ELEMENTARY TREATISE
ON
MECHANICS.
BY I. W. JACKSON,
PROlESSOR OF MATHEMATICS IN UNION COLLEGE6:
ALBANY:
PUBLISHED BY GRAY, SPRAGUE & C0O
1852.




Entered according to act of Congress, by ALEXANDER HOLLAND, in the
office of the Clerk of the Northern District of New-York, Jan. 1852.
RIGGS, PRINTER9 SCHENECTADY.




ADVERTISEMENTo
THE following exposition of the elementary principles of rational mechanics, is intended to be sufficiently brief and free from difficulties to
be thoroughly mastered by students of ordinary capacity, in the time
usually allotted to the subject, and yet comprehensive enough to be
made the basis of a course of general physics and practical mechanics.
It has no claims to originality; the methods employed being generally
those which have been long in use, and which may be found in many
of the best treatises on the same subject.
Of the works consulted in the preparation of this volume, most
assistance has been derived from those of POISsON and BOUCHARLAT:
occasional use has also been made of those of EARNSHAW, POTTER,
SAURI, and PUIssANT.
In the present emission, the subject of solids only is considered.
The few sections on fluids, necessary to complete the plan, will soon be
published.








TABLE OF CONTENTS.
INTRODUCTION.
Definitions,                                                   I.  1
Laws of motion,.                                      2
PART FIRST.
STATICS.
\ Definitions,                                                  6
Parallelogram of forces,  -                                   10
Methods of determining the position of a point,           -   13
Of forces situated in the same plane, and applied at the same point, 18
Of forces applied at the same point, and situated in different planes, 22
Of parallel forces,                                           25
Of oblique forces, applied at different points, and situated in the
same plane,  -.                        37
Of forces applied at different points, and situated in different planes, 48
Centre of gravity,..                      52
Machines,    -                    U                           69
The rope machine,   -. -   69
The lever,                        -.                         72
The inclined plane,            -.                      74
The pulley,                                                   75
The wheel and axle,                                           79
The screw,   -                                                81
The wedge,                                                    84
General principle of equilibrium in machines,   -    -    -   86
Friction,                                                     90




VI                         CONTENTS.
PART SECOND.
DYNAMICS.
Of the rectilinear motion of a material point,                 1
Of the motion of bodies upon inclined planes, -               11
Of the motion of a material point on a given curve,           15
Of the simple pendulum,                                    - 19
Of central forces,.                                    25
Of projectiles,    -..                   - 34
Measure of forces,  -...                 40
Principle of D'Alembert,    -                                 46
Moment of inertia,       -    -    -    -                     49
The compound pendulum,                                        57
Of the collision of bodies,           -.  -                65
Gravitation,    -.    -    -                              80




ELEMENTS OF MECIHANICS,
INTRODUCTION.
1. WHEN a body occupies successively different positions in space, it is said to be in motion.
2. Whatever produces, or tends to produce motion,
is called force.  The action of a force, whatever its
origin, may be conceived to consist in communicating to the body on which it acts, impulses, either
finite, or infinitely small.  The body is, in all cases,
supposed to be entirely inert, and subject only to
the influence of forces exterior to itself.A
3. It can readily be conceived that two or more
forces may be so applied to a body, that their effects
shall counteract each other, and no motion shall ens
sue: in this case the forces are said to be in eqzdlit4. Mechanics is the science which treats of equili
4. Mechanics is the science which treats of equilibrium  and of motion.  It is divided into two parts,
STATICS and DYNAMICS.
*Abstraction is thus made of vitality, and certain properties inherent
in matter, as gravity, and the chemical and electrical attractions, etc.
etc. When the effects of these are to be determined, they are regarded
as extraneous forces.




t2`                   MECHANICS.
5. In statics, the subject of equilibrium is considered. It is divided into two parts:
I. The statics of solids, called simply Statics;
II. The statics of fluids, called tHydrostatics.
6. In dynamics, the subject of motion is considered.
It also is divided into two parts:
I. The dynamics of solids, called simply Iygzamnics;
II. The dynamics of fluids, or Hyjdrocynamics.
7. The reasonings in mechanics are based upon
certain facts of great generality, derived from observation and experiment, called the laws of mnotion.  We
shall do little more than enunciate them, leaving it
chiefly to the instructor to supply the observations
and experiments by which they are proved and illustrated.
LAWS OF MOTION.
I. T7ie law of inertia.
The motion of a body, when left to itself; is rectilinear and uniform. Thus when a body receives an
impulse and then is abandoned to itself, according to
this law all its points will describe straight lines, and
will move over equal spaces in equal times.
At first view, this law appears to be contrary to
the most obvious facts; for the motions which we
observe in bodies on the surface of the earth are
neither rectilinear nor uniform, and they all soon
terminate.   In  these cases, however, the bodies
are really not left to themselves, but are constantly
acted upon by certain forces, as gravity, friction, etc.,
which in a greater or less degree interfere with their




INTRODUCTION. 
motions. But it is invariably found, that as these
interfering forces are diminished in intensity, the
motions take place more nearly in accordance with
the law. Thus in the case of a body moving on a
horizontal plane, as the surface of each is rendered
smoother, the more nearly does the motion become
rectilinear, and the longer does it continue. Again,
a pendulum, oscillating under ordinary circumstances,
is soon brought to rest; but if we diminish the friction at the axis of suspension, and cause the vibrations to be made in the vacuum of an air-pump, the
motion will continue for many hours.
From these and other similar experiments, as well
as from numerous analogous observations, we infer
that if the interfering forces could be annihilated,
the motions would take place in conformity with
the law.
II. Thle law of the coexistence or independence of motions.
The relative motions of a system of bodies are not
affected by any motion which is common to all the
points of the system.
To aid the student in the conception of this law,
suppose a number of bodies to be in motion on a
surface, plane or curved, and conceive the surface to
be at the same time moving parallel to itself; then,
according to this law, the motions of the bodies on
the surface, or their relative motions, will not be affected by the motion which they have in common
with the surface, but will take place in the same
manner as if the surface were at rest. The surface




4                  MECHANICS.
having been introduced merely to render the distinction between the relative motions of the bodies
and their common motion more obvious, it may now
be withdrawn, if in its place we give to each of the
bodies a motion identical with the common motion.
We shall thus have a system of bodies moving in
space in the manner supposed in the enunciation.
A case less general than the preceding, but one to
which it will be necessary frequently to refer, is that
in which we suppose a point to be moving on a
straight line, and the line to be carried at the same
time with a uniform  rectilinear motion parallel to
itself; the motion of the point, according to this law,
taking place as it would were the line at rest.
An apt illustration of this law is furnished by a
vessel sailing in a given direction, in which the motions of bodies relative to the parts of the vessel
take place in precisely the same manner as if the
vessel were at rest, though, in addition to their relative motions, the bodies all have the same motion as
the vessel. Since the vessel may obviously be regarded merely as the means of communicating motion
to the bodies, we may suppose it to be withdrawn, if
at the same time we suppose the bodies to retain
their common motion.
III. The law of the equality of aclion and reaction.
To every action, there is always opposed an equal
reaction.
According to this law, when a body rests upon an
immovable plane, the pressure of the body upon the




INTRODUCTION.                  5
plane causes in the plane a re-action, in virtue of
which the plane may be said to press upwards with
the same force that the body presses downwards.
In like manner, when a body falls upon an immovable plane, the plane reacts with a force equal and
opposite to that with which it is struck by the body.
Also when two bodies, moving in the same line,
come in collision, the effects upon the bodies, properly
estimaled,i are equal, and take place in opposite directions.
*The method of estimating the effects will be explained in a subsequent article.




PART FIRST.
STATICS.
8. By the term  nctericl point, or pcrticle, is meant
the smallest conceivable portion of matter. In mechanics, any one material point is conceived to be
exactly similar to every other material point. A
body may be regarded as a collection of material
points.
In considering the effects of forces, we first suppose the forces applied to a single material point.
This point we regard as entirely passive, and subject
only to the action of the forces under consideration.
In a force, three things are to be considered: its
point of application, its direction, and its intensity.
The point of application of a force is the material
point on which it acts.
The direction of a force is the direction in which
it tends to move its point of application.
The intensity of a force is its capacity of producing motion.
Two forces are regarded as equal in intensity,
when, acting separately upon two material points,




STATICS.                    7
they cause the points to describe equal spaces in
equal times.
Two equal forces applied to the same point, and
acting in the same direction, constitute a double
force; three equal forces, a triple force; four a quadruple force, and so on. Taking one of the equal
forces for the unit of intensity, and representing it
by 1 or by p, the intensities of the double, triple,
quadruple, etc. forces, will be represented by 2, 3, 4,
etc., or 2p, 3p, 4p, etc.  The intensities of forces may
thus be represented by numbers or algebraic symbols, and may be made the subject of arithmetical
and algebraic operations.
The intensities of forces may also be represented
by straight lines; a given straight line being assumed
to represent the unit of intensity, and a double, triple,
quadruple, etc. force being represented by a line of
double, triple, quadruple, etc. the length of the assumed line.
According to the second law of motion, double,
triple, quadruple, etc. forces  cause the material
points to which they are applied to describe, in a
given time, spaces double, triple, quadruple, etc. the
space due to the unit of intensity. The spaces described by the material points are thus directly proportional to the intensities of the forces; and consequently the same lines which represent the intensities of the forces may also be employed to represent
the spaces described by the material points to which
the forces are applied.




8                   STATICS.
If, for example, the lines AB, AC, AD [Fig. 11,
represent the intensities P, Q and R of three forces
referred to a common unit, they may also be employed to represent the spaces which these forces
severally would cause a material point to describe in
a given time.
The converse also is obviously true, viz. that if
AB, AC and AD represent the spaces which three
forces, whose intensities are P, Q and R, cause a material point to describe in a given time, the same
lines may also be employed to represent the intensities P, Q and R respectively.
9. Conceive now a material point to be moving
uniformly on the straight line A'B' [Fig. 1], and suppose the line to be at the same time moving parallel
to itself with a uniform rectilinear motion; then by
the second law of motion, the motion of the point on
-the line will be entirely unaffected by the motion
which it has in common with the line; and hence if
we represent the motions of the point and line during any time I by AB and AC respectively, at the
end of that time the point will evidently be at D,
having described the diagonal AD of the parallelogram constructed on AB and AC. But if we abstract
the line A'B', and the motion which the point has in
common with it, and suppose the point to be moving
alone in space in the direction A'B', at the same rate
as before, and to receive when it arrives at A an impulse, such as would cause it, were it at rest, to describe the line AC in the time 1, the final result will




STATICS.                   9
evidently be unaffected; and in virtue of the two
motions, the point, at the end of the time 1, will still
be found at D. Again, if, instead of supposing the
point to be in motion when it receives the impulse
in the direction AC, we suppose it to be at rest at A,
and to receive at the same instant another impulse,
such, as acting alone, would cause it to describe in
the time t the line AB, it is still obvious that in virtue of the two impulses the point will, as before, describe with a uniform motion the diagonal of the
parallelogram ABDC, and at the end of the time t
be found at D.
Let now the intensities of the forces to which the
motions AB and AC are due, be represented by P
and Q; and as the motion from A to D may also be
attributed to a single force, let the intensity of that
force be denoted by R: then the forces P, Q and R*
may be represented by the lines AB, AC and AD;
and instead of saying the forces P, Q and R, we may
say the forces AB, AC and AD. B, or AD, is called
the reszltalz of P and Q, or of AB and AC; and P and
Q, or AB and AC, are called 1i1e componzefnts of B, or
AD. The characteristic property of the resultant is
that it may be substituted for the components. Thus
the effect upon the point A is precisely the same
whether the components P and Q act upon it in the
directions AB and AC, or the resultant R acts upon
it in the direction AD.
*Hereafter, for the sake of brevity, we shall say simply the force P,
the force Q, etc., instead of the force whose intensity is P, Q, etc.
B




10                     STATICS.
The result at which we have arrived, may now be
thus enunciated:
The resultant of two forces applied to a material point,
and represented by lines measured from this point on their
directions, is represented in mnagnitude and direction by the
diagonal of the parallelogram constructed on these lines.
This proposition is called the parallelogram of forces.
The forces P and Q may evidently be supposed to
be either finite, or infinitely small.
If A be a fixed point, P and Q will exert upon it
a pressure of longer or shorter duration, the measure
of which is evidently R.*
10. In figure 2, in which AI is taken equal to AB
or P, as the angle BAC diminishes, ID tends constantly to become equal to BD or Q, and hence R to
become equal to P + Q.
In figure 3,- in which AI is taken equal to AD or
R, as the angle BAC increases to 180~, BI tends constantly to become equal to BD or Q, and R to become
equal to P- Q.
Hence, when the forces P and Q act in the same
straight line, it may be inferred from the above proposition that the resultant is equal to their sum or
difference, according as they act in the same or opposite directions. This result may'also be deduced
immediately from the second law of motion.
*When P and Q are forces of the kind commonly called impulsive,
such, for example, as that exerted in driving a nail with a hammer, the
duration of the pressure is very brief, though finite: when they are of
the same nature as gravity, the pressure is continuous.




STATICS.                11
When the forces act in opposite directions, we have
R = P- Q.
But P may be considered as the resultant of two
forces P' and p, both acting in the same direction as
P; and p as the resultant of two others P" and p',
both acting in the same direction as p, and so on.
The same is also true of Q; hence we have
R = P' + P"' + P"' + etc. - Q' — Q" - Q"' —etc......[1]
When the forces P', P", etc., Q', Q", etc., are in equilibrium, we have R = 0; and hence
P' + P" + P", + etc. = Q' + Q" + Q"' + etc.
is the equation which expresses the condition of
equilibrium of any number of forces which act in
the same straight line.
11. Resuming the consideration of the parallelogram of forces: Since [Fig. 4] BD is equal to AC,
we have the three forces P, Q and R represented by
the sides of the triangle ABD. Moreover we perceive
that the angle at B is the supplement of the angle
BAG formed by the directions of the components P
and Q, and that the angle ADB is equal to the angle
DAC which the direction of Q makes with that of R.
Hence of the three forces P, Q and R, and the angles
comprehended between their directions, any three
being given, one of the three at least being a force,
the remaining three can be determined by solving
the triangle ABD.




12                    STATICS.
Thus denoting the angles which the directions of
P and Q make with the direction of B,, by b and c,
and the angle which they make with each other by
a; if P, Q and a are given, and R and b are required,
to determine the latter, we have
R2 = P2 + Q2 _ 2PQ. cosB
= p2 + Q2 + 2PQ. cosa,
and          R  ~ Q:  sin a    sin b.
12. When three forces P, Q and R' [Fig. 4], are in
equilibrium about a point A, any one of them, as B',
must evidently be equal and directly opposite to the
resultant of the other two.  But employing the
same notation as in the preceding articles, and denoting the supplements of b and e by 1b and c', we have
R    P: Sin a    Sin 
sin a    since',
and          R: Q: sin a    sin b
sin a    sinl';
and hence, since R = R',
R'   P:~ Q:: sin a    sin c'   sin 6'.
That is,?whez three forces are in equilibrizugn about the sane
point, each of themn mnay be represented by the sine of the
angle comnprehended between the directions of the other two.
13. Any number of given forces P, P', P", P"', etc.,
applied at a point, and situated in the same or in
different planes, may, by means of the parallelogram
of forces, be reduced to a single force.  We first determine the resultant R of any two of them, as P and




STATICS.                 13
P'; then the resultant':, of R and any one of the
remaining forces, as P"; and so on till all the given
forces have been compounded: the last resultant
will be the resultant of the system.
Figure 5 presents a case in which four forces P, P',
P"' P"', are reduced to a single force B". An inspection of the figure shows that the resultant may be
determined by the following construction: The lines
AB, AB  AB AB   AB"', representing the four forces:
through the point B, draw BC equal and parallel to
AB'; through C, draw CD equal and parallel to AB'";
and through D draw DE equal and parallel to AB"':
the straight line joining the points E and A is the
resultant required.  This method of finding the resultant is applicable to any number of forces.
We have just seen that any number of forces
applied at a point may be reduced to a single force,
or resultant. The converse also is evidently true, viz.
that a single force applied at a point may be resolved
into any number of forces, or comnpoeWts, all acting
upon the same point, and producing the same effect
as the single force.
14. METHODS OF DETERMINING THE POSITION OF A POINT.
A convenient and elegant method of treating the
subject of forces, consists in decomposing the forces
into separate systems, parallel to certain assumed
straight lines. The application of this method requires a knowledge of the first principles of analytic




14                   STATICS.
geometry, of which we shall here give a brief exposition.
~ 1. JMetehod of deterinizig the position of a point on a
line.
The position of a point on a given straight line
XX' [Fig. 6], is evidently determined when we know
its distance from an assumed point 0 of the line, and
the direction in which the distance is to be laid off
fromn the assumed point.  The method of indicating
the direction is conventional, and consists in affecting
the distance with the sign plus or minus, according
as the point is situated to the right or left of O.
Thus, employing the letter x, as the general symbol
denoting the distance, the equation
x = ~- a
determines a point M, situated to the right of O, at
a distance from it equal to a linear units; and the
equation
x =- a
determines a point M', situated at the same distance
from O, in the opposite direction.
~ 2. Method of determiniing the position of a point on a
plane.
The position of a point on a plane is determined
when we know its distances, affected with their proper
signs, from two straight lines drawn in the plane at
right angles to each other.
Let XX', YY' [Fig. 7] represent these lines, forming by their intersection the four right angles XOY,




STATICS.                 15
YOX', X'OY', Y'OX. Consider the point M situated
in the angle XOY, and draw through it the straight
lines MP, MQ, parallel respectively to the lines YY',
XX', and intersecting them in the points P and Q.
Then it is evident that when P and Q are given, the
point M  can be determined by drawing through
them the straight lines MP, MQ, parallel to YY', XX'
respectively; it being at the intersection of these
lines. But P and Q are given when the distances
OP and OQ, or MQ and MP, are given: thus the
point M is determined by its distances from the lines
XX', YY'.
The line OP or MQ is called the abscissc, and OQ or
MP the ordincde, of the point M. The abscissa and
ordinate, taken together, are called the co-ordiczates of
M. The line XX' is called the axis of abscissas, and
YY' the axis of ordinates, and together they are called
the axes of co-ordinates. The point 0 in which the
axes intersect, is called the originz of co-ordicZates. The
general symbols, commonly employed to denote the
abscissas and ordinates, are x and y; and hence the
axis XX' is frequently called the axis of x, and YY'
the axis of y. The axis XX' is usually supposed to
be horizontal.
The quantities x and y are supposed to be capable
of assuming all possible values, both positive and
negative; and according to the method of the preceding section [~ 1], the positive values of x are laid
off on XX' from 0 to the right, the negative values
from 0 to the left; the positive values of y are laid




16                     STATICS.
off on YY' from 0 upwards, the negative values front
0 downwards. Thus, denoting the values of x and y,
in any particular case, by a and 6,
when x   + -   and y = + b, the point is in the 1st angle;
x =-  aand y  = + b,      " "       2   " 
x = -a and y —b,             " "    3d c;
x = -- a and y        =- b,      "    4th C6
Wherl  x = + a and y =    0, the point is on the axis of x;
x=    O and  y=+- b,           " "..     y;
x =    O and y _   0, the point is at the origin.
~ 3. heolod of deterteminy  tle position of ca point in
8 spc C e.
The position of a point in space is determined
when we know its distances, affected with their proper signs, from  three planes drawn through an assumed point at right angles to each other.
Let XOY, XOZ, YOZ [Fig. 8] be the three fixed
planes intersecting each other, when sufficiently produced, in the lines XX', YY', ZZ', and determining by
their intersection eight trihedral angles.  Consider
the point M  situated in the angle 0 - XYZ, and
draw  through it three planes parallel to the fixed
planes, viz:
MM/Q3i'"' parallel to the plane XOZ, and cutting the line YY' in Q;
1MM'PM" parallel to the plane YOZ, and cutting the line XX' in P;
and BIM"t"Rl"' parallel to the plane XOY, and cutting the line ZZ'
in 11.
Then it is evident that the point M  can be determined, when the points Q, P and R are known; for,
drawing through these points three planes parallel




STATICS.                   17
respectively to the fixed planes, they will intersect
each other at the point M. But the points P, Q and
R are known when the distances OP, OQ and OR,
or the equal lines MM"', MM", MM', are given: thus
the point M is determined in position by its distances
from the three fixed planes.
The distances OP, OQ and OR are called the co-ordinates of the point M; the lines XX', YY', ZZ', are
called the axes of co-ordinates; and the planes XOY,
XOZ, YOZ, are called the co-ordinate planes. The point
O is called the origin.  The co-ordinates OP, OQ and
OR are denoted by the general symbols X, y and z
respectively; and hence the fixed planes are denominated the planes of xy, xz and ye respectively. The
plane of xy is generally supposed to be horizontal.
The positive values of x, y and z are laid off, according to the conventional method, on the several axes
from O towards the points X, Y and Z respectively,
and the negative values from O in the opposite directions.
The signs of the co-ordinates of a point determine
in which of the eight angles the point is situated.
Thus, denoting the numerical values of x, y and z, in
a particular case, by a, b and c:
when x = a, y = b, and z = c, the point is situated in the angle
O-XYZ;
when x = - a, y =   b, and z = - c, it is situated in the angle
0 - X'Y'/Z'.
When one of the co-ordinates is zero, the point is
situated in the plane of the other two co-ordinates;




18                      STATICS.
thus when x = a, y = b, and z = 0, the point is in the
plane of xy.  When two of the co-ordinates are each
equal to zero, the point is on the axis of the third
co-ordinate; thus when x = a, y = 0, and z   0, it
is on the axis of x.  Whenx =0 y= 0, and z = 0,
the point is at the origin.
OF FORCES SITUATED IN THE SAME PLANE, AND APPLIED AT
THlE SAME POINT.
15. Let O [Fig. 9] be the point of application of
the forces; and through it, in the plane of the forces,
let the co-ordinate axes XX', YY' be drawn.
The direction of a force P  is determined by the
angles which it makes with OX and OY, the parts of
the axes on which the positive co-ordinates are laid
off, the angles being reckoned on both sides of these
lines from 0 to 1800.  Thus, considering the four distinct cases which can occur, as represented in figures
9, 10, 11, 12:
When the direction of the force is situated in the angle XOY, its position is determined by the angles XOM, YOM;
when in the angle YOX', it is determined by the angles XOM', YOM';
when in the angle X'OY', it is determined by the angles XOM",
YOM";
when in the angle XOY', it is determined by the angles XOM"',
YOM"'.
Let now the force P, in the several cases, be represented by the equal lines OM, OM', OM", OM"';
and let it be decomposed into the components OB
or OB', OC or OC', coincident respectively with the




STATICS.                19
two axes. Then we perceive that the components
coincident with XX' act, when the force is situated
in the 1st or 4th angle, from O to the right; when
in the 2d or 3d angle, from 0 to the left: also that
the components coincident with YY' act, when the
force is situated in the 1st or 2d angle, from 0 upwards; when in the 3d or 4th, from O downwards.
When, therefore, two or more forces, acting upon
the point 0, are decomposed in directions coincident
with the axes, there may be two sets of components
in the direction of each axis, the components of one
set acting in one direction, those of the other in the
direction opposite.  Considering the components
which act in the line XX': if we affect those acting
in opposite directions with opposite signs, as, for example, those which act from 0 to the right, with the
sign plus +, and those which act from 0 to the left
with the sign minus -; then the resultant will evidently be equal to the algebraic sum of the components. The same is true of the components which
act in the line YY'.
But the values of the components determined
by the usual method, will necessarily be affected
with the proper signs. Thus, denoting generally
the angles which the direction of P makes with OX
and OY by a and 3, when the force is situated in the
first angle, we have
OB = PcosBOM = + P cosa,
OC = P cosCOM = + P cosp/;




20                   STATICS.
when it is situated in the second angle, we have
OB' = P cosB'OM'  - P cosa,
OC = P Cos COM'= + P cos/;
when in the third, we have
OB' = P cosB'OM"   -- P cos a,
C00' = P cos COM" =   P cost3;
when in the fourth,
OB = P cosBOM"' = + P cosa,
0' = P cos C'OM"' = — P cosf.
Now let there be any number of forces P, P', PF",
etc. [Fig. 13], acting upon the point O, in the directions OM, OM', OM", etc.; and let the angles which
their directions make with the axes be denoted by
a %a x"/, etc., )/,', /3", etc.; then if we denote the
algebraic sum of the components which act in XX'
by X, and that of the components which act in YY'
by Y, we shall have
X =P cos a + P' cos a'  + P" cos c/" + etc.,...... [3]
Y= Pcosfi + P' Cos' + P" cos/3" + etc....... [4]
These equations are general. In a given case, the
values of the angles will determine the algebraic
signs of the terms of the second members. A common method of denoting the algebraic sum  of a
series of terms, all of which are of the same form, is
to write only a single term preceded by the Greek
letter z.  Applying this method to the above equation, we have
x = z (P cosa),
r = I (P cos/).




