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AN
ELEMENTARY TREATISE
ON
MECHANICS.
TRANSLATED FROM THE FRENCH OF M. BOUCHARLAT.
/a)DITIONS AND EMENDATIONS, DESIGNED TO ADAPT IT TO THE USE OF
THE CADETS OF THE U. S. MILITARY ACADEMY.
BY EDWARD H. COURTENAY,
PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE ACADEMY.
NEW-YORK;
PRINTED AND PUBLISHED BY J. & J. HARPER,
NO 82 CLIFF-STREKT,
AND SOLD BY THE BOOKSELLERS GENERALLY THROUGHOUT THE
UNITED STATES.
^^i ^ Mechanical Ensine^,
[Entered according to the Act of Congress, in the year 1833, by J. & J. Harper,
in the Office of the Clerk of the Southern District of New-York.]
• "^^TkO .00810/. ^«^^'■'''
Es^'neeriBg
Libiaiy
PREFACE.
In preparing a translation of Boucharlat's Elements of
Mechanics, it has been the principal object of the translator to
supply a suitable text-book for the use of the Cadets of the
United States' Military Academy. To accomplish this object
more effectually, it has been deemed necessary to introduce
several subjects which are not noticed in the original, and to
extend or modify others, where the methods of investigation
adopted by the Author appeared incomplete or obscure. It
was also judged proper to omit one or two subjects, the dis-
cussion of which is usually reserved for works of a less
elementary character.
These alterations were adopted with less hesitation as the
work was principally designed for a special purpose ; but it
is believed that they will render the work more generally
useful, by facilitating the comprehension of many of the more
difficult investigations, and by affording much valuable in-
formation in relation to those subjects which were not dis-
cussed in the original, but which are generally admitted to
form an essential part of an elementary course of Mechanics.
In supplying the deficiencies of the original, reference has
been had most frequently to the works of Poisson, Francœur,
Navier, Persy, Genieys, and Gregory ; and in some few in-
stances, the methods of investigation pursued by those authors
have been adopted with but slight alterations.
The works of Boucharlat have long enjoyed an unusual
share of public favour ; and the hope is therefore entertained
A2
4 PREFACE.
that the treatise now presented, in our own language, will
prove a useful introduction to the study of the higher
branches of Mechanics, and that it will be received with in-
dulgence by all those who are disposed to cultivate a taste for
the most interesting apphcation of Mathematical Science.
As the entire work may be found to constitute too exten-
sive a course for those students who can devote but a limited
time to the study of Mechanics, it was thought expedient to
indicate such of the more difficult subjects as might be
omitted. These subjects are designated in the table of con-
tents by being printed in italics ; and they will be found to
be unnecessary in enabling the student to comprehend those
which follow.
CONTENTS.
PART FIRST.
STATICS.
Introductory Remarks and Definitions 9
Of the Composition and Decomposition of Forces .... II
Of Forces situated in the same Plane and applied to a single
Point 20
General Remarks on Forces situated in any manner in Space 24
Of Forces situated in Space and applied to a Point .... 27
Of the conditions of Equilibrium of a Point acted upon by several
Forces, and subjected to the condition of remaining upon a
given Surface 31
Of the conditions of Equilibrium of a Point acted on by several
Forces, and subjected to the condition of remaining con-
stantly on two curved Surfaces, or on a Curve of double
Curvature 36
Of Parallel Forces 39
Of Forces situated in the same Plane and applied to Points con-
nected together in an invariable manner 48
Of Forces acting in any manner in Space 60
Theory of the Principal Plane, and Analogy existing between
Projections and Moments 71
Centre of Gravity 79
Of the Centrobaryc Method 97
Machines — Cords 99
Of the Catenary 103
Of the Lever 109
Of the Pulley 116
Of the Wheel and Axle 120
Of the Inclined Plane 126
Of the Screw 128
Of the Wedge .131
6 CONTENTS.
Page
Friction 132
Effects of Friction in certain Machines 136
Of the Stiffness of Cordage 142
On the Resistance of Solids 145
Of the Resistance to Compression or Extension 147
Of the Resistance of a Solid to Flexure and Fracture produced
hy a Force acting at right angles to the direction. if the
Fibres ,. 150
Of the Figure of the Solid after Flexure 164
Of Solids of equal Resistance , 175
Of the Principle of Virtual Velocities 177
Of the Position of the Centre of Gravity of a System when in
! Equilibrio 184
PART SECOND.
DYNAMICS.
Of the Law of Inertia 187
Of uaiform rectilinear Motion 188
Of varied Motion , 190
Of uniformly varied Motion 194
Of the Motion of a Body projected vertically upward . . .197
Of the vertical Motion of a Bod/ when acted upon by the
Force of Gravity considered as variable 199
Of the vertical Motion of a Body in a resisting Medium , . . 202
Of the Motions of Bodies upon Inclined Planes 205
Of curvihnear Motion 208
Of the Motion of a material Point when compelled to describe
a particular Curve . 221
Of the Motion of a rhaterial Point when compelled to move upon
a curved Surface . . . 229
Of the Motion of a material Point on the Arc of a Cycloid . . 236
Of oscillatory Motion . . . . . . . 238
Of the Simple Pendulum ..... . 240
Of the Centrifugal Force . . ... . . . . . , . 247
Of the System of the World 253
Of the Motions of Projectiles in Vacuo ........ 272
Of the Motions of Projectiles in a resisting Medium .... 278
Of the different Methods of measuring the Effects of Forces 288
CONTENTS. f
Page
Of the direct Impact of Bodies 293
Of the direct Impact of unelastic Bodies ....... 293
Of the direct Impact of elastic Bodies 294-
Of the Preservation of the Motion of the Centre of Gravity in
the Impact of Bodies 297
Of the Preservation of living Forces in the Impact of elastic
Bodies — Relative velocity before and after Impact — Loss
of living Force in the Collision of unelastic Bodies . . . 299
Principle of D'Alembert 301
Of the Motion of a Body about a fixed Axis 309
Of the Moment of Inertia 316
Of the Motion of a Body about a fixed Axis when acted upon
by incessant Forces 320
Of the Compound Pendulum 322
Of the Motions of a Body in Space when acted upon by im-
pulsive Forces 328
Of the Motions of a System in Space when acted upon. by in-
cessant Forces 333
General Equations of the Motions of a System of Bodies . . 388
General Principle of the Preservation of the Motion of the
Centre of Gravity ... ; 345
PART THIRD.
HYDROSTATICS.
Of the Pressure of Fluids 349
General Equations of the Equilibrium of Fluids 351
Application of the general Equations of Equilibrium to incom-
pressible Fluids 354
Application of the general Equations of Equilibrium to elastic
Fluids 359
Of the Pressure of heavy Fluids 36 J
Of the Equilibrium, Stability, and Oscillations of floating
Bodies 366
Specific Gravity — Hydrostatic Balance — Hydrometer . . . 379
Of the Pressure and Elasticity of Atmospheric Air .... 384
Of Pumps for raising Water 386
Of the Air-pump 391
Of the Barometer 398
b CONTENTS.
PART FOURTH.
HYDRODYNAMICS.
Page
Of the Discharge of Fhiids through horizontal Orifices . . . 407
Of the Motion of Water in Pipes 429
I
i
ELEMENTS OF MECHANICS.
PART FIRST.
STATICS.
INTRODUCTORY REMARKS AND DEFINITIONS.
1. Mechanics is the science which treats of the laws of
equihbrium and motion. Wlien appUed to solid bodies it is
divided into Statics and Dynamics ; the former discussing
the conditions of their equihbrium, and the latter those of
their motion. In the application of Mechanics to the consid-
eration of fluid substances a similar division is likewise made,
viz. Hydrostatics, which treats of the equilibrium of fluids,
and Hydrodynamics, which investigates the circumstances
resulting from their motions.
2. The object proposed in Statics being the determination
of the laws of equilibrium, this state of equilibrium may
always be regarded as resulting from the mutual destruction
of several forces.
3. The term force or power is applied to every cause which
impresses on a body or a material point a motion or tendency
to motion.
4. A force may act on a material point either by drawing
the point towards it, or by pushing the point in advance of
it. The first hypothesis will always be adopted, unless the
contrary is expressly indicated.
5. A material point being solicited by a single force will
naturally move in a right line, since there can be no reason
why it should deviate to the right rather than to the left of
this line.
10 STATICS.
6. The right line along which a force acts is called the line
of direction,
7. The effect of a force depends, 1°. On its intensity ; 2°.
On its point of application ; 3°. On its line of direction ; and
4°, On its pushing or pulling along this line.
8. By the intensity of a force, we understand its greater or
less capacity to produce motion.
9. If two forces directly opposed to each other sustain in
equilibrio a material point or an inflexible right line, the in-
tensity of either one of these forces may be assumed arbitra-
rily, provided we assign an equal intensity to the second force.
A similar remark is equally applicable to a system composed
of any number of forces ; and hence it appears that the con-
ditions of equilibrium will depend simply on the ratios of the
forces, and not on their absolute intensities.
10. Having assumed one force as a unit of measure, we
say that a second force is equal to it, when, if directly op-
posed to it, an equilibrium would ensue.
Two equal forces applied to a material point, acting along
the same right line, and in the same direction, constitute a
double force : in like manner a triple force may be regarded
as resulting from the union of three equal forces, &.c. ; so that
the number of these equal forces will constantly be propor-
tional to their joint intensity. It may hence be inferred that
if several forces solicit the point M {Fig. 1) in the same line
of direction from M towards B, we can add into one sum all
these forces, since their joint effect will be precisely the same
as that of a single force equal to their sum. For the same
reason we should subtract from this sum, or we should regard
as negative all the forces which tend to solicit the point from
M towards A.
11. The unit efforce being arbitrary, it may be repre-
sented by any portion of its line of direction.
12. When a force is applied to any point of a body whose
several parts are firmly connected together, this point cannot
be put in motion without communicating the motion to the
other parts of the body ; if, therefore, a force be applied to any
point A [Fig. I), it will have the same effect as though it
were applied to £m.y other point M, assumed on the line of
COMPOSITION OF FORCES. 11
direction AB, Moreover, if we drop the consideration of a
body, we may still regard the points in space situated on the
line of direction as mathematical points, no one of which can
be moved without imparting its motion to all the others.
13. It appears from Art. 12, that by interposing a fixed
obstacle on the line of direction of a force, its effect will be
entirely overcome.
14. Two equal forces P and Q, applied to the points A and
B of an inflexible right line {Pigs. 2 and 3), and acting along
this line, but in contrary directions, will sustain each other in
equilibrio : for if the force P tends to draw the point A from
A towards a, the point B, which is firmly connected with A
by means of the intermediate points, will have a tendency to
describe the space Bb, equal to Aa ; but by hypothesis, the
force Q, tends to move the point B over a space Bb' equal
to Aa ; and since B cannot yield to one of these influences
rather than to the other, it must remain immoveable, and aji
equilibrium will necessarily ensue (Art. 13). In like manner,
if the forces P and Q, had been supposed to exert a tendency
to push A and B, the same consequences might have been
deduced.
15. When the right line AB is reduced to a point, the two
equal forces, being directly opposed, are still in equilibrio ; but
if the forces are unequal, the point M {Fig. 1) will be moved
in the direction of the greater, by a force equal to the differ-
ence of the two unequal forces.
Of the Composition and Decomposition of Forces applied
to a Point.
16. When two forces act upon a moveable point in direc-
tions forming with each other an angle whose summit is the
point of application, the state of equilibrium cannot subsist.
For, if we suppose the two forces P and Q, {Fig. 4) to be
in equilibrio, we may introduce a third force P' equal and
directly opposed to the force P. The forces P and d being
supposed to destroy each other, the force P' must produce its
entire effect, and must consequently move the point M in a
direction from M towards P'. But P and P', being equal and
12 STATICS.
directly opposite, must likewise destroy each other, and the
force Q. will therefore act as though it were alone, soliciting
the point M in a direction from M towards Q, ; and since it
is impossible for the point M to move in two directions at
the same time, we cannot suppose that P and Q, are in equi-
Ubrio without involving an absurdity.
17. Since an equilibrium cannot subsist between two forces
whose lines of direction are not coincident, the point M will
tend to move in a certain direction MR, as though it were
solicited by a single force R. This force is called the result-
ant of the two others, and the original forces are called com-
ponents.
It may be observed that two forces which have a resultant
do not always intersect. For example, if two parallel forces
P and Q, be supposed to act on a body, and if a third force R
be found which shall produce the same effect, R will be the
resultant of the forces P and Q,.
18. Having examined the conditions of equilibrium of two
equal forces acting on a point, the most simple case which
next presents itself is that of three equal forces applied to the
same point. Let P, Q, and R represent these forces ; if they
produce an equilibrium, their directions will divide the cir-
cumference of a circle whose centre coincides with the point
of application, into three equal parts {Fig. 5) : for since the
same reasons may be adduced to prove that the point should
tend to move in the direction of each of these forces, it fol-
lows that it cannot yield to the influence of either in prefer-
ence, and must consequently remain at rest.
19. The equal angles PMa, PMR, and QMR {Fig. 5),
being measured by one-third of the entire circumference, each
of them is equal to f of a right angle, or 120°, Hence, if one
of the three lines PM, QM, or RM be prolonged through M, it
will bisect the angle formed by the other two. If MS, for ex-
ample, be the prolongation of the line RM, the angles PMS,
CIMS will be equal, being supplements of the equal angles
PMR and GlMR ; whence it appears that MS bisects the angle
PMGl.
20. Let us next suppose the two equal forces P and Q,
{Fig. 6) to be applied perpendicularly to the extremities A
COMPOSITION OP FORCES, 13
and B of a right line AB ; the resultant of these forces will
pass through the point O, the middle of the line AB, and will
be equal in intensity to the sum of the intensities of the two
forces P and Q,. For, draw through the points A and B the
four right lines AC, AD, BC, BD, each forming with AB an
angle equal to ^ of a right angle : the triangles ACB, ADB
will be isosceles, and will have the sides AC, CB, AD, DB
equal to each other.
The right lines AB, CD will intersect each other at right
angles, and the figure ACBD will be a rhombus : the sides
of this rhombus and their prolongations determine by their
intersections the four obtuse angles ACB, ADB, P'AC, Q'BC,
each of which is equal to | of a right angle ; for, the angle
CAD being by; construction equal to | of a right angle, its
supplement P'AC must be equal to | of a right angle ; and
since the opposite sides of the rhombus are parallel, the angle
ACB is equal to P'AC, and is consequently equal to f of a
right angle. The same may be proved of the angles CBQ,'
and ADB. Moreover, since the line CD bisects the angle
ACB, which was proved equal to f of a right angle, it follows
(Art. 19) that the three angles ACB, ACS, and BCS are equal
to each other. In like manner it may be shown that there
are three equal angles at each of the points A, B, and D.
21. We will now apply at the points A, B, C, D, which are
supposed firmly connected together, twelve equal forces, dis-
tributed as follows :
At the point A three equal forces P, P', P",
At the point B three equal forces Q,, Q,', Q,",
At the point C three equal forces S, S', S",
At the point D three equal forces V, V, V" ;
forming with each other angles equal to | of a right angle :
these twelve forces will sustain each other in equilibrio.
But the forces P' and V", Q' and V, being equal, and
directly opposed, will destroy each other, as also will the forces
P" and S', Q," and S". If, therefore, an equilibrium is main-
tained in the system, it must subsist between the four forces
P, Q, S, and V. The two last, acting in the same direction
along the line DC, are equivalent to a single force equal to
their sum, which may be applied at O, a point in their line of
2
14 STATICS.
direction. Thus, an equilibrium will take place between the
forces P and Q, and a force R whose line of direction passes
through the middle of the line AB, and whose intensity is
equal to the sum of the intensities of P and Q.
If we suppress P and Q, the equilibrium will be destroyed,
but it may again be established by applying at O a single
force R' equal and directly opposed to the force R. The force
R' must therefore produce an effect precisely equal to the joint
effect of P and Q, and will consequently be their resultant.
We hence infer that tlie resultant of ttvo equal and jmrallel
forces is equal to their sum, is parallel to them, and divides
equally the line AB, which is draivn j^erpendicular to the
co?nmou direction of those forces.
22. To determine the resultant of two unequal parallel
forces P and Gl applied to the extremities A and B of a right
line AB {Fig. 7), we will suppose p to represent the unit of
force, and make mp==V, 7ip=Q,. The ratio of m:n will be
the same as that of the forces P and d. Let the right line
AB be also divided in the same ratio at the point D, and we
shall have the proportion
P : Gl : : AD : DB («).
On the prolongations of AB, take AA'=AD, and BB'=BD ;
we shall then have, since A'D and DB' are double AD and DB,
P : a : : A'D : DB' : : w : n.
If then we divide A'D into m equal parts, DB' Avill contain n
such parts, and A'B' will contain one of these parts as many
times as p is contained in P+Q,. And since any two of the
points of division a', a", a'", 6cc. separate three equal parts, wh ile
three points separate four parts, &c.,the number of equal parts
in the line A'B' will exceed by unity the number of points of
division. A force being applied at each point of division,
there will remain one of the number m-{-n, of which one half
may be applied at A', and the other at B' ; the several partial
forces will thus be distributed throughout the line A'B'. But
the points A' and D being equally distant from the point A,
the force ^p applied at A' may be combined with one lialf of
the force p applied at D, and their resultant, which is equal to
their sum, will pass through A. The same remarks will
apply to the forces /> and p applied at a' and a^ to the forces
COMPOSITION OF FORCES. 15
p and p applied at a" and a,,, &-c. ; thus, the total resultant
of the partial forces distributed along A'D,will be equal to
their sum P, and will pass through the point A. In like
manner it may be shown that the forces applied to DB' may-
be replaced by Q, ; and the entire system of partial forces may
therefore be replaced by the two forces P and Q, applied at
the points A and B.
But these parallel forces may be otherwise compounded,
by combining them in pairs taken at equal distances from the
middle point O of the line A'B' ; and it may thus be easily
shown that the resultant of the whole system will pass
through the point O, and will be equal to P+Q,.
The position of the point O must now be determined. For
this purpose, it may be remarked that A'O {Fig. 7), being one-
half of A'B', is equal to AB ; and by substituting this value
in the equation
AO=A'0— A'A,
which results immediately from an inspection of the figure,
we shall obtain AO=AB— AA', or AO=AB— AD=DB. In
a similar manner it may be shown that OB=AD ; and by
substituting these values of DB and AD in the proportion (a),
there will result
Q : P : : AO : OB {b).
If P and Gt are incommensurable for the unit p, this pro-
portion which results from the division of A'B' into m-{-7i
equal parts, might seem to fail : but by diminishing indefi-
nitely the value of the unit p, and increasing in the same
proportion the number of these divisions, the demonstration
becomes applicable to all cases, since the equal parts Aa', a'a",
&c. being indefinitely small, the points of division will then
become continuous.
23. This proposition is equally true when the two parallel
forces P and Gl are applied to the extremities of an oblique
line CD {Fig: 8). For, by drawing AB at right angles to
the common direction of the two forces, and transferring the
points of application to the points A and B in their lines of
direction, the proportion {b) will evidently subsist ; but the
similar triangles AGO, EDO, give AO : OB : r OC : OD ;
whence we obtain
16 STATICS.
a : P : : OC : OD :
and we therefore infer that when two parallel and unequal
forces P a7id Q, are apjAied to the extremities of a right
line CD, their residtant will divide this line in the inverse
ratio of the intensities of tJie forces.
24. By the aid of this theorem we can readily demonstrate
that of the parallelogram of forces, which may be enun-
ciated as follows : — If any two forces P and Q, applied to a
"point A {F'g- 9) he represeiited in direction and intensity by
the lines AB and AC, their I'esultant ivill be represented in
direction and intensity by the diagonal of the parallelogram
constructed upon the lines AB a7id AC.
It is immediately obvious that the resultant will pass
through the point of application of the forces ; since the
forces conspire to solicit this point, and their resultant, which
may replace them, must therefore contain it.
25. The resultant of the two forces P and Q, will likewise
be contained in the plane of those forces. For, if it be situ-
ated above this plane, a position in all respects similar can be
selected below the plane : the same arguments may then be
advanced to prove that its direction coincides with either of
these lines ; and since the resultant cannot have two direc-
tions, we infer that it coincides with neither.
26. It may also be proved that the resultant of two equal
forces [Figs, 10, 11, 12) will bisect the angle included between
them.
For, if we suppose Km to represent the resultant of the two
forces P and Q, and draw AD bisecting the angle PAQ., a
line An may always be found, whose position with respect to
AD, AQ,, and AP shall be precisely similar to that of Am
with respect to AD, AP, and AGI ; hence, the same reasons
which would prove Am to b§ the resultant, become equally
applicable to An, and it might thence be inferred that there
are two resultants : this being impossible, we conclude that
the resultant coincides with AD.
27. Let the two unequal forces P and Q. be now supposed to
act upon the point A [Fig. 13), and let the parallelogram ABDC
be constructed, whose sides AB and AC are taken on the lines
of direction of those forces, and are proportional to theiE-
COMPOSITION OF FORCES. 17
intensities. It has already been shown that the resultant will
pass through A, and it remains to be proved that it will also
pass through D, the extremity of the diagonal AD. Having
taken DE=AB=P,* draw EF parallel to AB, and apply at
E and F, in contrary directions, the two forces Q,', Gl", each
equal to Q,. Since these forces will destroy each other, we
can substitute for P and Q, the four forces P, Q,, CI', and Q,".
But by regarding P and Gl' as two parallel forces applied to
the extremities of an inflexible line BE, and having obtained
by construction the proportion
P : a' : : DE : BD,
it follows immediately from the preceding theorem, that the
resultant R of P and Q,' will pass through the point D;
Again, if we transfer the force Q,, and apply it at F, in its line
of direction, the two equal forces Q, and Q," will have a re-
sultant S, which, bisecting the angle Q^FGl", will pass through
D, the opposite angle of the rhombus CDEF. We thus ob-
tain two forces R and S which are equivalent to the original
forces P and Gl ; aad since the forces R and S pass through
the point D, the resultant of P and Gl will likewise pass
through the same point.
28. It will now be proved that if the intensities of the forces
be represented by AB and AC, the diagonal AD will repre-
sent the intensity of the resultant {Pig. 14);
If at the point A {Mg. 14), and in the direction AD of the
diagonal of the parallelogram constructed on the sides AB=P,
AC=Gl, there be applied a force X equal and directly op-
posed to the resultant of P and Q,, an equilibrium will take
place between the forces P, Q, and X. But we may regard
Q, as equal and directly opposed to the resultant of the forces
P and X ; hence it follows, that if through the extremity B
of the line AB a line be drawn parallel to X, intersecting at
* It should be remarked that the expression AB=:P is merely intended as
an abridged method of stating that the line AB represents the relative intensity
of the force P, when compared with the unit of force whose intensity is likewise
represented by a line. In like manner, we speak of the " force AB," denoting
thereby that the line AB represents the line of direction and relative intensity
of the force. These abbreviations have been sanctioned by usage.
B
1 8 STATICS.
E the prolongation of the Une AC, which, as has been already
shown, coincides in direction with the diagonal of the paral-
lelogram constructed on P and X, the line BE, being a side of
this parallelogram, will be equal to the opposite side, which
must represent X : but BE, being also the side of the paral-
lelogram ED, is equal to the opposite side AD, which repre-
sents the diagonal of the parallelogram constructed upon P
and Q, ; whence X=AD, and the intensity of the resultant is
likewise measured by the length of the diagonal.
29. One of the simplest corollaries which may be deduced
from the foregoing proposition is the trigonometrical relation
existing between the components P and Q, and their resultant
R {Fig. 15). To obtain this relation, we will assume on
the directions of these forces the parts AB and AC propor-
tional to their intensities, and constructing the parallelogram
ABDC, we shall have the proportion
P : a : R : : AB : AC : AD.
And from the equality of the sides DD and AC, we shall have
in the triangle ABD,
P : a : R : : AB : BD : AD.
But the proportionality of the sides of the triangle to the sines
of tlie opposite angles gives
AB : BD : AD : : sin BDA : sin BAD : sin ABD.
Hence we deduce
P : a : R : : sin BDA, : sin BAD : sin ABD.
The determination of the relations between P, Q, and R is
thus reduced to the solution of a case in plane trigonometry.
30. K there be given, for example, the two components
AB and AC, and the angle BAC contained between them,
and it be required from these to determine the resultant, we
shall have, in the triangle ABD, the sides AB, BD, and the
angle B equal to the supplement of BAC. With these data we
readily obtain the value of the side AD=R, by means of the
formula
R3 =P2 -j-ds _ 2Pa COS B.
If in this formula we wish to introduce the angle included
between the two forces, since the angle B is the supplement
of the angle BAD, we shall have the relation cos B= — cos Aj
COMPOSITION OP FORCES. 19
whence by substitution the following equation is obtained
between the resultant, the two components, and the angle
included between them,
R2 =P- +0,2 +2Pa cos A (1).
31. When the angle A becomes equal to 90°, the parallelo-
gram ABDC {Fig-. 16) becomes a rectangle, and cos A=0.
The general relation between the resultant and its two com-
ponents is then reduced to
R2=p2+a^
The solution of the converse problem, or the resolution of
a single force R into two components P and d, having given
directions, is readily effected by constructing a parallelogram
upon the line representing the given force as a diagonal, the
sides of the parallelogram having the directions of the re-
quired components.
32. When there are several forces lying in different planes,
but all meeting in a single point, the resultant of the system
can always be determined ; for, by combining these forces in
pairs, and substituting each resultant for its two components,
the number of forces will be successively reduced, and we
shall finally obtain but a single resultant.
33. The method of compounding any number of forces
which has just been explained gives rise to a remarkable
graphic construction. Thus, let P, P', P", P'", &c. represent
any forces whose directions intersect at the point A {Fig:
17), and whose intensities are expressed by the hues Ap, Ap',
Ap", Ap"\ &c, assumed on the respective lines of direction ;
through the point p draw the line pr parallel and equal to
the line Ap', and complete the. parallelogram Aprp' ] the di-
agonal Ar=R will be the resultant of the forces P and P':
in like manner, by drawing rr' parallel and equal to Ap", and
forming the parallelogram Arr'p", the diagonal Ar' will be the
resultant of R and P", and therefore the resultant of the three
forces P, P', and P". By continuing this process, a polygon
Aprr'r" would be formed, having its sides parallel to the
directions of the forces, and their lengths representing the
intensities of those forces. The distances from the point A
to the angles of this polygon will be
B2
20 STATICS.
Ar=the resultant of P and P',
Ar'=the resultant of P, P', and P",
A?-"=the resultant of P, P', P", and P'".
And by repeating the construction for any number of forces,
the distance from the point A to the extremity 7-^''' of the last
side of the polygon will be equal to the resultant of the en-
tire system.
Of Forces situated m the same Plane, and applied to
a single Point.
34. Let P, P', P", &c. {Pig. 18) represent several forces
situated in the same plane, their directions intersecting at
the point A ; through this point let there be drawn the rec-
tangular axes Ax and Ky ; then, denoting the respective
intensities of these forces by AP, AP', AP", (fcc, let each be
decomposed into two components, whose directions shall
coincide with the rectangular axes.
For this purpose we will represent by «, «', «", (fee. the
angles included between the forces and the axis of a;, and by
/3, /3', /3", (fee. the angles which they form with the axis of y.
In the right-angled triangle ABC {Fig. 19), the^ side AC
being expressed by AB cos A, and the side BC by AB sin A,
the components of the forces P, P', P", (fee. in the directions
of the two axes are readily obtained : for the force P repre-
sented by AB, forming an angle ct, with the axis of x, and an
angle /3 with the axis of y, will have for its components along
these axes,
AC=P cos a, BC=P cos /3.
In like manner, the forces P', P", P'", (fee. will have for their
components in the direction of Aa:,
P' cos «', P" cos «", P'" cos «'", (fee,
and in the direction of the axis Ay,
F cos /3', P" cos /3", P'" cos /3'", (fcc.
If the sum of the components acting in the direction of x be
taken, as also the sum of those acting in the direction of y,
we shall have, denoting these sums by X and Y respectively,.
P cos «+P' cos «'-f-P" cos «"-f «fec.=X,
P cos |34-F cos |3'+P" cos /3"-f (fec.=Y;
FORCES APPLIED TO A POINT. 21
and the entire system will thus be reduced (Art. 10) to two
forces, of which one X is directed along the line Ax, the other
Y acting along the line Ay. Calling R the resultant of these
two forces, its value may be determined from the equation
Xa4.Y2=R2.
35. For the purpose of rendering the preceding determi-
nation of the value of the resultant general, we have attrib-
uted the positive sign to all the cosines which enter into the
expressions for X and Y ; but it will be necessary in practice
to regard the essential signs with which these quantities are
severally affected. The following considerations will serve
to explain the necessity of this distinction. Let a point M
{Pig. 20) be solicited by a force represented in intensity by
the line MP. By decomposing this force into two others
whose directions shall coincide with the rectangular axes
Mx and My, and calling « the angle which the direction of
the force makes with the axis Mx, its two components will
evidently become
MC=MP sin cc, MD=MP cos «.
The forces which are directed in the line Mx, being regarded
as positive when they act from M towards x, the component
MD will obviously be positive. If the force MP should as-
sume the position MP', the angle « would be increased, and
its cosine diminished ; and if the angle becomes greater
than 90°, the direction of the force will fall in the second
quadrant. In this case it will assume the position MP", and
the cosine of the angle will change its sign. But it is evident
that the component MD" of the force MP" becomes also nega-
tive, since it solicits the point M in a direction opposite to
that in which it was urged by the component MD. Thus
it appears that the signs of these two components result from
the signs of the cosine of a, and hence the forces MP, MP',
&c., which solicit a point, may be always regarded as essen-
tially positive, provided we attribute the appropriate signs to
the cosines of the angles which they form with the axes.
36. If the force under consideration fall below AB, as in
the position MP'", the angle x being measured by the arc
ALBP'", will be greater than two right angles. To avoid
this inconvenience, it has been agreed to reckon the angles as
22 STATICS.
and /J indiscriminately on each side of their respective axes.
Thus when the force falls beneath AB, the angle «e will be
measured, not by the arc ALBF", but by the arc AP'", which
has the same cosine. By this arrangement all the arcs em-
ployed are less than 180^. It is true that when the angle « is
alone given, the direction of the force would appear inde-
terminate, since this angle may be counted either from A to
Pj or from A to P'" ; but this ambiguity will immediately dis-
appear by considering the value of the angle /3, which is evi-
dently acute for the force MP, but obtuse for the force MP'".
37. Whatever may be the direction of the given force,
since it must necessarily lie in one of the four right angles
formed by the axes around the point M, its position must
correspond to some one of those given in Figs. 21, 22, 23, 24.
In the first quadrant,» and P being acute give cos« positive, cos P positive,
In the second, a, obtuse and/3 acute give cos «negative, cos /J aegBtwe, -
In the third, «obtuse and P obtuse give cosa negative, cos (3 negative,
In the fourth, «acute and /3 obtuse give cosas positive, cos (} negative.
Each of these angles will be less than 180°.
38. It may be observed that the signs of these cosines are
similar to those of the co-ordinates x and y of the point B-
For example, if the point be situated within the angle .r'Ay
{Fig. 22), X will be negative and y positive, while at the same
time we shall have cos « negative and cos j3 positive.
39. For the purpose of making an application of the pre-
ceding principles, let us determine the resultant of the five
forces P, P'j P", P'", P"j which are situated as represented
in Fig. 25, and solicit the point A. By attributing to the
components of the forces the positive or negative signs cor-
responding to the angles which are acute or obtuse, the com-
ponents of
P ^ C +P cos «, -fP cos/3,
F -f P' cos «', -F cos (3',
P" i will be I -fP"cos«", -P"cos/3",
P'" — P''"C0S«"',-P"'C0S/3'",
P- [ — P"'C0S«"',+P"'C0S/3".
Having taken the sum of the components which act in one
direction, we subtract from it the remaining components
which act in an opposite direction, and we thus obtain
FORCES APPLIED TO A POINT. 23
P COS «4-P' COS «'4-P" cos «"-F" cos «'"— P^ cos «"=X,
P cos /3 + P'' COS /3" — P'COS ^' — P" COS ^"— P'" COS /3"'=Y.
40. If we defer the determination of the signs of the cosines
until we wish to make an appHcation of the preceding equa-
tions, the several terms may be written with the positive sign,
and the general form of the equations will then become
P cos «4-P' cos «'+P"cos «"+&c.==X (2),
P cos /3+F cos 13' -fP" cos Ii"+ÔÙC.=Y (3).
41. The resultant being represented by the diagonal of a
rectangle, the lengths of whose sides are denoted by X and Y,
its value will be determined by the equation
R=^(X2+Y==) (4).
The position of the resultant remains to be determined. If
we denote by a and b the angles which the resultant forms
with the co-ordinate axes, we shall have
X=R cos a, Y=R cos b ;
whence
cos a=-^, cos 6=^5- (5).
K K
The positions and intensities of the forces being given, the
values of X and Y may be immediately deduced from the
equations (2) and (3). These values being substituted in the
equation (4), make known the value of the intensity of the
resultant, and its position may be determined from the equa-
tions (5).
42. Its line of direction passing through the origin A {Pig.
26), will liave for its equation
sin a
?/=a; tang: «i or V=a^ ;
•^ & > y cos a '
and by substituting cos b for sin a, since a and b are com-
plements of each other, the equation becomes
cos b
y~x ,
co^«
and by substituting in this equation the values of cos a and
cos b given in equations (5), we have
Y
^=x-^*-
43- When an equilibrium takes place, the intensity of the
24 STATICS.
resultant becomes equal to zero ; and the formula (4) then
assumes the form
V(X2+Y2)=0, or X2 4-Y2=0.
But since every square is essentially positive, the preceding
equation cannot be true, unless each of its terms is separately
equal to zero ; hence
X=0, Y=0.
Such are the equations which express the conditions of equi-
librium of any number of forces situated in the same plane,
and acting on a point.
44. If X alone were equal to zero, we should have
R=Y, cos rt=0, cos b=±l.
These equations prove that the resultant is equal to the com-
ponent Y, and is directed along the axis of y.
In like manner it might be shown that if Y were equal to
zero, the resultant would be equal to the component X, and
would be directed along the axis of a;.
General Remarks on Forces situated in any marmer
in Space.
45. If three forces solicit a point, their directions not being
confined to a single plane, a theorem analogous to that of the
parallelogram offerees will still serve to determine their re-
sultant. Thus, let any three forces P, P', and P" be applied
at the point A {Fig. 27), and let their intensities be repre-
sented by the lines AB, AC, and AD. If a parallelopiped be
constructed upon these three lines, the diagonal AE, of the
base of this parallelopiped, will evidently represent the re-
sultant of the forces AB and AC ; and by substituting the
force AE for its two components, the resultant sought will
be that of the forces AE and AD ; it will therefore be termi-
nated at the extremity F of the line EF drawn parallel and
equal to the line AD ; hence it will be the diagonal of the
parallelopiped DE.
46. If the three forces are rectangular, the angle ABE will
be a right angle, and hence we obtain
AE==AB2-|-BE2;
FORCES APPLIED TO A POINT. 25
but the triang^le AEF being also right-angled, we have
AF=»=AE2-fEF2.
And by substituting for AE* its value given above, we deduce
AF='=AB»4-BE='-fEF2.
Or by replacing BE and EF by their equals AC and AD, we
finally obtain
AF=^(AB« 4-AC^ +AD2),
or,
the resultant of the three forces being denoted by R.
47. It has been shown that any number of forces lying in
the same plane may always be referred to two rectangular
axes : in like manner, we may refer to three rectangular axes
those forces which are situated in different planes. Thus,
having assumed three co-ordinate axes passing through any
point O {Fig-. 28), we draw through A, the point of applica-
tion of a force P, the three rectangular axes Ax, Ay, and Az,
parallel respectively to the axes of co-ordinates ; and denoting
by X, /3, y the angles formed by AD, the direction of the force
P, with the three lines A.v, Ay, Az, the direction of the force
will be determined when these angles become known.
48. The values of these angles may also be employed to
determine the components of the force P, which act in direc-
tions parallel to the three co-ordinate axes. For, DC beins:
perpendicular to the plane i/Ax, the angle DCA will be a right
angle, and the triangle ADC, having the angle D=y, will give
DC=AD cos y (6).
In like manner, the components parallel to Ax and Ay will
be expressed by
AB=AD cos u, BC=AD cos |3 (7).
^nd replacing the line AD by the force P which it repre-
sents, we obtain for the three rectangular components of P,
P cos «, P cos /3, P cos y.
49. It is important to observe that the values of two of the
angles u, jâ, and y will serve to determine that of the third.
For, since the square of the diagonal AD is equal to the sum
of the squares of the three edges, we have
AB='+BC2-1-DC2=AD2 ;
and substituting in this equation the values obtained from the
o
O
26
STATICS.
equations (6) and (7), suppressing the common factor AD*,
there will remain
C0S2<«4-C0S2/3-}-C0S=y = l j
whence,
cos y=±y/(l — C0S = «— C0S'»/3) (8).
And since a similar value may be found for each of the other
cosines, it follows that the angle formed by the direction of a
force with either of the axes will become known, when the
angles formed with the other two axes have been previously
determined.
50. The radical in equation (8) being affected with the
double sign, the cosine of y may be either positive or nega-
tive. The first value will obtain when the angle is acute,
and the second when it is obtuse.
But the angle y will be acute or obtuse according to the
position of the force P ; in the first case, the force falls above
the plane xky, and the co-ordinates z of the points in the line
representing the force, will therefore be positive ; in the
second, it falls below xAy, and the co-ordinates z will then
be negative.
The same observations may be extended to the angles »
and /3 considered with reference to the axes o( x and y ; so that
in general the cosines will be affected with the same signs as
the co-ordinates x, y, z, reckoned from A.
51. The signs of the cosines may also be determined by a
rule which is founded on Art. 10. Thus, if Ax {Fig. 29)
represent the line of direction of a component, this compo-
nent will be positive when it acts in the direction from A
towards x, but negative if it acts from A towards x'. The
tendency of the force in the first case will be to remove the
point A from the origin O, but in the second to cause its ap-
proach. Hence, we derive the following rule : A compoiiem
is positive when it tends to increase the co-ordinate of th(
point of application, and negative when it tends to diniinisl
this co-ordinate.
FORCES APPLIED TO A POINT. 27
Of Forces situated in Space, and applied to a Point.
52. Let P, P', P", &c. represent different forces which so-
licit a point A, and let there be drawn through this point the
three rectangular axes Kx, Ay, Az ; represent by
«, /3, y, the angles formed by the force P with the axes of co-
ordinates,
«', |3', y', the angles formed by P' with the same axes,
«*',/3",y", the angles formed by P" with the same axes,
&,c. &c. &c.
By resolving these forces into components acting along the
three axes, we shall obtain (Art. 48)
P cos «, P cos /3, P cos y, compoueuts of P,
P' cos et'j P' cos Q', P' COS y', Components of P',
P"cos«", P" cos ô",P", cosy", Components of P".
If we defer, as in Art. 40, the determination of the signs of
the cosines of these angles until the formulas are applied to
a particular example, and denote by X, Y, and Z the com-
ponents of the resultant, directed along the three axes, we
shall have
P cos «+F cos a'+P" cos «"-F&c.=X (9),
P cos /3 + P' cos /3' + P" COS /3" + &c. = Y (10),
P COS y+P' cos y' + P" COS y"+&C. = Z (11),
53. But X, Y, and Z being the projections AB, BC, and
CD of the right line AD, which represents the resultant R
{Fig. 28), we shall obtain (by Art. 46)
AB^ -f BC^ +CD2 =AD%
and consequently,
X2-fY2+Z='=R^
The intensity of the resultant will thus be determined, being
expressed by the equation
R=^(X2+Y='-f-Z») (12).
Again, if we call a, 6, and c the angles formed by the result-
ant with the co-ordinate axes, the components of R directed
along the axes will be
R cos a, R cos b, R cos c ;
28 STATICS.
and since these components have been represented by the
quantities X, Y, and Z, we shall have
X=R cos a, Y=R cos 6, Z=Rcosc;
whence,
X , Y Z
cos «— p"> COS 0=-jîj cosc = :5- (13).
If the forces P, P', P", &.C., and the angles a, js, y, «', (3', y', dec
are known, the values of X, Y, and Z will result from the
equations (9), (10), and (11). These values being substituted
in formula (12), the intensity of the resultant will be deter-
mined, and its position will become known from the equa-
tions (13).
54. If an equilibrium subsists, the resultant becomes equal
to zero, and the equation (12) then assumes the form
X=+Y3-fZ2=0.
And since this equation cannot be true unless the terms are
separately equal to zero, we have
X=0, Y=0, Z = 0.
These values reduce the equations (9), (10), (11) to
P cos «-f-P' cos a' + P" cos <«"-}-&.C. = i
P cos i3 + P' cos /3'+P' COS /3"-l-&c.=0 > (14).
P COS y+P' COS y'-fP" cos y"-|-&c.=0 7
Such are the conditions of equilibrium of a system of forces
situated in any manner in space, and applied to a point.
55. If we determine the resultant of all the forces in the
system except one, the remaining force will be found equal
and directly opposed to this resultant. For, let R' represent
the resultant of all the forces except P ; X', Y', and Z its
three components, and a, b\ and c the angles which its direc-
tion forms with the co-ordinate axes ; we shall have
X'=P' cos «'+P" cos «"-f P" cos «'"-f&c,
Y'=P' cos /s'-f P" cos i3"4-P" cos /3 "+&C.,
Z'=P' cos y'-f P" cos y"-f P" cos y"+&C. ,
and by means of these values the equations (14) may be re-
duced to
P cos «-f X'=0,
Pcos/3-fY'=0,
Pcosy-fZ'=0;
FORCES APPLIED TO A POINT. 29
and eliminating X', Y', Z', by the equations
X'=R' cos a, Y'=R' cos 6', Z =R' cos c',
there results
P cos «=— R' cos d \
P cos /3= — R' cos 6' > (15).
P cos y = — R' cos c j
Taking the sum of the squares of these three equations, we
obtain
P*(cos2«-f cos»/3-f cos'y)=R'2(cos2a'4.cos26'4-cos2c') ;
and since the second factor in each member is equal to unity, / ^
this equation reduces to
P3=R2, or P=R'.
The force P is regarded as essentially positive, its position
being determined by the rule explained in Art. 35, &c
If the value of P be substituted in equations (15), the factor
R' being suppressed, those equations will become
cos «= — cos a (16),
cos /3=— cos h' (17),
cos y= — cos c (18).
The relation between the values of cos « and cos a indicates
that d and « are supplements of each other. For, if cos a'
1)6 represented by AC [J^ig. 30), cos a. will be represented by
AC'=AC ; whence a'=DAC, and «=DAO.
But these two angles are supplements of each other ; for,
AC being equal to AC, gives the angle DAC=D'AC'; whence,
by substituting this value in the equation
DAC+DAC'=2 right angles,
we get
D AC4-DAC=2 right angles,
or the angles a and « are supplements of each other.
In the same manner may it be proved by the equations
(17) and (18), that the angles h' and are supplements of
each other, as also are the angles c and y.
It results from what precedes that the forces P and R' are
directly opposed ; for, if R be supposed situated above the
plane of a:, y, having the co-ordinates x and y both positive,
30 STATICS.
P will be situated below this plane, and will have the co-ordi-
nates X and y both negative.
56. After reducing all the forces to three rectangular com-
ponents X, Y, Z, it was shown that the resultant R would
be represented by the diagonal of a parallelopiped, whose
contiguous edges were respectively equal to X, Y, and Z
{Fig. 27). The equation of this resultant, which is repre-
sented by AF, will therefore be that of a right line passing
through A, the origin of co-ordinates, and through the point
F, whose co-ordinates are equal to X, Y, and Z.
57. The case may be rendered yet more general by sup-
posing that the point of application of the forces has the
three co-ordinates x, y\ and z ; the co-ordinates of the point
F will then become {Fig. 31)
y-f X, y'-FY, z'-^Z.
And the equations of the resultant, being that of a right line
in space, will be of the form
z=ax-\-h, z=ay-\-b' (19) :
substituting in these equations the co-ordinates of the point
F; in place of the quantities x, y, and z, we find
z'-{-Z=ax'-\-aX+b, z'-\-Z = ay'+àY+b' (20);
but the co-ordinates of the point A should also satisfy the
equations (19), and therefore we obtain
z'=ax'+b, z'=ay-\-b' (21).
Subtracting these last from equations (20), we have
Z^aX, Z=a'Y;
whence,
Z , Z
a=X> «=Y-
Again, by eliminating è and h' between the equations (19)
and (21), we find
z—z'=a{x—x'y, z — z'=a'{y—y'):
and by substituting the values of a and a' previously ob-
tained, the equations of the resultant finally become
z-z~{x-xl z-z'=^{y-y'].
P
EQUILIBRIUM OP A POINT UPON A CURVED SURFACE. 3^1
Of the Conditions of Equilibrium of a Point acted upon
by several Forces, and subjected to the Condition of re-
maining upon a Given Surface.
58. The material point to which the forces P, F, P", &c.
were applied, has been supposed hitherto to submit freely to
the action which those forces exert ; but if, on the contrary,
the point were required to remain constantly on a given sur-
face, the equations (14) would no longer be applicable, and
the condition of the resultant being equal to zero, which was
then necessary, would, under this supposition, be replaced by
the condition that the resultant must be normal to the given
surface. For, if the direction of the resultant be oblique
to the surface, it can be decomposed into two forces, of which
one shall coincide with the direction of the tangent, and the
other with the normal : the first would cause the material
point to slide along the surface, while the second would be
overcome by the reaction of the surface. Hence, it follows
that the resultant of all the forces must act on the point in
the direction of the normal to the surface, and since the re-
sultant is destroyed by the resistance of the surface, we may
regard this resistance as a force directly opposed to the nor-
mal force, and denote its intensity by a quantity N.
If the intensity of the force N and the angles 6, 6', 6% which
it forms with the co-ordinate axes, were known, it, would be
sufficient to add to the equations of equilibrium the compo-
nents N cos 5, N cos 6\ N cos 6" of the force N ; we should thus
obtain the equations of equilibrium
N cos 6-\-V cos «s-f P' cos x-j-V" cos «"-|-&c.— 0,
N cos 6' + F cos /3-f-P' cos B' + V COS (3"-f &c. = 0,
N COS Ô' + P COS y + P' COS y+P'coS y "-{-&C. = 0.
59. These equations may be simplified by representing, as
in Art. 52, by X, Y, and Z, the sums of the components par-
allel to the three axes ; the equations will thus become
N cos ô-|-X=0, N cos o'-f Y=0, N cos â'-f-Z=0 (22).
60. To determine the values of the unknown quantities cos
ê, cos 6', COS ê", and N, we will suppose L=0 to be the equation
of the given surface, and x, y\ and z the co-ordinates of the
32 STATICS.
material point to which the forces are apphed, and which by
hypothesis is required to remain on this surface. The nor-
mal being a right hne passing through the point whose co-
ordinates are x\ y', z', its equations will be of the form
x~x=a{z—z\ y—y = h{z—z') (23).
The differences x — x\ y — y\ z — z, which enter into these
equations, represent the projections of the right line on the
axes of co-ordinates. To determine the relations existing
between these projections and the angles <!, 6\ 6", let MN {Fig.
32) represent the right line in space referred to the co-ordi-
nate axes whose origin is at the point O, and denote by x, y,
z, x\ y', z', the co-ordinates of the points N and M : if a plane
DF be passed through the co-ordinates MD—z and BD=y',
and a second plane EG through NE=2; and EC =y, these
two planes will be parallel to that of y, z, and the distance
between them will be measured by the part 'QC—x—x inter-
cepted on the axis of x : but since every parallel to this axis
is likewise perpendicular to the two planes, it follows that by
drawing through the point M, the extremity of the co-ordi-
nate z', the parallel MP to the axis of x, this parallel will be
perpendicular to the plane EG, and will intersect it at a dis-
tance MP=ar — x'.
But, by connecting the point P with N, the point at which
the right line MN intersects the plane EG, a triangle will be
formed right-angled at P, since MP is perpendicular to the
plane EG. Hence,
MP=MN cos M,
or,
x—x'^MN cos 6 ;
but MN being a right line passing through the two points
whose co-ordinates are x, y, z, x, y', z\ its length will be ex-
pressed by
^[{x-x-y +{y—yy Hz-zf].
Substituting this value in the preceding equation, we deduce
X — x'
^°^ ^^ Ai.x-xy-\-{y-yYHz-zYY
In like manner, by drawing planes through the co-ordinates
x\ z\ and a:, z, parallel to the plane of ar, z^ and through x. y\
EQUILIBRIUM OP A POINT UPON A CURVED SURFACE. 33
and X, y, parallel to the plane of x, y, we shall find for cos 6^
and cos 6", the similar expressions
, y — y'
cos Ô .— - ^ '^
cos 6"-.
z — z'
■ ^[[x-x'Y +{:y-yy +{z- z'Y]
by eliminating the values of x — x\y — y\ by means of equa-
tions (23), and suppressing the common factor z — z\ we obtain
a , h
cos 6 = , „ , , „ . ix ; COS 6 —■
']
COS 6''--
^(«=^+6=^+1)' -^{a^^ ^b- + iy \^ ^24^
61. These values, which serve to determine the direction
of the normal, contain the quantities a and b, which are yet
unknown. The values of these quantities will now be de-
termined. Let L=0 be the equation of the surface which
passes through the point x', y\ z ; if we draw through this
point a plane tangent to the surface, the equation of this
plane will be of the form
Kx+By+Cz-\-I>=0]
and since it must be satisfied by the co-ordinates x\ y\ z', we
shall have
Ax'+By'-^Cz'+B^O.
Eliminating D between these two equations, the equation of
the tangent plane to the surface becomes
A{x-x')+B(y~y')-{-C{z-z)=0 ;
and dividing by C, it may be put under the form
^(x~x'n^{y-y')-{-{z-z')=0 (25).
But if the plane be tangent to the surface whose equation is
dz' dz'
L=0, the values of -j^, and j^ deduced from that equation,
will be expressed as follows :
dz' A dz' B
rf?=~"C' dy'^~C ^^^^•
And from the known principles of analytical geometry, when
a plane whose equation is Ax-\-By-\-Cz+D=0 is pcrpen-
C
34 STATICS.
dicular to a right line represented by the equations x=az-\-»,
y=hz-\-^, the following relations between the constants
exist :
A B ^
the equations (26) will therefore reduce to
£:=_.|;=-......(2T).
