities are supposed to be given; and if x1, Y1, 81, .
be the coordinates of the centre of gravity of the entire body,
referred to the same axes as the preceding, we shall have, by
the theorem just cited,
Px₁ =рx+p'x'+p"x" + &c.,
Py₁ = py + p'y' + p"y" + &c.,
Pz₁ = pz + p'z' + p"z" + &c. ;
by means of which the values of x₁, y₁, 1, can be determined.
65. In these equations, the weights may be replaced by the
masses to which they are proportional. Therefore, if m, m',m",
&c., denote the masses of the different parts of the body to
which the weights represented by p,p′,p", &c., are proportional,
and if м denotes the entire mass, so that
AND CENTRES OF GRAVITY.
91
there will result
M = m + m' + m" + &c.,
Mx₁ = mx + m'x' + m"x" + &c.,
Mÿ, = my + m'y' + m"y" + &c.,
Mz₁ = mz + m'z' + m"z" + &c. ;
MZ1
(1)
from which it appears, that the centre of gravity is independent
of the intensity of gravity, and that it will be always the same
point of the body, in different latitudes and at different ele-
vations above the surface of the earth. It was the consider-
ation that this point does not imply the action of gravity,
and that it depends solely on the masses and on their respective
positions, which induced Euler, and other authors, to term it
the centre of inertia ; however the denomination of centre of
gravity has more generally obtained.
If the mass м be supposed to be divided into an infinite
number of infinitely small parts m, m', m', &c., we may
assume whatever point of each of these we choose for its centre
of gravity, since the coordinates of all the points of the same
element, parallel to the same axis, differ only from each other,
by infinitely small quantities. In this case, the second mem-
bers of equations (1) will consist of an infinite number of in-
finitely small terms, the sums of which will be definite inte-
grals, extended by the theorem of No. 13 to multiple inte-
grals. Consequently, we can always, by the rules of the
integral calculus, determine, exactly or by approximation, the
centre of gravity of any body whatever, without knowing that
of any of its parts.
In a body of which all the parts are homogeneous, their
masses are to each other as their respective volumes; we may,
therefore, substitute the volumes for the masses in equations
(1); and if the entire volume be represented by v, and the
parts corresponding to m, m', m", &c., by v, v', v", &c., we
shall have
1
•
92
CONSIDERATIONS RELATIVE TO HEAVY BODIES
v=v+v²+v" + &c.,
vx₁ = vx+v'x' + v"x" + &c.
vy₁ = vy + v'y' + v"y" + &c.,
Vz₁ = vz + v'z' + v"z" + &c.
The point which is determined by means of these equa-
tions, is the centre of parallel forces applied to all the points
of a body, and proportional to the elements of its volume;
this point is termed the centre of gravity of the volume,
although a volume has neither weight nor mass. In like
manner, the centre of parallel forces applied to all the points
of a surface, or a line, and proportional to their elements, is
termed the centre of gravity. Its coordinates are determined
by substituting in the preceding equations, for the volumes
v, v, v', v", &c., either the areas of the surface and its parts, or
the lengths of the line and its parts.
66. The masses M, m, m', m", &c., and the mutual dis-
tances of their centres of gravity, are connected together by
an equation which may be easily deduced from equations (1).
For this purpose, let the origin of the coordinates be placed
at the centre of gravity of м; and these equations will then
become
mx + m'x' + m"x" + &c. = 0,
my + m'y' + m"y" + &c. = 0,
mz + m'z' + m"z" + &c. = 0.
By squaring the first, there results
m²x² + m²²x²²+m/2x/12+ &c. 2mm'xx'—– 2mm"xx″
2m'm'x'x'
= —
&c.
If to the two members of this equation, there be added the
quantity
m (m'+m" + &c.) x² + m² (m + m”+ &c.) x²²
12
+ m" (m + m' + &c.) x'¹² + &c.,
there will result(a)
AND CENTRES OF GRAVITY.
93
}
M. (mx²+m²x²²+m"x¹²²+ &c.) = mm′(x − x')²+ mm" (x − x'')²
+ m'm" (x' — x'')²+ &c.
In the same manner, the second and third equations (1) will
give
M (my²+m'y'²+m"y''²+&c.) = mm' (y—y')²+ mm"(y—y″)²
+m'm" (y'—y'')²+ &c.
M (mz²+m'z'² + m's"² + &c.) = mm' (z—z')²+mm" (z — z″)²
+m'm'' (z' — z'')²+ &c.
Now, if these three last equations be added together, and
if we make
x² + y²+z² = r²,
12
x²² + y²²+z¹²= p²²,
112
x'¹² + y/¹² + 2/12 = poll²,
&c.,
(x − x')² + (y — y')² + (≈ — 2')² = p²,
(x − x″)² + (y — y′′)² + (z — z″)² = p²²,
х
112
(x′ — x′′)² + (y′ — y″)² + (2′ — 2″)² = p²²,
&c.,
we shall have
M (mr² + m²r²²+m'r'²+ &c.) = mm' p² + mm" p²²
112
+ m²m" p¹² + &c.,
for the equation which it was proposed to obtain, and in
which p, p', p', &c., are the mutual distances of the centres of
gravity of m, m', m", &c., and r, r', r", &c., the distances of
these points from the centre of gravity of M.
67. We can also infer from equations (1), a remarkable
property of the equilibrium of a material point entirely free.
It may be stated in the following manner.
Let o be the point in equilibrio (fig. 23); and let the
forces which solicit it be represented in magnitude and di-
rection by the lines OA, OA', OA", &c.; if their extremities
94
CONSIDERATIONS RELATIVE TO HEAVY BODIES
B
`A, a', a″, &c., be the centres of gravity of equal masses, the
point o will be the centre of gravity of this entire system.
In fact, if equations (1) be applied to these masses, and if
their number be supposed to be n, we shall have
nx₁ = x+x+x" + &c.,
ny₁ = y + y'+y"+ &c.,
nz₁ = z + z'+z" + &c.
On the other hand, if the angles which the force a makes
with three rectangular axes drawn through the point o, be
a, ß, y, and if a', ß', y', be what those angles become rela-
tively to the force oa'; and a", ẞ", y", what they become re-
latively to the force OA", &c., the equations of equilibrium of
these forces will be
OA cos a + OA' cos a' + OA" cos a"+ &c. 0,
OA Cosẞ+OA' cos ß'+ oa" cos ẞ”+ &c. = 0,
OA COS Y + OA' cos y' + OA" cos y" + &c. = 0,
Now, if the origin of the coordinates be placed at the point
o, the coordinates of the points A, a′, a″, &c., will be
Х – OA cos α,
xos' cosa,
x"OA" cos a",
y = OA cos ß,
≈ = OA COS y,
yos' cos B',
'OA' cos y',
yoA"cos B",
"OA"cos y",
&c.,
therefore we shall have, in virtue of the equations of equi-
librium,
hence we infer
x + x + x²+ &c. = 0,
y + y + y" + &c. = 0,
2 + 2 + z″ + &c. = 0;
x₁ = 0, y₁ = 0, 2₁ = 0,
for the coordinates of the centre of gravity of equal masses;
AND CENTRES OF GRAVITY.
95
consequently, this centre will coincide with the point o; which
was proposed to be demonstrated.
68. There are several particular cases in which the centre
of gravity is immediately known. Thus, the centre of gravity
of a sphere, or of an ellipsoid, is evidently the centre of the
figure; that of a parallellopiped is at the intersection of its
four diagonals, that of a cylinder whose bases are parallel, at
the middle point of its axis. The centre of gravity of a
circle, or of an ellipse, is likewise in the centre of the figure,
and that of a parallellogram at the intersection of the two
diagonals. The centre of gravity of a right line is in the
middle point of this line; hence may be inferred, without any
difficulty, the centre of gravity of the perimeter of any polygon
whatever, either by a series of proportions (No. 64), or by the
equations of the moments of parallel forces. It appears in
like manner, that when the centres of gravity of a triangle and
of a triangular pyramid are known, we can deduce, by one or
other of these methods, the centres of gravity of any given
polygon, or polyhedron, since they may be respectively de-
composed into triangles, or triangular pyramids.
But in general, the determination of centres of gravity
requires the application of the integral calculus, and in the
following chapter, we propose to furnish all the requisite de-
tails for this purpose.
"
CHAPTER V.
p-
:
DETERMINATION OF CENTRES OF GRAVITY.
Centres of Gravity of curved Lines.
This
69. LET s be the arc of the given curve, terminating at
any point м, and measured from a fixed point c. Likewise,
let x, y, z, be the three rectangular coordinates of M.
curve may be considered as a polygon of an infinite number
of sides, ds will be the side, or the element, of the curve
which corresponds to the point м; and in whatever point the
centre of gravity of this element may be, we can assume x,y,z,
for its three coordinates, as they cannot, in point of fact, differ
from those of M, except by infinitely small quantities.
Let 7 denote the length of the determinate part of the
curve, of which it is required to determine the centre of gra-
vity, and let so and s, represent the given values of s which
refer to the two extremities of l. Let x1,y1, 21, be the co-
ordinates of the centre of gravity of this arc l, referred to the
axes x, y, z. By the theorem of No. 13, the sum of the values
of each of the products xds, yds, zds, in the entire extent of l,
will be a definite integral taken from s so, to s=81, x, y, z,
s=80,
being considered as functions of s, depending on the nature
of the curve in question. Hence we shall have (No. 65)
S1
Lx₁ = S³¹ xds, ly₁
So
S1
So₂ yds, lz₁ = S³¹ zds,
Ss¹
So
So
(1)
for the three equations, by means of which x, y, z, can be de-
termined.
•
Suppose, for example, that the given line is a right one,
and that its part l terminates at the point c, so that we have
CENTRES OF GRAVITY OF CURVED LINES.
97
8001, 81=l. Let a, ß, y, denote the three angles which
this part 7 makes with the axes drawn through the point c, in
the direction of the positive xs, ys, zs. Likewise let a, b, c, be
the three coordinates of the point c; for any point м we shall
have
x=a+s.cosa, y=b+s.cosß, z=c+s.cos y.
By substituting these values in equations (1), integrating
and then dividing by 7, there results(a)
X1
x₁ = a+lcosa, y₁ = b+lcos B, z=c+cos y ;
which indicates that the centre of gravity of the line 7, is
situated at its middle point, as we know ought to be the case.
70. When it is proposed to find the centre of gravity of a
plane curve, if its plane be assumed to be that of the axes of
x and y, the two first equations (1) suffice to enable us to de-
termine the position of its centre of gravity in this plane.
Moreover, if the portion 7 of the curve be symmetrical on
each side of the point c, we shall have s。=-17 and s₁ = 141;
the centre of gravity will be situated on the normal raised at
the point c; and if this line be taken for the axis of x, it will
be sufficient to determine the value of x₁, which will be fur-
nished by the equation
la₁ = S
§¡¤ds.
The arc of a circle is comprised in this particular case, by
assuming for the axis of x, the diameter which passes through
its middle point. If the origin of the coordinates be at the
centre of the circle, and if we denote its radius by a, we shall have
X a.cos
S
α
a'
for the abscissa of any point м; hence we infer(b)
1
lx 2a2 sin
2a
and if c be the chord of the arc l, we shall have
O
+
98
CENTRES OF GRAVITY OF CURVED LINES.
c = 2a sin
1
2a'
lx₁ = ac;
!
from which it appears that x₁, the distance of the centre of
gravity of an arc of a circle from the centre of the circle, is a
fourth proportional to the radius, the chord, and the arc.
71. As the equation of a plane curve makes known one of
the two variables in a function of the other, if the value of y
is supposed to be given in a function of x, we shall have
dy²
ds = √1+
dx;
dx2
-and if a and ẞ denote the values of x, which refer to the two
extremities of the arc l, instead of the preceding equations,
we shall have
1=
= SB.
SBV
α
dy2
1+
dx²+dx.
a
So & V
x√ 1 +
dy2
dx2
5.dx,
(2)
dy²
dx2
lx1
=
& V
ly₁ = SB 3
So y√ 1+ d. dr.
α
If the given curve be a conic section, we can obtain by
the ordinary rules, the values of the integrals contained in the
two last equations (2), under a finite form. And in the case
of the parabola, we can likewise obtain the value of the in-
tegral of the first of these equations, so that the two(c) co-
ordinates of the centre of gravity of an are of a parabola may
be always obtained in functions of a and ẞ the abscissæ of its
extremities. By a theorem of Landens, the arc of the hyper-
bola
may be expressed by means of two arcs of the ellipse and
of an algebraic part; as to the arc of the ellipse, it must be
considered as a function irreducible to other simpler functions;
however, by means of very extensive tables of this function,
computed by M. Legendre, their numerical values may be
computed to a very close approximation. Hence, when the
་
!
centres of gRAVITY OF Curved lines.
99
numerical values of a and ẞ, and those of the axes of the
ellipse or hyperbola are given, it will be easy to compute the
value of 1, and consequently, x, and y₁, the coordinates of the
centre of gravity of an arc belonging to one or other of these
two curves.
72. Let the arc of the cycloid be taken for another ex-
ample of the application of equations (2). In this curve, the
length, the area, the surface, the volume generated by its
revolution, and the coordinates of their centres of gravity, can
be determined exactly. The drawing of a tangent at any
point of this curve is also extremely simple; its evolute is
another cycloid, and moreover it may demonstrated, that by
a series of successive developments, any curve whatever ap-
proximates more and more to coincide with the cycloid, and it
rigorously (d) coincides with it after an infinite number of de-
velopments. We shall likewise see in the sequel, that the
cycloid is a curve which possesses very remarkable properties
with respect to the curvilinear motion of heavy bodies. This
singular combination of such a number of curious proper-
ties of different descriptions, all appertaining to the same
curve, has rendered the consideration of it extremely useful,
and of very frequent application in geometry and mechanics.
Its equation may be obtained in the following manner.
As the cycloid is a plane curve ACB (fig. 24), generated by
a given point м of the circumference of a circle, while it rolls
without sliding on the right line AB, if the generating point
commences to move at the point a, when it arrives at the
point B of this line, the interval AB will be equal to the circum-
ference of the given circle. We may also perceive, that its
diameter will be equal to the perpendicular CB, let fall from
the summit of the cycloid on AB, and that it divides the curve
into two parts symmetrically equal. Therefore, if c denotes
the radius of the given circle, we shall have
AB = 2πc, CD = 2c.
Uorm
100
CENTRES OF GRAVITY OF CURVED LINES.
In
any position whatever of the circle, let нG be its di-
ameter perpendicular to the base AB, and H its point of con-
tact with this line. From the point м let the perpendiculars
MK and MP be let fall on GH and AB, and let
we shall have
APP, PM = 9;
AH = AP+MK = p + √ 2 cq — q²,
arc MH c.arc sin=
=
are
(sin=
√2cq — q²
S
C
-);
But as the generating circle rolls without sliding on the
line AB, it follows that we have constantly
AHarc MH
hence the required equation of the cycloid will be
p+ √2cq-q²= c.arc (sin=
√2 cq — q²
C
for the point of which the coordinates are p and q.
By differentiating, we obtain(e)
dp=
qdq
√ 2cq — q² ³
1
;
for its differential equation. From which it appears, that the
two chords MG and мH of the generating circle are the tan-
gent and normal to the cycloid at the point м. If the radius
of curvature at the same point be determined by the known
formulæ, it will be found equal to twice MH; hence it follows,
that if мH be produced so that HN = мH, the point N will be
the centre of curvature. If in the same manner, the line CDE
be made double of CD, the point E will be the centre of cur-
vature of the cycloid at c the summit of the cycloid; and hence
it is easy to perceive, that ANE, the evolute of the semi-cy-
cloid AMC, is an equal cycloid, reversed in such a manner, that
CENTRES OF GRAVITY OF CURVED LINES.
101
It fol-
its summit c is transferred to a, and its origin a to E.
lows also that length of AME, or of AMC, is equal to the right
line CDE, and consequently, the entire length of the cycloid is
equal to four times the diameter of the generating circle.
73. In the different applications of the preceding equation,
it will be more convenient to transfer the origin of the coordi-
nates to the summit c (fig. 25), and to take for the axes of x
and y, the lines cx and cy, which are respectively perpendicular
and parallel to the base AB. Hence, if from any point м a
perpendicular MP be let fall on cx, we shall have
CPX, MP= y.
By comparing these coordinates with the preceding, it
appears that
p = {πay, q=a-x;
(a denoting the diameter CD of the generating circle.) Hence,
if these values be substituted in the differential equation of the
cycloid, and if a be put instead of 2 c, it will become
(α - x) dx
from which we obtain(ƒ)
dy=
√ax
x²
ds=
de = √² dax
α
X
(a)
and if the integral of this equation be taken so that it may
vanish when x = 0, there results
s=2√ax,
for the length of the arc CM, of which the origin is at the
summit. At the point A we have xa; which gives, as
before, 2a for the length of the semi-cycloid CMA. It may be
remarked here, that the equation s²-4ax of the cycloid is
similar to that of the parabola, from which it only differs in
this, that the ordinate y is replaced by the arc s.
102
CENTRES OF GRAVITY OF CURVED lines.
If the two last equations (2) be applied to the determination
of the centre of gravity of the arc Cм, we shall have
SX1
=
= Sa√ da,
syi
h = Sy√ / dz
a
dx,
X
in which the integrals must be taken in such a manner that
they may vanish with x. By substituting for s its value, there
results
cydx
2 x ₁ √x
Sv
x dx,
2 y₁ √ x = √√√x
Hence we obtain
x₁ = {} x ;
from which it appears, that the centre of gravity of an arc
M'CM, which is symmetrical on each side of the summit c, and
which must therefore exist on the line CD, is distant from the
point c, by one-third of CP, reckoning from the point c.
integrating by parts, we obtain(g)
Syda = 2y√x - 2 S✓ady.
X
By
Therefore, if we substitute for dy its value given by equa-
tion (a), we shall have
y₁ √ x = y√ x − √ √ a −x dx,
and, consequently,
Y₁ =Y+3
= [(a − x)* − a√ã];
α
2
3√/x
and this combined with the value of x₁, completely determines
the centre of gravity of the arc CM. In the case of the semi-
cycloid, we have x = a and y = ½πa; hence there results
x₁ = α, y₁ =α (½π — }).
74. When a plane curve revolves about a line comprised
in its plane, which line may be taken as the axis of the ab-
CENTRES OF GRAVITY OF CURVED LINES.
103
scissæ, it will generate a surface of revolution, the area of
which can be expressed in terms of the length of this curve
and of the ordinate of its centre of gravity.
In order to demonstrate this, let x and y be the abscissa
and ordinate of м any point of this curve, and s the arc CM
terminating at this point, and measured from the fixed point
c; the element ds will generate the surface of a truncated cone,
and its middle point will describe a circumference equal to
2π (y+dy), or simply to 2у, because dy is infinitely
small. By the known rule, therefore, 2ryds expresses the
element of this surface. Hence if s。 and s₁ denote the values
of s, which answer to the two extremities of the generating
curve, and s the surface generated, we shall have by the theo-
rem of No. 13,
s = 2π √s¹ yds.
2T S$
So,
It should be observed here, that this expression implies
that the generating curve is not intersected by the axis of x,
if it is, then its parts situated on the opposite sides of this axis
will describe two different surfaces, of which s expresses only
the difference. With this restriction, the expression will
always obtain even when the generating curve is one which
returns into itself, and in order to apply it to this case, it
will be sufficient to substitute for s, the arc s, increased by the
entire circumference of this curve.
This being established, we obtain by comparing this for-
mula with the third equation (2),
s=2πly1,
which shews that s the surface generated is equal to 7 the length
of the generating curve, multiplied by 2πy, the circumference
described by the centre of gravity. This theorem will enable
us to determine the value of s at once, without any compu-
tation, and from the mere inspection of the curve, whenever
the centre of gravity of the generating curve is known; it
104
CENTRES OF GRAVITY OF SURFACES.
will not be of any use, if it is necessary to compute the or-
dinate y₁, since this computation would be the same as that of
s. Let us suppose, for example, that the generating curve is
a circle; if we denote its radius by a, the distance of its centre
from the axis of rotation by c, we shall have, on the supposition
that c is not less than a,
1=2πα, y₁ = C,
and, consequently,
s = 4π² αc.
When the circle touches the axis of rotation, we shall have
c= a, and the surface generated will be equivalent to a square,
of which the side is equal to 2πa, the circumference of the
generating circle(g).
II.-Centres of Gravity of Surfaces:
75. Let x, y, z, be as before, the coordinates of any point
м, and x₁, Y1, 21, those of the centre of gravity which it is re-
quired to determine; let z be considered as a given function of
x and y, and therefore
dz
dz
= P,
= 9,
dx
dy
W
then if o be the element of the given surface which corresponds
to the point м, we shall have (No. 21)
w= dxdy√1+p²+q².
In whatever point of w the centre of gravity of this element
exists, its coordinates differ by infinitely small quantities from
x, y, z; we may therefore take wx, wy, wz, for the moments of
w with respect to the three planes of the coordinates, and con-
sequently there will result (Nos. 13 and 65)
W
λ = SSW, Xx₁ =SSxw, Xy₁ = SSYw,
λ=SSw, ⋅ ssyw,
λα
λ=₁ = SS zw;
•
CENTRES OF GRAVITY OF SURFACES.
105
A denotes the area of the portion of the surface, of which the
centre of gravity is required, and the double integrals are sup-
posed to extend to all the elements of λ.
In the case of a plane surface, if it be assumed to be that
of the axes of x and y, the quantities p and q will be cypher,
and we shall have only to consider the three equations
λ=SSdxdy, λx₁ =SSxdxdy, λy₁ = SSydxdy.
If λ be terminated by the curve ABC (fig. 26); to each abscissa
x or op, there corresponds two ordinates PM and PN, which we
shall denote by y and y', and which will be given in functions
of x, by the equation of this curve; also if a and ẞ be the ab-
scissæ OD and OE of the points A and B where the tangents
are parallel to the ordinates, then the integrals should be taken
first from y' PN to y = PM, and then from x = a to x = ß;
by this means there will result
λ = So (y—y) dx,
λαι =
α
=SQ (y—y') xdx,
α
\y₁ = § So (y²—y'²). dx.
λψι
α
(1)
If the area λ, instead of being circumscribed by the reentrant
curve ABC, is comprised between two different curves and be-
tween two right lines parallel to oy the axis of the ordinates,
the value of y should be deduced from the equation of the su-
perior curve, and that of y' from the equation of the inferior
curye, and the distances of these two parallels from the point
o should be assumed for a and ß. In the case of most fre-
quent occurrence, ox, the axis of the abscissæ, replaces the
inferior curve; consequently we shall have y' = o, and simply
λ=Saydx, xx₁ = Sayxdx,
α
λαι
λα yxda, Xy₁ = & Soy'da,
P
α
(2)
106
CENTRES OF GRAVITY OF SURFACES.
by means of which the area and centre of gravity of a portion
of a plane surface comprised between a given curve, the axis
of the abscissæ and two ordinates of this curve, can be deter-
mined. It may be remarked, that equations (1) can be also
obtained in the following manner :-Let the area ABC be di-
vided into elements such as MNN'M' infinitely small and pa-
rallel to the axis oy. Let u denote the length of the line MN;
if through its two extremities there be drawn lines parallel to
the axis ox, then whether triangles, infinitely small of the se-
cond order, be added or subtracted from the element MNN'M',
its magnitude will not be changed, consequently this element
will be equal to udx. If v is assumed to denote the distance of
the middle point of MN from the axis ox, x and v may be taken
for the two coordinates of the centre of gravity of this element;
for it is evident that they only differ from it by infinitely small
quantities. Therefore, from the other notations already re-
ferred to, we shall have
SB
λ= = Soxudx,
· Souda, xx₁ = So anda, xy₁ = Sovudr.
α
λαι
α
α
(3)
Moreover, as y and y' denote always the coordinates PM and PN
which refer to the same abscissa, we have likewise
u = y-y', v = 1 (y + y') ;
from which it appears that these last formulæ coincide with
equations (1).
76. For the first example, let the centre of gravity of the
triangle ABC (fig. 27) be required.
Let the origin of the coordinates be placed at the vertix
c, and let the axis of a be perpendicular to the base AB; if we
represent this base by b, and the height CD by h, and if through
any point P in the line CD, there be drawn MN perpendicular
to this line, CP and MN will be the variables a and u, and we
shall have the proportion
u:x::b: h,
+
CENTRES OF GRAVITY OF SURFACES.
107
from which we deduce
bx
u = π⋅
h
We shall have besides, ao, ẞh. By means of these
values, the two first equations (3) will give
from which results
λα
λ=&bh, xx₁ = + bh²;
x₁ = zh.
It will not be necessary to calculate the value of y; for if E is
the middle point of AB, and if the line ce be drawn it will bi-
sect all the elements of the triangle parallel to AB, and will,
consequently, contain its centre of gravity. Hence, if on CD
there be taken a part
CF = CD = X1,
and if FG be erected perpendicular to CD, the point & in which
it meets CE will be the centre of gravity of the triangle. As
the line FG cuts CD and CE into proportional parts, we shall
also have
CG- // CE;
from which it appears that the centre of gravity of a triangle
exists on the line which joins its vertix with the middle point
of the base, and its distance from the vertix is two-thirds of
this line, and consequently its distance from the middle point
of the base is one-third of the same line.
77. This theorem may be also demonstrated without the
aid of the integral calculus in the following manner:
In fact, as it has been proved by the decomposition of the
triangle ABC (fig. 28) into elements parallel to AB that its cen-
tre of gravity exists on the line CD, which connects the vertix
c with D, the middle point of this side; so, by decomposing
the triangle into elements parallel to the side CA, it may be
shown in the same manner, that this centre of gravity exists
108
CENTRES OF GRAVITY OF SURFACES.
also on the line BE drawn from the vertix в to E the middle
B
point of ca; consequently this point must be at G, the inter-
section of the two lines CD and BE. But, if the line DE be
drawn, it will be parallel to CB, since it cuts CA and AB into
proportional parts, from which there results
DE: CB :: AD: AB::1:2,
DG: CG:: DE: CB :: 1:2 ;
so that DG will be half of CG, and consequently the third of
CD, which it was proposed to demonstrate.
It is evident from this, that the three lines which are drawn
from the three angles of a triangle to the points of bisection of
the opposite sides, intersect in the same point, which is agree-
able to a known theorem.
If the vertices A, B, C, of the triangle are the centres of
gravity of three equal masses, the centre of gravity of these
three bodies will coincide with that of the triangle; for, first,
the centre of gravity of the two masses which answer to a and
B is found at D, the point of bisection of the line AB; and then
the centre of gravity of these two masses and of the third will
be G, a point in the line CD, so situated, that GD is half of CG,
or a third of CD.
It follows from this, and from the theorem of No. 67, that
if there be applied to & the centre of gravity of a triangle,
forces represented in magnitude and direction by the lines GA,
GB, GC, drawn from this point to the three vertices, these three
forces will be in equilibrio.
78. Knowing the centre of gravity of a triangle, it is easy
to deduce successively those of a circular sector or segment.
Let CADB (fig. 29) be the sector, and c the centre of the
circle. If the arc ADB be considered as a portion of a polygon
of an infinite number of equal sides, the sector may likewise be
decomposed into equal triangular elements which will have all
these sides for bases, and their common vertix at the point c.
CENTRES OF GRAVITY OF SURFACES.
109
Let the force which acts on each of these elements be applied
at their respective centres of gravity; and as the distance
from the point c of each centre of gravity is two-thirds of the
radius of the circle, there will result a system of equal and pa-
rallel forces applied to all the elements of the arc A'D'B' de-
scribed from the point c as centre, and with a radius equal to
CD. Consequently, the centre of gravity of the sector will
be the centre of these parallel forces, that is to say, the centre
of gravity of this arc A'D'B'. Now, if a, l, c denote respec-
tively the radius CD, the arc ADB and the chord AB, the analo-
gous quantities corresponding to A'D'B', will be a, zl, c,
therefore if & be the required centre of gravity, we shall have
by the theorem of No. 70,
G
(x being equal to CG).
x =
2ac
37
Now let s, s', S1, denote the surfaces of the sector caDB, of
the triangle CAB, and of the segment ADBE; if their centres of
gravity, which will evidently be on the radius CD, terminating
at D, the middle of the arc ADB, be G, Gʻ, G₁; and if x, x', xɩ,
denote the distances of these three points from the centre c;
when parallel forces and proportional to s, s', s₁, are applied to
them, the first will be the resultant of the two others; therefore,
in considering the moments of these forces, we shall have
sx = s'x' + $1α1 •
Besides, we have
s = ½al, x =
2 ac
37°
And if CE, the altitude of the triangle of which the base is ab
or c, be denoted by h, we have likewise
Hence, as
s'ch, x'} h.
110
CENTRES OF GRAVITY OF SURFACES,
1
S₁ = s — s′ = { (al—ch),
the equation of moments will become
} a²c = {} ch² + { (al—ch) x₁ ;
by means of which, x₁ the distance of the centre of gravity of
the segment ADBE, from the centre of the circle, can be deter-
mined; for since
c=2a sinha.cos
2 a'
we can deduce (i)
Z
4 a² sin³
2a
X1 =
3(1-asin)
When the arc is the semi-circumference, we have rⱭ;
the sector and the segment coincide, as also the distances
and x₁, of which the common value is
4a
x = x₁ =
X1
3п
79. If the three conic sections be taken successively for
the curve to which formulæ (2) refer, the integrations can
be effected by the known rules, and the values of x and y₁,
the two coordinates of the centre of gravity, may be obtained
in a finite form. As this example has been merely adverted to
as an exercise of the calculus, we will not enter into any de-
tails, but
pass on to the determination of the centre of gravity
of the area of the cycloid.
Let CPM (fig. 25) be the segment of which it is required
to determine the centre of gravity; if the abscissa CP and the
ordinate PM be denoted by x and y, as in equation (a) of No. ·
73, it is necessary that the integrals contained in formulæ (2)
should vanish when xo; and these formulæ will become, by
integrating by parts,
CENTRES OF GRAVITY OF SURFAces.
111
λ = xy → Sxdy,
Xx₁ = { x²y — 1Sx²dy,
λy₁ = {xy² — Sxydy;
}
(4)
the new integrals likewise vanishing at the same time as x.
In virtue of equation (a), we have
Sxdy S√ ax-x² dx;
=
but if N be the point where the ordinate MP meets the circle de-
scribed on CD as diameter, this last integral expresses the semi-
circular segment CNP; therefore, if in order to abridge, the
area of this semi-segment be denoted by y, we shall have
x
λ = xy - Y•
༡.
In the case in which the point м coincides with the point
A, we shall have
x= CD = α, y = DA = πα, y = {πα²,
and consequently,
λ= πα
Hence it appears, that the area of the semi-cycloid is triple
of that of the semicircle CND, of which the radius is a, or in
other words, the area of the entire cycloid is three times that
of its generating circle. We shall have in like manner,
Sx²dy = Sxv ax = x².dx,
or, what is the same thing,
2
Sx²dy=&a S√ax - x² dx-S(α- x) √ ax — x² dx.
The last integral may be obtained at once, and because it
must vanish when x 0, we shall have(k)
λx₁ = {x²y — ‡ay + † (ax − x²)‡ ;
by means of which the value of x, may be obtained from that
of λ.
In the case of the semi-cyloid CAD, in which we have at
the same time
.. |
;
112
CENTRES OF GRAVITY OF SURFACES.
2=a, y= έπα, γ= επα", λ= πα,
x=α,
there results
7a
X1
x₁₂ = 1922,
for the distance of its centre of gravity from the axis cy.
Hence the centre of gravity of the area of the entire cycloid
is at the distance of seven-twelfths of the height CD from the
summit c.
With respect to any other segment, such as CMP, the ordi-
nate Y₁ should also be determined; this, however, requires a
much more complicated calculus.
80. In virtue of equation (a), we have
Sxydy=Syvax-x² dx,
and as the value of y given in No. 73 may be written as fol-
lows:
y=
if we assume
S(da—x) dx
√ ax
ax-x²
dx
Sva
+ -S
a
dx
2 J √
✓ax
ე;2
=2,
x²
this integral being supposed, as in the other cases, to vanish
when x = 0, we shall have
y = √ ax − x² + 1az;
from which results(7)
Sxydy = ax² - 4x³+‡a§z√ ax— x² dx.
Because we made
Y = S√ ax-x² dx,
we shall have by partial integration
-
Sz√ ax — x² dx = zy — Sydz.
The expression for y may be written as follows,
(5)
(6)
1
CENTRES OF GRAVITY OF SURFACES.
113
y = ±a² S
dx
Vax x²
( 1 α — x)² dx
Vax
Sa-
करे
and by integrating the second term by parts, there results
dx
y = ‡ a² √ √ a
ax
hence we infer (m)
x2
· (a−x)
(≥α-x) √ ax — x² -
Sv
ах
x² dx;
y = } a² z — } ({ a − x) √ ax — x²;
and because √ax-x2dz dx, we shall consequently have
ах
Sydz = 16 a²x² - † (ax — x²).
If these values of y and Sydz be substituted in equation
(6), there will result (n)
Sz V ax-
— x² dx = † 。 a² z² — — — — — x ) √ ax−x²+ } (ax—x)²;
16
2\x
which changes equation (5) into
Sxydy = {} a²x + § ax² — — x³
1 3
+3½ a³½² - ‡az († α — x) √ ax — x².
32
(≥a
By means of this value and that of z, namely:
reco
z = arccos =
a-2x),
α
}
(7)
the third equation (4) will not contain any unknown quantity,
and will, therefore, make known the value of y₁, for any
segment whatever such as CMP.
In the case of the semi-cycloid cad, we shall have
and formula (7) will be reduced to
and because
x = α,
z = arc (cos = − 1) = π ;
= a³
Saydy ≈ œ² († +357);
;
32
y=έπα, λ= πα,
114
CENTRES OF GRAVITY OF SURFACES:
the third equation (4) will give
+
πα
Y₁
4
9π²
which, together with the value of x₁ given in the preceding
number, will completely determine the position of the centre
of gravity.
81. Let s be the area of a zone of a surface of revolution,
comprised between two planes perpendicular to its axis of
figure. The centre of gravity of s will be on this axis; if
this axis coincides with that of x, and if x, denotes the distance
of this centre from the origin of the coordinates, a and ß being
the distances of the two planes which bound s, from the same
origin; the determination of the centre of gravity of this zone
will be reduced to that of the value of x₁.
Let s be resolved into elements of which each is the sur-
face of a truncated cone described by the infinitely small side
of the generating curve, as in No. 74; that which corresponds
to the point м of this curve, the coordinates of which are x and
y, will be equal to 2πу√ dx² + dy²; its centre of gravity will
be also on the axis of x, and the distance of this point from the
origin of the coordinates can be assumed to be equal to x, since
it can only differ from x by an infinitely small quantity. This
being so, (by Nos. 13 and 65,) we shall have
#SO Y V
s = 2π
α
y
2
dy²
1+
dx2
dx,
(8)
dy²
da2 dx,
1-
sx₁ = 2πS® xy √1+
SB
SX1
α
y being considered as a function of x, which is known from the
equation of the generating curve.
If, for example, this curve is the arc of a circle; by placing
the origin of the coordinates at the centre, we shall have
y = √ a² — x²;
(a denoting the radius,) from which there results(0)
:
CENTRES OF GRAVITY OF SURFACES.
115
and, consequently,
s= 2 πανβ -α,
sx = πα (β' - α?),
SX1 (B² a²),
x₁ = { (ẞ + a) ;
from which it appears, that the centre of gravity of a spherical
zone, is in the middle point of the part of the diameter comprised
between the two planes that terminate it, and which is perpen-
dicular to these planes.
82. The cycloid will furnish us with two examples of the
application of formulæ (8), by causing the arc CM to turn suc-
cessively about the axis cx, and the axis cy.
In the first case, in virtue of equation (a) of No. 73, we
shall have
S
dx
s = 2π √ @ Sy sx₁ = 2 = √ @ Sy√āde;
ay
√x
ay x x
the integrals being taken in such a manner, that they may
vanish at the point c, where x = 0. By partial integration,
and taking into account the value of dy, furnished by equation
(a), there arises
s = 4πу√ ax
4 π у √ ã x
− 4 π √ ã √ √ a − x dx,
Απ a
sx₂ =
3
SX1 Ayvaz -
4 π
yx ах
x
Vaava-a.da,
3
and, consequently,(p)
8π
ax+
3
s = 4 πy√ ãï+³™ √ ¯a (a−x)² — 8T a²,
α
—
4π
SX1
3
9
yx√ ax+³™ x√ã (a—x)²
a
+
45
16″ √a (a—x) * —
16π
a³;
45
by means of which the surface generated by the arc CM, which
is concave towards the axis of the figure, and the distance of
116
CENTRES OF GRAVITY OF SURFACES.
its centre of gravity from the point c, are determined. When
this arc becomes the semicycloid CA, we have xa and
y=ra, and, consequently,
Σπα
3
s = 2πα² (π — ½), sx₁ = (ㅠ​-).
In the second case, it is necessary, in order to be able still
to apply equation (a) of No. 73, to transpose x into y, in for-
mulæ (8), which by this means will become(q)
dy²
= 2 = S = √ 1 + dr dr
S = 2T
sy
Sy₁ = 2T Say
Say √1+
dx,
dx2
dy2
dx;
dx2
y, is the distance from the point c, of the centre of gravity of
s situated on the line cy, and the integrals are supposed to
vanish at the point c, that is to say, when x = 0. By means
of equation (a), we shall have
་
4π
s = 2r Se√ dr = 1 + 2 √ ax;
Sx
a
dx =
x
X
3
;
the value of sy, will be the same as that of sa₁ of the first
case, and by dividing it by this value of s, the distance from
the point c, of the centre of gravity of the surface generated
by the arc Cм, which is convex towards the axis of the figure,
will be obtained. When this are becomes the semi-cyloid ca,
the surface generated will be equal to 4πa²; in which case
also, the value of the distance y₁ will be
a
Y₁ = = (π
2
8
15
1).
It may be remarked here, that when the same arc of a
curve turns successively about two rectangular axes, which
pass through one of its extremities, the value of the second
member of the second equation (8) continues to be the same,
consequently the distances of the centres of gravity of the two
CENTRES OF GRAVITY OF SUrfaces.
117
surfaces which are generated, from this extremity, are in the
inverse ratio of the areas of these surfaces.
83. If the curve ABC (fig. 26) turns about the axis ox,
which is comprised in its plane, but does not meet it, its sur-
face will generate a solid of revolution, of which the volume,
denoted by v, may be expressed by means of the area of this
surface and of y, the ordinate of its centre of gravity.
For if all the notations of No. 75 be retained, it is easy to
perceive that we shall have
π SB (y² — y³) dx,
V=T
α
-
in fact, the infinitely small slice of this volume, generated by
MNN'M' the element of the generating area, will be equal to y³dx
- Ty^2 dx the difference of two cylinders of which the radii are
пуп
PM and PN, and whose common altitude is dx; for we may
neglect the infinitely small volumes of the second order, which
are generated by the triangles that are added or taken from
this element, by drawing through the points м and N lines
parallel to the axis ox. Now by comparing this expression of
v, with the third formula (1) of the number cited, we obtain(r)
v=2πλψι;
from which it appears, that the volume generated by λ, the area
of a plane curve, is equal to this area multiplied by 2πy₁, the
circumference of the circle described by its centre of gravity-
a theorem analogous to that of No. 74. By means of this ex-
pression the volume v can be determined when the centre of
gravity of A is known à priori. It will also subsist, when the
generating surface, in place of being circumscribed by a reen-
trant curve, is comprised between two different curves, and be-
tween two perpendiculars to the axis of the figure, provided
that this axis does not pass between these two plane curves.
If the generating area is a semicircle revolving about its
diameter, the distance of its centre of gravity from this axis
of rotation will be equal to
4 a
(No. 78), the radius being de-
3 п
118
CENTRES OF GRAVITY OF SURFACES.
A
noted by a; hence the length of the circumference described
8a
by this point will be; and as the area of the semicircle is
πα, we shall have
Απα
V
;
3
Ob
27
which is, in fact, the volume of the sphere.
If the reentrant curve ABC is an ellipse of which a and b
are the two semiaxes, and c the distance of its centre from the
axis of rotation; the area λ will be, as is well known, equal to
πab, and as its centre of gravity is evidently the centre of the
figure, we shall have y₁ = c; hence there will result
v = 2π² abc,
whatever may be the inclination of one or other of the axes of
the ellipse to the axis of rotation.
84. It is evident that the segment of the solid of revolution
comprised between two planes passing through the axis of
figure, is to the entire solid, as the angle contained between
these two planes is to four right angles, or, which is the same
thing, as the arc described between the two planes, by the cen-
tre of gravity of the generating area, is to the entire circumfe-
rence 2πy₁. Therefore, denoting the length of this arc by l,
and the volume of the corresponding segment by L, we shall
have(s)
L = 1;
A being always the generating area, which, by hypothesis, is
Anot traversed by the axis of rotation.
So. andy,
лубо
19
This formula may be extended in the following manner to
other segments which do not belong to solids of revolution.
Let us suppose, in fact, that a plane curve moves without
sliding or turning in its plane, and in such a manner that this
plane may be constantly perpendicular to a given line, which
may be either a plane curve or one of double curvature. In
this 'motion, the same point of this plane will always remain
CENTRES OF GRAVITY OF SURFACES.
119
on the directrix, and the other points will describe curves si-
milar to this line. Let A, L, 7, denote respectively the area of
the generating curve, the volume generated by this surface,
and the length of the curve described by its centre of gravity.
If I was the arc of a circle, L would be a segment of a solid of
revolution, but in all cases / may be divided into infinitely small
parts, each of which will coincide with the osculating circle
that corresponds to it. Let a denote one of these parts, and
v the volume of the corresponding segment of L; then if it be
supposed that the planes perpendicular to its direction, by
which v is terminated, intersect each other in a line, that does
not traverse the area of the generating curve, this element v of
L will be a segment of a solid of revolution; and by the pre-
ceding equation we shall have
υ παλ.
Hence, by taking the sum of all the values of v, and observing
that the factor X is constant, it will follow that the volume is
equal to the product of 7 and λ, as in the case of a solid of re-
volution. The rule that results from this equation LA,
is very useful in practice, and susceptible of a great number of
applications; however, we should never forget that it does not
obtain when the consecutive generating planes intersect on the
surface generated, and form by their successive intersections
what is termed by Monge, arète de rebroussement.
85. The consideration of the centre of gravity furnishes
likewise a rule for enabling us to determine the volume of a
prism or of a cylinder with any base whatever, which is cut by
a plane inclined to this base.
Let y be the area of a section of the cylinder perpendicular
to its sides, the area of the inclined section which terminates
it, ✪ the angle of these two planes, w any element whatever
of λ, & its projection on the plane of y, or the corresponding
element of the area y, which is itself the projection of λ. By
the theorem of No. 10 we have
·
:
120
CENTRES OF GRAVITY OF SURFACES.
y=λ cos 0,
ε = w cos 0.
This being agreed on, let λ be the surface to which the general
formulæ of No. 75 refer, and let be the inclination of its
plane on that of x and y. If the third of these formulæ be
multiplied by cos 0, we shall have, (by making this constant
factor to pass under the sign SS) in virtue of the values of y
and
E,
Y² 1 = SSZε.
Now, this double integral is the volume of the truncated
cylinder comprised between the two sections y and λ, de-
composed into elements infinitely small and perpendicular to
ولا
in which, however, it is always implied that these two sec-
tions do not mutually intersect; it follows, therefore, that the
truncated cylinder is equal to a right cylinder having the same
base y, and for an altitude, z₁ the distance of the centre of
gravity of the inclined section from this base.
This theorem is evident in the ordinary case, in which the
base of the cylinder is a circle, and the inclined section an
ellipse; for if through the centre of this curve a plane be
drawn parallel to the base, the volume of the cylinder is not
changed, for the segment which is cut off from it is evidently
equal to that which is added to it.
If the areas denoted by y and λ mutually intersect each
other, the volume will consist of two segments, of which the
integral SSzε will express the difference and not the sum.
When the cylinder is terminated by two inclined sections,
whose areas do not intersect, it may be always divided into
two parts, of which the common base and perpendicular to the
sides of the cylinder does not intersect either the one or the
other of these two sections; and as their centres of gravity
exist on the same right line perpendicular to this base, it is evi-
dent that the entire volume will be equal to the area of this
base multiplied by the mutual distance of these two points.
centres of gRAVITY OF VOLumes and of bodies. 121
III.-Centres of Gravity of Volumes and of Bodies.
86. The determination of the centre of gravity of a volume
depends, in general, on several triple integrals; but there are
some bodies for which the position of this centre is determined
by simple integrals. It is these which we propose first to
consider.
The centre of gravity of a cone or of a pyramid with any
base whatever exists on the right line which connects its sum-
mit with the centre of gravity of the base; for this line meets
all sections parallel to the base, in corresponding points which
are their centres of gravity, and which can also be taken for
the centres of gravity of the infinitely small elements of this
body, that are parallel to its base. Consequently, the line in
question contains the centre of gravity of the pyramid or of
the cone, and there only remains to determine its position on
this line.
Let 7 and x be the area of the base and that of a parallel
section; and let h and a denote the perpendiculars let fall
from the vertix on their planes; it is evident that
and, consequently,
x: 1 :: x²: h²,
X =
ac
lx2
h2
Moreover, we may take xdx for the element of the volume of
the cone or of the pyramid; and if v denotes the entire volume,
and x, the value of a corresponding to the section which con-
tains the centre of gravity, we can obtain, as in the preceding
questions,
X1
V =
Shxdx, væ₁ =
xxdx.
She ∞
By substituting for x its value, and performing the integra-
tions, there results
R
122
CENTRES OF GRAVITY OF VOLUMES AND OF BODIES.
from which we obtain
bh
bh2
V-
3
VX1 =
4
x₁ = h.
But if through the centre of gravity a plane be drawn parallel
to the base, it will cut the height h and the line drawn from
the summit to the centre of gravity of the base, into propor-
tional parts; it follows, therefore, that the distance of the
centre of gravity of a cone or of a pyramid with any base
whatever, from the summit, is three-fourths of this line, and
therefore one-fourth of it, reckoning from the base.
87. In the case of a triangular pyramid, this theorem may
be demonstrated without the aid of the integral calculus. Let
ABCD (fig. 30) be this pyramid. Likewise let E and F be the
centres of gravity of the faces ACD and BCD; let the lines BF
and AE be drawn, their productions will meet in Hн, the middle
point of the side CD; and then in the plane AHB, let the lines
AF and BE de drawn intersecting in a certain point G. This
point will be the centre of gravity of the pyramid ABCD; for
by resolving it into elements parallel to the base ACD, it is
evident, as in the preceding number, that its centre of gravity
must exist on the right line BE; and by resolving it into ele-
ments parallel to BCD, it is likewise evident that this point
appertains to the line AF, and as these two lines exist in the
same plane they must cut each other, consequently their inter-
section G will be the required centre of gravity.
Now, in the triangle ABH, the line EF is parallel to the
base AB, since it cuts the sides AH and вH into proportional
parts, that is to say, in the ratio of one to three, reckoning
from H; we shall have therefore
FG: GA:: EF: AB :: EH : AH.
and, consequently,
FG: GA::1:3:
CENTRES OF GRAVITY OF VOLUMES AND OF BODies. 123
so that FG is the third of GA, or the fourth of FA, which was to
be proved.
Hence it may be shown that if A, B, C, D, the four sum-
mits of the pyramid, be the centres of gravity of equal masses,
the point & will be the centre of gravity of these four masses,
for it was already shown in (No. 77) that the point F is that of
the three masses which are situated in B, C, D; and then the
point & so taken that GF is the third of GA, will be the centre
of gravity of these three masses and of the fourth.
It follows from this (No. 67) that if to the centre of gra-
vity of the pyramid, forces be applied, represented in magni-
tude and direction by lines drawn from this point to the four
summits, these four forces will be in equilibrio.
88. The centre of gravity of a triangular pyramid being
thus determined, it is easy to deduce that of a pyramid or
cone of any base whatever, by decomposing this base into a
finite or infinite number of triangles; the centre of gravity of
this pyramid or cone must exist at the same time in the line
drawn from the summit to the centre of gravity of the base, and
in that plane parallel to the base, which cuts all lines drawn
from the summit to this base, in the ratio of three to four,
measuring from the summit, which agrees with the result of
No. 86.
The centre of gravity of a spherical sector may be also
deduced from it. In fact, if this sector be decomposed into an
infinite number of pyramids, of which the common summit is
at the centre of the sphere, and whose bases are the infinitely
small elements of the base of the sector, their centres of gra-
vity must all exist on the base of a concentric sector, the
radius of which will be three-fourths of that of the given
sector; hence we infer that the centre of gravity of the given
sector will be the same as that of the base of the concentric
sector; by means of which consideration its position may be
determined.
Let the spherical sector be generated by the circular sector
CADB (fig. 29) revolving about the radius CD, which is drawn
}
124
CENTRES OF GRAVITY OF VOLUMES AND OF BODIES.
to the middle of the arc AB. The triangle CAB and the circular
segment ADB will generate, at the same time, a cone and a
spherical segment; and the centre of gravity of this segment
can be determined when those of the spherical sector and of
the cone are known.
19
For this purpose, let V₁, V, v', denote the respective volumes
of these bodies, and x, x, x', the distances of their centres of
gravity from the point c, we shall have
V=V'+V₁₂
vx=v'x' + V₁X₁•
(a)
CD, c the chord AB, and ƒ the sa-
Let a denote the radius
gitta DE of the arc ADB.
have(r)
Relatively to the cone, we shall
v' — 1½ πc² (a —ƒ), x' = {(a−ƒ).
The base of the spherical sector will be equal to the product
of the sagitta and of the circumference of the great circle, or
to 2raf,(s) and the value of its volume will be the product of
2 Ta²f. If from the point c as centre,
παρ
3
this base and of a, or
and with a radius equal to CD, an arc of a circle, such as
A'D'B', be described, the centre of gravity of the surface gene-
rated by this arc will be in the middle point of the sagitta D'E'
(No. 81); or, in other words, at a distance from the point c
equal to CD'D'E', the value of which is (a - žƒ) (t).
Therefore, as by what has been stated, this centre of gravity
is that of the spherical sector v, we shall have
'
v=
Σπαρ
3
x = }} (a− 1ƒ).
(a−1ƒ).
By substituting these different values in equations (a), it be-
comes
½ πа²ƒ= 1½πС² (a−ƒ) + V1,
{πа²ƒ (a− }ƒ) = f πc² (a−ƒ)² + V₁₁ ;
V₁×1;
by means of which the values of v₁ and x, can be determined.
1
If I denote the length of the arc AB, we shall have (v)
CENTRES OF GRAVITY OF VOLUMES AND OF BODIESbodies.
125
.
c=2 a sinƒ=a(1— cos),
hence there results
Σπα
V₁ =
3
2
2ra³ (1-cos- sin² cos).
xX1=
8 (1–
2a
4
3 a sin¹/1/
2 a
2a
COS 2- sin2 COS
2a
2 a
When the arc is the semi-circumferance, it is equal to πa,
and, consequently,
V =
2 πα
3
3 a
X1
X₁ =
8
•
89. The volume and centre of gravity of every body
which is symmetrical with respect to its axis, as for example
an ellipsoid, may likewise be determined by simple integrals.
Let x, y, z, be the three rectangular coordinates of any point
of the surface; let the axis of the figure be taken for that of
x, and let x denote the area of the section perpendicular to
this line, which corresponds to the extremity of the abscissa x.
If the volume be decomposed into infinitely small elements
perpendicular to the axis of the figure, xdx may be assumed
equal to the volume of any element, and x may be taken as
the distance of its centre of gravity from the origin of the co-
ordinates. Therefore, if we denote by v a slice comprised be-·
tween two sections corresponding to the given abscissæ a and
ẞ, and by x₁ the distance of its centre of gravity from the
origin of the coordinates, we shall have
V=
SBxdx, va₁ =
V1=
α
So xxdx.
α
In the case of the ellipsoid, the equation of the surface is
x² y² 22
3 + 1 + 1 = 1;
a²
b2
c²
126
CENTRES OF GRAVITY OF VOLUMES AND OF BODIES.
a, b, c, denoting the three semiaxes. Those of the section x
will be (x)
√/12/2011-2
x²
C
hence we shall have
x = wbc (1 — *-*),
— 22),
-
and, consequently,
v = rbe (B—a) (1 — a² + aß +ß"),
3a2
vz, = ↓ wbc (3º—aº) (1 — a² + B");
2a2
from which we deduce
X1
=
2
3(a+B) (2a2-a²-ẞ²)
4(3a²—a²—aß-B²)
If this formula be applied to the spherical segment which has
been considered in the preceding number, we should take
a = acos
which gives
za B = a;
2 a'
3a (1 + cos
2a
24) sin?
2 a
x1 =
;
2
4(1-cos+sin³)
this expression may be shown to coincide with the value of x₁
already found, by multiplying both its numerator and denomi-
nator by 1 - cos
1
2a
In order to obtain the entire value of the ellipsoid, it is
necessary to make ßa, and a = -a; which gives
V=
4πabc
3
CENTRES OF GRAVITY OF VOLUMES AND OF BODies.
127
This value of the volume is also furnished by the triple inte-
gral SSS dx dy dz, extended to all the elements of the space ter-
minated by the surface of the ellipsoid; but by making
x=ax', y=by', z=cs',
the equation of this surface becomes
x/2 + y/² + 2/2 = 1,
and the triple integral is changed into
abc SSS dx'dy'dz'.
This new integral should be extended to all the elements of
the space circumscribed by the surface which answers to the
preceding equation; consequently it will express the volume
of the sphere of which the radius is unity; and as this volume
4π
4 Tabc
is equal to ; it follows that
3
volume of the ellipsoid.
3 expresses, as before, the
90. Bodies which are symmetrical with respect to an axis,
comprise solids of revolution. In what follows respecting such
bodies, the axis of the figure will always be taken for that of
the abscissæ x. If then a solid of this nature be supposed to
be generated by a plane area comprised between two given
curves, and the perpendiculars to the axis of x which corres-
pond to xa and x = ẞ, and if y and y' denote the ordi-
nates of these curves with respect to the same abscissa x, it is
necessary to make
x = π (y² —у¹²),
X
in the formulæ of the preceding number; this gives
V=T
SB (y²-y¹²)dx, vx₁ =
π
S
(y²—y¹²) xdx.
α
a
In the most usual case, namely, that in which the interior
curve coincides with the axis of the figure, we shall have
y' = o, and simply
128
centres of grAVITY OF YOLUMES AND OF BODIES.
ན
V=π
Soyda, va₁ = T
Soyxdx.
T
α
a
(b)
The cycloid furnishes also, in this case, examples of the
application of these formulæ, in which all the integrations may
be effected under a finite form.
In the case of the convex solid generated by the revolution
of the area CMP (fig. 25) about the axis cx, there will result
by partial integration
v = πxу² — 2πSxydy,
·Sx²ydy;
Vx₁ = {πx²Ý² — πSx²уdу ;
VX1
the integrals being so taken that they may vanish at the point
c, that is, where x = o. Therefore, in virtue of equation (a)
of No. 73, we shall have (y)
V = πxу² - 2π SуVax - x² dx,
√x₁ = πx²у² — πSxу √ ax-x².dx;
VX1
and the calculations of these values can be effected by means
of transformations similar to those of No. 80. In the case of
the volume generated by the semicycloid CAB, we obtain
v=
2
πας (9 πε
رة
(971),
3 16
x1 =
(63π2-64)a
12(9 72—16)*
In the case of the convex solid, generated by the revolution
of the area CMP about the axis cy, it is requisite previously to
change x into y and y into x in equations (b); from which
there will result
v=πSx²dy, vy₁ =
x²ydy;
y₁ being the distance of the centre of gravity, which exists on
the axis cy, from the point c, and the integrals being supposed
to vanish at the point c. Therefore, in virtue of equation (a)
of the cycloid, we shall have
V
= Sx√ ax — x² dx, vy₁ = Syx√ax - x²dx.
π
Vi
ах
vy₁=#{yx√
Bj.
CENTRES OF GRAVITY OF VOLUMES AND of bodies.
129
L
The first integral can be obtained without any difficulty, and
the second by transformations similar to those of No. 80. In
the case in which cм is an entire semicycloid, we shall have
v=
· 16
Y₁ =
Yı
(-1.65
α
9
+4
+
91. Let now, x1, y₁, 21, be the rectangular coordinates of
the centre of gravity of a body of any form whatever, homo-
geneous or heterogeneous, of which the mass is represented by
M.
From what has been already stated in No. 65, it appears,
that in order to determine these three unknown quantities, M
must be divided into indefinitely small parts, consequently the
preceding sums must be changed into integrals in the second
members of equations (1) of this number. By this means, we
shall have
мx₁ = SSSxdm, My₁ = SSSydm, MZ1 = SSSzdm;
MX1
(1)
dm being the differential element of the mass of the body cor-
responding to the coordinates x, y, z. Naming p the density
of this same element, and dv its volume, we shall also have
dm = pdv.
The rectangular parallelopiped of which the adjacent sides
are parallel to the axes of x, of y, and of z, and respectively
equal to the differentials dx, dy, dz, may now be assumed for
the element dv of the volume; hence there will result
dvdxdydz.
If the body be homogeneous, its density will be constant; and
denoting its volume by v, we shall have
M = pv ;
thus equations (1) will become
vx₁ = SSSxdv, vy₁ = SSSydv, vz₁ = SSS zdv.
(2)
If the body be heterogeneous, two different cases may occur;
either this body will consist of homogeneous parts of a finite
S
130 CENTRES of gravity of volumes and of bodiés.
magnitude, and the density will only vary from one part to the
other, in which case we can apply equations (2) to each of them,
and then the centre of gravity of the entire body can be had
from knowing the centres of gravity of all its parts (No. 64).
Or the density will vary by insensible degrees in the interior of
the body; and then we should make use of equations (1), in
which p ought to be a given function of x, y, z.
We should however remark, that in the case both of a
homogeneous and heterogeneous body, the division of the mass
into infinitely small elements of which the densities are the
same, or vary only by insensible gradations, supposes that this
body is constituted of continuous matter. But this is not the
case in nature, in which the bodies are, on the contrary, made
up of detached material parts, and separated from one another
by empty spaces, which in point of magnitude are comparable
to the parts occupied by the matter. In the following chapter
we will revert to this observation, and we will show that for-
mulæ (1) and (2) may, notwithstanding, be applied to bodies
as they exist in nature, just as if matter experienced no dis-
continuity in its interior.
92. It will be sometimes necessary, in order to facilitate
the integrations, to employ the polar coordinates of each ele-
ment dm in place of the coordinates x, y, z. In this case, ifr
denotes its radius vector, the angle which it makes with the
axis of the positive xs, and the angle comprised between the
plane of these two lines and that of x and y; we shall have
(No. 9)
rsin.cosy,
x=r.cos0, y = r sin. cost, z=r.sin0.siny.
We should also, at the same time, express du by means of the
differentials of those new variables r, 0, . There are general
formulæ for the transformation of independent variables in
multiple integrals; but the expression of dv, namely,
dv=r2.sin@drdedy,
drd0d↓,
may be also found directly, as will be shown immediately.
CENTRES OF GRAVITY OF VOlumes anD OF BODies. 131
If pdv be put in place of dm in equations (1), and if the values
of x, y, z, in polar coordinates be also substituted in these
equations, they will become
MX1 = SSS pr³ sin 0.cos @drd0dy,
My₁ = SSS pr³sin20.cosy drdedy,
(3)
Mz₁ = S§§'pr³ sin²0.sin¥ drd0d¥,
to which should be joined the equation
M = SSS pr² sin @drd0dy.
As to the limits of these triple integrals, they will be different
according as the origin of the coordinates is placed without or
within the body. When this origin is itself one of the points
of м, we should first integrate from ro to r = u, in which
u denotes a function of 0 and given by the equation of the
surface; this being done, we should then integrate from 0 = 0
and 0 to 0 and 2π, beginning with either of the
angles or as we please. Generally speaking, the limits
will be more complicated, when the origin of the coordinates
does not belong to the mass M. In this case, let u and u' re-
present two given functions of 0 and 4, w and w', two functions
of 4, and a and a', two given angles; if the question was re-
specting a portion of a body comprised, on one part, between
two surfaces of which the equations are ru and r=u'; and on
the other part, between conic surfaces, which have for a com-
mon axis that of x, their common summit being also the origin
of the coordinates, and for equations 0 w, w'; finally,
between the two planes passing through this axis, and which
make the angles a and a', with the fixed plane from which
the angle is reckoned. We should integrate first from ru
tor=u', then from 0 = w to 0 = w'; and finally, from = a
to a'. For example, let us take for the two first surfaces
those of two concentric spheres having their common centre
at the origin of the coordinates, and of which the radii are
a and a'; at the same time, let the bases of the two cones be
&
\
132
CENTRES OF GRAVITY OF VOLUMEs and of bodies.
W
circles, or in other words, let the angles w and w' be constant;
moreover, let the density be solely a function of r; so that the
portion of the body which is considered, appertains to a sphere
composed of concentric strata infinitely slender, each of which
has the same density throughout its entire extent, and which
density varies from one stratum to another, according to a given
function of the distance from the centre. By making, in order
to abridge,
Sa pr²dr=▲, Sa pr³ dr =
B,
and performing the integrations relative to 0 and 4, we ob-
tain (z)
M = A (a'—a) (cos w—cos w^),
M≈₁ = √ B (a'—a) (cos²w—cos²w'),
MX1
My₁ = B (sina-sin a) (w'-w- sin 2 w'+ sin 2 w),
Mz₁ = B (cosa-cosa') (w'-w-sin 2 w' + sin 2 w);
by means of which equations, the values of x1, Y1, 21, will be
had, which, in this example, cannot be deduced from equations
(1).
If the mass м forms a complete ring, so that we have
a' = a +2π, there will result y₁ = 0 and z₁ = 0, that is to
say, the centre of gravity will be situated, as we know it
should be, on the axis of this ring; the value of x₁, its distance
from the centre of the sphere, of which this ring is a part,
will be (a')
B (cos w + cosw')
X1 =
2A
If the sphere be homogeneous, the density p being then con-
stant, we shall have
▲ = }} p (a¹³—a³), в = p(a^—a¹).
When the portion of the ring that is destitute of matter dis-
appears, we shall have w = 0; and finally, if it changes into a
CENTRES OF GRAVITY OF VOLUMES AND OF BODIES. 133
spherical sector, we will also have a = 0; hence there will
result
3a'
X1 =
8
(1 + cosw'));
this accords with the value of the quantity denoted by a in
No. 88, as appears by observing that the value(¿') of the sagitta
denoted by ƒ will be a'(1-cosw'), and that the radius is a'.
93. In order to obtain the expression of dv, the differential
of the volume, in terms of the differentials of the polar co-
ordinates, let м (fig. 31) be supposed to be the point of which
the coordinates are respectively r, 0, ↓; so that o being their
origin, oм will be the radius vector r, 0 the angle мox com-
prised between this radius and a fixed axis ox, and the
angle which the plane of these two right lines makes with a
fixed plane drawn arbitrarily through the second. Let м' be
any point situated on the production of om, of which let the
radius oм' be denoted by r'. From the point o as centre, and
in the plane m'ox, let the arcs MN and M'N', comprised be-
tween the two lines OMM' and ONN', be described, and let the
angle Nox be denoted by '; finally,
angle turn about the axis ox, and let
position, the angle which it makes
In this motion, the area MM'N'N will generate a volume
MM'NN'PP'Q'Q, which we shall represent by u. Now, this area
being the difference of the two circular sectors M'ON' and MON,
is equal to
☆ (20¹² — go²) (0' — 0).
let the plane of this
represent, in its new
with the fixed plane.
Denoting the perpendicular let fall from its centre of gravity
on the axis or by u, u(f'—4) will be the length of the arc
that this centre will describe about this line. Therefore, by
the theorem of No. 84, we shall have
u = '1/2 (r' + r') (r' — r') (0' — 0) (4'—4).
134
CENTRES OF GRAVITY OF VOLUMES AND OF BODIES.
This being established, if the three dimensions of u become
infinitely small, and if, in consequence, we make
r'-r=dr, 0'- 0 =d0, 4' — 4 = d¥,
the factor r'+r will, at the same time, be reduced to 2r; we
can also take for u the perpendicular мH let fall from the point
м on the axis ox, which is equal to r. sin 0, and which only
differs from u by an infinitely small quantity; finally, u will be
changed into dv, and its value will be
dv = r² sin 0 dr do df,
which it was proposed to determine.
In fact, it is evident that this volume dv may be consi-
dered as a rectangular parallelopiped, the three adjacent sides
of which are MM' or dr, the infinitely small arc MN, the centre
of which is at the point o, and whose length is rde, and the in-
finitely small arc MP, which has its centre at the point н,
and for length r.sino dų.
MNQP, the base of this parallelopiped, is the element of the
spherical surface of which the centre is at the point o and the
radius equal to r. Therefore, representing it by do, we have
do = r² sin 0 do dy, dv = dodr.
Denoting the element of the spherical surface, of which the
radius is taken for unity by dw, we shall also have(c')
dw= sin 0d0d, dv = r²dr dw.
By integrating this expression of do from 00 and
07 and 2π, we obtain 4 for the ratio of the surface
of the sphere to the square of its radius, which is, in fact, its
known value.
0 to
CHAPTER VI.
ON THE ATTRACTION OF BODIES.
I.-Formulæ relative to any Body whatever, and to a Sphere
in particular.
94. If a material point o (fig. 32) be subjected to the at-
tractions of all the points of a body of any form whatever; by
decomposing each of these forces into three others, acting in the
direction of rectangular axes drawn arbitrarily through the
point o, and then taking the sum of the positive or negative
components which act in the direction of each axis, we shall
obtain the three components, the resultant of which will ex-
press in magnitude and direction the entire attraction exerted
on the point o. These three components will be the sum of
an infinite number of infinitely small elements, extended to the
entire mass of the attracting body; they will be expressed by
means of triple integrals, and the calculation of these quantities
will be similar to that of the coordinates of the centre of gra-
vity of any body, with which we have been occupied in the
last chapter; and, in fact, this is the reason why the subject of
the attractions of bodies is introduced in this place.
This question is one which has engaged much of the atten-
tion of geometers, both on account of the difficulties which
occur in the analytical investigations, as also on account of its
connexion with the problem of the figure of the earth, and of
the law of gravity at its surface; but in the present treatise we
shall not enter on the considerations of these points in any de-
tail, but shall merely restrict ourselves to giving those formulæ
which are of most frequent occurrence, and some of their ap-
136 ATTRACTIONS—FORMULÆ RELATIVE TO ANY BODY.
1
plications. For their more complete development, the reader
is referred to the second volume of the Mechanique Celeste of
Laplace, and to the memoir of the author on the Attraction
of Spheroids inserted in the Connaissance des Tems for the
year 1829.
95. Let D be a fixed point assumed in the interior of the
attracting body; through this point let three rectangular axes
Dĩ, dy, dz, be drawn, these will be the axes of the positive
coordinates; let x, y, z, represent the coordinates of м, any
point whatever of the attracting body, and dm the differential
element of its mass corresponding to this point м; likewise,
let a, ẞ, y, and µ denote the three coordinates and mass of the
point o; and finally, let u be the distance oм, so that
μ
u² = (a−x)²+(B—y)²+(y−2)².
The attraction exerted by dm on μ will be in the direction of
the line oм. If this force be proportional to the product of
the two masses, and in the inverse ratio of the square of the
distance u, we shall have, by denoting it by F,
F = fudm
u2
fbeing a constant coefficient which expresses the intensity of
the attractive force, relative to the units of distance and of mass.
In order to obtain an accurate notion of this quantity f, let us
conceive two bodies of any form and dimension whatever, the
masses of which are equal, and respectively assumed for unity,
and suppose that the attraction does not vary either in inten-
sity or direction throughout the entire extent of these two
bodies, so that it must be the same between any two elements
whatever of their masses, equal respectively to dm and µ, as
between μ and dm the material points, which we are consider-
ing, when their distance oм is equal to unity; the force fis
the entire attraction which will then be exercised by one of
these two bodies on the other.
ATTRACTIONS—FORMULÆ RELATIVE TO ANY BODY. 137
:
The projections of the line oм, on the axes Dx, Dy, dz,
are a-x, ß-y, y−z; and by dividing them respectively by
u, the cosines of the angles which determine the direction of
the force F will be obtained; hence its three components are
α- X
U
F,
B-YE, Y-Z
И
U
F;
and, u being considered as a positive quantity, they will tend,
according as they are positive or negative, to diminish or in-
crease a, ß, y, the three coordinates of the point o. Therefore,
if we represent the three components of the entire attraction
exercised on this point, by A, B, C, we shall obtain, by substi-
tuting for F its value, and observing that μ and fare constant
quantities,
a = µf
SSS
α
X
dm,
ت 20
B-y
B = μf SSS B
C=
·Y dm,
ช
·SSS 2 dm,
IT SSS
µf
из
(1)
these triple integrals being supposed to extend to the entire
mass of the attracting body.
Denoting the density of the element dm by p, and its vo-
lume by dv, we shall have
dm = pdv.
In the general case, p will be a given function of the coordi
nates of the point м; in the case of the homogeneity of the
attracting body, it will become a constant quantity; du will be
expressed by means of the differentials of the coordinates of M,
and those are always selected which are most proper to facili-
tate the integrations.
96. The three triple integrals on which the values of
A, B, C, depend, may be reduced to one only, by a very simple
consideration.
T
138 ATTRACTIONS-FORMULÆ RELATIVE TO ANY BODY.
let
The limits being supposed the same as in these integrals,
T=
dm
· SSS am.
И
Since these limits are independent of the point o, if T be diffe-
renced with respect to its coordinates, these differentiations
may be effected under the sign S (No. 14); and as we have
also
d¹
น
da
there will result
X- α
из
da
dß
น
из
dy
из
SSS & 7 a
из
dm,
dB = SSSB dm,
SIS SE S&
dß
dr
dy = SSS === Y dm ;
из
by means of which, equations (1) may be changed into
µf
dr
A =
μf da
B
dr
C =
dß'
μf
;
dy
(2)
so that the calculation of the three components A, B, C, depends
only on one integral T. In determining it, it is material to
recollect, that the denominator u must have constantly the
same sign in the entire extent of the integration, and that it
should be positive, if we wish that the components A, B, C,
should tend to diminish or increase the coordinates of o, ac-
cording as their values furnished by equations (2) are positive
or negative.
If instead of an attractive, the point o is acted on by a re-
pulsive force, it will be only necessary to change the signs of
the values of a, B, C, or, what comes to the same thing, to re-
gard ƒ as a constant negative quantity. In the case in which
ATTRACTIONS—FORMULÆ RELATIVE TO ANY BODY. 139
1
the force, whether attractive or repulsive, that acts on the
point o, does not vary, as has been supposed, in the inverse
ratio of the square of the distance, and that, consequently, the
coefficient of µdm is, in general, represented by a given func-
tion of u, which we can denote by pu, if we assume another
function of u as pu, such that we may have
Афи
du
фи,
then this should be substituted in place of u, in the expression
of T. It may also happen that this force is attractive for one
part of the body that acts on o, and repulsive for another part,
in which case the function pu, in which is comprised the co-
efficient f, would change its sign, in the extent of the integral
represented by T.
The components of the action exercised on a body of any
form or dimension whatever, may be deduced from the pre-
ceding formulæ, by substituting for μ the differential element
of its mass, which corresponds to the coordinates a, ß, y, and
then integrating with respect to these three variables for the
entire extent of the mass; from this it appears, that the com-
ponents of the action which one body exerts on another, will,
generally speaking, depend on sextuple integrals.
The preceding are the formula, by means of which attrac-
tions or repulsions are computed; it is, however, necessary,
previously to our making any application of them, to explain
how they can be adapted to the intimate constitution of bo-
dies, and to examine the difficulty which was adverted to at
the end of No. 91.
97. It was already observed (No. 60) that different bodies
contain, under equal volumes, unequal quantities of ponderable
matter, and as these quantities vary, for the same body, with its
temperature and the exterior pressure to which it is subjected,
philosophers have been led to regard natural bodies as a col-
lection of material parts not in contact, but separated from
•
140
ATTRACTIONS-FORMULE RELATIVE TO ANY BODY.
each other by pores or spaces destitute of ponderable matter.
These material parts are termed atoms; in consequence of
their extreme minuteness, their dimensions, and those of the
pores, elude our senses, and all our means of measuring them.
The atoms are considered as indestructible, and the mass, the
form, the volume of each of them, as invariable. The dimen-
sions of the pores, on the contrary, vary with the different de-
grees of heat, which is either introduced into or expelled from
the body, and with the pressures to which it is subjected; and
as the changes in the volume of a body may be very great,
without its mass undergoing any increase or diminution, it
follows that the dimensions of the parts void of matter must be
comparable, and generally greater than those of the full parts.
Atoms of the same or of different natures combine in diffe-
rent proportions to constitute other parts of bodies, which are
likewise inappreciable by the senses, and which are termed
molecules. Bodies differ from each other in the nature and
proportion of the atoms which enter into the composition of
each molecule; and the atoms are regarded as invariable and
indestructible, as has been already stated, because that when
they are reunited in the same proportions, the same bodies, en-
dowed with the same properties, are invariably reproduced.
98. It is evident from this, that the division of the mass
into infinitely small elements, and the hypothesis of a density
in each element, which does not vary at all in homogeneous
bodies, and which, in the case of heterogeneous bodies, varies
by insensible degrees, is not the condition of bodies as they
actually exist in nature; however this does not prevent us
from making use of the formulæ which are founded on this
consideration, and even from applying them, when the bodies
have been divided into parts of a finite, but still insensible
magnitude.
:
In fact, the molecules are so small,
other, that a part of the mass of a body,
and so near to each
which contains im-
ATTRACTIONS-FORMULÆ RELATIVE TO ANY BODY. 141
.
•
mense numbers of them, may still be supposed extremely
small, and its volume regarded as insensible. Let v be the
volume of such a part, the magnitude of which is insensible,
and which, nevertheless, contains myriads of molecules; like-
wise, let m be the sum of their masses; and let м denote one
of the points of v, which can, if we please, be taken for its
centre of gravity. If we make
m
V
this ratio p will really express the density of the body at the
point м, (a) whatever may be in other particulars the masses of
the molecules and their manner of distribution, whether regu-
lar or irregular, through the extent of v.
In like manner,
denoting the number of the molecules that v contains by n,
and making
V
n
= &³,
this line ɛ, of inappreciable magnitude, may be termed the
mean interval of the molecules, which corresponds to the point
M and the density p. In a homogeneous body, this ratio
and this line do not vary with the position of the point м; in
a heterogeneous body these two quantities vary by insensible
gradations, and may be supposed to be given functions of the
coordinates of this point.
This being established, if it were required to know the mass
of a body, or more generally, the sum of the extremely small
parts of this mass, multiplied by u, a function of the coordi
nates of м one of its points; v the volume of this body, should
be divided into extremely small parts such as v, and then the
sum of all the products Upv, which we shall denote by
Συρυ,
should be taken, and extended to all the parts such as v of v.
It appears from the theorem of No. 13, that if the terms of this
142 ATTRACTIONS—FORMULÆ RELATIVE TO ANY BODY.
Je
sum are infinitely small, and if their number be infinite, its
value will be rigorously equal to the definite integral
Supdv,
extended to the entire volume v, of which dv is the differential
element. Now, we may conceive in general that the difference
between this sum and this integral will diminish more and
more according as the parts of the first become smaller, and
their number becomes greater; so that the magnitude of v
being insensible, but still always distinct from dʊ, we may
nevertheless assume, without sensible error, the integral, in
place of the sum. There is, however, one exception to this
general principle, namely, when u is of the species of functions
which vary very rapidly, and at the same time changes its sign in
the extent of the integration; this is the case, in point of fact,
in the calculation of the forces which arise from molecular at-
traction and from calorific repulsion, which are only sensible
at insensible distances. But for the present, it is sufficient to
observe, that this exception has no respect to the formulæ of
Nos. 91 and 95, relative to the centres of gravity of bodies,
and to attractions varying in the inverse ratio of the square of
the distances, and that we can consequently apply them to
natural bodies made up of detached molecules.
99. Let us now revert to the calculation of attractions. If
the distance of the point o from the attracted body, is very
great relatively to the dimensions of this body, we can, in the
expression for T of No. 96, develope the quantity in a con-
1
U
verging series, arranged according to the powers and products
of x, y, z. By making
we shall then have
18
1
+
a² + B² + y² = &²,
ax + By + y² + 3(ax+By+yz)²—(x²+y²+z²)8²
83
yz
+
+
285
&c.
(b)
ATTRACTIONS-FORMULÆ RELATIVE TO THE SPHERE. 143
If D the origin of the coordinates be in the centre of gravity
of the body, we shall have
SSS xdm = 0, SSS ydm = 0, SSS zdm = 0,
because these integrals, divided by м, the mass of the body,
will be the three coordinates of this point (No. 91). Therefore
if this mass be denoted by м, we shall have
T=
M
ठ
1
+
3
285 SSS (ax + By + y²)² dm
28³ SSS (x² + y² + 2²) dm + &c.
Dord
When the distance OD or d is so great that this value of
T may be reduced to its first term, equations (2) will become
umfa
A:
A =
B =
c =
C=
83
83
Now these components are the same as those of a force
equal to
82
acting at the point 0, in the direction OD; it
follows therefore, that the attractions exerted on the point 0,
by a body which is at a considerable distance from it, is very
nearly the same, in magnitude and direction, as if м the mass
of this body was condensed in its centre of gravity.
When the body is a homogeneous sphere, or one composed
of concentrical strata, we shall find that all the terms of the
value of T, except the first, destroy each other. In order to
demonstrate this, it is only necessary to substitute r, 0, 4, in
place of the coordinates x, y, z, as in No. 92; by means of
which the integrations relative to and can be effected.
Therefore, the theorem which has been stated above, will then
be altogether exact, if the distance & is only so great that the
1
development of may be a convergent series; and in fact, we
U
shall see in the following number, without having recourse
to a reduction into a series, that this theorem obtains, what-
1
144 ATTRACTIONS—FORMULÆ RELATIVE TO THe sphere.
1
ever may be the distance of the point 0 from the attracting
sphere, provided that it is not situated in its interior. It is
easy to infer from this, that the attraction of one sphere on
another is the same as if the mass of each sphere was con-
densed in its centre; for, denoting by м and м' the masses of
the two spheres, and by c and c' their centres, the attraction of
м on o, any point of м', is the same as if the mass м was con-
densed into the point c; moreover, this attraction of c on all
the points, such as o of м' is equal and contrary to the at-
traction of all these points, or of M'on c, which is the same as
if the mass M' was condensed in the point c'; consequently,
the attraction of the two spheres is the same as that of two
material points situated in c and c', and of which the masses
are м and м'.
100. The attraction exerted on the point o, by a spherical
stratum, which is homogeneous and of a constant thickness,
will evidently be reduced to a force acting in the direction OD.
Hence, if this line be made to coincide with the axis Dx, the
components B and c parallel to the axes Dy and De will vanish,
and there will only remain the value of a to compute. In
this computation, by making use of the polar coordinates
r, 0,4, as in No. 92, we shall have, since the axis Da coincides
with the line DO,
ODM = DOα,
0, DO = α, ß= 0, y = 0;
and as DM = r and Oмu, there will result
u² = a² 2ar cos 0 + r².
Because the angle is that which the plane ODM makes with
a fixed plane passing through the line Do, we shall have
(No. 93)
dv = r². sin. 0 dr do d↓,
for the element of the volume; and in the expression pdv=dm
the element of the mass, p can be regarded as a constant fac-
tor. After having substituted these values in the expression
for T of No. 96, we should integrate from rb tora, (a
ATTRACTIONS—FORMULA RELATIVE TO THE SPHERE. 145
and b denoting the exterior and interior radii of the spherical
stratum) and from 00 and 0 to 0 and 2π.
=
=
π =
As the variable does not occur under the sign S, the inte-
gration with respect to this variable, is in fact effected by sub-
stituting 2π for the differential dy. This being so, we shall
have
α
= 2πρ
Sa (Sov
r sin Odo
Va-2ar cos 0+2
rdr.
At the limits = 0 and 0=, the radical will have for-
values
±(a− r), ±(a + r) ;
but as it expresses the value of u, which ought to be always
positive (No. 96), it is necessary to assume a +r at the limit
0
=π, andra or ar at the limit 00, according as
the point 0 will be situated within or without the spherical
stratum. We shall see immediately how we ought to proceed,
when this point appertains to the stratum itself, in which case
we have r>a in one part, and ra in another part of this
stratum.
The indefinite integral being, relatively to 0,
Sva
r sin Odo
a²-2ar cos 0+r²
1
== a²-2ar cos 0+r²+const. ;
when the point is within the stratum, we must have(c)
sin odo
Sova²-2ar cos 0+73
α
1
—- [(r + a) — (r — a)] = 2 ;
0+r² α
consequently, the value of T will not all depend on a, and that
A
of a which is deduced from it by means of the first equation
(2), will be equal to zero. In the case where the point is
without the stratum, we shall have
sin odo
Soya²-2ar 008
a²-2ar cos 0+x²
and, consequently,
=
1
a
[(a+r) — (a− r)] = 2r,
α
U
146 ATTRACTIONS—FORMULÆ RELATIVE TO THE SPHERE.
T=
Απρ (α
a
Απρ
r²dr = 4πe (a³—b³);
Sa r² dr
or, what comes to the same thing,
3 a
M
T=
α
м being the mass of a spherical stratum, of which the volume
4 π (α³ —b³)
is
Hence we deduce
A =
a²
2
(3)
which is the same force as if the entire mass of this attracting
stratum was condensed in its centre.
101. These results may be immediately extended to the
case of a spherical stratum of a constant thickness, but com-
posed of other concentrical strata, the density of which varies
from one to the other, according to any law whatever, which
however, does not change in the extent of the same stratum;
for we can determine separately the attractions of these dif
ferent strata, and then take the sum of all these forces, which
in the case of an interior point will be cypher, and in the case
of an exterior point will be determined by formula (3); M
always expressing the entire mass of the attracting body.
Hence we infer
1st. That when the force varies in the inverse ratio of the
square of the distance, the attractions exerted by all the points
of a spherical stratum of a constant thickness, (which is either
homogeneous or composed of concentrical strata, the density
of which varies from one to the other according to any given
law,) on a point o, existing in the void space circumscribed by
this stratum, mutually destroy each other; so that this point will
remain in equilibrio, wherever it may be situated in this space.
2ndly. That the attraction of this same stratum, and con-
sequently also, the attraction of an entire sphere exerted on o a
point without the sphere, is the same as if the mass of the at-
ATTRACTIONS-FORMULE RELATIVE TO THE SPHERE. 147.
tracting body was condensed into its centre. If the point 0 makes
a part of the attracting stratum, or, in other words, if we have
a >b and a ▲ a, this spherical stratum may be conceived to
consist of two others, the exterior and interior radii of one of
which will be a and a, and those of other a and b; as the point
O is within the first of these two strata, it will exert no action
on this point, and if the mass of the second stratum, without
which the point 0 exists, be denoted by m, the attraction of
this stratum may be deduced from formula (3), by substituting
m in place of м. Therefore, the value of the entire attraction
exerted on the point 0 will be
A =
μm f
a2
α
If the spherical stratum changes into a sphere entirely full
of matter, and of which the density is every where the same,
we shall have
m =
4 πρα
3
A =
4πμέρα
3
that is to say, in the interior of a homogeneous sphere, the
attraction is proportional to the distance of the attracted
point from its centre.
The same theorems obtain in the case where the force is
repulsive, provided it varies in the inverse ratio of the square
of the distances.
102. The equilibrium of the point 0, situated within the
space bounded by a spherical stratum, and which is attracted
or repelled by all its points, may be easily verified.
For this purpose, let this stratum be first supposed infi-
nitely thin, if its thickness be denoted by ε, by decomposing its
surface into infinitely small elements, and denoting the area of
that which corresponds to the point P (fig.33) by w, the cor-
responding elements of the volume and of the mass of this
stratum will be ɛw and pɛw; hence denoting the distance of
by r, the value of the force acting in the direction of this line
will be
148 ATTRACTIONS-FORMULÆ RELATIVE TO THE sphere.
μέρεω
22
If we conceive the sides of a cone of which the base is w
and the summit 0, to be produced through o until they meet
the spherical surface again in p', a second element will be de-
termined on this surface; if we denote it by w', and its distance
from the summit o by r', the value of the force acting along
this line, in an opposite direction from the preceding, will be
μέρεω
2012
;
now it is easy to show that these two opposite forces are equal
to each other, that is to say, that
W
7.2
w'
2012
In fact, if POQ and P'oq' be sections of the two cones
made by the same plane passing through their common sum-
mit o, then the similar surfaces w and w' will be to each other
as the squares of the homologous lines PQ and P'q'. Besides,
as the triangles POQ and P'oq' are similar, we have

PQ: P'Q':: OP: OP';
therefore, by squaring the four terms of the proportion, we
shall obtain
w: w' :: r²: p/2,
from which it is evident that =
Hence it follows, that the actions exerted on the point 0,
by all the elements of the spherical stratum, destroy each
other two by two, consequently the total action of this stratum
will be cypher; and this will still be the case even when the
thickness of the stratum is finite; for in this case it
may
be
decomposed into an infinite number of infinitely thin strata,
the action of each of which on the point 0 is cypher.
ATTRACTIONS-FORMULE RELATIVE TO THE ELLIPSOID. 149
II. Formula relative to the Ellipsoid.
103. When the point 0 (fig. 32) belongs to the attracting
mass, the integrations will frequently be facilitated by placing
the origin of the polar coordinates at this point. The radius
vector of any point м will be then u; hence denoting, as in
No. 93, the element of the spherical surface of which the radius
is unity, by dw, we shall have,
dv=u'dudw, dm = pu²dudw;
and if the angles which the line oм makes with the parallels
to the axes Dx, Dy, Dz, drawn through the point o, be denoted
by g, h, k, we shall likewise, agreeably to the notation of No.
95, have
y-B
cos g=
x-a
U
cos h= B, cosk===Y;
U
this will change equations (1) of this number into the follow-
ing(d),
A = uf SSSP cos g dudo,
B=-μf SSSp cos hdudw,
c = -μf SSSp cos k dudw.
The integrals relative to u should be taken from u =o to u=r,
denoting by r the radius vector of any point of the attracting
surface which terminates the attracting body. If, for greater
simplicity, this body be supposed to be homogeneous, these
integrations can be effected at once, and there will result
A = -μfpsSr cos g dw,
B =
µfo SSr cos hdw,
c = µfp SSr cos kdw.
}
(a)
In order to determine the value of r, which should be sub-
stituted in these formulæ, let
F(x, y, z) = 0,
ぶー
​150 ATTRACTIONS—FORMULÆ RELATIVE TO THE ELLIPSOID.
be the equation of the surface of the attracting body, ex-
pressed in rectangular coordinates. At any point of this sur-
face, we have
x= a+r cos g, y=ẞ+r cos h, z= y + r cos k,
as is evident from the preceding values of cos g, cos h, cos k,
the three coordinates of the point o, of which the values are
given, being always a, ß, y. Hence, if these values of x, y, z
be substituted in the preceding equation, there will result in
general, two values of r, the one positive and the other nega-
tive, but the negative value should be rejected, since the
radius vector r is a positive quantity, the direction of which is
solely determined by the angles g, h, k, which may be acute
or obtuse. After the substitution of the value of r in equa-
tions (a), the double integral should be extended to all the
elements, such as dw of the spherical surface, described from
the point o as centre, and with a radius equal to unity.
104. Let these formulæ be applied to the case of an homo-
geneous ellipsoid, the equation of whose surface is
x²
a²
y2
+ 3/2 + 2 = 1;
b2
(b)
a, b, c, denoting the three semi-axes, and D the origin of the
coordinates, being at the centre of the figure. If the preceding
values of x, y, z, be substituted for them in this equation, there
results(e)
pr² + 2 qr = 1,
by making, in order to abridge,
cos² g
a²
cos² h
cos² k
+
+
b²
C²
P,
a cos g
B cos h
+
a²
b2
+
Y cos k
a²
1
B²
a²
= = 1.
b2
ATTRACTIONS—FORMULÆ RELATIVE TO THE ELLIPSOID. 151
火
​え
​L
Consequently we shall have
− q ± √ q² + pl
↑ =
p
Now, as the quantity p is positive, and as the quantity
is also either positive or cypher, (since the point o, which
answers to the coordinates a, ß, y, is situated in the interior
of the ellipsoid, or at farthest at its surface,) in order that the
radius r may not be negative, the radical must be affected with
the sign. Moreover, this radical may be suppressed in
formulæ (a). In fact, the corresponding part of the integral
contained in a, for example, will be
SS & VT + pl cos gdw;
but for each couple of elements such as do, of which the radii
are the productions the one of the other, the elements of this
double integral mutually destroy each other; for in passing
from one of these elements do to the other, each of the three
cosines cosg, cosh, cos k, changes its sign, the quantities p, l, q²,
remain the same, hence the values of the coefficient of do under
the sign S, are equal, but affected with contrary signs. Thus
all the elements of the preceding integral destroy each other
two by two, and the value of a becomes, having regard to the
value of q(ƒ)
A
β
cos 9 do + cos g.cos h
BSS
p
a = µfp
Mfp (a:
α
+ 2/2 SS
cos g cos k
dw).
γ
p
dw
for the lage
change by
p
Now, the two last of these three integrals are composed of
couples of elements which correspond to the same values of
h and of k, and to values of g, which are supplements, the
one of the other. Hence, each of these couples of elements
will be reduced to cypher, and consequently also, the entire
integrals. Therefore, by suppressing these integrals and by
These
J
152 ATTRACTIONS-FORMULÆ RELATIVE TO THE ELLIPSOID.
making the values of в and c undergo similar reductions, we
shall have simply
A =
g
ufpa SS cos² 9 dw,
a²
H/PB SS costh dw,
B =
2
=ufpy SS Cos
C=
Ρ
cos2 k
dw.
p
Now let be the angle comprised between the radius oм
and the parallel to the axis ox, drawn through the point D,
and the angle which the plane of these two lines makes
with a plane passing through the second and parallel to that
of the axes of x and y; we shall have (No. 8)
X
cos g = cos 0, cos h = sin 0.cos, cos k sin@.sin 4,
and at the same time (No. 93)
dw = sin 0d0dy;
from which there will result(g)
a² b² c² p = b² c² cos² 0 + (c² cos² + b² sin² 4) a² sin² 0,
0
A =
μέρα
uppa SS
cos² 0 sin Oded
P
In order to include the directions of all the radii oм, the
integrals should be extended from 0 = 0 and 4 = 0 to 0 = π
and = 2π; but because the coefficient of de has the same
value for and for π-0, it will be sufficient to integrate from
0 = 0 to 0 = 1, and to double the result, and since the co-
efficient of is the same for and for ±, it will likewise
suffice to integrate from 0 to π, and to quadruple
the result. This being agreed on, if we make
$
=tan, docos²
df
;
then since
ATTRACTIONS-FORMULÆ RELATIVE TÓ THE ELLIPSOID. 153
3
1
cos² + =
1++
sin²=
29
1+ +&²
+2
29
there results(h)
έπαψ
0 p
do
!= a²b²c²
0 (b'cos20+a²sin20)c²+(c²cos²0+a²sin²0)b²p
πabe
2 √(b² cos²0+a² sin² 0) (c² cos²0+a²sin²0)
A
A
by means of which the value of a will only depend on the
integral relative to 0. If we substitute ẞ in place of a, and
transpose the letters a and b, we can deduce в from a without
any new computation; and in the same manner, by substi-
tuting y in place of a, and by transposing the letters a and e
c may be deduced from A. In this manner, we shall have
finally
ㅎㅠ​.
0
77
3 = 4#µfpß³ ³
B=
bc cos² 0 sin Odo
√(b².cos20+a²sin20) (c²cos²0+a²sin20)
ac cos20 sin Odo
• √(a².cos²0+b²sin²0) (c² cos²0+b²sin²0)
c=4πµƒpy So√
ab cos20 sin Od0
0 √ (b².cos²0+-c²sin²0) (a² cos² 0-+-c² sin²0)
>
,
(c)
These values of A, B, C, being positive, it follows that each
of these three components tends to make the point 0 approach
towards the centre of the ellipsoid; the contrary has place
when the force is repulsive, in which case we should substitute,
in these formulæ, -finstead of ƒ.
105. If we denote by d a positive constant, and then sub-
stitute (1 +8) a, (1 + 8) b, (1 + 8) c, instead of a, b, c, in for-
mulæ (c), the factor 18 will disappear, and the values of
A, B, C will remain the same as before. But by this substi-
tution, the ellipsoid is increased by a part comprised between
its primitive surface and a similar surface, and as the compo-
nents A, B, C, do not change, it follows that the action of this
additive part on the point o within this part is cypher.
1