STATICS.                 21
We have now reduced the forces P, P', P", etc. to
the two forces X and Y, acting at right angles to
each other; and if we represent X by OE [Fig. 13],
and Y by OF, the resultant R of these two forces will
be represented by the diagonal OG of the rectangle
OEGF. This resultant will be the resultant of the
system P, P', P", etc., and will be determined in intensity by the equation
R = ~/X2 + Y2................. [5]
To find the angles a and b which the direction of
R makes with the axes, we have
x        Y
cos a =,  cosb =-................. [6]
The quantities R, a and b, are thus completely determined.
16. C'onditions of equilibriurn.
In order that P, P', P", etc. may be in equilibrium
about the point O, their resultant must evidently be
equal to zero; a condition which gives
X2 + 2, =0.
But as the square of a quantity is essentially positive, this equation can be satisfied only by making
X and Y separately equal to zero; thus the conditions of equilibrium are expressed by the equations
x= O, Y= o.




22                  STATICS.
OF FORCES APPLIED AT THE SAME POINT, AND SITUATED
IN DIFFERENT PLANES.
17. Let the three forces X, Y, Z, be applied at the
point O [Fig. 14], and be represented in intensity
and direction by the lines OS, OS', OS". On these
lines as adjacent edges, construct the parallelopiped
RS. Then it is evident that ON the diagonal of the
base OSNS' is the resultant of OS and OS', that is, of
X and Y; and that ON' the diagonal of the parallelopiped is the resultant of ON and OS", that is, of X,
Y and Z. Thus the resultant of the forces X, Y and
Z; represented by the lines OS, OS', and OS", is the
diagonal of the parallelopiped constructed on these
lines. If the directions of the three forces intersect
each other at right angles, the parallelopiped will be
rectangular; and denoting the resultant ON' by R,
we shall have
R = VX2 + Y2 + Z2.
The direction of a force P situated in space, and
applied at the point O [Fig. 15], is determined by
the angles which it makes with the lines OX, OY,
OZ, the parts of the co-ordinate axes drawn through
0, on which the positive co-ordinates are laid off, the
angles being reckoned from 0 to 180~. Thus if P is
situated in the angle 0 - XYZ, its direction OM is
determined by the angles XOM, YOM, ZOM, each
angle being less than 90'; if it is situated in the
angle O - XYZ', its direction OM, is determined ly




SIT'TIcsd                        23
the angles XOM,, YOM,, Z'OM,, the first two being
each less than 90, the third greater than 90~.*
If the intensity of P  be represented  by OM, its
three rectangular components in the directions of
the axes will evidently be OP, OQ and OR; and if
the angles which it makes with the axes be denoted
by a, 3 and y respectively, we shall have
OP = OM. cosMOP = P cosa,
OQ = OM. cos MOQ = P cos/3,
OR- = OM. osMOi      = P cosy.
Now let the forces P, PI5 P"', etc. be applied at the
point 0, and  let the angles which  their directions
make with the co-ordinate axes  be denoted by a, e,
Y; a', /', y!;  "l', /3", Y7', etc.; then the components coincident with XX' will be
P cosa, P' coso, PI" cosac", etc.;
those coincident with YY' will be
P cos /3, P cost/', P" cos/'!, etc.;
and those coincident with ZZ' will be
P Cos7, P' Cos Y, PI Cosy/", etc.;
and if the algebraic sums of the components in the
several directions be denoted by X, Y and Z respectively, we shall have
*The angles which the direction of P makes with the axes OX, OY,
fix its position on two conic surfaces, of which O is the common vertex;
and OX and OY the axes: these surfaces intersect each other in two
straight lines, and the third angle determines which of the two lines
is the direction sought. Thus a certain relation exists between the
three angles.  In accordance with our plan, this, like many other parts
of the general subject, is left to be developed either by the pupil or his
instructor.




24                      STATICS,
X = P cos a + P cos a' + P' cos a" + dc. = ~ (Pcos a),
Y= P cos, + P' cosi3' + P" cos/3" + etc. = y (PcosS),..[7]
Z = P Cosy + P' COS 7' + PI" cos y" + etc. =-   (P cos7),
If we denote by R the resultant of X, Y  Z, that
is, of P, P', P", etc., we shall have
R =  vX2 + Y2 + Z2;
and if we denote the angles which the direction of
R makes with the three axes by a, b and c respectively, we shall have
X           Y           Z
cosa =-     cos=b =    cosc =...    o.....o. s[8]
and thus the resultant will be completely  determined.*
18. Conditions of equilibrium.
In order that the forces may be in equilibrium, R
must be equal to zero; a condition which gives
X2 + Y2 + Z2 = 0,
and hence        X= 0, Y0, Z= 0.
If the point 0 is not entirely free, but subject to
the condition of remaining on a given surface, it is
not essential to an equilibrium that the resultant
should be equal to zero: it is only necessary that it
should act towards the surface, and at right angles
to it. The resistance of the surface is equal and
directly opposite to R.
19. As yet we have constantly supposed the forces
*The understanding of this article may be greatly facilitated by a
model of the co-ordinate planes, which the student himself can easily
construct of pieces of pasteboard.




STATICS.                     25
under consideration to be applied at the same point.
We shall now suppose them applied at different
points; the points being conceived to be so connected, that their positions with respect to each
other are invariable.  The lines of direction of the
forces may be parallel to each other, or oblique.
We shall first consider the case in which they are
parallel.
OF PARALLEL FORCES.
20. Let P and Q be any two parallel forces, applied
at the extremities A and B [Fig. 16] of an inflexible
straight line AB, and acting in the same sense* in
the lines AV and BS, and let their intensities be
represented by the lines AU and BT.
WTithout affecting the action of these forces, we
may apply at A and B in AB produced, two equal
and opposite forces AM  and BN.  The resultants
AD  and BI of the four forces AM, AU, BN, BT,
may then be transferred along their lines of direction
(produced backwards) to C their point of meeting,
and there be resolved into the four forces CG, CL,
CH, CK, equal and parallel respectively to the four
forces applied at A and B; but CH and CG being
*When the lines of direction of forces are parallel, we use the word
sense to indicate the relations of the directions: thus when two parallel
forces act towards the same parts of space, both to the right or both to
the left, for example, we say they act in the same sense; when they
act towards opposite parts of space, we say they act in opposite senses.
When we speak of the directions of parallel forces, we mean their lines
of direction.
D




26                   STATIGCS.
equal, and acting in the same line and in opposite
directions, destroy each other: hence the resultant
of the system is CL + CK, or P + Q, acting in the
same sense as the given forces, and In the line CO
parallel to their lines of direction.
The resultant may evidently be supposed to be
applied at 0, where its direction intersects the line
AB. To determine this point, we have from the
similar triangles AUD, COA, BTI, COB,
AU    UP   CO: OGA,
BT: TI:: CO     oB;
or            UD x CO = AU x OA,
TI x CO = BT x OB,
and hence      P x OA = Q x OB,
or            P:  Q::  o GB:OA
That is, 0, the point of application2 of the resltcant, divides
the line AB into parts reciprocally proportional to the iztensidies of the co2ponent forees,
The above proportion, it will be perceived, is independent of the angle which the directions of the
forces make with AB, and is true for any straight
line drawn through O and terminated by their directions.
Resuming the proportion,
P: Q: OB    G OA,
we get by composition,
P: P+ Q       OB O B  -B+OA,
or, denoting P + Q by R,
P    1 R    OB -    AB.




STATICS.                 27
In like manner we get
Q: R     OA    AB;
and hence we have
P      Q:       OB  OA    AB.
If the force P be represented by OB, the forces Q
and R will be represented by OA and AB respectively.
Thus each of the three forces mnay be represented by
the pacrt of AB intereepted by thle directions of tlhe other two.
The decomposition of a single force into two parallel forces, which shall act in the same sense and
satisfy certain conditions, is a very simple application of the foregoing results. Thlus, suppose it be
required to resolve the given force R into two parallel forces P and Q, one of which (P) is also given.
Here, assuming the points A, B and O [Fig. 17], as
the points of application of the forces P, Q and R
respectively, the given quantities are R, P and OA;
and the required quantities, Q and OB. To find Q,
we have R = P + Q; from which, Q = R- P; and
to find OB we have
Q    P:: OA    OB;
from which
P          P
OB =  x OA -         x OA.
21. We shall now consider the case in which the
forces P and Q, applied as before at the extremities
of the line AB [Fig. 18], act in opposite senses.
The force Q may be either greater or less than P:




28                  STATICS.
if Q is greater than P, let it be resolved into the
two forces P' and R, both acting in the same sense
as Q; the first equal to P, and applied at A in the
direction of P produced. We have, to determine
the intensity of R,
Q = PI +R;
from which,
R = Q - P' = Q -.P;
and to determine 0 its point of application, we have
R: P' =P: AB    OB;
from which,
OB =     p x AB.
Employing now in place of Q, its two components
P' = P and R, the two given forces P and Q may
be replaced by the three P, P' and R; of these, P
and P' destroying each other, there remains only
the force R as the equivalent of the given forces:
hence R = (Q _ P) applied at the point 0, and acting in the same sense as Q, is the resultant sought.
In like manner, it may be shown that if P is
greater than Q, the resultant R' is equal to P -- Q,
acting in the same sense as P, and applied at O' a
point situated to the left of A.
Thus when the given forces act in opposite senses,
their resultant is situated without tIhe forces (not between them, as in the preceding case), and on the side
of the greater, cand it acts in the salie sense as the greater.




STATICS.                  29
In this, as in the preceding case, it is evidently
true that
P     Q: R:: OB: OA: AB.
22. If in the value of OB, derived from the proportion
P    R:: OB: AB,
we substitute for R its value (Q- P), we get
OB    = Q- _x AB.
From this equation, it appears that as the difference
between Q and P  diminishes, the resultant, constantly diminishing, is applied at a point more and
more distant from B, till when Q = P its intensity
is reduced to zero, and the distance of the point of
application becomes infinite.  In this case, therefore, there is really no resultant. Such a system,
consisting of two equal and parallel forces acting in
opposite senses, but not in the same straight line, is
called a cotzple.
23. Parallelforees not in the same plane.
Let P,', PP", etc. be the forces, and let their
points of application be A, B, C, etc. [Fig. 19]; the
points being so connected by the inflexible straight
lines AB, BC, etc. as constantly to retain the same
relative positions. Required the intensity of the resultant, and its point of application.
We first compound any two of the forces, as P
and P', and find the point M at which their resultant
P + P' is applied, by the proportion
AB: AM:: P+ Pt     PI.




30                  STATICSo
We next compound the resultant P + P' with any
one of the remaining forces, as P"; and find the
point N at which their resultant P + P' + P" is
applied, by the proportion
MC: OM:: P +pIp'    P' P: P".
Proceeding in this manner, constantly compounding
the last resultant with one of the remaining forces,
we finally arrive at K the point of application of the
resultant R = P + P' + P" + etc. of the whole system.  The resultant evidently acts in the same
sense as the components, and parallel to their common line of direction.
24. In the case just considered, we have supposed
all the forces to act in the same sense. When some
of them act in one sense, and some in the opposite
sense, a circuwnslance indiecated by cifferenee of sign,
we find by the preceding method the resultants
R' and R" of the two systems, and their points of
application K' and K" separately; and then reduce
these resultants to a single force R, and find its
point of application K by the ordinary rules. When
the two resultants are equal, but not directly opposite, the case is that of Art. 22.
25. In determining the point of application of the
resultant of the system, we employ only the intensities of the forces, and the mutual distances of their
points of application; quantities which are not affected by giving to the directions of the forces a
common motion about these points. Hence whatever change we make in the common direction of a




STATICS,                  3
system of parallel forces, applied at points of which
the relative positions are invariable, by revolving
the directions of the forces about their points of
application by a common motion, the point of application of the resultant of the system  remains the
same. The point of application of the resultant of
a system  of parallel forces is called the cenlre of
parallel forces.
If the centre of parallel forces in any system is a
fixed point, the system is evidently in equilibrium
in all posiions abot -that point.
26. Let us now suppose the points of application
of the forces to be referred to three co-ordinate
planes, and let us seek the position of the point of
application of the resultant, or the centre of parallel
forces, with respect to these planes.
Let x, y, z, be the co-ordinates of the point A;
Xy, y17  S                    C      B?
etc.;
and z,, y,,,, the co-ordinates of the centre of parallel,
forces.
Let E [Fig. 20] be the point of application of the
resultant of P and P'; draw AI, EK and BL perpendicular to the plane XOY; join the points I and
L, and draw BG parallel to IL. We have [Art. 201
P +P: P:: AB      EB.
But the similar triangles AGB, EHB give
AB      EB:: AG: EH;




32                 STATIcs.
hence     P+PI   P + PAG  EH,
and        (P + P') E  = P x AG.
Adding to each member of this equation the product
(P + P') HK, we get
(P + P') (HK + En) = (P + P') EK + P x AG
= P (HK + AG) + P' x HK,
or     (P + P') EK   P x AI+P' X BL;
and denoting the ordinate EK by Z, we have
(P + P') Z = Pz + P'z,.
Conmpounding the resultant P + P' applied at E,
with the third force P", and denoting the co-ordinate of the point of application of their resultant by
Z' we get
(P + PI + P"l) Z' = (P + P') Z + P"I/";
and substituting for (P + P') Z its value Pz + P'z',
we have
(P + P' + P") Z' = Pz + P'z' + P/"zI".
Proceeding in the same manner with the remaining forces, and denoting as before the resultant of
the system by R, we finally arrive at the equation
Rz, = Pz  P' +  Z + P"z" + Pll/Z'"I + etc....... [9]
The first member of this equation is the product
of the resultant by the perpendicular let fall from
its point of application, upon the plane XOY; and
the second member is the sum of the products obtained by multiplying each of the given forces by
the perpendicular let fall from its point of applica



STATICS.                     33
tion, upon the same plane. These products are
called the moments of the forces with respect to the
plane XOY; thus ]z, P_, are called the moments of R
and P respectively referred to this plane. The result contained in this equation may then be thus
enunciated: The moment of the resultant of a system of
parallelforces, with respect to any plane whatever, is equal
to the sum of the moments of the forces with respect to the
samne plane.
Since the forces are affected with the positive or
negative sign, according to the sense in which they
act, and the signs of the co-ordinates of the points of
application are also positive or negative, according
to the positions of the points, the signs of the moments, determined by the ordinary rules, may be
either positive or negative.
If the moments of the given forces be taken with
respect to the planes XOZ, YOZ, we shall get
Ry, = Py + P', +- P y" + etc.,....... e...[10O
x, = Px + P'x' + P"x" + -etc.,............[11]
and thus we shall have, to determine the centre of
parallel forces,
Px + P'x' + Pl"x" + etc.
a,              R           9
Py + P'y' + p"y" + etc.
Pz + P'z' + P"z" + etc.
27. If' the plane with respect to which the moments are
taken, passes through the centre of parallel forces, the sum




34                    STATICS.
of' te mnomezts of tSie forces is eqtcal to zero; for if' we
suppose the plane XOY to be that plane, we shall
have
and hence    Pz + P'I' + P"z'" + etc. = 0.
28. Parcallel forces it tIke same plcaeo
It may happen that the points of application of
the forces are all in the same plane: when this is
the case, the centre of parallel forces, if it exist, is
also situated in this plane, and two co-ordinates are
sufficient to determine its position. Thus if we
suppose the plane XOY to coincide with this plane,
we shall have
z   O' =,     0   =- 0, etc.,
and hence           z, = 0;
and the position of the point will be determined by
the co-ordinates x, and y,, the values of which are
given by equations [10] and [11].
If the points of application are all situated in the
same straight line, the centre of parallel forces will
also be found in this line, and its position will be
determined  by a single co-ordinate.  Thus if we
suppose the axis OX to coincide with this line, we
shall have
y- = 0, zy = 0, y" = 0, etc.;
XZ=,  = O, zl " = 0, etc.;
and hence        y, = O and x, = 0,
and the value of x,, given by the equation
Rx/ = PF  + P'x' + P"xz" + et.,




STATICS.                  35
will determine the position of the centre of parallel
forces with respect to the point 0.
29. Comdilioms of eqtilibriwtmi.
Let the plane XOY [Fig. 211 be taken perpendicular to the direction of the forces; and let the forces
be reduced, as in Art. 24, to the partial components
R' and R", applied at the points K' and K". The
conditions of equilibrium will obviously be,
~ 1. That B' and R" shall be equal to each other,
and act in opposite senses.
~ 2. That their lines of direction shall coincide.
The first condition is expressed by the equation
R' = - R".......e         0 *  o e      [a]
But supposing P, P', P", etc. to be the components
of R', and P"'l PIV, P  etc. to be those of R", we have
R' = P  +  P' + P  + etc.,
and         BR" =  //"' + PIV + PV + etc.:
hence, by substitution, equation [a] becomes
+ p~J, P i,, + 1,, +  P pIv + Pv +   etc. = 0........ [12]
To express the second condition algebraically, let
the co-ordinates of K' and K", referred to the planes
XOZ, YOZ, be denoted by x,, y,, and x,,, y,,,: then
in order that the lines of direction of R' and R" may
coincide, we must evidently have
x,, = X,,/ and y,, = y'..u
Multiplying these equations by equation [a], we get
R'x,, = - R".................[b]




36                     STATICS.
But by the principle of moments [Art. 26], we have
R'x11 = Px + P'x' + —- P"x" + etc.,
RI'xI- = P"'x"' z+ PIVXTV + PVXV + etc.;
R',y = Py + P'y' + Pl/Y/ + etc.,
R"y,,, = P'y"'" + PIvyIv + PVyV + etc.
Hence equations [b] and [c] become, by substitution,
Px + P'x' -+ P/"x" + P"'x"' + P'Ix'v + PVxv + etc. = 0, [13].P + P/y, + P",y" + P///// + PIvyv + Pv V + + etc. = 0. [14]
Thus when the system is in equilibrium, equations
[12], [13] and [14] must be satisfied; and conversely
when these equations are satisfied, the system is in
equilibrium. These equations express,
~ 1. [Equa. 12] That the algebraic sum of the forces
vmust be equal to zero.
~ 2. [Equa. 13 and 14] That the sum of the moments
of the forces, taken with respect to two planes at right angles
with each other, and parallel to the common direction of the
forces, nmust also be equal to zero.
30. When the second of these conditions only is
satisfi'ed, that is, when equations [13] and [14] are
satisfied, but not equation [12], the given forces will
have a resultant, the direction of which will coincide
with the intersection of the two planes with respect
to which the moments are taken; for since the sum
of the moments of the forces is equal to zero with
respect to each of the planes, the moments of the resultant with respect to these planes are also equal
to zero, and hence the centre of parallel forces must
be in each plane; moreover the direction of the re



STATICS.                   37
sultant is parallel to the planes: consequently the
direction of the resultant must coincide with the
intersection of the planes.
If there be a fixed point in the intersection of the
planes, the resultant will be destroyed, and an equilibrium  will exist.  Hence when parallel forces which
have a rescltant are applied to a system of material points
rigidly connected, of whtich one is fixed, there will be an
equilibrium, if the sum of the mnoments of the forces takens
with respect to two planes drawn through the fixed point,
parallel to the direction of the forces, and at right angles to
each other, is ezqual to zero.
We shall next consider the subject of oblique forces
applied at different points. There are two cases:
10 When the forces act in the same plane; 2~
When they act in different planes.
OF OBLIQUE FORCES APPLIED AT DIFFERENT POINTS, AND
ACTING IN THE SAME PLANE.
31. Let the intensities of the forces be denoted by
P; P', P", P"', etc.; and let the plane which contains
their directions be taken for the co-ordinate plane
XOY [Fig. 22]. Let the points A, B, C, D, etc., connected with each other in an invariable manner, be
the points of application of the forces as represented
in the figure.  The resultant of the system, when it
exists, may be found in the following manner: Commencing with P and P', any two of the forces, produce their directions till they meet in G. Suppose
the two forces applied at this point; and taking Gm




38                    STATICS.
and Gaz to represent their intensities, construct the
parallelogram  G)mG'n: the diagonal GG' is their resultant.  In like manner, compound the resultant
GG' with P" any one of the remaining forces,
transferring their points of application to H their
point of meeting, and thus determine their resultant HH'.  Continue this process till all the forces
have been compounded: the last resultant will be
the resultant of the system. In the case of the
given forces P,.P P',, P"', the resultant is II'.
If, in any part of the operation, we arrive at forces
of which the directions are parallel, we apply to them
the rules for parallel forces.
If, in the final result, we find two equal forces acting in parallel lines and opposite senses, the system
has no resultant [Art. 22].
The above method is equivalent to applying the
given forces at the point I, in lines parallel to their
original directions, and then reducing them to a single force. For the resultant II', applied at I may
be resolved into the two forces In" = P"', and Lrn";
Irn" may be resolved into It, and It, equal and parallel to the forces Hn' = P" and I-n'; and so on till
in place of II' we have all the original forces applied
at I, and acting in lines parallel to their primitive
directions.
32. The resultant of the forces P, P', P", P"', etc.
may also be determined by the following method:
The co-ordinates of the points of application referred
to the axes OX, OY being given, and also the angles




STATIrcs.               39
which the directions of the forces make with the
axes, let each force be resolved into two components
parallel to OX and OY respectively. The whole
system of forces will thus be resolved into two systems of components, the one parallel to OX, the
other parallel to OY. Let these components be applied at the points where their directions intersect
the axes, and let the resultants X and Y of the two
partial systems be determined in intensity and position by the rules for parallel forces [articles 24 and
28]: the resultant of X and Y, or the resultant of
the whole system, may then be determined by the
ordinary rule.
Before we proceed to determine the conditions of
equilibrium, it is necessary to make known some of
the properties of what are called moments referred to
a point.
Of nomenzs referred to a point.
33. Let P and P' be any two forces meeting in A
[Fig. 23], and represented by AB and AC respectively, and let their resultant AD be denoted by R.
In the plane of the forces, assume any point 0; and
from it let fall upon the directions of P, P' and R,
the perpendiculars OI, 01', 0)". Through the points
A and 0, draw the straight line AO, and through A
draw AQ at right angles to AO. Also from the
points B, C and D, let fall upon AQ the perpendiculars BD', CD", DD"'; and through C draw CE parallel to AQ. The similar triangle AOI, ABD' give
AO     OI:: AB3   AD';




40                  STATICS.
or, denoting AO and OI by c and p respectively,
c: p     P  AD';
from which we get
AD'  P=
C
In like manner, denoting by p' and r the perpendiculars OI' and OI", and employing the corresponding triangles, we get
p''         or
AD"/ =      AD"' =~          ~
But the equal triangles ABD', CDE give
AD' = CE = D"ID"//
and hence we have
AD"' = ADD" + D"ID/'
= AD" +- AD';
and substituting in this equation the values of the
several terms found above, we get
Rr  Pp  Pp+
_1~  _    -..
C   C    C
or               r = Pp + P'op'.................[15]
We have supposed the point O to be taken without the angle BAC, or the opposite vertical angle
B'AC': when it is taken within either of these angles,
BAC for example, as in figure 24, we have, as in the
preceding case,
AD' = CE = D"/D"'
and hence    ID'D"' = AD".
Hence we have
AD/' = A D'- D'ID"' = AD' - AD",;




STATICs.                  41
and substituting as above, we get
Rr = pp -P p.. *......... [16]
The products Rr, Pp, etc. are called the mnoments of
the forces R, P, etc. with respect to the assumed point
0; and the point itself is called the centre of moments.
Thus the mzoment of a force witlz respect to cc point, is the
product of the force bI the perpenldictllar let fall fromn the
poinzt tpon the direction of the force. Employing these
terms, the principle involved in equations [15] and
[161 may be thus enunciated:
The moment of the resultant of any two forces is
equal to the sum or difference of the moments of
the forces, according as the centre of moments is
taken without or within the angle made by the
directions of the forces, or the opposite angle at
the vertex smade by their directions produced.
Another enunciation of this principle may be given
by introducing the idea of motion. Thus, suppose
the perpendiculars OI OI', OI", to be inflexible lines,
capable of motion in the plane of the forces, about
the point 0 regarded as a fixed point: then the
forces P, P' and R, which we may imagine applied
at the points I, I' and I", will tend to give to the perpendiculars a motion of rotation about 0; and we
perceive that in figure 23, in which the centre of
moments is without the angle BAC, or its opposite
angle at the vertex, these forces will tend to turn
their points of application in the same sense about
0: while, on the contrary, in figure 24, in which the
F




42                    STATICS.
centre of moments is within one of these angles, the
two components will tend to turn their points of
application in opposite senses about 0, and the resultant will tend to turn its point of application in the
same sense as the component which has the greater
moment.  We may therefore say that the mnomenzt of
tle rescltant of any tuwo forces is equal to 1/e sutm or difference of the moments of the forces, according as the forces
tend to turn, their points of application in the same sense, or
inz opposite senses, about the fixed point assutned in their
plane as the centre of momnents.
34. This principle can be shown to be true, whatever the number of forces.  Let us consider the case
in which the centre of moments is so situated that
the first three of the forces P, P', P", P"', etc. tend
to turn the system in one sense, and the remaining
forces in the opposite sense about that point.  Let
Q be the resultant of P and P', and Q' that of Q and
P".  Also let p,',n p", q and q', denote the perpendiculars let fall from  the centre of moments on the
directions of P, P', P", Q and Q' respectively. Then,
according to the principle just demonstrated, we shall
have
Qq = Pp + P'p',
Q'q' = Qq + P'~p";
and substituting in the second of these equations the
value of Qg derived from the first, we shall get
Q'q' = Pp + Pp1Y + P',p".
Again, denoting by Q, the resultant of the remaining




STATICS.                    43
forces P"', PlY, etc.; by q, the perpendicular let fall
from the centre of moments upon its direction; and
by P/i, pv, etc. the perpendiculars drawn from  the
same point to the directions of P"', Pv, etc., we shall
have
Qq, = Pl"'p"/ + PlIplY + etc.
Let now the resultant of Q' and Q, (that is, the resultant of the given forces P, P', P", etc.) be denoted
by R, and the perpendicular let fall from the centre
of moments upon its direction by r.  Moreover let
it be recollected that Q' and Q, tend to turn their
points of application in opposite directions. Then,
according as Q'q' is greater or less than Qg,, we shall
have, since Rr must be positive,
Pr = Q'q' -Q,q,
or                Rr = Q,q, -  Q'q'.
In the first case, the force R will tend to turn the
system in the same sense as the force Q', and consequently in the same sense as the forces P, P' and P".
Supposing this to be the case in question, and substituting for Q'q', Q,q,, the values found above, we
get
Pr = Pp J+ P/p' + P'" - P//'/p' - PIVpIV - et..... [17]
Thus, whatever the number of forces, the moment
of the resultant is eqgual to the sum of the moments of the
forces which tend to turn the system in the same sense as
the resztcant, minus the stmn of the  zoments of the forces
which tend to turn it in the opposite sense.