62. The values of these coefficients must now be deter-
mined from the equation of the surface. We obtain by dif-
ferentiating,
d\j , , dh , , dlj J ,,
dx dy dz
whence we infer that
dL^ dL
dz=z — '^dx — ^dy ; .
rfL dL ^
dz dz
and by applying this equation to the point of tangency, for
which the co-ordinates are x\ y\ z\ we find
dL_ rfL_
dz'_ _dx' dz _ dy'
dx' ^' ~d^'~~~dO
dz' d^
substituting these values in the equations (27), they become
dL dL
dL ' dL
dz dz
Replacing a and b in equations (24), by their values found
above, we obtain, after reduction,
dL
cos êz= ±
dx'
^/]&-m'^m
cos 6'= ±
dy'
^liÈr-ar^m'ï
EQUILIBRIUM OF A POINT UPON A CURVED SURFACE. 35
dh
COS ^ = ±
The double sign is here prefixed to the values of cos ^, cos ô',
COS 0", for the purpose of indicating that the resistance op-
posed by the surface may be exerted either in the direction of
the normal or along its prolongation, according as the body is
placed on the concave or convex side of the surface. The
form of these equations being inconvenient for the purposes
of calculation, they may be simplified by making
^ =V (28);
s/\m^m-{W\
which reduces them to
cosfl=V— ^, costf=V^— , cosfl =¥-—,;
ax ay' dz
substituting these values of the cosines in equations (22), we
obtain
NV^+X=0, NV'^.+Y^O, NV^4.Z=0 (29).
dx dy dz
63. The value of IN remains to be determined. If we
transpose X, Y, and Z in the equations (29), and take the
sum of the squares of the three equations, we shall obtain
and reducing by means of equation (28) there results
N2=X2+Y2+Z2,
whence,
N=^(X='+Y-+Z=') (30).
This value of N is precisely the same as that of the resultant
of the entire system ; but its components should be aftected
with signs contrary to those of the components of the result-
ant, since its action is exerted in an opposite direction.
Thus, having determined the resultant of all the forces P, P',
P", &c., the reaction of the surface will be equal to this re-
sultant, but will be exerted in an opposite direction.
C2
36 STATICS.
64. If the direction of the normal force be parallel to the
axis of 2r, we shall have
<j=90=, ^'=90°, 6"=0, or o"=180° ;
whence
cos 0=0, cos ô'=0, cos $"= ± 1 :
and the equations (22) will therefore reduce to
X=0, Y=0, N±Z=0;
which prove that the components in the direction of the tan-
gent plane destroy each other, and that the reaction of the
surface in the direction of the normal is equal to the sum of
the components directed along the axis of z.
65. The nature of the problem may also be such that
having given the forces P, F, P", &c. and the equation of the
surface upon which the material point should rest, it might
be required to determine x', y\ and z^ the co-ordinates of the
point at which the forces should be applied in order that the
material point should be sustained in equilibrio.
To resolve this problem, we first eliminate the quantity N,
by combining the equations (29) ; the factor V will likewise
disappear, and we shall then have
d-^ dz dy dz '
these equations, in conjunction with that of the surface, will
serve to determine the co-ordinates x', y\ and z of the point
of application.
Of the Conditions of EquiUhrium of a Point acted on by
several Forces, and subjected to the Condition of remain-
ing constantly on two Curved Surfaces, or on a Curve of
Double Curvature.
66. If a material point be retained on two curved surfaces,
it cannot remain in equilibrio unless the force which solicits
it can be decomposed into two components which shall He
respectively normal to the given surfaces ; for, if one of these
components had a different direction, it might be decomposed
into two forces, of which the first normal to one of the sur-
faces, should be destroyed by the reaction of the surface, and
and
EQUILIBRIUM OF A POINT UPON TWO SURFACES. 37"
the second tangent to the same surface, would move the body-
along the surface.
Let N and M represent the reactions of the two surfaces,
and 6, e', e", «, „', «s" the angles formed by their normals with
three rectangular axes drawn through the point to which the
forces are applied : by adoptmg the same course of reasoning
as in Art. 59, we shall obtain
N cos ê +M cos »» +X=0 ^
N cos 6' +M cos „' +Y=0 V (31>
N cos o'+M cos V'+Z=0 )
The equations of the surfaces L=0 and K=0 being differ-
entiated, make known, as in Art. 62, the values of the quan-
tities cos ê, cos s', cos 6", cos a, cos «', cos >)", and by adopting
abbreviations similar to those of Art. 62, making
± I _=u,
we shall find
„c?L ^-,dK
cos 6=\-T—, , COS t}=\J--—, ,
ax ax
,, T7-c?L , -r^dK.
cos^=V— ^. cos;î=IJ--;,
di/ dy
^^dL „ T,dK
cos 6 = V-— , , COS !,'=U— - :
dz dz
which values, being substituted in the equations(31), give
dx dx
NV^^+MU^-fY=0 y (32).
dy dy'
NV^-fMU^+Z=0
dz dz
From these three equations the unknown quantities M and
N may be readily eliminated ; and since U and Venter into
.them in the same manner as M and N, ihey will also disap-
38
STATICS.
pear in the elimination : or, to simplify the case, we may
regard MU and NV as the unknown quantities, which, being
eliminated between the three preceding equations, will give
an equation of condition including one or more of the three
variables. This resulting equation being combined with
those of the surfaces, viz. L=0, K=0, will determine the co-
ordinates .X-', y', z\ of the point sought.
It may be proper to remark that the radicals, which would
serve to complicate the expressions, disappear at the same
time as the quantities U and V.
67. When the point is subjected to the condition of re-
maining on a curve of double curvature, such curve may be
regarded as being formed by the intersection of two curved
surfaces. The equations of these surfaces being represented
as above by L=0 and K=0, the co-ordinates of the points in
which they intersect will necessarily appertain to both sur-
faces, and the quantities x\ y\ and z' may therefore be re-
garded as having the same values in each of these equations ;
but we have also the equation of condition referred to in
Art. 66 ; thus by eliminating the values of two of the co-
ordinates, the third will be expressed in functions of known
quantities : denoting by A, B, and C the values of the func-
tions corresponding to each of the co-ordinates x\ y\ and z\
we shall have
x'—K, y'=B, z—G.
68. It may occur that the equation resulting from the elim-
ination of M and N will not contain either of the variables.
This case presents itself when the surfaces become planes ;
their equations L=0 and K=0 may then be put under the
form Aa:+By+C;::+D=0, and the differential coefficients are
then constant. Under such circumstances the values of the
intensities M and N determined by the equations (32) become
independent of the co-ordinates x', y\ and z ; and since these
co-ordinates still apply to any points common to the two planes,
it follows that the conditions of equilibrium will be fulfilled,
if the forces be applied to any point whatever in the common
intersection of the two planes. A similar remark is appli-
cable to Art. 65.
PARALLEL FORCES. 39
Of Parallel Forces.
69. The forces which have been considered in the pre-
ceding paragraphs were supposed to have a common point
of apphcation ; but if they were appHed to different points
of a body or system of bodies, the points being retained at
fixed distances by means of their connexion with the inter-
mediate points, we might regard the forces as having their
points of apphcation united by means of inflexible right
hnes.
70. Let there be two parallel forces P and Q, applied to the
extremities of a right line AB {Fig- 34), which intersects at
right angles their common direction. It has been proved
(Art. 22) that the intensity of the resultant of these forces will
be equal to the sum of the intensities of the two components,
and that its point of application O will divide the line AB in
the inverse ratio of the two forces. This proposition may be
demonstrated in another manner, provided we admit that of
the parallelogram of forces, which is susceptible of direct
proof
Let the two parallel forces be represented by the right lines
AP and BQ, proportional to their intensities {Fig. 33) ; we
can add to the system, without changing the value of the
resultant, the two equal and opposite forces AM and BN,
and the four forces AP, AM, BQ., and BN may then he re-
placed by the two AD and BI, the diagonals of the rectangles
MP and NO.. But since these diagonals intersect at the
point C, the forces AD and BI may be conceived to be applied
at that point, and will be represented by CE=AD and
CF=BI. If the forces CE and CF be then decomposed into
rectangular components, by constructing the rectangles GL
and HK, having their sides respectively equal and parallel to
those of the rectangles MP and NQ,, we shall replace CE
and CF by the four forces CL, CK, CG, and CH. But the
last two are equal, being equivalent to the forces AM and
BN; which by hypothesis are equal, and being directly op-
posed, they must mutually destroy each other ; there will
tlxerefore remain at the point C, the two forces CL and CK
40 STATICS.
equal respectively to P and Q,, and having the common direc-
tion of the hne CO. The resultant of these two forces must
evidently be equal to their sum ; and if ii be denoted by R,
we shall have the relation
R=P4.Gl:
but since the resultant may be applied at any point in its line
of direction, we will consider it as acting at 0, the point in
which it intersects the line AB ; the position of this point
may be determined thus : the two similar triangles CAO,
CEL give the proportion
CO : AO : : CL : EL,
and the triangles COB, CKF give
BO : CO : : KF : CK ;
whence, by multiplication, suppressing the common factor
CO, we have
BO: AO:: CLxKF:ELxCK.
But KF and EL, being equal to BN and AM, which by hy-
pothesiG are equal to each other, the proportion reduces to
BO : AO : : CL : CK :
and since CL and CK are equivalent to the lines AP and BQ,
which represent the intensities of the given forces, the pro-
portion may be written
BO : AO : : P : a (33).
Hence we conclude that the point of application O of the two
parallel forces P and Q. divides the line AB into two parts,
reciprocally proportional to the intensities of those forces.
71. From the above proportion we obtain {Pig. 34)
BO+AO : AO :: P + Q : Q,
or,
AB : AO : : R : a (34).
And from the equations (33) and (34) combined, we find
P : a : R : : BO : AO : AB ;
from which we derive the following rule : T/ie jyarts AO,
BO, avd AB compj'ised between any two of the forces P, Q,
and R, will he co'nstanily 'proportional to the tJii)'d force. The
term R, for example, in the above proportion; corresponds: to
PARALLEL FORCES. 41
the portion AB, which is included between the forces P
and Q,.
72. If from the known values of P, Q, and AO, it were
required to determine that of BO, the proportion would give
a : P : : AO : BO ;
whence,
BO=P^_^.
a
73. Reciprocally, if there were given the force R applied at
O, and we wished to resolve it into two parallel components
whose points of application should be A and B ; by denoting
the unknown components by P and Q, the value of the first
would result from the proportion
AB : BO : : R : P ;
and that of the second would in like manner be obtained by
means of the proportion
AB : AO : : R : CI.
From these two proportions we deduce
p_RxBO RxAO
^~~AB~' ^ AB~'
In the preceding demonstration, the forces P and Q. have
been supposed perpendicular to the line AB ; but if they were
oblique to the direction of this line, we might draw through
O, the point of application of the resultant {Fig. 35), the right
line CD, perpendicular to the direction of the given forces,
and the force P applied at A would have the same effect as
though it were applied at the point C. In like manner, the
point of application of the force Q, may be transferred from
B to D : and since we have the proportion
P : a : : OD : OC,
we shall likewise obtain from the similarity of the triangles
OBD, AOC,
P : a : : BO : AO.
74. When the forces P and Q. act in opposite directions, the
resultant is equal to the difference of these forces. For, let
S {Pig. 36) be the resultant of the forces P and R, which are
supposed to act in the same direction, we shall then have
42
STATICS.
S=P+R (35);
and if we replace S by a force Q, equal in intensity, and
directly opposed to it, an equilibrium will subsist between the
three forces P, R, and Q, : we may therefore regard R as
beirig- equal and directly opposite to the resultant of the forces
P and Q, and the equation (35) will give for the intensity of
this resultant
R=S-P;
but S and Q, being equal in intensity, we have, by substi-
tuting the value of S,
R=Q— P.
The point O at which the resultant is applied, may be
found by the proportion
AB : BO : : R : Q,
whence we obtain
TJ/-V AL XQ,
R~'
or, replacing R by its equal Q— P, we have
QXAB
From this value of the distance BO, we infer that the point
O will be farther removed from B in proportion to the dimi-
nution of the quantity Q,— P ; if therefore Q and P become
equal, BO becomes infinite, and R becomes equal to zero :
hence, if two parallel and equal forces act in contrary direc-
tions, but are not directly opposed, the equilibrium cannot be
established except by the application of an infinitely small
force at a point whose distance is infinite ; it is therefore im-
possible in such cases to find a single finite force which shall
sustain in equilibrio the two forces P and Q, ; or, in other
words, the two forces P and Q, cannot be replaced by a single
resultant. The efifect of these forces will be simply to turn
the line AB about its middle point C.
75. These pairs of parallel and equal forces, acting in con-
trary directions, but not directly opposed, are called couples.
76. The results obtained in the preceding articles may be
applied to any number of forces. Thus, let P, P', P", F", P"',
{Fiff. 37) represent parallel forces applied to the points A, B,
PARALLEL FORCES. 4S
C, D, E, which are connected together by inflexible right
lines ; the point of appHcation and the intensity of the result-
ant may be readily found. For, the forces P and P' being
compounded, their resultant will be applied at a point M,
whose position may be determined by the following proportion,
AB: AM::P4-P':P'5
whence,
J^_ABXF
the line MC being then drawn, we can determine the point
of application N of the resultant of the forces P-fP' applied
at M, and of the force P" applied at C ; for we have
MC:MN::P+P'+P":P";
from which the value of MN results,
mcxp:^
P+P'+P"
By connecting the points N and D, the point of application
O, of the four forces P, P', P", P ", may be found in a manner
precisely similar, and lastly, by joining the points O and E,
we shall determine the point K at which the resultant of the
entire system must be applied.
77. When some of the forces of which the system is com-
posed exert their efforts in a contrary direction, we reduce the
components P, P', P", «fee, which are supposed to act in the
same direction, to a single resultant equal to their sum, and
likewise the components Q, Q.', Q,", (fee, which are supposed
to act in a contrary direction, to a second resultant equivalent
to their sum ; then, having determined the points of applica-
tion K and L {Fig. 38) of these two resultants, the system
will be reduced to two parallel forces, the one applied at
K, and equal to P+P'+P" (fee, the other at L, and equal
Q,+Q,+(^" «fee. : the resultant of these two forces may then
be determined by the method explained in Art. 74.
78. If the forces P, F, P", P'", (fee. {Fig. 39), retaining
their points of application, and continuing parallel, assume
the positions AQ, BQ,', CQ", DQ,",' «fee, the resultant will be
parallel to the new directions of the forces, but its intensity
and point of application will remain unchanged ; for, the
44 STATICS.
construction employed to determine this resultant, being-
dependent only on the intensities of the forces and their
points of application, the data of the problem will remain the
same.
79. If, for example, the forces P and P' should assume the
positions represented by the parallels Ad and BCi,' ; there
would be given P, P', and the line AB, to determine the posi-
tion of the point M ; and this would be determined from the
same proportion as when the forces were directed along the
lines AP and BP'.
The point through which the resultant of a system of par-
allel forces constantly passes, whatever may be the direction
of those forces, is called the centre of parallel forces.
80. To determine the co-ordinates of the centre of parallel
forces, let P, P', P", &c. represent the intensities of the several
forces, and denote by "
a;, y, z^ the co-ordinates of the point of application M of
the force P,
x\ y\ z' those of M',
x",y",z" those of M",
a:,, yi, Zi, those of the centre of parallel forces.
If we represent by N {Fig. 40) the point of application of
the resultant of the two forces P and P', we shall have
MM' : M'N : : P-f P' : P ;
and by drawing the line ML' parallel to HH', the projection
of MM' on the plane of x, y, the similar triangles ML'M',
NLM' will give
MM' : M'N : : ML' : NL ;
whence, by combining the two proportions,
ML' : NL : : P+F : P ;
from which results the equation
(P+P')NL=PxML':
adding to each member the product (P+P')LK, we have
(P+P )(NL+LK^ =P(ML'+LK)+P' X LK ;
and since
NL-{-LK=NK,
PARALLEL FORCES. 45
ML'+LK=MH,
LK=M'H',
the preceding equation may be reduced to
(P+F)NK=PxMH4-FxM'H'.
If we denote by Q, the resultant of the two forces P and P',
and by Z the co-ordinate of its point of application, this
equation may be written under the form
QZ=P;r+PV-
in Uke manner, representing by Q,' the resultant of the paral-
lel forces Q. and P", and by Z' the co-ordinate of the point at
which it is applied, we obtain
a'Z'=az-hPV;
and thence, by substitution,
Gi'Z':=Fz+l?'z'+V"z".
If the resultant of the entire system be represented by R^
and the co-ordinate of its point of application, parallel to the
axis of ;r, by z^, we shall obtain, by continuing the same pro-
cess, the general relation
Rz,=Pz-fPV+P"z"+&c (36).
81. The 7nom,ent of a force with reference to a 'plane
is the jiroduct of the intensity of this force hy the distance
of its point of application from the plane. The preceding
equation therefore expresses that the moment of the residtant
of the parallel forces P, P', P", ^«c, taken with reference to
the plane of x, y, is equal to the sum of the moments of the
several forces taken with reference to the same plane.
The moments being taken with reference to the other two
co-ordinate planes, we have
Ry.=Py+Py+PV'+(fec (37).
R2:,=Px-{-PV-f PV'-|-&c (38).
82. When the co-ordinates x, y, z, x, y\ z\ &c. of the points
of application, and the intensities P, P', P', &c. of the forces,
are given, the intensity of the resultant will become known,
being equal to the algebraic sum of the several intensities ;
and the values of the co-ordinates .t,, y,, and sr,, of the centre
of parallel forces, will be found from the equations (36), (37)i
and (38).
46 STATICS.
83. The forces are affected with the positive or negative
sign according to the directions in which they act ; and since
the signs of the co-ordinates are Ukewise determined by their
positions with respect to the origin of co-ordinates, the mo-
ments of the forces must be regarded as positive, when the
forces and co-ordinates have the same sign, but negative
when the two have contrary signs.
84. If the several points of application M, M', M", &c. were
situated in the same plane MM" {Fig- 41), the plane of x, y
might then be assumed parallel to that in which the forces
are applied, and the co-ordinates z, z', z", &c., being com-
prised between two parallel planes, we should have
z=z'=z"=<fcc. :
hence, if z, represent the co-ordinate of the centre of parallel
forces, its value will also be equal to z ; for, its extremity
must be found in the plane MM", being determined by a con-
struction similar to that in Art. 76. Thus the quantity z
becomes a common factor in the equation (36) which then
reduces to
R=P4-P'-fP"-i-&c.
85. If the points of application were situated on the right
line AB {Fig. 42), which we will suppose parallel to the axis
of X, we should have at the same time
z=z'=z"=&c., and y=y'=y"=éLC. ;
the equations (36) and (37) would then reduce to
R=P4-P'+P"-}-&c (39),
and there would remain but the single equation
R:r,=Pa:-f PV+PV'-f&c (40).
In this case, we may dispense with the consideration of the
three axes, it being only necessary to estimate the co-ordinates
X, x\ x'\ &c. along the line AB, to which the forces are
applied.
For example, if we had
a:=9, .t'=3, a;"=— 3, a:"'= — 4.
P=-iP, P"=-|P, P"'=2P;
by substituting these values in the equations (39) and (40)
we should deduce
PARALLEL FORCES. 47
R=p_^P_ |P-f 2P=2P,
Ra:,=9xP-3xiP+3x|P— 4x2P=lx2P;
whence,
.T,=l.
86. For the purpose of determining the conditions of equi-
hbrium of parallel forces, we shall adopt as most convenient
that position of the axes in which one of the co-ordinate
planes is perpendicular to the direction of the forces : let this
be the plane of .r, y. Having reduced all the forces which
act in the same direction to a single resultant R^ {Fig- 43),
and those which act in a contrary direction to a second re-
sultant R^^, an equilibrium will take place in the system when
the two resultants are equal and directly opposed.
The latter condition will be fulfilled when the distance
C'C" is equal to zero, which requires that the co-ordinates x,
and y, of the point C should be respectively equal to x,, and
y^, those of the point C".
Hence, we obtain
The condition of equality between the two resultants will be
satisfied when we have
R. = -R (41);
and we obtain by multiplication
'S.,x,=-\,x, (42),
R.y.=-R.y. (43).
If we denote by P, P', P", &c. the components of R,, and
by P'", P% (fcc. the components of R,„ the property of the
moments will give the two equations
R,*-, =P:r + V'x' ■\- V"x" ^ &c.,
R^^x,, = V"'x"' -f P'" X'' -f P^x^ -f- (fcc. ;
and substituting these values in equation (42), it reduces to
Pa; + PV-fP"^"-t-FV' + P"'x''+P'':c'-H&c.=0 (44).
By the same course of reasoning, the equation (43) may be
reduced to
Py+py +P'y'4-P'Y"+P'>" +py +&c.=:0 (45).
48
STATICS.
And finally, the values of R, and R„ being substituted in
equation (41), give
P + F + P" + F" + P"+P'' + <fcc.=:0 (46).
87. If the equations (44), (4.5), and (46) are satisfied, the sys-
tem of forces will be in equilibrio. The conditions expressed
by these equations may be enunciated as follows : Aji equi-
librium will subsist in a system of parallel forces, if the sum
of the moments taken with reference to each of two rectangu-
lar planes parallel to the cominon direction of the forces, is
equal to zero ; the sum of the forces being at the same time
equal to zero.
88. An equilibrium will also take place if the resultant of
the system be supposed to pass through a fixed point, since
the effect of this resultant will then be destroyed by the re-
sistance opposed by the fixed point.
Of Forces situated iji the same Plane, and applied to Points
connected together in an invariable manner.
89. Let P, P', P", P'", &c. {Fig. 44) represent several forces
situated in the same plane, and applied to the points A, B, C,
D, &c., which are supposed to be connected in an invariable
manner. If the system admits of a single resultant, its po-
sition and intensity may be readily obtained by means of the
following graphic construction : — Having assumed the por-
tions Aa, B6, Cc, and Dc? proportional to the intensities of
the respective forces, prolong the lines Aa and B6 until they
intersect at the point G, and apply the forces P and P' at
this point. Construct the parallelogram GG', having its
sides respectively equal to Aa and Bè, and its diagonal GG'
will represent in direction and intensity the resultant of the
two forces P and P' ; again, by prolonging GG' and Cc until
they intersect, and constructing the parallelogram HH', whose
sides shall represent the forces GG' and Cc, the diagonal HH'
will represent the resultant of these forces, and will therefore
be the resultant of the three forces P, P', and P". Lastly, by
finding the intersection of HH' and Drf, and forming a thiid
FORCES APPLIED TO DIFÎ'ERENT POINTS. 49
parallelogram, its diagonal 11' will represent the resultant of
the entire system.
90. If by this construction we should find one or more pairs
of parallel forces, the resultant may be determined by the
methods explained in Arts. (71), (72), and (74), and its intensity
will be equal to the sum or difference of the forces. If the
system contain two parallel and equal forces, acting in con-
trary directions, but not directly opposed, we may combine
one of them with the other forces, and the construction of
Art. (89) may then be continued ; but if the entire system
can be reduced to two equal resultants acting in parallel
and contrary directions, but not directly opposed, we con-
clude, as in Art. 74, that a single resultant cannot be obtained.
91. If the construction should give a resultant equal to
zero, an equilibrium would subsist throughout the system. ,
92. The preceding construction is equivalent to supposing
the forces applied at the point I, in lines parallel to their primi-
tive directions, and then compounding them into a single result-
ant. For, by considering the forces P, Q,, and S {Fig. 45),
the resultant DC of the forces P and Q, being applied at the
point D' in its line of direction, may there be decomposed into
the two components D'P' and D'Q,', parallel and equal to
P and a.
93. To determine the analytical conditions of equilibrium
in a system offerees disposed like the preceding, we will first
consider the case of three forces P, P', and P", applied to
points which are connected in an invariable manner ; and
we shall then find it necessary that the directions of the forces
should intersect in a single point. For, since the forces
P and P' {Pig. 46) are supposed to be sustained in equilibrio
by the third force P", it is necessary that this third force
should be equal and directly opposed to the resultant of the
two forces P and P'. But P and P' intersect in a point
D ; this point is therefore situated on their resultant, and
consequently in the direction of the third force P".
If, on tne contrary, the force P" were not applied at the
point of intersection of the other two, it would intersect the
direction of their resultant R at some point E {Pig. 47), and
the right lines RD and P"E being then inclined to each other
D 5
50 STATICS.
in a certain angle P"ER, the forces R and P" could not main-
tain an equilibrium (Art. 16).
94. When the directions of the three forces P, P', P" inter-
sect in a point, this point may be considered as their point of
application, and the conditions of equilibrium will then be
the same as if the forces had been originally applied at their
point of intersection.
These conditions are,
P cos « + P' cos a' + P" cos «" + (fec. = 0,
P cos /3 + P' cos /3' + P" COS |S" + &C. = 0.
To these must be added the equation which expresses the
condition of their intersecting in a point.
95. Let P, P', and R {Fig. 48) represent three forces whose
directions intersect at the point A. If through the point C,
assumed arbitrarily, a right line be dra\\ai to the point A, and
perpendiculars CI, CI', CI" be demitted on the lines of direc-
tion of the forces, the right-angled triangles CAI, CAF, CAI"
will have the same hypotheneuse CA : this condition of a
common hypotheneuse will establish that of the forces inter-
secting at a single point, since it results from the triangles
having a common vertex. Through the point A draw the
right line AB, perpendicular to CA, and from the extremities
of the lines AP, AP', and AR, which represent the intensities
of the forces, demit perpendiculars PD, P'D', RD" on the line
AB : the right-angled triangles ACI and APD will be similar,
having the alternate angles CAI and APD equal to each
other, and the following proportion will therefore obtain :
AC : CI : : AP : AD,
and by calling AC=c, CI =7), this proportion becomes
c : ^ : : P : AD ;
whence we obtain
c
denoting by p' and r the perpendiculars CI' and CI", we find,
in like manner,
c c
But if R be the resultant of P and P', the component of R
FORCES APPLIED TO DIFFERENT POINTS. 51
in the direction of AB will be equal to the sum of the com-
ponents of P and P'j directed along the same line ; we con-
sequently have
AD" = AD + AD';
and by substituting in this equation the values found above,
it becomes
Rr_Pp Py
or, by suppressing the divisor common to the terms, it
reduces to
Rr=Pp+Py (47).
96. If the point C were situated within the angle formed
by the directions of the forces, or in the opposite angle, the
product of the resultant by the perpendicular r would then be
equal to the difference of the products of the two components
multiplied by their respective perpendiculars ; we should
thus have
Rr=Pp— py (48).
97. The moment of a force with reference to a plane has
been defined (Art. 81) to be the product of the intensity of
this force by the perpendicular on the plane from the point
of application. By analogy, we call the moment of a fmxe
with reference to a j)oint, the froduct of the force hy the per-
pendicular de7nitted on the direction of the force from the
assumed point. The equations (47) and (48) will therefore
express that the moment of the resultant of two forces is
equal to the sum or difference of the moments of its compo-
nents, according to the position of the point C. This point is
called the centre of 7noments ; and if it be situated within the
angle PAP', or LAL' {Mg. 49), the difference of the moments
must be taken, but if it fall without these angles, the moment
of the resultant will be equal to the sum of the moments.
98. These two cases may be comprised in a single enun-
ciation, by attaching to the word sum its algebraic significa-
tion,! and the moment of the resultant will then be equal to
the sum of the moments of the two components, in which
expression the terms may be affected either with the positive
or negative signs. ^
D2
52
STATICS.
99. The condition of the forces intersecting in a point
gives rise to the preceding theorem of the moments : from
this theorem the third condition of equilibrium may be de-
duced.
For, if two forces P and P' {Fig. 50) are sustained in equi-
hbrio by a third force P", this force must be equal in intensity
to the resultant of the other two, and must act in a direction
exactly opposite. If, therefore, a perpendicular ja" be demitted
on the line of direction of the force P", which is also that
of the resultant R, the principle of the moments will furnish
the equation
%"=:P/> + Py;
and replacing R by — P", since the forces are equal, and act
in contrary directions, the equation becomes
Pp+py+py=o.
Thus the conditions of equilibrium of three forces situated
in the same plane, and applied to different points, will be ex-
pressed by the three following equations : —
P cos ^+F cos«' + P" cos cc"=Q (49),
P cos/3 + Fcos0'+P" cos/3"=0 (50),
P/> + Py + P>"=0 (51).
100. If the number of forces be greater than three, we
may regard P as being the resultant of the two forces P"' and
P"' : we shall then have
P cos «=P"' cos *"'+P'^ cos «'",
P cos /3 = P"' cos /3"' + P^ cos /3",
Pp=P"y"+P'y';
and by substituting these values in equations (49), (50), (51),
they become
P' cos «' + P" cos «" + P"' cos «'" + P"' cos «"=0,
F cos (3'+P" cos ^" + F" cos |3"'+P" cos ^"=0.
V'p' + p>" + vy + Yy =0.
101. The same principle may be extended to any number
of forces, and we shall therefore obtain for the general equa-
tions of equilibrium of forces acting in the same plane, and
applied to different points,
P cos « + F cos a' + P" cos«" + &c.=9 (52),
FORCES APPLIED TO DIFFERENT POINTS. 53
P C0S/3 + F C0S/3'+P" COS /3" + <fec.=:0 (53),
Pp + Py + Fy' + &c.=0 (54).
102. A more convenient notation is sometimes employed
to express the existence of these conditions, the equations
being written in the following form : —
r(P cos x)=0, 2(P cos /3) =0, s(P;?)=0.
The character s is here employed to denote the sum of any
number of quantities of the same form as those included
within the parentheses.
103. The process which has led to the equation (47) fur-
nishes an easy method of recognising the proper signs of the
moments. For, if the point C, the centre of moments {Pig.
51), be chosen without the angle formed by the directions of
the extreme forces, and the forces be supposed to act by push-
ing, being at the same time firmly connected with the per-
pendiculars p, ^'^, />"; <^c., these forces will all tend to turn the
perpendiculars in the same direction about the point C ; but
if, on the contrary, the centre C be situated within the angle
formed by the directions of the extreme forces {Pig. 52), or
within the opposite angle, the forces P, P', P", &c., situated
on the same side of the point C, will tend to turn the perpen-
diculars in one direction, while the forces P'", P'', &c., on the
opposite side, will tend to turn the perpendiculars in a con-
trary direction. But the expressions —^, — i-, — ^, &c., rep-
c c c
resented by the lines AD, AD', AD", <fcc., being affected with
signs contrary to those of AD'", AD'", (fee, it follows that all
the forces whose moments are positive will tend to turn the
system in one direction, while those whose moments are
negative will tend to turn it in a contrary direction.*
* This demonstration is perfectly conclusive when the directions of the several
forces intersect in a point ; but the property of the moments is equally true
when the forces are not directed to a single point. For, by prolonging the
directions of any two of the forces P and P' until they intersect, and joining
their point of intersection with the centre of moments, it may be proved by the
reasoning employed in Art. 103, that the moment of their resultant is equal to
the algebraic sum of the moments of the two forces P and P', the signs of these
moments being determined by the directions in which the forces P and P' tend
to turn the system about the centre of moments. We shall thus have
54
STATICS.
104. If the system of forces be not in equilibrio, the
moment of tlie resultant will be equal to the excess of the
sum of the moments of those forces which tend to produce
rotation in one direction, over the sum of the moments of
those which tend to turn the system in a contrary direction.
105. It appears from the preceding remarks, that the
equation s(P/>)=0, expresses the condition that the sums of
the moments of the forces wliich tend to produce rotation in
the two directions are equal to each other.
106. If, in the system supposed in equilibrio, we suppress
one of the components, P for example, the remaining forces
will have a resultant R ; and since this resultant should be
equal in intensity, but directly opposed to the force P, the
equations (52), (53), and (54) will be replaced by the following:
R cos a=F' cos a' + P" cos a" + F" cos «"'4-&c.,
R cos b^V cos /3' + P" cos /3" + P'" cos /3"' + (fcc,
Rr = Vy + V"p" + P'>'" + &c. ;
or,
R cos « = 2(P cos et) =X,
R cos b =!:.(? cos /3)=Y,
Rr=s(P/>).
The double sign is not prefixed to the moment P;?, since we are at liberty to
assume arbitrarily the sign of one of the moments. The moment Rr, deduced
from this equation, may have either a positive or negative value ; if positive, R
and P will tend to turn the system in the same direction ; if negative, in con-
trary directions.
The forces P and P', being then replaced by their resultant R, this resultant
can be combined with a third force P", and we shall obtain, in a similar maruier,
R'r'=Rr±PY';
in which equation Rr, whatever may be its essential sign, may be replaced by
P^^Py. The sign of the moment P"p" will be similar to that of Rr, if P"
and R tend to produce rotation in the same direction, and dissimilar in the con-
trary case. But the moments Pj) and Rr will have like or unlike signs, according
as the forces P and R tend to turn the system in the same or in contrary direc-
tions. Hence the signs of the moments Vp and P"p" in the equation R'r'=
T'pizP'p':izP"p'\ will be like or unlike according to the directions in which the
forces P and P" tend to produce rotation.
The same reasoning may be extended to a greater number efforces.
FORCES APPLIED TO DIFFERENT POINTS. 55
107. By means of these equations, the position and mag-
nitude of the resultant may be determined.
For, the two first equations give
R2(cos2a + cos26)=X2 +Y2 ;
and since the sum of the squares of the two cosines is equal
to unity, we have
R2=X2+Y2.
The inchnations of the resultant to the co-ordinate axes may
also be determined from the same equations ; for we have
X Y
cos a=— , cos 6 =-5-.
K Jti
108. To establish its position in the system, we first deter-
mine the position of a right line AB, passing through the
origin, and parallel to the resultant. If cos b be affected
with the positive sign, the line AB must form with the axis
of y an angle less than 90° : it will therefore assume one of
the positions indicated in {Fig. 53). But if, on the contrary,
this quantity should have the negative sign, the right line AB
would then be situated in one of the positions represented by
{Pig. 54). Thus, whatever be the sign of cos 6, the line AB
may assume two positions, one in which the angle formed
with Kx will be obtuse, and another in which this angle will
be acute. The sign of the cos a will determine which of
these positions the line AB must assume.
Having thus established the position of the right line AB, let
a perpendicular /■ be drawn to it through the origin A, equal to
s(P«)
ji • This perpendicular will be represented {Pig. 55) by
AO or by AO', according to the sign of the quantity r ; and
the line OR or O'R', parallel to AB, will represent the true
position of the resultant.
109. To. obtain the equation of this resultant, it may be
observed that its line of direction will, in general, intersect
the axis of y at a certain point B {Pig. 56), and that the form
of its equation will therefore be
y=x tang D-f AB (55) ;
56 STATICS.
and since the angle which the resultant makes with the axis
of a; is denoted by a, we have T>=a, and consequently
-r, sin a cos b R cos b Y
tanff D= = =c; =-— .
cos a cos a K cos a X
The value of AB may be obtained from the equation
OA=ABxcosOAB.
But the angle OAB is equal to the angle D, since they are
both complements of OAD. The angle OAB can therefore
be replaced in the preceding equation by D or a ; and since
the line AO is the perpendicular from the origin on the direc-
tion of the resultant, it will represent the quantity denoted
by r ; we shall thus obtain
?'=AB cos a ;
and consequently,
AB=-!1-.
cos a
Substituting the values of AB and tang D in the general
equation (55), it becomes
Y r _Y Rr__Y Rr.
'^"X'^'^co^ ~ X'^'^Rc^i^^X''^'^ X '
whence, by transposition and reduction, we find
yX-a:Y=Rr]
or, replacing Rr by its equal ^(Pj)), the equation of the
resultant finally becomes
yX-xY=i:(Pp).
110. When an equilibrium subsists, X and Y are equal to
zero, and the equation reduces to s(P/?)=0, corresponding
with the result previously obtained.
111. The data requisite for the determination of the
resultant being, 1°. The intensities of the several forces; 2°.
Tile angles on which their dii'eciions depend ; and 3". The
co-ordinates of their points of application, it will prove con-
venient to transform the equation (54) into another, in which
the quantities />, p', p", ôcc. shall be replaced by the co-ordi-
nates of the points of application. To effect this transforma-
tion, let the origin of co-ordinates be assumed at A {Fig. 57),
FORCES APPLIED TO DIFFERENT POINTS. 57
and let x and y denote the co-ordinates of the point M to
which a force P is applied : the intensity of this force being
represented by MP, its components parallel to the axes of x
and y will be respectively
MN=P cos «,
MQ,=P cos /3.
From the point A demit the perpendiculars AO, AF, and AE
on the prolongations of the force MP and its two components j
we shall then have
OAxMP=the moment of the force P,
AF X MN=the moment of the component P cos «,
AE X MQ,=the moment of the component P cos j3.
But if we regard the forces as pushing the point M, the
resultant MP and the component P cos « will tend to produce
rotation in the same direction about the point A. Their
moments may therefore be alfected with the positive sign ;
while the component P cos /3, tending to turn the system in a
contrary direction, must be affected with the negative sign.
We shall thus obtain the equation
Pp=yP cos » — xY cos /3.
For a similar reason,
P'jo'=3/'F cos «'-.r'F cos /3',
P"p"=y"P" cos a:'—x"Y" COS /3"j
<fcc. &c. <fcc. ;
and by substituting these values in the equation of the mo-
ments (54), it becomes
P(y cos tt—x cos |3) + P'(y' cos «' — x' cos /s') + &c. =0 . . . . (56) t
we shall therefore have for the equation of the resultant, when
the system is not in equilibrio (Art. 109),
2/X — a:Y=s[P(2/ cos a— a: cos/3)].
112. Jn determining the signs of the moments in equation
(54), we had recourse to the rule explained in Art. 103, which
is somewhat foreign to analytical considerations ; but when,
by a transformation, this equation takes the form indicated
above (56), the signs of the moments will be immediately
determined by an application of the rule in Arts. 37 and 38,
regard being had to the signs of the co-ordinates. Thus, let
68 STATICS.
P be a force whose position with respect to the co-ordinate
axes is that represented in {Fig. 58). The value of its mo-
ment, being in general V{i/ cos « — x cos/î), will become appli-
cable to the particular case, by making x negative, y positive,
cos X negative, cos p negative : thus, when the signs are con-
sidered, the moment becomes
P( — y cos x—x cos |8).
113. It should be remarked, however, that we here adopt
tacitly an hypothesis relative to the signs, which consists in
regarding a moment as positive, when the direction of the
force CD {Mg. 57) intersects the axis of y positive, and then
cuts the axis of x negative.
114. The equations of equilibrium (49), (50), and (51)
imply the condition that the system may be reduced to two
forces equal in intensity and directly opposite. For, if we
denote by P cos «, P' cos «', &c. the components acting in one
direction parallel to the axis of x, and by P" cos «", P'" cos «"',
&c. the components which act in a contrary direction, the
equation (49) may be put under the form
P cos x + V cos «'-f &C.=P" cos x" + F"' cos a'" +écC.
But the forces P cos «, P' cos «', &c., being parallel, may be
compounded into a single force X', equal to their sum and
parallel to them ; and the forces P" cos «", P'" cos a", <fec.
may in like manner be replaced by a single force X" : the
entire system will thus be reduced to the two forces X' and
X", parallel and equal, but having contrary directions.
By a similar composition, the forces parallel to the axis of y
may be reduced to two resultants Y' and Y", equal to each
other, and having opposite directions.
The forces X' and Y' being then applied at the point M,
where their directions intersect {Pig. 59), and the forces
X" and Y" at their point of intersection N, we can construct
the rectangles MA and NB, whose sides MC, MD, NE, and
NF shall represent the forces X', Y', X", Y" : and since the
homologous sides of these rectangles are equal, their diago-
nals MA and NB will also be equal and parallel.
The equations X=0, Y =0, therefore, express that forces
fiituated in a plane may be reduced to two INI A and NB, equal,
FORCES APPLIED TO DIFFERENT POINTS. &»
parallel, and acting in contrary directions ; but they do not
express the condition that the two forces are directly opposed.
That this may occur, the equation s(P^)=:0 is likewise neces-
sary : for, calling R' and R" the two equal forces AM and BN,
and /•', r" the perpendiculars OP and OGl demitted from the
point O, since R' and R" act in contrary directions, their mo-
ments must be taken with différent signs, and the equation
s(Pp) =0 will be replaced by the following :
RV— R"r"=0.
But the intensities of R' and R" being equal by hypothesis, the
common factor will disappear from the equation, and it will
then become
r'— r"=0;
thus, the difference of the right lines OP and OQ, will become
equal to zero, and the points P and Q, will therefore coincide :
hence, the forces MA and NB will be directed along the same
right line.
It also appears that when the condition 2(Pp) =0 is not
fulfilled, and we have simply X=0, Y=0, the system may
be reduced to two parallel forces similarly situated to those
considered in Art. 74.
115. If, on the contrary, the condition s(P/>)=0 were alone
satisfied, an equilibrium could not subsist ; for the quantities
X and Y having certain values, a resultant might be found
whose intensity would be determined by means of the
equation
In this case, the equation s(P/»)=0, or its equivalent Rr=0,
can only be satisfied by making the factor r equal to zero ;
hence, the centre of moments must necessarily be found on
the line of direction of the resultant R.
116. If there be a fixed point on the line of direction of the
resultant, the equilibrium will be still maintained, and the
centre of moments being placed at this point, the condition
2(P/)) =0 will be satisfied ; if, for example, the forces P, P', P'^^
&,c. be supposed applied to the different points of a solid body,
and if the point C through which the resultant passes be im-
moveable, the effect of this resultant will be entirely destroyed
60 STATICS.
by the reaction of the fixed point, and the condition 2(Py>) =0
will be alone sufficient to ensure the equilibrium. It will
appear hereafter that the intensity of this resultant is a
measure of the pressure sustained by the fixed point.
117. If the system can be reduced to two parallel forces,
equal in intensity, but not directly opposed, the addition of an
arbitrary force S will render it susceptible of a single result-
ant. For the new force S must necessarily be either par-
allel or inclined to the direction of the forces ; in the first case
{Fig. 60), it may be decomposed into two parallel components
P' and Q.' applied at the points A and B (Art. 73), and the sys-
tem of three forces P, Q,, and S will be replaced by the two
unequal forces P + P' applied at A, and Q, — Q.' applied at B ;
these two forces will obviously have a single resultant.
If the new force S is not parallel to the other two, its
direction may be prolonged {Fig. 61) until it intersects the
direction of one of them at A'. This point being then taken
as the point of application of the forces P and S, they may be
compounded by constructing a parallelogram on their lines
of direction, and the direction of their resultant will intersect
that of the force Q, with which force this resultant may be
combined.
Of Forces acting in any tnanner in Space.
118. Let P', P", P'", (fcc. represent different forces situated
in space ;
x', 2/, z', the co-ordinates of the point of application of P',
x'^, y'\ z", those of P",
x"', y'", z"\ those of P'",
&c. &.C. &c. ;
ce'j /3', y', the angles formed by P' with the axes of co-ordinates,
tt\ /3", y'', those formed by P" with the axes,
«'", 0'", y'", those formed by P'" with the axes,
&c. ■ (fcc. (fee.
Let us investigate the conditions of equilibrium iw this sj'-s-
tem, and endeavour to discover if these conditions cannot be
Ïi-ORCES ACTING IN STfACE. 61
rendered dependent on those which have been obtained in
the preceding cases. We first attempt to decompose all the
forces of the system into two groups, one of which shall con-
sist of parallel components, and the second of forces situated in
the same plane. Since the axes of co-ordinates may be as-
sumed arbitrarily, we will endeavour to decompose the forces
in such manner that a certain number of them may be in
the plane of a;, y, and the remainder be parallel to the axis
of z.
119. If in the given system there be no force parallel to
the plane of a;, y, the proposed decomposition may be readily
effected ; for, let one of the forces be represented by P', its
point of application being at W {Fig. 62); prolong the hne
of direction of this force until it intersects at C the plane of
.r, y, and transferring the point of application to C, decom-
pose the force P' into two others, one C'L parallel to the axis
of 2:, the other C'N in the plane of .r, y.
120. But if the force P' is parallel to the plane of a:, y, a
similar decomposition cannot be effected, and some other
mode of decomposing the forces must therefore be adopted.
For this purpose, let there be drawn through the point M'
{Fig. 63) a line parallel to the axis of 2;, and to the point M'
let there be applied along this line, and in contrary directions,
the two forces M'O and M'O', having intensities equal to g'
and —g' respectively. The introduction of these forces
cannot disturb the condition of the system, since the two
mutually destroy each other ; and we shall then have applied
at the point M' the three forces P', g\ and — g'.
The force P' may then be compounded with — g\ and by
calling their resultant R', we can replace in the system the
force P', by the two forces R' and g\ each of which must ob-
viously intersect the plane of a:, y.
121. Let the force R' be now applied at C, the point in
which its line of direction intersects the plane of a:, y, and let
it be decomposed into two components, one situated in the
plane of .r, y, and the other parallel to the axis of z. The
force P' will thus be replaced by a force applied at C, and
lying in the plane of a:, y, and by two others parallel to the
axis of 2;, one applied at C, and the other at M'.
122. The co-ordinates of the points of application being
6
62
Î3TATICS.
necessary to express the conditions of equilibrium, those of
the point C must be determined.
The equations of the resultant R' which passes through the
point .t', y', 2;', have been found (Art. 57) to be of the form
Z
z — z'=^{x — x')
(57);
in which X, Y, and Z represent the projections of R' on the
co-ordinate axes. These projections being equal to the com-
ponents of R' parallel to the axes, the quantities X, Y, and Z
may be replaced by the values of the three components. But
R' being the resultant of P' and— ^', we may substitute for
F its three components F cos «', P' cos /S', P' cos y' ; and R'^
will then be the resultant of the four forces
P' cos a, P' cos /3', F cos y', —g^.
These forces acting parallel to the axes of co-ordinates, we
shall have
X=F cos «', Y=F cos /3', Z=F cos y'-g';
and by substituting these values in equations (57), we obtain
for the equations of the resultant R',
, P' cos y' —ff, ,v 1
F cos «' ^ I
Fcosy'-^, ,- ^
(58).
F cos /3'
123. To obtain the co-ordinates of the point C (Pig. 63),
at which the right line R' intersects the plane of x, y, we
make z=0 in the equations (58) ; and denoting by a, and 6,
the other two co-ordinates of the point C, we shall have
^,_Fcosy'— ^'
—z'
Z=:-
P' cos «'
P'cos y' — £r'
F cos /3'
(a—x'),
(i-yO;
from which we deduce
a =x
h,=t/'
Z'V cos a'
P' cos y' — g'
z'V cos iS'
P' cos y'—g"
(59):
FORCES ACTING IN SPACE. 63
these are the values of the co-ordinates of the point C, at
which the resultant R' intersects the plane of x, y.
124. The force R', being represented in intensity by the
line M'R' {Fig. 64), may be supposed applied at C, in its line
of direction. Then making CD' = M'R', and decomposing
CD' into three rectangular forces, applied at C and parallel
to the co-ordinate axes, these components will be equal to
those of the force M'R' ; and the point C may therefore be
considered as solicited by the three forces F cos «', P' cos ^\
and P' cos y'—g\ the two former being situated in the plane
of a-, y, and the latter parallel to the axis of z. Thus, instead
of the force P' applied at M', we shall have
the force g applied at M', parallel to the axis of z,
the force P' cos y'—g applied at C, parallel to the axis of Zy
the force P' cos » applied at C, and acting in the plane of ar, y,
the force P' cos ^ applied at C, and acting in the plane of x, y.
125. By adopting a similar method of decomposition for
the forces P", P ", &c., employing the auxiliary forces g'\ g"\
&c., applied at the points M", M", (fee, the system will be
reduced to two groups of forces, of which one will have its
components parallel to the axis of z, and the other will be
situated in the plane of x, y.
The forces parallel to the axis of z will be
g\ g'\ g"\ &c.,
applied at the points M', M", M'", «fcc. ; and
P' cos y'—g, P" cos y"—g'\ F" cos y"'—g"', &c.,
applied at the points C, C", C", &c.
And the forces lying in the plane of x, y, will be
P' cos «', F' cos «", F" cos «'", (fee,
applied at the points C, C ", C ", &c. ^ and
P' cos (3', P" COS fl'', P'" COS /3"', &C.
applied at the same points C, C", C ", (fcc.
126. It will now be demonstrated that when an equilibrium
subsists in the system, it will be necessary, 1°. that the forces
parallel to the axis of z should be in equilibrio ; 2". that the
forces acting in the plane of a:, y, should also destroy each
other.
64 STATICS.
For since the equilibrium is supposed to subsist, the state
of the system will not be changed by supposing a line C'C"
assumed arbitrarily in the plane of x, y {Fig: 65) to become
immoveable. The forces situated in this plane will then be
destroyed by the resistance of the fixed line. For, every
force in the plane of .r, y must intersect the fixed line, or be
parallel to it. In the first case, let the force be represented
by AB, and prolong its line of direction until it intersects the
fixed line at a point O : this point being supposed immove-
able, the effect of the force AB, which is transmitted to the
point, must be destroyed. Again, if the force be parallel to
the line C'C ", its point of application E cannot be moved
without communicating a motion to the line C'C" which by
hypothesis is immoveable. The effect of this force must
therefore be destroyed by the fixed line. Thus, the forces
lying in the plane of x, y being destroyed, the system will be
reduced to the group parallel to the axis of z. These latter
forces would obviously tend to turn the system about the
fixed line C'C, unless the forces should be in equilibrio, or
their resultant should pass through the fixed line. But the
position of this line having been assumed arbitrarily, it cannot
happen that the resultant of the forces parallel to the axis of
z will always pass through this line. These parallel forces
must therefore be in equilibrio.
The group parallel to the axis of z being in equilibrio, the
forces lying in the plane of x, y must mutually destroy each
other, since the equilibrium of the entire system could not
oth-erwise be preserved.
127. The problem is thus reduced to finding the conditions
of equilibrium, 1°. of a system of fr .-_es parallel to the axis of
z\ 2°. of the forces acting in the plane of x, y.
Conditions of Equilibrmm of the Forces parallel to the
Axis of z.
128. These conditions being the same as those enunciated
in Art. 87, the following quantities must be equal to zero, —
FORCES ACTING IN SPACE. 65
1°. The sum of the forces parallel to the axis of z ;
2°. The sum of the moments taken with reference to the
plane of y, ^ ;
3°. The sum of the moments taken with reference to the
plane oi x\ z.
The first of these conditions gives
F cos y—g'+g'J^Y' cos y"—g"+g"
+ P"' cos y" —g" -]rg"' + «Sec. = ;
or, by reduction,
F cos y' + P" cos y" + F" COS y"' + &C.=0 (60).
The second condition requires the consideration of two dif-
ferent sets of moments.
1°. Those of the forces g\ g'\ g"\ &c., applied at the points
M', M", M'", &c.
2°. Those of the forces P' cos y —g'^ P" cos y'—g'\
P"' cosy'"— ^"', (fcc, applied at the points C, C", C", (fee.
The moment of the force g' applied at M' {Pig. 66),
taken with reference to the plane of y, z^ is g X M'N' : but
M'N'=B'D'=.t'' ; the moment therefore becomes g'x.