X
154 ATTRACTIONS-FORMULA RELATIVE TO THE ELLIPSOID.
Hence it appears, that a homogeneous stratum, comprised
between two similar elliptic surfaces, having the same centre,
and their axes in the same directions, does not exert any
attractive or repulsive action on a point o situated within the
empty space terminated by its interior surface; so that where-
ever this point may be situated in this space, it will remain in
equilibrio; a theorem which includes that which has been
already established in the case of a spherical stratum.
From this it follows, that the action of an ellipsoid composed
of homogeneous matter, on a point 0 of its mass, is reduced
to that which is exerted by the part of this mass, that is ter-
minated by the elliptic surface passing through this point,
similar and similarly placed to that of the entire body. And
it is evident from formulæ (c), that the component of this force,
parallel to each of the three axes of the ellipsoid, is proportional
to the ordinate of the point o parallel to this axis, and depends
solely on this variable(). In the general case, in which the
three semi-axes a, b, c, are unequal, the integrals relative to
0, which these formulæ contain, can be transformed into
elliptic functions; this enables us to compute their numerical
values, by means of the tables of M. Legendre. These same
integrals may also be obtained in a finite form, when two of
the constants a, b, c, are equal, which is, as we know, the case
of an ellipsoid of revolution.
106. Suppose, for example, that we have c=b; the form
of the integrals relative to will be different, according as the
ellipsoid is oblate or prolate, that is, according as b>a or
ba. If we suppose the first case to obtain, and make, on
this hypothesis(k),
b² - a² = a²e²,
4πρα (1 + e)
3
=m;
so that the fraction e may be the compression of the ellipsoid,
and m its mass.
There will result
A =
T
2
3 µfma C½r cos² 0 sin odo
a³
J01+ e² cos20
ATTRACTIONS—FORMULÆ RELATIVE TO THE ellipsoid. 155
and by performing the integration, we shall have
A =
3 μfma
a³ e³
[e. — arc (tang = e)],
for the component parallel to the axis of revolution. We shall
also have
B
C
B-
3 µfm
C&T
π cos2 0 sin Ode
0
a³ (1+e²) JO √1+e² sin² 0
As the components B and c are to each other as the or-
dinates ẞ and y of the point 0, it follows that their resultant
will act in the direction of the perpendicular, let fall from this
point on the axis of revolution. Naming A' this force, and a′
the length of the perpendicular, so that
A′ = √ B²+C², a′ = √ B² + y².
there results, by performing the required integration(7)
A=
3 fuma [
2 a³e³
[arc (tang = e) –
e
1+e².
7.
The resultant of the two forces A and B will express, in
Α
magnitude and direction, the entire action of the ellipsoid on
the point 0.
When e is a very small fraction, these values of A and A'
may be developed into very convergent series arranged ac-
cording to the powers of e. Because that
are (tang = e) ==
we shall have(m)
е
1+
A =
A'
e
e³
€5
+ &c.,
3
5
e — e³ + e³ — &c.,
= e−
ufma 3e2
-Hma (1-3²+&c.),
a3
α
µfma'
5
uma (1- Ge²+&c.).
απ
5
156 ATTRACTIONS-FORMULÆ RELATIVE TO THE Ellipsoid.
1
In the case of the sphere, in which e = 0, the resultant of
A and a' will be directed towards the centre, and it will have
the same intensity as in No. 101.
107. The determination of the attraction of an homo-
geneous ellipsoid on a point without the ellipsoid presents
much greater difficulties, but by means of a theorem, for
which we are indebted to Mr. Evory, this case may be re-
duced to that of an interior point; by which we are enabled
to express the components of the attraction by simple in-
tegrals similar to formulæ (c). The following is a demon-
stration of this important proposition. By making, in first
equation (1) of No. 95,
dm = pdxdydz,
and observing that p is a constant factor, we have
(a-x) dx dy dz
s = µf SSS [(a− x)² + ( B − y)² + (y —²)³] † •
If equation (b) is that of the surface, then by substituting
ax', by', cy', in place of x, y, z, it will be changed into
12
12
x¹² + y²²+ 2/2
= 1.
At the same time the value of a becomes
A
A
(a-ax') dx'dy' dz'
s = µfpabc SSS [(a—ax')² +(B—by')²+(y—cz')²]* ;
and if we denote by x₁, the values of x' equal and affected
with contrary signs, which are deduced from the preceding
equation, the integral relative to ' should be taken from
x=x₁, to x'x₁; which gives
x'
X19
A = µfpbc (SS
-SS
dy'dz'
[(a—ax₁)²+(ẞ—by')²+(y—cz')²] *
dy'dz'
SS [ ( a + ax₁)² + (b — by ' )² + (y—cz')³]
:).
Each of these two double integrals should be extended to
ATTRACTIONS-FORMULA RELATIVE TO THE ELLIPSOID. 157
all the elements of the spherical semi-surface of which the ra-
dius is unity, and whose centre is at the origin of the coordi-
nates; the product dy'dz' is the projection on the plane of the
axes of y and x, of any element whatever. Therefore, if de-
0
notes the angle which the radius that is drawn to this element,
makes with the axis of x₁, and if 4 is the angle comprised be-
tween the planes of these two right lines and the planes of
the axes of x and y, the area of this element will be sin O də d↓,
its inclination on the plane of the axes of y and z will be the
angle 0, and therefore there will result
dy'dz'= cos 0 sin 0 do dy,
cos Ꮎ
for its projection on this plane. We shall have at the same
time
x₁ = cos 0, y' sin 0 cos,
sin 0 sin.
The limits of the two integrals will be now 0 = 0, and
40,01 and 2π; but if in the second, is
π — ✪
T
substituted instead of 0, it is easy to see that these two in-
tegrals will unite into one, which will have the same limits
with respect to 4, and of which the limits relatively to ✪ will
become 00 and 0π;(n) so that by making, in order to
abridge,
R² — a²+ß² + y² — 2 (aa cos 0+ßb sin 0 cos &+yc sin 0 sin √)
+a² cos20+b² sin2 0 cos² + c² sin2 0 sin²,
and considering it as a positive quantity, we shall have simply
π (2π cos 0 sin 0 d0 d↓
A = μfpbc ST S²
STS
0
R
The two other components B and c may in like manner
be expressed by double integrals.
Let us now consider the attraction of another ellipsoid
having the same density p, the same centre, and its axes in
the same direction as those of the first, let a₁, b₁, c₁, denote
the three semi-axes corresponding to a, b, c, and let o, be the
158 ATTRACTIONS-FORMULÆ RELATIVE TO THE ELLIPSOID.
point which is subjected to this attraction, a₁, ß₁, Y1, its coor-
dinates, and A1, B1, C1, the components of this force parallel to
the three axes of the ellipsoid. If we denote as before the mass
of the attracted point by μ, we shall have
'π С2π cos 0 sin 0 d0 d↓
A₁ = μfor ST S2π
0
R1
C1
R₁ being what R becomes, when a, b, c, a, ß, y, are changed
into a₁, b₁, c1, α1, B1, 71, respectively. The values of B, and C,
may
be obtained in a similar manner from those of в and c.
If the two ellipsoids have the same foci, and consequently the
same excentricity, we shall have
2
2
2
b²=a²+h, c²=a²+k, b²=a₁²+h, c₁²=a₁²+k;
h, k, h—k, being either positive or negative quantities which
express, abstracting from the sign, the squares of the excen-
tricities common to these two bodies. Moreover, let us sup-
pose that the point o₁ attracted by the second ellipsoid exists
on the surface of the first, and the point o attracted by the
first, on the surface of the second. By equation (b) and that
of the surface of the second ellipsoid, we must have
2
a² + B²+212² = 1,
απ +1=1,
²² + B + 2 = 1
B² =1.
b2
(1)
Finally, if p and q be two given angles, and if we assume
a₁=a cosp, ß₁=b sinp cos q, y₁c sin p sin q,
a=α₁ cosp, ẞ=b, sin p cos q,
(2)
y=c, sin p sin q;
these values will satisfy the two preceding equations, and will
establish a particular relation between the coordinates of the
points o and o₁. If these values of a, ß, y, be substituted in
the preceding expression for R², there results by substituting
also the preceding values of b², c², b₁², c₁²,(0)
2
2
ATTRACTIONS-FORMULÆ RELATIVE to the eDLIPSOID. 159
J
2
R² = a₂² + a²+h (sin² p cos² q + sin² 0.cos²)
+ k (sin²p sin² q + sin² 0. sin² ¥)
2 (α₁ a cos p cos 0 + b₁b sin p cos q sin 0 cos
+ c₁c sin p sin q sin 0 sin 4).
2
Now, it appears without writing down the value of R₁²,
that it will be the same as that of R2; for it may be deduced
from it by the transposition of a and a₁, b and b₁, c and c₁,
h and k not undergoing any change, as being quantities com-
mon to the two ellipsoids; hence it is evident, that this last for-
mula is not changed by these transpositions. Since then
R₁ =R, the values of a and A₁, will contain the same double
integrals; therefore, by eliminating it, we obtain
A₁bc=ab₁c₁.
Similar results may be obtained relatively to the other
components, so that from the suppositions which have been
made about the two attracted points o and o₁, we shall have
finally
A1
b₁ c₁
A bc
B1 αici C1
a₁b₁
"
B
ac
C
ab
(3)
In order to enable us to state the theorem which these
three equations imply, let the two points on the surfaces of
the two ellipsoids, of which the coordinates are in the ratio of
the semi-axes, to which they are parallel, be termed corres-
ponding points. The point o, of the surface of the first ellip-
- soid, of which the coordinates parallel to the semi-axes a, b, c,
are a1, B1, 71, will have for its correspondent on the surface
of the second ellipsoid, the point o, of which the coordinates
parallel to the semi-axes a₁, b₁, C1, are a, ß, y, since by equa-
B₂r,
tions (2), we have
αι α
α
α1
ála
b
71
C
b₁
γ
C1
This being established, the following theorem results from
equations (3):
160 ATTRACTIONS-FORMULÆ RELATIVE TO THE ELLIPSOID.
If two ellipsoids have the same centre and foci, the at-
traction in the direction of each axis, which one of these two
bodies exerts on a point situated on the surface of the other, is
to the attraction of this last on the corresponding point of the
surface of the first, as the product of the two other axes of the
first ellipsoid to the product of the two other axes of the second.
108. When two ellipsoids have, as has been assumed, the
same centre and foci, one of the two falls entirely within the
other; consequently, if the point o is external with respect to
the first ellipsoid, the point o₁, will be internal with respect
to the second. In order to determine, by means of the pre-
ceding theorem, the attraction of a given ellipsoid on an ex-
ternal point o likewise given, the surface of a second ellipsoid
having the same centre and foci as the first, is made to pass
through this point; by the formulæ relative to internal points,
A1, B1, C₁, the three components of the attraction of this se-
cond body on the point o, of the surface of the first, corres-
ponding to the point o, can be determined, then equations
(3) will make known A, B, C, the components of the attraction
of the given ellipsoid on the given point. Thus the deter-
mination is reduced to finding the values of three semi-axes
a₁, b₁, c₁, of the second ellipse, from knowing those of the
first, (which are denoted by a, b, c,) and the coordinates a, ß,y,
of the given point o.
For greater clearness, let a be the least of the three quan-
tities a, b, c, then the quantities h and k of the preceding num-
ber will be positive. If we denote the square of a₁ by u, we
shall have
a₁ = √u, b₁ = √ u+h, q=√u+k;
and it only remains to determine this unknown quantity u,
which must be real and positive. Now, in virtue of the second
equation (1), we shall have(p)
´‚¸ã²+
B²u
you
+
u+h
u + k = u;
(4)
ATTRACTIONS FORMULE RELATIVE TO THE ELLIPSOID. 161
this equation which is of the third degree with respect to u
must have at least one real and positive root; for by sup-
posing & to increase from zero to infinity, its first member will
be at the commencement greater, and afterwards less than the
second; so that there must be at least one positive value of u,
which renders them equal; moreover, there is only one, for
if we suppose that there are two, u and u', then we must have
at the same time
α
a²
2
2
+
+
น
u + h
: 1,
u + k
a² 32
x²
+
+
=1;
ú
ú + h
u'+ k
7
and by subtracting these equations, the one from the other,
and suppressing the factor u'-u, which is common to all the
terms, there will result
a² 2
+
B²
༡
= 0;
uu' (u + h) (u' + h) + (u+h) (u+k) =
which is evidently impossible.
Therefore, there exists only one ellipsoid, which has the
same centre and foci as a given ellipsoid, and which besides
passes through a given point; and the quantity u, on which its
three semi-axes a₁, b₁, c₁, depend, is determined by equation
(4), which was proposed to be proved.
109. It may be remarked here, that the theorem of No.
107 is true for all laws of attraction, which are functions of the
distance, for the demonstration given above is founded on the
form that the expression of R2 assumes, which is found to be
identically the same for the two points o and o₁, and not
on the form of the function of R, which expresses the law of
attraction.
R²
If the two ellipsoids become concentric spheres, the at-
traction of each will be same on all the points of the surface of
the other, so that it will not be necessary that o and o₁ should
be correspondent points. Denoting the radii of these two spheres
Y
!
1
162 ATTRACTIONS-FORMULA RELATIVE TO THE Ellipsoid
''
by a and a₁, the attraction of the sphere of which the radius
is a, on a point of the spherical surface of which the radius is
a by D, and the attraction of the sphere of which the radius
is a, on a point of the spherical surface of which the radius is
a by D₁, these forces will act in the direction of the radii of
the attracted points, and we shall have(9)
2
D: D₁ : : α²: a₂²,
whatever function of the distance expresses the law of the
attraction.
It is easy to verify this proposition, in the case in which the
attraction is in the inverse ratio of the square of the distance.
In fact, it appears from the results of No. 101, that if we sup-
pose a >α₁, D the attraction of the sphere of which the radius
is a on an internal point situated at the distance a₁ from its
centre, and having a mass equal to µ, will be
4πμfαι
D=
;
3
and whose
and D, the attraction of the sphere of which the radius is a,
on an external point, of which the mass is also
distance from its centre is a, will have for value
Mg
D₁ =
4πμία
3 a²
3
;
and from a comparison of these values of D and D₁, it appears
that they are in the ratio of the squares of the radii a and a₁·
**
BOOK THE SECOND.
DYNAMICS.
FIRST PART.
CHAPTER I.
OF RECTILINEAR MOTION AND OF THE MEASURE OF FORCES.
I. Formula of rectilinear Motion.
110. THE simplest motion that can be impressed on a
material point, is that in which it moves in a right line, in
such a manner that the point describes equal spaces in equal
times. This rectilinear motion is termed uniform, and it is
made use of in comparing every other description of motion.
When the ratio of the spaces described to the times em-
ployed in describing them, is continually changing, the motion
is said to be variable; when this change takes place after
finite intervals of time, the motion will be a succession of uni-
form motions.
In any motion whatever, the space described by the move-
able, or more generally, its distance from a fixed point assumed
on the line that it describes, is a function of the time which
has lapsed from a given epoch. Thus, denoting this time by
t and this distance by x, we shall have in all cases,
x = Ft;
164
OF RECTILINEAR MOTION.
and the various descriptions of motion will differ from each
other by the form of this function Ft. The variable t may
be either positive or negative; its positive values will refer to
epochs posterior to that from which the time is reckoned, and
its negative values to anterior epochs.
In uniform motion, if a denotes the space described in each
unit of time, and b the distance from the fixed point at the
commencement of the motion, that is to say, the value of x
when t0, we shall have, at any instant whatever,
x = b + at ;
for, by the definition of this motion, the space x-b described
in the time t, must be equal to the constant space a repeated
as often as there are units in t.
III. Neither time nor space can be defined; however, this
is not of any consequence, because, for the purposes of geo-
metry and dynamics, it is sufficient that we are able to measure
the dimensions of bodies and the durations of their motions.
The measurement of lengths may be easily conceived, being
founded on the principle of superimposition; that of time,
however, requires some explanation.
It would be to reason in a circle, to say, on the one hand,
that uniform motion is that in which the spaces described are
proportional to the times; and on the other, that the measure
of time is the uniform motion, that is to say, that the time is
proportional to the spaces described in this motion. But the
notion of equal times and the measure of the time, does not
necessarily imply any particular law of motion, and we can
consequently suppose them in the definition of uniform motion,
and of any other description of motion whatever.
In fact, we may conceive, that bodies perfectly identical
move successively one after the other, and that, during the
entire continuance of the motion, each of the bodies is pre-
cisely in the same state as that which precedes it: it is evident,
that all these motions, of which the law is not given, are per-
OF RECTILINEAR MOTION.
165
formed in equal times, and that their number may be made use
of to measure the time. Thus, for example, if we suppose bodies,
such as the preceding, to be heavy, and to be retained by a fixed
horizontal axis, and if we make them all to deviate equally from
the position of equilibrium, and then remit them to themselves,
in such a manner, that the motion of the second may com-
mence when the first has returned to this position, and that of
the third as soon as the second has returned in the same way,
and so on of the rest; there will be no possible difference what-
ever between all these successive motions, which will be per-
formed in equal times. It will be proved hereafter, that it is not
necessary for this, that different bodies should succeed each
other, and, that the successive oscillations of the same body, on
each side of its position of equilibrium, are also isochronous, or
of equal duration; but the preceding consideration, which does
not suppose the solution of any problem of mechanics, is suf-
ficient for the object which it was adduced to explain.
The invariability of the apparent revolution of the celestial
sphere about the earth, has been established by repeated and
accurate astronomical observations; and in fact, theory does
not indicate any sensible inequality in the rotatory motion of
the earth, which is the cause of this apparent motion. The
constant duration of this revolution is termed a sidereal day,
which duration is less than the diurnal revolution of the sun.
This last is not exactly the same at all epochs of the year;
and it is its mean magnitude which is assumed as the unit of
time in ordinary usages, which on that account is called the
mean day. In this treatise, the division of the day into 24 hours,
of the hour into 60 minutes, and of the minute into 60 seconds,
will be adopted; so that the second will be the 86400th part
of the mean day. The sidereal day contains 86164,09 seconds;
hence it follows, that in order to express in sidereal days a por-
tion of time which is given in mean days, it should be multi-
plied by the ratio of 86400 to 86164,09, or by the constant
number 1,0027379.
!
V
166
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OF RECTILINEAR MOTION.
V'e wilt wetverts
t'e lund
112. One uniform motion differs from another, by the
magnitude of the space described in the unit of time. In each
uniform motion this constant space is what is termed the ve-
locity of the moveable; indeed, strictly speaking, this space is
not the velocity itself, but only the measure of the velocity.
The velocity of a material point in motion is a quality which
inheres in this point, by which it is actuated, and which dis-
tinguishes it from a material point at rest; it is not susceptible
of any other definition. The velocity, which in uniform mo-
tion is expressed by the space described by the moveable in
each unit of time, supposes that we assume for the unit of ve-
locity that of the moveable which describes the linear unit in
the unit of time.
In any variable motion whatever, the velocity of the
moveable varies by infinitely small degrees, and is a function
of the time which may be deduced, as will be shown presently,
from that which expresses the space described; but, it is ne-
cessary to know beforehand, the kind of motion which a mate-
rial point would assume in consequence of its acquired velocity,
if the force which impresses this velocity on it, by its action
continued during a certain time, should cease to act, and the
body be consequently abandoned to itself.
113. It is in the first place evident, that if the moveable
had previously moved in a right line, it will continue to move
along this line; for there is no reason why this material point
should deviate from the direction in which it moved, to one
side in preference to the other. But we cannot affirm, à
priori, that the velocity which has been impressed on it, will
not diminish of itself, and eventually vanish altogether; it is
only by experience and induction that this question can be
decided.
Now, according as the obstacles to the motion of bodies,
such as friction and the resistance of the media which they
traverse, diminish in intensity, they are observed to persevere
more and more in this state; and, as often as an alteration is
FORMULE OF RECTILINEAR MOTION.
167
observed to take place in their velocity, we at once perceive
that this effect can only be ascribed to a foreign cause. We are
therefore led to conclude, that if it was possible for a material
point, after being put in motion, neither to be solicited by any
other force, nor to meet with any obstacle, its motion would
be rectilinear and uniform, that is to say, the simplest of all
motions.
Thus, for example, if a mass of iron is made to move in a
vacuo, on a horizontal plane, without friction, by the sole
action of the pole of a magnet, and if the attractive power of
this pole was suddenly annihilated by placing in juxtaposition
with it an opposite pole of equal power, this mass would still
be directed towards this point; but its motion will become
uniform, and its velocity will be more or less considerable,
according as the attractive force has been permitted to act for
a longer or shorter time.
The impossibility which bodies are in, of exciting motion
in themselves, or of changing the motion which has been
communicated to them, is termed the inertia of matter. This
term does not imply that matter is incapable of acting; for on
the contrary, the cause of the motion of each material point,
is always found in the action of other points, but never in itself.
114. At the end of the time t, when the distance of the
moveable from a fixed point taken on the line that it describes,
is
X, let v be the acquired velocity, that is to say, the velocity
of uniform motion, which would have place if, at this instant,
the force which acts on the moveable should cease to act.
The action of this force continuing, the space do which the
moveable would describe in the instant dt, will be described
in virtue of this action and of the velocity v; vdt will be the
value of the part of dx corresponding to this velocity, which
would be described with a uniform motion. Therefore, de-
noting by ɛ the part of this space which answers to the action
of the force during the instant dt, we shall have
ε
dx = vdt + ε.
168
FORMULE OF RECITILINEAR MOTION.
7
Now, as the velocity varies by insensible degrees, and as
its variations are solely owing to the action of the force applied
to the moveable, it follows that in the time dt this action can
only produce a velocity infinitely small; consequently, this
same action can only cause to be described a space infinitely
small of the second order, and evidently less than that which
would be described uniformly by the body, if at the com-
mencement of dt, it had received all the velocity which would
be produced during this instant.
ε
We may, therefore, neglect & relatively to vdt in the pre-
ceding equation; and then we shall have
ย
dx
dt
for the expression of the velocity in any motion whatever. If
it were required to determine the part of the space described
by the moveable in the time dt, in virtue of the action of the
force which solicits it, the powers of dt superior to the first
should be retained. Now, denoting the distance of the move-
able from the fixed point at the end of the time t + dt by x',
we shall have by Taylor's theorem,
d²x
dt2
dx
x' — x
dt
dt + & de dt + &c.
for the complete expression of the space described in this in-
stant dt. The first term, equal to vdt, is the space due to the
velocity acquired at the end of the time t; therefore, if the
third and higher terms are neglected relatively to the second,
we shall have
d²x
ε = 3 de de²,
dt2
or, what comes to the same thing,
ε=dvdt,
for the part of the space x-x described by the action of the
force. As the velocity which is at the same time produced by
FORMULE OF RECTILINEAR MOTION.
169
!
this action is dv, it is evident that the space which the move-
able would describe uniformly during this time dt, if at the
commencement, it had received all this increase of velocity,
would be equal to the product of dv and dt, or to double of the
space & which it actually describes.
115. When the space described is given in a function of the
time, the corresponding velocity may be immediately deduced
For example; as the
by means of the equation v =
ds
dt.
moveables in Atwood's machine, describe spaces which in-
crease as the squares of the times, i. e., in which s is :: I to t²,
it follows that
ds
dt
·
is to t, hence their acquired velocities
must be proportional to the times during which these spaces
are described; which, indeed, this machine furnishes us with
the means of verifying. (Nos. 400, 401.)
Conversely, if the velocity be given in a function of the
time, by the definition of the motion, we can infer from it, by
integration, the expression of the space described. Thus, after
uniform motion, the simplest is that in which the velocity in-
creases or diminishes by equal quantities in equal times, and
which on that account is said to be uniformly accelerated or
retarded. Therefore, if we denote the constant increment,
whether positive or negative, of the velocity, in each unit of
time by g, and by a the velocity of the moveable when t = 0,
the velocity v at any instant whatever, in this description of
motion, will be,
v=a+gt;
yt qu
by multiplying by dt, and integrating, we shall have
x = b + at +½gt²,
for the distance of the moveable from a fixed point of the line
that it describes, b being this distance at the commencement
of the time t.
When the two constants a and b vanish, we shall have simply
v=gt, xgt2.
Ꮓ
f w
170
OF THE MEASURE OF FORCES.
Therefore, the space described is proportional to the square
of the time; and the velocity acquired at the end of any time
t is such, that in virtue of this sole velocity, the moveable
would describe in a time equal to t, a space vt double of that
which it actually describes. It follows, that if the space de-
scribed in the first unit of time be known, we can obtain by
doubling it, the value of the constant velocity g, by which one
uniformly accelerated motion differs from any other motion
of the same nature.
The motion of heavy bodies which descend in a vacuo is of
this description. In the same place, the velocity g is equal for
all their points; so that, in fact, all of them describe vertical lines
when actuated by a motion of this kind. This velocity varies
from one place to another; by accurate experiments, it has
been proved, that if the second is assumed for the unit of time,
**and the metre for the linear unit,
g=9m,80896
at the Observatory of Paris.
The force which produces equal velocities in equal times,
is considered to be a constant force; thus gravity is a constant
force, which implies here that it acts with equal intensity on
bodies already actuated with any velocities whatever, and not
merely as in No. 59, that its intensity is the same throughout
the entire extent of a body of ordinary dimensions.
116. The laws of equilibrium do not imply any particular
relation between the forces and the corresponding velocities ;
and to resolve the problems of statics, it is sufficient, if the
numerical relations of forces, such as they have been defined
in No. 5, be known. The laws of motion, on the contrary,
depend on the relation which should exist between the magni-
tudes of the velocities produced by given forces; and this re-
lation, the knowledge of which is indispensable for the solu-
tion of the dynamical problems, is the same as that of the
forces, as we proceed now to demonstrate.
Let x and v denote, as before, the space described and the
OF THE MEASURE OF FORCES.
171
!
}
velocity acquired at the end of the time t, and at this epoch,
let two given forces ƒ and ƒ' act simultaneously on the move-
able, in the direction of its motion; let u denote the infinitely
small velocity that the force ƒ would impress on the moveable,
if it acted solely for the infinitely small portion of time T,
and
let u' denote that which would be produced by the force f',
in the same time, if the force ƒ did not exist. I say, that the
circumstance of these forces being impressed simultaneously,
will not modify the velocities of which they are separately
capable, and that the velocity produced by the force ƒ+ƒ will
be u+u', that is to say, at the end of the time t +7, the velo-
city of the moveable will be v + u+u'. In fact, the aug-
mentation of the velocity of the moveable can only depend on
the time r, to which it will be proportional, and on the state
of this material point, or in other words, on its position and
velocity during this same time; therefore, it can only be by
influencing this state that the action of the force f' can modify
the velocity which will be produced by the force f. Now, as
during the timer, the distance of the moveable from a fixed
point and its velocity, can only vary by infinitely small quan-
tities, which may be neglected relatively to x and v; its vari-
ations of distances from other fixed or moveable points, from
which the forces fand f' may emanate, may in like manner
be neglected, consequently, the velocity which the force ƒ will
produce during this interval of time 7, cannot be modified in
any manner by the simultaneous action of the force f", and the
same will be the case, relatively to the velocity produced by the
force f', which will not be changed in any respect by the action
off. Therefore, the entire velocity impressed on the moveable
during the time by the force f+f', will be equal to u + u'.
It is likewise evident, that if the force facts in the di-
rection of the velocity v, and the force f' in the opposite di-
rection, the increase of velocity produced by the force ƒ — ƒ'
will be equal to u u'.
T
Whatever be the nature of each of the forces ƒ and ƒ', if
172
OF THE MEASURE OF FORCES.
they are capable of producing the same velocities in the same
infinitely small portion of time, we may consider them as
equal forces. If they are applied in opposite directions at the
same time, they will produce no change in the velocity of the
body, if it is already in motion; and if this material point is at
rest, it will continue so; this is conformable to the definition
of equal forces given in No. 5.
When the force which acts on the moveable in the direc-
tion of the acquired velocity becomes double, triple, qua-
druple, . . . . the velocity which will be produced in this time
T, will increase in the same proportion. On the other hand,
when this force is reduced to a half, a third, or a fourth, the
velocity produced will be diminished in the same manner; and
generally, the infinitely small velocities produced in equal
times, in the same or opposite directions from the acquired
velocity, or impressed on a material point at rest, will be as
the intensities of the corresponding forces.
It is on this general principle that the measure of forces in
dynamics is founded. It is usually presented as an hypo-
thesis, here we give it as a necessary consequence of the cir-
cumstance, that the velocities impressed by any force what-
ever in infinitely small intervals of time, are always infinitely
small, and of this, that in the same time, the displacements of
the moveables are also infinitely small.
117. If the forces to be compared together are constant
forces, so that each of them produces during the entire time
of the motion, equal velocities in equal times (No. 115), their
intensities will be to each other as the velocities which they
impress, in any equal times whatever, on the same material
point. When, therefore, these velocities are given by obser-
vation, the relation of the forces may be inferred, and con-
versely, where this relation is known à priori, we can assume
it for that of the velocities.
For example, let w, w', denote the intensities of gravity at
two different latitudes, if g, g', the velocities acquired in a
30
md when wit
售
​OF THE MEASURE OF FORCES.
173
{
second, by bodies which descend vertically in a vacuo, at these
places, be known; we shall have
w: w'::g:g'.
1
The relation between these forces w, w', will be also that of
the weights of the same body, or of two homogeneous bodies
having the same volume, at these latitudes. It appears from
observation, that the velocities produced by the action of
gravity increase from the equator to the pole, and that the
entire incrément is very nearly of the least weight, or of
that at the equator. It follows, therefore, that the weight of
the same body, transferred from the equator to the pole, will
increase by do, and that, in order there should be an equili-
brium between two homogeneous bodies placed in these two
places, the volume of the body situated at the equator should
exceed by that of the body situated at the pole.
2
W
1
Also if o denotes the intensity of gravity in the vertical
direction, and w₁ its component in the direction of a line which
makes the angle a with the vertical; then by the rule for the
composition of forces we shall have
w₁ = w cos a;
and if g, 91, denote the velocities which would be produced
in a unit of time by these two constant forces, acting sepa-
rately on the same material point, the proportion
will also give
g: g₁:: w: w1,
ω19
91 g cosa.
If this heavy material point rests on an inclined plane,
which makes with the horizontal plane an angle equal to
90° a, the force w should be decomposed into two others,
the one perpendicular to the given plane, and which will be
destroyed by its resistance, the other directed parallel to this
same plane, which will be the force w₁. It is this last force,
which, abstracting from the effect of the friction of the
174
OF THE MEASURE OF FORCES.
*
}
moveable against the inclined plane, will produce the motion
in a vacuo. This motion being caused by the action of a
constant force, will be uniformly accelerated, and if x, and v,
denote the space described and the velocity acquired at the
end of the time t, we shall have
v₁ = g₁t, x₁ = {19₁t²;
in which equations we should substitute for g, its value given
above.
This example points out the necessity of knowing, à priori,
the ratio of the velocities produced by the action of forces, of
which the ratio is known; for if g, could not be deduced from
9, the velocity given by the observation, and if it was neces-
sary, in order to apply the preceding equations, to determine
also by experiment the value of g₁, which answers to each
value of the angle a, dynamics would be almost entirely re-
duced to an experimental science.
118. In order to measure a variable force, we should con-
sider its effect for an infinitely small portion of time, during
which it may be considered as constant. Therefore, in any
rectilineal motion whatever, let o be the force which acts on
the moveable at the end of the time t, and which is considered
as positive, or negative, according as it acts in the direction of
the acquired velocity, or in the opposite direction. This ve-
locity being v at this same instant, it will be v + dv at the
end of the time t + dt; so that the force & will have impressed
the velocity dv on the moveable in the instant dt. Therefore,
if o denotes a constant known force, capable of impressing a
velocity g in a unit of time, and which can consequently im-
press the velocity gdt in the time dt, we shall have
p: w :: dv: gdt;
hence we deduce
wdv
Ф
gdt
When the linear unit and the unit of time are once arbi-
OF THE MEASURE OF FORCES.
175
1
dv
at the end
trarily selected, the constant g, and the value of
dt
of a given time may be expressed in numbers. This for-
mula will then make known, at the same instant, the nume-
rical ratio of the force to the known force w, and if this last
be that of gravity, this ratio will be that of the force to the
weight of the moveable on which it acts. So that if this ma-
terial point be acted on by gravity, and solicited by the force
4 in a direction opposite to that of gravity, it will be in equi-
librio, if we have
1 dv
g dt
1.
The preceding formula will be simplified if w and g be taken
for the units of their respective species, which will reduce it to
&=
dv
dt'
The unit of force will be then the constant force, which
would impress on the moveable, in the unit of time, a velocity
represented by the linear unit, so that if these two last units
are the second and the metre, the unit of force will, by the
value of g given in No. 115, be very nearly the tenth part of
the weight of the moveable.
It may be remarked here, that this measure
dv
dt
of the va-
riable force & is the velocity which would produce in the unit
of time a constant force, that would retain during this time
the same intensity, as the force o during the instant dt. Thus,
in the motion of a mass of iron towards the pole of a magnet,
which we have already taken as an example, No. 113, the force
$ depends on the distance from the pole, and is consequently
variable; but if we suppose, that at a given instant, the pole
recedes from the moveable, so as that the distance of the one
from the other may become constant, the force will become
so likewise, and the motion will be changed into one uniformly
accelerated, and the increase of velocity in the unit of time
φ
176
OF THE MEASURE OF FORCES.
will be the measure of this force at the instant it becomes
constant.
With respect to the value of & found in No. 114, it is evi-
dent, we can also write
2ε
&=
ቀ dt2°
It follows, therefore, from this and the preceding formula,
that a force may be expressed, either by the velocity which it
produces in an indefinitely small portion of time divided by
this time, or by twice the space which it causes to be de-
scribed, divided by the square of this same time. In motions
uniformly accelerated, these two equivalent expressions for the
force obtain also, when the time is finite, and not as in the
general case, infinitely small.
119. From what has been established in the preceding
numbers, it appears that the general formula of rectilineal
motion are
dx
x = Ft, v=
dv.
dt,
&=
dt
They point out the relations which, in any motion whatever,
exist between the space described, the velocity acquired, and
the force which acts on the moveable, and how these three
functions of the time may be deduced the one from other,
either by differentiation or by integration.
By eliminating v between the two last, we have
d²x
& = di²;
dt2;
which implies that t is taken for the independent variable, and
that its differential dt is constant, an assumption which we
shall make through this entire treatise, without having occa-
sion to repeat it again.
We shall also have by the elimination of dt,
ቀ =
d.v2
dx;
OF THE MEASURE OF forces.
177
which will enable us to determine v when the force o is given
in a function of x, and conversely, this force, when the velo-
city is known in a function of the space described.
In the following chapter, it is proposed to give different
applications of these general formulæ.
II. Measure of Forces having regard to the Masses.
120. Previously to our shewing how the masses should be
taken into account, in the comparison of forces which act on
different moveables, an inaccurate expression, which fre-
quently occurs, should be rectified, because it has a tendency
to produce a confusion of ideas.
1
Let us conceive, that a body is placed on a horizontal
plane, and that it is retained there without any friction. If it
was proposed to make it slide on this plane, it is necessary
nevertheless, on account of the inertia of matter, that some
effort should be made to effect this; and if to this body a
second is attached, then a third, &c., it is necessary, in order
to produce the same motion, that a more considerable force
should be exerted. In each case, a sensation of the effort,
which it is necessary to make, will be produced; but it ought
not to be inferred from this, that matter opposes any resist-
ance to this effort, and that there exists in bodies, what has
been very improperly denominated a force of inertia. When
such an expression is made use of, the sensation that is ex-
perienced, and which results from the effort that is made, is
confounded with the sensation of a resistance that does not
really exist.
When there is a friction of the body against the plane,
there is an actual resistance to the horizontal motion, and the
moveable cannot be displaced on this plane, unless an effort
is made superior to this resistance. In like manner, if it was
proposed to raise the moveable vertically, there is a resistance
to this motion which must be overcome by an effort that sur-
2 A
178
OF THE MEASURE of forces.
•
passes it.
it. In the two cases, no motion is produced as long as
the effort which is made is not greater than the weight, or
than its adhesion to the horizontal plane; but if it be sup-
posed that the body has no weight, or that it does not expe-
rience any resistance from friction, it can be put in motion,
however feeble the effort that is made, or however great the
mass of the body on which it is exerted; and if it is necessary
to make a greater effort, in order to communicate the same
motion to one body than to another, it may be inferred that
the first consists of a greater quantity of matter than the
second and if the magnitudes of these efforts could be accu-
rately compared together, their ratio will be that of the masses
of these two moveables. It is on a consideration similar to
this, that is founded, as we now proceed to show, the measure
of the masses deduced from the magnitudes of the forces,
which cause them to move, and conversely, the measure of the
forces, the masses and velocities being respectively taken into
account.
121. The masses of two material points, belonging to
bodies which may be of different natures, are equal or unequal
according as forces, which are assumed to be equal, impress
on them in the same time, equal or unequal velocities. Let
us suppose for greater clearness, that the forces applied to
these two points are vertical, and that when placed on the two
dishes of a balance, they are in equilibrio. These forces will
be equal on this hypothesis, and this being the case, if the
two points are rendered entirely free, and if the same forces
excite motion in them, their masses will be equal or unequal,
according as the infinitely small velocities with which they are
actuated in the first instant, are equal or unequal.
When, in this manner, the masses of different material
points are ascertained to be equal, other points of which the
masses may have any relation whatever, will be obtained by
uniting these together. Thus, denoting the mass of each
of the equal points by u, and by m, m', the masses of two
OF THE MEASURE OF FORCES.
179
other points made up of n and n' of the first, m and m'
will be to each other as these numbers n and n', and we shall
have
m = nμ, m'n'μ.
Now, let u, v, v', be infinitely small velocities, ¿ and ¿'
integral numbers, and
viu,
v'i'u.
If the two forces fand f' impress on the masses m and m',
the velocities v and v' in the same instant, we shall have
ƒ:ƒ':: mv: m'v'.
In fact, the force ƒ may be considered as the sum of n equal
forces, which would impress the velocity v on each of the n
equal points, of which m is composed, so that denoting one of
these equal forces by k, we shall have
f=nk.
Moreover, let h be the force which would impress the ve-
locity u on each of these equal points, during the same instant,
that the force k impresses on it the velocity v. These forces
acting on the same material point, will be to each other as the
velocities u and v (No. 116), and, because viu, there will
result(a)
We shall have also
kih.
ƒ'=n'k', k' = i'h',
ƒ being considered as the sum of the n' forces k', capable of
impressing the velocity v', on each of the equal points of which
m' is composed, and h' denoting the force which would im-
press on each of these same points the velocity u. Now, as
h, h', are forces which are capable of impressing in the same
instant, the same velocity u on two points of equal mass,
namely, on two points of which the common mass has been
represented by μ, it follows from what precedes, that we must
have h'h. Then by means of the preceding equations, we
shall have
1
180
OF THE MEASURE OF FORCES.
f=inh, f'= i'n'h';
and, if we take into account the values of m, m', v, v', there will
result the proportion which it was proposed to demonstrate.
122. This being established, let us consider a body of any
magnitude and form whatever, all the points of which describe
parallel lines with a common velocity, which may moreover
vary with the time. Let this body be divided into an infinite
number of material points equal in mass, such as they have
been just defined. The motion of all these points may be
ascribed to forces which are equal and parallel throughout the
entire extent of the moveable; their resultant for any part of
this body, is equal to their sum, and applied to the centre of
gravity of this part. The forces corresponding to any two
parts will be therefore to each other as their masses; conse-
quently, if ƒ be the entire force which acts on the moveable,
m its mass, and the force which answers to a part of this
mass, taken for unity, we shall have
f=mp.
With respect to the force p, it will be proportional to the
increase of the velocity of the moveable during an infinitely
small portion of time; and if v denotes this velocity at the
end of the time t, we may assume for its measure, as in No.
118,
Hence there will result
dv
$ = dv
dv
f=mat'
for the expression of the force in any motion whatever, the
mass of the moveable being taken into account, and all its
points being supposed to be actuated by the same velocity.
This force f, which is the resultant or sum of all the in-
finitely small forces, with which all the points of which the
body is composed, are actuated, is termed the motive force,
OF THE MEASURE of forces.
181
the factor of its value mp, is called the accelerating force,
φ
and is no other thing than the motive force referred to the
unit of mass.
The motive force becomes a pressure, when the mass on
which it acts, rests on a fixed plane perpendicular to its di-
rection. A pressure and motive force differ from one another
in the circumstance, that the velocities which a pressure tends
to produce, are continually destroyed by the resistance of the
fixed plane that supports it, whilst those which are actually
produced during each instant by the motive force accumulate
in the body, so that after a finite time there results in it a
finite velocity. Two pressures are to each other as the masses
multiplied by the infinitely small velocities, which they tend
to impress on them in the same instant, and which, in point of
fact, they would impress on them, if these masses were free.
123. If the motion that is common to all the points of a
moveable be a uniformly accelerated one, and if g denotes the
increase of velocity which has place in each unit of time, we
have
=g, f= mg.
Likewise, for another constant force f' acting on a mass m',
and producing a velocity g', in a unit of time, we shall also
have
f'm'g'.
Now, it appears from observation, that two heavy bodies,
whatever difference there may be in the matter of which they
consist, acquire the same velocity in falling in a vacuo during
the same interval of time. Hence, in the case of gravity, we
have g = g', and consequently, the weights fand ƒ' of any two
bodies are to each other as their masses m and m', as has been
assumed in No. 60. The sole fact, confirmed by daily ex-
perience, that heterogeneous bodies have equal weights under
different volumes is not sufficient to decide the question,
whether their masses are equal or unequal; it is necessary to
182
OF THE MEASURE OF FORCES.
know besides, whether gravity impresses the same motion on
them, to be able to infer from the equality of the weights, the
equality of the quantities of matter.
The weight of a heavy body which falls in a vacuo, is its
motive force, and the gravity its accelerating force. For the
sake of conciseness we give the name of gravity or weight to the
velocity g, which, however, is only the measure of this force.
124. If given forces act on the surface, or on other parts
of a solid body, and if these forces impress on all its points,
equal and parallel velocities, they must have a unique result-
ant, which will coincide, in magnitude and direction, with the
motive force, that has been defined above, and the accelerating
force is obtained by dividing it, by the entire mass of the
body.
If, for example, we conceive a heavy body to descend in air,
in water, or in any other fluid, and if it be symmetrical as to its
form and density, about a vertical axis, when it is not homo-
geneous, it is evident, that as every thing corresponds on
each side of this axis, all the points of the body will describe
vertical lines; in which case, since the body is solid, all its
points must, at each instant, be actuated with the same velocity.
The resistance of the medium, which acts on the surface of
the body, will consequently be reduced to a force acting in
the direction of the axis of its figure. Denoting its intensity
at any instant whatever by R, the corresponding part of the
accelerating force of the body by 4, and its mass by m; we
shall then have
&=
R
m
As this force acts, during the descent of the body, in a
direction contrary to that of gravity, the entire accelerating
force will be g-4. If the body is projected perpendicularly
upward, the two forces would act in the same direction, and
the total accelerating force will be negative and equal to
-g-4.
OF THE MEASURE OF FORCES.
183
1
The theory of the resistance of fluids is not sufficiently
advanced to enable us to determine, à priori, the value of R,
which will depend on v the velocity of the body, on its form,
and on the density and nature of the fluid. It is commonly
assumed to be proportional to the square of v, and to the den-
sity of the fluid, so that if this density be denoted by p, we
shall have
>
R = σpv²;
☛ being a coefficient depending on the form and dimensions of
the body, on the nature of the fluid (namely, whether it is
liquid or aeriform,) and on its temperature.
In the case of a sphere, the coefficient σ is assumed to be
proportional to its surface, or to the square of its diameter.
So that if we denote its radius by r, and its density by D, in
which case its mass m will be equal to
4π
3
Dr³, there will re-
sult,
4 =
Уруг
;
Dr
y denoting a numerical coefficient which will be the same for
all spheres, the actual value of which must be determined by
experiment for each particular kind of fluid. As this quantity
is of the same nature as g, it follows that if k denotes a
given velocity, we should have
Dr
k2
γρ
9
in order that the expression of may assume the form
gv²
4
k2
conformably to the principle of the homogeneity of quantities
(No. 23).
125. The same constant force acting successively on dif-
ferent masses, will produce uniformly accelerated motions, in
which the accelerating force, or the constant increment of
184
OF THE MEASURE OF FORCES.
the velocity in each unit of time, will be in the inverse ratio
of the mass.
Thus, for example, ƒ being the weight mg of a mass m,
if this mass be suspended to the extremity of a thread attached
at its other end to another mass m' laid on a horizontal plane,
it is evident, that if the friction and the weight of the vertical
part of the thread are not taken into account, these two masses
will both move with the same uniformly accelerated motion
produced by the motive force f. Therefore, if the accelerating
force of this motion be denoted by g', we shall have
g' =
f
m + m²
or, what comes to the same thing, as ƒ = mg,
g'=g.cos a,
in which a denotes an angle, such that
m = (m+m') cos a.
Consequently, the motion in question will be the same as
that of a heavy body on an inclined plane, which makes the
angle a with the vertical (No. 117).
All bodies being moveable and susceptible of acquiring
velocities which are in the inverse ratio of their masses, when
they are subjected, during the same time, to the action of the
same force, it follows, that no body can be considered as
really fixed; those which are said to be so, are bodies which
have very great masses relatively to those, on which the mo-
tive forces which are applied to them, depend, and which con-
sequently, only receive extremely small velocities from the
action of these forces. At the surface of the earth, bodies
attached to this surface, constitute one mass with that of
the terrestrial globe; and in fact, if m' in the preceding ex-
ample, be assumed to express this mass, it is evident that the
velocity g' which will be impressed on it, in the unit of time,
OF THE MEASURE OF FORces.
185
by a weight mg corresponding to a mass m of ordinary magni-
tude, may be considered as altogether insensible.
126. It is usual to term the product of the mass of a body
by its velocity, its quantity of motion. This expression will,
agreeably to custom, be still made use of, though it would be
more correct to substitute that of the quantity of velocity,
since it is the velocity that inheres in the body, whereas the
motion is only a subsequent effect.
of
No force whatever can produce instantaneously a finite
quantity of motion. The shock of a solid body in motion
against a solid body at rest can impress on this last, in a space
time which, though very short, is not infinitely small, a velocity
which may be sometimes very great; and, during this interval
of time, the two bodies do not suffer any sensible displacement.
Though they may be supposed to be ever so hard, they are
always susceptible of some compression, however little it may
be; thus the velocity is transmitted from the one to the other by
infinitely small degrees; and if the elasticities of the bodies is
not taken into account, their mutual action ceases, when their
velocities become equal.
This rapid transmission of velocity, without any sensible
displacement of the masses, is what is termed a percussion or
impulsion, it is equivalent, as appears, to a motive force acting
for a very short time, with a very great intensity. By con-
sidering the percussion in this manner, as the sum of the in-
finitely small actions of a motive force, it can be shown,
that it is resolvable into two other percussions acting in given
directions, by the rule of the composition of forces, as also
each of these successive actions. If, for example, there be
directed against the back of a wedge, a normal percussion
which we shall denote by P, it is resolvable into two other
percussions perpendicular to its two faces, and if q, q', repre-
sent these two components, K, K', the lengths of the faces to
which they refer, and H that of the back of the wedge, it is
easy to perceive that by the rule adverted to, we shall have
2 B
186
OF THE MEASURE OF FORCES.
3
Q:P::K: H,
Q': P::K': H;
hence we deduce
PK
PK'
Q=
Q' =
H
H
Thus, supposing that this percussion P arises from a mass
m which strikes the back of the wedge with a velocity a, its
two faces, or rather the fixed obstacle against which they are di-
rected, will be in the same circumstances, as if they were struck
perpendicularly by the same mass m, actuated by velocities
Κα
proportional to their lengths, and expressed by and
H
K'α
H
127. If a solid body at rest is struck at the same time, in
opposite directions, by two other bodies, of which the masses
are m and m', and the velocities v and v', then if these three
bodies are symmetrical about the same axis as to their form
and density, and if all the points of the two last move parallel
to this line, their percussions on the intermediate body will
constitute an equilibrium, when the quantities of motion mv
and m'v' are equal, that is to say, these quantities of motion
will in a very short portion of time be transmitted into the in-
termediate body, and will mutually destroy each other's effect,
without this body being at all displaced.
The equilibrium will equally obtain, if the intermediate
body be suppressed, and if the transmission of velocity takes
place at once from one body to the other. Thus, two solid
bodies which move towards each other in opposite directions,
will, if we abstract from a consideration of the elasticity, be
reduced to a state of quiescence, when their masses are in the
inverse ratio of their velocities; and conversely, the products
of the masses and velocities are equal, when there is an equi-
librium in the impact of two solid bodies.
In such a case, the two bodies are supposed to be, as was
stated, symmetrically disposed about the same right line, and
the directions of the velocities of all their points are assumed
OF THE MEASURE OF FORCES.
187
to be parallel to this line, which is the one that passes through
the centres of gravity of the two masses. The condition of
equilibrium in the impact of these bodies, is consequently, the
equality of their quantities of motion, or the equation
mv = m'v';
m and m' being their masses, and v and v' their velocities. In
a subsequent part of this treatise, we will determine the mo-
tions which have place after the impact, when these conditions
relatively to the magnitudes and direction of the velocities, and
with respect to the form of the bodies, are not satisfied, and also
when their elasticity is taken into account.
It results from this law of equilibrium in the impact of
two bodies, that percussion should furnish the most direct
means of measuring the mass of bodies. A known velocity a
being impressed on all the points of a body, the mass of which
is taken for unity, if we could determine exactly the velocity
v with which all the points of another body should be actuated
in order to constitute an equilibrium with the first, when they
impinge on one another, moving in opposite directions, the
numerical value of the mass of the second will then be the
α
ratio; but it is nearly unnecessary to state, that this means
v
is impracticable, and that we should always refer to the weights,
when it is required to measure their masses.
It also follows, that two percussions exercised on a solid
body must be deemed equivalent when they produce equal
quantities of motion; so that in the example of the preceding
number, the back and two faces of the wedge will experience
the same effects, or will be struck with the same energy, if
for the mass m and the velocity a, there be substituted a mass
m' and a velocity a', such that we may have ma = m'a'.
128. When two percussions, arising from velocities which
are in the inverse ratio of the masses, are simultaneously ex-
erted on the two dishes of a balance, there will be an equili
brium between them; in this case the balance supplies the
188
OF THE MEASURE OF FORCES.
place of the intermediate body which was considered in the
preceding number. For example, this will be the case of
two heavy bodies, of which the masses are m and m', and
which impinge at the same instant on these two dishes, having
respectively acquired velocities v and v' which are such that
we have mv = m'v'.
If the mass m is at rest in one of the two dishes, its
weight will exercise a pressure which will be generally over-
come by the percussion of the other mass; but it is not accu-
rate to say, as is generally done, that this will be always the
case, however great the pressure may be, or however small
the percussion. In fact, we can substitute for the percussion
of m' a motive force acting on one of the two dishes, without
sensibly deranging it, during an extremely short space of time,
such as 7. Denoting by m'udt the infinitely small quantity
of velocity, which this variable force is capable of producing
during
quan-
tity of velocity which will be communicated to the balance
during the time 7. During this same time, the weight of m
will produce a quantity of motion represented by mgr, g de-
noting the gravity. In order to an equilibrium in the sys-
udt should be the en-
the instant dt, the product m' Sudt will be the
tem, it is necessary that the integral S
tire velocity v with which the mass m' is actuated at the
instant the percussion commences, so that there may not
remain any degree of velocity when the shock has ceased, and
this being so, it is sufficient that the quantities of motion mgr
and m'
Studt
udt impressed in opposite directions on the balance
while the shock is going on, should be equal to each other.
Hence the condition of this equilibrium will be expressed by
the equation
m'v' = mgr ;
and according as we have m'v'> mgr, or m'v' mgr, the per-
OF THE MEASURE OF FORCES.
189
1
cussion will prevail over the pressure, or the pressure over the
percussion. Now, although the timer may be extremely
small, this last case is possible, the mass m being supposed to
be sufficiently great with regard to m': in order that it should
be impossible, it is necessary that the duration of the per-
cussion be infinitely small, which is not the case in nature.
As in dynamics there is a constant application of the prin-
ciples that have been explained in this chapter, it is of great
consequence to have accurate notions of them, before we pro-
ceed to the resolution of the different problems relative to the
motion of bodies.
"
:
Y
.
CHAPTER II.
EXAMPLES OF RECTILINEAR MOTION.
129. It appears from No. 119, that the equations of recti-
linear motion are
·
T
d
dv
d'x
v= dt' $ = dt'
$
& =
dt2,
(1)
(the last of which may be deduced from the two others;) in
these expressions a denotes the distance of the moveable at the
end of the time t from a fixed point in the line which it de-
scribes, v denotes the velocity which it has acquired, and & the
φ
force that solicits it, which is positive or negative according
as it acts in the same or the contrary direction from that of
the velocity v. These equations are applicable not only to an
isolated material point, but also to a solid body of any magni-
tude whatever, of which all the points describe parallel right
lines, and which consequently will be endowed with a motion
common to all its points; in this case will be the accelerating
force, equal to the motive force divided by the mass of the
moveable.
φ
The value of is supposed to be given in each problem; and
the question will be to deduce from the preceding equations,
the expressions of v and x in functions of t. They will contain
two constant arbitraries, the values of which can be deter-
mined from those of x and v at the commencement of the
motion, and which must be given in each example. In the
problems which follow, the time will be always reckoned from
this commencement, so that the given values of x and v
answer to to.
The integration will not be generally possible in a finite
form, except when 4 depends only, as in the following ex-
EXAMPLES OF RECTILINEAR MOTION.
191
When the given
amples, on one of the quantities t, v, x.
value of contains all the three, or even two, the values of ax
and v can only be expressed by a series.
130. Let the force be first supposed to be constant,
then if it was required to determine the vertical motion of a
body, which descends in a vacuo, in virtue of the force of gra-
vity, we shall have
d²x
9,
dt²
g denoting this force; hence we deduce,
v=gt, x=&gt²,
and, consequently,
v² = 2gx,
x being the distance from the point of departure of the move-
able, and the initial velocity being supposed to be cypher, so
that we have x=0 and v = 0, when t≈ 0.
If a denotes the velocity acquired in descending through
the height h, we shall have
a = √2gh;
which is a convenient expression for the velocity in terms of
the height, through which a heavy body should fall to acquire
this velocity, and of the constant velocity g. The time of
falling through this height being denoted by 0, we shall also
have
α
2h
Ꮎ
0 =
=
g
h=190² =
9
a²
2g
If the body is projected vertically upwards, the equation
of its motion in a vacuo will be
d²x
dt2
g;
g being the same constant velocity as in the preceding case,
since the action of gravity on bodies in motion is supposed to
be independent of the direction in which they move, as well
as of the magnitude of their velocity.
192
EXAMPLES OF RECTILINEAR MOTION.
F
1
If the initial velocity be denoted by a, we shall have
v = a - gt, x = at
19t2,
which express respectively, the velocity and space described
in any instant whatever. It is evident, that the moveable
will continue to ascend until this velocity vanishes. There-
fore, if ' denotes the time during which it moves until v
vanishes, and h' the height to which it will attain, we shall
have
a
•'==/3, x'=
-
a²
2g'
and as these values coincide with those of and h of the pre-
ceding case, it follows that a heavy body projected vertically
upwards with a velocity a, ascends in a vacuo to a height,
from which, if it fell, it would acquire this same velocity, and
the time it takes to attain this height is equal to that of its fall;
h is commonly called the height due to the velocity a, and
conversely, a the velocity due to the height h.
131. Whether the body ascends or descends, it will be suf-
ficient in order to obtain the equations of its motions on an
inclined plane, to substitute in the preceding equations g.cos a
in place of g, a denoting, as in No. 117, the complement of
the inclination of the given plane to a horizontal plane.
Hence, in the case of descent, we shall have
v = gt cos a, 30= &gt² cos a, v² = 2gx cos a;
now if I be the length of the inclined plane, and h its height,
we have
h = l cos a ;
therefore, if k denotes the velocity acquired in falling through
7 the entire length of the plane, we shall have also
k² = 2gl cos a = 2gh;
which shows that this velocity is the same, as if the body had
fallen through the vertical height h. If ABC (fig. 34) be the
EXAMPLES OF RECTILINEAR MOTION.
193
circumference of a circle situated in a vertical plane, and if
AB represents its vertical diameter, we can determine, by means
of the preceding equations, the time which a heavy material
point would take to describe the chord AC, which is terminated
at the superior extremity of this diameter. For if from the
point c the perpendicular CD be let fall on AB, we shall have,
in this case,
AC = 1, AD=h;
and if the required time be denoted by 0, then
1= 190² cos a =
902h
21
;
but by a known property of the circle, we have
12= hb,
b denoting the diameter AB; hence we can obtain
0=
0 = 1//212
gh
26
=
g
But this time is that of the fall through the vertical height
b; consequently it follows, that the chord AC will be described
in the same time as the diameter AB. The same result would
be obtained, if the time of describing the chord CB which is
drawn to the lower extremity of the vertical diameter AB be re-
quired, for this time is also equal to that of describing the vertical
diameter. As then it appears, that the time of describing any
chord drawn to either extremity of the vertical diameter, is
always the same, and independent of the length of the chord,
it will be the case, when the chord becomes indefinitely small;
which arises from this, that then the component of the gra-
vity in the direction of the infinitely small chord is no longer
a finite quantity.
132. Let now the motion of a solid body, which descends
or which is projected upwards, in a resisting medium, and of
which all the points describe right lines, be considered. In
2 c
194
EXAMPLES OF RECTILINEAR MOTION.
order that the accelerating force may depend solely on the
velocity, the density of the medium is supposed to be every
where the same.
In the case of a descent, we shall have
gv2
$ = 9 - 72'
the resistance being supposed to be proportional to the square
of the velocity (No. 124), and k denoting a constant and given
velocity. As this value of is a function of v, we must make
use of the second equation (1), and shall obtain from it(a),
k2
φ
k
dv
dv
+
k v.
:).
gdt = hdv = 132 (+v
k² — v²
k
By integrating and supposing that the initial velocity is
nothing, so that v
0 when t= 0, there results
k + v
gt = j k l og kv
v'
C
and, conversely,
hence we obtain(b)
k
บ
2 gt
=e
k
;
k + v
k
(e
k
e
ย
- gt
gt
k
e" + e
(2)
In these and similar expressions, e denotes the base of the
Naperian system of logarithms, and log a logarithm of this
species; however in formulæ, in which the base of these
logarithms does not occur, the letter e will be employed to
represent other quantities. Its value computed to an accu-
rate approximation is
é = 2,7182818;
and that of the constant modulus by which the Naperian
EXAMPLES OF RECTILINEAR MOTION.
195
logarithm of any number should be multiplied, in order to ob-
tain the vulgar logarithm of this number, is
0,4342945.
Since dx = vdt, we shall have by integrating and sup-
posing that x = 0 when t = 0(c),
We have also
k2
X =
g
log (e* + e
&
k
**).
(3)
k2vdv
and, consequently(d),
gdx = k2 v2,
k2
k2
x =
2g
log 2 v²
(4)
is the value of x, considered as a function of v.
133. The preceding formulæ enable us to solve the pro-
blem completely. It appears from a consideration of them,
that the time being supposed to increase continually, the mo-
tion approaches more and more towards uniformity, and that
it becomes sensibly uniform when gt, the velocity arising from
the action of gravity, is very great relatively to k. In fact,
K
"
if e which in this case is a very small fraction, be neglected,
we have(e)
k2
v=k, p ⋅
=0, x=ht
kt log 2.
g
As the resistance of the fluid is a force which acts on the
surface of the body, the motive force that results from it is
independent of the mass, and will be the same, whether the
body consists of a very dense matter, or whether the interior
matter be taken away altogether, and nothing left but a very
slender envelope. Now, as the accelerating force is equal to
the motive force divided by the mass of the body, it follows
that the first of these two forces will, every thing else being
196
EXAMPLES OF RECTILINEAR MOTION.
I
my -R--my.
the same, be in the inverse ratio of this mass, and conse-
quently(ƒ) k will be in the direct ratio of its square root.
This is the reason why the final motion in a resisting medium
is more rapid for those heavy bodies, whose density is greater,
the form and extent of surface being supposed to remain the
same. When the density of the medium is inconsiderable
relatively to that of the body, the quantity k is very great,
and in this case the motion will not approach to one of uni-
formity until after the lapse of a considerable length of time.
As long as the velocity gt does not become very considerable,
we obtain in converging series
¿ (e
k
gt
g³t³
e
+
k
+ &c.,
1.2.3k³
te
=1+ +
9212
gots
2k2 1.2.3.4k¹
+ &c.
log & (e
and formulæ (2) and (3) become
k
k
g2t2
+e
9414
2k2 1.2.3.2k¹
+ &c.(g)
v=gt
g³t3
1.3k2 + &c.,
x = 1 gt2
93.14
1.2.3.2k2+ &c.
They are reduced, as ought to be the case, to those which
a motion uniformly accelerated would give, when the density
of the medium entirely vanishes, in which case the quantity
k becomes infinite.
134. In the case in which the body is projected perpen-
dicularly upward, we have
gv2
k2.
If its superior surface is the same as its inferior surface, the
constant k will be the same as in the case of descent; but if
11
EXAMPLES OF RECTILINEAR MOTION.
197
these two portions of its surface are different, the values of k
will be so likewise; for example, if the projectile was a cone
of which the base was horizontal, the quantity k would in its
ascenscional motion be greater or less than in the descent,
according as its summit was situated above or beneath its base.
If for greater clearness the body be supposed to be a homo-
geneous sphere, of which the radius is r, D its density, p that
of the medium, then we shall have
k² =
Dr
YP
;
y being a constant arbitrary depending on the nature of the
medium, (namely, whether it be liquid or aeriform,) and on
its temperature. If the preceding value of o be substituted in
φ
the second equation (1), there will result
kdv
k² + v²
gdt
=
;
k
which, by integrating, and denoting the initial velocity of the
body by a, gives(h)
gt
arc tang= 21) = are (tang = 2) — 9t.
arc (ta
The value of which results from this expression may be
written under the form
v=
k (a cost — k sin &
a.sin+k.cost
gt
k
By multiplying this expression by dt, and integrating a
second time, so that when t= 0, x may be also equal to
cypher, there results (?)
x =
gt
= log (sin 2+ cos).
9
α gt
k
We shall also have
gdx=
gdø
k2vdv
k² + v² ›
1
198
EXAMPLES OF RECTILINEAR MOTION.
and, consequently(k),
x =
2g
k³
log
k² + a²
k² + v²°
1
If we make
a,
and then suppose a = 0, in order that
k
these formulæ may be applicable to the case of a body moving
0
in a vacuo, they will assume the form and by the known
0'
rule for determining the value in these cases, we find, as we
ought,
v=a gt, x = at
gt2;
a result which we would arrive at by expanding these formulæ
into a series as in the preceding number(1).
135. If h denotes the greatest height to which the body
can attain, and which corresponds to v = 0, we shall have
k2
h =
2g
log
k²+ a²
k2
Likewise if 0, denotes the time it takes to attain this height,
its value will be (m)
k
0₁ =
arctang=
%).
g
After having attained this height, the body will fall back,
and its motion will be expressed by the formulæ of No. 132.
If a' denotes its velocity, when it will have fallen back the
entire height h, we shall have, by means of equation (4),
k2
k2
h= log p² - a/2
2g
and by making this value of h equal to the preceding, there
results
k2
k2
a/2
k² + a²
k2
and, consequently,
+
a²k2
a/2 —
a² + k² ³
EXAMPLES OF RECTILINEAR MOTION.
199
hence it follows, that a' is less than a, so that the velocity of
the body, when it returns to the point of departure, is less than
its initial velocity.
Also, if 0' be the time of the entire descent, in which case
va', we shall have
k
0' =
2g
k+a'
log ka''
which, by substituting for a' its value, becomes
0' =
k
2g
√ a² + k² + a
log
√ a² + k²
a
;
it is evidently different from the value of 0, the time of ascent.
We obtain a simpler expression, by multiplying both the nu-
merator and denominator of the fraction comprised under the
logarithm, by Va² + k² — a(n), by which means we get,
k
0'--log
k
g √ a² + k²
α
and if ✪ denotes the entire time ′+ 0₁, which the body takes
in ascending and descending, we shall have
g0
k
90 = are (tang = 2) + log √²+k-a
k
α
If the body be a bullet shot into the air by a cannon di-
rected vertically upward, we can, notwithstanding the rapidity
of this motion, determine the time with some precision, and
if we knew likewise a the velocity of projection, the preceding
equation would enable us to determine the value of k with respect
to r, the radius of the bullet. Also, from a consideration of
the expression of k² given in the preceding number, it appears,
that if k' denotes what k becomes, with respect to another
bullet consisting of the same kind of matter, and of which the
radius is equal to r', we shall have(o)
зов
k' = k √// ?!.
r
is
200
EXAMPLES OF RECTILINEAR MOTION.
136. If the effect of the force of gravity is not taken into
account, the solution of the problem presents a remarkable
singularity, when the resistance of the medium is supposed to
be proportional to a power of the velocity which is less than
unity.
Suppose, for example, that
φ
29 √
V
k'
g and k representing, as before, the force of gravity and a given
constant velocity. The equation of motion will be
dv
dt
=-
29 √
v
from which, if the value of gdt be deduced, we obtain, by in-
tegrating and denoting the initial velocity by a,
and, consequently,
gt = √k (√ a — √õ),
gt
2
v = (√a-st).
Multiplying the members of this equation by dt, and in-
tegrating a second time, so that we may have x = 0 when
t=0, we obtain for the value of the space passed over at a
given instant(p),
== "vak
3g
-
+ 3gk(9t — Vak)".
It appears from the value of v, that the velocity dimi-
nishes from the commencement of the motion until the instant
Vak
g
in which t= ; at this instant, the velocity vanishes; im-
mediately after, the motion continues in the same direction
as before, and the velocity increases indefinitely. But as the
velocity vanishes at a certain instant, the accelerating force
vanishes at the same time; consequently the body should stop
at this instant and remain at rest. Now, it ought to be re-
EXAMPLES OF RECTILINEAR MOTION.
201
marked, that the equation of the motion admits a particular
solution v = 0; so that its complete solution is the sum of its
integral and of this equation v = 0; it follows, therefore, that
·
Vak
the problem is resolved from t = 0 to t = by the inte-
g
gral of the equation of motion, and beyond this value of t, by
the particular solution. During the first interval of time, the
body describes, with a motion continually retarded, a line
equal to
avak
3g
mains at rest.
>
at the extremity of which it stops and re-
This example, which is purely hypothetical, suffices to show
how necessary it is to take into account particular solutions of
the differential equations of motion, if there are such; it ap-
pears, however, from the expressions for the forces in func-
tions of the acquired velocity and space passed over, which
have place in nature, that no such case actually occurs.
137. We now propose to give some examples of motions,
in which the accelerating force varies with the space passed
over.
The simplest case is, that of a material point attracted
towards a fixed centre in the direct ratio of the distance from
this point, which is supposed to exist on the right line that
the moveable describes. Let z be this distance at the end of
the time t, and let the accelerating force at a given distance a
be supposed equal to the gravity g, by the given law, we shall
have
gz
Ф
α
for its value at any instant whatever.
If x is the space
described at this same instant, and if c be the distance of the
point from the centre of attraction at the commencement of
the motion, which is supposed to be directed towards this
centre, we shall have
2 D
202
EXAMPLES OF RECTILINEAR MOTION.
dz
x=c-Z, v=
dt'
and the third equation (1) will become
ď²z
gz
dt2
α
Its complete integral is
z= A cos t
√2 √
α
+ B sin t
A and B denoting two constant arbitraries.
α
If the initial velocity of the moveable is supposed to be
nothing, we shall have at the same time
dz
t = 0, z=c,
<= 0;
dt
hence we deduce
A = C, B = 0,
and, consequently,
z = c cos t
α
From this formula it appears, that the distance z is nothing,
or that the point will reach the centre of attraction at the end
of a time which is independent of c the distance(q) of its point
of departure, and equal to ¿π √ª; after that, it will per-
g
form oscillations on each side of this centre, of which the con-
stant amplitude and duration will be respectively equal to the
distance c and the time
T
138. For another example, let the motion of a heavy body
in a vacuo be considered, the height from which the body falls
being sufficiently great to require, that during its fall, the va-
riation of gravity should be taken into account.
EXAMPLES OF RECTILINEAR MOTION.
203
Let BAE (fig. 35) be a vertical great circle of the earth, D
the point from which the body departs in this plane, м its
position at the end of the time t, on the right line DC, which
is drawn from D to c the centre of the earth, and meets its
surface in A. Let CA the radius, be represented by r, the
height AD by h, DM the space passed over by the body by x,
and cм, its distance from the centre c, by z, so that we may
have
CM,
z=r+h
Xx.
The accelerating force will be the gravity at the point
M; if g denotes this gravity at the surface of the earth, that
is to say, at the point a; and if its intensity be supposed to
vary in the inverse ratio of the square of the distance from the
centre c, we shall have
hence we obtain
4:9::r²:22;
gr2
22
by means of which the third equation (1) will become
d²x
dt2 =
gr2
(r+ h − x)² °
If its two members be multiplied by 2dx, and then inte-
grated, we obtain
dx²
dt2
= 2gr2
1
1
= 2 gr² (= h ====== 1) :
+h-x
dx
the constant arbitrary being determined by making : 0,
dt
when t = 0; by means of this expression, the velocity ac-
quired by the body, at any distance such as x from the point
of departure, can be determined. At the point a, where x = h,
this velocity will be(r)
2gh V.
↑
r+h'
A,
204
EXAMPLES OF RECTILINEAR MOTION.
5
and consequently less, as it ought to be, than if the intensity
of gravity was the same, through the entire height h, as at
the surface. From the preceding equation we obtain(s)
√2 gr2
r+h
(r + h − x) dx
dt=
√(r+h) x − x²
Now, from a comparison of this differential equation with
the equation (a) of No. 73, it appears that if a semi-cycloid
DOC is constructed, which may have its summit at D, its origin
at the point o, situated on the line oc which is perpendicular
to the right line CD, and the diameter of the generating circle
equal to CD, then if through the point м, MN is drawn perpen-
dicular to the line DC, and meeting the cycloid in N, we shall
have
MN = t√ 2gr2
r + h
so that MN, the ordinate of the point N, makes known t, the
time in which the abscissa DM is described, and conversely. By
integrating and observing that x=0, when t = 0, we shall
have, in a finite form(t)
t√2gr2
r+h
√(r+h)x−x²+ {(r+h) arccos =
(co
r+h−2x
r+h
When the height h, and consequently the distance x, are
very small with respect to r, this formula differs very little
from that which is furnished when the gravity is supposed to
be constant. In fact, since
arc
COS =
r+h-2x
r + h
sin=
- arc
2√(r+h) x − x²
r+h
;
when the sine is very small, we may substitute it in place of
the arc, which renders the second term of the second member
of the preceding equation equal to the first. We
We may also
substitute the radius r in place of r+hx, and, conse-
EXAMPLES OF RECTILINEAR MOTION.
205
quently, reduce their sum to 2rx; by this means, the for-
mula in question will become
t
√2gr²
2√rx,
r+h
or simply, by neglecting h relatively to r,
x = gt².
As an example for the calculus, we shall suggest, before
we quit this subject, but without entering into any details,
the case in which the moveable is projected perpendicularly
upwards; and we now proceed to the consideration of the last
case of rectilineal motion that we propose to give, namely, that
of a material point attracted to two fixed centres situated on
the line which it describes.
139. Let A and B (fig. 36) be the two centres of attraction,
M the position of the body at the end of the time t, and D its
point of departure. Suppose, for greater clearness, that the
motion takes place between the two centres of attraction and
from a towards в; let
DMX, AM = 2, AD α, BMC 2 ;
the distance of the body
so that x is the space described,
from the point a, a the initial distance, and c the length of
the line AB. If the attractions be supposed to vary in the in-
verse ratio of the squares of the distances, and if the intensities
of the forces which emanate from the centres A and B, are
denoted at the unit of distance, by a² and b², we shall have
b2
for their respective intensities, when the body
(c — z)2,
is at M. The accelerating force will be the excess of the
second force which tends to augment the space x, over the
first which tends to diminish it; therefore, because dz dx,
the third equation (1) becomes
a²
22
and
d2z
b2
=
dt2
(cz)2
/ 。
229
(a)
206
EXAMPLES OF RECTILINEAR MOTION.
dz
and will express the velocity of the body at the point
M.
dt
By multiplying equation (a) by 2dz and integrating, we
obtain,
dz2 262 2a2
+
dt2
C- 2
2
ولا
(b)
y being a constant arbitrary. In order to determine it, let k
be the initial velocity which answers to z = a; then
Y=
262 2a2
+
k²;
α
C- α
and if this equation be taken from the preceding, there will
result
dz2
1
1
d² = x² + 28² ( _—_—-— - — _—_—_ _) — 2 a²° ( − 1); (©)
dt2
C
Z
с
this equation will determine the velocity of the body, in any
position whatever between the points a and B.
140. There exists on the right line AB, a certain point c,
in which the two forces of attraction are equal; so that if a
body be placed there, or arrives at it without any acquired ve-
locity, it will remain in equilibrio. Denoting the distance AC
by h, we have
a²
b2
(c — h)² — h²•
From this equation two values of h may be deduced, of
which one belongs to the point c situated between a and b,
and the other to a point in the production of AB, in the di-
rection of the centre of the least attraction. The first of these
two values is
ac
h=a+b⋅
Let ƒ denote the least initial velocity which must be im-
pressed on the body in order that it may reach the point c, so
EXAMPLES OF RECTILINEAR MOTION.
207
that when it attains to this point, its velocity may vanish; we
shall have at the same time
dz
k=f, z=h,
=0
0 ;
dt
and by virtue of equation (c) and of the value of h, there will
result(u)
262
2a2 2 (a + b)²
fr
+
С — α
α
C
(d)
If the initial velocity be less than f, the body will fall back
on a; if it is greater, it will pass beyond the point c, and will
fall on B. In the case of k=f, the body will take an infinite
time to reach the point c, because that at an infinitely small dis-
tance from this point, it will only be actuated by an infinitely
small velocity, and solicited by a force which is equally so.
141. If A and B are the centres of two spheres, which are
either homogeneous, or composed of concentrical strata, we
may suppose that the attractions which have been considered
are those of these two spheres; and then a² and b², the intensities
at the unit of distance, will be to each other as their masses
(No. 101). Supposing, for example, that A is the centre of
the moon, and в that of the earth, if the non-sphericity of the
earth is not taken into account, we shall have
a²
b2
75;
for the mass of the moon deduced from its action in raising
the waters of the sea, is of that of the earth.
73
Hence we
shall have(v)
C
h=
=(0,10352)c;
1+√75
so that the distance of the point which is equally attracted by
the earth and its satellite the moon, from the moon, is very
nearly the tenth part of their mutual distance.
Let r be the radius of the earth, then c, the distance of
208
EXAMPLES OF RECTILINEAR MOTION.
the moon from the earth, is about 60r, and if the body begins
to move from the surface of the moon, we shall have at the
same time, by the known ratio of the radius of the moon to
3r
that of the earth, a=II By means of these values of r and
a, and of a =
a=
b
√75
equation (d) becomes
262
ƒ² = (0,044894)
2
and since if g denotes the attraction of the earth at its surface,
then
b²=gr², km das trctiidua
is the expression for this force at the unit of distance; hence
if we make
there will result
(0,044894) r≈r',
f²= 2gr'.
Now, the attraction g may be assumed equal to the weight
of which it constitutes a principal part; consequently, ƒ is the
velocity acquired in falling through the height r', and since
g=9″,80896, πr=20000000",
its value is
f=2368m
As the atmosphere of the moon is not such, that its re-
sistance can diminish the velocity of bodies projected from its
surface, it follows, that if the earth and moon were at rest, a
body projected from the surface of the moon, towards the
earth, with a velocity greater than 2368 metres in a second,
will pass beyond the point of equal attraction, and at length
fall on the surface of the earth. In the motion of the moon
about the earth, the right line AB drawn from one centre
to the other always meets the surface of the moon(u) in the
same point, which must be the point D from which the
body would be projected in the direction DB; but, during a
.་་་
EXAMPLES OF RECTILINEAR MOTION.
209
T
second, the point D traverses on the circle described about the
centre of the earth, a length of about 1000m, consequently, the
absolute velocity of the body will be, in magnitude and direc-
tion, the resultant of a velocity in the direction of DB, and of
a velocity of 1000m in a second, perpendicular to DB. This
being so, the body will not remain on the moveable line AB,
but will describe a curve(x) in space; so that the preceding
formulæ are not applicable to its motion, neither will it fall on
the surface of the earth, as it would do, if the moon was im-
moveable.
142. If the equation (b) be resolved with respect to dt, we
obtain
dt=
z2.dz
V cz - z². dz
2a²c-(2a2-2b² + cy) z + yz²
z+yz²
The integral of this formula may be always expressed by
means of elliptic functions, so that when tables of these func-
tions are constructed, the time which corresponds to a given
distance such as z, may be computed, and reciprocally. But,
independently of the cases in which one of the two attractions
is supposed to vanish, there are also others in which the in-
tegral of the preceding formula may be obtained in a finite
form. These cases obtain when the quantity comprised under
the radical is a perfect square; this requires that we should
have
(2a² - 26²+cy)²= 8 a²cy;
from which equation we can obtain
2
y = = (a+b)².
If this value be put equal to that of y of No. 139, there
results(y)
k²=
262 2a2
+
2(a+b)²
α
C
C α
One of these two values of k² is that of f², the other is
2 E
210
EXAMPLES OF RECTILINEAR MOTION.
evidently much greater. It follows, therefore, if even neither
of the two quantities a or b is equal to cypher, that the time
can be expressed in a finite form as a function of %, when the
body is actuated by the least velocity ƒ with which it can
reach the point c, and also when a certain velocity greater
than this is impressed upon it(z).
By substituting the double value of y in the expression for
dt, there results
✓
dt =
√ cz — z² dz
C ac — (a± b) z³
which formula may be rendered rational, and integrated with-
out difficulty, by the ordinary rules. The differential of dt
must be always positive; the differential of dz is positive while
the body advances from D to B, and negative when it returns
towards A. In the first case, therefore, the sign of the radical
22 must be the same as that of the denominator
(a+b)z, and, in the second case, it must be affected
with a contrary sign.
√ cz
ac
143. Suppose that b= 0, or c= ∞, the body will then only
be subjected to the attraction of the centre a. The equation
(c) will be reduced to
dz2
=
-
dt2
k² — 2 α ² ( —
— — — ) ;
;
a
(e)
the value of dt which can be obtained from this, may be inte-
grated in a finite form, and will make known t in a function
of z.
If we make
dz
dt
= 0, we shall have
•
2a2
2a2
k² -
α
2
by means of which, the distance z at which the body will be
arrested, can be obtained. In the case of 2a2 k'a this dis-
tance will be infinite, which denotes that the body will never
be arrested in its motion. This is also the case, when 2a24k²a,
EXAMPLES OF RECTILINEAR MOTION.
211
聽
​from which there results for z a negative value that cannot
belong to any point of the indefinite line DB, in the direction
of which the body has been projected. In these two cases,
the motion of the body approaches more and more to uni-
formity, according as the distance of the body from a increases.
When the distance z becomes very great, and the motion sen-
sibly uniform, its velocity, as determined by equation (e), will
be very nearly equal to
√ 12
2a2
k2
,
α
or to √k² — 2ga, on the
supposition that a²= ga², that is to say, on the supposition
that the body is projected from the surface of a sphere, of
which the radius is equal to a, and of which the attraction is
equal to g. This shows that the diminution of the initial ve-
locity k will be so much the greater, as this force and this
radius are more considerable.
}
+
CHAPTER III.
OF CURVILINEAR MOTION.
I. General Formula of this Motion.
144. In curvilinear motion, the curve described by the
moveable is termed the trajectory of this material point. At
the end of any time t, let м (fig. 37) be the position of the
moveable. If s denotes the arc CM of the trajectory com-
prised between the moveable and a fixed point c arbitrarily
taken on this same curve, s will be a function of t; so that,
in any curvilinear motion whatever, we shall have
s = Ft.
If at the same instant, x,y,z denote the three rectangular
coordinates of the moveable, these variables will be also func-
tions of t, and we shall in like manner have
x=ft, y=ft, z=f"t.
When these three equations are known, we can deduce from
them, by the elimination of t, the two equations in x, y, z, of
the trajectory. By means of the equations of this curve, s can
be determined in a function of one of the three coordinates,
and, consequently, in a function of t; in this way the law
of the motion on the trajectory will be obtained. Each of the
three preceding equations is that of the rectilineal motion of
the projection of the moveable on one of the axes of the coor-
dinates; it follows, therefore, that the complete determination
of the curvilinear motion of a material point, is reducible to
that of three rectilinear motions, which will be the motions
GENERAL FORMULE OF CURVILINEAR MOTION.
213
!
of its projections on ox, oy, ox, the three axes of the coordi-
nates. When these three motions are uniform, that of the
moveable will be also uniform and rectilinear, and con-
versely(a).
145. During the instant dt, the moveable will describe, ds
the element of its trajectory: if in this infinitely small interval
of time, the action of the forces which solicit it be neglected,
its motion during this interval may be considered as uniform
and rectilinear. Therefore, denoting the velocity acquired at
the end of the time t by v, we shall have
V =
ds
dtⓇ
If these forces cease to act at the instant in question, the
moveable will continue to move with this velocity, and along
MT the production of the element ds, that is to say, along the
tangent to the trajectory, since, in consequence of the inertia
of matter, it cannot then change, either the direction of its
motion, or the magnitude of its velocity (No. 113). Hence,
a material point which describes any curve line whatever, may
be considered as being actuated at each instant, by a velocity
in the direction of a tangent to this curve, and expressed by
the ratio of its differential element to the element of the time.
If, at the end of the same time t, p, q,r denote the velo-
cities of the projections of the moveable on the three axes
of x, y, z, we shall likewise have, in these three rectilinear
motions,
dz
dx
p=
dy
dt
q=
dt
r =
dt
Now if a, ß, y be the angles which the tangent to the tra-
jectory, or the direction of the velocity v, makes with the pa-
rallels to the axes of x, y, z, we have (No. 17)
dx
dy
dz
cos a
ds'
cos ẞ=
ds'
cos y =
ds
214
GENERAL FORMULE OF CURVILINEAR MOTION.
hence we obtain(b)
p=v cosa,
q=v cos ß, rv cosy,
(1)
and also at the same time,
p² + q² + r² = v².
As the time t goes on continually increasing, its differen-
tial is always positive. The velocities p, q, r are positive or
negative, according as the coordinates x, y, z increase or de-
crease. In equations (1), the velocity v may be regarded as
a positive quantity. The direction of this velocity, or the
part мT of the tangent to the trajectory, along which it will
be directed, will then be determined by the signs of p, q, r,
which will indicate whether the angles a, ß, y are acute or
ds
obtuse. In the equation = the velocity v is considered
v dt'
to be positive or negative, according as the arc s increases or
decreases.
The components of v the velocity of a material point, are
p, q, r, the velocities of its three projections on the rectangu-
lar axes; and each of these three components is what is meant
by the velocity of the moveable parallel to the axis to which
it refers. From a comparison of equations (1) with those of
No. 31, it appears that this composition of velocities is per-
formed according to the same rules as that of forces. Hence
it follows, that if through the point м any line ma be drawn,
which makes with the parallels to the axes of x, y, z, drawn
through this same point, the angles a, b, c, which may be
either acute or obtuse, the general expression for the com-
ponent of the velocity v parallel to this line MA, will be
p cos a + q cos b + r cos c.
The quantity of motion (No. 126) of an isolated material
point, and that of a body all whose points are actuated by
equal and parallel velocities, can be decomposed into other
quantities of this nature, and these may be reduced to one,
GENERAL FORMULE OF CURVILINEAR MOTION. 215
*
according to the same rules as the velocities which they have
for factors.
146. Let p + p', q + q', r + r', be what the three com-
ponents of the velocity of the moveable parallel to the axes of
x, y, z, become at the end of the time t + dt; so that p',q',r'
may represent the infinitely small increments of the velocity,
which have place in these directions, during the instant dt.
The increment of velocity in the direction of the line MA
will be
p' cos a +q'cos b + r' cos c.
Now, whatever be the quantities p', q', r', if we make
12
u² = p²² + q²² + jo/2,
u being considered as a positive quantity, we can always find
three angles a', B', y', either acute or obtuse, such that
p' = u cos a', q' = u cos ß', r' = u cos y' ;
by means of which, the increment of velocity in the direction
MA will become
u (cos a cos a' + cos b cos ẞ' + cos c cos y´).
Moreover, we know that the quantity between the brackets
is the cosine of a certain angle σ. Therefore, the increment
in question is equal to u cos o ; consequently, u is its greatest
value, and it answers to the direction of the right line MA, for
which the angles a, b, c are the same as a', B', y', and thus
renders the coefficient of u equal to unity. In any other di-
rection whatever, the increment of the velocity will be equal
to the greatest value u, multiplied by the cosine of the angle
σ, which this direction makes with that of the greatest value;
hence it follows, that it will vanish with respect to all directions
perpendicular to those of its greatest value.
Whatever be the variation of the velocity of the moveable
in magnitude and direction, during the instant dt, there is
always a certain direction, for which the increase of velocity
216 GENERAL FORMULE OF CURVILINEAR MOTION.
is a maximum, and to which this property belongs, namely,
that for all directions perpendicular to this one, the velocity
is neither increased nor diminished.
147. The direction of a force which acts on a material
point in motion, is the right line along which it increases or
diminishes the acquired velocity, and perpendicularly to which
it does not produce any change. Thus, when we say that
the weight of a body which moves in any direction whatever
is vertical, like that of a body at rest, it is meant by this, that
this force increases the vertical velocity, and does not pro-
duce any change whatever in the horizontal velocity.
This being agreed on, let u, u', u", &c., denote the inten-
sities of the different forces which at the end of the time t
act on the material point, the curvilinear motion of which is
considered; a, b, c, a', b', c', a", b", c", &c., the angles which
their given directions make with parallels to the axes of
x, y, z; and x, y, z, the sums of their components in the di-
rections of these axes; we shall have (No. 32)
x= u cos a + u' cos a' + v" cos a" + &c.,
Y = u cos b + u' cos b'+u" cos b" + &c.,
z = u cos c + u' cos c' + u" cos c" +&c.
Let now u, u', u", &c., be the infinitely small velocities
which these forces u, u', u' would produce during the instant
dt, in their respective directions, on the supposition that each
of them acted by itself on the moveable actuated by the velo-
city v. It is evident, as in No. 116, that the circumstance
of the forces acting simultaneously will, in no respect, in-
fluence either the magnitudes or directions of the velocities,
which are actually produced; consequently, if we still denote by
p', q', r', the infinitely small quantities by which p, q, r, the
velocities of the projections of the point, are increased in the
instant dt, these quantities will be the sums of the components
of u, u', u', &c., in the directions of these three axes; so that
we shall have
GENERAL FORMULE OF CURVILINEAR MOTION. 217
p' = u cos a + u'cos a' + u" cos a" + &c.,
q' = u cos b + u' cos b' + u" cos b' + &c.,
r' = u cos c + u' cos c' + u" cos c" + &c.
Now, since it appears from No. 118, that the measure of
any force is the velocity which it is capable of producing, it is
evident that
u = vdt, u' = v'dt, u"v"dt, &c.;
hence, if the values of p', q', r', be compared to those of x, y, z,
there results
p' = xdt, q' = ydt, r' zdt;
which shews that the increment of the component of the ve-
locity in the direction of each axis, in the instant dt, is the
velocity produced during this instant, by the entire compo-
nent, in the direction of this same axis, of the given forces,
which act on this material point.
It is because the forces are proportional to the velocities
which they impress on the moveable in an infinitely small
portion of time, (which infinitely small velocities are the same,
whether these forces act separately, or simultaneously,) that
this result obtains. It likewise follows, that if three forces
not comprised in the same plane are applied to the moveable,
and that if, on the directions of these three forces u, u', v",
there be taken, reckoning from their point of application,
right lines of a finite magnitude, which are to each other as
the corresponding velocities u, u', u'; the resultant of these
forces will be represented in magnitude and direction by the
diagonal of a parallelopiped of which these three lines are the
adjacent sides, and its magnitude will be to that of each of
these forces as the diagonal is to the corresponding side.
148. If the forces which act on the moveable are inde-
pendent of its velocity and of its position in space, the motions
of its three projections on the axes of the coordinates will be
independent of each other; so that its projection on each axis.
2 F
218 GENERAL FORMULÆ OF CURVILINEAR MOTION.
&
at the end of any time whatever will be found at the same
point, and it will have the same velocity as if the forces and
the velocities parallel to the other two axes were cypher. In
general, this will not be the case when the given forces vary
in magnitude or direction, either with the position of the
moveable, or with its acquired velocity; its velocity and po-
sition can, however, be always determined at each instant, in
the following manner. Since all the forces which act on the
moveable are reducible always to one, let u, which is ca-
pable of producing the velocity u, be this unique force, and
let & be the space which it causes the moveable to describe in
the instant dt, in its direction, independently of v, the velocity
of this material point at the end of the time t. By what has
been stated in No. 114, we shall have
ε = ½ udt.
But, in virtue of this acquired velocity v, and of the action
of the force u, or of its components, the spaces traversed by
the projections of the moveable on the axes of x, y, z, during
the instant dt, will be
pdt + p'dt, qdt +q'dt, rdt +½r'dt;
therefore, because(c)
p' = u cos a, q'u cos b, r' = u cos c,
and in consequence of equations (1) and the value of e, we
shall have
x = w cos a + ε cos α,
·y = w cos ẞ +ε cos b,
y' — y
z'
z = w cos y + ε cos c ;
w being the space which the moveable describes in the instant
dt in virtue of the velocity v solely, and x', y', z', its three co-
ordinates at the end of the time t + dt, which were x, y, z, at
the end of the time t.
This being established, let м (fig. 37) be the point of the
GENERAL FORMULÆ OF CURVILINEAR MOTION. 219
trajectory of which x, y, z, are the three coordinates, and мT
the direction of the velocity v. Likewise let MA be that of
the force U. If on MA, MT, there be taken the lines Mн and
MK equal to ɛ and w, and if the parallelogram мHм´к, of
which these lines are adjacent sides, be completed, the extre-
mity м' of this diagonal will be, in virtue of the preceding
equations, the point of which the coordinates are x', y', z', or
the position of the moveable at the end of the time t+dt(d).
Naming v' the velocity of the moveable at the point м',
which velocity will be directed along мT' the production of
the line MM', its value will be the component of v in the di-
rection of Mм', increased by the velocity produced in this di-
rection, by the action of the force u during the instant dt. As
the space ɛ is infinitely small, relatively to w, it follows that
the angle ™мм' is also infinitely small, therefore if we neglect
quantities infinitely small of the second order(e), the compo-
nent of v will be the velocity v itself. Moreover, if ♪ denote
the angle AMM', which the direction of the force u makes with
the side MM' of the trajectory, u cos & will be the increment of
the velocity which will be produced by the action of this force,
hence there will result
v' = v + u cos d.
If v'dt - w', and if on M'T' we take a part м'K'equal to w',
and if M'A' be the direction of the force which acts on the
moveable when it arrives at м', by taking on this line a part
M'H' equal to the space that this force causes the moveable to
describe in an instant dt, and completing the parallelogram
M'H'M"K', M" the extremity of the diagonal will be a third
point of the trajectory. By commencing this series of con-
structions at the point of departure of the moveable, where it
is necessary to know its velocity in magnitude and direction,
it is evident that all the points of its trajectory can be suc-
cessively determined, whether it be a plane curve, or one of
double curvature; and also, at the same time, the velocity
220 GENERAL FORMULE OF CURVILINEAR MOTION.
with which it is actuated at each of these points. If the inter-
vals of time which have been supposed infinitely small and
denoted by dt, are only extremely small, a series of points
will be obtained, which will be the summits of a polygon, and
these will differ so much the less from the trajectory as its
sides are smaller. If the velocity on each side be regarded
as constant, and if for its value there be assumed the semi-sum
of the velocities with which the body is actuated at the two
extremities, the time employed in describing any portion of
the polygon may be computed; consequently we can in this
manner determine, to any required degree of accuracy, the
curves described by the moveable, and also its velocity and
position at any given instant on this curve.
But it is pre-
ferable to make the values of the coordinates of the moveable
in functions of the time, depend on differential equations,
which can be afterwards integrated, in these cases in which it
is possible.
149. These differential equations of curvilinear motion are
an immediate result of the principles established in No. 147.
In fact, the components of the velocity of the moveable,
parallel to the axes of the coordinates x, y, z, being at at' dt'
dx dy dz
,
at the end of any time t, their increments, during the instant
dx dy dz
dt, will be d. d.
dt' a. af,d.
dt dt'
and as each of them arises
from the component of the force, which acts at this instant on
the moveable, parallel to the corresponding axis, it is evident
that if x, y, z, denote the components of this force parallel to
the axes of the coordinates x, y, z, we shall have
dx
dy
d.
xdt, d.
= Ydt,
dt
dt
dz
d. = zdt,
dt
or, what is the same thing,
d²x
d'y
d'z
= x,
=Y,
dt2
dt²
Y, dt²
z.
(2)
•
GENERAL FORMULE OF CURVILINEAR MOTION. 221
The problem in each case will depend on the integration
of these three equations of motion; and the process of the
preceding number may be considered as a general method of
approximation, in order to this integration. Their integrals
will contain six constant arbitraries, which can be determined
by means of the three coordinates of the moveable at the com-
mencement of the motion, and of the three components of the
initial velocity, that is to say, by means of the values of the
dx dy dz
six quantities x, Y, Z, dt' dt' dt'
which will be given in the
case of t=0. These integrals, and their first differentials,
will then make known the position of the moveable at any in-
stant whatever, and also its velocity in magnitude and di-
rection. If the time t be eliminated between them, the two
equations of the trajectory will be obtained. If it is known
beforehand, that this curve is plane, its plane may be assumed
for that of the axes of x and y for example; which will reduce
the three preceding equations to the two first.
150. At the end of the time t, let a, b, c, be the coordi-
nates of a second material point, whose position it is proposed
to compare with that of the first. The axes of the coordinates
being those of x, y, z, let
x = a + x', y=b+y', x = c + z';
the variables x', y', z′, will make known at each instant the
position of the first point relatively to the second; and by
equations (2), we shall have
d²x'
= X
dt2
ď²a d²y'
dt², dt²
d2b
-Y
d²z!
dt2, dt2
d'c
= 2
dt2
,
by means of which they can be determined in functions of the
time.
When the motion of the second point is not known, but
there are only given A, B, C, the components of the force which
solicits it, parallel to the axes of the coordinates, we shall have
222
GENERAL FORMULE OF CURVILINEAR MOTION.
d2a
d2b
d2c
=A,
dt2
dt² = B, de² = c,
C,
dt2
and there will result
d²x²
dt2
— X — A,
ď³y'
dt2
d'z'
=Y B,
— z — C,
dt2
for the equations of the relative motion of the first point.
If the force of which A, B, C, are the components, acts at
the same time on the two moveables, these components will
also occur in the value of x, y, z, and will disappear from these
last equations. This will be the case, for example, with
respect to bodies moving at the surface of the earth, the po-
sitions of which are referred to determinate points of this sur-
face: the forces relative to these points, which arise from the
diurnal motion of the earth, do not occur in the equations of
the different motions which are considered on its surface; and
they should not at all be taken into account in the formation
of these equations.
However, it is not to be understood by this, that the mo-
tions which are observed are altogether independent of the
velocity of rotation of the earth, for it influences in a small
degree the intensity of gravity, and, consequently, the vertical
motions. Moreover, when a body falls from a considerable
height, the velocity of rotation with which it is actuated at
the point of departure, is somewhat greater than that of the
velocity at the foot of the vertical drawn through this point;
hence it is easy to perceive, that the moveable must deviate a
little from this line, and meet the earth at a small distance from
its lower extremity. This deviation, which has been actually
observed, proves by direct experiment the motion of the earth
about its axis(ƒ). The motions which are independent of this
rotation are those which are produced by the shock of two
bodies, and also those which arise from the muscular action of
men and other animals.
151. Equations (2) are those of the motion of a material
point entirely free; but it is easy to extend them to a material
GENERAL FORMULE OF CURVILINEAR MOTION. 223
point constrained to move on a given surface. For this pur-
pose, it is only necessary, as in the case of equilibrium
(No. 36), to join to the given forces which act on the move-
able, a force of unknown magnitude representing the resist-
ance of the surface. This force will be normal to the given
surface. If N denotes this force, and λ, u, v, the angles which
it makes with the productions of the coordinates x, y, z, the
equations of motion will then become
d2x
=x+N COSλ,
dt2
dy
dt2
= Y + N COs µ,
(3)
d²z
=Z+N COs v.
dt2
If the equation of the given surface be L = 0, and, if for
conciseness, we make
(11²
dL2 di
+ dz2
dx² + dy²
we shall have (No. 21), at the same time
cos λ = v
dL
dx
dL
dL
, cos µ = v
COS V V
dy'
dz'
If after having substituted these values in equations (3),
the product Nv be eliminated between them, the two equations
which result, combined with L=0, will enable us to determine
x, y, z, in functions of t. Then from one of equations (3), or
from any combination of these equations, the value of NV
may be deduced, and as N must be always a positive quantity,
the sign of this value will indicate that of v, by means of
which the normal force and the direction in which it acts can
be completely determined.
If the moveable is constrained to move on two given sur-
faces, or on their curve of intersection, it may likewise be
considered as entirely free, if with the given forces there be
ร
224 GENERAL FORMULE OF CURVILINEAR MOTION.
combined two unknown forces N, N', normal to these surfaces;
let λ, u, v, be the angles which determine the direction of the
first with respect to the axes of x, y, z, and X', µ', v', those
which refer to the second, then there will result
d²x
dt2
= x + N COS λ + N'COS λ',
d³y
dt2
= Y + N COS µ + N'cos µ',
(4)
d2z
dt2
= z + N COS v + N' cos v',
for the equations of the motion. If L = 0 is the equation of
the surface of which N expresses the resistance, and if L' = 0,
be that of the surface to which N' refers, the values of cos λ,
cos µ, and cos v will be the same as in the preceding case,
and those of cos λ', cos µ', cos v', may be deduced from them
by changing v and L into v' and L'. When both the one and
the other are substituted in equations (4), the products NV,
N'V' can be eliminated, and the equation which results from
this elimination, combined with the given equations L=0,
L' 0, will enable us to determine the values of x, y, z, in
functions of t. This being done, we can deduce from any two
of equations (4) the values of NV and N'v', the signs of which
will make known those of v and v'; and by this means, the
normal forces N and N' can be determined, and also the di-
rections in which they act; their resultant will be expressed
in magnitude and direction, by the resistance of the curve on
which the moveable is constrained to move.
152. A more simple form can be assigned to equations
(4) in the following manner, let m be the mass of the move-
able, and me the pressure, which in its state of motion it ex-
ercises on the curve it is constrained to describe, if w, w', w",
be the angles which the direction of this force makes with the
productions, on the positive side, of x, y, z, the coordinates of
this point, the resistance which the curve opposes to the mo-
GENERAL FORMULE OF CURVILINEAR MOTION. 225
tion of the moveable, considered as an accelerating force, will
be equal and contrary to P; by joining it to the given forces
X, Y, Z, which act on the moveable, we shall have, in place of
equations (4),
d²x
dt2
-X-P.cos w,
day
dt2
= Y — P.Cos w´,
(5)
d2z
Z — P. cos w".
dt2
The direction of the force P is not given à priori, it is
only known that it is normal to the given curve, from which
it follows that the cosine of the angle comprised between this
direction and the tangent to the trajectory must be equal to
zero; this gives
dx
cos w +
ds
dy
ds
cos w' +
dz
ds
cos w" = 0.
(6)
Moreover, the angles w, w', w" are connected together by the
equation
cos2 w + cos² w' + cos² w" 1.
P, w, w', w" can be eliminated between these equations, for
by adding together equations (5), after having first multiplied
dx dy dz
them by
ds' ds' ds respectively, and then taking into account
equation (6), and making, in order to abridge,
dx dy dz
X
+z
ds
ds
ds
= &,
we shall have
dxdx + dyd³y + dzď z
dsdt2
If the identical equation
= 8.
2
dx² + dy²+ dz²
2
ds²
dt2
dt2 ›
2 G
226 GENERAL FORMULE OF CURVILINEAR MOTION.
be differentiated, and then divided by 2 ds, it will appear that the
first member of the preceding equation is the same thing as
d's
1
dt²;
d; therefore we shall have simply
d's
dt2
p.
(7)
The force is the sum of the components of the given
forces acting along the tangent to the trajectory, which com-
ponents can be regarded as positive or negative, according as
they tend to increase or diminish s, the arc described by the
moveable. Consequently, equation (7) indicates, that in curvi-
linear, equally as in rectilinear motion, the force which acts on
the moveable in the direction of its motion, is equal to the second
differential coefficient of the space described; and because
dt,
ds
v= we may also state, that it is equal to the first differential
coefficient of the acquired velocity. This equation being
independent of the resistance of the curve, obtains equally
in a motion entirely free, and in that of a material point
constrained to exist on a given curve; but it is principally
in the case of a material point which moves on a given
curve, that this equation can be useful. The values of x, y, z,
in functions of s, may be deduced from the equations of this
curve, and after having substituted them in equation (7), it
only remains to integrate this equation of the second order be-
tween s and t. The two constant arbitraries which are intro-
duced in this integration, can be determined by means of the
ds
values of s and when t = 0, that is to say, by means of the
initial position and velocity of the moveable. When the three
coordinates x, y, z, shall have been determined in functions of
t, by means of the integral of equation (7), combined with
the two given equations of the trajectory; equations (5) will
enable us to determine at any instant whatever, the three
components of the pressure P, to which the curve, on which
dt
PRINCIPAL CONSEQUENces of preceding formulæ. 227
the moveable is constrained to move, is subjected. In the fol-
lowing chapter there will be given a simpler determination of
this force both in magnitude and direction.
II. Principal Consequences of the preceding Formula.
153. When the moveable is solicited by a force directed
towards a fixed centre, the three first integrals of equations
(2) can be obtained at once. For this purpose, let the origin
of the coordinates be placed at this point, let the radius vector
represent, in magnitude and direction, the force which solicits
the moveable, and let the parallellopiped, of which this radius
is the diagonal, and the axes of x, y, z, its three adjacent sides,
be constructed. The three coordinates x, y, z, of the move-
able, will be equal to these three sides respectively, and will
represent the three components of the given force, so that we
shall have
hence we deduce
X : Y: Z : : X : Y : z ;
xy = YX, ZX = XZ, YZ = ZY.
On the other hand, equations (2) can be replaced by the
following:
yd²x — xď³y = (xy — xx) dť²,
xd³z — zď²x — (zx — xz) dť²,
zďy – yď²z = (xz — zy) dť²;
}
(a)
but, in virtue of the preceding equations, their second mem-
bers vanish, and as their first members are the differentials of
ydx — xdy, xdz - zdx, zdy — ydz, we shall obtain by inte-
grating
ydxxdy cdt,
xdz
=
— zdx = c'dt,
zdy — ydz — c'dt,
c, c', c', being three constant arbitraries.
(b)
154. The theorem contained in these first integrals of the
228
OF CURVILINEAR MOTION.
*
equations of the motion, implies, that the area described during
each instant dt, by the radius vector of м, the projection of
the moveable on the plane of x, y, is constant and equal to cdt,
and the same is likewise the case for the projections of the move-
able on the planes x, z; y, z; for at the end of the time t, let
AMB (fig. 38) be the projection of the trajectory of the move-
able on the plane of the axes of the coordinates x and y, op and
MP, the abscissa x, and the ordinate y; c being the point where
this curve intersects the axis oy, let u denote the sector com,
the area copм, q the triangle OPM; we shall have
Ρ
u = p — q, q = xy.
X
If м' is the projection of the moveable at the end of the
time t+dt, MOM' will be the area described by the radius.
vector of this projection during the instant dt; likewise this
will be the differential of u, or of p-q; and because
we shall have
dp=ydx, dq = {xdy + žydx,
du = {(ydx — xdy);
hence it appears that the first equation (b) indicates, that the
area described during the instant dt, by the radius vector of M,
the projection of the moveable, is constant and equal to cdt;
therefore, also, the area described during t any time whatever,
is proportional to this variable and equal to act. The areas
described in this same time, by the radii vectores of the pro-
jections of the moveable, on the planes of x and x, and of y
and z, will be likewise equal to c't, c't.
It may be therefore inferred, that when a material point is
subjected to the action of a force constantly directed towards a
fixed centre, the areas described about this point by the radius
vector of its projection on any plane whatever, passing through
this same point, are proportional to the time employed to de-
scribe them.
Conversely, when this property obtains with respect to
three rectangular planes drawn through the centre of the
areas, it may be inferred that the force, or the resultant of the
PRINCIPAL CONSEQUENCES OF PRECEDING FORMULÆ. 229
forces which solicit the moveable, is constantly directed
towards this fixed centre.
In fact, if equations (b) are given, we shall obtain by dif-
ferentiating them,
yd²x - xd²y = 0, xd²z - zd²x = 0, zd²y-yd²z = 0;
consequently, in virtue of equations (a), which are those of any
motion whatever, we shall also have
XyYX, Zx=XZ, YZ = Zy;
therefore, the forces X, Y, Z, will be to each other as x, y, z,
the coordinates of the moveable; which is all that is required
in order for us to be assured that their resultant must be con-
stantly directed towards the origin of the coordinates. In fine,
this force may be either attractive or repulsive, that is to say,
it may act along either the radius vector of the moveable, or
along its production.
155. When a material point is subjected to the action of
a force directed towards a fixed centre, it is evident that its
trajectory is a plane curve, since there is no reason why it
should deviate towards one side rather than the other of a
plane passing through the direction of its initial velocity and
through the fixed centre. This may likewise be inferred from
equations (b), for if we multiply them by z, y, z respectively,
and then divide them by dt, there arises from their addition
cz + c'y + c" x 0.
As this plane may be assumed for that of x and y, it fol-
lows, that the area described by the radius vector itself of the
moveable, in the plane of its trajectory, will be proportional
to the time; and moreover, the preceding theorem is reducible
to this proportionality. In fact, if it obtains for the area
described on the plane of the trajectory, it will equally obtain
for the area described by the radius vector of the projection
of the moveable on every other plane, for this other area is
no other than the projection of the first on this plane; and
230
OF CURVILINEAR MOTION.
we know (No. 10) that the projection of a plane bears a con-
stant proportion to the projected area.
156. The infinitely small area мOм' may be also expressed
in polar coordinates. For this purpose, let r denote the radius
vector oм, the angle мox which it makes with the axis of x.
With the point o as centre, let there be described the arc of
the circle OMN, which cuts at the point N the radius vector OM'
corresponding to the angle ✪ — de, the length of which arc
will be rdo. The circular sector MON will be equal to r²de,
and may be assumed for the area мOм', by neglecting the in-
finitely small area MNM' of the second order. Consequently, we
should have
ydx — xdy = r²d0,
an equation, which may in fact be verified, by means of the
values
x = r cos 0, y = r sin 0,
and of their differentials,
dx = cos 0dr + r sin Ode, dy = sin Odr
r cos Odo,
de.
In this man-
because that of the angle 0 is in this case —
ner, the first equation (b) will assume the form
r²de = cdt,
which is that under which it is usually employed.
The element of the curve may be also expressed in polar
coordinates. Denoting the arc Cм by σ, and this element by
do, we shall have at the same time
MM' do, MN rde, NM' dr;
by considering MNM' as a rectilineal triangle, right angled at
N, we may therefore conclude
do² dr²+ r²d0²;
which we might also deduce from the formula
do² dx²+dy2,
by means of the preceding values of dx and dy.
I
PRINCIPAL CONSEQUENCES OF PRECEDING FORMULÆ. 231
It may be observed on this occasion, that in a plane tra-
jectory, the components of the velocity of the moveable in
the direction of мo' the production of the radius vector mo,
and in the direction of the perpendicular to this radius, are
respectively expressed by
dr
rdo
;
dt'
dt
for the angle O'MT which this production makes with the tan-
gent MT, is the complement of the angle м of the triangle
M'MN; by this triangle therefore we have
COS O'MT=
dr
do'
rdo
sin O'MT=
1.
do
and by multiplying this cosine and this sine by the velocity
do
dt
directed along MT, we shall have the components in ques-
tion. It is frequently convenient to employ them. They
differ from
dx dy
the components of the same velocity in this,
dt' dt
that the directions of these last are fixed, while those of the
preceding vary with the position of the moveable.
do
dt
The velocity with which the radius vector oм describes
the angle com reckoned from a fixed right line, is termed the
angular velocity of the moveable. It may be obtained, as is
rdo
evident, from its velocity perpendicular to oм, by di-
dt
viding it by the length of this radius.
157. Let us now revert to the differential equations of
motion. If equations (5) of No. 152 be multiplied by dx, dy,
dz, respectively, and then added together, there will be ob-
tained by taking into account equation (6) of the same number,
and obesvring that
dxd²x + dyd²y + dzď²z
dt2
ૐd.
ds2
dt2
= {d. √2 = { d. v²,
1d.v² xdx + Ydy + zdz.
}
(c)
232
OF CURVILINEAR MOTION.
If the expressions of the given forces X, Y, Z, do not involve
the time t or the velocity v, explicitly, and if, x, y, z, being
regarded as independent variables, this formula is an exact
differential, which consequently implies that
xdc +xảy + zdz = d.F(,,z);
in which F denotes a given function; then by integrating equa-
tion (c) we obtain
v² = 2F (x, y, z)+c,
in which c expresses a constant arbitrary. In order to elimi-
nate it, let a, b, c, k be the initial values of x, y, z, v; we shall
have
k² = 2F (a, b, c) + ¤,
and by taking this equation from the preceding, there results
v² = k²+ 2 F (x, y, z) — 2F (a, b, c).
(d)
As this result is independent of N, the resistance of the
curve, which resistance is equal and contrary to the force P,
that occurs in the equations from which it has been obtained,
it follows that it has place equally in the motion of a material
point entirely free, and in the motion on a given curve or
surface.
The immediate consequence of this equation (d) is, that
the velocity is constant and the motion uniform, as often as
the moveable is not solicited by any given force; for then the
function F is cypher, and we have v=k, whether the motion is
performed on a given curve or surface, or whether the moveable
is entirely free.
This equation shews us moreover, that in the hypothesis
which has been made respecting the nature of the forces X, Y, Z,
the increment of the square of the velocity of the moveable,
in passing from one point to another, is always the same,
whatever be the curve described, and depends solely on a, b, c,
x, y, z, the coordinates of the two extreme points. When this
curve is given, or when the moveable is only constrained to
PRINCIPAL CONSEQUENCES OF PRECEDING FORMULÆ. 233
move on a given surface, k should be assumed as the velocity of
the moveable, which is tangential to this curve or this surface.
If the percussion impressed on the moveable at the commence-
ment of its motion has not this direction, it can be resolved
into two other forces, the one normal, the other tangential;
the first will be destroyed by the resistance of the given curve
or surface; so that it is the second which will produce the ve-
locity k, and determine its direction.
c denoting a constant arbitrary, the equation
F(x, y, z) = c,
will be that of a surface which will be reached with equal velo-
cities by all moveables subjected to the action of the same forces,
which commence to move from a point of which a, b, c are the
coordinates, with the same velocity k, in different directions.
When, for example, these moveables are only acted on by the
force of gravity, this equation is that of a horizontal plane.
In the case of a given curve, if the values of x, y, z, be de-
duced from its equations in functions of the arc s, there will
ds
result by substituting them in equation (d), and putting in
dt
place of v(g),
dt = sds,
where s is a given function of s, consequently, in this case, the
determination of the time as a function of the space described,
will be reduced to the integration of a given differential. But
the supposition on which equation (d) is founded, and conse-
quently, this equation, has not place when the moveable expe-
riences any resistance in its motion through a medium, for this is
a force dependent on the velocity; neither will it obtain, when
the question is respecting the motion of a material point, which
is attracted or repelled by other points that are themselves in
motion; a circumstance which will introduce the time expli-
citly in the values of x, y, z. In these two cases, if the tra-
jectory be a given curve, equation (c) ought to be employed,
2 H
234
OF CURVILINEAR MOTION.
༈
ds
in which
dt
should be substituted in place of v, by which
means this equation will be changed into equation (7) of
No. 152.
158. The formula xdx + ydy + zdz will be an exact dif-
ferential as often as the moveable being attracted or repelled
by fixed centres, the intensities of these forces are expressed
by functions of the distance from the centres from which they
emanate.
In fact, let e, f,g be the three coordinates of one of the
fixed centres, referred to the same axes as x, y, z, and r the
distance of the moveable from this point, we shall have
r² = (e — x)² + (ƒf − y)² + (9 − z)²;
(g
and the cosines of the angles which this liner makes with the
axes drawn through the moveable in the directions of the posi-
tive xs, ys, zs, will be the ratios of e x, ƒ — y, g z, to r.
If R denotes the attractive force directed from the moveable
towards this fixed centre, the expressions of its three compo-
nents will be
z)
R(ex), R(fy), R (9—2);
T
go
and consequently, the part of xdx + ydy + zdz which arises
from R, will be
R
[(e — x) dx + (ƒ — y) dy + (g — z) dz].
But by differentiating the value of 2 we obtain
rdr — — (e — x) dx
— (e — x) dx — (ƒ — y) dy — (g — z) dz ;
which reduces the preceding quantity to Rdr. If the force
which emanates from the fixed centre was repulsive, it is only
necessary to change the sign of this quantity, which will then
become Rdr, R being regarded, in all cases, as a positive
quantity.
PRINCIPAL CONSEQUENCES OF PRECeding formulæ. 235
It follows from this, that if the moveable is solicited by
any number whatever of forces, such as R, R', R", &c., which
emanate from fixed centres, and of which the distances from
this material point, are r, r', r'', &c., we shall have
xdx+xdy + zdz= Rdr = R'dr' R"dr" &c.;
+
干
​the superior signs have place in the case of attractions, and the
inferior signs in the case of repulsions. Now, supposing that
each of these forces is a given function of the corresponding
distance, all the terms of this value of xdx + ydy + zdz will
be differentials depending on one sole variable, and, conse-
quently, this formula will be an exact differential, which was
to be proved.
It appears from this and equation (d), that the increment
of the square of the velocity arising from each of the forces
R, R', R", &c., will be the same as if it solely existed; for ex-
ample, with respect to the force R, this increment will be ex-
pressed by 25 Rdr, the integral being taken in such a man-
ner, that it may vanish for the initial value of r.
159. In the case of a heavy material point, which moves
on a given curve in a vacuo, and without any friction on this
curve, equation (d) will be reduced to
v² = k²+ 2g (z — c),
g denoting the gravity, and the direction of the axis of the
positive zs being supposed to be that of this force, so that we
may have
x = 0, y = 0, z=g.
Let ADBC (fig. 39) be the given curve, в its lowest and a
its most elevated point, which need not exist in the same ver-
tical as B, and D the point from which moveable sets out. If
the origin of the vertical coordinates z be placed in this last
point, and if the initial velocity k be due to the height h, we
shall then have
236
OF CURVILINEAR MOTION.
and, consequently,
c=0, k²=2gh,
v² = 2g (h + z).
It follows from this, that when the moveable attains the
point B, the maximum velocity will be the same as if it fell
from the height h, increased by that of the point D, above the
horizontal plane drawn through the point B. In virtue of
this acquired velocity, the moveable will ascend along BCA;
its velocity will diminish continually; and if h = 0, it will be
nothing at the point c situated in the same horizontal plane as
D. Having reached the point c, the moveable will redescend
along CB, and it will thus oscillate from D towards c, and from
c towards D. When the constant h does not vanish, the
moveable will ascend above the point c. If the elevation of
the point a above the horizontal plane, which comprises D and
c, is greater than h, the moveable will not reach the point a ;
its velocity will be cypher at a certain point c'; and if through
c' there be drawn a horizontal plane which intersects the curve
in another point D', the moveable will oscillate indefinitely
from c' towards D', and from D' toward c'. The oscillations
will be all isochronous, or of equal duration. This is evident
with respect to those which are performed in the same direc-
tion; and it appears also, that the duration of each oscillation
from c' to D' is the same as that from D' to c', for any element
whatever of the curve will be described with the same velo-
city in the two cases. This common duration of each of the
entire oscillations, will depend on the form of the curve and on
the magnitude of h
A
When the elevation of a above the horizontal plane which
passes through the point of departure is equal to h, the
moveable will approach indefinitely to the point a, but will
not attain to it until after the lapse of an infinite time. When
this elevation is greater than h, the moveable will pass beyond
the point a, and traverse the entire circumference of the given
PRINCIPAL CONSEQUENCES OF PREceding forMULÆ. 237
curve. When it returns to the point D, its velocity will be the
same as at the commencement of the motion; hence it follows,
that it will perform an infinite series of revolutions, the dura-
tion of each of which will be equal, and dependent on the form
of the curve and on the magnitude of h.
If the given curve(h) is comprised in a vertical plane,
which is a tangent to a cylinder of any base whatever, and if
this plane is enveloped on the cylinder, so that the given curve
may become a line of double curvature, the motion of the
moveable, whether it be oscillatory, or one of revolution, will
not be at all changed, it being always however supposed, that
the point of departure and initial velocity remain the same;
for then the value of t in a function of s, determined as has
been already stated (No. 157), will only depend on that of z
in a function of s, which will not be changed, whatever be the
base of the vertical cylinder on which the given curve is
traced.
160. In all cases in which equation (d) obtains, and in
which the moveable is not constrained to move on a given
curve, that which it describes in passing from one given point
A to another given point B, possesses the following remarkable
property. If the moveable be entirely free, the integral Svds,
taken from the point A unto the point B, is less than if it moved
along any other curve, terminating at these two points; if it is
constrained to move on a given surface, this property of the
trajectory obtains only relatively to all curves traced on this
surface, and which always terminate at the points A and B. In
these two cases, ds is the differential element of any curve
whatever, which corresponds to the coordinates x, y, z, and v
is a function of these three variables and of a constant k, which
is given by equation (d).
The demonstration of this theorem consists in proving that
in virtue of the equations of motion, the variation of Suds is
nothing, (the limits of this integral being supposed to be
fixed). In consequence of this, Svds will be either a maximum
238
OF CURVILINEAR MOTION.
J
or minimum, but it is evident that it must be a minimum in this
case, in which the moveable is entirely free; for the integral
Svds increases indefinitely with the length of the curve, and,
consequently, cannot be susceptible of having a maximum
value.
Now, by the rules of the calculus of variations, we have
S.Svds=58.vds, 8.vds Svds + vôds,
moreover, dt being the element of the time, we have ds = vdt,
therefore,
Svds = dtd.v².
If equation (d) be differentiated, and if the variations
Sx, dy, dz, be substituted for the differentials dx, dy, dz, we
shall have
đô.v = xô + xây + z8z.
By substituting for cosa, cosv, cos μ, their values given
in No. 151, and observing that(¿)
dL
dL
dL
Sx +
dx
dy
Sy + √z dz=
Sz = SL,
dz
equations (3) of same number will give
xô xây + z = Sx+
dt2
dr² dy +
d²x
dt2
dy sy+
d²z
Sz- NVSL.
dt2
Now if the moveable is entirely free, the term NvdL will
not occur in this equation, and when it is constrained to move
on a surface of which the equation is L=0, this term is
cypher, for as all the curves, which are compared to the tra-
jectory of the moveable, must also be traced on this surface,
we have SL = 0, therefore, in all cases this term must be
suppressed; it follows from this that
Svds = {dt8.v²
& dtd. v² = + 8x +
—
d²x
dt
day
d²z
Sy +
Sz.
dt
dt
With respect to vôds the second term of the variation of vds,
we have
PRINCIPAL CONSEQUENCES OF PRECEDING FORMULA. 239
1
and, consequently,
ds²= dx²+ dy²+ dz²,
dx
Sds = d² da + dy ddy + dz 8dx;
ds
ds
hence, because ds = vdt, by changing the order of the cha-
racteristics d and 8 in the second member, we shall have(k)
dx
vdds =
dy
ddx+ ddy +
dz
ddz;
dt
dt
dt
and by adding together these two parts of the value of §.vds,
there results,
dx
8.vds = d (dit 8x+
consequently we obtain,
dx
+dy
S8.vds= Sx + dy dy+
dt
dt
бу
Sy+
dt
dz 8z );
dz
Sz + constant,
dt
for the indefinite integral of 8.vds. But as the two extreme
points A and B are supposed to be fixed, the variations Sx, Sy,
Sz, which respect these points, must be nothing. Therefore,
the definite integral SS.vds, taken from the point A to the
point B, which is equal to the variation S.Svds, will be re-
duced to cypher; which it was required to demonstrate.
161. When the moveable, being constrained to move on
a curved surface, is not solicited by a given force, its velocity
is constant (No. 157), and the integral Svds is then the
product vs. Consequently, the arc s described by the move-
able is, in general, the shortest line between the points a and
B; and from the uniformity of the motion, it follows that, in
this case, the moveable goes from one point to another, in a
less time than if it was forced to describe on the given sur-
face any other curve than its trajectory. However, if this
surface is closed on all sides, as a sphere, for example, the
points A and B will be the extremities of two arcs of a
240
OF CURVILINEAR MOTION.
1
great circle, one of which will be less and the other greater
than all other arcs of lesser circles of the sphere, terminating
at the same points; and the moveable can describe either the
one or the other of these two portions of the same great circle,
according to the direction of the initial velocity k, which is
tangential to the sphere.
The differential equation of the trajectory may be presented
under a form, in which the property of the shortest line on any
surface whatever will be immediately apparent; which consists
in this, that its osculating plane in each point is normal to this
surface.
The forces X, Y, Z, being supposed to be cypher, equa-
tions (3) of No. 151 are reduced to
d²x
dt2
=N.cosλ, =N.COs µ,
day
d²z
dt2
=N.COS v.
dt2
Because v is constant,
and that vts, we have
d²x
d²x
=v2
= v2
dt²
ds², dt²
d'y day d²z d²z
=v2
ds2' dt2
d$2,
the arc s being assumed as the independent variable; and this
being the case, the preceding equations may be replaced by the
following:
dxd³y — dyd²x
S
N
ds3
v2\ds
(
dx
cos μ
ds
dy, cos λ),
dzđx
dxd2z
dz
dx
=
cos λ
COS
ds3
v²ds
ds
S
dyd²z dzď³y
N (dy
ds³
v2ds
(
dz
COS v
cos u
ds
µ),
which may be deduced from them without any difficulty. If
they be multiplied by cos v, cos μ, cos λ, respectively, and
then added together, the quantity N disappears, and we have
simply
dxd³y-dyd²x
ds³
·cos v + −
dzd'x-dxd'z
ds3
cosμ+
dyd²z - dzď'y
(e)
cosλ=0;
ds
PRINCIPAL CONSEQUENCES of preceding formulæ. 241
hence from the values of cos λ, cos µ, cos v, cited in No. 151,
we shall have
dzd²x-dxd2z du
dxd'y-dyd'x dL
+
ds3
'dz
ds3
dy
+
dyd²z-dzd'y di
(f)
dL
= 0,
ds³
dx
for the second differential equation of the trajectory. If in
this, there be substituted the value of one of the three coor-
dinates x, y, z, in a function of the two others, deduced from
L= 0, the equation of the given surface on which the curve is
supposed to be traced, and if the equation between the two
variables, which results from it, be then integrated, the two
constant arbitraries which the integral will contain being de-
termined, by subjecting the curve to pass through the two
points A and B of the given surface, the equation which will
be obtained in this manner will be independent, as is evident,
of the magnitude and direction of the initial velocity k, and
must be that of the shortest line between these two points.
Now if a, ß, y be the angles which the normal to the oscu-
lating plane of any curve whatever, at the point of which the
coordinates are x, y, z, makes with their productions in the
positive direction, and if, in order to abridge, we make
2
[(dxd³y—dyd²x)²+(dzd²x-dxd²z)²+(dyd²z—dzď³y)²]*—h,
we shall have by formula (3) of No. 19, in which these angles
are represented by λ, µ, v,
1
cosa
h
(dyd²z — dzď²y),
1
cosß = = (dzď³x — dxd³z),
h
1
cos y = √(dxd³y — dyď²x),
consequently in virtue of equation (e), we shall have
0;
cos à cos a + cos ß cos μ + cos y cos v = 0 ;
2 I
242
DIGRESSION ON THE MOTION Of light.
which shews that the normal to the osculating plane of the
trajectory, and the normal to the given surface are perpendi-
cular, the one to the other; hence we infer, that equation (f)
which belongs(g) to the shortest line, is also that of the curve
which has every where its osculating plane normal to the
given surface, so that these two lines are one and the same
curve traced on this surface, when both the one and the other
are constrained to pass through the same extreme points
A and B.
It follows from this, that when these two points belong
to one of the lines of curvature of the given surface, this line
is the shortest from one point to another; for its osculating
plane in any point whatever, contains two consecutive nor-
mals to the given surface, and is, consequently, normal to this
surface.
III. Digression on the Motion of Light.
162. The theorem of No. 160 is known by the denomina-
tion of the principle of least action, which was given to it,
from the metaphysical point of view in which it was first con-
sidered, but this has been since properly abandoned. How-
ever it may be useful here, to give one of the first applications
which was made of this principle, namely, that which is relative
to the reflexion and refraction of light in the system of
emission.
As long as a ray of light moves in a medium of uniform
density, its velocity and direction remain the same; but when
it passes from one medium to another, its direction is inflected,
and its velocity changes. At the instant of the transit, the light
describes a curve of inappreciable extent, which consequently
we may neglect without any sensible error. Therefore, the
trajectory of each luminous particle may be supposed to con-
sist of two lines, each of which is described with a uniform
velocity. Thus, if y, y' be the lengths of these lines, n the
DIGRESSION ON THE MOTION OF LIGHT.
243
velocity of light in the first medium, and n' its velocity in the
second, ny will be the value of the integral Svds taken from
the point of departure of the particle to its entrance into the
second medium; and n'y' the value for the part of this integral re-
lative to the second medium; consequently, this integral taken
in the entire extent of the trajectory, will be expressed by
ny + n'y', and it is this sum, which, by the principle of least
action, should be a minimum.
Before we proceed further, it is necessary to observe, that if
the second medium is a diaphanous and crystallized substance,
the velocity of light in this substance will, in general, depend
on the direction of the luminous ray; so that it will be con-
stant for the same ray, but variable from one ray to another.
The phenomenon of double refraction which is observable in
Iceland spar, and the greater number of transparent crystals,
arises from the difference of velocity of the different luminous
rays which traverse them. The velocity n' should then be re-
garded as a function of the angles which determine the direction
of each ray; and the law of refraction depends on the form of
this function. Laplace succeeded, by means of the form which
he assigned to this function, in deducing from the principle of
the least action, the law of double refraction, which was
originally discovered by Huyghens, and afterwards confirmed
by the experiments of Malus; but as this is not the place to
enter into a detail of this theory, we shall restrict ourselves
to the consideration of the case in which all the rays move
with the same velocity, whatever be their directions. There-
fore, in the following investigation the velocities n and n' will
be considered as given quantities for each medium in parti-
cular, and independent of the direction of the luminous rays.
163. Let now A and B (fig. 40) be the two extreme points
of the trajectory, and through them let a plane pass, inter-
secting the surface of separation of the two media, which is
supposed to be plane, in the right line CD. Let the line AEB,
the parts of which are AE and EB, represent the projection of the
244
DIGRESSION ON THE MOTION OF LIGHT.
trajectory on this plane, and through the points A, B, E, let
AF, BG, HEK be drawn perpendicular to the line CD. Since
the position of the points A and B is given, the three lines
AF, BG, FG are known, but the position of the point E, and
the angles AEH, BEK are unknown, and must be determined by
the condition of the minimum. Therefore, let AF α, BGb,
FG = C, AEH= x, BEK = x'.
From the right angled triangles AFE and BGE, we obtain
EF = a tang x, EG = b tang x';
consequently, there results
a tang x+b tang x'= c.
(a)
The luminous ray traverses the surface which separates
the two media, in a point of which E is the projection on the
plane of the figure. If z denotes the distance of this unknown
point from the point E, y will be the hypothenuse of a right
angled triangle, of which z and AE are the two lesser sides,
and y' will be the hypothenuse of another triangle, of which
z and BE are the two lesser sides, but from the triangles AEF,
and BEG, we obtain
AE=
hence we shall have
a
COS X
BE =
,
b
cos x'
a²
y = √ 2² + 008²x²
cos²x
2
b2
y = √ z²+ cos²x²
2
If these values be substituted in the expression ny + n'y',
there will result a function of z, x, x' which should be a mini-
mum with respect to these three variables, the two last of
which are connected together by equation (a). It is, therefore,
in the first place necessary, that the differential of this func-
tion taken with respect to z, should be equal to zero, hence
we obtain
ndy n'dy' nz n
+
+
0.
dz
dz
y
y'
U w W Z W W X ud
"
3
DIGRESSION ON THE MOTION OF LIGHT.
245
But this condition can only be satisfied by assuming z = 0;
which shews that the luminous ray traverses, at the point E,
the surface that separates the two media, and consequently
that it does not deviate from the plane drawn through the
points A and B perpendicular to this surface. Therefore,
making z = 0, we shall have simply,
na
nb
ny+ny' =
+
cos x
cos x
and by putting the complete differential of this quantity equal
to cypher, there arises
na sin xda
+
cos² x
n'b sin x'dx'
cos² x'
= 0;
but by differentiating likewise equation (a), we have at the
same time
dx'
and if
dx
results
adx bdx'
+
=0;
cos² x cos²x²
2
be eliminated between these two equations, there
n sin x n' sin x'.
(b)
X
This and equation (a) will determine the values of x and
x', which answer to the minimum of ny + n'y'. The value
of a being obtained, the point E can be constructed, by taking
EF= a tang x; then the lines AE and BE can be determined;
and the line AEB, the parts of which are AE and EB, will be the
path of the luminous ray in passing from the point a to the
point B.
The angle AEH contained between EH which is normal to
the surface of separation of the two media, and the incident
ray AE, is what is termed the angle of incidence; the angle
BEK contained between EK, the production of this normal and
the refracted ray BE, is the angle of refraction. These two
angles have been denoted by x and x'. Hence equation (b)
will make known the angle of refraction when the angle of
246
DIGRESSION OF THE MOTION OF LIGHT.
incidence is given; and from this equation it appears, that the
sine of the angle of incidence is to the sine of the angle of re-
fraction in a constant ratio.
This is, in fact, the known law of ordinary refraction, which
was first discovered by Des Cartes. The ratio of the two sines
depends on that of the velocities n and n' relative to the media
which are considered; and for this reason, it varies with the
different descriptions of transparent media.
164. If the luminous ray, instead of penetrating the second
medium, is reflected at the surface of separation, its velocity
will be constant for the entire length of the trajectory, which
then exists altogether in the same medium. Hence the inte-
gral Suds will be equal to the entire length of the trajectory,
multiplied by this constant velocity, consequently, in virtue of
the principle of least action, this length should be a minimum.
Let us therefore suppose, that, as in the preceding num-
ber, the surface of separation is a plane. Let A and B (fig. 41)
be the two extreme points of the trajectory, through these
points let a plane be drawn perpendicular to this surface, and
intersecting it in the line CD: each particle of light will, in its
passage from the point A to the point в, describe the line AEB, the
parts of which are AE and EB, and it will be the shortest of all
those that are reflected from the surface of separation. Now
it is immediately evident, that this line must exist in the plane
perpendicular to this surface, for every other trajectory is ne-
cessarily longer than its projection on this plane. Moreover,
it is easy to show without the assistance of the calculus, that
the shortest line between A and B, which meets the surface of
separation, is that which makes equal angles with the line CD,
that is to say, if
А
AEC BED,
the line AEB will be shorter than any other line such as AE'B,
of which the point E', like the point E, appertains to the line
CD.
DIGRESSION OF THE MOTION OF LIGHT.
247
In fact, from A let the perpendicular AF be let fall on this
line, and produce it so that AF' may be equal to AF, and let
the lines A'E, A'E' be drawn. The two angles AEC, AEC will be
equal; hence the two angles A'EC, BED will be so likewise on
account of the preceding equation; consequently A'EB will be
a right line, and we shall have
and as
A'E + BE A'E' + BE';
A'E AE and A'E' = AE'
there will result
AE+BE AE' + BE';
which was to be proved.
If at the point E, the perpendicular EH be erected to the
line CD, AEH and BEH will be the angles of incidence and re-
flexion of the luminous ray which issues from the point a to
the point B. These angles will be equal, because they are
the complements of the equal angles AEC and BED; hence re-
sults the known law of the reflexion of light, which consists
in this, that the angle of incidence is always equal to the an-
gle of reflexion.
165. If the theory of the emission of light be admitted,
the laws of reflexion and refraction can be inferred from the
expression of the square of the velocity of a point subjected
to the action of attractive forces (No. 168), in a more di-
rect manner, than by making use of the principle of least
action. As this question presents us with an example of the
motion of a material point, which is interesting, as well from a
consideration of the nature of the forces which operate, as
from its physical application, we will give the solution of it in
the ordinary case, in which the two media traversed by the
light are not crystallized.
In this theory, each luminous particle is supposed to be
subjected to the attraction of all the material points of the me-
dium which it traverses, and this force is regarded as an un-
248
DIGRESSION OF THE MOTION OF LIGHTS
1
known function of the distance, for all that is known respecting
it is, that it decreases with extreme rapidity when the dis-
tance increases, so that when the distance is of a sensible mag-
nitude, it becomes altogether insensible. Thus, for example,
ifr denotes the distance of the attracted from the attracting
point, a a line of a finite but insensible magnitude, and e the
base of the Naperian system of logarithms, then a force of this
nature may be represented (n) by a. e; A being the intensity
relative to an infinitely small distance r; when this distance is
of a sensible magnitude, and is, consequently, a considerable
multiple of the value of this function will be no longer
sensible.
α,
If the luminous ray moves in a homogeneous medium of a
uniform density, the attractions which it experiences mutu-
ally destroy each other's effect, and its motion becomes uni-
form and rectilinear. But if it attains the point m (fig. 42)
situated at an insensible distance from the surface CD, which
separates the two media, and which, for greater clearness, we
will suppose to be horizontal: let a perpendicular мP be
drawn from this point on CD, and then in the upper medium
let there be drawn two planes C'D' and c"D" parallel to CD,
the mutual distance of which may be equal to MP, and let the
first pass through the point м; it is evident that the attrac-
tions exercised on the luminous rays at the point м, by the
two strata of the upper medium, which are comprised, the one
between CD and c'D', the other between c'D' and c"D", will be
equal and contrary; therefore they will destroy each other's
effects, and the moveable will only be solicited by the part of
the medium it traverses, which is above c"D", and by the entire
attraction of the inferior medium. These two forces will be
perpendicular to CD; and they will vary with the distance MP,
according to unknown laws, but such, that each of these forces
will be insensible when MP is not so, and they will attain their
maxima when this distance vanishes; in which case the move-
DIGRESSION ON THE MOTION OF LIGHT.
249
able will have attained to the surface which separates the two
media.
At the end of the time t, let z be the distance MP, and
z, z' the unknown functions of z, which express the accelera-
ting forces arising from the attraction of the inferior medium,
and from the part of the other medium above c"D". The entire
accelerating force tending to diminish z, will be the difference
z-z'; consequently, we shall have, in the superior medium,
d²z
dt²
+z-z'=0;
(1)
for the equation of the vertical motion of a luminous particle.
When the moveable, after traversing the surface CD in a
point E, has penetrated into the lower medium as far as a point
M', which is such that M'P', the perpendicular to CD, may be
also represented by s, it is easy to see that the accelerating
force, which will tend to diminish this variable, will be then
the difference z'-z, so that we shall have
d²z
dt2
+z'—z=0,
for the equation of vertical motion in the inferior medium.
(2)
As to the horizontal motion or that parallel to CD, it will
be uniform, and the horizontal velocity will undergo no change
in passing from one medium to the other; for the attractive
forces of each medium parallel to CD, will destroy each other's
effect, so that a luminous ray is not subject to any accelerating
force in this direction. Thus, if k be the velocity of light at
a, a point of the superior medium, situated at a sensible dis-
tance from CD, and a the acute angle that the direction of this
velocity makes with the vertical, k sin a will be the velo-
city parallel to CD at any instant whatever. If the luminous
ray penetrates by a sensible quantity into the inferior me-
dium, and if k' and a' denote what k and a become at a', any
point of this medium, situated at a sensible distance from CD,
2 K
250
DIGRESSION ON THE MOTION OF LIGHT.
we can likewise represent the horizontal velocity of the move-
able by k' sin a'; so that we should have
k sin a = k' sin a'.
(3)
It appears also, a priori, that the trajectory of the moveable
will be a plane and vertical curve; therefore it only remains to
determine its velocity perpendicular to CD, whether in the su-
perior or inferior medium, at z any distance whatever from this
surface CD.
166. Let this velocity be denoted by u, so that we may
dz2
have =u² for the two media. By multiplying equation
dt2
(1) by 2dz, then integrating and denoting the constant arbi-
trary by c, we shall have in the superior medium
u²=c+2Sz'dz — 2§zdz.
Let these two integrals be supposed to vanish with z, and let
their values at a sensible distance from cb be h and h'; these
integrals h and h' can be extended from zero to infinity, for
by hypothesis beyond a sensible value, the functions z and z',
and consequently, the corresponding parts of Szdz and §z'dz
vanish, or become insensible.
Therefore, we may write if we please,
h =
Sozdz, h'=So2'dz.
Moreover, for any sensible value of x, we have
u²´ = k² cos² a ;
hence in that case we shall have
k². cos² a=c+2h'-2h;
and by eliminating c from the general value of u², there will
result at any point whatever,
u² = k² cos²a + 2h−2h' + 25z'dz – 25zdz.
Let k, be the velocity of the moveable at E any point of the
DIGRESSION ON THE MOTION OF LIGHT.
251
αι
2
2
surface CD, and let a₁ be the angle which its direction makes
with the vertical. At this point we shall have u² = k₁² cos²α₁;
and as in this case z=0, the two last terms of the preceding
formula will vanish, and it will be reduced to
2
2
k₁² cos²α₁ = k² cos²a + 2h — 2h'.
(4)
In order, therefore, that the luminous ray may reach the
surface which separates the two media, it is necessary that the
second member of this equation should be a positive quantity,
and, consequently, that
h' 4 h + ½ k² cos² a.
If this condition be not satisfied, in which case the attrac-
tion of the upper medium should surpass that of the inferior,
the vertical velocity will be destroyed before it reaches the
plane CD; therefore there will be a point of the trajectory, at
which the tangent will be horizontal. Having attained this
point, the moveable will retrograde, the two branches of this
curve, which terminate in this same point, will be similar,
since they are described by the action of equal forces, for the
same value of z; and when this distance z is of a sensible
magnitude, these two branches will be changed into right lines,
which make equal angles with the vertical, or, in other words,
the angles of incidence and reflexion will be equal. If, on
the contrary, the attraction of the inferior medium surpasses
that of the superior, and if the preceding condition is satisfied,
the luminous ray will penetrate into the inferior medium with
a velocity the direction of which will be perpendicular to CD,
and which will be determined by equation (4). In this hypo-
thesis we shall have, by equation (2), relative to this medium,
u² = k₁² cos³α₁ + 25zdz — 2§z'dz,
2
the integrals being always supposed to vanish when z0.
At a sensible distance from CD, u² k'2 cos²a'; therefore we
shall have
=
k/2 cosa² = k cos a2+2h - 2h';
•
252
DIGRESSION ON THE MOTION OF LIGHT.
2
and eliminating k₁2 cosa,² by means of equation (4), there will
result
k2 cos a² k². cosa²+4h-4h'.
(5)
Therefore, in order that the luminous ray, after having tra-
versed the surface CD, may penetrate to a sensible depth into
the inferior medium, it will be necessary that
h' 2h+k² cosa,
and this is the only condition that is necessary.
It may happen that h', although less than h+k² cos²a,
surpasses hk2 cos2a, in which case the moveable will not
penetrate into the inferior medium a sensible distance beyond
CD; but it will return into the superior medium, and the two
branches of the trajectory which it describes will be similar
on each side of the point at which it commenced to retrograde.
Consequently, the light will be reflected, as in the preceding
case, making the angle of incidence equal to the angle of re-
flexion; so that there are two distinct cases of reflexion in the
theory we are considering.
167. Let us now suppose that neither the one nor the
other of these two cases obtains, so that the luminous ray must
be refracted. By equation (3) we have
12
k²² sin²a′ = k² sin²a
a;
and by adding the corresponding members of equation (5), and
of this, there will result
k²² = k²+4h-4h';
(6)
which shows that the increase of the square of the velocity of
the moveable, in passing from the point a of the superior me-
dium, to the point a' of the inferior medium, is (as it ought to
be (No. 157)) independent of the route along which it moves.
Likewise, from equations (3) and (6) there results
sina'
k
sin a
√ k² + 4 (h — h')
;
(7)
DIGRESSION ON THE MOTION OF LIGHT.
253
(
from this formula it appears, that the sine of the angle of in-
cidence is to the sine of the angle of refraction in a constant
ratio, and it is evident that the value of this ratio is a function
ofk, the velocity of light in one of these two media, and of h—h',
the difference of their refractive powers h and h'.
If the inferior medium is terminated by two parallel planes,
and if the medium below is the same as that above it, experi-
ment shows that the light, after having undergone two refrac-
tions, and traversed the two faces of the intermediate medium,
resumes a direction parallel to that which it had in the supe-
rior medium. This likewise results from equation (7). For
if a" be the angle which the luminous ray makes with the ver-
tical after it emerges from the intermediate medium, it is neces-
sary, in order to determine sin a", to interchange among
themselves the quantities, h and h′, and to put k', a', a″ in-
stead of k, a, a'. Consequently, we shall have
sin a"
sin a'
12
k'
=
√ k¹² + 4 (h' — h)
or, in consequence of equations (6) and (7),
sin a"
√ k² + 4 (h — h')
k
sin a
sin a´
sin a
which gives in fact
a" = a.
The phenomenon of dispersion which arises from a diffe-
rent value of the angle of refraction a', for the differently
coloured rays, of which the same incident ray is composed,
may be attributed, agreeably to formula (7), either to an in-
equality in their velocity k, or to a different action in each
medium on these different rays, from which results unequal
values of h - h'.
168. Every thing else being the same, it appears from equa-
tion (7) that the ratio of the sine of the angle of incidence to the
sine of the angle of refraction, must change with the velocity of
254
DIGRESSION ON THE MOTION OF LIGHT.
NO
{
the light. Now, for a star situated in the plane of the ecliptic,
there is an epoch in the course of the year, in which the di-
rection of the velocity of the earth is contrary to that of light,
and another epoch in which the direction of the first mentioned
velocity is the same as that of the second; this renders the
velocity of light, relatively to the medium in which it moves
along with the earth, sensibly greater in the first case than in
the second. The ratio in question must consequently be dif-
ferent at these two epochs; but from extremely accurate expe-
riments instituted by M. Arago, it appears that this ratio
does not vary in a sensible manner, during the entire course of
the year, and moreover, that its magnitude is the same for the
sun and for the different stars, from which light on the theory
of emission emanates.
Whatever theory of light be adopted, it is a very remark-
able fact, that the composition of its velocity with that of the
earth, which is indicated in the apparent motion of the stars,
termed aberration, has, notwithstanding, no appreciable in-
fluence on the refraction of the light, which is transmitted to
us from them on different days of the year.
In a vacuo, the motion of light, whether direct or reflected,
is always uniform, and its velocity is independent of the source
from whence it emanates. This velocity is so great, that
light traverses in 493,34 seconds, the mean distance of the sun
from the earth: from which it follows, that it describes 30950
myriametres in a second.
A luminous ray emitted from the sun or a star, should,
like any other projectile, experience a diminution in its velo-
city, caused by its gravity towards this star, that is to say, by
the attraction in the inverse ratio of the square of the distances
from its centre, which the mass of the body exerts on each
material particle of light; but this diminution is a very small
fraction of the final velocity of light. Thus, for example, as
it will be shown hereafter, that the intensity of gravity at the
surface of the sun is twenty-seven times and a half greater
DIGRESSION ON THE MOTION OF LIGHT.
255
than the intensity of the terrestrial gravity, and as the sun's
radius is 110 times the radius of the earth, it follows from what
has been observed in No. 143, that the velocity of light, in
order to be 30950 myriametres per second at a considerable
distance from the sun, must, when issuing from the surface,
be greater by a little less than(k) two millioneths only.
•
{
CHAPTER IV.
OF THE CENTRIFUGAL FORCE.
169. THE pressure of a material point on a curve which it
is constrained to describe, is not the same as when it is in
equilibrio on this curve. In consequence of the motion, a
particular pressure arises, which is termed the centrifugal
force, because it was first considered in the circle, where it is
directed along the production of the radius, and continually
tends to increase the distance of the moveable on which it acts
from the centre. It is this force which we now proceed to
consider in any curve whatever.
Let M₁M and MM' (fig. 43) be two equal and consecutive
elements of the given curve, H and н' their middle points,
MT and M'T' their productions. Their plane, and the angle
TмT', will be the osculating plane and angle of contact of the
curve at the point м; and if in this plane the line мo be
drawn, dividing the angle MMM' into two equal parts, its di-
rection will coincide with that of the radius of curvature at this
same point м; and consequently the centre of curvature will
exist in a certain point of this line such as o. Let ds denote
M₁M the element of the curve, which will be also equal to
Hмн'; moreover, let ♪ be the infinitely small angle тмт', and
the radius of curvature mo, we shall have (No. 18)
P
ds
d =
P
This being established, let us first abstract from the conside-
ration of the given forces which may act on the moveable, and
let us suppose, that at the end of the time t, it reaches the
OF THE CENTRIFUGAL FORCE.
257
point м, with the velocity v. If it was entirely free it would
continue to move on the line MT with the same velocity; but
it is by hypothesis forced to describe a given curve, this pro-
duces a change in the direction of its motion, which thus be-
comes мT'. Now if there be erected to MT' the perpendicular
MK, such that it may exist in the osculating plane, and fall
without the concavity of the curve, we can substitute for the
velocity v, the direction of which is мT, two other velocities,
of which one is equal to v cos d, and directed along мт', and
the other equal to v sin S, and directed along мK, the effect of
the curve will be to destroy the last of these two velocities, so
that the first only remains, or in other words, this effect will
be the same thing as if there was impressed on the moveable
a velocity equal and contrary to v sin 8. Therefore if the
given curve be replaced by an infinitismal polygon, its re-
sistance consists in impressing on the moveable at each sum-
mit of this polygon such as M, an infinitely small velocity
v sin 8, in a direction opposite to that of MK.
In order that this resistance may be completely assimilated
to f, a motive force which acts incessantly on the moveable,
the velocity v sin & may be supposed to be produced while this
material point moves from н to H', and dt may be assumed as
the time during which this action continues. The change in
the direction of this force may be also neglected in this inter-
val, and it may be assumed, for example, parallel to the line
MO. Then the measure of the corresponding accelerating
force will be, like each of the forces u, u', u", &c. of No. 147,
the velocity v sin d, which is produced in the instant dt, di-
vided by dt; and if the mass of the moveable be denoted by m,
there will result, for the value off,
mv sin d
f=
dt
Consequently, if & be substituted for sin d, and
shall obtain, since ds = vdt,
ds
for d, we
P
2 L
258
OF THE CENTRIFUGAL FORCE.
mv r
f=
P
The pressure that the curve experiences, and which is solely
due to the state of motion of the material point that describes
it, or the centrifugal force that acts on this moveable, is equal
and contrary to this force f. It follows therefore, that, at any
point whatever of the given curve, such as м, the centrifugal
force exists in the osculating plane, and is directed from the
concavity of this curve along MN, the production of its radius
of curvature, and that its intensity is in the inverse ratio of
this radius, and in the direct ratio of the mass of the moveable
and of the square of its velocity.
170. As this velocity along the side м,м is v, and as it be-
comes v cos d along the following side MM', it follows that its
magnitude is not affected by the curve; for the quantity
v (1-cos d) may be neglected, being an infinitely small quan-
tity of the second order, from which there can only result an
infinitely small diminution of velocity on a part of the curve of
which the magnitude(a) is finite. Hence, then the motion
on any curve whatever is finite, when the moveable is not
solicited by a given force. This has been already observed in
No. 157; but moreover, the reason why this is the case, is
because the angle of contact is infinitely small, for in a point
where two different curves intersect at a finite angle, the
moveable will experience a finite loss of velocity in passing
from one curve to another, which loss will be equal to the pri-
mitive velocity multiplied by the versed sine of this angle.
When the moveable is solicited by one or more given
forces, its velocity varies with the components of these forces
that are tangential to the trajectory, and their normal com-
ponents exert, as in a state of rest, a pressure on this curve,
which must be combined with the centrifugal force.
In general, let mr be the resultant of the given forces which
act on the moveable when it attains the point M. If this motive
force be resolved into two others, the one, in the direction of the
OF THE CENTRIFUGAL FORCE.
259
tangent, the other normal to the trajectory, and represented
respectively by mr and mo; the first force is that which causes
the velocity to vary, and the second will produce the part of
the pressure that is independent of the state of motion of the
moveable. If by the rule for the composition of forces, the
mv2
resultant of mo and of the centrifugal force for be taken,
ρ
the entire pressure exerted on the point м of the given curve
will be obtained, both in magnitude and direction. This
force, divided by the mass of the moveable, or the resultant of
the accelerating forces a and
Q
22
,
ought to coincide with the
ρ
force P of No. 152. It is in fact this, which we now proceed to
verify.
171. We may substitute for equations (5) of this number,
the following, which can be immediately deduced from them.
dxd²y - dyd²x
dsdt2
dsdt2
dzdx dxd2z dz
w),
dx
dy
dx
=Y
X-
P
cosw'
dy cos w
ds
ds
ds
ds
dx
=X
Z
ds
ds
P (d
dz
dx
COS W
ds
ds
de cosa"), (1)
dx
=2-
Y
P
ds
ds
(dy
dz
cos w"
ds
ds
cos'),
dyd²z - dzd³y dy
dsdt2
whichever of these is considered as the independent variable,
we have
dy
dxd³y - dyd²x
dt2
dx2 dx
d.
dt;
dt2 dt
we have also, at the same time,
d.
dy
dx²
dx² ds2
==
dt2
ds2 dt2
>
dx
dt
and because v =
ds
dt'
there will result(b)
d dy
d.
dx ds
ds dt
260
OF THE CENTRIFUGAL FORCE.
}
dxd³y — dyď'x
dsdt2
d.
dy
dx2 dx
v2. — ¿² (dxd³y — dyd²x)
ds² ds
v2
In the same manner, there may be obtained,
dzda dxd2z (dzd'x-drdz)
اتر
ds³
v2
dsdt2
ds3
dyď³z — dzď³y
=v2
(dyď³x — dzď³y)
dsdt2
ds3
Q
If q, q', q″, denote the angles that the force o makes with
the parallels to the axes of x, y, z, we shall likewise have, X, Y, Z
being the components in the direction of these parallels, of o
and of the tangential force T,
dx
X=T
ds
dy
ds
z=r!
dz
ds
Q
+Q cos q, Y=T +Q cos q', z=T+Q cos q";
and by means of these, and of the preceding values, equations
(1) will become
v2
(dx
´dxd³y — dyd²x`
ds, dyd-x)
P
dy
= Q
(da
cos q'
dx
cos w’-
ds
ds
dz
(da
2² (dzd³½ — ded°z) = a (ds
v2
'dzd2x -
ds.3
—
P
(dz
ds
COS w
v² (dyd³z — dzdy) = a
22
ds³
Q
ds
dx
ds
.w),
COS W
day cos q) -
ds
dx
cos q - cos q¹) -
w"),
COS
"
ds
dz
q"
q″ -decos g')-
(dy cos q" -
ds
dy
Р cos w".
ds
ds
dz cosw').
ds
COS
(2).
Now, if y, y', y" be the angles which the direction of the
centrifugal force, that is to say MN, the production of the ra-
dius of curvature Mo, makes with the parallels to the axes of
OF THE CENTRIFUGAL FORCE.
261
x, y, z, drawn through the point м, and if x', y', ', be the co-
ordinates of the centre of curvature o, we shall have
x - x' = p cosy, y — y' = p cos y', z- z' = p cos y",
z
and by combining equations (2) with the formulæ of No. 20,
we may without difficulty deduce from them (c)
y=
[dy (dx
ds\ds
v2
cos y = Q
cos q'-
ρ
P
[dy
dy (dx
dy
cos w
v2
ρ
-ds\ds
ds
ds
COS W
sw).
cosy' =Q[dz (dy cosq" - dx cosq') da da
ds
d-cosq') - dz
dsds
dy cosq) - dz (dis
dz/dz
ds\ds
dx
cos q
C
ds
dz (dz
dx
COS W
dsds
ds
cosq")]
cosw")],
dz
~cosq'c
dz (dy
dz
dx
P
cos w"
w
ds
cos')-da (de cos u'-dy cos )],
w
W
22
ρ
cosy" =Q[da (dz
cosy"—a[da
P
[da
dx (dz
dsds
·ds\ds cosq
COS w-
dx
dsds
ds
da cos q") - dy (dy cos q" - dz cos q')]
ds
dsds
de cos") - dy (dy
ds
ds
dz
cos "
COS
s
ds
sw')].
But because the forces P and Q are perpendicular to the
tangent of the trajectory, we have
រ
dx
dy
dz
cos q +
ds
ds
ds
cos q' + cos q' = 0,
qu
dx
dy
dz
cos w +
cosw'+ cos w"= 0;
ds
ds
ds
this reduces the coefficients of Q(d) in the three preceding
equations to
cos w,
—
cos q, - cos q', cos q", and those of P to
cos w', and cos w"; therefore we shall have
22
lela Bla
cos y + Q cos q = P cos w,
cos y' + q cos q' = P cos w',
cos y" + cos q" P cos w".
Uor M
262
#
OF THE CENTRIFUGAL force.
from which it appears, that the force P is the resultant in mag-
v2
nitude and direction of the two forces and Q, as may be easily
verified.
Չ
172. When the moveable is merely constrained to move
on a given surface, it is necessary that the resultant of the
mv2
motive forces mo and which we know already is perpen-
P
dicular to its trajectory, should be moreover normal to this
surface. Therefore, denoting this resultant by mn, and the
angles, whether acute or obtuse, that its two components
make with a determinate part of the normal to the surface, at
the point where the moveable exists, by and, we shall
have
φ
v2
N = ± (Q cos &+
cos).
ρ
The force N will act in the direction of this part of the
normal, or in the direction of its production, according as
the quantity comprised between the crotchets is positive or
negative, and in order that N may be always a positive
quantity, we should take the superior sign in the first case,
and the inferior sign in the second. This accelerating force
N should be equal and contrary to that which occurs in equa-
tions (3) of No. 151; in fact, these last differ from equa-
tions (5) of No. 152 only in this, that they contain N, λ, µ, v,
in place of P, w, w', w", and by the preceding analysis we
can deduce from them the components of the force N, which
will be equal and contrary to those that have been found for
the force P.
་
In this same case of a given surface, if p', ' denote the
angles that the forces mo and make with an axis, drawn
mv2
P
through the point where the moveable exists on the curve,
tangential to this surface, and perpendicular to the trajectory,
so that we may have
2
cos²+cos²=1, cos²+cos² = 1,
-
OF THE CENTRIFUGAL FORCE.
263
it is necessary that the sum of the components of these two
forces acting in the direction of this tangential axis, should be
equal to cypher, because their resultant is normal to the same
point of the surface; consequently we shall have
Q cos o' +
22
cos = 0;
P
by means of which equation, the inclination of the oscu-
lating plane of the trajectory on the plane which touches the
given surface, can be determined.
' =
When the moveable is not subjected to the action of any given
force, or, more generally, when it is only subjected to the action
of a force tangential to its trajectory, we shall have Q=0; hence
there results cos 0 and 90°; so that the osculating
plane of this curve will be constantly perpendicular to the
given surface. As this is in general the property of the
shortest line between two given points on this surface, it is
this line that the moveable will describe, as has been already
stated (No. 161); but we see now besides, that a force which
is tangential to a surface, such as friction against the given
surface, or the resistance of a medium, will not cause the
moveable to deviate from the shortest line between the two
points, from the one to the other of which it passes.
173. Finally, if the moveable be entirely free, it is neces-
sary that the component of the motive force mR, which is nor-
mal to the trajectory, should be in equilibrio with its centri-
mv2
fugal force
since in this case, there is nothing in the
P
curve, or given surface, which can destroy the normal result-
ant of these two forces. It is necessary, therefore, in the first
place, that the osculating plane of the trajectory should be
that which passes through the tangent and the given direction
of the force mR; naming 0 the angle which this direction, at
any point whatever, makes with the radius of curvature Mo,
this angle should be acute, in order that the normal compo-
nent of the force mr may act in a direction opposite to the
Uor M
264
OF THE CENTRIFUGAL FORCE.
centrifugal force which is directed along MN; and this being
so, we should have
R COS =
v2
P
(a)
When the accelerating force R, to the action of which the
moveable is subjected, is a central force directed to a known
point, and when the curve described about this fixed centre
is known from observation, the radius of curvature p and the
angle ✪ that it makes with the direction of the force R may be
deduced from the equation of this curve; there may likewise
be obtained from this equation and the proportionality of the
areas to the times (No. 155), the expression of the velocity v
at any point whatever of the trajectory; consequently, equa-
tion (a) will determine the value of R, or the law of the central
force, by the action of which the moveable is made to describe
the given curve(f). It is in this manner that Newton dis-
covered the law of the force directed towards the centre of the
sun, which causes each planet to describe an ellipse of which
this point occupies one of the foci; but it will be shown in the
sequel, that proceeding from the same data, this determination
can be effected by a much simpler process.
174. Huyghens, to whom we are indebted for the measure
of the centrifugal force, deduced it from the consideration of
circular motion; and although this method is less direct than
the preceding, it may, notwithstanding, be useful briefly to
explain it here.
Let м (fig. 44) be a material point attached to a fixed
point c, by an inextensible thread cм; if by means of a per-
cussion a velocity a be impressed on it in a direction perpen-
dicular to the length of the thread; and if in order to simplify
the question, no other given motive force be supposed to act
on the moveable; this material point then describes a circle
AMB, of which the centre and radius are the fixed point and
the length of the thread. During this motion, the thread
which retains the moveable will experience, in the direction
OF THE CENTRIFUGAL FORCE.
265
1
of its length, a certain tension which is, in fact, the centri-
fugal force. By applying to the moveable a force equal to
this tension, and constantly directed towards the fixed centre,
we may abstract from the consideration of the thread, and
consider the material point as entirely free. It is therefore, in
virtue of this central force, the magnitude of which is un-
known, combined with the velocity a, that the circle will be
described.
It follows immediately, that the circular sectors traced by
the radius of the moveable, will be proportional to the times
(No. 155); this requires that the arcs of the circle which are
described should be so also. Hence the circular motion will
be uniform; and if s denotes the arc described in the time t,
we shall have s = at. Let m be the mass of the moveable,
ma the central force, and consequently, a the accelerating
force which is to be determined, Whatever the nature of this
force may be, we can consider it as constant in magnitude
and direction for an infinitely short interval of time. Thus,
while the moveable describes MM', an infinitely small arc of the
circle, the force a may be supposed to be constant and pa-
rallel to CM, the radius which is drawn to the origin of this
arc; hence it follows, that if the moveable was not actuated
by the velocity a, the central force would cause it to describe,
in an infinitely short interval of time, the versed sine mn,
or the projection of the arc MM' which it actually describes, on
Cм. Now, the measure of every accelerating force is twice
the infinitely small space, which it is capable of making the
moveable describe in an infinitely short time, divided by the
square of the time (No.118); therefore if a denotes the versed
sine MN, and the time in which the arc мM' is described,
we shall have
2ε
ɑ =
Uor M
but if we denote this arc by σ, and the radius cм by r, we
have
2 M
266
OF THE CENTRIFUGAL FORCE.
ε =
2
2r'
the arc being taken for the chord; hence, because σ = ar we
shall have
a =
a²
r
This value of a is therefore that of the centrifugal force re-
ferred to the unit of mass, in a circle described with a uniform
motion. It follows at once, that this force in any curve what-
ever, will be equal to the square of the velocity divided by the
radius of curvature; for since in the trajectory any two con-
secutive elements are common with its osculating circle, we
can suppose that, during an infinitely small portion of time,
the moveable will move circularly about the centre of curva-
ture, and that consequently it has the centrifugal force which
corresponds to this motion. If this accelerating force be mul-
tiplied by m, we shall obtain the same value as for the force
denoted by fin No. 169.
175. In order to compare the centrifugal force in the circle
with the gravity, let a be the velocity acquired in falling through
the height h, so that we may have a² = 2gh (No. 130), g de-
noting the gravity; then there will result
α
g
2h
which shews that the centrifugal force is to the gravity, as
twice the height through which the moveable should fall to
acquire the velocity a, to the radius.
If the moveable is a body of which the dimensions are very
small with respect to its distance from the point c, the value
of a may be considered for its entire extent, as nearly constant,
α
and consequently we may assume as the expression of the
g
ratio of the centrifugal force arising from circular motion, to
the weight of the body on which it acts.
Maou
#
OF THE CENTRIFUGAL FORCE.
267
When the motion is not performed in a horizontal plane,
the velocity of the moveable, the centrifugal force and tension
of the thread attached to the point c, will be variable. If the
motion takes place in a vertical plane, and if 2gh denotes the
square of the velocity when the moveable is in the horizontal
plane passing through the point c; let z be its distance from
this plane, at any instant whatever, which is considered as
positive when the moveable is situated below, and as negative
when it lies above this plane, 2g (h + z) will at this instant
be the expression for the square of the velocity (No. 159), and
2mg
for the centrifugal force. In order to obtain the
entire tension, there should be added to this force the component
of the weight acting in the direction of the production of its
radius, which component it is easy to shew is equal to
it is easy to shew is equal to mg (g).
the total tension of the thread at any in-
(h+ z)
Therefore, naming
stant, we shall have
0 =
mg(2h+3 z)
p
r
This force expresses likewise the pressure which the point
c experiences at any instant in the direction of the radius ter-
minating at the moveable. It will attain its maximum, when
the moveable is at the lowest point of the circle, i. e. when
z=r, and its minimum, when it is at the highest point, i. e.,
3r
2
when 8= r. If h is less than the tension will become
negative, or will be converted into a contraction (h) during a part
of the motion. It is therefore then necessary, that the thread
should be inflexible in order that the circular motion may have
place. In this discussion, the weight and centrifugal force of
the thread are neglected, because its mass is considered as ex-
tremely small with respect to that of the moveable. It will be
shewn hereafter, how this ought to be taken into account if it
should be necessary to do so.
JorM
268
OF THE CENTRIFUGAL FORCE.
176. Let us now revert to the consideration of the uniform
motion performed in a circle, and let T denote the time in which
the moveable describes the entire circumference; we shall then
have
and, consequently,
2πι
α=-
T
4π²r
α
T2
from which it appears, that the centrifugal force is in the di-
rect ratio of the radius of the circle, and in the inverse ratio
of the square of the time of an entire revolution. When a solid
body revolves about a fixed axis, all its points describe, in the
same time, circles, the planes of which are perpendicular to the
axis; their centres lie in this axis, and their radii are the per-
pendiculars let fall from each point on this same axis; conse-
quently, the centrifugal forces of their several points are to
each other as these perpendiculars. Thus, for example, the
centrifugal force of bodies at the surface of the earth, and
which revolve with it about its axis, is proportional to the radii
of the parallels which they describe; and moreover, this force
at each place acts in the direction of the production of the ra-
dius of the parallel drawn to this point.
177. The force which causes bodies to descend to the sur-
face of the earth, and which is termed weight, is due princi-
pally to the attraction of the terrestrial spheroid on these
bodies. But whatever be the cause of it, there can be no
doubt, that the centrifugal force diminishes this tendency of
heavy bodies; so that except at the pole, where the centri-
fugal force is nothing, the weight is in every other place less
than if the earth had no rotatory motion. At the equator, the
centrifugal force and weight act in opposite directions the one
to the other; therefore, the weight is equal to the excess of
the attraction of the earth over the centrifugal force; conse-
quently we have
Maou
OF THE CENTRIFUGAL FORCE.
269
9= G
4 пър
T2;
g being this weight, & the terrestrial attraction, or the weight
which would have place if the earth was immoveable, r the
radius of the equator, and T the time of the earth's rotation.
As the second term of the second member of this expres-
sion is very small relatively to the first, we have, very nearly,
v = (1–
G(1-4).
g= G
2
In order to find the numerical value of
4 T²r
9T2,
the radius of
the meridian may be assumed in place of r the radius of the
equator, from which it differs but little; we shall then have
2πr 40000000m.
Assuming the second for the unit of time, and neglecting,
in this computation, the small variation of gravity at the surface
of the earth, we have also (No. 115)
g= 9,80896.
Moreover, by (No. 111)
T86164;
hence, we obtain very nearly,
4π²r
9T2
= 285.
Hence, at the equator, the weight is diminished by, in con-
sequence of the motion of rotation of the earth about its axis.
If this motion should become more rapid, T would diminish, and
the centrifugal force would differ less from gravity. Since 289
is the square of 17, it is evident that if the rotation was per-
formed in the seventeenth part of a day, the centrifugal force
would be equal to that of gravity; in this case the weight
would be equal to cypher, and bodies remitted to themselves
would remain in equilibrio.
270
OF THE CENTRIFUGAL FORCE.
In this computation, we have only considered the centri-
fugal force arising from the motion of heavy bodies about the
axis of the earth; and in fact, it is easy to conceive, that the
motion of translation about the sun, which is common to the
earth, to its axis, and to all these bodies, cannot influence
their tendency to deviate from this moveable line. For if we
suppose, for example, a thread parallel to the equator at-
tached to this axis, and terminating in a body situated at the
surface, it is evident that its tension will not be in any respect
changed by the effect of a motion which carries along, at the
same time, the axis, the thread, and the body, without changing
their relative positions.
178. The centrifugal force diminishes the weight in all
points of the earth's surface, but by a greater quantity at the
equator than at any other point, both because the centrifu-
gal force decreases in passing from the equator to the poles,
and also because the angle which it makes with the vertical
increases. Naming r the radius of the equator, μ the latitude
of any place on the surface of the earth, and u the radius of
the corresponding parallel, we shall have
u = r cos µ ;
the non-sphericity of the earth being neglected, the angle µ
will be that which the production of u, or the direction of the
centrifugal force makes with the vertical; therefore the vertical
component of the centrifugal force will be obtained by multi-
4π²u
plying its intensity
T2
by cos u; this gives
4π²r cos² μ
T2
for the diminution of the centrifugal force arising from the ro-
tation of the earth; and, by what precedes, the actual value of
this quantity will be
cos μ
289
OF THE CENTRIFUGAL FOrce.
271
This would be all the diminution that the weight would ex-
perience, if the earth was a homogeneous sphere, and as it is
proportional to the square of the cosine of latitude, the entire
diminution from the pole where μ 90°, to the equator where
µ = 0, amounts to 7. But the earth is a spheroid flattened
at its poles; and for this reason, the attraction which it exerts
on bodies situated on its surface, diminishes from the poles to
the equator; this diminution in each point of the surface is
also proportional to the square of the cosine of latitude; it
should be added to that which is produced by the centrifugal
force; and by this addition the coefficient
very nearly. Therefore it is this fraction
press, as has been observed (No. 117), the total increment of
the weight of a body transferred from the equator to the pole.
becomes (e)
which will ex-
·
t'
CHAPTER V.
EXAMPLES OF THE MOTION OF A MATERIAL POINT ON A
CURVE OR ON A GIVEN SURFACE.
I. Oscillation of the simple Pendulum.
179. A pendulum is, in general, a solid heavy body, which
oscillates about a fixed and horizontal axis. But, in order to
facilitate the comparison of the durations of the oscillations of
different pendulums, and the corresponding intensities of gra-
vity, geometers have devised an ideal pendulum, that is termed
the simple pendulum, and which consists of a heavy material
point, attached to a fixed point, by means of an inflexible in-
extensible string, that is supposed to be void of gravity and
of uniform density; the length of this string is that of the
pendulum.
In a subsequent chapter it will be shewn, that there always
exists a simple pendulum of which the oscillations coincide,
both as to their durations and amplitudes, with those of any
other pendulum whatever. It will also be shewn there, how,
when the form and dimensions of the second description of
pendulum are given, the length of the first can be determined,
and in the discussion on this subject it will appear, that if this
agreement obtains between the motions of two such pendu-
lums in a vacuo, it will also subsist in a resisting medium,
whatever be the function of the velocity which expresses the
resistance. Thus, it will be sufficient to consider the motion
of the simple pendulum either in a vacuo, or in a resisting
medium, which we propose to do in this first section.
Let c (fig. 45) be the point of suspension, CB the vertical
OSCILLATION OF THE SIMPLE PENDULUM IN A VACUUM. 273
passing through this fixed point, and CA the initial position of
the pendulum. If the material point at the extremity of ca,
moves from the point a in a direction perpendicular to ca,
and in the plane of the lines CA and CB, with a velocity repre-
sented by k, it is evident that it will not deviate from this ver-
tical plane, and that it will describe arcs of acircle, of which
the centre is a and radius ca.
After any time such as t, let м be the position of the
moveable, from м and A, let the perpendiculars MP and AD be
let fall on the vertical CB, and let us make
CP = 8, CD = C.
Then, if g denote the gravity, and v the velocity of the
moveable at the point м, we shall have, when the motion is
performed in a vacuum, (No, 159),
v² — k²+2g (z − c),
and if s denotes the arc described by the moveable, we have
ds
dt
— v, consequently there results(a)
dt=
ds
√ k²+2g (z — c)*
If 0 denotes the angle мCB, which will be positive when the
pendulum exists to the left of CB, and negative when the
pendulum lies to the right of the vertical; likewise if a be the
angle ACB, or the initial value of 0, we shall have
ds
de
= a (a — 0), v =
=
α
dt
dt'
a representing CM or CA the length of the pendulum. We shall
have also
z=a.cose, c = a.cos a ;
by means of these values, that of dt will become
dt =
ade
(1)
a)*
Vk²+2ga (cos - cos a
2 N
274 OSCILLATION OF THE SIMPLE PENDULUM IN A VAcuum.
This is the expression which it is proposed to integrate either
exactly or by approximation.
181. There is only one case in which the integration can be
effected in a finite form, and that is when(b)
k² = 2ga (1 + cos a) ;
this equation obtains, when the moveable departs from the
point a with a velocity acquired in falling through a height
equal to ED; E being the most elevated point of the circle in
which the pendulum moves. Making 0=24, and observing that
1+ cos 242 cos²,
we then have(c)
dt
Va dif
g cos f
If this expression be integrated, and the constant arbitrary
determined, so that ↓ = a when t = 0; it becomes, by sub-
stituting in place of,
t=}.
√ a log