4 4                  STATICS.
35. If, in the plane of a system of forces such as
we have been considering, there is a fixed point with
which the points of application are connected, and
about which they may revolve in the plane of the
system, the forces will obviously be in equilibrium if
their resultant passes through this point. Suppose
their resultant thus to pass, and take the fixed point
for the centre of moments; then we shall have
r - 0,
and hence           Rr = 0,
and equation [171 will become
Pp + P'2p' + P"p" + etc. - 0.
Hience, in order to an equilibrium in this case, it is
only necessary that the sum of the moments of the
forces which tend to turn the system in one sense
about the fixed point, should be equal to the sum of
the moments of the forces which tend to turn the
system in the opposite sense, the moments being
taken with respect to the fixed point.
36. If, in tl:i e case of a system of forces P, P', P",
etc. in which there is a resultant, that is, in which.R
is not equal to zero, we have
Pp + Pop' + Pp" + etc. = 0,
then                 Rr = 0,
and hence             r = 0,
and consequently the resultant passes through the
point assumed as the centre of moments.




STATICS.                 45
37. Conditionzs of equilibrium.
Let P, P' and P" be any three forces situated in
the same plane, and applied at the points A, B and
C, as represented in figure 25. It is essential to an
equilibrium, that their directions should meet in the
same point; for in order that any two of them, as
P and P', may be in equilibrium with the third P",
the latter must act in the same line with the resultant R of the two former, and hence its direction
must pass through through the point A at which the directions of P and P' intersect. But P may be supposed
to be the resultant of two other forces P"' and Pv
applied at the point E, as in the figure; PIV of two
others Pv and PLv, and so on. Hence in order that
any number of forces situated in the same plane, and
applied at different points, may be in equilibrium,
they must be reducible to three forces which meet in
the same point. To express this condition algebraically, let the forces P, P' and P" be supposed to be
applied at A [Fig. 26] their point of meeting, and let
their intensities be represented by AB, AC and AD,,
AD, being equal to AD or R the resultant of P and
P'. Also from a point O assumed in the plane of
the forces, let the lines OI, OI', OI", be drawn perpendicular to the directions of P, P' and P" or R.
Recurring to article 33, and employing the same
notation as in that article, we perceive that the relation between these forces and perpendiculars is expressed by the equation
Rr = Pp Z Pp',




46                   STATICS.
or, substituting for Rr its equal P'P"" by the equation
P"p3  =  P  1 P'p.;                L[e   e 0   aa]
the upper or lower sign of the second member being
employed, according to the position of the point 0.
This equation expresses that the moment of the
force which tends to turn the system in one sense
about the centre of moments, is equal to the sum of
the moments of the forces which tend to turn it in
the opposite sense; and a little consideration renders
it apparent that when this condition is fulfilled, the
forces P, P' and P" must meet in the same point.
Equation [a] may therefore be taken for the equation of condition in the case of three forces, the forces
necessarily meeting in the same point when it is
satisfied. To deduce from it the equation for a
greater number of forces, let P be considered as the
resultant of the two forces P"' and P'V applied at
some point E taken in the direction of P: we shall
have, by the principle of moments,
pp = p,,i p,,, - pivpIV,
(it being supposed, as represented in the figure, that
P"' and P'v tend to turn their points of application
in opposite senses about the point 0, P and P"' in
the same sense;) and substituting this value of Pp in
equation [a], and employing the upper sign of the
second member, we shall get
p,,pI, = pIIpI/, -, PIVpIV + Pip,
or          Pp"I, + PIVpIV=  P'p' + P/,p/"',




STATICS.
for the equation of condition in the case of four
forces. By thus successively regarding one of the
forces as the resultant of two others, the equation of
condition may be found for any number of forces.
The general equation is commonly written thus,
Pp + Pp' + PIp/i + P"lp"l + etc. = 0;
the signs of the terms being determined by the directions in which the several forces tend to turn their
points of application.
Since this equation indicates that the forces P,
P', P", P"', etc. are reducible to three forces which
meet in the same point, when it is satisfied, we may
suppose all the forces applied at the point of meeting
of the three forces in lines parallel to their primitive
directions. The conditions of equilibrium will thus
be reduced to those of article 15; and adopting the
notation of that article, we shall have for the remaining equations of condition,
P cosa + P' cos a' - P" cos al" + PI"' cosa"' + etc. = 0,
P cos/  + P' cos/3' + PI' cos" + P"' cos PI' + etc. = 0.
Thus the equations of condition for the equilibrium
of any number of forces situated in the same plane,
and applied at different points, are
_V (PP3)    =9 0. [..B.......201




48                     STATICS.
OF FORCES APPLIED AT DIFFERENT POINTS, AND SITUATED
IN DIFFERENT PLANES.
38. This is the most general case: We shall not
discuss it in detail, but merely indicate the principal
steps in the several processes.
The forces being given in intensity, direction, and
the position of the points of application (these points
being conceived to be rigidly connected as in the
preceding case), to determine their resultant, when
it exists, we proceed as follows:
1~ We reduce the given system to two partial
systems (b) and (c): the one consisting of forces situated in the plane of xy;* the other, of forces perpendicular to that plane, and hence parallel to the axis
of z.
20 By the foregoing methods [Articles 32 and 23],
we determine the resultants of the systems (b) and (c).
3~ When these resultants are situated in the same
plane, we reduce them to a single force by the ordinary rule.
If the resultants are not situated in the same plane,
the given forces are obviously not reducible to a single force.
39. Conditions of equilibrium.
If the systems (b) and (c) are separately in equilibrium, the given system must evidently be in equilibrium.  The converse is also true, viz. that if the
given system is in equilibrium, the systems (b) and
* The position of the co-ordinate planes being arbitrary, we may
give to the plane of xy any position whatever.




STATICS.                 49
(c) must be separately in equilibrium. For if an
equilibrium exists among the given forces, it will
evidently not be destroyed by supposing any line of
the plane of xy, connected with the points of application; to become immovable; but in this case, the
forces situated in the plane of zy will be destroyed
by the resistance of the fixed line; and hence the
forces parallel to the axis of z, unless in equilibrium
amongst themselves, will tend to turn the plane
about this line: therefore, since the equilibrium
must continue, the forces parallel to the axis of z
must destroy each other. The system [c] being thus
necessarily in equilibrium of itself, the system [b]
must be also. The conditions of equilibrium  for
these systems have already been found [articles 29
and 37]; those for the system [b] are expressed by
the equations [18], [19] and [20]; those for the system [c] by the equations [12], [13] and [14].
40. When the system contains a fixed line, about
which the points of application may revolve, without
being capable of moving in a direction parallel to it
(in the same manner as the material points of a solid
body about an axis on which the body is prevented
from sliding), the conditions of equilibrium can be
expressed by a single equation. This case of equilibrium, being one to which we shall have to refer
hereafter, we will treat it more fully than we have
treated the preceding case.
Let OZ [Fig. 27] be the fixed line or axis, about
which the points of application may revolve in the




50                 STATICS.
manner just supposed, and let it be taken for the
axis of z. Let AK be the direction of any one of the
forces, as.P, and A its point of application. Through
AK let a plane be drawn perpendicular to the plane
of xuy; and let P, represented by AK, be decomposed
in this plane into the two forces AH and AG, the
one parallel to the plane of xy, the other parallel to
the axis of z. Since the point A can by hypothesis
have no motion parallel to the axis of z, the second
component must be destroyed. The first, which we
will denote by Q, is expended in tending to give to
the point A a motion of rotation about the axis.
We will now show that this force may be replaced
by another, equal and parallel to it, but applied in
the plane of wy. For this purpose, let the line GA
be produced to meet the plane of wy in A'; and
through A', let the line A'H' be drawn parallel to
AH. Also through the axis OZ, let the plane ON'NN"
be drawn perpendicular to the parallels Alh, A'H',
and meeting these parallels produced in N and N'.
Without affecting the system, we may apply at the
point A' two forces 8 and 8', each equal to Q, and
acting in the opposite directions A'lH', A'E; thus replacing the force Q by the forces 8, 8' and Q. But
conceiving the plane ON'NN" to be rigidly connected
with the points of application, the forces Q and 8',
which we may suppose applied at N and N', and
which tend to turn this plane in opposite directions,
will, as we shall presently show, destroy each other
by means of the fixed axis; and we shall have, as




STATICS.                   51
the equivalent of Q, only the force S, equal and
parallel to Q, and acting in the same sense as it. Each
force of the system is capable of a similar reduction.
Thus the given forces P, P', P", etc., applied at the
points A, B, C, etc., may be replaced by the components Q, Q' Q", etc., Q,, Q,,, Q,,,, etc., applied at determinate points of the plane xy. Hence since the
point 0 is fixed, in order to an equilibrium, it is only
necessary that the sum of the moments of the forces
which tend to turn the system in one sense, may be
equal to the sum of the moments of the forces which
tend to turn it in the opposite sense about the point
0 taken as the centre of moments.  If then we denote the perpendiculars drawn from 0 to the directions of the forces, by q, q', q", etc., q,, i,,, q,,,, etc. respectively, and suppose that the forces Q, Q', Q", etc.
tend to turn the system in one sense, and Q, Q,,, Q,,,,
etc. in the opposite sense, we shall have for the equation of condition,
Qq + Q/q' + etc. - Qq -,,,, -- etc.  0........ [21]
To show that the forces Q and S' are in equilibrium
with each other, by means of the fixed axis, at the
points 0 and N", let the forces S" and S/", S'V and
Sv be applied, each equal and parallel to Q; S" and'IV acting in the same sense as Q, S"' and Sv in the
opposite sense.  These forces will obviously not
change the state of the system; and hence we may
consider the two forces Q and S' as replaced by the
six forces Q and Li", I' and S, SYl"' and S'w. But of




52                   STATICS.
these, Q and S", S' and SV, may be reduced to two
equal and opposite forces applied at I the point of
intersection of the diagonals ON, N'N", and are therefore in equilibrium; and the remaining forces S"'
and SNv, acting upon the fixed points 0 and N", are
destroyed by the reaction of the points. Consequently the six forces are in equilibrium by means
of the fixed axis, and hence the equivalent system
Q and 8' must also be in equilibrium.
CENTRE OF GRAVITY.
41. The force which causes a body, when not supported, to fall to the earth, is called gravity.  We
learn from experiment, 10 That gravity acts with
equal intensity upon the particles of all bodies, however the bodies may differ in size, form, or nature;
2~ That it acts in directions perpendicular to the
surface of the earth, or the surface of a liquid at rest.
The line of direction of gravity at any place, is
called the vertical at that place; and any plane perpendicular to the vertical, is called a horizontal plane.
Since the form of the earth is nearly spherical, the
directions of gravity at different points will converge
towards its centre; but the length of the earth's
radius is so great, compared with the dimensions of
the bodies usually treated of in mechanics, that we
may neglect this convergence, and consider the
directions of gravity, for all the points of the same
body, as parallel.




STATICS.                 53
From experiments and observations which will be
made known hereafter, it has been found,
1~ That the intensity of gravity at'the surface of
the earth, though always the same at the same place,
varies with the latitude of the place; being least at
the equator, and increasing as we approach the poles,
in the ratio of the square of the sine of the latitude.
2~ That its intensity varies from one point to
another of the same vertical; diminishing as the
distance of the point from the centre of the earth
increases, in the ratio of the square of the distance.
But the variations of intensity from these two
causes, for small changes of distance, are so minute
that the action of gravity upon all the particles of a
body of ordinary size may be considered as equally
intense.  Since then within the requisite limits,
gravity may be considered as constant in both intensity and direction, we may regard the particles of a
heavy body as the points of application of a system
of equal and parallel forces, acting vertically and in
the same sense. The resultant of these forces, which
is equal to their sum, and presses the body vertically
downwards, will evidently constitute what is called
the weight of the body. If then we denote the weight
of the body by W; the number of material particles
composing it, by M; and the effect of gravity upon
a single particle, by g, we shall have
~W = M   veeee.   ~ eb B ea *ee ~e[22]




54                  STATICS.
An immediate inference from this equation is, that
in homogeneous bodies, the weights are proportional
to the volumes; a deduction constantly verified by
experiment.
In heterogeneous bodies, this relation between the
weights and volumes does not hold; the weights of
equal volumes, when compared, being found unequal.
Thus a cubic inch of gold is found to weigh about
nineteen times as much as a cubic inch of distilled
water at a certain standard temperature; a cubic
inch of silver, eleven times as much.
Since the weights of equal volumes of two substances must be directly as the numbers of material particles which they contain, equal volumes of gold and
silver must contain, the one nineteen, the other
eleven times as many material particles as the
same volume of distilled water. Thus the numbers 19 and 11 express the relative numbers of material particles, or the relative quantities of matter
contained in equal volumes of gold and silver; the
number of particles, or the quantity of matter, in an
equal volume of water, being assumed as the unit.
The numbers which thus express the relative quantities of matter referred to a common uniut, contained
in equal volumes of heterogeneous bodies, are called
the densities of these bodies; thus 19 and 11 are the
densities of gold and silver respectively, the density
of distilled water being taken for the unit. If now
we denote the density of a, body by, D; its volune,
expressed in terms of the unit' of volume, by V; and




STATICS.                   55
the number of particles that it contains, that is, its
quantity of matter, or mnzass, by J, we shall evidently
have
V = VZ e D..... o...e.o.. [23]
If we substitute this value of M in equation [221, we
shall get
W =   ).. o e.  O o.o e s O  O e o o... s [24]
Since the value of g the general symbol denoting
the intensity of gravity, varies with the position of
the place, we take its value at some determinate
place for the unit of intensity, and consider its
general value as expressed in terms of this unit. We
thus have for the place at which g = 1,
w= VD;
and for any place whatever,
T/F= VDg,
g being in this equation the ratio of the intensities
of gravity at the two places.  If the unit of-volume
be the cubic inch, the unit of weight is evidently the
weight of a cubic inch of distilled water at the place
where g = 1, and W7 designates the number of these
units which the weight of the body contains.
42. Since the weight of a body is the resultant of
a system of parallel forces applied at points of which
the relative positions are fixed, it must have a determinate point of application.  This point, which is
the centre of parallel forces for the case in question,
is called  /ezc centre qof gravity of the body.  The positioni of the centre of gravity, -with respect to the




56                   STATICS.
points of the body, is invariable; remaining the
same, while we suppose the body to take all possible
positions in space; for a change in the position of
the body is merely equivalent to a revolution of the
directions of the forces about their points of application, an operation which [Art. 25] does not affect the
position of the centre of parallel forces.
When the centre of gravity of a body is supported,
the body will be in equilibrium in all positions about
that point; for, in all positions, the direction of the
weight of the body will pass through the point of
support.
43. Conceive a body now to be divided into any
number of parts, and suppose the weights and centres
of gravity of the several parts to be known. Let the
weights of the parts be denoted by w, zv', w", etc., and
the co-ordinates of their centres of gravity by x, y and
z; x', y', and z'; x", y" and z", etc. respectively; and
let the weight of the entire body be denoted by W,
and the co-ordinates of its centre of gravity by x1, y,
and z,. Then since Wis the resultant of the parallel
forces w, w', w"Z etc. applied at the respective centres
of gravity, we shall have [Art. 26] to determine the
co-ordinates x,, y,, z,, the equations
_wx + Uw'xt + w':x" + etc.
XI-    -      TV,WY + w'y' + w"y" + etc.        [25]'wz + Vw'z' + WI'z" + etc.
ZI -- ~  - t




STATICS.                  57
When the centres of gravity of the several parts
are situated in the same plane, the plane of xy for
example, the centre of gravity of the entire body
will be found in that plane: when they are situated
in the same straight line, as the axis of x, it will be
found in that line [Art. 28].
The above equations are evidently true, whatever
the number of parts into which we suppose the body
divided.
44. If we denote the masses corresponding to the
weights W, w, w', w", etc. by i rn, mn', in", etce., we
shall have
WJ =  g", w = mg, w' = m'y, w"I = m"'g, ete.;
and by substituting these values in the above equations, and omitting the common factor g, we shall get
mx + m'x' + m"x" + etc.
x/ =          _/ 
my +- m'y' + m"y" + etc........[6]
mz - mtz'' + m"z" + etc.
From these equations, it appears tha;t the position of
the centre of gravity is independent of the intensity
of gravity.
45. If we suppose the body to be homogeneous,
and denote its volume and density by V and D respectively, and the volumes of its several parts by'V v', V/ etc., we shall have
1- = VD7, m = vD, m' = v/D, m/" = v/LD;




58                    STATICS.
and by substituting these values in equations [26],
and omitting the common factor D, we shall get
vx + v'x' + v"x" + etc.
y + v'y' -+ v"y" + etc.,      o. [27]
vz +- VIZ' + v"z" + etc.
If we suppose the parts into which the body is
conceived to be divided, to be infinitely small, we
may derive fronm equations [26] and [27] the following theorem: The sum of the products obtained by nmultliplyiny either the nzasses or the volmnes of the elements, or
infnitely small parts of a body, by their respective distances
from any plane whatever, is equal to the product of the entire
mass or volume by the distance of its centre of gravity from
the same plane.
If the elements of the body are all situated in the
same plane, or, in other words, if the body itself is a
material plane, we may consider the plane with
respect to which the moments are taken as reduced
to its line of intersection with the plane of the elements, the two planes being supposed to be perpendicular to each other, and the moments may be supposed to be taken with respect to this line.
It will be recollected that these products, or mosments of the masses or volumes, as they may be
called, nmust be affected with opposite signs, according as the perpendiculars are situated on the
same or opposite sides of the plane or line,




STATICS.                   59
It is obvious, that if in any case the sum of the
moments of the masses or volumes, with respect to
the plane or line, is equal to zero, the centre of
gravity of the entire body must be situated in the
plane or line.
46. Determnination of the centres of gravity of particttlar
bodies.
The application of the preceding equations to the
determination of the centres of gravity of particular
bodies, requires in general the use of the integral
calculus.  In very many cases, however, these points
can be determined by the most elementary processes.
The examples which follow will illustrate the methods commonly employed.
We shall first consider some of the simplest cases
of material or physical lines and surfaces: the lines
being conceived to consist of single series of material particles; the surfaces, of single laminae of particles, the particles being supposed in both cases to be
uniformly distributed.
10 The straight line.
The centre of gravity of a mnaterical straight line is its
middle point.  For, drawing through this point any
line or axis whatever, the sum of the moments of the
particles on one side of the axis is obviously equal
to the sum of the moments of the particles on the
opposite side; and these sums are, moreover, affected
with opposite signs. Their algebraic sum is therefore
equal to zero, and hence the centre of gravity of the
line must, by the last paragraph of the preceding




60                   STATICS.
article, be in the axis; but it is also in the line itself,
consequently it must be at the point of intersection
of the line and the axis.
The same result may be found more directly by
applying the ordinary rule for parallel forces. Thus,
let the weights of the particles be regarded as a system of equal and parallel forces acting in pairs, the
components of each pair being applied on opposite
sides of the middle point and at equal distances from
it, and let the point of application of the resultant
of the whole system be sought. The resultant of
each pair, and hence the resultant of the whole system, will be found to pass through the middle point
of the line.
20 The perimeter of a polygon.
To determine the centre of gravity of the perimeter of a polygon: in the plane of the polygon,
draw the co-ordinate axes OX and OY, and determine the co-ordinates of the centres of gravity of the
several sides referred to these axes. Then in the
first two of equations [27] Art. 45, substitute for the
volumes v,',", etc. the lengths of the sides; for x
and y, X' and y', x" and y", etc. the co-ordinates of
their centres of gravity respectively; and for the
volume V, the entire perimeter of the polygon. The
values of x, and y,, thus determined, will be the values
of the co-ordinates of the centre of gravity of the
polygon.
Another method consists in regarding the weights
of the several sides, applied tt the respective centres




STATICS.                 61
of gravity, as a system of parallel forces, and determining the point of application of the resultant by
the ordinary rule.
3~ The are of a circle.
Let AFB [Fig. 28] be an arc of a circle, and let
MN be one of the infinitely small parts or particles
of which we suppose it made up. Draw the diameter LR parallel to the chord AB; and from Q the
middle point of MN, draw QP perpendicular to LR.
Draw also the radius QO, the line MI parallel to AB,
and the lines MH and NK perpendicular to ABo
The similar triangles MNI and QOP give
MN:MI:1: QO: QP,
or            MN x QP =M T x QO
= HK x QO.
The first member of this equation is the moment of
MN with respect to LR; and the second is the product of the projection of MN on the chord AB, by the
radius of the circle. Thus the moment of each particle of the are is equal to the product of its projection on the chord, by the radius of the circle: consequently the sum of the moments of all the particles
of the are with respect to LR, is equal to the product
of the chord by the radius, or to AB x OQ. But the
sum of the moments of the particles of the are, with
respect to the radius OF drawn to its middle point,
is evidently equal to zero, and hence OF must contain the centre of gravity of the are. If then we
suppose the centre of gravity to be at C, the expres



62                    STATICS.
sion of the moment of the arc with respect to LR
will be, arc AFB x CO; and hence [Art. 45] we shall
have
are AFB x CO = AB X OQ,
or        are AFB: AB       OQ: CO.
Hence the centre of gravity of the are of a circle is on the
radius which bisects the arc, at a point, the distance of which
from the centre of the circle is a fourth proportional to the
length of tlhe are, its chord, and radius.
40 The area of a parallelogram.
The centre of gravity of the area of a parallelogram is
at the intersection of its diagonals.  For the sum of the
moments of the particles of the parallelogram, with
respect to each of its diagonals, is evidently equal
to zero, and hence each diagonal must pass through
its centre of gravity: consequently the centre of
gravity of the parallelogram must be at the intersection of these lines
Another method of determining the centres of
gravity of plane figures, consists in regarding them
as made up of physical lines.  To apply this to the
case of the parallelogram, conceive it made up of
lines parallel to one of its diagonals; the other diagonal, bisecting these parallels, and hence passing
through their centres of gravity, will contain the
centre of gravity of the entire figure; but this property is common to the two diagonals: consequently
the centre of gravity of the parallelogram must be
at their intersection.