The moment of the force F cos y — g applied at C, taken
with reference to the same plane, is evidently (F cos y — g'")
XE'C, or (P' cosy— ^')a, ; and the sum of the moments of
the two forces will therefore be represented by
g-'x' + (P' cos y'—g')a,.
Substituting in this expression the value of a, (59) deter-
mined in Art. 123, we obtain
gx^(^ cosy— ^) I X'— ,5; — -1;
\ P COSy— ^/
performing the multiplications indicated, and reducing, we get
.-pT'cosy'— «T'cos»'.
By a similar process, the moments of the parallel forces
applied at M", M'", C", C", (fee. may be obtained, and being
collected into one sum, the equation expressing the second
condition of equilibrium becomes
Y{x cos y — 2;' cos «') •\-Y\x" cos y — 2;" cos «")
+P'"(a:"' cos y"'—z" cos «"') +(kc.=0 (61).
To obtain the third condition of equilibrium of parallel!
E
66 STATICS.
forces, we find tlie moment of the force g' applied at M',
talccn with reference to the plane of a;, z, and that of the force
P' cos y — g' applied at C, taken with reference to the same
plane : the first of these will be equal to ^'xM'L'=^' xB'G'
=g y^y ; the second will be (P' cosy' — g')h^ ; and their sum
will be expressed by
^y+(Fcosy'-^')5,.
Substituting for b, its value (59) found in Art. 123, and re-
ducing, we obtain
yV cos y' — z'P' cos /3'.
And by finding the moments of the other parallel forces,
taken with reference to the plane of x, z^ we shall have for
the third condition of equilibrium,
F(2/' cos y'—Z cos /S) +P "(2/'' cos y"—z" COS |3")
+F"(y" cosy'"— ^"' cos/î"')+&c.=0 (62).
Conditions of Equilibrium of the Forces situated in the
Plane of x, y.
129. These conditions being such as arise when the forces
act in the same plane, it is necessary,
1°. That the sum of the components parallel to the axis
of X should be equal to zero.
2°. That the sum of the components parallel to the axis
of y should be equal to zero.
3°. That the sum of the moments of the forces taken with
reference to the origin should be equal to zero.
The first two conditions are expressed by the equations,
F cosa'+P" cos «" + P'" cos«'"+(fcc.=0 (63),
P'cos /3' + P" C0S/3" + F" cos/3"' + &c.=0 (64).
"With regard to the third,, it may be observed, that the two
forces F cos a! and P' cos /3' are applied at the point C {Fig-
67) ; the moment of the fir«tj being taken with reference to
the origin A, will be
F cos «' X AE'=F cos «' X C'F'=F cos «' . 6, ;
in like manner, the moment of the force P' cos /3', taken with
reference to the origin A, will be
F cos ^'XAF'=:F cos p'xE'C'^rF cos/3', a,.
FORCES- ACTING II» SPACE. 67
These moments should be taken with contrary signs, since
the two components P' cos «' and P' cos /s' tend to turn the
system in contrary directions about the point A. Thus, by
regarding that momeni as positive in which the component
P' cos «' enters, the sum of the moments may be written
F cos u X b^—V cos /3' X a, ;
substituting in this expression the vahies of a, and Z>, (59),
we get
•n, > t ' ^'P' cos /3' \ ^, ,/ , z'Y cos <*' \
F cos a! ( y — _ —'— ) — P cos /3 ( x'—^, -, ) ;
\ PcoSy— o-/ V Pcosy— ^/
and by performing the multiphcations, and reducing, we-
obtain
y'Y cos x—x'Y cos /3'.
The moments of the forces apphed at C", C", &c., being found
in a similar manner, the third condition of equilibrium of the
forces which lie in the plane of x, y becomes
F(y' cos «' — x cos /3') + P"(y" cos x' — x" cos /3")
+ P"'(2/"' COSa:"—x" COS|3"') + (fec.=0 (65).
130. The six equations of equilibrium (60), (61), (62),
(63), (64), (65), may be written under the following form :
2(P cos«)=0 ^
x(P cos /3) =0 V . (66).
2(Pcosy)=0>
5:[F(y cos at, — X cos (S)] =0 ^
s[P(x cosy — z cos«)]=0 > (67).
2[P(y cos y — z cos /3)] =0 3
131. If there be a fixed point in the system, the six equa-
tions will not be requisite to express the conditions of equi-
librium. For, if the origin be placed at the fixed point, the
equilibrium will subsist between the forces acting in the plane
of X, y. when the system has no tendency to turn about this
point. This condition will be fulfilled when we have
2[P(2/ cos « — X cos /3)J=.0.
It remains to discover the conditions of equilibrium of the
forces parallel to the axis of z. Let .t„ y,, and be the co-
ordinates of the point at which the resultant of the parallel
forces intersects the plane of a-, y ; the moment of this result-
E2
68 STATICS.
ant taken with reference to the planes of x, z, and y, z, will be
equal to the sum of the moments of the several forces taken
with reference to the same planes ; whence we have
'Rx=-l\P{x cosy— 2; COS a)],
Ry^ = 2.[P(y COS y — Z COS /s)].
If an equilibrium subsists between the parallel forces, their
resultant must pass through the fixed point, which, by hy-
pothesis, coincides with the origin of co-ordinates, and we
therefore have :r,=0, y=0. The preceding equations will
thus be reduced to
Y.\P{X cos y — Z cos «)J=0,
5;[P(y cos y—z cos/3)]=0.
We therefore conclude that when the system contains a fixed
point, the equilibrium will subsist, if the equations (67) are
alone satisfied, the origin being taken at the fixed point.
132. Wlien the system contains two fixed points, one of
the co-ordinate axes may be drawn through them ; this axis
will thus become fixed, and the system can only be subject to
a motion around it. A similar case will be examined in the
succeeding paragraph.
133. When there exists a fixed axis about which the system
may turn, this axis may be assumed as the axis of z, and the
forces parallel to it will produce no effect. The remaining
forces are situated in the plane of x, y. But the condition of
equilibrium of these forces requires that their resultant should
pass through the point A {Fig. 67), which point is immove-
able, being on the axis of z ; and the condition of the result-
ant'spassing through A is expressed, as above, by the equation
2:[P(y cos a—x cos /3)]=0.
This equation expresses that the system is in equilibrio,
when the axis of z is supposed fixed.
134. If we suppose, successively, the axes of y and x to
become fixed, it may in like manner be demonstrated that
the system will be in equilibrio, in the first case, when
5;[P(.'r cos y—z cos «)]=0,
and in the second, when
5;[P(y cos y—z cos /3)]=:0.
135. When the body is capable of shding along the fixed
FORCES ACTING IN SP^ACE. 69
axis, supposed to be that of z, an additional condition of
equilibrium becomes necessary ; this condition is expressed
by the equation
2:(Pcos y)=0.
136. By comparing the conditions of equilibrium of a sys-
tem moveable about a fixed axis, with those which obtain
when the system turns about a fixed point, we iiifer. That
an eqniUhriiirii icill take place about tJie fixed point tvlien, by
vegarding the axes passing through this point as fixed in
succession, the equilibrium is tnaintained with reference to
each of them.
137. If the forces be supposed to act against a fixed plane,,
which may be assumed as the plane of x, y, the components
perpendicular to it will be destroyed by the reaction of the
plane, and the conditions of equilibrium will thus be reduced
to those of forces acting in a plane ; we consequently have
^(P cos «)— 0,
5:(P cos iS)=0,
2[P(y cos »—x cos /3)]=0.
138. If a body be supposed placed on a fixed plane, being
at the same time liable to be overturned by the action of thei
forces exerted upon it, we must add to these three equations
the condition, that the résultant of the perpendicular forces
shall pass through a point in which the body touches the
plane, or that it shall intersect the plane within the polygon
formed by connecting the points of contact.
139. The discussion of this subject will be terminated by
the solution of the following problem : To find the analyti-
cal condition expressive of the existence of a single result^
ant of any number of forces situated in space. The system
will admit of a single resultant, when the resultant of the
components parallel to the axis of z intersects the plane of
X, y, in a point situated on the resultant of the forces lying
in that plane. To express this condition, we remark, that in
case of an equilibrium, the following relations must subsist
between the forces parallel to the axis of z (Art. 128) :
P cos y-}-P' cos v'+P" cos y"-]-&:c,=Q.
70 STATICS.
P(:r cos y— Z COS «)-f-P'(a;' cos y' — z' COS «')
+P"(a;" COS y"-z" COS «")+<fcc.=0.
P(y COS y— 2; COS /3)4-P'(2/' COS y' — 2;' COS /S}
+P" {y" COS y"— 5;" COS ô")+<fcc.=0.
If we consider P cos y, the first of these forces, as equal
and directly opposed to the resultant Z of all the others, we
shall have Z= — P cos y, and
— P cos y=P' cos y'+P" cos y"+&.C,
— P(ar cos y—Z COS «)=P'(a:' cos y — z' COS «')
+ P"(a:" COS y"-z" COS /)+«S6C.
P(2/ COS y — 2; C0S;S) = P'(2/' COS y' — 2;' COS/3')
+P"(a:" COS y"— ;3" cos^")+(fcc.
The point of application of the resultant being supposed in
the plane of x^ y, let x;, y^, and be the co-ordinates of this
point ; these values, being substituted in the first members of
the preceding equations, give
— P cos y 3= P' COSy'-fP" COS y"-f tfec,
— P COS y.r, = P'(x' COS y'~Z COS a')
-\-V'{x' COS y" — Z'' COS «") + &c.,
P' C0Syy, = P'(3/' COS y'—z' COS S')
4-P"(y" COS y"—z" cos^'')4-<fcc. ;
and denoting by M and N the second members of the two
last equations, and replacing the factor — P cos y by its value
Z, we obtain
Z=P'cosy'-f&c.,
Zy;=N;
whence we deduce
_M _N
^~T ^~Z'
Having thus obtained the values of the co-ordinates of the
point at which the resultant of the parallel forces intersects
the plane of x, y, it remains to express the condition that this
point shall be found on the direction of the resultant of those
forces which are situated in the plane of .r, y ; the equation
of the latter resultant (Art. Ill) is
THEORY OF THE PRINCIPAL PLANE. 71
Xy— Yx=:s[P(y cos x- x cos /s)] ;
and putting-, for brevity,
j:[P(y cos K—x cos P)]=L,
it becomes
Xy— Yx=L;
replacing x and y in this equation by the values of x^, and y„
determined above, the required condition will be expressed,
and we shall obtain
XN YM T
or, by reduction,
XN=^LZ+MY (68).
If this equation be satisfied, the system will admit of a single
resultant, except in the case when
X=0, Y=0, Z=0.
140. When the forces are situated in the same plane, the
system will in general admit of a single resultant ; for the
quantities M and N which represent the sums of the moments
taken with reference to the planes of x, z, and y, z, being
equal to zero, as also the quantity Z which expresses the
sum of the components P' cos y', P" cos y", &,c., the equation
(68) will be satisfied.
141. It appears from Art. 114 that the equations X=0
and Y=0 express the condition that the forces lying in the
plane of x, y may be reduced to two equal resultants R' and
R", parallel to each other, and acting in contrary directions.
By a similar process, the forces parallel to the axis of z may
be reduced to two, Z' and Z", equal and acting in contrary
directions. Hence, when we have simply the conditions
X=0, Y=0, and Z=0, the system maybe reduced to four
forces R', R", Z', Z". These may be still further reduced to
two equal forces, having parallel and contrary directions.
Theory of the principal Plane, and Analogy existing be-
tioeen Projections and Moments.
142. The theory of the principal plane, which presents
results so nearly allied to those obtained in the theory ofino-
72 STATÎCS.
merits, is of such importance m the higher branches of me^
chanics, as to forbid its omission in an elementary treatise.
It is founded on a theorem demonstrated in the elementary
treatises on the Differential Calculus, which may be enun-
ciated as follows : The projectmi of a plane surface upon a
plane is equal to the area of this surface multiplied by the
cosine of the angle of inclination.
It follows, from this theorem, that if <p represent the angle
formed by two planes, and a the area of a surface situated in
the first plane, the projection of this area on the second plane
will be expressed by x cos <p. But the angle ç included be-
tween the two planes MF and EN {Pig. 68) is equal to that
included between the two perpendiculars demitted from a
point C on these planes. If one of these planes, EN for ex-
ample, be supposed that of x, y, the perpendicular All will
become parallel to the axis of z. Thus the angle formed by
the plane MF with that of x, 2/,is measured by the angle in-
cluded between the perpendicular BK and the line AH par-
allel to the axis of z.
In general, if «, /3, and y represent the angles formed by
the perpendicular to a given plane with the three co-ordinate
axes of a:, y, and ;r, these angles will measure the inclinations
of the assumed plane to the planes of y, 2;, .r, z, and a-, y, re-
spectively.
143. Let a, |3, y, and f, «', .-", represent the angles formed
respectively by any two planes with the three co-ordinate
planes, these angles being equal to those formed by the per-
pendiculars to the given planes with the axes of co-ordinates.
By introducing the cosines of these angles in the formula
expressing the value of the cosine of the angle included be-
tween two lines, the value of their inclination ç> may he
determined.
If we draw through the point C {Pig. 69) the lines CA
and CB perpendicular to the given planes, these lines will
contain between them the angle ^, and its value will result
from the formula
cos p=cos£ cos «-f cos t cos /î 4- COS î" cos y (71).
144. When the angle ^ is a right angle, its cosine will be
equal to zero, and the equation becomes
THEORY OF THE PRINCIPAL PLANE. 73
COS t COS «-fcOS s COS jS-fCOS t" COS y = 0.
145, From the formula (71) we deduce a very remarkable
property of projections. For, let there be two planes, the first
of which forms with the co-ordinate planes the angles a, 6,
and c, and the second the angles «, /3, y ; the angle <p included
between these planes being deduced from the formula (71),
we have
cos ^=cos a cos <« + cos h cos /S+cos c cos y.
But if we represent by x the area of a plane surface situated
in the first plane, the preceding equation being multiplied
by A, gives
A cos <p=x cos a cos «+ a cos h cos /3+a cos c cos y (72).
The product a cos ç> is equal (Art. 142) to the projection of the
area x on the second plane, and the products x cos a, x cos 6,
and X cos c are, in like manner, the projections of the same
area on the co-ordinate planes.
146. The equation (72) therefore gives rise to the following
theorem : The j)rojection of a plane surface on any plane is
equal to the simi of the jiroducts of its projections on each of
the co-ordinate planes, nudtiplied respectively by the cosines
of the angles o, (S, and y, which measure the inclinations of
the plane of projection to the co-ordinate planes.
This theorem becomes much more general, if, instead of the
area x lying in a single plane, we consider several areas
X, a', a", «fee. situated in different planes, and projected on a
plane whose inclinations to the co-ordinate planes are de-
noted by u, j3, and y : to avoid repetition, let us call the plane
of projection x, /3, y, and denote by
ç and ) the inclinations of the ( to ttie plane «, /j, y, and
a, b, c, ) area a ( to the co-ordinate planes,
the inclinations of the \ to the plane «, ,3, y, and
area a' ( to the co-ordinate planes.
ç" and } the inclinations of the C to the plane u, /s, y, and
a", b", c", ) area a" I to the co-ordinate planes,
<fcc. &C. &.C.
By a method similar to that in which equation (72) was
obtained, we can obtain similar expressions for the projections
of the different areas ; thus, ^
<p' and )
a', b', c', S
74 STATICS.
A COS ^=A COS a COS a + A COS h COS /3 + A COS C COS y,
a' cos ^'=A' COS o! COS a + A' COS i' COS /S + A' COS c' COS y,
a" COS ç" =>^' COS a" COS <* + /' cos 6" cos /S + a" cos c" cos y,
(fcc. (fcc. &.C. <fcc. ;
whence, by addition,
A COS(p+A' C0S<8'+A" COS^" + (fcc. 'l
=(a COS a + A' COS a' +x" cos a"4-&c.)cosrt I _
+ (a cos b+x' cos 6' + A" cos è" + <fcc,)cos ^ T ^ '^'
+ (a cos c + a' cos c' + x" cos c"+&-c.)cos y J
The first member of this equation is the sum of the pro-
jections of the areas a, x', x", <fcc. on the plane a, 0,y] and the
terms included within the brackets express the sums of the
projections of the same areas on the co-ordinate planes. We
therefore conclude that the enunciation of the theorem in
Art. 146 will, in the present case, require to be so modified,
that we may substitute in the place of the plane area ?., a sur-
face composed of any number of plane areas x, x', x", éôc.
situated in different planes : this modification renders the
theorem much more general.
147. For the purpose of simplifying the last equation, let
us denote by P the sum of the projections of the areas
\, \', x'', &c. on the plane «, /3, y, and by A, B, and C the re-
spective sums of the projections of the same areas on the three
co-ordinate planes ; the equation will thus be reduced to
P=A cos<t-fB C0S/3 + C cos y (74)
148. It should be observed, in taking the sums of these pro-
jections, that the cosines of the angles which enter into the
expressions are positive or negative, according to the values
of a, b, c, a', b', c', &c ; thus, these sums will occasionally be
changed into differences. For this reason, we should under-
stand the enunciation of the general theorem as being appli-
cable to the algebraic sums of the projections.
149. Let the areas ?>, >,', x", &c. be now projected on two
other planes which form with the co-ordinate planes the
angles «', /3', y', a.", li", y" ; and denote by P' and P" the sums
of the projections of x, x', x", <fcc. on the planes «', ii\ y',
«", ft y", respectively ; we shall obtain equations similar to
THEORY OP THE PRINCIPAL PLANE. 75
(74), and if we represent, as above, by A, B, and C, the sums
of the projections of k, k', \", &c. on the co-ordinate planes,
we shall have
P =Acos« +Bcosi8 -fCcosy ^
F = A cos *' 4-B cos /3' -f-C cos y' > (75).
F =A cos a"+B cos 0"-\-C cos y" }
150. If the planes upon which the projections P, P', and
P" are made be supposed rectangular, their intersections
will be perpendicular to each other, and may therefore be
regarded as three rectangular axes, which intersect at a
point O ; consequently, by representing these new axes by Ox'
Op', and Oz', they will be respectively perpendicular to the
new planes of co-ordinates ; but the axes of x, y. and z were
likewise perpendicular to the primitive co-ordinate planes ;
hence, the angles formed by the primitive axes with the new,
will be measured by the inclinations of the primitive co-
ordinate planes to the new. These angles of inclination are,
by hypothesis *, /?, y ; *', ^\ y ; «", /2", y" 5 and since each of
the primitive axes corresponds to the same letter although
differently accented, we find that
The axis of x forms with the new axes the angles «, «', <»",
The axis of y forms with the new axes the angles /;, /3', /3",
The axis of z forms with the new axes the angles y, y', y".
The following relations will therefore subsist between the
cosines of these angles,
COS^ « -{-cos'' *'-|-C0S2 a"=l ^
C0S2 /8 -fcos* yfi'-f cos2 ^"=1 > (76).
COS^ y-4-COS' y'-fC0S2 y" = l ^
Again, since the angle formed by any two of the primitive
axes is a right angle, we shall obtain (Art. 144)
cos * cos yg-fCOS *' COS i^'-j-cos a' COS (S"=0 ^
cos tf COS y-l-COS «' COS y'-f COS a' COS y"=0 > ^'l^)'
COS ^ COS y-f-COS /j' COS y'-f COS ^" COS y"=0 J
151. If we take the sum of the squares of the equations
(75), reducing by means of (76) and i^l)^ we shall obtain the
relation
P» 4.F3 -j-prrs ^p^i -}-B^ -f C^ (78) ;
76 STATICS.
which expresses that the sum of the squares of the projec-
tions of the areas a, a', a", <fec. on any three rectangular planes
is a constant quantity.
152. Several important consequences may be deduced
from this theorem : thus, if we resolve the equation (78) with
reference to P, we find
P=^(A^ H-B^» +C* — ?'== — P"2).
The value of P will evidently be greatest when P' and P'''
are equal to zero. In this case, the sum of the projections of
A, a', a", &.C. on the plane «, /3, -y, will be given by the equation
P-v^(A2+B2+C^) (79).
But the angles t, a', a", being the angles formed by the primi-
tive axis of a;, with the three new axes, we must have the
relation
A=PC0S «+ P' cos a'+P" cos cl" ]
and by considering the other angles, we obtain in Hke msuiner,
B=P cos /3+P' cos /S'+P " cos B",
C=P cos î'+P' cosy+P" cos/'.
If we suppose, as above, the quantities P' and P" to be equal
to zero, the preceding equations reduce to
A=P cos a, B=P cos 0, C=P cos V (80).
whence,
A ^ B C
COSce = — -, C0S/3 = — , 008?=—,
and by substituting for P its value given in equation (79)^
we find
A
B
^^'''"^/(A^+B^+C^)
C
cos>-^^^2_|_Bs_j_C3)
These angles express the inclinations of the plane of maxi-
mum projections^ which is called the principal plane.
The determination of this plane being dependent only on
the angles «, 0, y, the same property will be enjoyed by every
parallel plane.
(81).
THEORY OP THE PRINCIPAL PLANE. 77
153. It may also be demonstrated that the sum of the pro-
jections of the areas a, a', V, (fee, on every plane which is
equally inclined to the principal plane, will be equal to a con-
stant quantity. For, let Q, be the sum of the projections on
any plane whose inclinations to the co-ordinate planes are
denoted by a, 6, and c : if we represent, as heretofore, by A,
B, C the projections of these areas on the co-ordinate planes,
we shall have
Cl=A cos a-f B cos b + C cos c ;
but if d, /3, y denote the inclinations of the principal plane,
the equations (80), which are
A=Pcos'', B=Pcos/3, C=Pcosy,
will reduce the preceding equation to
Q=P(cos a cos *-fcos b cos /3-1-cos c cos y).
The quantity within the brackets being equal to the cosine
of the angle included between the principal plane «, (3, y and
the assumed plane a, b, c, we shall have, by calling this in-
clination 6,
Q,— P cos 6;
and since P represents the sum of the projections on the prin-
cipal plane, which, by Art. 152, is equal to .y/iA' -f B" +0^),
the substitution of this value gives
a=v/(A2-fB='-fC2)Xcos<» (82).
But the projections A, B, and C remaining the same, it follows
from the equation (82) that the value of Q,, the sum of the
projections on any plane, will be constantly the same for all
planes having the same inclination to the principal plane.
It also appears that this sum will increase or diminish in
the same ratio as cos $.
154. Lastly, it may be remarked that the sum of the pro-
jections on every plane perpendicular to the principal plane
is equal to zero ; for 6=90° gives cos fl=0, and Q,=0.
155. The several theorems relative to projections which
have just been demonstrated are likewise applicable to the
case of moments. For, let the centre of moments be sup-
posed to coincide with the origin of co-ordinates, and con-
ceive the plane «, 0, y to pass through the origin : if from the
78 STATICS.
points of application of the several forces we take upon their
respective hnes of direction, portions wliich shall be propor-
tional to the intensities of these forces, these lines may be
represented by the letters P, P', P", &.c. The centre of mo-
ments may then be regarded as the common vertex of several
triangles, of which P, P', P", (fee. represent the bases : the
projections of these triangles upon the plane «, /3, y, and on
the co-ordinate planes will likewise be triangles, their bases
2^, ])', p", (fee, being the projections of the lines P, P', P", (fee,
and their altitudes h, A', h", (fee, being the perpendiculars de-
mitted on the lines p, p', p", (fee. from the centre of moments.
These values behig substituted in eq^nation (73), which
may be written under the following form :
2(the projections on the plane «, /3, y) =
{ The projections on the co-ordinate planes multipHed )
( respectively by the cosines of the angles of inclination y
convert it into
Iph + \p'h' + \p"h!' -1- (fee. =
( The projections on the co-ordinate ^
s ^ planes multiplied respectively by the j> (83)..
1^ cosines of the angles of inclination J
The second member of t'his equation will contain similar pro-
ducts, and the factor \ will therefore be common to the two
members ; this being suppressed, the first member will re-
duce to
ph-{-p'h'^p"k!'^6i.c.
But p, p', p", (fee, being the projections of the right lines P,
P', P", (fee, the products j^h, p'h', p"h", tfee will be the mo-
ments of the lines />, p', jy", (fee, taken with reference to the
origin of co-ordinates. The same remarks being applicable
to the second member of equation (83), it follows that the
sum of the moments of the projections of the forces on the
plane «, /3, y, which passes through the origin of co-ordinates,
is equal to the sum of the moments of the projections of the
same forces on tlie three co-ordinate planes, multiplied re-
spectively by the cosines of the angles of inclination.
156. By making similar substitutions in equations (78), it
may likewise be proved that the sum of the squares of the
CENTRE OF GRAVITY. 7*^
moments of the different forces, when projected on three
rectangular planes, is a constant quantity.
The equations (80) make known the position of the plane
in which the sum of the moments will be the greatest pos-
sible. And the equatian (79) determines the sum of the
moments on the principal plane.
Centre of Gravity.
157. The particles of matter are constantly subjected to the
action of a force which tends to draw them towards the earthy
in directions perpendicular to its surface. This force is called
the force of gravity.
The earth being nearly spherical, the lines of direction in
which material points tend to move, will converge towards
its centre ; and since the distance of this centre from the sur-
face is exceedingly great when compared with the dimen-
sions of those objects which we usually consider, the direc^
tions of the forces which act on the different particles of the
same body^ may, without sensible error, be regarded as
parallel.
158. It is known from observation that, as we recede from
the centre of the earth, the intensity of gravity diminishes in
the inverse ratio of the square of the distance included be-
tween the centre and the place of observation. For example,
if a body be placed at a certain distance from the centre of the
earth, assumed as unity, and be subsequently transported to
distances represented by 2, 3, 4, &c., the intensity of the force
of gravity will become _,_,_, &c., or -^ ^, --, &c., of
what it was at the distance of unity.
159. The earth being flattened towards the poles, and pro-
tuberant at the equator, it follows, that in going from the
equator towards the poles, we must necessarily approach the
centre of the earth, and the intensity of gravity will therefore
increase. It will appear hereafter in discussing the subject of
centrifugal forces, that from another cause, the intensity of
the force of gravity is greater at the poles than at all other
places on the earth's surface.
80
STATICS.
160. The action of gravity being- exerted on all the particles
which compose a body, these particles may be regarded as
solicited by forces whose directions are parallel ; the resultant
of these forces is equal to their sum, and constitutes what is
called the weight of a body. Hence, if the bodies considered
are homogeneous with each other, their weights will be pro-
portional to their volumes.
161. The term density is used to express the gi*eater or
less number of particles contained in a body of a given
volume, when compared with the number of particles con-
tained in some other body assumed as a standard. If we
assume as the unit, the quantity of matter contained in a
cubic foot of a given substance, distilled water for example,
and compare this quantity with that contained in a cubic
foot of any other substance, their ratio will express the density
of the second substance. Let this ratio be denoted by D.
If the second substance considered were gold, by calling D
the density of gold, we should have
The quantity of matter in ) _ ( D X T/ie quantity of matter
a cubic foot of gold ) ( in a cubic foot of water ;
whence
y^_ (Quantity of 'matter in a cubic foot of gold
Quantity of matter in a cubic foot of water
162. In the preceding article we have considered bodies of
the same volume ; but if we wish to estimate the quantity of
matter contained in a homogeneous body whose volume is V,
the quantity D must be taken as many times as there are
units of volume in the volume V ; we shall thus have
M=DV (84).
The quantity M is called the mass, and evidently expresses
the relation between the quantity of matter contained in the
body, and that contained in the unit of volume of the sub-
stance assumed as the standard.
163. If the intensity of gravity were the same at all places,
the weight of a body would be proportional to its mass, and
might be represented by the same quantity. For, if ^^ denote
the effect exerted by gravity on the unit of mass, or the
weight of the unit of mass, and W the weight of the body, we
CENTRE OP GRAVITY. 81
shall have, from the definition of the weight, W=M^ ; in
which expression the quantity g will be constant, and may
be assumed as the unit ; we shall thus obtain the relation
W=M (85).
This equation merely expresses that the number of units of
weight is equal to the number of units of mass.
But, if by transporting the mass to different distances from
the earth's centre, the intensity of gravity be subject to varia-
tion, the quantity g will be variable, and the equation ex-
pressing the relation between the mass, weight, and intensity
of gravity, must then be written under the general form
W=M^ (86).
164. From the equations (84) and (86), we deduce
which indicates that the weight varies proportiorially to the
gravity- g, the volmne V, and the density D.
165. If, for example, two bodies of the same volume be
subjected to the action of the same force of gravity, their
weights will be in the direct ratio of their densities.
The intensity of gravity varying only with change of
place, it follows that^ will be constant for all bodies at the
same place.
166. If there be any number of points firmly connected
together, and solicited by the weights P, P', P", &c., we may
regard these weights as parallel forces ; and denoting the
co-ordinates of the respective points by x, y, z, a/, y', z\
x",y", z"j 6cc.,we shall obtain, from Art. (80) and (81)," the
expressions for the co-ordinates of the centre of parallel forces ;
these co-ordinates being represented by X/, yi, Zj, we find
'Px + F'x' + V"x" + ôcc.
x,=
P-fP' + P"+<fcc. '
Py-fPy + P^y^ + &c.
P + P' + P"+&c. '
Pz-i-P'z'+P"z"+&c.
P+P'+P" + &c.
167. When the forces are exerted, as in the present in-
stance, by the action of gravity, the centre of parallel forces
is called the centre of gravity. Let m, ml\ m", (fcc. represent
F
WS STATICS.
the masses corresponding to the weights P, P', P", &c., we
shall have
P=mg, P' ='m'g; P" = m"g, <fec. ;
and by substituting these values in the preceding equations,
omitting the factor g-, which is common to the numerators
and denominators of the fractions, we obtain
mx^'m'.T'-^m"x" + &^c.
m-^-m'-^-ni" +ÔÙC. '
_my-^on'y^-\-Qn"y" -{-éùc.
'm-^m' +?n"-\-âcc. '
_mz-\-'m'z'-\-m"z"-{-&:-c. ^
■m-j-ni' -i-m" +ÔCC. '
whence it appears that the position of the centre of gravity
is independent of the intensity of the force of gravity.
168. If the bodies are composed of a homogeneous sub-
stance, the density of which is represented by D, we shall
have, by denoting their volumes by v, v', v", ôcc. (Art. 162),
m=vD, m'=v'D, m" =v"T>, <Scc. ;
and by a substitution and reduction similar to the preceding,
we find
vx-\-v'x' +v"x"-{-&c.
xr-
Zi —
v-\-v'-{-v"-\-6lc.
vy + v'y' + v"y" -f- &c.
v-\-v'-\-v"-\-&,c.
vz + v'z' -f- v"z" + <fec.
or calling V the volume of the entire system, these equations
become
vx-\-v'x'-\- v"x" + <fec.
x,-
yi
_vy-\-v'y'-\-v"y"+ècc.
vz 4- v'z' + v"z" + &c.
Z/= ^
169. To determine the centre of gravity experimentally,
we suspend the body by a thread CA {Fig. 70), and the pro-
longation AB of the direction of this thread wiU necessarily
CENTRE OP GRAVITY. 83
pass through the centre of gravity. The point in the Une
AB at which the centre of gravity is situated, may then be
found by suspending the body from a second point E ; the
vertical line EF, passing through this point, must likewise
pass through the centre of gravity, which will consequently
be found at the point G, the intersection of the two lines AB
and EF.
In this experiment, the body is sustained by that point to
which the thread is attached : the resultant of all the actions
of gravity upon the particles of the body must therefore pass
through this point, and its direction must coincide with that
of the thread.
170. The centre of gravity of a right line AB {Fig. 71) is
situated at its middle point C : for, by regarding the hne as
composed of heavy material points, each particle m situated
on one side of the point C will correspond to a particle m' on
the contrary side, and equally distant from the same point :
the moments m X Cm and m' X Cm' are therefore equal and
have contrary signs. The same remarks are applicable to
all the other points of the line AB, taken by pairs ; hence it
follows, that the algebraic sum of the moments of all the par-
ticles taken with reference to the point C is equal to zero ;
the moment of the resultant taken with reference to the same
point is therefore zero, and the direction of the resultant must
pass through the point C, situated in the middle of the
line AB.
171. The centre of gravity of a 'parallelogram AD {Pig.
72) is at the intersection G of the right lines EF and HK,
which bisect the parallel sides.
For, if we conceive the particles which compose the paral-
lelogram to be situated on lines parallel to AB, the centres of
gravity of all these lines will be found on the hne EF drawn
through the middle points E and F of the opposite sides AB
and CD, since EF will bisect all these parallels. Hence, the
centre of gravity of the entire parallelogram will be situated
on the line EF. In like manner, it may be proved that the
centre of gravity lies on the line HK which bisects the sides
AC and BD ; it will therefore be situated at the point G, the
intersection of the two lines EF and HK.
F2
84 STATICS.
172. The centre of gravity G of the area of a triangle ABC
{Fig- 73) is found by drawing a line CD from the vertex to
the middle of the opposite side., and taking a part DG equal
to one-third of the whole line CD. For, since the line CD
passes through the middle of all the lines parallel to the base
AB, it contains the centre of gravity of the area of the tri-
angle : for a similar reason, this centre must lie on the line
AE drawn through the middle of the side CB : hence, it is
found at the point G, the intersection of these two lines. But,
by connecting the points D and E, we form the triangle BED,
which is similar to the triangle BCA, since the two triangles
have a common angle, and the sides adjacent directly propor-
tional : the line DE is therefore parallel to AC, and the tri-
angles ACG and DEG are likewise similar ; hence,
CG : GD : : AC : DE : : AB : BD : : 2 : 1 ;
from which results
CG=2GD,
and, consequently,
CDorCG + GD=3GD,
or,
GD = iCD.
173. To find the centre of gravity of a triangular pyra-
mid, ive draw through the vertex and the centre of gravity
of tlte base, the line AG {Pig. 74), and take the distance
GO = iAG ; the jmint O will be the centre of gravity of the
pyramid.
For, if we conceive the pyramid divided into an infinite
number of elements by planes parallel to the base BCD ; the
line AG will pass through the centres of gravity of all these
elements, and will therefore contain the centre of gravity of
the pyramid. In like manner, by drawing the line DG'
through the vertex D and the centre of gravity G' of the oppo-
site face, this line will also contain the centre of gravity of the
pyramid. But, since the lines AG and DG' are situated in the
plane of the triangle AED, and are not parallel, they will
intersect, and hence the centre of gravity of the pyramid will
be found at O, their point of intersection.
The points G and G' being connected, the triangles EGG'
CENTRE OF GRAVITY.
85
and EDA will be similar, since they have a common angle E,
and the sides containing it directly proportional ; hence, GG'
is parallel to AD, and the triangles AOD, GOG' are likewise
similar ; from these we obtain
GG' : AD : : GO : OA ;
but the similar triangles EGG' and EDA give
GG' : AD : : EG : ED ;
whence, by comparing these two proportions, we have
GO : OA : : EG : ED : : 1 : 3 ;
and from this proportion we, find
3G0=0A ;
adding GO to each member of the equation, there results
4G0=0A+G0=AG,
or,
GO = iAG.
174. In general, the centre of gravity of any pyramid (Fig,
75) is situated on the right line SF, drawn from the vertex
to the centre of gravity of the base, and at a distance
FO = |^SF. Tf we draw through the point O thv ; deter-
mined, a plane parallel to the base of the pyramid, this plane
will contain the centre of gravity of the pyramid. For, if
through F, the centre of gravity of the polygonal base, the
lines FA, FB, »fcc. be drawn to the several angles of this poly-
gon, we shall form as many triangles as the figure has sides,
and these triangles may be regarded as the bases of triangu-
lar pyramids having a common vertex S. The lines drawn
from the vertex S to the centres of gravity of the several
triangles will be cut proportionally by the plane parallel to
the base, and the points of intersection will therefore be situ-
ated at distances from the base, equal to one-fourth of the
distance from the base to the vertex of the pyramid. Hence,
these points of intersection will be the centres of gravity of
the several triangular pyramids. But the centres of gravity
of all the partial pyramids being situated in the same plane
parallel to the base, it follows that the centre of gravity of the
whole pyramid will likewise be situated in this plane. It
must also be found on the line SF, which contains the centres
8
86 STATICS.
of gravity of all the sections parallel to the base, and we
therefore conclude that the centre of gravity of any pyramid
is situated on the line drawn from the vertex of the pyramid
to Hie centre of gravity of the base, and at a distance from,
the base equal to one-fourth of the entire distance to the vertex.
175. To find, the centre of gravity of the area of a polygon.
Let the polygon be divided into triangles {Fig. 76), and de-
note by a, a', a", (fee, the areas ABC, ACD, ADE, «fee. of
these triangles : let weights proportional to a, a', a", (fee. be
supposed applied at the centres of gravity G, G', G", (fee, of
the several triangles. The centre of gravity of the area
ABCDA may then be found by the proportion
a+a' : a : : GG' : G'O.
In like manner, the centre of gravity K of the area ABCDEA
may be found by determining the resultant of a-\-a" acting
at O, and a" acting at G". Its position will be ascertained
by the proportion
a + a'+«": a":: OG": OK;
and the same process may be continued for any number of
triangles.
176. This problem may also be solved by means of the
equations of parallel forces. For let Xj and y, denote the co-
ordinates of the centre of gravity of the polygon {Fig. 77) :
from the theory of parallel forces we obtain the equations
R=P+F-fP"4-P"',
'Rx,^Vx-^V'x'+V"x"+V"'x"', *
%^ = Py -f V'y' + V"y" -^ V"'y"'.
And denoting as above by a, a', «", a'", the areas of the tri-
angles ABC, ACD, ADE, AEF, we shall have, since the areas
may be substituted for the weights to which they are pro-
portional,
P=a, F=a', P"=a", P"'=a"' ;
and the preceding equations become
R=a-|-a'-f-a" + a"',
ax-\-a'x'^d'x"-k-a"'x"'
^'~" a^a:^a:'-\-a:" '
ay-{-a'y'+a" y"+ al'Y'
^'"^ a+a'+a"-^a"'' *
CENTRE OP GRAVITY. 87
Thus, having- taken the part OP=a:,, we draw the line PG
parallel to the axis of y and equal to y, ; the point G will be
the centre of gravity.
177. To find the centre of gravity of the perimeter of a
polygon. We proceed in the present case in a manner similar
to that adopted in the preceding example, merely observing
that the centre of gravity of each side will be situated at its
middle point, and that these points may be regarded as
loaded with weights proportional to the sides.
178. To find the centre of gravity of the arc of a jylane
curve. If the curve be divided into elementary portions,
the value of the element mm' {Fig. 78) will be expressed by
*/{dx^ -\-dy^\ and since this element is indefinitely small, its
centre of gravity may be regarded as coinciding with its
middle point o, and having the same co-ordinates x and y as
the point m ; the moment of TnmJ with reference to the axis
of :r,will therefore be
op X tnrr^ =y X ^{dx^ + dy^ ),
and its moment with reference to the axis of y, will be
oq X 'mm'=x X y' (dx^ -\- dy~ ).
If X, and y I represent the co-ordinates of the centre of gravity^
£ind s the length of the curve MM', the moments of this arc
supposed concentrated at its centre of gravity, taken with
reference to the axes, will be respectively sx^ and sy^ : and
since these moments must be equal to the sum of the mo-
ments of the elements, we shall have
sx,—fx^{dx^ -\-dy^)^
sy,=fy^{dx^-\-dy-)\
and the length of the arc MM' will result from the formula
s^f^idx^^dy^).
179. Let it be required, for example, to determine the centre
of gravity of the arc BO of a circle {Pig. 79). The co-
ordinate axes being selected in such a manner that the arc
shall be bisected by the axis of abscisses passing through the
centre of the circle, the arc will be divided symmetrically by
this axis, and the centre of gravity of the arc will then be
88
STATICS.
found on this line; hence, we shall have 2/,=0. It will
therefore be only necessary to determine the absciss AG=a:,
of the centre of gravity of the arc BO. But the value of x,
results from Art. 178 ; thus,
sx,=fx^{dx''-]-dy') (87).
To integrate the second member of this equation, we elimi-
nate one of the variables by means of the equation of the
circle, which is
y^-=a^—x^ (88);
and by differentiating this equation, we obtain
ydy^= — xdx ;
whence,
X^
and by substituting this value in the expression '^{dx^-\-dy^\
we have
Vkdx-^-^dy-^^s/ i^^^dy^y,
which, reduced by means of equation (88), gives
ady
^{dx--\-dy-)=-^\
this value being substituted in equation (87), we find, by
integration
fx^{dx^-Vdy^)=ay^^ (89),
the quantity B representing an arbitrary constant.
If we denote by c the chord of the arc BO, and wish to
determine the centre of gravity of the arc which it subtends,
we must integrate between the limits 2/=ic, 2/= — \c. But
since the arc extends from O to B, this integral will become
zero at the point O, the ordinate of which is y— — \c. This
supposition reduces equation (89) to
0= — iac+B;
by eliminating B between this equation and (89), we find
fx^{dx''-\-dy'')—ay-\r\ac\
and making y=\c^ for the purpose of taking the entire
integral from the point O to the point B, we obtain
CENTRE OP GRAVITY. 89
fxy/{dx^ •\-dy^)=ac ;
which vahie substituted in equation (87), gives
sx,=ac,
or
radius X chord
x.=-
(90);
arc
the absciss of the centre of gravity is therefore a fourth pro-
portional to the arc^ the chord, and the radius.
180. To find the centre of gravity of a curve of double
curvature, or, in general, that of any line situated in space.
The expression for the element of a curve of double curva-
ture being
^{dx^-\-dy^-\-dz') (91),
let the moments of this element be taken with reference to
the co-ordinate planes. The co-ordinates x, y^ and z repre-
sent the distances of this element from the planes of y, z, x, z^
and xy, and the respective moments will therefore be
x^{dx''-\-dy'^-\-dz'')
y^\dx^-^dy^-\-dz^) \ (92) ;
z^ {dx^ ■\-dy'^ -\-dz2)
consequently, if we denote by x^,y„ and z, the co-ordinates
of the centre of gravity, and by s the length of the arc, these
quantities will be determined by means of the equations
s-=- f^idx^^dy^^dz^) \
sx,=fx^{dx^-^dy^^dz^) I
sy,=fy^{dx^+dy--\-dz^) ^ • • • • V^-^^
sz,=fz^{dx^-\-dy^-\-dz'^) J
181. Let it be required to apply these formulas to the
case of a right line situated in space. Assume the origin at
one extremity of the line ; the equations of the line will then
be of the form
x=«z, y=fiz (94);
whence,
dx = ecdz, dy —^dz.
These values substituted in the expression (91), give
^{dx^ -{-dy^ -\-dz^)=dz^{\+cc^ ■^^^)',
CO -4- M .
90 STATICS.
and putting, for brevity, the radical equal to A, we shall have
Substituting this value in the equations (93), and likewise
those of X and y given by equations (94), we find
s=fA.dz=A.Zy •
SX I =fKcczdz = ^Aaz^,
syi =fA^zdz = ^Aisz^;
sz, =fkzdz = 1 Az 2 .
Let h represent the ordinate z of the point M {Fig. 80).
To determine the centre of gravity of AM, we must integrate
between the limits z=0 and z=/t, and we shall thus find
s=AA,
5a:y = |A«A^,
5y; = |A/3/z%
Eliminating s^ and reducing, we obtain
^i = \^K yi=ï^h, z, = \h.
These values correspond to the co-ordinates of the point O,
the middle of the right line AM ; for, if AO be the half of AM,
the similar triangles AOQ, AMP will give
GlO = iMP=iA;
which value being substituted in equations (94), we find
x=^i>^h, y = ¥^^-
182. To find the centre of gravity of a plane surface,,
bounded by the arc of a cvjrve^ and the axis of abscisses.
Let Xj and yj be the co-ordinates of the centre of gravity
of the entire surface, and let G be the centre of gravity of an
element MP' {Pig. 81); the area of this element being equal
to ydx, its moment with reference to the axis of x will be
GNxy^dx, and that with respect to the axis of y will be
ANxydx. But since the element MP' may be regarded as
a rectangle whose side PP' is mdefinitely small, we shall have
PM
AP=AN=a:, and GN=— — =^y : hence the moments with
reference to the two axes become \y^dx^ and xydx. If we
represent by x the surface DBMP, its area and the co-ordi-
CENTRE OP GRAVITY. 91
nates of its centre of gravity will be determined by means of
the equations
x=fydx, ^
xxt=fxydx, V (95'),
183. To apply these formulas, let it be required to find the
centre of gravity of a circular segment CDE {Pig. 82). The
origin being assumed at the centre of the circle, and the axis
of abscisses AD a line bisecting the arc CE, the centre of
gravity of the segment will evidently be situated upon this
line ; it will therefore be only necessary to calculate the value
of the absciss AG=X;. If g and g' represent the centres of
gravity of the semi-segments, they will be found at equal dis-
tances from the axis AD, on a line gg' perpendicular to this
axis, since the entire segment is divided into two symmetri-
cal portions ; the line gg' will therefore intersect the axis of
abscisses at a point G, the centre of gravity of the entire
segment.
The question is thus reduced to determining the absciss of
the centre of gravity of the semi-segment CDB, and, its value
may be foimd by integrating the equation (95).
For the purpose of eliminating one of the variables in this
expression, we assume the differential equation of the circle,
ydy— — xdx ;
from which, by substitution in equation (95), we obtain
xx^^f-y^dy (96);
and by integrating, and introducing a constant A, we have
f-y^dy=-\y'+K (97).
To determine the value of this constant,, the integral must be
taken from the point C to the point D ; or, if we denote by
c the value of the chord CE, the limits of the integral will be
2/=ic and 2/=(X Thus, if we suppose the integral to become
zero, when y=\c, the constant A will result from the equation
and the equation (97) will therefore become
Oj2 STATICS.
Putting y=0, to obtain the value of the entire integral from
C to D, we have
This value substituted in equation (96), gives
""'-24^ =
but since x represents in this expression the area CDB, we
have
A=i area CDEB,
whence,
-___£!___ •
^'""12 area CDEB'
and we therefore conclude, that the distance from the centre
of gravity of a circular segment to the centre of the circle is
equal to the cube of the chord divided by twelve times the area
of the segment.
184. To find the centre of gravity of a circular sector CAE
{Fig. 83). The centre of gravity is evidently situated on
the radius AB which divides the sector into two equal parts ;
it will therefore be only necessary to determine the value of
the absciss AG. If we regard the sector CAE as composed
of an infinite number of elementary sectors, the centre of
gravity of each will be situated at a distance from the point
A equal to two-thirds of the radius AC, since these sectors
may be considered triangular. Hence, if from the centre A,
with a radius equal to two-thirds of AC, we describe the arc
HK, the centres of gravity of all the elementary sectors will
be distributed uniformly along this arc ; and consequently,
the centre of gravity of this arc will coincide with that of, the
circular sector. But if X) denote the absciss AG, we have, by
Art. 179,
_ AH X chord HK,
*' arc HK '
and from the similarity of the sectors AHK and ACE, we
find
CENTRE OP GRAVITY. 93
AH=|AC,
chord HK=| chord CE,
arcHK=|arc CE;
which values substituted in the preceding equation give by
reduction,
_ |ACx chord CE
*' arc CE
185. To find the centre of gravity of an area OBO'
{Pig. 84) comprised between tioo branches of a curve.
Let y and y' represent the two ordinates PM and PM' cor-
responding to the same absciss AP=x : the element MN' of
the surface, being the difference of the areas PN and PN', will
be expressed by
ydx—y^dx—{y — y'^dx ;
and if we represent by a a portion of the area included
between the chords MM' and 00', we shall have
The element MN' being regarded as a rectangle having one
of its sides indefinitely small, its centre of gravity will be
situated in the middle of the line MM' ; and the ordinate of
this point will therefore be
PM' + iMM'=y' + i(y-2<')=i(2/+y');
hence, the moment of this element with reference to the axis
of X will be
\{y^y'){y—y')dx=\{y ^ —y'^)dx ;
and the moment with reference to the axis of y will be
^{y—y')d^-
Thus, if Xj and y, denote the co-ordinates of the centre of
gravity of the entire surface, their values will become known
from the equations
xx,=fx{y—y')dx,
>^yi^My'-y")dx.
186. To find the centre of gravity of a surface of revolution.
Let the surface be supposed generated by the revolution of
the curve AM {Fig. 85) about the axis of x. The element of
the surface, or the zone generated by the elementary arc Mwz,
94 STATICS.
will be expressed by 2iryds : hence, by calling x the entire
surface, we shall obtain
x-=f2Tryds.
But since the centre of gravity is evidently situated on the
axis of revolution, the co-ordinate x, will be alone necessary.
To determine its value, we take the sum of the moments
with reference to the plane yz, which sum being equal to the
moment of the whole surface supposed concentrated at its
centre of gravity, we find
xx=fxy,2fryds]
whence,
f2fryxds
x= - ;
substituting for x and ds their respective values, and suppress-
ing the factor 27f common to both terms of the fraction, we
obtain for the absciss of the centre of gravity,
fxy^{dx^ +dy^)
X.—
(98).
~fyV{dx^-^dy^)
187. For the purpose of applying this formula, let it be
required to determine the centre of gravity of the surface of
a spheric segment. This surface being generated by the
revolution of a circular arc BC {Fig. 86) about the axis of
a:, we may eliminate one of the variables in the preceding
formula by means of the equation of the circle ^
which gives, by differentiation,
, , x'^dx''
^ y^
hence,
This value being substituted in the integrals of equation (98),
we find
fxy^{dx'^ + dy' ) =frxdx — \ rx' + C,
fy^{dx'-\-dy^ )=frdx =rx-\-C'.
Taking the integrals between the limits a;=AD=a, and
x==AB=r, we obtain
CENTRE OF GRAVITY. 96
Sy^idx^ -\-dy^)=r{r—a).
These values transform the equation (98) into
x,^\{r-{-a)=a-{-\{r—a) ;
thus, the centre of gravity is situated at the middle of the
line DB.
188. To find the centre of gravity of a solid of revolution
Mjhounded hy tivo planes perpendictdar to the axis, (Fig:87).
The centre of gravity being necessarily situated upon the
axis of revolution, which is supposed to coincide with the
axis of X, it will be sufficient to determine its absciss x,.
The element of the solid is expressed by Try dx, and we
therefore have
M=fry*dx (99).
The moments being taken with reference to the plane ofy,z,
we shall obtain
Mx=f7ry''xdx (100) ;
and by dividing this equation by the preceding, we find
We must eliminate one of the variables in this formula, by
means of the equation of the curve, and then integrate be-
tween the limits a;=AP and a.-=AQ,.
189. This formula being applied to the determination of
the centre of gravity of a cone, it will be necessary to obtain
the two integrals
fy' dx and fxy^ dx.
Eliminating y^ by the equation of the generatrix y=ax, we
obtain, after integration,
fy^ dx—fa^x^ dx=^^,
-, - , «^a*
fy^xdx=fa^x^dx=—j—.