α
g
(1 sin 0) (1 + sin a)
(1 + sin ½ 0) (1 − sin § a)°
If the point a coincides with the point E, we shall have
a =π; and this will render the preceding value of t infinite,
whatever may be the magnitude of the angle 0. This indi-
cates that the moveable does not leave the point E; in fact, in
this case, its initial velocity will be nothing, aud as the tangent
at the point E is horizontal, it will remain at rest.
At the point B, 0= 0, therefore in every other case,
1 + sin a
.log
g
1- sin a'
expresses the time that the moveable takes to describe the arc
AB. It will ascend on the semi-circumference BA'E with the
velocity acquired at this point; but it is evident from what has
been stated in No. 159, that it would require an infinite time
to reach the point E; and this is in fact evident from the pre-
OSCILLATION OF THE SIMPLE PENDULUM IN A VACUUM. 275
>
ceding expression, in which when π, too. What-
0 — —
ever be the initial velocity k and the angle a, formula (1)
may be integrated by elliptic functions; so that the times of
oscillations, or of revolutions of a pendulum, may be always
determined by means of the numerical tables of these functions,
but as in practice, it is only necessary to know the durations of
very small oscillations, we shall here restrict ourselves to the
consideration of such.
182. In order that the pendulum may make only small
oscillations on each side of the vertical CB, it is necessary that
the angle a and the velocity k should be inconsiderable; as
this velocity may be rendered entirely evanescent(d), by making
the moveable depart from a point a little more elevated than a,
that is to say, by a suitable increase of the angle a, the gene-
rality of the question will not therefore be affected by sup-
posing k = 0; this supposition reduces equation (1) to
do
9 √2 cos 0 – 2 cos a
dt=-√a
α
By known formulæ, we have
A²
04
cos 0 = 1
+
&c.,
2
1.2.3.4
a²
a4
cos a = 1 ∙
+
&c.
2
1.2.3.4
(2)
And since by hypothesis, the angles a and ✪ are very small,
their fourth powers may be neglected, therefore we shall have
dt
Va
α
do
9 Va²
2
α
02
By integrating and observing that t=0 when 0=a, there
results
α
t=1
hence we can deduce
√ are (cos = 9),
9
mood man sholt as follows
got 2
ས་
2
276 OSCILLATION OF THE SIMPLE PENDULUM IN A VACUUM.
0 = a cos t
Va do
a' dt
α
9. sin t
9
a
α
These formulæ indicate, agreeably to what has been already
pointed out (No. 159), that the pendulum will make an inde-
finite series of equal and isochronous oscillations on each side
of the vertical CB; the velocity vanishes, and the moveable
will return to the point a, where 0=a, whenever t√ will
α
be a multiple of 27, and to the point a' equally elevated as a,
and where = -a, as often as 0 will be an odd multiple of π.
If T denotes the time employed in passing from one of these
extreme points to the other, that is to say, the time of an entire
oscillation, we have(e)
T=T
√
1810
The durations of the two semi-oscillations, the one descend-
ing, and the other ascending, will be respectively equal to each
other and expressed by.
In general, at two instants separated by an interval of
time equal to T, the pendulum will exist on opposite sides of
the vertical CB, in points which are equally distant from this
line, and will be actuated by equal and opposite velocities, for
do
dť
if in the values of 0 and t + T be substituted in place of
t, it is evident that the only change which they undergo is a
change of sign(ƒ).
The pendulum coincides with the vertical when = 0, in
T
which case t is an odd multiple of; hence it follows, that
do
dt == a
and, consequently,
v=±a√ga,
OSCILLATION OF THE SIMPLE PENDULUM IN A VACUUM. 277
expresses the velocity of the moveable at the point B. Naming
b the height DB of its point of departure above B, we shall
have
b=a.(1-cos a) = aa²,
because the fourth and higher powers of a are neglected.
Consequently, abstracting from the consideration of the sign,
the velocity acquired at the lowest point will be(g)
v = √2gb,
which expresses, as it ought to do, the velocity acquired in
falling through the height b.
183. The value of T is, as we have seen, independent of
the angle a; it will even subsist, and be rigorously exact, when
this amplitude a is infinitely small. Therefore, if the pen-
dulum deviates by an infinitely small quantity from the ver-
tical, it will take a finite time equal to π
α
a to return to it.
9
In this movement, the moveable will describe an infinitely
small space in a finite time, this arises from the circumstance
of the intensity of its accelerating force being infinitely small.
In fact, this force is that of gravity resolved in the direction
of a tangent to the trajectory; now in the extent of the in-
finitely small arc that is terminated at the lowest point of
this curve, the tangent makes with the vertical an angle
which differs from a right angle, by an infinitely small
quantity, consequently the cosine of this angle, by which the
gravity must be multiplied in order to obtain its component,
is infinitely small, therefore, this component must be also infi-
nitely small.
This result may be extended to the oscillations of a heavy
material point on any curve whatever, of which the osculating
plane at the lowest point is vertical; for within an infinitely
small extent, the curve coincides with its osculating circle,
and in an extent which is only very small, it deviates from it
very little; hence it follows, that if c be the centre of this
4
278 OSCILLATION OF THE SIMPLE PENDULUM IN A VACUUM.
circle, the duration of the very small oscillations on the curve
on each side of its point B, is the same as for a simple pendu-
dulum whose point of suspension is c, and of which the length
will be CB, the radius of curvature corresponding to this point
B. Therefore, the duration of very small oscillations on all
vertical curves, which have the same curvature at their lowest
point, is the same and independent of their amplitude. When
the osculating plane in this point is not vertical, we must sub-
stitute in the expression for T, in place of the gravity g, its
component in this plane, which is equal to g sin i, i denoting
the inclination of the given plane on the horizontal plane.
184. When the angle a is of a finite magnitude, but very
small, then the preceding value of T is only an approximate
one. In fact, if the fourth powers of a and 0 are retained in
the values of cos a, cos 0, and if they are substituted in formula
(2), we shall have(h)
α
dt =
= -
do
9 Va² ² — 0² √ 1 — 1½ (a² +0³)
√1 (a²+0²)'
12
At this degree of approximation, we must assume
(1 − 1½ • (a² + 0²))~* = 1 +34 · (a² + 0²) ;
2
therefore we shall have
dt = -
Va. (.
2
do
+
(a² + 0²) 10),
a² 02
24 √ a²
which formula may be integrated by the known rules. By in-
tegrating from a to 0 = a, in order to obtain T the du-
0 = −
ration of an entire() oscillation, we find
α
T=T
√= (1 + 1);
;
16
from which it appears, that this duration is a little increased
by the magnitude of the amplitude.
It follows from this expression, that if n denotes the num-
OSCILLATION OF THE SIMPLE PENDULUM IN A VACUUM. 279
!
ber of infinitely small oscillations of any pendulum in a given
time, and n' the number of oscillations of the same pendulum
and in the same time, when their amplitude a is only very
small, we shall have(k)
a
~=n' (1+~~);
16
for this number n' must diminish in the same ratio as the du-
ration of each oscillation is increased by the magnitude of this
amplitude.
185. Although, in the different applications of the pendu-
lum, philosophers always take precautions that the amplitude
of the oscillations should be very small, by which means the
correction relative to the magnitude of a, which has been de-
termined above, will be always sufficiently accurate, it may,
nevertheless, be useful to know the converging series, by means
of which the duration of an oscillation may be expressed,
whatever be its amplitude.
For this purpose, let x and ẞ be the versed sines of the
angles 0 and a, so that
Ꮎ
1. cos 0=x, 1- cos a = ß;
and also, at same time,
dx
do=
√2x
The formula (2) will become
dt =
α
dx
9 √ ßx−x²√1 — 1 ∞
;
and, in order to deduce from these the duration of T, a semi-
oscillation, we must integrate from x = ß, (which answers to
0 = a,) to x = 0, (which answers to 0 = 0.)(1)
Now, developing by the formula for expanding a binomial,
we obtain
(1 − {x) ˜¯ * = 1 + 1 +
1.3 x² 1.3.5 23
+
2.44 2.4.6'8
+ &c. ;
280 OSCILLATION OF THE SIMPLE PENDULUM IN A VACUUM.
in this series, the general term is
1.3.5..
2.4.6
2n − 1
....2n
(
and it always converges, because x is constantly less than 2.
If, therefore, the order of integration be reversed, which we are
permitted to do, by changing at the same time the sign of dt,
and if we make
S
C
xndx
√ Bx-x²
Ani
(n being either cypher or any number whatever) there will
result
T=
1.3
A₂+
2.4
= √². (+-+23++ 1.3.8. J₁₂+ &c.),
g
2.4.6
T being the duration of an entire oscillation.
·}A3+
The values of the definite integrals A。, A1, A2, A3, are con-
nected together in such a manner, that one of them being
known, it is easy to deduce successively all the others. In
fact, we have(m)
S
xndx
Bx - x²
Se
1B).xn-¹ dx
√ Bx — x²
ac ẞ) x²-1dx
√ Bx-x²
= — xn−1
2
+ BS
உ ல
xn-1 dx
√ -
Вx — x²
´ ¯x—x² + (n−1) San—² √ √ ẞx—x³dx,
¹√ ẞx—x²+(n-
B
San² √ Bx—a² dx = B.
xn-1dx
√ Bx - x²
xn dx
´Вx-x²,
hence we infer
SVB
xn dx
x²-1 √ Вx-x² - (n
(n−1)
√ ßx − x²
S√
(2n-1)
+
2
BS
xn-1 dx
√ Bx-x²
and, consequently,
xn dx
Bx-x²
OSCILLATION OF THE SIMPLE PENDULUM IN A VACUUM. 281
xn-1 dx
(2n-1) BS - Ba
S√ √ Bx² + (2n−1)
Вx-x²
At the two limits x=0 and x = ß, we have √ẞx-x²=0;
and, therefore, by taking the definite integrals we shall have,
by means of this last equation,
(2n − 1)ß
An=
An-1 •
2n
If in this formula we make n=1, n=2, n=3, &c., suc-
cessively, we obtain from it
A₁ =
BA0,
A₂ = BA₁ =
1.3
2.4Ao
1.3.5
33
Ao
A3 = &BA2 = 2.4.6
&c.;
consequently, we shall have, generally,
1.3.5..
2n-1
An=
2.4.6...... 2n
B" Aoi
and as to the value of a。, it is, when taken between x = o and
x = B,
A。=
SO
dx
=T.
Bx-x²
By substituting these values of Ao, A1, A2, &c., in that of
T, there will result
2
1.312
π== √ √ ~ [¹ + ( ) ( ) + (-_-) '()*+ (1.35)*()*+&c.],
TET
2.4/
2.4.6
for the required value of T, and which necessarily converges,
since ẞ is always less than unity.
If the fourth power of a be neglected, we shall obtain
Ba²; hence this series will be reduced to its two first
terms, and the value of T will coincide with that of the pre-
ceding number.
+
20
282
OSCILLATION OF THE SIMPLE PEndulum
186. Let us now proceed to consider the motion of a simple
pendulum in a resisting medium. If the preceding notations
be retained, the force of gravity resolved in the direction of the
tangent MT will be g. sin 0, because the angle which this line
makes with the vertical MN is the complement of the angle
MCB or 0. As the accelerating force arising from the resist-
ance which we denote by v, acts in a direction contrary to
g.sin 0, the equation of the motion will be (No. 152)
d's
dt2
dr² = 9 sin 0 - v.
g
(3)
(in which s denotes the arc AM). We may make different
hypotheses as to the value of v, considered as a function of
the velocity of the moveable; the simplest is that in which it
is supposed to be proportional to this velocity, in which case,
we have
V=
9 ds
k'dt'
k denoting a given constant velocity.
We have also
03
s=a (a-0), sin 00-
+ &c.;
1.2.3
if, therefore, be considered, as heretofore, a very small angle,
and if its third power be neglected, equation (3) will become
ᏧᎾ
хо
do
de + 2 ⋅ at + 2.0 = 0,
dt2
dt
of which the complete integral is(n)
= V
9
0 = ( ccos by √2+ c' sin ty √2),
ty
α
a
1)
e
in which c, c', denote two constant arbitraries, e the base of
the Naperian system of logarithms, and also, for the sake of
abridging,
1
Vi-ga
472 = Y,
ولا
4
*
283
IN A RESISTING MEDIUM.
cand c'are determined by the conditions 0 a and
t = 0, which gives
c = α,
c' =
a√ ga
2yk
consequently, we shall have
9
0 = a(costy √2+ Ya sin
and, by differentiating,(0)
a
2yk
sin ty Ve
√ga ty
do
α
a
√2 (sin ty √2)
Vo
2k
e
do
0, when
dt
2k
e
dt
for the expressions, by means of which we can determine at
any instant whatever, the position of the pendulum and its
angular velocity.
do
At the end of each oscillation, we have = 0; this is the
9
case as often as ty√ is a multiple of .
α
dt
is a multiple of π. It follows, there-
fore, that the oscillations are isochronous, as in a vacuum, and
that we have(p)
π
T=
γ g
for the duration of an entire oscillation, so that it is increased
by the resistance of the medium, in the ratio of unity to the
fraction 7. As to the amplitudes of the oscillations, it is evi-
-gt
dent from the form of the exponential e, that they continu-
ally diminish. Naming an the amplitude of the nth oscillation,
that is to say, supposing that 0 = (− 1)" an when t — nT,
there will result(q)
an=αe
-nn√ ga
2yk
which shews, that the successive amplitudes constitute a
284
OSCILLATION OF THE SIMPLE PENDULUM
decreasing geometrical progression, of which the ratio is
-πVga
e 24k
This oscillatory motion supposes that T is a real quantity;
and, in fact, this is the case in all experiments of the pendulum,
in which the length is never very considerable, its density
being at the same time always very great, compared with that
of the air in which it moves; and as the velocity k is pro-
portional to the ratio of the first density to the second, it is
very considerable compared with ✔ga, and, consequently, y
is a real quantity, that differs very little from unity. How-
ever, if 2k was less than √ga, y would be imaginary, and of
the form ẞ1, ẞ denoting a real quantity; by the known
formulæ, the sines and cosines which occur in the expression
for 0, will then be transformed into exponentials, and when
this transformation is effected, it is evident from inspection,
that the time required to lapse, before ✪ vanishes, is infinite(r),
so that the pendulum will approach indefinitely near the ver-
tical CB, without ever passing or even attaining to it.
187. It appears from experiment, that according as the
amplitudes of the oscillations performed in the air diminish,
they tend more and more to decrease in a geometric progres-
sion; for example, they deviate little from this progres-
sion, when the angle a is the third of a degree, or somewhat
less. Experiment likewise proves, that this decrcase is very
slow; thus, in the experiments of Borda, in which the ampli-
tudes of the oscillations constitute, as to sense, a geometric
progression, the amplitude was reduced to about two-thirds,
after 1800 oscillations. Therefore, if to this example, the ex-
pression for an, given above, be applied, we shall obtain
and, consequently,
1800TV ga
2k
e
-1800π/ga
2yk
= 3/3,
= y logy (0,40546);
IN A RESISTING MEDIUM.
285
but we have
ga
= 1-7²;
4/2
1
hence there results
(1800)² π² (1 — y²) = y2 (0,40546)²;
from which we obtain
y= 1,00000000257 ....
or y= 1, q.p.; we are therefore permitted to neglect the con-
sideration of the resistance of the air in computing the value of T.
It follows from this, that when the oscillations are very small,
the resistance of the air may be assumed to be proportional to
the velocity, as has been supposed, and also that this resist-
ance does not sensibly influence their duration. But when the
amplitudes are of some magnitude, it appears from observa-
tion, that they do not decrease in a geometric progression, so
that it becomes necessary to make some other hypothesis on
the law of resistance.
188. Let it be assumed that this force is proportional to
the square of the velocity, and let
g ds2
V=
k² dt²
k being a given constant velocity, which will always be very
great; so that if we make
2 ga
k2
= μg.
μ will be a very small fraction. Since dsade, equation
(3) will become
d20
dt2 α
+2 sin 0 = 1 μ
d02
dt2
multiplying this equation by 2d0, integrating and making
(4).
Sa
dt2
Sdope do
do²
dy
d0 = y,
dt2
do’
we shall have
286
OSCILLATION OF THE SIMPLE PENDULUM
dy
2g
cos
do α
y=0 ;
this being a linear equation of the first order, its complete in-
tegral is(s)

у = семь +29
(sin 0 — µ cos 0)
(1 +¸µ²) a
c being a constant arbitrary, and e the base of the Naperian
d02
system; differentiating with respect to 0, and substituting dt2
in place of
dy
; there results

d02
0
(cos + u.sin 0)
dt2
= мсемь +29-
;
(1 + μ²) a
which is a first integral of equation (4) given in a finite form.
In order to determine c, let us assume, as has been done
do
above, that
= 0, when = a; then there will result
dt

(cos au sin a)
με - 2g
сна,
(1 + μ²) a
consequently, we shall have, at any instant whatever,
d02
2g
dr² + (1+u²) a [cos 0+usin✪ — (cos a+usina) e¯µ(a−0)]. (5)
Therefore, at the lowest point, where = 0, we shall have
a²d0²
dt2
2ga
1+μ²
2
½ [1 − (cos a + µ sin a) e̟-μа],
for the expression for the square of the acquired velocity,
which is evidently less than in a vacuum.
In virtue of this
velocity, the moveable will ascend on the arc BA' to a point A
less elevated than A', and for which we shall have
If the corresponding value of @ be denoted by
will result
do
= 0.
dt
a₁, there
IN A RESISTING MEDIUM.
287.
(cos α1
μ
sin a₁) eμа₁ = (cosa +µ sin a) e-μa,
and if the exponentials be developed according to the powers
of the square
με
of μg which is a very small fraction, being
neglected, we shall obtain(t)
còs α¡— µi (sin α1 — α1. cos a₁) = cos a + µ (sin a a cos a).
The value of a, which can be deduced from this equation,
differs very little from a ; therefore, by making a₁ = a -8, and
neglecting the square of ♪ and the product ud, there results
so that we shall have(u)
8
S sin a = 2µ (sin a
a cos a);
2μ
a₁ = a
(sin a
a cos a),
sin a
as an expression, abstracting from the sign, for the magnitude
of 0 at the end of the first oscillation.
This result does not imply that the oscillations are very
small; however, if they are so small, that the fourth power of
a may be neglected in the expression for a, it will become(v)
α π α
2 μαζ
3
After the moveable attains the point A₁, it will descend
again, and will thus continue to oscillate from one side to the
other of the point в, until the amplitudes of these oscillations
to all appearance vanish. If a₂ denotes the amplitude of the
second ascending demi-oscillation, it is evident that it can be
deduced from a₁, as a has been deduced from a, so that we
shall have
a2
2 μαι
2
α2 = α
3
and in like manner, if aз, a4, &c., be the successive amplitudes
of the other ascending demi-oscillations, we shall have
}
;
༔
288
OSCILLATION OF THE SIMPLE PENDULUM
2 μα,
3
Σμα
2
04 - аз
&c.;
3
which shews that they no longer decrease in geometric pro-
gression, as in the case of a resistance proportional to the
velocity.
189. In order to determine the time in which the angle
is described, the value of dt deduced from equation (5) must
be integrated, and this can be always effected by the method
of quadratures, when the numerical values of a, µ, 0, are
known. But when the oscillations are small, the value of 0
considered as a function of t, may be obtained in a converging
series, and vice versa.
The initial velocity of the moveable is always assumed
equal to cypher; and the value of 0 at any instant whatever
will be a function of a and t, which should become cypher
when a 0; therefore it can be represented by
0 = a01 + a²02 +a³03 + &c. ;
01, 02, 03, &c., being coefficients independent of a. By sub-
stituting this series in equation (4), and then developing the
two members according to the powers of a, we shall obtain by
putting the coefficients of the same powers respectively equal
to each other, a series of differential equations of the second
order, by means of which the unknown quantities 01, 02, 03,
&c., can be determined. Moreover, in order that we should
de
dt
have 0 = a and = 0, when t=0, whatever may be the
magnitude of a, it is necessary that the initial values of 02, 03,
do₂ do
2
&c.,
do
dt
3
dtdt, &c., should all vanish, and that those of 0, and
should be unity and cypher; it is by means of these con-
ditions that the constant arbitraries which are contained in the
complete integrals of these series of equations, can be deter-
mined. In this manner, we may compute as many terms as we
IN A RESISTING Medium.
289
please, of the preceding series. If the approximation be limited
to the square of a, in which case the cube and higher powers
of this quantity should be neglected, we obtain
d20
ad20,
dt2
dt² + a²
2
ď²02
dt2,
sin 0 = a0₁ + a²Ð½
2
α
d02
d0,2
dr² = a²
dt2
;
dt2
and if these values be substituted in equation (4), there results,
by equalling the coefficients of a and of a² in its two members,
ᏑᎾ, g
+ 2/0₁₂ = 0,
dt2 α
d²02, g
d0¸²
2
+ 12/02 = 1 μ
dt2 α
dt2
If the first of these two equations be integrated, and if the
two constant arbitraries be determined in such a manner that
0₁ = 0 and
de,
dt
=0, when t=0, we shall have
0₁ = cos t
9
a
Hence there results
d0₁²
2
g
2
dt2 α
2 sin' t√2=2(1-co
2 (1—
a
V
1- cos 2t √2)
consequently the second equation will become(t)
2t
de: +20,= 24 (1- cos 21 √2).
dt2 a
a
;
and then
02=
μ
g
-c
cos t
++μ cos 2 t
9
a
will be its complete integral subject to the conditions 0₂ = 0
d02
dt
and = 0, when t = 0.
A
2 P
290
OSCILLATION OF THE SIMPLE PENDULUM
By means of these expressions for 0, and 02, that for @
becomes
α
2
9
g
0 = (a− a) cost √2+ a² + a² cos 21 √2
3
a
a
αμ αμ
4 12
ado
and because v
we shall, at the same time, have
dt
α
v = (a− a ) √ ga sint√2+ again 21 ?
(aa) v.
α
6.
2t V
a
these formulæ make known the position and velocity of the
moveable at any instant whatever.
190. If in the last equation, we substitute in place of
g
g
sin 2t √2
its value 2 sin t√ cost√2, the equation v=0,
α
which obtains at the end of each oscillation, will assume the
form
αμ
αμ
(1 - a + cos t√2) sint √
13
十
​α
0.
As the angle a is very small, the first factor cannot vanish ;
the second is cypher as often as t√ is a multiple of 7. It
a
follows, therefore, that the interval of time which lapses be-
tween two consecutive velocities, respectively equal to cypher,
or T the duration of an entire oscillation, is(u)
T=T
T = 7
= √
α
hence it appears, that when the resistance of the air is pro-
portional to the square of the velocity, the duration of the
oscillation is not affected by it. However, it increases the
time which the moveable takes to attain the point B. In fact,
if we denote it by ť, and make 0 = 0, we have
cost' √2 + 4 + al cos 21' √2
(1 − a) cos
9 αμ αμ
12
α
t'
α
=
= 0.
IN A RESISTING Medium.
291
The least value of t√, which satisfies this equation,
t'
α
differs little from ; let therefore,
T
+ √ ½ = & π + d ;
ť
α
neglecting the square of 8 and the product ad, we obtain(v)
and, consequently,
δε γαμ,
a
αμ
C = = √ ½ ( 1 + 1 ),
t'
3п
hence it appears that the resistance of the air increases the
duration of the first descending semi-oscillation in the ratio of
αμ
1+auto unity, and since it does not influence the duration
3п
of an entire oscillation, it must diminish in the same ratio, the
duration of the ascending semi-oscillation.
By substituting this value of t' in that of v, and neglecting
the cube of a, we obtain(x)
v= (1-)avga.
αμ
3
α
hence we infer, that the velocity acquired at the lowest point
is diminished by the resistance of the air, in the ratio of 1 –
to unity.
aft
3
If α denotes the value of 0 at the end of the first entire
oscillation, which answers to that of t
=π, we shall ob-
α
tain as before(y)
α = α
2ua®
These different results are independent of the magnitude
of μ, the coefficient of the resistance, and only suppose that the
angle a is very small; they apply equally to the motion of a
Madr
292
OSCILLATION OF THE SIMPLE PENDULUM
3
pendulum in an aeriform fluid and in a liquid, provided that
the coefficient μ be determined for each medium in particular.
When a is very small, it is unnecessary to consider the case of
a resistance proportional to the cube or any higher power of
the velocity, because in the values of 0 and v, there can only
result terms depending on powers of a higher than the square,
which we have shewn in the preceding calculations ought to
be neglected. Therefore, if what has been stated in No. 187
be considered, it will be manifest, that the resistance of the air
does not influence the duration of very small oscillations of the
pendulum, for which the correction relative to the magnitude
of the amplitude is neglected (No. 184). When this cor-
rection, however, is taken into account, the resistance has a
small influence, because it causes the amplitudes to vary during
the continuance of the motion.
191. It does not follow from this, that the duration of the
oscillations of a heavy body, however small they may be sup-
posed to be, is the same in the air and in a vacuum; for this
fluid, by the pressure which it exercises on the moveable, in-
creases this duration, by diminishing the gravity. It can be
shewn by experiment, and it will be hereafter demonstrated
when we come to treat of Hydrostatics, that a body at rest,
when plunged in a fluid, loses a part of its weight equal to
the weight of that portion of the fluid which it displaces.
Thus, p being the weight of this body in a vacuum, p' its weight
in air, п the weight of a volume of air equal to that of the
body, we have
PP-II.
Naming p the ratio of the density of the air to that of the
body, g the gravity in a vacuum, g' what this force becomes in
air, and m the mass of the body, we have
P
PP,
п=Рp, Р= mg,
P'= mg';
consequently there results
g′ = g(1 − p).
IN A RESISTING MEDIUM.
293
Now, if T, T' represent the durations of the small oscilla-
tions of the same pendulum, when the accelerating forces are
respectively g and g', we shall have TT
唔
​√ T = √
α
,
gis
and, consequently,
T
T
1 p.
Likewise if a' be the length of the pendulum which, actu-
ated by the gravity g', makes oscillations isochronous with the
pendulum actuated by the force of gravity g, and of which the
length is a; we must have
g
ges
and, consequently,
a' — a (1 − p).
Therefore, if the loss of weight which a body sustains in a
state of repose be solely taken into account, the durations of
oscillations made in the air will be increased in the ratio of
ρ
unity to 1-p when the length of the pendulum is sup-
posed to be the same, and if the duration be the same in the
air and in the vacuum, the length will be diminished in the
ratio of 1 — p to unity. Moreover, M. Bessel has shewn by
experiment, that the loss of weight which any body sustains
in the air, is not the same when the body is at rest, and when
it has an oscillatory motion. The loss is greater in the second
case; it is therefore necessary, in the preceding formulæ, to
muitiply p by a factor ƒ greater than unity, and depending on
the form of the moveable. The same result was also obtained
by M. Poisson in a memoir, published in the eleventh volume
of the Transactions of the Academy, On the simultaneous Mo-
tions of a Pendulum and of the ambient Air; by his analysis he
found fwhen the pendulum consists, like that of Borda,
of a sphere attached to the extremity of a very slender thread,
its length being very considerable compared with the diameter
294
OSCILLATION OF THE SIMPLE PENDULUM
of this sphere; so that the correction relative to the density
of the air, which, previously to the observation of M. Bessel,
was applied to the duration of small oscillations and length of
the simple pendulum, must be increased by one-half. In all
cases, the coefficient ƒ is independent of the density of the pen-
dulum, and also of the density and nature of the fluid in
which it vibrates, so that we may always determine it by ex-
periment, by comparing the durations of the oscillations of
two pendulums of the same form and of different densities, in
the same fluid, or even of the same pendulum in two dif
ferent fluids, such, for instance, as air and water.
192. Now let n be the number of infinitely small oscilla-
lations that any pendulum makes in a vacuum during a given
time T. In order to deduce this number, by the rule of No.
184, from that of very small oscillations, which is given by
observation, and at the same time to take into account the
variation of the amplitudes during this timer, we must assume
for the angle a the mean of the extreme amplitudes which are
also furnished by observation. This being premised, T the
duration of an infinitely small oscillation of this pendulum
will be
T-
T
N
and the error into which we may fall, in measuring the time
7, will have so much less influence on the value of T, as the
number n will be more considerable. If the form and di-
mensions of the vibrating body be known, we can determine,
by means of a formula that will be given in a subsequent
chapter, the length of the simple pendulum, of which the
motion is the same as that of this body; and then this length
can be reduced to what it would be in a vacuum, in the way just
now explained. Hence, if a denotes what it becomes after
this reduction, we shall have
π
N
IN A RESISTING MEDIUM.
295
(where g denotes the force of gravity in a vacuum) from which
we obtain
π²n²a
g
(a)
It is by means of this formula, that the measure of the gra-
vity, or g, the velocity which heavy bodies acquire in falling
vertically in a vacuum, during a unit of time, is determined.
From experiments made by Borda at the Observatory of Paris,
with a pendulum, the length of which was about two metres,
there results
a = 0m,993855,
(the unit of time being assumed equal to a second,) hence we
deduce,
g= 9m,80896,
in the latitude of Paris, which is equal to 48° 50' 14". M.
Bessel having made bodies of every species of matter to vi-
brate, such as metals, ivory, marble, meteoric stones, &c.,
invariably found that the values of g were sensibly equal; the
greatest differences, on one side or the other of the mean value,
scarcely amounting to the hundredth thousandth part of this
value, and even this may be attributed to the inevitable errors
of observation. There can therefore exist no doubt as to the
perfect equality of the attraction exercised by the earth on all
bodies, whatever may be their nature, which are situated at
the same place of its surface; for this equality results from
that of the values of the gravity g, since this force is the excess
of the terrestrial attraction over that part of the centrifugal
force common to all bodies, which is resolved in the direction
of the vertical.
193. It is demonstrated in the Mechanique Celeste, that
if the surface of the earth is the same as that of the level sur-
face of the sea when at rest, the variation, at this surface, of
the length of the simple pendulum, which vibrates in a unit
of time, is proportional to the cosine of double the latitude;
296
OSCILLATION OF THE SIMPLE PENDULUM
"
so that if a denote this length, in a place of which the latitude
is, we should have(z)
W
λ = 1 (1 — w cos 24);
W
(b)
I and a being constant quantities which must be determined
by observation. It is likewise shewn, that the coefficient w is
connected with the compression of the terrestrial spheroid by
the equation
2w+d={r,
in which is this compression, (so that the radius of the
equator and that of the pole are to each other in the ratio of
1+d to unity,) and r denotes the ratio of the centrifugal
force to the force of gravity at the equator; the numerical
value of this last is (No. 177)
r = 289.
The formula (b) is, in fact, confirmed by experiment, when
we abstract from the consideration of all those local circum-
stances, which, as we shall see in the sequel, may influence
the attraction of the earth and the length of the pendulum.
From a comparison of a great number of observations, made
at different latitudes, we obtain
w = 0,002588;
from this it follows, that 8 is very nearly equal to r. The
constant 7 is the value of a corresponding to = 45°; it dif-
fers but little from that value of λ, which corresponds to the
latitude of Paris; and by this last, we have
0,993855 = 7 [1 + 0,002588 . sin (7° 40′ 28″)] ;
hence we obtain
1=0m,993512.
T=
If in formula (a) we make n = 1 and T = 1, and then sub-
stitute successively 7 and λ in place of a, we shall have, if
p and p' denote the corresponding values of g,
5
IN A RESISTING MEDIUM.
297
p = r²l, ´ ´p' = = πλ;
consequently, we shall have
p=9m,80557,
and for any latitude whatever,
Since
p' = p(1 — 0,002588 cos 24).
cos 242 cos²↓ — 1,
it is evident that the diminution of gravity, as we go from the
pole to the equator, is proportional to the square of the cosine
of latitude, conformably to what is stated in No. 178.
If the same pendulum be transferred to different places on
the earth, it is evident from equation (a), that n the number
of its oscillations, in the same time 7, will vary proportionably
to the square root of the gravity. Thus, for example, a clock
regulated at Paris, by the diurnal motion of the earth, and
then transferred to the equator, will lose.
pen-
Naming n and n' the number of oscillations which this
dulum performs in a sidereal day, in these two places respec-
tively, we shall have
n = 86164, n' = n
and, consequently,
1 — 0,002588
1+0,002588 sin (7° 40′28″)'
n'= 86037;
so that a clock, when transferred to the equator, loses about
127 seconds in 24 hours. It was from observing that a clock
lost when carried to the equator, that philosophers first esta-
blished the variation of gravity on the surface of the earth.
II. Motion on a Cycloid.
194. Let ABC (fig. 46) be the trajectory of a heavy mate-
rial point, the plane of which is vertical. Let us suppose that
this moveable departs from any point D without initial velo-
J

2 Q
298
OSCILLATION OF A SIMPLE PENDULUM
འ
1
city, and at the end of the time t, let it be at м; from D and M
let fall the perpendiculars DE and MP on the vertical passing
through the point B, which is the lowest point on the curve;
then by making EP = 2, and denoting the velocity acquired at
the point м by v, and the gravity by g, we shall have (No. 159)
v=√2gz,
provided that the gravity is the sole force which acts on the
moveable. Likewise let s denote the arc BM; since it decreases
as the time increases, we shall have
and if we make
there will result
ds
บ
;
dt
EB h, PB = xhz,
ds
√2g dt =
(1)
√h-x
This curve being
whatever be the nature of the given curve.
by hypothesis a cycloid, we shall have (No. 73)
s²-4ax,
a denoting the diameter BF of its generating circle. Hence
we obtain
and, by integrating,
29 dt =-
dx
√ hx — x²³
α
1√34 = are (cos = 2x=h):
t
α
a constant arbitrary is not added, because we have t = 0, at
the origin of the motion, or when xh.
If t' denotes the time that the moveable takes to reach the
point в which corresponds to x = 0, we shall have
2g
t'
= arc (cos=-1) = π,
a
IN A CYCLOID.
299
and, consequently,
α
t'= π
T
2g
Hence it appears, that this time is independent of h the height
of the point D, (from whence the moveable commenced its mo-
tion,) above the lowest point B. So that this property that
obtains only approximately, for all curves of which the height
h is very small, is rigorously true for the cycloid, whatever this
height, within the limit a or BF, may be. Hence it follows,
that all moveables which commence to move simultaneously
from different points of the cycloid, will arrive at the lowest
point in the same time.
It appears, therefore, that ✔
π
expresses the time of an
9
entire oscillation from one side to another of the point в ; now, it
B
is evident, that this is the time in which very small oscillations
of a pendulum, whose length 2a is the radius of curvature of
the cycloid at this point, are performed (No. 72). And it thus
accords with the result of No. 185, relative to the duration of
very small oscillations on any curve whatever, which duration,
in the case of the cycloid, is the same as those of oscillations
of any amplitude whatever.
195. The time of describing the arc DB of the cycloid is
likewise independent of the length of this arc, when the mo-
tion is performed in a resisting medium, provided that the
resistance is proportional to the first power of the velocity.
gv
In fact, if we denote this force by as in No. 186; the
force of gravity resolved in the direction of the tangent MT is
dx
I ds'
dx
because is the cosine of the angle TN that this line
ds
makes with the vertical MN, therefore the force which acts at
the point м, and which tends to diminish the arc MN or s, is
the difference g
dx
ds
V
-9; consequently, we shall have for the
equation of the motion,
300
OSCILLATION OF A SIMPLE PENDULUM
+
d's
dx
dt2
g
ds
127) >
or, what is the same thing,
because
d's
g ds
gs
dt2
k dt
+ + = 0,
:
2a
ds dx
s
บ
dt'
ds
2a'
If at the commencement of the motion, i. e., when t = 0,
the velocity is nothing, and if then sa, by determining the
two constant arbitraries by means of these conditions; and
making, for the sake of abridging,
1-
ga
2k2 2,
the integral of the preceding equation will become (No. 186)
g
α
s=a. (costy √ / +
V
√2ga
2yk
sin ty V e
2k
2a
g
2a
Therefore, if t' denotes the time when the point в, at which
s=0, is attained, we shall have
cos t'y
costy√+V
2a
2yk
√2ga sin t'y
sint'
g
=
=0;
2a
by means of which equation we can deduce a value of t' inde-
pendent of a ; which was required to be done. If the resist-
ance be very small, or the velocity k very great; we shall
have y 1, very nearly; and the preceding equation will
give(a')
ť
g
2a
√2ga
όπ+
2k
which shews that the time t is somewhat increased by this re-
sistance.
IN A CYCLOID.
301
196. If the line BF be continued to o, so that of may be
equal to BF; this point o will be the centre of the circle which
osculates the cycloid at the point B ; and if two equal semi-
cycloids oa and oc be described, touching the lines OB and AC,
and having the diameter of their generating circle equal to
of, oa will be the evolute of AB, and oc that of BC (No. 72);
consequently, if a thread of a constant length, equal to oв, or
2a be attached to the point o, and be then made to envelope
the two curves oa and oc successively, its other extremity will
trace the cycloid ABC.
This suggests a means of constructing a cycloidal pendulum.
For this purpose, let the curves OA and oc be traced in relief,
and let oв be an inextensible and perfectly flexible thread
attached to the fixed point o; if we also attach a heavy body
to its other extremity B, and cause this thread to deviate from
the vertical position in such a manner, that it may envelope
entirely, or in part, either of the curves oa or oc, and that
the part which is not enveloped may be a tangent to this
curve, then when the moveable is remitted to itself, the infe-
rior extremity of the thread will describe the curve ABC, and
from No. 194 it follows, that the duration of the oscillations
of this pendulum, in a vacuum, will be rigorously independent
of their amplitude. But in practice this method is not suscep-
tible of much precision; and besides, when the body oscillates
in the air, the durations of the oscillations will not be equal
when the amplitudes are considerable, for the resistance of
this fluid is not then proportional to the first power of the
velocity.
197. Those curves are termed tautochronous, on which
if a material point moves, it will always arrive at the lowest
point at the same time, from whatever point of the curve it
may have commenced to move. Thus in a vacuum,
the cycloid
is a tautochronous curve, and moreover, it is the only curve
to which this property belongs in a vacuum, as we now proceed
to demonstrate. Ift' denotes the time, which the moveable takes
or Wing/


forthen 4
302
H


21/3/2
for the tie d
OSCILLATION OF A SIMPLE PENDULUM
if si
$ - k
to go, without any initial velocity, from the point D to the low-
est point B, on ADB any curve whatever, the value of t' √2g
will be given by the integral of formula (1), taken from x=h,
to x = 0, or, which is the same thing, from x=0 to x = h,
the sign of this formula being changed; consequently, we shall
have
t'
ť √2g = So √ h
Sh
ds
Vh X
X
and in order that the curve should be tautochronous, it is ne-
cessary to determine s as a function of x,
sion for t'√2g may be independent of h.
such that this expres-
Now, if we suppose
that this unknown function is developed according to the
ascending powers of x, so that we may have
Bxß
s=Axª + вx² + cx² + &c. ;
A, B, C, &c., α, ß, y, &c., being indeterminate coefficients and
exponents, then as the abcissa x and the arc s have their origin
at the same point в, we must have x = 0 and s=0 at the
same time; it is therefore necessary that all the exponents
a, ẞ, y, &c., should be positive, and that none of them should
be cypher. It is likewise evident, à priori, that the least of
them should be less than unity, for as, by hypothesis, the
point of в is the lowest point of the given curve, the tangent
there is horizontal or perpendicular to the axis of x; this re-
quires that we should have
∞ when ∞ 0(b). If the
ds
dx
differential of this series be taken, and then substituted for ds in
the preceding formula, there results
12
人
​t' √2g=sa
Sh
Chxa- dx
Ch x-1dx
+BB
• √h-x
•h x-1dx
+undo.
0 √h-x
+&c.
• √h-x
And if we assume xhx' and dx = hdx', the limits of the
integrals relative to this new variable x will be zero and
unity; for example, we shall have(c')
4110x
In troben les 154, let 5=gl/
kish-m
موم
4/nson = /W
1inth + oh)
h+th)
IN A CYCLOID.
303
Sh
h xa-1 dx
ha~ →S!
x'α-1 dx'
So √h-x
and if, for the sake of abridging, we make,
la-1
Stav
Sta
x'α-1 dx'
1 x'B-1 dx'
= A',
= B', &c.;
• √1 — x²
there will result
t' √2g= ass'ha→++ẞBB'h² + уcc'hr→ + &c. ;
it is important to observe, that none of these integrals A',B', c',
&c. can become equal to cypher, for the values of the dif
ferentials, of which they constitute the sums (No. 13), do not
change their signs between the limits of the integrations,
therefore, as these values are all positive, the values of the
integrals will be so likewise. Now, it is evident, that the
value of t cannot be independent of h, unless all the terms
of the preceding series vanish, with the exception of that in
which the exponent of h is zero, or which corresponds to that
one of the exponents a, ß, y, &c., which is equal to . Let
us suppose, that this term is the first, and that we have a=.
In order that the second term may disappear, it is necessary
that the product ẞBB' should be cypher, this implies that
B should be zero, because neither ẞ nor B' do vanish. In
like manner it may be shewn, that the other coefficients C, D,
&c., are also equal to zero, so that the equation of the tauto-
chronous curve is reduced to the following,
s = Ax³, or s² = A²x,
this indicates that it is a cycloid, whose base is horizontal, and
whose summit is at the point B which the moveable always
attains in the same time.
If a denotes the diameter of the generating circle, we shall
have a² = 4ɑ, and, consequently,
t'√2g=s'√a.
('Jag. Wheth (123)
(170)
: 2
12.
K
= (0)
2K (2 (1467)/h = 4(4(110)
a
112
0.
304
OSCILLATION OF A SIMPLE PENDULUM
Besides, as a = 4, (d') we have,
hence we obtain
as in No. 194.
A' =
S₂
dx'
X/2
t'
√
α
" = π V.
2g'
=T;
198. It is also the equation of the cycloid that we arrive
at when we wish to determine the brachystochrone curve in a
vacuum, that is to say, the curve AMB (fig. 47), which a heavy
material point must describe, in order to pass in the least
time, and without any initial velocity, from a given point
A to another given point B.
In order to determine this curve, let x, y, z, be the three
rectangular coordinates of the point м, where the moveable
exists at the end of the time t; likewise let s denote the arc
AM which it has traversed. If the axis of x be vertical, and
in the direction of gravity, and if a denote the value of x at
ds
the point A, the velocity acquired at the point м, will be
dt
equal to the velocity acquired in falling through the height
Х a; therefore, if g represents the gravity, we shall have
ds
√2g(x − a);
dt
and if in order to abridge, we make
dy2 dz2
√1+ +
dx2 dx2
- И,
in which case ds = udx, there will result
√2g dt =
udx
√x α
Therefore, if ẞ denotes the value of x at the point в, and
t' the time which the moveable takes to pass from the point à
to the point в, we shall have
IN A CYCLOID.
305
udx
t'√2g=
SB
a Va
x-α
Consequently, the question is reduced to the determination
of the curve, for which this integral is a minimum, but, for
greater generality, let the integral
U=
CB
So
a
xudx
be considered, in which x is a given function of x, for this
will be useful in the sequel, in the resolution of another pro-
blem of the same kind; in the one we are now concerned with,
we shall assume (x a) for x.
199. Let i denote a constant quantity infinitely small, and
let Sy and Sz denote two arbitrary functions of x, subject
solely to the condition of vanishing when xa and x =ß,
and of not becoming infinite for the intermediate values of x.
Let u' and u' be what u and u become when y+idy and
z+idz are substituted in place of y and z, so that we shall
have
Ư
=SB xu'dx;
α
an integral which will correspond to another curve AM'B,
passing, in like manner as the required curve AMB, through
the points A and B, and deviating almost insensibly from this
last. We shall have by this means
U=
·S® x (u' — u) dx ;
α
X
and, from the property which the curve AMB is supposed to
possess, this difference u'- u must be positive, whatever may
be the values of Sy and of Sz, and whatever may be the sign
of i. Now, if the difference u'-u be developed according to
the powers of i, and if idu be the first term of its development,
the first term of that of uʼ— u will be i Soxdu da; hence it
u'
follows, that we must have
α
2 R
306
OSCILLATION OF THE SIMPLE PENDULUM
SB xdudx = 0,
α
(a)
otherwise the difference u'- U would change its sign at the
same time as i.
This condition must be satisfied, both when u is a maxi-
mum and a minimum. When it is fulfilled, the difference
u'— u will be, in general, an infinitesmal of the second order,
and it will have the same sign as the coefficient of 22 in its de-
velopment, consequently, it will be a maximum or a minimum
according as this coefficient is negative or positive. But, it is
evident from the nature of the question, that the time t' is not
susceptible of having a maximum value, consequently this co-
efficient will be positive in the problem of the brachystochrone,
and it will be only necessary that the condition expressed by
equation (a) be satisfied.
The quantity idu is in fact the differential of u, taken with
respect to y and to z, and in which the increments of these quan-
tities are represented by idy and idz. If the factor i, which is
common to idu and its value, be suppressed, we shall have
1 dy ddy
Su=
+
u dx dx
1 dz d8z
;
u dx dx
so that equation (a) will become
- 、
ẞx dy ddy
a u dx dx
dx + SB
x dz dồz
a u dx dx
dx = 0.
If we integrate this equation by parts, we shall obtain,
because by hypothesis dz and dy are cypher at the two limits
x = a, x=ẞ(e')
Bxdy động
a u dx dx
dx =
Pau
x dz dƐz
SB x dz
a u dx dx
dx=
x dy
u dx
FB d (242) By dr.
α
dx
Sy dx,
SB d ( ) &z dx;
α
u dx
dx
Sz
hence the preceding equation will be changed into
IN A CYCLOID.
307
x dy
u
dx
d dz)
u dx
S² [ d ( 2 ) 2 + ( 2 ) 2 ] dx = 0
α
dx
dx
8z]dx = 0.
But, as dz and dy are arbitrary functions of x, this integral
cannot be equal to cypher, unless that the quantity contained
under S the sign of the integration, be of itself equal to cypher;
we shall consequently have
dy
x dz
d ( 2 ) 2 + d ( 2 ) &z = 0,
dx
dx
u dx
Sy
dz
Sx
(b)
200. If the sought curve AMB, and any other curve am′b,
were required to be traced on a given surface, the equation
of which was L 0, then the values of y and z considered as
functions of x, which it is required to determine, and these
values respectively increased by idy and idz, must successively
satisfy this equation; hence we infer
dL
dy
dL
8y + √ & z = 0;
dz
by means of which, one of the two quantities dy and Sz may
be eliminated from equation (b), and as the other will disappear
at the same time, we shall have
u
dɩ d. (I dy)
dL
d. C
x dz
u dx
dz
dy dx
dx
= 0.
In this case the two equations of the required curve will
be L=0 and this last equation, by means of which we can
determine the curve of quickest descent on a given surface.
If, on the contrary, the question was to determine the mini-
mum of u among all the curves which are terminated at the
points A and B, without restricting it to exist on any particu-
lar surface, then the quantities Sy and Sz will be arbitrary and in-
dependent of each other. Their coefficients must consequently
be separately equal to cypher in the preceding equation (b),
and thus it will be decomposed into two others, namely,
•
308
OSCILLATION OF THE SIMPLE PENDULUM
1.
d. (½ dy)
u dx
d (2 dx)
dz
0,
=0;
dx
u
dx
we shall restrict ourselves to the consideration of this last
case.
If we integrate and denote by a and a' the two constant
arbitraries, which are introduced by the integration, we obtain
x dy
x dz
a,
u dx
= a',
u dx
and, consequently,
dy
dz
a
α = 0;
dx
dx
(c)
integrating again, and denoting by y a third arbitrary quan-
tity, there results
a'y —'az = y;
which shews that the required curve is one of single curvature,
and comprised in a plane perpendicular to that of the axes of y
and z. For greater simplicity, let the plane of this curve be
assumed to be that of the axes of x and y, we shall have then
u = √ 1+
dy²
dx2
and it will be only necessary to consider the first equation (c),
which will become(g)
xdy = a √ dx² + dy²;
hence we obtain
adx
dy=
a²
(d)
It only now remains for us to integrate this formula, which
will depend on the form of the function x, and then to de-
termine a and the new constant arbitrary introduced by this
integration, from the condition that the required curve passes
through the given points a and B.
201. Before we proceed farther, let us suppose that c is
IN A CYCLOID.
309
any constant whatever, and that x + c is substituted in place
of x in the preceding formulæ. The integral u will become
U-
dy²
dy2
1+ dar dx+c.
dx²
α
SB √ 1+ dy dr;
dx²
and the value of y, which renders this function a minimum, will
be furnished by the equation
dy=
adx
√(x+c)² — a²
(e)
Now, as this sum of the integrals that u represents, is a
minimum, for all the curves which are terminated at the points
A and B, it is evident that the first integral
SB
α
dy²
X 1+ dx
dx²
will be a minimum, if we only consider, among all these
curves, those which answer to the same value of the second
integral.
This simple remark enables us to extend immediately to
problems of relative maxima and minima, the solutions of pro-
blems of absolute maxima and minima; in the sequel, another
instance of the application of this principle will occur. As in
the present case, the second integral contained in u is the
length of the required curve, it follows that by means of equa-
tion (e) we can determine, among all isoperimetrical curves,
that for which the first integral is a maximum or a minimum.
If I denotes the given length, common to all these curves, we
shall have
S
1+
α
dy2
dx2
. dx = 1;
which condition will be satisfied by means of the indeterminate
constant c, that has been introduced into the formula (e).
310
OSCILLATION OF THE SIMPLE PENDULUM
202. Let us now apply formula (d) to the curve of quickest
descent.
Since
1
x=
we shall have
1
dy=
X α
(x − a) dx
√ a (x − a) — (x — a)½³
Ja
being substituted in place of a.(g'). Now, this differential
equation is that of a cycloid (No. 72) passing through a, (the
point from which the moveable commences its motion,) the
base being horizontal and the diameter of the generating curve
being equal to a; which establishes what was proposed to be
proved in No. 198.
By integrating, we obtain (h')
ya'
a' = § a are (cos
COS =
-
-2x+2a) — √ a (x− a) —
a (x—a)—(x—a)²;
a
a' being what y becomes when xa. If ẞ' denotes the value
of y when xß, we shall have
α
B'-a'aarc (cos =
a-26+2a) -
√ a(ß—a)—(B—a)².
a
As the coordinates a and a', ẞ and B', of the points A and B,
are given, the constant a can be determined by this last equa-
tion; and then the preceding value of y will not contain any
unknown quantity.
By means of the value of dy, we obtain()
И
W =
dy2
1+
dx2
therefore we shall have (No. 198)
Va
√α-x+a
Va dx
1' √ 2g = Sa √a(x − a)
ť' √2g=
;
a √a (x − a) — (x — a)²
:
IN A CYCLOID.
311
and, consequently,(k')
t' =
α
:√. are (cos=-26+2a),
2g
α
α
expresses the shortest time in which the moveable can pass
from the point a to the point B.
If these two points exist in the same vertical, we shall have
B' a'; a condition which can be satisfied by assuming a=;
for we have(l')
-
(co
arc cos=
α
α
+2a)=arc(sin=2
a(B—a)—(B—a)²
(B—a)³),
α
and, when a∞, we can substitute this arc for its sine,
which reduces the preceding value of ẞ'a' to cypher; and as
at the same time, the value of y becomes equal to a', it follows
that the moveable will not deviate from the vertical direction.
The value of t' will likewise be in this case
t' =
a 2√a (ẞ — a) — (B — a
Va 2√a (B
2g
α
√2 (ẞ — a)
√ g
which is in fact the time that the moveable takes to traverse the
height B a, in descending from the point a situated above
the point B.
As the determination of the line of quickest descent is a
problem of pure curiosity, we have restricted ourselves to the
consideration of the simplest case, namely, that in which the
motion is performed in a vacuum, the extreme points being
given. If these points A and B are not fixed and given, but
only subjected to exist on the given curves DAE and FBG, or
on surfaces that are given, the brachystochrone, when the
motion is performed in a vacuum, will still be a cycloid, and,
by means of the rules of the calculus of variations, we can de-
termine, in all these cases, the coordinates of these two points.
In a resisting medium, this line will be a different curve, the
differential equation of which may be obtained by the rules of
312
MOTION ON A GIVEN SURFACE.
this calculus; and as this equation depends on the law of the
resistance with respect to the velocity of the moveable, it fol-
lows that the curve must be different for each law.
III. Motion on a given Surface.
203. The simple pendulum treated of in No. 179, will fur-
nish us with an example of the motion of a material point on a
given surface, provided we suppose, that after it is drawn out
of the vertical CB, and transferred to CA (fig. 45), there is im-
pressed on it a velocity, the direction of which does not exist
in the vertical plane ACD. The pendulum will then deviate
from this plane, and the material point by which it is termi-
nated will move on the surface of a sphere described from
the point c as a centre, and with a radius equal to a the length
of this pendulum. The percussion that the moveable will
experience at the commencement of the motion, may be re-
solved into two forces, the one acting in the direction of Ac, or
of its production, and which will be destroyed by the resistance
of the fixed point c, the other will be perpendicular to AC,
and will produce the initial velocity of the pendulum, which we
shall denote by k. As the motion is supposed to be performed
in a vacuum, the gravity is the only given accelerating force
that acts on the moveable.
This being established, let CM be the position of the pendu-
lum at the end of the time t, and let x, y, z, denote the rectan-
gular coordinates of the point M. Likewise, let m denote the
mass of the moveable, and mN the unknown tension of the
thread Cм, acting in the direction of its production. If the
origin of the coordinates x, y, z, be at c, the components of
the accelerating force N in the direction of their productions,
will be
х
2
N,N, N.
a
a
a
Now, if to the moveable a force' equal and contrary to N,
1
།
MOTION ON A GIVEN SURFACE.
313
be applied, we may abstract altogether from the consideration
of the thread Cм, and consider the moveable as entirely free;
therefore, if we suppose that the axis of the positive zs is ver-
tical, and its direction to be that of gravity, the three equa-
tions of the motion will be
+-N=0,
d2x
dt2
α
d²y
dt2
+2/N
+ N = 0,
(1)
a
d2z
dt2
+²²=g=0.
which accord with equations (3) of No. 151. By eliminating
the unknown N, they will be reduced to two; and these com-
bined with the equation of the sphere, namely,
x²+ y²+z²= a²,
will furnish us with three equations, by means of which x, y, z,
may be determined in functions of t.
204. Multiplying equations (1) by x, y, z, respectively, and
then adding them together, there results
xd²x+yd²y + zd²z
dt2
+ Na
gz=0.
The first differential of the equation of the sphere will give us
and the second
xdc +ydy + zdz = 0,
xď³x + yd²y + zď²x + dx² + dy² + dz² = 0.
(2)
Hence, if v denotes the velocity of the moveable, at the end
of the time t, so that we may have
dx² + dy²+ dz²
= v²,
dt²
there will result(m)
N
α
2/18
gz
+
;
α
2 s
314
MOTION ON A GIVEN SURFACE.
and, in fact, the tension my must be the sum of the centrifugal
mv2
force and of mgz the component of the weight which acts
a
a
in the direction of the radius CM.
Likewise, if equations (1) be multiplied by dx, dy, dz,
respectively, and then added together, the unknown quantity
N will disappear, in virtue of equation (2), and we shall have
dx d²x + dy d³y +dz dz
dt2
If this be integrated, we obtain
=gdz.
dx² + dy²+ dz²
=2gz+b₂
dt2
(3)
b denoting a constant arbitrary. As the initial value of the
first member of this equation is k², if y denotes that of z, we
shall have
and, at any instant whatever,
as we know already.
k² — 2gy = b,
v² — k² + 2g (≈ — y);
Finally, if the first equation (1) be multiplied by y, and
then subtracted from the second multiplied by x, we obtain
xd²y yd²x
dr² = 0;
dt2
dt2
hence, by integrating, there results
xdy — ydx = cdt,
c denoting a constant arbitrary.
(4)
In this manner, the solution of the problem will only depend
on the three differential equations (2), (3), (4), which are all of
the first order, and with respect to the first of these, we know
already, that its integral is the equation of the sphere.
We can separate the variables and reduce the question to
quadratures by means of the following calculus.
motion on a given surface.
315
205. From equation (2) we obtain
xdx +ydy =
zdz;
if the two members of this equation, and also those of equation
(4), be raised to the square, we obtain, by adding the resulting
equations together,
(x²+ y²) (dx² + dy²) = z²dz² + c²dt².
If a² 22 be substituted instead of x²+ y², and if dx²+ dy²
be eliminated by means of equation (3), we shall have
(a² — x²) [(2gz+b) dť² — dz²] = z²dz² + c²dť² ;
hence we obtain
dt =
adz
√ (a² — z²) (2gz + b) — c²
(5)
Let r denote the radius vector of the projection of the move-
able on the horizontal plane of the axes of x and y, and the
angle that this radius makes with the axis of x, we shall have
x=rcos, y = r sin, xdy — ydx = r²d↓ ;
and, because r² = a² ², equation (4) will become
(a²-22) d↓ = cdt;
by substituting for dt its preceding value, we deduce
dif=
cadz
z²)
(a² — z²) √ (a² — z2) (2gz+b)
(6)
The integrals of these expressions of dt and d will make
known the values of t and in functions of z, they may be
always reduced to elliptic functions, and cannot be obtained
under a finite form, unless the quantity of the third degree
with respect to s, contained under the radical, has a double
factor. The value of 4, and the equation of the sphere, will de-
termine the trajectory of the moveable, and the value of t as a
function of z, or of z as a function of t, will then make known
the position of the moveable on this curve, at each instant.
316
MOTION ON A GIVEN SURFACE.
The constantb may be known from the given quantities kandy.
The constant arbitraries which are introduced by the integra-
tions of dt and dy, are determined by the conditions t = 0 and
0, when zy, the second of these conditions, supposes
that the axis of x exists in the vertical plane ACB, from which
the pendulum commences its motion. There remains, there-
fore, only the constant quantity c to determine. Now, as
the direction of u, the velocity of the moveable, is perpen-
dicular to CM, the radius of the sphere on which the body
moves, if it be resolved into two, one perpendicular to the
vertical plane MCB, and the other comprised in this plane, the
first component will be the velocity of the horizontal pro-
jection of the moveable, perpendicular to its radius vector r, con-
sequently, if it be represented by u, we shall have (No. 156)
u =
rdy
dt,
we have also in virtue of equation (4)(n')
นะ
==
Va²
C
therefore, if ɛ denotes the angle that the initial velocity k makes
with the perpendicular to the plane ACB, so that we may have
u = k cosε at the commencement of the motion, there will
result
2
c = k √ a² — y² cos ε.
When the velocity k is cypher, we shall have c = 0,
b=2gy, and, consequently,
dt=
adz
2g √ (a² — z²) (≈ − y)
this expression coincides with the value of dt given in No. 185,
because a z and a y are what have been denominated ax
and aß in that value.
206. Let us consider the case in which the pendulum has
MOTION ON A GIVEN SURFACE.
317
烹
​received a very small initial velocity, and, therefore, deviates
very little from the vertical CB. If the direction of this ve-
locity be horizontal, it will be perpendicular to the plane ACB,
and, consequently, we shall have = 0. Let ẞ denote a very
small fraction, and let
k=ẞvga.
Likewise let a and 0 represent the angles ACB and MCB,
their fourth powers be neglected, we shall have
y=α-aa², z = a -1a0²,
2
b=-2ga+ga (a²+ẞ²), c²= ga³a²ß²;
and formulæ (5) and (6) will become(o')
if
Odo
9 √ (a²-02)
0²) (0² — ẞ²)³
dt = -
Va
α
df = -
aßdo
0 √ (a² — 0²) (0² — B²)
(a)
The angle will make known the position of the vertical
plane MCB, in which the pendulum exists at each instant; it
evidently may increase indefinitely. The angle ✪ will deter-
mine also at each instant, the position of the pendulum in this
variable plane; it is considered as a positive quantity, and the
positions of the pendulum, which are equidistant from the
vertical Cв on opposite sides, correspond to the same angle 0,
and to values of which differ from each other by 180°.
do
From the value of
dt'
it appears that the angle
deduced from the first equation (a),
will be always comprised between
the limits a and B. If ßa, we shall have 0 always a;
if equations (a) be divided the one by the other, there results,
in all cases, (p')
df = √9 aß
0 = =
dt;
a f²
therefore, when a ẞ, we shall have
(b)
318
MOTION ON A GIVEN SURFACE.
¡
+= 1 √ ²;
α
consequently, the pendulum will then describe uniformly the
surface of a right cone with a circular base, and the time
of an entire revolution will be 2= √(9) that is to
say, the
same as the time of a double oscillation in the vertical plane
ACB. Thus, two pendulums having the same length a, and
which depart together from the same line ca, the one without
any initial velocity, the other with a velocity perpendicular
to the plane ACB and equal to a ga, will return after the
lapse of the same time, to this line ca.
207. The value of dt may be expressed under the following
form,
dt = − 2
-
2 √a
α
Odo
g√ (a² — B²)² — (20² — a² — ẞ²)²
and if, for greater simplicity, we make
2 0² — a² — ß² — (a² — ß²) x, 40d0 = (a² — ß²) dx ;
(a2-62) VI-2; and there results
the radical becomes
dt == √ a
dx
章
​x²
Since 0 = a, and x = 1, when t = 0, we obtain from this
and conversely,
α
t=±√ a arc (cos = x),
9
x = cos 2t √//9.
a
Therefore, at any instant whatever, we shall have
— t
9
0² = § (a² + ß²) + § (a² — (3²) cos 21 √2;
α
MOTION ON A GIVEN SURFACE.
319
which shews that the oscillations of the pendulum, of which
the two extremities correspond to a and 0 ẞ, in the va-
riable plane MCB, will be isochronous, and their duration will
be &π α which is equal to half the time of an oscillation
ğ'
in the fixed(r) plane ACB. Substituting this value of 02 in
equation (b), and observing that
g
√√
g
cos 21 √2 = cost √2 - sin² t√?
2t
there results
g
df =
a
α
V
α
aẞdt
a² cos² t√ E+ẞ²sin³t√
and, because 40 when t≈ 0, we infer(t')
a tang.ẞ tang t
9
α
a
α
This being the case, the motion of the plane мCB will be
no longer uniform as when a = ß; but it is evident that this
plane will perform successively the four quarters of its entire
revolution, in times respectively equal to each other and to
έπ ν
α
9
which is half the time in which an oscillation is
performed in this variable plane.
From this last equation we deduce