OTATI 5.
50.  The arecz of a r&zanyle.
To find the centre of gravity of a triangle ABD
[Fig. 29]: From the vertices B and D drawv the lines
DE and BF to the rmiddle points of the opposite sides
AB and AD, and join the points E and F; the centre
of gravity will be at C.  For, regaldin g the triangle
as made up of physical lines parallel to AB, its centre
of gravity mzust be in the line lD, -,whh;iseCts these
parallelso it must also be in BFE? which bisects the
lines drawn parallel to  B; consequently it is at the
point C.
To determine the position of this point, we have,
from the similar triangles BCD and FCE, ABD and
AEF,
CD  -  CL?D  El
oAB3    AE.  2      p
hence                2 CE =  D,
3 CE =ED,
and                    CE =      o
Hence tle centre of gravity of the areaof  a trinise is on
/ie strcahl/t line drCat.    from any oll    e of is vertices, to 1/ie
mniddle poin  of tze opposite side, ct a distciZee from  1/is
point equalc to one-third of the lengtih of ite line.
6~. 2/Ie  area of a polygozn.
To determine the centre of gravity of the area of
a polygon, divide  the  polygon into triangles, and
find the centre of gravity of each triangle by the




64                     STATICS.
preceding method; then proceed as in the case of
the perimeter.
70. The area of a circular sector.
Let AOB [Fig. 30] be a circular sector, and let it
be supposed to be divided into an infinite number of
infinitely small equal triangles, having their bases
in AB, and their vertices at the centre 0. The
centre of gravity of each of these triangles will be
in the radius drawn to the middle of its base, at a
distance from the centre of the circle equal to twothirds of the length of the radius: hence the centre
of gravity of the sector will be the same as that of
the arc FG described with a radius O1, equal to twothirds the radius of the sector.  It will therefore
[No. 30 of this Art.] be on the radius OD drawn to
the middle of the arc AB, at a point C, the distance
of which from O will be given by the proportion
arc FIG  FG: ~  OI  OC;
or,       arc ADB: AB  -O20D          OC.
Hence the centre of gravity of the area of a circular sector
is on the radius which bisects the are of the sector, at a
point, the distance of which from the centre is a fourth proportional to the arc, its chord, and two-thirds of the radius.
80. The surface of a spherical zone.
The centre of gravity of the surface of a spherical zone
is at the middle point of its axis, or the line joining the centres of its bases. For, conceive the entire zone divided
into an infinite number of infinitely small zones of
the same altitude, by planes parallel to its bases;




STATICS.                  65
the centres of gravity of these infinitesimal zones will
be on the axis of the entire zone; but these zones
are equivalent in surface: hence their common centre of gravity, or the centre of gravity of the entire
zone, will be at the middle point of the axis.
9~0 The parcdlelopipedon.
Let AH [Fig. 31] be a parallelopipedon.  Through
the opposite edges BG and EI, AF and DIH, let the
planes BGIE, ADHF be drawn; and through C the
middle point of 00' the intersection of these planes,
let the plane A'B'D'E' be drawn at right angles to
00'.
The matter of the solid is obviously so disposed
on the opposite sides of these planes, that the sum
of the moments of its particles with respect to each
plane is equal to zero: consequently the centre of
gravity of the parallelopipedon is at C the common
point of intersection of the three planes. This point
is obviously the middle point of one of the diagonals
of the solid.
Another method of determining the centre of
gravity of solids, consists in regarding them as made
up of physical planes. Thus, in the case just considered, we may suppose the solid made up of planes
parallel to its base ABDE: the centre of gravily will
obviously be at C the middle of the line 00' which joins the
centres of gravity of the two bases.
1Oo  hie pyramid.
Let A -BED [Fig. 32] be a triangular pyramid.




66                   STATICS.
From the vertices A and D, draw to F the middle
point of the edge BE, the straight lines AF and DF;
take FK equal to one-third of AF, and FH equal to
one-third of FD, and draw DK and AH: the point C
in which DK and AH intersect is the centre of gravity of the pyramid. For, conceive the pyramid to be
made up of planes parallel to the face BED: the line
AlE will pass through the centres of gravity of all
these planes, and will therefore contain the centre
of gravity of the pyramid. In like manner it may
be shown that the line DK will also contain this
point; consequently it must be at C the point of
intersection of these lines. To determine its position,
joining the points H and K, we have, from the similar
triangles ACD, HCK, AFD, KFH,
CH0   CA:: hK: DA:  FH: FD::   1: 3;
hence              3CH = CA,
4CH = — A,
and                 c    AH = -
4.4
Hence the centre of gravity of a triangular pyramid is on
the straight line drawn from the vertex of any one of its
angles, to the centre of gravity of the opposite face, and at a
distance from the face equal to one-fourth of the length of
the line.
It can now be easily shown that the centre of gravity
of any pyramid whatever is on the straight line drawn from




STATICS.                  67
its vertex to the centre of gravity of its base, at a distance
from the base equal to one-fozurth the length of the line.
Let A - BDE.... [Fig. 33] be the pyramid. From
the vertex A, draw the straight line Al to I the centre of gravity of the base; and divide the base into
triangles, by joining its vertices with the point I.
From the centres of gravity of the triangles, draw
lines to the vertex; and taking IC equal to onefourth of AI, through C draw a plane parallel to the
base of the polygon. The point C will be the centre
of gravity of the pyramid. For, since the plane
drawn through C cuts all the lines drawn from A to
the base proportionally, it will contain the centres of
gravity of the triangular pyramids A - IBD, A - IDE,
etc., and consequently will also contain the centre of
gravity of the entire pyramid; and since the straight
line AI passes through the centres of gravity of all
the planes drawn parallel to the base, it also will
contain the centre of gravity of the entire pyramid:
therefore the centre of gravity of the entire pyramid
will be at the intersection of the line and plane.
The result just obtained, being independent of the
number of sides of the base of the pyramid, is true
when the number becomes infinite, and the base a
re-entering curve: it is therefore true for a cone of
any base whatever.
Any polyhedron being divisible into pyramids, its
centre of gravity may be found by conceiving it to
be thus divided, and determining the centre of
gravity of each pyramid thus formed, with respect




68                      STATICS.
to three co-ordinate planes, and then applying equations [27] of article 45.
11~ The spherical sector.
The centre of gravity of a spherical sector, or the solid
generated by the revolution of a circular sector about one of
its sides, is on the axis of the sector, at a distance from the
centre of the sphere equal to three-fourths of the radius,
minuls three-eighths of the altitude or axis of the corresponding spherical calotte.*
For conceive the entire sector to be made up of
an infinite number of infinitely  small equivalent
pyramids, having their vertices at the centre of the
sphere.  The centre of gravity of each of the pyramids will be on the radius drawn to the centre of
gravity of its base, at a distance from  the centre of
the sphere equal to three-fourths of the radius.  The
centre of gravity of the entire sector will therefore
be the same as that of a spherical calotte concentric
with the calotte of the sector, and having its radius
equal to three-fourths the radius of the sphere: hence
the centre of gravity of the entire sector will be at
the middle point of the axis of this concentric calotte.
If we denote the height of the calotte of the sector
by h, the height of the concentric calotte will be ]h:
hence, denoting the radius of the sphere by r, the
distance of the centre of gravity of the entire sector
from the centre of the sphere will be ]r -h.
* "S herica calotte," or "cap," a 8pherical zone of one base.




STATIcs.                   69
MACHINES~
47. A ncehine is an instrument, by means of which
a force may be made to act upon points that lie
without its direction.
The sipie     hik mzaclimnes, of which all. others are composed, are the cord or  fope machine, the lever, and the
inc2lined pclaze.  Certain modifications or combinations
of these are frequently ranked among the simple
machines, viz. the pzlley, the zueel acnd Caxle, the screw,
and the wedge.
The fbrce employed in working a machine, is
called the potwer; and the force to be overcome, the
resislance.
In discussing the theory of machines, we seek only
the conditions of equilibrium  of the power and resistance.  The consideration of the motion which ensues
when the power is increased beyond what is required
for an equilibrium, belongs to dynamics.
In the first investigations, we omit the consideration of the weight of the machine, the stiffness of
cords, the flexibility of rods, friction, etc. etc.
THE ROPE MACHINE.
48. The simplest form of the rope machine is that
represented in figure [34], in which three cords AB,
AC and AD, lying in the same plane, are firmly united
at a point A; and the forces P, Q and R are applied
at their extremities B, C and D. The conditions of
equilibrium, in this case, are evidently expressed by
the proportion [Art. 12],
K




70                   STATICS.
P    Q    R       sinp  ~ sin q    sin r;.....[ 28]
p, q and r denoting the angles made by the cords, or
the directions of the forces as represented in the
figure.
49. If the forces P and Q be suppressed, and the
extremities B and C be attached to fixed points, the
conditions of equilibrium will still be given by proportion [28]; and P and Q will, in this case, denote
the re-actions of the points respectively.
50. WThen the extremities B and C are fixed, and
tilhe cord AD is attached to a ring which is capable
of moving freely upon BAC, it is evident, that as the
ring slides along this line, the point A will describe
an ellipse.  Consequently in order to an equilibrium,
the direction of AD must coincide with the normal
to the curve at the point A, and hence must bisect
the angle BAC of the radii vectores.  We shall then
have
p = q and P = Q,
and          P::: sin p    sinll r
51. The common intensity of two equal and opposite forces applied at the extremities of a cord, in the
direction of its length, so as to extend it, is called the
lension of the cord; thus,.P, Q and R are the tensions
of the cords AB, AC and AD  respectively. The
effect upon the cord will obviously be the same,
whether the two equal forces thus act at its extremities, or one of them be suppressed, and the extremity
to which it was applied be attached to a fixed point.




STATIcs.                  71
52. The case in which there is any number of,cords united at the same point, and solicited by
forces lying in the same plane, is immediately reducible to that just considered.
53. Another form of this machine, called the funicular polyyon, is represented in figure [35]; in which
EABCF, AG, BHi, CI, are supposed to be cords
united at the points A, B and C, and solicited at
their extremities by the forces P, P', P", etc. acting
in the same plane.  When these forces are in equilibrium with each other, it can readily be shown:
10. That the forces must be such that they will
be in equilibrium when applied at a single point (C,
for example, as represented in the figure), parallel to
their primitive directions, and hence that the conditions of equilibrium are the salme as those of Art.
116].
20. Denoting the tensions of the portions AB and
BC of the cord EABCF by ~1 and Q, and the angles
at A, B and C by s and p, q' and s', pIV and q, as represented in the figure, and applying the principle of
proportion [28], that the relations of the tensions P
and Pl" will be given by the proportion
P. PIv:: sinp X sins' X sinq q   sinls X sin q' X sijpIv
30. That when the forces P', P", P"' are parallel
to each other [Fig. 36], and hence the sums p + I'd
s' + plv, are each equal to 1800,
P: P:: sin ~  sins.
40~ That when the cord EBF [Fig, 37], sustained




72                   STATICS.
by the forces P and P'v applied at E and F, is subjected to no other additional force than gravity,
P:::: sin FMG: sil EG: sin EMF;
R denoting the weight of the cord, and EM and FM
being tangents to the cord at the points E and F,
and MG a vertical line drawn through their point of
meeting.
THE LEVER.
54. A lever is a bar or rod of any form whatever,
capable of motion about a fixed axis or support
called  ike fulernm.  In investigating the properties
of the lever, we conceive it reduced to a material
line.
Let the line ACBD [Fig. 38], represent a lever, and
F its fulcrum; and suppose it to be acted upon by
the forces P, P', etc. tending to turn it in one sense
about the point F, and P", P"', etc. tending to turn
it in the opposite sense about this point, all the forces
being supposed to act in the same plane. Also let
the lines p, p', etc., p", p"', etc. be drawn from F perpendicular to the directions of the forces. Then the
conditions of equilibrium  of the forces will be expressed by the equation [Art. 35],
Pp + P'p' + etc. = PI'p" +  PI"/'" - etc.
In this equation, the forces P, P', etc. may be supposed to represent the powers applied to the lever,
and.P" P"' etc. the resistances to be overcome.
The weight of the lever may be regarded as a vertical force applied at its centre of gravity.




STATICS.                   73
55. The case most frequently occuring in practice,
is that in which only two forces act upon the lever.
The preceding equation is then reduced to
= Pp,,,
and we have
P: P":: p/": p.
Thus, in this case, the power and the resistance are inversely as the perpendiculars drawn from the futlcrum  to
heir directions.
56. When the lever AB [Fig. 39] is straight, and
the two forces act in parallel directions, we get from
the similar triangles FAM, FBN,
p: p        FB: FA,
and hence we have
P: PI      FB: FA;
that is, the power and the resistance inversely as the distances
qf their points of application from  thze flcrcum.  These
distances are called the arms of the lever.
57. The pressure upon the fulcrum  is evidently
equal to the resultant of all the forces which act
upon the lever.  When the lever is not retained by
a fixed axis, but only rests upon a fixed support, it
is essential to the equilibrium that the direction of
the resultant should be perpendicular to the lever.
58. According to the relative positions of the
power, the resistance and the fulcrum, levers have
been divided into three kinds.  A lever is said to be
of the 1st kind, when the power and the resistance
are applied on opposite sides of the fulcrum, as in




STATICS.
Fig. 40; of the 2d kind, when these forces are applied
on the same side of the fulcrum, the resistance being
the nearer to the fulcrum, as in Fig. 41; and of the
3d kind, when these forces are on the same side of
F, and the power is the nearer to the fulcrum, as in
Fig. 42.
THE INCLINED PLANE.
59. This machine consists essentially of a plane,
inclined at any angle whatever to a horizontal plane.
Conceive a body to be in equilibrium on an inclined
plane: it is evident that the force P which prevents
the body from sliding down the plane, and P' the
gravity of the body, must have a resultant perpendicular to the plane, and meeting it within the limits
of the base of the body. The plane of these forces
will be vertical, and also perpendicular to the inclined
plane. The section of the inclined plane and the
body, by the plane of the forces, is represented in
Fig. 43, in which AB is the section of the inclined
plane, and MLN that of the body, and BC and AC are
horizontal and vertical lines meeting each other at
the point C; AB, BC and AC being the length, base,
and height of the plane respectively. Let the line
KE be the direction of the force P; KF, which we
suppose to pass through the centre of gravity of the
body, that of P'; and KG perpendicular to AB, that
of the resultant R. Then denoting the angles as
represented in the figure, we shall have
P: P': R:  sinp   s sinp'   sin r;




STATICS.                   75
or, since p is equal to i the inclination of the plane,
and sin i=
P  P~  P  h.  I sinp'   I lsinr.  I.... [29]
1~. When the direction of P is parallel to the inclined plane,p' = 900, and we get from  the above
proportion,
P  ~ P      h    1;
that is, the power to the weight of the body, as the height of
the plane to its length.
20. When the direction of P is parallel to the base
of the plane, p' = (90~- i), and we have
P    P''  A    lcosi
h    b;
that is, the power to the weight, as the height of the plane
to its base.
We have regarded the plane as inclined with respect to the horizon, and supposed the force P' to be
gravity; but the above results will obviously be
true if we consider the plane as inclined with respect
to any assumed plane, and substitute for gravity any
force acting perpendicular to the latter plane.
THE PULLEY.
60. The pulley is a small grooved wheel, capable
of motion about an axis, and having its circumference partly enveloped by a cord, to the extremities
of which the forces are applied. The axis is supported by a frame called the block; and according as
the block is fixed or capable of motion, the pulley is
said to be fixed or movable.




76                   STATICS.
61. lThe fixed pulley.
Let IGHK [Fig. 44] represent a fixed pulley, and
EIKHF a cord enveloping the are IKII of its circumni
ference; and suppose the two forces P anrd Q to be
applied at the points E and F, in the directions IE
and HF tangent to the circumference at the points
I and H. Produce EI and FH, and let the points of
application of P and Q be transferred to A the point
of meeting of these lines. Then since 0 is the only
fixed point in the pulley, in order to an equilibrium,
the direction of the resultant IR of P and Q must
pass through that point: hence it must bisect the
angle EAF, and consequently the forces P and Q
must be equal to each other.  Let the resultant be
represented by AD, and complete the parallelogram
ABDC: then P and Q will be represented by the
equal lines AB and AC respectively, and we shall
have
P    R   ~ AB    AD.
But from the similar triangles ABD, IOH, we get
AB: AD:  IO  ~ I I;
consequently we have
P: R::  10   III.
Thus, in the fixed pulley, each of I/e forces alpplied to
the cord or rope is to their resultan-t, or the pressure tponi
the point of support, as the radius of the pulley is to the
chord of the are with which the rope is in contact. The
fixed pulley is employed, when it is desired to change
the direction of a force without affecting its intensity.




STATICS.                   77
62. Movacle pulley.
In the movable pulley, one extremity of the cord
is attached to a fixed point F [Fig. 45]; and a power
P applied to the other extremity, holds in equili
brium a force R applied in a direction passing through
the centre 0.  The resistance R is usually the weight
of a body suspended from  the centre.  In place of
the re-action of the fixed point F, we may substitute
a force Q, and consider the machine as in equilibrium
under the action of the forces P, Q and R, applied at
points entirely free.  The condition of equilibrium
will obviously be, that the resultant of P and Q must
be equal and contrary to R. Hence it may be readily inferred that P and Q are equal, and, as in the
preceding case, that
P: R        IO     I II.
Thus, in the movable pulley, the power is to the resislance, as the raditus of the pulley is to the chord of the are
in contact with the rope.
1~. When the cords El and FH are parallel, the
chord II becomes a diameter, and we have
P: R     1  2;
that is, the resistance equal to twice the power.
2". When the are IKH  in contact with the cord
is equal to 60~0, we have
P = R;
and when this arc is less than 600,
P> R.




78                   STATICS.
63. Systems -of plleys.
1~. Let C, C', C", be a system of movable pulleys,
connected with each other in the manner represented
in Fig, 46, in which F, F', F", are fixed points, and
the power P applied in the direction I"E holds in
equilibrium the weight R suspended from C. Denoting the tensions of the cords IC', I'C" by P" and P'
we have
PI: R        IC      IH,
P:  P'''::'  IH',";
and hence
P' R     ICx:'C'x"C"    I  I  I  IH x I'H' x I"H
That is, in this system of pulleys, the power is to the
resislance, as the product of the radii of the pulleys is to the
prodclct of the chords of the arcs in contact with the ropes.:When the cords are parallel, as in Fig. 47, the
above proportion is reduced to
P.R       1    23
Hence, in a system of this kind, in which n movable
pulleys are employed, we shall have
P    R      I    2X.
By means of movable pulleys arranged in this
manner, a great weight may be moved by a small
force; but it will readily be perceived that the gain
in power will be attended with a loss in respect to
time.
20. A combination more convenient than the pre



STATICS.                 79
ceding, is that in which several pulleys, both fixed
and movable, are embraced by a single cord as represented in Fig. 48, in which the several parts x, y, z,
etc. of the cord are supposed to be parallel to each
other. In this arrangement, we may obviously consider the tensions of these portions of the cord as a
system of equal and parallel forces acting in the
same sense, of which the resultant is equal to the
weight or resistance R. Thus in the case of three
movable pulleys, in which the cord has six branches,
we have
P    R 1'    1   6;
and hence denoting the number of pulleys by n, we
have generally
P    R:: 1: 2n.
The method of determining the power necessary
to put in equilibrium the weight of the several movable pulleys, is sufficiently obvious.
THE WHEEL AND AXLE.
64. This machine consists of a wheel AA' [Fig. 49],
and a cylinder or axle BB', so adjusted as to have a
common axis, and firmly connected with each other.
The cylinder is supported at its extremities in such
a manner as to admit of only a motion of rotation
about the common axis. The power P and the resistance R are applied at points of the wheel and,axle, respectively, in the direction of tangents to
their circumferences.  lhe motle of operation of the
jmachine is indicated in the figure. The case of an




s0                   STATICS.
equilibrium  between P and R in this machine is
obviously comprehended in the more general case
already considered in Art. 40. Referring then to
this article, and denoting the radius of the wheel by
p, and that. of the axle by r, we get
Pp = Rr,
or             P    R      r ~ p.
That is, in the equilibrium of the wheel and axle, the
power is to the resislance as the radius of tIe axle is to the
radius of the wheel.
When P and R act upon the machine by means
of cords, we must suppose these forces applied to the
axes of the cords, and increase the radii of both the
wheel and axle by the radii of the cords respectively.
65. A combination of wheels and axles is sometimes used. One of these compound machines, in
which three of the simpler machines are employed,
is represented in figure 50. To find the relation in
this case between the power P and the resistance
P"' denote the tensions of the intermediate cords
iMN, M'N', by P' and P"; the radii of the several
wheels by p, p' and p"; and those of the corresponding axles by r, r', r": then we shall have
P     P'      r: p,
p/l   p/  r"  Pp;
and hence
P   J"       r X r'x        p X p' X p"




STATICS.                   81
that is, the power to tle resistance, as the prodzct of the
radii of Ize axles to the prodtcet of the radii of the wzveels.
66. In these combinations, the connexion between
the simple machines is frequently effected by means
of teeth or cogs projecting from the several convex
surfaces, as represented in figure 51. The relation
between the power and the resistance is not altered
by this modification of the mode of connexion,
THE SCREW.
67. This machine consists, 1st, of the interior screw
(usually called simply the screw), a cylindrical solid,
around whose convex surface passes a uniform bland
or fillet, oblique to the axis, and constantly inclined
to it at the same angle; and, 2d, of the exterior scretw,
or nut, a hollow cylinder of the same diameter as the
solid one, on the concave surface of which is a groove
exactly adapted to the fillet of the interior screw.
The mode of operation of this machine is indicated
in figures 52 and 53, in which AB represents the
interior screw, inserted in the nut CD; and OS, 0'S',
two arms or levers to which the forces are respectively applied, according as it is desired to communicate nmotion to the nut or to the interior screwThe investigation of the properties of the screw is
best conducted by first considering the manner in
which the fillet may be conceived to be generated.
Let ABDC [Fig. 54], be a cylinder, and BMd[ a rectangle, of which the base DiM is equal to the circumference of the base of the cylinder. Let the sides
BD and NM be divided into the equal parts BF, Fli,




82                   STATICS.
etc., NG, GI, etc.; and let the oblique lines FN, HG,
etc. be drawn. Then if the rectangle be applied to
the convex surface of the cylinder in such a manner
that the line NM shall coincide with BD, the points
N, G, I, etc. will coincide with the points B, F, H, etc.,
and the oblique lines FN, HG, etc. will form on the
surface of the cylinder a continuous curve ss'. This
curve is called a helix'; and the constant interval
NG, its pitch. If now we conceive a triangle, whose
plane constantly passes through the axis of the
cylinder, to revolve about that axis in such a manner
that its base, never greater than NG, shall constantly
be in contact with the cylinder, and have one of its
extremities in the helix ss', it is evident that each
point of the revolving figure will describe a helix
similar to ss', and that the assemblage of helices thus
described will form the fillet of a screw. The generating surface is usually a triangle, as we have above
supposed it to be: it is sometimes a rectangle, and
may be of any form whatever.  The conditions of
the problem  being the same, whether we suppose
the screw to be movable and the nut fixed, or the
converse, we will adopt the latter hypothesis.
Let then the screw be supposed to be fixed, and
the nut resting upon it to be subjected to the action
of two forces: the one, the resistance R, acting directly upon it in the direction of the axis; the other,
the power P, acting at S at right angles to the axis,
and also to the lever OS. In the revolution of the
nut, each of its points in contact with the screw de



STATICS.                 83
scribes, as we have seen, a helix, and may obviously
be regarded as moving on an inclined plane whose
height is equal to the pitch of the screw, and whose
base is equal to the circumference of the circle having for its radius the distance of the point from the
axis of the screw. Hence, considering a single point
of contact M [Fig. 55], whose distance from the axis
is OM, and denoting the forces which act upon it by
p and r, both being supposed to act directly upon
the point, the first parallel to the base of the plane,
the second perpendicular to it, we shall have, in the
case of an equilibrium [Art. 59, 20],
p: r:: NG: 2r x 01.
3ut if the power be applied at S instead of M, at the
distance OS from the axis, we shall have, to determine the force P which, applied at 5, will be equivalent top applied at M, the proportion
P: p       OM: 0  OS;
whence we get
P ~ p:: 2~r x OM    2 xOS.
Comparing this proportion with the first, we have
P: r  NG  2r x OS;
and hence
P rx  NG
J~-2 — r   X
2,xx OS'
For the other points of contact M', M", etc., at which
the resistances r', r", etc. are put in equilibrium by
the powers P', P", etc., applied at the same distance
from the axis as the power P, we have




834                     STATICS.
pNG
Pt1  v/   X__
2~ x  0'
P"1      xNG='   x 2 X     OS'
etc.
H[-ence, denoting the sum of the powers P, P', etc. by
P, and the sum of the resistances r, r', etc. by R, we
have
NO
2~ x  OS'
whence,
P.   R?       NG o   2v X OS.
That is, in the equilibrium  of the screw, the power is
lo Mhe resis'ance, cs the pitch of the screw, or the cdislanzce
betlzeee Mwe /hreads, is to Mhe eircumtference of the chircle describ-ci /)ct y ll poild of Cqppeliaion of the wpowzer.
THE WEDGE.
68. The wedge is a solid body of the shape of a
triainguilar prism.  The surface CDEF [Fig. 56], is
called its back; ABED and ABFC, its sides; and AB
the intersection of the sides, its edge.  The use to
which it is most commonly applied, is to separate
the parts of a body, by introducing the edge AB into
a small cleft, and applying an impulsive force to the
back.  Since the resistance, or the force which the
parts of the body oppose to the separation, is always
unknown, it would be useless to investigate the conditions of equilibrium  between it and the power.
We shall, therefore, seek merely the relation between