There are no constants introduced by integration, since tlie
volume is equal to zero at the origin A (Pig: 88). These
values, being substituted in the formula (101), give
96 STATICS.
a-x*
3
from which we conclude that the centre of gravity of a cone
is at a distance from the vertex equal to three four tits of the
altitude Ax.
190. As a second example, let the required centre of gravity
be that of the volume of a paraboloid generated by the revo-
lution of the parabolic arc AM {Pig. 85) about the axis Ax.
The equation of the curve being y^ =px, we have
fy'^ dx =fpxdx = \px^ ,
fy- xdx=fpx^ dx=-\px^ :
these values substituted in formula (101), give
x,='^^^ = ^x.
\px^ '
The constants introduced by integration are equal to zero in
the present instance, for the reasons assigned in the preceding
paragraph.
191. Let the solid of revolution be an ellipsoid, the equa-
tion of whose generatrix is
this value ofy^ being substituted in the integrals of equation
(101), we obtain, since the constants are equal to zero,
/y^dx=—/ (a^dx — x'^dx) = — la^x——-j,
r, J ^" /? , J ,j N h"" (a'^x^ x^\
ly^xdx=—:; l{a^xdx—x^dx)=-^ I — ^ -r- I .
These values reduce equation (101) to
_\a'X — \x^ _&a'^x — 3x2
'~ a^ — ix2 12a2— 4x- '
and by taking the integral between a;=0 and x=a, we find,
for the absciss of the centre of gravity of the semi-ellipsoid,
x, = ^a.
192. To find tlie ceîitre of gravity of a volume generated
CENTROBARYC METHOD. Ô7
by the revolution of an area embraced by a curve BMCM'
{Fig> 89) about the axis of x, this axis being situated entirely
without the curve.
Represent by y and / the ordinates MP and M'P : the
volume generated by the revolution of the element Mot', will
be equal to the difference of the volumes generated by the
elementary rectangles Mp and M'p ; thj expressions for these
volumes being ^y'^dx and vy'^dx, that of the element of the
solid will be 7r(y2 — y'^)dx] hence, if we denote by M the
entire volume of the solid generated, we shall have
M=-!rf{y'' —y"')dx.
By taking the moments with reference to the plane of y, z, we
obtain
Ma:, = vrfiy '^ —y'^) xdx.
The value of x^ will be alone necessary, si ice the centre of
gravity must be situated on the axis of abscisses.
Of the Centrobaryc Method.
193. Let X, and y, represent the co-ordinates of the centre
of gravity of a plane surface MPP'M' {Fig. 90), the area of
which is represented by x. The moment of the element of
this surface, taken with reference to the axis of re, is, by
Art. 182, {yxydx ; and by making the sum of the moments
of all the elements equal to the moment of the whole body
supposed concentrated at its centre of gravity, we have
f\y^dx=y^x.
The two members of this equation being multiplied by the
quantity 25r, it becomes
f'>Fy^dx=2-7ryi\:
The expression f^ry^ dx represents the volume generated by
the revolution of the given surface about the axis of x, and the
second member 2xy/ is the product of the generating surface
by the circumference described by the centre of gravity;
hence, we deduce this general theorem : The volume of every
solid of revolution is equal to the product of the generating
wea by the circuTnference described by its ceidre of gravity.
194. Let it be required, for example, to determine the
G 9
98 STATICS.
volume of the solid generated by the revolution of an isosceles
triangle ABC {Fig. 91) about the axis of x. Denote CD
by A, and AB by a ; the generating area will then be ex-
pressed by ia/i. But the centre of gravity of the generating
triangle being at a distance from C equal to fCD, the circum-
ference described by this point will be |/iX2îr. Hence, the
volume will be expressed by the product |/iX2jrX^a/i =
fw-a/i-.
As a second example, let us determine the volume of a
right cone generated by the revolution of the right-angled
triangle ABC [Fig. 92) about the line AB. The area of the
generatrix will be iAJBxBC. The line CE being drawn to
the middle of the side AB, the centre of gravity G of the
generating area will be situated upon this line at a distance
from the point E equal to iEC (Art. 172) ; its ordinate GD
will therefore be determined by the proportion
3 : 1 : : EC : EG : : CB : GD ;
whence,
GD=iCB.
The path described by the centre of gravity will therefore be
expressed by f^rxCB; which, multiplied by the area of the
generating triangle gives the volume of the cone equal to
|«-xCB=^ XiAB=:iABx^xCB2.
195. Again, let the volume be that of a right cylinder :
the ordinate GE of the centre of gravity of the generating
rectangle {Mg. 93) being equal to ^AC, the path described
by this point will be ;rAC. This expression being multiplied
by the generating area which is equal to AB X AC, we have
srxAC^ xAB for the volume of the cylinder.
196. The area of any surface of revolution may be found
by a rule analogous to the preceding. For, if we consider
the surface generated by the revolution of any curve MN
[F\g. 94) about the axis of abscisses, and denote by y, the
ordinate of its centre of gravity G, we shall have, by Art.
178,
fy^{dx"- +c?y=*)=y,Xarc MN (102) ;
and by multiplying each member by 2v, this equation be-
comes
MACHINES — CORDS. 99
f27ry^{dx^ + dy')=2vy, X arc MN.
The expression f2Try^{dx^-\-dy'') representing the area of
the surface generated, we conclude, that the area of a surface
of revolution is equal to the product of the generating arc
by the circumference described by its centre of gravity.
197. Thus, to determine the surface of a conic frustrum
generated by the revolution of the right line CD {Mg. 95)
about the axis of x, we have the ordinate EG of the centre of
, , AC+DB , „ AC+DB , , ,,
gravity equal to ^ ; and 25rX ^ equal to the
circumference described by this point : hence, the product of
this expression by the length of the generatrix CD gives
271- X — X CD=25r . GE . CD for the convex surface of
the conic frustrum.
198. The two preceding theorems may be included in a
single enunciation, viz. : Every solid or surface of revolution
is equal to the product of its generatrix by the circumference
described by the centre of gravity of the generatrix.
Machines.
199. Machines serve to transmit the action of forces in
directions different from those in which the forces are applied,
and to modify the effects of those forces.
The force applied to a machine is called the power, and
that which tends to oppose the effect of the power is called
the resistance.
The most simple machines are the cord, the lever, and the
inclined plane. To these are sometimes added the pulley,
the wheel and axle, the screw, and the wedge, which may
be formed by very simple combinations of the first three.
These machines are usually called the Mechariical Powers.
Cords.
200. We shall adopt the hypothesis that cords are perfectly
flexible, that they are inextensible, without weight, and re-
duced to their axes. If the extremities of a cord be sohcited
G2
100 STATICS.
by two equal forces P and Q, {Pig. 96), Avhich tend to stretch
it, the tension of the cord will be measured by one of these
forces ; for, since the equilibrium subsists, we may regard A,
the middle of the line PQ,, as a fixed point, and drop the con-
sideration of that portion of the cord included between A and
Q, ; thus, the force P, acting alone against the fixed point A,
will measure the tension of the cord PQ,.
201. When the force Q, exceeds P, a portion of Q. equal to
P is employed to stretch the cord, while the remaining part
of the force tends only to move the cord in the direction from
P towards Q, : thus the tension will be measured by the least
of these forces.
202. If three cords be united by a knot, the conditions of
equilibrium are similar to those which obtain when any three
forces act on a point. The force acting in the direction of
each cord must be equal and directly opposed to the resultant
of the other two ; hence, the conditions of equilibrium require
that the three forces be situated in the same plane, and bear
to each other the following relations {Fig> 97),
P : Q, : R : : sin 7> : sin g- : sin r.
203. This proportion will be insufficient to establish the
equilibrium, if the cords are united by a sliding knot. For,
by regarding P and R as fixed points {Fig. 98), to which the
cord PCR is attached, if the force Q, be supposed to act upon
this cord by means of a ring or sliding knot, the point C will
describe an ellipse, the plane of which will pass through the
points P and R. But the revolution of this ellipse around
the axis PR will generate an ellipsoid, having its transverse
axis equal to PC |-CR, and the point C will necessarily be
found upon the surface of the ellipsoid, or, in other words,
at some point of the moveable ellipse ; but the point C being
only subject to motion when the force Q. has a component in
the direction of the elliptical arc, the equilibrium will be
maintained when the direction of the force Q, is normal to
the ellipse. If the line T^ be drawn tangent to the curve,
wo shall have, from the well known property of the ellipse,
Z.TCP=ZRC^;
and by subtracting these angles from the right angles TON,
if ON, there will remain
CORDS. • 101
z:pcn=zncr;
thus the angle PCR must be bisected by the direction of the
force Ct, and the proportion
P : R : : sin NCR : sin PCN
becomes, in the present case,
P : R : : sin NCR : sin NCR;
whence, P and Q, are equal to each other.
204. The funicular machine consists of a number of cords
united to each other at several knots, and maintaining an
equilibrium between the forces applied to these cords.
205. When several forces P, R, S, T, &c. (Fig: 99), act
conjointly at a single knot, their number will be reduced
by unity, if we substitute for any two forces P and R their
resultant R'; and by a repetition of the same process the
entire system may always be reduced to three forces united
at a single knot.
206. Let there be several forces P, P', P", F", P'^, (fee.
{Fig. 100), acting at the knots A, B, C, &c. of the cord ABC.
The conditions of equilibrium of these forces may be reduced
to those of a system acting on a single point : for, let R
represent the resultant of the forces P and P' ; since its effect
must be destroyed by the third force acting in the line AB,
the direction of this resultant must coincide with the pro-
longation of AB : but the point of application of a force may
be assumed any where on its line of direction, and hence we
may transfer the force R to the point B. If it be there de-
composed into two components parallel and equal to P and
P', the effect will be the same as if the two forces P and P'
had been transported parallel to their original directions, and
applied at the point B. In hke manner, by transporting the
forces P, P', P", (fcc, which are supposed to be applied at B,
to the point C, the entire system may be considered as acting
on this point. Thus the conditions of equilibrium are,
(Art. 54),
2(P cos «) =0, 2(P cos /3) =0, s(P cos y)=0.
To determine the ratio of the extreme tensions P and P,^,
we will denote by t and t' the tensions of the portions AB and
BC, and by
102 • STATICS.
a the angle PAP', a' the angle ABP", a" the angle BCP'",
b the angle P'AB, b' the angle P"BC, b" the angle P"'CP" ;
we shall then obtain, Art. 202,
P : ^ : : sin 6 : sin a,
t : i' : : sin b' : sin a',
i : P" : : sin b" : sin a" ;
whence, by multiplication, suppressing the factors which are
common to the two first terms, we have
P : P" : : sin 6 xsin 6'xsin b" : sin a Xsin a' Xsin a".
We may, in like manner, determine the relations between
aiiy other two forces.
207. If the forces P', P", P'", (fcc. be supposed parallel, we
shall have
b+a'=lSO% 6' + «"=180°;
and since the sine of an angle is eaual to the sine of its sup-
plement, we must have
sin 6= sin «', sin b'—sin a" ;
and the preceding proportion will then reduce to
P : P" : : sin b" : sin a.
If the forces P', P", and F" represent weights {Mg. 101), the
entire system will be situated in the same vertical plane ; for,
the right line AF being vertical, the plane of the forces
P, P', and t will be vertical. For a similar reason, the plane
of the forces t, P", and t', will be vertical ; but the line AB
not being vertical, it is impossible to pass more than one ver-
tical plane through it : hence, the forces P, P', t, P", and i'
will be situated in the same vertical plane. The same rea-
soning may be extended to a greater number of forces.
208. The extreme forces P and F" being required to sus-
tain the resultant of all the others, this resultant must be
directly opposed to that of the forces P and P", and must
consequently f>ass through the point G, at which the direc-
tions of those forces intersect. Moreover, its direction must
be vertical, being parallel to the components P', P", and P"',
and it will therefore be represented by the vertical line GH
drawn through the point G.
209. If we regard a heavy cord as a funicular polygon,
CATENARY. • 103
loaded with an infinite number of small weights, it results
from what precedes that the effect produced on the fixed points
by the weight of the cord may be estimated by drawing the
tangents PG and QG {Fig. 102), and applying at G a weight
equal to that of the cord ; since if we denote this weight by
G, we shall then have
P : Gl : G : : sin LGa : sin LGP : sin PGQ.
Of the Catenary.
210. The catenary is the curve which a perfectly flexible
cord assumes when it is suspended from two fixed points
A and B {Fig. 103), and subjected to the action of the force
of gravity. We will suppose that the cord is uniformly
heavy, and that the force of gravity is exerted on every
particle : it will readily appear, as in Art. 207, that the curve
will be situated in a vertical plane. Let the origin of co-
ordinates be assumed at A, the horizontal line AC being the
axis of abscisses ; the co-ordinates of a point M will then be
AP=a:, and PM=y. Through the point M, and through the
origin A, let tangents AH and MH be respectively drawn,
intersecting at the point H, and through this point draw the
vertical line HL. If we consider the portion of the cord
MA, we shall have, by Art. 209,
tension at A : weight of the portion AM : : sin LHM : sin AHM (103).
Let s denote the length of the arc AM ; A the tension of the
cord at the point A, which is exerted in the direction of the
tangent AH ; and « the angle included between this tangent
and the horizontal line AC. The quantities A and « will
remain constant.
The tension at A, being a quantity of the same kind as that
contained in the second term of the preceding proportion,
will necessarily be expressed by a weight ; and if we repre-
sent by p the weight of a portion of the cord whose length is
equal to unity, sp will express the weight of the part AM,
and the tension at A will be of the form ap. Thus the two
first terms in the above proportion will be replaced by the
ratio ap : sp, or by its equal a : s] hence,
a : 5 : : sin LHM : sin AHM (104).
104 • STATICS.
211. To determine the analytical expressions for the sines
which enter into this proportion, we remark, that in the
elementary triangle 7nMn, we have
Mm X sin mMn =mn, Mm X cos m'M.7i=M.n ;
or,
__ mji _^ M;t
sm mMfi= ^r-r— 5 cos mM?i=r — — ;
Mm' Mm '
and replacing these elementary lines by their analytical values,
these equations become
dx di/
sin ?n'M.fi——r-, cos nïM.n=—r (105).
as as
Bat the angle mMn included between the vertical and the arc
of the curve, is equal to the angle LHK formed by the
vertical with the tangent at M ; hence,
sin LHK=4^ , cos LHK=4^ (106).
as as
The first of these equations may be reduced to
sinLHM— (107);
as
for the angles LHK and LHM being supplements of each
other, we have
sin LHK=sin LHM.
Again, the angles AHK and AHM being supplements of each
other, we obtain
sin AHM^sin AHK=sin (LHK-LHA) ;
and from the well known trigonometrical formula for the sine
of the difference of two angles, we have
sin AHM=sin LHK cos LHA— sin LHA cos LHK ;
eliminating sin LHK and cos LHK by means of the equations
(106), we find
sin AHM^'^-^cos LHA-^'sin LHA (108).
as as
The triangle LAH being right-angled at L, the angles LHA
and HAL are complements of each other, and the latter hav-
ing been denoted by a, we obtain
cos LHA=sin «, sin LHA=cos «.
CATENARY. 105
212. These values substituted in equation (108) give
» sin AHM=^sin «-"^cos « (109) ;
as ds
and the equations (107) and (109) convert the proportion (104)
into
dx dx . dy
a: s '.: -y : -y-sin « — fcos *,
as ds ds
From this proportion we deduce the equation
5=asin« — a-^cos» (110).
dx
This equation contains three variables, one of which may
be eliminated by means of the relation
ds=^y/{dx'i-\-dy^).
For, by differentiating equation (110), regarding dx as con-
stant, we find
ds= — a cosa— -^ ;
dx
and by equating these values of ds, and dividing each mem-
ber of the equation by dx, we obtain
\/(l + -r^ )=— a cos«— ^,
^ \ dx^/ dx""'
or, by division,
d'^y
— a cos«~
dx^
v/O+g)
This equation will become integrable, if we multiply its two
members by 2dy ; we shall thus obtain
2dy^^
2dy— — a cos «__ ^ ; ^ 4. L
whence, by integration, jV^'^
y=-acos«^(l-F^)+c.
This equation being multiplied by dx gives
{c—y)dx=a cos*y/{dx^-\-dy'^) ;
106 STATICS.
and by reduction
dy^ ^/[{c-yy—a^ cos»c«] .^^^^
dx a cos «
213. The constant c may be determined by the consider-
ation that at the point A,
x=0, y=0, and ^=tang«.
dx
These values reduce the equation (111) to
tang« — ^ ^ i.;
a cos«
from which
we deduce
but
a tangae cos »=^{c^—a^ cos* «) :
whence,
tang* cos «= sin a;
a^ sin^ *=c2 — a^ cos^ *,
and consequently
c^=a2 (sin' a+cos^ «)=a'.
Thus the constant c is equal to a, and by substituting its
value in equation (111), we find for the differential equation
of the catenary,
dy_ ^[{a—yY-a^ cos^ a\ ,^^^y
dx a cos <*
214. It appears from a comparison of this equation with
(110), that the catenary curve is rectifiable ; for, if the pre-
ceding value of Y ^^ substituted in equation (110), we shall
obtain
s=a sin a— ^[{a—yY —a^ cos^»] (113) :
from this expression the value of s may be readily found in
terms of y, when the constants a and » have been determined.
. 215. To integrate the differential equation of the catenary,
we make
a—y=z, a cos et=b (114) ;
and we then obtain
dy = — dz ]
these values substituted in equation (112) give
MACHINES.
107
dx = --4^—— (115);
this expression becomes integrable by making
^{z^-h^)=z—t (116):
which by squaring and reducing, gives
By the differentiation of this equation, we obtain
xdt-\-tdz=tdtj
or,
dz _ dt
z—t~~T'
This relation, in connexion with that assumed above (116),
converts the equation (115) into
, bdt
dx — — J
t
which gives, by integration,
x=-h log^+e;
and by substituting for t its value expressed in terms of sr, we
obtain
x=h\o%\z—^{z''—h^-)\^e\
or finally, by replacing the quantities h and z, by their values
given in equations (114), we find
ar=acos<«log|a— y— ^[(a— y) = — a^ cos2<«]| +e (H^).
216. To determine the value of the constant e, we observe
that at the point A, a;=0, and y=0 ; which conditions reduce
the equation (117) to
e=— «coscelog |a[l— y/(l— C0S2 «)]|.
This value substituted in equation (117) gives
a;=acos«log[a— y— -v/(a— y)"— a^cos* « ]
—a cosee log [a(l — v/l— cos= «)] ;
or by reduction.
Such is the equation of the catenary.
217. The values of the constants c and e have been deter-
mined in functions of a and «f ; but these two quantities are
108
STATICS.
Still unknown. To determine their values, we will suppose
that jc' and y' represent the known co-ordinates of the second
point of suspension B, and I the length of the curve AMB ;
these values being substituted in the equations (113) and
(118), we obtain
l=a sin »— ^[{a—y'Y —a^ cos* «],
X —a cos«»
^\ â[l— /(l-cos^«)] )
218. These equations, in connexion with the relation
C0S2 «+sin' «=1,
determine the values of a, cos «, and sin ec,in functions of^.y',
and /. But another difficulty still presents itself; this consists
in the proper choice of the signs with which to affect cos «,
and the radicals which in the preceding expressions have not
received the double sign. To resolve this difficulty, we will
determine the co-ordinates of that point to which the max-
imum ordinate appertains. The characteristic property of
this point is that -^=0, which reduces equation (112) to
dx
^/[{a-yy—a'' cos'' ^] ^Q .
acos« '
and consequently,
a — 2/=acos« (H^)-
To establish the condition that this equation belongs to a
maximum, rather than to a minimum value, we attribute the
proper sign to the second differential co-efficient -j-^.
But by squaring the equation (112), we obtain
dy'^ _{a—yY —a^ cos» «
dx'^ a^ cos 2 a. '
and by differentiating, and dividing each member by 2c?y, we
find
d^y__ a— y .
dx^ a'cos'^tt'^
substituting in this equation the value oia—y determined in
equation (119), we obtain
dry^^ l__
dx' a cos <*'
LEVER. 109
219. This equation indicates that the condition of a maxi-
mum will be fulfilled by attributing the same sign to a and
cos « ; but these signs must be positive ; for, if they were
negative, the value of y determined hj the equation (119)
would be also negative, which is evidently inadmissible in
the hypothesis adopted, that the positive ordinates are reckoned
from the line AC downwards. From the equation (119) we
likewise infer, that the quantity a exceeds the maximum
value of y, and therefore that it exceeds all other values.
Let EF represent the maximum ordinate {Fi^. 103); it is
evident that between the limits â;=0 and a-=AE, as y in-
creases, the arc of the catenary will likewise increase. But
it appears from equation (113) that the increase of y will not
necessarily involve that of the arc 5, unless the radical in that
formula be affected with the negative sign. For, as y in-
creases, the quantity a — y will decrease, and the value of the
radical will therefore decrease ; but the smaller the value of
this radical, the less it will diminish the positive part of the
expression a sin «, and the greater will.be the value of the
arc. The equation (113) is therefore in perfect accordance
with the hypothesis that the co-orJinate has not attained its
maximum value. But from :2;=AE to a;=AD, the arc 5
should increase while y diminishes, and since this decrease
in the value of y augments the value of the radical expres-
sion, the required condition can only be fulfilled by affecting
the radical with the positive sign : thus, between the limits
ar=AE and ar=AD, the sign of the radical must be changed
in the formula (113).
Of the Lever.
220. The lever is a bar of wood or metal moveable around
a fixed point, which is called the fulcrum. To simplify the
considerations which relate to this machine, we shall regard
the lever as destitute of thickness, and will therefore represent
it by a simple line, either straight or curved. Let a lever AB
{Fig. 104) be sohcited by the two forces P and P' ; the effect
of these forces cannot be destroyed by the resistance of a fixed
10
110 STATICS.
point C, unless they are situated in a plane passing through
this point. If this condition be fulfilled, the equilibrium will
be maintained, when the sum of the moments taken with
reference to the point C is equal to zero.
221. If the lever is capable of sliding along its point of
support, it will also be necessary that the resultant of the
forces acting on the lever should be perpendicular to the lever
at the point of support.
222. When the lever is straight and the two forces parallel
to each other, ifp and p' represent the lengths of the portions
AC and BC {Pig. 106), we shall have from the theory of
parallel forces (Art. 73),
V :V' ::p' :p]
from which we infer, that when the forces are in equilibrio,
their intensities will be inversely proportional to the arms of
the lever.
223. If the lever be curved, and a right line ED {Fig. 105)
be drawn through the fulcrum C, the forces may be conceived
to be applied at the points E and D taken on their respective
directions ; we shall thus obtain
P : F : : CD : CE.
224. Levers are divided into three kinds. In the first kind,
the fulcrum C {Fig. 106) is situated between the power and
the resistance : in the second kind, the resistance R {Fig.
107) is situated between the power and the fulcrum ; and in
the third kind {Fig. 108), the power is between the fulcrum
and the resistance.
The balance and steelyard are examples of the first kind
of lever ; a bar of iron used in raising weights and having
its fulcrum at one extremity, forms a lever of the second
kind ; the treddle of a turning lathe is a lever of the third
kind.
225. The effect produced by the weight of a lever may be
readily estimated by regarding it as a force S applied at the
centre of gravity of the lever. For example, let P and P'
{Fig. 109) be two weights suspended from the extremities of
the lever AB, whose centre of gravity is situated at G ; we
LEVER. Ill
shall have, by virtue of the principle of the moments,
P'xCB+SxCG=PxAa
This equation will determine either P or F; and the weight
sustained by the fixed point will be
P + P' + S.
If the power and resistance act in opposite directions, regard
must be had to the directions in which they tend to turn the
lever; thus, in Fig. 110, the equation of the moments be-
comes
PXCA + SXCG=P'XCB (120);
and the weight sustained by the fulcrum is
P+S-F.
226. Let the lever CB {Fig. 110) be supposed homogeneous,
and of uniform weight throughout its length : represent by
tn the weight of a portion of the lever whose length is one
foot. If X represent the length of the lever expressed in feet,
its weight S will be expressed by mx^ and should be regarded
as a force acting at its centre of gravity, which corresponds
to the middle point G : thus, if we make CA=a, the equation
(120) will then become
Va + \xXmx=.V'y.x]
from which we deduce
F==— -fima: (121).
X
\i, therefore, x be assumed arbitrarily, this formula will make
known the value of P' ; but it may be required to assign the
value of X which shall render P' the least possible ; we must
then regard P' as a function of x, and make the differential 0^
co-efficient -r- equal to zero ; we shall thus obtain ,J^ \
dx
whence,
— — 4-im=0; ^^
2Pa , //2Pa
x'=-
and X
m
-^m
By substituting this value in equation (121), we obtain
112
STATICS.
P =
Va
or, by reduction,
2Pa
^ \ in /
227. The cornmon balance is an important application of
the lever. It consists essentially of a lever having equal arms,
from the extremities of which are suspended scales of equal
weight. The lever of the balance, which is called the beam,
is sustained by a horizontal axis perpendicular to its length,
which rests upon a firm support, and the substance to be
weighed, being introduced into ne of the scales, is counter-
poised by the addition of known weights in the opposite
scale. The figure of the beam is so chosen that its centre
of gravity will be found immediately beneath the axis, or
centre of motion, when the beam has assumed a horizontal
position ; and the weights suspended from its two extremities
are known to be equal when they will retain the beam in
this situation. If the centre of gravity were found upon the
axis, the beam would obviously rest in any position, and there
would be nothing to indicate the equality of the weights io
the two scales ; and if this centre were situated above the
axis, the beam would have a tendency to overturn if deranged
in the slightest degree from the horizontal position.
228. When the balance has been constructed with such
accuracy that the lengths of the arms are exactly equal, the
beam will assume the horizontal position if equal weights be
introduced into the two scales ; but in the false balance, where
the lengths of the arms are unequal, the weights necessary to
maintain the beam in this position are likewise unequal. In
this case, the weight of the body may be obtained by counter-
poising it successively in the tv/o scales : the true iceight will
be a geometrical mean between the two ajiparent weights.
For let 2^ and p represent the lengths of the two arms, and
W the true weight of the body. Then, if a weight P, sus-
pended from the extremity of fhe arm y>, be supposed to
sustain the weight W when suspended from the extremity
LEVER. 113
of the arm jo', the conditions of equilibrium in the lever
(Art. 220) will give
But if the weight W be transferred to the extremity of the
arm j», it will be necessary to apply a dijEFerent weig-ht P' to
the extremity of the arm p', in order that the equilibrium
may be preserved. Thus we shall have
and by multiplying the corresponding members of these two
equations, we obtain
or, by reduction,
W=y(PP');
hence, the truth of the proposition enunciated becomes appa-
rent.
229. It is frequently necessary that the balance employed
should possess great sensibility, or should be capable of indi-
cating very minute differences in the weights of the substances
placed in the two scales. The sensibility of the balance is
measured by the smallness of the weight necessary to produce
a given inclination of the beam, when the scales are charged
with a given load.
The sensibility depends upon the following particulars.
1°. The beam should be as light as is consistent with a
proper degree of strength, in order that the friction at the
axis, which is proportional to the pressure, may oppose the
least possible resistance to the motion of the beam.
For the same reason the axis is constructed of hardened
steel, and has the form of a knife-edge, or triangular prism,
the lower edge of which rests upon polished steel or agate
planes.
2°. The lengths of the arms should be as great as possible,
other things remaining the same, since the moments of the
weights introduced into the scales, taken with reference to
the centre of motion, will be directly proportional to these
lengths. Thus, the same weight, placed at twice the distance
from the centre of motion, will exert a double effort to turn
the beam.
H
114
STATICS.
3". The sensibility will be increased by diminishing the
distance between the centre of gravity of the beam and the
centre of motion. For, when the beam has been deranged
from the horizontal position through a given angle {Fig: 111),
the weight of the beam W, which acts at its centre of gravity
G, will exert an effort to restore it to its former position,
which effort will be directly proportional to the moment of
the weight W, taken v ith reference to the centre of motion
D; this moment will be expressed by Wxdg. But the
derangement of the beam having been made through a given
angle, the distance dg will evidently be proportional to DG,
the distance between the centre of gravity of the beam and
the centre of motion. Thus, in proportion as the distance
DG is diminished, the tendency of the weight of the beam to
counteract the derangement which would be produced by ah
inequality of the weights in the two scales will likewise be
diminished, or the sensibility will be increased.
4°. The line joining the points of suspension of the two
scales should pass through the centre of motion. For, if the
centre of motion be found at C above the line AB, and the
beam be supposed to have assumed the inclined position
represented in Fig. Ill, the effective arm of lever CE' of the
scale P' will evidently be greater than the arm CE of the
scale P. Thus the beam may have a tendency to return to
the horizontal position, although the weight P' be less than P.
And if, on the contrary, the centre of motion be placed at a
point C below the line AB, the lever-arm C'F of the scale P
will exceed that of the scale P', and the beam would therefore
have a tendency to overturn, although the weights in the
scales were equal to each other. When the centre of motion
is situated at the point D, the equality of the two arms will
be preserved, whether the beam be in a horizontal or inclined
position.
5°. The sensibility of the balance will be increased by
diminishing the load with which the scales are charged, since
the friction at the axis will be diminished in the same pro-
portion.
230. A very accurate balance will be sensibly affected by
the addition of jg^ part of the load with which the scales
are charged.
LEVER. 115
231. The steelyard, represented in Pig. 112, is a balance
having unequal arms, and is so constructed that a moveable
weight P, applied successively at different points of the longer
arm, shall sustain in equilibrio different weights suspended
from the extremity of the shorter arm. The longer arm
GB is so graduated as to indicate the weight which will be
supported by the moveable weight P^ when placed at each
of these divisions.
232. To discover the law according to which this arm
should be graduated, we will denote by
W, W, W", &c., the weights suspended successively
from the extremity of the shorter arm ,
j9, p', p", &c., the corresponding distances at which the
weight P must be placed to maintain the equilibrium,
r, the length of the shorter arm ,
IV, the weight of the beam,
r', the distance of its centre of gravity from the fulcrum^
Then, if the centre of gravity of the beam be supposed to lie
on the side of the longer arm, as usually happens, the con-
ditions of equilibrium will give
Wr=wr' + P/?,
Wr—tvr' + Fp\
W"r=ur' + Fp",
&c. &c. &c. ;
and by subtracting each of these equations from that which
follows, we obtain
(W'-W)r=(p'-p)P,
(W"-W')/-=(p"— _p')P,
( W" — W")r = (p'" —//')?•
If the weights W, W, W", &c. be supposed to increase in
arithmetical progression, we shall have
W— W=W"— W'=W"'— W"=&c. ;
and therefore
2)' —2i=2}" — p'=p"' — ;y' = (fcc. ;
thus the distances ^?, />', p", <fcc. will likewise increase in
arithmetical progression.
If, for example, the moveable weight P when placed at a
point F should be found to support a weight of 10 pounds,
H2
116 STATICS.
and if when placed at the point E,the weight supported should
be found equal to 20 pounds, we might divide the distance EF
into ten equal parts, and the points of division will cor-
respond to the weights 11 pounds, 12 pounds, 13 pounds, &c.
The zero of the scale will evidently be found at that point
from which the weight P is suspended when it merely serves
to counterpoise the weight of the lever. The steelyard is
frequently constructed in such a manner that the two arms
of the lever counterpoise each other : the zero of the scale
will then coincide with the fulcrum.
Of the Pulley.
233. The pulley is a wheel having a groove cut in its cir-
cumference for the purpose of receiving a cord which par-
tially envelopes it : when a motion is imparted to this cord it
is immediately communicated to the pulley, causing it to turn
about an axis which passes through its centre, and is usually
supported by a curved piece of iron terminating in a hook
{Mg. 113).
Pulleys are distinguished into two kinds, the fixed and the
moveable. In the fixed pulley, the hook is attached to an
immoveable point, as in {Fig. 113); and in the moveable
pulley the resistance R {Fig. 114) is applied to the hook.
234. The conditions of equilibrium in the fixed pulley
require the equality of the power P, and the resistance Q,
{Fig. 113) ; for, if the intensities of these forces were unequal,
the greater of the two would prevail.
This property'may also be demonstrated in the following
manner : we prolong the directions of the two forces which
act tangentially, until they intersect at the point E ; their
resultant will pass through this point ; and since the effect of
this resultant is destroyed by the resistance of the axis of the
pulley at 0, the resultant must likewise pass through this
point. But the triangles EP'O, EQ,'0 being identical, the
angle P'EQ,' is bisected by the direction of the resultant;
whence it follows that the force P is equal in intensity to
the force Q,.
235. Let there be now taken the equal parts 'Eg and EA,
PULLEY. 117"
and construct the parallelogram ^gfh ; the forces P and Q.
being represented by the lines E^ and E/i, their resultant R
will be represented by E/: we shall thus have the proportion
P : Gl : R : : E^ : E/i : E/;
and from the similarity of the triangles E^/ and P'OQ',
whose sides are respectively perpendicular to each other, we
obtain
FO : Oa' : P'a' : : E^ : E/i : E/j
hence,
P:Q :R::FO : OQ,' : FQ':
from which we conclude, that in the fixed pulley, each of
the forces is to the resultant, or the pressure upon the point
of support, as the radius of the 'pulley to the chord of the arc
with ivhich the rope is in contact.
The equality of the forces P and Q having been demon-
strated, it follows that the advantage of the fixed pulley con-
sists only in changing the direction of the power.
236. Let the cord QABP {Fig. 114) be supposed to em-
brace the arc AB of a moveable pulley, one extremity of the
cord being attached to the fixed point Q, ; and let a power P
be applied to the other extremity, for the purpose of sustain-
ing a resistance R. The reaction exerted by the fixed point
Q, will be similar in its effect to a force Q,, and the conditiorus
of equilibrium between P, Q., and R will be the same as in the
case of the fixed pulley, except that the resistance which was
then denoted by (i,will in the present case be represented by
R. Thus, the relation between the power and resistance will
be determined from the proportion.
P : R : : radius : chord of the arc AB.
As the intensity of the power may be less than that of the re-
sistance, the moveable pulley may effect a gain of power.
When the cords are parallel, the preceding proportion be-
comes
P : R : ; radius : diameter : : 1 : 2,
and the power is then equal to one-half the resistance.
If the chord of the arc be equal to the radius, the power and
resistance will become equal ; and when the radius exceeds
118 STATICS.
the chord, the use of the moveable pulley will induce a loss of
power.
237. By the combination of a number of moveable pulleys
we may succeed in raising enormous weights by the applica-
tion of a very small force : the pulleys may be arranged in the
following manner :
The weight R {Fig. 115) is suspended from the hook of the
moveable pulley ABD, around which a cord is passed having
one of its extremities attached to the fixed point K, and
the othej: to the hook of the pulley A'B'D'. This second
pulley is in like manner supported by a cord, attached at one
end to the point K', and at the other to the hook of the pulley
A"B"D" ; and the same arrangement is continued to the last
pulley, which is embraced by a cord connected at one end with
a fixed point K", the force P being applied to the other. If
an equilibrium subsists throughout the system, the tensions
of the cords AE, A'E', (fcc. being denoted by T, T', <fcc., we
shall have, by supposing there are three pulleys,
R : T : : AB : AC,
T : T' : : A'B' : AC,
T' : P : : A"B" : A"C".
These proportions being multiplied together give
R : P : : AB X A'B' x A"B" : AC X A'C'x A"C" ;
from which we conclude, that the power is to the resistance
as the continued product of the radii of the pulleys is to the
continued product of the chords of the arcs embraced by the
ropes.
When the ropes are parallel these chords become diameters,
and the proportion is reduced to
R : P : : 2=» : 1 ;
and, in general, for a number of pulleys denoted by n,
R : P : : 2" : 1.
238, This arrangement of pulleys is seldom adopted, on
account of its requiring too great a space. For, if the ropes
be parallel, as represented in Pig. 116, and the centre of the
pulley BOC be raised through a height denoted by h, the
line BC being brought into the position he, each branch of the
PULLEY. 11^
cord D"CBX must be shortened by the quantity Bb=Cc=h,
and the whole rope will therefore be shortened by the quan-
tity 2h : consequently, the pulley AE will rise through the
distance 2h ; for a similar reason, the third pulley will rise
through a distance Ah, equal to twice that described by the
second ; the same may be said of any number of pulleys :
and the power P applied to the extremity of the last rope
must rise through twice the distance which the last pulley
ascends. Thus with a number of pulleys represented by n,
the power will rise through a distance expressed by 2"h, and
we therefore lose in the space described, in the same propor-
tion that we gain in power.
To estimate the pressures sustained by the fixed points D,
D', D", (fee, we will represent them by Q., Q.', Q,"; then, calling
S and X the tensions of the cords SA and XB, we shall have
p=Q, s=a', x=a'';
which values substituted in the proportions
. P : S : : 1 : 2,
S : X : : 1 ; 2,
give
a-2P, a"=4P.
239. The nviiffie is a combination of several pulleys, all of
which are disposed in the same block, and have a common
cord passing around their respective circumferences.
To determine the relation between the power and the
resistance in the muffle, represented in Fig. 117, we remark
that the several branches of the rope must be equally stretched,
and that these tensions acting conjointly must produce
an equilibrium with the resistance R, which may therefore
be regarded as solicited by six equal and parallel forces. The
force d will be measured by the intensity of one of these
equal forces, and will consequently be equal to one-sixth of
the resistance. Or, in general, the power will he to the resist-
ance as unity to the number of cords which support the resist-
ance.
240. In the use of either system of pulleys, a certain force
will be necessary to overcome the weights of the moveable
pulleys. The value of this force may be readily estimated
120 STATICS.
by regarding the weight of each pulley as an additional force
applied to its hook. Thus, in the system with separate ropes
represented in Pig, 116, the weight of the pulley BOC may
be considered as applied to the hook, and will be equally sup-
ported by the cords BX and CD" : and since the addition of
every moveable pulley reduces the power one-half, it follows,
that the power will support one-half the weight of the upper
pulley, one-fourth of the weight of AE, and one-eighth of the
weight of BC. In the muffle {Fig. IIT'), the weight of the
moveable block being equally distributed among the cords,
the power will sustain one-sixth of this weight.
Of the Wheel and Axle.
241. This machine is composed of a wheel firmly con-
nected with a cylindrical axis. To the circumference of the
wheel a cord is attached, by means of which we can impart
to it a motion of rotation, the effect of which is immediately
communicated to the cylinder ; a second cord being wrapped
around the cylinder in a contrary direction, communicates
motion to the resistance which is to be overcome. The axis
is supported at its extremities by two cylindrical pivots which
are of less dfômeter than the cylinder itself, and permit it to
turn freely about the points of support.
242. To investigate the relation between the power and
resistance in this machine, let us suppose its axis AB (Fig.
118) to have a horizontal position, and let a horizontal plane
be drawn through this axis, intersecting the direction of the
power P at the point F. Represent the intensity of the force
P by the portion FP of its line of direction, and decompose it
into two forces FL=P' acting in a horizontal direction, and
FK=P" acting in a vertical direction. The direction of the
force P' being prolonged will intersect the fixed axis, and the
effect of this force will be destroyed by the reaction of the axis.
If motion be communicated by the force P, the point of
application F of the vertical component P" will descend, and
the resistance R will ascend, while the point M, the intersec-
tion of the line HF with the axis of the cylinder, will remain
immoveable. The point M may therefore be regarded as the
WHEEL AND AXLE. 121
fulcrum of a lever HF, to the extremities of which the forces
R and P" are applied ; we shall consequently have, by the
property of the lever, when an equilibrium subsists,
P" : R : : MH : MF.
Again, the planes of the wheel and of the section EOH being
perpendicular to the axis of the cylinder, the triangles HIM
MCF are right-angled and similar : hence,
MH : MF : : HI : CF.
From these proportions we deduce
P" : R : : HI : CF.
Let (p represent the angle FPK {Fig. 118 and 119), we shall
have
FPK=DFC=.p,
and consequently,
FK=FPxsin^, DC=CFxsin^;
P"=P sin <p, CF=4— ;
sm <f
these values bdng substituted in the preceding proportion,
give
Pxsin*:R::HI:-Ç^; ^"^
sni <p
whence,
PxDC=RxHI:
and from this we deduce the following proportion,
P : R : : HI : DC (122).
It thus appears that the conditions of equilibrium in the
wheel and axle require that the -power shall he to the resist'
mice as the radius of the cylinder to that of the loheel.
243. The pressures sustained by the pivots A and B arise
from three distinct causes, viz. : the power, the resistance,
and the weight of the machine. If T represent the value of
this weight, the centre of gravity of the machine being situ-
ated at the point G, we may regard the weight T as sus-
pended from the point G : the machine being symmetrical
with r€spect to its axis, this point will be situated upon the
axis. Then, if the power P be replaced by its components
122
STATICS.
P' and P", it will be simply necessary to substitute for the
four forces F, P", R, and T, two others applied at A and B
respectively.
The forces R and T having been determined by experi-
ment, P' and P" may be expressed in functions of R. For,
we have {Pig. 118 and 119)
P'=FL=P cos FPK, P"=FK=P sin FPK ;
or,
P'=P cos <p, P"=P sin ^ (123).
But the angle p being equal to the angle CFD, we obtain
1 : cos <5 : : CF : DF, 1 : sin <5 : : CF : CD ;
whence,
DF CD
cosp=_, sinç=-.
Substituting these values in equations(123), there results
CF' CF'
and replacing P by its value given in the proportion (122),
we obtain
p, R.HI.DF Rjn
"" DC.CF ' "" CF •
The vertical forces R and P" being regarded as acting at the
extremities of a lever whose fulcrum is situated at the point
M, their resultant will pass through this point, and its value
will be expressed by R+P".
If Z and Z' denote the effects produced by this resultant
upon the points A and B, their values will be determined by
the proportions
AB : BM : : R+P" : Z,
AB : AM : : R+P " : Z'.
Representing in like manner by U and U', the components of
T acting on the points of support, we shall have
AB : BG : : T : U,
AB : AG : : T : U'.
The forces U and U' being vertical, they must be added to
Z and Z' respectively. The horizontal force P', which acts
at C, the centre of the wheel, being likewise decomposed into
two components Y and Y' applied at the points A and B, the
TVHEEL AND AXLE. 123
values of these components Y and Y' will result from the
proportions
AB : CB : : F : Y,
AB : AC : : F : Y'.
Thus, having constructed two rectangles, the first of which
shall have a height Z + U and a base Y, and the second a
height Z'+U' and a base Y', the diagonals of these rectangles
will represent the pressures on the points of support ; and the
angles formed by the diagonals with the sides of the rectangles
will make known the directions in which these pressures are
exerted.
244. If regard be had to the thickness of the cords, we must
consider the effects of the powers as transmitted through the
axes of the cords ; thus, the radius of the cylinder and that
of the wheel must be increased by the semi-diameter of the
cord, and we shall then have the proportion : the power is to
the resistance as the sum of the radii of the cylinder and
cord to the sum of the radii of the wheel and cord.
245. The capstan is a variety of the wheel and axle, in
which the axis of the cylinder has a vertical position.
246. Let it now be supposed that we have a system of
wheels and axles arranged in the following order :
The power P applied to the circumference of the wheel
AD {Pig. 120) communicates motion to the cylinder BC,
from which the motion is transmitted to a second wheel
A'D', by means of the cord BA'. The wheel A'D' turns the
axle OB', to which is attached the cord B'A", and a similar
arranarement is continued to the last axle, from which the
resistance R is suspended.
When the system is in equilibrio, if we denote by T, T', T",
&c., the tensions of the cords BA', B'A", &c., we shall have
For the first wheel and axle, P : T : : OB : OA,
For the second . . . . T : T' : : O'B' : OA',
For the third T' : R : : 0"B" : 0"A".
These proportions being multiplied together, there results
P ; R : : OB X OB' X 0"B" : O A x O'A' x O "A" ;
whence,
P OBxO'B'xO"B"
R OAxO'A'xO'A
7/ J
124
STATICS.
irom which we conclude that the -power is 'o the resistance
as the continued product of the radii of the axles to the con-
tinued product of the radii of the ivheels.
If the radius of each axle be supposed equal to the 7i" part
of the radius of its wheel, the preceding proportion will
become
p : R : : 2^x — X^^' ": OA X O'A' X 0"A",
)i n 2i
which reduces to
P : R : : 1 : n\
247. The different parts of a system of wheel-work are
frequently caused to act upon each other by means of teeth
projecting from the several circumferences. These teeth
perform the same office as the cords in Pig. ISO. Each
toothed-wheel is traversed by an axis bearing a smaller wheel
which is called a pinion^ and the teeth of this pinion are
called leaves. The first wheel turns its own pinion, both
being firmly connected with the same axis, and the leaves of
the pinion catching into the teeth of the second wheel, com-
municate a motion to it in a direction contrary to that o f the
first wheel, hi a similar manner, the pinion of the second
wheel transmits a motion to the third wheel, and the same
arrangement is continued throughout the system. The
pinions replace the axles of the preceding combination, and
hence the condition of equilibrium is, that the power shall be
to the resistance as the continued product of the radii of the
2nnions to the conti/tued jjroduct of the radii of the wheels.
248. Let D, D', D", &c. represent the numbers of teeth in
the wheels A, A', A", &c. [Pig. 121), and d, d', d", <fcc. the
numbers of leaves in the pinions a, a', a", &c. ; and let us
suppose that while the wheel A makes N turns, the wheels
A', A", (fcc. make respectively N', N", &c. turns. At each
revolution of the wheel A, the pinion a will engage in suc-
cession all its leaves in the teeth of the wheel A' ; so that in
N revolutions it will engage with A', a number of teeth ex-
pressed by Nrf : in like manner, the wheel A' making N' turns
must engage with the pinion a, a number of teeth expressed
by N'D'j and since the numbers of teeth and leaves which the
WHEEL AND AXLE. 125
wheel A' and the pinion a mutually interlock are equal to
each other, we must necessarily have
for a similar reason, the other wheels will furnish the
equations
N"D"=N'c;', N"'D"'=N"<i", (fee.
These equations being multiplied together, there results
N"'D'D"D"'=Ndd'd"',
whence,
D'D'D"*
For example, if it were required to determine the number of
teeth which should be employed in order that the wheel A'"
should make one revolution while the wheel A performs 60,
we should have
N"' = l, N=60, l=60^j^ (124).
The numbers d, d' and d" being assumed arbitrarily, we will
suppose tZ=4, d'=5, d" =7 ; this supposition will reduce the
last of the equations (124) to
D'xD"xD"'=60x4x 5x7=8400.
The number S400 being divided into the three factors 12, 25,
and 28, will evidently furnish a solution to the problem, since
the quantities D', D", and D'" may be made respectively
equal to these factors. The problem obviously admits of an
indefinite number of solutions.
The quantity N'" must be assumed less than N, since we
have supposed d<D', d'<D", d"<D"', and the wheel A'" will
therefore make a less number of revolutions in a given time
than the wheel A.
249. The theory of the jack-screw is likewise to be referred
to that of the wheel and axle. There are two varieties of
this machine, the simple and the compound. The simple jack
is composed of a toothed bar of iron AB {Pig. 122) which
slides in a case CD. The teeth of this bar work in the leaves
of the pinion EF, which is put in motion by means of a
crank G ; thus, the teeth of the bar being subjected to a
pressure from the leaves of the pinion, the bar will move in
136 STATICS.
the direction of its length, and will overcome a resistance at
A. In this machine, the crank and pinion perform the
offices of the wheel and the axle in the common machine, and
the conditions of equilibrium may therefore be stated thus :
the power is to the resistance as the radius of the pinion to the
radius of the crank.
250. In the compound jack-screw, the motion is cormnu-
nicated by means of a crank to a pinion, the leaves of which
work into the teeth of a wheel ; the axis of this wheel carries
a second pinion, which in its turn communicates motion to
a second wheel, and the same arrangement is continued to
the last pinion, whose leaves act on the teeth of the iron bar.
The condition of equilibrium in this machine obviously is,
that the power shall he to the resistance as the continued pro-
duct of the radii of the pinions to the continued product of
tht radii of tlie wheels and the radius of the crank.
Of the Inclined Plane.
2b\. This machine consists of a plane inclined to the
horizon : its object is to support in part the weight of a body
placed upon it.
Let M represent a body {Pig. 123) the weight of which is
supposed concentrated at its centre of gravity, and exerted in
the vertical direction MP. In order that this body may be
sustained in equilibrio upon the inclined plane by the appli-
cation of a force Q,, it is necessary that this force Q, and the
weight of the body represented by P, should have a single
resultant ; this condition can only be fulfilled when, the
directions of the forces intersect at some point M : but the
hne MP being vertical, and passing through the centre of
gravity, the plane of the forces PMQ, must likewise be ver-
tical, and must contain the centre of gravity. Thus the first
condition of equilibrium requires that the direction of the re-
sultant be situated in a vertical plane passing through the
centre of gravity of the body. The second condition is, that
the resultant MN of the two forces P andQ, shall be destroyed
by the resistance of the-inclined plane, which condition can
only be satisfied when the direction of this resultant is per-
INCLINED PLANE. 127
pendicular to the plane, and intersects it at some point within
the polygon formed by connecting the extreme points of can-
tact of the body and the plane.
252. The preceding conditions being fulfilled, we will sup-
pose KL to represent a body {Fig. 123) retained in equili-
brio upon an inclined plane by the application of a force
d. Let the lines ME and MF be taken proportional to
the weight P and the force Q,, and let the parallelogram
FMER be constructed : the diagonal MR will represent the
pressure exerted by the body against the plane, and if this
pressure be denoted by R, we shall have
a : P : R : : sin PMR : sin dMR : sin PMQ, (125).
The triangles APO and OMN being similar, the angles PMR
and CAB will be equal to each other, and therefore
sin PMR=sin A=-— ;
AO
this value being substituted in the proportion (125), we ob-
tain
a : P : R : : CB : AC xsin aMR : AC Xsin PMQ,.
253. If the direction of the power be parallel to the plane
{Fig. 123), the triangles MER and ACB will be similar, since
the angles C and E are then equal to each other, and we have
the proportion
ER : ME : : CB : AC ;
from which we conclude that when the power acts parallel
to the plane, the power Q, is to the loeight P as the height of
the plane is to its length.
254. When the power becomes parallel to the base of the
plane {Fig. 124), the similar triangles MER and CAB give
the proportion
ER : EM : : CB : AB,
or,
a : P : : CB : AB ;
thus, in this case, the poiver is to the weight as the height of
the plane is to the base.
255. The angle Abeing supposed equal to 45°, and the power
applied parallel to the base, the weight and power will be-
come equal ; if the angle A be less than 45°, the weight will
126 STATICS.
be greater than the power, and if A be greater than 45°, the
power will exceed the weight, or the use of the machine will
occasion a loss of power.
256. If a body be sustained in equilibrio between two in-
clined planes, the conditions of equilibrium will require that
the weight of the body be susceptible of being resolved into
two components which shall be respectively perpendicular to
these planes, and shall intersect them at points situated within
the polygons formed by joining the points of contact of the
body with each plane. The line of direction of the weight
being vertical, the plane of its components will likewise be
vertical : and since these components are respectively per-
pendicular to the inclined planes, their plane will be perpen-
dicular to the common intersection of the inclined planes :
hence, this intersection must be a horizontal line.