cos²=
sin² =
we have also
g
a² cos² t√5
a² cos² t√ ₂+ẞ² sin² t
cost√ t√
α
B' sin't √
&
a
a² cos²t √ E + B² sin² t
a
t√
2109
a

a
320
MOTION ON A GIVEN SURFACE.
*
x² = (g² — z²) cos² ↓ = a² 0² cos² ¥,
= a²
y² = (a² - 2²) sin2 a2 02 sin² 4,
23
by substituting for ≈ its approximate value; therefore, since
we obtain
9
9
0² = a² cos't √ √ ½ + p' sin' t √ ½,
α
α
x²= a²a² cos², y2 = a² ß2 sin2,
and, consequently, (v)
x²
y2
a²
B²
= a²;
hence it appears, that the trajectory of the projection of the
moveable on the horizontal plane passing through the point c,
is an ellipse which has its centre in this point, and one of its
axes in the plane ACB, from which the pendulum commences
its motion with a velocity, the direction of which is perpendi-
cular to this plane.
CHAPTER VI.
EXAMPLES OF THE MOTION OF A BODY ALTOGETHER FREE.
1. Motion of Projectiles.
208. THE theory of projectiles, which are discharged by
artillery with great velocity, and are at the same time sub-
jected to the action of gravity, and also to the resistance of
the air, will constitute the subject matter of this section.
In the first place, let us abstract from the consideration of
this resistance, and let us consider the motion of a heavy ma-
terial point which moves from the point o (fig. 48), with a
velocity a in the direction of the line oa. It is evident, that
the moveable will not deviate from the vertical plane passing
through (a) this line. Let OMD be its trajectory in this plane,
to which oa will be a tangent. If in this same plane, two axes
be drawn, the one horizontal and the other vertical and di-
rected from the horizon, these axes may be taken for those
of the coordinates, and at the end of any time, such as t, let
M be the position of the moveable, its absciss OP, and y its
ordinate PM, then if a denotes the acute angle Aox, which
the direction of the initial velocity a makes with the axis ox,
its components will be, a cos a in the direction of this axis,
and a sin a in the direction of the axis oy; the angle a will
be negative, if the right line oA falls below ox.
From what has been established in No. 148, it is evident
that the motions of the projections of the moveable on the two
axes ox and oy will be independent of each other; the mo-
tion of its horizontal projection will be therefore uniform, and
•
2T
322
MOTION OF PROJECTILES.
हमें इसको
equal to the velocity a cos a, and as that of its vertical pro-
jection arises from the initial velocity a sin a, and from g the
constant force of gravity acting in an opposite direction to this
velocity, we shall consequently have
x= ta. cos a,
y =
ta, sin a — § gť²;
-
eliminating t between these equations; and substituting √2gh
for a, (h being the height through which the moveable must
fall to acquire the velocity a) there will result
y=x tang a
x2
4 h cos² a'
for the equation of the trajectory.
Therefore, this curve is a parabola, of which the axis is
dy
vertical; its summit determined by the equation
=0, has for
dx
coordinates(b)
x = 2h cos a sin a, y = h sin² a ;
and it meets the axis ox in a second point B, which is obtained
by making y = 0 in the equation of the curve, hence, if b
denotes the distance OB, we obtain
b = 4h sin a cos a = 2h sin 2 a.
This distance b is called the amplitude of the projection.
If the motion is performed in a vacuum, it is a maximum, when
a = 45°; in this case it is equal to 2h, or to twice the height
through which the body must fall to acquire the initial ve-
locity.
Naming v the velocity of the moveable at the end of the
time t, we shall obtain, by substituting the differentials of the
preceding values of x and y in the equation(c)
v2
dx² + dy²
dt2
2
v² a²-2agt sin a + g² t².
=
+
*
MOTION OF PROJECTILES.
323
'As t, the time which the moveable employs to reach the
point B, while describing the curve OCB, is the same as it takes
to describe the line OB with the velocity a cos a, we must
consequently have
and since h=
t=
α
a²
2g'
,
b
a cos a
4h sin a
;
α
there results gt = 2a sin a; hence we have
v² a²; therefore the velocity of the moveable in this point
B, is the the same as at the point o, its direction is along the
tangent BE, and EBx, the angle which the direction of the
moveable makes in its descent with OB, is the same as aox,
the angle of projection. If the moveable, instead of being a
material point, be a solid body of any form and dimensions
whatever, it will be shewn in a subsequent chapter, that these
equations of parabolic motion ought to be referred to its centre
of gravity.
209. The velocity a being given, if it were required to de-
termine the angle a such, that the moveable may reach a de-
terminate point, of which the coordinates are x = B, y = y,
these values should be substituted in the equation of the tra-
jectory, and we shall have by this means,
to determine a. By making,
y= ẞ tang a
B²
4h.cos²
,
α
tang a=z,
cos² a =
1
1+229
this equation becomes
hence we obtain
4hy + ß² - 4hẞz + ẞ² z² = 0;
2 =
2h
+
1
√ 4 h² — 4hy — ẞ².
B
As there are two values of z or tang a furnished by this
equation, it follows that when 4h2 is greater than 4hy + ß²,
12
324
MOTION OF PROJECTILES.
!
the proposed object will be struck, if the elevation of the pro-
jectile be either of the two angles which are deduced from
this equation; if the radical part of this equation vanishes, the
two values of z are equal, and these two directions coincide;
and when the radical part is impossible, in which case 4h² is
less than 4hy + B², the mark cannot be reached under any
direction whatever.
Hence, if in the vertical plane which passes through the
initial direction of the moveable(d), a parabola be described of
which the equation is
4 hy + ß² = 4h²,
this curve will divide the plane into two parts such, that no
mark without this curve and in the vertical plane can be struck
by a body projected with the given velocity, while for all
points which lie within this curve, there are two different di-
rections, along either of which, if the body be projected, it will
reach the mark; and if the object to be struck exist on this
parabola itself, there is only one elevation at which the pro-
jectile can be discharged to reach the mark.
210. It appears from what has been stated in the two pre-
ceding numbers, that the theory of the motion of projectiles
would be extremely simple, if the resistance which the air
opposes to their motion could be neglected; but when the ve-
locity is very great, as in the cases which we are particularly
concerned with, this force is too considerable not to be taken
into account; in fact, as we shall presently see, it changes
altogether both the form of the trajectory, and the law of its
motion on this curve.
It will be proved in a subsequent chapter, that whatever
be the form and dimensions of the projectile, its centre of
gravity will have the same motion, as a heavy material point
whose mass is equal to that of the moveable, the direction and
magnitude of the initial velocity being the same in both, and to
which besides are applied, parallel to the directions in which
MOTION OF PROJECTILES.
325
they act, the forces, which, arising from the friction and the
resistance of the air, act on the surface of this solid body. It
will likewise be shewn, that the motive force which results
from these resistances, when transferred to the centre of gra-
vity, may sometimes cause this point to deviate from the ver-
tical plane passing through the direction of the initial velocity;
but here we suppose that this is not the case, and that the motive
force in question acts always in the direction of a tangent to
the trajectory described by the centre of gravity.
This being established, in order to obtain the equations of
its motion, let the preceding notations be retained, and let them
be supposed to refer to figure 49, in which the trajectory OMD
is no longer to be considered a parabola. Let s be the arc oм
described by the moveable at the end of the time t, and let R
denote the motive force arising from the resistance of the air,
and which acts in the direction of the part мT of the tangent
at M. The cosines of the angles which this line MT makes
with the axis drawn through the point м in the directions of the
dx
dy
positive xs and ys, will be
therefore, ifm denotes
ds'
ds
the mass of the projectile and g the gravity,
d²x
dt²
R dx d'y
m ds dt2
R dy
g
m ds'
will be the equations of the motion of its centre of gravity.
Let this projectile be an homogeneous sphere, or one com-
posed of concentrical strata each of which will be homogeneous,
then if its mean density be denoted by D, and its radius by r,
we shall have
m =
4ᎠᏑᏰ
3
Likewise if we suppose, agreeably to the hypotheses which
are generally admitted, that the force R is proportional to the
square of the velocity of the centre of gravity, to the surface of
the projectile, and to the density of the air, there results(e)
326
MOTION OF PROJECTILES.
R
ηρ
ds2
X
m
Dr
dt2;
p being this density, and n a numerical factor to be determined
by experiment. This expression satisfies the condition of the
R
homogeneity of the quantities, for and the ratio of
m
ds2
dt2
to r,
are two quantities of the same nature as g the gravity, and the
factors n and P
nience, let
are abstract numbers. For greater conve-
D
ηρ
Dr
C,
1
so that may be a line of a given length, which, when we do
с
not take into account the change of density of the mass of air,
traversed by the projectile, may be considered as constant.
R
211. By substituting in place of, its value c
equations of motion become(ƒ)
m
ds2
the two
dt2,
d²x
ds dx
+c.
= 0,
dt2
dt' dt
(1)
d²y ds dy
dt2
+c.
+g=0.
dt dt
The integral of the first is
dx
= a cos ae―c³,
dt
dx
for = a cosa, at the point o where s = 0, and e is the base
dt
of the Naperian system of logarithms. As the form of the
second equation differs from the first only in its last term, we
may assume, in order to integrate(g),
dx
dy
dt
= Pdt
p being a new unknown. If we substitute this value of
MOTION OF PROJECTILES.
327
1
dy
dt
in the second equation (1), we obtain, by taking into
account the first(h),
dx dp
dt dt
g.
If the members of this equation be respectively divided by
dx
the square of and its value given above, there will result
dt'
dp
dx
9
e2cs.
dt dt
a² cos² a
If y and p be regarded as functions of x, we shall have
p =
dy dx dy dp dx
=
dt dt dx dt dt
dp
;
dx
therefore, if we substitute 2 gh for a², the preceding equation
will become
dp
1
e2 cs;
dx
2h cos2
α
(2)
and this will be the differential equation of the trajectory.
Since
√1+ p². dx= ds,
if each member of the preceding equation be multiplied by
these quantities respectively, we shall obtain
ds
2
√1+p². dp =—
-e2c8;
2h cos2
α
hence, if we integrate and denote the constant arbitrary by y,
there results
1
p√T+p²+log (p+√1+p²)=Y−2ch cos² e2cs. (3)
α
In order to determine y, let s = 0 and p = tang a at the same
time; this gives
328
MOTION OF PROJECTILES.
1
Y=2ch cos²a+tanga√1+tanga+log(tanga+ √1+tang²a);
however it is better to retain y in place of this value, as it is
a more concise expression.
From the preceding equations, we obtain(¿)
dx =
2h cos² a e-2c8 dp, dy =p dx, gdť²
=
dx dp.
Hence, if the exponential be eliminated by means of equa-
tion (3), there will result
cdx =
cdy =
√cg.dt =
dp
pvI+p²+ log (p + √1+p²)—y
pdp
p√1+p²+ log (p + √ 1+p²)−y
dp
[y−p√1+p²—log(p+√1+p²)]
15
(4)
formula which cannot be integrated under a finite form; in
the last, the radical should be considered as a positive quan-
tity, because the angle of which p is the tangent, diminishes
when the time increases.
212. Naming w this angle, that is to say, the inclination
MTX of the tangent to the trajectory, on the horizontal axis ox,
we shall have
dw
p = tang w, dp= cos2 W
w
The values of x, y, t, deduced from equations (4), will be
of the form §dw; the integral being taken, so that it may
vanish at the point o where = a, and denoting a given
Q
function of w. These three values should be calculated for
each point м, by the method of quadratures (No. 15). In this
manner, the trajectory can be constructed by points, and t the
time that the moveable takes to describe each arc Oм, of
which the length s is given by equation (3), will be known.
To determine the velocity of the moveable at the point M,
we have(k)
MOTION OF PROJECTiles.
329
dx2
dt2
v² = (1 +p²)
dt2
= g² (1+p²)
dpa,
and, consequently,
cv² =
g (1 + p²)
y—p√1+p² — log (p + √1 +p²)
(5)
If these integrals be extended to = 0, the abscissa and
ordinate of c, the most elevated point of the trajectory, can be
obtained. Then by assigning to w negative values, the points
of the descending branch CBD of the trajectory can be deter-
mined. When we arrive at a value, such as - a', of w, for which
y the ordinate of the trajectory vanishes, the corresponding
value of x will be equal to OB the amplitude of the projection;
which will be no longer, as in the case of a vacuum, double of
the ordinate of the point c when it is a maximum ; in fact a is
then equal to an angle less than 45°, and depending on the
magnitude of the initial velocity. The angle a', or EBx, and
the velocity at the point в will likewise differ from a and a.
Thus it appears, that all the circumstances of the motion
will be known, and the problem completely resolved, when
the values of the three constant quantities h, a, c, contained
in the preceding formulæ, are given; however the numerical
calculations which must be performed in each particular case,
are extremely tedious.
213. The motion of the projectile on the descending branch
of the trajectory, continually tends to become vertical and
uniform; in fact, if we transfer the origin of the coordinates
to c the summit of the curve (fig. 50) by making
x=x1+x', y = y₁ — y', t = t₁ + t' ;
then x', y', are the abscissa and ordinate of any point м' of the
'descending branch, referred to the horizontal axis cx', and to
the axis cy', which is drawn in the direction of gravity, and t'
denotes the time employed in describing the arc Cм'. Like-
wise if p' represents the tangent of the angle M'T'x', which the
2 U
330
MOTION OF PROJECTILES.
NO
tangent to the curve at м', makes with the axis cx', we shall
have (1)
p' =
dy'
ཧྰུཾཚེ་
- P,
and because
log (√1 + p'² — p') = − log (p' + √1+p”),
— -
the first equation (4) will become
dp'
cdx' =
P'
in which p' denotes, for conciseness,
y+p'√1 + p² + log (p' +√1+p²½.
As the acute angle M'T'x' continually approximates to a
right angle, the variable p' will increase indefinitely; but this
will not be the case with respect to x'. When p' is very great,
we can substitute p' in place of √1 +p²², and then if y + log2
be neglected with respect to p²², we shall have(m)
p′ = p²² + ½ log p”,
or simply p′ = p²², since the logarithm of a very great quan-
tity p²², and, à fortiori, ½ log p²², is very small with respect
to this quantity, therefore, for these values of p', we shall
have
dp'
dx' =
cp/2•
12
If this equation be integrated there will result
1
x' = c -
cp'
c being an arbitrary constant; and from this integral it ap-
pears, that the values of x' do not increase indefinitely with
those of p'. This being the case, if
1 x
dp'
q = = 5 % dr
0 ;
MOTION OF PROJECTILES.
331
9 will be a line of finite magnitude, which can be calculated
by the method of quadratures; and if we take on ca a part ca
equal to this line, the vertical AB drawn through this point
will be an asymptote of the part CD of the trajectory; so that
the motion of the projectile on this descending branch ap-
proaches indefinitely the vertical direction. It may also be
observed, that when the values of p' are very great, the two
last equations (4) become
cdy' =
dp'
p',
Veg.dt' = dp
;
p'
consequently, there results
dy'
dť
hence it appears, that when the final and vertical motion of
the projectile becomes uniform, the velocity of this motion will
be that which a heavy body acquires in falling through a height
1
equal to in a vacuum; this is also evident from formula
2 c
(5), by substituting - p' in place of p, and then considering
p' as a very great quantity (n).
If in the first equation (4) we make
p = tang w, dp=
and, for conciseness,
2
dw
cos2w
[y-tanw√/1+tang³w-log(tangw+V1+tan²w)] cos²w=
we shall obtain
1
X1
X₁ =
C JO
Sa
α
Qdw,
I
for the abscissa of the point c.
(fig. 49) a point F, such that
If, therefore, we take on ox
OF = x₁ + 9,
the vertical FG, drawn through this point F, will be the asymp-
tote of the descending branch of the trajectory.
332
MOTION OF PROJECTILES.
214. Let on be the production of the trajectory OCD; o
being the point from which the projectile commences to move,
the motion of the projectile will not take place in this part of
the curve; but, notwithstanding, we may wish to know its
form. Now, if it be constructed by points by means of the
two first formulæ (4), taking care that the values of p are po-
sitive and greater than tang a, it is easy to be assured that it
also has an asymptote, which however is not vertical, as in the
case of the descending branch.
For this purpose, we may observe that it appears from the
value of y of No. 211, there is always given an acute angle ß
greater than a, which is such, that p= tang ẞ renders the
common denominator of these two formulæ cypher, that is to
say, an angle ẞ which satisfies the equation
y-tangẞV1+tang²ß-log(tangẞ+1+tang²ß)=0. (6)
This being so, it appears from the value of dp deduced from
either of the two first equations (4), that though the abscissa
x and ordinate y increase indefinitely (abstracting from the
sign), in this part on of the curve, the quantity p ceases to in-
crease, when the difference between it and tang ẞ is infinitely
small, so that p can never pass, nor even rigorously attain to
the value p tang ẞ; this indicates, that the branch of the
curve on has an asymptote, which intersects the axis ox when
produced, in an angle equal to ß; and its distance from the
point o may be determined in the following manner; let an
axis be drawn through the point o, which may make with the
production of ox, an angle equal to the complement of ß, and
which is, consequently, perpendicular to the asymptote of on.
Let u denote the abscissa of any point of the curve, reckoning
from the point o on this axis; as the coordinates of this point
with respect to the axes ox and oy are always x and y, we
shall have
uy.cosẞ-x sin ß.
By taking the differential of this expression, and by sub-
MOTION OF PROJECTILES.
333
stituting for de and dy their values furnished by the two equa-
tions (4), we shall obtain
cdu=
(tang ẞ-p) cos ẞ.dp
(y−p√1+p²) — log (p+√1+p²)
in this formula, values should be assigned to p greater or less
than tanga, according as the point in question is on the part
ON, or on the part om of the curve. If from its denominator
the first member of equation (6) be taken, and if, moreover,
we make
do
p = tang w,
tang w, dp
dp=
;
cos2 W
and, for conciseness,
tang ẞ1+tang²ß + log (tang ß+ √1+ tang²ß
- tang w√1+tang² - log (tang w + √1 + tang²w) = u,
there will result
du =
(tang ß-tangw) cosẞdw
Now, if we assume
CU Cos² w
r =
C
α
cos BCB (tangẞ-tangw)dw
U cos² w
;
r will be a line of a finite magnitude, which can be calculated
by the method of quadratures, and it will express the value of
u with respect to the asymptote of ON, that is to say, the
length of the perpendicular let fall from the point o on this
line; which was required to be determined. The equation of
this asymptotic line will be
u cos ẞ — x sinẞ=r;
so that if on the production of ox a point н be so taken, that
we may have
он
OH =
r
sin Bi
the asymptote of the branch ON will be the line Hк, drawn
through the point н, and making with ox produced, an angle
Kнo the supplement of B. The two asymptotes FG and HK,
cheme
dou
at
~ex
a a d e
atv
+ Calze
in pent. For with from to ton & we sch
334
MOTION OF PROJECTILES.
24
esdy
+9=0
==
et
Zen
produced above the axis ox, will meet in a point L, so that the
entire curve will be comprised within the angle KLG, the com-
plement of ß, the angle that was determined by equation (6).
215. When Oax or a the angle of projection is very small
(fig. 51), the height of the projectile above the horizontal axis
ox, drawn through the point from whence it was discharged,
is inconsiderable. Now, in this case, we can obtain by an
approximation that is sufficiently accurate, the equation in x
and y of the part оCB of the trajectory, that is situated above
ox; and we can even extend this equation to a point D, the
distance of which from this axis is not very great. In fact, in
all this part оCB, or even OCD of the trajectory, the tangent to
this curve will be very nearly horizontal, and the quantity p
will be very small; therefore, if the square of p be neglected,
we shall have
ds = dx, s= X,
and equation (2) will become
= 0
dp
=
d²y
1
e2cx.
2h cos2a
dx dx²
By integrating twice successively, and determining the arbi-
when x = 0, we obtain (p)
dy
trary constants, so that we may have
tang a and y = 0,
dx
1
y = x tanga -
- 2cx-1),
8c²h cos²a (e²cx
for the approximate equation of the trajectory; which was pro-
posed to be determined. If the exponential that it contains
be developed, this equation becomes by reducing and then
making c = 0, the exact equation of this curve in a vacuum.
From the equation gdť² = - dxdp of No. 212, compared
with the preceding value of we can infer (q)
dp
dx'
1
dt=
.ecx dx,
√2gh cos a
and, consequently,
and the solution & tree in byretingos
di
4
9
+
of Man
1
品
​MOTION OF PROJECTILES.
t =
1
· (ecx — 1);
335.
c√2gh cos a
by means of which, the time t that the moveable takes to tra-
verse oм, any portion of the curve OCD, can be determined.
216. If we suppose that the projectile falls on the ground
at a point D, and if λ denotes the depression of this point below
the horizontal plane drawn through the point o, or DQ the per-
pendicular to the axis ox; also, if I be the distance oq, and
the time employed to go from the point o to the point D, we
shall have, at the same time,
x = l, y = −λ, t=r;
and, for greater simplicity, by substituting unity for cos²a,
in the preceding formulæ, there will result
8c2h (+7tang a)
e2cl2cl-1,
Tc√2gh = ecl—1.
(a)
When, therefore, the two constants h and c are given, and
the angle a, and λ the elevation of the point o above the
ground, are measured, these equations will make known the
horizontal range 1, and the time in which the projectile de-
scribes this range. Conversely, when a, λ, 1, 7, are known by
direct measurement, these equations will enable us to deter-
mine c the coefficient of the resistance, and h the height due to
the initial velocity. If h be eliminated, we obtain
4(λ+7tang a) (ecl — 1)² = gr² (e² cl — 2 cl− 1);
an equation from which the value of c may be deduced, and
then one of the two preceding will immediately give the value
of h.
There still exists some uncertainty respecting the magni-
tudes of the ranges and of the initial velocities. According to
Lombard, for a twenty-four pounder, of which the charge is a
third of the weight of the bullet, the initial velocity is about
336
MOTION OF PROJECTILES.
463 metres for each second; and for a twelve-pounder, of
which the charge is also a third of the weight of the projectile,
this velocity amounts to 494 metres. According to the same
author, the corresponding ranges relative to λ = 0, are 700
metres in the first case, a being supposed = 1° 15′ 6″, and
660 metres in the second case, in which we suppose that
a = 1°5′ 36".
In place of the time, we may employ for the determi-
nation of h and c, a second range of the same cannon at a dif-
ferent elevation above the point where the projectile reaches
the ground. Thus, if we suppose that the weight of the pro-
jectile, that of the charge, and the angle a remain the same,
the quantities h and c will likewise remain unchanged, and if
A and I become λ' and l', we shall have
8c²h (X' + l'tang a)
hence there results
e2cl'-2cl'—1;
(λ +ltang a) (e²cl′ — 2cl'—1)
=(\'+l'tang a) (e2cl2cl-1),
h being eliminated by means of the first equation (a).
(6)
Those authors who have treated of the Ballistic pendulum,
are by no means agreed as to the magnitude of the number n
which occurs in the expression of the coefficient c, namely
(No. 210),
C =
ηρ
Dr
9
It has been inferred from a very imperfect theory of the
resistance of fluids, that this number n is; but the result of
all experiments that have been made, assign a less number
than this, and Lombard made it equal to Equation (b)
would furnish a means, the most susceptible of precision of any,
for determining c, if a the angle of projection was supposed to
be known and invariable.
MOTION OF THE PLANETS.
337
II. The Motion of the Planets.
217. The laws of the motion of the planets about the sun,
are generally known by the denomination of the laws of Kep-
ler, because they were discovered by that astronomer, who
deduced them from observation. They are three in number,
and are expressed as follows:
1. The planets move in plane curves, and their radii
vectores describe areas proportional to the times, about the
centre of the sun.
2. The orbits or trajectories of the planets are ellipses, of
which the sun occupies one of the foci.
3. The squares of the times of the revolutions of the
planets, are to each other as the cubes of the greater axes of
their orbits.
Kepler does not appear to have been aware of the great
importance of these laws; it was Newton who applied them
to determine the force which retains each planet in its orbit,
that is to say, the direction of this force, and the variations of
its intensity, from one position to another of the same planet,
and also from one planet to another. In fact, it will be proved
in this section, that each of these three circumstances is a ne-
cessary consequence of the three laws of planetary motion
which have been just stated.
As these laws refer to the motion of the centre of gravity
of each planet, it is the motion of this point which we proceed
to consider, and in every question respecting the position or
velocity of a planet, it is the position and velocity of this
centre that is always referred to.
218. Let AMBD (fig. 52) be the ellipse described by a planet,
AB its greater axis, c its centre, o and o' its two foci, o that
which the centre of the sun occupies, в the perihelion, or the
point of the orbit nearest to the sun, a the aphelion, or the
point which is most remote from the sun. After the lapse of
B
2 x
338
MOTION OF THE PLANETS.
time t which is reckoned from the moment that the planet
passes through the perihelion, let м be the position of the
moveable. Let r denote its radius vector, the angle мOB,
which is termed in astronomy the true anomaly. The sector
described by the radius during the instant dt will be r²d0 (No.
156), therefore, in consequence of Kepler's first law, we have
r²de = cdt,
(1)
c being a constant arbitrary equal to twice the area described
in the unit of time, and ct twice the area MOв described in t
any time whatever. Likewise, let O'Mr', CB CA = α,
=
coco'=ae. By a property of the ellipse we have
r+r' = 2a;
we have likewise, in the triangle OʻMO,
p2= p²+4rae cos 0 + 4a²e²;
and if r' be eliminated between these two equations, there
results
a (1 − e²)
1+e.cos 0'
for the equation of the trajectory.
(2)
For all the planets which were known previously to the
present century, the excentricity e is a very small fraction,
with the exception of Mercury, the excentricity of which is
more than a fifth; in the case of the orbit of the earth
e=0,01685318,
or, very nearly, a sixtieth. The greatest was that of Mars,
which exceeds nine hundredths, it was consequently for this
planet that the elliptic motion ought to differ most from the
motion of a planet in an excentric(a) circle, in which the
planets were supposed to move previously to the time of Kep-
ler; and in fact, it was from the observations of Ticho-Brahe
on this planet, that Kepler first recognized the difference be-
tween these two motions. If the values r and be developed
◊
MOTION OF THE PLANETS.
339
into series arranged according to the powers of e, by means of
the equation of areas proportional to the times, combined with
that of the elliptic trajectory, or with that of the trajectory of the
excentric circle, we shall find that when the time t is the same,
the corresponding expansions of these two curves differ only
in terms, that depend on the square or higher powers of e;
from which it is evident, that in Kepler's time it was extremely
difficult, in consequence of the imperfection of the observa-
tions, to discover the difference between these two motions.
219. If T denotes the time of the revolution of a planet,
and if we assume
2π
T
n=-
this constant quantity n will be the mean angular velocity,
and nt the mean motion of the planet.
0
If we conceive a fictitious star whose motion is uniform, to
set out from perihelion, and to complete its revolution in the
same time as this planet, then its radius vector will describe
nt, while that of the planet describes the angle 0, and the
angle - nt contained at any epoch between these radii, is
what astronomers term the equation of the centre; it is posi-
tive, and the planet precedes the fictitious star in going from
perihelion to aphelion(b), the contrary takes place in re-
turning from the second point to the first. The maximum of
the equation of the centre depends on the magnitude of the
excentricity. If we assume the mean day for the unit of
time, we shall obtain in the case of the earth's orbit, by sub-
stituting 360 for 2π,
T365d, 256374, n = 0,59', 8".
This value of T is the duration of the sidereal year, or
the interval of time which lapses between two consecutive
returns of the sun to the same fixed star, in its apparent motion
about the earth. The interval of time between two conse-
cutive returns to the same equinox, is shorter, because the equi-
rdo
at
but an Ame
& Ara
at
when 7:0
EC Area.
Area
アル
​MOTION OF THE PLANETS
340
sun.
~ C=2 Tab
noctial points have a retrograde motion on the ecliptic, that is,
a motion the direction of which is contrary to that of the
If we suppose that the annual precession in 1800 was
50/23427, then since the radius vector of the sun describes
this small angle in 04, 014158, there results 365, 242216,for
the length of the equinoctial year at the commencement of
the present century. The sidereal year is constant, but as
the precession of the equinoxes varies a little, the equinoctial
year must participate in this inequality; its length diminishes
very nearly half a second in a century.
220. The value of the constant quantity c will be equal
to twice the area of the ellipse divided by T; therefore, as
the semi-axis minor is a VI-e², and the area of the ellipse
Ta² V1-e², we shall have
T
2πa²√1-e²
C =
T
By means of this value and of that of n, equation (1) becomes
equation (2) gives(c)
r²d0 = na²√1— e² dt.
a ( 1 − e²) — r
er
a√1 — e².dr
0
= arc (cos
COS=
de=
therefore, we shall have
r √ a²è² — (r — a)²
rdr
nadt =
√ a²e² - (r—a)²
In order to integrate these formulæ, let
and, consequently,
r = a (1 − e cos u);
–
•
dr = ae sin u du, ndt = (1— e cos u) du ;
(a)
MOTION OF THE PLANETS.
341
because at the point в, r = α (1-e), it is necessary that u
should be equal to cypher at this point, where we have also
t=0; therefore, by integrating we obtain
nt = u — e sin u.
(b)
By substituting in the expression of de, for r its value,
and observing that cos u
cos2u - sin² u, there results
✓ 1— e² du
de=
1 − e cos² ¦ u + e sin² ½ u
and if we assume
du
tang u=z,
=2dz,
this value becomes
2
cos² u
2 VT-e² dz
d0 = 1 − e + (1 + e) z² °
If we integrate and observe also that 0 and u vanish at
the same time, that is, at the point B, we shall have
10 arctang = z
e (t
1+
;
1- e
and hence we can deduce, by substituting for z its value,
tang 10 = √ ¹+e
Vite tang
tang u.
(c)
These three equations (a) (b) (c) express in a finite form,
the values of r, nt, 0, by means of the variable auxiliary u,
which is called the excentric anomaly. By eliminating u
between them, we shall obtain the polar coordinates r and 0 of
the planet in functions of the time, under the form of series
arranged according to the powers of the excentricity, which
will, consequently, converge very rapidly in the case of the
planets known previously to the present century.
In the series which will thus arise, the powers of cos nt,
which occur in the development of r, and those of sin nt,
:
342
MOTION OF THE PLANETS.
}
which are contained in the development of 0-nt, can be re-
placed by the cosines and sines of multiples of nt.
If we sup-
pose that these developments of the radius vector and of the
equation of the centre, are then arranged according to the sines
or cosines of the increasing multiples of nt, the values of the
coefficients of these two series in functions of the excentricity,
may be determined directly by the following analysis.
221. Let
r = ª¡ + ª¡ cos nt + a₂ cos 2nt +……….. + à¿ cos int + &c.,
✪ — nt — B₁sin nt + B₂ sin 2nt + ……………. B; sin int + &c.;
Ao, A1, A2, &c., B1, B2, &c. and generally, Ai, Bi, being the co-
efficients which it is required to determine.
If i and ¿' be any two positive integral numbers different
from each other, we shall have, by performing the integra-
tions(d)
ST
cos int cos i'nt d. nt = 0,
ST
sin int sin i'nt d. nt = 0;
and when ii', we shall find,
So d.
ST
cos² int d . nt = ½/π,
sin² int d. nt = ½π.
ㅠ​.
These last formulæ cannot be applied when i = 0; in
this case, the first integral is equal to π, and the second to
zero. This being agreed on, if the development of r be mul-
tiplied by cos int d. nt and that of 0-nt by sin int d. nt; and
if the results be respectively integrated from nt=0 to nt=π,
then all the terms vanish, except those of which the coeffi-
cients are a; and B; respectively; hence we infer,
A¿ =
2/ ST
T
r cos int d. nt,
MOTION OF THE PLANETS.
343
1
Bi=
2
·ST CO
nt) sin int d. nt.
In the case of i = 0, we shall have, in this particular case,
A0=
1
π
ST
rd.nt,
that is to say, the general value of A; is reduced to the half.
If we integrate by parts, and at the same time observe, that
―nt=0, at the two limits nt=0, and nt =π, the integral
of B; may be replaced by the following expression(e)
●
2
Bi
=
ST
cos int d (0-nt).
Since, from what precedes, it is evident that
do
√ 1 — e²
d.nt
= 1 1-e.cos u,
du
1-e.cos u
du
it follows, that if in place of r, nt, 0, their values in functions
of u, deduced from the equations (a), (b), (c), be substituted,
we shall have(f)
2
2a
ST (1 — e cos u)² cos (iu
= 2ª ST
A₂ =
Ai
π
e cos u)² cos (iu - ie sin u) du,
cos (iu ie sin u)
B; = — ST [√ 1,— e² − (1 − e . cos u)²]
Bi
iT
1 — e. cos u
du;
because u = 0 and u = correspond to nt = 0 and nt = π ;
these formulæ will make known the numerical values of the co-
efficients A; and B, either by the method of quadratures, or by
reducing them into series. In order to effect this reduction, we
shall have by Taylor's theorem
12e2
cos (iu—ie. sin u) = (1 — — sin²u+
2
24e4
2.3.4
sinu-&c.) cos iu
i3e3
+(ie sinu-
sin³u + &c.) sin iu ;
2.3
344
MOTION OF THE PLANETS.
and hence there will result for A and B series which are integrable
with respect tou, the exact values of which can be obtained either
immediately, or by means of known formulæ, so that we may
continue these developments of A; and B; as far as we please.
We can even obtain their general terms in functions of i and
e; but we will not insist longer on this subject here, as it be-
longs especially to astronomy.
In the case of i = 0, we have(h)
до
A。=-
π
(1 — e. cos u)² du = a ( +$ e²),
for as was observed above, only half of the general value of A;
is taken, when i = 0; this is the only one of the coefficients
Ao, A1, A2, &c., Bo, B1, B2, &c., the exact value of which can
be obtained.
222. If v denotes the velocity of the planet at the end of
the time t, and 8 the angle which its direction makes with the
radius vector r or oм produced, by No. 156 we shall have
v2
dr² + r²do²
dt2
do
v sin 8=r.
dt
!
Eliminating dt by means of equation (1), there results
v² = c²
(
de
:)
2
c2
+
2.2,
v sin d ==.
From equation (2), we likewise obtain
1
1
1 + e cos 0
d.
a (1 − e²) '
do
e sin
a (1 — e²);
hence we have(i)
a² (1 − e²)² v² = (1 + 2e cos 0 + e²) c²,
-
and, consequently,
MOTION OF THE PLANETS.
3.45
c²
a
(2-1), sin8=
2
v² =
2
a² (1 — e³) (2ª − 1 ),
—
a√1 — e²
; (d)
2a
1
go
hence it appears that in case of elliptic motion, the velocity
and direction of the moveable at each point may be deter-
mined by means of its radius vector. Substituting the value
of c of No. 220, that of v² may be written
2 وح
2
4π²a² (2a -1).
T2
These formulæ, when combined with those of the preceding
number, will completely determine the motion of a planet in
the plane of its orbit; but when the motions of two or more
planets are considered at the same time, then it is necessary
to refer the position of each of them to another plane, which
is generally the plane of the ecliptic, or of the earth's orbit.
223. Let NON' (fig. 53) be the intersection of the plane of
the orbit of a planet with a plane passing through o the
centre of the sun, OE a right line drawn in this second plane,
Oм' the projection of oм, the radius vector of the planet, on
this same plane. Let us denote by y the inclination of the
two planes, by a the angle NOE, by w the angle BON which the
radius vector oв drawn to the perihelion of the planet makes
with the line on. These three angles a, w, y, being given, they
determine the plane of the orbit, and the position of the ellipse
in this plane. Also, let & and represent the variable angles
MOM' and M'OE, which the radius vector oм makes with its
projection oм', and this projection with the line oɛ, these
angles will determine, at each instant, the direction of om the
radius drawn to the planet.
φ
This being established, in the solid angle, whose vertex is
at the point o, and which is contained by the three sides
OM, Oм', ON, as the true anomaly, or the angle MOB is re-
presented always by 0, the three faces of this solid angle
will be
2 Y
346
MOTION OF THE PLANETS.
MON MOB + BON = 0 + w,
M'ON— M'OE — NOE —— a,
=
MOM'
the first is opposite to a right angle, and the third to the angle
7. By the rules of spherical trigonometry we have
sin sin y sin (0 +w),
tang (4 — a) = cos y tang (0 +w) ;
and, as by what precedes the angle ✪ is known in a function
of t, each of the angles & and will be also given by means
of these formulæ.
When the given plane, on which the angle is reckoned,
is the ecliptic, then if the line OE, from which this angle is
measured in the direction of the earth's motion, is the line
drawn from the sun to the vernal equinox, the angles and $
are called the longitude and latitude of the planet in question.
The line NON' is the line of the nodes of its orbit; if it passes
into the northern hemisphere, when it traverses the plane of
the ecliptic at the point N, this point is the ascending node,
and N' is the descending node. According as the planet
exists in the northern or southern hemisphere, the latitude o
is said to be positive or negative, and the angle MON, or
+w is less or greater than 180(k). The angle & extends
from-90° to 90°, and the angle MON, and also the longi-
tude M'OE, from zero up to 360°.
W
If the point o be replaced by the centre of the earth, and
if the plane of the equator be that on which the angle is
measured; then when this angle is counted from the line
drawn from this centre to the first point of Aries, the angles
and will denote respectively the right ascension and de-
clination of the planet. When these formulæ are applied to
the apparent motion of the sun about the earth, we have a=0,
will express the obliquity of the ecliptic, and 0 + w will be
Y
MOTION OF THE PLANETS.
347
the longitude of this star, hence, denoting it by λ, we shall
have
sin 4 = sin y.sin λ, tang = cos y tang λ,
and, consequently (7),
sin
sin o
Y. tang
&=
✓ cos²y+tang²
This angle y
The greatest northern and southern declinations corres-
pond to λ= 90 and λ= 270°, and are ±y.
is also equal to the angle which the axis of rotation of the
earth makes with the perpendicular to the plane of the ecliptic;
it is subject to a slight inequality called the nutation, the period
of which is about eighteen years, and which at its maximum
amounts only to 9"4. Its mean value, at the beginning of this
century, was
y = 23°,27',55";
it diminishes by 0"45692 for each year.
224. In what precedes, no reference has been made to the
force which acts on each planet; in fact, all the circumstances
of its motion have been determined by data furnished by ob-
servation, and without having any recourse to the principles of
dynamics; it is now requisite to determine the laws of this
force as they have been already stated in No. 217.
It follows from the first law of Kepler, that the force which
retains each planet in its orbit, is constantly directed towards
the centre of the sun; although this necessary consequence of
the proportionality of the areas to the times has been already
deduced in No. 155 from the equations of motion, it will not
be deemed superfluous here to give a synthetic demonstration
of it.
Let M₁M (fig. 54) be the side of the trajectory described by
the moveable in an indefinitely small portion of timer. When
it arrives at the point м, if no force acts on this moveable, it
will describe in a portion of time equal to 7, a part мm of мT,
the production of M, M, and equal to MM₁, but, in consequence of
348
MOTION OF THE PLANETS.
the force which acts on it, in this second instant, it is trans-
ferred to another point м'. Let мк be the direction of this
force at the point м; we may suppose, that during the time
r it remains parallel to itself; hence, if we draw the line mm',
it will be parallel to мK (No. 148). Now, if c be the fixed
centre about which the radius vector describes areas propor-
tional to the times, the triangles м,CM and MCм', which are
the areas described in two equal portions of time, will be
equal to each other. But the triangles M,CM and мcm are
also equal to each other, since their vertices are at the same
point c, and their bases MM, мm, are equal and exist on the
same line; consequently, the triangles мcm and мCм' are
equivalent; and as they exist on the same base мc, the line
M'm which joins their vertices must be parallel to this base;
hence the line мк, parallel to mм', coincides with мc. There-
fore, in each point м of the trajectory, MK the direction of the
force, will be that of the radius vector mc.
Conversely, if the force which acts on the moveable at
any point м, is directed along MC, the right line mм' will be
parallel to this radius vector, the two triangles M'CM and мсm
will be equal, and, consequently also, the two triangles M'CM
and M₁CM.
The areas described by the radius vector about
the point c, in two consecutive instants, being equal, and this
being the case for the entire trajectory, it follows, that if the
force which acts on the moveable be constantly directed to this
point, the areas described in equal times will be equal, and, in
any time whatever, proportional to these times.
225. Let, as in No. 218, м be the position of the planet
at the end of the time t (fig. 52), and let, as in preceding
numbers, r denote the radius vector oм, and the angle MOB,
moreover, if x and y represent the two rectangular coordi-
nates OP and PM, referred to the axes ox and oy, of which the
first passes through the perihelion, we shall have
x=r.cos 0, y = rsin 0, x²+ y² = r².
MOTION OF THE PLANETS.
349
Also, let R denote the accelerating force, acting on the
planet, the intensity of which is unknown.
We have seen that this force acts along the radius vector,
and, because the concavity of the trajectory is turned towards
the sun, it is directed from м to the point o; therefore, the
cosines of the angles which it makes with x and y pro-
and-2; consequently, the equations of the mo-
duced, are
X
tion will be
d²x
RX
d²y
Ry
=
dt2
dt2
;
r
(1)
v being the velocity at the point м, we shall have
dx2 dy2
v² = +
dt2 dt2,
2
and, by differentiating,
ď²x
day
dt2
1 d.v² = dx+ dy,
dt2
consequently, if equations (1) be multiplied by dx and dy
respectively, and then added together, we shall obtain, by
observing that xdx + ydy = rdr,
d.v² =
Rdr.
But, when the motion is performed in an ellipse, we have
(No. 222)
2 ս.
v2 =
°
μ
218
μ being made equal to
4π² a³
T2
Consequently, we shall obtain,
R =
μ
from which it appears, that the force which acts on each
planet, varies in the inverse ratio of the square of the distance
from the centre of the sun.
350
MOTION OF THE PLANETS.
The magnitude of this force at the unit of distance is u,
if u' be what it becomes for any other planet, of which the
semi-axis major and time of revolution are represented by a'
and T', we shall have
4 π2a/3
13
μ'=
T/2
But, by the third law of Kepler, we have
from which results
a³
T2
T²: T/²:: a³: a¹³;
=
a/3
Τ2, μ= μ';
T/2,
consequently, at the unit of distance, and, generally, at the
same distance from the sun, the accelerating force R is the
same for two different planets.
The motive force of each planet is therefore independent
of its particular nature, and, like the weight of a body at the
surface of the earth, is proportional to its mass. It varies
from one planet to another, according to the same law, as it
does from one position to another of the same planet; and if
two planets were situated at the same distance from the sun,
and then remitted to themselves, without any initial velocity,
they would descend towards this star with the same velocity,
and reach it in the same interval of time.
Thus it appears, that the three laws of Kepler completely
determine the force which retains the planets in their orbits;
the law of the areas proportional to the times, shews that this
force is constantly directed towards the centre of the sun; that
of the elliptic motion, or the expression for the velocity, which
is deduced from this law, combined with the preceding, proves
that its intensity varies for the same planet in the inverse ratio of
the square of the distances from the sun; finally, from the law
of the squares of the times of revolutions proportional to the
cubes of the greater axes, we infer that at equal distances from
the centre of this star, the intensity of the motive force is pro-
MOTION OF THE PLANETS.
351
portional to the mass of each planet, and independent of its
particular nature.
226. Newton extended to the motion of comets about the
sun, and of the satellites about their respective primaries, the
laws of Kepler, and the consequences that follow from them,
with respect to the force which acts on these moveables.
Comets in their motion differ only from planets in this, that
they are not constantly visible, in consequence of the great
distance of their aphelia, which renders the determination of
their orbits extremely difficult. Notwithstanding this, there
are three comets, of which the orbits and times of revolution
are known almost as accurately as those of the planets. With
respect to other comets, astronomers have computed their
motion in an approximate manner, by assuming for their tra-
jectory, in the small extent for which they are visible, a para-
bola of which the focus exists in the centre of the sun, and
supposing always that the areas described by the radii vectores
about this point, are proportional to the times for each comet.
This case is embraced under the preceding formulæ for elliptic
motion, by making
a=x, a (1-e) = b;
b denoting the perihelion distance оB, which is a finite quan-
tity.
The masses of the comets are very small compared with
those of the planets, and they seem to be of an entirely different
nature(m). By the third law of Kepler, the motive forces of two
comets, or of a comet and a planet, at the same distance from
the sun, are to each other as their respective masses, and their
accelerating forces are equal; this is likewise true for several
satellites of the same planet, but not for satellites of two dif-
ferent planets, or for a satellite and a planet ; for the law of the
squares of the times of the revolutions proportional to the
cubes of the greater axes of the orbits, obtains only for bodies
which revolve about the same centre; in the sequel we will
352
MOTION OF THE PLANETS.
shew the relation which exists between the motive forces of
two satellites belonging to two different planets, or between
those of a satellite and a planct.
It ought to be observed here, that within the last few
years, the laws of elliptic motion have been extended to those
double stars, in which a periodic motion of one of the stars
about the other, has been recognized, and their relative posi-
tions, computed by means of these laws, agree with their ob-
served positions as accurately as could have been expected(n).
227. Let us now consider the changes which the resistance
of an extremely rare ether diffused through the heavenly
regions would produce in the elliptic motion of the planets
about the sun. If these bodies are not perfectly spherical,
this circumstance, combined with the friction of the fluid
against their surface would cause the centre of gravity to de-
viate from the plane of its orbit. However we will at present
abstract from the consideration of these circumstances, and
form the equations of the motion of this point, from the con-
sideration of a central force varying in the inverse ratio of the
square of the distance, combined with a tangential force arising
from the resistance of the medium.
Let this resistance be supposed, as in the motion of pro-
jectiles in the air, to be proportional to the square of the ve-
locity, to the density of the medium, and to the surface of
each planet; the accelerating force which results will be in
the inverse ratio of the mass of the moveable. If ds repre-
sents the velocity of the moveable, then we can denote this
ds2
force by p. p being a very small coefficient, and for the
pdt²
dt2,
,
same planet proportional to the density of the medium. Since
this force acts always in a direction contrary to the velocity of
the moveable, if the principal force directed towards the centre
of the sun be denoted by μ at the unit of distance, and by
μ
2.2
at the distance r, equations (1) will be replaced by the fol-
lowing(0),
MOTION OF THE PLANETS.
353
+
d²x
мх
ds dx
dt2
+ 23
ρ
dt dt'
d²y му
+
ds dy
(2)
dt2
203
P
dt dť
If we make use of polar coordinates, we, may without dif-
ficulty, deduce from them(p)
d (dr²+r²d0²)
dt2
1
2μd. =2p
2p (dr²+r² d0²)
ds,
dt2
(3)
μα
d.r2d0=- pr² do ds,
which are the equations into which the preceding may be
transformed.
228. Equations (2) become, when their second members
are neglected,
d²x
dt2
de + 4 = 0,
+4=0,
d³y
di² + My
+ My =
=
0,
(4)
and equations (3) are reduced to
d. (dr² + r²d0²)
dt2
1
2ud.= 0, d.r²d0 = 0.
(5)
We can satisfy equations (5) by means of formulæ (a), (b),
(c), of No. 220; however, as these formulæ contain only two
constant arbitraries, a and e, they are not the complete inte-
grals of these equations; but if we consider that equations (5) do
not contain explicitly the variables ✪ and t, but only their
differentials do and dt, it is evident that the formulæ of the
number cited will still satisfy these equations, by adding to t
and 0, certain constants. In this manner, the complete in-
tegrals of equations (5), and, consequently, of equations (4),
will be expressed by the following system of formulæ,
ra (1-ecos u),
nt + ɛ - w = U
e sin u.
+ e
tang 3 (0 – w) = √1± tangu;
I - e
(a)
22
354
MOTION OF THE PLANETS.
a, e, ɛ, w, being four constant arbitraries, and n a constant
connected with a by the equation
a³ n² = µg
2π
4π² α³
2
which results from =n,
T
T2
=μ, by the elimination of T.
When the variable u is cypher, the value of r is a minimum,
which is the case at the perihelion B (fig. 52). In the case of
u = 0, we have ✪=w, so that in this case 0 will represent the
angle MOE, measured from the right line OE, which makes the
angle BOEw, with OB. The value of 0, expressed in a series,
will be of the form
0 = nt + ε + 01,
Ε
in which 0₁ denotes its periodic part, arranged according to
the sines of the increasing multiples of nt + w. This
angle ✪ will be the true longitude of the planet in the plane
of its orbit, at the end of t any time whatever; nt + ɛ will ex-
press its mean longitude at the same instant, & its mean longi-
tude at the epoch from which the time t is reckoned, and w the
longitude of its perihelion.
229. This being established, when the complete integrals
of a system of differential equations, such as equations (4),
are known, the integrals of another system of differential
equations, such as equations (2), which only differ from the
first by very small terms, may be deduced from them by a
method that has been most successfully applied by geometers
to different questions in celestial mechanics, and of which we
proceed to explain the principles, for the sake of the problem
which we are at present occupied with. The values of x and
y which satisfy equations (4), are of the form,
x = ƒ (t, a, e, ɛ, w), y = F (t, a, e, ɛ, w) ;
fand F denoting given functions. In order that these values
may likewise satisfy equations (2), a, e, ɛ, w, should be con-
sidered as new variables, which it is required to determine.
MOTION OF THE Planets.
355
But as the number of these unknown is four, while equations
(2) are only two in number, we may assume arbitrarily two
auxiliary equations, and, in consequence, make
df
df
df
df
da +
de +
dɛ +
dw = 0,
da
de
dε
dw
(b)
dF
dF
dF
dF
da +
de+ dε +
dw = 0,
da
de
dε
dw
that is, in other words, the parts of dx and dy, which arise
from the variations of a, e, ε, w, are made equal to zero. By
dx dy
dt dt
this means, the complete values of
and are simply
dx
df dy dr
dt
dť dt
dt
By differentiating again, we obtain,
d²x_ d²ƒ, d²f da ď²f de
+
d²f dε d²f dw
dr² = dt²+dtda dt + dtde at dtde at dtdw・ dt
d'y d²F
dt²- dt²
diat
dt
dt
+
d°F da der de der de d²F dw
dtda dtdtde dt+dtds dtdtdwat
Now, by hypothesis, the preceding values of x and y satisfy
equations (4), a, e, &, w, being considered as constant arbitra-
ries, consequently we have
MOC
d'f
dt2
+= 0,
p³
dt2
d²F му
+ = 0;
23
hence, if the complete values of
d²x d²y
and be substituted in
dt2 dt2
equations (2), we shall have
ď²f_da+
dtda
ď²ƒ
dtde
de + atde
df dε + dtdw
d²ƒ
dw=
ds dx
Pdt dt
dt,
(c)
ď² F
d'F
d2F
d²F
ds dy
·da +
de +
dε +
dw=
dt;
P
dtda
dtde
dtdε
dtdo
dt dt
and thus, by means of the four equations (b) and (c), we can
determine a, e, ɛ, w, in functions of t.
#
356
MOTION OF THE PLAanets.
230. In general, this substitution of four differential equations
of the first order, in place of the two equations (2) of the second
order, will not be attended with any advantage. But as the va-
lues of da, de, dɛ, dw, which are deduced from equations (b) and
(c), have for a factor p the coefficient of the resistance, which
is a very small quantity, the variable parts of a, e, ɛ, w, will be
also extremely small; and if the square of P be neglected,
ɑ, e, ɛ, w, may be considered as constant, in the expressions of
da, de, dɛ, do; hence the determination of the variable parts of
ɑ, e, ɛ, w, may be reduced to quadratures. By the method of
succcessive approximations, the values of these quantities may
be thus obtained, arranged according to the powers
of
and
continued to whatever degree of accuracy we please; how-
ever, we shall stop at the first power of p, in the present in-
vestigation(q).
of pr
If in equations (a), the variable values of a, e, ɛ, w, be sub-
stituted, they will make known, as in the case of elliptic mo-
tion, the values of r and 0 in functions of the time. The
trajectory which in this case is described, will be still an ellipse,
but its elements continually vary. If at each instant, the
constant ellipse which satisfies the values of the elements at
this same instant, be constructed, the ordinates x and y, and
their differentials de and dy will be common, in virtue of
equations (b), to this ellipse, and to the trajectory, which
consequently will be the osculating curve of all the constant
ellipses. For the same season, the velocity of the moveable
and its components will have the same expressions in the el-
liptic motion, and in the motion altered by the resistance of
the medium, and will be determined by formulæ (d) of No.
222.
231. As we have identically
nt = Sndt + Stdn;
by comprising Stdn in the unknown ε, the second equation (a)
may be written as follows,
MOTION OF THE PLANETS.
357
Sndt + ɛ — w = u
e sin u ;
(d)
Then, if in the equations of elliptic motion, there be substi-
tuted for the constants a, e, e, w, their variable values, we should
by the integral Sndt, wnich is
at the same time replace nt,
supposed to vanish when t=0.
The quantity n that it involves,
may be deduced from a, by means of the formula
n
α
ava
μ
which is furnished by the equation a³n² = µ of No. 228. This
integral Sndt expresses the mean motion of the planet (No.219),
altered by the resistance of the medium, and thus, the diffe-
rential of the mean motion will be ndt, in the disturbed motion
as well as in the elliptic motion.
ɛ
At the perihelion, the angle -w is either cypher, or a
multiple of 360°; and this will be also the case with the angle
u, in virtue of the first equation (a); therefore, in the time
which lapses during two consecutive passages of the planet
through its perihelion, the quantity Sndt + ε - w will be in-
creased by 360°; this will enable us to determine this interval,
when n, ɛ, w, are given in functions of t. The time of a revo-
lution, or the interval which lapses between two returns of the
planet to the same fixed point, will in like manner be that in
which its true longitude 0 is increased by 360°(r).
232. Equations (b) and (c) may be replaced by equivalent
ones, from which the values of da, de, dɛ, dw, can be inferred
with greater facility. For this purpose, it is to be observed,
that if any equation
p (nt, r, 0, a, ɛ, w) = 0,
obtains in the case of elliptic motion, it will likewise subsist
when the motion is altered by the resistance of the medium,
ɑ, e, ɛ, w, being regarded as variables determined by equations
(b) and (c), and Sndt being substituted instead of nt. There-
fore, the differential of the function will be equal to cypher,
г
358
MOTION OF THE PLANETS.
whether it be taken in the first case with respect to nt, r, 0, or
in the second case, with respect to Sndt, r, 0, a, e, ε, w. Now,
as r and are functions of x and y, their differentials are the
same in the two cases, in virtue of equations (b); consequently,
аф
if in the complete differential of p, the part
аф
аф
d.nt
d.nt +
dr
dr + do do, which is separately equal to cypher, be suppressed,
we shall have
аф аф
do da + do de+do de +
da
de
dε
аф
dw = 0.
dw
W
Now, if in equation (2), of No. 218, 0w be substituted
in place of 0, there results
r+re.cos (0-w) — a (1 — e²) ;
and, if this be differentiated in the manner already pointed out,
we shall have
r cos Od. e cos w + r sin Od . e sin w = d.a(1-e²). (e)
If the first equation (a) and equation (d) be also differenced,
we obtain
(1 — e cos u) da a cos u de + ae sin u du — 0,
dε do + sin u de (1 — e cos u) du = 0,
u being regarded as function of ɑ, ɛ, e, w.
By eliminating du between these two(s) equations, we
obtain
(1 − e cos u)² da + a (e — cos u) de + ae sin u (dɛ — dw) = 0.
But, if in the formulæ
cos u =
1- tang² 1 u
1 + tang² u
sin u =
2 tangu
1+tang² u
"
we substitute for tang u, its value which is given by the third
equation (a), we have(t)
MOTION OF THE PLANETS.
359

e + cos (0 w)
cos u =
√ 1 — e² sin (0 — w)
1+e.cos (0 - w)'
—
by means of which, the preceding equation becomes
sin u =
1+e.cos (0 — w)
(1-e²) da
1+e.cos(0—w)
acos (0−w) de+
√ 1 − e²
ae sin (0—w) (dɛ—dw)=0. (f)
What has been stated with respect to the equation = 0,
may likewise be applied to the case in which the function
contains the first differentials of r and 0. Thus, in the case of
elliptic motion, we have
dr² + r² do² 2μ
dt2
r²d0 = √ µa (1— e²) dt,
μα
√μa being substituted in place of a²n in the expression for
r²d0 of No.220(v). Now, since the differentials dr, d0, as also
r and 0, remain the same when a, e, ɛ, w, become variable, it
follows that these two equations will also subsist on this hy-
pothesis; this being so, if their complete differentials be com-
pared with equations (3) of No. 228, we can infer(u)
d. 1 = 2p (1) ds,
a
d. √ a (1 — e²) = − p √ a (1 — e²) ds.
(g)
Now, the values of da, de, dɛ, dw, may easily be inferred
from the four equations (e), (f), (g); by substituting for r its
value, namely,
r =
a (1-e²)
1 + e. cos (0 — w)'
in order to express them in functions of the angle solely, we
find(x)
360
MOTION OF THE PLANETS.
1
da=
2 pa
1-2 [1+2 e cos (0 — w) + e²] ds,
de=2p [e+cos (0 - w)] ds,
edw=- 2p sin (0 — w) ds,
de = + 2 pe sin (0—w) [√1—e²—e²— e cos(0—w)],
[1+ e cos (0-w)] (1+VI—e²).
(h)
The value of ds which should be substituted in these formulæ,
is
ds= √ √² +
dr²
do;
d02
and, by substituting in this expression the value of r, we obtain
ds=
a ( 1 − e²) √ 1 + 2 e cos (0) — w) + e²
[1 + e cos (0 — w)]²
2
do.
The second members of these equations should be inte-
grated on the supposition that a, e, ɛ, w, are constant, as has
been already stated; and when the coefficient p is given in a
function of r, and consequently of 0, the values of the variables
ɑ, e, ɛ, w, may be deduced by the method of quadratures, or by
reduction into a series, which values should be substituted in
the equations of elliptic motion.
233. If the excentricity e be a very small fraction, formulæ
(h) reduced to their principal part, become
da = − 2 pa²d0, de
edw= 2 pa sin (0 - w) do,
2
pa cos (0 — w) d0,
dɛ = 2 pae sin (0
dɛ = 2 pae sin (0 — w) d0 ;
and the coefficient p may be regarded as constant. Hence,
by integrating and denoting the variable parts of a, e, ε, w, by
Sa, de, Sɛ, Sw, we shall have
бе
Sa=2pa²0,
de = -
2 pa sin (0-w),
edw = 2pa cos (0 — w),
δε Ξ
= - 2 pae cos) 0-w).
MOTION OF THE PLANETS.
361
If the corresponding part of n, or of
be expressed by
ava
Sn, so that we may have
Sn =
3 √μ
2a² Va
δα,
α
there will result
Sn = 3 pano.
Hence it appears, that the effect of the resistance of an
extremely rare medium, on the motion of a planet in an orbit
of very small excentricity, will be to cause the greater axis to
decrease indefinitely, also to increase n the angular velocity,
and to produce in each of the three quantities e, w, ɛ, an in-
equality of which the period is the same as that of the planet's
revolution. Not only is the angular motion more and more
accelerated, but also the absolute velocity; for it is very nearly
equal to an; its increment is therefore adn + nda; which is a
positive quantity and equal to pa²n0.
If the excentricity be altogether neglected, we have
r = α, 0 = Sndt + ε;
=
therefore, if dr and 80 denote the parts of the radius vector,
and of the longitude which are produced by the action of the
resisting medium, we shall have to the same degree of approxi-
mation,
Sr=2pa²0, 80=
2 pa²0, 80 = 3 pa0².
In consequence of this continual diminution of the radius
vector, which amounts to 4pa² at the end of each revolution
of the planet, this body must eventually fall on the surface of
the sun.
Finally, if there exists in the regions of free space, an
ether which affects the motion of the heavenly bodies, its in-
fluence will be particularly sensible in accelerating the motion
of the comets, on account of the extreme smallness of their mass,
and because that, every thing else being the same, the coefficient
is in the inverse ratio of the mass of the body. And, in fact,
p
3 A
362
MOTION OF A MATERIAL POINT
f
we have not been able as yet to recognize any traces of an
ether which resists the motion of the planets and satellites; but,
according to the calculations of M. Enke, this resistance ap-
pears to have an appreciable influence on the motion of the
comet recently discovered, the time of whose revolution is about
1200 days.
III. Motion of a material Point acted on by a central Force.
234. The problem which we now proceed to consider is
the inverse of that of the preceding section; there the tra-
jectory and the law of motion being supposed to be furnished
by observation, it was proposed to determine, in magnitude
and direction, the force which produced this motion; in
the present case, a constant force directed towards a fixed
centre, and given in a function of the distance of the move-
able from this point, is supposed to be applied to this move-
able, and it is required to determine from thence the trajectory
and the law of the motion.
This curve DMB (fig. 55) will be contained in the plane
passing through the fixed centre c, and through DAa, the di-
rection of the initial velocity. Let two rectangular axes ca,
cy, of which the first passes through D, the point from which
the body commences its motion, be drawn in this plane,
through the point c, and let them be the axes of the coordi-
nates. At the end of the time t, which is reckoned from the
departure at D, let the moveable be supposed to be in м, and
let x and y denote its coordinates CP and PM, r its radius
vector CM, R its accelerating force directed from м towards
c, and given in a function of r; the equations of the motion
will be
d²x
dt2
R,
d²y
dt2
y
R;
T
(1)
if the force R acts in the direction of the production of CM, it
will only be necessary to change the signs of their second
ACTED ON BY A CENTRAL FORCE.
363
members. We can immediately deduce from these equations,
the two following integrals of the first order,
dx² + dy²
xdy — ydx = cdt,
dt2
25Rdr+b,
in which b and c are the constant arbitraries introduced by
the integration; and if ✪ denotes the angle мca, so that we
may have
x=rcos, y = r sin 0;
these integrals will become
dr² + r²d0²
r²deedt,
dt2
2SRdr+b;
(2)
if the values of dt and do be deduced from these equations,
they will be of the form
dt = frdr, d0 = Frdr,
which it will be only necessary to integrate either accurately
or by approximation.
If dt be eliminated between equations (2), we obtain
2
c²
co ( 4 ) + 2 + 25ndr = b,
c2
ᏧᎾ
22
for the differential equation of the trajectory.
(3)
If v represents the velocity of the moveable at the point
M, we shall have
v²=b― 25Rdr ;
(4)
and 8 denoting the angle that its direction makes with MC, its
components will be
v cos 8 =
dr
dt'
v sin d=
rdo
dt'
in the respective directions of MC, the radius vector, and of MH
perpendicular to MC.
The two constants b and c, and those which are introduced
by the integration of the values of dt and de, can be determined
364
MOTION OF A MATERIAL POINT
by means of the initial position and velocity of the moveable.
For this purpose, let y represent the initial distance CD, a the
angle CDA which may be either acute or obtuse, and h the
height due to the initial velocity, so that this velocity may
be √2gh, g denoting the gravity. If the integral SŔdr,
which occurs in the preceding formulæ vanishes, when ry,
we shall have, as is evident from the general expression for v²,
b=2gh.
In virtue of the equation r²d0 = cdt(y), the value of v sin 8
is the same thing as
с
; consequently, we shall have
c=y√2gh.sin a.
As to the two other constant arbitraries, they are deter-
mined by the supposition that 00, and ry, when t = 0,
and thus, the problem will be completely resolved.
235. When the force R is proportional to the distance r,
the variables x and y are separated in equations (1), and there
is no occasion to recur to polar coordinates, or to equations
(2). In fact, let k denote the value of R which corresponds
tory, and
will be its general value.
d²x
dt2
kr
R=
γ
Equations (1) will become
kx
d²y
ky
༧ dt2
Y
and their complete integrals will be(z)
x = A cost V
y= B cos t
γ
+A'sin t
V
༧
k
√ + B'sint √;
Y
γ
A, A', B, B', being the four constant arbitraries that are intro-
ACTED ON BY A CENTRAL FORCE.
365
r
duced by the integration. In order to determine them, we
have, by the preceding suppositions,
x=y, y=0, dx
= √2gh.cosa, = √2gh.sin a,
dy
dt
dt
when t=0; hence it follows that
k
A = 1,
Α'
√2gh cos a,
k
B=0, B'
=
√2gh. sin a,
γ
and, consequently,
k
ky
cos.
cos a sin t
•
sin a sin t
ky
√
= 7 (cost √ √2gh
x=Y
Y
y = y V2gh
It appears from these formulæ, that the revolutions of the
moveable about the point c, will be isochronous, and their
common duration equal to 2π √(a^). We can deduce from
them
y sin a sin t
V
k
=y√ ky
γ
2gh'
Y
√ = x.sina + y cosa;
y sin a cost
hence we obtain(b′)
ky
2gh
y²+(x sin a + y cos a)² = y² sin² a,
for the equation of the trajectory, which is, evidently, an ellipse
of which the centre is at the point c.
circle when a = 90°, and ky = 2gh.
This ellipse becomes a
In this case, the motion
is uniform, for by the expressions for x and y, we have
dy
dx
dt
Vyk
=-7k sin t√ √ cost;
Vyk V.
=
7' dt =
366
MOTION OF A MATERIAL POINT -
hence the velocity v is equal to vyk. The central force R and
the centrifugal force
v2
Y
are constant, and respectively equal tok.
If the force R be repulsive instead of being attractive, as
has been supposed, k must be changed into k in the pre-
ceding formulæ. The trajectory will be then an hyperbola,
and the body in its motion will never return a second time to
the same point.
236. Let the force R be supposed to be proportional to the
inverse cube of the distance, and, consequently, that
R =
ky³
2.3
k being always its value at the point D.
In this hypothesis, we shall have,
= ky (1 − 2²),
2SRdr = ky
because the integral vanishes when ry.
If we take into account the values of b and c, and make
yd.— __ dz
=
ช
= 2,
go
do do'
equation (3) will become
dz2
d02
+(1-2gina
ky
2gh.sin a
1
ky
22
sin² a
2gh sin²
a
As the coefficient of 22 may be either positive or negative, let
1-
ky
2gh.sin² a
= ±n²;
consequently, we have(c')
dz2
do²
n²z²= cot² a±n²,
from which we obtain
<
ACTED ON BY A CENTRAL FORCE.
367
:
ndz
ndo=
√ cot² a±n²
干 ​n² z²
When the superior signs are employed, we shall have(ď)
n0 =
no arc (sin =
(sin
nz
arc (sin=
✓ cot² a+n²
(sin
N
;
√ cot² a +n²
and, when the inferior signs are made use of,
no = log
nz + √ cot² a n² + n² z²
n+cot a
taking care to observe, that ry and z = 1 when 0.
From the first value of n0, we deduce
nz = cot a sin n✪ + n cos n✪.
The maximum of z, or the minimum of r, corresponds to the
value of 0, deduced from the equation dz = 0, or(e')
1
tan no =
cot a,
N
for which we obtain
2 = 2 = √ 1 + 1/2 cot² a
n
Beyond this value of 0, the distance of the moveable from
the point c will increase indefinitely, and its radius vector r
will be infinite, for the least value of 0 deduced from the equa-
tion z = 0, or(ƒ)
tang non tang a;
and to attain this value of an infinite time is required. The
value of t as a function of 0, may be obtained without any
difficulty, by substituting in the first equation (2), in place of
r, the value of 2.
2
When the value of no is a logarithmic function, we shall
have, by passing to numbers, and denoting the base in Napier's
system of logarithms by e,
.
368
MOTION OF A MATERIAL POINT
་
nz + √ cot² a
n² + n²z² = (n + cot a) ene ;
hence we obtain (g)
Y
2=
2 = 121 (n+cot a) en² + 1/2 (n
(n − cot a) e-no ;
2n
2n
it appears from this expression, that the value of r diminishes
indefinitely, consequently, the curve described by the move-
able body about the point c, will be a spiral, and it will not
reach the centre until after it has performed an infinite number
of revolutions.
If, in order to simplify, we make a 90°, we shall have
λ=
27
eno +e-nos
for the equation of this spiral. The first equation (2) becomes
√2gh dt =
4 ydo
(eno+e-no)2;
and, by integrating, we obtain
(eno
e-no)
eno + e-no
nt √2gh=y
237. Let us suppose, for the last example, that the force
R varies in the inverse ratio of the square of the distance,
which is the law that obtains in nature, in which case we
have
=hy, Sudr=ky (1-2);
R =
ky2
2.2
k being the intensity of this force at the point D, where the
preceding integral is supposed to be equal to nothing.
If we make
1
= = p, 2ky — b = ß,
equation (3) of the trajectory will become
ACTED ON. BY A CENTRAL FORCE.
369
ર
2
dp²
d02
-
from which we obtain(h)
de=
2ky2p-c²p² - ẞ;
cdp