STATICS.                 85
the power and its components perpendicular to the
sides of the wedge.
Let MNO [Fig. 57] be a section of the wedge by
a plane perpendicular to its edge. Let the power
P, which we suppose to be perpendicular to NO, be
represented by IK, and be resolved into the two
components IR and IS perpendicular respectively to
NM and OM. These components represent the effects of the power upon the sides of the wedge, and
tend directly to separate the parts of the body. Denoting the power and its components by P, P' and
P" respectively, we have
P    P  Pf':"  IK         IS.
But from the similar triangles MNO and IKR, we
have
IK: IR. IS.: NO    MN    MO::
hence
P     P'   P"'    NO' MN  ~MO;
and multiplying the last three terms of this proportion by the line DE [Fig. 56] we get
P  ~ P: PI   N Ox DE    I MN x DE: MO X DE.
The products composing the last three terms of
this proportion represent the respective surfaces of
the back and sides of the wedge. Consequently, in
the wedge, the power coapplied at right angles to the back,
and the eforts exerted at the sides, are respeetively propose
tional to the swfaces of the back and sides'1




8(i 61,         ~STATICs
GENERAL PRINCIPLE OF EQUILIBRIMU ITN MIACHINtESo
6 9. By combining the sinzple machines above de=
scribed, an endless variety of cormpounld  machines
may be -ormed.  )in these compouTd im-achines, the
ratio of the power to the reisitance can be deter-minerd w'hen'the tensions of the cords which connect
the various parAts are known, as has been doine in the
case of the systems of pulleys of article 63, and that
of wheels and axles of article 65.  But in every case,
however complex the machinie, this ratio can also'be
determilned  by the follow;-U4g   rule:Leo  the equilibriumn of the machine be supposed
to seufer an infinitely small dist-urbance: the points
of ppllcation of P and B$, the p ower and. the resistance, will describe infinitely small arcs. If the directions of? and a.d are tangent to these arcs, let the
cores be denoted by i and v respectively: if they are
not, let the projyectioz.s of Ike ares upon tle directions
of 2  and?1 be denoted by these letters; then will.
the relation between P and R be expressed by the
proportion
P  ~ R       v    u o.
This rule is a particular case of a general principle
of- mechanics, called fte principle of vsirtlcd've1oeitzies
7 0. APPLICATIONS.
1~o Y2ke levern  Ist. Let AB [Fig. 58] represent the
lever; ancn fron F the fulcrurn, let the perpendiculaprs p and p' be drawn to the directions of the two
forces F and P'o Let M and N, the feet of the per



STA TICSo                  87
pendiculars, be taken as t'he points of anplication of
P and P'; and conceive an infinitely small motion
-to be communicated to the lever, causing these
points to describe the infinitely small arcs M2  and
Nn.  The directions of P  and P' being tangent to
-these arcs, denoting them by ib and v, we shall have,
by the ruleT
P    P':  v    u;
but              V    u      p'    P:
hence           P'::' 
That is, the forces are inversely as the perpendiculars dXrawn from  the fulcrum  to thleir directions as
found in article 55n
2d. Let the forces P and P' be supposed to be applied at the points A and B [Fig. 59] the extremiLies of the lever. In the case of motion, the directions of P and P' will not in general be tangent to'the arcs AE and BG described by A and B; and in
applying the rule, we must consider ui and v as denotf
ing the projections of AE and BG uvpon these directions.
To find the values of it and v, draw EBC and GD
perpendicular to the direetions of P  and P', andi
consider the infinitely small ares AE and l BG  as
straight Aines perpendiculal.r to the radiii.A and FBo
then we have the triangles MlA, C AIEJ similar to
each other, and also the triangles NFB, DBG; and
hence we get




88                  STATICS.
FA  ~ FM:: AE: AC oru,
and       FB  FN:: BG: BDorv;
and hence
AE               BG,A XFM, ancd v=     X FN.
Applying the rule, we have
BG          AE
P:  P'''  x FN ~          x FIxM;
but         AE: BG:: FA: FB,
or                BG   AE
FB   FA'
consequently
P' P  FN: FM,
or           P: P  p'    p.
20. TIVzeel aczd axle. In the wheel and axle, the directions of the forces P and R [Fig. 48] are tangent to
the arcs described by their points of application:
hence, denoting the arcs by ut and v respectively, we
have, by the rule,
P: R:: v: u;
but            v    u  -  r    p
hence         P:  R:: r   p.
30. Tuze serew. In the screw, the direction ST [Fig.
52] of the power is not tangent to the arc of the
helix described by S its point of application. Hence,
in the proportion
P    R: v    tu,




ST'ATICS.                 89
we must regard uz as denoting the projection upon
ST of an infinitely small arc described by the point
S; v being the corresponding space described by
any point of the nut, in the direction of the axis of
the screw. Conceive a plane to pass through OS
perpendicular to the axis, and let the helix described
by the point S be projected upon this plane; the
projection will be the circumference of the circle of
which OS is the radius. Now since ST is tangent to
the circumference of the circle, the projection of the
infinitely small arc of the helix upon this line will
be equal to its projection upon the circumference,
and the one may be taken for the other.  Considering then iu as an arc of the circle, it is evident from
the nature of the helix, that v is to ut, as the pitch
NG [Fig. 55] of the screw is to the circumference of
the circle of which OS [Fig. 52] is the radius, or that
V::I: PNG: 27r X Os;
whence we get
P::R  NG    2 X OS.
4~,. Syslemns of pulleys. We will consider the system
of pulleys represented in figure 48. Denoting the
spaces described by the points of application of the
forces P and R by ut and v respectively, we have, by
the rule,
P    R      II,.
But when, by the action of P, the weight R is elevated through the space mn each of the six branches
of the cord embracing the movable pulleys is short



90                    STATICB.
ened by the same quantity; and l hence the point of
application of the power must descend through the
space 6z&. Hence  we have
v    ~       6,
anid consequently
P    R      I    6.
FRICTION.
71. A perfectly smooth body, placed on an inclined
plane also perfectly smooth, would move, if abandoned to the action of gravity, lowever slight the
inclination of the plane; but in practice this motion
never takes place, till the angle of incliat ion reaches
a certain magnitude, greater or less accordin4g to the
circumstances of the case.  The reason of this is, that
the surfaces of even the most highly opolished bodies
are only comparatively smooth, fund consequently
the sliding of one surface upon another is always attended with a certain resistance.  This resistance is
called friction.
Let the body MN  [Fig. 60]  be placed on a plane
AB, and let the angle of inclinatlion of the plane be
gradually increased till it reaches the magnitude iZ
at which the body just begins to move.  In this
position of the plane, let P the weigght of the body
applied at its centre of gravity C, and represented
by CF, be resolved into the tw o componenits
@G = Pcosi, and CE  -P sini,
th.e one perpendicular to the plane, the other parallel to it.  The first will obviously measu-re the pres



STATICS.                     91
sure of the body upon the plane; the second, its
friction.  If now  while the surface in contact with
the plane remains the same, the weight D, and
hence the pressure P cos i, be made to vary, the
angle of inclina-tion at which the body begins to
move will be found to be unaffected, retaining con
stantly its first value i: whence we infer,
1~0 T/-ac the friceion is direely proortionalz to tie presDenotinrg the friction by?F, the pressure by P',;nd
the ratio of the two by f, we have
f  -- = tang i,
and                  - PI X tang i.
The angle i is called {he acgyle of friction:, and the corn
stant f the coefflienZ of friction.  The latter expresses
-the friction for the unit of pressure, and is taken as
the measure of friction.
If the'body 1WN be a polyhedron whose faces are
unequal in extent, but equally polished, the angle i
at which it will begin to imove will be found to be'the same, on whichever face it may rest upon the
plane: whence it appears,
20.o  hat the friction is independent of the extent of the
stface iz cotacet.
It is also found,
3~0. That tze friction is inclepenzdent of the velocity; that
is, that the friction is the same, whatever the velocity with which the one surface moves over the other,








PART SECOND.
DYNAMICS,
OF THE RECTILINEAR MOTION OF A MATERIAL POINT.
I. Of unifornz rectilinear motion.
1. UNIFORM rectilinear motion is that in which a
material point, moving in a right line, passes over
equal spaces in equal times.  It is the simplest kind
of motion, and takes place whenever a body is acted
upon by an impulsive force, and then abandoned to
itself.
The velocity of the point is the space which it
describes in the interval of time arbitrarily chosen
as the unit.  It evidently expresses the rate  at
which the point moves.
If the velocity be denoted by v, then
The space described at the end of the 1st unit of time, is v,
"C  LC  C"  2d "4 "2        2v,
66 C' 9   6  3d  C"  cc   3v
eo denn  te  0 e b  e ae  eIe
" C       6.C    tth'n  I  tv;
and denoting the whole space by s, we have
S = V1t. 9 e 0 a sc    a 9  0  a 0 a 0 la a e  a. a B      a a   0 I




2                   DYNAMICS.
~ 1. From the preceding equation, we get
S
from which it appears, that in uniform motion, the
velocity is the ratio of the space described to the
time employed in describing it.
~ 2. If we denote by s' and s" the spaces passed
over in the times I' and t" by two points moving
uniformly with the velocities v' and v", we have
SI = V! f 
$i/  =   vl!I;
and hence    s':  s"    VT:    t"
that is, the spaces described are as the products of
the times by the velocities.
When the times are equal, we have
$8:       V 
or the spaces as the velocities.
When the velocities are equal, we have
or the spaces as the times.
When the spaces are equal, we have
et  tI: ~    v  i   V-'
or the times inversely as the spaces.
2. A more general form may be given to equation
[1]o Thus [Fig. 11 suppose the material point to be
moving on the line AB from left to right, with a
velocity v, and denote by s its distance at any instant




DYNAMICAS                   3
from the point 0 arbitrarily assumed on AB, and let
the time t be reckoned from the instant at which the
point is at 0'; then when the point is at D to the
right of 0', we have
OD   001 + 0'ID,
or, putting,    00' = b,
s = vt + b................... [2]
In this equation, the variables s and t may be either
positive or negative. The positive values of t refer
to times posterior to the instant at which the point
is at 0'; the negative values, to times anterior to
the same instant. The positive values of s must be
reckoned from O to the right; the negative values,
from O to the left.
In this general form, the equation will enable us
to determine, at any instant whatever, the position
of the material point on the indefinite straight line
ABR
If we suppose another material point to move in
the line AB with a velocity v', and to be at O"-at the
instant at which the first point is at O', its motion
will be determined by the equation
s' = v't + b';................. [3]
s' denoting its variable distance from 0, and b' the
distance 00".
By means of equations [2] and [3], any problem
may be solved, which depends upon the relative mo



4                  DYNAMICS.
tions of the two points. Thus, to determine when
the two points will meet each other, in which case
S -st
we have          vt + b = v't' +',
b6 — b
and hence get        t   -
II. Of motion unnif(ormnly varied.
3. A force which acts without intermission, and
with a constant intensity, is called a constant accelerating force. The motion of a material point, subjected
to the action of a constant accelerating force, is
called a cuniforinly varied notion.
Let a material point be supposed to be acted upon
by a constant accelerating force; then if at any instant the accelerating force cease to act, the motion
will evidently become uniform, and the point will
move with the velocity which it had at that instant.
Hence in motion uniformly varied, the velocity at
any instant is the space which the point would describe in any succeeding unit of time, should the
accelerating force at that instant cease to act. The
velocity evidently depends upon the intensity of the
force, and the time during which it has been acting.
Since a constant accelerating force acts at every
instant with the same intensity, it must generate
equal velocities in equal times. We assume as the
measure of a constant accelerating force, the velocity
which it generates in the unit of time.




DYNAMICS.                    5
Let us now suppose a material point to move from
a state of rest, under the action of a constant accelerating force, of which the intensity is denoted by
g; and let us seek the velocity v, and s the space
described, at the end of the time t.*
To facilitate the investigation, conceive the time
divided into an infinite number of infinitely small
equal intervals, or instants; and suppose the force to
act upon the point at the commencement of each instant. Then during each instant the motion will be
uniform, and the variable motion will be resolved
into a series of uniform motions.
Denote by n the number of instants in a single
unit of time, and by k the number of instants in the
whole period t; then
k = nt.
Also let i equal the space which the point describes
during an instant, in virtue of the constant velocity
which the force communicates to it at the beginning
of each instant; then the spaces described by the
point during the successive instants of the time t,
will be
i,   2i,    3i,  ni,  ki.................. e[4]1
If at the end of the first second the accelerating
force should cease to act, the point, having during
the nth instant described the space ni, would, during
the following second, describe the space nz. ni = n2i;
* The time t is given in terms of the unit of time; that is, if the second be taken as the unit, t denotes a certain number of seconds.




6                   DYNAMICS.
and this being the velocity communicated during
the preceding second, or the measure of the accelerating force, we shall have
g = n2i.
If at the end of the time I we suppose the accelerating force to cease acting, the point, which in the
kth instant has described the space ki, will, in the
following second, describe the space n x ki = nii';
and this will be the velocity acquired at the end of
the time 1. Denoting this by v, we shall have
V = nki;
but                 k = nt;
hence               v = n2it
= gtA @ees...........eee [5]
To find the space s, we have only to determine
the sum of the series [4]: we thus have
s = (i+ ki) -
(1 +k)-k;
but k being infinite, ( 1 + k) becomes k, and we have
ik=  n2it2  gt2
=................. s[6]
4. We have supposed the material point to be at
rest when the accelerating force begins to act upon
it: if we suppose it to be in motion, and to have al



DYNAMICS.                     7
ready described a space b with a constant velocity a,
we shall have the more general equations
I V  a + glt,.f........... c............ [7]
3 =  b + at + g2 gt............2........ [8]
If the accelerating force act in the direction opposite to the primitive impulse, g must be affected with
the negative sign, The motion in this case is said
to be un~ifornzly retarded.
If we suppose g = o, the equations are reduced to
those of uniform motion.
5. If a material point, moving from a state of rest
under the action of a constant accelerating force y,
describe the spaces s and s' in the times t and t', and
acquire the velocities v and v', equations [5] and [6]
give
v = gt,  VI  g=
2yt, SI = 2yt 
vs~;gt2,  v'=- gt';
and hence we have the proportions
S    I   t2  tl 2
v:   v'  t   o   t
V    Vo  VS
that is, in motion uniformly accelerated, the spaces
described are as the squares of the times, and the
velocities acquired are as the times, or as the square
roots of the spaces.
6. If in the equation s = ]gt we make t = 1, we
get
s$=  gy, or g =2s;




8                   DYNAMICS.
that is, the velocity acquired in the unit of time is
equal to twice the space described during that time;
and hence, as the unit of time is arbitrary, it follows
that a constant accelerating force communicates to a
material point, in any time whatever, a velocity
equal to twice the space which it causes the point to
describe in the same time.
7. It can be shown by direct experiment, that
bodies, near the surface of the earth, fall with a uniformly accelerated motion. Terrestrial gravity may
therefore be considered a constant accelerating force;
and hence equations [7] and [8] may be employed
to determine the circumstances of the motion of falling bodies. To find the numerical value of g when
it represents the intensity of gravity, we have recourse to an indirect process, which will be explained
hereafter. It has thus been found, that in the latitude of the city of New-York, the second being assumed
as Mhe unit of time,
g - 32,1598 feet, or nearly 32- feet.
8. If from the equations
s =- gt2 and v = yt,
we eliminate t, we find
v- =   2gs,.ee.e...ee.......e.. e e[9]
an equation which gives the velocity acquired in
falling through a given height, or, as it is usually
expressed, the velocity due to a given height.




DYNAMIS.                    9
To determine the time in which a body will fall
through a given height, we have the equation
t          e.   ee.  e.  0......  [10]
9. To determine the circumstances of the motion
of a body projected vertically upwards: in equations
[7] and [8], we make b = 0, and affect g with the
negative sign; we thus get
v = a - gt,....0 a e Is......              [a]
s = at -  yt2,...............]..... [b]
in which a denotes the velocity of projection.
To find the time during which the body is ascending: in equation [a], we make v = 0; we thus get
a
g
To find the greatest elevation of the body, we substitute this value of I in equation [b]: we thus get
a2
s ='
The body having attained its greatest elevation,
to find the velocity which it will acquire during its
descent: in the equation v = V2gs, we substitute for
a2
s the expression a; we thus get
V = a.
From  this last result, it appears that the velocity
acquired during the fall is equal to the velocity of
projection.
B




10 DYNAMICS,
10. A force acting without intermission, and constantly varying in intensity during the time of its
action, according to some law, is called a variable acceleraling force.
The velocity of a point moving under the action
of a variable accelerating force, is measured in the
same manner as in motion uniformly varied.
The measure of the intensity of a variable accelerating force, at any instant, is the velocity which it
would generate, should it act during the unit of time
with the intensity which it has at that instant. Thus
a point being in motion under the action of a variable accelerating force, to determine the measure of
the force at the end of the time t, conceive the force
at that instant to cease varying, and to act during
the succeeding second with the intensity which it
has at that instant; the velocity acquired by the
point during the second is the measure of the force.
Let this velocity be denoted by c; then the velocity
acquired during the infinitely small interval of time
I', immediately succeeding t, is evidently Pt/; and
denoting this velocity, which is also infinitely small,
by v', we have t1' = v', and hence
V/
For the case of a constant accelerating force, we
have
V
t
Hence the measure of an accelerating force is the




DYNAMICSo                     11
same, whether it be constant or variable; only in
the latter case the time and velocity, of which the
ratio is taken, are infinitely small.
Experiments made near the surface of the earth
indicate, as already remarked, no variation in the
intensity of gravity at different points of the same
vertical.   A  more  extended  induction, however,
shows that gravity is really a variable force. The
general law is, as will be seen in a subsequent
article, that the intensity of gravity depends upon
the distance of the point at which it acts, from the
centre of the earth; the intensities at any two points
of the same vertical being inversely as the squares
of the distances. It can be shown, by a very simple
calculation, that according to this law, the motion of
a falling body near the surface of the earth should
not differ sensibly from a uniformly accelerated motion.
OF THE MOTION OF BODIES UPON INCLINED PLANESo
11. The motion of a body placed on an inclined
plane, and abandoned to the action of gravity, may
evidently be reduced to that of a material point
under the same circumstances.*
To consider the motion of a material point on an
inclined plane, conceive the inclined plane to be
* It may be proper to remark, that throughout the following pages
the body whose motion is the subject of investigation is considered to
be under the influence of the enumercted forces alone. Thus, in the
present case, it is supposed to be acted upon by gravity only, abstraction being made of friction and the resistance of the medium in which
the motion takes place.




12                  DYNAMICS.
intersected at right angles by a vertical plane, and
suppose the section to be represented by AB [Fig.
2.]; and from A and B let the lines AC and BC be
drawn, vertical and horizontal respectively. Denote
the angle at B by i, and the height and length of the
plane by h and I respectively. A material point,
placed on the plane at A, will evidently describe the
line AB.
Suppose the point to be at M; and let g, the intensity of gravity, be represented by the vertical MP.
Let MP be resolved into the two components MR
and MQ, the one in the line AB, the other perpendicular to it; the latter will be destroyed by the reaction of the plane, and the former will cause the
point to describe the line AB.
The angle MPR being equal to the angle at B, we
have
MR = g sini.
Thus the force which urges the material point down
the inclined plane is of the same nature as gravity,
differing from it only in intensity. If, therefore, in
the equations of uniformly varied motion,
v = gt, s=gt2 and v = V2gs,
we substitute g sini in place of g, we shall have, to
determine the circumstances of the motion of the
point, the equations
v = gt sini,........................ [11]
S = ~gt2 ini,......*.........   [12]
v _=  /2gs sini................ [13]




DYNAMICS.                13
If the point be made to ascend the plane by an
impulse in the direction BA, the motion will be determined by the equations
v = a - gt sini,
s = at - ~gt2 sini,
a denoting the velocity due to the impulse.
12. To determine the velocity acquired by the
material point in describing the line AB, we have
the equation
v = ~V2gs sini
- V2gl sini
- V 2gh.
But      is the velocity which the point would acquire in falling through the height A or AC: hence
it appears that a material point acquires the same
velocity in descending the length of an inclined
plane, that it would acquire in falling freely through
the height of the plane; and, consequently, that if
several material points, setting out from the same
point A [Fig. 3], and moving on different inclined
planes, describe the lines AB, AB', AB", on arriving
at the points B, B', B", situated in the same horizontal plane, they will all have acquired the same velocity.
13. To find the relations of the times t and i' employed in describing the lengths 1 and i' of two inclined planes of which the heights are AI and t', we
get from equation [12]




14                 DYNAMICs.
t2    S
}g. sini
1   1   12
— X-=
and hence          tHence, for the second plane, we have
and consequently we have
t   t/     h
that is, the times employed by two material points
in describing two different inclined planes, are as the
lengths of the planes divided by the square roots of
the heights. Hence when the heights of the two
planes are equal, the times are simply as the lengths.
14. Let AB [Fig. 4] represent the line described
by a material point in descending an inclined plane;
and from D any point whatever of the altitude AC,
draw DE perpendicular to AB. Then denoting the
angle at B by i, and by I the time in iwhich a material point will describe the line AE, we have [equa.
12]
AE = -gt2 sini
but              AE = AD. sini;
hence        2gt2 sini = AD. sini,
and                t= -  AD
let'1




DYNAMICS,                 15
Again, denoting by I' the time in which a material
point will describe the line ED, we shall find by a
similar operation,
t( =J.
But AlD is the expression of the time [eqlua. 103 in
which a material point, moving freely, will descend
from A to D; and the point E is in the circumference
of the circle described on AD as a diameter: hence
a material point, moving in the plane of a vertical
circle, will describe any chord drawn from  either
extremity of the vertical diameter, in the same time
that it will describe the diameter itself.
OF THE MOTION OF A MATERIAL POINT ON A GIVEN CURVE,
15. From  any point M [Fig. 5] of the circumference of a circle ABD, let the straight line MP be
drawn perpendicular to the diameter AB; then
AP    PM      PM:   PB.
If we suppose the arc BM to be infinitely small, we
shall have PM  infinitely small with respect to AP,
and hence PB infinitely small with respect to PM;
that is, PB will be an infinitely small quantity of the
second order. But PB is the versed sine of the arc
BM; hence the versed sine of an infinitely small are
is an infinitely small quantity of the second order.
16. Let AB, BC [Fig. 6] be two contiguous sides
of the perimeter of a polygon; and let a material




16                 DYNAMICS.
point, which is constrained to' describe this perimeter,
be supposed to have arrived at B with a velocity v.
Let v be represented by BD a portion of AB produced, and be resolved into the two velocities BN,
BM, the one in the direction of BC, the other perpendicular to it: the latter is destroyed by the reaction of the side BC; the former is the velocity
with which the point describes this side.
The velocity lost by the point in passing from the
side AB to BC is evidently equal to BD - BN. But
denoting the angle NBD by cu, we have
BN -= v cosu;
hence we have
the velocity lost = v - v cosu
= v (I - cosu)
= v. vers sinu.
If now we suppose the given polygon to become a
a polygon of an infinite number of infinitely small
sides, or a curve, the angle tc, which any side makes
with the adjacent side produced, will become infinitely small; and hence (Art. 15) the velocity lost
in passing from the one side to the other will be an
infinitely small quantity of the second order. When,
therefore, a material point is constrained to describe
a curve, under the action of either impulsive or
accelerating forces, it retains all the velocity which
is communicated to it in a direction tangent to the
curve; for since the velocity lost at each angle of
the polygon is an infinitely small quantity of the




DYNAMICS.                  17
second order, the whole velocity lost in passing over
a curve of finite length can only be the sum of an
infinite number of infinitely small quantities of the
second order, that is, an infinitely small quantity of
the first order.
17. Let ABCD, etc, [Fig. 7] be a vertical curve, or
polygon of an infinite number of infinitely small
sides AB, BC, CD, etc.; and let the sides be produced
to meet the horizontal line KH in A, E, TI, etc. Then
a material point starting from A, and constrained to
describe the perimeter ABCD, etc., under the action
of gravity, will have, when it arrives at the point B,
the same velocity as if it had described the line EB
[Art. 12]; and as by the preceding article it loses no
velocity in passing from AB to BC, when it arrives
at C, it will have the same velocity as if it had described the line EC. In the same way it may be
shown that at D it will have the same velocity as if
it had described the line HD, or the vertical GD.
Hence, generally, a material point which descends
under the action of gravity, along a vertical curve,
has, at any point whatever, the same velocity as if it
had fallen freely from a height equal to that of the
arc described. It is also evident that this velocity is
independent of the form of the curve; so that several
material points, setting out from the same point O
[Fig. 8], and describing in the descent different
curves OM, OM'S OM", etc., will, on arriving at the
same horizontal plane MM", have the same velocity.