The pressures sustained by these planes may be readily
determined by constructing the parallelogram of forces, whose
diagonal shall represent the weight of the body, and whose
sides shall be perpendicular to the inclined planes.
Of the Screio.
257. Let the sides of the rectangle AM' {Fig. 125) be
divided into equal parts by the parallel lines BB', CC, (fee,
and let the diagonals AB', BC, &c. be drawn. If the rectan-
gle M'A be then applied to the surface of a right cylinder
with a circulai base, the circumference of which is equal to
the line AA', in such manner that the right lines MA and M'A'
shall be caused to coincide, the points A, B, <fcc. will fall upon
the points A', B', (fee. respectively, and the diagonals will trace
upon the surface of the cylinder PQNM {Fig. 126) a curve
PRSTUV (fee, which is called a helix.
258. The characteristic property of this curve is that the
tangent at every point is equally inclined to the element of
the cylinder passing through that point : this is obvious from
the manner in which the curve is generated.
The distances mn, 7?i'n', m"n", (fee. {Fig. 125) being equal,
their equality will be preserved when the rectangle is applied
to the surface of the cylinder : consequently, if we assume
SCREW. 129
mn as the base of an isosceles triangle mno, the plane of which
passes through the axis of the cyhnder, and cause the triangle
to move around the cylinder, in such manner that the points
9n and 7t shall constantly remain on two adjacent helices, the
plane of the triangle continuing to pass through the axis of
the cylinder, there will be generated by this motion a project-
ing fillet which will completely envelop the cylinder MQ,.
The cylinder and fillet taken conjointly constitute the screw,
and the latter is usually called the thread of the screw. This
thread is sometimes generated by the motion of a rectangle,
instead of a triangle.
259. The nut is composed of a hollow piece, having a
spiral groove cut in its interior, in which the threads of the
screw work. It may be regarded as forming the mould of a
portion of the screw.
The screw can be readily turned within the nut, and at
each revolution passes over a distance in the direction of its
length equal to the distance between the threads.
Since the conditions of the problem are precisely the same,
whether we regard the nut as turning on the screw, or the
screw as turning within the nut, we will adopt the first
hypothesis.
260. To determine the conditions of equilibrium in this
machine, we will suppose the nut to be placed on its screw,
and the axis of the screw to have a vertical position. Let
the nut be divided into any number of particles, whose weights
are denoted by m, m', m", &.C., each of which rests on some
point of the screw ; and let us determine the force necessary
to sustain any one particle m {Mg. 127).
The particle m, being connected with the axis of the screw
in such manner that its distance from the axis shall remain
invariable, must, if unsupported, descend along a helix, every
point of which will be at the distance mC from the axis.
Thus, by regarding this helix as an inclined plane, the height
of this plane will be the distance between the threads, and
its base will be the circumference described with mC as a
radius.
Let us suppose a horizontal force P {Pig: 128) to be ap-
plied immediately to the particle w, for the purpose of
I
130 STATICS.
sustaining it in equilibrio upon the inclined plane. By con-
structing the right-angled triangle KHw, whose height shall
be the distance between the threads, and its base the circum-
ference described with the radius mC, we shall obtain by the
principle of the inclined plane (Art, 254),
P : m : : height : KH ;
or,
P : m : ; mH : circumference Cm (126).
But if the point of application of the power be transferred
from the point m to the point D, the extremity of the lever
CD, the force d, which applied at this point will produce the
same effect as the force P applied at m, can be determined
from the following proportion,
a : P : : Cw : CD ;
or,
Q, : P : : circumference Cm : circumference CD.
And by comparing this proportion with (126), we obtain
Q, : m : : mH : circumference CD.
Thus, for the particle m, the power is to the weight as the
distance between the threads is to the circumference described
by the power.
This proportion being true, whatever may be the distance
of the particle m from the axis of the cylinder^ we shall ob-
tain for the other points in the surface of the screw, which
support the weights m, m", (kc, by means of the ^oroes
Q,', Q.", (fcc, applied at the same distance CD,
: : mH : circumference CD,
: : mH : circumference CD,
: : mH : circumference CD,
(kc. &c. &.C.
From these proportions and the preceding, we deduce
„_ 771 XmH ^,_ m'XmH ^„_ 7;i"XmH
"circumf CD' circumf CD' ""circumf CD' ' ^ ^'
These values are independent of the distances of the points
m, m', m", ôcc, from the axis of the cylinder ; and since the
forces Q,, CI', Q,", &.c, were supposed applied at equal dis-
tances from the axis, they will communicate to the nut the
a
: m
a"
: m'
a"
:m'
WEDGE. 131
same motion of rotation as would be imparted by a single
force equal to their sum, and acting along the line DQ,.
Thus, by adding the equations (127), we find
circumf. CD
(w+m'+m"+&c.)=(Cl+Q,'+a"+&c.)-
mH
and since the sum (m+y'i'+w"+&c.) represents the entire
weight M of the nut, we shall have, after replacing the sum of
the forces Q, Q,', Gl", (fee, by a single force Q,; ,
jj^ circumfCD-
ma.
whence,
Q.y : M : : mH : circumference CD :
or, the power is to the toeight as the distance between the
threads is to the circumference described by the power.
It thus appears that the machine will be rendered more
powerful by applying the force at a greater distance from the
axis, or by diminishing the distance between the threads
of the screw.
Of the Wedge.
261. The wedge is a triangular prism, one of whose edges
is introduced into the crevice of a body, for the purpose of
enlarging the opening.
All cutting instruments, such as knives, scissors, razors,
&.C., may be regarded as wedges.
262. The power is usually applied by communicating an
impulse to the back of the wedge, in a direction perpendicular
to it : if the direction of this impulse be oblique, it may
always be resolved into two components, of which one shall
be perpendicular to the back of the wedge, and the other shall
coincide with it. The first will produce its entire effect, the
second will only tend to move the point of application of the
power along the back of the wedge.
Let ABC {Fig. 129) represent a profile of the wedge;
AC and BC are sections of its faces, and AB a section of its
back, upon which the power is applied in a perpendicular
direction.
12
132
STATICS.
To determine the relation between the power appUed to the
back of the wedge and the pressures exerted at the faces, we
will suppose the power F to be represented by the line DE,
and draw DM and DN perpendicular to the faces AC and BC :
then, by constructing the parallelogram DIEK, the compo-
nents DI and DK will represent the pressures exerted against
AC and BC. Denoting these pressures by X and Y, the
similar triangles ABC and IDE give the proportion
DE : DI : IE : : AB : AC : BC ;
or,
F : X : Y : : AB : AC ; BC ;
and by multiplying the three last terms in this proportion by
the line GH {Fig. 130), we have
F : X : Y : : AB X GH : AC X GH : BC XGH.
The products AB X GH, AC X GH, and BC X GH express the
surfaces of the back and faces of the wedge, and we therefore
conclude that in this machine, the poiver F applied to the
back, and the efforts X and Y exerted by the sides, are respect-
ively propoî'tio?ial to the surfaces of the back and sides of
the wedge.
The power of the wedge will evidently be augmented
either by decreasing the back of the wedge, or by increasing
the lengths of its faces.
Friction.
263. If a body be placed upon a horizontal plane, the action
of gravity exerted upon it will be entirely counteracted by
the resistance of the plane, and the least possible impulse
will communicate a motion to the body, if it be not retained
by physical causes which oppose motion. The most efficient
of these causes is friction. This term is applied to the force
which tends to prevent a body from sliding along the surface
of a second body, and which arises from the slight inequalities
in the two surfaces ; the projecting points of one surface en-
tering the cavities of the second give rise to a passive force
which tends to assist or oppose the power, according as this
power is employed to sustain or move the body.
FRICTION. 133
The eiFect of friction is found to be sensibly proporiional
to the pressure, so long as this pressure is retained within
moderate limits. Thus, if we denote by / the friction ex-
erted by a homogeneous body AB {Pig. 131), the weight of
which is equal to unity, and if AB' be supposed equal to twice
AB, the corresponding friction will be expressed by 2/"; if
AB" be triple AB, the friction will be equal to 3/, &.C.; so that
if F denote the friction exerted by the body AM, which con-
tains a number N of units of weight, we shall have
F=N/- (128).
264. The friction may be measured in the following
manner :
Let AB (Fig. 132) represent the body which exerts by its
weight the unit of pressure on a horizontal plane LK. To
the body is attached a thread CDE, which passes over a fixed
pulley, and sustains the weight M : this weight being grad-
ually increased, its intensity at the moment when it is about
to overcome the resistance which the body opposes to motion,
will measure the friction /, corresponding to the unit of pres-
sure.
265. There is another method of measuring the friction,
which results from the following theorem : If a body MN he
placed upon an inclined plane AC {Fig. 133), a7id if the
angle A wliich this j)lane forms with the horizon he grad-
ually augmented until the body is about to commence sliding
upon the plane, the nmnerical value of the unit of friction
toill then be equal to the tangent of the angle ichich the
inclined -plane forms with the horizon.
To demonstrate this fact, let the lines GD and GK be
drawn, respectively perpendicular to AB and AC ; the centre
of gravity of the body being supposed situated at the point
G. Represent by GD the weight of the body, and decom-
pose GD into two forces GH and GK, parallel and perpen-
dicular to the inclined plane : we shall then have
GH=DK=GD sin DGK,
GK=GD cos DGK ;
12
134
STATICS.
but the angles DGK and CAB are equal to each other ; and
hence, the preceding equations may be written thus,
GH=GDxsinA,
GK=GDxcos A;
or if N expresses the weight of the body,
GH=NsinA,
GK=N cos A.
The pressure sustained by the inclined plane being expressed #
by GK=N cos A, the corresponding friction will be expressed
by N cos A/; but since the effect of friction is to counteract
that tendency which the body has to move along the plane
when there is no friction, it follows, that an equilibrium will
subsist between the force of friction and the component of
the force of gravity, GH=N sin A, which acts in the direc-
tion of the plane ; whence we obtain
N cos A./=N sin A.
From this equation we deduce
/=tangA (129).
266. The angle thus determined is called the angle of
friction ; its value will remain constant only when we adopt
the hypothesis that the friction varies proportionally to the
pressure. For, the relation expressed in (129), has been
deduced by employing (128), which expresses this law ; and
the law, as has been already remarked, exists only for mode-
rate pressures.
267. Since different substances have pores of very unequal
magnitudes it happens that the friction is not the same for all
bodies ; hence, experiments have been instituted for the pur-
pose of determining the friction peculiar to each.
The following results which express the relation between
the friction and the pressure, have been obtained by Coulomb :
Iron against iron /=0.28,
Iron against brass .... /=0.26,
Oak against oak /=0.43,
Oak against fir /=0.65,
Fir against fir /=0.56,
Elm against elm /=0.47.
FRICTION. 135
These last results were obtained when the friction was
exerted in the direction of the fibres ; but when the direction
of the fibres formed a right angle with that of the motion,
the friction was found to be much less, but still in a constant
ratio to the pressure ; the results in this case were as follows :
Oak against fir /=0.158,
Fir again&t fir /=0.167,
Elm against elm /=0.100.
It also appears from the experiments of Coulomb, that
the friction exerted by a body m motion is very nearly inde-
pendent of the velocity of the body.
The polish of the body and the introduction of an unc-
tuous substance between the rubbing surfaces contribute to
lessen the effect of the friction.
268. When one body is caused to roll upon another, a cer-
tain degree of resistance is still offered by friction, but this
resistance is much less intense than in the case of a sliding
motion. This result appears to be a consequence of the dis-
engagement of the inequalities in the surfaces, which the
motion of rotation tends to effect.
269. The general laws of friction, as deduced from the
experiments of Coulomb, may be summed up as follows :
1°. Priction varies loith the polish of the surface : Thus,
the resistance opposed by friction may be reduced by diminish-
ing the asperities of the rubbing surfaces.
2°. The friction between bodies of the same kind is greater
than between bodies of different kinds.
3°. Priction does not depend on the extent of surface iti
contact, the entire pressure exerted betiveen the bodies remain-
ing the same.
4°. Priction is proportional to the pressure.
5^. Priction is diminished by interposing a substance of
an unctuous nature betioeen two surfaces which slide upon
each other.
6°. The friction is greatly diminished by substituting a
rolling for a sliding motion.
270. The adhesion which takes place between the surfaces
of bodies is another physical cause opposed to their motion.
136 STATICS.
It is difficult to estimate, in a precise manner, the proper
measure of this effect, in consequence of its being hable to a
very great increase with time in those machines which are at
rest ; and, on the contrary, to undergo occasional changes in
those which are in motion.
The law which this force usually follows is that of being
sensibly proportional to the extent of the adhering surfaces.
Thus, by denoting the adhesion of a superficial unit by the
quantity i^, the adhesion of a surface whose area is a will be
expressed by a-^.
*
Effects of Pi-iction in certain Machines.
271. Let P and S {Fig. 134) represent two forces applied
to a material point which rests in equilibrio on an inclined
plane AB, and let « and «' denote the angles which the direc-
tions of these forces make with the plane. If we disregard
the effects of friction and adhesion, the conditions of equi-
librium will require the relation
P cos a=S cos «' (130) ;
but if friction and adhesion be considered, since these two
forces are opposed to the motion which the power P tends to
impress in a direction from ni towards B, it will be necessary
to add these forces to the component of S in the direction of
the plane, which is expressed by S cos «'. To determine their
values, we remark that the pressure exerted upon the inclined
plane is produced by the normal components of the forces
P and S. These coniponents are expressed respectively by
P sin cc and S sin «' ; and their sum will be equal to the entire
pressure which is denoted by N in equation (128), Thus, the
force arising from friction is expressed by (P sin «+S sin x)f.
If we denote by a the area of the surface in contact with the
plane, the adliesion will be represented, as has been before
stated, by the quantity ai'. Consequently, by adding these
forces to the second member of the equation (130), we shall
obtain for the condition of equilibrium
P cosa — S cos*' + S sin «/+P sin o/'+aT//;
from which we deduce
FRICTION. 137
p_S cos et'+ S/sin ei'+a-'P ^^l^
cos a— /sin e» ^ ''
272. If, on the contrary, the power be only required to
retain in equiUbrio the point m, the friction and adhesion,
being still opposed to motion, will tend to assist the force P,
and the algebraic signs of these quantities must therefore be
changed. Representing by P' the force necessary to support
in, upon this hypothesis, we shall have
^,^S c osx-Sfsmu—a-^
cos <* +/ sin « ^ ^'
It is evident that the equilibrium may be preserved by the
application of any force P" in the direction Pm, provided the
intensity of this force be intermediate between the intensities
P and P' given by equations (131) and (132).
273. The eifect of friction in modifying the conditions of
equilibrium in the lever and pulley will now be considered.
Let the lever be perforated by a circular hole, through
which is passed a cylinder having a vertical position. Since
the circumstances will be the same, whether we regard the
lever as turning about the cylinder, or the cylinder as turning
within the lever, we shall adopt the first hypothesis, and con-
sider the point m of the lever {Mg. 135), which, being in
contact with the cylinder, is subjected to the action of the
force of friction. Let the cylinder be intersected by a hori-
zontal plane passing through w, and let this plane be assumed
as the co-ordinate plane of x, y. For the purpose of simplify-
ing the question, we shall suppose the resultant R of all the
forces applied to the lever to be situated in the plane of a:, y.
The intersections of the cylinder and lever by the plane of
.r, y will be represented respectively by the circle mBE, and
the plane curve GIL. The cylinder being immoveable, the
point m can be subject only to a circular motion about the
point C, at which the axis of the cylinder is intersected by
the plane of x, y. If the point m remain immoveable, the
equilibrium must result from the combined actions of the
resultant R of the several forces applied to the lever, the fric-
tion, and the resistance opposed by the axis. The direction
of this resistance being normal to the surface of the cylinder,
y-
138 STATICS.
we may drop the consideration of the fixed cyHnder, and con-
sider the point as perfectly free, and sustained in equihbrio by
the three following forces : 1°. the normal force, which acts in
the direction from C towards m ; 2°. the friction, which acts
along the tangent wD ; 3°. the resultant R of all the forces
in the system.
274. It should be remarked that although two of the three
forces are applied at m, the third force may be applied at any
other point, provided its line of direction passes through m.
If we regard the point of application of the third force as
unknown, the conditions of equilibrium of the three forces
will be expressed by the equations (52), (53), and (54).
275. To express these conditions, we will suppose the origin
of co-ordinates to be placed at C, and represent by N the nor-
mal force, which forms with the axes angles equal to « and /3 :
denote by F the friction, the direction of which forms Avith
the axes the angles <*' and ^', and by h the radius of the cylin-
der which is supposed to be nearly of the same size as the
circular hole through which it passes. The components of
the force R, parallel to the two axes, will be represented by X
and Y respectively, and the perpendicular distance of this
force from the point C by the letter r.
This being premised, the condition expressed by equation
(52) requires that the sum of the components parallel to the
axis of X shall be equal to zero ; hence,
N cos ^ + X + F cos «'=0 (133).
For a similar reason, the components parallel to the axis of
y give
N cos /3 + Y + F cos /3'=.0 (134).
And the third equation of equilibrium, which expresses the
relation between the moments, gives
Rr + F/i=0 (135);
which becomes, by substituting for F its value deduced from
equation (128),
Rr + N/A=0 (136);
276. Before employing equations (133) and (134), it may
be remarked that any one of the four quantities cos «, cos «',
cos /8, cos /3', which appear in those expressions, will serve to
:zl\ (1^^)-
FRICTION. 139
determine the remaining three. For, the angle yCx {Pig.
135) being equal to a right angle, we shall have
cos /3=sin « ;
and if we draw the line FK parallel to Cm {Fig. 136), we
shall obtain
wFH=rmFK+KFH;
or,
consequently,
cos *'=cos 90° cos *— sin 90° sin *=— sin «,
cos /3'=sin «'=sin 90° cos «+cos 90° sin «=cos «.
By means of these values of cos ji, cos <*', and cos /3', we reduce
the equations (133) and (134) to
N cos «+X— F sin «=0
Nsina + Y+F cos
277. These equations admit of a further reduction, from
the consideration that the friction exerted at the point w is
proportional to the normal pressure N ; thus, by replacing F
by its value N/in the equations (137), we find
X=N/sina— N cos»
Y=— N/cos«— Nsin^
But X and Y being rectangular components of the force R,
we must have the relation
R2=X2+Y2,
Substituting in this equation the values of X and Y found
above, we obtain
R2=N2(sin2 «+cos'<i)-{-N3/2(sin" a+cos^ x),
or,
R2=N2(l+y2) (139).
From this equation taken in connexion with (136), we find
r=±—Ih. (140).
This value of r will always be less than that of A, since the
f
fraction — - — ~ is less than unity ; but h represents the
■v/(l+/')
radius of the cylinder, and hence it follows that the equi-
(138).
140
STATICS.
librium is only possible when the distance r of the point C
from the direction of the resultant does not exceed the radius
of the cylinder. The direction of the resultant will there-
fore intersect the surface of the cylinder. This condition,
without which the equilibrium of the lever, maintained by
the effect of friction, becomes impossible, is not alone suffi-
cient ; for the value of r must not exceed that determined by
equation (140) ; otherwise the condition of moments could
not be fulfilled.
278. It may be remarked, that the equation of the moments
expresses the condition that the friction and the resultant of
all the forces applied to the lever, acting conjointly, will pre-
vent any tendency to rotation. For since the direction of
the normal force passes through the origin, it can have no
tendency to produce rotation. If, therefore, an equilibrium
subsists, it must be produced in consequence of the forces
R and F exerting equal efforts to turn the system in con-
trary directions. But this is precisely the condition expressed
by the equation (135), since the moments of the forces are
equal and have contrary signs.
279. We can also determine the relation which must sub-
sist between the power and the resistance. For this purpose
the preceding results must undergo certain modifications.
Let P and S represent the power and resistance [Fig.
137), which form with each other an angle è ; the result-
ant of these two forces will be determined by the equation
(Art. 30)
R2=P2+S2-}-2PScos<i.
By substituting this value of R» in equation (139), it becomes
P2+S2-1-2PS cosô=N^(l+/2) (141).
280. Let the value of N be now expressed in functions of
the quantities P and S. For this purpose, let the perpen-
diculars f and s be demitted on the directions of the forces
P and S respectively ; the moment Rr of the resultant can
then be chanored into Pp — Ss, or S5 — Pp, according to the
direction in which the resultant tends to turn the system ;
thus the equation (136) will become
±(Pp-S5)-f-rsyA=o
FRICTION. 141
whence,
and by substituting this value in equation (141), we find
This result may be simplified by making
P=S0, and — ^=^'' (142).
The quantity S^ will then disappear, being a common fac-
tor, and the equation will reduce to
z'-{-2z cos 6-{-l=-^(pz—sy.
From this equation we deduce
z^h- +2zh' cos 0+^2 =k^^ {p'z^ —2pzs-\-s^) ;
and by transposition,
{k^'p^ —h^)z^ —2{psk^ +/i2 cos 6)z^k^s^—h^=(i\
or, by division,
, 21 psk^ -\-h'' cos 6) k'^s^'—h^ ^
z — ~ z A =U.
The value of z deduced from this equation is the ratio of the
power to the resistance ; and since z has two values, it is
obvious that the first will apply to the case in which the
power is about to overcome the resistance, and the second to
that in which the resistance is about to overcome the power.
By resolving the equation, we find
_ psk'>-]-h'cos6±V[{psk2+h'^cos6y—{k^p^—h'>){k''s^ — h^)] ^
and by developing and reducing the terms contained under
the radical signs, we obtain
ps k^ -\-h^cos6±h^[k^{2r- -\-2pscos6-\-s'')—h''{\—cos''6)] ^
^'^~~ k'p^-h' '
and finally, by substituting for z and 1 — cos^ e their respect-
ive values, we shall have
P psk'' +h- cos 6 ±hy[k''(p'' +2ps cos 6 + s'')—h' sin'' O ]
S l^2p2_h,2
142 STATICS.
281. If the radius of the cyUnder be very small, its square
h^ maybe neglected, and the preceding ratio will then become
P _5 A^(jJ^+2jJgCOS0 + 5')
If the perpendiculars p and 5, demitted from the point C on
the respective directions of the power and resistance, become
equal to each other, the results will apply to the case of the
pulley ; and by still neglecting the quantity h^, we shall find
P^ AvW+cos^] .
S kp ^ ^*
282. Finally, when the power and resistance act in par-
allel directions, the angle 6 becomes equal to zero ; whence,
sin ^=0, cos^=l ;
and the equation (143) then reduces to
?=1±?^.
S kp
283. The same principles will serve to determine the con-
ditions of equilibrium in the other mechanical powers, when
regard is had to the effects of friction ; but the results obtained
would in general prove much more complicated.
Of the !Stiffness of Cordage.
284. In employing the cord as a means of transmitting the
effect of a force to a machine, we have hitherto supposed the
cord to be perfectly flexible. But as this hypothesis is inad-
missible in practice, it becomes necessary to estimate the ad-
ditional force that will be necessary to overcome the rigidity
of the cord.
Let P and d {Fig. 138) represent two weights which are
applied to the extremities of a cord passing over a fixed pul-
ley : if the weight P be supposed to prevail, and the cord be
regarded as perfectly rigid, the extremity Q. will evidently be
brought into a position Q,', such that the vertical line Q'O
will intersect the horizontal line CO drawn throusfh C, at a
distance CO from the centre, greater than the radius CG.
The extremity P will at the same time assume the position P',
STIFFNESS OF CORDAGE. 143
such that the vertical Hne drawn through P' will intersect
the radius CF. Hence the arm of the lever to which the
force Q, is applied will now be longer than that of the force
P, and the condition of equilibrium will therefore require that
the force P shall exceed Q.,
285. If the cord be supposed imperfectly rigid, similar
effects will be produced, though in a less degree ; and in
practice, it is found that the decrease in the arm of lever, to
which the preponderating weight is applied, is wholly insen-
sible. Hence, in estimating the effects produced by the rigid-
ity of a cord employed in a machine, it will simply be neces-
sary to increase the arm of the lever to which the resistance
Q is applied, by a proper quantity q.
286. To determine the value of q^ we remark that the re-
sistance to flexure opposed by a given cord arises from two
distinct causes, — viz. 1°. The tension of the cord, or the force
Q, which is employed to stretch it ; and, 2°. The materials
used in the construction of the cord, and the degree of twist
which has been given to it. The resistance arising from the
tension of the cord is found to be proportional to this tension,
and may therefore be represented by an expression of the
form 6Q, in which h represents an indeterminate constant.
The resistance produced by the second cause may be repre-
sented by a quantity a.
Thus, for the same cord bent over the same pulley, the
expression (« + 6Q,) may be supposed to represent the effort
necessary to bend it. But if we suppose the diameter of a
second cord to be greater, the force necessary to bend it will
become greater, and we can assume that this force will
increase according to some power n of the diameter D. The
force will also increase as the curvature increases, or as the
radius of the pulley is decreased, and hence (a + 60,) may
be taken as an expression for the force necessary to overcome
the rigidity of the cord. This expression represents the
increment that must be given to the power P, in order that
it may be on the point of overcoming the resistance Q, : but
we also have
Vr=Gi{r-\-q) ;
144 STATICS.
and since the forces P and Q, become equal when the cord is
supposed destitute of rigidity, P— Q, or Q i will also express
r
the value of this increment. By making these values equal
to each other, we obtain
D"(a+6a)=%;
whence,
q^^l.{a-\-hGi) (143 a).
287. This equation should only be regarded as furnishing
an approximate value of the quantity q, since the above rela-
tion has been obtained by considerations of a very general
character. It moreover contains certain unknown quantities
a, b, and ?i, which vary with different cords.
For the purpose of verifying the truth of the preceding
formula, and at the same time determining the values of the
unknown constants, we proceed as follows.
Having selected a cord, we pass it over a fixed pulley, and
attach to its extremities two equal weights : we then increase
one of these weights until it is about to prevail over the other,
and the difference k will give one value of the quantity
r
By repeating the experiment several times, changing the
weights, the cord, or the pulley, we can obtain a number of
similar equations, in which the quantities a, b, and n will be
the same, and the quantities D, r, and Q., although different,
will be known by observation. Three such equations will
serve to determine a, b, and 7i, and their values being sub-
stituted in the general relation expressed by formula (143 a),
the accuracy of the formula can be tested by comparing it
with the results furnished by other experiments.
The quantity n was found by Coulomb to be usually about
1.7 or 1.8 ; and the resistance to flexure must therefore vary
nearly as the square of the diameter of the cord : but the
quantity n is itself subject to some variation, becoming nearly
1.4 when the cord has been long used.
The following results, expressed in French poinids, were
obtained in the experiments of Coulomb.
RESISTANCE OP SOLIDS. 145
ibs. Tin lbs.
J)rt iOS. jy„
30 threads in a yarn . . — Xa=4.2
White rope < 15 threads
f 6 threads
Tarred rope
30 threads in a yarn . . — Xa=6.6
15 threads
6 threads
i4.2 .
. _ixl00=: 9
r
1.2 .
. . . . " 5.1
0.2 .
« 2.2
lbs.
;6.6 .
D» lbs.
. —6X100=11.6
r
2.0 .
« 5.6
0.4 .
" 2.4
On the Resistance of /Solids.
2S8. The particles of every solid body are found to oppose
a certain resistance to any force which tends to separate them.
This resistance arises from the mutual actions exerted by the
particles upon each other ; and if the nature of these actions,
as well as the arrangement of the particles which compose
the body, were accurately known, it might be possible to
estimate the force necessary to separate the particles, or to
produce a given change in the figure of the body. Bat as we
are entirely ignorant of these particulars, it becomes neces-
sary to adopt some hypothesis relative to the manner in which
bodies are constituted, and the nature of the actions exerted
by the particles upon each other. Then, by reasoning upon
such hypothesis, we can obtain results which, compared with
those derived from experiment, will serve to test the accuracy
of the supposition.
289. The hypotheses most generally adopted are — 1'^, That
of Galileo, which supposes all solid bodies to be niade up of
fibres, disposed parallel to the length of the body, and sus-
ceptible of being ruptured without undergoing flexure, ex-
tension, or compression ; or, 2°. That of Leibnitz, modified
by Bernoulli and others, which regards the fibres of all bodies
as elastic ; being susceptible of extension and compression, and
capable of opposing a resistance directly proportional to their
extensions or compressions. The force required to produce
a given extension is, moreover, supposed to be equal to that
which is capable of producing an equal compression.
290. It is very certain that neither of these hypotheses is
strictly correct ; but as the results given by the latter difier but
K 13
146 STATICS.
little from the truth, when the extensions or compressions are
inconsiderable, we shall adopt it, and apply it to the investiga-
tion of the resistance which a solid will oppose under different
circumstances.
291. The kind of resistance which the body offers will de-
pend in a great measure upon the manner in which the force is
applied. Thus, the force may exert an effort to extend or
compress the solid in the direction of its length, or it may
tend to produce a flexure of the solid, or it may operate as
a force of torsion ; and in each of these cases it may be
required to determine the force necessary to produce a rupture
or separation of the particles, or simply that necessary to
effect a given change in the figure of the solid.
The cases which more generally occur are, 1°. That in
which the solid sustains an extension or compression in the
direction of its length, without undergoing sensible flexure ;
and, 2°. That in which flexure is produced by the applica-
tion of a force perpendicular to the length of the solid.
As it is the object of the present article merely to exhibit
the general methods in which the hypothesis assumed may
be applied to the determination of the strength of bodies, or
the resistance which they are capable of opposing, we shall
confine our investigations to the consideration of these two
cases.
292. The resistance of a body to a change of figure de-
pends upon its force of elasticity, which is measured by the
effort necessary to compress or extend the body by a given
quantity. Its resistance to rupture depends upon its force of
tenacity, or upon the effort necessary to rupture or crush the
body.
The values of these forces having been determined experi-
mentally for a body composed of a given substance, and
having a simple form, we can calculate the compression, ex-
tension, or flexure produced in another body, of the same
substance, by the application of a given force. The methods
of effecting this calculation will now be explained.
RESISTANCE OF SOLIDS. 147
Of the Resistance to Compression or Extension.
293. When a solid is stretched or compressed in the direc-
tion of its length, being at the same time prevented from
experiencing flexure, the lengths of its fibres are found to
undergo very slight variations, and we can therefore assume,
in conformity with the hypothesis adopted, 1°. That the
extensions or compressions of all the fibres will be equal to
each other, and uniform throughout the extent of each fibre ;
and that the force necessary to produce a given extension will
be capable of producing an equal compression. 2°. That
the variations in the lengths, and the resistances opposed by
the fibres, are constantly proportional to the forces which
produce them ; and that this proportion obtains even for those
forces which rupture or crush the body.
294. Let a cubical mass of any substance be placed upon
a horizontal plane, and subjected to the action of a weight
which rests upon its upper surface, compressing the substance
in the vertical direction. Denote by
a, the length of one of the edges of the cube ;
a', the quantity by which its vertical dimension is com-
pressed, and which is always extremely small in
comparison with a ;
P, the force which produces the compression.
Then, since the compression of each fibre is supposed uni-
form throughout, or since the particles which compose any
one fibre are supposed to approach each other equally at every
point of such fibre ; it is obvious that the entire compression
a', sustained by any fibre, will be directly proportional to its
length a. For example, if the length of another solid be
supposed equal to 2a, its transverse section remaining the
same, and if the same force P be applied to its upper surface,
the number of particles in the length 2a will be twice as great
as the number contained in a ; and each pair of consecutive
particles being caused to approach each other to within the
same distance, in order that the resistance of the fibre may
be uniform throughout, the whole variation in the length 2a
will evidently be twice as great as that which was produced
K2
148 STATICS.
in the length a, and will therefore be expressed by 2a'. And,
generally, the compression of the solid, whose length is 7ia,
and whose transverse section remains the same, will be ex-
pressed by im', when the same force P is applied to its upper
surface. Let the quantity a be supposed equal to the linear
unit, — one foot, for example ; then 71 will express the number
of feet contained in the length of the second solid, and 7ia'
will express the variation produced in the length of a solid
whose transverse section is equal to one square foot, and
whose length is equal to 71 feet.
295. The preceding remarks have been confined to the
case in which the solid suffers compression, but from the
nature of the hypothesis, they must apply with equal force to
the case in which the eflbrt is exerted to extend the body.
296. If the transverse section of a second solid, whose
length is likewise equal to 7i, be supposed greater than that
of the first, the number of its fibres will be increased in the
same proportion, and the total effort exerted by these fibres
when compressed to the same degree will evidently be pro-
portional to their number : thus, if P' represent the force
necessary to compress a prism whose length is 71, and whose
transverse section contains ??i square feet, by a quantity equal
to na, we shall have the proportion
section 1 : section w^ : : P : P' ;
whence,
P'=wP.
297. If the force P' be increased, the solid will undergo a
greater compression, and the quantity by which the length /*
of the fibre is compressed will no longer be represented by
'iia\ but by an unknown quantity 7ia". To determine this
quantity, we recur to the hypothesis which assumes that the
compressions are proportional to the forces which produce
them ; hence, by calling P" the value of the force which pro-
duces the compression 71a", we shall have
7ia' : 7ta" : : P' : P",
and therefore,
P"=P'Z^-
na' '
RESISTANCE OP SOLIDS. 149
or, replacing P' by its value mP, we have
p„^P^m^" ^^3jj
d n
p
298. The quantity — is called the coefficient of the elas-
a
ticity : its value will depend only on the elastic force of the
substance of which the prism is composed, and will therefore
be independent of the dimensions of the particular prism
under consideration. If we denote this coefficient by A, we
shall obtain, for the entire compression of the prism,
„ nV"
na = — r-.
mA
This expression will determine the quantity by which a given
prism will be compressed under the influence of a given force,
when the coefficient of the elasticity has been previously
ascertained. It should be remembered, however, that this
formula is only applicable when the compressions are exceed-
ingly small ; and that the solid is ruptured or crushed before
its length undergoes a very sensible change.
299. The preceding expression is equally applicable when
the force P" tends to stretch the solid.
300. To determine the force necessary to rupture a given
prism, when exerted in the direction of the length of the
prism, we shall denote by B the force necessary to rupture
a prism of the given substance whose transverse section is a
square foot. Then, if the transverse section of the given
prism be supposed to contain m square feet, the number of
its fibres will be m times greater than the number contained
in the prism whose section is equal to one square foot ; and
since each fibre in the two prisms must oppose the same
resistance at the instant of rupture, we shall determine the
force P" necessary to rupture the given prism, by the propor-
tion
section 1 : section w : : B : P" ;
whence,
P"=mB.
301. The quantity Bis called the coefflciejit of the tenacity,
and depends only on the nature of the substance under consid-
150
STATICS.
eration. Having determined ihis quantity by experiment, we
can readily calculate the force necessary to rupture a given
prism of the same substance. This investigation is equally
applicable whether the force be exerted to compress or extend
the solid. The methods of determining experimentally the
coefficients of the elasticity and tei.acity will be explained
hereafter.
Of the Resistance of a Solid to Flexure and Fracture produced
by a Force acting at right angles to the direction of the
Fibres.
302. When the length of a solid body bears a certain pro-
portion to its thickness, the body is found to undergo a cer-
tain degree of flexure before breaking. Th^s flexure becomes
more perceptible as the length of the solid is increased : thus
a bar of wrought iron whose length does not exceed twelve
or fifteen times its thickness gives very slight indications of
flexibility ; but when its length is increased to forty or fifty
times its thickness, it yields readily to an effort exerted to
bend it, and becomes susceptible of taking a very consider-
able flexure before breakingf.
303. If a force P be applied in a direction perpendicular
to the length of the solid AB {Fig. 139), which is supported
at its two extremities, and if this force be supposed to produce
a certain degree of flexure in the solid, causing it to assume
the form represented in Fig. 139 a, the fibres aa, &c. situated
on the convex side will be extended, their lengths being in-
creased, and those situated on the concave side will suffer a
compression, and will undergo a diminution in length. This
effect is readily observed : for, if the force P be gradually
increased until it become capable of breaking the solid, the
rupture will be found to commence at a point D on the con-
vex side, thereby indicating that the fibres aa on that side
have been most extended ; and if some of the fibres situated
on the convex side be previously separated by cutting them
through transversely, it will be found that a smaller force
than P will be required to fracture the solid. But if, on the
contrary, the fibres bb situated near the opposite side of the
RESISTANCE OP SOLIDS. 161
solid be cut transversely to a certain depth EF (Fig. 139),
and if a thin plate of some unyielding substance be intro-
duced into the cut EF, so as to fill it entirely, it will be found,
upon subjecting the solid to the action of the force P, that
the thin plate will be -retained by a strong pressure tending to
compress it, and that the strength of the solid will not be
diminished, the rupture commencing at the convex side,
when the force P has been increased in the same degree as
was necessary to rupture the solid before severing any of its
fibres.
As we proceed from the convex towards the concave side
of the solid, the extensions of the fibres will gradually dimin-
ish, and at a certain distance from the surface, their lengths
will undergo no variation ; beyond this distance the exten-
sions will be changed into compressions, and these will again
increase until we arrive at the concave side.
304. The flexure of the fibres being supposed to take place
entirely in planes parallel to the axis of the solid and the
direction of the force applied, it is evident that the change of
figure experienced by the solid will require that those fibres
whose lengths undergo no variation should be contained,
previous to the flexure, in a plane perpendicular to the direc-
tion of the force which produces the flexure ; and that, after
the flexure, these fibres will form a cylindrical surface,
whose elements will be parallel to the same plane. More-
over, the fibres situated at equal distances from this plane
will undergo equal extensions or compressions.
305. Let us now conceive a right prism AB to be firmly
fixed at its extremity A, in such manner that its axis shall be
horizontal, and that a vertical plane passing through the axis
shall divide the solid into two symmetrical parts. Let a
weight P be applied at the other extremity of the solid,
causing it to undergo a certain degree of flexure, and to
assume the form represented in Fig. 140. If two planes,
ariv, a'u'v', be drawn infinitely near to each other, and normal
to the curve Auu'B assumed by the fibres whose lengths re-
main invariable, such planes will include between them an
elementary portion of the solid, and if the system be sup-
posed in equilibrio, the state of equilibrium will not be dis-
162 STATICS.
turbed by regarding the portion of the solid included between
the sections ACD and uav as absolutely immoveable, and
the portion of the solid included between the sections BEF
and 2iav as constituting a distinct system. The conditions
of equilibrium in this system will evidently require that the
force Pj together with the force necessary to retain the part
DCAïiav in its position, shall be just capable of sustaining
the efforts arising from the compressions and extensions of
the fibres, or, in other words, that all these forces should reduce
to two that are equal to each other and directly opposite.
306. If we assume any two rectangular axes Ax and Ay
situated in the vertical plane passing through the axis of the
solid, we can resolve each of the several forces into two com-
ponents respectively parallel to these axes ; since these forces
are all situated in planes parallel to the plane of the axes.
Moreover, since the solid has been supposed to be symmetri-
cally divided by the vertical plane passing through the axis,
the forces of elasticity arising from the extensions or com-
pressions of the different fibres will be symmetrically disposed
with respect to this plane, and the conditions of equilibrium
will therefore be the same as though the forces were all situ-
ated in this plane. These conditions are, 1°. That the sum
of the components parallel to each axis shall be equal to zero ;
and, 2°. That the sum of the moments of all the forces taken
with respect to any line perpendicular to the plane of the
forces shall be equal to zero.
307. We shall assume the origin of co-ordinates at the fixed
extremity A of the solid, and refer the points in the curve
Ann' Bto the axes of x and y, which are respectively hori-
zontal and vertical.
308. The normal plane auv intersects the cylindrical sur-
face which contains the fibres of an invariable length, and the
vertical plane passing through the axis of the solid, in two
lines au and nv, at right angles to each other ; and the points
in the section anv will be referred to two rectangular axes,
one of which au will be called the axis of ii, and the other,
parallel to irv, and passing through the origin a, will be desig-
nated as the axis of v. Thus the two co-ordinates of the
point 7)1 will be ao=u, and om=v. The moments of the sev-
RESISTANCE OF SOLIDS. 153
eral forces will be referred to the line au, which is frequently
called the axis of equilibrium.
309. This being premised, we shall denote by
A and B, the coefficients of elasticity and tenacity (Arts.
298 and 301),
R, the radius of curvature wr, of the curve of flexure, at
the point u,
s, the length of the arc Au of the curve of flexure,
X and y, the co-ordinates Ap and jm of the point u re-
ferred to the origin A,
x' and y', the co-ordinates of the point ^ referred to the
same origin,
U and U', functions of the absciss ao=t(., expressing the
values of the corresponding ordinates ol and ol' of
the curve of intersection, reckoned from the axis
of equilibrium au, towards the convex and concave
sides of the solid,
a, the dimension of the solid estimated along the axis
of equilibrium,
V, the greatest value of U or U', or the distance from the
axis of equilibrium to that fibre which is most
stretched or compressed at the instant of rupture.
Then, if we consider an elementary portion of the solid, in-
cluded between the consecutive normal planes, whose base is
represented by the element inm" = du. dv, of the normal section
auv, its original length will be equal to uu'=ds ; and after
the flexure, this length will be increased or diminished,
according to its position with reference to the axis of equili-
brium, and will be represented by mm' or nn' {Fig. 141).
But from similarity of the figures rmm\ rim', run', we have
the proportion
rii : rm : m : : uu' : mm' : nn' \
or,
R : R + v : R — v : : uu' : inmf : w/i' ;
and therefore,
R : V : r : : uu' : mm' — uu' : uu! — nn' \
whence,
, v.uu' vds
mm —uiv=tiu—nn= -5— = -5-.
R R
154 STATICS.
This expression will represent the variation in the length of
the element whose base is equal to du.dv {Pig. 140) and
whose original length was equal to ds. To determine the
resistance opposed by this element when thus extended or
compressed, we employ the expression (143 b) in which we
replace 7ia", the variation in length, by mm'—uu', or nu'—nn' ;
the transverse section ni, by dii . dv ; and the length ii, by
ds : we shall thus obtain an expression for the resistance P"
opposed by the element,
jy„_vds dudvK _Avdvdu ,-...y v
"^ ~R^ ds KT" ^ ''^''
and the moment of this resistance taken with reference to the
axis of equilibrium au, will be
A A
-vdvdu'><.v = -v^dvdu (143 d).
SX K
310. The other elementary portions of the solid included
between the consecutive normal planes will give similar
expressions for the resistances and their moments ; and by
taking the sums of these expressions, we shall obtain the
value of the entire resistance, and that of its moment with
reference to the axis au. To determine the value of these
sums, we must integrate the expressions (143 c) and (143 d)
throuo^hout the limits of the section a7iv. This intégration
is effected, first with reference to one of the variables, v for
example ; and its value being then substituted in terms of u,
we integrate a second time with reference to the other varia-
ble. The limits of the first integration will evidently be
v=0, and ^=11, for those fibres which sufier extension ; and
v=Q, v—V, for those which suffer compression. The limits
of the second integration will be 7i=0, and u=a.
311. This being premised, the sum of the resistances
arising from the extensions of the several fibres will be ex-
pressed by
* An expression of the form y du is intended to indicate that the integral of
da is to be taken between the limits u=0, and u=a. In like manner, f vdv sig-
nifies that the integral of vdv shonld be taken between the limits v=0, and v=U.
RESISTANCE OP SOLIDS. 155
and the sumofthe resistances arising from thecompressions of
the fibres will be
The sum of the moments of these resistances, taken with
reference to the axis au, will be
£(/"''"/''"■*+/"''"/"'"'''') (1*3^)-
312. For the purpose of resolving the resistances (143 e)
and (143/) into components parallel to the axes of x and y,
we must multiply them respectively by -— and--^, the cosines
CiS (tS
of the angles which their directions form with the axes : but
as the curvature assumed by the solid is always found to be
exceedingly small even at the instant when the rupture takes
place, the expression -— will be very nearly equal to unity,
as
and the components in the direction of the axis of x may there-
fore be assumed equal to the entire resistances. These being
the only forces in the system which have components par-
allel to the axis of x, the condition of equilibrium which
requires that the sum of the components parallel to this axis
shall be equal to zero, will be expressed by the equation
/a /»U pa /»U'
duj vdv—J duj vdv^O (143 /t).
The negative sign is given to the resistances offered by those
fibres which suffer compression, because they are exerted in a
direction contrary to the resistances of the extended fibres.
This equation will determine the position of the axis of
equilibrium au when the figure of the transverse section is
known.
313. A similar condition may be obtained for the com-
ponents parallel to the axis of y] but as it will not be required
in the succeeding steps of this investigation, it will be unne-
cessary to express it analytically.
314. The moment of the force P taken with reference to
the axis au will be expressed by V{x'—x), and since this
force tends to turn the system about the axis au, in a direô-
156
STATICS.
tion contrary to that in which the resistances of the fibres
would cause it to turn, the condition that tfie algebraic sum
of the moments of all the forces taken with reference to the
axis of equilibrium shall be equal to zero, will be expressed
by the equation
^{jduj v^dv^jdaj v^'dvS —V{x' —x)^{) .{\^Zi).
315. When the radius of curvature becomes equal to unity
the expression (143 g) becomes
KyJ duj v^dv^-J dvj v-dv) (143 A-).
This quantity is called the moment of elasticity of the solid,
and will depend upon the elasticity of the substance, and the
figure of the transverse section. Its value will evidently
determine that of the force P, which, acting at the extremity
of a given arm of lever, will be necessary to produce a
given curvature in the solid ; thus, the moment of elasticity
becomes a proper measure of the resistance to flexure opposed
by the solid.
316. If the flexure of the solid be supposed such that the
extreme fibre, or that which undergoes the greatest extension
or compression, is about to be ruptured or crushed, the resist-
ance opposed by this fibre will be that due to the tenacity of
the substance : hence, if dudv denote, as in Art. 309, the base
of an elementary portion of the solid included between the
consecutive normal sections, and if the distance of this ele-
ment of the solid from axis of equilibrium au be equal to V,
that of the fibre which is most extended or compressed, the
resistance opposed by such element will be expressed by
Bdiidv ;
B denoting the coefficient of the tenacity.
This element being at the distance V from the axis of
equilibrium, its original length ds will undergo a variation
represented (Art. 309) by
Yds
R '
tind the corresponding variation in the length ds of the ele-
RESISTANCE OF SOLIDS. 157
ment, whose distance from the same axis is denoted by v,
will be
vds
~K'
But the resistances opposed by the two elements being by
hypothesis (Art. 293), proportional to their extensions or com-
pressions, we shall have the proportion
Yds vds r»j J r»//
—5- : -5- : : Bdudv : P",
SX JK
P" denoting the resistance opposed by the element at the
distance v from the axis of equilibrium. From this propor-
tion we deduce
T?"=l.Bdudv.
317. Similar expressions may be obtained for the resistances
offered by the other elements ; and by taking their moments
with reference to the axis of equilibrium, and adding them
mto one sum, we shall obtain for the moment of the entire
resistance, at the instant when a fracture commences.
This expression is called the moment of rupture, and will
depend upon the tenacity of the substance, and the figure of
the transverse section. This moment must evidently be
equal to the moment V[x' — x) of the force P, which is just
capable of causing rupture. Thus, we shall have
Y{/,"di/\^^dv+fJduf\^dv'^=V{x'-x)...{U3m).
The value of the moment of rupture will serve to determine
that of the force P, which, acting at the extremity of a given
arm of lever, will be just capable of producing fracture.
Thus, the moment of rupture becomes a proper measure of
the resistance to fracture opposed by the solid.
318. By comparing the expression (143 k), for the moment
of elasticity, with (143 I), which represents the moment of
rupture, we shall perceive that the latter may be deduced from
the former by merely substituting — for A.
41 ^
158 STATICS.
319. When the transverse section can be divided sym-
metrically by a horizontal line, that line will be the axis of
equilibrium, since the equation (143 A) will evidently be satis-
fied by regarding that line as the axis of ti. The moment
of elasticity will then be expressed by
2Af dut v-clv]
t/ t/
and the moment of rupture by
— - / dut v^dv,
320. In other cases, it will be necessary to determine the
position of the axis of equilibrium by the condition (143 /i),
and then to calculate separately the two integrals which enter
into the expressions for the moments of elasticity and rup-
ture.
321. To apply these principles, we shall determine the
moments of elasticity and rupture for those solids whose
transverse sections are such as are more commonly adopted
in practice.
322. Let the transverse section be a rectangle {Fig. 142),
whose breadth and height are denoted respectively by a and b.
The value of the moment of elasticity will then become
2 At dut V'dv.
pa pib
At dut V'
•/ t/
and by integrating with reference to v=om, between the limits
v=0, and v=ot=^b, we shall obtain double the sum of the
moments of all the elements, whose bases constitute the ele-
mentary rectangle oq. Performing the integration, we have
2Ar duy."^ =2Ar duy.^^l=i-^Ab^r du.
Integrating a second time, with reference to w, between the
limits M=0 and u=a, we shall obtain for the moment of
elasticity «,
«=y_Aa&' (143 w).
Hence it follows that the resistance to flexure opposed by a
solid whose transverse section is rectangular, will be propor-
tional to the breadth and the cube of the depth.
T>
323 If we replace A in this expression by —, we shall
RESISTANCE OF SOLIDS. 159
obtain the moment of rupture $ of the rectangle ; and since
V is in the present case equal to 16, we shall have
i3=JBa6= (143 o).
Thus the resistance to fracture is proportional to the breadth
and the square of the depth.
324. If the solid be disposed in such manner that the
dimension a shall become vertical, and the dimension b hori-
zontal, the expressions for the moments of elasticity and rup-
ture will become respectively
and by comparing these expressions with those obtained,
when the dimension a was supposed horizontal, we shall
deduce the proportions
«:»':: ab^ : ba^ : : b* : a*,
/3 : /3' : : ai^ : ba' : : b : a.
It thus appears that the resistance to flexure when the broader
face b is placed vertically, will be to that exerted when the
narrower face a is vertical, as the square of the broader face to
the square of the narrower. But that the resistances to frac-
ture in similar cases are proportional simply to the first powers
of the same quantities.
325. If in the expressions (143 n) and (143 0), we make
az=b, we shall obtain for the moments of elasticity and rup-
ture of a prism with a square base,
«=^TVAa% 0=lBa^ (143 jo).