V-B-(-)
·B — (ky² — cp) **
Hence if we integrate and denote the constant arbitrary by
w, we shall have
ky² — c²p
0 = w + arc
(co
COS =
;
√ k³y¹ — c²ß
4
by means of which we can obtain
4
ky²r = c² — r √ k²y¹ — c²ß cos (0 - w),
(a)
w+being substituted in place of w, in order that w may
be the value of 0 which corresponds to the least value of r,
that is to say, to the point of the trajectory where the move-
able is nearest to c.
In order to obtain the equation of this curve in rectangular
coordinates, let
x' = r. cos (0 — w), y' r. sin (0 — w) ;
x' and y' being the rectangular coordinates of the moveable
referred to the axes cx' and cy', so that x'cx = w; we shall
have evidently
x¹² + y¹² = r²;
and if the two members of equation (a) of the trajectory be
raised to the square, it will become(¿')
k²y¹y¹² +ẞc²x²² = c² 2c²x' √k²y ¹ — c²ß.
—
Now, it is evident from inspection of this expression, that it
belongs to a conic section, which will be an ellipse, a parabola,
or hyperbola, according as the constant ẞ is positive, cypher,
or negative. It appears also, that in all cases, the point c
will be a focus of this curve; for, in the equation (a), the
radius vector r is a linear function of the abscissa x'; now in
3 B
370
MOTION OF A MATERIAL POINT
the three conic sections, this is only the case, when the origin
of the coordinates is one of the foci.
Since b2gh, we shall have
B = 2ky - 2gh;
hence it follows, that the sign of B, and consequently, the
nature of the conic section described by the moveable, de-
pends only on the initial distance and velocity of projection,
and not at all on the direction of this velocity; so that if se-
veral material points commence to move from the same point
c with equal velocities, all of them will describe conic sec-
tions of the same nature, whatever may be their initial di-
rections. If for example kg, the curve described will be
an ellipse, a parabola, or hyperbola, according as the height
due to the initial velocity is less than CD, equal to this dis-
tance, or greater than it.
238. In the case of the ellipse, equation (a) shews that the
greatest and least values of r correspond to 0 = ∞ + and
w; if we denote them by a (1+e) and a (1 − e) res-
pectively, then a is the semi-axis major, and e the excentri-
city, and we shall have
4
(ky² — √/k²y¹ — c²ß) a (1 + e) = c²,
4
(ky²+ √/k²y¹ — c²ß³) a (1 − e) = c²,
or, what is the same thing,
Ba (1+e) = ky²+ √ k²yª — c²ß,
Ba (1 − e) = ky² — √/ k²yª — c²ß•
-
If we add these equations together, and if we also multiply
their corresponding members, we shall obtain
Baky², Ba² (1 — e²) — c² ;
and, by substituting for ẞ and c their values,
B = 2 (ky — gk),
we obtain
γ
c=y√2gh sin a,
ACTED ON BY A CENTRAL FORCE.
371
yk VI
2 (ky — gh) a = ky³,
e22
--
yh V1 - 8 = 2 Vgh (ky — gh) sin a ;
}
(b)
by means of which, the semi-axis major, and the excentricity,
may be obtained. The angle w is determined by making, at
the same time, 0=0 and ry in equation (a). Thus, when
the position, initial velocity and direction of the moveable are
given, the dimensions of the ellipse, and the position of its
greater axis, will be completely determined. With respect to
its motion on this curve, that is known by formulæ (a) (b) (c)
of No. 220. It appears from formula (4) of No. 234, that
the square of the velocity at any instant is expressed by the
equation
v² = 2gh — 2ky + 2ky²,
or, what comes to the same thing,
2
v² = μ
(3-4),
(c)
on account of the value which we have found for a, and by
making ky² =µ, so that here, as in the formula of No. 225,
μ expresses the intensity of the central force at the unit of
distance.
239. As the motions of comets are observed to approximate,
during the time of their apparition, to that of a body moving in
a parabola, it will not be irrelevant to discuss here this parti-
cular species of motion. Since we have in this case ẞ= 0,
or ky≈ gh, equations (b) give a = ∞ and e=1; which is in
fact the case in the parabola. Formula (c) becomes
-
2μ
v² =
and if u denotes the velocity in a circle, of which the radius is
equal to r, we shall have in virtue of the same formula,
u² = 12;
1
372
MOTION OF A MATERIAL POINT
consequently, at equal distances from the sun, the velocity of
a comet is to that of a planet, which describes a circle, as
1/2 to 1.
In general, if the two members of the last of equations (b)
be raised to the square, and if they then be respectively múlti-
plied by those of the first, we shall obtain
kya (1 − e) (1+e) = 2ghy sin²
—
a,
hence, if p denotes the least distance of the comet from the
sun, so that(k')
p = a(1 − e),
and if we make ky = gh and e = 1, we shall have
p=y. sin ²a;
from this it appears, that the perihelion distance is determined,
when the initial distance and angle of projection are known.
If, in equation (a), we make ẞ = 0, and ky = gh, it will be-
come, by substituting for c² its value 2ghy². sin² a,
r = 2y sin 2a
r cos (0 — w) ;
hence there results
2p
(d)
1+ cos (0 — w)
for the equation of the trajectory. If we make 0=0, and
r=y, we obtain (l')
y (1+ cos w) = 2p, cos &w= sin a ;
by means of which, we can determine the angle w that the
radius vector drawn to the perihelion, makes with the radius
drawn to the point from which a body is projected.
If we substitute for the values of c and r in the first equa-
tion (2) of No. 234, and if, in order to abridge, we make
γ
✓gh sin a
p²
N,
there results
ACTED ON BY A CENTRAL FORCE.
373
and because
4 do
[1+cos (0-w)]2
= n√2.dt,
1 + cos (0 - w) = 2 cos² (0 — w),
0-w=24, de 2d,
=
if we make
we shall have
d↓ ndt
cos*
1/2
hence by integrating and denoting the constant arbitrary by ɛ,
we obtain(m')
(3 + tang²) tang 4 + ε =
3nt
√2
In order to determine this constant, it is to be observed
that we have, at the same time,
and because cos
t = 0, 0 = 0, =ngu;
& w
sin a, there results
ε = (3 + cot² a) cot a.
Naming t' the interval which lapses from the moment the
body begins to move until it passes through the perihelion,
we shall have at once
and, consequently,
t=t', 0=w, 4 = 0,
ε √2
t' =
3n
This being so, if
denotes the time reckoned from the
3nt
; (e)
instant of this passage, so that t = t' +7, we shall have
[3 + tang² 1 (0 - w)] tang 1 (0 — w)
√2
and if this equation, which is of the third degree with respect
to tang (0-w), be resolved, we shall obtain tang 1 (0 — w)
374
MOTION OF A MATERIAL POINT
:
!
in a function of 7, and, consequently, we shall have the values
of r and 0 at any given instant; the timer is evidently positive
after the body has passed through the perihelion and negative
previously to this passage.
Because
ghy=ky² =µ,
√7.sina =√p,
the preceding value of n is the same as
μ
n =
;
PVP
μ
therefore, it appears from the equation a³n² µ of No. 228,
that n expresses the mean angular velocity of a planet, of
which the greater semi-axis is equal to p; and if i denotes
that of the earth, and its semi-axis major, so that we may
have
we shall obtain(n')
for the value of n.
i=
n =
ιντ'
il vī
PVP'
240. From the preceding analysis it appears, that if the
determination of the motion of a comet be considered as a mere
problem of dynamics, and if, in consequence, its initial posi-
tion, its direction, and velocity, are supposed to be known,
we can, from these data, determine p the distance of the sum-
mit of the parabola from its focus, the instant of the passage
of the body through this summit, or the value of t', and the
position of the axis which depends on the angle w; the velo-
city of the comet, and its position on its trajectory at any
instant whatever, are known by means of equations (c), (d), (e),
and as the plane of this curve is that which passes through
the centre of the sun, and through the direction of the initial
velocity, it follows that the motion is completely determined.
But the astronomical problem is altogether different. When
ACTED ON BY A CENTRAL FORCE.
375
a comet is first discovered, neither the plane of its orbit, its dis-
tance from the sun, its velocity, nor its direction at the instant
of its apparition, are furnished by observation, so that if its
position at this instant be taken for its point of departure, the
constant quantities y, h, a, will not be given as in the pre-
ceding problem. The question then consists in deducing
from observations, the five following quantities, namely, the
inclination of the orbit, and the longitude of its ascending
node on the plane of the ecliptic, (which two quantities will
determine the plane of the orbit,) the longitude of the peri-
helion, and its distance from the sun, (from which two, the
position of the orbit in its plane may be known ;) and lastly,
the time corresponding to the passage of the comet through
its perihelion. When these five unknown quantities are de-
termined, equations (c), (d), (e), represent, as in the preceding
numbers, the motion of the comet in its plane. Now, from each
observation of the comet, we can deduce its right ascension
and declination; therefore, three observations furnish six data,
and, consequently, six equations, which are more than suffi-
cient to determine the five unknown quantities mentioned
above; and this circumstance enables us to replace two of
these equations by that combination of them which is most
proper to diminish the influence of the errors of observations.
Having thus deduced approximate values of the five ele-
ments adverted to above, from three observations made at the
epoch of their apparition, subsequent observations serve to
correct these first values, and to verify formulæ (d) and (e).
We can only in this treatise thus generally advert to this
problem, which is one of pure astronomy, of which different
solutions have been proposed.
CHAPTER VII.
DIGRESSION ON UNIVERSAL ATTRACTION.
241. The material points of all bodies attract each other
mutually, in the direct ratio of the masses, aud in the inverse
of the square of the distances.
This great law of nature, which was discovered by Newton,
is a necessary consequence of observation and calculation. In
fact, it is shewn in the exposition of the system of the world,
how, by setting out from experiment, we are conducted,
without assuming any hypothesis, by an uninterrupted train
of rigorous reasoning, to the principle of universal gravitation.
The developments of this principle constitute the especial
object of the celestial mechanics. We shall restrict ourselves,
in this chapter, to a brief exposition of its principal conse-
quences.
242. The force that retains the planets in their orbits is
the resultant of the attraction, which all the material points
of the sun exercise on all those of each planet. Considering
the smallness of the dimensions of the sun and the planets re-
latively to the distances which intervene between them, it is evi-
dent that these attractions may be regarded, with an approxi-
mation sufficiently near to the truth, as equal and parallel forces
in the entire extent of each planet; their resultant is then equal
to their sum, and, the distance remaining the same, the mo-
tive force of each planet is proportional to the product of its
mass into that of the sun; the circumstance of the form of
these bodies being very nearly spherical, renders this con-
clusion still more exact, when their mutual distance is as-
sumed to be that of their centres of gravity, (No. 99.)
ON UNIVERSAL ATTRACTION.
377
In order to express the intensity of this force numerically,
let a certain distance, for example, that of the sun from the
earth, be assumed as the linear unit, also let a determinate
mass and interval of time be selected as the respective units
of these two descriptions of quantities; and finally, let the
unit of force be, as in No. 118, the constant accelerating
force, which produces, in the unit of time, a velocity equal to
the unit of length. Let us now suppose two bodies, whose
masses are respectively equal to that which has been se-
lected to represent unity, to be distant from each other by a
quantity equal to the linear unit, then if ƒ denotes the at-
tractive force of one of these two bodies on the other, that is
to say, the numerical ratio of its intensity to that of the force
which has been selected to represent unity, м and m being the
masses of the sun and planet, the motive force of the planet
will be fмm at the unit of distance, and at any other distance
r, it will be fìm
p.2
The magnitude of the quantity which we have denoted
by f, depends on the attractive power with which matter is
endowed; when the mass and distance are equal, this power
is the same for all bodies; nothing hitherto observed leads us
to suppose that it increases or diminishes with the time; and
we have every reason to think that it has been and will re-
main constantly the same.
243. The motive force of the mass м, due to the attraction
fMm
of m, is also represented by, so that the reaction of each
planet on the sun, is equal and contrary to the action of this star
on the planet; but the motive forcem
acting on the two masses
зов
м and m, will impress on them, in each instant, infinitely
small velocities, which are reciprocally proportional to these
fm
masses, or, in other words, their accelerating forces are and
22
Hence it follows, that if these bodies are remitted, with-
ƒM
дог
+
3 c
378
ON UNIVERSAL ATTRACTION.
Low for
1PS force
In plamb
ん
​out any initial velocities, to their mutual attraction, they will
move towards each other, passing over, in the same time,
spaces which are in the inverse ratio of their masses; and they
will meet at their common centre of gravity, which we know
divides their initial distance in the inverse ratio of their masses.
In general, if a planet is projected in any direction whatever
in space, and if it be required to determine its apparent mo-
tion about the centre of the sun, considered as a fixed point,
we must conceive that in each instant, there is impressed on
this star, an infinitely small velocity, equal and contrary to
that which it receives from the attraction of the planet; but,
in order that the relative motion of these two bodies be not
altered, we should, at the same time, impress this velocity on
the planet; which is the same thing, as if there was applied to
it an accelerating force equal and contrary to that of the sun;
therefore, in the motion which we are considering, the acce-
lerating force of the planet m will be constantly directed
towards the sun м, and equal to the sum of the two forces
fM
2.2
and fm; hence if it is represented by, as in No. 225, we
should assume
µ =ƒ(M + m).
Therefore, this value ought to be substituted in the diffe-
rent equations of elliptic motion which have been given in the
preceding numbers, and the equation
T
4π2a3
Tμ
of that number will then give
T2
472
a³¯¯f(M+m)
(1)
T being always the time of the planet's revolution, and a the
şemi-axis major of its orbit.
It appears, therefore, that the ratio inasmuch as it de-
T2
a³,
pends on m, the mass of the planet, will not be the same for
4
ON UNIVERSAL. ATTRACTION.
379
two planets when their masses are unequal. However, as the
observations which establish the third law of Kepler prove, that
this ratio, if not exactly, is, at least, very nearly constant; it
follows, that the masses of the planets are very small rela-
T2
tively to that of the sun, and this is the reason why the ratio
a³
α
of the square of the time to the cube of the mean distance
varies very little, in passing from one planet to another. In
fact, the mass of Jupiter, which is the most considerable of
them all, is less than the thousandth part of the mass of the
sun.
244. It is on this account that the mutual attraction of
the planets produces only perturbations, which are either very
slow, or very inconsiderable, in the elliptic motion produced
by the attraction of the sun. In fact, the masses of two planets
being m and m₁, the motive force directed from the one to
the other, is expressed by
fmm1
2
at the distance p; therefore,
ρ
p²
the accelerating force of m arising from the attraction of m₁,
will bem, ; and as the distance
2
p²
never becomes very
ρ
small
relatively to r the distance of m from the sun, it follows, that
if m₁ be a very small fraction of Mм, the motion of m produced
by the solar attraction, ought to be very little modified by the
attraction of m1.
The planetary perturbations may, therefore, be deter-
mined by the method of the variation of constant arbitraries,
which has been already explained in No. 229. They are of
two different species. The one consists of periodic inequalities,
which are, for the most part, very small, and of which the pe-
riods comprise multiples of the revolutions of the disturbing
and disturbed planet, that in general are inconsiderable.
However, when their mean motions are nearly commensurable,
these periods may become much longer, and the inequalities
much more sensible. Thus, as the mean motions of Saturn and
380
ON UNIVERSAL ATTRACTION.
Jupiter are very nearly in the proportion of 2 to 5, Laplace
proved, that an inequality results from the mutual attraction
of these two planets, whose period is 929 years, and of which
the maximum is about 48'in the longitude of Saturn, and very
nearly 20′ in that of Jupiter.
The other perturbations of the planets are-1st, the pro-
gressive motions of the perihelia and nodes of their orbits,
in which these points traverse the entire circumference, in
periods of such length, that they may exceed thousands of
centuries; 2ndly, the secular variations which affect the
excentricities and inclinations of these orbits, and also the
mean longitudes of the planets; the periods of these are
similar to the preceding, and their amplitudes, though con-
tained within very narrow limits, are not yet very well
known.
But while these different elements of elliptic motion simul-
taneously vary in virtue of the planetary attraction, it is very
remarkable, that this force produces no change in the greater
axes of the orbits and the mean motions of the planets, which
will be the same at all epochs, as also the times of revolutions,
that are connected with the greater axes by equation (1).
Nevertheless, the secular variations of the mean longitudes
produce corresponding ones in the intervals between two con-
secutive returns to the same fixed point; they are insensible
in the motion of the planets; but this is not the case in the
motion of the satellites, and particularly in that of the moon,
which, on this account, is accelerated from century to century.
As the accelerating force which arises from the attraction of a
planet m₁, on another planet m, is independent of the mass
of m, and proportional to the mass m₁, it is easy to conceive
how the perturbations due to this force, and which are ob-
served in the motion of m about the sun, may enable us to
determine the ratio of the mass m₁ to that of this star. Thus,
for example, by means of the great inequality in the motion
of Saturn produced by the action of Jupiter, we find that the
ON UNIVERSAL ATTRACTION.
381
mass of this last planet is equal to the Tooth of that of the
sun.
In the next number we will point out another means of
estimating the mass of those planets, which are accompanied
by one or several satellites. The comets, on account of the
smallness of their masses, do not produce any appreciable
effect on the planets; but their motions are deranged by the
attractions of the planets, and their perturbations, which have
a considerable influence on the epochs of the reappearance of
each comet, that is to say, on the interval of time comprised
between two consecutive passages through its perihelion, are
also determined by the method of No. 229.
245. Let m' and m denote the masses of a satellite and its
primary, and the distance of their centres. The motive
force of the satellite, directed towards the centre of the planet,
fmm'
2/2
will be expressed by at this distance r'; the coefficient f
being the same as in the former expressions. The accelerating
force of the satellite in its apparent motion about the planet,
in which
will be expressed by
μ'
µ' =ƒ(m+m').
If a' denotes the semi-axis major of the orbit of the satel-
lite, and T' the time of its revolution, we shall obtain, by ap-
plying equation (1) to its motion,
T/2
4π2
a¹³ = ƒ (m+m')'
F
13
and if these two equations be respectively divided by each
other, in order to eliminate the coefficient f, there will result,
T2. d'³ 3 m+ m²
T/2.
a³
M+m
Now, with the exception of the moon, the masses of the
satellites are very small relatively to those of their respective
382
ON UNIVERSAL ATTRACTION.
primaries; for example, the mass of a satellite of Jupiter is not
the ten thousandth part of that of this planet; therefore, in this
last equation, we may put m in place of m + m'; and as a, a',
T, T', are given by observation, this equation will enable us
to determine the ratio m to M. It was in this way Newton
found that the mass of Jupiter was the 17th of that of the
sun, which differs very little from the fraction 1070, that has
been since obtained by another method.
67
246. If a planet is attended by several satellites, their
mutual attraction, combined with the inequality of the action
of the sun on each satellite and on the planet, produces in the
elliptic motions of the satellites, perturbations analogous to
those we have already adverted to in the case of the planets.
The perturbations which arise from the reciprocal action of
the satellites, will enable us to determine the ratio that their
masses bear to that of the planet, whose attraction produces
their elliptic motion. But as the moon is the only satellite
which revolves about the earth, we cannot apply this me-
thod to determine its mass; however it may be obtained from
other considerations, one of which is the action of this satel-
lite on the waters of the ocean, and which we now proceed to
detail.
Let c (fig. 56) be the centre of the earth, a that of the
moon, м any point whatever of the terrestrial spheroid, also let
CA = α, AM = p,
CM=r,
then if we denote the angle ACM by λ, we shall have
p²=a2-2ar cos λ +r²;
and if from the point м, the perpendicular MB be let fall on
the line ac, we shall also have
MB = r sin λ, AB = α-
r cos λ.
At the point м, the value of the accelerating force arising
from the attraction of the moon, and acting in the direction
Ma, will befm'
Q2,
m' denoting the mass of this satellite and ƒ the
ON UNIVERSAL ATTRACTION.
383
same coefficient as in the preceding expressions. The com-
ponents of this force in the direction of the perpen di cular MB
and of MD parallel to ac, will be therefore
fm'r sinλ f'm' a fm'r cosλ
3
3
p³
If in these expressions we substitute for p its value given
above, we may neglect the square of r, the radius of the earth,
as it is about the sixtieth part of a; hence if we make(a)
fm'r sin λ
2 fm'r cosλ
3
α
= 8,
3
= &',
α
the two components of the lunar attraction will be
f'm'
2
a²
α
P
and
+p'. Therefore, all the points of the earth will be soli-
cited in a direction parallel to ca, by a constant force equal to
fm'
a²
α
; and besides, by the forces and p', of which the result-
ant varies in magnitude and direction, from one point м to
another, and at the centre c it vanishes. Now, it is evident,
1
that in virtue of the force
fm'
a²,
the entire mass of the earth is
α
urged towards the moon, by a motion which is common to all
its parts, without the points of the fluid parts undergoing
any change in their relative position; therefore, the flowing
and ebbing of the sea, produced by the action of the moon,
arise from the forces & and 'applied to different points of the
If м be the mass of the sun, and a its distance from
the centre of the earth; and if, moreover, μ,, ', denote what
λ, p, p', become relatively to this star, we shall also have
ocean.
fur sin u
sin μ, '=
f=
ป
a³
2fмr cos μ
a³
"
for the components of the force arising from the action of the
sun, which contribute to produce the phenomena of the tides.
384
ON UNIVERSAL ATTRACTION.
If they be compared with the forces & and p', it will appear,
that for a point of the ocean, whose radius vector makes
the same angle à or μ with the radius vector of the moon's or
or sun's orbit, the actions of these two stars, which produce
the oscillations of the sea, are to each other as their masses,
divided by the cubes of their respective distances from the
centre of the earth. Now, we may conceive that, when every
thing else is the same, the magnitudes of these oscillations
must be to each other, as the corresponding forces; if, there-
fore, w represents the ratio of the lunar to the solar tide, in
the same place of the earth and for similar positions of these
two stars, we shall have(b)
m'
WM
α
a³ ;
in which equation we assume, that a and a denote the mean
distances of the moon and sun from the earth, hence we obtain
m'
a³ M
= @
W
m
a³ m
m denoting the mass of the earth.
As the lunar and solar tides may be perfectly distinguished
from each other, by means of the different laws to which they
are respectively subject, their ratio in each place of the earth
may be determined. From a mean of a great number of ob-
servations made in the harbour of Brest, we obtain
w= 2,3533,
W
for the value of this ratio. The distance a is very nearly 400
times the distance a, and the mass м is also, as we shall see
immediately, very nearly 355000 times the mass of m. By
substituting these values in the preceding formula, we find
the mass of the moon to be equal to the 7th of that of the
earth. Besides the oscillations of the fluid part of the earth,
the actions of the sun and moon produce also in the motion of
the terrestrial spheroid, about its centre of gravity, in conse-
1
ON UNIVERSAL ATTRACTION.
385
quence of its not being perfectly spherical, perturbations, the
nature of which will be investigated, when the motion of rota-
tion of solids comes to be treated of.
247. We may remark here, that if each of the forces and
'be resolved in the direction of ME, the production of the
radius vector cm, their sum will be p′ cos λ — p sinλ; so that
its value is(c)
(2 cos² λ — sin²)) fmr
75a³.
=
2
This is the diminution of gravity at the point м, produced
by the action of the moon. Now, if м be supposed to exist
on the surface of the earth, we have fm gr² very nearly,
g denoting the gravity at this point; moreover, 2 cos2 A-sin² A
is a maximum when λ = 0, in which case its value is equal to
2. Consequently, the greatest value of this diminution of gra-
vity will be
2 gr3
75a3,
a quantity which is very nearly equal to the
eight thousandth part of g, on the supposition that the ratio
a is equal to 60. Therefore, in order that the influence of
the action of the moon, on the length of a pendulum that
vibrates seconds should be appreciable, the approximation
ought to be continued as far as the second decimal beyond the
hundredth thousandth place, at which point we generally stop
in expressing the measure of its length. This influence will
produce, in the measure of the time, an inequality depending
on the motion of the moon, which, when a maximum, does not
amount to half a hundredth of a second in a day.
248. If we do not take into account the centrifugal force
that arises from the rotation of the earth, the weight which is
observed at its surface is the resultant of the attractions that
all the points of the spheroid exercise on each material point,
which resultant depends solely on the position and mass of
this point, and not at all on the nature of the body to which
it belongs; and this, in fact, has been fully confirmed by ex-
3 D
386
ON UNIVERSAL ATTRACTION.
в
ز معا
9 = (60)² =
#
1757
60€
periment. The intensity of this force should diminish ac-
cording as we ascend above the surface of the earth; this di-
minution is indicated by observations of the pendulum made
at different elevations. Besides, the terrestrial gravity, dimi-
nished in the ratio of the square of the radius of the earth to
the square of the radius of the lunar orbit, should be equal to
the accelerating force, which retains the moon in its orbit.
But, as the distance of this satellite is very nearly sixty times
the radius of the earth, it follows, that if the moon fell from a
state of rest, it would move towards the earth in a minute,
through the same space that any body would in a second fall
through, in a vacuo, at the surface of the earth. This quan-
tity is, in fact, the versed sine of the arc which the moon
describes in a minute in its orbit, or very nearly the square of
this arc divided by the diameter of this curve; and as the cir-
cumference of the orbit is 60 times that of the earth, it fol-
lows, that the quantity in question is equal to 40 millions of
60π
metres, multiplied by n denoting the number of minutes
n2,
contained in the time of a revolution of the moon. Therefore,
it appears from the numerical value of g, which is determined
by experiments made with the pendulum, that this product
must be very nearly equal to 4m, 90; it has been found, in fact,
equal to 4m,88, n being assumed equal to 39343(d); and the
difference will be still less, if we take into account the various
circumstances, the consideration of which we omitted in order
to simplify the demonstration. It follows from this, that the
terrestrial weight is only a particular case of universal attraction;
and, on this account, this general force has been likewise termed
the weight, or universal gravitation.
249. As the earth deviates very little from the spherical form,
the attraction which it exercises on a point of its surface is very
fm
nearly I'm
p2;
; which is the expression for that of a sphere, in which
m denotes its mass, r its radius, and ƒthe coefficient of universal
ON UNIVERSAL ATTRACTION,
387
attraction. This approximate value should be altogether
exact for points which exist on a certain parallel; and, it
appears from the theory of the attraction of spheroids, which
differ little from spheres, that this parallel is that of which the
square of the sine of the latitude is (e). On this parallel, the
measure of the gravity is 9m, 79386 (No. 193); but, in order
that it may be accurately equal to the terrestrial attraction, it
should be previously increased by the vertical component of
the centrifugal force, which component is, under this parallel,
equal to the fraction 3:2 of the gravity (No. 178).
Consequently if we make
g = (9m, 79386) (1+ 3,289) = 9,81645,
we may regard this value of the gravity, thus corrected, as
equal to the attraction of the earth, and assume
fm
g p.2
By multiplying the members of this equation by the corres-
ponding members of equation (1) of No. 243, applied to the
motion of the earth about the sun, we shall obtain
m
M + m
9T22
4723
which formula enables us to determine the ratio of the mass of
the earth to that of the sun.
If we conceive a right angled triangle, of which the base is
the radius of the earth, and whose height is its distance from
the sun, then the small angle at the sun opposite to the base
is the parallax of the sun, which can be determined either
directly by astronomical observations, or theoretically by a
certain inequality produced in the motion of the moon by the
action of the sun, and which has been termed the parallactic
inequality. The magnitude of the parallax depends on the
the radius of the earth, and on the distance of the sun from
3
388
ON UNIVERSAL ATTRACTION.
the earth; for the mean distance a, and for the radius drawn to
the parallel of which the square of the sine of the latitude is §,
its value is 8″,60. Consequently, we have
g
α
=tang.8",60, a=(23984) r.
If we assume the compression of the earth equal to zō,
we have under this same parallel,
r=6364551m,
for the value of the radius r. The time of its revolution
about the sun, expressed in seconds, is
T = (86400) (365,256374).
By means of these values, and that of g, in which it is
also assumed that the second is the unit of time, we find
m=
M
354592*
250. The sun is a sphere, of which the radius is 110 times
that of the earth; therefore, as the ratio of the volumes of these
two bodies is known, and also that of their masses, we may
infer at once that of their mean densities. It appears by this
method, that the density of the sun is, very nearly, the fourth
part of that of the earth. At the surface of this star, the at-
traction is expressed by
ƒM
R2,
R representing its radius. And since
R 110r, g=
this quantity is equal to
fm
дм
(110)² m²
which, by substituting for
M
m
its numerical value, becomes
ON UNIVERSAL ATTRACTION.
389.
(29,5) g. As the duration of the sun's rotation on its axis is
25,5, the centrifugal force at its equator is only the sixth
part of this force at the equator of the earth. Therefore, if
we neglect the diminution which it produces in the weight at
the surface of the sun, it is evident that the weight of a body
at this surface is 29 times and a-half the weight of the same
body at the surface of the earth, and that bodies descend there
through 135 metres in the first second of their fall.
If equation (1) of No. 243 be applied successively to the
earth and to any other planet, the quantities m, a, T, relatively
to the earth, being supposed to become mi, a1, T1, with respect
to the planet, we shall obtain from it, by eliminating ƒ,
2
3
M+ m₁ T₂²
a³
M + m T2
If the value of a be known from an observation of the solar
parallax, and if likewise, m the mass of the earth, and T the
duration of the sidereal year be known, this equation will en-
able us to determine a₁ the semi-axis major of any planet,
when m, its mass, and T, the time of its revolution are known.
In the method given in No. 245 for determining this mass, it
is only supposed that an approximate value of the semi-axis
major is known.
251. The attraction which a considerable mass, such as a
high mountain, exercises at the earth's surface, causes heavy
bodies to deviate from the vertical direction, consequently,
the production of the plumb-line will not in such a case meet
the heavens in the zenith. It will be deflected from it, in con-
trary directions, at the two opposite sides of the mountain; so
that if every thing corresponds on one side and the other,
with respect to the form of the mountain, and the quantity of
the deviation of the plumb-line from the vertical, the angular
distance of the two stars through which its production passes,
will be double of its deviation(ƒ). This effect has been observed
at Peru and in Scotland; but, because the masses of the highest
↑
390
ON UNIVERSAL ATTRACTION.
mountains are very small, relatively to the mass of the earth,
the deviations in question are also inconsiderable, and amount
only to a small number of seconds. We proceed to give an
example of the calculation of the deviation of the plumb-line,
caused by the attraction of a given mass.
Let A (fig. 57) be the centre of a homogeneous sphere sus-
pended at the extremity of an inextensible and inflexible
thread, of which the other extremity is attached to the fixed
point c; also let o be the centre of another homogeneous
sphere which acts on the first. The thread CA will be
drawn from the vertical CB, without, however, deviating
from the vertical plane which passes through this line and co;
and, when it is in equilibrio, the resultant of the weight of
the first sphere and of the attraction of the second, should
pass through the fixed point c; now, these two forces will be
applied to the point A, the one in the vertical direction AD, and
the other along the line Ao; and their tendency is to make
the thread ca turn in opposite directions about the point c.
In order, therefore, that their resultant should pass through
the point o, it is necessary that their moments with respect
to this same point, should be equal (No. 46); consequently,
if p and q denote respectively the weight of the first sphere
and the entire attraction of the second, and if p and q denote
the perpendiculars CE and CF, let fall from the point c on the
productions of DA and oa, we shall have
Pp Qq,
for the equation of equilibrium, by means of which the un-
known deviation BCA can be determined. If x denotes this
angle, y the given angle всo, a and c the distances CA and
co which are also given, and y the unknown distance AO; we
shall have
and, besides,
sin COA =
y² = a² + c² — 2ac cos (y — x),
a sin (y — x)
y
ac sin (y — x)
q=
p = a sin x.
y
;
ON UNIVERSAL ATTRACTION.
391
Likewise if m denotes the mass of the earth, m₁ that of
the attracted sphere, and m' that of the attracting sphere, the
motive forces P and Q will have for values
fm'mı
P =
fmm,
2.2
Q =
;
y²
fand r denoting, as before, the coefficient of universal attrac-
tion, and the radius of the earth; also, if p be the mean density
of the earth, p'that of the attracting sphere, and r' its radius,
we shall have
m' =
mp'r/3
pr³
By means of these different values, the equation Pp=Qġ
will be changed into(g),
pry³ sin xp'r'³e sin (y-x),
in which it is only necessary to substitute the value of y given
above, to determine that of x.
If we assume that CA, the length of the plumb-line, is very
small with respect to the distance co, which is generally the
case, we may neglect a relatively to c in the values of y, and
we shall have simply y = c; hence there results.
sin x
sin (y-x)
=
p'p/3
prc²·
The density p' and the radius r' of the attracting sphere
remaining the same, the value of x, which is deduced from
this equation, will be so much the greater, according as the
distance c is less, and the angle y approaches to a right
angle; and as c can never be less than the radius r', it follows,
that the greatest deviation of the plumb-line, which the at-
traction of a given sphere can produce, will be obtained by
assuming c=r' and y = 90; from which suppositions we
deduce
tang x =
p'p'
pr
392
ON UNIVERSAL ATTRACTION.
If, for example, we suppose p'=p, and that it were re-
quired to know the magnitude of the radius r', when the devi-
ation x amounted to one second, we would have r'rtang 1";
and, because 2πr the circumference of the earth is 40 millions
of metres, it follows that r30m,856..... Hence it appears,
that a homogeneous sphere, of which the radius is about 31
metres, and whose density is equal to the mean density of the
earth, produces only a deviation of one second, at the most,
in the direction of the plumb-line, and in order that it may
produce this, it is necessary that it touch the inferior extre-
mity of the string, and that its centre should be situated in the
horizontal plane passing through this extremity.
252. This mean density of the earth, which has been in-
ferred from the deviation of the plumb-line produced by the
attraction of mountains, has been estimated at four or five
times that of water. Cavendish, who determined it by the at-
traction exercised by two leaden spheres, eight inches in dia-
meter, which he rendered sensible by means of the balance of
torsion, found it equal to five times and a-half this density.
Without entering herein to all the details of this beautiful
experiment, the different precautions which it requires, and the
computations that must be made in order to produce an exact re-
sult, we shall only briefly indicate the principal points of these
computations. The balance of torsion is the most exact instru-
ment which we have for measuring very small forces. Coulomb,
to whom we are indebted for its invention, has chiefly employed
it, in measuring the attractive and repulsive forces of electrified
bodies; and, on this account, it is also known in physics, by
the name of the electric balance. It consists principally of a
very fine vertical metallic thread, attached to a fixed point
and having a horizontal(h) lever suspended at its extremity.
This lever consists of a slender rod aca′ (fig.58) divided into
two equal parts at the point c, to which the thread is attached,
and it is terminated by two spheres of a small diameter, whose
centres are A and a'. From the point c as centre, and with a
ON UNIVERSAL ATTRACTION.
393
radius equal to CA, let a horizontal circle BA B'A' be described,
and let its circumference be divided into a great number of
equal parts. When the lever turns about the point c, its
extremities a and A' will trace this circumference, and the
points of division to which they correspond at each instant,
will make known the arcs that they have described. As long
as the suspended thread which terminates at the point c, is not
twisted, the lever quiesces in a certain position. Suppose
that then it exists on the line BCB', if we cause it to deviate
from this line, and make it assume any other position aca',
the suspended thread will be twisted on itself, and this torsion
will tend to bring the lever back towards the line BCB'. Let
us suppose that, in order to retain it in the direction ACA',
there is applied to its two extremities equal and opposite
forces, directed in the horizontal plane, and perpendicular to
its length; the common value of these two forces will be the
measure of the force of torsion with which they constitute an
equilibrium. Now, Coulomb has proved by experiment, that,
when the thread of suspension remains the same, this force of
torsion is proportional to the angle BCA; therefore, if we take
the right angle to represent unity, and h to denote the force of
torsion which corresponds to this angle, ✪ the angle BCA, this
force will be equal to ho, in the position ACA' of the lever; thus,
in this position, the torsion of the suspended thread is equiva-
lent to two horizontal forces, equal to h0, applied at the points
A and a′ perpendicularly to Aca', and which tend to bring
back the lever to the line of repose BCB'.
This being premised, let us cause two equal homogeneous
spheres, consisting of the same matter, and symmetrically
placed on opposite sides of the line всв', to approach the
lever, and let o and o' be their centres, situated in the hori-
zontal plane which contains the lever, equally distant from c,
and existing on oco', the line drawn through this point. The
attraction of these two bodies will cause the lever to deviate
from the line BCB', and, because every thing is similar about
3 E
394
ON UNIVERSAL ATTRACTION.
the centre c, the line ACA will turn about this point, which
will remain immoveable. According as the lever deviates from
the line of repose the force of torsion will increase, and there
exists a position in which this force constitutes an equilibrium
with the attraction of the two spheres; but as the lever attains
this position with an acquired velocity, it passes beyond it,
and oscillates on one side and another in the same manner as
a horizontal pendulum. The duration of an entire oscilla-
tion may be known from observation, and by comparing the
length of this pendulum with that of a common pendulum,
which vibrates in the same time, the ratio of the force of
attraction of each sphere to gravity may be inferred; and, con-
sequently, the ratio of the mass of this sphere to that of the
earth. It is easy to obtain the equation, by means of which
this ratio can be determined, as we now proceed to shew.
A
253. Since the two spheres whose centres are in a and a′,
are solicited by the same forces, and have the same motion
about the fixed point c, it will be sufficient to consider the
motion of the centre of one of them, the point a for example;
if, therefore, as in the preceding problem,
CA = α, CO = c, BCO = y;
and if, as before, m' denotes the mass of the attracting sphere,
and ƒ the coefficient of universal attraction, we shall have
z² = a² + c² - 2 ac cos (y-0),
z denoting the distance ao, and @ the angle ACB at the end of
any time t; also the accelerating force arising from the attrac-
fm'
tion in the direction AO will be. If this force be resolved
into two other forces, one in the direction of the production of
CA, and the other perpendicular to CA, this last component will
fm
be equal to sin cao, that is to say, to
ལྟ
fm'c
23
sin (y-0), by
substituting for sin CAO its value deduced from the triangle coa.
ON UNIVERSAL ATTRACTION.
395.
1
If from this component, which acts in the direction of a tan-
gent to the trajectory, we take the force of torsion h0, which is
directly opposed to it, we shall have, as the arc BA described
by the sphere is equal a0,
ad³0 __fm'c sin (Y
=
dt2 23
sin (y — 0) — ho,
for the equation of the motion (No. 152).
As the attraction of the mass m' is a very small force, the
angle 0, by which the lever ACA' deviates from the line of re-
pose, is very small. Denoting the distance Bо, or the value
of z which corresponds to 00, by b, in which case we have
b² a² + c² 2ac cos y,
there results by developing according to the powers of 0(i)
___
A
sin (y— 0) — siny — [(a²+c²) cos y — 2ac — ac sin³y] +&c.
z3
63
Therefore, if in order to abridge, we make
[(a²+ c²) cos y — 2ac — ac sin²y]
b5
ac sin²y] f'm'c
გა
+ h = g',
f'm'c sin y
b3
=Bg',
and if powers of higher than the first are neglected, the equa-
tion of the motion will become
d20
a af² = g'B' - g′0;
dt2
hence, by integrating, we obtain
0 = ß + k cos (t √ ? + k');
k and k' being two arbitrary constants.
α
It appears from this value of 0, that the least and greatest
deviations of the lever ACA', reckoning from the line BCB',
will be ẞk and ẞ+k; and, if the line DCD' be drawn in
396
ON UNIVERSAL ATTRACTION.
such a manner, that the angle BCD may be equal to ß, the
lever will perform on each side of this line, equal and iso-
chronous oscillations, whose amplitude will be equal to the
constant k; the angle ẞ is determined by measuring the
least and greatest deviation of the lever, and then, taking
for this angle, half the sum of these extreme values of 0. The
line DCD' that answers to 0 = ß, is the position of the lever
in which it would remain in equilibrium, if it attained to it,
without any acquired velocity. The duration of each entire
oscillation of the lever, on one side and the other of this line,
will be the time in which the angle t
a
+k' increases by
180°; therefore, if we denote it by T, we shall have
T=T
and this duration T will be likewise given by observation.
Now, if g denotes the force of gravity, and 7 the length of
the simple pendulum, which makes infinitely small oscilla-
tions in the time T, we have (No. 182),
consequently,
and hence, because,
T=T
gag'l;
fm
9
2.2,
g' =
fm'c sin y
Bb³
m'
Bab³
M
clr² sin y
we shall have, finally,
m being the mass of the earth and r its radius.
As all the quantities contained in this formula are known
in each experiment, it will enable us to determine the ratio of
the mass m' to that of the earth; and as besides, the volumes
ON UNIVERSAL ATTRACTION.
397
of these two bodies and the density of m' are known, we can
deduce from it the mean density of the earth.
254. It is demonstrated in the Celestial Mechanics of
Laplace, that in order to insure the stability of the equili-
brium of the sea, it is necesssary and sufficient that the mean
density of the earth should surpass that of water. It is in
consequence of this condition being fulfilled, that the forces
arising from the simultaneous actions of the sun and moon
produce only very small oscillations; if this was not the case,
and if the earth, for example, while its mean density continued
to be what it is, was covered over by a sea of mercury, the
action of the least forces extraneous to the terrestrial spheroid
would produce in this fluid a progressive motion, in conse-
quence of which it would traverse the entire surface of the earth.
It may be also proved, from various considerations, that
the density of the concentrical strata of the terrestrial spheroid
must increase as we approach the centre; hence it follows,
that its mean density must surpass that of the superficial stra-
tum; a condition which is found in fact to be established; for
with the exception of the metals, that constitute a very small
part of this stratum, the density of the other materials of which
it is composed, are all of them much less than five times and
a-half the density of water.
But it should be observed, that this increase of density
does not imply the existence of substances entirely different
from those which are observed at the surface, and of which the
actual density is excessively great; it may be assumed, that
all the strata of the earth are composed of the same kind of
matter, a little compressible, or of a variety of different sub-
stances, as is the case at the surface; and on this hypothesis,
which appears to be the most natural, their increase of density
would arise from the condensation, produced by the pressure
of those above them, and which continually augments from the
surface to the centre.
In the interior of the earth, the law of attraction depends
398
ON UNIVERSAL ATTRACTION.
t
on the unknown law of the densities; beyond the surface, it
varies on the production of each radius, very nearly in the
inverse ratio of the square of the distance from the centre;
and it experiences at the same time, from one radius to
another, a variation proportional to the square of the cosine
of the angle which each radius makes with the axis of figure
of the terrestrial spheroid(k). It follows from a consideration
of this last variation, that at equal distances from the centre
of the earth, the force applied to the centre of the moon, and
arising from the attraction of this spheroid, is not the same in
all directions of the radius vector; so that this force may be
considered as composed of two others, the one arising from
the spherical part of the earth, and which may be regarded as
constant, when the distance from the centre of the earth re-
mains the same, the other is due to the protuberance, or excess
of matter at the equatorial parts of the earth, and this varies
with the direction of the radius with respect to the axis of the
poles. Laplace has determined the small inequality in longi-
tude and in latitude, which this second force produces in the
motion of the moon; assuming that its magnitude depends on
the compression of the earth, and comparing it with that fur-
nished by observation, the compression of the earth is found
to be equal to 33, which differs very little from that which
results from taking the mean of a great number of measures
of the pendulum and of degrees of the meridian(7).
At the surface of the earth, the variation of gravity arising
from that of the attraction and of the centrifugal force, fol-
lows the same law as at any distance whatever from the
centre, that is to say, it is, as has been already stated, (No.
178), proportional to the square of the cosine of latitude.
When it is proposed to verify this law by taking the lengths
of pendulums which vibrate seconds, we should take care not
to make the observations near to a mountain; for the horizon-
tal component of the attraction of the mountain will cause
the pendulum to deviate from the vertical, in its position of
ON UNIVERSAL ATTRACTION.
399
equilibrium, at the same time that the vertical component
of this force diminishes the gravity, and consequently,
the length of the simple pendulum, that vibrates seconds.
Even when this cause of anomaly is avoided, it is still found,
that in certain places, the length of the pendulum which
vibrates seconds deviates from the law given by theory; this
must be owing to the density of the earth in those places,
being for a considerable depth and extent, greater or less than
the general density of the superficial stratum; hence arises
an increase or diminution of the total gravity, and, conse-
quently, of the length of the simple pendulum, which is pro-
portional to its intensity. The pendulum may, in this point
of view, be considered as a geological instrument, that in-
dicates, by its anomalies, variations of great extent in the na-
ture of the strata.
Finally, it should be observed, that the law of the de-
crease of gravity, proportional to the square of the cosine of
latitude, as we proceed from the pole to the equator, supposes
that we assume for the surface of the earth, the production of
the level of the seas; and as the elevations of those places on
land, where observations have been made, above this level,
are different, the observed lengths should be reduced to those
which they would have at this level itself in each vertical.
This reduction is commonly made by increasing the gravity
and the length of the pendulum which vibrates seconds, in the
ratio of the square of the distance of the place of observation
from the centre of the earth, to the square of this same dis-
tance diminished by the height of this place above the level
of the sea; it thus appears that the attraction of the stratum
of earth comprised between the surface of the continent and
the production of the sea is neglected. But this correction is
too great by nearly one-half, as is evident from the following
considerations.
255. Let AM'B (fig. 59) be the surface of a continent,
DAMBE the level of the sea and its production, c the centre
400
ON UNIVERSAL ATTRACTION.
{
of the earth; also let м' be the place of observation, and м the
point where the radius CM' meets this production; let м'м, the
height of the point м' above the surface of the sea, be repre-
sented by h, (this can be determined either by levelling, or by
barometrical measurements.) If м' was very near to the sea,
the gravity would be a little diminished, and its direction a
little deranged, because the density of the water is less than
that of the earth; but for the present we shall suppose that
this is not the case, we shall also assume that the surface of
the earth about m' is horizontal, or sensibly perpendicular to
the radius cm', and that its density is uniform. The question
then is to compute the attraction exercised at the point m', by
the stratum Aм'BM elevated above the level of the sea. In
this computation, it is not necessary to take into account either
the curvature of this stratum, or the variation of its thick-
ness, i. e., in other words, we may consider the thickness of
this stratum as constant and equal to h, through the entire
extent in which its thickness can be sensible. Let c denote
the radius of this extent, and p' the density of the stratum.
The volume
2πydy for its
This being premised, let к be any point whatever of the
attracting stratum, z and y its distances from the surface of
the earth and from its radius CM'; and let two cylindrical
surfaces be described which may have мM' for their common
axis, and whose radii may be y and y+dy.
y+dy.
comprised between these two surfaces will have
base and dz for its height; and if it be decomposed into hori-
zontal rings, whose thickness is infinitely small, the volume
of the ring corresponding to the point K, will be 2πуdy dz,
and its mass 2πp'у dy dz. The attraction of this ring on a
material point situated at m', will be reduced to a force acting
in the direction MM', which will be equal to the sum of the
vertical components of the attractions of all its points; and
since for any point K, we have
2
KM' = √ y²+z², COS KM'M
√
y² + z 2
ON UNIVERSAL ATTRACTION.
401
the expression for the accelerating force arising from the at-
traction of the entire ring, will be
2πfp'yzdy dz
(y² + z²) i
;
falways representing the coefficient of universal attraction.
Consequently, in order to obtain the attraction of the stratum
in question, this formula should be integrated from 20 to
z = h, and from y = 0 to y=c, which gives(m)
k' = 2πƒp' (c+h— √ (c² + h²)),
=
k' denoting this force. But in general, the vertical thickness
of the attracting stratum is small, relatively to its horizontal
radius; therefore, if h² be neglected with respect to c², we shall
have simply,
k' = 2πpƒ'h.
Let k denote the attraction exercised at the point м by
the
part of the earth which is terminated by the level of the
sea, and the radius CM; this attraction will become at the
point м'
kr2
(r+h) 2°
Denoting the weight and the vertical component of the
centrifugal force at the point м by g and y, respectively, and
by g' and y' at the point m', we shall have therefore
g=k-y, g' =
kr2
(r+h)²
+k'-y'.
Developing the first term of the value of g' according to
the powers of h, and then taking g' from g, we obtain, by
neglecting the square of h, and the small difference y'—y,(n)
2kh
g-g':
k'.
h
Since the factor is very small, we may assume ky' in
T
3 F
402
ON UNIVERSAL ATTRACTION.
the first term of this formula; in the value of the small quan-
tity k', we may likewise suppose
4πpfr
3
=g',
p denoting the mean density of the earth, and then assuming
4 прг
its volume equal to
there will result
3
and, consequently,(0)
3
k'³p'hg'
2pr
9=g(1+24-3p/h)
pr
Therefore, it appears from this, that it is by the factor
comprised between these parentheses, and not by the factor
2h
до
1+
as is usually done, that we ought to multiply the
weight g', which has place on the continent at the height h
above the level of the sea, in order to reduce it to this level.
In general, we may estimate p' as equal to the half of p, and,
5h
consequently, assume 1+for this factor(p). At Paris, h, the
Ar
elevation of the point of the observatory where the barometer is
placed, is about 63 metres; hence it follows, that the gravity
and length of the pendulum which vibrates seconds, is less
there than at the level of the sea in the ratio of one to
1,0000125.
BOOK THE THIRD.
STATICS.
SECOND PART.
CHAPTER I.
OF THE EQUILIBRIUM OF A SOLID BODY.
256. THERE is no body whatever that is not more or less
compressible, and which does not consequently change its
form, when it is subjected to the action of the forces which
constitute an equilibrium. But when the solid body which
we now proceed to consider has assumed the suitable form, then
the points of application of the forces which solicit it, may be
regarded as a system of an invariable form, and it is to this state,
that the coordinates of these different points, which occur in the
equations of equilibrium and are assumed to be known, refer.
Let м, м', м", &c., be this system of material points. In
the case of each point, there are seven quantities to be con-
sidered, namely, its three coordinates, the force which solicits
it, and the three angles which determine its direction. Let
P denote the force which is applied to the point м, MD its di-
rection (fig. 60), x, y, z, the three coordinates oG, GH, HM of
the point м referred to the rectangular axes ox, oy, oz ; a, ß,
the angles, either acute or obtuse, which the line MD makes
,لا
404
OF THE EQUILIBRIUM OF A SOLID BODY.
'
with the parallels to these axes drawn through the point м.
Relatively to the other points м', м", &c., the analogous
quantities are represented by the same letters with correspond-
ing accents.
This being premised, previously to investigating the con-
ditions of equilibrium of the given forces P, P', P", &c., we shall
proceed to transform this system of forces into three others, of
which one will be the resultant of forces parallel to the axis
oz, another of forces parallel to the axis oy, and acting in the
plane of the axes of x and y, and the third of forces acting in
the direction of the axis Ox.
257. Let each of the forces P, P', P", &c., be decomposed,
without changing its point of application, into three forces
parallel to the axes of x, y, z ; P cos a, P' cos a', p" cos a", &c.,
will be the forces parallel to the axis ox; pcos ß, p' cos ß',
p" cos ẞ", &c., the forces parallel to the axis oy; pcos y,
P' cos y', p" cos y", &c., the forces parallel to the axis oz ;
the given forces may at once be replaced by these three groups
of parallel forces.
We can, without at all altering the system of forces un-
der consideratlon, apply to the same point, two equal and
parallel forces. Therefore, at the point м, let two forces g
and - g, equal and directly opposed, be applied, parallel to
the axis of z. Let the force g which acts in the direction MC,
be compounded with the force P cos a acting in the direction
MA parallel to ox, and let ME be the direction of their re-
sultant, and K the point where its production meets the plane
of the axes of x and y; if its point of application be trans-
ferred to the point к, and if it be then decomposed into two
forces parallel to the axes of x and z, the forces P cos a and
9
will be reproduced; but the force P cos a is now directed
along the projection of its first direction on the plane of the
axes of x and y, and the force g is applied perpendicularly to
this plane, at the point K of this projection, the coordinates of
which are easily determined.
OF THE EQUILIBRIUM OF A SOLID BODY.
405
X
In fact, H being the projection of м on the plane of the
axes of x and y, its coordinates will be x and y, and y and
x - KH will be those of the point K, since these two points
belong to the same parallel to the axis of x. Now, since in
the rectangle KNMH, its diagonal KM is the direction of the re-
sultant of the forces g and P cos a, which act in the direction
of the sides KN and KH, we have
KH: HM:: P cos a: g;
hence we obtain, because Hм = S,
KH =
Z P COS a
g
consequently, the coordinates of K, the point of application of
the force g in the plane of the axes of x and y, are
y and x
Z P COS a
9
By performing the same operations on the forces P cos B
andg, the first will be transferred on the plane of the axes
of x and y, along the projection of its first direction, and the
coordinates of the new point of application of the force -g,
in this same plane, will be
2 ZP COS B
y +
and x.
g
If in the same manner, all the forces p'cos a', P" cos a",
&c., P'cos ẞ', p" cos ẞ", &c., be transferred in the plane of
the axes of x and y; each of these forces will act along the
projection on this plane, of its primitive direction, which may be
either above or below this same plane, and besides, there will
be as many couple of forces, g' and — g′, g" and — g', &c., as
there are points м', м", &c. The expressions for the coordi-
nates of the points of application of these last forces, in the plane
of the axes of x and y, may be inferred from those which express
those of the forces gand -g, by accenting the letters x, y, z, g,
P, a, B.
406
OF THE EQUILIBRIUM OF A SOLID BODY.
258. Now, if a similar operation be performed on the
forces P cos a, P' cos a', P″ cos a", &c., parallel to the axis of x,
and comprised in the plane of the axes of x and y, they will
also be transformed into two groups of forces, one of which will
be composed of forces parallel to the axis oy, and the other
of forces acting in the direction of the axis ox.
Thus, let two forces parallel to oy, and represented by h
and — h, be applied to the point н (fig. 61), at which the
force P cos a acts in the direction HF, and let the force h,
acting in the direction HB, be compounded with the force
P cosa, then if the point of application of their resultant be
transferred to the point o, where the production of its di-
rection HK meets the axis or; and if it be decomposed into
two forces in the rectangular directions ox and oy, the forces
P cos a and h will be reproduced at this point o; moreover we
shall have
GQ:GH:: P Cos a: h,
and, because OG = x and GH = y, we shall obtain
OQ = x
y P cos a
h
for the abscissa of the point Q.
Therefore, the force P cos a, the direction of which was HF,
will be replaced by a force P cos a, acting in the direction of
the axis ox, and two forces h and -h, perpendicular to this
axis, and applied to the points Q and G, the positions of
which are known. The same will be the case for the other
forces p'cos a', p" cos a", &c., parallel to the axis of x, and
comprised in the plane of the axes of x and y, for these will be in
like manner replaced by the forces p' cos a', p' cos a“, &c.,
acting in the direction of the line ox, and by the couples of
forces h' and -h', h" and -h", &c., parallel to the axis oy.
259. It appears, therefore, that by means of these two
successive operations, the given forces will be transformed, as
has been stated, into three groups of forces, acting in the di-
OF THE EQUILIBRIUM OF A SOLID BODY.
407
rection of the axis of x, in that of a perpendicular to this axis,
and comprised in the plane of x and y, and in the direction
of a perpendicular to this plane.
In this transformation, any of the forces such as p will be
replaced by six others, namely,-1st, the three forces P cos y,
g, -g, parallel to the axis of z, the coordinates of whose points
of application on the plane of the axes of x and y, and referred
to the axes ox and oy, will be, for the first, x and y; for the
second, x-
ZP COS a
g
and y; for the third, x and y +
X
ZP COS a
9
2ndly. The two forces P cosẞ— h and h, parallel to the
axis of y, comprised in the plane of the axes of x and y, and
which may be supposed to be applied to the axis of x; the first
at the distance x from the point o, the second at the distance
Y P cos a
h
3rdly. The force P cos a acting in the direction of the axis
of x, the point of application of which may be transferred to o.
260. It is easy now to form the equations of equilibrium
of the given forces P, P', P", &c., or of the three groups of
forces which have been substituted for them.
It ought, however, in the first place to be remarked, that
this equilibrium cannot exist, unless it obtains separately for
each of these three groups of forces. In fact, if the forces
parallel to the axis of z do not destroy each other's effect, and
if, notwithstanding, the equilibrium of all the forces was pos-
sible, we could, without disturbing this equilibrium, fix a line
traced in the plane of the axes of x and y; but then the
forces comprised in this plane will be destroyed either because
they will meet this fixed axis, or because they will be parallel
to it. It is, therefore, permitted to suppress them; and if
this is done, the equilibrium will be deranged, contrary to
hypothesis, since there is nothing to prevent the forces per-
pendicular to the plane of the axes of x and y, from causing
the solid body to revolve about the fixed axis; consequently,
•
408
OF THE EQUILIBRIUM OF A SOLID BODY.
the equilibrium will be impossible, unless these last forces
separately destroy each other's effects. In the same manner
it may be shewn that the equilibrium cannot subsist between
the forces comprised in the plane of the axes of x and y, unless
the forces parallel to the axis of y mutually destroy each
other's effects. For if it obtained, and this condition was not
at the same time satisfied, a point might be fixed in the axis
of x, which would destroy the effect of all forces acting in the
direction of this line, nothing would then prevent the forces
perpendicular to this line, from making the system to turn
about this point, so that the equilibrium would be destroyed
by fixing a point in the system, which is absurd. This being
established, it is necessary, in the first place, (No.57), if the
solid body, which is considered, be entirely free, in order for
the equilibrium of the parallel forces P cosy, P'cosy', p″cosy",
&c., g and g, g' and -g', g" and -g", &c., that their sum
should be cypher, hence we have
P cos y + P'cos y′+ P″ cos y" + &c., = 0.
It is likewise necessary, that the sums of their moments
with respect to the plane of the axes of x and z, and to that of
the axes of y and z, which are parallel to these forces, should
be also equal to nothing. Now, with respect to the first plane,
we have
y P cos y + y'P' cos y' + y" p" cosy" + &c.,
for the value of the sum of the moments of the forces, P cos y,
P' cos y', p" cos y", &c., that of the moments of the forces
9, g', g", &c., is
gy+g'y' +g"y" + &c.;
and the sum of the moments of the forces
9, -g', - g",
&c., is, in consequence of the expressions of the coordinates of
their points of application,
ZP
- 9 (y+ COS B³) - 9' (1' + ''Cos (3)
g
z'P' B'
g'
&c.,
OF THE EQUILIBRIUM OF A SOLID BODY.
409
therefore, by adding together these three sums and concinna-
ting, we obtain
P (y cos y − z cos ẞ) + P' (y'cos y' - z' cos ẞ') + &c., = 0;
and in the same manner, if the sum of the moments of the
same forces, with respect to the plane of the axes of y and z be
formed, and put equal to cypher, we shall have
P(x cos y − z cos a) + r′(x' cos y' — z' cos a') + &c., = 0.
With respect to the forces P cos B-h, P'cos B' — h',
p" cos ẞ" — h", &c., and h, h', h", &c., parallel to the axis of
B"
y, as they are comprised entirely in the plane of the axes of x
and y, there are only two equations of equilibrium (No. 57),
it will suffice, therefore, that their sum should be equal to
cypher, which will give
P cos ẞ+P'cos ß' + r" cos ẞ" + &c., = 0,
and, that the sum of their moments with respect to the plane of
the axes of y and z, should be also equal to cypher. Now,
with respect to this plane, the sum of the moments of the first
forces is
x (P cos ẞ − h) + x′ (P′ cos ẞ′ — h') + &c., = 0 ;
that of the moments of the forces h, h', h", &c., is, at the same
time,
h (x - Py cos a) +h'(x_P'y' cos a
h
(x²
h'
+ &c.,
as is evident from the values of their distances from the axis of
y; consequently, if the entire sum be put equal to cypher,
we shall have
P (x cos ẞ — y cos a) + P' (x' cos ẞ' — y' cos a') + &c., = 0.
Finally, for the equilibrium of the forces acting in the di-
rection of the axis of x, it will suffice if their sum be cypher,
consequently,
P cos a + P' cos a' + p″ cos a" + &c., = 0.
A
་
3 G
410
OF THE EQUILIBRIUM OF A SOLID BODY,
The preceding are the six equations which are necessary
and sufficient to secure the equilibrium of a solid body en-
tirely free, and solicited by any forces whatever.
261. If, in order to abridge, we make
P cos a + P′ cos a' + p" cos a" + &c. = x,
PCOS ẞ+P'cos ẞ' + p" cos ẞ" + &c. = Y,
P cos y +P cos y' + p" cos y" + &c. = z,
P (x cos ẞ-y cos a) + P′(x' cos ẞ'—y'cos a') + &c. = L,
P (z cosa―x cos y) + P′(z' cos a'—x'cos y) + &c. = M,
P (y cos y―zcos ß) + r′(y' cos y'—z'cos (3') + &c. = N,
these equations of equilibrium will become
x=0, Y=0, z=0, L=0, M=0, N= 0.
(1)
It may be remarked here, that these quantities, L, M, N, as
also z, y, x, may be deduced the one from the other by the
rule of No. 22.
These six equations contain the conditions of equilibrium
which belong to all systems of material points, that are en-
tirely free; for whatever be the nature of such a system, or
the mutual connexion of the material points which compose
it, it is evident that, if their coordinates and the forces which
solicit them remain the same, the equilibrium will not be dis-
turbed by making their distances invariable. Consequently,
the equations of equilibrium of a system of an invariable form,
which obtain between these quantities, must also subsist for
every other system; but then they are no longer sufficient,
and it is necessary to combine with them other conditions that
are peculiar to each system in particular, which, as we shall
see in the sequel, will enable us to determine the relative po-
sitions of its different points in the state of equilibrium.
262. When all the given forces are parallel to each other,
the angles which they make with each of the axes ox, oy, oz,
are either equal or supplementary, according as these forces
OF THE EQUILIBRIUM OF A SOLID BODY.
411
act in the same or in opposite directions; they may, however,
all be supposed equal, if, at the same time, those forces which
act in one direction are considered as positive, and those which
act in the contrary direction, negative, (No. 11); therefore
in this case we shall have
a=a′ = a", &c., ß=ß' = ß", &c., y = y' = y", &c. ;
in consequence of which the three first equations (1) are re-
duced to one, namely,
P+ P+P + &c. = 0,
and the three others will become
(Px + P'x'+P''x''+&c.) cosẞ=(ry+P'y'+P"y"+ &c.) cosa,
(Pz + P'z'+ P''z"+&c.) cosa=(Px+ P'x'+P'x''+&c.) cosy,
(Py+ P'y'+P"y"+&c.) cosy = (Pz+P'z'+ P"z" + &c.) cosß.
But as in the case of the equilibrium of parallel forces, the
number of equations is only three, these three last equations
must be reducible to two; and in fact, if they be added toge-
ther, after having been multiplied by cosy, cos ß, cosa, re-
spectively, there results an identical equation, so that one of
these may be deduced from the two others.
When all the given forces exist in the same plane, this
plane may be taken for that of the axes of x and y; in this case
the angles y, y', y", &c., are right, and the coordinates z, z', z″,
&c., are equal to cypher, which will cause the third and also
the two last equations (1) to disappear. In this particular
case, as in that of parallel forces, there are only three equa-
tions of equilibrium, which are
x = 0, Y = 0, L = 0.
263. When the given forces do not constitute an equili-
brium, it may be required to know what condition they should
satisfy, in order that they may have an unique resultant, and
what is the value of this resultant. In order to determine this
question, let R denote this force, and a, b, c the angles which
412
OF THE EQUILIBRIUM OF A SOLID BODY.
its direction makes with lines parallel to the axes ox, oy, oz,
drawn through one of its points, which may be taken as its
point of application, and whose coordinates parallel to these
same axes are represented by x1, y1, 21. If this force be
taken in an opposite direction from that in which it acts, it
will constitute an equilibrium with the given forces. There-
fore equations (1) will obtain, if to P, P', P", &c., there be
joined a force equal and contrary to R; consequently, we shall
have
X=R cos α, YR cosb, Z=R COS C,
(2)
and, besides,
L= R(x, cosб — y₁ cosa),
M = R(z₁ cosα- x₁ cosc),
N = R(y, Cosc
Z₁ cos b),
that is to say, in virtue of the three first equations,
XY₁ - Yx₁ + L = 0,
ZX1 XZ1 +м= 0,
Ꮓ
M
YZ1 - ZY1+N=0.
(3)
As the coordinates x1, y₁, z₁ may belong to any point what-
ever of the right line along which the resultant is directed,
these three last equations will be those of its projections on
the three planes of the coordinates. Therefore, in order that
this line may exist, these equations must be reducible to two;
now, if after having multiplied them by z, x, x respectively,
they be added together, the three variables x₁, y₁, 21 will
disappear, and there results
ZL + YM + xn = 0;
(4)
consequently, in order that the given forces may have an
unique resultant, it is necessary, and it suffices, that equation
(4) be satisfied; when this is the case, this force will be de-
termined both in magnitude and direction, by equations (2).
If the three sums of the components parallel to the axes of
OF THE EQUILIBRIUM OF A SOLID BODY.
413
こ
​x, y, z are respectively cypher, equation (4) will be satisfied,
but then the resultant will be an infinitely small force, si-
tuated at an infinite distance from the point of application
of the given forces, or more correctly, these forces are reducible
to two, equal and parallel, acting in opposite directions, but
not directly opposed, and therefore not reducible to an unique
force, (No. 44).
When the three sums L, M, N are respectively cypher,
equation (4) will also be satisfied; and, from equations (3) it
appears, that the resultant will then pass through the origin of
the coordinates.
264. When the condition expressed by equation (4) is not
fulfilled, it may be satisfied by joining a suitable force to the
given forces. For greater simplicity, let it be supposed that
it passes through o, the origin of the coordinates; if it be de-
noted by Q, and if λ, µ, v represent the angles which it makes
with the axes ox, oy, oz, the quantities L, M, N will not be
changed by the addition of this force, and the sums x, y, z
will be increased by the terms a cosλ, Q cos µ, Q cos v.
Equation (4) will therefore become
Q (LCOS v + M Cosμ + N cosλ) + LZ + MY + NX = 0;
so that it may be satisfied in an infinite variety of different
ways, by means of the force Q and of the angles λ, µ, v, which
determine its direction.
The intensity and position of R, the resultant of the forces
Q, P, P′, p″, &c., will be determined by means of equations (2)
and (3), in which x + Q cosλ, Y+Q cosμ, z + Q cosy should
be substituted for x, y, z. Consequently, the given forces.
P, P′, p″, &c., may be replaced by this resultant R and by a
force equal and directly opposite to the force Q (a); hence we
may infer that when the given forces are neither in equilibrio,
nor reducible to an unique force, they may always be reduced,
in an infinite variety of different ways, to two forces only, which
however will not exist in the same plane, for otherwise, if they
414
OF THE EQUILIBRIUM OF A SOLID BODY.
did, they would be reducible to one sole force contrary to the
hypothesis. This is, moreover, rendered immediately appa-
rent by the transformation effected in No. 257; for the given
forces P, P′, P″, &c., may be replaced by the resultant of the
forces parallel to the axis of z, and by that of the forces com-
prised in the plane of the axes of x and y: and we may then,
without any difficulty, transform these two resultants into two
other forces in an infinite variety of ways. If the condition
which should be satisfied in order that their directions should
meet, be investigated, we will light on equation (4) relative
to the existence of an unique resultant.
265. From what has been now established it will be easy
to determine the conditions of equilibrium of two solid bodies
A and A' (fig. 62), which being solicited by given forces, touch
at a point K, and press against each other.
For this purpose, let the six quantities X, Y, Z, L, M, N
of No. 261, refer to the body A, X', Y', z', L', M', N', being
what these quantities become with respect to the body a', and
let x1, y₁, z₁ be the coordinates of the point K, referred to the
same axes as those which occur in these different quantities ;
through the point K let the line нкн' be drawn perpendicular
to the plane which touches the two bodies, and let a, b, c
denote the angles, which KH, the part of this line comprised in A,
makes with lines drawn through K parallel to the axes of x, y, z;
all these quantities are given, and it is only necessary to form
the equations of equilibrium which they ought to satisfy.
Now, the body a will exercise on A', in the direction of
KH', an unknown pressure, which we shall denote by R;
it will experience from it, at the same time, a resistance equal
and contrary to this normal force. If, therefore, to the given
forces which act on a, there be joined a force R acting in the
direction KH, we may then abstract altogether from the consi-
deration of a'; and, in the same manner, if to the forces ap-
plied to a' there be joined a force R acting in the direction
Kн', ' may be considered by itself as detached from a. It
OF THE EQUILIBRIUM OF A SOLID BODY.
415
follows from this, and from equations (1), that the twelve fol-
lowing equations will be required in order to the equilibrium
of these two bodies
x + R cos a = 0,
(
Y + R cos b = 0, z + R cos c = 0,
X-R cos a 0, Y'R cosb 0, z'-R cos c 0,
= =
=
L+R (x₁ cosb−y₁ cosa) = 0,
M+R (≈₁ cos α-x₁ cos c) = 0,
N+R (Y, COSC-≈₁ cosb) = 0,
L' R (x₁ cosb-y₁ cosa) = 0,
M'-R (2₁ cosα-x₁ cosc) = 0,
N' R (y₁ cosc-z₁ cosb) = 0,
which will be reduced to eleven by the elimination of R. After
that these eleven equations of equilibrium shall have been
verified, one of the preceding will make known the value of
R, which must be a positive quantity, in order that the two
bodies may press the one against the other.
From these twelve equations there results immediately
x+x'=0,
Y+Y' = 0,
Z+z' = 0,
NN' = 0;
L+L' = 0, M+M'= 0,
which may be also deduced from the conditions of equilibrium
which belong to all systems entirely free, like that of the two
bodies a and a' (No. 261).
In the same manner, the equations of equilibrium of any
number of bodies, which press against each other, may be
found, and it is easy to perceive that the number of these
equations will be equal to six times that of the bodies minus
the number of their points of contact (b).
266. The equations of equilibrium of a solid body subject
to given conditions must be comprised among those of a body
entirely free; for the equilibrium of this last will not be dis-
turbed if it is subjected to these particular conditions, pro-
*
416
OF THE EQUILIBRIUM OF A SOLID BODY.
vided that no new equation of equilibrium be introduced by
these conditions. But, on the contrary, one or more of equa-
tions (1) will become superfluous; and it will be of use to de-
termine, in the different cases that may occur, which of these
equations will be necessary. It is this which it is proposed to
do, in the following cases that are discussed in this number, it
being always supposed that the given forces P, P′, p", &c., are
replaced by the three groups of forces of No. 259.
1. If the solid body which is in equilibrio, contains a fixed
point, we may take this point for the origin of the coordinates.
The forces acting in the direction of the axis ox will be de-
stroyed by this point; this will cause the equation x = 0 to
disappear. In order that the forces parallel to the axis oy,
and comprised in the plane of the axes of x and y should be
in equilibrio, it is not necessary that we should have y = 0,
it will suffice if their resultant coincides with the axis oy, or
that L, the sum of their moments with respect to the plane of
the axes of y and z, should be equal to cypher (c). Finally,
for the equilibrium of the forces parallel to the axis of z, the
equation z = 0, will not be necessary, it will suffice, if their
resultant coincides with the axis oz; this will require that the
sums of their moments with respect to the planes of the axes
of y and z, and of x and z, which are represented by M and N,
may be equal to cypher. Thus, in this first case,
equations of equilibrium which will be necessary, are
L = 0, M=0, N= 0.
the three
In fact, they indicate that the given forces have an unique
resultant, and that this resultant passes through the fixed
point o.
This force will express in magnitude and direction the
pressure exerted against this point, and it will be determined
by equations (2).
2. Let the solid body be retained by a fixed axis, about
which it is constrained to turn, without being able to slide in
OF THE EQUILIBRIUM OF A SOLID BODY.
417
the direction of its length. Let this axis be assumed to be.
that of z; the forces parallel to this line oz cannot produce
any motion, and the three equations z=0, м= 0, N = 0,
relative to their equilibrium, will be no longer necessary.
Neither will equations x = 0 and y = 0 be required for the
equilibrium of the forces comprised in the plane of the axes of
x and y; so that in this case, there will be only one sole equa-
tion of equilibrium, which will be L = 0, that is to say,
P(x cos ẞ—y cosa) + P′(x'cosß'—y' cos a') + &c. = 0. (5)
But if the body is free to slide along the fixed axis, it is,
moreover, necessary, in order to hinder this motion, that z,
the sum of the forces parallel to oz, should be equal to cypher;
and in this case there will be two equations of equilibrium,
Z=0, L= 0.
The pressure which the fixed axis will experience perpen-
dicularly to its direction, will be the resultant of forces com-
prised in the plane of the axes of x and y, determined both in
magnitude and direction, by the two first equations (2), and
passing through the point o in virtue of equation (5). The
forces parallel to this axis will at the same time tend to make
it revolve on itself.
It appears from a comparison of the quantities м and N
with L, that the equation of equilibrium about the axis oy will
be м=0, and that it will be N=0 about the axis ox. It
likewise results from what has been established, that the condi-
tion of equilibrium about a fixed point consists in this, that
the equilibrium obtains successively about three fixed rect-
angular axes, drawn arbitrarily through this point. Conse-
quently, if the equilibrium subsists about three rectangular
axes which intersect in the same point, it will also obtain about
every other line passing through this point.
3. If in a solid body, three or a greater number of points
which do not lie in the same right line, are constrained to
3 н
418
OF THE EQUILIBRIUM OF A SOLID BODY.
exist on a fixed plane, of which the position is given, and if
this plane be assumed to be that of the axes of x and y ; as
the forces parallel to the axis of ≈ cannot produce any motion,
the equations relative to their equilibrium will not have place,
but the three equations
x = 0, y = 0, L=0,
which obtain in the case of forces comprised in the plane of
the axes of x and y, will be necessary, to prevent the body
from sliding or turning parallel to this fixed plane.
The force z will express the entire pressure which the
fixed plane experiences. If the body is merely placed on this
plane, so that it may act, for example, as a polyhedron, one
of whose faces is in contact with the plane of the axes of x
and y, the sign of z must be such, that this force may press
against this plane. It is, moreover, necessary that this resul-
tant of the forces parallel to the axis of z, should meet the
plane of the axes of x and y, within the area of the base of the
body, otherwise it will detach it from this plane, by causing it
to turn about one of the sides of this base. Now, if x, and y₁
denote the coordinates of the point, where this resultant meets
the plane of the axes of x and y, its moments with respect to
the planes of the axes of x and z, and of y and z, will be zy,
and za₁; they should be equal to the sums of the moments of
the components with respect to the same planes; and from the
values of these two sums which have been already found,
(No. 260), we shall have
X1
1
Zx₁ =
M, Zy₁ = N.
It is necessary, therefore, in each particular case, to shew
that the values of x, and y₁, deduced from these equations,
belong to a point within the base of the body, this condition
of equilibrium cannot be expressed by equations, no more than
that which is relative to the sign of z.
4. If the number of points of a body which is constrained
}
OF THE EQUILIBRIUM OF A SOLID BODY.
419
to rest on the fixed plane of the axes of x and y be only two,
or if they are all situated on the same line, we may assume
this line for the axis of y, the resultant z must then meet
the plane of the axes of x and y, in a point situated on this
axis; and, independently of the three equations of the pre-
ceding case, we shall have a fourth equation of equilibrium,
namely, M=0.
5. Finally, when the solid body touches the fixed plane of
the axes of x and y in only one point, at which, if o the origin
of the coordinates be placed, it is easy to perceive that there
will be five equations of equilibrium, namely,
x=0, Y = 0, L=0, M=0, N= 0.
The force z will always express the pressure exerted on
the fixed plane at the point o, and it must consequently be
affected with a suitable sign.
This result coincides with that of the preceding number;
for if the body a' be supposed to be fixed and terminated by a
plane which may be taken for that of the axes of x and y, and
if the point K (fig. 62) be taken for the origin of the coordi-
nates, we should make in the equations of this number,
x₁ = 0, y₁ = 0, ≈₁ = 0, a = 90°, b = 90°; by means of
which we can reduce to the five preceding equations, a like
number of the six equations which refer to the equilibrium of
the body A. The sixth of these equations will, at the same
time, become
R+ z = 0,
by supposing that c is equal to cypher, or which is the same
thing, that the part RH of the normal coincides with the posi-
tive axis of z; consequently, the pressure exerted on a', which
is equal and contrary to the resistance R, will be the force z,
in magnitude and direction.
From a consideration of the preceding enumeration of the
different cases of equilibrium, it is evident that the number of
equations relative to a solid body constrained by fixed ob-
•
1
420
OF THE EQUILIBRIUM OF A SOLID BODY.
:
stacles, may be any number less than six, which is the num-
ber in the case of a body entirely free.
267. As equation (5) relative to the equilibrium about the
axis of & supposed fixed, contains neither the components of
the given forces P, P', P", &c., parallel to this axis, nor the
coordinates of M, M', м", &c., their points of application pa-
rallel to the same axis; the equilibrium will not be deranged, if
these forces and their points of application are replaced by their
projections on the plane of the axes of x and
X
easily demonstrated a priori.
this
y;
may be
Let Q, Q', Q", &c., represent the forces P, P', P", &c. pro-
jected on the plane of the axes of x and y, that is to say, re-
solved parallel to this plane, and transferred to the projections
of the points M, M', м", &c. on this same plane. Let q, q', q',
&c. denote the perpendiculars let fall from the origin of the
coordinates, which is supposed to be fixed, on the directions
of the forces Q, Q', q", &c., and, for greater clearness, let the
effect of the forces a, a', q", &c. be to cause the body to turn
in the same direction, and the effect of q", QI, &c. be to
make the body turn in the opposite direction. For the equili-
brium of all these forces, it is necessary, by No. 47, that we
should have
IV
Qq + o'q′ + Q"q" — Q""q""' — Q¹ q¹V — &c. = 0 ;
(6)
q, q', q″, q''"', q¹º, &c. being considered as positive quantities,
as also q, q', q", "", &c.; consequently, this equation must
coincide with equation (5), which it is easy to verify in the
following manner:
Let H (fig. 63) be the projection of the point м, OG and
HG its coordinates x and y, HA the direction of the force a,
λ and μ the angles which this line makes with parallels to the
axes ox and oy, drawn through the point H. Through the
point o, let two other axes ox, and oy, be drawn, the first in
the direction HA, and the second perpendicular to this line,
and such that the angle yoyi, may be acute or obtuse at the
OF THE EQUILIBRIUM OF A SOLID BODY.
421
same time as xox₁; now if x and y₁ denote the coordinates
OF and FH of the point H, referred to these new axes, we shall
have
x₁ = y cosμx cosλ, y₁y cosλ
x cosμ.
But, as the perpendicular ok or q, let fall from the point o
on HA, must be a positive quantity, we shall have
Q = ± y₁ = ± (y cosλ x cosμ),
according as the ordinate y, is positive or negative, or what
comes to the same thing, from the direction which has been
assigned to the axes oyi, according as the force tends to
make the body turn in one direction or the contrary, about the
point o. Moreover, we have
and, besides, (No. 8)
Q = P sin y,
cos a=siny cosa,
=
siny cosλ, cos ẞ siny cosμ;
hence there will result (d)
.
Qq = ± P(y cos a
x cos B).
As, by hypothesis, the forces o' and o" tend to make the
body turn in the same direction as Q, we shall have also
±
Q'q' r' (y' cos a' x' cosẞ'),
Q"q"p" (y" cos a"-x" cos B");
and as the other forces Q"", Q, &c. tend to make the body
turn in the opposite direction, we shall have
B'''),
Q""'q'"' = = P'"' (y"" cosa""-x"" cos ẞ"),
QIVqIV = = Q¹V (yIV cosa¹V
&c.
α
XIV cosẞir),
Therefore, in all these values, the superior signs or the in-
ferior must be taken at the same time, and by substituting
them in equation (6), it will become equation (5); which was
required to be verified.
422.
OF THE EQUILIBRIUM OF A SOLID BODY.
268. The body in equilibrio being always acted on by
gravity, the weight of a body acting in the direction of a
vertical passing through its centre of gravity, must be sup-
posed to be included among the given forces P, P′, p", &c.
For example, if we suppose that the heavy body rests on an
inclined plane, and is sustained by one sole force, (fig. 64 re-
presents a section of this body passing through the centre of
gravity G, and perpendicular to the inclined plane,) the
length of this plane is AB, its base BC, and its height AC. Let
o, the origin of the coordinates, be in the vertical GH passing
through the centre of gravity, and let the axes oz and ox be,
the one perpendicular, and the other parallel to AB; the third
axis oy, which is not represented in the figure, will be perpen-
dicular to the plane of the figure. The force p will be the
weight of the body, the vertical GH its direction, and нor the
angle a. Moreover, we shall have x = 0, y=0, B=90°.
Hence, if p' represents the given force which sustains the
heavy body, the equations of equilibrium of the third case of
No. 266 will be reduced to
=
P cosa + P′ cos a' 0, P' cos B' 0,
P′(x'cos ẞ'— y'cosa′) = 0.
=
From the two last, we obtain B′ = 90°, y' = 0; which shews
that the force p' must exist in the plane of the axes of x and
z; and, in fact, this is evidently necessary, in order that this
force and the weight of the body may have an unique result-
ant, perpendicular to the inclined plane. Let o be the point
where the direction of p' meets the vertical GH, and let OD re-
present this direction. The angle a' or Dox must be obtuse,
in order to satisfy the first of the three preceding equations;
let & denote the acute angle Dox', which the force p' makes
with the production of ox, so that we may have
cos a' = cos 8.
The angle a or нox is the complement of ABC, the inclina-
OF THE EQUILIBRIUM OF A SOLID BODY.
423
เ
tion of the plane; therefore denoting the height AC by h, and
the length AB by l, we shall have
cos a =
h
hence there will result the equation of equilibrium, (e)
Ph
Τ
= P'cos 8;
by means of which one of the two quantities p' and ♪ will be
determined, when the other is given.
For example, when the force p' is parallel to the inclined
plane, we shall have 8= 0, and consequently,
P': P::h: l,
or, what comes to the same thing,
P' P sini;
Q
i denoting the inclination of the plane. If a denotes the pres-
sure that the plane experiences, and which in this case will be
the weight e resolved in the direction of the perpendicular oz,
we shall have, at the same time,
Q = P cos i.
does
269. In the preceding discussion no account has been
taken of the friction, which combines its effect with that of the
force P' parallel to the inclined plane, in preventing the body
from sliding along it. If this force p' vanishes, the friction
alone may retain the body, as long as the inclination
not attain a certain limit. If this limit be λ, i. e. what i be-
comes when the equilibrium commences to give way, and if at
this instant the friction is the fraction f of the pressure Q, the
force fo must be in equilibrio with P sin λ, the component of
the weight of the body resolved parallel to the inclined plane.
Consequently, we shall have at the same time
Q = P cosλ, fa= P sinλ;
424
OF THE EQUILIBRIUM OF A SOLID BODY.
hence we obtain
ƒ = tang λ;
which will enable us to determine the value of f, by ob-
serving the angle at which the body commences to move;
this has been termed the angle of friction.
Q
It is proved by experiment, that, every thing else being
the same, at the instant the equilibrium gives way, the friction
is proportional to the pressure, so that the coefficient ƒ and
the angle λ are independent of the pressure (ƒ), and conse-
quently of the weight P. This coefficient depends on the na-
ture of the body and the smoothness of the surfaces; it has
been also observed, that it does not attain its maximum value
until the body and the plane have been some time in contact,
(which time varies with the nature of the body), and that it
is only when this maximum is attained, that the friction is
proportional to the pressure. Assuming this experimental
law to be correct, it follows, that if several bodies of the same
nature, and whose surfaces have the same degree of smooth-
ness, are placed on a horizontal plane, and if after the lapse
of a certain time this plane is gradually inclined, all these
bodies will commence to slide at the same inclination λ, what-
ever be their weights, and the extent of their surfaces in con-
tact with the plane.
270. When a body is placed on a horizontal plane the
force with which, in consequence of its weight P, the plane is
pressed, is distributed among the points of support of this
plane; but when their number is more than three, this distri-
bution seems at first view to be indeterminate. The diffi-
culty which appears to occur in this case we now proceed to
examine.
For greater clearness, let this horizontal plane be supposed
to be the surface of a table, the feet of which are vertical.
Let two rectangular axes ox and oy (fig. 65) be drawn in
this plane, let c be the projection of the centre of gravity of
the body on this plane, and a, a', A", &c. the points of this
OF THE EQUILIBRIUM OF A SOLID BODY.
425
plane where the legs of the table meet it; and let x, and y₁,
x and y, x' and y', x" and y", &c. denote the coordinates of
the points c, a, A', A", &c. referred to the axes ox and oy. In
order that the table may not be overturned, the point c must
lie within the polygon A A' A" A", &c. This condition being
satisfied, the weight P, applied to the point c, may be re-
solved into a number of parallel forces acting in the direction
of gravity, and passing through the points of support a, a', a”,
&c., which forces are the loads that are supported by the legs
of the table. If q, a', a", &c., denote these unknown loads,
by the theory of parallel forces, we shall have
";
P=Q+Q'+q" + &c.
PX₁ = Qx + Q'x' + Q″x″ + &c.
PY₁ = Qy + Q'y' + q″y" + &c.
Now, if there are only three points of support, a, a', a″,
these three equations will be sufficient to determine the loads
q, q', q″; but if there be a greater number than three, the
problem will be indeterminate, and the values of all the un-
known pressures, minus three, may be assumed to be what we
please, provided that for these three, the values which result
for them are positive. This indetermination would in fact
obtain, if the table was rigorously inflexible; but this is ne-
ver the case: and, though this flexibility may be supposed
ever so little, still it will so far cause a slight displacement,
the effect of which will be, that the table will be unequally
pressed in its different parts. Now, the figure that it will
assume, and, the force with which it will be pressed, in each
point, will depend not only on the weight P, but also on
the number and disposition of the points of support a, a′, a″,
&c.; and both the one and the other, as also the pressure
which has place in each of these points, will be completely
determinate in each particular case. However, this determi-
nation is an extremely difficult problem, the general solution
3 I
426
OF THE EQUILIBRIUM OF A SOLID BODY.
+
of which belongs to the department of mathematical physics,
and has not yet been given. We shall therefore restrict our-
selves here to one sole observation which is suggested by this
subject, namely, that in nature every thing is necessarily deter-
minable, and that when any thing appears to us indetermined,
it is because we have not taken into account some one of the
data of the problem, i. e., some property of matter, like the
degree of flexibility in the table in question.
CHAPTER II.
THEORY OF MOMENTS.
271. THE moments which we propose to treat of in this
chapter are those that have been already considered in No. 42,
where the moment of a force P was defined to be the product
of this force and of p the perpendicular let fall from the centre
of the moments on its direction. Therefore, if c be this centre
(fig. 66), and if the force P be represented by the line ma taken
on its direction, its moments will be expressed by twice the
triangle CAM, which has this force for its base, and its vertex
at the point c. From this it appears, that the theorem of
No. 46, relative to the moment of the resultant of two forces,
is a geometrical proposition extremely easy to demonstrate.
In fact, if MA and MB be the two components, MD the di-
agonal of the parallelogram MADB will be their resultant; and
if the point c be without the angle AMB and its vertically op-
posite, it is easy to demonstrate that the triangle CMD is
equal to the sum of the triangles CMA and CMB. For in the
first place, we have
CMD = CMA + CAD + MAD ;
and if a perpendicular CE be let fall from the point c on the line
MB, this will meet DA which is parallel to MB in F, and we shall
have
and since
CMB = .MB.CE, CAD.AD.CF.;
there will result
MB = AD, CF = CE — EF,
and, consequently,
CAD = CMB MAD,
CMD = CMA + CMB :
which was required to be proved.
428
THEORY OF MOMENTS.
In the figure, the line EF is supposed to be the difference
of the perpendiculars CE and CF; but it may be equal to their
sum, and it is easy to modify the preceding demonstration so
as to apply it to this case. In the same manner it may be
proved, that, when the point c lies within the angle AMB, or
its vertically opposite, the triangle CMD is equal to the differ-
ence between the triangles CMA and CMB.
272. Through the centre of moments (fig. 67), let any
plane whatever be drawn, and let the right line AB, which
represents the force P in magnitude and direction, be projected
on this plane, also let o be the force represented by A'B' the
projection of AB; the moment of the force P will be twice the
triangle CAB, and that of the force Q twice the triangle CA'B';
consequently, the centre of moments remaining the same, the
moment of the projection of a force on a plane passing through
this point, is the projection of the moment of this force on this
same plane.
If H and K denote respectively the moment of the force P
and of its projection ; and if the perpendiculars CD and CE
be erected to the planes of these respective moments, the
angle contained between these perpendiculars will be equal to
the inclination of H on к, and by No. 10 we shall have
KH COS 8.
The force p remaining the same, the angle & and the mo-
ment H will vary with the position of the point c on the line
CE; but if the position of this line is not altered, the product
H cos & will not vary; for K or the triangle CA'B' will only be
displaced parallel to itself, without undergoing any change in
its magnitude.
273. In place of one force P, let any number of forces
P, P′, P", &c., be considered. If н, н', н", &c., be their mo-
ments with respect to the point c (fig. 68), S, &,d", &c., the
angles which the perpendiculars CD, CD', CD", &c., to the planes
of these moments make with the same axis CE ; Q, Q', Q", &c.,
the projections of P, P′, P", &c., on the plane drawn through
THEORY OF MOMENTS.
429
1
the point c, and perpendicular to this axis; K, K', K", &c., the
projections of H, н', н", &c., on this same plane, we shall
have
K=H COS 8, K'H' cos &, K" H" cos 8", &c.
If it is merely proposed to determine the areas of the pro-
jections from knowing those of the projected surfaces, the in-
clinations 8, 8, 8", &c., must be considered as acute angles; but
in the applications which we shall make of the projections of
the moments, these angles will be regarded either as acute or
obtuse, or in other words, it being agreed on to consider a cer-
tain direction as positive, and the opposite as negative, we will
assume for the lines CD, CD', CD", &c., the parts of the perpendi-
culars to the planes of the moments H, H', н", &c., which make
acute or obtuse angles with the axis CE, according as q, qʻ, q",
&c., the projections the forces of P, P', P", &c., tend to produce
a revolution in a certain direction previously agreed on, or the
contrary. Thus, in the figure, the angles DCE, D'ce, d'ce,
being acute, and the angles D'"'CE, DICE, &c., being obtuse, this
implies, that the forces Q, Q', Q", tend to produce a revolution
in the same direction, and the forces q''', q¹, &c., in the op-
posite direction. As the lines CD" and CD"" are the produc-
tion the one of the other, this indicates that the forces p" and
p''' are both comprised in the same plane, passing through
the point c, but that they, as likewise their projections o” and
Q''', tend to produce revolutions in opposite directions.
Denoting the sum of the values, whether positive or ne-
gative, of K, K', K", &c., by s, we shall have,
SH COS 8+ H' cos d+H" cos 8" + &c.;
and, abstracting from the consideration of the sign, s will be
the sum of the moments of the forces Q, a', Q", &c., which
tend to produce a revolution in one direction minus the sum
of the moments of those which tend to produce a revolution
in the opposite direction; therefore, by the theorem of No.
47, the quantity s will express the moment of their re-
430
THEORY OF MOMENTS.
sultant, which will tend to produce a revolution in the di-
rection of the forces that refer to the acute angles S, S', 8", or
to the obtuse angles "", SI, &c., according as the preceding
value of s will be positive or negative.
If all the lines CD, CD, CD", &c., are at the same time
changed into their productions at the other side of c, the
angles S, S', d", &c., will be changed into their supplements,
and s will be changed into s. This will also be the case
when for the axis CE there is substituted its production CE'.
Each of the parts which make up the sum s, and conse-
quently s itself, will be independent of the position of the
point c on the axis CE; it will depend only on the system of
forces P, P', P", &c., on the position of this axis, and on its
direction perpendicular to the plane of projection. Hence-
forth this quantity s will denote the moment of the forces
P, P', p", &c., with respect to the axis CE.
274. From this definition it appears, that the three quan-
tities L, M, N, of No. 261, will be the moments of the forces
P, p′ p″, &c., with respect to the axes of the positive coordi-
nates of their points of application.
In order to prove this, let o be the projection of the force P
on the plane of the axes of x and y, and q the perpendicular let
fall from the origin of the coordinates on its direction, so that
the value of its moment with respect to this point may be oq.
Let us suppose that the force o acts from A towards B (fig. 69),
and that AC and AD are the coordinates x and y of its point of ap-
plication a, referred to the rectangular axes ox and oy. Like-
wise let λ and μ be the angles BAC' and BAD' which the force
o makes with the productions of x and y; o cos λ and o cos µ
will be the components acting in the direction of Ac′ and ad',
and yo cosa, xq cos μ, will be their moments with respect
to the point o; it appears from the figure, that they will tend
to make the system to turn in opposite directions, and that
the force o tends to produce a revolution in the direction of
Q cos µ; consequently, we shall have
μ
THEORY OF MOMENTS.
431
Qq = xQ cos μ
yQ cos λ.
Q
It is easy to perceive from a consideration of the different
positions which the point a can have, and of the different di-
rections which may be assigned to the force Q, that this equa-
tion will subsist, whatever be the signs of x, y, cos λ, cos µ,
provided that the force a transferred to the point E or F, where
its direction meets the axes of the coordinates x or y, tends to
make ox the axis of the positive coordinates as to turn within
the angle of the positive xs and ys, and, consequently, oy,
the axis of the positive ys, without this angle, as is indicated
by the sagitta s and s'. If the contrary be the case, that is
to say, if the force Q, thus transferred, tends to make the axis
of the positive ys to turn within the angle of the positive xs
and ys, and, consequently, the axis of the positive as without
this angle, we shall have
yq cos λ — xq cos µ.
Qq = yQ
whatever may be the signs of x, y, cos λ, cos μ.
It follows from this, that ifs be the moment of the forces
P, P', P", &c., with respect to the axis of the positive zs, and
if the angles S, S', d", &c., of the preceding number are con-
sidered to be acute or obtuse, according as the projections
Q, Q', Q'', &c., of these forces tend to make the axis of the
positive as to turn within the angle contained between the
positive coordinatės x and y, or without this angle, we shall
have
s = Q(x cos μ- y cos λ) + Q'(x' cos μ'y' cos λ)
+ Q″ (x" cos µ"— y″ cos \") + &c. ;
x', y', λ',µ', x'', y", \", µ", &c., being what x, y, λ, µ, become
relatively to the forces Q', Q", &c.
Moreover, if a, ß, y, a', B', y', a", B", y", &c., be the
angles which the directions of the forces P, P', P", &c., make
with parallels to the axes of x, y, z, we shall have
432
THEORY OF MOMENTS.
-
Q=P siny, Q'=r′sin y', Q″= p″ sin y″, &c.,
cosa sin y così, cos a'siny'cos', cosa" siny" cosλ", &c.,
cosẞ=sinycosμ, cosß'=siny' cosμ', cosß"=siny" cosµ", &c. ;
and from these values, it is evident that the expression for s
will coincide with the value of L of No. 261(a). Hence, Lis
the moment of the forces P, P', P", &c., with respect to the
axis of the positive zs: and, according as it is positive or ne-
gative, this system of forces will cause the plane of the axes
of the positive xs and zs, to turn within the solid angle of the
positive coordinates, or without this angle.
Now, if the axes of the positive zs, xs, ys, be substituted
for those of the positive xs, ys, zs, L will be changed into M ;
it follows, therefore, that м is the moment of the forces
P, P′, p″, &c., with respect to the axis of the positive ys, and
that according as it is positive or negative, this system of
forces will tend to make the plane of the axes of the positive
zs and ys to turn about this axis, within the solid angle of the
positive coordinates, or without this angle. In like manner,
if the axes of the positive ys, zs, xs, be substituted for those
of the positive zs, as, ys, м will be changed into N; conse-
quently, N will be the moment of the forces P, P′, p″, &c.,
with respect to the axis of the positive rs; and, according as
this moment is positive or negative, this system will tend to
make the plane of the positive ys and xs to turn about this
axis, within the solid angle of the positive coordinates, or
without this angle.
The three quantities L, M, N, are consequently, as has been
stated, the moments of the same system of forces with respect
to the three axes of the positive coordinates of their points of
application, and the signs of their values, such as they are
written in No. 261, refer to a known direction of rotation,
about each axis supposed to be fixed.
275. It appears from the preceding number, that the first
value of oq is the same thing as(b)
Qq
1
H
THEORY OF MOMENTS.
Qq = x P cos ẞ — y P cos a.
433
Hence if н denotes the moment of P with respect to the
origin of the coordinates, and 8 the angle contained between
a part of the perpendicular to the plane of this moment and
the axis of the positive zs, we shall have (No. 272)
H COS & P (x cos ẞ — y cos a) ;
which implies, that this part of the perpendicular to the plane
of н, is that which makes an angle with the axis of the posi-
tive zs, that is acute, or obtuse, according as the quantity in-
cluded between the brackets is positive or negative. If d
and d₂ be the angles which the same part of this perpendi-
cular makes with the axes of the positive ys and xs; we shall
have in like manner
H cos d₁ = P(z cos a
COS
H cos d₂ = P (y cos y
COS
x cos y),
sy − z cos ß).
If, therefore, in order to abridge, we make
(x cosẞ-ycosa)²+(z cos a− x cosy)²+(y cosy z cosß)²=p²,
a·
there results, (p being regarded as a positive quantity,)
H = PP,
because
cos² + cos² 81+ cos² d₂ = 1;
consequently, we shall have
1
cos &=
(x cos ẞy cos a),
p
1
P
cos d₁ ==(z cos a x cos y),
cos d₂ = = (y cos y − z cos ß) ;
Ρ
by means of which the three angles 8, 81, 82, can be deter-
mined without any ambiguity. The angle & will be acute or
obtuse, according as the sign of x cos ẞ — y cos a is positive
3 K
434
THEORY OF MOMENTS.
or negative, and the angles &, and 82, according to the signs of
≈ COS a x cos y, and y cos y
γ z cos B.
It is easy to verify these formulæ. In fact, if the equation
of the plane which comprehends the origin of the coordinates
and the force P, be denoted by
Au + BV + cw = 0;
u, v, w, being any coordinates whatever, then if for these we
substitute x, y, z, the coordinates of the point of application
of this force, this equation will become
Ax + By + cz = 0 ;
moreover, the équations of a right line drawn through the
origin of the coordinates and parallel to this force, will be
u cos ß, w cos a = u cos y ;
v cos au cos ß,
and as this parallel is likewise comprised in the plane which
we are considering, there results from it this second equation
of condition
A cos a + B cos ẞ + c cos y = 0.
From these two equations we deduce(c)
C=
A ▲ (x cos ß — y cos a)
y cos y ― z cos ß
A (z cos a x cos y)
B = y cos y — z cos B
and by substituting these values in the equation of the plane,
it becomes
u (ycosy―zcosẞ)+v(zcos a-x cos y)+w(x cos ẞ-ycosa)=0.
α
Now, by known formulæ (No. 17), the cosines of 8, 81, 82,
the angles that the normal to this plane makes with the axes
of u, v, w, which are likewise those of x, y, z, will have for
values the formula which it was required to verify.
In virtue of the equation HPP, the quantity p is the
perpendicular let fall from the origin of the coordinates on the
THEORY OF MOMENTS.
435
direction of the force P. This may be also verified without
any difficulty, by taking the foot of this perpendicular for the
point of application of P; for, denoting the radius vector of
this point, which will then be this perpendicular, by r, and
the angles which its direction makes with the axes of x, y, z,
by λ, u, v, we shall have
x=r cosλ, y=r cos µ, z = r cos v;
and if these values be substituted in that of p2, in consequence
of equations (Nos. 6 and 9), namely,
cos² a + cos²ß + cos² y = 1,
cos² + cos² μ+ cos² v = 1,
cos a cosλ + cos ẞ cos μµ + cos y cos v = 0,
we shall find(d)
p²= r², or p = r.
276. The moments of the same system of forces with res-
pect to different axes, possess remarkable properties, which
being an immediate consequence of those of the projections
of plane surfaces on different planes, we now proceed to in-
vestigate.
Let ox, oy, oz, be three rectangular axes which intersect
in the point o (fig. 70). Through this point let three other
retangular axes ox', oy', oz', be also drawn. In order to de-
termine the directions of these new axes with respect to the
first, let
xox′ =a, yox′ = ß,
zox′ =y,
xoya, yoy' = B',
zoy' = y',
zo2 = ;
xOz = a, y0z =0”,
a, ß, y, a', &c., being considered as nine given angles, which
may be either acute or obtuse. Their cosines will be con-
nected together by six equations, for with respect to each of
the three lines ox', oy', oz', we shall have evidently,
436
THEORY OF MOMENTS.
cos² a + cos²ß + cos³ y = 1,
2
cos² a' + cos²ß' + cos³ y' = 1,
cos² a"+cos² ẞ"+cos2y"=1;
(1)
and because x'oy', x'oz', y'oz', are respectively right angles,
we shall likewise have
cos a cosa + cos ß cos B'+ cos y cos y = 0,
cos a cos a" + cos ß cos B" + cos y cosy"=0,
cos a'cos a" + cos B'cos ẞ" + cos y'cos y″ = 0.
}
(2)
By means of the nine angles a, a', a", &c., we can reciprocally
determine the directions of the first axes ox, oy, oz, with re-
spect to the second ox', oy', oz', in which case we shall have
cos² a + cos² a' + cos² a" 1,
cos²ß + cos²ß' + cos²ß″ = 1,
(3)
cos² y + cos² y' + cos² y″ = 1,
and, besides,
cos a cosẞ + cos a' cos B' + cos a“ cos ẞ” = 0,
α
cós a cos y + cos a' cos y' + cos a“ cos y″ = 0,
cos ß cos y + cos B' cos y + cos ẞ” cos y″ = 0;
(4)
these equations are evidently equivalent to the six preceding,
and may be substituted for them.
If a denotes the area of a plane surface bounded by any
outline whatever, and situated in a plane passing through the
point o; and if a perpendicular OD be erected at this point to
this plane, then by making
XOD=q, yoDq', ZOD=q",
these three angles, which may be either acute or obtuse, will
determine the direction of on and that of the plane of a; if
each of these three be changed into their respective supple-
ments, the right line oD will also be changed into its pro-
duction, but the plane of a will remain the same as before.
THEORY OF MOMENTS,
437
.
Also, if the projections of a on the planes of yoz, xoz, xoy,
be denoted by p, p', p", we shall have (No. 10)
p=a cos q, p'= a cos q', p"= a cos q″.
Finally, if b denotes the projection of a on any fourth
plane, as for instance the plane y'oz', and if c denotes the
angle x'OD, we shall also have
b = a cos c,
and, by formula (2) of No. 9,
cos c = cos q cos a + cos q' cos ẞ + cos q″ cos y,
hence we infer,
b=p cos a +p' cos ẞ + p" cos y ;
(5)
(6)
by means of which the projection of an area a on any plane
whatever, will be determined, when its projections on three
rectangular planes are known.
As equation (5) obtains only when the signs of the cosines
which it contains are taken into account, it follows that it is
also necessary to have regard in equation (6), to the signs of
the projections p, p', p", and to treat them as positive or ne-
gative, according as OD the perpendicular to the plane of a,
makes acute or obtuse angles with the axes ox, oy, oz.
277. This being established, let any number of plane
areas whatever, such as a, a', a", &c., existing in different
planes, be projected on the three planes xoy, xoz, yos, and
let the sum of the projections on each of these planes be
taken, respect being had to their signs in the manner above
specified, then if the three sums, which are obtained in this
manner, be denoted by a, a', a”, and if in like manner в de-
notes the sum of the projections of a, a', a", &c., on the plane
y'oz'; by forming for each of these areas, an equation similar
to equation (6), we shall obtain, by adding together all these
equations,
B = A cos a + a'cos ß + a″cos y.
438
THEORY OF MOMENTS.
If again the sum of the projections of a, a', a", &c., on the
plane x'oz' be denoted by B', it is evident that the value of B′
may be deduced from that of B, by substituting the axis oy'
perpendicular to this plane, for the axis ox' perpendicular to
the plane y'ox', that is to say, by substituting in the preceding
formula, a', B', y', for a, ß, y; by this means we obtain
B′ = A cos a + B cos ẞ'+ c cos y'.
If, in like manner, the sum of the projections of a, a', a″
&c., on the plane x'oy' be denoted by B", its value may be de-
duced from that of B, by substituting a", ß", y", in place of
a, ß, y; from which there will result
B" — A COS a" + B cos " + c cos y".
We can likewise deduce conversely from these values of
B, B', B", and by having regard to equations (3) and (4),
A =B cos a + B' cos a' + B" COs a",
A′ = B COSß + B' cos B' + B" cosẞ",
A"=B cos y + B' cos y + B" cos y".
(7)
From a consideration of these different equations, it is evi-
dent, that the projections of plane surfaces on different planes,
are subject to the same laws as those of right lines on other
right lines.
278. If the sum of the squares of the values of B, B', B",
be taken, there results, by equations (3) and (4),
12
1/2
2
12
B² + B²² + B'¹² = A² + A¹² + A1⁄2;
.(8)
from which it appears, that the sum of the squares of these
three quantities B, B', B", does not vary with the direction of
the three rectangular planes of projection to which they are re-
ferred. In the particular case, in which all the areas a, a', a",
&c., exist in the same plane, this sum is in fact the square
of the entire area a + a' + a' + &c.; and, if for example,
this plane be assumed to be that of the axes oy and oz, we
shall have
THEORY OF MOMENTS.
439
A = a + a + a" + &c., a' = 0, A"=0.
In this case all the areas a, a', a", &c., lie in the same plane,
but if they should exist in different planes, then it is easy to
prove, that this sum is equal to A2+ A2 + A2, in fact, by
equation (8) we have
12
B = √ A² + A²² + A
112
12
Β΄ B/2
from which it appears, that the sum B which varies from one
plane of projection to another, is greatest when B′ = 0, B″= 0;
in which case it is equal to √A²+A+A", this is, therefore,
2
12
the expression for the greatest sum of the projections
projections on the
same plane, of the plane areas a, a', a", &c., considered as ex-
isting in different planes.
279. The plane y'oz' that answers to this greatest pro-
jection, possesses remarkable properties, which will be advert-
ed to in the sequel of this treatise. It is easy to determine
its position by means of the equations B' = 0, в"= 0, which
characterize it.
In fact, equations (7) are then reduced to
A B COS a,
A B COS ẞ, A"= B COS Y ;
from which we obtain
cos a =
2
A
12
√ A² + A
cos B =
√
cos y =
2
A'
12
+
a² + A²² + A
2
A"
12
A² + A²² + A
7/2
712
112
Hence, when A, A', A", the sums of the projections on any
three rectangular planes yox, xoz, xoy, selected arbitrarily, are
known, the direction of y'oz', the plane of greatest projection,
may be immediately ascertained, by means of the angles a, ß, y,
which determine the line or' perpendicular to this plane. With
respect to its absolute position in space, that is evidently unde-
440
THEORY OF MOMENTS.
termined, for the projections of each of the areas a, a', a'', &c.,
and, consequently, the sum of these projections, are the same
on all parallel planes.
280. The sum of the projections of the areas a, a', a", &c.,
on all planes equally inclined to the plane of greatest pro-
jection, is always the same.
In order to demonstrate this, let the plane perpendicular
to the line OD be taken, let c denote the sum of the pro-
jections of a, a', a", &c., on this plane, q, q', q", the angles
which this line OD makes with the axes ox, oy, oz, &c., c the
angle x'oD, which measures the inclination of this plane on
that of the greatest projection. By what has been esta-
blished in No. 277, we shall have
c = a cos q + a' cos q′ + a″ cos qʻ.
Hence, by substituting for A, A', A", their respective values
B cos a, в cos ẞ, в cos y, we shall have
c = B (cos a cos q + cos ẞ cos q' + cos y cos q″),
that is, in virtue of formula (5), c=B cos c, and, substituting
for B its value,
2
12
C = √ A² + A²² + A' 112 cos c;
consequently, the value of c, is the same for all planes, which
make the same angle c, with y'oz', the plane of greatest pro-
jection.
This value diminishes according as the angle c approaches
to 90°; and it is cypher for all planes perpendicular to y'oz'.
281. In order to apply these propositions relative to the
projections on plane surfaces to the theory of moments, it is
only necessary to suppose that the areas a, a', a", &c., are the
doubles of triangles which have for their common vertex the
point o, and for bases the lines, which represent, in magnitude
and direction, the forces P, P', P", &c., that have been con-
sidered above. Their moments L, M, N, with respect to the
THEORY OF MOMENTS.
441
axes oz, oy, ox, of the positive coordinates of their points of
application (No. 274) will be then the sum of the projections
of a, a', a", &c., on the planes xoy, xoz, yoz. The following
consequences result from the propositions which have been
now demonstrated:
1st. Denoting by E the moment of the forces P, P', P", &c.,
with respect to an axis passing through the point o, and
that makes with the axes ox, oy, oz, the angles ε, ε', ε", which
may be either acute or obtuse, we shall have
Sε".
EN COS ε + M COS έ' + L COS E
2ndly. Among all the directions which the axis of the
moment E can assume about the point o, there is one for which
this moment is the greatest possible and equal √12+M²+N².
With respect to every other axis passing through the point o,
and perpendicular to that of the greatest moment, the moment E
is cypher, and it is equal to VL2+ M² + N2 cos 8, relatively to an
axis which makes the angle & with that of the greatest moment.
3rdly. Finally, if a, ß, y, denote the angles that the axis of
the greatest moment passing through the point o, makes with
ox, oy, oz, the axes of the moments N, M, L, and if G repre-
sents the magnitude of this greatest moment, we shall have
N
COS α =
G'
and, at the same time,
M
L
cos sB = 1/1, cos y = 1/1,
G'
G = √ L²+ M²+N²;
G
hence it follows, that if on the axes ox, oy, oz, we take, rec-
koning from the point o, lines proportional to the moments
N, M, L, and if the parallellopiped, of which these lines are
the adjacent sides, be completed, the length of its diagonal
will represent the magnitude of the greatest moment, and this
line will be the axis of this principal moment.
Euler was the person who first announced these remark-
able theorems. They establish a complete analogy between
the composition of moments and that of forces; an analogy
3 L
442
THEORY OF MOMENTS.
4
which arises from this, that forces being represented by right
lines, moments are expressed by plane surfaces, which are
projected on different planes, in the same manner as lines are
projected on different lines (No. 277).
282. The point o and the system of forces P, P', P", &c., being
given, their greatest moment & is termed the principal moment
of these forces. If all these forces be transferred parallel to
themselves, to this point o, their resultant R will have for
components parallel to the axes ox, oy, oz, respectively, the
three quantities x, y, z, of No. 261. The consideration of
this resultant and of the principal moment, furnishes us with
a very simple means of stating the results of the preceding
chapter.
In order to the equilibrium of the forces P, P', p″, &c.,
applied to a solid body entirely free, it suffices if the resultant
R and the principal moment & are respectively equal to cypher,
for since
R² = x² + y²+z², G² = L² + M² + N²,
the equations R = 0, &= 0, imply that the six equations of
equilibrium of No. 261 have place.
Hence it is easy to infer, that when one system of forces is
in equilibrio with another, it is necessary, and it suffices, 1st,
that the resultants R of the forces of the two systems should
be respectively equal and contrary; 2ndly, that for the same
point o, their principal moments should be equal, and should
refer to two axes, drawn in opposite directions, or of which
the one is the production of the other. The resultant R and
its direction, the principal moment and the direction of its
axis, will remain the same, in all the transformations which
the same system of forces can undergo, and, generally, for any
two systems of equivalent forces.
If a, b, c be the angles which the force R makes with the
axes ox, oy, oz, we shall have
cosa =
X
R
Z
cos b =
R
COS C = -
R
THEORY OF MOMENTS.
443
Likewise, if o be the angle comprised between its direction
and the axis of the principal moment, we shall have, (a, ß, y,
being the angles which this axis makes with ox, oy, oz),
cos w = cos a cos a + cos b cos ẞ + cosc cosy,
or, what comes to the same thing,
COS @=
XN + YM + ZL
RG
It follows, therefore, that the condition of an unique re-
sultant which is expressed (No. 263) by the equation
XN+YM + ZL = 0,
consists in this, that the axis of the principal moment &, and
the direction of the resultant R, should be at right angles to
each other. This is what can in fact be verified by observing,
that if the forces P, P′, P", &c., in their actual position, have
an unique resultant, this force must be equal and parallel to R,
and that its moment with respect to the point o, must also be
the principal moment G, so that the axis of the principal mo-
ment is then perpendicular to this resultant transferred to the
point o in a direction parallel to itself; but this reasoning
does not suffice to prove conversely that, when the preceding
equation obtains, the given forces have an unique resultant.
283. If the point o be transferred to any other point 01,
and if ~1, Y1, 21, be the coordinates of the point o, with re-
spect to the axes ox, oy, oz; and if L1, M1, N1, be what
L, M, N, become relatively to this point o₁, the values of these
last quantities may be obtained from the first (No. 261), by
substituting x-x1, Y-Y₁, z-z₁, in place x, y, z, from which
there will result (e)
L₁ = L+XY1 — YX1,
M1
M₁ = M + ZX1
X21,
(a)
N₁ = N+ YZ1-ZY1.
From these formulæ it appears that when P, P', P", &c. are
444
THEORY OF MOMENTS.
reduced to equal and parallel forces, which though acting in
contrary directions, still are not directly opposed to each
other, in which case we have x = 0, y = 0, z = 0, the
quantities L₁, M₁, N₁ are independent of the coordinates of the
point o₁; so that the magnitude of the principal moment and
the direction of its axis do not vary with the position of this
point. In fact it is evident, that, whatever be the position of
the point o₁, the axis of the principal moment of the two pa-
rallel forces which may be substituted for the given forces
P, P′, P″, &c., is perpendicular to their plane; and we know
from other considerations (No. 48) that the sum of the mo-
ments of these two forces, which will be the principal moment
of the given forces, is a constant quantity.
In every other case, the principal moment varies with the
position of the point o₁; and if it were required to determine
the point, or series of points, for which this moment is a
minimum, by making generally
we shall have
2
2
2
2
G₁² = L₁²+M₁² + N₁².
G₁² = (L+ XY1—¥æ₁)² + (M + Zx1−X≈1)²+ (N + ¥2,—ZY₁)².
If in order to determine the minimum value, its three par-
tial differences with respect to 1, 31, 21, be put equal to
cypher, and if we observe that
2
R² = L² + M² + N²,
three equations result, which it is easy to write under the fol-
lowing form, (ƒ)
R²x₁ = x(xx₁ + YY₁ + 221) + YL — ZM,
X
R²y₁ = Y (Xx₁ + Yy₁ + zz₁) + ZN — XL,
R²%₁ = Z (Xx₁ + YY₁ + Z≈1) + XM — YN.
Now, if these three equations be multiplied by x, y, z
respectively, and then added together, there will result an
THEORY OF MOMEnts.
445
identical equation; it follows, therefore, that one of them is a
consequence of the other two; and as the coordinates x1, Y1, 21
do not exceed the first dimension, they appertain to a right
line, which is therefore the locus of the centres of the mo-
ments, with respect to which the principal moment is a mini-
mum. It is evidently unnecessary to examine whether the
preceding equations determine a maximum or a minimum,
for the value of G is not susceptible of being a maximum,
inasmuch as it increases indefinitely with the variables x₁, Y1, 21.
284. If the quantity xx₁ + Yy₁ + zz₁ be eliminated be-
tween the preceding equations taken successively two by two,
there results (g)
X1 Y + L=2
(NX + MY + LZ)
R2
,
NX
MY LZ
Ꮓ
ZX1-X21+ MY
,
R2
(b)
Y21-Zy₁+N=X
(NX + MY + LZ)
R2
;
which are the equations of the projections on the three planes
of the coordinates, of the loci of the centres of the principal
minima moments.
By squaring, and then adding them together, we obtain
NX+ MY+ LZ
G₁ =
G1
Ꭱ
(c)
for the value of the principal minimum moment, which is con-
sequently the same for all the centres 01.
Denoting the angles that the axis of the moment G₁ makes
with parallels to the axes ox, oy, oz, drawn through the point
01, by a1, B1, Y1, we shall have
M1
cos α₁ =
N1 cos B₁ = M₁₂,
cos y₁ =
Yı
G1
G1
515
wherever the centre of the moments may be; and, from equa-
1
444
THEORY OF MOMENTS.
tions (a), (b), (c), there will result in particular, for a point 0,
which belongs to the right line determined by equations (b)
X
Y
Z
cos aj =
cos B₁ =
R
R'
cosy₁ ==/;
R
which shews that the axes of all the principal minima moments,
whose common value is given by formula (c), are parallel to
each other and to the direction of the force R.
When the given forces have an unique resultant, it is evi-
dent that the value of G is the least of all when the point o
exists on its direction, in which case this value is cypher.
Conversely, if the value of G, is cypher with respect to any
point o₁, it follows that the given forces P, P', p", &c. have
an unique resultant passing through this point; for if they
can be reduced to two forces not existing in the same plane,
one of them may be made to pass through the point 01,
and thus reduce their principal moment to that of the other
force, which would not be cypher, contrary to the hypothesis.
From this it follows, that the condition which is necessary and
sufficient in order that the given forces may have an unique
resultant consists in this, that their principal moment may be
equal to cypher. This moment being then a minimum, the
condition in question will, in consequence of formula (c), be
expressed by the equation
LZ+MY + Nx = 0;
and as the point o, to which it refers, belongs to this resultant,
the equations of the line in the direction of which it acts will
be, in virtue of equations (b),
XY1 — YX1 + L = 0,
ZX1 - X21 + M = 0,
YZ₁- ZY₁+ N = 0.
These results coincide with those of No. 263, that have been
obtained from other considerations.
CHAPTER III.
EXAMPLES OF THE EQUILIBRIUM OF A FLEXIBLE BODY.
1. Equilibrium of the Funicular Polygon.
285. EVERY system of cords connected together by fixed
knots, or merely run into rings which can slide along these
cords, is termed a funicular machine. Any number of strings
may terminate at the same knot, but for greater simplicity we
will suppose that at each knot there are never more than three
strings, and, in the first place, we will exclude the consideration
of moveable rings.
Let then A and B (fig. 71) be the two extremities of a cord
perfectly flexible, and of any length whatever, and, м, м', м',
&c., being different points of this cord, let the strings
MC, M'C', M'c", &c., drawn in the direction in which the given
forces P, P', P", &c. act, be attached to these points; likewise
to the point м let there be applied a given force н, acting in
the direction of the string MA, and to the last of the points
M, M', M", &c., let another given force K, directed towards the
point B, be applied. In the state of equilibrium, this flexible
cord will constitute a polygon, the summits of which will be
the points A, M, M', M,". . . . . B, and it is termed a funicular
polygon. The problem to be solved with respect to it, is to
find the conditions, which the given forces H, P, P′, P", . . . K,
must satisfy, in order that this equilibrium may be possible,
and to determine the figure of the polygon which suits this
state.
In order to find these conditions, this self-evident principle
must be assumed, namely, that when the equilibrium obtains,
each of the strings MM', M'M", &c., must be drawn at its
448
EQUILIBRIUM OF THE FUNICULAR POLYGON.
two extremities by equal forces, acting in the directions of its
productions; for if these two forces had not the same directions
as the string, nothing would prevent them from deflecting it ;
and, unless they were equal and contrary, they would cause
the string to move in its direction.
It follows at once, that the resultant of the two forces H
and P, applied to the point м, must coincide with MD, the
production of the string м'м. We may therefore transfer the
point of application of this force to the point m', which exists
on its direction (No. 41); if it be then compounded with the
force p' applied to this point, this second resultant, which will
be in fact that of the three forces H, P, P', must coincide with
M'D', the production of the string м"M'; and, consequently, it
may be transferred to the point м". If, in the same manner,
the resultant of this force and of p", which acts at this same
point м", be taken, the force which draws the string м"M"" at
its extremity M", and which should act in the direction of its
production M"D", will be obtained. This force is evidently the
resultant of the forces H, P, P', p"; by similar reasoning it
may be shewn, that the force which draws the same string at
its extremity м"", and which must coincide with its other pro-
duction M'''D''', is the resultant of the forces P"", p¹, . . . . . K;
therefore these two resultants are equal, and directly opposed
to each other; and consequently, the resultant of all the given
forces H, P, P', P",.... K, must be equal to cypher. We would
evidently arrive at the same result, if the forces which act at
the two extremities of any other side of the polygon were
considered.
M
Hence, the forces applied to the funicular polygon should be
such, that when they are transferred to the same point respec-
tively parallel to themselves, they may constitute an equilibrium.
This furnishes us, as we know, with three equations, between
the magnitudes of these forces and the angles which their di-
rections make with three rectangular axes drawn through this
point. These equations are (No. 35)
EQUILIBRIUM OF THE FUNICULAR POLYGON.
449
H COS α + K cose + P cos a + P' cos a' + &c. = 0,
Hсosb+K Cosƒ+ P cosẞ+ P'cosß' + &c. = 0,
HCOSC + KCOsg+P cosy + P' cos y' + &c. = 0;
(a)
a, e, a, a', &c. denoting the angles relative to one of the axes;
b, f, B, B', &c. the angles relative to another axis; and c, g,
Y, Y', &c. those which refer to the third.
286. When the forces H, P, P', P", ..... K, and the direc-
tions of the strings along which they act, do not satisfy these
equations, an equilibrium cannot be effected between them by
means of the funicular polygon, whatever be the figure it is
made to assume; but whenever these equations are satisfied,
we may assign to the polygon a figure suitable to the existence
of the equilibrium. As the magnitudes and directions of the
forces H, P, P', P", .... K, are given, this figure is necessarily
determinate, and its construction results from the series of
compositions of forces which we proceed now to point out.
In fact, the directions of the strings MA and мc along
which the forces H and P act, being known, the magnitude
and direction of their resultant will be determined. On the
production of this direction, reckoning from the point м, let
there be taken the given length of the side MM', and then to
the point m' let there be applied the resultant of н and P act-
ing in the direction of the line M'м, and the force p' acting in
the given direction of the string M'c'. Let the resultant of
these two forces be taken, and on the production of its direc-
tion, let there be laid off, reckoning from the point м', the
given length of the side м'м", and at the point м”, let a con-
struction similar to that indicated for the point м' be per-
formed, and then at the point м" let there be applied this last
resultant in the direction of the side м"m', and the force p" in
the given direction of the string м"c", these two forces must
then be compounded into one, and, on the production of the
resultant, the given length of the side м"M' should be taken.
This operation should be continued until we arrive at the
M
3 M
450
EQUILIBRIUM OF THE FUNICULAR POLYGON.
IV
last of the knots M, M', M", &c., if in the present case we sup-
pose the last of them to be MV, then мIVB will be the last side
of the polygon. Its direction is known, being that of the last
force K, which by hypothesis is given. It is therefore neces-
sary that the produced direction of the resultant of the two
forces applied to the point м, in the direction of the side
м”м”, and in the direction of the string мICIV, should co-
incide with the given direction of the side мIB. This is, in
fact, what will always be the case; for, by our construction,
the force acting in the direction of MM"", is the resultant
itself of the five forces H, P, P′, P", p””, transferred to the point
MIV
м¹ parallel to their directions, and if it be compounded with
the force p¹, acting in the direction мIVCI, the resultant of
all the given forces, except the force K, will be obtained; and,
in consequence of equations (a), which are supposed to be sa-
tisfied, this resultant is equal and directly opposite to the force
K (No. 35).
polygon intercepted between
They may be determined in
lengths of the sides of the
If through the point A, the three axes to which the angles
a, e, a', a', &c., b,ƒ, ß‚ß', &c., c, g, y, y', &c., are referred,
be drawn, the coordinates of each of the summits of the poly-
gon, with respect to these axes, will be the projections on
these same axes of the parts of the
the point a and this summit.
functions of these angles, of the
polygon and of the given forces; the general formulæ which
are obtained in this manner, will enable us, in each particular
case, to construct directly all the summits of the polygon, or
one or more of these points; but it is simpler to determine
them, one after another, in the manner pointed out above (a).
287. When the given forces satisfy the conditions expressed
by equations (a), and when the polygon has been made to
assume the figure suitable to the equilibrium, the common in-
tensity of the two equal and opposite forces which draw each of
the sides in the direction of its production, is the tension which
this string experiences; it is therefore of consequence, to be
EQUILIBRIUM of the FUNICULAR polygon.
451
able in practice to compute this tension, and to be assured, by
experiment, that it is not greater than that which a string of
the same diameter and material can bear without breaking.
Now, from what precedes it appears, that this tension will
vary from one side to another of the polygon; the tension of
the side MM' will be equal to the resultant of the forces H and
P, or to that of the forces P', P", P"", &c. . . . . K; the tension
of the side M'M" will be equal to the resultant of the forces
H, P, P′, or to that of the forces p", P"", K, and so on. It
will therefore be easy, in each particular case, to determine
the tensions, which all the sides of a polygon in equilibrio
experience, when the magnitudes and directions of the forces
H, P, P', P", .... K, are given.
If A and B, the extreme points of the polygon, are fixed,
the forces H and K will represent both the tensions of the
strings that terminate at these points, and also the pressures
which these points sustain. In this case, the values of н and
K, and of the angles a, b, c, e, f, g, which determine the direc-
tions of the two extreme sides of the polygon, will be no
longer given; but we shall have eight equations, by means of
which these eight unknown quantities can be determined ;
these will be, equations (a), equations (No. 6), namely,
cos²a + cos2b + cos c = 1, cose + cos2f+cos³g = 1,
and three equations resulting from the consideration that the
position of the two fixed points A and B is given. These are
obtained by computing the values of the three coordinates of
one of these points referred to axes passing through the other
point, that is to say, the projections of the entire polygon on
these three axes, and then putting them equal to the given
values of these same coordinates.
In general, the determination of these eight unknown
quantities is extremely complicated, but after that the funi-
cular polygon has assumed of itself the figure suitable to the
equilibrium of the forces, which are applied at its summits,
452
EQUILIBRIUM OF THE FUNICULAR POLYGON.
LETT
the tensions of these different sides may be obtained without
any difficulty; and this is sufficient in practice. Thus, if the
force P applied to the point м be resolved into two others act-
ing in the directions of the productions of the sides AM and мM',
the components, which are given immediately by the rule of
the parallellogram of forces, will be the tensions of these two
sides. That which will act along the production of AM, must
be equal to the force acting in the direction of this first side,
when the point A is free; and when it is fixed, it will express
the pressure exercised on this point. In like manner, the com-
ponents of the force p' in the directions of the productions of
MM′ and м'M” will express the tension of мM', already known
by the decomposition of P, and that of the adjacent side м'M" ;
and so on.
288. As the strings which constitute the different sides of
a funicular polygon are always a little extensible, the length
of each of them is increased by a small quantity, in conse-
quence of the tension which it experiences in the state of
equilibrium; and when this tension is known, the correspond-
ing increase of length may be calculated.
In fact, it appears from experiment, that when the tension
of a homogeneous thread of a constant thickness is inconsider-
able, compared with the force necessary to break it, its increase
of length is proportional to its length and to the tension to
~(Q-RW which it is subjected; and in different threads, it varies with the
J
W
تاو
丌
​thickness and the material of which the thread consists. This
being so, if to a fixed point a thread is attached of the same
thickness and material as the string AM, and if at its inferior
extremity there be suspended a given weight I that is very
great, relatively to that of the thread, and if 7 and 7. (1 + w)
be its lengths before and after the suspension of the weight II,
this quantity w will be a very small fraction, independent of 1,
and proportional to п, (the weight of the thread being ne-
glected); so that if, in another experiment, the three quantities
l, w, I become l', w', n', we shall have
EQUILIBRIUM OF THE FUNICULAR POLYGON.
453
WI
w
whatever may be l and l'. Now, it is evident that a thread
attached to a fixed point, and drawn at its other extremity by
a force acting in the direction of its production, is in the same
condition as if it was drawn in the directions of its two pro-
ductions by this same force. If therefore, T be the tension of
the string AM, and if its length be increased in the ratio of
1+ to unity, we shall have
WT
T= II'
by means of which, this increase of length can be determined;
and in the same way the value of r for all the other sides of
the polygon can be obtained.
T
289. Whether the extreme points A and B of the polygon
be fixed or free, if one or more of the knots м, м', м", &c. are
replaced by rings, this circumstance will give rise to new con-
ditions of equilibrium. If, for example, м" is a moveable ring
which can slide along the chord м'м"M"", it is evident that
in this motion the sum of the distances M'M" and м"M"" of
the point м", from the points м' and м””, will remain constant.
Now, if the equilibrium obtains, this state will not be deranged
by fixing these two last points; but then the point м" will be
in the same circumstances as if it was constrained to exist on
an ellipsoid of revolution, of which м' and м"" are the two foci,
and whose major axis is equal to the given length of the
string MʼM"M", therefore this point cannot be in equilibrio
(No. 36), unless the force p" which is applied to it is perpen-
dicular to this surface; hence, by a known property of the
ellipse it follows, that the direction of this force must divide
the angle between the two radii vectores м'M" and м"M"" into
two equal parts.
When, therefore, in performing the construction indicated in
No. 286, we come to a moveable ring such as м", if it is found
454
EQUILIBRIUM OF THE FUNICULAR POLYGON.
that, after we have taken the resultant of the two forces acting
in the directions M" M' and м"c", of which the side м"M" is
the production, the angles c"M"M′ and c″м"M" are not equal,
it follows that the equilibrium does not subsist. In general, it
is necessary that the direction of the string м"c" attached to
the moveable ring, should not be given beforehand, in order
that we may be enabled, by a suitable determination of it, to
satisfy the condition of the equality of the angles м'м"C" and
M""M"C".
It is evident that when there is an equilibrium, the tensions
of the two sides adjacent to a moveable ring will be equal to
each other; this follows from the circumstance that these two
sides make equal angles with the direction of the force applied
to this ring, and that their tensions are equal to this force re-
solved in their respective directions; but, indeed, this equality
of tension may be regarded as self-evident, inasmuch as the
two sides, in the directions of which the ring can slide, consti-
tute parts of the same string, which necessarily experiences
an equal tension throughout its entire extent.
290. What has been stated relatively to a ring constrained
to slide along an inextensible and perfectly flexible thread,
may be applied to all the points of a system of material points
in equilibrio. In whatever manner these points are connected
together, it is evident that this equilibrium will not be de-
ranged, by fixing all the points of the system except one.
Now, if the connexion of this point with the others is such
that it may besides describe a surface, or only a curved line
about these fixed points, it is evident that the moveable point
will be in the same condition as if the surface or curved line ac-
tually existed; consequently, the direction of the force which
is applied to it must be normal to this surface or this line.
Hence it follows, that in any system of material points in
equilibrio, the force applied to each of these points is perpen-
dicular to the surface or line on which this point would be
EQUILIBRIUM OF THE FUNICULAR POLYGON.
455
constrained to exist, if, for an instant, all the other points to
which it is connected, were considered as fixed points.
When this condition relative to the direction of the forces,
and to the connexion of the parts of the system, is not satisfied,
we may be certain that the equilibrium does not exist, but on
the other hand it is not of itself sufficient to secure the equili-
brium of the system.
А
291. If all the forces which act on a funicular polygon
suspended at the two fixed points A and B, are given weights, it
follows, from the construction of No. 286, that this polygon
must exist altogether in the vertical plane passing through
these two points; this is indeed evident of itself, for there is
no reason why it should deviate from this plane to one side
rather than to the other. Hence, if the perpendicular to this
plane be assumed for the axis to which c, g, y', y', &c. are
referred, all these angles will be right, and the third equa-
tion (a) will disappear; the two others will become
H COS α + K Cos e = 0,
H cos b+K cosƒ+= 0;
(b)
in which the angles a, e, a, a', &c. refer to an horizontal axis,
and the angles b, ƒ, ß, ß', &c. to an axis drawn in the direc-
tion of gravity, п denotes the sum of the weights P, P', p", &c.
which are applied to the polygon.
The equilibrium of this polygon will not be deranged, by
making its form invariable; consequently, the force п must be
equal and directly opposed to the resultant of the forces H
and K.
From equations (b), we know already that it is
equal and contrary to this resultant, it must therefore pass
through the point o (fig.), where the productions of the 73
extreme strings AM and BN intersect; hence this point may
be taken for the common point of application of the two forces
H and K. Thus, in the state of equilibrium, II the resultant
of the vertical forces P, P', P", &c., will be directed along the
vertical OD, and, consequently, we shall have (No. 29), (b)
༣?
456
EQUILIBRIUM OF THE FUNICULAR POLYGON.
H:II:: sin BOD : sin AOB,
K: II:: sin AOD: sin AOB;
by means of which, the tensions of the extreme strings, or the
pressures H and K on the two fixed points A and B, will be
known, when the angles AOD and BOD are measured.
292. The same remark is applicable to the tensions of
strings which support a given weight, as has been already
made relatively to the pressures experienced by the points of
support of a horizontal plane, on which a weight is placed,
(No. 270).
P
Let us suppose that the three strings attached to the fixed
points A, B, C (fig. 73) are reunited at the point м, and that
at this point a weight P is suspended, which acts in the direc-
tion of the vertical MD. On the production of this line, let a
point D' be assumed, and let a parallellopiped be constructed,
of which MD' is the diagonal, and whose three adjacent sides
are MA', MB', MC', taken on the directions of the three strings.
If the force p be represented by the line MD', its components
in the directions of the lines MA', MB', MC', will be represented
by these lines respectively, and they will express the tensions
of the three strings MA, MB, MC, or the pressures on the three
fixed points A, B, C, which will, in this case, be completely
determined. But, when the number of strings which terminate
at the point м is four, or a greater number, the force P may
be resolved in their directions, in an infinite variety of different
ways; so that their tensions and the pressures of the fixed
points, will be no longer determined, and one or more of them
may be either cypher or any arbitrary magnitude whatever.
Now, this indetermination really obtains in the abstract ques-
tion, when the extensibility of the strings is not taken into
account; but it no longer exists, when regard is had to this
property of the material of which the string consists; then
all the strings are lengthened by some small quantities,
they may be ever so small; they depend on their num-
·
EQUILIBRIUM OF THE FUNICULAR POLYGON.
457
ber and on their relative positions; and by measuring the
small increase of length for each of them, the tension of each
string, or the pressure of each fixed point, which actually has
place, may be determined.
Thus, if we suppose that the string AM, for example, is
lengthened in the ratio of 1+d to unity, and if, besides, we
know that a string of the same material and diameter is
lengthened in the ratio of 1+w to unity, when being sus-
pended vertically from a fixed point, the weight P is attached
to its inferior extremity; it follows from No. 288 that the
tension experienced by this string, or the pressure which the
point a sustains, is equal to the product – P.
8
W
If we denote by w' and 8', w" and 8", &c., what the frac-
tions and 8 become relatively to the strings MB, MC, &c.,
and by y, y', y", &c. the acute angles which the strings MA,
MB, MC, &c., make with the vertical MD', we must have
8
&
cosy +
cosy' +
811
w"
13
cos y" + &c., = 1,
in order that the sum of the vertical components of all the
tensions may be equal to the weight P. If the same strings
are projected on a horizontal plane drawn through the point м,
and if n, n', n', &c., denote the angles which the projections
of MA, MB, MC, &c. make with a line Mo arbitrarily traced in
this plane, we shall also have(d)
co 13 col
W
sin y sin n +
ท
&
13 20
8-
şin yʼsin n' + siny" sin ŋ”+ &c. = 0,
siny cos n+siny' cos n' +
S"
siny" cos n" + &c. = 0;
which indicate that the resultant of all the tensions is a vertical
force.
When there are only three strings, these three equations
are sufficient to enable us to determine the relations which
exist between their tensions and the weight P, or the values of
3 N.
458
EQUILIBRIUM OF A FLEXIBLE THREAD.
E
& & &
ய ய ய
by means of the angles which these three strings
make with the vertical MD', and of the angles comprised be-
tween the planes of this line and their directions. When there
are only two, their directions and this vertical exist in the
same plane; this reduces the two last equations to one.
II. Equilibrium of a flexible Thread.
A
293. The case which we propose to consider first, is that
of a homogeneous thread, which we suppose to be acted on by
gravity, and to have a constant diameter. If it be perfectly
flexible, and attached at its extremities a and c (fig. 74) to two
fixed points, the figure that it assumes in the state of equili-
brium is termed the catenary, all its points evidently exist
in the vertical plane which passes through the two fixed
points a and c, for there is no reason why it should deviate to
the one side rather than to the other of this plane.
Let there be drawn through a point o situated in this
plane, the two rectangular axes or and oy, which will be those
of the positive coordinates, let or be the horizontal axis drawn
on the same side as the point A, and oy the vertical axis drawn
through B, the lowest point of the curve, in a direction the
opposite to that of gravity. Let x and y be the coordinates op
and PM referred to these two axes, of м any point of the cate-
nary, s the arc BM, measured from the point B, and terminating
at this point; and let x', y', s' denote what x, y, s become, rela-
tively to another point of this curve which is so situated that
s' > 8.
If p be the weight of the unit of length of the thread,
when it exists in a horizontal plane, p(s'—s) will, in this
state, be the weight of any length ss of this curve, since
it is assumed to be homogeneous and of a constant diameter.
If it be suspended at the two fixed points a and c, its different
parts will be unequally elongated, on account of their re-
EQUILIBRIUM OF A FLEXIBLE THREAD.
459
spective tensions; and, at the same time, since their masses do
not undergo any change, either their densities, or their thick-
nesses must diminish; consequently, the weight of this length
s' - s will be no longer the same as before; however, if the
material of which the thread consists is very little extensible,
so that the small dilatations of its parts may be neglected,
p(s's) may still be considered as the weight of the arc мM'
of the catenary.
Moreover, let T, T', be the unknown forces that act at its
extremities м and м', which arise from this, that these points
are connected with the parts Cм and Aм'. By joining these
CM am'.
forces to the weight p(s' — s), we may consider мм' as en-
tirely free; consequently, if a and ẞ denote the angles which
the direction of the force T makes with the productions of x
and y, the coordinates of its point of application, and if a'
and B' denote corresponding angles relatively to the force T',
we shall have
T cos a + T'cos a′ = 0,
T cos ẞ+T'cos ß' — p(s′ — s),
T(x cos ẞ—y cos a) + T′(x'cos ẞ'— y'cos a') = p (s'—8) X1,
(a)
for the equilibrium of these three forces existing in the same
plane (No. 262); x, being the horizontal abscissa of the centre
of gravity of the arc мM'. These equations will obtain, whatever
be the length of this arc; if it be infinitely small, then the in-
finitely small quantities of the second order may be neglected
in these equations, but the quantities of the first order must
be retained; this, however, will not prevent us from consi-
dering the force T as acting in the direction of MH, which is a
tangent to the curve at the extremity M, and the force T'as
acting in the direction of M'H', the tangent at the other ex-
tremity м'.
In order to be satisfied of this, let there be taken on мм' a
point m, such that the arc мm may be an infinitely small quantity
of the second order; this will enable us to neglect the weight
460
EQUILIBRIUM OF A FLEXIBLE THREAD.
of this part of the catenary. The equilibrium will not be de-
ranged by fixing the point m, but as the thread is supposed to
be perfectly flexible, there is nothing to prevent the force T
from causing the arc мm to turn about m, if it did not act in
the direction of its production MH. In the same way it may
be shewn, that the force T' must act in the direction of м'H'.
In consequence of this, we shall have
dx
dy.
COS α =
ds'
cos ß =
ds
dx'
cos a' =
ds"
cos B'
dy'
=
;
ds'
and by neglecting infinitely small quantities of the second
order, these last values will become
dx
cos a' =
ds
+d.
ds,
dx
cos B'=
dy
ds
+d.dy
ds
It may be likewise shewn, that T'T+dr. In fact, the
quantity T is a function of the coordinates of any point м to
which it refers, which, consequently, becomes at the point м'
T+dr; at this point, it expresses the force which acts in the
direction of MH1, the production of H'M', on Am' the upper
part of the catenary. But, if m' be a point of the curve, the
distance of which from м' is an infinitely small quantity of
the second order, the force which acts at m' on the part am',
will be the same, in magnitude and direction, as that which
acts at м′ on AM'; consequently, the part м'm' of the catenary
is drawn in opposite directions, along м'H' and m'н₁, by the
forces T' and T + dr, which must be equal in order that м'm'
may continue in equilibrio.
19
This being established, if these different values are substi-
tuted in the two first equations (a), they will become, by
makings's ds(e),
=
d.rdx = 0,
dy
=0, d.T
d.r='pds.
(b)
EQUILIBRIUM OF A FLEXIBLE THREAD.
461
As to the third, it will assume the form
d.r(a
x (x dy
dx
y
=pxds,
ds
in which a is substituted in place of x, in the second member,
which substitution we are permitted to make, as infinitely
small quantities of the second order are neglected. Now, this
equation being the same thing as
dy
dx
xd.T
ds
yd.T =pxds,
ds
it is evidently a consequence of the two others. In point of
fact, the problem cannot depend on more than two equations,
inasmuch as there are only two unknown quantities to be de-
termined in functions of x, namely, y and T. The first makes
known the equation of the curve, the second determines the
tension in any point м, that is to say, the magnitude of the
equal forces which draw the element мm along its two pro-
ductions.
294. The integral of the first equation (b) is
dx
T
= c,
ds
in which c denotes a constant arbitrary. At the point в,
dx
we have
ds
= 1, and T= c; if, therefore, the tension in this
lowest point be denoted by the weight of a certain length,
such as h, of the thread, we shall have c ph, and, in any
point whatever,
ds
T=ph Tx
dx
The second equation (b) will therefore become
dy
hd.
= ds;
dx
from which we deduce
dy
s = h
dx'
462
EQUILIBRIUM OF A FLEXIBLE THREAD.
there is no constant arbitrary, as we have, at the same time,
s=0 and = 0, at the point B.
dy
By means of these
dx
equations, the arc s and the tension T will be immediately
known, when the ordinate y shall have been determined in a
function of x.
By substituting in the preceding equation, in the place of
ds, its value
we obtain
ds dx V1+
Vi
dy²
dx2,
hd dy
dx=
dx
1+ y²
dy?
dx2
dy
point B, we have x = 0,
dx
By integrating this expression and observing that at the
y = 0, there results
x= h log
dx
dy + √ 1+
dy2
dx²
and, consequently,
dy + √ 1 +
dy
:ež;
dx
dx²
e denoting, as usual, the base of the Naperian system of loga-
rithms. If this equation be multiplied by
(1
dy² dy
1+
dx²
dx
e
there results
h
e
小
​dy² dy
1+
dx2
consequently, we shall have(ƒ)
da;
ds = § (eå + e¯¹) dx, dy = † (ei — e¯î) dx,
1/
hence we deduce
EQUILIBRIUM OF A FLEXIBLE THREAD.
463
h
h
s (en e î),
= ¦ ¦ (ci – c´¯¡), v = ¦ ¦ (ei + €¯ï),
(ež
h
2
e
B,
(c)
in which we suppose that s=0 and x = 0, at the point в, and
that o the origin of the coordinates, is taken at a distance
below this point equal to h, so that we have y = h when
x = 0.
From equations (c), we obtain 'sh
dy
dx
as before. The
second is the equation of the catenary in its simplest form; it
shews that this curve is symmetrical on each side of its lowest
point.
The preceding value of r will become
T
ds
T=ph
= ph
=py;
dx
from which it appears, that the tension in any point м is ex-
pressed by the weight of a length of the thread, equal to mp,
the perpendicular let fall from this point on the horizontal
line, passing through the point o. Therefore, the tension is
least at the point в, where it is equal to ph, as has been al-
ready supposed.
295. It only remains to determine the constant h which
occurs in these formulæ. The expression for y will then
make known the figure of the catenary; but, in order to know
its position in the vertical plane passing through a and c, it is
necessary also to determine the distance of the axis oy from
one of these fixed points.
For this purpose, let there be drawn through the point a
a horizontal line cutting the axis oy in a point Q, and through
the point c, a vertical line meeting AQ in in the point D. The
position of the point c, with respect to the point a being known,
the distances AD and DC will be given. If these lines be de-
noted by a and b, and if k denotes the distance aQ, we shall
have
AD = α,
DC=b, AQ=k, Oв=h;
464
EQUILIBRIUM of a FLEXIBLE Thread.
a and b are given, and k and h are the two unknown quan-
tities which it is proposed to determine.
Denoting the distance QD by k', the given length of the
curve ABC by l, its parts AB and BC by g and g', and the sa-
gitta BQ by f, we shall have
k + k' = a, g + g'= l,
A
in which k' and g' are considered to be positive or negative,
according as the point c belongs to the production of AB or to
AB itself. The coordinates of the points a and c will be h+ƒ
and h +ƒ—b, in which b is to be considered as positive or
negative, according as c falls below or above the horizontal
line, drawn through the point a.
If, in equations (c) we first make,
and then
x=k, s=9, y=h+f,
x=-k', 8=-g', y=h+f—b,
there will result
s=
9=4(eå—e“), h+ƒ="4" (ch+e¯†),
g=
h
kl
g` = '" (e" — e *), h + s − b = " (c + e¯);
g'
2
(ež
е
from which we obtain
k
k
1 = {} (ei — e ¹ + e
に
​1=
h k
k
k!
k'
h
− e− ),
k'
b = ¦ (cî + e¯Â — ex — ¸¯ ).
b=
en
eh e h
Hence and because k + k' = a, we obtain(g)
a
e.
r_v=r(ci+e¯¡—2),
and, consequently,
b² -
12/1, (~ — c • ) = n,
a
:(ea-
(d)
EQUILIBRIUM OF A FLEXIBLE THREAD.
465
in which we make, in order to abridge,
12 — b²
α
= α,
=n,
2h
a²
As n consists entirely of given quantities, equation (d) will
make known the value of a, and, consequently, that of h. In
general, this equation can be resolved by trials; and the nu-
merical value of a can be deduced from that of n as accurately
as we please. If n differs little from unity, the value of a will
be very small, and, in this case, if the exponentials be deve-
loped, we shall obtain, by neglecting the fourth power of a,
a² = 6 (n − 1). (h)
·
·
If in like manner, we make
we shall have
k - k'
h
= 26,
±
k = a+hẞ, k' = ža — hß;
and the preceding value of b will become(¿)
6
2h
= ½
½ (eï — e¯ï) (eª³ — e−º) ;
2h (eß
(e)
by means of which, the value of ß, and, consequently, the
quantities k and k' will be known, when that of h is determined.
The sign of k' will determine on what side of oy, the point c is
situated.
The simplest case will be that in which the fixed points a
and c are situated on the same horizontal line. In this case
we have b = 0, and equation (e) will give ß = 0, and, con-
sequently, kk'a, as we know it ought to be. At the
same time, we shall have
h
h + f = ½ ½ (e + e¯ï);
by means of which, when the value of h shall have been calcu-
lated, the tensions at the points A and C, or the pressures
which these fixed points will have to sustain, can be deter-
30
466
EQUILIBRIUM OF A FLEXIBLE THREAD.
་
mined. In the general case, these extreme tensions can be
deduced from the values of y corresponding to x = k and
x =
- k.
296. Among all the curves of the same length, which ter-
minate at the given points A and c, the catenary is that
whose centre of gravity is the lowest.
In fact, if through the point a (fig. 75) an horizontal axis
Ay be drawn, and a vertical axis Ax' in the direction of gravity,
and if x and y' be the coordinates of any point м, referred to
these axes, we shall have
lx₁ = Sox' √1+
dy/2
dx' 12
dx';
in which ₁ denotes the distance of the centre of gravity of
AMC any curve whatever, from the axis Ay', b the value of x'
which refers to the point c, and the given length of this
curve, the value of which we know is
dy/2
1=
So V1+ dys
dx'.
dx/2
•
Now by formula (e) of No. 201, the differential equation
of the curve, for which, among all curves of the same length,
the first integral is a maximum, is
dy' = -
c'dx'
√(x + c)² — c'²
³
c and c' being two constant arbitraries. The integration of
this gives, as x and y' are equal to cypher at the same
time(k)
y' = c' log
x²+c+√(x'+c)²—c′
12
and, consequently,
x² + c √ (x' + c)² — c¹² —
yec',
in which, for the sake of abridging, we make
EQUILIBRIUM OF A FLEXIBLE THREAD.
467
c+√ c²¬c²² = y.
Hence it is easy to perceive that
2
x' + c = √(x + c)² — c¹² = y'c
in which, in the same manner, we suppose
=
Consequently,
c — √ c² — c¹² = y.
=y:
x² + c = 1/1/1 yec' + 1 y'e
(f)
will be the equation of the curve which possesses the required
property. At the point c, we shall have
α
a
¿ye¯~;
b + c = 1 ye²² + 1 ye
in which a is the given distance of this point from the axis ax,
so that we have at the same time x' b and y' = a. By
means of this particular equation, and of 7 the length of the
curve, the two constant quantities c and c' can be determined.
Now, in order to make equation (f) to coincide with that of
the catenary, let & denote an indeterminate constant, and let
the ordinates x and y' be changed into two others, such
that
x² + c = y, y' = ε - x;
so that these new coordinates x and y must be drawn in a con-
trary direction from that of x' and y', and also be referred to
another origin. By this change, equation (f) will become
€
y = − 1 y e ˜ e
ε can be determined by putting
€
е C'
- y'e deal.
yea = y'e?
дест
and denoting their common value by -h, so that we may
have
ye" = -h, y'e-&= − h.
468
EQUILIBRIUM OF A FLEXIBLE THREAD.
As yy' = c'²(1), there will result hc'; and the preceding
equation of the curve will become
e¯ž);
y = ¿h (ež — e¯‡ ) ;
which evidently coincides with the second equation (c) that
has been found for the catenary.
297. If the vertical force which acts on each element of
the thread suspended at the points a and c, (No. 74), in place
of being proportional to the element ds, was proportional to
its horizontal projection dx, the second equation (b) will be-
come
dy
d.T = pdx;
ds
p being a given constant, that denotes the weight of a prism,
the altitude of which is the linear unit. From the first equa-
tion (b) which does not undergo any change, we shall always
have
ds
r = ph dx'
T
in which h denotes a line of unknown length, and ph a weight
equivalent to the tension at B, the lowest point of the curve.
From hence there will result
and, consequently,
dy
hd.
dx
= dx;
dy
h
=x, 2hy = x²;
dx
the origin of the coordinates x and y being placed at the point
B. In this case, the curve is evidently a parabola, the summit
of which is at the lowest point, and the expression for the ten-
sion at any point whatever will be (m)
T = p√ h² + x².
EQUILIBRIUM OF A FLEXIBLE THREAD.
469
By employing the notations of No. 295, we shall have, at the.
points A and C,
2hf-k², 2h(f-b)=k'², .
and because k+k'a, there will result
2hb = a(k − k');
by means of which f, k, k', will be known, when h shall have
been determined, and the value of this last can be deduced
from knowing that of l, the length of the thread. In fact,
we shall have
ds
dx =
1
{√ h² + x², hl= S_k √h³+x³dx;
which, by integrating, gives
√ h² + k² + k
2hl=h² log
+ k √ h² + k² + k'√ h²+k¹².
√ h² + k¹² — k'
If, for greater simplicity, we suppose that the two points a
and c exist in the same horizontal line, we shall have
b= 0, k = k' = ž a ;
and the preceding equation will be reduced to (n)
hl=h² log
k + √ h² + k²
h
+ k√ h² + k²;
from which, when the numerical values of 1 and k are given,
an approximate value of h may be deduced by trial. This un-
known h can be determined with greater facility, when 7, the
length of the curve, differs little from a its projection; for then
the value of h will be very great relatively to that of a. In
this case, we shall have in very convergent series
√ h² + k² = h + ½ - 173 + &c.
k2
R4
1/2 h
h³
k + √ h² + k²
k
k³
11
log
h
h
€ 13
+ &c.
470
EQUILIBRIUM OF A FLEXIBLE THREAD.
Phang iss
*
Lauk
Beverly 200
By means of these values, the preceding equation becomes (o)
q. p.
h² (l — 2k) — } k³,
and, consequently,
h =
av2a
4 √3 (1 − a)
This example has been selected, because it may be usefully
applied in the construction of suspension bridges, in which it
is important to be able to calculate the tension of the chain of
suspension, and also the pressure on its points of support.
298. If all the points of the thread are supposed to be
solicited by any forces whatever, the figure which it will as-
sume in consequence of the action of these forces, will be, in
general, a curve of double curvature; the equations of equili-
brum of each of its points will be three in number; and if the
thread be perfectly flexible, these equations can be obtained
in the manner which has been explained in detail in No. 293.
In this way, we obtain
dx
d.T
+xεds = 0,
ds
dy
d.T +Yεds = 0,
ds
dz
d.T +zɛds = 0;
ds
(1)
x, y, z being the rectangular coordinates of м any point what-
ever of the curve, ds the differential element of its length, ɛ
the product of the density of the thread and of the section
made by a perpendicular to the thread at the point м, so that
eds is the element of the mass of the thread, T the tension at
this same point, or the force, of unknown magnitude, which
draws this element ɛds in the direction of each of its produc-
tions; x, y, z the forces referred to the unit of mass, parallel
to the axes of x, y, z, the coordinates of the point м, which will
be given functions of these three coordinates.