18                  DYNAMICS.
18. Let AMA' [Fig. 9] be a curve, symmetrical
with respect to the vertical MG, in which the points
A and A' are in the same horizontal line, and MT is
the horizontal tangent; and suppose a material
point, placed on the curve at A, to be abandoned to
the action of gravity. It will acquire, in descending
to the point M, a velocity due to the height GM,
and, in virtue of this velocity, will rise in the branch
MA'. But the velocity with which it will describe
the arc MA' will be constantly diminished by the
action of gravity, and evidently in such a manner
that, at any point, as in', its velocity will be the same
that it was at mE, in and qn' being in the same horizontal line: hence at A' its velocity will be reduced to
zero. From A' the point will descend to M, and acquire as before the velocity due to the height GM;
and this velocity will just serve to carry it back to
the point A from which it set out, and thus it will
continue to oscillate for an indefinite time between A
and A'. Since any element of the curve is described
with the same velocity, whether the point is ascending or descending, it is clear that the time of the
ascent in either of the branches is equal to the
time of the descent in the same branch: it is also
evident that the two branches are described in equal
times. The point will therefore occupy the same
time in going from A to A', as in returning from A' to
A; and thus all the oscillations will be made in
equal tines.  Oscillations which are thus made in
equal times, are said to be isoclzhroozots




DYNAMICS.                19
OF THE SIMPLE PENDULUM.
19. A pendulum is an apparatus consisting of a
solid body attached to one extremity of a rod,
through the other extremity of which passes a horizontal axis, about which the whole system is capable
of oscillating.
When the vertical, drawn through the centre of
gravity of the system, meets the axis of rotation, the
pendulum is at rest; but if it be removed from this
position, and abandoned to the action of gravity,
it makes a series of oscillations, which may be
shown to be isochronous, and consequently may be
employed to measure time. Such a pendulum is
called a conozsund penc&lu2m. To acquire a knowledge
of its properties, we first investigate those of a purely
ideal pendulum, called tie si2mple pendulzwn. In this
pendulum, the solid body of the compound pendulum
is supposed to be reduced to a heavy material point,
and the rod to a line inextensible and inflexible, and
Without gravity.
Let CB [Fig. 10] be a simple pendulum, suspended
at C; and let it be withdrawn from the vertical position CK, and made to assume the position CB. Let
the intensity of gravity, represented by BN, be resolved into the two components BQ and BP, the one
in the direction of CB, the other perpendicular to it.
Of these two components, the latter alone communicates motion to the pendulum, the former expending itself in producing a pressure upon the point C.
The case is then precisely similar to that of a heavy




20                 DYNAMICS.
material point compelled to describe a curve, the
reaction of the fixed point taking the place of the
resistance of the curve.
Suppose the pendulum, or material point, to have
already descended from B to M [Fig. 11], and to
have acquired in the descent a velocity zu. Draw
the horizontal line BD, and the vertical diameter
HK. Also from the extremities of the arc MM', supposed to be infinitely small, draw MP, M'P' perpendicular to HK; and on AK as a diameter, describe
the circumference ANO. Draw also M'O' perpendicular to MP, and join C and M'.
Let AP' = x,   M'P'  y,   PP' =s,    AK = b
MC =-; and denote the time in which the pendulum describes the arc MM' by T'.
On account of the smallness of the arc MM', we
may suppose it to be described with a constant velocity it: hence we immediately have
MM'
t'
But the triangles M'CP', MM'O', having their sides
mutually perpendicular, are similar, and give
M'P': M'C    M'O M'O'';
M'O' X M'O
and hence   MM' =    MP
PP' x M'C     Z
MI'P-; 
and [Art. 171     U =  2ygx,