326. Let the transverse section of the solid be a rhombus,
{Fig. 143), whose diagonals are represented by 2p and 2q,
and let the diagonal 2q be placed vertically. If we first
determine the moment of elasticity of the triangle oBG, that
of the rhombus can be immediately deduced by simply multi-
plying by the number 2. The limits between which the first
integration with reference to the variable v=ow, should be
effected, are -«=0, and v=ot. But from the similarity of tri-
angles, we have the proportion
ao : ot : : oD : DC,
or,
u: ot ::p : q]
100 STATICS.
whence,
ot=— ;
P
and the limits of the first integration will therefore be v=o,
and v=—. Making these substitutions in the general for-
mula for the moment of elasticity, we shall obtain
2Af''duf"^v'>dv=2Af"duX^(^-^) '=tA^' f\=du.
t/ot/o «/o \p / ]J^»/
Integrating a second time, with reference to the variable u,
between the limits 71 = 0, and u=p, the moment of elasticity
of the triangle aBC becomes
and by doubling this expression, we find for the moment of
elasticity « of the rhombus,
T>
327. If in this expression we replace A by —, we shall ob-
tain the value of the moment of rupture &, which, since
Y=q, will become
^ = L-Xpq= = ^Bpq='.
328. If we make 2y=q, the rhombus will become a square,
and the values of » and p will reduce to
or if the side of the square be denoted by a, we shall have
the relation a^=2p'', and therefore
«=iAx^=-i-Aa*, '5=iBx 2^ =-g-^ Bad-
aud by comparing these expressions with those obtained
(Art. 32o) for the moments of elasticity and rupture of a
prism with a square base, when the sides of the base are
respectively vertical and horizontal, we shall find that the
resistance to flexure will be the same whether the diago-
nal or side of the square be disposed vertically; but that
the resistance to fracture when the side is vertical, will be
RESISTANCE OF SOLIDS. ^61
greater than when the diagonal is vertical, in the ratio of
v^(2) to 1.
329. When the section is a circle whose radius is equal to
r, the integration with reference to the variable v must be
effected between the limits v=0, and v=^{2ru — u"); and
the second integration, with reference to u, between the limits
u=0, and u=2r. Thus, the expression for the moment of
elasticity will be
«=2A/ duj v^dv = \kj ^ {2ru—u''ydu..{U3q).
For the purpose of effecting the second integration, we
make r—u—z, which gives
du= — dz, 2ru — w^ =r^ — z^.
Substituting these values in the expression for «, and observ-
ing that the limits u=0, and ?«=2r, correspond to the values
z=-{-r, and z=—r, we shall obtain
{2ru—u^ydu=—J ^ {r'—z^ydz =
-f[^{r^ —z^ )(r= -z^ fdz \
or,
r i^ru—u'' y du =J z^ {r' —z' fdz
-J^j'ir^ -z^Ydz (143 r).
The first term of the second member, being integrated by
parts, gives
f'z^ (r« —z^fdz^-r^r^ —z^Y2zdz=
-{r^-z^f .'^^■^\fl{r^-z^fdz (143 s).
3Z
The quantity {r^—z")"'^ will reduce to zero, when z=-f-r,
or z=—rt this term will therefore disappear; and the last
term being resolved into factors will reduce equation (143 s)
to
[[z^r^ —z^fdz=lj*jr* -z^fr^ —\J_J^'' -z^Vz'dz'r
whence, by transposition and reduction, we obtain
16^ STATICS.
This value being substituted in (143 r) gives
J \2ru—u''ydu— — lr^j\r^ —z'^fdz.
But the integral / (7'^ —z^ydz represents the area of a
«/
semicircle whose radius is equal to r. This area being ex-
pressed by ^?rr", we shall have
•2r
{2ru—u^ydu=—^^^\
and by substituting this value in the expression (143 q) for
the moment of elasticity «, it will become
330. To determine the moment of rupture j8, we replace
A by — or —, and thus obtain
V r
331. By comparing these values with the expressions
(143 jo),we shall find that the moments of elasticity and rup-
ture of a square are to those of the inscribed circle as
1 "> Î6-
332. The moment of elasticity of a tube or hollow cylin-
der whose exterior and interior diameters are represented by
r' and r", will be determined by taking the difference of the
moments of the exterior and interior sections. Thus we
shall have
«=iA5r(r"'— r"*),
and the moment of rupture /3 will be found by replacing A by
B B ,
^or~; hence,
(8-iBT ~ —
r'
333. If the section of the hollow cylinder be supposed
equal tothat of a solid cylinder, the radius of the latter being
denoted by r, we shall have the relation
f2 ^=,f'2 y>"3.
RESISTANCE OP SOLIDS. 163
and the resistances to fracture opposed by the two will be to
each other as
J— : {r'^-— r"^y, or as ~^~ : {r'^—r'"')' ;
replacing r'^ _ r"^ by its value r^, this ratio will be reduced to
r
The first term of this ratio must always exceed the second :
thus the resistance to fracture opposed by the hollow cylinder
will always be greater than that offered by the solid cylinder ;
and since the value of the first term may be increased indefi-
nitely without affecting that of the second, it follows that
the resistance of the hollow cylinder may likewise be in-
creased indefinitely without changing the area of its section.
334. Let a and b represent the breadth and height of a
rectangle inscribed in a circle whose diameter is denoted by
D : we shall have the relation a=^ -|-6=^ =0^ ; and therefore,
«62=«^D2— «3).
But the moment of rupture of a rectangle being proportional
to the breadth and the square of the depth (Art. 323), if we
wish the resistance to fracture to be a maximum, we must
differentiate the preceding expression with reference to a, and
place the first differential coefficient equal to zero : we shall
thus obtain
da
and therefore,
:I)'-—3a'=0:
Hence, the strongest rectangular solid which can be cut from
a given cylinder will be that in which the diameter of the
cylinder, the depth of the rectangular section, and its breadth,
shall be to each other as the square roots of the numbers 3,
2, and 1.
L2
16^ STATICS.
Of the Figure of the /Solid after Flexure.
335. We will now consider the form of the curve Aiiu'B
{Fig. 140) assumed by the fibres whose lengths remain inva-
riable. For this purpose, let AM {Fig. 144) represent the
solid which is firmly fixed at its extremity A, and subjected
to the action of the weight P, applied at the other extremity,
in a direction perpendicular to the original direction of the
axis of the solid. Then denoting by « the moment of elas-
ticity, the equation (143 i), which expresses a condition of
equilibrium, when the solid merely undergoes flexure, with-
out being ruptured, will become
or by substituting for the radius of curvature R its general
value J '^ , this equation will reduce to
_d^
d^
(^-'1^)
336. In Uke manner, when the solid is about to be rup-
tured, if we substitute /? for the moment of rupture, in equa-
tion (143 m), we shall obtain
li=V{x'—x) (143 w).
337. Let c denote the horizontal distance AB between the
extremities of the solid,
/, the ordinate BM,
s, the length of the arc AmM^
», the angle included between the tangent to the curve
at the point M and the horizontal line.
Then, since the curvature is supposed to be extremely small,
even at the instant when fracture takes place, the expression
-^, which represents the tangent of the angle formed by the
RESISTANCE OF SOLIDS. 165
element of the curve with the axis of x, will also be extremely-
small, and its square may therefore be neglected in compari-
son with unity. Thus the equation (143 1) will be reduced to
Multiplying by dx, we obtain
»--^dx=Vlc — x)dx ;
dx^ ^
and by integration, we have
-l=K— Ï) (^*^''>-
The arbitrary constant introduced by integration is equal to
zero; since, when ar=0, —^, which represents the tangent of the
dx
angle included between the element of the curve and the axis
of abscisses, is likewise equal to zero.
Multiplying again by dx we have
»-~dx~Vl ex — ^ \dx:
dx V 27 '
and performing a second integration, there results
the constant will be equal to zero, since x—0 gives y=0.
338. If in this expression we make x—c, the ordinate y
will become equal to/; hence we shall have
^-(2-6)=.-^ 3 ("^">-
In like manner, by making x—c in equation (143 v), we shall
have '^= tang <», and therefore
dx
P/ , c2\ P ça
P 3/*
or, replacing — by its value — deduced from the preceding
equation, we have
3/-
tang «= ;j- (143 x).
i6ê STATICS.
339. To determine the length s of the arc A^nM, we take
the general expression for the element ds of this arc,
which, being developed, rejecting all but the two first terms
as inconsiderable, gives
and by replacing — ^ by its value (143 v), this equation
ax
becomes
pa
Integrating, we obtain
F- {c^x^ cx'^ , x'\
and by making x=c, the value of the entire arc AwiM becomes
pa /^6 f.s c5v pa c«
^='=+^i6-8+ro)='^+^^r5'
pa
or, replacing — by its value deduced from equation (143 w\
this expression reduces to
5=c + |: (143 2/).
5c
340. When the weight P is just sufficient to fracture the
solid, the rupture will take place at the supported end ; since
the moment V{c—x) of the force P will be the greatest when
a:=0: the equation (143 ii) will then become
/3=Pc (143 z) ;
diJ^
or, if the curvature be still supposed so small that — - may be
neglected in comparison with unity, the equation of the curve
will be the same as when the flexure was extremely slight,
and we shall therefore have
/3
P=
3/^
^ 5c
341. Let it now be supposed that the solid is loaded with
RESISTANCE OF SOLIDS. 167
weights distributed uniformly throughout its length. Denote
by z the absciss of any point between M and wi, and by p the
weight supported by a portion of the solid which corresponds
to a unit of length of the absciss : then since the distribution
of the weights is supposed uniform, we shall have the
proportion
\: J) w dz '. pdzj
the weight supported by the element of the solid whose pro-
jection on the axis of x is represented by dz. The moment
of this weight, with reference to the point w, will be
pdz{z — x), and the sum of the moments of all the weights
supported between M and m, taken with reference to the
same point m, will be
.fp{z — x)dz.
This integral should be taken between the limits z=c and
z=x, the quantity x being regarded as invariable : thus we
shall have
•/ X
q2 ^2
p{z—x)dz=p — ~ px{c—x) (143 a').
2
But the condition of equilibrium requires that the sum of
d^V
these moments shall be equal to « -^, the sum of the mo-
dx''
ments of the resistances offered by the several fibres. Hence,
we obtain
«-^ =p (t-^\ —px{c—x) = i/?c3 —pcx+lpx^ .
d integrating, we obtaii
Multiplying by dx, and integrating, we obtain
dy_
dx
and multiplying a second time by dx^ and integrating, there
results
»y=p{\c^x^ —\cx^-[-yc').
Making x=c, y=f-, and -^=tang <v, we find
/=^(ic«-ic*+ic^)=f .'^ (143 h%
tang «=%c»-ic3+ic=')=|^.
1G8 STATICS.
342. When the weights distributed along the soHd are just
capable of producing rupture, the fracture will take place at
the supported end, since the expression (143 a') which repre-
sents the sum of the moments of these weights will evidently
be the greatest when x~0. This sum being then equal to
the moment of rupture /3, we shall have
2/3
/3 = i;.c^ cp = -j (143 c');
the expression pc represents the entire weight distributed
along the solid.
343. If we make cp=V, and compare the values (143 6')
and (143 w) of the ordinate /, it will appear that the depres-
sion of the point M below the horizontal line Ax, produced
by the action of the weight P applied at the point M, will be
greater than the depression produced by an equal weight
distributed uniformly along the solid, in the ratio of 8 to 3.
And by comparing the values of — in equations (143 c') and
(143 ;r) we shall perceive that the weiglit necessary to frac-
ture the solid, when distributed uniformly, will be double that
required when it is applied at the extremity M.
344. Tt frequently occurs that the weight of the solid forms
an important part of the load which it is required to sustain.
The eiFect produced by this weight is readily calculated by
regarding it as uniformly distributed throughout the solid.
Thus, if the solid be loaded with its own weight V=pc, and
a weight P applied at its extremity M, the sum of the
moments of the weight P, and the weight of that portion of
the solid which lies to the right of the point w, taken with
reference to that point, will, by Arts. 337 and 341, be
P(c— :r) +p{l c2 — ca.- + |.r2) ;
and in case of equilibrium, we shall have
«^=P(c-^)+Kic=>-c.r+i^-^) (143 <^');
or, if the solid be supposed on the point of being ruptured,
the fracture taking place at the point A, for which a;=0, the
condition of equilibrium will be
/3=Pc-f ipc^».
RESISTANCE OF SOLIDS. 169
345. The expression (143 d') gives, by two successive
integrations,
and by making x=Cj y=/, and ~^= tang «, we obtain
tano-a;=— (iP + i«c')= ^P+P' 4/
,5=c(P+i;^c) = c(P+iF)
346. When the soHd is supported in a horizontal position
at its two extremities M and M' (Fig: 145), and loaded with
weights at its middle point A, the results obtained Arts. 337-
340 will apply to each half of the curve assumed by the
solid ; for we may regard either half as perfectly immoveable,
and suppose the other portion to be solicited by a force acting
at its extremity and equal to the resistance offered by one of
the points of support. Hence, if we denote by
2P5 the weight suspended at the middle point,
2c, the distance between the points of support,
25, the length of the curve,
/, the sagitta CA,
a, the angle included between the line MM' and the
tangent to the curve at M or M' ;
the resistance exerted by each fixed point in the vertical
direction will be equal to P, one-half the weight applied at
A, and the formulas (143 w), (143 x), (143 y), and (143 z) will
become immediately applicable to the present case. Hence,
P c2j2c^ 2P
•^ «'3 « -48 (^"^^^h
tang .=g
25=2c+^^
oc
/î=cP (143/).
15
170 STATICS.
The value of / indicates that the depression of the soHd at
tlie middle point, or the sagitta AC, will be proportional to the
weight 2P, and the cube of the distance between the points
of support.
347. The expressions deduced in the preceding article have
been obtained upon the supposition that the resistances op-
posed by the fixed points were exerted in a vertical direction ;
whereas, the resistance is actually exerted in the direction
of the normal to the curve at the point M or M' ; and in
some instances the inclination of this normal to the vertical
line is too great to be neglected. This circumstance will
seldom occur except in the case of fracture, the curvature
of the solid being then greater than in the case of a mere
flexure. If we represent the resistance exerted at M' by the
hne M'F, and resolve this force into two components which
shall be respectively vertical and horizontal, the latter com-
ponent ME will be equal and opposite to the similar compo-
nent of the resistance at the point M, and the vertical
component M'D will be equal to P, or to one-half the weight
supported at the ?Tiiddle point of the solid. The value of
the horizontal component M'E may be readily found ; for we
have
M'E =DF=M'DX tang DM'F=P .tang«.
When the equilibrium subsists, and the solid is on the point
of being ruptured, the moment of rupture must be equal to
the sum of the moments of the vertical and horizontal com-
ponents. The moment of the former, with reference to the
point A, has been found equal to cP ; that of the latter will
obviously be P tang a xAC=P tango- ,/; thus, the con-
ditions of equilibrium will become
j8=Pc + P tang*./;
or, if we suppose the curve to be represented by the same
3f
equation as in Art. 337, in which case tang*'=^, this rela-
tion may be written
348. If the weight be uniformly distributed throughout the
RESISTANCE OF SOLIDS. 171
length of the solid, we may regard each half as firmly fixed
at the point A, and solicited at the same time by a system of
parallel forces applied at every point of the solid, and acting
downwards ; and by a single force equal to their sum, or to
the resistance offered by the point of support, applied at the
extremity of the solid, and acting upwards. Thus, the case
will be the same as that considered in Art. 344, with the ex-
ception that the forces arising from the weights uniformly
distributed along the solid are exerted in contrary directions.
The equations obtained in that case will therefore become
applicable to the present one by simply changing the signs
of the moments of these forces, and replacing P by pc ; we
shall thus obtain
a,^=,Cp{cX — \X')—p{\c^X — \cX^-{-\x'^) {^^'^ g')'\
ay—cp{\cx''—\x^)—p{\c''x''—\cx''-\-^^x^) ..... (143 /i');
ç>—cp . c — cp .\c (143 Ï) ;
making x=c^ y=-fi -f— tang a», we obtain
^■x C»=-
24'
tang. = ^(l-i-i+^-i)c3=i?Ç=|',
^=cp.\c (143/:').
By comparing this value of/ with that obtained in equa-
tion (143 e'), it will appear that the depression of the solid at
its middle point produced by a weight 2pc uniformly dis-
tributed throughout the solid, will be less than that produced
by the same weight suspended at the middle point, in the
ratio of 5 to 8. And by comparing the values of — given by
equations (143 k') and (143/') we shall perceive that the solid
will be equally liable to fracture by the action of the weight
2pc distributed uniformly, or by half that weight applied at
its middle point.
349. The preceding expressions, like those in Art. 346,
have been obtained upon the supposition that the resistances
offered by the fixed points are exerted in vertical directions.
172 STATICS.
In the case of riiplure, the hne of direction of the resist-
ance may deviate so far from the vertical as to render the
above supposition inadmissible. We then resolve this resist-
ance, as in Art. 347, into two components respectively vertical
and horizontal ; the former will be represented by pCy and the
latter by jjc • tang a. In case of equilibrium, it will simply
be necessary to add to the second member of equation (143 i')
the moment jjc . tang « X/, of the horizontal component; thus,
we shall have
^ = cp .c—cp. Ic-jrcp . tang « ./=rc/v(|e+/tang<v),
Sf
or, by replacing tango; by its value p^, we have
oc
and therefore.
4^
2cp=-
(-1")
we here suppose that the equation of the curve remains the
same as in Art. 337.
350. If the solid be loaded at the same time with a weight
2P at its middle point, and its own weight 2pc=z2F' uni-
formly distributed, the case will be similar to that considered
in the two preceding articles, with the exception that the
force applied at the extremity of the solid will now be repre-
sented by P+/>c=P+P': thus, when we suppose the resist-
ances exerted by the fixed points to act vertically, we shall
obtain, by substituting P-f-i>c for pc in the first terms of the
second members of equations (143 h') and (143 i'),
«y=(P+pc)(ic.r2 —}z-^)—p{lc^x- —}ca;^+^\x*),
^ = (P+pc)c—cp . ic (143 r) ;
which give, by making a:=c, y=f, and pc=P',
351. But, if regard be had to the oblique direction of
the resistance, as may be necessary in the case of rupture,
we must add the moment of the horizontal component
RESISTANCE OP SOLIDS. 173
to the second member of equation (143 1% which thus
becomes
^=(P+2?c)c— cp . ic+(P+pc) tang a, ./;
and therefore,
gp_ 2^-F(c+2/tang^) ^
c-\-f tang a
The equation (143 g') likewise gives, by replacing pc in the
first term of the second member by P+^c, and making
|=tang »,
3P + 2P' 4/
and this value of tang a may be regarded as sensibly equal
to that employed in the preceding expression for the value
of2P.
352, To apply the several results which have been ob-
tained to particular cases, it will be necessary to substitute
the values of the moments of rupture and elasticity apper-
taining to the figure of the transverse section. We must
likewise assign to A and B the coefficients of elasticity and
tenacity, their particular values which depend upon the nature
of the substance, and which are supposed to have been pre-
viously determined by experiment,
353. The best method of determining the values of A and
B consists in supporting a prismatic solid at its two ex-
tremities in a horizontal position, loading it with weights at
its middle point, and observing the sagittas which correspond
to different weights ; or simply, the weight and sagitta at the
instant when the fracture is about to take place.
If the transverse section of the solid be a rectangle,
whose breadth and height are denoted respectively by a and
6, we shall have (Arts. 322 and 323),
et=j\kab'', ^=iBa62 ;
and if we neglect the weight of the solid (Arts. 346 and 347),
and by eliminating « and /3, we obtain, for the case of simple
flexure,
174 STATICS.
/=2P-^?^, orA=2pi?$4. (143^0;
and for that of fracture
«=^Pa^O+i?) ("=>"')'
2c being the interval between the supports, and 2P the weight
with which the solid is loaded.
The values of A and B are thus expressed in functions of
quantities which are readily determined by observation.
354. If the weight of the solid 2P' be likewise taken into
consideration, it will simply be necessary (Art. 350) to add
1 . 2P' to 2P in equation (143 m% and to replace equation
(143 n') by the formulas of Art. 351 : we shall thus have, i»
the case of flexure,
/=(2P + |.2P')-i?^, A=(2P + f.2F)-^^')'
and for that of fracture,
6g _ (2P+2FXc+/.tang<^)— P^c
3P + 2P' 4/
*""^^=8P+5P'-T
355. If the solid be loaded with a weight 2Q,, and if the
corresponding sagitta be denoted by f, we shall obtain a
value for/' similar to that of /in the preceding article : thus
"we shall have,
and by taking the difference between / and /', the weight of
the solid 2P' will disappear, and we shall obtain
/'_/=(2a-2P)_^, A=(2a-2P) — (^^^'
4Aa63' ^ -^ ' 4.abHf'—f)
Thus, it will only be necessary to observe the increase /'— /
in the sagitta, which corresponds to a given increase 2Q,— 2P
in the weights suspended at the middle point.
RESISTANCE OF SOLIDS. 175
Of Solids of equal Resistance.
356. When a solid having the prismatic form is subjected
to an eflbrt which tends to break it, there will always be a
particular point at which the fracture will be most likely to
take place. For, the moment of rupture will be the same at
every point, whilst the moment of the force applied will de-
pend upon its distance from the point with reference to which
the moments are taken. Hence, if the strength of the solid
be sufficient at that point where a rupture is most likely to
occur, it will be unnecessarily great at other points.
357. It becomes an object, therefore, to determine the
figure of the solid which shall be uniformly strong through-
out, since the adoption of such a figure may frequently effect
a material reduction in the quantity of materials employed.
Solids having such figures are called solids of equal resist-
ance.
358. As an example, let a body ABM {Fig. 146), whose
upper surface AB is horizontal, and. whose two lateral faces
are vertical, be firmly fixed at its extremity A, and subjected
to the action of a weight P suspended from its other extrem-
ity. It is required to determine the form of the under surface
BmM such that the solid may be equally strong throughout,
or that the moment of the weight P taken with reference to
any point in the length of the solid, shall be equal to the mo-
ment of rupture of the transverse section at the same point.
Denote by a the breadth of the solid, h the height AM, c
the length AB, x the variabl-e absciss Bjo, and v the corre-
sponding ordinate jmi : the moment of rupture of the section
ah^
AM will be (Art. 323) B— - ; and since this must be equal to
the moment of the force P, we shall have
In like manner, the moment of rupture of the section pm
av^
will be B-^ , and the moment of the force P with reference
6
to a point in this section will be Vx. These moments being
176 STATICS.
equal by the conditions of the problem, the general relation
between the quantities v and a: will become
P:r=B-— , v^ = — .
D C
This equation evidently appertains to a parabola, the axis of
which will be the line AB.
359. To determine the figure of the curve assumed by the
solid when bent, we observe that the moment of elasticity of
the section p?n will be (Art. 322) A— — =A —, Hence, if
^^ 12c^
y denote the ordinate of the curve of flexure corresponding to
the absciss Ap=c—x, the conditions of equilibrium in case
of flexure will be (Art. 337)
A X — 4=P^.
Performing two successive integrations, and remarking that
when x—c^ -/-=0, and y=0, we obtain
ax
and by making a;=0, and y=/, we find, for the depression of
the extreme point B,
P 8c^
''~'A.'ab^'
By comparing this expression with that obtained in equa-
tion (143 w), it will appear that the depression / is twice as
great in the present instance as when the solid had the pris-
matic form.
360. If the weight supported by the solid be distributed
uniformly along its length, each unit of length being sup-
posed to support a weight jh '^he sum of the moments of
these weights, taken with reference to the point A, will be
(Art. 342) pc.ic; and the condition of equilibrium will
therefore be
B-—=pc.\c.
PRINCIPLE OF VIRTUAL VELOCITIES. 177
In like manner, the sum of the moments of the weights sup-
ported between the points p and B, taken with reference to
the point /?, will be fx . \x. Hence, we shall have
„«v^ , hx
b ^ ^ c
the equation of a right line.
361. The preceding examples will be sufficient to illus-
trate the manner in which the form of the solid of equal re-
sistance may be determined when the distribution of the load
is previously known.
Of ike Principle of -Virtual Velocities.
362. The principle of virtual velocities, which was dis-
covered by Galileo, and very fully developed by John
Bernouilli and Lagrange, may frequently prove of great
utility in stating the analytical conditions of statical problems.
Indeed, it is regarded by Lagrange, who has adopted it as the
basis of his "Mécanique Analytique," as so essential, that he
considers all the general methods which can be employed in
the solution of questions relating to equilibrium, as being
nothing more than applications more or less direct of this
general principle.
363. A virtual velocity is the path described by the point
of application of a force, when the equilibrium is disturbed
in an infinitely small degree. Thus, by supposing that the
point of application vi of a force P {Fig. 147) is, by an
instantaneous derangement of the system, transferred to ??,
the small line mn which it describes is called the virtual
velocity of the point m.
364. If this virtual velocity be projected upon the direction
of the force, it will occupy thereon the small space ma, and
the product of the force P by this projection ma is called the
moment of this virtual velocity, or, sometimes, the moment
of the force ; it should however be observed, that the term
moment is here employed with a very different signification
from that usually implied.
The principle of virtual velocities, as will be demonstrated,
M
178 STATICS.
consists in this, that when the system is in equihbrio, the sum
of these moments is equal to zero ; thus, if P, P', P", &.C.,
represent different forces appUed to a system, and p, p', ;j",
&c., the projections of the virtual velocities on the directions
of these forces, we must have in case of equilibrium,
Fp + P'p'+F'Y +Ôcc.=0 (144).
It is necessary to remark that when any one of these pro-
jections p, p\ p'\ (fcc, falls upon the prolongation mh
{Fig. 148) of the force P, applied at m, this projection must
be regarded as negative ; and since the forces P, P', P", <fcc.,
are all considered as having the positive sign, the moment
corresponding to this negative projection, must likewise be
affected with the negative sign ; thus, the equation (144) will
express that the algebraic sum of the moments is equal to zero.
365. This principle will first be demonstrated for that case
in which the forces are applied to a single point. Let P, P',
P", &-C., represent any number of forces applied to the point
m {Fig. 149), and sustaining it in equilibrio ; if, by the effect
of an infinitely small derangement, the point rti be trans-
ported to «, the line tnn being infinitely small, may be
regarded as a right line. Let the axis of x be supposed to
coincide in direction with the line mn, and denote by «, «', <«",
&c., the angles formed by the several forces with this axis ;
we shall have, since an equilibrium subsists in the system,
Pcosa + P' cos a'+P" cos4" + (kc. =0:
multiplying the several terms of this equation by the line W7i,
which will be denoted by z, we shall obtain
Vz cos «+PV cos *'-fF'z" cos «"+&c. =0 (145).
But it is evident that z cos «, or mn . cos nml^ is equal to the
small line 'ml, the projection of mn on the direction of the
force P. Thus z cos « represents the same quantity as the
letter p in equation (144). The same remarks being appli-
cable to the other forces, the several products z . cos cl, z cos «",
&c., may be replaced by- p\ p", <fcc., the projections of the
virtual velocity of the point 771 upon the directions of these
forces, and the equation (145) will then become
P;)+P>'+P>"+(fcc.=0 ;
from which we conclude that the principle of virtual velocities
is true when the forces are applied to a single point.
PRINCIPLE OP VIRTUAL VELOCITIES, 179
366. The most general case of this principle which usually
presents itself, is that in which the several forces P, P', P",
(fcc, are applied to different points of a body or system of
bodies : these points preserving their distances invariable,
may be regarded as connected with each other by inflexible
right lines. Before examining the general state of the system
when the equilibrium has been slightly disturbed, we will
consider singly one of these inflexible right lines mw', at the
instant when the point m has been brought into the position
denoted by ?i. The other extremity m! of this right line
will at the same time change its position, and may be situated
either above mm' {Fig. 150), or beneath it {Fig. 151) : let
it be first supposed above mm', and the line 7nm' will then
assume the position mi' {Fig. 152) : the lines mn and on'n'
may be regarded as infinitely small when compared with the
lines min' and nn', since the derangement of the system is
supposed infinitely small. If the points m and n' be con-
nected by a right line we shall form a triangle m/m'n', in which
the side m'n' being infinitely small, the angle n'mm' will like-
wise be infinitely small, and the arc w'a, which measures this
angle, may therefore be regarded as a right line. But this
arc beinof described with a radius ma, if we assume mb=zma
{Fig. 153), the angle bn'a being an angle in a semicircle, will
be equal to a right angle, and may be considered equal to the
angle mn'a. For, since the angle ii'nia is infinitely small, the
angle m,7i'b must be so likewise, and the angles bn'a, mn'a, will
therefore differ by an infinitely small quantity. Thus, the
triangles mn'a and ii'la {Fig. 152) being right-angled and
having a common angle a, will be similar, and we shall there-
fore have the proportion
m,a : n'a : : n'a : la.
But n'a being infinitely small with respect to ma, la must be
infinitely small with respect to n'a ; and since n'a is an in-
finitely small quantity of the first order, la will be one of the
second order. Hence, the quantity la may be neglected, and
mrî may be regarded as equal to ml ; thus we shall have
mn'=mm'-±m'l.
In a similar manner may it be proved that if with the point nf
M2
180 STATICS.
as a centre, and radius n'm, we describe the arc ma', we shall
obtain
tmi'=7in'-^7ih,
and by placing these values of m?t' equal to each other, we find
vnn' -\- ni'l=7in' + nh ;
but the right line min' being supposed inextensible, it must
preserve its length invariable in its new position ; hence,
mm'—nn' ; and by suppressing these equal terms in the pre-
ceding equation, we obtain
'}n'l=nh.
Again, the lines Qnm' and nn' form with each other an infi-
nitely small angle ; for, if they intersect at a point o {Fig. 154),
we shall have a triangle m'on', two of whose sides are of finite
extent, the third side m'n' being infinitely small ; thus, the
angle o will likewise be infinitely small. It results from the
preceding remarks, that if the perpendicular iik be demitted
on the side oTim' (Fig. 152) we shall have
nh=9nk]
and by substituting this value of nh in the preceding equation,
we find
'm'l=mk,
which proves that the projections mk and m'l of the virtual
velocities mn and m'n' of the points m and m' are equal to
each other.
367. Let us now suppose that the point w {Fig. 155) is
transported to n, and that the extremity m' falls at n' below
Tnm'. It may be proved as in the former case, that the angle
is infinitely small, and consequently that the projections ol
and oh may be regarded as equal to o?i' and 07i ; whence,
on'=om'-{-m'l, on =07n — mA ;
by the addition of these equations, we obtain
07i'-{-o7i—om'-\-om-\-m'l— mh ;
or,
7in' =mm'-\-m'l — mh ;
but 7in' and mtn' are equal to each other, and therefore
7n'l—7nh,
PRINCIPLE OP VIRTUAL VELOCITIES. 181
which proves that the projections of the virtual velocities are
still equal.
368. In this demonstration it has been supposed that the
derangement of the system is such as to preserve the lines
mm' and nn' in the same plane. This restriction is however
entirely unnecessary. For,if we suppose that mm' and nn'^re
not contained in the same plane, we can draw through the
points n and 7t' {Fig. 152) planes perpendicular to the line
mm', intersecting this line at the points k and I, the projections
of n and n'. Then, if a line be drawn through any point of
m>m' parallel to nn', and terminated by the perpendicular
planes, such line will evidently be equal to nn', and its ex-
tremities will likewise be projected on the line mm', at the
same points k and I. Hence, if the property be true for the
parallel line which intersects mm', it will likewise be true for
the line nn'.
369. It should be observed, that in each of these cases, the
projections will be affected with contrary signs, one falling
upon the line mm', the other upon its prolongation.
This appears from an inspection of the figures 152 and
155, and it likewise results from the consideration that if the
two projections fell upon the line or upon its prolongations,
the length of nn' would necessarily be greater or less than
that of mm', which by hypothesis, is impossible.
370. It follows from the preceding remarks, that if we sup-
pose two equal and opposite forces to act in the direction of
the line mm' on the points m and m', and denote by v and v'
the projections of the virtual velocities mn, and m'n' on the
line of direction of the forces, we shall have
V— — v' ;
and consequently, that if we represent by {mm!) each of
these equal forces, we shall obtain
{mm')v + {mm')v'=0 ;
which proves that the forces represented by {mm') being
applied at the extremities of the right line, and being regarded
as sustaining those points in equilibrio, the sum of the
moments of the virtual velocities of these points will be equal
to zero.
371. By the aid of this proposition it will be easy to
16
182 STATICS.
establish the principle of virtual velocities in the case of any
number of forces applied to different points. For, let P, P',
P", «fee. {Fig. 156), be several forces applied to the points
m, in\ m", <fec. If we regard these points as firmly con-
nected by inflexible right lines, these lines may be considered
as the directions of equal and opposite forces acting on the
points w, m', m", «fee, and if we denote these forces by {mm'),
{7n''ni"), (fee, the equilibrium will be maintained
at the point m, by the forces {Tnm/), {mm")^ {mm'"), and P,
at the point m', by the forces {m^'m), {in'ml^, (m'm'"), and P',
at the point m", by the forces {m"m), {tn"m'), {m"m"') and P",
at the point m'", by the forces im"'m), {r)i"'m'), {m"'7?i"), andP'",
&c. (fee. (fee.
Since the equilibrium subsists at each of these points, the
equation of the moments obtained in Art. 365, will manifestly
be satisfied. Let the following notation then be adopted, viz ;
v=projection of the virtual velocity of one of the points m
?«', in", &c., the point to which this velocity refers being
designated by the manner in which v is written in the ex-
pression for its moment ; thus, v{m?n') represents that v in
this moment applies to the point m, while v{m'm) denotes
that V applies to 7?i'.
The character v will thus represent quantities which may
be equal or unequal, according as the projections of the
virtual velocities fall upon the same or upon different lines.
372. Having adopted this notation, the equations of the
moments as given by Art. 365, may be expressed as follows :
for the point m, P2^-}-v{7nm') -\-v{?7im") +v{mm"')=0,
for the point m', ¥'p' -{-v{7}i'm) +v{7)i'7n")-^v{m'm"')=0,
for the point 7Ji"j V"p"-\-v{m"7n)-\-v{7?i"77i')-^v{77i"7n'")=0,
for the point m"', V"'p'"+v{7n"'m)+v{?Ji"'?7i')+v{??i'"7?i")=0.
The sum of these four equations being taken, we remark
that the moments appertaining to the same right line mutu-
ally destroy each other ; thus, the term v {771771') will cancel
the term v{m'77i), <fcc., and by continuing the process, the sum
will be reduced to
rp -f P'p' -i-p>" + P"'7/"=o.
PRINCIPLE OF VIRTUAL VELOCITIES. 183
The same demonstration is evidently applicable to a greater
number of forces.
373. As an example of the manner in which the conditions
of equilibrium in any machine may be inferred from the
principle of virtual velocities, we will suppose the relation
between the power and resistance in the lever to be unknown.
The forces exerted upon the lever are the power P, the
resistance P', and the reaction of the point of support. If a
slight motion be communicated to the lever, causing it to turn
about its fulcrum, this fulcrum will remain immoveable, and
the moment of the reaction exerted by this point will there-
fore be equal to zero. Hence, the principle of virtual velo-
cities will give
or,
Pp = —Fy (146).
This being premised, let the values of the quantities jo andp'
be now determined. Let C represent the fulcrum of a lever
mm' {Pig. 157), which being slightly removed from its po-
sition of equilibrium has assumed the position mi' ; the angles
at C being equal to each other, the arcs mn, m'n' will be pro-
portional to the radii with which they are described, and we
shall therefore have
mn : m'n' : : Cm : Cm' (147).
But if through the points n and n' perpendiculars be drawn
to the directions of the forces P and P', we shall have
mr = — p , m'?-' —p ',
the negative sign being prefixed to p, because it falls on the
prolongation of the force P. The arcs being regarded as in-
definitely small right lines, the right-angled triangles mm,
m'r'n' will be similar ; for the isosceles triangles 7nCn, 7n'Cn'
give
angle 7imC= angle n'm'C :
and by subtracting these equal angles from the right angles
rmO, r'm'C, there will remain
anffle 7-mn=an2fle r'm'n'.
Thus, the triangles rmn, r'7n'n' will be similar, and will give
the proportion
184 STATICS.
mn : m'n' :: mr : mY ;
or,
mn : m'n' : : — p : p' ;
and therefore the proportion (147) may be converted into
Cm : Cm' : : — p : p'.
But the equation (146) which expresses the principle of vir-
tual velocities gives rise to the proportion
F :P:: -p :p';
whence, by the equality of ratios,
Cm : Cm' : ; P' : P,
or the forces are in the inverse ratio of the arms of the lever.
Of the Position of the Centre of Gravity of a tSystem when
in Equilibrio.
374. Let m, m', m" (fee, be the centres of gravity of different
bodies which are connected together in an invariable manner ;
let perpendiculars z, z', z", &c., be demitted from these points
on the plane of xy, supposed to be horizontal ; the weights
P, P', P", &.C., of the several bodies, which may be regarded
as suspended from the points m, m', m"y ace, will act along the
directions of these perpendiculars. If z/ denote the co-
ordinate of the centre of gravity of the whole system, we
shall have (Art. 166)
P^-fP^^^-fP^s'^+&c.
^/-- p+P'4-P"+&c. •
When the system of bodies changes its position, the ordi-
nate z becoming z + hyOr z — h, the increment of 2; will affect
the values of z', z'\ z", (fee, since the points m, m\ ?n", (fcc^
being connected in an invariable manner, the value of z.
cannot change without the values of z', z", (fee, undergoing
a corresponding alteration. Although we are generally unac-
quainted with the law of dependence which exists between
the positions of the different bodies composing the system,
the preceding equation may nevertheless be written under
the form
_ Pz-^V'<pz+'P"Fz-\-6cc.
^'~ P-i-F+P"+(fec. '
in which ç, F, (fee. denote certain indeterminate fimctions.
POSITIONS OF THE CENTRE OF GRAVITY. 185
If the value of zi be a maximum or a minimum, the dif-
ferential of the second member will be equal to zero, hence
Vdz + V'd<pz -f V'dFz + &c. = ;
or,
Vdz+V'dz'+V"dz"-\-6LC.^Q.
But this equation is necessarily satisfied when the system
receives an infinitely small derangement from its position of
equilibrium. For, when the centres of gravity m, m, in, &.C.,
change their positions and are transferred to ??, n\ n\ &c.,
the paths described will be the lines mn, m'n, m'n\ &c. If
therefore, these paths be projected on the primitive directions
z, z\ z", (fee, of the forces P, P', P", (fee, we shall obtain the
values of the projections of the virtual velocities. Thus,
m/i [Fig. 158) the ])rojection of mn upon the co-ordinate z,
is equal to nk, the increment which the value of z has re-
ceived in consequence of the derangement sustained by the
system : the sign of this increment may be either positive or
negative. We shall therefore have, without reference to the
signs, p~dz ; and by applying the same considerations to the
other co-ordinates, it appears that the differçntial Vdz +
Vdz' -\-Vdz" + 6cQ.., will represent the same quantity as the
expression Pp+P'jo'-}-P"^"-f (fee. ; and since the latter quan-
tity becomes equal to zero when the system is in equilibrio,
according to the principle of virtual velocities, we must like-
wise have
Vdz-{-V'dz'-V'P"dz"-\-ôi.c.=Q] *
hence, dz=0, which proves that the centre of gravity is in
general situated at the highest or lowest point, when the
system is in a state of equilibrium. But this proposition
will not always be true, since dz=0 will not always indicate-
the existence of a maximum or minimum.
375. The converse of this proposition is always true, viz:
Jf the centre of gravity of the system he situated at the
highest or lowest point, the system loill necessarily he in equi-
librio ; for, dz, will then be equal to zero, and the sum of the
moments of the virtual velocities will also be equal to zero.
PART SECOND.
DYNAMICS.
OF THE LAW OF INERTIA.
376. Dynamics has been defined to be that part of Me-
chanics which treats of the laws of motion of sohd bodies.
We shall, in the first place, establish as a principle the general
law of nature, that every body will continue in the state of
rest or motion in which it may be placed, unless it be acted
upon by some external force. This indifference of matter to
a state of motion or rest is called inertia. It is a conse-
quence of this principle of inertia that one body when struck
by another, exerts an effort of resistance to the impulsion,
whilst acquiring a portion of the motion of the striking body.
By this same principle, a body having received an impulse,
must move uniformly in a right line, if not opposed by any
obstacle : for there can be no reason why the body should
deviate to one side rather than to the other, nor that its
motion should be accelerated rather than retarded. It is
true, that the nature of the force being unknown to us, we
cannot foresee whether its effect will be such as to preserve
the motion of the body invariable : thus, the law of inertia
should be regarded as a simple result of experience and
analogy.
If we do not perceive the motions of bodies to continue
unchanged, it is merely because these motions are constantly
affected by the resistance of media, by the action of gravity,
or by other similar causes. The most simple kind of motion
which can be conceived is that which takes place uniformly,
and in a right line.
188 DYNAMICS.
Of Uniform Rectilinear Motion.
377. A body is said to have a uniform motioîi when it
passes over equal spaces in successive equal portions of time :
thus, if V denote the space which it describes in a unit of
time, it will have described a space 2V at the end of two
units of time, 3V at the end of three units of time, (fcc. Con-
sequently, if we represent by t the number of units of time
necessary for the body to describe a space s, this space will
be equal to ^xV ; we shall thus have
s=Yt.
Such is the equation of uniform motion. The coefficient V,
or the space passed over in a unit of time, is called the
velocity, and it evidently expresses the rate of a body's
motion. For, if a body M move n times as rapidly .as
another M', the space V described by the first in a unit of
time, will obviously be n times greater than the space V,
described by the second in the same time.
378. For the purpose of comparing the circumstances of
motion of two bodies which depart at the same instant from a
point A, with velocities represented by V and V", we will
denote by s' and s" the respective spaces passed over by these
bodies, at the expiration of the times i' and t" : we shall then
have
whence we deduce
s' _ Y't' .
s"~Y"ï" '
which proves that the spaces passed over are proportional to
the products of the times and velocities. When the times are-
equal, this equation reduces to
7'~V"''
and the spaces described are then proportional to the ve-
locities.
379. The body may have alteady passed over a space S,
previous to the instant from which the time t is reckoned :
UNIFORM RECTILINEAR MOTION. 189
we shall then have the more general equation of uniform
motion
s=S+Yt,
in which s represents the distance of the body from the
origin of spaces. The quantity S is called the initial space,
and evidently represents the distance of the body from the
origin, at the commencement of the time t.
380. By the aid of this equation we can readily solve all
the problems of uniform rectilinear motion.
For example, if the distance of a body from the origin of
spaces at the end of the time t\ be supposed equal to s' ; and
if this distance become s" at the end of the time t", we can
thence determine the velocity V, and the initial space ; for
we shall have the equations
s=S+Yt', s"=S-{-Yf;
from which we obtain
~ t"—t' ~~ t"—t'
381. As a second example, let it be required to determine
the time of meeting of two bodies M' and M {Pig. 159),
which depart at the same instant from the two points A and B,
having the respective velocities V and V. Let C be their
point of meeting : the spaces actually passed over by the two
bodies will be
BC=V^, and AC=V^
If we denote by h the distance AB between the bodies at the
commencement of the motion, and reckon their distances at
the end of the time t from the point A as an origin^ we shall
have the equations
Each of the spaces 5 and s' will then be represented by the
line AC ; and by placing the second members of the above
equations equal to each other, we deduce
_ b
382. Since the space & constantly varies with the time /,
190 DYNAMICS.
we can differentiate the equation s=zS-\-Yt with reference to
these two variables, and we shall thus obtain
dt
Hence it appears, that in uniform motion the velocity is the
differential coeificient of the space, regarded as a function
of the time : it will presently appear that the same is true in.
varied motion.
Of Varied Motion.
383. When the motion of a body is such that it passes over
unequal spaces in equal successive portions of time, the body
is said to have a varied motio?i. This kind of motion cannot
be produced by the action of a single force of impulsion,
since by the law of inertia the velocity imparted by a single
impulse should constantly remain unchanged ; and hence the
motion would continue uniform : whereas, we have in the
present instance supposed it variable. It therefore becomes
necessary to suppose that the body, having received the first
impulsion, is subsequently subjected to the action of a second
impulse, a third, &-c., which, by constantly changing its
velocity, produce a variable motion. If the force acts without
intermission, the impulses will be communicated at intervals
which are indefinitely small, and the force is then called an
incessant force. If the force tends to increase the velocity
of the body, it is called an acceleratiîig- force, and when it
tends to diminish the velocity, a retarding force.
384. The velocity of the body being supposed constantly
variable, we can only estimate its value at any particular
point of the path described, by supposing it to become con-
stant at this point. Thus, to measure the velocity of a body
which has arrived at B {Fig: 159), at the end of the time t,
we suppose the action of the incessant force to be suddenly
arrested, and the body will then move uniformly with the
velocity which it has acquired at the point B. The space
BC described in a unit of time, with this uniform motion, is
the measure of the velocity at the point B.
VARIED MOTION. 191
385. The second is usually adopted as the unit of time.
Hence, the velocity of the body at the expiration of the time t
will be the space which this body would describe in the second
which succeeds the time t, if, at the end of the time t, the
incessant force should cease to communicate new impulses to
the body.
386. To determine the analytical expression for the ve-
locity, we will suppose the body to have arrived at the point
B, at the expiration of the time t ; the space AB which it has
already passed over being dependent on the length of time
which has elapsed, the former will evidently be a function
of the latter. Thus, we may regard the space 5 as the ordi-
nate of a curve whose absciss is equal to t ; consequently,
when t becomes t + dt, s will become s-\-ds ; hence, the space
passed over in the time dt will be represented by ds. This
being premised, let it be supposed that when the body has
arrived at the point B, the incessant force ceases to act ; the
body will assume a uniform motion with the velocity ac-
quired at the point B, and will describe in the instant dt
succeeding the time t, the indefinitely small space ds : in the
next succeeding instant dt it will describe a second space ds,
and the same will continue until the body has described a
space BC, which will correspond to the unit of time. This
space BC will therefore contain ds as many times as dt is
contained in unity ; but — will express the number of times
which the unit of time contains the quantity dt ; hence, the
1 ds
space BC will be expressed by ds X -^, or by —, since the dif-
(it (it
ferential is taken with reference to the variable t ; but the
space BC represents the quantity v ; we shall therefore have,
for the expression of the velocity in varied motion
ds
dt
387. It may also be observed that the space passed over,
after the expiration of the time t, will be {Fig. 159),
B6=c?5 at the end of the time dt,
Bb'=2ds at the end of the time 2dt,
192 DYNAMICS.
Bb"=3ds at the end of the time 3dt.
BC=iids at the end of the time ?i . dt.
And since the time elapsed during the passage of the body
from B to C is by hypothesis equal to unity, we may suppose
7uit~l\ whence n=--. This value beinff substituted in
dt
the expression nds=v, the space described in a unit of time
we shall obtain, as above,
"4: •••■("«>
388. Before investigating the expression for the value of
the incessant force, it will be necessary to discover the rela-
tion which exists between the force and the velocity.
If a force P be supposed to communicate a velocity v to
any body, a force 7i times as great will communicate to the
body a velocity equal to nv. The truth of this proposition
might well be questioned, since the nature of forces being
entirely unknown, we cannot affirm that a double force will
necessarily produce a double velocity ; or, in general, that a
single force equal to the sum of two others, will necessarily
produce a velocity equal to the sum of the velocities which
the two forces would separately produce. But the fact being
confirmed by universal experience, we adopt it as a principle.
Thus, by supposing different forces applied to the same body
or material point, their relative intensities can be estimated
by comparing the velocities which they would severally com-
municate.
The proper measure of an incessant force will be the
velocity which it can generate in a given time ; but the in-
tensity of the force being constantly variable, we must sup-
pose the force to become constant at the instant when we
wish to estimate its value, and the measure of the force will
then be the velocity generated in the unit of time succeeding
this instant. The velocity communicated by this incessant
force during the unit of time, when it is supposed to retain a
constant value, will obviously be unequal to that which
would have been communicated b]f the variable incessant
force, in the same time.
389. The preceding remarks indicate the method of meas-
VARIED MOTION. 198
uring the incessa,nt force ; since they determine the ratio in
which the intensity of the force varies in different times.
If, for example, at the expiration of the times t and t\ the
incessant force, having become constant, can generate in a
second of time velocities represented by the numbers 60 and
20, we infer that the intensity of the force at the end of the
time t is triple its intensity at the end of the time t' .
390. To deduce from the above definition the analytical
expression for the incessant force, let v represent the velocity
acquired by the body at the end of the time t ; then, at the
expiration of the time t-\-dt^ the velocity will become v^dv ;
consequently, dv will be the velocity communicated during
the time dt ; but if at the end of the time t the intensity of
the force be supposed to become constant, there will be com-
municated to the body in the instant dt which succeeds the
time /, a velocity represented by dv ; and the same effect will
be repeated during any number of succeeding instants ; so
that the velocities communicated after the expiration of the
time t^ in the instants dt,2dt, 3dt, &c., will be expressed by dv,
2dv, 3dv, &.C. : and consequently, the velocity communicated
in the unit of time which succeeds the time t, will be equal to
dv repeated as many times as dt is contained in unity. This
number being expressed by -— , it follows that — x dv, or — ,
dt dt dt
will express the effect of the force or the velocity generated in
a unit of time. If, therefore, we denote this force by ^,we
shall obtain for the second equation of varied motion,
^4 ("«)•
The character ç will hereafter be used to designate the inten-
sity of the force ; the force being represented by the effect
which it produces.
391. From the preceding equation we obtain
çdt—dv]
thus, if the incessant force be given, the increment to the
velocity in the time dt can be readily calculated.
392. By eliminating dt between the equations (148) and
(1 49), we obtain a third equation of varied motion,
çds=vdv.
N 17
194
DYNAMICS.
Of Uniformly Varied Motion.
393. The incessant force imparting at each instant a new
impulse to the body, if these impulses are equal in intensity,
the body will acquire the same velocity in a unit of time after
the expiration of the time t, as it would after a time t'. Let
this velocity which is constantly generated in a unit of time,
be denoted by ^ ; we shall then have
Substituting this value in the equation
^ dv
'^=dt^
we shall obtain
dv=gdt]
and by integrating and denoting by a the constant which will
thus be introduced, we find
v=a+gt (150).*
We have likewise obtained for'the value of the velocity
ds
hence, if we eliminate v between these two equations, we
shall have
ds={a-i-gt)dt,
from which, by integration, we find
s=b + at+^gt' (151),
the quantity b being an arbitrary constant.
* This equation might also have been obtained from the following considera-
tions : Let it be supposed that a body in motion has acquired a velocity a: if it
then be solicited by a constant force which communicates to it a velocity g in
each second of time, the velocity of the body will become
a-\-g, at the end of one second,
a-\-2g, at the end of two seconds,
a-{-3g, at the end of three seconds,
a-\-tg, at the end of t seconda :
thus, if we represent by v the velocity of the body at the expiration of the time /,
we shall have
UNIFORMLY VARIED MOTION. 195
If ^ be supposed positive in this equation, the motion will
be uniformly accelerated, but if negative, the motion will be
uniformly retarded.
394. If we make ^=0, we find h=s\ thus, h will represent
the initial space, or the distance of the body from the origin,
at the instant from which the time is reckoned.