~1
EQUILIBRIUM OF A FLEXIBLE THREAD.
471
In virtue of the tension T, the element ds will experience
an extension, and the quantity & a diminution, such, however,
that the mass ɛds remains the same; consequently, if ds' and
' denote what these quantities were in the natural state of
the thread, we shall have
ɛds = &'ds';
and if the extension be proportional to the force which pro-
duces it, (No. 288), we shall, at the same time, have
ds = (1+wT) ds';
w being a very small coefficient, depending on the material and
thickness of the thread at the point м. When the thread is
homogeneous, and of a uniform thickness throughout its en-
tire length, ɛ' and w will be constant quantities; but, in
general, these two quantities may be considered as given
functions of the arc s', reckoned from a determinate point of
the thread, and terminating at the point м.
299. If the thread, whatever be its nature, is only acted
on by gravity, and suspended vertically at a fixed point a, the
two last equations (1) will disappear, and the third will be
reduced to
dr+gɛdx = 0,
the axis of the ordinate x being supposed to be vertical, and
to act in the direction of g which denotes the force of gravity.
If the origin of the ordinates x be placed at a, and if o de-
notes the value of T when x = 0, that is to say, the load
which it will have to support, we shall have for м any point
whatever,
T=QgSεdx,
in which the integral vanishes at the same time as x.
If to B, the lower extremity of the thread, a weight p is
attached, and if I denotes the length of AB, it is evident that
the tension at the point в will be equal to P; consequently,
472
EQUILIBRIUM OF A FLEXIBLE THREAD.
K
we shall have at the same time, x = l, and T = P, by means
of which we obtain
and, consequently,
Q=P + 9 So edx,
g
+ 9 Soeda 9 Seda
T=P+
But as the second and third terms of this formula are evi-
dently the weight of the entire thread and of its part am, it
follows that the tension at any point such as м, is equal to the
weight of the part вM, increased by the weight P; which in-
deed is evident of itself.
The law of the extensibility of the thread throughout its
entire length, depends on the nature of the material of which
it is composed and on its thickness. If, for example, it was
homogeneous, and of the same thickness throughout, the co-
efficient w would be constant, and if we denote the length of
the part AM before the thread is stretched by x', which length
becomes x by the effect of the tension, and if, in consequence,
dx' and dx be substituted for ds' and ds, in equation (2), we
shall have
W
dx = (1 + wT) dx'.
Likewise, if l' be the entire length of thread before it is
stretched, and p its entire weight, the weight of the part BM
will be P(x), and the value of the tension at the point м
will be
ľ
p (l' — x')
T=P+
ľ
By substituting this expression for T in the preceding equation,
and then integrating, we shall obtain, because x = 0, x = 0,
at the point a, (p)
x
x' = wpx' +
wp (2l'x' — x'²)
21'
EQUILIBRIUM OF A FLEXIBLE THread.
473
which is, therefore, the quantity by which the length of the
part AM is increased. The entire increase of length will be
obtained by making x'= l', x'= 1, this gives
1 − l' = wl' (P + 1p) ;
from which it appears, that if, in calculating the increase of
length of the thread, its own weight is taken into account,
half of this weight must be added to that which is attached to
its inferior extremity.
300. In the general case, if, after having multiplied equa-
dx dy dz
tions (1), by a'd's respectively, they be then added to-
,
gether, there will result(q)
because
dr + ɛ (xdx + xdy + zdz) = 0,
dx
ε
dx
dx², dy², dz²
+ +
ds2 ds2 ds² = 1,
dy, dy
dsd. ds + d.
2
dy+dz d.dz = 0.
ds ds ds
ds
(3)
If the thread be homogeneous and of a uniform thickness,
and if the small dilatation of its parts be neglected, the quan-
tity & will be constant; moreover, the formula xdx+xdy+zdz
is, in general, an exact differential of a function of the three
variables x, y, z, considered as independent, therefore, by
making
xác + xdy + zdz = d. p (x, y, z),
we shall have
and, consequently,
dr = ɛd. p(x, y, z),
T = εp (x, y, z),
in which, the constant arbitrary that is introduced by the inte-
gration, is supposed to be comprised in the function . This
constant will disappear in the expression for the difference of
the values of T relative to two points of the thread; it follows
3 P
474
EQUILIBRIUM OF A FLEXIBLE THREAD.
{
therefore, that it is not necessary to determine the figure
which the thread assumes in the state of equilibrium, in order
to know the increment of the tension from one point to
another; so that the tension will be known throughout the
entire length of the curve, if it be known in any one deter-
minate point of it. The curve which the thread assumes will
be determined by two of the three equations (1), or by any
two combinations of these three equations, in which the pre-
ceding value of T should be substituted; so that in general it
will be necessary to integrate the system of two differential
equations of the second order, in order to have the equation
of this curve.
-
Its radius of curvature at M, any point whatever, will be
obtained by means of the following differential formula, which
does not surpass the first order, and in which it is only assumed
that the direction of the tangent at this point is known.
Equations (1) may be replaced by the following:
dx
d.T
dx
dy-dy d.T = ε (xdy — xdx),
ds ds ds ds
dz d. r dx - dad.rdz = € (zdx
ds ds ds ds
dz
dyd.rd - dzd.
ds
xdz),
dy = ε (xdz — zdy) ;
ds ds ds
€
which, by performing the differentiations, and assuming the
arc for the independent variable, are the same as(r)
ds2
dxd³y — dyd²x = (xdy — Ydx) T
dzdx — dxdz = (zda — xdz)
eds2
(4)
T
εds²
T
dydď'z — dzď³y = (xdz — żdy)
Now, if the radius of curvature at the point м be denoted
by p, we have (No. 18)
EQUILIBRIUM OF A FLEXIBLE THREAD.
475
ρ
ds3
(dxd³y — dyd²x)² + (dzď²x − dxd²z)² + (dyďz — dzď²y)²;
which by means of the preceding equations, and by substituting
the value T, becomes
p =
p (x, y, z) ds
(xdy — xd) + (zda – xdz) + (xdz − zdy)
In the case of the catenary,
x = 0, Y = −9, z = 0, $
p = gy,
(5)
the axes and origin of the coordinates being assumed to be the
same as in equations (c) of No. 294. We shall therefore have
ds
p = y ;
dx
which it is easy to verify, by means of these equations.
301. Let these formulæ be applied to the case of a string
stretched on the surface of a solid body, and, for greater sim-
plicity, let it be assumed that it is not subjected to the action of
any given force, so that the only force which acts on its dif-
ferent points is the unknown resistance of the solid on which
it is stretched.
Let Nds be the magnitude of this force applied to ɛds the
element of the thread at any one of its points M, its three com-
ponents will be xɛds, Yɛds, zɛds; its direction will be normal
to the surface of the solid, and directed from without inwards.
The pressure on the part of the solid corresponding to ds,
will be equal and contrary to this force Nds, so that N will
express the measure of the pressure referred to the unit of
length. If λ, µ, v, denote the angles which the exterior part
of the normal at м makes with lines drawn through this point,
parallel to the axes of x, y, z, we shall have
EX= N cos λ,
εY = N COS µ,
EZ N Cos v.
Moreover, if L = 0, is the equation of the surface of the solid,
and if, for the sake of abridging, we make
476
EQUILIBRIUM OF A FLEXIBLE THREAD.
14
V =
(d12 di2 dL
(+
+
\dx²+ dy dz²
we shall likewise have (No. 21)
cos λ = v
dL
dL
dx'
cos µ = v
dL
dy'
COS v=V
dz'
a suitable sign being given to v.
have(s)
This being so, we shall
xdc + xdy + zdz = NVL = 0 ;
in consequence of which, the value of dr furnished by equation
(3) will be cypher. Hence, whatever be the form of the solid
body, the tension will be the same throughout the entire length
of the thread. Let its value be supposed to be given, and to be
represented by k, then if the thread is attached at one of its ex-
tremities to a point of the body, and if, at its other extremity,
there be suspended vertically a weight such as p, that is con-
siderable relatively to that of the thread, which may, therefore,
be neglected, this weight will be the tension k and the pressure
which the fixed point will experience. If the thread is free
at its two extremities, and if considerable weights are sus-
pended from them, they will express the extreme tensions;
consequently, they must be equal, and each of them will be
the tension k. Finally, if the two extremities are supposed
fixed, its tension k will be deduced from its extension, which
will be constant throughout its entire length.
302. If X', µ', v', denote the angles which the perpendicu-
lar to the osculatory plane at the point м, makes with parallels
to the axes of x, y, z, then the radius of curvature at this point
being p, we shall have (No. 19)
dxd³y - dyd'x
ds³
=p cos v's
dzdx
dxd2z
=
ds³
p cos µ',
dyd²z — dzď³y
=
ds³
p cos X'.
EQUILIBRIUM OF A FLEXIBLE THREAD.
477
If, therefore, equations (4) be multiplied by cos v, cos µ,
cos A, respectively, and then added together, there will re-
sult, by taking into account the values of x, y, z, in the case
which we are considering(t),
cos v cos v' + cos µ cos µ' + cos à cos λ'= 0 ;
consequently, the normals to the surface of the solid body and
to the osculating plane of the curve formed by the thread at
each point м, are perpendicular the one to the other; this
we have proved to be the characteristic property of the line,
the length of which is a minimum or maximum on a given
surface (No. 161). It follows, therefore, that a thread
stretched on a solid body, traces, in general, the shortest dis-
tance from one point to another on the surface. Strictly
speaking, it is possible that this distance may be, on the con-
trary, a maximum. Thus, for example, two given points on
a sphere are the common extremities to two arcs of great
circles, of which one is the shortest distance between these
points, and the other the plane curve which is the longest;
now, it is evident, that the equilibrium of the stretched
thread will be rigorously possible on these two arcs of the
circle, since if it be placed on one of them, there is no reason
why it should deviate from it to the one side rather than to
the other; but on the small arc the equilibrium will be stable,
and on the great only instantaneous, so that, physically, it
cannot subsist except by means of the friction of the thread
against the solid body.
Likewise, if the values of ɛX, ɛY, ɛZ, of the preceding num-
ber, be substituted in formula (5), we shall have
da cos μ)" + (cos v-
2 dx
ds
ds
N
[(dy
dx
dz
cos λ
-de cos x)"
ds
ds
dz
+
ds
COS μ
dy
2.
- de cosv)'] ==
ds
COS
because ε (x, y, z) k. We have at the same time
=
2
478
EQUILIBRIUM OF A FLEXIBLE THREAD.
dy2 dz²
+++==--
2
2 ds²
1,
cos λ²+ cos µ²+ cos v² = 1;
and, as the normal to the surface of the body, and the tangent
to the curve of the thread, at each point м, are perpendicular
the one to the other, we have also
dz
dx
dy
cosλ +
cos μ + cos v = 0;
ds
ds
ds
but, it is easy to shew by means of these three last equations,
that the coefficient of N in the preceding expression is unity,
hence we have simply(u)
N=
P
from which it is evident, that the pressure, referred to the unit
of length, exerted by a thread stretched on the surface of a
solid body, is, at each point of the curve, equal to the tension
divided by the radius of curvature of the thread, that is to say,
by the radius of the section which is normal to the surface,
and to the tangent to the curve which the thread assumes.
303. These results will be modified by the friction of the
thread against the surface of the body on which it is stretched.
In order to show how this force should be taken into account
in the equilibrium of a flexible thread, we proceed to consider
the equilibrium of a cord ABMCD (fig. 76), whose part BMC is
applied to the throat of a fixed pulley, and which is drawn in
the directions of BA and CD, the productions of this part, by
given forces. Let the pulley and the line AB be supposed to
be vertical, and let the force acting in the direction of BA be
represented by the weight k, and that in the direction of CD by
F, it is evident that k and F denote the tensions at the points
в and c, in the directions of the tangents BA and CD. If, in like
manner, in order to simplify the question, the pulley is supposed
to be circular, its radius to be denoted by c, and the centre
EQUILIBRIUM OF A FLEXIBLE THREAD.
479
0, to be the origin of the coordinates; the axis of z will be per-
pendicular to the pulley, the axis of y vertical and directed up-
wards, and the axis of a horizontal and passing through the
point B. Finally, let c be the origin of the arc s, terminating
at м, any point of the cord, so that CM = s.
This being premised, if there is no friction, it is necessary
that in the case of equilibrium we should have k = F, but, in
consequence of the friction, the equilibrium may subsist as
long as the difference between these two forces k and F does
not pass a certain limit. Let us suppose, therefore, that the
equilibrium is on the point of giving way in the direction of
the weight k, this implies that k>F. At this instant, the
friction of the cord against the pulley, which has place at any
point м, will act in the direction of the part мH of the tan-
gent at this point. If its intensity be denoted by μ, and if,
as before, the normal resistance which has place at the point
M in the direction of м'o, the production of мo, be denoted
by N, then uds and Nds will be the tangential and normal
forces which act on eds, the element of the cord terminating
at the point M, and μ and N represent these forces, referred to
the unit of length. If through this point м, Mx' and мy' be
drawn parallel to the axes or and oy, we shall have
hence we infer,
COS x'MH ==",
y
cos y'MH = 2,
C
cos a'MO' = 2, cos y'MO' =
2
C
εX=
C
NX му
C
εY =
Ny + Mat;
C
C
for the values of ex and EY, which should be substituted in
equations (1). The force ez will be evidently equal to cypher;
the third equation (1) will disappear, and the two first will
become
480
EQUILIBRIUM OF A FLEXIBLE THREAD.
dx Nxds
d.T
μyds
ds
+
= 0,
C
C
d.T +
ds
dy, Nyds
Nyds, µxds
+
= 0.
C
C
As the point м belongs to the circumference of the pulley,
we have
tỷ mơ, xdx + ydy=0;
by means of which the two preceding equations may be changed
into the following(v)
dx
xd.T +yd.T
ds
dy
+Ncds = 0,
ds
dy d.T
dy
(yde — xdy) = 0.
} «
(6)
dx dx
d.T +
ds ds ds ds
But (ydx-xdy) is the differential of the sector described
by the radius om reckoning from a fixed line (No. 156), oc
for example. This sector being circular and corresponding to
the arc s, its value will be cs; hence we have
ydx − xdy = cds.
Moreover, we have likewise
dx
dy
dx
X
ds
+y
= 0, xd.
ds
ds
+yd.dy
dx2 dy2
+
ds2 ds2
dx dx dy, dy
= 1, die d. + d. = 0;
ds
this reduces equations (6) to(x),
hence we obtain
ds ds ds
T = CN, dT = µds;
cdNµds.
ds,
As the pressure at the point м, on the throat of the pulley,
is equal and contrary to the force N, if the friction be propor-
tional to the pressure (No. 269), we shall have
µ =ƒÑ;
EQUILIBRIUM OF A FLEXIBLE THREAD.
481
ƒ being a coefficient which will depend on the nature of the
surfaces in contact. Consequently, we shall have
and, by integrating,
cdN=fnds,
fs
N = Aec;
A being a constant arbitrary, and e the base of the Naperian
system of logarithms. We shall also have at the same time,
f8
T = Aceº,
με Afer.
At the point c, s = 0 and T = F, hence we have
F
A =
C
and denoting the length of the arc CMB by l, we shall have
s=l and T =k, at its other extremity B.
shall have finally at м, any point whatever,
Therefore, we
N = ec,
F js
C
fs
T=Fec,
fr
μπ e
C
and moreover, the equation of equilibrium will be
fl
k = Fec.
Denoting the total friction through the entire length of
CMB by F', we shall have
r'=S?
fl
= So μds = x (e." — 1),
F
and the equation of equilibrium may be written thus,
k=F+F'.
By assuming
fi
ec
1=ƒ',
we shall have
r'=ƒ'e, ƒ=k−1;
F
3 Q
482
EQUILIBRIUM OF A FLEXIBLE THREAD.
from which it appears, that the total friction F' is equal to the
least of the two forces k and F, multiplied by a coefficient f',
which varies not only with the quantity f, but also with 7, the
extent of the surface in contact, and c the radius of the pulley.
The difference of the forces k and F, at the instant the equi-
librium gives way, will make known the value of F', and their
ratio, diminished by unity, will be the value of the coefficient ƒ“,
from which it is easy to deduce that off. When F is a weight
as well as k, we should, for greater accuracy, include as be-
longing to these weights, those of BA and CD, the vertical
parts of the string.
304. It is easy, by means of the three equations (1), to shew
that the six general equations of equilibrium (No. 261) obtain
in the case of a perfectly flexible thread.
For this purpose, let к and к' be the two extremities of
the thread, and 7 its length; also let the origin of the arc s be
fixed at the point K. By integrating the first members of
equations (1) from the point к to the point K', we shall have
dx
(T) [T+ Sxeds = 0,
- da]
ds
dy
T
ds
ds
dy
T
(d) - [d] + So
ds
dz
(1 dz) - [ 1 de ] + So
T
ds
T
ds
Yɛds = 0,
Zɛds = 0;
the quantities comprised within the crotchets refer to the
point K, and those which are contained within the parentheses,
to the point K'. Besides the forces X, Y, Z, which act through
the entire length of the thread, particular forces given in mag-
nitude and direction, may be supposed to be applied at its two
extremities; let k be that which acts at the point K, and a, ß, y,
the angles which its direction makes with lines drawn through
this point, parallel to the axes of x, y, z; and let k', a', B', y'
be the corresponding quantities relatively to the point K'.
These forces k and k' will be the extreme tensions, in magni-
EQUILIBRIUM OF A FLEXIBLE THREAD.
483
tude and direction; and as their directions must coincide with
parts of the tangents to the curve, which the rod assumes at
K and K', we shall have
dy
dz
[rd] = -k cosa, [ry] = -k cos B, [1] =―k cosy,
T
ds
dx
Τ
ds
(Tdz) = X' cos a', (rdy)
ds
ds
= k' cos B', (rdz) = k' cos y';
ds
consequently, the preceding equations will become
k cos a + k' cos a' +
h' cos a' + So
xɛds = 0,
kcosß + k'cosß'+So xeds = 0,
·(7)
(8)
kcosy + k'cosy' + Sc
Zɛds = 0;
and it is evident, that they express the conditions of equilibrium
contained in the three first equations (1) of No. 261.
As the
following equations
dy
dx
xd.T
ds
ds
yd.T =d.
= d. x (x
dy
dx
y
ds
Y as
ds
dx
dz
zd.T
ds
ds
xd.T =d.T
d.T z
r (2
dx
ds
-x de),
dz
ds
dz
yd.T ás
zd.r dy
ds
ds
=d. Tyds
r (y
dz
2
ds
are evidently identical; by equations (1) of No. 298 we shall
have
dx
d. r (zdy da) + (xx - yx) eds = 0,
(x
ds ds
d.T(2
d. x ( z dz
A) L.
ds
dz
dz
x—4) + (2x − xz) eds = 0,
ds
dy) + (yz — zx) eds = 0.
d. T (yd/ -214)
Hcnee, if the first members of these respective equations be
484
EQUILIBRIUM OF A Flexible THREAD.
integrated from the point к to the point K', and if a, b, c de-
note the values of x, y, z, relative to K, and a', b', c', those
which refer to K', we shall obtain by means of equations (7),
k(acosß—bcosa)+1′(a'cosß'—b'cosa)+So (xx—yx)ɛds=0,|
k(ccosa—acosy)+k’(c'cosa′—a'cosy')+So(zx—xz)eds=0, } (9)
k(bcosy—ccos(B)+k'(b'cosy'—c'cosß')+So (yz—2x)eds=0,
these express the conditions of equilibrium relative to the
moments of the given forces, which are contained in the three
last equations (1) of No. 261.
305. Equations (8) and (9) will, in general, enable us to
determine the coordinates a, b, c, a', b', c', of the two extreme
points K and K'; nevertheless, there are cases in which some of
these quantities must remain indeterminate. If, for example,
the given forces which act on the thread are the force of gra-
vity and other forces that are independent of the coordinates
of their points of application, it is evident that the absolute
position of the thread in space cannot be determined, hence the
three coordinates of one of the points K and K' may be arbi-
trarily selected. Equations (9) will determine the three coor-
dinates of the other point; and, in order that the equilibrium
may be possible, it is necessary that the given forces should
satisfy equations (8). If one of the points K and K' be fixed,
the first, for instance, equations (8) and (9) will still obtain,
provided that the force k is considered to be unknown, in mag-
nitude and direction, and as representing the pressure which
the point K will have to sustain. In this case, the values of
a, b, c will be given; equations (9) will determine those of
d', b', c', and equations (8) will make known the three compo-
nents of the force k. When the two points K and K' are fixed
and given in position, their coordinates are determined, and
equations (8) and (9) will enable us to determine k and k', the
pressures exerted on K and K', in magnitude and direction.
EQUILIBRIUM OF AN ELASTIC ROD.
485
In all cases, whether the coordinates of K and K' are given,
or whether they have been deduced from equations (8) and
(9), the curve formed by the thread must pass through these
two points; this will enable us to determine the four constant
arbitraries which the complete integrals of these two diffe-
rential equations of the second order contain. With respect
to the constant arbitrary which the function of No. 300
contains, its value can be obtained from the given length of
the thread, that is to say, from the equation
Si++
dz2
dx = 1,
dx2
in which y and z are regarded as functions of x. By this
means, the problem will be completely solved.
III. Equilibrium of an Elastic Rod.
306. By this term we understand a straight or curved rod,
the figure of which cannot be changed, without applying one or
more forces to it, while at the same time it resumes its natural
form when these forces cease to act; on the contrary, a per-
fectly flexible thread retains, without the aid of any force, the
curvature which it has been made to assume, and is elastic
only in the direction of its length. In order that a rod may
be elastic with respect to its flexion, the material of which it
is composed should be very little extensible and contractible;
but this is not solely sufficient; it is likewise necessary, that
the dimensions of its thickness, although very small rela-
tively to its length, should nevertheless be of a determinate
magnitude; for, of whatever material the rod consists, its
thickness may be always so diminished, as that it cannot have
any sensible tendency to resume the figure from which it has
been deflected, by which means it may be reduced to the state
of a perfectly flexible thread.
486
EQUILIBRIUM OF AN ELASTIC ROD.
When an elastic rod is deflected from its natural form by
given forces, each of the longitudinal filaments of which it
consists may experience three different effects; each part (the
length of which may be as small as we please) may be con-
tracted or dilated, its natural curvature may be either in-
creased or diminished, and this part may have been twisted on
itself. The tendency of each part to resume its natural state,
depends on the mutual attractions and repulsions which obtain
between the molecules of all bodies, and which extend only to
insensible distances. It is the province of mathematical phy-
sics to compute the total effect of all the forces which result
from them, and which must constitute an equilibrium with the
given forces. For information on this subject, the reader is
referred to a memoir by the author on the Equilibrium and
Motion of Elastic Bodies, which is inserted in the Mémoires
de l'Académie des Sciences, tome viii. In the present treatise,
the equations of the equilibrium of an elastic rod will be
formed, by setting-out from such secondary principles as are
generally admitted.
By an elastic plate is understood a rectangular parallelo-
piped of small thickness, which is bent in the direction of its
length, so that it is always comprised between two cylindrical
surfaces, whose heights are equal to its breadth. This di-
mension may be of any magnitude whatever; by dividing it
by planes very near to each other, and perpendicular to its
direction, the plate will be distributed into rectangular elastic
rods. James Bernoulli was the first who determined the
figure of an elastic plate in equilibrio, from considerations that
we proceed to develope, and by means of which we shall then
be enabled to solve the problem completely, in the case of any
elastic rod whatever.
307. Let us consider an elastic plate made fast at one of
its extremities, that is to say, fixed in such a manner that one
of the two small rectangles which terminate it perpendicularly
to its length, cannot be moved in any direction. If it be supposed
EQUILIBRIUM OF AN ELASTIC Rod.
487
to be bent in the direction of its length by means of a force
applied to its other extremity, and which is the only one that
acts on the plate, in order that the plane may assume a cylin-
drical form, as has been stated, it must be terminated at its
free extremity by an inflexible rectangle, to the middle of
which the given force is applied in a plane perpendicular to
the breadth of the plate. All the longitudinal sections, that
is, those which are perpendicular to this breadth, will be
equal; that which contains the direction of the given force is
represented by fig.77; and the curves AMB and A'м'B' are sec-
tions of the two cylindrical surfaces of the plate, which its two
faces form in its natural state. All the points which, in this
state, appertain to the common perpendicular to these two
faces, are still supposed, after the plate has been bent, to exist
on the common normal to the two cylindrical surfaces, which,
in fact, agrees with what has been observed to take place in
its change of figure. Hence it follows, that if MM' is normal
to the curve AMB, it will be so likewise to A'M'B', and will
contain all those points of the plate, which, previously to the
bending of the plate, existed on a common perpendicular to
its two faces; it appears also, that if the plate, in its natural
state, is divided into longitudinal filaments, and if the curve
CND represents what one of those filaments becomes after the
change of figure, it will intersect the normal Mм' at right
angles in N.
If m be a point of the curve AMB, infinitely near to M ;
and if mnm' be drawn normal to the three lines AMB, CND,
A'M'B', and intersecting them in m, n, m', respectively, the
productions of MNM' and m n m' will meet in a point o, which
will be the common centre of curvature of these three curves.
Let p denote the radius of curvature of the mean filament, that
is, of the filament which is equally distant from AMB and
A'M'B', σ the part of this filament contained between the two
normals MNM' and mnm'; u the distance of any filament what-
ever, such as CND, from the mean filament, and o' the length
ཝཱ
488
EQUILIBRIUM OF AN ELASTIC ROD.
of Nn; u is to be considered as positive or negative, according
as CND exists, relatively to the mean filament, nearer to AMB,
the convexity of the plate, or on the side of its concavity
A'M'B', NO the radius of curvature of CND will be equal to
p+u, and the infinitely small lengths o' and σ will be to each
other as p + u to p, so that we shall have
o'=o+
σ
υσ
P
In consequence of the bending of the plate, the longitudinal
filaments undergo small contractions or dilatations, which will
destroy the equality that previously existed between ☛ and ☛'.
If their primitive magnitude be denoted by y, we shall have
。 = y (1+8), o' = y (1 + 8');
σ
in which 8 and 8' are very small fractions, positive or negative,
according as the mean filament and the filament CND are
W
lengthened or contracted. The fraction is likewise
P
posed to be very small; hence if the product of 8 and
neglected, we shall have
sup-
И
be
ρ
8=8+
018
;
which shews that, when the length of the mean filament is not
changed, the filaments situated on the side of its convexity
are all lengthened, and the filaments situated on the side of the
concavity are all contracted, and each proportionably to their
respective distances from the mean filament (a).
This being established, let the form of each of the two
parts of the plate, which correspond to AMM'A' and вmm'B', be
rendered invariable, and, for the sake of abridging, let them be
denominated by H and K. The part н will be immoveable,
H
the part K will be drawn towards H, or repelled from it, by the
tendency of the intermediate part мmm'm' to resume its natural
EQUILIBRIUM OF AN ELASTIC rod.
489
as y·
state, and become again a section of a constant thickness such
The filament Nn of this slice will tend to contract or
dilate itself according as it shall have been lengthened or short-
ened, that is to say, according as the quantity & will be posi-
tive or negative. Consequently, the part K will be drawn in
the first case, and pushed in the second case, by a force applied
to the point n; but as this force arises from the action of Nn,
we may suppose it to be proportional to the quantity d´and
normal to mnm', as if this filament Nn was detached. By
adopting this hypothesis, the force in question referred to
the unit of surface, may be represented by ad, and, conse-
quently, the normal force exercised on the transversal element
of the surface K, which corresponds to the point n, may be
represented by adλdu; a being a constant depending on the
material of which the plate consists, λ its breadth, and Adu
the area of this element. Hence denoting the thickness of
the plate by 2ɛ, and representing the entire force which draws
or pushes K, according as it is positive or negative, by T, we
shall have
T=
= aλ S_&& du;
E
ε
S+ u
sie
Τ = 2αλεδ.
ེ་
area strain & E
and, by substituting for &' its value,
Moreover, if μ be the moment of the forces normal to the
surface of K, taken with respect to a transversal axis equally
distant from the two faces of the plate, we shall likewise have
and, consequently (b),
=
με αλ
ad S__& d'udu ;
2 αλε
μ
3p
:
Hence it appears, 1st, that the force T, which tends to con-
tract or dilate any slice of the plate, is proportional to the
3 R
490
EQUILIBRIUM OF AN ELASTIC Rod.
positive or negative extension of the mean filament, and inde-
pendent of its curvature; 2ndly, that its moment µ is, on the
contrary, independent of this extension, and in the inverse
ratio of its radius of curvature; 3rdly, that, the material and
breadth of the plate remaining the same, the value of r is
proportional to its thickness, and that of μ, to the cube of this
dimension.
T
When the length of the mean filament does not undergo any
change, we have 80, T=0; the parallel forces which draw
or push K are reduced to two, equal and contrary, but not
directly opposed, the moment of which, relative to the trans-
versal axis perpendicular to these forces, is always equal to µ.
This quantity μ is what is termed the moment of elasticity; in
each point, it is proportional to the curvature of the plate, or
to the angle of contact of its mean filament (c).
308. It is easy now to obtain the equations of the equi-
librium of this plate. In the first place, if T' be what the
force T becomes at the point м, it is evident that the infinitely
small slice which corresponds to мmm'м', will be drawn or
pushed on one side by this force r', and on the other by a
force equal and contrary to T ; and since by hypothesis, no given
force is supposed to act on this slice, we must have T=T'.
Hence it appears that the force T is constant throughout the
entire length of the plate, and, consequently, it is equal to the
given force which acts at its free extremity, resolved in the di-
rection of this length. In like manner, the dilatation & will
be constant, proportional to this force, and positive or negative
according as this force tends to lengthen or contract the longi-
tudinal filaments. It will not influence the figure of the plate;
but by measuring it, it will enable us to determine the value
of the constant a, with reference to the material of which the
plate consists. If we suppose that w denotes a weight equiva-
lent to the force which draws the plate in the direction of its
length, and that w represents the area of each transversal sec-
tion of the plate, we shall have
EQUILIBRIUM OF AN ELASTIC ROD.
491
ως 2λε, τ= w = αωδ,
พ
α=
wd
In order to determine the figure of the plate, let there be
drawn through the point a, in the plane of the mean filament,
two rectangular axes ax and Ay, of which let the first be a
tangent to the curve AMB, and drawn in the direction of the
plate in its natural state, and let the second be directed from
the side of its concavity. Let x and y be the coordinates of
any point of the mean filament, referred to these two axes,
a and b those of its free extremity, which is supposed to be
the point of application of the given force that retains the
plate in equilibrio; P and Q the components of this force in
the direction of the productions of a and b. Let there be
drawn through the point indicated by x and y, an axis per-
pendicular to the plane of the figure to which the moment
denoted by μ refers, and let there be made a section perpen-
dicular to the mean filament. In order that the part of the
plate contained between this section and its free extremity
may be in equilibrio, it is necessary that the sum of the mo-
and the moments of P and Q with respect to the same
axis, should be equal to zero, regard being had to the direction
in which the forces, of which u is the moment, and the forces
P and Q, tend to make this part of the plate to turn; in this
manner we shall have
ment
µ + P(b− y) — Q (α — x) = 0.
By assuming the abscissa x for the independent variable,
and observing that the plate is convex towards the axis Ax, we
shall have
1
day
!= d²x² + (1+dx)²
P
÷
in which the radical is considered to be positive. If, there-
fore, this value be substituted in that of μ, and the resulting
expression, in the preceding equation, and if, in order to abridge,
we make
•
1
492
EQUILIBRIUM OF AN ELASTIC ROD.
3 αλε = β,
we shall have
Ᏸ
| dy
dx²
= [q(a−x) — p (b—y)] (1+
P
dy),
,
(1)
for the equation of the curve formed by the elastic plate in
equilibrio.
Its integral will contain two constant arbitraries, which
are determined by the conditions y = e and dy
dy
dx
=0, when
x=0, or, if we prefer it, y=0 and 0, for this value
dx
of x, in consequence of the extreme smallness of ε. If then x be
made equal to a, and y equal to b in this integral, an equation
between a and b will be obtained, and this combined with that,
which is furnished by the given length of the plate, gives the
two equations that are necessary to determine the unknown
coordinates a and b ; and the elastic curve, properly so called,
will be completely determined.
309. If the plate, instead of being fixed, is entirely free at
its extremity a, then in order to maintain it in equilibrio, it is
necessary to apply to this extremity a force, the components
of which are equal and contrary to P and Q; by taking the
corresponding extremity of the mean filament for its point of
application, it is, moreover, necessary that the resultant of P
and a should pass through this point; in order that this should
take place, we must have
Q
Qa − P (b − ε).
This equation will be sufficient, when the plate is retained by
a fixed axis, passing through this extremity of the mean fila-
ment, and drawn in the direction of its breadth. If it is
merely placed on a plane perpendicular to its length, which
does not prevent it from turning about the edge of one of its
two faces, it is necessary that the friction of this edge against
the plane, or some other force, should prevent the plate from
EQUILIBRIUM OF AN ELASTIC ROD.
493
sliding. When the plate is not firmly fixed, the direction of its
plane which is a tangent to the curve at the point a, will not
be known; if, however, the origin of the coordinates x and y,
be placed at this point, we shall still have y = ɛ, or y = 0,
when x = 0, but we can no longer take the axis of x on the
tangent at a, since its direction is not known a priori. This
axis will then be the given direction of the force P, and in the
determination of the constant arbitraries, the equation
dy
dx
= 0,
when x = 0, must be replaced by the preceding equation rela-
tive to the moments of the forces P and Q, which can be re-
duced to qa = rb.
310. If P be supposed equal to cypher, the plate will be
bent by a force a perpendicular to its primitive direction;
which is, for example, the case of a horizontal plate, one end
of which is fixed, a given weight a being attached to the
other.
If in this case we make
B = c² Q ;
c being a line, the given length of which is generally very
great, unless the weight a is also very considerable, equa-
tion (1) will become
C2
d³y
dx2
(1+ dy³) ::
dx²
α x;
(2)
and by integrating it so that we may x=0 when x=0, we
dy
dx
shall have
dy
2c2
=
dy2
dx
1+
2ax — x².
dx2
Hence we obtain
dy=
ds=
(2ax — x²) dx
√ 4c¹ — (2 ax — x²)²²
2 c2dx
√ 4 c¹ — (2 ax − x²)²
;
494
EQUILIBRIUM OF AN ELASTIC Rod.
ds being the element of the curve. These formulæ may be
exactly integrated by means of elliptic functions; but in con-
sequence of the magnitude of c, we have s = x very nearly,
and the value of dy may be reduced to
1
dy=
2c² (2αx
(2ax - x²) dx;
hence we obtain
=
6 c²y 3 ax²- x³,
for the equation of the curve.
If the plate deviates very little from the horizontal direc-
tion, the abscissa a may be taken for its length, and the ordi-
nate b will express its greatest deviation. Because
3 Qc² = awε²,
αωε,
if, as before, we make 2ɛλ = w, we shall have
awε³b = a³Q,
when xa and y = b. Hence it follows, that the nature of
the plate remaining the same, the quantity b, by which it will
be deflected, will be proportional to the weight a and to the
cube of the length a, and in the inverse ratio of the square
of its thickness & and of the area of w its transversal section.
W
W
If for aw, its value given in No. 308, be substituted,
8'
and if h denotes the total lengthening ad of the plate, produced
by a weight w, we shall have
b:
ha² Q
b =
If w be supposed equal to Q, it follows from this that when
the same weight Q, applied to the free extremity of an elastic
plate, acts successively in the direction of its length and per-
pendicularly to its length, the extension h and flexion b, which
EQUILIBRIUM OF AN ELASTIC ROD.
495
79
are supposed to be very small in respect to the length a, will
be as the squares of the thickness and of this length(e).
311. Whatever be the nature of the forces P and Q, a first
integral of equation (1) may be always obtained, by reducing
it to the form of equation (2), by the transformation of co-
ordinates. We shall restrict ourselves to the consideration of
the case, in which the plate being pressed against a plane, and
not fixed in it, deviates little from its natural form. This
will be, for example, the case of a spring, the lower extremity
of which is laid on a horizontal plane, its upper extremity
being at the same time loaded with a given weight. Let us
suppose that in bending under this load, the spring deviates
very little from the vertical AB, and that throughout its entire
length, the tangent to the curve, which it forms in its state of
equilibrium, makes a very small angle with this right line.
Figure 78 represents the different forms which it can assume
in this state.
dy
dx
Let there be taken for the axes of x and y, the vertical ax
drawn in a direction opposite to that of gravity, and the hori-
zontal line Ay. The quantity being very small by hypo-
thesis, its square may be neglected in equation (1); we shall
likewise have Q-0, since the force which acts at the extremity
B is vertical; in virtue of the equation Qarb of No. 309, it
follows that b = 0; and as the weight P acts in the direction
from в towards a, the sign of this force must be changed in
equation (1), in which it is supposed to act in the contrary di-
rection. This equation will thus become simply
dzy
c² = — π³y,
dx2
by making, in order to abridge,
B = fawe² = = = P.
In this equation w denotes the area of the section of the
496
EQUILIBRIUM OF AN ELASTIC ROD.
spring perpendicular to its length, & its semi-thickness, in
the direction in which it is bent, and a a quantity depending
on the material of which it is composed.
These three quan-
tities are supposed to be constant, and, consequently, c is a
line of a constant and given length.
Since y = 0, when x = 0, we obtain from this equation
y= k sin
TTX dy πk
dx C
TX
COS
:
C
k being a constant arbitrary, which must be either cypher, or
very small relatively to c.
When k = 0, the spring will remain straight, and its
length AB will be a little diminished by the weight P. When
this coefficient k does not vanish, the spring will bend; at
the point B, we shall have xa and yb0; therefore, i
denoting any whole number, it is necessary we should have(ƒ)
a = ic
for the value of a or AB. Naming 7 the length of the spring,
we shall likewise have
b = Sa
Sa√ 1+ dye
dx²
dx = Sa
·Sa√1+
T2k2 TX
cos2 dx;
C²
C
k
which, by neglecting the fourth power of, and substituting
its value for a, becomes(g)
from which we obtain
1=ic (1+k²)
ic (1 + 7743)
4c2
с
k = 2c √ ?
1..
π
ic
(3)
Thus then, the coefficient k will either vanish, or be ex-
pressed by this formula.
312. The following remarkable consequences follow from
this result:
EQUILIBRIUM OF AN ELASTIC ROD.
497
}
1st. As long as l is less than c, formula (3) will be ima-
ginary for every value of the whole number i; the coefficient k
cannot then be different from cypher, and the spring will not
be bent by the weight P.
2ndly. When by increasing the length of the spring, or
by diminishing the quantity c, (which last is effected by in-
creasing the weight P,) 7 is made to surpass c; the value of
k, which is different from cypher, and which corresponds to
i-1, will be real, and the spring can be bent by this weight.
If ƒ denotes a very small fraction, and if we make
we shall have
71″
2
1= c(1 + 1),
4
i=1, a=c, k=fɑ;
and the equation of the curve of the spring will be con-
sequently
TX
y=fa sin
a
from which it is evident, that it does not intersect the curve
between the two points A and B.
l
3rdly. The ratio still continuing to increase, if it sur-
C
passes 2, the value of k which corresponds to i2 will be
real, and the spring can assume a figure different from the
preceding. For if f' denotes a very small fraction, and if we
make
we shall have
l = 2c (1 + π²ƒ¹²),
i=2, a = 2c, k=f'a;
hence there will result,
2πα
y=f'a sin
;
a
from which it is evident, that in this case, the curve will intersect
3 s
498
EQUILIBRIUM OF AN ELASTIC ROD.
the vertical at the middle point of AB, which corresponds to
x = a.
4thly. By continuing in this manner, it may be proved,
that if I surpasses ic by a small quantity, and if, ø denoting a
very small fraction, we have
l=ic (1+²²²p³),
4
we can assume
a=ic, k = pa;
by means of which we obtain
ίπα
y= pa sin
;
α
which is the equation of a curve that will intersect AB in the
number i + 1 of equidistant points, A and B being reckoned
in this number(h).
When 7 surpasses a multiple of c, by a quantity which is
not very small, the value of k, furnished by formula (3),
ceases to be very small relatively to c; and as the value of
dy
is then no longer a very small fraction, the figure of the
dx
spring cannot be determined by the preceding analysis. It
should be observed, that in all cases, the rectilineal figure,
which corresponds to k=0, is possible; but it is not necessary
and stable except in the case of l≤ c.
313. By the force of a spring, (which, for greater clear-
ness, we suppose to be vertical,) is understood the greatest
weight which it can support without bending. This weight
P is determined by the equation cl, which gives(i)
P=
παωε
312
2
from which it appears, that, every thing else being the same,
the force of a spring is in the inverse ratio of the square of
its length. It likewise appears, that when the spring is a
rectangular parallellopiped, if its adjacent faces be bent, its
EQUILIBRIUM OF AN ELASTIC rod.
499
force will be proportional to the square of the thickness which
is perpendicular to the face that is bent. With respect to the
absolute magnitude of P, it may be computed by substituting
in the preceding formula the value of a, which may be de-
duced either from h the extension of this spring, or from its
flexion 7, produced by a weight w; now, it appears from Nos.
308 and 310, and because ad = h and a = 1, that these values
are
a =
wl
α=
a
wh'
consequently, we shall have
2
wl³
ωε
T2wl
P=
P =
,
3lh
зь
314. The results of No. 307 may be easily extended to an
elastic rod, when it is supposed to be straight, or of a single
curvature in its natural state, and that, when it is bent, it
continues to be of single curvature, and does not experience
any torsion.
In this case, the mean filament will be assumed to be that
which passes through the centres of gravity of all sections
perpendicular to its length, which may be either constant, or
variable, provided that in each point their dimensions are very
small, with respect to the radius of curvature of the rod. Let
w be the area of one of these sections, made through any point
whatever of the mean filament; if it be divided into elements
perpendicular to the plane of this filament, and if vdu be the
area of the element that is at the distance u from this same fila-
ment; in this expression the variable u may be either positive
or negative, and v denotes a given function of u. Likewise,
if k and k' denote the extreme values of u, we shall have
Sk
vdu = w,
- k'
Sk ki
vudu = 0
the second equation obtains because the origin of the variable u
is at the centre of gravity of w.
500
EQUILIBRIUM OF AN ELASTIC ROD.
Let o, o', 8, 8, p denote the same quantities as in No. 307,
and let y, y', r, be what σ, o', p, were in the natural state of
the elastic rod; we shall have for the two states of this rod,
UY
υσ
y=x+"/7/1, 0=0+ P
and, for the passage from the one to the other,
o=y(1+8), o'=y'′(1+d').
Su би
we shall ob-
go
ρ
Hence by neglecting the products and
tain(k)
8 _ 8 + u ( + - ) ;
a value that coincides with that of the number cited, in the
case in which the rod is naturally straight, when we have
r=x.
μ
Moreover, let T be the sum of the forces perpendicular to
ως which draw or push one of the two parts of the rod that are
separated by this normal section. Naming u the moment of
these forces with respect to the axis passing through the centre
of gravity of w, and perpendicular to the plane of the mean
filament; we shall have by the hypothesis of No. 307,
Sk
ΤΞα
a Sh1, &'vdu,
μπα
=aSk₂, &'vudu;
-k'
a being a quantity dependent on the material of the rod,
which is supposed to be constant through the extent of each
section w,
but which may vary from one point to another of
the mean filament. By substituting for &' its preceding value,
and making, in order to abridge(7)
k
S½
vu²du=}wq²;
-k'
there will result
Τπαωδ, μπ
T=awd,
awq²
3
•
1
EQUILIBRIUM OF AN ELASTIC ROD.
501
!
When the elastic rod is a curve of double curvature in its
natural state, or after its change of figure, the expression for
the force T will be still the same; moreover, the mean fila-
ment being always that which passes through the centres of
gravity of all the normal sections, and r, p, denoting the radii
of curvature in the same point, before and after this change,
this value of μ may be taken to express the moment of elas-
ticity with respect to an axis passing through this point, and
perpendicular to the osculating plane of the mean filament;
but it will be necessary, besides, to take into account the tor-
`sion of the rod, as will be done immediately.
315. It appears from a comparison of this value of µ with
μ
that given in 307, that the second differential equation of
the plane curve formed by the mean filament of an elastic rod
which does not experience any torsion, differs from that which
refers to the elastic plate properly so called, only in this, that
1
1
1
it will contain in place of, and the quantity q instead
P
ρ
of the semi-thickness ε. If the rod is homogeneous, and if, in
its natural state it is either a prism or a cylinder of a small dia-
meter, the three quantities a, w, q, will be constant, and we
shall have r = ∞. Hence it appears, that the flexion of a rod
which is naturally straight, produced by a weight perpen-
dicular to its direction, and the force of this spring, may be
deduced from the values of b and P found in Nos. 310 and 313,
by substituting q in place of ɛ. By means of this substitution,
we shall have, (7 being the length of this rod,)
b
1³Q
awq2,
P
π²awq2
312
or, what comes to the same thing,
Shy vu³du.
T²lQ
πα
b
P
3P
12
-Sk
- k
In the case of two different rods, having the same length,
502
EQUILIBRIUM OF AN ELASTIC ROD.
ī
the flexions produced by the same weight will be in the in-
verse ratio of the forces of the spring, so that we will be en-
abled to compare together the magnitudes of these forces, in
the different hypotheses made respecting the contour or out-
line of the normal section.
Let, for example, the normal section be an isoceles triangle,
and let it be proposed to bend the rod so that the face cor-
responding to the base of this triangle might become a cylin-
drical surface, either convex or concave. Let a and c denote
the base and height of this triangle. When the surface is that
of a convex cylinder, towards which the positive values of u
(No. 307) are directed, we shall have
α
v = 2 (fc+u),
k=}c, k'=zc, v=
C
and there will result from this
P=
T
3613
In the case of the concavity, we shall have
k=fc, k'=fc, v=2(}c+u) ;
C
hence we obtain(m)
2
Π
P=
12♫²;
which shews, that in this second case, the force of the spring
is triple of what it is in the first.
If the normal section is a square represented by Ļ, and if
it is proposed to bend the spring in such a manner that two of
its opposite faces may become cylindrical surfaces, we shall
have
2
k =k'=\ƒ, v=ƒ, P=
π³aƒ
12/2
If it is a circle, the radius of which is k, we shall have
k'=k, v=2√ k² — u²,
u², P=
T ³ak4
412;
and, if the area of the normal section be supposed to be equal
EQUILIBRIUM OF AN ELASTIC ROD.
503
{
in this and the preceding case, so that we may have ƒ²—ñk²,
it is evident that the force of the spring in the first case, is to
the force of the spring in the second, in the ratio of π to 3(n).
Let the cylindrical spring be supposed to be a hollow tube,
of which the radii of the interior and exterior concentrical sur-
faces are g and g'. In order to obtain the force of the spring,
and g′ ought to be successively substituted in place of k in
the last value of P, and then the results should be taken, the
one from the other, this gives
g
P=
12
3
ñ³a (g¹² + g²) (g/² — g²)
4/2
If the area π (9'2 — g²) of the normal section is equal to
πk², we shall have(o)
3
π³ak² (k² + 2g²)
P =
472
hence it follows, that the volume, the length and the material
being the same, the force of a hollow spring is greater than
292
that of a full spring in the ratio of 1
1+ k2 to unity; 2g being
the interior diameter, and k² the area of the normal section.
316. Let us now proceed to form the equations of equili-
brium of any elastic rod whatosever, all whose points are soli-
cited by given forces.
Let A and B be the two extremities of the mean filament,
x, y, z the three rectangular coordinates of any point м of this
curve, s the arc AM, w the normal section of the rod made
through the point м, y its density at this point, and conse-
quently, ywds the mass of the infinitely slender slice of the
rod. If x, y, z, be the forces referred to the unit of mass,
parallel to the axes of x, y, z, then xywds, Yywds, zywds, will
be the given forces which act on this mass. The sum of these
forces resolved in the direction of the tangent to the mean fila-
ment at the point м, and tending to increase the arc s, will be
504
EQUILIBRIUM OF AN ELASTIC ROD.
(x dx + x dy + z dz) ywds.
ds
ds ds
Likewise, if T denotes the force arising from the action of
one part of the rod on the adjacent part, applied to one of the
faces of the slice ywds, perpendicular to w, and tending to
increase or diminish the arc s, according as it is positive or
negative, the other face of ywds will be drawn or pushed in a
contrary direction by a force equal to T + dr; consequently,
in order that this slice may be in equilibrio, the force dr
must be equal and contrary to the given tangential force, hence
we must have(p)
di+yo (xd + xdy + zdz)=0;
which agrees with equations (3) of No. 300.
W
1
(a)
As the material of which the rod consists is very little ex-
tensible, y and in this equation (a) may be assumed to be
the density, and the normal section of the rod at the point м,
in its natural state. If these two quantities are constant, and
if the formula comprised between the parentheses is an exact
differential, the value of T will be had by an immediate inte-
gration, and, since T awd (No. 307), it is easy to infer the
positive or negative dilatation of the element ds, which will
be lengthened in the ratio of 1 + 8 to unity; but this will not
make known the dilatation of the normal section w, nor the
change of density of the rod at the point м. Now, by what has
been established in the memoir cited at the commencement of
this paragraph, it appears that the lengthening or shortening
of ds is always accompanied by a diminution or increase of w,
but such, that the volume wds will increase and diminish with
ds, and the density y will vary in the inverse sense. It follows
from this, that when a prismatic or cylindrical homogeneous
rod is fixed at one end, and drawn at its other extremity by a
force acting in the direction of the production of its length, it
will experience at the same time an extension and increase of
volume, proportional to this force, which, in point of fact has
EQUILIBRIUM OF AN ELASTIC Rod.
505
been confirmed by experiment. On the other hand, if this
rod is placed vertically on a horizontal plane, and loaded with
a weight at its upper part, which does not cause it to bend, it
will be shortened, and, at the same time, its volume will be di-
minished in proportion to the magnitude of this weight.
317. If on AM the arc of the mean filament, a point m in-
finitely near to м be taken, and through it a normal section
be made, and if the part of the rod comprised between this
section and the extremity A, be rendered immoveable, and the
form of the part comprised between the other end в, and the
section made through м, be merely rendered invariable; the
conditions of the equilibrium of this second part, which we
shall denote by K, may be determined in the following man-
ner.
In virtue of the torsion of the rod, the points of the slice
comprised between the two normal sections made through м
and m, will be solicited by forces which will tend to untwist
its different longitudinal filaments, and will act in planes per-
pendicular to мm, that is to say, to the tangent to the mean
filament at M.
These forces will tend to make к turn about
this line, in a direction contrary to that of its torsion. Let r be
their moment with respect to this line, it will be what is meant
by the moment of torsion of the rod, corresponding to the point
If through this point lines be drawn parallel to the axes
of x, y, z, then since the axis of this moment makes with
M.
dx dy dz
T
these lines, angles whose cosines are
we shall have
ds' ds' ds'
(No. 281)
dx
dy
dz
·T
T
T
ds
>
ds'
ds'
for the moments relatively to these three parallels, of the forces
which act on K in the direction of the torsion.
μ
Let u denote the moment of the elasticity with respect to
the point м, that is to say, the moment of the forces of which
T is the sum, relatively to an axis drawn through this point,
3 T
506
EQUILIBRIUM OF AN ELASTIC ROD.
and perpendicular to the osculating plane of the mean fila-
ment; we shall have (No. 314), (r and p being respectively
the radii of curvature at this same point, in the natural state of
the plate, and after its form has been changed, and ẞ denoting
a positive quantity depending on the material and normal sec-
tion at the point M,)
~=B (1 − 1);
and if f, g, h, be the angles which the axis of this moment
makes with parallels to the axes of x, y, z, drawn through the
point M, the moment of elasticity relative to these lines will
be
μ cos f, μ cos g, μ cos h.
Let м' be any point whatever of the arc MB, x', y', z', its
three coordinates, s' the arc AM', and y', w', x', y', z', what
y, w, x, y, z, become relatively to the point m'. If we denote
the entire length of the mean filament by l, and make
S', [x' (x' — x) — x' (y′ — y)] y'w'ds' =
— = zı,
S', [x′ (z' — z) — z′ (x' — x)] y'w'ds' = ¥1,
S', [x' (1' — y) − x' (~' — 2)] y'w'ds' = X1,
·
these three quantities will be the moments of the given
forces which act on K, with respect to the axes drawn through
the point м, in the directions of the axes of x, y, z.
Finally, if any particular forces be supposed to act at the
free extremity of K, and if P, Q, R, denote the sums of their
components parallel to the axes of x, y, z, and a', b', c', the
coordinates of the point of application of their resultant; their
moments relatively to the same axes as Z1, Y1, X₁, will be
Q (a' — x) — P (b′ — y),
F (c' — z) — R (a' — x),
R (b' — y) — Q (c′ — z) ;
EQUILIBRIUM OF AN ELASTIC ROD.
507
and if a, b, c, be the coordinates of the extremity в of the
mean filament, these moments may be replaced by
Q (α — x) — r (b − y) + R′,
P (c — z) — R (α — x) + q',
R (b − y) — Q (c − z) + p′,
in which, for the sake of abridging, we make
P
Q (a' — a) —
x ( b' — b) — R',
P (c′ — c) —
R (a′ — a) = Q',
R (b′ — b) —
Q (c' — c) = P'.
In general, the coordinates a', b', c', will be distinct from
a, b, c, for the extreme forces P, Q, R, will not be immediately
applied to the elastic rod, but will act at the extremities of the
arm of a lever. Whether these forces have, or have not, an
unique resultant, the quantities p', Q', R', will be their mo-
ments with respect to axes drawn through the point в parallel
to those of x, y, z, therefore, if at this point, we assume
dx
cos a',
dy=
= cos B', dz=
cos y's
ds
ds
and if we make
P' cos a' + Q' cos B'+R'cos y'L,
this quantity L will express the moment of the extreme forces
with respect to the tangent at the point B (No. 281); hence it
appears, that L will be the moment of the extreme torsion, or
the value of relative to this same point.
T
This being established, in order to the equilibrium of the
part K of the elastic rod, it is necessary that the sum of the
moments with respect to each axis, of all the forces which
act on its different slices and at its extremities, be equal to
cypher; in consequence of which the three following equations
result:
508
EQUILIBRIUM OF AN ELASTIC ROD.
da
µ cosƒ—ht + x1 + P' + R (b − y) − Q (c — z) = 0,
ds
dy
μ cos g-T-
ds
dz
ds
+ x₁ + Q' + P (c — 2) — R (a — x) = 0, } (b)
μ cos h—τ + 41 + R' + Q (a − x) − p (b − y) = 0.
P
318. By the formulæ of No. 19, we have
d²z
dy ď³z — dz d³y
cosf=
cos f=
λας3
dzd²x
dx d'z
cos g=
λας 3
dx d³y — dy d²x
cos h=
;
λds³
in which λds³ represents the square root of the sum of the
squares of the three numerators. Hence we obtain
d.μ cos f=dyd.
μd²z
dzd. μdzy
λds³
λds3'
d.μ cosg=dzd.
μd²x
dxd μd²x
μαχ
Ads3
λds3
d. μ cos hdxd.
May
Ads3
dyd. ;
μαχ
Xds3
and, consequently,
dx d.μ cos ƒ +
dy
dz
ds
d.µcosg+d.μcos h = 0,
ds
u
moreover, since
dx² dy² dz2
+ + = 1,
ds2 ds2 ds²
dy
dx.d.dz + dy. d. by + dz. d. dz = 0,
ds ds ds ds ds ds
we shall obtain, by multiplying the differentials of equations
(b) by
dx dy dz
ds' ds' ds
and reducing(p)
respectively, then adding them together
EQUILIBRIUM OF AN ELASTIC ROD.
509
1
:
dr = dx dx₁ + dy dx₁ + dz dz₁;
ds
ds
ds
but because the quantities subjected to integration in the ex-
pressions of X₁, Y1, Z1, vanish at the limit s's, in order to
obtain the values of dx1, dy₁, dz₁, it is sufficient (No. 14) to
differentiate under the signs S with respect to x, y, z; hence we
have simply
dx₁ = dz S', x'y'w'ds' — dy S' z'y'w'ds',
dx₁ = dx S' z'y'w'ds' — dz S' x'y'w'ds',
dz₁ = dy S' x'y'w'ds' — dx S' x'y'w'ds';
s
and if these values be substituted in the preceding equation,
it will become dr=0.
Thus it appears, that the moment of the torsion is constant
throughout the entire length of an elastic rod in equilibrio ;
consequently, its value will be the same for all its other points
as at the two extremities of the rod; and it is easy to show
that at the point B, we have 7L, as has been stated above.
In fact, at this point, we have xa, y=b, z=c; the inte-
grals X₁, Y1, Z₁, vanish, and equations (b) become(q)
τ cos a' =µ cos ƒ + p′,
T cos B'μ cos g + a',
T cos yμ cos h + R'.
As the normal to the osculating plane of the mean filament,
and the tangent to this curve, are perpendicular the one to the
other, we have at this same point B,
cos a'. cos ƒ + cos B'. cos g + cos y'. cos h=0;
hence, if the preceding equations be multiplied by cos a', cos ß',
cos y', respectively, and then added together, the quantity μ
will disappear, and we shall have
510
EQUILIBRIUM OF AN ELASTIC Rod.
1-
7. (cos² a' + cos² ß' + cos²y') =T
= P' cos a' + Q' cos ß' + R' cos y' = L.
It is only the moment of the torsion which we are enabled
to determine by the equations of equilibrium; with respect to
the torsion itself, it varies along the rod, when the material of
which it consists, or the normal section varies from one point to
another. When the rod is homogeneous, and the normal sec-
tion constant, the difference of the angles of torsion is the
same at the extremities of any two parts of the rod of equal
lengths, and proportional to their lengths, when they are dif-
ferent. For greater clearness, let us suppose that a prismatic
or cylindrical homogeneous rod, is firmly fixed at one ex-
tremity, and that at its other extremity, two equal forces,
parallel and opposite are applied, this rod will remain straight,
but it will be twisted on itself proportionably to its length, and
to the moment of these two forces with respect to its mean
filament, which moment will be the value of the quantity L. M.
Poisson proved in the memoir already cited (No. 306), that
if the normal section of this rod be a circle, the quantity of
torsion will be proportional (every thing else being the same)
to the fourth power of its diameter, which accords with ex-
periment.
319. Two of equations (b), or any two combinations of
these equations, will enable us, when its value is substituted
for μ, and L is put for r, to determine the figure of the rod in
equilibrio. If in its natural state it is straight, and if all the
forces which are applied to it, exist in the same plane, the
three equations (b) will be reduced to one which will be that
of the plane curve formed by the mean filament.
If the plane of these forces be taken for that of the axes of
x and y; we shall have
z=0, cos f=0, cos g=0,
c=0, c'=0, R=0,
cos y' = 0;
EQUILIBRIUM OF AN ELASTIC ROD.
511
A
hence there will result
X₁ =0, Y₁ =0, p'≤0, q'=0, T=L=0;
and the two first equations (b) will vanish.
the value of u will be; in like manner we
μ
ρ
Because r = ∞,
shall have cos h
=±1; but as we should take into account the direction of
the action of T on the part K of the rod (No. 314), it is easy to
perceive that we must assume cos h = − 1 in the third
tion (b), which will consequently become
S's [x' (x' — x) — x' (y' — y)] y'w'ds'
3) = 2;
+ R' + Q (a − x ) — P (b − y) =
equa-
}
(c)
and we may remark, that if we retain the notations of No.
314, the value of the coefficient ẞ will be
B = as htt
α
Sk
-k
vu²du.
When the forces x and y vanish, this equation (c) will coin-
cide with equation (1) of No. 308, for then the forces P and Q
in equation (c) act at the very extremity of the rod, which
renders their moment R' equal to cypher. In all cases, the
integrals that occur in this equation (c), can be made to dis-
appear by successive differentiations, by means of which it
will become a differential equation of the fourth order.
The figure of the rod being determined by equation (c),
is necessary moreover that the given forces which are applied
to it should satisfy the conditions of equilibrium of No. 261,
which are reduced to three, because all these forces are com-
prised in the same plan. Therefore, if we denote by D and E
the sums of the particular forces which act at the extremity a
of the rod, parallel to the axes of x and y, so that D, E, F', may
be, with respect to this point, what P, Q, R', are relatively
512
EQUILIBRIUM OF AN ELASTIC Rod.
to the other extremity B; the three equations in question
will be
D + P + S₂ x'y'w'ds' = 0,
E + Q + S₂ x'y'w'ds' = 0,
F' + R' + Q(a−x) — r(b−y)
+ S₂ [x'(x'—x) —x'(y'—y)]y'w'ds'=0 ;
(d)
in which the coordinates of the point a should be substituted
for x and y.
When the two extremities of the rod are entirely free, the
extreme forces and their moments will be given. If the rod
is firmly fixed at the extremity A, the forces D and E, and
also their moment F', will be undetermined; but the values of
dy
X, Y, relative to this point a, will be known. If the
dx'
rod is merely retained by the fixed point a, the forces D and
E will be still undetermined; their resultant will be equal
and contrary to the pressure at this point of support, of
which it will express the resistance, and we shall have
for their moment F' 0; the values of x and y will then be
known, but not that of The same remarks are applicable
to the point B.
dy
ཙུ|✺
dx
320. Let us suppose, for example, that the rod is homo-
geneous, and in its natural state either of a prismatic or cylin-
drical form, the three quantities y, w, ẞ will then be constant.
Moreover, let us suppose that it is only subject to the action
of forces perpendicular to its length, which cause it to deviate
very little from its primitive position, and let the mean fila-
ment in this position be taken for the axis of x; we shall then
have
D= 0, x = 0, P=0;
EQUILIBRIUM OF AN ELASTIC ROD.
513
in consequence of which the first equation (d) will disappear.
If the square of dy be neglected, we shall also have
dx
dzy
dx2
ds= dx, = ;
ρ
and equation (c) will be reduced to
d²y
B = R' + Q (α − x) + yw
dx2
S' x' (x'—- x)ds'.
(e)
B d³y
dx³
- Q — yw
• yw S₂xa
y'ds'.
By differentiating this we obtain
Likewise, by No. 14, we have
d
'a Sox'ds'
=
yds;
therefore by differentiating a second time, and substituting de
for ds, we shall have
d³y
B√24 = ywx.
dx¹
γων.
(f)
The four constant arbitraries which the complete integral
of this last equation will contain, can be determined by
means of the conditions relative to the two extremities of
the rod, and by observing that the value of y deduced from
this equation must satisfy the two preceding for all values of
x. Now, as equation (f) results from the two others by dif-
ferentiation, it will be sufficient for this purpose, that this
value of y should satisfy these two for a particular value of x,
hence it will be sufficient if we have
Bdy = R', Bdy
ß
- Q
β
dx²
(g)
for x = a; conditions which result from equation (e) and its
first differential, by ascribing to a this particular value. If we
3 U
514
EQUILIBRIUM OF AN ELASTIC Rod.
ascribe to x the value relative to the point a, we shall have, in
consequence of equations (d),
β
d²y
dx2
=
d³y
F', B =E,
dx³
(h)
but these equations do not express new conditions distinct
from those contained in equations (d) and (g), so that they
may, if we please, be replaced by the system of equations (g)
and (h).
321. These formulæ include the case of a heavy rod.
The point a is then supposed to be fixed, and to be the origin
of the coordinates x and y, and also the axis of x, which re-
presents the natural direction of the rod, is supposed to be
horizontal. If the direction of the positive ys be that of gra-
vity, which we suppose to be represented by g, we shall have
y=g, and the integral of equation (f) will be (q)
By=
gyw x² + cx³ + c²x² + c´x ;
24
(1)
c, c', c", denoting three constant arbitraries, the fourth being
equal to cypher, because at the point a we have x = 0 and
y = 0.
If the rod be firmly fixed at this extremity, we must
also have
dy
O when x 0, hence there results c"0. If,
dx
moreover, we suppose that the weight o is directly attached
to the other extremity B, so that its moment R' may be cypher,
then in consequence of equations (g) which belong to this point
or to x = a, we shall have
1
½ gywa² + 6 ca + 2 c′ = 0,
gywa+6c — Q.
By means of these equations the values of c and c'
may be
obtained; and if they be substituted in equation (1), of which
the term c" may be suppressed, there results by naming q the
weight of the rod, in which case we have g = gywa, (r)
EQUILIBRIUM OF AN ELASTIC rod.
515
qx4
By
24a - } (Q + q) 20³ + 1 (Q + 1 9 ) ax²;
x
if in this equation, the weight of the rod be neglected, and if
Qc² be substituted in place of ß, it will coincide with that of
No. 310.
In the two cases of q = 0, q = 0, we have
b =
qa³
b =
86'
Qa³
36'
for the expression of the ordinate of the point в, which ex-
presses the entire flexion of the rod. Hence it
Hence it appears, that
if we suppose Q=q, the flexions produced by a weight o sus-
pended at the free extremity of a horizontal rod firmly fixed
at its other extremity, is to the flexions produced by the same
weight distributed uniformly over the entire length of this rod,
as 8 to 3.
322. If the point в is, like the point a, fixed and situated
on the same horizontal line, we must have y = 0 when x = a,
in consequence of which, equation (1) is changed into the
following (s):
qx
24 a
By= (x³—a³) + cx (x² — a²) + c'x (x—a) ; (2)
q denoting as above the weight of the rod. The determination
of the two constants c and c' furnish the following cases.
1st. When the rod is firmly fixed at both ends, we must
dy
have = 0, for x = 0, and x = a, hence there results (†)
dx
c'
C= √29, c² = 24 aq;
and equation (2) becomes
(x-a)²
By = qx²
24a
If ƒ denotes the sagitta of the curve formed by this rod,
516
EQUILIBRIUM OF AN ELASTIC ROD.
that is to say, the value of y at its middle point, when
a
x= we shall have
2'
ƒ=
q a3
16.24.6
2ndly. If the rod is merely retained by the fixed points a
and B, the forces E and Q taken in an opposite direction from
that in which they act, will express the pressures exerted
against these points of support, and their moments F' and R'
will be cypher, (No. 319). In consequence of the first equa-
= 0 for x = 0 and x=a;
tions (g) and (h), we shall have
hence we infer that
dy2.
dx² 2
and also
c = 129, c'=0;
By = 9x (a — x) (a² + ax − x²)
24 a
and the sagitta ƒ will be
5 qa³
f=
16.24.ß
that is to say, quintuple of what it is in the first case.
means of the last equations (g) and (h), we shall also have
By
E =Q=
*9;
which values also obtain in the first case, as is evident of
itself.
3rdly. Finally, when the rod is firmly fixed at its extre-
mity a, and merely retained at its other extremity, we have
313
= 0 for x = a; and, consequently,
dy
d²y
= 0 for x=0,
0, and
dx
dx²
5q
C =
C =
48
qa
16
by means of which, equation (2) becomes
EQUILIBRIUM OF AN ELASTIC ROD.
517
By=
qx² (a−x) (3 a−2x)
48 a
The second equations (g) and (h), at the same time, give
Q=
E=
which shews that the weight of the rod is unequally distributed
between the two points of support, and that the load at the
extremity which is firmly fixed, is to that at the other extre-
mity, in the ratio of five to three.
323. The points A and B being always assumed to be fixed
and situated on the same horizontal line, and the rod being
also homogeneous and prismatic, let us consider the case in
which the other points are loaded with weights unequally dis-
tributed throughout its entire length.
Let therefore
γων
q
Ywx ==px;
α
q denoting the entire weight, and pr being a given function,
that vanishes when x = 0 and x = a, which implies that
Sa pada
=α.
α.
This function pa may be either continuous or discontinuous,
that is to say, its analytic expression may change once or se-
veral times between the extreme values x = 0 and x = a; or,
in other words, if it be represented by the ordinate of a line of
which a is the abscissa, this line may consist of several por-
tions of different curves. If 8 denotes the length of a line ever
so short, we may, for example, suppose that px=0 from
x = 0 to x = 1 a 8, and from x = a + 8 to x = a, so
that this function has not values different from zero except in
a very small extent, such as 8, on each side of x=a. This
case will be that of q, a weight acting at the middle of the
elastic rod, which we propose to examine particularly imme-
diately.
1
J
518
EQUILIBRIUM OF AN ELASTIC ROD.
Whatever be the nature of the function pa, whether it be
continuous or discontinuous, provided that it is equal to cypher
for x=0 and for xa, we shall have from x = 0 to x=a
inclusively,
px = 2/2 = (Sa
Σ
(So sin
ηπα
α
nwu̸′ px'dx'). sin 277;
NπX
α
a
(a)
n denoting any entire and positive number, and the charac-
teristic indicating a sum which extends to all values of n,
from n=1 to n = x. We are indebted to Lagrange for this
formula, he first announced it in Les Anciens Mémoires de
l'Académie de Turin, tome iii. p. 261; a demonstration of it
is given in No. 325 of this section. Substituting this value
px, equation (f) becomes
of
ηπα
B dy = 24 = (Sa sin wwx"
β
dx¹
:)
Σ sin px'dx' sin
α
NπX
;
a
and by integrating and observing that y=0 for x=0 and
x=a, we shall obtain(u)
ηπα
2002 sin pr'de') sin
4-(Sa
By=
2qa² 1
4
π
Σ
a
фх
+x(a−x) [cx+c' (a−x)];
NπX
a
(b)
c and c' being the constant arbitraries which can be deter-
mined as in the three cases of the preceding numbers.
324. Let us examine in detail the case in which the weight
g is suspended at the middle of the rod, that is to say, the case
in which, as has been stated above, the function pr' is cypher
for all values of x' that differ, ever so little, from a. We
can then make x' - a in the factor sin
ηπα
α
the integral relative to x'; this will give(v)
Sa
which contains
>
sin px'dx' = sin "So pa'dx' = a sin
ηπα
a
NT
ппСа
2
Sa
пп
2'
and cause all the terms of the sum Σ, in which n is an even
EQUILIBRIUM OF AN ELASTIC ROD.
519
number, to disappear. Let i denote any number odd or even,
then if we make n2i-1, and if the sum Σ includes all va-
lues of i from i=1 to ix, since sin
(2 i — 1) π
2
—— (-1),
= — (— 1)',
equation (b) becomes (x)
;
By=x (α- x) [cx+c' (a−x)]
2 qa³ (−1)¿ (2i_1)
Σ
4
4
(2¿ — 1)¹
sin
α
But by a known formula we shall have, as will be shewn in
No. 327,
Σ
(-1)ε
1)¹
3
(2i — 1)4 sin (2 i—1) w =
πω
πω
24
32
for all values of w, from w = 0 to w =
α
2'
7. If therefore x 42
TX
by making w=
S
,
we shall have (y)
α
(−1)i
(2i_1)
4
π
Σ
sin
(2i-1)4
α
96a³
(4x³ — 3a²x);
if, on the contrary, x> a, let w=
π (α-x); and as
a
(2i-1)T(α-x)
(2i_1)
sin
= sin
a
α
it follows that
sin
4
W
96 a³ [4 (a—x)³ — 3a² (a— x)].
(-1) (2 i —1)πx
Σ (2i-1) +
a
In this manner we shall obtain one or other of the following
equations :
q
By = x(a—x) [cx + c′(a — x)] — — (4x³ — 3 a²x)
48
By=x(a—x)[cx+c′(a—x)]—— [4(a—x)³—3a²(a—x)].
(1)
It only remains to determine the constants c and c' in the
three following cases:
520
EQUILIBRIUM OF AN ELASTIC ROD.
dy
dx
1st. The condition =0, for x=0 and xa, which
obtains when the rod is firmly fixed at its two extremities,
gives
q
c' = c = - 16'
Equations (1) will become
By = 2/4 (3 ax² — 4 x¹³),
48
By = 1 [3 a (a—x)² — 4 (a—x)³] ;
we shall have
dy
dx
48
0, at the middle point of the rod, as well
as at the extremities; and the sagitta f, or the ordinate corres-
ponding to x = a, will be
Я аз
f=4.48.6'
that is to say, double of what it is in the first case of No. 322.
In like manner we shall have, in consequence of the second
equations (g) and (h),
E
which we know ought to be the case.
9,
2ndly. In the case of a rod merely retained at its two ex-
d²y
tremities, in which case = 0 for x=0 and for xa, there
dx2
results
and, consequently,
c=0, c' 0,
—
By = 1 (3 a² — 4x³),
Չ
By=
[3a² (a—x) — 4 (a—x³)].
48
The tangent at the middle point of the curve is horizontal(z),
and the values of Q and E are -19, as in the preceding case
but the expression for the sagitta ƒis
qa³
f=
48 ß'
EQUILIBRIUM OF AN ELASTIC rod.
521
so that it is quadruple of the preceding, and greater in the
ratio of 8 to 5, than that of the second case of No. 322. If a
tangent be drawn to the elastic curve, through one or other of
the points A and B, and if we denote its inclination by a, and
byƒ' the vertical ordinate of the point of this line, which cor-
responds to the abscissa equal to a, we shall have(a)
tang a=
qa²
163'
ƒ' = a tanga;
hence we infer
ƒ' = {ƒ.
f" f.
In the second case of No. 322, the ratio of ƒ to ƒ would
be 9.
3. Finally, if the rod is firmly fixed at the extremity a, and
merely supported at the other extremity B, we shall have d=0
dy
dx
= 0 for xa; from which there results
for x = 0, and
d³y
dx2
q
c' -
C=
16'
1
q
32;
:0
and equations (1) will become
q
By= 1/6 (9αx² - 11x³),
96
By=2(5x³— 15 ax²+12a³x−2a³).
They give when xa, the same value for y, namely
y=
7 qa³
8.96.B
but this is not the greatest ordinate. We shall likewise have
E=
-11q
16
-5q
Q =
;
16
so that the weight q will be distributed between the points of
support A and B in the ratio of 11 to 5.
3 x
522
EQUILIBRIUM Of an elastic rod.
325. We now proceed to demonstrate the formula of La-
grange, adverted to in No. 323.
For this purpose, it may be remarked, that if the quantity
1 — h²
1-2h cos 0+h²
which is a rational fraction with respect to h, and in which 0 is
a real angle, be developed with respect to the powers of h, its
expression will be
1+2hcose + 2h² cos 20+ 2 h³ cos 30+ 2 h¹ cos40 + &c.;
which is easily verified; for if this infinite series be multiplied
by the denominator of the fraction, i. e. by 1 — 2h cos 0 + h³,
the result will be its numerator, as will be evident at once from
the consideration that
2 cos n0.cos 0=cos (n + 1) + cos (n-1)0,
whatever be the number n. If, abstracting from the sign, h
be less than unity, this series will be convergent, and the frac-
tion will be rigorously equal to its development continued
ad infinitum; hence since
1-2h cos 0+h²= (1-h)2+4h sin² 10,
we shall have, on this hypothesis,
1- h2
(1 − h)² + 4h sin² ≥ 0
=1+2Σ hr cos n✪ ;
in which, under the sum Σ, are included all values of the
integral number n, from n=1 to no. Therefore, whatever
be the nature of the function ƒ0 and of the real constant a,
we shall have also
=
So
(1-h²) fode
1 — h² +4h. sin² 3 (0 — a)
So fode +22h™ STƒ0. cosn (0 — a) d0.
Let g be an infinitely small positive quantity, this equation
EQUILIBRium of an ELASTIC ROD.
523
J
will still subsist if we make h=1-g, since it subsists for all
values of h less than unity. For all finite values of n, we shall
have
h² = (1 − g)" =1;
for infinite values of this exponent, he may differ from unity,
but by partially integrating, we obtain
1
Sare
Sƒ0 cosn (0—a) d0 = = ƒ0 sinn (0 — a) — — sinn (0—a)d0;
Sfocosn
n
N
So that if fe does not become infinite, between the limits
00 and 0π, nor for these limits, the integral
STƒ0 cosn (0 – a) do
by which h" is multiplied,will vanish, when n; hence it fol-
lows that we can always replace hr by unity under the sign Σ.
In the numerator of the fraction comprised under the sign S, we
shall have 1-h2-2g, g2 being neglected with respect to
2g; in the second term of the denominator, we can substitute
unity for h or 1 - g; and, by this means, we shall have
h²=2g,
Sãƒodo + ≥ Sãƒ0 cosn (0 − a)d0
Sog²
+4 sin² 1 (0 — a)'
(1)
The coefficient of de in this last integral is infinitely small,
except for values of 0, that differ from a by infinitely small
quantities, which render its denominator infinitely small; con-
sequently this integral is infinitely small or cypher, when the
difference 0-a is a finite quantity; which will be the case
throughout the entire integration, when we suppose a ≤ 0, or
a>π; hence whenever the constant a falls without the limits
zero and π, the following equation will have place:
π
Saƒ0d0+≥STƒ0 cos n (0 − a) d0 =
d0 = 0.
(2)
1
524
EQUILIBRIUM OF AN ELASTIC rod.
་་་
If, on the contrary, we have a > 0 and 7, there will be
values of 0 which differ infinitely little from a; therefore by
making
=
0 = a+u, do du,
the integral in question will still vanish for finite values of u,
but not for infinitely small values of this variable, whether
positive or negative; with respect to these, we shall have
ƒ0=fa, sin (0—a) = ½ u ;
consequently, the second member of equation (1) becomes
S
gdu
Jas
fa g² + u²
2
2
when a falls between zero and 7. But, as this integral vanishes
for every value of u which is not infinitely small, it may now
be extended, without altering the value, to any values what-
ever of u, whether positive or negative, so that we may take
it, if we please, from u = ∞ to u = ∞; in which case, we
shall have
and, finally,
S_
gdu
∞ g² + u²
- По
Пя
STƒode + ≥ STƒ0 cos n (0 − a) d0 = nƒa.
(3)
This reasoning will also suit the case in which a coincides
with one of the two limits zero or ; but if a = 0, then we can
assign none but positive values to u, and only negative values,
when aπ, in order that in these two cases, the variable 0
which is made equal to a+u, may, not pass the limits of the
integration. In this manner, the integral relative to u will be
reduced to the half of its value in the former case, or to ;
2
Π
so that if ẞ and y be the values of fa in the case of a = 0 and
a=π, there will result
+ STƒodo + STƒ0 cos node = §πß,
f0
STƒdo + ≥ (− 1)" STƒo cos n0d0= }πy.
(4)
EQUILIBRIUM OF AN ELASTIC ROD.
525
Now if we make
πα
0 =
Tdx'
d0= ;
α
a
and also
πα
ƒ(Tx) =
= &x';
α
and if, the quantity z being positive and less than the constant
a, there be substituted
and
TX
α
пх
a
in the place of a, in equation (2),
for a in equation (3), we shall obtain by observing,
that the limits relative to x' are cypher and a,
1
2a
2 4 5 0 px d x + 1 = Sa
Nπ (x² + x)
Σ px' cos
dx'=0,
a
α
1
1
2
So
za S0 px'dx' + = Sa px' cos
Nπ (x′—x)
= Σ
= 0x,
}
(5)
a
α
and, by subtracting the first equation from the second, there
results
2
α
² (So paʼsin.
ηπα
ппх
da sin.
=фх;
a
a
which was to be demonstrated(b').
326. This formula represents the values of the function
px, for all values of the variable x, which are positive and less
than a, and even for x = 0 and ≈
x = 0 and x = a, when pa vanishes for
these extreme values. It is important to observe, that the
series indicated by Σ, will be always convergent, when con-
tinued to a considerable number of terms; for when the
values of n are very great, the integral relative to x' will be-
come a very small quantity, which will diminish more and
more according as n increases; and when n∞, it will be-
come cypher, as was already shewn by means of partial inte-
gration. This remark is necessary, as it shews that we are
justified in the application which we propose to make of the
preceding formula.
1
526
EQUILIBRIUM OF AN ELASTIC ROD.
The different formula, by means of which portions of arbitrary
functions, either continuous or discontinuous, may be thus ex-
hibited in series of periodical quantities always converging, can
be deduced from equations (5), which have been demonstrated
above. We shall restrict ourselves in this chapter to give two
of these formula, which will be useful hereafter; the reader is
referred for more extensive developments on this matter to the
author's Memoirs on the Integral Calculus, inserted in the
Journal de l'Ecole Polytechnique, where will be found a com-
plete theory of this kind of transformations.
If equations (5) be added together, and also if the first be
subtracted from the second, and if 27 be substituted for a, then
x+l and x' + 1 for x and x', and afterwards px and px' in
place of p (x+1), p (x + 1); the limits of the integrals rela-
tive to x' become±l, and these equations will be replaced
by the following:
1 cl
8x=
¿x'dx' +
_1px'cos
21
nπ (x²+¹²) dx') co§¹π(x+1),
COS
21
фх
Σ
1)
(x
px = = = (S_', pe′sin "= (x + ") de') sin ** (1+1).
21
21
Let each sum Σ be divided into two others, of which the
one may contain even, and the other odd numbers. For this
purpose, let i be any integer number whatever, then by making
n successively =2i, and 2i — 1, we shall have
COS
2iπ (x+1)
21
iπx
= (-1) cos
2iπ (x + 1)
πα
sin
= (-1) sin
21
−
cos (2 i — 1) r (x + 1) − (− 1)' sin (2-1) me,
21
i
sin (24 -1) x (x +
1)
21
=
21
= (-1) cos (21-1)=x;
;
EQUILIBRIUM OF AN ELASTIC ROD.
527
and, in the same manner, we can express the sines and cosines
comprised under the signs S; consequently, we shall have
px'cos dx') cos
iTx
27
21
1
iπx'
фх
-1
фос
-1
1
+ 1/2 = (S 1/2
(2i – 1)
px'sin
da') sin
(2-1)πα
πω
T
da') sin
sin 7
Σ
i
ФХ = = = (S ²,
Σ
-Z Þæ′sin -
πα
ī
+ } = (S_1pz'cos (25 −1) =*' dr′ ) cos (21-1) **;
21
(6)
in which the sums Σ extend to all values of i, from i=1 to
i=∞. These equations will obtain for all values of x that
are comprised between the limits ±1.
This being established, if the function px is such that
4(−x) =—px, there will result
-¿ Px'dx' = 0,
Spa' cosi' da' = 0,
πα
S'px' cos (21—1) ™z' da' = 0;
21
dx
da' = 2 Sope' sining" da",
dx',
and, moreover,
~7 px'sin -
Spa' sin (2-1) wa da' = 2 Sox's
(2i—1) ** dr';
πα
pa' sin
2/
πα
by means of which the second equation (6) will coincide with
formula (a), by changing in it, a into 7; and the first will be re-
duced to
2
фх =
Σ
px'sin
(2i−1),
21
-1)* dr) sin (21-1)*2. (7)
(2i−1)
sin
21
If, on the contrary, the function or is such, that ø ( − x)
= px, we shall have
i —1)
pe′sin (2-1)**"' da' = 0, S_, pa'sinin da' = 0;
S_₂ paʼsin
528
EQUILIBRIUM OF AN ELASTIC ROD.
and the other integrals may, by doubling the results, be ex-
tended from x = 0 to x = l. If l-x be substituted for x, and
px for ø (1 − x), the second equation (6) will coincide with
equation (7). The first equation (6) will become
1
πα
itx
4x=/ So pa'da + 2(Spa'cos de') cos. (8)
px'dx' 2
φα
0
COS
These formulæ (7) and (8) will represent the values of px, from
x=0 to x = 1; those which can be deduced from them, by
differentiating with respect to x, will express the values of
dpx,
dx
within the same interval. Formula (7) supposes that
афх
px = o for x = 0, and dò̟x=0 when x=1; in formula (8)
афх
dx
= 0 for x=0 and xl. When these conditions are not
dx
satisfied, neither these formulæ nor their differentials have
place for the extreme values of x(c').
327. Conversely, formulæ of this kind make known the
sums of numerous periodical series, which have been obtained
by different means. Thus, for example, in order to infer
from them the sum of the series made use of in No. 324, let
a be substituted in place of a in equation (2), and then let
it and equation (3) be added together; there will result
= cos nºão) c
·ST Sodo + 2 (STƒ0
cos
na
cos nα = πfα.
If then we assume ƒ0 = 0, we shall have
T
Sore
ƒl cos nod✪ =
COS Nπ
1
n2
which is equal to cypher for all even numbers, and to
2
(2¿ — 1)²
becomes
for n=2i-1. Hence the preceding equation
Σ.
cos (2i - 1) a
(2 i — 1)²
11
π
¦ (π — 2a) ;
EQUILIBRIUM OF AN ELASTIC ROD.
529
71
in which the sum Σ extends to all values of the whole num-
ber i, from i=1 to i∞.
Multiplying by da, and integrating, we obtain
Σ
π
sin (2 i — 1) a — — (π — a) a.
(2 i — 1)³
=
8
It is not necessary to add any constant arbitrary, for
the two members of this equation vanish for a = 0, and
for aπ; so that this equation obtains for all values of a,
from a = 0 to aπ inclusively; if we make a = π+w, we
shall have
sin (2 i — 1) a = − (− 1)i cos (2 i — 1) w,
and, consequently,
Σ
π
—
(— 1)'. cos (2 i — 1) w — 7 (w² — ‡ π²),
(2i-1)³
from w to wπ. If this be multiplied by dw and
W = 1π
integrated again, there results
x (−1)i sin (2¿ − 1) w
(2 i — 1 )¹
4
which was required to be obtained.
πως
3
πω
24
32
328. If 2a be substituted in place of a, and then x + a,
x' + a in place of x and x' in the second equation (5), we shall
obtain, by making 4 (a + x) = FX,
1
1 α
FX =
Fx'dx' +
4a-a
+2α a F
=S
a
nπ (x'—x)
Fx' cos
dx'
2 a
for all values of a comprised between ±a.
By making
nx
π
= &,
<=nɛ=U,
2a
2a
this equation may be written as follows,
€
FX =
2π
= = - Sa
α
α
Fx'd x² + 1 = [ S a a
Π
Σ
Fx' cos u (x' — x) dx]ɛ,
3 Y
530
EQUILIBRIUM OF AN ELASTIC Rod.
ε
in which we suppose that u is a multiple of ε, and that the
sum Σ extends to all values of u, from u= ε to u = ∞.
ɛ
But
if the constant a becomes infinite, & the difference of the conse-
cutive values of u will become infinitely small, and the sum Σ
will be changed into an integral which should be taken from
u=ɛ, or u=0, to ux. Consequently, making a = ∞,
ɛ= du, and substituting the sign S for Σ, we shall obtain, by
suppressing the first term of the preceding formula,
FX=
1
+ S3 [S2000
Fx′ cos u (x' — x) dx'] du.
This important formula, which extends to all real values of the
variable x, positive or negative, and is applicable like the pre-
ceding, from which it has been deduced, to any function Fx,
whether continuous or discontinuous, was first given by
Fourier.
CHAPTER IV.
PRINCIPLE OF VIRTUAL VELOCITIES.
329. In the simplest cases of the equilibrium of machines,
the power and resistance are reciprocally proportional to the
spaces which their points of application would simultaneously
describe, if the equilibrium was destroyed. But, since in
consequence of the connexion between the points of applica-
tion, the spaces which these points would describe, if they
were entirely free, is different from the spaces that they ac-
tually describe, in order that this relation may always obtain,
the infinitely small spaces which are described in the first
instant, should be replaced by their projections on the direc-
tions of the forces. This relation has been a long time recog-
nized in the case of simple machines; John Bernouilli afterwards
extended it by induction to any system whatever of material
points solicited by given forces; and, under the denomination
of the principle of virtual velocities, it is now become the ge-
neral principle of equilibrium. We propose to demonstrate it
here in all its generality, after having first verified it, in the
following examples:
1st. Let (fig. 79) a, A', A“, . . . . be a series of pulleys con-
tained in the same block, and constituting a fixed system, and
B, B', B″, . . . . another series of pulleys also contained in one
block, and constituting a moveable system. Let us suppose
a thread attached to the inferior extremity of the fixed system,
to be successively rolled round all the pulleys by passing alter-
nately from one system to the other. To the free extremity of
this thread let a weight P be suspended, which may constitute
532
PRINCIPLE OF VIRTUAL VELOCITIES.
an equilibrium with a weight R suspended at the inferior
pulley of the moveable system. The tension of the thread
will be the same throughout its entire length, and equal to
the weight P ; moreover, if the diameters of the pulleys be very
small, relatively to the distance which separates the two sys-
tems, the strings which pass alternately from the one to the
other will be sensibly parallel and vertical; consequently, the
force which sustains the weight R will be equal to the sum of
their tensions, or to n times the weight P, n being the number
of these strings; hence, in the case of equilibrium, we shall
have
RNP.
Now, if the equilibrium is destroyed, and the weight R
ascends or descends by a quantity a, all the strings which ter-
minate in the moveable system will be shortened or lengthened
by this same quantity. As the entire length of the thread
remains the same, the part to which the weight P is attached
will be lengthened or shortened by n times this quantity a;
hence if ẞ denotes the space through which the weight P
ascends or descends, we shall have ßna, and, consequently,
Ra = pß;
which contains the principle of virtual velocities already ad-
verted to.
2ndly. ABC (fig. 80) represents the wheel, and a'b'c' the
intersection of the vertical plane of this wheel and of the
surface of the cylinder, in the machine called the axle in the
wheel; o is the common centre of these two circumferences,
and aoc, a'oc' are their horizontal diameters. A thread coiled
round the wheel is attached to one of its points; another
thread attached to one of the points of the cylinder, is in the
same manner coiled round its surface. A weight P is sus-
pended from the first thread, and a weight R from the second;
these two weights tend to turn the machine in opposite direc-
PRINCIPLE OF VIRTUAL VELOCITIES.
533
tions, and are supposed to be in equilibrio. This being agreed
on, if to the point c' there be applied two vertical forces R'
and R", equal, and acting in opposite directions, the equili-
brium will not be disturbed; if, moreover, these forces are
respectively equal to R, the force R", and the weight R, will
constitute an equilibrium, for there is no reason why their si-
multaneous action should cause the machine to turn in one
direction rather than in the opposite; there must therefore
be likewise an equilibrium between the weight P and the force
R', which act perpendicularly to AOC' at the extremity of this
lever, of which the point o is the fulcrum. Hence, r denoting
Ao the radius of wheel, and r' oc' the radius of the cylinder,
the equation of equilibrium will be (because R' R)
Pr = Rr';
Now, if the equilibrium is destroyed, and if the weight R
rises or falls by a quantity a, while the weight P falls or rises
by a quantity ẞ, it is evident, from the nature of the machine,
that we shall have ẞr' =ar; hence we infer
Pß = Ra;
conformably to the principle which it was proposed to verify.
3rdly. Let a vertical screw be loaded at its upper extre-
mity by a weight R, and let a horizontal wheel, having its centre
in the axis of this screw, be adapted to its inferior extremity,
let then a thread be wrapped on this wheel, and fixed to its
circumference by one extremity, while at its other extremity
a horizontal force F acting in the direction of a tangent to the
wheel, constitutes an equilibrium with the weight R. By
placing a fixed vertical pulley in the direction of this tangent,
we can give the thread a vertical direction, and thus replace F
by p a weight equal to this force, and attached to the free ex-
tremity of the vertical part of the thread. Denoting the in-
terval between the threads of the screw by h, and by c the cir-
P
534
PRINCIPLE OF VIRTUAL VELOCITIES.
cumference of the wheel, we shall have, by the known condition
of equilibrium in this machine,
PC Rh.
The two weights R and P tend to make the screw to turn
in opposite directions; if the equilibrium is destroyed, one of
the weights will rise while the other falls; and if the weight
R rises or falls by a quantity equal to h, the interval between
the threads of the screw, the weight P will fall or rise through
a space equal to c the circumference of the wheel; hence it
follows, that, in general, denoting the spaces simultaneously
traversed by the two weights R and P, by a and ß, we shall
have ac = ßh, and, consequently,
Pẞ= Rα,
conformably to the principle in question.
4thly. Let us consider the case of two weights, P and R,
placed on two inclined planes, and connected together by a
thread passing over a fixed pulley situated above the two
planes, which are supposed to rest against each other. Figure
81 represents a vertical section of this system; AC is the
length of the plane on which the weight R is placed, BC
that of the plane which supports the weight P, AB is a hori-
zontal line, and CD a vertical line, equal to the common
height of the two planes. Let
AC =α, BC=b, CD = h;
h
the component of R in the direction of CA will be R
and that
α
a'
h
of P in the direction CB will be p In order that there may
Põ
be an equilibrium, these two components should be equal, con-
quently, we must have
Pa = Rb.
If the equilibrium is destroyed, and the weight R slides on
PRINCIPLE OF VIRTUAL VOLOCITIES.
535
the plane CB by a quantity equal to y, the weight p will slide
by the same quantity, but in a contrary direction, on the plane
AC; denoting by a the vertical height by which the weight R
is elevated or depressed, and by ẞ, that by which the weight
P is depressed or elevated, it is easy to perceive, that we
shall have
a
yh, B = yh
a
and, consequently,
PB = Ra,
as in the preceding examples; but in this case a and ẞ are the
vertical projections of the spaces simultaneously described by
the weights R and P, while, in the preceding case, a and ß
were those spaces themselves.
330. It appears from No. 49, that two forces which con-
stitute an equilibrium, through the intervention of any lever
whatever, are in the inverse ratio of the infinitely small spaces,
described in the same time by their points of application, pro-
jected on the respective directions of these forces.
This relation, which exists in the case of the lever between
two forces in equilibrio, is true also when two forces, applied
to any other machine, are in equilibrio. Thus, if we denote
the power and resistance which are in equilibrio, by the in-
tervention of any machine whatever, by P and R, when an
infinitely small motion is impressed on this machine, we shall
always have
Þß
PB = Ra,
ẞ and a being the projections on the directions of these forces,
of the spaces which would be simultaneously described by
their points of application; it is necessary besides to take
notice that one of these projections must be taken in the di-
rection of the corresponding force, and the other on its pro-
duction, as is the case in the lever.
It is only requisite in practice, that the motion impressed
536
PRINCIPLE OF VIRTUAL VELOCITIES.
on the machine be very small. By measuring the lengths of
the projections ẞß and a, the ratio of the power to the resist-
ance may be immediately obtained, without knowing anything
of the particular construction of the machine.
331. This mode of expressing the relations between the
power and resistance when they are in equilibrio, is not only
true in the case of every machine, but it may be also extended
to any number of forces whatever in equilibrio. Therefore,
let M, M′, M″, &c., (fig. 82), be a system of material points con-
nected together in any manner whatever; let us suppose that
the forces P, P', P", &c., act on these points in the directions
Ma, M′a′, M"a", &c., if these points be made to undergo infi-
nitely small displacements compatible with the conditions of
the system, and if in this way they are transferred to N, N′, N",
&c., and if finally N, N', N", &c., be projected on the lines
MA, M'A', M"A", &c., to a, a', a", &c., so that
мap, м'ap', M"A"=p", &c.;
then these projections p,p′, p", &c., being considered as po-
sitive or negative, according as they fall on the directions of
the corresponding forces, or on their productions, we shall
have, when the equilibrium obtains,
Pp+P'p'+P"p" + &c. = 0,
and conversely, there will be an equilibrium, when this equa-
tion subsists for all displacements compatible with the con-
ditions of the system.
The infinitely small lines MN, M'N', M"N", &c., are termed
the virtual velocities of the points M, M', M", &c., because they
are the spaces that would be simultaneously described by the
points of the system, in the very first instant in which the
equilibrium is destroyed.
It should be observed, that the principle of virtual velo-
cities contained in the formula stated above, furnishes merely
the conditions of equilibrium which may be expressed by
·
PRINCIPLE OF VIRTUAL VELOCITIES.
537
equations, but not those which are relative to the directions
of certain forces, and to the extent within which they should
meet a fixed plane (No.266). The motions compatible with the
conditions of the system, which furnish the equations of equi-
librium, are such, that motions directly contrary are equally
possible. For example, if a material point is placed on a
fixed plane, the motion will be possible in every direction
taken in this plane, and also in the corresponding opposite di-
rection; but perpendicularly to this plane, it can only have
place in one sole direction. Now, the consideration of the
motions performed in the plane, will furnish us with conditions
of equilibrium which may be expressed by equations, and the
consideration of the perpendicular motion will only determine
the direction of the normal force, which ought to be contrary
to that of the possible motion. In the general statement of the
principle of virtual velocities, it is implicitly supposed, that
each of the motions compatible with the conditions of the sys-
tem, and the directly contrary motion, are equally possible;
and if the preceding equation be successively applied to these
two motions, all the quantities p, p', p", &c., will change their
signs, and there will only result but one equation of equilibrium.
If the force P is the resultant of several given forces
Q, Q', Q″, &c., and if q, q', q', &c., denote the projections of
MN on their directions, we shall have (No. 34),
Pp = Qq + Q'q' + Q″q″+ &c. ;
so that we can replace in the preceding equation, the term pp
relative to the force P, by this sum of terms of the same nature,
which refer to its components; and we may make the same
substitution in case of the forces P, P', P", &c., if they are also
the resultants of several other forces.
It appears from No. 39, that in the case of an isolated
point, the principle of virtual velocities is a consequence of
this last equation, whether the point in question be entirely
free, or whether it be constrained to remain on a given curve
3 z
538
PRINCIPLE OF VIRTUAL VELOCITIES.
or surface. We proceed now to demonstrate this general prin-
ciple in the case of any system whatever of material points
M, M', M", &c.
332. Let us suppose that these points are connected to-
gether, either by inflexible rods, or by flexible threads, of
which the first must be firmly attached to these points,
while the second traverse them like moveable rings. In this
last case, these points or rings are free to slide along the
threads that traverse them, which, therefore, must be per-
fectly flexible. It is evident, that, after the given forces
P, P′, P″, &c., are applied to the points M, M', м", &c., and
the equilibrium is established, the threads which connect
these points two by two, must each of them experience a
particular tension, that is to say, each of these threads will
be drawn at its two extremities by equal and opposite forces,
acting in the directions of its productions, as has been al-
ready stated in the case of the funicular polygon (No. 285).
The intensity of this force will measure the unknown tension
which this thread experiences. Any thread that is not
stretched, will contribute nothing to the equilibrium, and,
therefore, need not be taken into account.
The tension may vary from one thread to another; but in
the case of two threads, that are the production, the one of
the other, through a ring, the tension is the same, in these
two parts of the same thread, which must necessarily expe-
rience the same tension throughout its entire length (No. 289).
Thus, for example, if м is a ring traversed by the thread
M'MM", the tension of мM' is the same as that of мм".
When several threads cross each other in passing through
the same ring, the tension is the same in the two parts of
each thread, but it may vary from one thread to another. If,
therefore, beside the thread м'мM", the thread м""мM" like-
wise passes through the ring M, the tension will be the same
in the two parts мM"" and мм of this last thread, and, in
general, it will be different from that of the two parts мм' and
PRINCIPLE OF VIRTUAL VELOCITIES.
539
MM" of the first thread. And, if another thread, such as MM"
should terminate at the same ring м, to which it is firmly at-
tached, its particular tension will, generally speaking, be dif-
ferent from the tensions of the other threads that terminate at
the same point M.
It is likewise to be observed, that if м' is a ring as well as
M, and if the thread м"мм', after having traversed the ring
M, passes likewise through the ring м', and terminates at the
point м"", the tension will be the same in the three threads
M″M, MM', M'M'"; for then these three threads constitute only
one м"MM'M"". In general, when a thread is distributed into
several parts by moveable rings, the tension is the same in all
its parts.
With respect to inflexible rods, when the equilibrium ob-
tains, they are drawn or pushed in the direction of their length,
by equal and contrary forces, acting at their extremities. The
common intensity of these two forces, in the case of each rod,
is the measure of the tension or contraction that it under-
goes. If there be any rods in the system which are neither
stretched or contracted, they do not contribute to the equi-
librium, and, being useless, they may be suppressed. Hence,
in what follows it is assumed, that the rods or threads which
connect the different points of the system, are stretched or
contracted in the direction of their length by unknown forces.
The advantage of the principle of virtual velocities, con-
sists in this, that it furnishes the equation of equilibrium in
each particular case, without requiring us to compute these
interior forces; but as the demonstration which we propose to
give of this principle, is founded on the consideration of these
forces whose magnitude is unknown, the following notation
may be advantageously employed to represent them.
The tension or contraction of the flexible or inflexible
thread which connects any two points м and м' of the system,
will be denoted by [m, m']. In this manner [m, m"], [m', m"],
4
540
PRINCIPLE OF VIRTUAL VELOCITIES.
&c., will represent the tensions or contractions of threads
which connect м and м", м' and м", &c.
333. We must likewise consider the infinitely small vari-
ations which the distances of the points M, M', M", &c., taken
two by two, undergo, when only one of these points changes
its position, and also when they are both displaced simul-
taneously. Denoting the distance between any two points
м and м', м and м", м' and м", &c., by (m, m'), (m, m'),
(m' m'), &c., if we employ the characteristic d₁ to denote
the variations of these distances relatively to the displacement
of the point м, the characteristic di' to denote those which
obtain when it is the point m' that is displaced, the charac-
teristic d" to indicate the variations arising from the displace-
ment of м", and so on, and finally, if the characteristic & be
reserved to denote the variation of the distance of the two
points, resulting from their mutual displacements, we shall have
♪ (m, m′) — MM′ — NN′,
Sı (m, m') = MM' — NM′,
S₁' (m, m') = MM' — MN';
for м has been supposed to have been transferred from м to N,
and м' from м' to N'.
It is of consequence to observe, that the entire variation
indicated by 8, is equal to the sum of the partial variations
indicated by d₁ and d'; so that for any two points whatever,
we have
♪ (m, m') = 8, (m, m') + Sı' (m, m') ;
this equation obtains, because the displacements of м and м′
are supposed to be infinitely small, and it is only true on this
hypothesis. In fact, (mm') is a function of the coordinates of
these two points; and when м and M' are transferred to N and
N', these variables experience infinitely small increments,
positive or negative; now, if the powers of these increments
higher than the first, be neglected, it is evident that the entire
PRINCIPLE OF VIRTUAL VELOCITIES
541
increment of any function whatever of these coordinates, is
equal to the sum of the partial increments arising from the
variation of each coordinate separately; consequently, the
entire variation of (m, m'), indicated by the characteristic 8,
must be equal to the sum of its partial variations which refer
to d₁ and Sí'(a).
334. What precedes being admitted, let any point м, to
which the given force P is applied, be considered. This point
is connected with the others by the threads мM', мM", &c. ;
it is, therefore, drawn or pushed in the direction of each of
these threads by a force equal to the contraction or tension
which this thread experiences; so that besides the given force
P, the point м is likewise subject to the action of as many other
forces as there are threads terminating at this point. These
interior forces being thus taken into account, we may abstract
altogether from the consideration of the threads which connect
M with the other points of the system, and regard it as an
isolated point, which is in equilibrio, in consequence of the
action of the force P, and of the forces [m, m], [m, m'], &c.
If м is a fixed point, no equation of condition will result from
them, but if it is entirely free, or if it is only constrained to
remain on a given curve or surface, we shall have between
these forces, the equation of virtual velocities, which has been
already demonstrated to obtain in the case of a material point
in equilibrio.
In order to form this equation, let a point be taken in-
finitely near to M, and appertaining to the curve or surface on
which м is constrained to exist, if it is not entirely free. If
p, t, t', t", &c., be the projections of MN on the directions of
the forces P, [m, m'], [m, m'], [m, m""], &c.; by No. 39 we
shall have
Pp + [m, m'].t + [m, m"]. t' + [m, m'"'].t" + &c. = 0.
But, as the line MN is by hypothesis infinitely small, it is
easy to perceive that its projection on MM', is q. p. the dif-
542
PRINCIPLE OF VIRTUAL VELOCITIES.
ference of the two distances MM' and NM'; for if from the
point N (fig. 83) the perpendicular NH be let fall on мм', the
line мH will be this projection, and we shall have
MH = MM' HM'.
We have also, by neglecting infinitely small quantities of
the second and higher orders,
hence, therefore,
HM'= √(NM')² — (NH)² = NM',
MHMM'
NM'.
According to the notation explained above, this equation is
t = 8, (m, m') ;
and in the same manner we shall have
t' = d₁ (m, m''), t" = d₁ (m, m"), &c. ;
consequently, the equation of equilibrium will become
pp + [m, m'].S₁ (m, m) + [m, m'].dı (m, m')
+ [m, m″] dı (m, m") + &c. = 0.
In the case of each of the other points м', м", м"", &c., of
the system, similar equations may be obtained, these equa-
tions will be
P′p' + [m', m].Sı′ (m², m) + [m′, m′].di″ (m′, m″)
+ [m', m'"] &,' (m', m") + &c. = 0,
P″p"+ [m", m] di″ (m", m) + [m", m] di″ (m", m')
+ [m", m″'] d,″ (m", m''') + &c. = 0,
p"p" + [m", m] d," (m", m) + [m", m′] S″"″ (m", m′)
+ [m'", m'] &,"'' (m", m") &c. = 0;
p', p", p'"', &c., being the respective virtual velocities of
M', M", m'"', &c., projected on the directions of the given
forces P', P", p'", &c., which act on these material points.
PRINCIPLE OF VIRTUAL VELOCITIES.
543
We shall obtain, by adding these equations together, and
by observing, that [m, m'] and (m, m') are the same thing as
[m', m] and (m', m), and that the same is likewise true for the
other points m", m'", &c.,
Pp+P'p' +P"p" + P""p"" + &c.
+ [m, m']. 8 (m, m') + [m, m']. ♪ (m, m")
+ [m, m''']. § (m, m''') + &c.,
+ [m', m'] . ♪ (m', m'') + [m' m''']. § (m', m''')
+ [m", m'''] . § (m", m''') + &c.,
+ &c. = 0.
(a)
in which the total variation of each distance is substituted in
place of the sum of its partial variations.
335. In what precedes the displacements MN, M'N', M"N",
&c., (fig. 82), are supposed to be independent of each other;
and in equation (a) it is only implied that these points have
remained on the given surfaces or curves, on which they are
obliged to exist; but, if we suppose, in addition, that, in
consequence of these displacements, the points of the system
which are connected by a rod, or stretched thread, have pre-
served the same respective distances, we shall have
8 (m, m') = 0, 8(m, m") = 0,
8(m', m") = 0, &c.,
and equation (a) will be reduced to the following:
Pp + P'p' + P'p" + pp"" + &c. = 0,
(b)
which is precisely that of the principle of virtual velocities
(No. 331).
If in the displacements of the points м, м', м", &c., those
which are rings slide along the threads which traverse them,
equation (b) will still obtain, provided that the total length
of these threads does not vary.
that м is a ring which slides along the thread M'мM", then we
shall have no longer separately 8 (m, m′) and ♪ (m, m") = 0,
but we shall always have
Let us, for example, suppose
544
PRINCIPLE OF VIRTUAL VELOCITIES.
+
S (m, m') + d (m, m") = 0,
since the entire length of the thread continues constant. But,
in this case, as the tensions [m, m'] and [m, m'] of the two
parts of this thread are equal; the terms which contain these
tensions in equation (a) may be written thus,
[m, m] . [8(m, m′) +8(m, m')],
and, consequently, they destroy one another.
In general, when a flexible thread passes through any
number whatever of rings, the equal tensions of its different
parts will disappear from equation (a), as often as the entire
length of this thread does not vary. We can, therefore, finally
infer,
1st. That the equation resulting from the principle of vir-
tual velocities obtains in the case of all infinitely small mo-
tions which can be impressed on a solid body, whether free or
constrained by fixed obstacles; for in all these motions, the
respective distances of the points of this body are invariable.
2ndly. That this equation obtains also in the case of all
the infinitely small motions which a system of points or rings
connected together by flexible threads can acquire, provided
that these threads remain straight or stretched. When this con-
dition is not satisfied, all the tensions would not disappear in
equation (a), and, consequently, equation (b) no longer obtains.
336. It is necessary likewise to demonstrate that, con-
versely, when equation (b) obtains for all the infinitely small
motions which can be impressed on a system of points M, M', M",
&c., the given forces P, P', p", &c., are in equilibrio, as has been
already stated, (No. 331).
If for one instant it be supposed that the equilibrium does not
obtain, the points м, M', м", &c., or at least some of them, will
begin to move, and in the first instant they will describe simul-
taneously right lines such as MN, M'N', M"N", &c.; therefore, all
these points may be reduced to a state of rest by impressing on
them suitable forces, acting along the productions of these lines,
PRINCIPLE OF VIRTUAL VELOCITIES.
545
in a direction opposite to that of the motions produced; conse-
quently, if we denote these unknown forces by R, R', R“, &c., there
will be an equilibrium between the forces P, P', P", &c., and
R, R', R", &c. ; so that if r, r', r", &c., denote the virtual velo-
cities projected on the directions of these new forces R, R′, R“,
&c., we shall have, by the principle of virtual velocities that
has been demonstrated,
Pp+P'p'+P"p" + &c. + Rr + R'r' + R"r" + &c. = 0,
or simply
Rr+R'r'+R'7" + &c. = 0,
(c)
in virtue of equation (b), which is supposed to have place.
As equation (c) obtains for all infinitely small motions
compatible with the conditions of the system of points м, M', м",
&c., we may select for their virtual velocities, MN, MʼN′, M"N",
&c., the spaces actually described in the same instant; but as
these lines are laid off on the productions of the directions of
R, R', R", &c., it follows that all the projections r, r', r", &c.
will be negative, (No. 331), and, abstracting from the sign,
equal to these very lines MN, M'N', M"N", &c. In this case,
then, all the terms of equation (c) have the same sign, and
consequently, their sum cannot vanish, unless that each term
in particular is equal to cypher; hence we shall have
R.MN 0, R'. M'N' 0, R". M"N"= 0, &c.
=
=
Now, if the product R.MN= 0, we must have either R≈ 0,
or MN 0; in both which cases, it follows that the point M
cannot move; it is the same with respect to all the other
points; consequently the entire system is in equilibrio; which
it was proposed to demonstrate.
337. In the case of fluids, it will be shewn hereafter, by
means of their fundamental property, that the principle of
virtual velocities obtains likewise in the equilibrium of a sys-
tem of forces, the actions of which are transmitted by the in-
tervention of a fluid contained in a canal or vessel of any form
4 A
546
PRINCIPLE OF VIRTUAL VELOCITIES.
whatever. So that this principle of equilibrium, being appli-
cable in every case, will have all the required generality; for
when material points are detached from each other, the only
way in which the action of these forces can be transmitted
from one point to another, is either by means of inflexible rods,
of stretched threads, or of fluids contained in canals; and
moreover, if some of these points are immoveable, others per-
fectly free, and others constrained to exist on given curves or
surfaces, the system of material points thus constituted will be
the most general which there will be any occasion to consider.
Nevertheless, it will be perhaps not unnecessary to give
another demonstration of the same principle, for which we
are indebted to Lagrange, and which is grounded on notions
that are more elementary than the preceding; in fact, it is
founded on the possibility of our being able to replace all the
forces applied to any system whatever of material points, by
one weight acting in a manner which we now proceed to
explain.
338. If a point м (fig. 84) is solicited by a force p acting
in the direction of the line MA, we may, in the first place,
suppose that this force is applied to the point a, and that it acts
by means of a cord MA attached to this point м.
We may
then substitute for this cord, a thread which is alternately
rolled round a fixed and moveable system of pulleys, and is at-
tached by one of its two extremities to one or other of these
two systems; that which is fixed being supposed to refer to
the point a, and that which is moveable to the point м. If at
the free extremity of the thread, a weight к is suspended ver-
tically, its tension throughout its entire length will be equal
to K. If the dimensions of the pulleys be supposed to be in-
finitely small, the tensions of all the parts of this thread, which
terminate at the moveable system, will have the same direc-
tion; and, denoting their number by i, their resultant will be
equal to ik, which will act at the point м in the direction of
PRINCIPLE OF VIRTUAL VELOCITIES.
547
=
MA; consequently, if iK P, the action of the force p may be
replaced by that of the weight K.
The same will be the case with respect to the other forces
p′, p", &c., applied to the points м', м", &c., and acting in
the directions M'A', M"A", &c.; each of them can be replaced
by a weight equal to a submultiple of its intensity, and acting
in the manner that has been explained with respect to the
force P. Moreover, it is easy to perceive that, as is repre-
sented in figure 85, we can always make the same thread to
pass successively over all the fixed systems at a, a', a“, &c.,
and over all the moveable systems attached to the points
M, M', M", &c. Hence if we make
K=P, ŻK=P, K=p, &c.
M
(d)
i, i, i", &c., being whole numbers, we can, by suspending the
weight K at the free extremity of this thread, replace the system
of given forces P, P', P", &c., by this weight, the action of
which will be transmitted to the points м, м', м", &c., through
the intervention of this thread, and of the fixed and moveable
systems. Indeed, it is implied in equations (d), that the forces
P, P', p", &c. are commensurable; but this hypothesis is always
admissible, because their common measure к may be a weight
as small as we please, and even, if necessary, infinitely small(c).
339. Let us now conceive that there is impressed on the
points м, м', м", &c., a motion which, as also the directly
contrary motion, may be consistent with the conditions of the
system; after an infinitely small portion of time, let N, N′, N",
&c., be their positions, and let, as before, p, p', p", &c., denote
the projections of MN, M'N', M"N", &c. on the directions of
P, P′, p″, &c., or on their productions. The point N being
projected at a on the right line MA, each of the cords which
issue from a to м will be shortened by a quantity AM AN,
which, if infinitely small increments of the second order be
neglected, may be considered as equal to ma; on the other
hand, this cord will be lengthened by a quantity equal to мa,
548
PRINCIPLE OF VIRTUAL VELOCITIES.
་
if the point a falls on the production of AM; hence it follows,
that in consequence of the displacement of M, the weight K
will descend in the first case, and ascend in the second, by a
quantity equal to the product of Ma and i; which implies,
agreeably to what was stated relative to the sign of p, (No.
331), that, in consequence of this sole displacement, the posi-
tive or negative variation of its vertical height will be ex-
pressed by ip. The same will be the case with respect to all
the other points м', м", &c.; consequently, if & be an infi-
nitely small quantity that represents, according as it is positive
or negative, the entire quantity by which the weight к de-
scends or ascends in consequence of the simultaneous dis-
placements of all the points of the system, we shall have
¿
"=ip+ i'p' + i'p" +&c.
Now, when the weight K tends to descend, as it is the only
force which acts on the system, it is evident that nothing will
prevent it from producing the motion in question, if this value
of is positive; and that, if it is negative, nothing will pre-
vent the weight K from producing the direct contrary motion,
which is supposed to be equally possible, and for which the
sign of must be changed. It is therefore necessary, in or-
der that the equilibrium may obtain, that should be equal to
zero. Conversely, as the weight к cannot produce any mo-
tion whatever, without descending by an infinitely small quan-
tity in the first instant, it follows that it will not produce any,
and that if % = 0, the equilibrium will obtain for all the infi-
nitely small displacements of the points м, M', м", &c., which
are compatible with the conditions of the system.
Now, if к be multiplied by the equation
K
ip + i'p' + i"p" + &c. = 0,
which is necessary, and suffices to insure the equilibrium, it
will be changed, by having regard to equations (d), into
equation (b), which is that of the principle of virtual velo-
cities, it was proposed to obtain.
PRINCIPLE OF VIRTUAL VELOCITIES.
549
་
340. In this demonstration it is not assumed, as in the
former, that the principle has been previously demonstrated in
the case of an isolated material point. If the system be re-
duced to one sole point м, to which the forces P, P', P", &c.,
given in magnitude and position, are supposed to be applied,
we thould substitute for their simultaneous action that of one
sole weight K, as in No. 338; and, in the case of the equilibrium
of these forces, the principle of virtual velocities may be de-
duced from this substitution by the same mode of reasoning
as in the case just considered. Now, this principle will at
once furnish the equations of equilibrium of the point м, con-
strained to remain on a curve or a surface, or which may be
entirely free, (No. 39). In this last case, if one of the forces
be considered as being equal and contrary to the resultant of
all the others, the rules of their composition and resolution,
and also the theorem of the parallelogram of forces may be de-
duced from it. By applying this principle to the equilibrium of
three parallel forces, one of which is consequently equal and
contrary to the resultant of the two others, the rules of the
composition and resolution of parallel forces may, in like man-
ner, be deduced.
We may also, without difficulty, infer from the general
principle of virtual velocities, the equations of equilibrium of
a solid body entirely free, which have been already obtained
in another manner in No. 260.
In fact, we can, in the first place, suppose that all the
points of this body describe right lines mutually equal to each
other, and parallel to one of the axes of the coordinates. De-
noting the length of these lines by h, and by a, a', a", &c.,
the angles which their common direction makes with the di-
rections of the given forces, we shall have
=
p=hcosa, p' = h cosa', p" h cosa", &c.,
for the virtual velocities of M, M', M", &c., the points of the
solid body, projected on the directions of the forces P, P′, P",
•
550
PRINCIPLE OF VIRTUAL VELOCITIES.
*
&c., which are applied to these points; therefore, if these va-
lues be substituted in equation (b), we shall obtain, by sup-
pressing the factor h, which is common to all the terms, the
equation of equilibrium
P cosa + P'cos a' + p" cos a' + &c. = 0.
If the motions of the body parallel to the two other axes
of coordinates be respectively considered, two other equations
of equilibrium similar to this may be obtained in the same
manner.
each of these points will de-
We may also cause the body to turn about one of the axes
of the coordinates. In order to obtain the equation which
corresponds to this motion, let the coordinates of the points
M, M', м", &c., and the angles which the directions of the
forces P, P', P", &c., make with those of these coordinates, be
denoted by the same letters as in No. 260. If the rotation
takes place about the axis of z,
scribe an arc of a circle parallel to the plane of the axes of x
and y, the radius of which will be the perpendicular let fall
from this point on this axis. Moreover, as the body is sup-
posed to be solid, and therefore all its points to be firmly con-
nected together, by the nature of the solid body, the angle
described by this perpendicular will be the same for all its
points. If, therefore, it be supposed to be infinitely small,
and denoted by w, and if r, r', r'', &c. denote the distances of
the points м, M', м", &c., from the axis of z, their respective
virtual velocities will be rw, r'w', r'"'w", &c., and if S, d', S",
&c., denote the angles, either acute or obtuse, which the di-
rections of these velocities make with those of the forces
P, P′, P", &c., there will result
prwcosd, p'r'w cos d', p"r"w cos d", &c.;
for the expressions of the projections of these same velocities,
on the directions of these forces, or on their productions.
Moreover, if a, b, c be the angles comprised between the
PRINCIPLE OF VIRTUAL VELOCITIES.
55k
}
direction of the velocity ro and the parallels to the axes of
x, y, z, drawn through the point м; the same angles relative
to the direction of the force P being a, ß, y, we shall have
cosd = cosa cosa + cosb cosẞ + cosc cosy ;
but because ro is in the direction of a tangent at the point M
to the circle, the centre of which is in the axis of z, it is easy
to perceive that
cosb=
+1
8
y
cos a =
,
cosc = 0,
and, consequently,
p = rwcosd= (x cosß-y cosa) w.
In like manner we shall have
p' = ±(x'cosẞ' — y'cos a') w,
p" = ±(x" cosẞß”—y'' cosa") w,
+ &c.
The signs will depend on the direction of the rotation; we
should take the superior or the inferior signs in all these
values at the same time; therefore, if they be substituted in
equation (1), we shall obtain, by suppressing the factor±w,
which is common to all the terms,
p (x cosẞ — y cosa) + p' (x'cos ẞ' — y' cosa') + &c. = 0.
This equation of equilibrium is that of the moments with
respect to the axis of z, about which the motion has place; the
equations of the moments, with respect to the axes of the co-
ordinates of x and y, may be obtained in the same manner, by
making the solid body to turn successively about these two
right lines.
341. Equation (b) may be made to assume another form,
which will render it of easier application in particular cases.
For this purpose, let x, y, z be the coordinates of the point
M. in its position of equilibrium; x + Sx, y+dy, z + Sz,
552
PRINCIPLE OF VIRTUAL VELOCITIES.
what these become, when this material point is transferred to N
a point infinitely near; x, y, z, the components of the force P
in the directions of the productions of x, y, z, estimated posi-
tively; these infinitely small quantities Sx, dy, Sz will be the
projections of the virtual velocity MN on the directions of
X, Y, Z; and if p denote, as before, its projection on the di-
rection of P, we shall have (No. 331)
Pp=x} + xây + z8z.
If the corresponding quantities which refer to the points
M', M", &c., be denoted by the same letters with accents, we
shall also have
P'p'—x'x' + y′dy' + z'dz',
P"p"x"Sx" + x"dy" +z"Sz",
&c.
By adding these equations to the preceding, we can write
Pp + P'p' + P"p" + &c. = Σ(xdx + vdy + zdz):
the sum Σ being supposed to extend to M, M', M", &c., all the
points of the system, and it, consequently, consists of a num-
ber of similar parts, equal to that of these points. By this
means, equation (b) will assume the form
Σ(xdx + vdy + zdz) = 0,
which it was proposed to give it.
(e)
Now, whatever be the connexion between the points of the
system, it may be always expressed by one or more equations
between their coordinates. Therefore if L, L', L", &c., be given
functions of x, y, z, x', y', z', x", &c., or of a part of these co-
ordinates; and if we suppose that these equations are
L=0, L'=0, L"=0, &c.,
(f)
then as the simultaneous displacements of all the points of the
system must be compatible with the conditions to which it is
subjected, it is necessary that the coordinates x, y, z, x'y', &c.,
PRINCIPLE OF VIRTUAL VELOCITIES.
553
of м, M', M", &c., and x+dx, y + dy, z + Sz, x' + dx', &c., the
coordinates of N, N', N", &c., should successively satisfy these
equations; consequently, if infinitely small quantities of the
second order be neglected, we shall have
dL
dL
Sx +
dx dy dz
бу +
dL8z+
dL
Sx' + &c. = 0,
dx'
dL'
d
dL'
di
·Sx +
Sy +
·82 +
Sx' + &c. = 0,
dx
dy
dz
dx'
(g)
dL"
dL"
dL"
di”
Sx +
dx
dy
dz
dx'
&c.
Sy + 8z+ Sx + &c. = 0,
If the direction of the displacements of all the points of
the system be changed simultaneously, the signs of dx, dy, dz,
Sa', &c., will be all changed at the same time, and these equa-
tions will be still satisfied; so that the infinitely small motions
to which they refer, and the motions directly contrary, are
equally compatible with the given conditions, as is implicitly
supposed in the general statement of the principle of virtual
velocities, (No. 331).
This being established, by means of these equations (g),
there can, in each case, be eliminated from equation (e) a num-
ber of the quantities Sx, dy, Sz, dx', &c., equal to that of equa-
tions (f); those of these quantities which will afterwards remain
in the first member of equation (e), will be independent of each
other; consequently their coefficients may be put separately
equal to cipher; this will furnish all the equations of equili-
brium of the system, the number of which will be equal to
three times that of the material points M, M', M", &c., minus
the number of equations (f). When the positions of these
points, that is to say, the values of their coordinates x, y, z, x',
&c., are given, it is necessary that the components of the
forces P, P', P", &c., should satisfy these equations of equili-
brium; when, on the contrary, these forces being given in
magnitude and direction, the positions of the points of the
4 B
554
PRINCIPLE OF VIRTUAL VELOCITIES.
system are unknown, these same equations, combined with
equations (f), will enable us to determine all their coordinates.
342. As equations (e) and (g) are linear with respect to
Sx, Sy, Sz, dx', &c., the elimination of a part of these quan-
tities may be effected by the usual method, i. e. by adding
these equations together, after having multiplied equations (g)
by indeterminate factors, and then, making in this sum, the
coefficients of the quantities dx, Sy, dz, dx', &c., which it is
proposed to eliminate, equal to cipher. As the coefficients of
the remaining quantities must be likewise equal to cipher, it
follows that the coefficients of all the quantities dx, dy, Sz, dx',
&c., should be indiscriminately put equal to cipher, in the
sum in question; hence there will result a number of equa-
tions equal to that of the coordinates, between which there will
remain, in each case, indeterminate factors to be eliminated, in
order to obtain the equations of equilibrium of the system.
If we denote the factors by which equations (g) should be
multiplied, by λ, λ', λ“, &c., we shall have, by what precedes
dL
dL'
x+λ
dL"
dx
dx
+λ"
dx + &c. = 0,
dL
dL'
x+1.
dL”
h
dy
dy dy
+λ"
+ &c. 0,
(p)
dL
dr
dz
dz
di"
+λ"
dz
+ &c. = 0,
for the equations arising from the coefficients of da, Sy, dz; in
like manner we shall have
dL
x' +λ
dɩ'
dx'
dx'
dx'
+ &c. = 0,
di
dr'
dy
+1
di"
dy
+1"
dy'
+ &c. = 0,
(h')
dL
z' + λ
dL'
dz'
+ d
+1
dz'
dz + &c. = 0,
for those which arise from the coefficients of dx', dy', Sz'; and
so on.
PRINCIPLE OF VIRTUAL VELOCITIES.
555
Instead of simply eliminating λ, λ', λ", &c., their values
may be obtained by means of these equations; and we now
proceed to shew how we may deduce from them, in mag-
nitude and direction, the forces arising from the connexion of
the points of the system, which act on all these points, and
constitute an equilibrium with the given forces P, P', p", &c.
The determination of these unknown forces is an important
part of the problem of equilibrium, the general and complete
solution of which will be found thus comprised in equations (f),
(h), (h'), &c., taken together.
343. If all the points of the system, minus the point м,
be rendered fixed, the equilibrium will not be disturbed. In
virtue of equation L 0, the point м will be then con-
strained to move on the surface of which L = 0 is the equa-
tion, and in which the coordinates x, y, z, are the sole vari-
ables. Now, if the resistance of this surface be denoted by μ,
it will act in the direction of one of the two parts of the nor-
mal at M, and we can substitute for this surface, or the equa-
tion of condition L = 0, this unknown force. In like man-
ner, we can replace L' 0, by a force μ, acting perpendicu-
larly to the surface which belongs to this equation; L" 0
by a force μ, normal to the corresponding surface, and so on;
therefore, if these normal forces µ, µ, µ,,, &c., be joined
with the given force P, or its components x, y, z, the point м
may be regarded as altogether free and isolated. Conse-
quently, if a, b, c, denote the angles which the direction of
the force μ makes with lines parallel to the axes of x, y, z,
drawn through the point м; and if a₁, b₁, c₁, be the same angles
relative to the force µ₁, and so on; we shall have
Mill
x + μ cos a + µ, cos а, + µ,, cos a,, + &c. = 0,
x +μ cos b + µ, cos b, + µ,, cos b,, + &c. = 0,
z + µ cos c + µ, cos c, + µ,, cos c,, + &c. = 0,
for the three equations of equilibrium of the point м.
over, if, in order to abridge, we make
More-
556
PRINCIPLE OF VIRTUAL VELOCITIES.