DYNAMICS.                21
I   S
whence          t.
~~~2y x
Now        (M'P') 2= P'K x P'i,
or               y =  (b — x) x (21 —(b-x));
and if we suppose the oscillations of the pendulum
to be made in very small arcs, (b - x) may be neglected in comparison with 21, and we shall have
y =   (b - x) 21.
Substituting this value of y in the expression for t',
we get
is
t2gi.  21 (b —x)
g x (b-_X)
bVg  Vx (b x)
But, applying the result contained in equation [c] to
the circle ANKO, we get
PP' x'AK   PP' x ~AK s_ 
NNP= -NAP' x P'K   Vx (b-x)'
and hence we have t' =  x NN'
Now for the time of describing each of the indefinitely small arcs composing the finite arc BMK, a
similar expression may be found; thus, for t" the
time of describing the indefinitely small are M'M",
we should get
l VI  N'N"
tn ~ - X — ~




22                 DYNAMICS.
Hence if we denote the time of describing the whole
arc BKD by T, we shall have
V/1  are ANK
and         T =I  2ae x   rcANK
V/g    b
=. -..............e..e..e.e..e..e..eea[14]
I being the ratio of the circumference of a circle to
its diameter.
From this expression, it appears that when a pendulum oscillates in very small arcs, the time of an
oscillation is independent of the height AK which
determines the extent of the arc described; and
hence that oscillations of small extent will be isochronous, though the amplitudes should vary.
20. Let I and 1' be the lengths of two pendulums
which oscillate in the times T and T', under the action of gravity of different intensities denoted respectively by g and g'; then we shall have
T =       and T',,
and hence T    T' /
that is, the times of oscillation are directly as the
square roots of the lengths of the pendulums, and
inversely as the square roots of the intensities of
gravity.
~ 1. If the intensities of gravity are the same, we
have




DYNAMICS.                 23
T  T':: v/1: Vl';
and if n and zn' denote the number of oscillations
made by the two pendulums in a given time k, then
k         k
T = -, and T' =
n         nn'
n    n
or         n2:v n2:,
oO ~ ~ ~or  n   n2
and             1='
n2
By means of this equation, we can calculate for
any place the length of the pendulum which will
make a given number of oscillations in a determinate
time, when we know the number of oscillations which
a pendulum of given length makes at the same place
in the same time. Any small error that may be
made in determining the time le, may be rendered
insensible by taking n the number of oscillations sufficiently large. It is thus that the length of the
seconds pendulum in the latitude of the city of NewYork, that is, the pendulum which in that latitude
makes 86400 oscillations in a mean solar day, in
vacuo, has been found to be
39,in'10168, or 3,ft-25847.
~ 2. The measure of the intensity of gravity (ab
ready given Art. 7), may now be found, For this
purpose, we employ the equation




24                  DYNAMICS.
12. 
From it we get  =  T
and making T - 1"/,  = 3,1415926, and 1 = 39,in'10168,
we find     g - 385,in'9183 = 32,ft'1598.
The value of 1, and consequently that of g, has been
found to vary with the latitude of the place.
~ 3. If we suppose the lengths of the two pendulums to be equal, proportion [d] becomes
T: T'::,/g': g;
and denoting by n and n' the number of oscillations
made by the pendulums in any given time, we have
n': n::  g:.  g,
or          g: #:: n            n2;...............[e]
that is, the intensities of gravity at any two places
are to each other as the squares of the number of
oscillations made at the places in any given time,
either by the same pendulum, or by two pendulums
of equal lengths. Hence by causing the same pendulum to oscillate at places in different latitudes, the
relative intensities of gravity at these places can be
obtained. It has thus been found that the intensity
of gravity increases as we proceed from the equator
towards the poles. The relation between the ellipticity of the earth, and the intensity of gravity at
different points of its surface, are such that the one
can be found by means of the other: experiments




DYNAMICS.                 25
with the pendulum thus furnish a mneans of determining the figure of the earth.
The principle expressed in proportion [el, is employed in determining the law of the intensities of
the electrical and magnetic forces.
OF CENTRAL FORCES.
21. Let a material point be supposed to be in motion, in virtue of a primitive impulse, and an accelerating force which constantly solicits it towards a
fixed point. Let the accelerating force be supposed
to act at infinitely small equal intervals, or instants;
communicating to the point, at the commencement
of each instant, an infinitely small velocity. Let C
[Fig. 12] be the fixed point through which the direction of the accelerating force constantly passes; and
suppose the material point to be moving in the line
MN, with a velocity which would cause it to describe
the infinitely small space M'N in an instant. When
the point arrives at M', let the accelerating force be
supposed to communicate to it a velocity which,
were it at rest, would cause it to describe the space
M'G in an instant; then the point will describe, during the instant, the diagonal M'M" constructed on
M'N, M'G. Arrived at M", the point, if left to itself,
would move in M'M" produced, and in the following
instant describe the line M"N' equal to M'M"; but
at M", the accelerating force' again acts; and if we




26                 DYNAMICS.
represent the space which it alone would cause the
point to describe in an instant by M"G', we shall
have M"M"' for the space actually described during
the instant.  It may thus be shown, that as long as
the accelerating force continues to act, the point will
move on the perimeter of a polygon, the sides of'which, being infinitely small and all in the same
plane, form a plane curve. The plane of the curve
is evidently that which contains at the same time
the fixed point, and the direction of the primitive
impulse.
The velocity of the point, at any instant, is the
space which it would describe in the direction of the
element of the curve in which it is then moving,
should the accelerating force at that instant cease to
act.
The line drawn from the fixed point C to -the position of the material point at any instant, is called
the radiuts vector.
22. The areas described by the radius vector, in any
two consecutive instants, are equal. For the triangles M'CMz", M"CN' having equal bases and a common
vertex C, are equal; and so also are the triangles
M"CN' and M"CM"', which have the same base M"C,
and their vertices situated in the same straight line
parallel to the base. Hence the triangles M'CM",
Ml"CM["', that is, the spaces described by the radius
vector in two consecutive instants, are equal. But
the areas in any two consecutive instants being equal,
the areas described in any two equal intervals of




DYNAMICS.                   27
time must also be equal, since each interval contains
the same number of instants; and, hence, generally,
the areas described in any two intervals of time are
proportional to the intervals. Thus when a material
point describes a curve, in virtue of a primitive impulse, and an accelerating force which constantly
solicits it towards a fixed point, the areas described by
the rccdiizs vector are proportional to the times emn2loyed zin
describinig' them.
Conversely, when a material point, which  describes about a fixed point a plane curve, moves so
that the radius vector describes areas proportional
to the times, the direction of the acceleraCrtig force cozstctdly passes throztgh the fi:ced point.
For, employing the preceding construction, the
areas being proportional to the times, and hence the
triangles described by the radius vector in any two
consecutive instants being equal, we have the triangle
but we also have CM'M" = CMG"N';
therefore        CI"/M'" = CM"N',
and hence the line N'M"' which joins the vertices of
these triangles must be parallel to the common base
CM".  But N'M"' must be parallel to the accelerating force which acts at M"; consequently the accelerating force at M" must act in the direction M"/C.
23. Conceive a material point at M  [Fig. 13], to
be connected with a fixed point C by a thread MC




28                 DYNAM1CS.
inextensible and without mass, and suppose it to receive an impulse in the direction MT at right angles
with MC. In virtue of this impulse, it will describe
the circumference of a circle of which the fixed point
is the centre and MC the radius; and during the
motion, the thread will suffer a certain tension in
the direction of its length. But if we suppose
applied to the point an accelerating force equal to
this tension, and constantly directed towards the
fixed centre C, we may consider the string withdrawn, and the point to be moving freely under the
action of the primitive impulse and the accelerating
force. Then as the areas described by the radius
vector CM, in equal times, will be equal [Art. 21],
the arcs described in equal times will also be equal,
and the motion of the point will be uniform; and if
we denote the velocity of the point by v, and by s
the space described during the time t, we shall have
S = vt.
Let MN be the arc described by the material point
during the infinitely small interval 1', and denote the
velocity of the point as before by v; then MN = vi.
The arc MN being infinitely small, the direction of
the accelerating force may be supposed to remain
parallel to MC while the point moves from M to N;
and hence during the time t' this force may be regarded as constant in direction, as well as in intensity.
If then the accelerating force were to act alone
upon the point at M, it would cause it to describe




DYNAMICS.                  29
the line MO during the time I'; and hence denoting
the force by f, we shall have [Equa. 6]
MO=  ft'2
But considering the arc MN as coinciding with its
chord, we have
MO X MD = MN2,
and hence             MO   MN2-;
MD'
and denoting the radius MC by r, and substituting for
MN its value vit', we have
MO = v2t'2
2r
Hence we have      ft'2     2tr'
and                  f =  v.      *. e......    0....[15]
The tension of the string to which the force f is
equal and opposite, is called the centrifugal force, and
f itself is called the central or centripetal force; each
having for its measure the square of the velocity
divided by the radius of the circle.
We have considered the radius CM constant; but
we may suppose it to vary in length, so that the
point shall describe any curve whatever. In that
case the centrifugal force will vary from one point
to another; but at any point of the curve, it will be
equal to the square of the velocity divided by the
radius of the osculating circle. For, the osculating
circle at any point coinciding for an infinitely small




30                   DYNAMICS.
space with the curve itself, the material point may
be considered as moving, at each point of the curve,
on the arc of the osculating circle at that point.
24. To find the relation between gravity and the
centrifugal force, when the material point revolves
on the circumference of a circle, let h be the height
from which a body must fall in order to acquire the
velocity v; then v2 = 2gh, and substituting the value
of v in equation [15],
V2
f =2
we find             f   2k
that is, the centrifugal force is to gravity, cas twice the
height duce to the velocity of the material point, is to the
radiuts of the circutmfereze described by the point.
25. If we denote by T the time in which a material point, moving uniformly with a velocity v, will
describe the entire circumference of a circle of which
the radius is r, we shall have
2~r
V -=;
and substituting this value of v in the equation
V2
r
4qr2r
we get              f=  T2
For a point which describes a circumference of which
the radius is r', in the time T', we in like manner get




DYNAMICS.                     31
f — T12;
and hence we have
r      rf
f'        fi.' T~' T'"f'......[16]
fY: J      T2    T/2 2f
that is, the central forces in the two cases are directly as
the radii, and inversely as the squares of the limes.
26. An illustration of the foregoing results is furnished by the revolution of the earth on its axis.
Assuming the figure of the earth to be that of a
sphere, let PP' [Fig. 14] represent its axis, and the
semi-circumference PEP' the terrestrial meridian of
any place M on its surface.  From the centre C draw
CE perpendicular to PP', and draw MN parallel to it.
Draw also CM.  Then as PEP' revolves, the point E
will describe the equator, and M  a parallel of latitude; and as each circle is described in the same
time, denoting the centrifugal forces at E and M by
f and f', and the latitude of the place by n, we shall
have (proportion 16)
f:  f':: EC: MN,:  EC: CM. cosn,.
or             f: f':  1: cosn,
and hence           f   f. cosn;........... o. [17]
that is, the centrifugal force at any place on the earth's
surfcace, is equal to the centrifugal force at the equator, multiplied by the cosine of the latitude of the place.
27. The effect of the centrifugal force due to the




32                   DYNAMICS.
rotation of the earth, is evidently to diminish the
intensity of gravity at all points of the earth's sure
face except at the poles. At the equator, the centrifugal force and gravity are directly opposed to each
other. Hence if we denote the intensity of gravity
at the equator, as determined by observation, by g;
the intensity which it would have, did the earth not
revolve on its axis, by G, and the centrifugal force
at the equator by f, we shall have
g = G —f, andG-g =+f.
The value of g has been  found to  be equal to
32,f"0861. The general value of f [Art. 25], is given
by the equation
4qr2r
f / =
and for the case in question, we have r, or the equatorial radius of the earth = 20920300 feet; and T
the sidereal day, or the time of the earth's revolution
on its axis = 0,997269 of a mean solar day = 86164"
of mean solar time; and, hence, by substitution, we
get
f = O,2ft.11l12............   o   o.....e e        [a]
We thus have
G- = 32,ft0861 + O,ft1112 = 32,ft.1973,....... [b]
28. Dividing equation [a] by [b], we get
f    O,ft.1112    1              1'G - 32,ft.1973 = 289'           89 henf = 
that is, at the equator, the centrifugal force is the




DYNAMICS.                  33
2% th of the intensity which gravity would have, had
the earth no motion of rotation.
29. At any point of the earth's surface, not on the
equator, the directions of gravity and the centrifugal
force are oblique to each other. Thus at M [Fig. 14],
gravity acts in the direction CR, and the centrifugal
force in the direction NO.  To determine the effect
of the latter in the direction CR, let the centrifugal
force be represented by the line MO, and be resolved
into the components MR, MQ, the one in the line CR,
the other at right angles to it; then denoting the
latitude of M as before by n, we get
MR = MO. cosn.
But [equa. 17]    MO = f' =fcosn,
and hence         MR = f cos2 n;
that is, the diminution of gravity, due to the centrifugal force at any place on the earth's surface, is
equal to the centrifugal force at the equator, multiplied by the square of the cosine of the latitude of
the place.
Experiments with the pendulum show that the
intensity of gravity at the poles exceeds the intensity at the equator by 200th of G: of this increment,
the >Ath is, as we have seen, due to the centrifugal
force; the remainder is accounted for by the spheroidal figure of the earth.
30. As the centrifugal force depends upon the
tiu.e of rotation, It may be proposedl to determine in
E




34                ]DYNAMICSO
what time the earth should revolve on its axis, i
order that the centrifugal force may be equal to
gravity. For the solution of this problem we have
recourse to proportion [16,]
of    f  r    3/r
f. fi::
T2   T,2
In this proportion we make r = r'= the equatorial
radius of the earth, and we suppose T and T' to represent respectively the present and required periods
of rotation, andf and f' = G the corresponding centrifugal forces. We thus get
fv G  T'2   T2,
and hence         T =Jf
or                T'P / _     _ =. T.
Thus the required time is slth of the present period
of rotation.
OF PROJECTILES.
31. A body moving near the surface of the earth,
in virtue of a primitive impulse and the action of
gravity, is called a _projectile. In what follows, we
shall consider the projectile as reduced to a material
point, and the motion to take place in a vacuum.
Suppose then a material point, situated at A [Fig.
15], to receive an impulse in the direction AD, and
then to be abandoned to the action of gravity. Let
the space which it would describe in an instant in




DYNAMICS.                35
virtue of this impulse be represented by AD, and let
the effect of gravity during the instant be represented by AE; then at the end of the first instant,
the point will be found at B, having during the instant described the diagonal AB of the parallelogram
constructed on AE and AD.
Again, on AB produced, take BG- AB; and on
BG, and the vertical BF = AE, construct the parallelogram BFCG: at the end of the second instant
the point will be at C, having described the diagonal
BC. In like manner, during the third instant, the
point will describe the side CH; during the fourth,
the side HI, and so on. But each diagonal being
infinitely smalll the series of diagonals forms a curve;
and since each of the parallelograms has its contiguous sides in the vertical plane which contains the
preceding parallelogram, all the points of this curve
are in the same vertical plane. This curve is called
the tra/ectory of the material point.
32. To find the equation of the trajectory, through
the point A [Fig. 16] from which the material point
is projected, draw the axes AX, AY, horizontal and
vertical respectively. Let AK be the direction of
the primitive impulse, and denote the velocity due
to the impulse by v. Let the curve described be represented by AMC, and let M be the position of the
point at the end of the time I. Draw MP perpendicular to AX, and produce it to M'. Put AP = x,
PM =y, and the angle M'AP =i. Then let the




36                    DYNAMICS.
initial velocity v, represented by AM", be resolved
into the components
AP' = v cosi, and AQ = v sini.
The material point may evidently be regarded as
having two motions, the one parallel to AX, the
other to AY.  The motion parallel to AY  is the
same as that of a body projected vertically upwards
with a velocity v sin i; and hence we have [Art. 9],
y - v sin e t - ~yt2................. [18]
The other is due to the horizontal component v cos i,
and gives
or                   t =
V COS i
Substituting this value of I in equation [18], and putting 2gh in place of v2 [Art. 8, equa. 9], we get
4h1y cos2 i = 4hix sin i cos i - x;.......... [19]
and hence,
x = 2h sin i cos i - V4h cos" i (h sin2 i -).  )... [20]
33. The distance from  the origin A to the point
C, where the curve intersects the horizontal line
AX, is technically called tl/e rcange.  To determine its
value, in equation [20], we make y = 0: we thus get
AC = 4h sin i cos i
= 2h 2sin i cos i
= 2h sin 2i...........,... [21]
This is the general expression for the range.




DYNAMICS.                 37
When i = 450 or 2i = 90, we have
AC = 2k.
This is evidently the maximum value of AJC. Hence,
for a given initial velocity, the range is greatest
when the angle of projection is equal to 45.0  Denoting this value of the range by B1', we have
R'  2k,
and hence            L =;
from which it appears that the height due to the
initial velocity is equal to half the maxiimum range.
Hence, to determine h, we have only to measure the
range when the angle of projection is 45~.
34. Resuming equation [21], and denoting AC by
B, we have
R = 2h sin 2i;
and substituting for 2h its value A', we have
i = BR' sin2i,
an equation which gives the range corresponding to
any angle of projection, when the maximum range
for the same initial velocity is known.
35. Let us now suppose two material points to be
projected with equal initial velocities, and denote
the angles of projection by i' and i", and the corresponding ranges by A"' and A'IV  We have, equa. [21],
I"' = 2h sin2i',
RI = 2k sin 2i";




DYNAMICS.
and if i" = (90~ - i'), we get
RIV = 2h sin (180~ - 2i')
= 2h sin 2i',
and               R1"' = RIV;
that is, when two material points are projected with
equal initial velocities, at angles of projection which
are complementary to each other, the range is the
same for each.
36. From equation [20], we learn that the greatest
elevation of the point, or the maximum value of y, is
h sin2 i; for if y be greater than h sin2i, the radical
becomes imaginary. The value of x corresponding
to the maximum value of s, is 2h sin i cos i. Supposing AE and ED to be these values, we have
AE = 2h sin i cos i,
ED = hj sin2 i.
37. IfU when the initial velocity is given, it be required to determine what must be the value of the
angle of projection, in order that the projectile may
reach a given point N, for which we have
x = AQ' = x', and y = Q'N = y,
we have recourse to equation [19]. Substituting in
this equation the given values of x and y, and dividing each member by cos2 i, we get
sin i    X/2
4hyl _ 4hzxi
coS i   coS2 i
and since     -  = see2 i = 1 + tang2 i,
we have   4hy' = 4hx' tang — x' - x'2 tang2 i,




DYNAMIcs.                  39
4h        _ 4ayn — x'2
Or.         tang2 i --   tang  -- x
2h Z     4h2 _ 4hy' _ X12
and hence    tangi = 2ht  4 —4h y' —x'2
We learn from  this equation, that in order that
the problem may be possible, we must have
4hy' + x'2 < 4h2;
and that when this condition is satisfied, there are
two directions in which the projectile may be
thrown, so as to reach the given point.
38. The time employed by the projectile in describing any portion of the trajectory, may be found
from the equation
x = V coS i t.
39. In equation [19], the points of the curve are;
referred to the axes AX, AY. Let us now refer
them to the axes DE and DY' [Fig. 17], drawn
through D, the highest point of the trajectory, parallel to AY  and AX  respectively.  Denote DR, the
new abscissa of the point M, by x'; and MR, the
new ordinate, by g': we have [Art. 36]
x = 2h si-ni cos i - y,
and             y= h sin2 i -- x.
Substituting these values of x and y in equation [191
4y cos2 i = 4hx sin icosi — x2,
we have
4a ( sin2 i - x!) cos2 i = 4h (2h sini cosi - y') sini cosi'
- (2h sin i cos i - yl)2;




40                      DYNAMICS
or, reducing,
41/2 sin2i 2os2 i - 4-  os2i x' -- 812 sin2  eos2 -- 4h sill i cosi y
-- 4h2 sin2 i cos2 i q+ 4h sinzi cosi y' - y' 2
and, by a farther reduction,
y/ 2 _ 4h cos2i xI.
From  this equation, it appears that the trajectory
is a parabola, having its vertex at D, and of which
the parameter to the axis is 4h cos'2:i.
MEASUIJE OF FOtRCES.
40. In article 8 of Part First, we have shown how
the intensities of forces may be represented by algebraic symbols; and assuming the forces to act upon
single material particles, we have also shown that
the spaces described  by  different  particles in  the
same time, or their velocities, are directly  proportional to the intensities of the  forces; and, conversely, that the forces are directly as the velocities.
Thus denoting the forces which act separately upon
two material particles by f andf', and the velocities
which they communicate to the particles by v and v',
we have shown that
f. f'  v~    VI.....        [a]
10 Let us now  substitute for the particles, two
"' That the trajectory is a parabola, may also be shown by referring
the points of the curve to the oblique co-ordinates AX. AK. The
method in the text is employed, because it illustrates the simplifcatiomn
which frequently results froni a change of co-ordinates,




DYNAMICS.                41
bodies of equal masses, each body consisting of 1f
particles; and let us suppose each particle of the
first to be acted upon by a force equal to f, and each
particle of the second by a force equal to f'; the
impulses being supposed to be given in the same
sense, and in parallel directions. The bodies will
evidently move with the velocities v and v' (all their
particles describing parallel straight lines); and
their motions may be considered as due directly to
the resultants, equal respectively to 3f and 3f/f', of
the Al equal and parallel components which act
upon the particles of each body. Denoting these
resultants by Q and Q', we have
Q - _f,arid Q' =  ff';
and hence
Q: Q,    M: lf   f,::   f ~  f'
or, equa. [a], Q:       vQ::: v'..........0....[b]
Thus the intensities of any two forces are to each
other as the velocities which they would respectively
communicate to two bodies of the same mass; or,
more briefly, when the masses are equal, the forces
are as the velocities.
2~ Referring again to the two bodies, let us suppose them to have unequal masses, the one to have
a mass mn, the other a mass 31; and let us conceive
each of their particles to be acted upon by a force
equal to f. The bodies will evidently move with
the common velocity v; and their motions may be
considered as due to the resultants, equal to mf and




42                 DYNAMICS.
Mf respectively, of the parallel components which
act upon their particles. Denoting these components by F and Q respectively, we shall have
F:  Q::f: Mf
m    ]........... [c1
Thus any two forces are to each other as the masses
to which, if respectively applied, they would communicate equal velocities; or when the velocities
are equal, the forces are as the masses.
30 Having ascertained the relation between the
intensities of two forces, lst, When the forces communicate unequal velocities to equal masses, ana,
2d, When they communicate equal velocities to unequal masses, we will now consider the general case,
that in which the forces communicate unequal velocities to unequal masses.
Let _F and F' denote the forces, m and m' the
masses, and v and v' the velocities respectively. Also
let Q and Q' denote two auxiliary forces, whose intensities are such that they would communicate to
two bodies of the same mass 31 the velocities v and
v' respectively.
We have [1~ of this Art.],
Q: Q.  v: V,.
We also have [20 of this Art.]
F- Q-, m        M,
F  Q'    m'   M;
and eliminating the quantities Q, Q' and M1 we gec
P                oil PV  4 a 0 t)'s 61d]~:~nQBB~~~




DYNAMICS.                  43
The product of the mass m of a body by its velocity v, is called the quantity of motion of the body. Employing this term, the proportion just found may be
thus enunciated:
f/wo forces are in general to each other as the quantities
of motion which they would respectively generate in any two
bodies during the same time.
4~ If in proportion [d] we suppose mn' to become
the unit of mass, and v' the unit of velocity, F' will
become the force which will communicate to the
unit of mass a velocity equal to the unit of velocity;
and if we take it for the unit of force, we shall have
E' = mv, a s o o e o * 0 * a e a a eO  * o e * e o * a 0 Ee]
my being the ratio of the force F to the unit of force.
50 If in proportion [d] we suppose F = F', we get.m  = m'v', and hence
v =-   o..eO............es....[h]
Thus the quantity of motion m'v' due to a force F
being given, to find the velocity which the same
force would communicate to a body Kwhose mass is
m, we have only to divide the given quantity of motion by this mass.
6~ The preceding relations are evidently true,
whatever the value of v: whether it be a finite velocity, due to a force of the kind called impulsive; or
an infinitely small velocity, such as we conceive a
force acting at infinitely small intervals, like gravity,
to communicate at the commencement of each interval; or, lastly, whether it be the finite velocity pro



44                  DYNAMICSo
duced by the latter kind of force, that is, an accelerating force acting during a given time.
Reserving v to denote the velocity due to an impulsive force, and employing g to denote the velocity
due to an accelerating force acting uniformly during the unit of time, we shall have, for the two cases,
F = my,. we....a.. =.. e. [k']
If in these equations we make n = 1L we get
F= v, F= 9;
and F becomes in each case the force which acts
upon a material particle, or the unit of masso  For
the sake of brevity, q is commonly called ithe acceleraling force; and mg the general expression of F in
equation [k'], is called Mte qoving force.
41. 2Talure of an inzpulsive force.
We learn from  experiment, that in all cases of
collision, the bodies concerned suffer a sensible compression, of greater or less extent, according to their
degree of hardness.  This compression is evidently
effected in a finite, though very short time, and by
infinitely small degrees. Hence in the production
of motion by an impulsive force, we may conceive
the transmission of the motion from the one body to
the other also to take place by infinitely small degrees. Thus in this case, as in that of an accelerating or moving force, we may conceive the time during which the force acts, to be divided into an infinite number of instants, and suppose an infinitely
small velocity to be communicated at the commence



DYNAMICS.                 45
ment of each instant; the essential difference in the
two cases being that the infinitesimal velocities communicated by the impulsive force must be supposed
to be vastly greater than those communicated by
the moving force. An impuldsive force macty therefore be
regarded as a imoving force, acltin for a very short time, with
a very great intensily. Since then the finite impulse
given by an impulsive force may be supposed to be
made up of an infinite number of infinitely small
impulses, we may simplify our first notion of the
mode of action of a force (Art. 2 of Statics), and say
that the action of a force may always be conceived
to consist in communicating to the particles of the
body on which it acts, infinitely small impulses of
greater or less intensity.
42. To give a simple example of the application
of some of the preceding results (Art. 40), let mn and
mn' denote the masses of two bodies, suspended at the
extremities of a cord which passes over a fixed
pulley, and let the velocity which gravity will communicate to the bodies during a unit of time be denoted by g. The moving forces put in operation by
this disposition of the bodies will be mny and n'gq;
and supposing in > in', their resultant will be mng —n'g.
This resultant, acting upon the sum (m + in') of the
two masses, will cause the one body to descend, the
other to ascend; and the velocity g' which it will
communicate to the system in the unit of time, will
be given (50 of this Art.) by the equation
m - m'
M +M= + g.
m -~ml




46                  DYNAMICS.
The simple arrangement just considered, is the
skeleton of a machine called, from the name of its
inventor, Atwood's machine; in which, by rendering
the difference between the masses sufficiently small,
the motion of the bodies is rendered so slow that the
velocity acquired, and the space described, in any
given time, can be determined by actual observation.
It has thus been found that bodies near the surface
of the earth fall with a uniformly accelerated motion,
and hence that terrestrial gravity is a constant accelerating force.
PRINCIPLE OF D ALEMBERT.
43. In the preceding sections, we have constantly
supposed the body whose motion was the subject of
investigation, to be reduced to a single material
point. We purpose now to consider some [of the
more elementary cases of the motion of systems of
points, or bodies of sensible magnitude. But the
student must first be made acquainted with a general
principle of great utility in the solution of problems
of dynamics, called, from the name of its discoverer,
the principle of D'Alembert.
Let A, B, C, D, etc. [Fig. 18], be a system of material points, connected with each other in any manner whatever, and acted upon by the accelerating
forces f, f', f", f"', etc. respectively.  The velocities
which the points will actually acquire in an infinitely
small time or instant, under the action of these forces,
will, in consequence of their mutual connexion, be
different in both intensity and direction from those




DYNAMICS.                    47
which they would acquire were they free. Thus if
the velocity which f would communicate to A in an
instant if that point were free, be represented by AC,
the velocity actually acquired by A will, in consequence of the connexion of the points of the system,
be represented by some other line AD. Let the velocity AD be considered a component of AC, and let
the parallelogram  {BCD be completed: we shall then
have the velocity A.C resolved into the two velocities
AD and AB. To distinguish these velocities from each
other, we call AC the imprnessed velocity, AD the efeclive velocity, and AB the velocity lost.  Let the velocity
which f would co%?m municate to A in the unit of time
if it were free, be denoted by u; then the velocity
which it would conmmunicat,e in the instant 1', or the
impressed velocity, will be expressed by ut'; and if
we denote the components of iu by q and p, the effective velocity and the velocity lost will be expressed
by qt and pt respectively.  It moreover we denote
the mass of the point A by Rev then for this point
0-ile impressed quantity of motion will be expressed by mout';
the effective qncftity of motion by............... mqt',
and the quantity of motion lost by.............. mt'.
If for the points B, C, D, etc., we denote the quan-;ities corresponding to t, q, p and m, by these letters
accented, t has; is by I'U/, q m' p', i; u", q", p", m", etc.,
the several qa!ii.tes of motion for the entire system
will be tbh.uS expressed:
The impressed quantities of motion by
mut', mu'tt', m"u" t', etc.;.......... [a]




48                   DYNAMI1S.
the effective quantities of motion by
mqit, m'q't', m"q"t', etc.;........   o  [b
the quantities of motion lost by
rnpt', m'P t', 2n'p"t,', etc. e *   [C]
Now since the impressed quantities of motion are
reduced, in consequence of the mutual connexion of
the points, to the effective quantities of motion, it is
obvious that the Cquantities of motion lost mnust be in equilibriumw anmong the~mselves.  In this consists the principle
of D'Alembert. The manner in which the equilibrium will take place, will of course depend upon
the nature of the system.
44. This principle may  also be enunciated in
terms of the impressed and effective quantities of
motion. To get this latter enunciation, produce
DA: and taking AD' equal to AD, complete the
parallelogram  ACBD'.  We perceive that AB the
velocity lost is the resultant of AC the impressed
velocity, and AD' the effective velocity taken contrary to the actual direction of the motion. Thus
for the point A, the quantity of motion lost may be
resolved into the impressed quantity of motion, and
the effective quantity of motion, the latter being
taken contrary to its actual direction. The same
being true for all the points of the system, we may
substitute for the quantities of motion [c], the quantities of motion [a] and [b], the latter [b] being taken
with the above modification.  But the quantities of
motion [c] are in equilibrium among themselves:
consequently an equilibrium must also exist between
the quantities of motion [a] and [b], the latter being




DYNAMICS.                 49
taken contrary to the actual motions.  Thus, in any
system whatever, there will be an equilibriumn between the
impressed and efective quantities of motion, the latter
being taken contrary to the actual motions; regard being
had, in forming the equations of equilibrium, to the
nature of the system.
This enunciation might have been obtained independently of the former; for the truth of it is obvious, the moment the terms impressed and efective quantitles of motion are understood in the sense in which
they are here used. The preceding method has
been employed to ensure a greater familiarity with
the principle, than could be acquired by regarding it
from a single point of view.
We have supposed the particles of the system to be
acted upon by accelerating forces only, but the principle is equally true when the forces are impulsive.
MOMENT OF INERTIA.
45. The sum of the products obtained by multiplying the masses of the particles of a body by the
squares of their respective distances from any line
whatever, is called the moment of inertia of the body
with respect to the line.
The moment of inertia of a body is represented
by the expression m(r2m); in which m denotes the
mass of a particle of the body, and r the distance of
the particle from the axis with respect to which the
moment is taken. When the moment of inertia of
a body with respect to an axis which passes through
G




5f03              3L~~~DYNAsICS.
its centre of gravity is known, its moment of inertia
with respect to any other axis can be readily determined.
Let C [Fig. 19] be the centre of gravity of the
body, and FF' an axis passing through it, with respect to which the moment of inertia of the body is
known: and let its moment of inertia with respect
to any other axis KK', parallel to FF", be required.
Let the point C be taken for the origin of co-ordinates, and the axis FF' for the axis of z; and let CX
and CY, the axes of x and y, be drawn. Let N be
the place of any particle of the body, and draw
through it the plane NKF parallel to the plane of
zy, and meeting the axes of moments in the points
F and K. Draw NE perpendicular to the plane of
zy, EP and GD perpendicular to the axis of x, and
ED' parallel to that axis; and join the points C and
E, C and G, E and G. Let CD and DG, the co-ordinates of G, be denoted by a and F; CP and PE, the
co-ordinates of E, by a and y; and CG, the distance
between the two axes, by a. Also let NK = EG be
denoted by r, and NF = EC by r,. We have
CG2 = CD2 + D3)G2, and CE2 -2 CP2 + PE2 
or      a2   2 +    anld r,2 - - 2  y..........[a]
We also have
EG.'2 - /G7,
or               r   (   )  (a 2y    ) 
= ziz 5x. 9  - - __,J _  2 > l
orn, 2eduang bgy entlim IVE.




DYNAMICS.                    61
r2 = r2 - 2ax ~- 2y + a2;
and multiplying each member of this equation by
the mnass m of the particle, we get
T2m   r 2r — 2axm - 2fym q- a2m.
If, for other particles of the system, we denote the
quantities corresponding to r, r,, x and y, by these
letters accented, we shall have
ra2n: r,,2m -- 2cx'm -- 2ySn +- a2m,
11'2m = r,,,2  -m 2UX//m 2-13y"m' +  a2m,
etc.;
and adding these equations, supposed to extend to
all the particles of the body, we shall get
(r2m") = z(r,2M) - 2sot(xm) -- 2~;(ym) + a2:m.
Now z(x@m) is the sum of the moments of the masses
of the particles with respect to the plane of yU, which
passes through the centre of gravity of the body;
and hence we have z(xnm) = 0. For a similar reason,
we have Iy(nv) = 0. Also zm expresses the sum of the
masses of all the particles of the body, or its entire
mass A. Consequently we have
1(r2M) -= 1(r,2n) + a2A... e    [oeeo  b]
In this equation, the term =(r,2rn) expresses the moment of inertia of the body with respect to the axis
FF' which passes through its centre of gravity; and
z(r2nm) expresses its moment of inertia with respect to
the axis KK' parallel to the first axis, and at a distance from it equal to a. Hence the moment of inertia
of a body with resaoect to any axis, is equal to its moment of




52                   DYNAMICS.
inertia with respect to an axis passing through its centre of
gravity, parallel to the first, plus the product of the mass of
the body by the square of the distance between the two axes.
The second member of equation [b] may be put
under the form
w((r,2m)+2
Hence we have
Z(r2m)  ((r) + a2);
or, denoting
(r,2M) by k2,
1(r2m) = M(k2 + a2.)................. [c]
46. The determination of the moments of inertia