The constant a is equal to the initial velocity of the body,
as appears by making ^=0 in equation (150).
395. When the initial space and initial velocity are each
equal to zero, the equations (150) and (151) become
•^^gt (152),
s^\gt^ (153),
and the body then moves from rest, under the action of the
incessant force.
396. Let s and s' represent the spaces described in the
times t and t', under the action of a force g ; the equation
(153) gives
s=^\gf, and s'==\gV* (154) ;
whence we obtain the proportion
s:s' \:t^ '.t'^ (155).
Consequently, the spaces described by a body in different
times, when it moves from rest, being solicited by a constant
accelerating force, are proportional to the squares of those
times.
397. The equation (152) gives
v=gt, and v'=gt',
whence,
V :v' ::t: t',
and by comparing this proportion with (155), we have
v:v'::^s: ^s'.
Hence it appears that the tim,es elapsed are constantly pro-
portional to the velocities, or to the square roots of the spaces
described in those times.
398. If we make ^=1, the equation (153) becomes
sz=ig.
In this case, s represents the space described by the body
in the first unit of time, and it appears that this space is
N2
1^ DYNAMICS.
equal to one-half the quantity g, which represents the mea-
sure of the accelerating force. It has been found, for example)
that a body subjected to the action of gravity, would describe
in the first second of time, in the latitude of New-York a
distance equal to
16.0799 feet, or nearly 16j^ feet ;
this value being substituted in tht? place of s in the preceding
equation, we find
^=32.1598 feet, or nearly = 32i feet.
399. The equation (153) will determine the space described
in a given time ; for example, if t=6", we shall have
5 = 1^^2^1(321") x36=579 feet;
thus a body being elevated to the height of 579 feet, would
require six seconds to fall to the surface of the earth.
400. The velocity acquired by this body, when it has reached
the surface, may be determined from equation (150), in which
we make
a=0, g=32i feet, ^=6".
We thus find
v=32i"x 6=193"-.
401. If it be required to determine the height from which
a body must fall to acquire a given velocity, we eliminate (
between the equations
and we thus obtain
^=v/(2^*) (156).
Let it be supposed, for example, that we wish to détermine
the space through which a body would fall in acquiring a
velocity of 386 feet per second ; we shall have
386"=.,/(2 X 321"- X5)=</(64i"- X5) ;
whence,
(386")^_148996;;;_
The velocity acquired in falling through a given height is
called the velocity due to that height.
402. To determine tlie time in which a body will fall
UPWARD VERTICAL MOTION. 197
through a given height s, we employ the equation (153),
which gives
'=v/(|)
403. The general equations of variable motion
ds dv ,1 -^.
ir"' *=* (1'^'
will now be applied to the investigation of the circumstances
of varied motion under different hypotheses. This investi-
gation is reduced to the determination of the relations which
exist between the time elapsed, the space described, and the
velocity acquired, since, if the two latter can be expressed in
functions of the time, we shall be able to discover the place
of the body, and the velocity with which it moves at any
given instant. Thus, the circumstances of motion will be
entirely known.
Of the Motion of a Body projected Vertically upward.
404. When the action of gravity is alone exerted on a body^
we have the relation
v=gt,
in which v expresses the velocity at the end of the time t :
but if we suppose the body instead of moving from rest, to
be projected vertically in a direction opposed to that of gravity^
with a velocity «, this velocity will have been diminished at
the end of the time t, by a quantity equal to the velocity
which gravity could impart in the same time ; consequently,
the velocity of the body at the expiration of the time t will
be represented by a—gt] and if we represent this velocity
by V, we shall have
v=a—gt (158) :
ds
substituting for v, its value -^, we find, by integration,
s=at—\gtK
The initial space being supposed equal to zero, no constant
has been added in this integration.
198 DYNAMICS.
This equation being placed under the form
if we substitute for t its value deduced from equation (158),
we shall obtain
a + v a — V
2 ^- '
or,
405. The equations (158) and (159) make known all the
circumstances of the motion under consideration. Thus,,
the equî^tion (158) indicates that the velocity constantly de-
creases as the time increases ; and the equation (159) proves
that the velocity decreases as the space described becomes
greater : hence, the velocity constantly becomes less as the
body rises. When this velocity becomes equal to zero, the
body has attained its greatest elevation : if we denote this
elevation by h, the equation (159) will give, by making v=0,
■ *=J (160).
To determine the time corresponding to this elevation, we
make v=0, in equation (158), and thence deduce
t=- (161).
§■
The velocity due to the height h is found by making h—»
in the formula
and by substituting the value of h deduced from equation
(160), we obtam
hence, the body acquires the same velocity in descending, that
it lost in ascending.
406. Let it be required to determine the greatest height to.
which a body will rise when projected vertically upward with
a velocity of 100 feet per second : we shall find from equa-
tions (160) and (161), that the greatest height is 155 -^Vô feet.
VERTICAL MOTION OP A BODY. ISS'
and that the time of rising or falling is equal to 3i seconds,
nearly.
407. The preceding equations may likewise be applied to
the case in which the body is projected downward, by simply
changing the sign of the quantity g ; we shall thus have an
expression for the velocity f ,
v=a-{-gt.
Of the Vertical Motion^of a Body when acted upon by the
Force of Gravity considered as variable.
408. Gravity is a force whose intensity varies at different
distances from the earth's centre. The law of this variation
has been discovered to be that of the inverse ratio of the
square of the distance ; that is to say, that at distances from
the centre of the earth represented by 2, 3, 4, dec, it becomes
•—, --, —, &c., of its value at the distance unity. Thus,
/i Ô 4'
although a body falls through a distance of 1^-^^ feet in the
first second of time at the surface of the earth, it would fall
through a much less space in the same time if the distance
of the body from the centre were greatly increased.
409. Let a body be supposed to depart from rest at the
point A {Pig- 160), and let it be required to ascertain the
velocity of the body when it has reached the point B. Denote
by g the intensity of the force of gravity at M, the surface of
the earth, and by ç> its intensity at the point B ; by r the
radius of the earth CM, and by x the distance from B to C :
for the purpose of simplifying the calculation, let the known
distance AC be assumed as the linear unit. The force being
supposed to vary in the inverse ratio of the square of the
distance from the earth's centre, we shall have
g i<p :ix^ : r^ ;
whence,
But the general expression for the incessant force being
dv
200 DYNAMICS.
we shall obtain, by placing these values of ^ equal to each other,
^=^-11- (162).
Again, the velocity being equal to the differential of the space
divided by the differential of the time, it will be represented by
dt '
or by its equal
"=-'7 (i«^)-
Multiplying the terms of this equation by the corresponding
terms of equation (162), we find
and by integration,
, „dx
vdv= — ^r^ — ,
The constant may be determined from the consideration that
when a:=AC=l, i;=0; hence.
This value substituted in the preceding equation gives
j=^'-il~') (^«^)-
This equation determines the value of the velocity at any
given point of the line AC.
410. To determine the time employed by the body in
describing the space AB, we eliminate v, between this equa-
tion and the equation (163), and we thus obtain
2di^-^'' \x V'
whence,
dt'=- ^x
2^r^''l_^'
X
and consequently,
dx
"-v^va-o'
VERTICAL MOTION OF A BODY. 201
by the integration of this equation we shall obtain
--^4/^1^ (-)■
To effect the integration which is here only indicated, we
reduce the fraction to a simpler form, thus
The radical in the denominator may be caused to disappear,
by making
l—x=z'.
We deduce from this equation,
These values substituted in the preceding formula give
ng by parts, we find
•(1-4
Integrating by parts, we find
z'^dz
But we likewise have the identical equation
rj /-. .\ P dz /* z^dz
Adding these equations, and dividing by 2, we obtain
fdz^{i-z^)==^,z^{i-zn+^f^^;^^^)
=^z^{l—z'^} + \ arc (sin=z) ;
consequently,
—2fdz^{l—z') = —z^{l—z'^)—a.rc{smz=z),
and by substituting this value in the equation (167), we shall
obtain the integral of (166) ; hence, the equation (165) will
become
i=±~./hz^(l-z')+ arc (sin=2:)] (168).
The constant will be equal to zero, since a-=:1, when ^=0 ;
and therefore, 2; = y/(l—.r)=0; this supposition causes the
202 DYNAMICS.
second member of the equation to vanish. Moreover, the
time being essentially positive, we use only the inferior sign
in the preceding equation ; and by observing that z' =i — x=
the distance AB which the body has described, we shall have,
by representing this distance by s,
^=-\/2i-'^x/V(l-*)+arc (sin=^5)].
411. This last equation is much simplified by supposing
the distances AB and AM to be exceedingly small when com-
pared with the distances AC and MC ; for, the quantity
^(1 — s) may then, without sensible error, be supposed equal
to unity; and the arc (sin = y'5) may likewise be considered
as equal to its sine ; hence, by changing r into unity, the
preceding expression will reduce to
and from this we deduce the relation
or, the motion is then similar to that which would take place
if the intensity of the force remained invariable.
Of the Vertical Motion of a Body in a resisting Medium.
412. It has been ascertained that a body when moving in
a fluid experiences a resistance which is proportional to the
square of the velocity. Thus, by calling w the intensity of
this resistance when the velocity of the body is represented
by unity, the resistance will be expressed by mv^, when the
body has acquired a velocity v.
413. This force being opposed to that of gravity when the
body descends, we shall have, by supposing the intensity of
gravity constant,
<p=g—mv' ;
dv
and by substituting for ç its general value —, we obtain
dv
VERTICAL MOTION OF A BODY. 203
whence,
dt=- ^"^ (169).
g — mv'
To integrate this equation, we decompose the denominator
into factors, and thus have
If we then suppose, according to the method of rational frac-
tions,
^" =^dv( ^— + — ? \ (170),
g—mv^ \^g + v^m y/g—v^m;
we shall find, by reducing the terms of the second member
to a common denominator, and placing the coeiRcients of
the like powers of v equal to each other,
A=B=-— ;
these values substituted in the equation (170), give
dv 1 / dv dv \
g — mv'' ~~ 2v^ V ^g -\-v^m ^g — ■y-/m / *
Multiplying and dividing the second member of this equation
by y/m^ we shall obtain a value, which substituted in equa-
tion (169) will reduce it to
. 1 / d'Oy/m dv^m \
~ 2»/7n^g\^g-^v^m ^g—v^m) '
and by integration, we obtain
^^ 2^(mg) i^^^ (v^^+^^^Z»^) -log(v/g--i'v^m)) -f C ;
or,
t= -J— loff ^/S+'^^/m ,y^^.
^\/{^g) \^g-vx/m ^ ^'
The constant may be suppressed, since when ^=0, ^=0.
414. If the two members of the equation (171) be multi-
plied by 2^mg, and the first member by the logarithm of
the base e of the Naperian system, which is equal to unity,
we shall have
2V(^ff).logg=log. ^^+^^^ ,
204 DYNAMICS.
or,
'/g—Vy/m
and by passing to the numbers, we have
415. This equation being written under the form
. 'i-__ x/g—Vx/ni ,^^2^
e^Vc^e) ^g+v^7n
it is obvious that if t be supposed to increase indefinitely, the
value of the first member will approach to zero ; and conse-
quently, when t becomes infinite, we shall have
^g—Vé/m^O (173)
From this equation we deduce
v—^!-^=di constant quantity.
Hence we conclude, that as the time increases the velocity
becomes more nearly constant.
416. To determine the space described in functions of the
velocity, we multiply the corresponding terms of the equations
dv , ds
and we thus find
vdv = {g — mv'^ )ds ;
whence,
J vdv .^^..
ds= (174).
g — niv^ '
This equation may be rendered integrable by making
g — mv^ —z.
For, we obtain by differentiation,
, dz
and these values substituted in equation (174), transform it
into
MOTION UPON AN INCLINED PLANE. 205
the integral of which is
or, by replacing z by its vakie g — mv^, we have
*=-2^1og(^-mt72)+C.
The constant C may be determined by making 5=0, and
t;=0; whence,
C=ilog^;
which value being substituted in the preceding equation,
gives
"»= 2^[log g—^og (g-mv')] ;
or, finally,
'5=-7r- log ( ^^-vl-
2m \g—mv^ /
Of the Motions of Bodies upon Inclined Planes.
417. Let a body be situated upon an inclined plane, and
let the weight of this body, considered as a vertical force ap-
plied at its centre of gravity, be resolved into two components,
which shall be respectively parallel a id perpendicular to the
surface of the plane. The perpendicular force, being sup-
posed to pass through a point of contact, will evidently be
destroyed by the resistance of the plane, while the parallel
component will cause the centre of gravity to describe a line
parallel to the plane. The question will thus be reduced to
the consideration of the motion of a material point upon the
inclined plane.
418. Let m represent the material point {Pig. 161), and g
the velocity which gravity can impart in a unit of time : if
the force of gravity, represented by the vertical line wiB, be
resolved into two components nîD and mC, respectively parallel
and perpendicular to the plane, the latter will be destroyed by
the resistance of the plane, and the former will cause the
material point to slide along the plane.
18
206
DYNAMICS.
But, since forces are proportional to the velocities which
they communicate in the same time, if we denote by g' the
velocity communicated in a unit of time by the component
which acts in the direction of the plane, we shall have
mB : 7nD '.: g : g'.
The ratio between mB and niD being the same as that between
the length and the height of the plane, we shall have, by
representing these quantities by It' and ]i respectively,
g:g'::h':h]
whence,
^'=f (175).
419. From this equation it appears, that the velocity g',
which is generated in a unit of time by the component of gra-
vity parallel to the plane, is equal to the velocity^, multiplied
by the constant ratio — ; and we therefore conclude that the
force which urges the body along the inclined plane differs
from the force of gravity only in its intensity. Hence, if we
denote by t' the time requisite to describe the entire distance
mA=h', the same relations will exist between the quantities
g\ h', and t', as have been already obtained between g,h, and
t, in investigating the circumstances of uniformly varied
motion : we shall therefore have
h'=^ig'r- (176) ;
and the velocity acquired by the body at the point A will be
v'=g't';
or by eliminating the time i', we shall find
v'=^{2g'h').
If in this equation we substitute for g' its value found in
equation (175), we shall obtain, after reduction,
v'=^{2gh).
The expression for the velocity being independent of the
angle /«AE, which the inclined plane forms with the horizon,
it follows that if several bodies be allowed to descend from
the same point m upon different inclined planes ?nA, ?nA',
mA", <Scc. (Mg. 162), they will all have ac(]^uired the same
MOTION UPON AN INCLINED PLANE. 207
velocity when they shall have arrived at the same horizontal
plane.
420. Although the velocities acquired at the points A and
E are equal, the times of descent will be unequal ; for, if t
and i' represent the times of describing mE and mA, their
values will result from the equations
'2h .. /2h'
▼ o- ~ rr'
but we have
g>g', or-<--;
ff g
and from these inequalities we deduce
2h 2h'
7 F'
which proves that the value of t' exceeds that of t.
421, In general, if t' and t" represent the times of describ-
ing two inclined planes A' and h", having a common altitude
A ; and if g' and g-" represent the components of gravity
respectively parallel to these planes, we shall have
whence,
or by replacing g' and g" by their values (Art. 418), we
obtain
Thus, the times of describing different inclined planes hav-
ing a common altitude will be proportional to the lengths of
those planes.
422. The motions of bodies upon inclined planes give rise
to a remarkable mechanical property of the circle : it consists
in this,— that if the plane of the circle be supposed vertical,
the body will require the same time to describe a chord AC
(Fig. 163), as is necessary to fall through the vertical diam-
208
DYNAMICS.
eter AB. For, the equation (176) gives, for the time of
descent through AC,
^ g
and by substituting for g-' its vaUie ^ , this equation will be-
come
t'=x/^ (177).
gh
But if the diameter of the circle be denoted by d, we shall
have, by the property of the circle,
AB : AC : : AC : AD ;
or,
d:h':: h' : h ;
and consequently,
This value substituted in equation (177), gives, after reduction,
^ g
but this value is precisely the same as that which has been
found for the time t, in which the body would fall through
the diameter AB : for, the height AB being expressed by d,
we shall have
whence,
/2d
Of Curvilinear Motion.
423. We have hitherto supposed the motion under consid-
eration to be rectilinear ; but if it be curvilinear, the space
described, and the velocity acquired in a given time, will be
insufficient to determine all the circumstances of the motion:
it will likewise be necessary to know the nature of the curve
described by the body, and tlie point of this curve at which
the body is found at the end of a given time.
CURVILINEAR MOTION. 209
424. In the resolution of this problem, we employ the prin-
ciple of the parallelogram of velocities, which is similar to
that of the parallelogram of forces. It may be enimciated as
follows : If two forces P and Q, {Fig. 164) communicate^ in
a imit of time, to a m,aterial point m, velocities represented
by wB a?id mC respectively, the resultant R q/" P and Q, will
communicate to the point, in the same time, a velocity 7wD,
which will he represented hy the diagonal of the parallelo-
gram constructed on the lines mB and mC. The truth of
this proposition may be thus established : — Let the force P be
represented by the line ?nB ; then, since forces are proportional
to the velocities which they communicate in a given time,
the force Gi will be represented by the line mC. But, by
regarding mBDC as the parallelogram of forces, the diagonal
mD will represent the resultant of the forces P and Q, ; and
it is required to prove that the velocity resulting from the
composition of the two velocities wiB and tnC is the same as
that which is due to the force R. Let a; represent the velocity
which the force R can communicate to the point m in a unit
of time ; then, since forces are proportional to the velocities
which they generate, we shall have
V :R::mB:x.
But from the parallelogram of forces, we deduce
P : R : : »iB : mD ;
hence,
mB : niD : : mB : x ;
and therefore,
x=mD.
425. In the preceding remarks the forces P, Q., and R have
been supposed to act incessantly, communicating new im-
pulses at each successive instant of time. The results
obtained will however be equally true if we regard P, Q,,
and R as impulsive forces which communicate their effects in-
stantaneously, since the velocities imparted by such forces
are proportional to the intensities of the forces.
426. The composition of three velocities by the construc-
tion of a parallelepiped, results immediately from the pre-
ceding principle ; for, let P, Q, and R {Pig. 165) represent
O
210 DYNAMICS,
-three forces which communicate the velocities mp^ mq, and
mr to the material point m ; let the velocities mp and mq be
compounded into a sino;le velocity mp\ which, by the pre-
ceding demonstration, will be the same as that communicated
by the force P', the resultant of the two forces P and Q, : in
like manner, the resultant ms of the two velocities mp' and mr,
will represent the velocity communicated by the force S, the
resultant of the two forces P' and R, or of the three forces
P, Q, and R ; hence, the diagonal of the parallelepiped con-
structed on the lines representing the three velocities will
represent the velocity communicated by the resultant of the
three forces P, Q,, and R.
427. We will now examine the circumstances in which a
material point will describe a curvilinear path. For this
purpose, let the material point m {Pig. 166), at rest, be sup-
posed to yield to the effect of an impulsion which causes it
to describe the right line mK in the time 6, and at the end of
this time let it receive a iiecond impulsion capable of making
it describe the line AB in the same time 6 ; the material point
will not entirely yield to the action of this second force, which
tends to draw it in the direction of the line AB ; since, by the
law of inertia it would have described the line AC=mA in
the time 6, if the second impulsion had not been communi-
cated to it ; but it will describe the diagonal AD of the paral-
lelogram ABDC. If it should receive at D a third impulse
capable of moving it over the line DG in a third time 6, it
will, for a similar reason, describe the diagonal DP of a
parallelogram constructed upon DG, and DE the prolonga-
tion of AD, (fee. ; thus, at the end of a time equal to 116, the
material point will have described a polygon having n sides.
The velocity being constant so long as the material point
remains on the same side of the polygon, it follows, that if
at its arrival at the extremity of either side, it be not sub-
jected to a new impulse, it will continue to move in the
direction of this side, with a constant velocity.
42S. If the time 6 be supposed indefinitely small, the im-
pulsions will be communicated in consecutive instants, and
the polygon will then be transformed into a curve.
The time é being supposed indefinitely small, it may be
CURVILINEAR MOTION. 211
represented by dt, and the side of the polygon which is
passed over in this time, will become the element of the
curve : consequently, to determine the velocity, which will
be measured by the space which the body would pass over
in the direction of the tangent, in a unit of time, if the in-
cessant force should cease to communicate new impulses,
we must multiply ds, the element of the curve, by the num-
ber of times that dt is contained in unity ; that is, we mul-
tiply ds by—-, and we thus obtain
^ ^ ^ dt
_ds
~di'
429. Let the body be supposed to describe the polygon
m, 711', m", m'", &c. {Fig. 167), receiving increments to its
velocity at the points 7n, m', J7i", m'", &c. Let v, v', v", v'",
&c. represent the velocities which the body has acquired at
the points m, m', m", m'", «fcc, and ê, ô', ê", ô'", (fcc. the times
employed in describing the sides mm', m'm", m"m"', &c.
Since each of these sides is supposed to be described with a
constant velocity, we shall have, by the principles of uniform
motion,
m.m'=v0, m'm"=v'6', m"m"'=v"6", &c. ;
and the perimeter of the polygon will therefore be expressed by
vê+v'6' + v"6"-^6cc.
If we project the sides of this polygon on the co-ordinate axes,
denoting by «, /S, y, «', /3', y', (fcc. the angles formed by the
sides mm', m'm", m"m"', &c. with these axes, the projections
of the sides will be expressed by
v6 cos «, v'ô' cos «', v"ô" cos «", <fcc., on the axis of x,
v6 COS /3, v'ê' cos jS', v"6" cos jS", «fcc, on the axis of y,
vê cos y, v'ô' cos y', v"e" COS y", &c., on the axis of z ;
and the projection nn'n"n"', <fcc. of the perimeter mm'm"m"'^
on the axis of x, will be expressed by
v6 cos ci + v'ê' cos «,'-f f'T' cos «"-f &c (178).
It thus appears that while the material point m describes the
polygon mm'm"m"', &c., its projection n will describe the space
nn'n"n"', &c. But if the point n were merely solicited by a
02
212
DYNAMICS.
force X directed along the axis of .r, and of such intensity
that the point should describe the spaces nn', n'n", n"n"\ ôùc,
in the times e, ê', 6", &.c., with the velocities v cos x, v' cos «',
v" cos cc", &c., the space passed over on the axis of x would
be expressed by
V cos u9-\-v' COS u'ê'+v" COS ct"ê" + &,c (179).
In obtaining the expression (179), no reference has been had
to the components of the velocity parallel to the axes of y
and z ; and the identity of the expressions (178) and (179)
therefore proves that when the point 7n is transported in
space, its projection moves on the axis of x; as thougli the
other two components of the velocity did not exist.
The same remarks being applicable to the other two axes,
and the polygon becoming a curve when the number of its
sides is increased indefinitely, it follows that when a material
point solicited by an incessant force describes a curve in
space, each projection of the point moves independently of
the motions of the other two.
Thus, by calling X, Y, and Z the components of the in-
cessant force ?>, parallel to the three axes, we can regard
these components as forces which impress on the projections
of the material point motions which are entirely independent
of each other.
430. To determine the analytical expressions for these
incessant forces, we remark, that while the material point
describes the space ds, its projections describe the spaces dx,
dy, and dz respectively : the velocities of the projections will
therefore be represented by — -, -^, and-- : and since the
dt dt dt
incessant force is equal to the differential coefficient of the
velocity considered as a function of the time, we shall have,
by regarding dt as constant,
^^— X
'dt^~
dt^
d'z „
(180)
CURVILINEAR MOTION. 213
Such are the equations which serve to determine the circum-
stances of the motion of a material point describing a curve.
431. When the functions X, Y, and Z are given by the
lature of the problem, and if the integrals of the equations
^^180) can be obtained, these integrals will give three relations
between the four variables x, y, z, and t : the quantity t being
eliminated, there will remain two relations between x, y, and
z, which will represent the equations of the trajectory, or
curve described by the 7naterial point under the influence of
the incessant forces.
When the forces are situated in a single plane, which may
be taken as that of x, y, the trajectory will he contained in the
same plane, and it will then only be necessary to use the two
equations
dt^~~ ' dt^~ '
When, by the nature of the problem, the quantities X and Y
are known, and if the integrals of these equations can be
obtained, they will contain no other variables than x, y, and t ;
thus, by eliminating t, we shall find a relation between x
and y, which may be written under the following form,
this relation will be the equation of the plane curve described
by tha material point.
432. The velocity of the material point at any instant is
expressed by
ds
dt
hut the element ds of the arc of a curve situated in space,
being considered as an indefinitely small right line, whose
projections on the co-ordinate axes are represented by dx, dy,
and dz. the value of this element will be
^{dx^+dy'^+dz'').
Substituting this value in the preceding equation, we have
v=\ ^{dx^ ^dy^ +dz.^),
dt
or, since the difierentials are taken with reference to ; as a
variable,
214
DYNAMICS.
,;=V^(^^ + ^>^\ (181).
The angles formed by the direction of the motion with the
co-ordinate axes will result from the equations
dx
V cos « = — -,
dt'
dy
V cos & = —-,
^ dt'
dz
V cos y=-^-.
dt
433. The velocity may likewise be determined in the
following manner. Let the equations (180) be multiplied
respectively by 2dx, 2c/y, and 2dz ; the sum of these pro-
ducts will give
2dx.d^x-\-2dy.d''y + 2dz.d'z ^,^, , ^ , , v-j ^
^ = 2(Xc?x + Ydy + Zrfz) :
and si^ce the first member is the differential of dx'-^-dy"
•{■dz^t^ divided by df^, we shall have
or, rejjîacing dx' +di/' +dz- by its value ds-, and integrat-
ing, wi^Éibtain
'■Jl iÇ--m^d.v-hYdy + Zdz) + C;
^- ds
and by substituting î? for — , we find
v'=2f{Xdx-{-Ydy-{-Zdz)+C (182).
434. It thus appears that the determination of the velocity
will depend on the integration of the expression
J\Xdx + Ydy-hZdz) (183).
When this integration is possible, the integral will be a
function of the variables x, y, and z, and the equation (182)
may be written under the form
v"-^2F{x,y,z)-\-C (184).
To determine the value of the constant, we must know the
velocity of the moveable point, at a given point of the trajee-
CURVILINEAR MOTION. 215
tory. Thus, if V be the velocity at that point which cor-
responds to the co-ordinates ar=a, y = 6, z=c, we shall have
The value of C being deduced from this equation, and sub-
stituted in equation (184), we shall obtain
v--N-=2F{x, y, ^)-2F(a, 6, c).
435. The expression (183) is integrable when the move-
able point is subjected to the action of a force which is
constantly directed towards a fixed centre. To demonstrate
this proposition, we will represent the resultant R of the
several forces acting on the material point by CD, a portion
of the line CM drawn from the point to the fixed centre
{Fig. 168) ; let this centre be assumed as the origin of co-ordi-
nates, and denote by x the distance of the point M from the
origin, and by a, /3, y the angles formed by CM with the axes
of co-ordinates : the direction of the resultant forming the
same angles, we shall have
X = RC0S«, Y=RC0S|3, Z=Rcosy,
and consequently
X COS» Y_C0S/3 Z_C0S7 (\QK\
COS /3 Z COS y X COS a.
But if X. y, and z denote the co-ordinates of the point M, we
shall have
a;=ACOS«, y=Acos/3, 2;=Acosy;
whence, by division,
a:_cos« y cos (S r cos "/
y cos /s' z cos y X cos « '
these values substituted in equations (185), give
yX— .rY=0, 5;Y-yZ=0, .rZ— ;^X=0.
If in these equations we replace X, Y, and Z by their vn1nf>«!
deduced from equations (180), we shall find
d^x d^u _
y- :r— ^=0,
^dr- dt^ '
d^y d-z „
d' z d'X ^
x^ - — z-, — =0.
dt^ dr~
216 DYNAMICS.
Multiplying the first of these equations by dt, integrating and
reducing, we obtain
ydx-xdy^^
dt ^ '
The other two equations being treated in a similar manner,
we find
ydx — xdy = Cdt,
zdy — ydz = Cdt,
xdz — zdx= Cdt.
If we multiply each of these equations by the variable which
it does not contain, and take the sum of the products, there
will result
d^Cz-\-C'x+C"y)=0,
or,
C^ + C'.r + C"y=0.
This equation being that of a plane passing through the ori-
gin of co-ordinates, or centre of attraction, it follows that the
point will describe a plane curve.
In the resolution of this problem it will therefore be unne-
d'^ z
cessary to employ the equation Z=-—, and it will simply be
necessary to integrate the equation (186), which may be
written thus :
ydx — xdy=Cdt\
and from this we deduce
f{ydx-xdy)=--Ct+C' (187).
To determine the value of this integral, we remark that ydx
being the element of a surface bounded by a curve, we can
suppose this surface to be included within the limits x=0 and
x=-CV {Pig. 169); thus, the expression Tydr will be repre-
sented by the area LCPM. If from this area we subtract
the triangle CPM, there will remain
sector LCM=area LCPM-triangle CPM,
or,
sector LCM=yyo?ar—'^;
differentiating and reducing, we find
CURVILINEAR MOTION. 217
</(sectorLCM)=^i^:=^;
and again integrating,
2 . sector 'LCM.=J{ydx—xdy) :
hence> the equation (187) can be reduced to the following:
2 . sector LCM=Ci (188) ;
the constant C is here suppressed, since we may always re»
gard the times as reckoned from the instant when the move-
able point is situated at the point L, in which case the sector
will become equal to zero.
If we make C=2A, the equation (188) will become
sector LCM=A^ ;
from which we conclude, that when a material point solicited
by a force which is constantly directed towards a fixed centre^
describes a curve LM about this centre^ the area of the sector
LCM described by the radins vector drawn to the material
point is co7istantly jiroportional to the time which the point
employs in describing the curve. This property is called
the principle of areas proportional to the times.
436. The formula (183) i. always integrable when the
forces are directed towards fixed centres, their intensities
being at the same time functions of the distances of the
material point from these centres.
Let M represent the place of the material point {Fig. 170),
which is attracted by the forces P, P', P", (fcc. towards the
fixed centres C, C, C", &c. : denote by
A-, y, z^ the co-ordinates of the point M,
a, 6, c, the co-ordinates of the centre C,
a', 6', c', the co-ordinates of the centre C,
a", 6", c", the co-ordinates of the centre C",
&c. (fee. (fee.
p, p\ p'\ (fee, the distances CM, CM, C"M, (fee. ;
«, 0, y, the angles formed by p with the axes of co-ordinates,
a', /3' y', the angles formed by p' with the same axes,
«", |8", y", the angles formed by p" with the same axes,
&c. (fee. (fee. (fee.
The total resultant of the attractive forces will have the fol-
lowing components parallel to the three axes,
19
218 DYNAMICS.
X=Pcos«4-P'cos«'+P"cos«" + (fcc. ^
Y=P cos /3+P' COS /s' + P" COS 0" + (fee. V (189).
Z = P cos y + F COS y' + P" COS y " + (fcc. ^
The projection of the right Hue CM on the axis of .r being rep-
resented by BD {Fig. 170), we have
BD=AB-AD;
and by observing that AB and AD are the co-ordinales x and
a of the points M and C, and that BD, being the projection
of MC on the axis of x, is expressed by j) cos «, we shall find,
by substituting these values in the preceding equation,
p cos ct=x — a ;
the same remarks being applicable to the projections on the
other two axes, we shall have
2^ cos u^=x — a, J) cos/3=y — h, j) cosy=2r — c
And in like manner,
p' cos oc'=x — a', p' cos i3'=2/ — b', p' cos y'=.z — c',
J)" cos ct"=x — a", jj" cos (i"=y — b'\ p" cos y"=-z — c",
&c. &c. (fee.
By eliminating the cosines of these angles, the equations
(189) become
X=P^^^+P'^^+P"^^'+&c.,
p p' p"
Y=pfc-VP'^+F'^ + &c.,
J) J) p
Z^-p^^+F'^^+V^^ + acc.
p p' p'
These values substituted in the formula (183) give
f{Xdx+Ydy + Zdz)=^fp{^^^dx-\-'^^dy+''-^dz\
+(fec. (fee. (fee (190).
But the distances of the point M from the centres C, C, C",
(fee. being given by the equations
{x-aY->r{y—hy-{-{z-cY=p^,
&c. (fee. (fee,
CURVILINEAR MOTION. 219
we shall obtain, by differentiating,
dx-\-- ay-\ az=ap,
P P P
ax •\-- dy-\ dz = dp .
p' P P'
&c. &c. &c. ;
and substituting these values in equation (190), we find
J{Xdx-\-Ydy->rZdz)^f{Vdp-^V'dp' + V"dp"^&cc.)....{l<èl).
But the forces P, P', P", (fee. are, by hypothesis, functions of
the distances jo, p', p", (fee; the expression Vdp-\-Vdp'-\-
V'dp" will therefore contain but a single variable in each
term, and its integral may be effected by the method of
quadratures.
It should be observed that the factors dp, dp', dp", (fee.
may become negative, if the expressions x — a, y — h, z — c,
X — o', (fee. should be transformed into a—x, b—y, c—z,
a'—x, (fee.
437. For the purpose of making an application of the pre-
ceding theorem, let it be required to determine the velocity
of a material point which moves from rest, under the influ-
ence of a force of attraction which is constantly directed
towards a fixed centre, and which varies in intensity in the
inverse ratio of the square of the distance from the position
of the point to the fixed centre. Let the direction of the
force be supposed to coincide with the axis of z : the co-
ordinate axes being disposed as in Pig. 171, the intensity of
the force and the co-ordinate z will increase together, and we
shall have
p=AG—AM=c—z, dp——dz.
If g represent the intensity of the force at the distance r from
the centre C, and P its intensity at the distance p, we shall
have the proportion
P . 1 1.
^■^''■7^''p^^
whence,
but dp being negative, the quantity Vdp should be replaced
28d DYNAMICS.
by —- — df ; integrating, we reduce the equation (191) to
r
This vahie being substituted in formula (182) gives
v2-?ll!4-C (192).
V
To determine the value of the constant C, we suppose the
body to commence its motion at a point whose distance from
the centre of attraction is represented by a ; the velocity at
this point being equal to zero, we have
a
or,
2g^r2
0=
the equation (192) will therefore become
..=2^r»(i_i).
If a be regarded as the unit of distance, the value of v^ will
become identical with that determined in Art. 409.
438. To apply the formulas (180) we will first investigate
the trajectory described by a material point which moves
under the influence of a single impulse. In this case, the
incessant forces being equal to zero, we shall have
X=0, Y=0, Z=0;
and the equations (180) reduce to
^=0 ^-0 '-^=0^
multiplying by dt^ they become
^=0 ^=0 — =0
dt ' dt ' dt
The integrals of these equations are
dx dy dz
-T-=a, -~=h, -i-=c (193).
dt ' dt ' dt ^ ^
Substituting these values in equation (181), we find
MOTION UPON A GIVEN CURVE. 221
v=^{a^ +b' -\-c')=Si constant;
and denoting this constant by A, we have
ds .
dt '
consequently,
s=At-{-B]
and the motion of the material point will be uniform.
The motion is likewise rectilinear ; for the equations (193)
give, by integration,
x=at-{-a\ y=ht-\-b\ z=ct-\-c',
whence, by eliminating t,
az , a'c — ac' bz , b'c — be'
c c ^ ^ c c
These equations evidently appertain to the projections of a
right line on the planes of x, z and y, z.
Of the Motion of a Material Point when compelled to
describe a partictdar Curve.
439. When a material point m, without weight, has received
an impulse K (Fig. 172), and is subjected to the condition of
moving upon a particular curve, we can resolve this impulse
into two components, one mN=K' normal to the curve, the
other mT=K" in the direction of the tangent : the normal
force will be destroyed by the resistance of the curve, and the
tangential component will produce its entire effect in com-
municating motion to the material point.
If we regard the curve as a polygon mm'm"m'", &c. {Fig.
173), having an infinite number of sides, the angle tm'm"
formed by the prolongation of the side mm' with the consecu-
tive side m'm" is called the angle of contact ; it will be
denoted by « ; the plane tm'm" is the osculatory plane at the
point m', and in plane curves coincides with the plane of the
curve.
The material point m, being solicited by a force K, receives
a primitive velocity v, causing it to describe the side mm' ;
but having arrived at the point m', it is deflected from its
course, and describes the side ?n'm". By this deflection it
^^
DYNAMICS.
necessarily undergoes a loss of velocity which will now be
estimated.
For this purpose, let the velocity v be represented by the
line m'q. This velocity being resolved into two components
9?i'n and m'l, respectively parallel and perpendicular to the
side 7ii'j}i", we shall have
m' 1=971' q . sin tm'm", 'm'n=m,'q . cos Vnilml'^
or,
m'l=v . sin », tn'n^v . cos t».
The component v . sin » being destroyed by the resistance of
the polygon, the velocity v will be reduced to v . cos a ; and
consequently, the velocity lost, being equal to the primitive
velocity diminished by the velocity actually remaining, will
be expressed by ^(l— cos i").
When the polygon is supposed to become a curve, the
angle tm'm" becomes infinitely small, and the quantity
v(l — cos u) is at the same time an infinitely small quantity
of the second order.
To prove that this is the case, we observe that 1 — cos»
represents the versed sine DB of an angle » {F^g- 17^4),
measured by the arc BC ; and we have the proportion
AD : CD : : CD : DB.
But when the arc CB becomes infinitely small, CD will be so
likewise ; and since CD is then infinitely small with respect
to AD, it follows from the above proportion, that DB must
be infinitely small with respect to CD, or that it is an infi-
nitely small quantity of the second order. Thus, the velocity
lost at each side ot the polygon being an infinitely small
quantity of the second order, it may be neglected, since the
sum of these velocities, although infinite in number, will con-
stitute but an infinitely small quantity of the first order, which
may be neglected in comparison with the original velocity v.
Hence, we conclude, that a material point which is compelled
to describe a curve, preserves undiminished the velocity
which was originally communicated to it.
440. The component of the velocity v . sin a with which
the material point is pressed against the curve, and which is
destroyed by the curve's resistance, varies constantly as the
MOTION UPON A GIVEN CURVE. 223
point changes its position, since sin * is constantly variable :
we may regard this resistance exerted by the curve as an
incessant force constantly acting upon the point and deflecting
it from the tangent along which it would otherwise tend to
move.
When there are several forces acting on the material point,
we resolve each in a similar manner, and the sum of the nor-
mal components must then be added to the pressure arising
from the component of the velocity.
441. Let it be supposed that a force N equal and directly
opposed to the resultant of all the normal forces is applied to
the material point : this force will produce precisely the same
effect as the resistance offered by the curve, and the latter will
therefore be represented by N. Let «, /s, y be the angles
formed by the direction of the force N with the co-ordinate
axes ; the components of N in the direction of the axes will-
be respectively
N cos «, N cos /3, N cos y,
and should be added to the components of the incessant forces
in the general equations of motion (180) : we shall thus obtain
- — =X4-N cos «
^=Y+N cos ,3 ^ (194).
- — =Z+N cos y
dp
To these equations may be added two others which result
from the relations existing between the angles u, /3, and y ;
the first of these equations is
cos- «+C0S2 /3-{-COS2 y=l (195).
The second is
cos « . cos a'-f-COS /3 . COS /3'-fC0S y . COSy' = 0,
in which », /3', y' represent the angles formed by the tano-ent
to the curve with the co-ordinate axes. The cosines of these
last angles may be expressed as follows :
, d.v , dy , dz
cos «'=--, C0S/3'=-/, COSy'=--:
as as ds
224 DYNAMICS.
these values substituted in the preceding equation convert it
into
-^ cos u-{-^ cos /3+-^ cos y=0 (196).
as as its
442. To determine the velocity of the material point, let
the equations (194) be multiplied respectively by 2dx, 2dy,
and 2dz : their sum being taken, we shall obtain
2dx'^+2dy^ + 2dz^^2(Ldx^Ydy+Zdz)
dt'' dt^ dP
4-2N(c?a; . cos a+f/y . cos ^-l-dz . cos y) :
the last term of this equation being equal to zero, by formula
(196), there remains
2dx'^-V2dy'^+2dz'^^=2{X.dx-^Ydy+Zdz) ;
or,
d{dx^+dr 'rdz^)^2{Xdx^Ydy-^Zdz) :
whence, by substitution and integration, we find
1^ =2f{Xdx+Ydy^Zdz)+C ;
or,
v'=2f{Xdx-^Ydy+Zdz)+C (197).
443. When the material point merely receives an impulse,
without being acted upon by incessant forces, we have
X=0, Y=0, Z=0;
and consequently,
v' =a constant.
Thus, when the material point is compelled to describe a
curve, being acted upon only by an impulse, its velocity will
remain invariable. This result accords with that which has
been already obtained (Art. 438), on the supposition that the
motion is perfectly free.
444. Let the material point which is supposed to describe
the curve, be acted on by the forceof gravity ; we shall then
have
X-0, Y=0, Z=^;
and the equation (1 97) will be reduced to
v''=2fgdz-{-C.
MOTION UPON A GIVEN CURVE. 229
If the velocity v be supposed equal to V, when «=0, we shall
find
This value substituted in the preceding equation gives
whence,
v=^{2gz+N^) (198).
This expression for the velocity being independent of the
relations which may exist between the co-ordinates x, y, and z^
it follows that the velocity will be the same for the same
value of z, whatever may be the form of the curve.
To determine the expression for the time employed by the
material point in describing a given portion of the curve, we
ds
replace v by its value -r^, and thus obtain
whence,
<"=7(^JW,---<^«^>^
or, by substitution,
^'^ VC^gz + Y^) (''^^-
To integrate this equation, it will be necessary to reduce it, by
means of the equations of the curve, to one which shall con-
tain but two variables ; thus, if the equations of the curve
are
f{x,z)=0, At/,z)=^ (201),
we may, by the aid of these equations, in connexion with
equation (200), eliminate two of the three variables x, y, and
z ; and it will then only be necessary to integrate an equation
expressing the relation between dt and one of the co-ordi-
nates of the moveable point.
445. If, for example, the curve be supposed to become a
right line, the equations (201) will be of the form
x=az+u, y=bz+^ (202):
from which we deduce
dx^adz, dy=bdz\
P
226 DYNAMICS.
and by substituting these values in the formula (200), it is
transformed into
dz y{l + a'- \-b')
If the point be supposed to move from rest, its initial velocity
V will be equal to zero, and we shall have, by division,
dt dz
whence, by integration,
^ ]■^/{2gz) (203).
The constant introduced by integration becomes equal to zero,
since, by hypothesis, when ^=0, -y^O, and z--0 (Art. 444).
446. To determine the space passed over in the time t, we
suppress V in equation (199), which then becomes
and eliminating z by means of equation (203), there results
, g-t .dt
and by integration,
10-/3
9— "- 4-C •
which proves that the motion is similar to that of a body on
an inclined plane, as might have been anticipated.
447. The co-ordinates .r, y, and z are readily determined
in functions of the time ; for the latter is given by formula
(203), and this, taken in connexion with equations (202), will
determine a; and y in functions of t.
448. If, as in the present instance, the point be supposed
to describe a plane curve, and if the incessant forces act en-
tirely in this plane, we may, by placing the axes of t and y
in this plane, dispense with the consideration of the third of
equations (194) ; the formulas (195) and (196) will then be
reduced to
cos'«-l-cos2j3=l, ^-cos«4— /cos/3=0;
as as
MOTION UPON A GIVEN CURVE. 227
and the two equations of the curve will be replaced by the
single relation
449. The velocity being given by formula (198), without
the aid of equations (201), we conclude that the velocity ac-
quired by the moveable point is independent of the form of
the curve, being determined by the value of the vertical ordi-
nate. Consequently, if from the point O {Fig. 175), where
^=0, and v— V, we draw the arcs of different curves OM,
OM', OM", (fcc, terminated by the horizontal plane KL, the
ordinates z of the first and last points of all these arcs being
equal, it follows that different bodies departing from the point
O with the common velocity V, and describing the several
curves, will all have acquired the same velocity when they
shall have arrived at the points M, M,' M", (fee, situated in the
same horizontal plane.
450. In general, whatever may be the number of forces
acting on the moveable point, if the equation (197) be inte-
grable, we can determine the velocity v without knowing
the nature of the curve described. For, the values of the in-
cessant forces X, Y, and Z, expressed in functions of the co-
ordinates X, y, and z. being substituted in equation (197), if
the expression
f{X.dx + Ydy + Zdz)
becomes integrable, we may represent it by
/(^j y, z) ;
and the equation (197) will then reduce to
'v'=2f{x,y,z)+C.
If we denote by a, h, and c the values of x, y, and z which
correspond to the velocity V, the value of C will become
known ; thus,
C=V2-2/(«, 6,c);
and replacing C by this value in the general expression for
the velocity, we find
v^ =V* -\-2f{x, y, z)-2f{a, b, c) ;
an expression which depends only on the initial velocity, and
the co-ordinates of the first and last points of the curve
described.
P2
228 DYNAMICS.
451. It has been explained (Art. 440) that the normal
pressure exerted against the curve arises in part from the nor-
mal components of the incessant forces, and partly from the
normal force due to the velocity. To determine the value of
the latter, let perpendiculars on and o?i' be erected at the
middle points of the equal consecutive sides wtwt' and m'tn"
(FS,g. 176) of the polygon having an infinite number of
sides : the angle tm'm,", formed by one of these sides with the
prolongation of the other, W\\\ be the angle which we have
represented by •». But the angles n and n' being right anglesj
we have
non' ■\- ii'm'n' = 180° = tm'm" -\- nm!iï ;
and therefore,
trn'm" = » = non' = 2nom'.
The angle nom' being infinitely small, its sine may be re-
garded as equal to the arc which measures it ; but this sine
, , m'ît' m'n' . j , i
IS expressed by , or , smce 7io and mo may be con-
m'o 710
sidered equal ; hence,
2m'n mm'
û/= = .
710 710
If we now return to the consideration of the curve which is
the limit of the polygon, the side mw' becomes the element
of the curve, and 7io the radius of curvature : the preceding
relation will therefore be transformed into
ds
0,= — ,
y
y denoting the radius of curvature.
Let ç> denote the intensity of the incessant force which is
due to the normal component of the velocity : this intensity
being in general expressed by the quotient of the element of
the velocity, divided by the element of the time, and the ele-
ment of the velocity being represented in the present instance
by V . sin «, we shall have
V .sin a
or, since the infinitely small arc may be substituted for its
sine, this expression becomes
MOTION UPON A CURVED SURFACE. 229
Vu
replacing « by its value found above, we have
vds v^
^ = -^' or^=— .
yat y
The normal pressure resulting from the other forces may be
determined by the parallelogram of forces, and this pressure
must then be combined with that due to the velocity, in order
to obtain the total pressure,
452. Let it be supposed, for example, that the material
point describes a plane curve, and that the incessant forces
are directed in the plane of this curve : let these forces be
reduced to their resultant R, and denote by 6 the angle formed
by the direction of the resultant with that part of the normal
which lies on the concave side of the curve : the component
of the resultant in the direction of the normal will be ex-
pressed by R cos ^, and will act in the same or in a contrary
direction to the pressure arising from the velocity, according
as 6 is obtuse or acute. The pressure arising from the velo-
city being always directed /rowi the centre of curvature, the
entire pressure will be expressed by
N=— — RcosO;
y
this pressure will be exerted /rom the centre of curvature so
long as the quantity N is positive, and towards the centre in
the contrary case.
Of the Motion of a material Point when compelled to move
itpon a Curved Surface.
453. When a material point which is compelled to move
upon a curved surface is subjected to the action of inces-
sant forces, these forces, and that resulting from the velocity
of the point, will exert a pressure against the surface, which
will be counteracted by the resistance of the surface. If we
denote this resistance by N, the material point may be re-
garded as moving freely in space, provided we include the
components of the force N in the general equations (180),
which express the circumstances of motion of a point under
20
230
DYNAMICS.
the influence of incessant forces. Let «, /3, and y represent
the angles formed by the direction of the force N with the
co-ordinate axes ; its components in the directions of these
axes will be expressed by N cos «, N cos /3, and N cos y : con-
sequently, the equations of the required motion will be
- — =X4-N cos «
^=:Y + NC0S/3
^=Z+1S cosy
dt^
(204).
The angles «, /2, and y will become known when the equation
of the surface L=0 is given, for we have, (Art. 62),
dx
cos ct— ±
^m^m^it)
cos /3=±
dy
dL
dy
cos y= ±
dL
d^
^{Èï-m'HÈ)
the double signs prefixed to the values of the cosines, indi-
cate that they may refer to the direction of a force which
tends to pull the point, either along the normal to the surface,
or along its prolongation.
If we put, for brevity,
1
± — - — =Y,
^{Èï^Èï^Èy
dy
the preceding equations will become
.dL
cos «=V
dx'
COS/S:
dy^
COS»/— Y
M.
dz
MOTION UPON A CURVED SURFACE, 231
these values substituted in equations (204), reduce them to
dt'' dz
(205).
If N be ehminated between these equations, V will likewise
disappear, and we shall thus obtain two relations, which,
taken in connexion with the equation of the surface L=0,
will determine the co-ordinates of the moveable point in
functions of the time.
454. As an example : — Let it he required to determine the
circumstances of the motion of a material point on the
surface of a sphere : let the origin of co-ordinates be as-
sumed at the centre, the plane of x, y being horizontal, and
the co-ordinates z being reckoned positive downwards ; these
co-ordinates will then be affected with the same sign as the
force of gravity.
The equation of the sphere being
lj=x^-\-y^-\-z'—a^=Q (206),
we obtain by differentiation,
dl.=2xdx-^2ydy^-2zdz=0 (207),
and consequently,
g-=2x, ^=2y, -=2z.
y , 1 ,\.
~v'(4a^ +4^2+42^)-"^ 2a'
or,
COS«=±-, 008/3= ±?^, cosy=±- (207 a).
a a a
Again, the force of gravity being the only incessant force
acting on the material point, we have
X=0, Y-0, Z=g;
these values reduce the equations (205) to
232 DYNAMICS.
^=±N?, !^=±N^, '^. = ±^-+e (208).
at- a at- a at^ a
The positive signs should be taken together, and evidently
correspond to Hke signs in the vahies of the cosines of «, /î)
and y ; a similar remark is apphcable to the negative signs.
We ehminate ± N between the two first of these equations,
by miihiplying them respectively by y and i\ and taking
their difference ; we thus obtain, after muhiplying by dt^
yd^x—xd^y _^ or ^(y^-^~^^y) -o .
dt ' dt '
whence, by integration, dt being regarded as constant,
ydx-xdy=Cdt (209).
A second relation between the variables may be found by
multiplying each ot the equations (208) by the differential of
the variable which it contains ; the sum of these products will
give
dx.d-ix-\rdy .d^'i/-\-dz .d^z , N, , , , , , . , ,
dP ^ -{xdx-\-ydy-\-zdz)-\-gdz ;
and since the quantity included within the brackets is
equal to zero, by equation (207), the preceding result will be
reduced to
dx .d~x-\-dy .d^y-\-dz . d^z _ ,
W^ ~^ ^'
multiplying by 2, and integrating, we have
^.^^l±^p:^=2gz+G' (210).