บ.
1,=
=
+ &c.
dL 2
√(dt)² + (dt)Ⓡ
dx
dL 2
dx
八​2
dL\2
(
318
+
dy dz

dr/\2
dy
+(税​)
2 dL 2

√(d'") + (d") + (d')",
dx
dy
dz
we shall have also, by known formulæ (No. 21),
cos a =
1 dL
v dx
cos b =
1 dl
v dy
1 dl
COS C =
v do
1 dl
1 d'
cos α =
a
v, da
cos b₁ =
b,
ν
v, dy
cos c, =
v, dz
1 dL'
บ
(i)
1 dí"
cos a,, =
cos b₁=
บ
V
dx
1 dL"
V₁dy
1 di"
cos c₁ =
dz
VII
&c.;
by means of which, the three equations of equilibrium will
be changed into the following:
μ
v dx
dL
x+ +
μι
dL'
V₁ dx
+
μα
μ,, dĹ!
Vil
dx
+ &c. = 0,
Y+ +
μ dL
μι dL'
v dy
v, dy
+ Mudy"
vdy
+ &c. = 0,
z+
ματ
v dy
+
μι
dL
+
μι
dL"
ν
dz
V₁ dz
+ &c. = 0.
Now from a comparison of these equations with equations
(h), with which they are identical, it appears that
μ πνλ, μπολ, μα = ν λ", &c.
Thus, relatively to the point м, the forces which arise from
its connexion with the other points of the system, are expressed
by the products vλ, v‚λ', v„\”, &c.; as these forces must be
positive quantities, the radicals v, v,, v, &c., must have the
PRINCIPLE OF VIRTUAL VELOCITIES.
557
same signs as the quantities A, A,, λ„, &c., and their directions
will be completely determined by equations (i).
If, in the same manner, µ, µ, µ,,", &c., denote the forces
arising from the connexion of the system, which act on the
point м', and are normal to the different surfaces on which it
is constrained to exist, when all the other points м, M", M"",
&c., are fixed, we shall find
µ'=v'λ, µ₂ = v{'X', _µ₁ = v,,'X", &c. ;
μι μια
in which, in order to abridge, we suppose

v' =
v =
&c.
dL
2
2
(赤​)+(赤​)+(赤​)・
2
dy'

2
√ (dr)² + (dr)² + (d)²;
dx' dy' dz'

2
2
=√()+(税​)+(水​)
dz
2
Similar expressions may be obtained for the forces relative
to the points M", M"", &c.
344. It appears from a comparison of the values of µ an
μ', that
µv' = µ'v ;
so that they are as the quantities v and v'. When, therefore,
two material points м and м' are connected together, and also,
if we please, with any number whatever of other points, by
an equation L = 0, there results in the state of equilibrium,
the forces μ and μ' applied to м and M', the magnitudes of
which are as v and v', and these forces make with the axes of
the coordinates, angles whose cosines are
1 dl
1 dl 1 dL
v dx'
v dy
v dy'
v dz
vdzs
for the force
fly
and
1 dL 1 dl 1 dl
v'dx'' v'dy'' v'dz'
558
PRINCIPLE OF VIRTUAL VELOCITIES.
་
for the force u'. The direction and magnitude of these forces,
depend on the sign and magnitude of λ, which in each case may
be deduced from the equations of equilibrium.
The consideration of the surfaces on which each of the
points of a system are at liberty to move, when all the
others are supposed fixed, determines the normal directions of
the forces arising from the connexion of these moveables, for
each of the equations by which this connexion is expressed
(No. 290); but we cannot deduce from them any relation
between the forces relative to two material points connected
together by the same equation; and it is the principle of vir-
tual vclocities, or equations (h), (h'), &c., that have been de-
duced from it, which enables us to determine this ratio à priori,
in the case of equilibrium.
345. For an application of these formulæ, let us consider
the case of the funicular polygon, which has been already dis-
cussed in the first section of the preceding chapter; and let us
suppose that the material points м, м', м", &c., are the suc-
cessive summits of this polygon.
If l, l', l'', &c., be the given lengths of the sides MM', MʼM”,
м"M"", &c., equations (f) will in this case be
L = √(x−x')²+(y—y')² + (z—2)² — 1 = 0,
2
L' = √(x' — x")² + (y'—y") + (z' —z")² — l'′ = 0,
L″ = √ (x" — x''')² + (y'' — y''')² + (z″ — z''')²— 1″ — 0,
&c.;
hence there will result(b)
dL
di
x-x' dı'
dL'
x' — x''
di
dy
di
dz
SE SIS SI
11
&c.,
dx'
dx'
dx"
dL
y—y'
y' du'
di'
y'—y"
=
&c.,
dy'
dy'
dy"
V
dL
·z' dı'
dı'
&c.,
dz'
- dz'
dz"
ľ
PRINCIPLE OF VIRTUAL VELOCITIES.
559
and all the other partial differences of L, L', L", &c., which
occur in the preceding formula, will be equal to cipher.
In the case of the two points м and м', we shall have
v=v=±1, μ=µ'= ±λ,
in which the superior or inferior signs should be taken, accord-
ing as the value of A is positive or negative. From this and
the preceding equations, we may infer that the points м and
M', will be solicited by equal and opposite forces, acting either
in the direction of the line MM' or of its productions, and the
quantity λ will, abstracting from the sign, express the com-
mon intensity of these forces. The same will be the case
relatively to the points м' and м", м" and м"", &c.; so that
in the state of equilibrium, the quantities A, A', λ", &c., will
express the contractions or tensions of the successive sides
MM', M'M", M"M"", &c. Since, by equations (i), we have
XC
x'
cos a = ±
cosb=+
y — y'
"
cos c = ±
=±
in which the superior or inferior signs should be taken, accord-
ing as the value of A is positive or negative, it follows, for
example, that the force applied to the point м will be directed
from M towards м', and will express a contraction of the side
MM' when this value is negative, and, when this value is posi-
tive, this force will act in the opposite direction, and express a
tension, and will indicate that the string is stretched. One
or other of these cases is possible, if the sides of the polygon
are inflexible rods, joined by hinges; and the second case can
alone obtain, if the sides are flexible threads.
Equations (h), (h'), (h"), &c. may be written as follows:
x = λ
x()
y
x = 1 (1/2),
入
​z = λ
=),
>
560
PRINCIPLE OF VIRTUAL VELOCITIES.
x'
x² + 1 (207
') = x (~
x'
V
y'
~ +1 (13) = x (~~ ~~),
2)=x(
z'
λ
x² + 1 ( — — 2 ) = x (~ ~ ~ ~ ' ),
"+x(2
λ
V
x" + " ("*'" =—= "") = x " ("*"" => "'"),
y"—y'
で
​Y!!
+
+x(2
=
で
​+x(²
)=\"
z" +λ
الح
Z!!
で
​,
It appears from a consideration of the three first, that the
tension A is the resultant of the forces x, y, z (c). By adding
them to the three following, we shall obtain
x + x = x² (= = = x^)
= x²(x'"'—x'),
"y"
V
x + Y' = \'(''" —
Y')
z + z' = x²(½″ ~
~ ~ ' ) ;
入
​い
​which shews that the tension X' is the resultant of x', Y', Z',
and of the forces X, Y, Z, transferred to the point м, parallel
to themselves. By proceeding in this manner, we shall ob-
tain, for the tension of any side whatever, the same value as
in No. 287.
The number of summits M, M', м", &c., being denoted by
n, that of the preceding equations will be 3n, and that of the
tensions λ, X', λ”, &c., will be equal to n−1(d). Therefore if
these quantities be eliminated, there will result 2n +1 equa-
tions of equilibrium, which combined with 7, l', l", &c., the
n - 1 given lengths of the sides of the polygon, will be suffi-
cient to determine the 3n coordinates of its summits, and,
PRINCIPLE OF VIRTUAL VELOCITIES.
561
consequently, its figure of equilibrium. But this mode of de-
termining the figure, &c., is attended with no practical advan-
tage; it is therefore preferable, as has been done in No. 286,
to trace the sides of the funicular polygon successively, by
means of the given magnitudes and directions of the forces that
act at its different summits.
346. In the case of м, м', м", &c., any system whatever of
material points, if the given forces which are applied to these
points, arise from their mutual attractions or repulsions, or
from similar forces which emanate from one or more centres,
we shall have
Σ (xdx + ydy + zdz) = d.6(x, y, z, x', y', z', &c.) ;
in which denotes a given function of the coordinates of
M, M', M", &c., dependent on the law of these forces in a
function of the distances.
In fact, with respect to forces emanating from fixed cen-
tres, this follows from what has been established in No. 158.
Moreover, let u express the mutual action of м and M', which
we will suppose to be attractive; let also u be their mutual
distance, so that u may be a given function of u; it will be
expressed by the equation
u² — (x' — x)² + (y'− y)² + (z'—z)².
The cosines of the angles which мм' makes with lines drawn
through the point м, in the directions of the positive axes of
x, y, z, will be
x' — x y'-y
И
и
;
W
if these be respectively multiplied by u, we shall obtain the
components of this force applied to the point м, and acting in
the direction MM'. Those of the same force u, applied to the
point м' in the direction M'M, will be equal and contrary;
hence it follows that the part of the sum Σ which arises from
the action and reaction of м and M', will be equal to
4 C
562
PRINCIPLE OF VIRTUAL VELOCITIES.
U
[(x'—x) (dx—dx') + (y'—y) (dy—dy') + (z' — 2) (dz−dz')].
But, by differentiating the value of u², there results
udu=(x' — x) (dx' — dx) + (y'—y) (dy'—dy) + (z' —z) (dz'—dz);
from which it is evident that the preceding quantity is equal
toudu, that is to say, to the differential of a function of u.
Similar expressions may be obtained for the parts of the sum
Σ, which arise from the mutual actions of the other points of
the system; consequently, its entire value will consist of terms,
all of which will be exact differentials, and this value will be
also the differential of a given function of the coordinates of
all these points.
In virtue of equation (e), this function, which we will de-
note by p, will be a maximum or a minimum, relatively to the
values of the coordinates that belong to a position of equilibrium
of the system; and, conversely, if the maximum or minimum of
the function & be determined, regard being had to the equations
(f) which may exist between the coordinates, the values that
will be obtained for these variables, will refer to positions of
equilibrium. Hence it follows, that when the system of points
M, M', м", &c. is in motion, in which case their coordinates,
and consequently, the quantity are functions of the time,
this function will attain its maximum or minimum, whenever
the system passes through a position in which it would remain
in equilibrio, if the points of which it is composed had not
any acquired velocity.
φ
347. There is an essential difference between the maxi-
mum and minimum of the function p, which, as it is of conse-
quence to take into account, we now proceed to explain.
The state of equilibrium of a body or a system is said to
be stable, when, if the moveables be made to deviate, ever so
little, from their positions, they tend to revert to them, by
making small oscillations which eventually, the frictions, and
the resistances of the media in which they move, render
PRINCIPLE OF VIRTUAL VELOCITIES.
563.
insensible, or finally extinguish. The equilibrium is instable
or instantaneous, when the body, or system of bodies, being
made to deviate a little from it, recedes more and more from
this position, until it finally terminates by being turned upside
down. If we assume that there is no friction, which does to
a certain extent retain the bodies in their positions, this second
state of equilibrium is a case purely mathematical, that can
never be observed, since the least disturbing force is sufficient
to destroy it.
This being established, the equations furnished by the
principle of virtual velocities, or, what is the same thing, by
the condition of the maximum or minimum of the function 4,
are common to these two states; but the maximum refers to
the case of stable, and the minimum to the case of instan-
taneous equilibrium; and this is, in fact, what will be shewn in
a subsequent chapter, when the nature of the motion that has
place when a system of material points is made to deviate from
any state of equilibrium will be considered. At present, we shall
content ourselves with furnishing examples of these two states
of equilibrium in the case of a system of heavy bodies, and
with pointing out a remarkable property of its centre of
gravity.
348. Let gravity be supposed to be the sole force applied
to the points M, м', m", &c., which are the centres of gravity
of bodies, whose weights are denoted by w, w', w", &c., then
as this force is vertical, and acts in the direction of the weight,
we shall have
7=W, z' = w', z"=w", &c.;
the other components will all vanish, and there will result
dp = wdz + w'dz' + w'dz" + &c.
But if n denotes the sum of the weights w, w', w', &c.,
and z₁ the vertical ordinate of their centre of gravity, drawn in
the direction of the gravity, we have likewise (No. 64),
-
564
PRINCIPLE OF VIRTUAL VELOCITIES.
112₁ = w²+w'z' + w"z" + &c. ;
consequently, there results
dė =
døпdz₁, = c + п11,
c being an arbitrary constant.
8
From this it follows-1st, that the ordinate z₁ is a
quantity which should be either a maximum or minimum,
when the system is in equilibrio, and conversely; 2ndly, that
the maximum of a corresponds to the case of stable equi-
librium, and its minimum to the case of instantaneous equi-
librium.
Thus, the condition of equilibrium of any system what-
ever of heavy bodies, consists in this, that the centre of gravity
of the entire system is the lowest or highest possible; the
lowest, when the equilibrium is stable, and the highest when
it is only instantaneous.
349. It follows from this theorem, that if a heavy chain
attached by its extremities to fixed points, is in equilibrio, its
centre of gravity will be the lowest possible; this agrees with
the result of No. 296.
If a heavy material point is placed on a curve, and if in
several points of this curve, the tangent is horizontal, the ver-
tical ordinate of the material point, estimated in the direction
of gravity, will be a maximum for those of its points, in which
the concavity is turned upwards, and a minimum in the case
of those points for which the concavity is directly opposite;
consequently, the first will be positions of stable, and the last
positions of instantaneous equilibrium.
If a homogeneous heavy ellipsoid rests on a fixed hori-
zontal plane, its centre of gravity or of figure will be the
lowest possible, when the ellipsoid touches the fixed plane in
one of the two extremities of the least of its three axes; and
in this case the equilibrium will be stable. When it touches
it in one of the two extremities of the greatest of its three
axes, its centre of gravity will be the highest possible, and the
equilibrium will be only instantaneous
PRINCIPLE OF VIRTUAL VELOCITIES.
565
Finally, if the point of contact is an extremity of its mean
axis, the elevation of the centre of gravity will be a minimum
for one part of the sections of the body, and a maximum for
the other sections; consequently, the equilibrium will be
stable or instable according as the displacements have place in
the direction of the first or last sections. As what has been
now stated is evident à priori, it will enable us to verify the
theorem of the preceding number.
Let us now suppose that two homogeneous heavy liquids
are poured into a vessel; if the surface of separation and that
which terminates the upper liquid are respectively hori-
zontal, and if this liquid be the one which has the least den-
sity, the centre of gravity of these two liquids will be the
lowest possible; for it is easy to perceive, that if one or other
of these two surfaces be inclined or curved, the centre of
gravity of the system will be always elevated. These two
surfaces being always horizontal, if the less dense liquid be
below the other, in the same manner, it will appear that the
centre of gravity of the system will be the highest possible.
Consequently, in order that these fluids may be in equilibrio,
it is necessary and sufficient that each of them be terminated
by a horizontal plane; but in order that the equilibrium may
be stable, it is moreover necessary that the densest liquid
should occupy the inferior part of the vessel. When the dif-
ference of the two densities is inconsiderable, it is possible,
by taking proper precaution, to make the denser liquid float
on the other, but this instable equilibrium can only maintain
itself sufficiently long to be observed, in consequence of the
friction of the two liquids against the sides of the vessel.
:
:
NOTES.
INTRODUCTION.
(a) As there are ninety degrees in the quadrant of the circum-
ference, there must be 90, 60, or 5400 minutes, and 5100.60, or
32400 seconds in it.
(b) By stating this proporition we have
π
32400".2n
w: 32400"::n:
∞ =
2
T
and by substituting the value of given in the text, we shall obtain
the expression for w.
(c) In fact, if X, X, X", be the projections of a given areal on
these planes, and if i, i', i", be the respective inclinations of on these
planes, by what as been just established, we shall have
λ=lcosi, xl cosi', "l cos i"; .x² +λ²² +λ²=
x"
·
1² (cos² i + cos² i' + cos² ¿'') = l², since cos² i + cos² ¿' + cos² ¿"= 1.
See No. 281.
(d) See on this point Le Journal de l'Ecole Polytechnique,
18th number, page 320.
(e) By substituting in the expression for Fb Fa, the values
which have been deduced for its second and third terms, we obtain as
♪³, ♪4, &c. are neglected.
Fb—Fa=dΣƒ(a+ið)+¦§‹ƒb−ƒa)—{d²(ƒ′b—ƒ'a)+¿d.
(ƒ'b—ƒ'a),
which is evidently equal to the value of Fb-Fa given in text; and if
f(a+id) be expressed in a series, there results by obliterating the
8
Safada
term fa, the expression for farda given in the text.
568
NOTES.
(ƒ) When one of these equations is taken from the other, their
difference, which is equal to cipher, becomes
2(x − x)dx+2(y' − y) dy+2(z' — z) dz + dx²+ dy²+dz²=0,
but dx², dy³, dz², being infinitely small quantities of the second order,
may be neglected.
(g) By Taylor's theorem, if u = Fx, when x receives an incre-
ment dø, and u becomes u', the value of w expressed in a series is
du
u'=u+ +
d2u
d³u
1.2.
1.2+
1.2.3
+ &c.;
if u cosa, when a receives an increment and becomes a', then
d.cos a ď² cos a d cos a
cos a' = cos a +
1
+
1.2
+
1.2.3
+ &c.
Now if in the value of sin² ♪ given in the text, we substitute for
cos a', cos B', cos y', their values, and omit all terms in the resulting
expressions after the third, as involving infinitely small quantities of a
higher order than the second, we obtain
sin² 8 = 1- [cosa (cosa +
B
d cosa
+
de cos a
2
d².cos
cos 8( coss+d.com+d+cos 5) - cosy (cosy+do+dos)]
1
2
2
Now since cos² a + cos² ß + cos² y = 1, and therefore cos ad cos
a+cos ẞd cos ẞ+ cos yd cos y = 0, the expression between brackets
becomes equal to
-[1+ cos a d² cos a + cos ẞd cos ẞ+ cos yd cos y]',
that is to
— 1 — cos ad² cos a
-
cos ßd² cos ß - cos yd² cos y,
as infinitely small quantities of a higher order than the second are
neglected; hence it is evident, that the values of sin²d is that given
in the text.
1
(h) By substituting in the expression for d.cos a, the values of
ds² and dsd's we obtain
d.cos a=
(dx²+dy²+dz²) d²x − (dx d²x + dyd³y + dz dz) dx,
ds³
which is equal to the value of d cos a, as will be evident by oblite-
rating dada, and concinnating the other terms.
:
NOTES.
569
(i) If the squares of d cos a, d cos ß, d cos y, be taken, and then
added together, we obtain
(d.cos x)+(d.cos B)+(d.cos y)=
dx² + dy
dso
(dy d²x — dx d³y)²+
dx² + dz²
dy²+ dz²
dso
(dz d²x — dx d²z)² +
2 dy dz
(dz d²y — dy d²z)² +
ds
ds³ ds3
dx dz
(dy d³x—dx d³y).(dz d³x— dx d²z) +233 · · (dx d³y—dyd²x) •
ds3
ds³•
(dz d³y—dy d²z)+2
dx dy
ds³ ds3
(dx d²z — dz d²x) (dyd²z — dz d³y).
Now if the factors in the three last terms be respectively multi-
plied together, they will be equal (by concinnating and obliterating
the quantities which mutually destroy each other) to
dz2
dy2
dx²
· (dyd²x—dx d²y)²+
ds
156 ·
(dz d²x — dx d²z)²+
•
(dz d²y — dy d²z)²,
ds
hence then we obtain
´dx²+dy²+dz²\
+dy+dz²)
(d cos a)*+ (d cos B) + (d cos y) =
[dyd²x—dxd²y)²+(dzd²x—dx d²z)²+(dz d²y—dy d²z)²],
which is evidently the same as the expression for & given in text.
(k) By substituting these expressions of A, B, C, it is easy to per-
ceive that the resulting expressions will be identical.
(1) When the equation
(x' — x) dx + (y' − y) dy + (≈' — ≈) dz = 0,
is differenced with respect to x, y, z, there results
(x' — x) d²x + (y' — y) d²y + (≈' — z) d²z — dx² — dy² — dz²—
(x' — x) d²x + (y' — y) d²ÿ + (≈' — z) d²z — ds² = 0.
Now if the equation
(x-x)dx + (y'—y) dy + (≈'—z) dz = 0
be multiplied by day, and if from the product, the value of
(x' — x) d²x + (y' − y) d³y + (≈' —z) d²z = ds²,
-
multiplied by dy, be subtracted, there will result by concinnating
4 D
570
NOTES.
(x' —x) (dxd³y dyd¹x) = (z' — z) (dyd²z— dzd²y) + ds'dy, (1)
and if the same equations be multiplied by dx and de respectively,
and then the second be taken from the first, we obtain in like man-
ner,
(y' y) (dyd²x— dxd²y) = (z' — z) (dxd²z—dzd²x)+ds²dx. (2)
Now if equation (1) be multiplied by dyd²z-dzd²y, and equation (2)
by dxd²z-dzda, there results
(x' — x) (dyd²z — dzd³y) (dxd³y—dyd²x) = (≈' — z) (dyd²z— dzd³y)²+
ds²dy (dyd²z — dzd²y) ;
(y' —y) (dxd³z—dzd²x) (dyd²x— dxd³y)=(z' —z) (dxd²z— dzd²x)²+
ds dx (dxd²z - dzd²x);
and if the second be taken from the first, we obtain
[(_x)(dy ~ _dzdy)+(y−y)(ddz_dzda)](ddy_dydr)=
(z' —z)[(dyd²z-dzd²y)²+(dzd²x— dxd²z)]²+ds²[(dyd²z — dzd²y)dy
- (dzd²x — dxd²z)dx];
-
but from the equation of the osculating plane given above, it is
evident that
X
[( – )(dy z – dxdy)+(y−y) (dd cdxd z)](ddy_dydr)=
— (z' —z) (dxd³y — dyd²x)² ;
therefore by substituting this last quantity for that to which it is
equal, we obtain
(z' — z) [(dyd²z — dzd²y)² + (dzd²x— dxd²z)² + (dxd²y — dyd²x)²] =
[(dyd²z—đzd³y)²
ds²[dy.(dyd²z — dzd³y)— dx (dzd³x— dxd²z)];
and, consequently,

ds² [dy(dyd²z-dzd²y) — dx (dzd²x— dxd²z)]
(dxd³y—dyd²x)²+(dzd²x—dxd²z)* + (dyd²z—dzd²y)²
p²
and if for the denominator of this expression its value be substi-
dsб
tuted, the value of z'—z given in the text will be obtained; the ex-
pressions for x' —x, y'—y, may be obtained in the same manner.
(m) When the values of (x'—x)², (y' —y)², (z' —z)², are added
pt
together there results, after all reductions, a factor of equal to
dss
02
dss
as is evident from the value of 8 determined in note (i).
NOTES.
571
BOOK I.
CHAPTER I.
(a) This is, in fact, a particular case of that axiom which was used
by Archimedes in demonstrating some of the fundamental propo-
sitions of mechanics; it was Leibnitz however who first announced
it as a general principle of philosophy in the following manner:
"Nothing exists in any state that is not determined by some reason
to be in that state rather than in another." Hence if, as in the pre-
sent case, there are two conditions, and no reason to determine a
subject to be in one of them rather in the other, we are to conclude
that it is in neither. It has been denominated the principle of suffi-
cient reason.-See King de Origine Mali Bishop Law's Notes, and
also Theorie Analytique des Probabilities, Introduction, page 2. This
principle is frequently assumed in the following treatise.
(b) As cosẞ
2
cos² ß - sin² 6-2 cos 6-1,
=
(4/6)² = 2 cosẞ+2= 4 cos²ß.
(c) For by performing the multiplication we have
R. COS RMO = R cos a cos g + R cos b cos h + R cos c cos k=
x cosg+Y cos h + z cos k
;
and if equations (c) be multiplied by cosg, cosh, cosk, and then
added together, there results
x cos g + Y cos h +z cos k=
P (cos g cos a + cos h cos B + cos k cos y + &c.) + :
P' (cos g cos a' + cos h cos B' + cos k cos y' + &c.) +
p'' (cos g cos a" + cos h cos B" + cos k cosy" + &c.);
that is,
R. COS RMO P COS PMO + P' cos P'MO + &c.
(d) By means of these formulæ (b) we have
dL
dx
dL
dL
Y
dy
dx
di
X
dy
CHAPTER II.
(a) Since q, q', the components of s' are less than s or s', and as
the effect of a' is destroyed, because it is supposed to pass through
the
prop, the only forces that remain will be s and Q, and since they
act in opposite directions, their resultant will be s— Q.
572
NOTES.
CHAPTER III.
(a) By similar triangles we have
NG:M/H::MN:MM::P:P+P….(P+P)NG = MH.P,
and by adding the identical equation to both members of this, we
obtain
(P+P) (NG + GK) P.MQ + P'(M'H + HQ') =
(as NG+GK)= NK, and MQ = ≈, M'H + HQ' = M'Q' = ≈',)
the expression in the text.
(6) It is not implied in the proof that these planes are at right
angles to each other, and it is easy to demonstrate that if this condi-
tion is satisfied with respect to any two planes parallel to the com-
mon directions of these forces, it will be so with respect to all
others.
(c) This follows from equations (1) of No. 55.
CHAPTER IV.
(a) By adding these equations together, we obtain
x/2
m(m+m' +m'' + &c.) x² + m' (m + m' + m'' + &c.) x¹² +
m" (m + m' +m" + &c.) x" &c. = mm' (x² + x¹² — 2xx')+
mm" (x² + x/12
that is,
2xx'')+m'm'' (2012 + 20/12
2x'x'') + &c. ;
(m+m' +m" + &c.) (mx² + m'x'² + m''x!!²) =
M. (mx² + m²x²² + m¹¹² xll² + &c.) =
mm' (x − x¹)² + mm" (x — x")² + m'm'' (x' — x'')² + &c.
CHAPTER V.
(a) By making this substitution, in the first of equations (1), we
obtain
lx₁ = Sads+ Ssds cosa;
and by integrating between the limits s。 = 0, s, = l, there results
lx₁ = al + 1/2 c COS, ···x₁ = a +
$
(b) x = a cos—;
cos;
a
1
COS.
2
xdsa cos-ds, and Sæds = a²sin-; when
a
this integral is taken between the limits
S
α
1
2a'
2a'
we shall evidently
NOTES.
573.
is equal to twice the sine of half the arc, c = 2a sin
obtain the value of l☛, given in the text; and as the chord of an arc
I
2a
lx,=ac. When /
2 a
is a quadrant, and ·.· equal to
an
2
then ca√2, ·.· x' =
; when lis
T
2a
equal to ar, then ₁
and a=
=
1
T¹, i. e. in this case a = quadrant ·
2
π
of the circle whose radius is x,; and when l = a, x₁ = c.
1
(c) In all conic sections, the two last equations (2) may be inte-
grated in a finite form, as is immediately evident by substituting for
√1+ dye
dx2
its value in function of x, for example, the two last
equations become, in the case of the parabola, in which y²= px.
dx,
вх
lx₁ = Sox (1 + 2) *da
1
by = S&px ( 1 + 2)* dx;
α
which equations can be integrated in a finite form; consequently, if
the value of I was also given in a finite form, x, and Y₁ will be ob-
tained in functions of a and ẞ; in the case of an ellipse, of which
α
1
b2
the equation is y²
a² (a² — x²), ds =
dx √ a* — (a² —b²) x²
ανα
which
X2
dx V1 — e²x²
√ I — x²
by making a 1, a² — b² = e², can be reduced to
and in the hyperbola of which the equation is y² =
ds=
dx√ (a² + b²) x²
a. V x2 - a²
a4
be
a² (x — a²),
which by making a²= 1, a² + b² =
1+b²=e², can be reduced to
Sdr Vo
dx Ve²x²- 1
√ x² - 1
; now these values of
ds, in the case of the ellipse, cannot be obtained in a finite form,
though when they are respectively multiplied by x and y, the inte-
grals of the resulting expressions can be obtained in a finite form, and
therefore the values of Y₁ and X'1 •
(d) See the Journal of the Polytechnic School, 18th Number,
page 431.
574
NOTES.
尊
​(e) By differentiating this equation we obtain
dp +
dq=
(c-g)
cdq
√2cq-q² √2cq-q²
qdq
··· dp
********
√2cq-q²
(f) By substituting the value of dy² in the equation ds² = dy²+
da², there results
a²
ds² = dx²
2αx + x²
ax x²
+ dx²,
that is,
ds2 =
=("
a²
2αx + x² + ax
x2
ax X2
2º) daº,
a (a−x)
a
11
dx³ : ds =
.dx.
X
X a-x
(g) In the equation 2x,√x=§√x dx, there results by inte-
3
X
X
grating, 2x₁√x = 3 x² •.• x¸ =
and in the equation 2y, √x =
igo
Syd, by partially integrating we
obtain
2y₁ √x=2y√ x − 25√xdy;
and by substituting for dy, we obtain
25 √x dy = 2
x.(a-x).dx
√x. √ a― x
SVx.
=2√ a―x.dx;
the integral of this last quantity, taken between the limits a and 0,
is ((a — x) — a‡,) consequently the value of y, will be that given in
+
the text.
(h) When a semi-circle revolves about its diameter as in this case,
Y₁ =
1
2 a
T
(see note (b) ·.· 2πly₁ = 4πα², i. e. the surface of the sphere
is four times a great circle of a sphere.
(i) In the equation a²c
their respective values 2a sin
comes
and a cos- be substituted, it be-
α
‡ a' sin = + a' sin cos² + + (al — 2 a' sin cos);
} †
ch² + (al-ch) x,, if for c and h,
เ
2 a
1
เ
,
2a
2a
2a
2a
1
1
,
and as cos²
= 1 — sin²-
2a
2a
2 sin 2a
COS
= sin
there results
2a
a³
a³
2a
2a
し
​§ œ sin — = faª sin — — fœª sin² + (al — a² sin²) x, ;
§
1
2a
1
NOTES.
575
by obliterating the quantities which are common to both sides of
the equation, and dividing by a, there results
¥ a' sin' ' = 1 (1 — a sin²) ~",-
2a
—
1
when 7=α, the first member of this equation becomes a², and
the second
аX{•
(k) By equation (a) of No. 73, we have dy =
(ax-x²)
X2
x dy
AX
(α — x)
dx, and
Vo
ax X2
x² dx
dx = √ ax—x². dx; and ·.· Sx²dy=5x√ ax
= 1 / α√ √ αx — x². dx
= as
Ꮊ
and because √ √ ax-x² dx = y,
1
SG -
a
x√αx
ax x2 dx;
we obtain
xx₁ = }} x²y — } { x²dy = x²y − 1 ay + † (ax— x²)†,
— 1
(1) Since Sxydy = Sy√ax - x²dx, by substituting for y its va-
lue, there results
α
Sxydy = §(ax—x²)dx + — Sz√ ax − x²dx =
α
IN
x2
100
ax — x² dx.
+ - Sz√ ax-
X3 a
3
a²
dx
2
(-)
dx,
(m) dy =√ ax—x².dx =
y=
S
V
dx
ax X2
8 10
Vax
Vax
X2
x² - √ √ ax — x². dx;
α
a)
x vax
—
S
dx
there results by
Vax
X2
hence as y√ ax x² dx, z=
=
substituting y and z for these values, and concinnating
a²
2y=
x vax
·.· as dz =
dx
Vax
229
if the first member of this equation be mul-
tiplied by dz, and the first term of second member by dz, and the
dx
second term by its equivalent.
there results by dividing
√αx
by two, and integrating Sydz=
a²
Z2
(ax
x²).
16
576
NOTES.
(n) Now if these values of ydz and y be substituted in equation.
(6), there results
§ z√ ax — zº dx = fa²z² — — (§a—x)√ ax—x²—¡¡a²x²+‡(ax—x²);
from which it is easy to obtain equation (7).
-xdx
16
a²
' a² = x², dy² + dx² = ada², and
√ a²x²› dy²+
(0) In this case as dy
s=2π
So, adx, `sx₁ = 2x Sexdx, ··s=2xd(ß—a), sx¸=ra(ß³—a²).
dx,
α
α
1
(p) By performing the integration, we obtain as dy=x-x²
a
Vax
s=4πу√ αx—4π√ √ ax.dy=(−4π√㕧√α-x.dx,
=4πу√ax+¹‚²π√ a (a−x)³+ c.
4.2
3
α
4.2
Now as s0 when x 0, c must be equal to
πα, hence
3
the value of s is that given in the text. In like manner we shall
have
SX1
T
3
a
- ÷ ñ√ˆã уæ³ —‡Ã√ a§æ³ dy (= − §x√ ã§x√ a—x.dx)
+ž:
·.·(as Sx √ a—x.dx = − zx (a−x)# +÷÷÷ (a−x)§),
V
sx¸=‡π√a.yx$ + §.3 xx√ a (a−x)} + ¿:t z√ a (a− x)§ +c',
•.• when x = 0, c'=
in the text.
-
4:4a³, and the value of sæ, is that given
(9) In this case, the arc of the generating curve whose value is
given in the first of equations (8), becomes (as y is changed into a)
equal to the expression for s given in the text.
(r) The third equation of No. 75 is λy, = 1/
So
2
S (y² — y¹²)dx,
- ...
we have 2xxy' = √ (y² —y¹²) da = v. This is called the Centro-
B
α
baric method of determining centres of gravity, it was discovered by
Pappus, but from the frequent applications made of it by Guldin, it
is more commonly denominated Guldin's method. It follows from
it, that if several solids are generated by a revolution of surfaces
about a common axis, the sum of them all is equal to the circum-
ference described by their common centre of gravity, multiplied into
the sum of the generating surfaces.
NOTES.
577
1
(s) In this case we have the following proportion:
L: 2πλу¸ :: 1:2πy, '.' L= lλ.
1
(r) If the area of the base of the cone v' be denoted by b and
hb
3'
the height by h, we shall have v' = in this case b
h = a — ƒ • . • v'
1
ƒ···v' = = 13xc² (a—ƒ).
4.3
TC2
4'
and
(s) As s is equal to 2≈a (ß—a), (see No. 81), in the case of
a sphere, ẞ — a a the difference between the radius and cosine
of the spherical segment, and therefore equal to f, the sagitta of this
segment. It appears from this expression 2af, that the entire sur-
face of the sphere is equal to four times the area of a great circle of
the sphere, as we know from other considerations should be the
case.
(t) It appears from No. 81, that the distance of the centre of
gravity of the surface, generated by the revolution of the arc A'D'B',
from the centre, is equal to (CD'+CE') = CD-D'E' = {(a−ƒ),
for D'E' = f.
if.
α
(v) If in the equation ra²ƒ="c² (a -ƒ) + V₁, their values
be substituted for c and ƒ, there will result
πας (1
-1),
COS
2a
=4a²sin2 a cos
1
1
+ v₁
2a
2a
1
,
COS
2a
.sin COS
2a 2a
·
1
Bra (1-
πα1
and if these values be substituted in the equation
we shall have
} xa²ƒ (a− }ƒ) = tf πc² (a−ƒ)² + V₁‚時‚
f a
(as a -2 = 24. (1+ cos
2'
1
2a
2a
1
Επα
§≈α². (1
·(1–
COS
2α
(a) · ba² ( 1 +
cos
1/4)
==
75
4a² sin²
• a².cos²+v₁₁,
2a
that is
¿wa.sin"
2a
= 4. wa'. (sin' — — sin') + V,,;
2a
and, if the terms which are common to the two sides of this equation
be obliterated, there will result by substituting for v, its value, the
expression for æ, which is given in the text.
1
4 E
578
NOTES.
(x). Each section of an ellipsoid, made by a plane parallel to one
passing through two of the principal axes, is an ellipse, the major and
minor axis of which are the ordinates of the ellipsoid corresponding
to x the distance of the plane from the centre of the solid, when the
cutting plane is parallel to the plane of the axes b and c, the axes of
the ellipse made by the section are
b
a
C
- for
Va²x² and -√ a² — x²,
α
they are evidently the ordinates of the ellipsoid corresponding to the
x²dx
abscissa x, and in this case xdx = bc. ( dx
a²
which taken between the limits a and ß, is πbc.
>
the integral of
α 3a²
a²+aß+ß²), because ß³ — a³ — (a—ß) (a² +
==bc. (8-a) (1_a²+as+9"),
(1–
væ, bc(xdxd) = (when the
aß + B²). In like manner V = zbc\ | d
integral is taken between the limits a and 6) rbc (6ª — a² _ ßt at
2
4a²
evidently equal to the value of væ, given in the text, from which, by
substituting for v its value obtained above, there results the value
of x1.
(y) In this value of v, if for Syda its value (ax+}az)dx
be substituted, we have
S
v=xxу² — 2x √(ax— x²) dx — añ § z Vax - x².dx,
V
-
i. e., by integrating and substituting for this last quantity, its value
given in No. 80,
V=ñxу²—2π
'αx2 x3
2 3
απα?? από
16
-+~2~[(§ a−x) √ ax−x²+4(ax—x²)]
πα
equal when ≈=a and y=2
πα
x²α2
4
a3
2F
(ร
3
a semicycloid.
Vax
azx Vax
а33
16'
the value of v when the arc is
With respect to the
x²+az, we obtain væ₁ =
value of v₁, then as y =
x²y²
x²у² — √x(αx − x²) — §}
π§x(αx
x³dx, in this case we must make y = 5x.√ ax
J
x2dx
= √ √ ax³ — x^dx, and the value of y may be written as follows:
NOTES.
579
S
x2dx
(¤¸x − x²)²dx
; the second term can be
√ ax³
X4
a2
treated as the corresponding term in Number 80, and the first
xdx
α
=
and if we make
4'
x=z we obtain ax x²=
-
ax
X2
2
a²
α
22 and xdx =
dz
zdz, by means of which the value of væ,
4
2
may be obtained.
A
(~) M=SSSpr² sin◊ dr de dų — A §§ dų d◊ sin ◊ = ▲ (a' — a).§sin0d0—
▲ (x' — a) (cos w—cos w'), in like manner as sin 0.cos e de
cos² 0,
(between the limits = w, and w') (cos² w-cos² w'), it fol-
◊ /
lows that the value of Mx, is that given in the text; likewise, as
S sin desin 0.cos 0+0=- sin 20 +0,
0
1
(between the
limits w, w'.) — sin 2w'+ sin 2w+w-w, it is evident that
the value of my, is also that given in the text.
(a) When M is a complete ring, that is, when a'= a + 2x, it is
evident that the factor sin a'- sin a in the value of My,, and cos a-
cos a' in the value of мz, are respectively cipher, consequently, in
1
α
this case, y, and z, are cipher, and x₁
B
2M
becomes by substituting for m its value =
(a' — a) (cos² w
cos² w')
B
2A
(cos w + cos w').
(b') When w= 0 and a = 0, then вcal, Apa³, conse-
3a'
quently, a, = (1 + cos w').
1 8
(c') The base of the parallellopided=rdyrsine de = r²sine de dy,
and when r1, the integral of this expression is cos 0., which
taken between the prescribed limits, gives the integral = 2π.2, in
the same way the volume of the sphere is 4π.
раз
3
CHAPTER VI.
(a) The body whose mass is m and volume v being supposed to
be condensed into its centre of gravity, p the density which is equal
m
to must be that of the centre of gravity; now if the shape of all
V
the molecules be the same we shall have when their number is n,
V
v = ne³,
ε.
n
580
-
NOTES.
(b) u² = (a−x)² + (B − y)² + (y −2) becomes, by expanding,
= a² + ß² + y² – 2(ax + By + y²)+x² + y²+ ≈² ;
and if we make
a² + B² + y² = d and 2(ax+By+yz) — (x² + y² + z²) = p,
we shall have
1
02
U²
1
and
52-p И
+ + +
p
1.3
2.4 85
• 203² + &c.
Now if there be substituted for ♪ and p their respective values, there
results when no powers of x, y, z, higher than the square, are taken
into account
1
W
||
1
+
2(ax+By+yz)—(x²+y²+z²), 1.3 (2(ax+By+yz)
28
+2.4
1
(2(
By+y*))",
2
>
გა
which is evidently equal to the value of given in the text; and if
น
this value be substituted in the expression T =
dm
SSS & m
И
it is evi-
dent, that when the origin of the coordinates is at the centre of gra-
vity, the second term of its value given in the text will vanish. It
likewise appears, that when OD or ♪ is so great that the value of T is
reduced to its first term, then
dr
da
Μα
and. A =
3
umfa.
(c) A threefold integration is to be performed relatively to
r, 0,; with respect to y, as neither this quantity nor any function
of it occurs under the sign of integration, the integration relative to
dų may be effected by writing 2 without the sign §, for it should be
integrated between the limits 0,2π, and then there only remains the
double integration to be performed relative to and r.
With re-
spect to 0, the radical Va² - 2arcos +r², becomes, when 0=x,
√ a² +2ar + r³, and when 0=0, Va²-2ar+r²; now as the in-
definite integral relative to ◊ is
taken between the limits ◊ =
π
1
α
√ a² — 2arcose +² + const. when
and 0 = 0, it becomes, when the at-
tracted point is within the stratum,
1
— [(r' + a) − (1 − a) = 2,
a
− −
consequently in this case the value of T will not depend on a, and
NOTES.
581
.A=
1
ирова
da
value of a²
and 0, is
α
= 0.
0. When the point is without the stratum the
2ar cost + r², taken between the same limits
2r
- [(a + r) — (a − r)] = 2¹¹,
α
α
if this value be substituted in the expression for T, there results
T=
Απο
α
α
So redr,
which, integrated between the limits a and b, gives
Απρ
T= (a³-b³),
Βα
and as the masses of the spheres, whose radii are a and b, are respec-
tively
4πρας 4πρό
3 3
Απρ
masses of these spheres.
X
(a³ — b³)
3
ƒSSSa = * dm = (
= M is equal to the difference of the
(d) A=µf
as dm = gu²dudw, cosg=
(*=α))
µƒSSS p cosg dudw, and when p is constant, by integrating with respect
to du, from u = 0 to u=r, we obtain a = µfpsSr cosgdw.
(e) By making this substitution we obtain
cosg\2
(a + reos)' + (3 + rcosh) + (y + roosk) * = 1,
α
b
this becomes by developing and concinnating
a²
|。
+
+ 2 (a cosg+
B2 22
+
B cos h
y cosk
b2
+
C2
b2
cosk),
(cosig +
b2
cos'h cos2k
+
= 1,
c2
a2
which, by substituting l, p, q for their respective values, becomes
the expression in the text.
(ƒ) Since the parts of the value of r under the radical sign mu-
-9
tually destroy each other, there remains only the quantity to be
p
taken into account, and if for q its value be substituted, there results
the expression in the text.
582
NOTES.
(g) If p and its value given above, be multiplied by a² b² c², there
will result by substituting for cos g, cos h, cos k, their values, in terms
of the polar coordinates, the expression in text; and if in value of a
the expressions for dw and cosg be substituted, we obtain this value
of A.
dy
p
(h) When &=4(=tang↓) is ∞, hence when is integrated
75
between 0 and the equivalent expression
a²b²c².
2'
do
(b² cos²0 + a² sin²0).c² + (c² cos²0 + a² sin²0) 6²¿²
2
must be integrated between the limits 0 and . It is easy to show
that the two expressions are equivalent, for if there be substituted for
dy and p their respective values, namely,
do
1+ ¤²
b²c² cos20+(c² cos2y+b² sin2y) a² sin20
and
a²b2c²
1
and also for cos² its value.
and
1 + $2³
$2
1+φ2
for sin, there re-
sults the second expression given above.
Now if (b² cos²0+ a² sin²0)c²=m, and (c² cos² + a² sin²0)b² = n,
then a²b²c²
So
аф
(b² cos²) + a² sin²0) c² + (c² cos²0 + a² sin²0)b²p²
аф
a2b2c2
So
0 m +nq²²
the integral of this last expression is equal to
a² b² c²
tang.
√ mn
n
m
which, when taken between the prescribed limits, becomes, by sub-
stituting for m and n their values, the integral given in the text.
(i) That is, if there be taken any two points in the ellipsoid, the
force acting on these points, resolved parallel to an axis, is proportional
to the ordinates of the respective points.
(k) Since b=c, the part under the radical in the value of a is
equal to
a² cos20+ a²e² cos20+ a²sin20 = a²(1 + e² cos²0);
and by substituting this value in A, we obtain
:
583
NOTES.
3
A =
4xufça
S
T
and as
b² cose sine de
0 a²(1+e²cos²0)
b²=a² (1 + e²) and 4πp=
3m
a³ (1 + e²)
it is evident, when these values are put for b² and 4p in the value of
A, that the expression given in the text will be obtained.
Now in order to integrate this expression, let e² cos² = y², then
cos² ◊ sin Ade
1+e² cos² 0
y²dy
e³ (1+ y²)'
of which the integral is
1
(y
1
arc tang=y) =
031
(e cose arc tange cos 0),
15
and this expression, when taken between the limits 0 and becomes
2'
1
e- arc tange), hence we obtain at once the value of a.
(1) If we suppose as in the last note, that e cosy, then we
have
3 µfma!
0 √1+ e² sin² 0
π
2
y❜dy
Sufmal Så vidy
a³ (1+e²)√0 e³ VI+e²—y²
is evidently equal to
a³ (1+ e²)
S
cos² e sin de
3 µfma!
2
Now the integral of
y²dy
√ 1+ e² — y²
y
1+ e²
Y
1+ e²
√1+ e² - y²+
arc sin =
=
arc tang =
2
2
Vite-7)
√ I + e² — y²
and if for y its value e cose be substituted, and if the integral be
then taken between the limits = 0, 0
75
=, it becomes
2
1
•12
e
1+ e²
+
arc tange, and ·.·
3 µfma'
2
2
a³ (1 + e²) J 0 e³ √ ]+e² −y²
3µfma!
e²
2
are tang = e),
a²e (1+)-++ arc tang =
a³é³ e²)
which when concinnated is the expression in the text.
A
(m) By substituting for arc tange we obtain
^ =
and
€3 e5
e+
5
µfma 3e2
3 fuma (e
( - + - + &c. ) = ( (1—3+ &c.),
(e-et
a³é³
3
a³
5
584
NOTES.
A' =
3ufmat (
2 a³é³
+
3100
es
5
− &c. — e + e³ — e³ + &c. ),
− &
equal, by dividing as before the terms of the series by e³
µfma! (1
(1-
6e2
+ &c.
&c.).
5
a³
A
(n) If in the value of a there be substituted for x,, y', z', dy', dz',
their values in terms of 8 and y, it is evident that when in this ex-
pression, π — ◊ is put for 0, it will coincide with the first term, so that if
the second be integrated between 0 and the result is the same
as if the first was integrated between
2
2
765
2'
and 7.
(0) a²=a,& cos²p, B2=b'sin2p cos'q=asin'p cos'q+hsin'p cos'q,
y² = c,² sin²p sin²q = a,² sin²p sin²q + k sin²p sin²q,
2
a² cos²0+ b² sin20 cos2+ c² sin² sin²;
= (by substituting for b² and c² their respective values, namely,
a² + h, a² + k,) a².cos20+ a2 sin20+h.sin20 cos2y+k.sin20.sin²;
hence by concinnating these expressions, and substituting for z, ß, y,
their respective values a,cosp, b,sinp cosq,, c,sinp sing, we obtain
the value of R² given in the text.
(p) If u be zero, the first member of equation (4) is reduced to
a², and the second to zero, and if u be so the first member becomes
a²+B+7², while the second member is ; consequently in the
passage from zero to infinity there must be a value of u, which ren-
ders the first member equal to the second.
(9) This follows at once from the equation a,bc ab,c,.
· NOTES.
585
(a) We have
BOOK II.
CHAPTER I.
k: h : : v : u ·.· as v = iu, k = ih.
CHAPTER II.
k² dv
=
(a) 4 = de = g(k² = m).. kdo = gdt ='( 2+ dv.)...
dv
— v²)
k dv
dt
k2
k² - v2
2k+v
k
k
2
gt=5[log (k + v) − log (k − v)]
— =
2
k
log (1+0).
k-
=)
(b) If both sides of the equation
k
-2gt
v
e
k+v
gt
be multiplied hy ek, there results
(+)•
k + v
gt
k
e =e
е
-gt
k
>
and ·.· (k − v) e
gt
k
(k − v) e* = (k + v) e
% ( 5 ) = 1
e
gt
gt
(² + + + );
k
)
= 0 (0%
ekte
k
hence we obtain the value of v given in the text.
(c) If both members of the equation v =
27
gt
k
and
gt
—gt
e
e
be multi-
gt
-gt
ek te
k
plied by dt, and then integrated, we shall obtain
k2
x = Svdt
·
g
gt
-gt
k
log(+)+ const.,
k
and as by hypothesis x = 0, when t = 0,
k2
k2
k2
const. =
·log (e⁰+eo) - log 2 =
log;
g
ggg
g
(d) gdx=vgdt
k²vdv
k² - v²
hence it is evident, that the value of x is that given in the text.
k2
k2 k² — v²
log
29
k²
k2
2g
log k² — v³°
4 F
586
NOTES.
(e) From note (d) it appears that
x=
ka
-gt
which when e
g
log (e.
gt
ek + e
k
kt
log 2,
g
gt
k
k is neglected, becomes, as log e* = the value of
≈ given in the text.
(ƒ) It appears from No. 124, that the accelerating force varies
1
(every thing else being the same) as but as in this case it varies
1
k²'
as it follows that mass varies as k, hence as k is the uniform
mass
velocity to which the body continually approximates, the greater the
mass or the density, the greater will be this final velocity.
(g) Generally log (1 + u) = u
Ha
u2
из
+
2 3
&c. hence as
k
e
te
k
g2f2
=1+
2k2 +
8414
1.2.3.44
›
, log (e + é
gt
gt
k
e
-
g2+2
0414
gsts
g-212
QK2
1.2.3.4k*
8/4
+ &c. =
gs th
+ &c.
2k2
12+
kdv
gdt
k² + v²
k'
we obtain, as va when
(h) By integrating
t = 0, i. e., at the commencement of the motion,
e(t
v
arctang =
k
a
arctang=
(tang
=)
gt
=
k'
hence from the known expression for the tangent of the difference
of two arcs we have
gt
k
ak — vk
k²+av
gt
sin &
tang .
k
cost
k.a.cos
(a.cos st
gt
k.sin
k
k
k
() = v. (a sin St+k.cos
gt
k
from which it is easy to obtain the value of v given in the text.
(i) The integral of the differential equation
da=vdt=
gt
k (a cos — k sin) dt
k
gt
u siu + k cos gt
k
NOTES.
587
is
k2
x =
g
(log. (a sin &+k.com
gt
gt
k
+ const.,
when t = 0, and a = 0 this expression then becomes
k2
.log.kconst. = 0,
By
consequently,
k2
const. =
.log.k,
g
k2
k2
1
hence as
.log k=
g
g
.log.'
it is easy to infer the value of x
given in the text.
(k) By integrating we obtain
k2
X =
2g
· log. (k² + v²) + const.,
when = 0, then va, hence
k2
const. = 2•log (k³ + a³),
and, consequently, by substituting this expression for const., we ob-
tain the value of a given in the text.
(1) By expanding into a series the value of v given above, we
obtain
gatз
k [a (1 − 6*+ &c.) − k ( € - 8.3 + &c.) ]
2k*
gate
1.2.3 k*

+ &c.)+k.(1–67+&c.)
;
v=
α
1 (st
1.2.3 k³
1
this expression becomes, when
<= 0,
k
k (a = st)
gt
α
gt.
k
And by a like expansion of the expression for x, we obtain
[a (gt
k2
log
kk
g
(
and as generally
gusts
1.2.3k³
+ &c.) +1
c.)+1
grå 12
1.2³ +1.2.3.4—&c.];
W
log (1 + u) = ī
this value of a becomes, when 1 = 0,
Из
u³
3
+5
&c.,
588
NOTES.
k² (a gt
9212
gtz
g
k k
= at
1.2.k2
2
all the higher powers being neglected as being equal to cipher.
(m) In this case as v = 0, we have evidently
α
arctang =
4) = 80,
and ·.· 0₁
(tang
k
2. arc. (tang = %).
α
1 g
(n) When the numerator and denominator are multiplied by this
quantity, we obtain
k2
(√ a + k² − a)²'
which is evidently equal to .log.
k
k
0' =
2g
log.
k
g
Dr!
and k/2=
we have
"
YP
YP
(0) In fact, as k² =
Dr
√ a² + k²-a
k² : k12 :: r: r'.
2
(p) vdt = dr = } ({√ ak — gi)'dt :: # = ( Vak — gt)" +
-
const., and since x = 0, when t = 0,
· 1
Vakconst.;
34 (Vak)"="
3gk
1
-
3
3gk
+
consequently,
a√ ak
X=
+ -32k (81 - Vak)",
1
and as v=
= ½• (Vak
3g
2
gt
(√āk – gt) the sign of the second member is always
the same, ·.· direction of motion is the same as before.
2018
==andt=
α
π
α
18100
(q) z=ccost
g
... when z=0, t
α
d²x
(") The integral of the value of
is in this case
dt2
dx²
1
1-194
= gr².
+ c.
dt2
r + h
X
Now as
dx
dt
=0 when x = 0, we must have c =
when x=h, the expression for
dx2
dt²
becomes
2gzz
and
r+h'
NOTES.
589
2 gr². ( — — — 1) = 2gr².
h
=2gh.
r.(r+ h)
r+h
r+
+h
(s) By reducing to a common denominator we obtain
dx2
dt2
2gy2
30
(v + h). (r+h-
· h — x)
·.· dx² (r+ h − x ) __ 2gr². dt²
IC
=
r+h'
and if numerator and denominator of the first member of this equa-
tion be multiplied by r+ h − x, and then the square root be taken,
we shall obtain the expression in the text.
(t) Since
r+h-2x
.dx+
2 √ (r+h) x − x²
r+h
2√(r+h)x−x2
dx=
r+h-x
√(r+h)x−x²
• dx
we shall have by integrating
√2gr³.dt,
√ (r+h) x − x² + 1 (r+h)
зав
r+h
arc
(cos
COS =
r+h-2x
r+h
2gr²
.t;
r+h
and because