of particular bodies requires in general the use of
the integral calculus, but in many cases they can be
found by more elementary processes.  We will give
a few examples.
1~ The straight line.
Let it be required to find the moment of inertia
of a material straight line, with respect to an axis
which passes through the centre of gravity of the
line, at right angles to it.
Let AB [Fig. 20] be -the line, and C its centre of
gravity; and let it be supposed to be divided into
2n indefinitely small equal parts or elements, by sections perpendicular to its length. Let its length be
denoted by 1, and the area of a section by s. The




DYNAMICS.                      53
volume of an element will be expressed by ~s; and
as the mass is here supposed to be proportional to
the volume, we may also take this expression as the
measure of the mass of the element.  Moreover the
respective distances of the elements of BC (half of
the line) from C, that is, from the axis of moments,
will be expressed by
1     1     1                  1
-72. -)3                   n
2n' 2.n'  2n'                 n
Consequently the moment of inertia of BC will be
sl  (1  )2      s    212               sl  (nl)2
2'   +             ~n +  +~. ~n,           2n
or          2n'
s 1    n 1 3 n2   n 
~or        8-~ I;  T3 +  2 + 6'
*To find the sum of the series
5a2   22 + 32 4+ 4-  bbo  n2
in the identical equation
(X-  )3 = (X -1)3,
or           (x - 1)3 = 3    32 + 3X —:
substitute successively for x each term of the series
1, 2, 3, 4,.      e.....  e.. n.
We shall thus get the following equations:
0 = 13  3. 12 + 3.1 -- 1,
13= 233.22 + 3.2 -1,
23  = 33   3. 3a + 3.3-1,
(n —- 2)3 = (n-1)3 - 3 (n — 1)2 + 3 (n —  1) —1
(n —1)3 =n3 —3n2 + 3n — 1;
and if we add the corresponding terms, cancelling those which are common, and denoting the sum of the series 1 + 2 + 3 + 4...... + n




54                    DYNAMICS.
or (since wb must suppose n  — om   )
siZ3
24
Hence the moment  of inertia of the entire line will
sl3
be expressed by  o2r; or, denoting the mass of the
line by   1,f by M 42-o
The nmaoent of inertia of AB with respect to any
other, axis parallel to the first, and. a aa dstance from
it denoted by a, will be e-xpressed VA-t. 4r5A by
/ 1
20 fhe relctangle.
Let it be required to determine the moment of
inertia of an infinitely thin rectangular plate.
Ist. Let ABDE [Fig. 21] be the plate, and let the
axis HI with respect to which the moment is to be
taken be supposed to pass through C the centre of
gravity of the plate, and to be parcdlel to one of its
sides, as AB.  Conceive the plate to be  nade up of
by s', and that of the series 12 + 22 + 32 + 42......    by
s", we shall have
O0  n3     s" - 3 s'   2;
and hence
n3       n
s" - + s'/
3    3
=    + ( +   n   r
3,2   6
0+2+




DYNAIMICS.                  55:naterial lines perpendicular to HI.  The monient of
inertia of each line with respect to this axis will be
expressed by 11, i12  denoting the mass of the line
and I its length. Hence if we suppose JV to denote
the sunn of the minasses of the lines, or the mass of the
enltire plate, the moml0ent of inertia required will be
expressed by bM 12
2d. Let the axis be supposed to pass through C,
at rigihi angyes to I/e sturface of Itie plate. As in the preceding case, conceive the plane to be made up of
material lines or elements parallel to the side AE:
and let the number of these elements, or the indefinitely small equal parts into which we suppose the line AB to be divided, be denoted by 2n.
Let AB and AE be denoted by 6 and I respectively,
and the thickness of the plate by i. The volume,
and hence the mass of each element, vill be mea61i
sured by;-n and the moment of inertia of an element will be expressed by
-               112
being cany -na-mber of the n.Atlral series. Hence
the su-m of thee moseomsents of inertia of all the liniL es
CoMnP081img Ihe en:- ectangee Ai'".    will 1e
7r     $.b, r    5'~ @__!6 23 2  




56                 DYNAMICS.
or    bi-   + () (12  + 22 + 32 + 2+
or   2E+ (2 ).(- +-+ 6))
bli /(n12    b 2 1 3   2 n   n))
or    2n 12~(2n)    
bli   12  b2 12 + b 2
or                               I
bli 12~62
or             2 -- 12
Consequently the moment of inertia of the entire
rectangle, its mass being denoted by M and its diagonal by d, will be expressed by dM 12
30 The rectangular parallelopipedon and triangular
prism.
Since a rectangular parallelopipedon [Fig. 22] may
be supposed to be made up of a series of equal rectangular plates, its moment of inertia with respect to
an axis CC' passing through the centres of gravity of
any two of its opposite faces will be expressed by the
formula M 12; Mbeing supposed to denote the mass
of the solid, and d the diagonal of either of the faces
to which the axis is perpendicular.
The same formula also expresses the moment of
inertia of either of the equal triangular prisms into
which the parallelopipedon is divided by the diagonal plane ADD'A', with respect to the axis CC'
which joins the middle points of the hypothenuses
AD and A'D' of their triangular bases; M being
supposed to denote the mass of either of the two
prisms, and d the hypothenuse AD or A'D'.




DYNAMICs.                 57
THE COMPOUND PENDULUM.
47. The simple pendulum has only an ideal existence. It is essential therefore, to the practical
application of the results obtained in articles 19 and
20, relative to this pendulum, that the relations
between it and the compound pendulum should be
known. These relations we are now to investigate.
Let a compound pendulum be conceived to oscillate about a horizontal axis. During its motion, the
material points, or particles of which it is composed,
will describe arcs of circles, the planes of which will
be perpendicular to the axis. Let OGFHI [Fig. 23]
be one of these planes, and 0 the point in which it
is intersected by the axis; the axis being supposed
to be perpendicular to the plane of the paper.
The pendulum being in motion, suppose that
at the expiration of the time 1, the particle which
describes the are HFG has arrived at the point A,
having during that time described the arc HA. Denote by c the gzgtZlar velocity of the whole system at
this instant; that is, the velocity of any point, as I,
at the distance from the axis which we assume as
the linear unit; and denote the distance OA by r:
then the absolute velocity of the element at A will
be expressed by ra. In the following instant, this
velocity will be increased by the action of gravity.
Let y the intensity of gravity be represented by the
vertical line AD, and be resolved into the two components AE and AB, the one in the direction of OA,
and the other perpendicular to it. The first being
H




68                  DYNAMICS.
destroyed by the re-action of the fixed point C, we
have to consider only the latter. This component
causes the angular velocity to vary, and, if we represent the angle BAD by 6, will be expressed by g coss.
If we denote the increment which the angular velocity receives during the instant t', by i', the corresponding increment of the velocity of the particle
at A will be expressed by rw'; and at the end of the
time t + I', the velocity of the particle will be expressed by ra- + rco'.
If the element at A were unconnected with the
other particles of the system, the accelerating force
g cos  would communicate to it, during the instant
t,; the velocity g cos tI'; and at the end of the time
t + t', its velocity would be rw + y cos1 t'; the direction of the Velocity being AB, the same as at the
end of the time t.
Now these increments of velocity g cosa t' and ro'
are what we have called, in Art. 43, impressed and
efective velocities respectively. Hence denoting the
mass of the particle at A by m, the impressed quantity of motion of this particle will be # cos  tI'm, and its
effective quantity of motion rw'm. All that we have
shown to be true of the particle at A, being equally
true of all the other particles of the system, we shall
have for the impressed quantities of motion of the
whole system a series of terms each of the form
g cos6 t'mn, the sum of which, employing the usual
synlbol, may be expressed by
I(g cos 6 t'm).




DYNAMICS.                 59
We shall also have for the effective quantity of motion of the system a series of terms of the form rwtm,
the sum of which may be expressed by. (rwlm).
Now according to the principle of D'Alembert
[Art. 44], these two quantities of motion will be in
equilibrium with each other, if the latter be taken
in directions contrary to the actual directions of the
motions. The case then is exactly similar to that of
Art. 40 of Statics; and the condition of equilibrium
is, that the sum of the moments of the forces which
tend to turn the system in one sense about the fixed
axis, must be equal to the sum of the moments of
the forces which tend to turn the system in the
opposite sense, the moments being taken with respect
to the axis itself.
To obtain the moments of the forces or quantities
of motion, we have only to multiply the quantities
of motion (impressed and effective) of each element,
by the distance of the element from the axis. We
shall thus get for the equation of equilibrium,
Z (r2z m) = Z (rg cos6 t'm);
or since t' is common to all the terms of the first
member, and I' and g to those of the second,
WII (r2m) = gt'tF(r cos  m);
and hence we shall have
co'   I(r cos Z m)




60                 DYNAM1CS.
Conceive now a vertical plane to be drawn through
the fixed axis, and perpendiculars to be drawn to it
from all the particles of the pendulum. Let AN =y
be the perpendicular drawn from the particle at A;
then we shall have
y = r cos6;
and hence, substituting y in place of r cos  in the
preceding equation, we shall get
Wi l    (ym)
t/  Z(r2m)                     [a]
The expression i(ym) represents the sum of the
moments of all the particles of the pendulum with
respect to the vertical plane drawn through the axis;
and if we denote by y' the perpendicular drawn from
the centre of gravity of the pendulum to this plane,
and by 3 the entire mass of the pendulum, we shall
have [Statics, Art. 45],
y(ym) = y'l;
and hence
y'M
t' -   ( 2.     e
Let C [Fig. 24] be the centre of gravity of the
pendulum, and O'G'F'lH' the plane drawn through it
perpendicular to the axis, and let 0' be the point of
intersection of the plane and axis; then the line CN',
drawn perpendicular to the vertical 0'F', will be the
line which we have denoted by y'; and denoting
O'C by a, and the angle 0'CN' by 6', we shall have
y' = a Cosa 61




DYNAMI>CS.                61
We also have [Art. 45],
(r2m) = ) (a2 + 72).
Hence, by substitution, equation [b] will become
WI   g#a cos 6 M   yga cos8 
-t -1 (a2 + k2) - a2 + k2'
The second member of this equation is obviously
[Art. 10] the expression of the angular accelerating
force for the whole pendulum.
Let us now seek the expression of the angular
accelerating force for a single particle, situated at R
on O'C produced; O'R being supposed to be without
mass, and inextensible and inflexible, so that the
line and particle together constitute a simple pendulum.
On the supposition that the compound pendulum
is reduced to a single particle, equation [a] becomes
tl  ram;
and applying this result to the particle at R, and
denoting the line O'R by 1, we get'/  gi cos ~;
tI    12
g cos&'
The second member of this equation is the expression required.
48. If now it be proposed to determine the value
that must be given to 1, in order that the simple
pendulum (or the point at R) may oscillate in the




62                 DYNAMICS.
same time as the compound pendulum, we have
only to put the expressions of the accelerating forces
in the two cases equal to each other. We thus get
ga cos J'  g cos'
a2 +1 2-   1
a     1
and hence
a2 +- k2
and                 1
a
k2
- a - +.
a
49. To apply this formula to a particular case, we
must determine the mass X of the given pendulum,
and the distance a of its centre of gravity from the
axis of suspension. We must also calculate the moment of inertia of this mass, with respect to the axis
passing through the centre of gravity parallel to the
axis of suspension. Then dividing this moment by
the mass, we shall have the value of k2; and substituting the values of a and k2 in the second member
of the equation, we shall have the length I of the
simple pendulum which will oscillate in the same
time with the compound pendulum.
If through R, considered as a point of the compound pendulum, a line be drawn parallel to the
axis of suspension, all the points of the line will
obviously oscillate in the same manner as the point
R. This line is called the axis of oscillation of the pendulum, and the points 0' and R are called the centres
of suspension and oscillation respectively.




DYNAMICS.                  63
50. The axes of oscillation and suspension are reciprocal;
that is, when the axis of oscillation is made the axis of suspension, the latter becomes the axis of oscillation.
For let AB [Fig. 25] be the section of a compound
pendulum by a plane passing through its centre of
gravity, perpendicular to the axis of suspension;
and suppose C to be the centre of gravity, and O'
the centre of suspension. To determine the length
of the corresponding simple pendulum, we have the
equation
Z=a+;
a
or, since a = OTC,
= o' +  PO
O'Cr
If then on O'C produced we lay off CR equal to n/0
the point R will be the centre of oscillation.
Now let the pendulum be inverted, and the point
R [Fig. 26] be taken for the centre of suspension:
then denoting the length of the corresponding simple
pendulum by 1', and the distance between the centres
of gravity and suspension by a', we have
k2
a = a' +V
k2
siCR + c 
or, since       C R   /C2




64                    DYNAMICS.
k2    k2
O'C + k2
O'C
Hence              1 ='.
Thus the lengths of the corresponding simple pendulums are the same, and the oscillations about the
axes of suspension and oscillation are made in equal
times.
51. Conversely, if the duration of the oscillations of a
compound penduluzm  is the samne about two parallel axes
which lie in the samce plane with the centre of gravity of the
pendulum, and at unequal distances from that point, the distance between the axes will be equal to the length of the simple pendulum which will oscillate in the same time.
For representing by AB [Fig. 27] the section of
the pendulum by a plane which passes through C its
centre of gravity, and supposing the points 0' and 0,
to be made the centres of suspension alternately, we
have for I and 1', the lengths of the corresponding
simple pendulum,
ik2            k2
l=a+-  and l' = a' +  -,
a              a"
a and a' denoting the distances O'C and O,C respectively. But if the oscillations of the two pendulums
are made in the same time, we shall have
k2      ik2
a   -- = a' +,
fa        a      n"
from which we get for a' the values a and -.   But
a




DYNAMICS.                  65
0'1, the distance between the axes is equal to the
sum of a and a'; hence we have
0'0, = a + a';
or, employing the latter value of a',
k2
00, = a + -
Hence 0'0, is equal to the length of the simple pendulum which oscillates in the same time as the compound pendulum.
OF THE COLLISION OF BODIES.
52. We learn from experiment, 10, that all bodies
in nature are more or less compressible; and, 20,
that when in any case a body suffers compression, it
tends in a greater or less degree to resume its primitive figure.  The tendency of a body, when compressed, to resume its primitive figure, is called its
elasticity. The elasticity of a body varies with the
nature of its substance. The limiting cases of the
elasticity of bodies are, Ist, the case in which the
body tends to resume exactly its primitive figure;
and, 2d, that in which it has no tendency in any
degree to recover its figure. In the first case the
body is said to be perfectly elastic; in the second, to
be inelastic. Neither of these cases is found in nature:
no known substance being either perfectly elastic,
or entirely destitute of elasticity. All known subI




66                 DYNAMICS.
stances are hence said to be imzperfectly elastic. There
are, however, some substances in which the elasticity
approaches very near to the extreme cases.
53. Inelastic bodies.
Let A and B [Fig. 281, be two inelastic hornogeneous bodies of spherical form, moving from left to
right on the straight line A'B' which joins the centres C and C'; and let A be supposed to have the
greater velocity. When A overtakes B, a mutual
compression will commence, which will continue for
a very short time, till the bodies have acquired a
common velocity. The two bodies will then cease
to act upon each other, and will move with the common velocity as one body.
Let the masses of A and B be denoted by m and m',
and their velocities before meeting by v and v' respectively, and let the common velocity which they
have after meeting be denoted by u: then the velocity lost by A will be v - u, and the velocity gained
by B will be u - v', and the quantities of motion lost
and gained will be mn (v- i) and m' (t — v') respectively. Consequently, since by the third law of motion the force or quantity of motion lost by A must
be equal to that gained by B, we shall have
m (v - u) = M' (U - v');
and hence
my + m'v'
The= equation for the case in which. the two bodies
The equation for the case in which the two bodies




DYNAMICS.                  67
move in opposite directions, may be immediately
deduced from equation [a] by making v' negative.
We shall thus have, for this case,
mV - m/v'
% = -    -.............    o e..... [b] 
m + m'[b                      ]
54. Applicction of equedions ta] and [b] to particzular
cases.
10 Let one of the bodies, as B, be supposed at
rest: then v'= 0, and we shall have
rev
Ub = - -'
m + m'
In this case, we perceive that as mn' increases, v
and m being supposed to remain the same, u diminishes; and that when mn' is so great that it may be,considered infinite with respect to an, u is zero.
2~ Let the bodies be supposed equal in mass;
then m = m'; and accordingly as they move in the
same or in opposite directions, we shall have
U= - ( + vI), or   U = 1 (v  VI).
30 Let the bodies be supposed to be equal in
mass, and one of them, as B, to be at rest: then
= Mn' and v'-= O and we shall have
55. Elastic bodies.
When a perfectly elastic body of spherical form
impinges at right angles upon a fixed plane, its velocity is gradually diminished, and, when the compression reaches its limit, is reduced to zero; but the
instant the compression is completed, the body, in




68                  DYNAMICs.
virtue of its elasticity, begins to resume its primitive
figure, and, in the operation, acquires a velocity in
the opposite direction, exactly equal to that which
was lost. Let us apply this result to the collision of
two perfectly elastic spherical bodies A and B [Fig.
29], supposed to be moving from left to right on the
line joining their centres; and let the same notation be employed as in the preceding case, Art. 53.
In estimating the effect of the collision, it is evident
that until the elasticity begins to act, we may consider the bodies as inelastic.  Consequently at the
instant of greatest compression, the velocity u common to each body will be given by the equation
mV + m'v'
U =  m           E               [el
m -t+ m'
and the velocities lost by A and gained by B during
the compression, will be
v-    and   - vi.
But in the process of resuming its primitive figure,
each body may obviously be regarded as acting
aupon a fixed plane, supposed to pass through the
point of meeting of the two bodies, at right angles to
the line on which they are moving. Hence during
this process A will lose, and B will gain, the additional velocities
v-u  and       v'1
respectively, and the total velocities lost and gained
by the two bodies will be
2(v - u)  and  2(u Zv')




DYNAMICS.                   69
respectively. If then we denote the velocity of A
after collision by v,, and the velocity of B by v,, we
shall have
v, = v - 2(v - u),
or               v, = 2u —v;
and             v/,,= vI + 2(u -v),
or              v,, = 2u- v';
and substituting in these equations the value of nz
given by equation [c], we shall get
V   2(mv + m'v')
~v,         --- -  v~
m + m'
2(mv + m'v') __v
and           v,-    -,  -m'
m + Mn'
or, reducing,
v(m - m') + 2m'v'
V, =         m']
v  + m/    [d]
v' (m' -- m) + 2mv
and           van = I/->0   [e]
m + m'
To adapt these equations to the case in which the
bodies move in opposite directions, we have only to
suppose the velocity v' negative. We shall thus get
v(m -- m') - 2m'v'
V/ =;  m'i;. []'(m - m') + 2mv
V/,         -     =/    **e **v @ [g]
m + m'
56. Application of equcdiozs [d] and [e].
1~ Let nm = n': then
v,  v',  and  v,, = V.




70                  DYNAMICSo
Thus when the masses are equal, the collision will
cause the bodies to exchange velocities.
20 Let rn =  Bm' and v' = 0: then
v1-=0,  and  v,, = v.
That is, the body A will be brought to rest, and B
will acquire its entire velocity; a result immediately
deducible from the preceding case.
From this it is evident, that in the case of a series
of elastic balls, A, B, C,...... P, Q, R, of equal mass,
in contact with each other, and having their centres
on the same straight line: if the first ball (A) be
made to impinge with any velocity directly upon
the second (B), the only visible effect will be to
cause the last of the series (R) to move in the same
direction with an equal velocity, all the intermediate
balls remaining at rest.
30 When v' = 0, but rn is not equal to m' the direction of the motion of the impinging ball, after collision, depends upon the relative values of mn and mn'.
Thus when m > m', v, and v,, are both positive, and
both balls will move after collision in the original
direction; but when n < n', v,, is positive, but v,
negative; and the impinged ball only will move in
the original direction, while the impinging ball will
rebound.
57. Application of eqtations [f] and [g].
10 Let in = in': then
v, = -v',/  and  V,, V.




DYNAMICS.                    71
Thus, in this case, the bodies will exchange both velocities and directions.
20 Let v = v': then
v(m - 3m')        v(3m -- i')
+ml-, and o,,              
and we perceive that when m = 3m', we have
v, = 0,  and v,, = 2v.
58. Relative velocity before and cafter collision.
From the equations v, = 2u - v, and v,, = 2u   v
we get by subtraction
/ —, = - (v --');
from which it appears that the relative velocities of
the two bodies before and after collision are equal
to each other.
59. Imperfectly elastic bodies.
In the collision of an imperfectly elastic ball with
a fixed plane, the ratio of the force or quantity of
motion destroyed by the compression of the ball (or
the force of compressionz) to the force generated by its
restoration to its primitive figure (or the force of restitution), is called the modulus of elasticity of the substance
of the ball If, then, in any case, we denote the
modulus by e, the mass of the ball by rn, and the
velocities corresponding to the forces of compression
and restitution by c and r respectively, we shall have
mr  r
me   C
and hence           r = ec..o..............eo.. oa a [h]
The value of e will obviously always lie between zero




72                  DYNAMICS.
and unity; the former limit answering to the case of
an inelastic body, the latter to that of a body perfectly elastic.
To determine the velocities after collision, in the
case of two imperfectly elastic balls moving on the
line joining their centres, we proceed in the same
manner as in Art. 55.  Thus, employing the same
notation as in that article, the velocities lost by A
and gained by B during the compression, being
v -u and u -vt',
the velocities lost and gained by the balls in the process of recovering their figure, will be [equa. h],
e(v - u)  and e(u — v),
and the total velocities lost and gained will be
(v -- u) + e(v - u) and (u - v')     e (u -- v').
Hence we shall have
vV - (v - u) - e(v - u)
- (1 + e) u - ev
-  (+- e)     -  - eV,.  e   v    le   l. -{- m'
and, = v' + (u- v) + e (u -v')
= (1 + e) u- ev'
my + m'v'
(1+ e) m+' M  — ev'.......... [k]
If we suppose the balls to have equal masses, and
A to impinge upon B at rest, we have
m = m' and v'  O;
and hence       v,,  (1 + e) 2;




DYNAMICS.                 73
from which we get
e  2v1, 1[]
V
To determine in a given case the modulus of elasticity, we suspend at the points C and C' in the manner represented in Fig. 30, two equal spherical balls
A and B of the substance in question; and withdrawing the one (A) from its position of equilibrium, suffer it to descend and impinge upon the other (B) at
rest. The arcs described by the respective balls'being MNG, M'N'G', the velocities v and v,, with
which A impinges upon B, and which B acquires
from the impulse, will be given [Art. 17] by the
equations
MG2
2 =2g x PG = 2g x FG2
FG'
v,   2g X P'G' = 2g x FIG,
"Hence, since FG -F'G', we have
v,,  /P'GI  M'G'
v'=   PG    MG 
and substituting this value of vL Lin equation [mn], we
V
get
e=/pG -  1
M'G'
=2X         1.
MG
The arcs IMNG,'N'G' being known, their versed
sines PG P.-'2, or their chords,1G, M.'G', can be
K




74                   DYNAMICS.
easily calculated, and hence the value of e determined.
60. Loss of living force iz the collision of ineclstic bodies.
The product of the mass of a material point (or of
a body, all the points of which have the same velocity), by the square of its velocity, is called the living
force, or vis viva, of the point or body.
Thus, employing the same notation as in article
53, the suni of the living forces of the inelastic
spheres A and B, moving in the manner supposed in
that article, is, before collision, nv2 + n'v'2; and after
collision, (n + mn')u2O.  Let the latter of these expressions be subtracted from  the former, and let their
difference be denoted by d; then we shall have
d = mv2 + m'Iv2   (M +'v'  m +')2.
But multiplying equation [a], (Art. 53,) by (nz + m')u,
we have
(m + rz')U2 = muv + m'uv',
and hence
0 = muv + m'uv' — (m + rm')2; [eoEI[l
and adding these two equations, we get
(m +  m')u'2 = 2muv + 2m'zv' - (m  + m')u2.
Consequently we have
d = mv2   nm'v'2 - 2mzuv- 2m'uv' +- mu2 + m'u2
-= Wz(v -  t) 2 +,M' (U - v')2 
The second member of this equation being positive,
it follows that in the case of the -two inelastic spheres
A and B, the sum of the living forces after collision is




DYNAMICSo                    75
less than their sum before collision. It is also evident
from the form of the second member, that the difference of the two sums, or the loss of living force by
the collision, is equal to the sum of the living forces
due to the velocities (v - u) and (t - v') which are
lost and gained by the bodies respectively.  This is
a particular case of the general principle, that the
sudden changes of velocity which occur in the collision of inelastic bodies, whatever their number and
forms, and whether the collision takes place among
themselves or with fixed obstacles, are always attended
Vwilh a loss of livizlg force.
61. Conservatiolz of living force in the collision of perfectly elastic bodies.
If, as in article 55, we suppose the two spheres A
and B to be perfectly elastic, employing the same
notation as in that article, we shall have
= 2u - v and v, =2u —v';
and hence
mv,2 =m(2u. -v)2  and myv2     m_(2u -v)o 2
Consequently we shall have
mv,2 +?mv,,2 = m(4ua — 4uv + v2) + m'(4u2 - 4uzv + v'2)
- 4(mu2 + m'u2 -  amv - m'lv') +- mY2 + m'vI2;
and since [Equa. n] the expression within the parenthesis is equal to zero, we shall get
mv,/2 + mV,,2 = mv2 + m'v'2
Thus in the case of the two perfectly elastic spheres
A and B, the sum  of the living forces before and




76                  DYNAM1CS.
after collision is the same.  This is a particular case
of the general principle, that in the collision of perfectly
elastic bodies, there is no loss of living force.
62. Conservation of the motion of the centre of gravity
in the collision of bodies.
Let the distances of the centres of the two spheres
A and B from  any assumed point of the line on
which they are moving, at any instant before collision, be denoted by x and y; and let the distance of
their common centre of gravity from the same point,
at the same instant, be denoted by z: then, retaining
the previous notation, we shall have (Art. 45. Statics)
(m + mn')z = mx + my.
Again, let the corresponding distances from the same
point, for the instant immediately following, be denoted by x', y' and z': we shall also have
(m + m')z' = mx' + my'o
Hence, subtracting the first equation from the second, we shall have
(m + m') (z' - z) =  (x' -- x) + m'(y' -y);
and denoting the infinitely small interval between
the two instants by t', and dividing each member of
this equation by it, we shall get
(n + MI) (Z -   =   ( x -(  -  t + rn' 
If we suppose the second instant to be that which
immediately precedes the collision, t'  and.   t
will respectively express the velocities which we




DYNAMICS.                 77
have denoted by v and v': Also  t'  will express
the velocity, which we will denote by V, of the common centre of gravity of the two bodies. Hence we
shall have
(m + m') V = my + m'v',
and                   V   my + m'v'
m + m'
If we denote the velocities of A and B immediately after collision, by w and w', and the velocity of
their common centre of gravity at the same instant
by VI, we shall find, by an operation similar to the
above
mw. mI'w'
We will apply these results to the two extreme
cases.
10 When the two bodies are inelastic, they have,
after collision, the common velocity ut. Hence, for
this case, we have
W   -= Wi  U,
and              V   ( +   ).
(m + m')
m + m'v
But [Art. 53]    UI   m=+m;
hence [Equa. o]      V= U,
and consequently    V= V.
2~ When the bodies are perfectly elastic, we have
w = v, = 2u - v, and w' = v,=  2u -v';




78                   DYNAMICS.
and hence
V- _n(2u - v) + m'(2u- v')
m + m'
= 2u -_my + n'v'
m + m'
= 2u -- u = u.e
But              V= u;
consequently     V= F.
Thus whether the two bodies are inelastic or perfectly elastic, the velocity of their common centre of
gravity, immediately before and after their meeting,
is the same; the collision, though changing the velocity of each body, producing no alteration in the velocity of that point. This is a particular case of the
general principle called "the principle of the conservation of the motion of the centre of gravity," that
the multual action of the bodies of any system does not alter
the motion of the centre of gravity of the system.
63. Oblique collision.
We shall consider only the simple case of the collision of a spherical ball with an immovable plane.
Let AB [Fig. 31] represent the plane, and CI the
line described by the centre of the ball.  On CI produced, let IE be taken to represent the velocity of
the ball; and let it be resolved into the components
IF and IM, the one in the plane AB, the other perpendicular to it. Denoting the velocity of the ball
by v, and CIL the angle of incidence by a, we shall
have
IF= v sin a, and IM =v cosa.




DYNAMICS.                   79
1~ If the ball is inelastic, there will be no component perpendicular to the plane after collision, and
the ball will move in the plane with the velocity
v sin a.
2~ If the ball is perfectly elastic, it will tend to
rebound in the direction IL with the velocity IN,
equal to IM or v cos a; and in virtue of the two velocities IF and IN, will move in the direction IG with
a velocity ID, equal to IE; the angle of refXection GIL
being equal to the angle of incidence.
30 If the ball is imperfectly elastic, it will tend to
rebound with a velocity IN' less than IN, and such,
that if e denote the modulus of elasticity, we shall
have
IN'
IN'
and hence          IN' = e X IN.
In this case the ball will move, after collision, in the
direction IG', and with a velocity ID' which will be
given by the equation
ID' = VV2 sin2 a + e2v2 cos2 a;
and if we denote the angle of reflection LIG' by a',
this angle will be given by the equation
v sin c   tang a
tang a' = - -  =
ev cos c    e




80                   DYNAMICS.
GRAVITATION.
64. It has been found by observation, that the
planets, in their revolutions about the sun, observe
the following laws:
1st. The areas described by the radius vector of a
planet (or the line drawn from the centre of the sun
to the centre of the planet), are as the times employed in describing them.
2d. The orbit of a planet is an ellipse of small ece
centricity, having the centre of the sun at one of its
foci.
3d. The squares of the periodic times of any two
planets (or the times of the complete revolution of
each about the sun), are to each other as the cubes
of their mean distances from the sun (or the semitransverse axes of their orbits).
The satellites, in their revolutions about their primaries, are found to obey the same laws.
These laws, from the name of their discoverer, are
called Kepler's laws.
According to the first law of motion, the tendency
of a planet, at each point of its orbit, is to move in
the straight line which is tangent to its orbit at that
point: therefore, since its motion is curvilinear, it
must constantly be acted upon by an accelerating
force.  The object of this section is very briefly to
investigate the laws which regulate this force. The
laws of Kepler refer to the motions of the centres of
gravity of the planets; and in all that follows, we




DYNAMICS.                  81
shall consider the masses of the planets as reduced
to these points.
10 The direction of the force.
Since the areas described by the radius vector of
a planet are proportional to the times employed in
describing them, the force which deflects the planet
from the straight line in which it tends to move,
must (Art. 22), coincide in direction with the radius
vector, and constanlly solicit the body towards the centre of
the csun.
20 -Law according to whiich the intensity of the force
varies.
Let AMA' [Fig. 32] represent the elliptical orbit
of a planet, and F the focus occupied by the centre
of the sun. Let MM" be an arc described by the
planet in the indefinitely short time 1'; and to its
extremities M and M", draw the radii vectores FM
and FM". Draw MY tangent to the curve at M;
and from M and F, draw MC' and FY perpendicular
to MY. Draw also M"N, M"K perpendicular to MC'
and MF respectively; and take MC' equal to the
radius of curvature at the point M.
The arc MM" being infinitely small, the direction and intensity of the accelerating force, during
the time t', may be supposed to remain the same.
During this time, then, the accelerating force acting
alone would cause the point to describe the line MI.
Hence denoting the intensity of the force by F, we
have [Art 3],
MI ='Ft'2
2~~~~~~~~~I




82                     DYNAMICS.
and taking t' for the unit of time, we get
F = 2MI.                        [      ee n  es   a]
To find an expression for MI, we get from  the
similar triangles MNI, M"KI,
M"K    M"I        MN    MI;
and hence         MI = MN x M —  K....   a  e   [b]
But since MM" may be considered the arc of a circle
of which MC' is the radius, we have
M"N2 = (2MC' -MN)MN;
or, neglecting MN in comparison with 2MC',
M"N2 = 2M0' x MN;
M"N2
and hence         MN =
2MC'
Moreover, we have
MC'        xFM) 
(p denoting the parameter of the ellipse); or, observing that the similar triangles M"KI, FYM give
FM'FY        M"I    M "K,
FM M"I
or                   FY = M/K'
We have          MC' =- X (M/t  3
* NI may be neglected in comparison with M/"I, since it is a qnantity of the same order as MN; that is, aln infinitely small quantity of
the second order~
t Jackson's Conic Sections, Chap. VI, Prop. I, Cor. 2.




DYNAMICS.                  83
hence substituting this value of the radius vector
in equation [c], we get
1   M I"K3
p    M"I
Lastly, substituting this value of MN in equation [b],
we have
MI =- X M"K2. *0....0 0.   0 0..   [qel
p
If now we denote the area of the elliptical sector
FMM" by s, we have (considering MM" as a straight
line),
FM X M2"K
2 
and hence         MIK2 4- 2
FM2;
and substituting this value of M"K' in equation [el,
we get
1   4S2
MI =- X
p   FM2'
hence [Equa. a] we have
2   4X s2
Again, considering the action of the accelerating
force at some other point of the ellipse, as M', and
for an equal interval of time (t'), we have, denoting
the intensity of the force by F',
2   4s2
p  FM'2'
consequently we have




84                    DYNAMiCS.
PF    F':: FM'2    FM2.
Thus the force which retains the planet in its orbit,
varies in the inverse ratio of the square of the distance of
the planet from the centre of the sun.
65. Variations of the force from one planet to another.
Let T and T' denote the periodic times of two
planets; s and s', the areas described by their radii
vectores in the time t'; and a and b, a' and /', the semiaxes of their orbits: then the area described by the
radius vector of the first planet, in the time 7T will
be Ys; but this area, being the entire surface of the
ellipse, will also be expressed by scab; hence we shall
have
Ts = sab,,tab
and                    T
$
We shall also have
cralbl
consequently, according to Kepler's third law, we
shall have'r2a2b2    qr2a'2b/2
__.  a3  a't3;
2          8t2
or, multiplying the antecedents by s  and the conSt 2
sequents by a-3
b2    b'2
-.. S2    s'2;
or  denoting the parameters of the ellipses by  and 
or, denoting the parameters of the ellipses by p and p',




DYNAMICS.                  85
2    12    P    _p
S%    S     -
2    2
But denoting the accelerating forces which act
upon the two planets at the points M and M, of their
orbits, by F and F,, we have (Equa. f; Art. 64)
1   s2      1    S/2
F:F,:  -X:    x
p   FM2.p XFM,2
Combining then this proportion with that immediately preceding, we get
X ~ F,:: FM,2   FM2.
From this we infer that the force which solicits
the planets towards the centre of the sun, varies
from one planet to another according to the same
law as that which governs in different positions of the
same planet; the variation depending only upon
the distance, and not upon any peculiarities that
may exist in the constitution of the planets themselves.
66. Since the satellites, in their revolutions about
their primaries, also conform to Kepler's laws, the
force which retains them in their orbits must solicit
them towards the centres of their primaries, according to the same law as that which obtains in the case
of the sun and planets.
67. From the foregoing results, it is inferred that
the matter of the bodies composing the solar system
is endowed with a peculiar property, in virtue of
which these bodies exert a mutual attraction according to the above law.




86                   DYNAMICS.
With respect to the relation between the attraction exerted by a body, and the mass of the body,
the most obvious supposition, viz. that the force is
directly proportional to the mass, is found to agree
with the results of observation and calculation.
Lastly, it can be shown, by methods not adapted
to the present work, that if bodies, considered in
the aggregate, attract each other in the inverse ratio
of the squares of their mutual distances, then their
ultimate particles must attract each other according
to the same law. We thus arrive at the general law,
that the particles of all bodies attract each other in the direct
ratio of their masses, and the inverse ratio of the sqzuares of
their distances.  This law is called the lazo or theory of
universal gravitation. Its full development in its application to the bodies of the solar system, constitutes
the science of pphysical astronomy.  The agreement
of the deductions of this science with the results of
observation, is such as perfectly to establish the truth
of the law.












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