1 –
r+h−2x\²
r+h
=21
(r+h)x-x²
r+h
it is evident that
are (co
Cos =
r+ h − 2x
r+h
2 √(r + h
− x2
= arc sin
r+h
therefore we shall have
2gr²
t
r+h
=√(r+h)x−x²+(r+h)are(sin=
c(si
2 √(r+h)x− x²`
r+h
now as the sine is very small, it may be assumed equal to the arc,
consequently
t
✓
2 gp2
r+h
= √(r
(r + h) x − x² + (~ + h).
1 (r
2√(r+h) x − x²
r+h
2√(r+h) x − x² = 2√(r+h)−x) x = (when r+h− x is reduced
to r) to 21rx.
dz
(u) Since
dt
=0 and z=h, we have by formula (c)
262
ƒ²= k² =
c - α
262
c – h
2a2
2a2
+
α
h'
590
NOTES.
Now as
cb
Qb*
c-h-
2b (a + b)
2α*
and
a+b'
c-h
C
h
(a+b);
= 2a (a + b)
с
and, consequently,
2.62 2a2
+
c - h h
hence we obtain
(v) h =
ac
f2
a + b
2a (a + b) + 2b (a + b) — 2 (a+b)²;
262
c - a
+
ac
C
2a2 2.(a + b)²
α
a (1 + √75)
C
C
1+ √75
с
(w) This is the case, because the mean motion of the moon in its
orbit is equal to its motion about its axis. See Vol. II. No. 433, note.
(x) The projectile describes a curve, because it is acted on by
dissimilar forces. See No. 144, note.
(y) From the equation
(2a² — 2b² + cy)² = 8a²cy,
we obtain by solving it for y,
27 (6³ — 00²)
Y-2α
2a
-2
с
consequently,
C
2
and ".:
a = ±
-b;
C
с
y = 2 (a±b)²;
and if we substitute 8acy for (2a² — b² + cy) in the expression
under the radical, it becomes
√2a²c−2a√/2cy•z+yz²=√(a√/Zc−z√7)²=a√πc—z√7;
now, if for√7, its value √2. (a±b), be put, the expression be-
comes a√2c − (a + b);
C
from which it is easy to perceive
how the denominator in the expression for dt, may be rendered
rational.
(*) See equation (c). of No. 139.
(a') If 2a²= k³a, the value of %, when velocity is cipher, is
¤; and, consequently, in th's case, a body projected at the dis-
tance a from the centre with a velocity a² =
k² a
2
will ascend to ∞,
NOTES.
591
i
where its velocity becomes cipher; if 2a² k'a then the body, when
it has attained an infinite distance, has still a finite velocity equal to
k²a — 2α², it is evident also, that when 2a² > k²a, that z can never
become infinite.
If a body falls freely without any initial velocity from a distance
a, towards a centre of force varying inversely as the square of the
dz2
dt2
distance, the square of the velocity, is = 2a².
then
dz2 2a2
dt2
if a is
α
now it is evident from what will be established in
No. 239, that a²= the square of the velocity in circle at unit of
distance from the centre of force, and ·.·•
a²
= the
the square of the ve-
locity of a circle whose radius =, consequently, we have, when a
body falls from an infinite distance to the centre of force, the velo-
city at any point of the descent, to the velocity in a circle at the
same distance :: √2:1. Hence, if a body be projected from the
centre of force with a velocity which is to that in a circle at the
same distance :: √2: 1, the body will ascend to an infinite distance.
This indeed is evident from the expression 2a² = k²a, or k² =
a²
α
2a2
for is the square of the velocity of a circle at distance
2, and
the converse of this is true, namely, if a body falls freely from an
infinite distance, when it arrives at the distance a from centre, it
will have acquired the velocity k. If 2a² is > than k², then ≈ the
2 a²α
2a² — ak²
distance at which the body is arrested =
CHAPTER III.
(a) If the forms of the functions ft, f't, f''t are all the same, so
that
x=aft, y=bft, z= cft,
it is evident that the point will move in a right line, and
a
√ a²+b² + c²
b
с
√ a²+b² + c²² √a²+b²+c²²
express the cosines of the angles which the direction of this line
592
NOTES.
makes with x, y, z. The constant quantities a, b, c depend on the
nature of the function ft, if, for example, ft
sent the uniform velocities parallel to x, y, z,
t, then a, b, c repre-
and the uniform velo-
city of the point Va²+b+c²; if ftt, then a, b, c are pro-
portional to the accelerating forces parallel to x, y, z, and the point
will be moved with a motion uniformly accelerated, and represented
by √ a² + b² + c².
If x = aft+bf't, y = cft + df't, z= eft + gf't, the point will
move in a curved line, which however is of single curvature, for by
eliminating t, we obtain an equation of the form a'x + b'y+c'z,
which is the equation of a plane. The simplest case of this form is
x = at + bť², y = ct + dť². z = et + gť², and eliminating t between
the two first eqnations, we obtain an equation of the second order
between and y, which from the relation that exists between the
three first terms, is evidently that of a parabola.
If, as in text, x=ft, y=f't, z=f't, the curve described will
be of double curvature,
(b) v=
ds
dt
cosa =
dx
ds'
• v cos α=
ds dx
dt ds
=
dx
= p.
dt
ds
dx
dx ds
(c) w = vdt =
dt
dt, cosa = ;
ds
w cos α =
•
dt =
ds
dt
dx
dt = dx = pdt, by equations (1); in like manner
dt
ε cos a = u cos adt
= (a
as u=
cos a
a) $p'dt,
consequently
wcosa +ɛ cosa = pdt + ½ p'dt=x'
X.
(d) If each of the three preceding equations be squared, and then
added together, there results
(x' — x)² + (y' − y)² + (≈' — ≈)² = w' (cos²a + cos²ß + cos²y) +
ε²(cos²a+cos²b+cos²c) + 2 wɛ(cosa cosa + cosß cosb+cosy +cosc),
and it is evident that the factor of wɛ is equal to the cosine of the
angle contained between MH and MK.
(c) Let & denote the infinitely small angle TMM', then the component
c.),
of v in the direction MM' vcosd, = v.
=
·(1-
+ &c.
1.2
and as by
NOTES.
593
}
hypothesis is infinitely small, its square and higher powers may
be neglected, and this renders the component of v in the direction
MM' v.
(f) See Harte's Translation of the System of the World, Book 2,
chapter iii., and Mechanique Celeste, Book 10, No. 15. When the
body is let fall freely at the equator, the deviation is ÷÷l to h³, h being
the height of the tower, for if r be the radius of equator, and a, a',
the arcs described by the bottom and top of the tower during the
fall, we have
a : a' : : r : r + h ··· a' — a =
ah
now a is÷l to the time of the fall, and it varies as √h, conse-
I
quently a'—a, or the deviation is
a, or the deviation is to h³, for is constant.
J'
(g) In equation (d) by substituting
ds2
dtc
for, and multiplying
both sides by dť², we obtain
+
ds2
dt2
dt² (= ds²) = k² + 2F (x, y, z) — 2 F(a, b, c) đť² :
dt =
ds
√k²+2F(x, y, z)-2F(a, b, c)
in which by substituting for x, y, z, their values in functions of s, we
obtain the expression given in the text.
(h) When the elevation of the point a above the horizontal plane
passing through D and c is equal to h, the general expression for the
velocity at this point, namely, √2g(hz) becomes then 2g.(h-h),
ds
i. e. at
√2g.h—h, and ·.· dt =
ds
√2g.(h—h)'
which is infinite;
but if the elevation of this point above the plane is less than h, then
the velocity after passing this point becomes negative, which indi-
cates that the moveable must descend towards the point of departure,
where it is in precisely the same circumstances as at the commence-
ment of the motion.-See No. 181.
dL
dz'
(i) As cosλ= v cos μ = V
di
di
COS y = V there results
dy
dz'
by multiplying equations (3) by dx, dy, dz respectively, and then add-
ing them together, the expression in the text; for
4 G
594
NOTES.
d² x
dy
S.x +
Sy +
dt
dt2
d2z
dt2
d = xả + xây + z8z + N(cosAde + cosurg
+cosydz) i. e. (by substituting for cos λ, cos μ, cos v,)
(k) öds = de dx + dy ôdy + dzodz. Multiplying both sides by
d2x d2y
Sx +
dez
Sy +
Sz = x5x + ydy + zdz +
dt2
dt2
dt2
dL
dL
NV Sx +
dx
dy
8y + dr
dL
Sz
dz
).
dx
ds
ds
v =
ds
dt'
and observing that ddx
dx ds
v.dds=
ds dt
dz ds
ddz,
ds dt
dx ds
dx
which
?
dt dt
(as ds at
dox, &c., we obtain
dy ds
ddx + ddy +
ds dt
¿c.), becomes the expression in the text.
(n) If r be infinitely small, then e
=1, consequently the ex-
pression of the force is then equal to a; on the other hand, when r
r
is a considerable multiple of a, e
a is cipher.
(k) When is considerable, the expression for
2
dz
dt
the velocity of
a body that has been projected from the surface of a sphere is, by
No. 143,
... in this case
and as g
dz
dt = √k²-2ga,
30950m = √ k² — 2ga,
27½ the terrestrial gravity G, and a = 110r, r denoting
the radius of the earth, we obtain
k² = (30950m)2 + 55G. 110.r,
by means of which it is easy to compute the diminution in the text.
CHAPTER IV.
(a) The velocity along the side MM' v cosd, therefore the ve-
locity lost is equal to v.(1—cos d) = v.
(ร
S4
1.2.3.4,
+ &c.),
which
is evidently an infinitely small quantity of the second order.
NOTES.
595
dx2
dx² ds2
d dy
dx
(b) By substituting for
its value
and for
dt2
ds² dt?'
dt
d.
dy
dx
ds
ds dť
ds dy
dx
we
its value
obtain (as
(as v=de and d. d = d'yda dandy)
dt
dx
dyda–dady _ (@ydz_Fady).
d²ydx-dx dy
dx2
d dy
= m
dx
dt2
.ds,
d²ydx-d²xdy
ds2 ds
ds.dt2
(c) The value of xx given in No. 20, is
02
X* —- X¹
X-x' =
—
´d²ydx-d²xdy'
ds³
aşı (dy(dxd³y — dyd²x)—dz(dzd³x— dxd³z)),
ds4
now if for dady-dyd2x, dzd²x-dxd2z, we substitute their values,
which may be deduced from equations (2), there results
p² ds3
dx
x-x' = pcosy =
dst
v²
ds
1. do [a {dy (de cosq' - dy cosg) -
dz
dz
ds
ds
cos q
ds
dx cos
cosq")}
– P { dyda
dy
dz
dx
cos w
COS W
ds
dz cos w
ds
cos w
ds
")}]·
ds
hence, eliminating quantities which
nominator, we obtain the value of
712
-
occur both in numerator and de-
cosy given in the text; and cor-
responding expressions may be obtained for
(d) The coefficient of Q in the value of cos y is equal to
v2
2.2
cos y',
cos y".
?
P
V2
cosy
P
dy dx
ds ds
dy2
dz2
dx dz
cosq'-
ds
cos q-
ds2
cos q+
cos q';
ds ds
and as
dy
dz
dx
cos q' +
ds
cosq" =
ds
ds
cos q,
we have
dx dy
ds ds
dx dz
dx²
cos q'+
ds ds
cos q" =
ds2
cos q,
consequently the coefficient of q
=
dx² dy² dzi
+ +
ds2 ds2 ds2
cos q
=
cos q;
in the same manner it may be shown that the coefficient of - p is
596
NOTES.
cos w, &c. Now it is easy to verify that P is the resultant of the
v2
forces and Q, for by squaring these equations, and then adding them
P
together, we obtain
2
("=") "(cos³y + cos³y' + cos³y'') +
2v2
Q. (cos y cos q + cos'y' cos q' +
P
cosy"cosq")+Q."(cos'q+cos'q'+cos'q")= p² (cos w+cos²w'+cos²w");
spectively equal to unity, and as the factor of
now as the factor of
(~-²);
ક
>
and also the factor of q² and p², are re-
2v2
P
Q is cipher, it is evi-
22
and Q.
P
dent that the force P is the resultant of
1
(ƒ) As the areas are proportional to the times, the velocity v
varies as, p being the perpendicular let fall from the centre of force
p
on the tangent; it is easy to show that this expression coincides with
the general expression for central force given in terms of the perpen-
dicular and chord of curvature, for if this chord be equal to c, and if
the radius vector be denoted by r, we have by hypothesis,
h²
h2
ບ
v2 -
c=pcos ·.·R =
- '.'R
p2.c
(as
as c=
See Newton's Principia, Book I., Sections 2, 3.
2pdr\ 2p² dp
dp p³ dr.
(g) As the cosine of the angle which the radius makes with the
vertical is equal to the part of the weight mg which acts in the
2
2
direction of the string is equal to mg., and this must be added to
2
the centrifugal force in order to obtain the entire tension; as long
as the moveable is below the horizontal plane passing through c,
2
mg is positive, and directed from the centre, but in the upper part,
mg
"
-
is negative, which shows that it is there directed to the centre;
on the other hand,
2mg (h+*) which expresses the centrifugal force,
I'
is always directed from the centre; hence in the lower half of the
circle, the tension is the sum of
2mg (h+ z)
mgz
and
and in the
2
J'
2*
upper half it is the difference of these two expressions. If the pen-
NOTES.
597
dulum falls freely from the horizontal plane, h vanishes, and the en-
3mgz
tire tension is •.• the entire tension is thrice that produced by
the weight, when sustained by a force equal and contrary to the tan-
gential force, and the tension arising from the centrifugal force is to
that produced by the weight :: 2 : 1. If h=r, which is the same
as if the moveable descended freely from the highest point of the
2r + 3≈
is at the lowest point where zr, equal
circle, mg.
P
5mg, i. e. the entire tension is five times the weight; when ◊ = mg,
i. e. when the entire tension is equal to the weight, mg (2+3)
r-2h
3
=
mg, z=
hence if there is no initial velocity, i. e. when the
body falls freely, the strain is equal to the weight when the move-
able has fallen freely through one-third of the height, ◊ will evidently
2h
vanish in the upper half of the circle when ≈= but as z can ne-
3'
ver exceed r, it is evident that the value of h may be such that will
never vanish, e. g. if r ▲ the tension will never vanish, the
2h
3
moveable will revolve about the centre, and the tension will be the
sum of 2mg. (h+*) mgx.
and in the lower half, and the difference
ՈՐ
of these quantities in the upper half of the circle.
2h
3
(h) If z = is less than r, the weight at this point resolved in
1
the direction of the radius is equal and opposite to the centrifugal
force, beyond this point the tension due to the weight is greater than
the centrifugal force, and therefore if the rod was not inflexible, the
2h
moveable would part from the circle when ≈ = and describe a
3
parabola (No. 209) having a common tangent with the circle at the
point of departure. If the moveable was made to describe an entire
cycloid in its oscillations instead of a circular arc, it would be easy to
show that if it moved from its horizontal base without any initial ve-
locity, the tension at any point arising from its centrifugal force is
equal to the component of its weight acting in the direction of the
string, and at the lowest point the entire tension is equal to twice
the weight of the moveable.
598
NOTES.
(i) In proceeding from the poles to the equator, as the attraction
varies in the inverse ratio of the square of the distance, it must dimi-
nish, and this diminution is nearly proportional to the increment of
distance. For if the attraction at the pole be denoted by a, and the
radius by m, the attraction at any other point by a', and its distance
from the centre of the earth by m + dm, we have
A': A:: m²: (m + dm)²
and
- 1) v
m²
A'A
A
(m + dm)²
2dm
q.p.
m
'.' A
-
A' or the decrement of gravity is proportional to dm, but dm
varies as the square of the cosine of μ. q. p. See Nos. 254 and 193,
note.
CHAPTER V.
ds
= √
dt
(a) If both sides of the equation k²+2g (z−c) be mul-
tiplied by dt, we obtain
ds
dt
dt = ds = √ k²+2g (z—c).dt.
(b) For ED is equal a (1 + cos α), ·.· if k²= 2ga (1+ cos a) ;
the initial velocity is equal to that acquired in falling through the
height ED, and if we substitute for k² this value, the expression for
ade
√2ga.(1+cose)
dt given in the text, will be dt =
(c) By hypothesis, 1+ cose=1+ cos 24 2 cosy, and de=2dy,
2ga(1+cose)=4ga cos²4,·.· dt =
α
-2ady
Va dy
2√ga.cos
g cos
cosi
d&
dx
now let sinx, then
the integral of this last
cos
1-x
1- sin 10
quantity is log +*+c=\log · 1 + sin 30
have log
1−sinh
1+ sina
α
+c=0;
+c; when t=0), we
substituting the value that may
be obtained in this way for c, we obtain the expression in text.
(d) It is evidently the same thing whether the moveable sets out
from the point M, which is very near to B, with a small initial velo-
city denoted by k, or whether it falls without any initial velocity
NOTES.
599
from a point so much higher than M, as that when it has reached
M it would have acquired the velocity k.
(e) When t
√ε
is a multiple of 2,0 = ∞, and when t
α
do
√
g
2018
α
=α.cost
雪
​.COS T
α
an odd multiple of π, ◊ = a, and in each case = 0.
dt
(ƒ) By making this substitution, we have 'a cos (t+T)1
&
g
which expres-
(t + r) √
a
V-a sint√Esin TV
a
α
α
sion, when T=π V
√
becomes
α cos t
أمة
a
ade
dt
(g) As ad is the differential of the arc of the circle, ex-
de
dt
presses the velocity, and therefore as =+α
point, the velocity at this point = ±a Vi
have a =
26
and ·.· v=√2gb.
& at
α
at the lowest
ga, and as b½ɑɑ², we
(h) Retaining the fourth powers of a and 8, the expression for
dt=
-Va
α
de
a4 049
8 √ a² - p² - 12+12
which is evidently reducible to the expression in the text; and as
2
a
[1 −7· (α² + 0)] ~~* = 1 + 2 + (x² + 0²),
the expression for dt is evidently that given in text.
(i) The integral of
do
Ө
a²
is arc cos-,
that of
02
α
24
do
α
Ө
arc cos -
that of
24
; ·.· the integral of
•
α
Ardo
24√ a²
Ө
√ α² - 0²+
α 02
48
a²
Ө
arc cos
48
α
Ө
3a²
Ө Ө
dt =
a
arc cos - +
α
48
g
α
the last term vanishes at the limits a, a, and the other terms become
at these limits
arc cos - +
+ 2/4 √ a² - 6²],
600
NOTES.
α
cos=1+2
(2 are cos = 1 + 2 arccos=1,) = √ ; : (+1)
16
α
α
=T
1+
16
T
(k). Let 7 be the given time, T' the duration of each oscillation
when the amplitude is infinitely small, and T the duration of the os-
cillations where the amplitudes are only very small, we have
nr'= n'T = 7 i. e., nø
a²
√ 1 = n = √ 2 (1 + 21 )
V
a
•
(1) cos²f=1–2 x + x² · · · sin² 0 = 2 x
•
:: dt = √ a
—
Va
α
dx
926-2x.√2x-x²
16
n'::
a²
···n : n' : : 1 +
: 1.
16
dx
x², and de=
√2x-x²
which is evidently equal to
the expression in the text; now when x = ß, 1- cos = 1 − cos α,
...a, and when x = 0, 1 cos 6 = 0 ·.· ◊ — 0. · ·
(m) If in the expression 2-2 √x-x dx, we multiply and di-
ßx x²
x²-1dx
vide by vBx-x, we obtain Sa"-2.dx = ß
✓ Bx-x2
·S;
x"dx
Bx-x²
(n) For the manner in which this integral is obtained, see La-
croix's Traite du Calcul Differential et Integral, tom. 2, page 314.
(0) By differentiating with respect to t, we obtain
Esin
ag
g
sin ty√ + costy √ con by V-
· dt
α
de = [-ay √ & sin ty √ E+08
g ag
a 2k
ag V ga
4yk²
sin ty V
امد
α
a
-gt
Jef
2k
е
2k
ty
a
Now in this expression the factors of the cos mutually destroy each
other, and those of the sine are
αγ
√
g og√ ga
4yk
a
- ay √9
a
=(as
ga
= (as 4 k² = 20)
a (1−y³) √
γ
9 a 9
a
Y
a
NOTES.
601
() When the first oscillation is completed,
ΤΥ
TY√ ₂ =
α
T=
γ
g
α
Now in a vacuum the value of T is √, consequently, it is increased
9
in a resisting medium in the ratio of 1:7.
gt
(9) In the expressions for the amplitude, the quantity by which
2k is multiplied becomes the same at the end of each oscillation,
gt
but e 2 evidently diminishes as t increases, therefore the amplitudes
continually diminish; now as
Π
α
9
T=
Y
;, nTy
ηπ
a
therefore, the value of 0 after the nth oscillation is equal to
a. COS NT.e
nav ga
2yk (as cos na =(−1)"), a. (− 1)" e
nxVga
2yk
;
but the value of 0 in this case is by hypothesis (1) an, hence then we
obtain the value a given in the text; from which it evidently appears
that these values of a constitute a geometrical progression of which the
ratio is e
TV ga
2yk
(r) In this case when the cos and sin are expressed in exponen-
tials, the value 0 is equal to
tp
e
g tß V
• te
2
√ga.
2Bk
e
bola
e
e
2
一枝
​{
from which it is evident, that if 2k ← √ga, ◊ will be cipher when t
is infinite.
(s) The integral of the equation
is
d20
2
2g
doe
do +
dt
sin Ode = μde,
α
d2
2g
+
И
dtª
and as by hypothesis, Sape
cos & = 4 Safe do,
d02
dy
d◊ = y, and
this equation be-
dt²
dt2
do
comes, by substituting these values,
4 H
602
NOTES.
dy - 2g
cos f
My=0.
α
dy their respective values ☺z and
do
do
Let y = ☺z, ☺ being a function of 6, then substituting for y and
Odz zdo
do
>
+ the preceding equation
becomes, by multiplying by de,
2g
Odz+zdo-
α
cos edo — µ☺zdo=0.
Now if we assume (Lacroix's Traite Elementaire, No. 257),
2g
Odz μezde 0, zde cos ode = 0,
тв
α
from the first equation we obtain zeme, and substituting this value
of z in the second equation, there results
+
do=
2g
cos de e-μo
= =
2g -тв
sine
mg
a
α
α
2m# S sin bdb6-00 =
-тв
e
(-
2µg
mo
cos le
a
α
2g \
cos edee-μe
+ c;
and, by continuing the partial integration in this manner, the value of
will be found equal to
2g
e
a
6
'sin @ [(1 −µ²+μ¹ —µ°+&c.)—µ.cos @ (1 −µ²+µª—µº+&c.)];
but we know from the expansion of a binomial, that
therefore,
1
-
1 + μ² = 1 − µ² + µ³ — µ³ + &c. ;
2
Ⓒ = 2ge¯µÐ
(
sin 0
μ cos
1 + μ²
+c;
2g sin
μ cos
a
2
2050) + Colo.
семь
consequently, y=☺z=☺.еμe
(t) If in equation (5), — a、 be substituted for 4, then as the ve-
locity is by supposition cipher, the second member of this equation
becomes cipher, by means of which, the expression in the text can
be deduced; now, when in this equation, the square and higher
powers of μ are neglected, ea=1+ μa, consequently, the members
at each side of this equation become
µ
cosa, (1+μa,) - μ sin a, cosa (1-pa) + μ sin a,
µa) µ
which is evidently the expression of the text.
NOTES.
603
µ (sin a₁
μ
(u) Asd is very small, cos &= 1, and cos a₁ cos a + d sin a,
and sina,sina+cosa.d,. a, cosa, (=(a-8) cos (a-d))acos α-
α, cos α₁) = cos a + ♪ sin a —
d cos a + ad sin a, ·.· cos α,
*.*
μ sin a+μa cosa cos a + sin ɑ —
α
*
•
μacosa,
♪ sin a=2µ (sin ɑ —
acos a).
α
a³
(v) sina ==
α
1
1.2.3'
and a cosa=1
૨૩
1.2
1
1
powers), sin a
a-1
a cos α =
3' sin a
α
as
1.2.3
(neglecting the fourth
+
1 1.2.3'
a-2a³
8=2μ
sin a
a cos a
μας α
-
sin a
3
1.2.3
μα
(as the fourth power is neglected) 2μa, consequently,
3
a - d = a
2μα
3
(t) For the integral of this equation, see examples on the dif-
ferential and integral calculus, page 387.
(u) By substituting for sin 2: √, and dividing all the terms
2t
α
by a and √ga, which occur as factors in all the terms, we obtain
the expression in text when v≈ 0, and as the factor sin t
α

, it is evident that for this value of t, v=0.
9
a
sin ™,
when t=
7
(v) If t√2 = +3, then cos t'√
g
COS
α
cos 2 t'
a
a
=
- d,
= cos(x+28) = − 1, ... the given equation becomes
-(1-a) 3+= 0;
αμ
3
αμ
αμ
4 12
which, as the product as is neglected by hypothesis, gives
(x) By substituting for t
d=
8=ale.
αμ
6
αμ
its value, and neglecting
α
the cube and higher powers of a₁, we obtain
604
NOTES.
αμ αμ
3
3
sine√e=sin(+9)=1-a in 2√2=sin(x+2)=
α
αμ
6
αμε
1.2.6'
2ť
α
and if these values be substituted in the general expression for v, we
obtain, inasmuch as the cube of a is neglected,
+= (aa) √ga.
=(a- *)
3
(y) In this case the general value of 6 becomes, by substituting
# for t
√, equal to
a
αμ
αμ
Σμα
(a_ak) cos +ª²ºμ+ª²º½ • cos 2 π, '.' α, = α — ²µª²
4 12
a
3
(*) See Mechanique Celeste, Book III. Nos. 24, 25, 38.
(a') When y = 1 this equation gives tang t'
√
2k
√2ga
2k
but
=
√2ga
tang. (+
√2ga
for
2k
tang+
√2ga
2k
(23
as tang
is infinite
2
•)
tang.
tang+tang.
2
1 - tang tang
2
Voga
√2ga
2k
√2ga
2k
; but when k is very great,
2k
√2ga
√2ga
1
2k
g
π
√2ga
tang
=
...t'
+
2k
2k
tang 2k
√2ga √2ga
2a-
2k
(b') If any of the exponents of this series were negative, then
when x=0,s would be infinite; and if any of them were cipher when
ds
dx
x=0, s would be infinite; also as = the secant of the angle which
a tangent to the curve makes with the axis of x, at the lowest point
dy ds
dx'
when =
dx
ds
dx
this tangent is perpendicular to the axis of x, and
therefore is ∞, hence it is at once evident, that the least of the
exponents a, ß, y, &c., must be less than unity.
xα-1 dx
ha-1 xlu-l hdx'
(c') As x=hx', dx = hdx', we have
√h−x
Vh-ha'
||
NOTES.
605
ha~}}
x'α-³ dx'
and unity.
and the limits of the new integral are evidently zero
(d') It has been proved in a former note, that one of the expo-
nents, a, ß, y, &c., is less than unity, it appears from what is es-
tablished here, that the value of the least exponent is; hence
then it follows, that
S
x'ɑ—¹ dx'
-1
0
=S₁₂
dx'
and the integral of
2arc
2 arc
√ x01 - 2012
dx¹
dx
√ x' — x¹²
(tang=)=(when
taken between the limits of 1 and 0), as stated in the text.
(e') Asu=√1+
rdy Sdy
su = [dy da
dy2
dx2 dx²›
dz2
+ би
x
now the entire integral of I dy day
u dx dx
and as at the limits B, a, dy is cipher,
x dy doy
Sex dy
day, da
u dx dx
dx=-
(f') The first equation (c) gives x
dz 8dz 1
+
dx dx dx dx U
dx = x dy
u dx
x dy
u dx
sy — 5d.(
x dy
Sy,
u dx
dy vanishes, .. we have
dy
Så d. (2. d4) sy.
u dx
a.
dy
#dx
➡u.a, and as u =
V₁+
1+ dyz
dy²
dx²
by multiplying both sides by da, we obtain
dy
X
dx = √ dy²+ dx².a.
dx
(g') It is evident that this equation may be made to coincide
with that given in No. 72, by changing the origin of the cordinates
on the axis of x.
Z
(h') Let x-aaz, then dx=2adzz, and
2 az² dz
VI
= (by
(x−a) dx
√ α ((x−α)—(x —α)²)
the integral of this last is — az√1-2³ + a.arc sin =
+c, by substituting for ≈ its value
z
α
√x-a
Vā
³) -
XC α
a
(Vi
IC a
+a. arc sin=
a
V
a
X-α a
+c=−√/a.(x−α) — (x —α)²
•
606
NOTES.
a-x+a
α
x + a
α
+ a.arc cos=
+c, but arc cos =
α
Vä
a
2α-2x+2α
COS =
cos
a
·1):
arc
a
=
COS =
2
α
2x+2x
quently, the integral is evidently that given in the text.
(i') As dy =
(x-a)dx
dy
+1 = u² =
—α)³ dx³
√ α(x—a) — (x — α
a.(x —α) — (x − a)² + (x−α)² ·
a.(x —α) — (x — α
(k') Leta—a=y²√a and dx=2√a.ydy hence
; conse-
arc
2'

a
=
a—(x —α)
Va.
a
Va.dx
√α.(x-α)-(x-α)³
2aydy
√ ay² — y'a
2√ady
√ I—y³
= 21a are cos=√1-y=2a. arc
α
cos =
vi
2α-2x+2α
α
α
+2α _ 1) = v
√a arc cos=
the expression given in text.
1.
1—*—¤, 2√α.arc cos = √a-x+a
=
a-2x+2α
α
✔a arccos
α
equal, when xß,
α

(1) Now arc cos=
a-26+2a
arc sin
'a² (a-2ẞ+2α)³
α
a²

=2
a. (B — a) — (B− a)³
a²
a² — a² + 4a (B—a) — 4(B—a)²
a²
when the radius is infinite the sine equal the arc, and if in
the expression for ẞ'a' given above, there be substituted for
a−26+2a 2√a.(B-a)-(6-a), the terms which
a
2 arccos =
α
α
express the value of ẞ'-a' will be equal, and have contrary signs,
therefore, B'-a' in this case is cipher, and the value of y becomes
evidently equal to a', for the second member of the equation
vanishes; likewise in this same case when a is, the part of
the expression for t', in which a does not occur as a factor, vanishes,
relatively to the other part, so that it becomes equal to
Vä
α
√28
2√ a(ẞ—a)
√2 (B—a)
=
a
√g
+
NOTES.
607
(m') By substituting for
xd³x + yd³y + zd²z
dt2
its value
´dx² + dy² + dzª
+ d₂³)
= – v³,
dt2
we obtain the value of N given in the text.
(n) rd↓ is the arc of a circle described with a radius = r in dt,
rd↓
dt
and ... =u,
u, and xdy - ydx = r²d↓ = cdt, ·.·
C
(as r = √ a² — 2º) √
√ a² — 28′
Az
2'
a
rdy
с
=2=
dt
18
(o') ≈ = a cose, cose = 1−2, ':' z = a− 2 0²; k² = b+2gy,
... by substituting for k² and y, their values, we obtain ß².ga = b +
2ga-ga a², ... b = −2ga+ga(x² + B²), c²=k²(a²—y²) = ß³ga (a² —
(a — — aα²)²) = ẞ²ga (a² — a² + a²a²) = ẞ²ga³ a², for by supposi-
tion the fourth power of a is neglected. Now dz―ade, and
a² — z² — a² —
(2gz + b) — c²
( a − 1 a0²)² = a²0², 2gz 2ga-gab², ·.· (a² — z²).
(a —
(a —
=
= a²0²(2ga — gal² — 2ga + ga(x² + ẞ²)) — ga³a²ß² ;=
=
ga (a² 0² (a² + ß² — (²) — a² a² ß²),
a² a² B²), ··· √ (a².
√ (a² — ≈²) (2gz + b) — c²
a √ g (a² — 0²) (0² — ẞ²) a, ··· as dz=-a0de, by substituting for
√ (a² — ≈²) (2gz+b)—c² and also for dz their values, we obtain the
expression for dt given in the text; and as
d↓ =
cdt
a² — z³
√ ga³.aßdt
a²G2
1
by substituting for dt its value, we obtain for d the expression in
text.
(p') It appears from equation (b) that the increment of the angle
will be always positive, it will consequently increase indefinitely,
g (a² — 6²) (0² — ß³)
Ө
with respect to 0, as
doe
= — —
dt2 α Ө
O always lies between
a and ß, therefore when these two quantities are equal, è must also
be equal to each of them.
(q′) The base of the cone will be circular because is constant,
and the motion will be uniform, because is proportional to the
time t; when a revolution is completed we have
27
9 and . T= 2″
2x = T√2 and
a
α
= 2 √
g
608
NOTES.
(~') When ◊ = a, cos 2t^
cos 2t
√2= - 1, ·.· 2t
α
By substituting for cos 2t
sinat
امة
α
√2 = 1,
2 = 1, ·.·• t = 0, and when 0 = ß,
α
√2 = (when 0 = 6) x, . . t =
α
2, and observing that cost
√2,
α
sia +
√ ½ +
α
= 1, we obtain 6ª = § (a² + 6º) (sin² t√+ cost√5)
+1 (a² — ß²) ( cos' t
+(a² - (cos²
sin² t
α
2018
α
g
a² cos² t
√E+B'sin't
+ B² sin² t
√E.
a
(t') Dividing this value of dy by a cost
dt
容
​α
there will result
t
g
diy = √ √ + (1 + 2. tang' (√/2);
dt √ E
g
α
o ප
αα
cos²
d.tang.tv
B.dx
?
2018
α
•
if we call tang. t
and arc tang =
018
α
X
;
18018
=x, we
now
cost v
shall have
d↓ =
consequently,
1+
tang ↓ =
when
山​二
​B
α
x and a tang = ß tang t
g
√5=3, and t =
(u') From the equation
2'
α
V
a tang ↓ = ẞ tang t√5,
g
1 2018
a
we obtain
1
cos2↓
sin² t √g
cos²
B2
α
cos² t√
consequently,
NOTES.
609
1
et √E = (a² cost √E +5 sin. (√) cos'
a² cos² t
g
a
g
α
t
α
but the quantity within the brackets=0², consequently,
90
a² cos² t
α
²
cos²;
02 cos², and x2 a2 02 cos2a2a2 cos2t^
in like manner it may be shown, that
*
y² = a²0² sin² ↓ = a² ß² sin² t
g.
X2
+
y2
B2
= a².
امة !
CHAPTER VI.
(a) As the medium is uniform, and every thing consequently the
same on both sides of the vertical plane, there is no reason why the
projectile should deviate to one side rather than to the other of this
plane.
XC
a cosa
and if this value of t be substituted in the ex-
(b) t
X2
pression for y, we obtain y=x tanga – g
g
1
as
a².cos²a
a²
2h
313
dy
dx
there results for y the expression given in the text, and as
X
at the vertex where
dy
dx
tang &
0,x=2h sina cosa, y =
2h cos² a
h sin'a; it appears from these expressions for the coordinates of the
vertex of the parabola described by the projectile, that when the ve-
locity of projection or h is given, the locus of the vertices of all para-
bolas which can be described, is an ellipse whose minor axis is vertical
and equal to h, and whose major axis is horizontal, and equal to 2h,
for we have — 4h³. sin²a (1 − sina) 4hy — 4y² ·.· 4y² + x² —
x² = — —
4hy =0, which evidently is the equation of such an ellipse as has
been described, in which the origin of the coordinates is at the lower
extremity of the minor axis. From the equation = ta cos a,
y=ta sinagt, it appears that when t the time is given, the locus
of the points arrived at in a given time is a right line parallel to the
direction of projection, and intersecting the axis of y at a distance
gt2
from the origin equal to As the direction of projection bisects
2
the angle between vertical and line drawn from point of projection
4 I
610
NOTES.
to the focus, and as h is always equal to the distance of the origin
from focus, if " and y' denote the coordinates of the focus, and a,
as before, the angle between the horizontal axis and direction of pro-
jection, ah.cos (2a - 90°), y" sin (2x-90°), i. e., x'' — h.
1/2 =
sin 2a, y" = h cos 2a, ··· x² + yll² — h²,
y"
-
cot 2a, the first
equation shows, that when the velocity of projection is given, the locus
of the foci is a circle radius equal to h, the second proves that when
the angle of the projection is given, this locus is a right line; likewise
in the same circumstances, that is, when direction of projection is
given, it appears from the equations a
from the equations = 2h sin a cos a, y = h sin² a,
that the locus of the vertices of the described parabolas is a right
line of which the equation is y
X
tang t.x.
dx
dy
(c)
a cos α,
dt
dt
dx² + dy²
asina-gt, ·.· v² =
gt,`.`v²
= a² (sin² a
dt2
+cos² a) - 2a (gt sin a) + g² t² = a² - 2 agt sin a + get, and as
a² = 2gh, 2 agt sin agt 2gy, v² = 2g. (hy), as we know
t² =
from other considerations. If the equation y = x tang a
x2
X2
4h cos² a
hence we can
be solved for h, we obtain h =
4. cos² a (x tang a—y)’
determine the velocity, with which a body should be projected in a
given direction, in order to reach a given point.
(d) It appears from inspection of the value of %, that the part
under the radical is the equation of a parabola, and it follows from
what is established in the text, that all parabolas described, with a
velocity of projection equal to √2gh, will touch the concavity of this
parabola.
It is evident, that the maximum range on a given plane, passing
through the point of projection, is when the direction of projection
bisects the angle between this plane and the vertical, consequently,
the given plane passes through the focus of the parabola, which is
described; and if & be the inclination of the given plane to the ho-
rizon, we have = 2α 90°, and tang.<= tang. († + 90°), ·.·
2h
2h
tang a tang (+90°)
and the greatest range which is equal
B=
β
2h
2h
cos &
&
cos tang(+90)
+90) = i 1 + sin &
>
for tang.(+90) =
NOTES.
611
1 + tang
sin+cos 2
as cos & cos³
94
--sin³
221, cos 4.
1
tang
cos-
sin 2
2
tang (+90)=1+2 sin
in옴​.
COS
=1+ sin 4.
(e) m =
4DR3
3
and the surface of the sphere is equal to 4r², *.
R
if these values be substituted in the value of
the result will be of
m
the form in the text.
R
d²x
R dx
(f) By substituting for
in the equation
we ob-
m
dt2
m ds'
tain
d2x
dt2
ds dx ds
ds dx
C
C
dt ds dt
'dt' dt'
dx
log.
dt
multiplying both sides by dt, and integrating, we obtain
cs + log. A... =A.e˜c8.
dx
dt
dx
Now when s = 0,
a
dt
= a cosa, ··· ▲ = a. cos a, consequently, the
dx
value of
is that given in the text.
dt
R
(g) In fact, by substituting for
in the value of
m
dzy
dt2'
there results
day
ds dy ds
dt2
= c
dt ds dt
ds dy
g=cat dt g;
and it is evident, from inspection of these equations, that we may
assume
dy dx
dx dt' p being a new unknown.
=P
•
(h) Differentiating the equation
dy dx
dt =P and dividing by dt,
dt'
there results
dry_dp dx
d²x
+p
C-
dt2 dt dt
dt2
ds dy
dt dt
g=cp
•
ds dx
dt dt
d²x
-g=P¯dť²
g,
dx dp
g.
dt dt
(i) Multiplying both sides of equation (2) by dx, we obtain
dp=
dx
2h cos² a
dy
2c8,・・ dx = 2h cos² ae—2c³ dp; also .dt=
dx
p.dt; i.e., dy=pdx; and .dt.
dx
dt
dt
dp. dt (= dxdp) = — gåt³.
gdť².
612
NOTES.
dx² + dy²
dx2
(k) y²=
dt²
= (1 + p²)
dt2
=(2
(as ap
dx²
dt2
ge
dt² dp²
6),(1+p³).
dt2
ga.
dp²
(1) (√1+p²² — p'). (√1+p²² +p') = 1, therefore, the sum
of the logarithms of the two factors is cipher, consequently, one of
them is equal to the other affected with a contrary sign.
cda
by sustituting for p and dp
dp'
-p'√1+p+ tog (√1+ p²² — p') — y
dp'
− p' √1+ p²² — log (p' + √1+p² - y
12
dp'
p/
(m) For very great values of p', p' may be assumed equal to
√1+p¹², for in this case when √p+1 is expanded, the first term
is p', and the powers of p' which occur in the second and following
terms are very small with respect to it; therefore,
y+p²² + log2p', or y + p²² +
log p²² + log 2 = P';
and from the equation e≈≈1+ x + x² + &c., it appears, as is stated
in the text, that the logarithm of a very great number is always very
small relatively to this number.
(n) In this case the expression becomes, by substituting — p'
for P,
cv² =
g(1+p²²)
y+p' √1+ p²²+ log (p' + √1+ p²²)'
which becomes, by neglecting y + log 2 and log p²,
g
cv²g, and ·.· v²
v² = b.
с
(0) By taking the differential of du, we have evidently
du— dy cos ẞ — dx sin ß — (dy — dx tang ß). cos ß,
ß
and if the values of dy and dæ be substituted in this equation, and
the sign of the common denominator be changed, we have the ex-
pression for du given in the text.
(p) By integrating this equation, we obtain
dy
dx
-1
4 ch cos² a
e2cx
- 1
+c, and because when x, y = 0, tang a=
+c, c=tang a
4ch cos² a
NOTES.
613
1
>
by integrating a second time, we obtain y——
+
4ch cos² a
ec2x+tang a.x+
tion, we have c' =
Ꮖ
4ch cos² a
1
8c2h
1
8ch cos² a
+c', and when x and y=0 in this equa-
hence by concinnating, we obtain for y the
the expression given in the text. As e2cx=1+ 2cx +4c²x² +8c³
x3, &c.
e2 cx
2 cx
8 c²h cos² a
X2
2h cos² a
vacuum.
1
X2
2h cos² a
•
when c=0; hence y=tang ax
which is, as we know, the equation of a projectile in a
See on this subject the twenty-first Number Du Journal
de l'Ecole Polytechnique, page 191, and Vol. II. Nos. 358, 359.
(9) From this equation we obtain -
ecx dx
√2gh.cos a
1
√2gh c cos a
andt-
ecz
√gh.c cos a
gdt dp e2cx
dx²
dx
2h.cos² a
+A, when t0 x=0 •.
the value of t is that given in the text.
dt-
(a) The planet was in this case supposed to move in the circum-
ference of a circle of which the sun occupied a position at some dis-
tance from the centre, it was on this account that the motion was
said to be performed in an eccentric circle.
(b) It is easy to determine when ◊ nt, i. e. the equation of the
centre is a maximum, for if about the focus of the ellipse, which the
sun is supposed to occupy, as centre, a circle be described whose ra-
dius is a mean proportional between the semiaxes of the ellipse, the
area of this circle is equal to that of the ellipse, and if we conceive a
body to move in this circle with the mean angular velocity n from
the radius which coincides with the perihelion distance, at the instant
the earth passes through the perihelion, and in the same direction,
it is evident that as the earth's velocity in this point is greater than
the mean angular velocity, the earth will precede the body which
moves in the circle, and it will continue to precede it, until the an-
gular motion of the body becomes equal to that of the earth, i. e.
until the earth's angular motion becomes equal to the mean angular
motion; after this, the angular motion of the earth being less than the
1
·
614
NOTES.
mean angular motion, the body will begin to gain on the earth, and
will evidently overtake it when the earth arrives at its aphelion; hence
it is evident that the earth precedes the body by the greatest quan-
tity, and the equation of the centre is a maximum when the angular
motion of the earth is equal to that of the body; this is evidently,
when the radius of the circle, is a mean proportion between the
semiaxes, or r=a (1-e²). For if n be, as in the text, the mean
p
pin
2
angular velocity, we have = the area described in an indefinitely
small portion of time in the circle, and if r' and n' be the corresponding
pl2n'
2
quantities in the ellipse, we have = the indefinitely small area de-
scribed in the ellipse in the same time, consequently we have n: n' : :
as the synchronous areas in circle and ellipse divided by the squares
of their corresponding radii r and r', but in the circle and ellipse the
synchronous areas are equal, being as the whole areas divided by the pe-
riodic times, i.e.in a ratio of equality, hence the angular motionsn, n'are
equal, and 0—nt the equation of the centre a maximum, when r=r',
i. e. at the points where the circle, which has been described with a
radius equal to a(1-e), intersects the ellipse. It appears from this
value of r, that the greatest value of the equation of the centre de-
pends on the eccentricity, as is stated in the text.
(c) From equation (2) there results, cos ◊ =
a(1 — e²)
er
√ a²e² — (r—a)²) (VI—e²)
d.cose=
sinode=
a(1-e²)dr
ere
therefore as sin²0 = 1 − cos²0
e²z² — a² (1 — e²)² —p² + 2ra (1—e²)
,sin@=
e2 r2
and, consequently, d◊ = —
d.cose
sin
=
which is evidently equal to the
12 do
a.√1-e²
nadt =
a (1 − e²) dr
er
r. Va¹e (ra)². √ I — e²'
—
expression in the text; and as
rdr
√a²e²-(r-a)
a²e² — (r—a)³
their values, there results nadt =
a²e.(1—e). coзu. sinudu
ae.sin u
; by substituting for r and dr
a.(1 − e. cos u). ae sin udu
Va¹e²-a²e². cos²u
...ndt (1-e) cosudu; likewise, by sub-
NOTES.
615
stituting for r and dr their values in the expression for de, it becomes
a². √ 1 — e². e sin uðu
do=
a.(1 — e cosu). ae sinu
√1_e².du
VI-e.du
1-e.cosu 1—e.cos²u+e.sin² u
now if both the numerator and denominator of this expression be
divided by cos²u, and if we observe that cos² + sin² u = 1,
we shall obtain dê = √/I — e².
du
costu
÷ by
sin2u cos² u
+
cos² u
cos² u
e.cos2u esin2u
du
+
(by substituting for tanu and for
their
cos²u
cos² u cos² u
respective values), the expression in the text. Now in this value of
de, let 1—e=m, 1+e=n, then
21/1-edz
I—e+(1+e)≈²
2√mn.dz
m + nz²
(by making
√ ñ
√ m
y
2 dy
1+y²²
consequently we have
= arc tang
2
vite
y= arc tang = z
and tang 10 =
tangu.
√ I-e
(d) Cosint.cosi'nt=
cos(i + i')nt+cos(i-i)nt, sin int. sini'nt
2
cos(i+i')nt — cos(i-int, and it is evident that if these values
2
be substituted for cosi'nt.cosint, and sini'nt.sinint, and if the inte-
grals be taken between the limits 0 and 7,
indicated in the text, and as cos²int =
1-cos 2int
2
the results will be those
cos2int +1
2
sin i²nt =
; it is evident when these values are substituted in the
cosi³nt.d.nt, So
expressions Socosi'ni
П
each case
ནྭ་
sin'int.d.nt, that the results will be in
Sad.nt, and therefore
When i = 0, ST cos int.d.nt = Sod.nt,
s.ST dnt = s₂ == Stro
r.d.nt.
(e) The integral of (0—nt) sin int.d.nt is =
(0 — nt). cosint
+
cosint.d (0 — nt), and when taken between the limits nt = 0,
nt=, it is evident, from the value of 0~nt given above, that it is
+
616
NOTES.
reduced to
Scosint.d(―nt), hence it appears that the value of
B, is that given in the text.
(ƒ) As int = iu ie sinu, we have cosint = cos (iu— ie sinu),
and if we remark that d.(— nt)=d.0 — d.nt =
-
e2
√l-e²
1- e cosu
du
(1 e cosu) du, it is easy to deduce the value of B; given in the text.
(g) As cos (iu—ie. sinu) = cosi.u. cos (ie sinu) + siniu. sin (ie sinu)
by expanding cos.(ie sinu), sin.(ie sinu) into a series according to the
formula expressing the sine and cosine of an arc in terms of the arc,
we will obtain the expression for cos (iu-ie sinu) given in the text.
(h) By substituting for r and d. nt in the value of A, we obtain
a
a
0
A。 = -
T
Soa
e. cos u)²du ==
cosuđu
π
S
1-2ecosu++cos 2u du,
№² —
c2e2² sin20
a² (1 − e²)²
+
(1 + 2e cose +e²) c², but
c²(1 + 2e cose + e² cos²)
a² (1 — e²)²
2a.(1-e²)
α
equal at the limits u = 0, u = 7, (≈ +½e²π), i. e. a(1+½e²).
21%
(i) By substituting these values, we obtain, v c²
-, ··· v².a² (1 — e²)² =
1
d. ↑
+
ds
2+2e cose, and
2a(1 − e²)
r
-1+e²=1+2 ecos + e...
r
(2ª − 1). (1 − e) = 1+2ecos)+eº,
—
2 a
and ·.· v². a² (1— e²)² =c².
(1—e²)²
2a
с
1, ·.· sin d =
..
=
γ
vr
a. √ 1 — e²
; it appears from this ex-
pression, that at the mean distance the velocity is equal to velocity
in circle at same distance, and also that at this same distance the
angle & is a minimum.-See page 374.
(k) Since sin &= siny sin (0+w), as sin y is always + when sin &
is +, then + w is 180.
(1) From the equation tang cosy.tangλ we obtain tan² ↓ cos²λ
= cos²y. sin²λ, i. e. tan²↓.(1—sin³λ) = cos²y sin²λ, •·.· sin³λ =
− 1 ) . (1 − e²),
C
e²), and
v=
V =
a√ I — e²
r.V. 1
2a
*tan
cos² + tan²
and sin & sin y sin λ =
siny tangy
cost+tang
NOTES.
617
(m) The comets do not appear to have exerted any sensible at-
traction on the planetary masses, or in any way to have deranged
their motions; when they are observed through very powerful teles-
copes, and under circumstances in which we ought only to perceive
a part of the illuminated atmosphere, we are not able to discover any
phases; stars are said to have been seen through the densest part of
the nucleus.
(n) See Dorpat Catalogue of Double Stars, by Struve; Philoso-
phical Transactions, 1802 and 1803, Connaissance des temps, 1830.
(0) The force
ds2
dt2
acts in the direction of the tangent, and this
force resolved in the directions of the axes of x and y, is equal to
ds² dx ds² dy
dx ds dy ds
§
dt²・ds' Pdt²・ds'
which since
are respectively equal
ds dt' ds dt'
dx dy
dt
dt'
to and become the second members of equations (2) given
in the text.
(p) If equations (2) be respectively multiplied by dx and dy,
and then added together, there results, as
dx
dx =
ds
.dt, ds =
dt, &c.
dt
at
dxd²x + dyd²y
dt2
ri+
x+ydy)
=
P.
´dx² + dy²
dt2
・) ds,
which, since
dx² + dy
dt2
dr² + p²d G²
dt2
dr² + r²do²
μ.d
dt²
t.d.(dro +
and in like manner, if the first equation (2) be multiplied by y, and
then taken from the second multiplied by x, we obtain
1
P
2.
이
​and r = √x²+ y², is equal to
'dr² + p²df²·
ds;
dt2
dt2
xd³y yd²x
dt2
=d.
xây d
dt2
=
Pat
(
ds (ydx yxd
dt dt
xây d
do
and as
... we obtain
dt
dt
dt'
do
ds
d.r2
P
r²də.
dt
dt
(9) See No. 232.
..
4 K
618
NOTES.
*.་
ક
(r) 0 = nt + + 0,; now 0, is a periodic function arranged ac-
cording to the sines of the increasing multiples of nt+Ew, when
◊ is increased by 360, it is evident that we arrive at the same point
as before. If there was no perturbation the value of the time lapsed
between two consecutive returns to the perihelion would be equal to
that of a return to the same fixed point.
(s) If the first of these equations be multiplied by 1-e cos u,
and the second by ae sinu, we obtain by adding them together,
(1-ecosu)³da — a.cosu(1―eeosu) de + ae.sinu (1 − e cosu) du+
aesinu(de-dw)+ae sin²ude-ae sin u(1 — e cosu) du,
which as ae(sin³u+cos²u) = ae, is evidently the expression in the text.
(t) As by the third cquation (a)
tang² u =
1 – e
1+e
tang² 1 (0-w),
if this value be substituted in the expression for cosu, there results
1
cos u=
1 +
(1-e) sin (0—w)
(1+e).cos³ (0—w)
(1 − e).sin'}(0 — w,
(1+e) cos³ {(0—w)
(1+ e) cos²¹ (6—w) — (1 − e) sin²(0 — w)
=
(1+e).cos² (0—w) + (1 − e) sin² (0-w)
which as cos²(0-w) + sin²1(0—w) = 1, cos² (0—w) — sin²1 (0—w)
= cos(8-w) is evidently equal to the expression given in the text;
hence we have
(1 − e cosu)² =
(1-e²) 2
(1+ecos (0-w)²)³
a.(e—cosu) — (e² — 1).acos(0—w)
ae. sinu =
1+e.cos(0-w)
ae √ 1 — e² sin(0—w)
1+e.cos(8-w)
now if these values of (1 —e.cos u)², a(e—cosu), ae sinu be substi-
tuted in the equation
(1 —e cosu)ºda + a(e—cosu)de + ae sinu(dɛ—du) — 0,
we will obtain, by multiplying by 1+e cos-w, and dividing by
1-e², the equation (f).
(v) It appears from the equation
r²de = √µa(1-e²)dt
NOTES.
619
that the synchronous areas described about different centres of forces,
vary as the square roots of the parameters of the conic sections de-
scribed, multiplied into the square roots of the sum of the masses of
the attracting and attracted body; see Princip., Book I. sec. 3,
prop. 16.
2
´dr²
1
(u) v² = dr² + rede = p. (~ - ') ; d. (dr+d) - 2µd. =
dt2
r²
a
dt2
1
α
(-2p. (dr² + 10) ds) = - p. d. 1. d. = 2p. ( -
dt2
¹²)ds) :
a.(1
α
2μα.
p
)de.
d.r²de = — pr²deds =√p.d.Va. (1— e²)dt, ·.· — p√ μa(1—e²). dt ds
√p.d√ a(1—e²)dt.
2
(2a=r)ds;
)ds;
ds; and if in
(*) d. 1=2p (1)de da = 2p. (2ª—")de;
0.01 20 ds
a
α
α
this equation there be substituted the value of r, we shall obtain the
expression given in the text.
d.√ a(1— e²) = − p √ a(I —e²)ds, i. e.
da (1 — e²) — 2aede
21/a. (1—e²)
§Va(1— e²)ds ·.·• 2ede =
2ça(1-e²)ds+da(1—e²)
=
α
(by substituting for da its value)
a
2p(1 — e²)ds — 2p(1 + 2c. cos(0 – w) + e²)ds,
which, by obliterating the quantities that destroy each other, becomes,
by dividing by e, equal to the value of de given in the text.
Equation (e) becomes, by performing the differentiations,
r cose cos wde re cose sin wdw + rsin 0. sin wde + re sin cos wdw =
da(1 − e²) −2 aede,
that is, by concinnating and substituting for da and de in the second
member,
r.cos((—w)de + re. sin (0 — w)dw = −2pa(1 + 2e cos(0 −w)+e²)ds
+4pae(e + cos(♦ — w)ds (= — 2pa(1—e²)ds);
now if in the first member of this equation we substitute for r and de
their values, the preceding equation becomes by dividing both sides
by the common factor, a.(1—e²)
2p(e.cos(0—w) + cos(0 ~w)²)ds + sin(0—w)edw
1+e cos(0-w)
2pds,
620
NOTES.
and consequently
.edwsin(0—w)=—2p(1+ecos(0–w)ds+2p(e.cos(0—w)+cos(8—w)²)ds
=-2p.(I—cos(0-w)²)ds = — 2psin(e—w)ºds ···
edw=- 2psin (0-w)ds.
If in equation (f) there be substituted for da, de, and edw their
values already obtained, there results
[— 2pa (=
1 + 2e cos( 0 -w) + o² ) + 2pa(e cos(§—w)+cos (§ — w)º) +
1+e.cos(0—w)
2pa sin(-w)2
2º)]ds +
aesin(-w)
de,
√ I—e²
√ 1 — e²
reducing the two first terms to the same denominator they become

w)*+e.cos(0—w)) 3
—2pa.(1+2e.cos(0~w)†e*)+2pa(e.cos(0-w)+cos(0-w)*)+-2pa(e*cos(0—w)+e.cos(0–
1+e.cos(0—w)
=(as e.cos(e-w)se.cos (0-w) (1-sin(-w)2)

2pa.(1 + e²). sin²(0—w) + e.cos(0—w) sin(0—w)²
and thus equation (f) becomes
-2pa [[(1+e³)
1+e.cos (0-w)
[(1+e²)+ecos(0—w)] sin(0—w)=
1+e cos(0-w)
2pa.sin(-w)
+
ds
√ I— e²
+ ae.
sin(0 — w).dɛ
= 0,
√ 1— e²
dividing by sin(0-w), and then reducing to a common denominator,
we obtain
— 2pa[sin(0—w) (1.+ e.cos(0—w) + e²)√ I—e² — (1+ecos(0—w) sin
(0-w)]ds+ae.(1 + e. cos(-w))de=0,
multiplying both sides by 1+ √1-e², there results
—2pa.sin((—w) [e² VI—e—e³.cos(w)—e^]ds +ae (1+e(0—w).
(1 + √I—e²)) dɛ = 0,
hence we obtain the value of dɛ given in the text; and in order to
obtain the expression for ds we have
NOTES.
621
do
$183
dr
a(1—e³)esinė —w)
drr
(1+e.cos(0—w))²
do2
(a. (1— e²));¹ + 2e.cos(8—w) + e°.(sin³(0—w) + cos²(ê—w)
(1+e cose-w) +
•·.· ds = √ r² +
d02
e²)
e
(1 + e cos(0—w))²
It appears from the value of da that the radius vector diminishes
while in the angular velocity increases; and the variations of the
quantities e, w, a are only periodic; the variation of the absolute ve-
locity becomes by substituting 3pant for dn, and —2ga²0 for da, the
expression in the text. Likewise, as dr = -2pa20, the sum of these
continual diminutions from = 0 to 0 = 2π, is πρа².
(y) From the equation de cdt, it appears that when the areas
are proportional to the times of their description, the angular velocity
cdt
do=
varies inversely as the square of the distance, and con-
versely, when the angular velocity varies inversely as the square of
the distance, the areas are proportional to the times, for then cdt
=r²de = a constant quantity, and generally, in different orbits the
cdt
angular velocity do = varies as the synchronous area divided by
the square of the distance.
rde
dt
r²de = cdt·.· : (v sin d) ——, and at the initial distance where
с
r=y, d = a, v = √2gh we have √2gh sin a = -.
γ
(*) Multiplying both sides by dx and integrating, we obtain for
the first equation
dx2
dt
kx2 dy
ky2
+c,
+ c', hence it is
Υ
dt2
γ
X
easy to obtain by the known rules the values of x and y given in the
text.
(a') When t is increased by 2, the values of x and y remain
the same
as before, for in this case, the value of x is y.cos
k
2gh
2π
γ
((t + 2×)√ })√5 - √2. cosa.sin (1 + 2-√3) √
½) 1
ky
k
k
622
NOTES.
=y.(cos 1√k.-
COS t
√/2gh
.cos a.sin
t
γ
ky
V
the value of y.
k
>
the same is true for
It appears from this, that when the force is proportional to the
distance the periodic time varies as 1
γ
(b) By squaring both sides of these equations, and then adding
them together, this result is obtained; if a = 90 and ky ≤ 2gh, the
trajectory will be an ellipse, of which y will be the major semiaxis,
but if ky > 2gh, y will be the minor semiaxis.
When the force becomes repulsive, i. e., when R is changed into
-R, the values of x and y will not be expressed in periodic functious
of t, but will increase indefinitely.
++ 25Rdr=b, becomes, by
d.
2 c²
(c) Equation (3) c²
T
do
ন
substituting for c², SRdr, and b, their respective values, y² 2gh. sin² a.
dz
(d. (;;)"'+ 1 ) + ky (1 − 2²) = 28h, = (as y. = da),2,hain'.
(2)+1)+(1-2)
dz
do
(de** + * ) + ky (1 − 2) = 2gh, hence dividing by 2gh sin³ a, and
2²
−
concinnating, we obtain the expression
ky
dz
de
+ (1-
2gh sin² a
a).
22
ky
2gh. sin² a
which, by substituting n² for 1-
1
ky
sin³ a 2gh.sin² a'
1
and observing that
sin² a
in the text.
cota + 1, becomes the expression given
(d') In order to integrate this differential in the case where the
superior signs are employed, let cot² a +n²
dz
n²
of which the integral is arc sin =
when 0=0, ≈ = 1 · c = arc sin =
lue of ne is that given in the text.
✓ cot² a + n²
l², and then nd) =
nz
+c, and
n
>
hence the va-
√ cot² a + n²
±
In the second case, the integral is n = log (± nz +
✓ cot² a
c =
-
n² + n²x²) + c, and when 90, and z = 1, we have
log(±n+cota), and therefore,
NOTES.
623
nolog
+nz± √ cot² a n² + n² z²
+(n + cot a)
From the first value of no we obtain
sin.no =
cos.no =
nz.cota—n√ cot² a + (1 — ≈²) n²
cot² a + n²
n²z + cot a√ cot² a + (1— 2²
α
cot² a + n²
n
2
hence, multiplying the first of these equations by cota, and the
second by n, we shall have
nz cot² a
cot a. sin. no =
n.cot a√ cot² a + (1 − x²). n²
cot² a + n²
α
n.cos.no =
n³z+n.cot a√ cot² a +(1−z²).n²
and by adding them we obtain
cot² a + n²
'cot2a + n²
cot a sin noncos no = nz
=nz.
cot²a + n²
ndz
(el)
do
n.cota.cos no n² sin no, ... when ≈ is a maximum
we have
ndz
de
=0, and. cota. cos non sin ne0, consequently,
cotan tang ne.
n tang no. It is easy to perceive, that this value of tang ne
gives z a maximum, and, consequently, r a minimum, substituting
this value of cot a in the expression for nz, we obtain
nzn.
sin² no
cos no
'sin² no + cos² no
+n.cos no = n.
=
cos.no
✓
1
cot² a + 1.
n²
dz
dr 1
do
= 0,
n . √ tang² no + 1; and ≈ =
At the point where r is a minimum, we have
dr
therefore, at this point 70
at this point.
n
=0, consequently, the curve has an apse
(f') When ≈= 0, r is x, the value of tang no for z=0, is
cot a
n tanga, consequently, if from the centre we draw a line
making with the initial distance y an angle ◊, such that tang.no =
624
NOTES.
―n tanga, the line drawn making this angle will meet the curve at
an infinite distance; and the curve has an asymptote parallel to this
line, for if p be a perpendicular let fall from centre on tangent, we
1 dz²
have in general
= +2², in the case of an asymptote z va-
p² de
dz 1
nishes, and the value of
= cot a²+n², consequently, if
do
p
angles to the direction of
such, that tang ne- n
a line be drawn from the centre at right
the radius which makes with y an angle
tang a, and equal to p, and if through the extremity of p a line be
drawn parallel to the infinite radius, this line will be an asymptote.
(g') In the case of the logarithmic value of ±ne, if we suppose n
to be affected with a negative sign, we will have
nz + √ cot a-
ne = log.
and therefore
(n+cot a) e
but we have also
-no
=
(
n+cot a) e¹º
пв
n+cot a
n² + n²x²
nz +√ cot² a − n²+n²x²,
a)e" =nz + √ cot² a
n² + n²x²,
hence we obtain, by taking the first equation from the second,
пв
(n + cot α) e¹º + (n − cot a) e
пв
— ne z =
* = 2n x = 2 ny;
p
;
consequently, by substituting for e", e-ne, their respective values, we
(n+cota) (1+
shall have
пв
+
Ι
n2f2
+
23f3
+ &c.,
1.2
1.2.3
2nz =
(n-cota) (1
(1 –
пв
n² fr
n3f3
+
+ &c.,
1
1.2
1.2.3
and from the inspection of these it is evident that continually
increases.
dz
The value of
do
:(2 cot a)
'n + cot a
2
его
(
n
cot a
-по
e if in order
2
>
to determine the greatest value of ≈, we put this expression equal to
по
1
n — cot a
cipher, we will obtain e¹º —
and z = √
n² cot² a. if
√n + cota.
n
n> cot a this value is real, and ≈ is a minimum for this value of eno
NOTES.
625
and beyond this value of e, ≈ increases, and the curve which is
traced has two values that are symmetrical on each side of this
value of ene. The curve has an apse at this point, for
rdo
dr
држав
do
dr dz
= in this case, but is the tangent of the angle under the
tangent and radius vector, and therefore at this point this angle is
right. If the motion commenced from this point, we would have
a=90, and the equation of the spiral would be r=
27
eno + e no
from inspection of this equation it is evident, that the branches at
each side of the apse are symmetrical, for the curve remains the
same when no is changed into ne, the branches intersect in an
indefinite number of points, all of which exist on the line drawn from
the centre to the apse, the distance of any one of these intersections
such as the m from the centre is determined by the equation
th
r =
27
emx + e-mx'
From the general expression for the time it follows, that the
time to the centre, when the number of revolutions is infinite,
γ
n √2gh √2gh
If in the preceding value of ≈ =
Y
ky
√ n²
cota, we suppose
n
cot² an², then the value of z is impossible, and the curve does
not admit of either a maximum or minimum for z, however, the
curve has an asymptote, for we shall find that the value of
dz
1 (= x² + days)
when ≈0 or r = ∞, is equal to (cot² a
n²).
From the preceding discussion it appears, that in the equation
1
1
2
2. (n+cot a) ene +
(n
2n
2n
N-
cot a) cne, the curve which it
denotes is different according as n is > or
cot a the curve has an apse, in the second
than cot a, when n>
case it has an infinite
radius and asymptote. In both cases we have 2gh. sin² a
ky, but
as 2gh - kyky cot² a — 2nigh; in the first case also we have
ky, and in the second 2gh > ky.
2gh
4 L
626
NOTES.
If in the equation
n + cot a
(1+
no
n2f2 n303
+ + + &c.
1 1.2
1.2.3
2nz=
n cot a
(1
па
n202
n3f3
+
&c.
1.2 1.2.3
r
we suppose n=0, i. e., 2gh sin³ a=ky, we shall have ~==
cot a. 0, and y = cot.a.r., which is the equation of the reciprocal or
hyperbolic spiral. In order to determine the time, if in the equation
12d0cdt, we substitute
22
cotza.02
for r²,
we obtain
72.do
cot² al. 2
cdt. ct
— 7º 1
cot2a 0
+c, and between the limits 0, 0', we have
22
ct=
(−1)
cot² a
If T denote the entire time to the centre, we shall have, com-
mencing with
2,
22 1
1
CT=
+
cota 2 | 1.2
2.3
·]=
3.4 + &c. •] = Y², tang² a
2π
The asymptote to this curve is determined by the equation
202
= x² +
dz2
d02
—in this case z² + cot² a, ·.· when ≈0, or r➡∞,
we have p the perpendicular from the centre on the asymptote
tang. a.
If in the expression for z, we suppose n cot a, i. e., 2gh=ky,
the value of z becomes e cota., from which it is evident, that the
curve described is the logarithmic spiral, and if the base a=e‡cota,
the equation of the curve will be ra in order to determine the
time, if in the equation cdtride, we substitute for r², and inte-
grate between the limits 0=0, 0 — — 2π, we obtain ct =
:
>
=(1-
1
2π
and in like manner between the limits 0 = 2x 0 — — 4π, we find
1
ct!=
2.a2
:(1.
1
and so on for successive revolutions, therefore,
T
if r denote the time to the centre, it will be expressed by the series
#1
1
1+
2c
α2π
+
1
a¹π + &c.],
αλπ
NOTES.
627
by substituting for c, and taking the sum of this series
cos ect.a
7.2.√2gh
(
(h') This value is evidently equal to
do =
c²dp
√ k²y*— ßc²—k²y*+2ky²c²p—c+p²
which by assuming y➡ c²p — ky², and by concinnating becomes
dy
√ k²y* — Bc² — y²
and the integral of this is equal to
W
w+ arc cos
equal by substituting for y,
w + arc cos=
-y
√ ]²y+ — Bc²
ky² c²e = 0.
—
√ k²y* — Bc²
The value of r deduced from this equation is
.r=
c2
:
ky² — √ k²y^ — c²ß.cos (0-w)
consequently, when 0 = w, r is a maximum ; when 0 = w+x, r is a
minimum. w+T is substituted for w to embrace the case of the
comets.
(i) By substituting for r and r. cos ( — w), their respective
values in equation (a), and squaring both sides of the resulting ex-
pression, we obtain
k²y¹. (x¹² + y¹²) = c²+ x¹². (k²y* — c²ß) — 2c²x². √//k²y² — c²ß.
(k) If in general 4p denotes the principal parameter of an
ellipse, we will have ae
Va-2apa-p+(
a (1 − e) the shortest distance =p+(
when a is ∞, we shall have a (1 − e) = p.
)=&c.
a
1
&c., consequently,
α
(V) Substituting for p and dividing by y, we obtain
1+cos w=2sin³a, i.e., 2cos² = 2sin² a, for 1+ cos w = 2cos²½w.
(m²) Since
S
d↓ sin
=
cos² cos'
, by partially integrating, we obtain
628
NOTES.
S
d↓ sin
cost↓
=
dy
-2S cost
cos'
+S.
sin 2 sin.cos d
cos
2dy
cos²
>
cost
sind
=
(2 in) =
=tang &, we have
and as
S
dy
cos²
S
cos¹
= tang 4. (1+ tang² 4) + 2 tang 4-2
dy
"
have So
by integrating this
last quantity in the same way, we shall obtain a similar result multi-
plied by 2, and by continuing this partial integration, we will arrive
at the following series,
tang.(1+tang²)(1-2+4-8+&c.)+2tang.v(1-2+4-8+&c.)
+e, but (1−2+4−8+&c.) = (1+2)~¹= }};
hence then we shall have this integral,
tang. (1+tang²) + 2 tang
3
nt
+e=
(n) It appears from equation (e), compared with n =
PVP
that the times, in which comets moving in different parabolic orbits,
describe equal anomalies, vary in the sesquiplicate ratio of the peri-
helion distance.
Also by means of equation n =
√, i =
PV p
Vie
ιντ'
the synchro-
nous areas described by the earth and comet may be compared; for
the sector described by the comet at the perihelion distance p,
is to the synchronous sector, described by a body moving in a circle
of which the radius = p with an angular velocity equal to n, as
√2:1. See Mechanique Celeste, Book II. Chap. IV.
CHAPTER VII.
(a) When ² is neglected in consequence of its minuteness, we
have p³ (a²-2ar cos x)=a3+3a-r cos λ, consequently,
=
fm'r sin λ fm'r sin λ fm'a fm' 3fm'r cos λ
therefore,
03
a³
;
+
p³
fm'a — fm'r cos λ ___fm' 2fm'rcos λ
p³
a²
23
03
A
NOTES.
629
(b) See Mechanique Celeste, Tom. IV., and also the translation
of the System of the World, Vol. I. Book I. Chap. XV., and Vol. II.
Chap. XI.
(c) The force 'resolved in the direction Cм is equal to q'cos λ,
for λ is the angle between CM and AC, and as the direction of the
force is perpendicular to AC, the angle which it makes with CM
must be the complement of A, consequently this force resolved in
the direction of CM = 4 sin λ, and if we substitute for ò̟'and & their
2fmrcosa fm'r sin λ
respective values
and if we observe, that these
components of p' and act in opposite directions along the radius
CM, and consequently must be affected with opposite signs, the re-
03
sult of the two must be equal to (2 cos² à
m
75'
sin³ λ)
fm'r
which, by
substituting for m' is the expression in the text. These results
may be applied to the disturbance produced in the lunar motions by
the sun; and if we suppose that r is equal to the radius of the moon's
orbit, and a the mean distance of the sun from earth, it appears
from this expression, that when λ= 0, i. e. in the syzygies, the force o
vanishes, and the force o' is a maximum and equal to
; on the
contrary, when λ=90, i. e. in the quadratures, the force o' vanishes,
fmr
75 a37
2 fmr
75x3
and &=
value of as 2:1; but we should always keep in mind, that the
force increases the gravity, while it is diminished by the action of
the force d'; as they act in opposite directions, and as they respec-
tively vanish, o' in the quadratures and in the syzygies, there must
be some intermediate position where they are equal; this is easily
determined; for since 2 cos². A sin² = 3 cos² A 1, the value of
hence the greatest value of ' is to the greatest
$
the force 'cos λ- sinλ is (3 cosλ1)
fmr
75 a³
G
consequently, when
1
3 cos² λ-1=0, i. e., when cos λ =
'cos sin λ, and from
'3'
1
sin λ, or the additious force pre-
√3
syzygies to an elongation =
dominates over 'cos λ, or the ablatious force, and from this point
to the quadratures, the latter exceeds the former; and therefore,
during an entire revolution, the ablatious force predominates; it
6.30
NOTES.
would be easy to compute the actual diminution, for by multiplying
(3 cos² λ − 1) f'm'r by dλ, and then integrating, the result is
a³
(† λ − λ + § sin 2^), f'm'r; which is the effect of the disturbing
a³
forces, while the angle is described, and for an entire circumfe-
rence this becomes (because the periodical part vanishes),
fmr
03
π;
and therefore, the mean disturbing force acting on the moon in the
direction of the radius vector is
m'r
it is easy to show that this
quantity is equal toth part of the moon's gravity to the earth,
for if T', T, denote the periodic times of the moon and earth, we
have
M
: F the central force of the moon ::
α r
a²
T²°T/
mr
03
T/2
=}.
T2
T/2
1 mr
Now =
T2 179
force =
F
358
F
179'
and consequently, the mean disturbing
The effect of this diminution of the moon's gravity
by ath part is to increase her distance from the earth, and as the
force by hypothesis acts in the direction of the radius vector, the
mean area is not altered, but the angular velocity which varies in-
versely as the square of the distance is diminished by a 4th part,
and the periodic time is increased by the same quantity.
(d) Let c = 2πr' = circumference of the moon's orbit, a the
arc described by the moon in a minute in her orbit, v its versed sine,
the radius of her orbit, then as she is supposed to revolve uni-
formly, we have
A2 22p!
A:C::l":n, ·. A
=
=
n
2pl
n2
and by substituting for , we obtain the expression in text; now if
the same space v be described at the surface, and at the distance of
the moon, and if ƒ and t denote the force and time of descent at the
surface of the earth, and f',t', the corresponding quantities at the
moon, we have
v=ft² =ƒ't²²; ··· t²:t²²: f':ƒ,
i.e., whenƒ:ƒ'::(60)²:1, t:t'::1:60;
NOTES.
631
$
i. e., a body at the earth's surface would in a second fall through a
space equal to what a body at the distance of the moon would fall
through in a minute. The corrections or various circumstances, the
consideration of which are omitted in order to simplify the demon-
stration, are—Ist. The disturbing action of the sun, which was ad-
verted to in the preceding note. 2ndly. The circumstance of the re-
volution of the moon not being performed about the centre of the
earth, but about the common centre of gravity of the earth and
moon. See No. 243, and Harte's Translation Du Systeme du Monde,
Vol. II. page 381.
(e) If the figure of the earth be supposed to be that of an el-
lipsoid generated by a revolution about its lesser axis, and if the
equicapacious sphere be supposed to have the same centre, the
greatest elevation of the sphere above the spheroid is half the de-
pression beneath it, when the spheroid differs little from a sphere,
for if r be the radius of the sphere, a and b the major and minor
semiaxes of the spheroid, then if a r = p, r bs, we have
s=2p.q.p, for
Απγ3 4″.
a²b=
3
3
Απ
4 * (r + p)². (r − s), = ¹ * (r³+ 2r²p — r²s),
3
3
when the squares and products of p and s are neglected, conse-
quently, we must have 2gs; we can by means of this relation
obtain the value of r in terms of a and b, for we have evidently
2a + b
; the elevation of this equicapa-
3
2a—2r—r—b, ··· r =
cious sphere above the sphere inscribed in the ellipsoid, and whose
radius is therefore equal to b, is =
2a + b
3
-b= (ab); now in
figure 60 let AB be the minor axis, and CD the major axis of the
spheroid, ABB a section of the inscribed sphere, let d be any point of
the spheroid, do, da lines drawn to the centre and parallel to oc,
meeting the inscribed sphere in the points m and n, we have by the
nature of the ellipsoid,
CR: dn::oc: dx: rad: cos doc,q p:
and by similar triangles ndm, dxo, nd: dm :: od: da: radius to cos doc ;
·.· : RC: dm :: co² : dx², i. e. as square of radius as to the square of the
cosine of doc, or of the latitude λ q.p, .' as RC = a
as RC = a − b, we have
1
632
NOTES.
dm = (a - b) cos; now the depression of a point of the equi-
capacious sphere below the spheroid, and existing in the radius od,
is evidently equal to the difference between dm and rb, or to the
difference between dm and & (a - b) which by substituting for dm its
value (a - b). cosλ, is equal to (ab). (cos2 λ), consequently,
when cos A, this depression vanishes, or the equicapacious sphere
intersects the spheroid; from the value of dm it is evident the in-
crease of the force of gravity in going from the equator to the poles,
varies as the square of the cosine of latitude, as was stated in No. 193.
The value of the latitude resulting from the above equation, namely,
is remarkable also as being the distance from the quadratures
√3'
at which the additious is equal to the ablatious force.
1
f) As lines drawn from places on the opposite sides of the
mountain to the same fixed star, are supposed to be parallel, and as
the deviation on each side is supposed to be equal, the angular dis-
tance of the two stars must be double the deviation.
(g) By substituting for p and q their respective values, we ob-
Qac sin (y — x)
fm'm₁
tain P. a sin x =
and then as P='
fmm,
y
2.2
y2
we have
fmm,
a. sin x=
fm'm, ac sin (y — x)
p2
уз
which by substituting for m' its value, and dividing by m, is the ex-
pression in the text; now in the equation
sin x
pl2/3
evident that
sin (y — x
=
it is
prez
sin x = (sin y cos x
cos y sin x)
pr13
and
p.rc²,
1 = (sin y. cotx
cos y)
prez
.cot x=
prc²
p'³.sin y
+ coty,
·.· when c is least and y = 90, cot x is least, and ·.· x the greatest.
(h) It is necessary that the lever should be accurately horizontal,
otherwise the effect of gravity would be complicated with that of the
force of torsion.
(i) By substituting for cos (70) its value, we obtain
≈² = a² + c² 2 ac cos y cos
2ac. sin y sin 0,
= (if the square of @ be neglected in the series expressing sin 6, cos 0,)
NOTES.
633
2་3
~~3 = (b²
a² + c² - 2ac cosy - 2ac sin y.0, b2ac sin y. 0,
and therefore,
- 2 ac siny.e)-b-3+ Sac sin y.b-5, ·.·
23
sin (7-0)
(sin cosy.) (b-3 + 3ac sin y.0b-³) =
siny
cosy. Sacsin³y.
+
b3
3 3
now if the numerator of
cosy.8
b3
be multiplied by the value of b²,
and if the denominator be multiplied by b², we obtain
0
cosy. A (a² + c²) cos y. ◊ — 2 ac cos³ y.
b3
hence then we have
-
sin (7 6)
for
22
b5
siny — [(a² + c²) cos y — 2 ac — ac sin³y].
b3
2ac. (sin²y+cos²)=2ac.
-
(k) See note (e) of this Chapter, and No. 193.
Ө
65,
(1) See the Mechanique Celeste, Livre 7; also Harte's translation
of the System of the World, Vol. II. Chap. V., and notes.
·(y² + ≈²) 3,
yz dy dz
(m) When 2 fap'.
is integrated with respect to z, it
becomes
2xft (v
ydy
+x), the value of x is determined
by
ydy
√ x² + y²
integrated
x²+ y²
making = 0, which gives Y = c; and-2f'
with respect to y, gives y = — √ ≈² + y² + z, in which z is evi-
h,
dently equal to ǹ, hence then the value of — 2-ff'S
yz dy dz
be-
(y²+ ≈²) §}
tween the limits o and h for z, o and c for y, is the value of k' given
in the text.
kr2
(n) g — g'(=by neglecting y'-y and h²) k·
k =
(~ + h)²
2kh
k'.
2°
Now as k differs from g' by a small fraction, if in the fraction
2kh
2 kh
g' was substituted for k, the result would differ from
by
2
ԴՐ
h
the product of into the difference between g' and k, which as it is
ጥ
T
1
4 M
634
NOTES.
a very small quantity, the product is a very small quantity, and con-
sequently, may be neglected.
(0) As k' = 2xfp'h, if in this expression we substitute for ƒ its
value derived from assuming
namely, k =
3 p'hg'
2pr
3
will differ from the actual value by a very small
quantity of the second order, hence then if in the expression
4πpfr = g' namely,
3.gl
Απρο
the result,
2kh
g-g'=
k' we substitute g' for k, and
3p'ho
for k', we shall
グ
​2 pr
2h
obtain g - g′ = gl
3p'h
2pr/
p'
(p) If p′ = { then the value g — 9' is g'. (2h — 34)
3h
5h
g'
=g'. Ar
NOTES.
635
BOOK III.
CHAPTER I.
(a) It appears from what has been just established, that R is the
resultant of the given forces P, P', p", &c., and of Q, consequently
these forces may be replaced by R, and a force equal and directly
contrary to Q.
(b) As each body furnishes six equations, namely, X, Y, Z, L, M, N,
respectively equal to cipher, the entire number of equations which
thus result, will be six times the number of bodies; but for every
point of contact, the elimination of R and of the corresponding quan-
tities of the other bodies reduces this number by one, therefore, the
entire number of distinct resulting equations will be six times the
number of bodies minus the number of points of contact.
(c) When L = 0 the resultant of the forces parallel to oy must
coincide with it, and therefore be destroyed by the fixed point.
(d) Q = P sin y, q=± (y cos λ—x cos µ), therefore, qq = ±
P (y sin y cos x x sin y. cos μ) equal ± P (y cos a
a P' cos a'; therefore substituting
x cos B.)
h
Τ
(e) P cos α =
for cos a, and
cos d for cos a', we obtain the expression in the text.
l
(f) Since the friction is by experiment to the pressure, it
follows from what has been just established, that the coefficient f,
and consequently λ, are independent of the pressure, and therefore
of the weight.
CHAPTER II.
(a) The expression for L in No. 261 is
P (x cos ẞ — y cos a) + P(x' cos a' - y' cos ẞ') + &c. = L;
now by substituting for Q, Q', &c. in the value of s, P siny, P' sin y',
&c., we obtain the value for L, for cos ẞ sin y cosμ, cos a = sin y
cos λ, &c.
=
(b) If the squares of the three quantities H cos &, H cos d', H cos d'
be taken, and if they be then added, there results
H². (cos + cos 8, + cos² 8,) = r². p²,
636
NOTES.
*
på being substituted for its value given in the text; therefore we have
pp.cos d = P(x cos ẞ — y cos a),
consequently,
1
cos d
(x cos ẞ — y cos a).
p
(c) If in the equation au + By + cw, we substitute for v, w,
their values
u cos Bucosy, and then multiply by cos a, we obtain
cos a
cos α
the equation a cos a + B cos ß + c cos y = 0; now from the equation
Ax+cz
(^2+ C≈) = − (^~
-(
Ax+BY+CZ=0, we obtain B——
consequently,
and therefore,
A cos a+c cos
ß
cos B
Ax cos ẞ+ cz cos ßay cos a + cy cos y,
C = A
x cos ẞ - y cos a
y cos y — z cos B
in a similar manner we may obtain the value of B.
-
(d) By expanding the binomials in the value of p² we obtain
a² cos² B+ cos y + y² cos² a + y² cos² y + z² cosa + cos² ß -
2xy cos a cos B 2xz cos a cosy 2yz cos ẞ cosy, now x²(cos²ß+
cos² y) = x²x² cos² a, y² (cos² a + cos y) = y² - y cos² ß, x²(cos² a
+cos² B) = 2²² cosy, consequently as 2+ y²+22= r², the
y²+≈²=
value of p² will become
x+2
p¹³ — r² cos² a cos²λ-r²cos²ß cos³µ—r² cos³y cos²v — 212 cos a cosλ cos ẞ.
cos μ-2r² cos a cos λ cos y cos v-2r2 cos ß cos μ cos y cos v.
ре
Now it is evident, that the parts of this expression which contain the
cosines multiplied by r2, are the square of cos a cosλ + cos ß cos µ +
cos y cos, and consequently it vanishes, therefore the value of
p²= r², as was proposed to be verified.
(e) By making this substitution, the fourth equation of No. 261
becomes
P(x cosẞ-y cosa)+P'(x'cos B'-y'cos a')+&c.-(Pcosẞ+P'cos B'+
p" cos 3" + &c.) x, + (P cos a + P' cos a' + P" cos a' + &c.) y₁,
now the two first terms of this expression are equal evidently to L,
and the two last terms are equal to xy,-Yx,, hence the value of L, is
that given in the text.
(ƒ) By taking the partial difference of the value of G² with res-
pect to a, we obtain, when G, is a maximum,
NOTES.
637
— 2Y (L+ XY, — Yx₁) + 2z (M + zx, − xx,) = 0,
·
consequently,
x₁ (y² + z²) = x (Yy, + zz₁) + YL — ZM,
and by adding x,x2 to each side of this equation, there results.
x₁ (x² + y² + z²) (= x,R²) = x (xx, + vy, + zz₁) + YL ZM.
1
The expressions for Ry,, R2%,, may be obtained in a similar manner.
i. e.,
1
(g) From two of the preceding equations we obtain
R²x₁+ ZM YL R²y, + XL ZN
X
Y
R² (xy, — Yx₁) + (x² + Y²) L = Z (NX + MY),
and by adding z²L to each member of this equation, there results.
R². (xy₁ — Yx₁) + (x² + y² + z²) LZ (NX + MY+LZ),
which, by dividing by R², and observing that x² + y² + z² = R², be-
comes the expression in text.
CHAPTER III.
(a) It is evident from the construction, that if m and n be the
angles, which any of the forces such as P, makes with the adjacent
sides MA, MM', of the polygon, the tension along the side MA is equal
to
P sin n
and that along the side MM'
P sin m
sin (m+n)'
; now if
sin (m+n);
m' and n' be the corresponding quantities at the summit м' of the
polygon, we have the tension along M'M evidently equal to
p' sin n'
p' sin n'
sin(m'n')
hence when the direc-
P sin m
consequently,
sin.(m + n) sin (m' +n');
tions of the forces P, P', P", &c., applied to the summits of the poly-
gon bisect the angles contained between the sides of the polygon,
the tension in each side is the same throughout the entire polygon,
and equal to any of the forces P, P', p", &c., divided by twice the
cosine of the angle which it makes with the side of the cord to
which it is applied; when P, P', P", &c., are all equal, the polygon is
a regular figure; and conversely, when the polygon is a regular
figure, the forces P, P', P", &c., must be equal, and each of them is
to the tension as a side of the polygon to the radius of the circum-
scribed circle.
638
NOTES.
(b) As a, a', a'', &c., and also e, g, y, y'y', &c., are in this case
90°, and ß, ß', ß", &c., are cipher, it follows that the first equation
(a) will be equal to
and the second will be
H COS α + K Cos e = 0,
н cos b + and н COSƒ+ P + P' + p'', &c. = 0,
and all the terms of the third equation will vanish; now if both
tions (b) be squared and added together, we have
H² + R² + 2HK (cos a cos e + cos b cosƒ) = II²,
equa-
and the factor of 2 HK is evidently equal to the cosine of the angle con-
tained by the sides H and K, therefore, II is equal and contrary to the
resultant of these forces; it follows from the equation н cos b + à cosƒ
+П=0, that the directions of the extreme forces H and K meet in
the vertical drawn through the centre of gravity of P, P', P", &c. In
this case it is evident that sin n = sin m', consequently if the equa-
tion which we obtained above in note (a), namely,
p' sin n'
sin.(m'+n')'
becomes
P sin m
sin.(mn)
=
&c., be multiplied by sin n and sin m' respectively, it
P.sin m. sin n
sin.(m + n)
p'. sin m'. sin n'
=
sin.(m'+n')
p/
p//
P
cot m'cot n' cot m" + cot n'
+ &c.,
cot m+cot n
and it is remarkable, that in this case, namely, when all the forces are
vertical, the horizontal tensions on each side of the polygon will be
respectively equal to these fractions, consequently, the horizontal
tension is constant throughout the entire polygon. If this equal
horizontal tension be denoted by a, and if the tensions along the
sides of the polygon be denoted by T, T', T", &c., we find T. sin m =
A = T'sin m' =T" sin m", &c., so that the tension on each side varies
inversely as the sine of the inclination of the side to the vertical.
(d) Since the sum of the tensions of all the strings resolved in
the direction of the vertical is
.w
ठ
w
cosy + cosy +
8"
w"
y"
cos y” + &c.] P ;
y''
P;
and since this is by hypothesis equal to P, the quantity by which P is
multiplied must be equal to 1.
NOTES.
639
(e) By substituting their values for cos a, cos a', T, T', y', x', the
first equation (a) becomes
dx
dx
T + T
ds
ds
+Td + dr = d(
dx dx
ds ds
dx
T
ds
The second equation (b) may, by similar substitutions, be derived
from the equation (a), and with respect to the third there results by
these substitutions,
dy)+(x+dx) [(x+dx) (y+ddy) - (y+dy)
dx
T
Y
X
ds
ds
T
dx
dx
ds ds
(d+dda)] = peds,
ds
ds
)]=
which by obliterating equal quantities affected with opposite signs, is
evidently equal to
x(:
d. Tx
dy
$13
ds
dx
y
ds) = pxds.
h
h
(ƒ) By adding together the values of e² and e and observing
that ds = 1 +
Vi
dy2
dx2
•
dx we obtain the value of ds, and if the value
h
of e be taken from e, and if we then multiply by dx, we obtain
the value of dy.
(g) By adding and subtracting the values of 7 and b we obtain
k
k'
k'
k
l + b = h
h
e
e
l-b=h
h
е
(k+k')
··12 b² = h²
h
te
h
(k+-k') −2) ;
=by substituting a for k+k', the expression in the text; and since
α
(ea—e—a)³ — e2a+e-2a - 2, and
there results
a²
1ª — b² = aa² (e«
4a² (e« — e—a)², ·· 1
—
2α,
h
b2
1
(ea
e-a)= n.
a²
2a
a
a²
α
+ &c., e a =1
1.2 1.2.3
a³
1
a²
1.2.3.
2α
&c., '.' (ea —e—a) (= n) = 1 +
6
ဝ
(h) ea=1+it +
+
a²
1.2
a² = 6`(n − 1).
640
NOTES.
α
α
(i) k + k' = a, k — k' = 2 ßh, ·.· k = 2 + ßh, k' = 2 — ßh,
k
a
k'
a
h
+ B,
h
2h
k k
h'h
2h
B, and if these values be substituted for
in the value of b, the result will be equation (e).
(k) The integral of the value of dy' is clog(x'+c)+√(x'+c²) — c²²
+c, when x' = 0, y' = 0, ·.· c = c log (c + √c²— c¹²), ··· y'=
the expression in the text.
(¹) y = c + √ c— c¹², y' = c — √ 2² — ¿¹², '.' yy' = d'² = h², for
e
exe¯ = = 1.
C
есхе
(m) By substituting for T its value ph
dy
ds
dx
there results phd.
dy ds
ds dx
=pdx, i. e., h.d.=dx,·· h = x, 2hy = x², hence as T = ph
ds
dy
dx
=ph√ 1+
dy?
dx2
we obtain by substituting
20
for
h2
dy2
dx2
the value of T
given in the text.
(n) √ h²+x²dx =
h2dx
√ h² + x² √ h² + x²²
taken between the limits k, k', is
x²dx
+
the integral of
h³dx
√ h² + x²
√ h² + k² + k
h² (log √ h²+k²+ k) — log (h² + k'² — k') = h² log
√ h² + T¹² — k'
in like manner as
S 2+ 75
k
x2dx
=k
•h² + k²
+ k
2
Th² + k/²
h2dx
k
h²dx
· K' √ h² + x²
by substituting for +4
-k' √ h² x²
its value, and multiplying by 2,
we obtain the expression for 2hl in the text; now if both numerator
and denominator of the logarithmic function be multiplied by
√ h² + k²+k, there will result, when k = k',
2
√ h².
h² + x²
2hl=h². log.
√h+k²+k\²
2
+2k √ h² + k²,
h
from whence we can easily deduce the value of hl given in the text.
NOTES.
641
(0) If for √h²+k² its value in a series be substituted, we shall
obtain
k + √ h² + k²
log.
h
ka
k + h + 3/1/20
h
2 − 1
k4
k
k2
h³
=log.
now as in general log (1 + u) = u
h
= log 1+t
[ 3
2 h2
-&c.],
U²
из
+
&c., we have
2
3
k2
k2
2/3 K3
h
log [1 + 2 + 3 / &c. ] = [ / ++
&c.
h2
2h2
2.2.h³
3h³
+37 &c.]
k
k³
h
- 6h³ + &c.;
hence, by substituting this value in the expression for hl, there results
hl
=h2
h²
砂 ​[一​]
kc3
+kh +
2h'
and if we multiply by h,
h²l = 2hºk + ‡ k³, ·.· h² =
α
by substituting for k its value 2.
K3
23
3 (l — 2k) ¯¯¯ 8.3. (l—a)'
(p) By substituting this value of T we obtain
dx=dx'+wPdx'+wp
V
い
​~) dx
dx', '.' x=x'+wpx' + wpl'ar
wpl'x
wpx12.
V
21
(9) By performing this multiplication, there results by adding
dx dx dy, dy dz dz
d.-
+ T
+ d. +
d +
ds ds ds ds ds ds
dr
(
ds2
dx2 dy² dz2
+ +
ds2 ds2
X
ε
dx
ds
dy
ds + Y
ds
ds + z dz ds)
³) =
= 0,
ds
which from what is stated in text is evidently equal to equation (3).
dy
ds
(r) If the first of equations (1) multiplied by be taken from
dx
the second multiplied by
we shall have, as
ds
dx
dx
d.T
= dr.
ds
ds
+ I.d. &C.,
dx
ds
dy dx dx dy
dr.
dx,dy dy dx
+T
ds ds ds ds
dy
dy d. de) + e ( x dads - xly de
dsds ds ds
Y ds-xds
ds=0,
ds
consequently, as ds is the independent variable, we can obtain from
this expression the first equation (4).
·
4 N
642
NOTES.
di
(8) As ex=n cos λ= (by substituting its value for cos λ) NV ;
dx
by performing the same operation for ɛY, ɛZ, then multiplying by
dx, dy, dz respectively, and finally adding them together, we obtain
dL
dL
ε (xdx + ydy + zdz) = Nv
= NV
dx +
dx
dy
dy+dz dz) =
dz) = NvdL = 0.
(t) By substituting their values for X, Y, Z, in the second mem-
bers of equations (4), they will become
ds2
cos µ dx) dș², N (cos v dæ
N (cos ady - cos dx)
T
N (cos µ dz cos v dy)
ds2
vdy) T
т
ds2
cos λdz)
and if these be multiplied by cos v, cos μ, cos à respectively, and then
added together, the result is evidently cipher, hence the expression
p (cos y cos v' + cos μ cos µ' + cos à cos λ′),
to which the preceding is equal must be cipher.
(u) The coefficient of N in the expression for
panding
A
k
-
2
becomes by ex-
P
dy2
dx²
ds2
(cos + cos y) +
dz2
dş² (cos² µ + cos² v) +
ds2
(cos²λ + cos² μ)
2 dy
dy dx
•
ds ds
cos λ cos μ
2
dx dz
-.cos λ.cos
• cos λ. cos y — 2
ds' ds
dy dz
ds ds
cos μ cos y,
μcos v,
which, as cos² + cos² µ + cos2y= 1, becomes
μ
dy² dx2
dz2
ds2
+ +
ds2
ds2
(dy
ds2
dx2
ds
cos² μ + cos2.λ+
dz2
dy dx
•
.cos² v +2
ds2
dsds
dx dz
cos λ.cos μ +2
•
Cosλ cosy + 2
dy dz
ds ds
ds ds
cos μ
cos
»).
Now the expression between the brackets is equal to cipher, and the
k
other expression is equal to unity, hence N.
P
(v) If these equations be multiplied by x and y respectively,
there results by adding them together,
dx
xd.T +yd.T + N. (2º + y²) ds= 0,
ds
dy
ds
C
and as x² + y² = c², this equation coincides with the first equation
NOTES.
643
(6), and if these equations be respectively multiplied by
dx dy
there
ds' ds
results from their addition the second equation (6), for xdx+ydy
= 0.
(x) The first equation (6) is evidently equal to
ds
(xda+y
dy
ds
Jar
dr + (x.d.+y.d.
dy
T+Ncds 0;
ds
dx
+y.d.dy
=ds, this equation
ds
ds
ds
ds
dx dy
hence as x +y =0, x.d.
is reduced to T = CN; in like
manner the second equation (6) is
equal to, by substituting cds for xdy — ydx,
dx² dy²
+
ds2 ds2
1) dx
dx dx
.d.
+(
ds
dr+
consequently we have druds.
ds
+dy.d.dy).
T — µds = 0,
ds
Notes to Paragraph III.
~+25, by substituting for ♂', a, their respective values,
(a) σ =σ+
P
we obtain y (1+8)=√(1 + 8) +
иб
u.y (1 +³), dividing by Y,
y, and
neglecting we obtain, by omitting a common quantity on both
ρ
น
sides of the equation, d'+~; and, when the length of the mean
P
น
filament is not changed, ♪ = 0 and 8', i. e. the lengthening and
Р
contracting of the filaments, is proportional to u their distance from
the mean filament.
(b) Substituting for & in the value of T, it becomes
ΤΞ αλ
S_
δου + αλ
du
S
ε udu
ε ୧
E
and performing the integration between the limits a,, the first
term becomes 2aλde, and the second becomes
αλεξ
αλεξ
=0; and
2P
it becomes
20
in the same manner, substituting for ♪' in the value of
αλ
ε u²du
µ = aλS__ dudu+ax S₂
ax =
E
و سلام
644
NOTES.
*
integrating these quantities between E, E, the first term vanishes,
ɛ,
as in the preceding case, and the second becomes
αλεύ
P
From the value of T it appears, that in the case of the same
plate, it is proportional to ♪ the extension of the mean filament, and
when 80, i. e. when the length of the filament is not changed,
T 0, and the forces which in this case act on K must be parallel
forces reducible to two, but not directly opposed.
(c) The angle of contact is always equal to the element of the
curve divided by the radius of curvature, and when the element is
given, this angle or the curvature varies inversely as the radius of
curvature, consequently μ varies in this case as the angle of contact.
dy
(d) Making dx
=≈, equation (2) will become
c2dz
(1 + 2²) €²
=adx xdx,
and integrating so that we may have 20 when a 0, we obtain
2 c²z
=2ax
p
x²,
consequently,
and,
4c+z² (2ax - 2·²)² (1 + ≈²) ;
=
(4c¹ — (2ax − x²)²) ≈² = (2ax − x²)²;
hence by substituting for z its value, we obtain
(2ax - x²)
√ 4c¹ — (2 ax — X²)¿
14c - a
dy
=
dx
ds² =
4c* — (2 ax
(2αx — x²)² dx²
x²) 2
4c4dx²
+dx² =
4c⭑* — (2ax 22)2
= (when c is
G
very great, in which case 2ax-x² may be neglected relatively to c,)
dx², hence in this case we must have 2c²dy=(2ax
...
=3ax² — X³.
6c²y=3ax²
6c²y=3ax²
x²) dx, and
(e) 6c²y = 3ax² — x³, which, when xa and yb, becomes
=
3cb=a³, and from the equation 3qc² = awɛ², we obtain 3c² =
and awɛb = a³q; by substituting for aw its value
αωε
Q
W
we will have
W
= eb =
εba³Q and b =
азба
ε2 w
therefore, as ad = h, we shall obtain
NOTES.
645
arhQ
b =
ε² W
•
w being by hypothesis a weight equivalent to the force which draws
the plate in the direction of its length, if we suppose it equal to Q,
we will have b =
a²h
£2
consequently b: h:: a²: º.
(f) When k=0, y is always equal to cipher, therefore the
dy
spring cannot be bent; when k does not vanish, y and will have
dx
finite values, therefore the spring will bend; at the point B, x=α,
y =0=k sin, therefore a = ic, i being a whole number.
k
(g) If the fourth power of be neglected, we shall have
C
T²k2
TX
2k2
πX
1+
cos2 =
1 + 1/1/20
cos2
с
с
TX
2πχ
1/1/2
с
C
= as cos²==+ cos
π2k2 72/2 2πχ
++
1+
4 c²
COS
;
C
if this value be substituted in the expression
1=
Sa
π2k2
TX
VI 1+
c²
cos.dx,
с
and then integrated, there will result
π2k2
1 = a + α,
4c²
which, by substituting for a its value ic, becomes the expression in
the text.
TX
α
(h) In the equation y=fa sin y=0, when x = 0, and x = a,
therefore, the curve will not cut the vertical between the two points
A and B. In the equation y = f'a sin
α
2'
2πχ
α
-, y=0, when x =0; x =
and xa, therefore the curve will cut the vertical in the middle
point between A and B. In the equation y = da sin
iπx
sa
,y=0when
646
NOTES.
x=0,
x = 0, x = α, x=
a, hence it appears, that the curve
α
will cut the vertical AB, in the number i + 1 points.
When k = 0, y=0, therefore the figure is rectilineal, but the
least increase of k above this gives a finite value for y, in which case
the figure is not rectilineal, but when 7 is ▲c, k is impossible,
therefore y cannot have a possible value, consequently, the figure must
be rectilineal.
(i) In general we have P
αωε
3 c²'
now it appears from No. 312,
that as long as lisc, the spring cannot be bent, but when the
quantity c is so diminished that 7 may surpass c, then the spring may
be bent by the weight, consequently, lc is the limit after which
the spring is bent, hence, the value of P determined by the equation
P=
αωε
8× »
gives the greatest weight which the spring can support
without bending.
?
σ
(k) σ'=~+20=, = (by substituting for its value y (1+♪), and
neglecting du), y (1+8) + Yu, in like manner o' = y′ (1 + ♪') =
(by
из
y'
из
by substituting y'its value y + and neglecting
and neglecting 14), y++yö',
2'
hence we obtain, by dividing by y and obliterating common terms
зи
uy
in the equation y (1 + ♪) + y² = y + u?!? +7d', d'= 8+ w
r
γ
(Substituting for d' its value given above, we obtain
k
=aS.
k
-k'
Svdu+avudu.
--
the integral of the second term vanishes, and that of the first = awd;
in like manner by a similar substitution in the value of μ, we obtain
=aS_20% svudu + a Sk
Evudu + a Sz vudu (
α
# = Sk
(-3),
in this case, the integral of the first term vanishes, and that of the
second is equal to
awq²
3
(m) In the first case, as the distances of the centre of gravity
from the vertex and base of the triangle are respectively c, c, we
NOTES.
647
}
have in this case k = c, k' = c, now any line parallel to the base
such as v, which is distant from the centre of gravity by a quantity
=
u, which may be either positive or negative, is to the base a:: c +
α
u: c, . v= == (‡c+u), and consequently
au³
3
a u¹
S vu²du = +
c
C 4'
2 ac³ ac3
+
9'33 4.34'
and when
this quantity becomes equal to
с
when u=
2c
2a 23c3 a 24c3
it is equal to
u =
น
πα
12
>
3
+
9 33 4 34
; hence when the quantity
.vu³du is integrated between these limits c, c, which are those
of k, k' in the present case, it becomes equal to
Je
** [2 (1₂+370)
.ac3
+(²
24
4.34
T
x²αac³
• ac³ =
3612
9.33
In the case when the form of the base is concave,
α C
v=
= = [ + ],
+u], :· Svw³du=
C
au³ aut
+
9
c4,
which becomes, when u c, c, which correspond to k and k' in
this case,
23 ac³
9.33
24 ac3 ac3
+
+
4.34 9.33
ac³
4.34'
hence we obtain,
Па
[2
a S_11/01
k
(1 + 2³) ac³
vu²du =
+
k'
9.33
(241) ac
4.34
π²α ac²
=
2122
When the normal section is a square, as the centre of gravity is at
the intersection of the diagonals, k, k', the limits between which the
integral should be taken are respectively equal to, and, as all
2'
lines at any distance u from the centre of gravity, drawn parallel
to a side of the square are equal to the side, we have
vu²du= 2 f
-k
2ƒ ƒ³ _ƒ¹
3 23 12'
P =
π³aƒ*
12/2
In the case of a circle, v the line drawn at the distance u from the
centre of gravity which is that of the circle, is evidently equal to
་
648
NOTES.
2 √k²—u², and
S
-k'
-k'
vw'du = 2 { _ % √ k—w. wdu = by multi-
k² u².
plying and dividing by √ k²— u²,
2
·S.
k k2u²du
k
u¹du
·k' √ k² — u²
·k' √ k² — u³´
now if we make u=kx, these expressions become respectively
2/4
S
k
x²dx
-2k4
>
-k' √ I — x²
S
k
xdx
·k' √ I— x²
dx
the integral of the former is
— k¹x √ I — x² + k⭑
·
and that of the latter
} k*x°³√] − x² + {k+x√ T − x² − Z k*
S
dx
Now when this integral is taken between the prescribed limits k, -k',
kt
dx
all the other terms will vanish but the term
which is evi-
4
/1−x²
dently equal to
k*T
4
hence the value of P becomes that in the text.
(n) If the area of the normal section in the case of the square
and circle were equal, then fak², consequently, by substituting
this value of ļ in the expression for P in the case of the square, it
Tak¹
12/2
becomes =
π³ak¹
:: 7:3.
4/2
which is to the value of P in the case of the circle
(0) By making this substitution we obtain
T
П³а
P= ·(g'-g₁) =
412 04)
T-3
4/2 (~12 +80) (0-12 — g²);
now if (g'-g³), the area of the normal section, be supposed equal
to ʊk², then g¹² = k²+g², consequently if k² be substituted in place
of g²² —g², and k²+g² in place of g", we shall obtain the expression
in the text.
dx
dy=
(p) As ds is supposed to be the independent variable, dx=ds
dy ds, &c., consequently xdx + xdy + zdz =
ds
ds
NOTES.
649
dx
x de ds + x dy ds + z
ds
+z da ds
dz
ds
ds
(9) Since in this case all the quantities in equation (b) vanish but
cosμ, cos f, cosg, cosh, P', ', R', and T, ds' ds' ds
dx dy dz
; and as these three
last quantities are equal to cosa', cos ß', cosy', it is evident that
the values of T cosa', Tcos ß', T cosy are those given in the text.
(r) From equation (1) we obtain
dy
4gyw
B
x²+3cx²+2c'x, ß
d³y __ 3.4.gywx²+2.3.cx+2c',
dx2
24
dx
24
B
ß das
d³y
2.3.4.gryw
x + 2.3c;
24
now since by hypothesis 30, 6
d³ y
= Q, when xa, there
d.x³
results gywa² + 6ca + 2c′ = 0, gywa + 2.3.c =
From the second of these equations we obtain, by substituting q
for gywa, c = (+9), and by putting this value of c in the first
equation, and substituting for gywa its value q, there results
(@
Q
=
=
Iqa − (e + q) a +2c' = 0, ... c' } (@+ £q)a,
consequently if these values of c and c' are substituted in equation
(1), there results the value of By given in the text, for it was shown
by the conditions of the question that c" 0. When Q = 0, this
equation becomes
=
By =
qxs 9
24a 6
x³ + qax²,
and when x = a, in which case y = b, there results
Bb = 21/14a³ - † qa³ + ‡ qa³ =
03
24
1
and when q0, this equation becomes
By — — — Qx³ + ½Qax²,
which, when x = a, and ·.· y = b, becomes
даз
;
8
Qa³
Bb =
3
;
and when q = 0, the preceding values of b are evidently in the ratio
of 3:8.
4 0
650
NOTES.
1
(s) Since in this case y = 0 when xa, equation (1) becomes,
because gyw =
q
=
0:
=
•
aª + ca³+ c'a², hence if it be multiplied
a
24a
by a, and then taken from equation (1), we obtain the expression in
the text.
dy
(t) p¹y = 1 (x³¬a³)+c(x²— a°)+c'(x−a)+
dx 24a
when x = 0, this expression becomes
dy
Bay = -24a² - ca² - c'a = 0,
dx
—
when a = a, this expression becomes
x
β
dy__ qa³
dx 8a
Здхо
24a
+2cx²+c'x,
c/=
(12/14 + c) a ;
+c)as
(+2c)a;
+2
+ 2 ca² + c'a = 0, •·.· c' =
hence a comparison of these two values of c' gives c =
consequently c'aq, by substituting these values of
equation (2) becomes
qx
I
T2
T', and
c and c',
By= ·(x³ — α³) — 7½ qx.(x²—a²) + aqx.(x —a),
24a
which may be made to assume the form
qx
By=
24a
(x—a)[(x² + ax + a²)−2a(x + a) + a²] =
qx — =
(x — a).(x² — ax) — £x². (x—a)².
24a
d³y___ 3qx²
In the second case ß
dx² 24a
24a
•
9x2
+2cx + c² +9 + 4 cx +
24a
c'=1, when x = 0, hence it follows that c' = 0, and when ≈ = a,
6c=
124, and ... c=g; consequently, by substituting for
24
c, equation (2) becomes
2
By:
qx
=
24α (׳ — a³) — 1'z qx.(x² — a²) =
qx
24α
(x − a). [x² + ax + a²—
2a. (x+a)]= · (x—a) [x² — ax-a²], which by changing the
qx
24a
signs of the two factors of which this product is composed, becomes
α
the value of ẞy given in the text; and when x =
the value of y,
2'
which in this case is f, becomes =
a', No. 324.
q a
a 5a²
24a 2 2 4 24.16
5qa³
-
See note
NOTES.
651
In this case also we have
β
d³y 2.3.qx
2.9.qx
+2c+
+4c,
dix3
24.a
24a
d³y
dx³
which, when x = 0, becomes ẞ = 6c = − 1 9 ; and when
2 1/1/
+2c+39 + 4 c = q — 1/2 g =
d³y
x = α, ß
a,
dx3
dy
In the third case, as
=0 for x = 0, we have
dx
dy
β
ga2
ca2
c'a = 0;
dx
24
day
and as in this same case B
=0 when x = a, we have
dx2
qa
22 + 6 ca+2c' = 0,
consequently, comparing the two values of c' derived from these two
equations, we obtain
24
[3/4 + c ] a = [2/4 + 3 c ] a
5q
· c =
-
and ·.· c' =
qa
5q
+
48'
169
a
qa
16
; and, if these values be substituted in equation (2), we obtain
By=
qx
24a
(x³ — α³) –
5qx
48
(x² — a²) + qα.27 ( x − a)
qax
16
qx
qx
48a(x− a) [2x² + 2ax + 2a² −5a.(x + a) + 3a²]
= (x− a) [2x² —3ax] = (by changing the signs of the factors)
48a
(a−x)[3α-2x]. The value of ß
722
48a
this case B
d³ y
dx3
3q
this case
5q
d³y
Ewhen x = 0), is in
dx3
d³y
B-
dx (= ~ Q when x = a) is in
—
5q
59 = 32 =
3q
- Q.
8
B- =6c= 54, and B
8
2 +34 +6c, i. e. q
4-
-
NÆX
(u) By performing the integration with respect to sin which
a
is the variable of the second member of this equation, we find
a
Ssin
NπX
a
dx
NT
·
a
NxX
COS-
and
>
a
ПП
S
ηπ
a2
nxx
COS-
da =
sin
;
α
a
652
NOTES.
hence if this operation be continued four times, the last integration
gives
a¹
NAX
sin
a
from which, and by a suitable determination of the arbitrary con-
stants introduced by the integration, from the consideration that
y = 0 for x = 0, and for x = a, we obtain equation (b).
α
(v) When .r' = sin
2'
nπx!
ηπα
a
NT
= sin
2
and as So pa'da' = a the
NT
sin px'd' in this case = a sin 2'
α
expression So
1)2
(x) In this case as n = 2i — 1, and sin (2i — 1).
= (− 1)i,
we have the first term of the second member of equation (b) equal to
2 ga³
Σ
(-1)
(2i-1)+
sin
(2 i — 1) πx
α
(y) If in this equation we substitute for w its value
obtain
(-1)i
Σ
sin (2¿ −1)
π-4203
Х
πX =
(2 i — 1)¹
α
24 a³
32 a
75X
>
we shall
α
which, by reducing to a common denominator, becomes the expres-
sion in the text.
(*) By differentiating we obtain
α
dy
β 1. · (3a² — 3.4.x²),
dx 48
which, when x = i. e. at the middle point of the curve, is equal to
2'
cipher, therefore, at this point the tangent is horizontal, and the
sagitta then becomes equal to
9
3a³ 4a³
=
B.48'
2
8
9.a³
B.48*
(a') dy
=
q
dx 6.48
a
(3a² −3.4.x²) (= when x = 0, tang. a) =
α
and as f'::: tang a: rad, ƒ'=tang.a; but f=
2
qa²
16.ß
ga³ a
48.B
=tang a,
f':f:: 3:2, and in the second case of No. 322, as
NOTES.
653
dx
equal when x = 0,
qa³.
24a
Bay
dy q(a−x)(a²+ax— x²) — qx(a²+ax—x²)+qx(a—x)(a—2x)
24 a
... in this case ƒ'
qa³
9.8.a³
48 16.24
>
conse.
quently it is to the value of ƒ in that No.:: 8:5.
(b') By multiplying the infinite series given at the commence-
ment of this number by the denominator, there results
1+2h cos 0 + 2h² cos 20+ 2h³ cos 30+ &c.
1-2h.cos + h².
= 1 + 2h cos 0 + 2 h² cos 20 + 2 h³ cos 34 + &c.
2h.cos - 4h² cos² 0 4h³ cos 28.cos
Ө
4ht cos 30.cos
+h² + 2h³ cos + 2h+ cos 20 + 2h³ cos 38 + &c.
Now since 2 cos²= cos 20+ 1, 2 cos 26. coscos 36+ cos 0,
2 cos 30: cos = cos 40+ cos 20, by substituting these values in place
of cos²0, 2 cos 20. cos 0, 2 cos 30. cos 0, and adding these lines together
their sum will be equal to 1+ h².
This theorem is proved, a priori, in the Theorie de la Chaleur,
par Poisson, No. 93.
When the values of u are infinitely small, 4 sin² 1 (0 — α) = u²,
and if in the expression
gfede
we substitute for fe, de,
g+4 sin(u)'
and 4 sin² (-a) we will obtain the expression in the text; and as
the integral of this function between the liimits and 0, is and
2'
૭|s
between the limits 0 and
∞ is also
2'
the integral is π.
between the limits and
When a = ″, then cos n (♪ — a) = cos nã. cos nl, (= as cos nã
= (− 1)".) (—)". cos n§.
παι
Since 0 =
>
α
and as the limits relative to ♦ are ≈ and 0, the
limits relative to x' must be a (when 0 =) and when = «, it must
be cipher.
'x±x²·
Since cos na
COS-
a
ηπα
a
NAX
.COS- + sin
a
ηπα
α
NTX
sin
when
a
the first equation (5) is taken from the second, we obtain the expres-
sion in the text.
(c') When fo= 0, this function becomes
654
NOTES,
SETE
n9.0
по
cas ne ode = sin
no
+ cos
n
222
which taken between the limits and 0, is
COS
NT 1
n2
•
consequently, since when n is an even number of the form 2i, cos nã=
1, and when ʼn is an odd number, of the form 2i-1, cos në =−1, in
the former case cos
ПП
n²
1
=0, in the latter it is equal to
hence the given equation becomes as
72 cos (2i-1) a
-4Σ
2
Sofo.
So fo.do =
(2i — 1 )²
α = πα
πα,
Σ
now in the expression sin (2-la
T
(2, − 1)³
3
T
cos (2i-1) a
¢{ {
π
8
(2i — 1)²
2
T2
2'
(2i — 1)²'
and fa= a,
(π-2α);
+ w₁
d)
(π-α)α, when a =
as cos(2i −1) = 0, sin(2i−1) = (− 1), (x−2).a = (———w²),
=(
we have evidently, by changing all the signs, Σ(-1) sin
4
(2i-1)w
(2i — 1)³
= (w²—‡x²), from w = tow, for these limits w =
~ ~
π
- 1 π, w =
π
2 correspond to a=0, a = ñ.
l
7
END OF VOL. I.
Treatise of Mechanics.
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3
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1

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