If two of the three variables .t, y, and z be eliminated between
the relations (206), (209), and (210), ihe result will be an
equation which, being integrated, will give a relation between
the third co-ordinate and the time / : this result will evidently
be independent of the normal force, which has already disap-
peared from these three equations.
455. The equations (207) and (209) being squared, give
x^dx-+ 2xydxdy + y"" dy^ =z^ dz^^
y^ dx^ — 2xydxdy + x^ dy- =C^ dt^ .
The sum of these equations being taken, the middle terms of
the first members will disappear, and we shall have
MOTION UPON A CURVED SURFACE. 233
substituting for {x^ +y") its value deduced from equation
(206), there results
, , , , , C^dP-ifZ^dz^
dx^' -\-dy= — ;
and eliminating dx'^ -\-dy^ between this equation and (210),
we find
dt= — (211).
V'[(a^-z^)(2^z + C')-C^l ^ ^
The integral of this equation, which can only be obtained by-
approximation, will give the value of z in functions of the
time.
456. To determine the expressions for the other co-ordinates
in functions of the time, we will suppose ft to represent the
approximate value of z determined from the integration of
the preceding equation : if this value be substituted in equation
(210), we may, by combining the resulting equation with that
designated as (209), obtain two relations, the first between x
£md t, the second between y and t : but as the variables in
each of these equations would not be separated by this pro-
cess, we adopt another method of determining the values of
x and y in functions of t.
Let KC=x, DC=y, iiiD^z {Fig. 177) be the three co-
ordinates of the point tn on the surface of the sphere ; if for a
given value of z, the angle CAD, formed by the projection
AD of the right line AM with the axis of x, were known, the
corresponding values of x and y might be readily determined
in functions of z. For, the angle CAD being denoted by 6,
and the radius Am by a, we shall have KD=^{a^ —z^)]
and the triangle ACD right-angled at C, will give
AC=AD . cos CAD, CD=AD . sin CAD ;
or,
x=^y/{a^—z'') Xcos Ô, y=^{aP- -z^) Xsin (212).
These two equations establishing a relation between x, y, and
z, may be considered as replacing the equation of the sphere,
which can be obtained by taking the sum of their squares.
An additional variable 6 is here introduced, but the number of
relations is likewise increased by unity.
234 DYNAMICS.
From the equations (212) we obtain by differentiation,
...(213):
zdLz
dx=—s'med6^(a'' — z") — ,cos Ô
zdz
di/=cos eds^ia' — z^) r- rMn 6
multiplying the first of equations (213) by the second of
(212), and the second of (213) by the first of (212), and taking
their difl'erence, we obtain
]/dx—xdy=~-{a^—z''){sm''ô + cos^6)d6',
or,
pdx —xdy={z^ — a^)d6.
This value substituted in equation (209), gives
{z''—a^)d6=Cdti
and consequently,
z^ -a" '
or, replacing dt by its value deduced from equation (211), we
obtain
, a.C .dz
-\z^-a^^)^[{a--z')[2gz+C')-C^]
This equation, being integrated by approximation, will deter-
mine the value of 6 : we thence deduce the values of cos ê,
and sin 6, which substituted in equations (212), determine the
values of x and y in functions of z, and consequently in
functions of the time t.
457. The equation (210) proves that the velocity is inde-
pendent of the normal pressure ; for, we deduce f om that
equation,
or,
*» ='°^+*^>
v^=2gz-^0:
and consequently,
v=^{2gz-[-C').
To determine the value of the normal pressure, we must
recur to equations (208) : these being multiplied respectively
by X, y, and z, and added, give
MOTION UPON A CURVED SURFACE. 236
^,J = ±-{x'-^y^+z')+gz (214).
But the diflerential of equation (207), xdx-\-ydi/ + zdz=0,
being taken, we find
xd''x+yd''i/-i-zd''z_ dx^ ■\- dy'' -\-dz^ __ j,
dT^ df^
and this value substituted in equation (214) gives, after re-
placing .r» -^-y^ +2;2 by «3,
or,
a a
458. If the moveable point be supposed situated at any
instant below the horizontal plane passing through the
centre of the sphere, the ordinate z will be positive, and the
value of ± N becomes negative : and since N, which denotes
the intensity of a force, is by hypothesis an essentially posi-
tive quantity, the inferior sign must be taken in order that
— N may be essentially negative. Hence, it will be neces-
sary to take the inferior signs in equations (208), and also in
equations (207 a). The resistance of the surface will there-
fore be directed towards the centre, or the material point must
be regarded as situated upon the concave surface of the sphere.
When the material point rises above the horizontal plane
of a-, y, the ordinate z will become negative, and the quantity
— tj3 — gz may then become positive. In such case, the
superior signs must be taken in equations (207 a) and (208),
and the resistance of the surface will be exerted from the cen-
tre, or the body must be supposed to be on the convex surface.
The pressure exerted against the surface will be equal and
directly opposed to the resistance which it offers, and will
therefore be represented by ^— without reference to the
sign of z.
If the moveable point be retained upon the surface of the
sphere by an inflexible thread connecting the point with the
centre, this thread will experience a tension so long as v^ -\-gz
is positive ; but, on the contrary, there will be a tendency to
compress the thread whenever v^ -{-gz becomes negative.
236 DYNAMICS.
Of the Motion of a material Point on the Arc of a Cycloid.
459. Let a material point M [Pig. 178) be supposed to
move fram rest on the arc of a cycloid, under the influence of
the force of gravity : the initial velocity being by hypothesis
equal to zero, the equation (198) is reduced to
v2 =2gZj
or
whence we deduce
dt= ^
Let the origin of co-ordinates be assumed at the point E, the
absciss ED of the point M' being denoted by w, and the absciss
EC of the point of departure by A : we shall then have
CD=EC-ED;
or,
z—h — u.
This value being substituted in the preceding equation givesi
dt=^—--^^ (215).
This equation contains three variables ; we must therefore
eliminate one by means of the equation of the cycloid. For
this purpose, let 2a represent the diameter BE of the gene-
rating circle, and x and o/ the co-ordinates AP and PM'
of the point M', reckoned from A as an origin ; the equation
of the curve will then be
"^^TW^^ ^^^«>-
But if s denote the arc AM', we shall have the relation
ds=^{dx'' +dj/'');
or,
MOTION UPON A CYCLOID. 237
substituting in this equation the value of -7- deduced from
the relation (216), we find
or,
^='y\^^y (^^^-
But from an inspection of the figure, we have
2a — y=u ;
and hence,
dy= — du.
By substituting these values in (217), we obtain
ds=^ — du\^/ — .
V u
The difierentials of 5 and u have contrary signs, since the
first is a decreasing function of the second.
The preceding value of ds will reduce equation (215) to
du
g' ^{hu—u^)
460. This equation may be integrated by the formula
dx
dt=-y/^. ,/^ ^, (218).
lis eq
/ y/o"^ ^3^ =arc (versed sine ~x) ;
for by making x=—, this formula becomes
— 7x r=arc I versed sine =— | (219) ;
^{2az—z') \ a F v />
and consequently, by referring the integral of equation (21 8)
to this formula, we obtain
I^-k/-- . arc f versed sine= ^\-\-C (220).
^ g \ \h}
To determine the constant, we remark, that since the time is
reckoned from the instant when the body is at the point M,
we must then have
/=0, and «=EC=A;
this supposition reduces the equation (220) to
238 DYNAMICS.
0= — \./ — • 3-rc (versed sine =2)4-C.
^ g
But the arc whose versed sine is equal to 2, being a semi-
circumference, if we denote by v the semi-circumference of a
circle whose radius is unity, the preceding equation will
become
This value will reduce equation (220) to
i=>v/— fw— arc (versed sine =-t-|.
This expression gives the time of descent to the point M', the
absciss of which is equal to u. To obtain the entire time
of descent to the vertex E, we make w=0, and the value of
t is then reduced to
'-^\/~t
g
This value of the time being independent of the height h
of the point of departure, we conclude that the time necessary
for a material point to deseend to the vertex E of the cycloid,
under the influence of the force of gravity, is constantly the
same, whatever may be the position of its point of departure.
Of Oscillatory Motion.
461. Let OBC {Fig. 179) represent a continuous curve,
intersected at the points O and C by a horizontal line, and
supposed to contain no angular points that might occasion a
loss of velocity to a body or material point moving upon it.
Let the tangent BT at the point B be supposed horizontal,
the co-ordinate plane of x, y being likewise horizontal. If
the co-ordinates z be reckoned positive downwards, we
shall have the following equations to determine the circum-
stances of the motion of a material point sliding along the
curve under the influence of gravity :
OSCILLATORY MOTION. 239
To determine the velocity of the moveable point, we proceed
as in Art. 433 : multiplying these equations by 2da:, 2dy, and
2dz respectively, and adding, we find
2dxd^ X + 2dyd^ y + 2dzd'' z
2gdz',
dt^
and by integration,
——de ^^^+^'
or,
Replacing-— by its value v^, there will result
If V denote the velocity at the point O, when ;$r=Oj the pre-
ceding equation will become
and consequently, by substituting this value of C, we shall
obtain
'V^=Y^+2gz (221).
462. Since the ordinates increase from the point O to the
point B, it appears from equation (221) that the motion will
be accelerated while the material point is describing the arc
OB, and that its velocity will be a maximum at the point B :
the ordinates decreasing beyond this point, the velocity of the
moveable point will likewise be diminished. This diminution
must be such that the material point will have at the point
m!. the same velocity as it previously had at the point m^
situated in the same horizontal plane ; for the vertical ordi-
nates of these points being equal, their values substituted in
equation (221) will necessarily give the same values for the
two velocities.
The velocity diminishing constantly with the arc Om, we
shall find on the prolongation of this arc a point A at which
this velocity will have been equal to zero ; and the moveable
point may therefore be considered as moving from rest at this
point. If through the point A a horizontal line AA' be
drawn, intersecting the second branch of the curve at A', the
240 DYNAMICS.
velocity at A' will likewise be equal to zero. Thus, the
motion will cease at the point A', and the action of gravity,
causing it to descend from A' to B, will augment the velocity
in the same manner that it was before diminished. At the
point B the velocity will again become a maximum, and the
moveable point will then ascend to the point of departure A,
its motion being retarded in the same manner that it was
before accelerated in descending from A to B.
The same efiects being repeated by the action of gravity,
the point will continue to oscillate indefinitely.
If the arcs AB and A'B are similar, the times of describing
them will evidently be equal. When the oscillations of a
body or material point are all performed in equal times, they
are said to be isochroiial.
463. Let B'OBO' {Fig. 180) represent a curve returning
into itself, and symmetrical with respect to a vertical axis
passing through the points B and B' at which the tangents are
horizontal. If the material point descend from a point O,
with an initial velocity such that upon arriving at B it can
ascend from B to B' on the second branch of the curve, it will
descend a second time on the arc B'OB, the force of gravity
restoring the velocity lost during the ascent on the arc BO'B'.
The same effects being repeated, the body will continue to
revolve indefinitely.
Of the Simple Pendulum.
\ 464. The simple pendulum is composed of a material
heavy point M {Fig. 181), suspended by an inflexible right
line MC devoid of weight, and oscillating about a point C.
In this motion the point M describes the arc of a circle about
C as a centre, and the velocity of M will be given (Art. 444)
by the equation
v^ = V2 -[■2gz (222).
ds
Replacing v by its value —, we find
dt= ^^/\ , (223).
SIMPLE PENDULUM. 241
The origin being acsiimed at the point of departure, z will
represent the ordinate M'P' {Fig. 182) of the point M', at
which the material point is found after the lapse of a certain
time, and V* will represent the square of the velocity which
the body has at the point M, where xr=0. If h denote the
height due to this velocity, we shall have the relation
and the equations (222) and (223) therefore become
-v/[2=^(A+.)], *=;7i^$+;)] (224)
465. To express the quantity z in functiors of the co-ordi-
nates of the circle described with the radius CM, we demit
the perpendiculars MB and M'D on the vertical line CE, and
denote by a the radius CE, by h the vertical distance EB,
and by x the absciss ED of the point M' referred to the point
E as an origin ; we shall then have
- = BD = 6— .r.
And by introducing this value into equations (224), they be-
come
.=^[2^(, + 6-.)], d.=_^^^*__^ (223).
From the first of these relations we obtain the velocity of the
material point at the point M', corresponding to the absciss
X ; the second, being integrated, will determine the time em-
ployed by M in descending to the point M', To effect this
integration, we must eliminate one of the variables contained
in the second member, by m-eans of the relations
ds=^iclx^-\-dij^) (226),
y^=2ax—x^ (227).
The latter being differentiated, gives
ydy={a — x)dx ;
and consequently,
dy^=i^^IIpldx'.
This value being substituted in equation (226), we find
a 21
242 DYNAMICS.
or, replacing y^ by its value given in equation (227), and re-
ducing, we obtain
, , /a^ , adx , adx . adx
as=dx^ / — = ± = ± — -r r==t — ft^ r— t 5
V y2 y ^(2ax—x') ^[(2a—x)x]^
whence,
~~ ^[{2a-x)x]^[2ff{hi-b-x)]'
The negative sign is here prefixed to the second member, be-
cause the co-ordinate a; is a decreasing function of the time t.
466. If the initial velocity be supposed equal to zero, we
shall have
h=0]
and if at the same time the arc through which the oscillation
is performed be supposed extremely small, we can neglect x
in comparison with 2a, and the value of dt will be then re-
duced to
, adx
^~^{2ax)^[2g{b~x)]-
This equation may be put under the form
dt=—l\/- X .^f"" , , (228).
The value of i will be immediately obtained by an integration
of the formula
dx
:^{bx—x^)
which, by a comparison with (219), gives
a=^b, z=x:
and by substituting these values in equation (219), which is
f:77^) (^«>^
/ r- =arc I versed sine = — | ,
J^(2az-z'') \ a/'
^{2az-z'')
it becomes
/! :.^£- — -=arc (versed sine =^ |
^{bx-x') \ \b}
=arc (versed sine=-^ j .
But, in general, the cosine of the arc corresponding to the
SIMPLE PENDULUM. 243
versed sine c and radius unity, being equal to 1 — c, we shall
have
arc
(versed sine =— j=arc (cos=l — —\
)•
/ b
=arc| cos= —
V b
This value of the expression (229) being substituted in equa-
tion (228), we shall find
^=-i\/-Xarc(cos=^::^Wc (230).
467. The constant may be determined from the consider-
ation that when t=0, x=b ', these values reduce the equation
(230) to
0=-!^/- Xarc(cos=_l)-f C.
If «• denote the semi-circumference of a circle whose radius
is unity, we shall have (Pig: 174)
arc (cos = — 1) = ar c BC A = sr ;
and consequently.
By substituting this value in the equation (230), we obtain
The integral being taken between the limits ar=6, which cor-
responds to ^=0, and x equal to any assumed absciss, will
make known the time of descent from the point M {Fig: 182)
to the point M' corresponding to the assumed value of x.
468. When we wish to obtain the time of descent to the
lowest point E, we make x=0, in the preceding expression ;
and since the arc whose cosine is unity is equal to zero, we
shall have
<=iTv/-^ (232).
^ g-
469. When the material point arrives at the point E, it
Gt2
244 DYNAMICS.
will have acquired its maximum velocity ; for the velocity
being expressed by
it will evidently be a maximum at that point of which the
ordinate z is the greatest. Thus, in virtue of the velocity
acquired at E, the moveable point will describe the arc EN ;
and since this arc changes its sign in passing through zero,
we find for the expression of the time requisite for the point
to arrive at N'
^=i\/|[-+arc(cos =1-^)] (233).
If from this expression we subtract that given by (232),
which expresses the time of descent from the point M to the
point E, there will remain
I S/— X arc / cos= 1 — — V
an expression Ibr the time of ascent from E to N': this time
is equal to that employed in descending from M' to E, as may
be proved by taking the difference between equations (231)
and (232).
Finally, when the material point shall have arrived at the
point Nj situated in the horizontal line passing through M, we
shall ]iave j:— 6, and the expression arc |cos=l — —I will
then become arc (cos =— l)=5r; thus, the equation (233)
will be reduced to
^ g
Such will be the value of the time required by the moveable
point to describe the whole arc MEN. This time being de-
noted by T, we have
T='^\/- (234).
The velocity of the material point upon its arrival at the point
N will be equal to zero ; for, since the initial velocity was
supposed equal to zero, we have
A=0;
SIMPLE PENDULUM. 245
and this value, taken in connexion with that of x=b, reduces
the equation v=^[2g{k-^b—x)] to
v=0.
The motion of the material point being entirely destroyed
when it arrives at the point N, the force of gravity will cause
it again to descend, and since the circumstances of the
motion are precisely similar to those presented when the
point commenced its motion at M, a second oscillation will
be performed in the same time, and a similar motion will
continue indefinitely.
470. The equation (234) being independent of the quantity
b which expresses the vertical distance MK, it follows that
if the point of departure had been taken at M', instead of at
M, the time of oscillation would have been the same ; and
consequently, that if several material points depart from the
different points M, M', M", (fcc, they will all perform their
oscillations in the same time. It should be recollected, how-
ever, that this result has only been obtained on the supposition
that the arcs described are extremely small.
471. These oscillations of equal duration are called iso-
chrmial. But if the length of the pendulum be supposed varia-
ble, the time of vibration will likewise vary : for, if I and V
represent the lengths of two pendulums, whose oscillations
are performed in the times T and T', we shall have
g ^ g
hence,
T : T' : : v^Z : v^Z' (235).
Thus, if the time of oscillation T of one pendulum be accu-
rately known, we can determine by the preceding proportion
the length V of a pendulum which shall vibrate in an arbi-
trary time T'.
472. To ascertain with greater precision the time of a
single oscillation, we will represent by N the number of oscil-
lations made by the pendulum whose length is I in a time 6,
and by N' the number of oscillations of the pendulum I' in
the same time « : we shall then have
t=Vf' '^-Vl'
246 DYNAMICS.
T=^, andT'=^, (236).
By means of these values, the proportion (235) is reduced to
N'^ : N= : : ^ : l\
whence,
When the number of oscillations made by a pendulum of a
given length, in a given time, has been ascertained from
observation, we can calculate the length of the pendulum
which will oscillate in a second of time.
If an error be committed in observing the time Ô, this error
will he greatly reduced by being divided by the number of
Dscillations, and if this number be taken large, the effect of
the error upon the time of a single vibration may be regarded
as insensible.
473. It is jXi this principle that the length of the seconds
pendulum, which makes 86,400 oscillations in a mean solar
day,in vacuo, and at the latitude of New- York, has been found
■equal to
in. ft.
39.10168=3.25847, nearly.
474. To determine the value of g, the measure of the in-
tensity of the force of gravity, we employ the equation (234)j
which gives
and by making in this equation
in. ft.
T=l", Z=39.1G168, and «-=3.1415926,
or,.
a-2 =9.8696046;
sire find
in. ft.
^=385.9183=32.1598.
475. If g and g' represent the intensities of gravity at dif-
ferent places, and I and V the lengths of two pendulums
which oscillate in the times T and T', we shall have
T=-V'^-, T^'s/!:'
g' ^ g"
CENTRIFUGAL FORCE. 247
ffom which we deduce
T:T':: v/|: V^l (237)-
Let N and N' represent the numbers of oscillations made by
these pendulums in the time 6 ; T and T' will be given in
functions of 6 by equations (236), and their values being sub-
stituted in the proportion (237), will give, after reduction^
If the same pendulum be used at the two places, Z and V will
be equal to each other, and the preceding proportion will
become
whence,,
1 -1 •• /I- /L-
N ■ N' ■ ' V ^ ■ V ^'
N'
Of the Centrifugal Force.
476. If a material point be supposed to move around a
fixed centre C, describing the curve LMK {Pig. 183), and if,
upon its arrival at the point L, the connexion with the centre
be suddenly destroyed, the material point will, in virtue of
the law of inertia, continue to move in the direction of the
tangent LT. But if we conceive the point to be compelled
to describe the curve, it will leave the tangent, and will after
a certain time arrive at the point M. The arc LM being sup-
posed indefinitely small, the angle LCM will be so likewise,
and the lines LC and MC may be considered as parallel.
Thus, replacing CM by the parallel CM, and constructing
the parallelogram LDMN, it appears that the material point,
if free, would describe the side LD, while by its connexion,
with the fixed centre it is caused to describe the diagonal
LM ; the effect of the force which draws the point towards the
centre has therefore been to move it through the space MD.
The point may be supposed to be retained on the curve
LMK, either in virtue of a force of attraction which is con-
stantly directed towards the centre C, or by the resistance
248 DYNAMICS.
opposed by the curve regarded as material ; or, finally, by
being connected with the point C, by means of a cord of
variable length.
Whilst the point is describing the elementary arc LM, we
can regard it as moving upon the equal arc of the osculatory
circle, and can suppose it to be retained on this arc by means
of a thread of an invariable length, attached to the centre of
the osculatory circle. Moreover, since this thread will ex-
perience a tension only in consequence of the resistance
offered by the material point to the force which tends to
deflect it from the tangent, this tension or the resistance op-
posed by the point will be precisely equal to the force which
causes it to deviate from the tangent. This resistance is
exerted in the direction of the radius of curvature, and its
constant tendency is to remove the material point from the
centre of curvature. Hence, it is called the centrifugal
force ; and the force which constantly urges a body towards
any fixed centre is called a centj'ipetal force.
The centrifugal force evidently corresponds to the quantity
represented by — in Arts. 451 and 452.
7
477. To determine directly the expression for the centri-
fugal force, we replace the infinitely small arc LM by the
chord of the osculatory circle at the point L {Pig. 184).
Then, the versed sine LN will represent the space through
which the point would be drawn in virtue of the action of
the centrifugal force, during the time occupied by the point
in describing the arc LM. From the known property of the
circle, we have
LN : LM : : LM : LE ;
or, by substituting the arc for its equal the chord,
hence,
and by substituting for ds its value vdt, we find
LN^"?^ (238).
2y
CENTRIFUGAL FORCE. 249
A second expression for the value of LN may be obtained in
the following manner. The time required to describe the
arc LM will be represented by dt, since this arc is itself
denoted by ds ; hence, dt will likewise represent the time
in which the material point would be caused to describe a
space equal to LN under the influence af the centrifugal
force alone. Moreover, the centrifugal force acts incessantly,
and during the infinitely short time dt, its intensity may be
considered invariable. If, therefore, we regard this force as
constant, and denote its intensity by/, the circumstances of
motion of the point, under the influence of this force, will be
expressed by the equations
dv „ ds _
di~^' ir''''
and by integration,
But the space LN being that which corresponds to the time
dt, if in the preceding equations we make LN=5^ t will be-
come dt ; we shall thus have
LN = ^c?PX/.
This value of LN being substituted in equation (238) gives,
after reduction,
y
478. If the material point be supposed to have a circular
motion, — as, for example, when a stone is whirled round in
a sling, V will become the radius of the circle described,, and
the expression for the centrifugal force will then be
/=^ (239).
Let h represent the height due to the velocity v ; the follow
ing relation will then subsist (Art. 401),
eliminating v^ between this equation and that which pre-
cedes, we obtain
f^2h
250 DYNAMICS.
from which we conclude, that the centrifugal force is to the
force of gravity J as twice the height due to the velocity is to
the radius of the circle described by the material point.
479, If a semicircle EAF {Pig. 185) be supposed to re-
volve about its diameter EF=2R, the point A, the middle of
arc EAF, will describe a circumference equal to 25rR ; if this
motion be performed uniformly in the time T, with the ve-
locity V, we shall have the relation
vxT=2^R;
and by eliminating v between this equation and (239), we
find
/=^ (240).
In like manner, if/' represent the centrifugal force of a point
which describes uniformly the circumference of a circle
whose radius is R', in the time T', we shall have
T'
and consequently.
From this proportion we immediately conclude, that when the
radii R atid R' are equal, the centrifugal forces will he in the
inverse ratio of the squares of the times of revolution ; and
that when the times are equal, the forces will be directly as
the radii.
480. The effect of the centrifugal force at the equator,
caused by the revolution of the earth upon its axis, can now
be estimated. For, the equatorial radius of the earth being
20920300 feet, we replace R by this value in equation (240),
substituting at the same time the values of sr and T. But
we have, approximatively,
«■=3.1415926, ^='=9.8696046.
The time T is determined from the consideration that the
earth performs a revolution upon its axis in 0.997269 days,
the day being composed of 86400 seconds. Thus we shall have
T=0.997269 x 86400" ==86 164".
CENTRIFUGAL FORCE. 251
Substituting this value and that of R in equation (240), there
results
/=0?Ï112 (242).
481. Having found the value of /, we can determine the
intensity G of the force of gravity which would be observed
at the equator if the earth were immoveable. For, since the
force / is directly opposed to the force G, a portion of the
latter will be destroyed by/; and hence, if g denote the in-
tensity of gravity as determined by observation, we shall
have
^=G-/;
or,
substituting in this expression the value of/ given by equa-
tion (242), and that of ^, which at the equator is 32.0861 ft.,
we find
G-32.086H- Oil 12=32.1973 (243).
To determine the relation between the centrifugal force and
the force of gravity, we divide equation (242) by equation
(243), which gives
/ 0.1112"- 1 , ,„,,^
^=321973^-^2-89 "^"^^^ ^^^^^'
482. The proportion (241) will furnish a solution to the
following problem :
To find the time in which a revolution of the earth should
be performed, in order that the centrifugal force at the equa-
tor may he equal to the force of gravity.
Let T' represent the required time of revolution, and/' the
corresponding centrifugal force ; we shall then have, by the
nature of the problem,
/'=G, and R'=R ;
these values substituted in the proportion (241) reduce it to
* * r£â ' fjvâ •
whence we obtain
252
DYNAMICS.
f
If the fraction -- be now replaced by its value (244), we shall
G
find
T T
v/2S9 17'
Thus, if the earths rotation were seventeen tinier more rapid
tliati it actually is, the centrifugal force at the equator would
be equal to the gravity.
483. To find the diminution of the gravity produced by
the centrifugal force at any other point on the earth's surface,
it will be necessary to determine the effect of the centrifugal
force in the direction of the vertical BZ {Fig. 185) drawn
through the point under consideration.
For this purpose, we will regard the earth as spherical, it
being nearly so : the latitude of the point B being then repre-
sented by the arc AB, it will be measured by the angle
BOA=ZBC=^^.
Denoting by R the radius AO of the earth, and by R' the
radius BD of the parallel of latitude passing through B, we
shall have
R'=R cos OBD ;
or,
R'=R cos ^.
Let the centrifugal force at the point B, which is exerted in
the direction of the radius DB, be represented by the line BC,
and resolve it into the two components B6 and Be. The
force BC will, by Art. 479, be expressed by -^- — , and the
component /' in the direction of the vertical BZ, which is
represented by B6, \vill be given by the relation
/=->p,-Xcos^:
and by substituting in this relation the value of R', we shall
obtain
/--Tj^XCOS^'r/..
4 3f?
The factor -=^ represents the centrifugal force / at the
GRAVITATION. 253
equator ; this equation may therefore be transformed into tlie
proportion
/:/:: 1 : cos- ^]
from which we conchide, that the dimhiations of gravity at
different places on the earth's surface, arising from the action
of the cetitrifu gal force, are proportional to the squares of the
cosines of the latitudes.
484. The latitude of New-York beinor 40° 42' 40", its
cosine will be
0.7580 ;
and by multiplying- the value of/ (242) by the square of this
number, or by
0.5746,
we find
/=o!b639.
If G' represent the value of the force of attraction, or that
which the observed gravity would have in the latitude of
New- York, if the earth were immoveable, the gravity actually
observed being denoted by g', we shall ha-'e, as in Art. 481,
G'=g'+f.
The observed gravity g', in the latitude of New- York, being
32!l598,
we find, by substituting this value and that of/' in the preced-
ing equation,
G'=32?i598+o'.0639=32!2237 (245).
Of the Systeiyi of the World.
485. In discussing the properties of the centre of gravity,
we have already had occasion to consider that remarkable
force exerted by the earth, in virtue of which all bodies are
solicited in directions perpendicular to its surface. The ex-
istence of this force was not entirely unknown to the ancients :
Anaxagoras, and his disciples, Democritus, Plutarch, Epi-
curus, and others, admitted the existence of such a force ; and
similar opinions were entertained by Kepler, Galileo, Huy-
gens, Fermât, Roberval, <fec., in modern times. The celebrated
22
254
DYNAMICS.
Kepler distinctly affirms, in his work De Stella Martis, that
the force of attraction is not confined to bodies situated upon
the surface of the earth, but that it extends to the most distant
stars.
This bold conception remained \on^ unimproved, from the
difficulty of verifying its truth. The eifects of gravity at the
earth's surface were measured by Galileo.
Lord Bacon, suspecting that the intensity of this force must
vary with the distance from the centre of the earth, endeav-
oured to verify the truth of this conjecture by observing the
distances through which bodies would fall, in a given time,
at different elevations above the surface of the earth. But,
however great were these elevations, they proved too small to
render the variations in the intensity of gravity perceptible.
Newton extended his views yet further ; and not satisfied
with the mere conjecture that the intensity of gravity was
subject to variation, he endeavoured to measure the law of its
diminution. He adopted, as the most probable law of diminu-
tion, that of the inverse ratio of the square of the distance ;
such being the law according to which light and other emana-
tions were known to be propagated. To test the truth of this
supposition, he endeavoured to obtain a measure of the inten-
sity of gravity at the distance of the moon, and the only
obstacle to this determination arose from an imperfect know-
ledge of the moon's distance, and of the dimensions of the
earth ; but more exact determinations of these elements having
been supplied by Picard and others, he was enabled to base
his calculations on more accurate data.
486. The first element to be determined in this investi-
gation, is the intensity of gravity at the surface of the earth.
The method of obtaining this quantity by the oscillations of
a pendulum has already been explained in Arts. 474 and
484 : it was thus found, that in the latitude of New- York,
and on the supposition that the earth was immoveable,
G'=32.2237 (246).
This quantity is nearly the same for all places on the surface
of the earth.
To ascertain the diminution which the intensity of gravity
should sustain at the distance of the moon, according to the
GRAVITATION. 255
supposed law of Newton, it will be necessary to know the
distance of the moon from the centre of the earth. This dis-
tance depends on the horizontal parallax of the moon.
487. Let CL and HL {^Pig. 186) represent two lines drawn
from the moon to the two extremities of the terrestrial radius,
the line HL being perpendicular to this radius. The angle
HLC is called the horizontal parallax of the moon, and its
mean value, according to Delambre, is 57'. If therefore, the
radius of the earth be taken as unity, we shall have
CLsinL=CH = l,
and consequently,
CL=-^— =60.314;
sin 57'
this value differs but little from that employed by Newton,
who supposed the mean distance to be 60|.
488. If we denote by y the intensity of gravity at the dis-
tance of the moon, upon the hypothesis that it decreases ac-
cording to the law of the inverse ratio of the square of the
distance, and by s' the space which it would cause a body to
describe in the time t^ we shall have
G':y :: (60.314)3 : 1» ;
whence,
_ ^'
^"" (60.314)»'
Such is the expression for the velocity which should be im-
parted by gravity, at the distance of the moon, in a second
of time, if the hypothesis assumed be correct.
489. By substituting this value for g in the general formula,
s^\gt\
and replacing 5 by ^', we shall obtain the space described in
the time t.
Thus, if we suppose the time to be one minute, or 60", we
shall have for the space s\ which the body would describe in
a minute of time,
K.60)^' (247).
* (60.314)3 ^'^'^'f'
256 DYNAMICS.
490. If we nefflect the decimal fraction, in the denomina-
tor, the equation reduces to
from which we conclude that the space described by a body-
moving from rest, in a minute of time, at the distance of the
moon, should be equal, according to Newton's hypothesis, to
the space passed over in a second of time, at the surface of the
earth.
But if we take account of the decimal fraction, the equation
(247) will give by reduction,
5'=JG'x0.9896;
and by substituting the value of G' (246), we have
5'= ix32!2237x 0.9896;
or, by performing the multiplications indicated,
5' = 15.9443 (248).
Such would be the distance described by the body in a
minute of time, at the distance of the moon^ if the body
were supposed to move from a state of rest.
491. Let us now examine whether this result is confirmed
by experience. For this purpose, let the moon when at its
mean distance be supposed to describe the arc LM {Pig. 187)
in a minute of time : if the lines LQ, and Q,M be drawn
respectively parallel to the sine and versed sine of the arc
LM, we may regard LM as the diagonal of a parallelogram
of which LQ. and LP will be the sides. If the moon were
not solicited by the earth's attraction, it would describe the
tangent LQ,, and if solicited by this attraction solely, it would
describe the line LP in the same time : this line LP will there-
fore serve to determine the inten??ity of the earth's attraction,
and it will evidently be equal to the versed sine of the angle
LCM.
But since the mean radius r of the moon's orbit undergoes
but a very slight variation in a minute of time, this portion
of the orbit may be regarded as the arc of a circle described
with the radius r ; and since the moon, when at its mean dis-
tance, moves with nearly its mean velocity, we shall have, by
GRAVITATION. 257
calling T the time of a sidereal revolution, or the time
required by the moon to return to the same point of the
heavens,
T : 1 minute : : 360° : angle LCM ;
whence,
, T^,, 360°
angle LCM— -.
This time of revolution being known from observation to be
27 days 7 hours 43 minutes, or 39343', we will replace T by
this value, at the same time reducing the degrees to seconds,
to render the division possible ; we thus obtain
angle LCM=~:=32''.94.
492. The question is thus reduced to finding the versed
sine of an arc of 32".94, in a circle described with the mean
radius of the lunar orbit.
To effect this, let the perpendicular CI {Fig. 187) be drawn
to the middle of the chord LM : the right-angled triangles
LMP, LCI, having the common angle L, will be similar, and
will give the proportion
LC:IL::LM:LP;
or,
LC:IL::2IL:LP:
whence,
LP=^ (249).
Let e represent the angle LCI equal to |LCM, and r the mean
radius LC, we shall have
YL=r. sin 6,
and the equation (249) will become
LP=2r.sin2 tf;
or, by substituting the value of the angle 6,
LP=2r.sin''16".47 (250).
If a denote the mean radius of the earth, the mean radius
of the lunar orbit will be expressed by
r=(60.314)a.
R
258
DYNAMICS.
493. But the mean radius of the earth determined by the
measurement of a degree upon its surface, being equal to
20886500 feet,
we shall have, by substitution,
LP=60.314 x2x 20886500 Xsin2(16".47),
and changing
sine whose radius is unity . ^ tabular sine
^ mtO — ; — ,
1 tabular radius
for the purpose of using logarithms, we find
Log 60.314 --------- 1.7804181
Log 2 - 0.3010300
Log 20886500 -------- 7.3198657
Log sin3 (16".47), or2.log sin (16".47) 11.8041388
Corresponding number =16.0492 ft. - 1.2054526
494. It thus appears that the moon falls towards the earth
in a minute of time a distance of 16.0492 ft., corresponding
very nearly with that deduced on the supposition that the
intensity of gravity varies inversely as the square of the dis-
tance from the centre of the earth. The diiference between
the two results amounts only to about 0.15 ft. in tlie space
fallen through by the moon in a minute of time, and will
consequently become nearly insensible in the space described
in one second. Moreover, this slight difference might fairly
have been anticipated, since mean values of the several quan-
tities which enter into the calculation have alone been em-
ployed.
495. The remarkable accordance exhibited by the preced-
ing calculation between the results of theory and experience,
justifies us in concluding that the force of gravity exerts
an influence at the distance of the moon, but that its inten-
sity is less than at the surface of the earth, in the inverse
ratio of the squares of the distances from the centre of the
earth. The truth of this supposition has been uniformly
confirmed by experience ; astronomical tables calculated upon
the hypothesis of Newton assign the positions of the celestial
bodies such as they are determined by direct observation.
GRAVITATION. 259
without presenting a single exception. This hypothesis may
therefore be regarded as fully established by experience.
Those general truths which are designated tJie laws of Kep-
ler, and which have been repeatedly verified by observation^
serve to establish the hypothesis of Newton in the most clear
and decisive manner. These laws may be enunciated as
follows :
1°. The planets describe ellipses, having the centre of the
sun at one of their foci,
2°. The areas of the elliptical sectors described by the
radius vector draw7i from the planet to the centre of the sun
are constantly j)7'oportional to the times of description*
3°. The squai'es of the times of revolution of the several
planets are proportional to the cubes of their mean distances
from the sun.
496. The first of these laws, as will be demonstrated, is a
particular case resulting from the more general law of nature,
which requires that a body subjected to the action of a force
which varies inversely as the square of the distance from a
fixed point, should necessarily describe a conic section.
The second law has already been noticed (Art. 435), and
subsists in general for every body which is constantly at-
tracted towards a fixed point. The question is thus reduced
* In the general course of reasoning which is here applied to the motions of
the planets, these bodies are regarded as mere material points. The propriety
of making this supposition will not fully appear until after we have discussed
the circumstances of motion of a solid body whose several particles are acted
upon by incessant forces. It will then be found that the motion of the centre
of gravity of such a body will be precisely the same as though the mass of the
body were concentrated at its centre of gravity, and the several forces appliffd
directly to ihat point. Thus the case will be reduced to that of the motion of
a material point. It is, however, quite obvious that this hypothesis cannot
differ much from the truth ; for, since the dimensions of the planets are exceed-
ingly minute when compared with their distances from the sun, it follows that
every particle in the planet will be acted upon by a force which is very nearly
equal and parallel to the force exerted upon that particle which coincides with
the centre of gravity of the planet. Thus, the particles, being acted upon by
parallel and equal forces, will have the same motions as though they were uncon-
nected with each other; and the reasoning may be applied to any one of these
particles, the central one, for example.
R2
260 DYNAMICS.
to proving the truth of the first and third of Kepler's laws,
after adopting the hypothesis of Newton.
497. Let the origin of co-ordinates be placed at the centre
of attraction (Fig: 188), which corresponds to the centre of
the sun, for the planetary system, and let R denote the value
of the force of attraction exerted by the sun upon one of the
planets, and r the radius vector drawn to the planet.
The force R coinciding in direction with the radius vector
mA, if we represent by <p the angle mAP, which the radius
vector forms with the axis of a;, the components of the force
R in the directions of the axes will be
X=R cos (p, Y=R sin ç.
But in the right-angled triangle AwP, we have
AP .V . mP y
cos<p— — r=— J sm<z'= — -=^.
mA r niA r
Thus, the components X and Y of the force R will be ex-
pressed by
X=R-, Y=R^:
r r
and since the incessant force is supposed to act in the direc-
tion from m towards A, it will tend to diminish the co-ordi-
nates AP=:r, and Pm=y, of the point m : hence, the compo-
nents of the incessant force should be affected with the
negative sign (Art. 51) ; the two preceding equations will
thus become
■ r r
or, replacing X and Y by their values given in equations
(180), we obtain
^-_R^ ^ R^ r25n
498. For the purpose of integrating these equations, let the
first be multiplied by y, and the second by x : taking the dif-
ference of the products, and multiplying by dt, there will
result
d'x d'y „
^ dt dt '
GRAVITATION. 261
the integral of which is
ydx — xdy _
(252),
dt
the arbitrary constant introduced by integration being denoted
by a.
499. To obtain a second integral, we multiply the first of
equations (251) by 2dx^ and the second by 2c/y, and take their
sum ; we thus obtain,
2dx . d^x-\-2dy • d'^y _ p„ Jxdx-\-ydy\ /okq\
■ 11^ \ r / ^ ''
The second member of this equation containing the three
variables x, y, and r, we eliminate two of them by means of
the relation x^ -\-y^ —r'^ , which gives, by differentiation,
xdx-\-ydy=rdr ;
this value substituted in the second member of equation (253)
reduces it to
2dx . d'x-\-2dy . d'^y__nnr.,
df' "~ ''
or, since dt is regarded as constant,
Integrating, and denoting by b the arbitrary constant, we ob-
tain
^^l^^=b-2fRdr (254).
500. The quantity Rdr is affected with the sign of integra-
tion, the intensity of the force R being supposed a function of
the distance r ; the nature of this function will remain arbi-
trary, so long as we do not adopt a particular hypothesis.
501. This equation still containing three variables, we re-
duce the number to two {<p and r), by introducing the values
ofx and y, expressed in functions of r, and the angle ç included
between the radius vector and the axis of x ; these values
are given by the formulas,
x=r . cos <p, y=r . sin ^ (255).
By differentiating, we have,
dx = —r sin (pd(p-{-cos çdr ) /qka\ .
. dy= r cos ^c?^+sin <pdr S '
262 DYNAMICS.
and the values of x, y, dx, and dy given by equations (255)
and (256), being substituted in equation (252), transform it
into
-^"|=« (257).
The sum of the squares of equations (256) gives, after re-
duction,
dx''-\-dy''=r^dç>''-\-dr^ (258)
and by substituting this value in equation (254), we obtain,
'l^^^Èl=h-2f^dr (259).
502. To determine the equation of the curve described by
the moveable point, we eliminate dt between equations (257)
and (259) : the first of these gives
- r-dçi
dt = ;
a
this value, introduced into the second, transforms it into
a^r^dç)'^ +a^dr
r*d<p^
whence we deduce,
'^^r^{br^—a= -2r-'fRdr) ^ ^*
This equation being integrated, and the values of the con-
stants being determined, we shall have a relation between the
radius vector r and the angle ^.
503. To determine the constants a and b, we will resume
the integrals,
ydx—xdy r^dç>- -^-dr"^ , o/»t)j /oai\
—dt"' S3— =''-2-^^'- (^•'l'-
The integral of the first of these equations is, by Art. 435,
2 . sector LAm=a^ (262) ;
consequently, by making ^=1, we shall find that a is equal to
twice the sector described in a unit of time.
The same result may be obtained from the equation
r'f^^a (263);
for d/p being the infinitely small arc described in the time dt^
= 6— 2/Rrfr;
GRAVITATION. 263
by a point on the radius vector whose distance from the centre
of attraction is equal to unity, rd(p will be the arc described in
the same time with the radius r ; hence \r . rd<p, or —~-, will
represent the infinitely small sector described by the radius
vector in the time dt ; but since the areas described are propor-
tional to the times of description (Art. 435), we can find the
area described in the time 1, by the proportion
—^ : dt :: area described in time unity : 1 ;
lit
whence,
area described in time unity = — -^ ;
' Idt '
r"^ d0
and consequently, —r-) or its equal a, will be double the area
described by the radius vector in the unit of time.
It may be remarked that the change in the sign of the first
member of equation (257) which converts it into (263), is
merely equivalent to a change in the position of the fixed line
from which the areas described are reckoned : in the first
case they are reckoned from the axis of y, and in the second
from the axis of x.
From the equation (263) we deduce
^==^ (264.)
dt r^ ^ '
The quantity — expresses the angular velocity of the body,
or the velocity of that point on the radius vector which is at
the distance unity from the centre of attraction ; and it ap-
pears by the preceding relation, that the angular velocity
varies in the inverse ratio of the square of the radius vector.
504. From the first of equations (261), we may infer that
the quantity a is independent of the law according to which
the attractive force is supposed to vary ; but the quantity b,
which appears in the second equation, will evidently depend
on the attractive force, which likewise appears in the same
equation. It will therefore be necessary to adopt some
hypothesis respecting the law of this force, such, for example,
264 DYNAMICS.
as the law of Newton, which supposes that different bodies
attract each other in the direct ratio of their masses, and the
inverse ratio of the squares of their distances.
Let the force exerted by the unit of mass, at the distance
>t, be denoted by 1 ; the force exerted by the sun upon a
body placed at the distance k will then be expressed by the
mass M of the sun, or, in other words, by the number of units
which its mass contains : but the mass of a planet attracted
by the sun being denoted by m, this planet will exert an
attraction upon the sun, which will, for a similar reason, be
expressed by its mass m : moreover, since the two forces
M and m tend to cause the approach of the two bodies, their
effect upon the relative motion of the bodies will be the same
as if the force M + w were concentrated in the sun, and acted
on the planet at the distance k. When this distance varies
and becomes equal to r, the intensity of the force will like-
wise vary. Let R denote its intensity at the distance r ; the
assumed hypothesis will give the proportion
M+7;ï:R::l:i;
le- r^
whence,
R=^!1M+^) ^2g5)_
Such is the value of the attractive force which, acting at the
distance r, will cause the bodies to approach each other.
505. The value thus determined corresponds to that of the
incessant force which we have hitherto represented by R :
we therefore have
/RA.=yiii(M&.
putting, for brevity,
/v2(M+m)=M' (266),
the preceding equation will be reduced to
'Wdr
fRdr=fl
■ ■ (267) ;
but since the quantities M and m and the distance k remain
invariable, the quantity M' will be constant ; the equation
(267) may therefore be readily integrated, and will give
GRAVITATION. 265
—M'
replacing /Rc/r by this value, and 6— 2c by b', the equations
(259) and (260) will become
r^d^'+dr' 2M'
V ^^^^+7^ ^^^^^'
, adr
^^r^{b'r2-a'+2Wr) ^ ^'
506. To determine the value of the constant b', or its equal
6— 2c, we observe that the equations (258) and (268) give, by
comparison, *
dx^-^-dy^ j'^dp^ ^dr^ __ 2:vr
dt^ dt^ ^ + r '
and since ^{dx"" ■\-dy'') is equal to ds, the element of the
curve, it appears that the quantity — i- — - — is equal to
(ds\ ^
— j , or equal to the square of the velocity estimated in
the direction of the tangent to th^^ curve ; thus, denoting this
velocity by v, the equation (268) will become
2M'
,;2=:6' + f^ (270).
If V represent the velocity at a given instant, and x the cor-
responding value of the radius vector, the equation (270)
will contain but a single unknown quantity b\ whose value
will result
507. The constant a may also be determined in functions
, by replacing
of the initial velocity ; for, by replacing -— in the formula
dV
)
by its value — deduced from equation (264), we shall obtain
v2=^+^ (271).
dt= ^r= "^ ^
23
266 DYNAMICS.
The quantity dr represents the infinitely small difference ml
{Fig. 189) between two consecutive radii Am and Kn] and
by regarding the triangle mnl as rectilinear, and right-angled
at /j we shall have
7nl—mn . cos mnl,
or,
dr=ds . cos wmZ;
(Is
substituting this value of dr in (271), and changing — into v,
we shall find
v^=v'^ cos'^ nml-i — .
But if * denote the value of the angle nml, when v and r
are transformed into V and a, we shall have the relation
Y''=Y^ .cos' a-\-~:
whence,
«2=^272(1— cos^ «)=A2V2sin"'«;
and consequently,
a=A. V.sin«.
508. Having determined the constants which enter into
equation (269), we proceed to integrate it, for the purpose of
discovering the nature of the trajectory described by the ma-
terial point.
To facilitate the integration, make r= -, and the equation
(269) will then become
■~~^[b'-{a''z=' -2M'z)] '
or,
adz
d^=
making
az =p, and 6'-t— — =Aa,
a a'
the preceding equation will be reduced to
GRAVITATION. 267
, _ —dp
and by integrating, we find
^+ constant = arc |COS=^).
Replacing p and A by their values, suppressing the common
factor a, and denoting by 4- the arbitrary constant, we obtain
/ a^z—W \
whence.
=cos(<?) + V')5
and by restoring the value of z in terms of r, we finally
obtain
a=»— M'r=r^(a«6'+M'2) .cos (^+^^) (272).
509. The arbitrary constant ^^ serves merely to change the
direction of the axis with which the radius vector forms the
variable angle : if, for example, the angle CAm or <(> {Fig.
190), formed by the radius vector with the primitive axis AC,
be supposed successively equal to 1°, 2°, 3", &c. and if the
variable angle be reckoned from the axis AB, which forms
with the axis AC an angle CAB=^^, the angle included be-
tween the radius vector km and the axis AB, will be succes-
sively equal to
1°+^^ 2°+>/., 3°+^/.,&c.;
or, in general, to
510. The angle <p-\-i^ will disappear from equation (272),
when the polar co-ordinates are transformed into rectangular
co-ordinates,by means of the formulas
r- =xs +y% x==r cos(^-l-'<^), y—rsin(<p+4) (273) ;
for the first two of these formulas reduce the equation
(272) to
a^— MV(a^'+2/')=^\/(«'6'+M'2) ;
which gives, by transposition,
MV(^"+y')=a'-^\/(«'*'+M") (274):
squaring and reducing, we find
268 DYNAMICS.
M'2y= —b'a\v' =a^ -2a^-a;^{a'b' + 'M.'-) (275).
This equation appertains to a conic section, or curve of the
second degree : it will be the equation of an ellipse or hyper-
bola, according as b', upon which the sign of the second term
depends, is negative or positive ; for, in the first case, the
terms containing the squares of the co-ordinates will have
similar signs, whilst in the second, tliey will be affected with
contrary signs : when b' becomes equal to zero, the term con-
taining .? - will disappear, and the equation will then apper-
tain to a parabola.
511. If we resolve equation (275) with reference toy, there
will result
y=±:^^y[a'-\-b'x'-2x^{a'b' + W)];
which proves that every rectangular ordinate is equally
divided by the axis of x, and consequently that this axis must
necessarily be the greater or lesser axis of the curve : but by
introducing into equation (274) the value of the radius vector
given by the first of equations (273), we shall obtain
M'r = «2 _;r ^ (a= 6' + M'^ ) ;
hence it appears that the radius vector is constantly expressed
in rational functions of the absciss x, and that the origin
therefore corresponds to the focus. Thus the co-ordinate
axis of X will coincide with the greater axis of the curve.
512. The second law of Kepler is thus demonstrated to be
a consequence of the hypothesis of Newton, and admits of a
generalization wholly unknown to its discoverer ; lie was in-
duced, judging by analogy, to assign elliptical orbits to the
planets, whereas it appears from the preceding demonstration,
that they might have described either hyperbolas ox parabolas.
If amongst the comets hitherto observed we have found no
examples of a hyperbolic motion, it results from the fact
that the chance of a body's describing a curve which shall
be sensibly hyperbolic is found to be extremely small. " I
have found," says Laplace, " that the chances are at least six
thousand to one that a comet which comes within the sphere
of the sun's action will describe an extremely elongated
ellipse, or a hyperbola, which, by the magnitude of its trans-
GRAVITATION. 269
verse axis, will be sensibly confounded with a parabola, in
that portion of its orbit which can be observed ; it is not sur-
prising, therefore, that the hyperbolic motion has not yet
been observed."
513. If in equation (275), we make x=0, and y—\j*, we
shall obtain for the ordinate passing through the focus, or
the semi-parameter,
1 =^
514. The equation (275) admits of simplification, by
making
^{a^h'-\-W')=n (276),
and transporting the origin to the centre of the curve : for this
purpose we make x=x'-\-»^ and dispose of the arbitrary quan-
tity ct by the condition that the coefficient of the first power
of x' shall vanish. Making these substitutions in equation
(275), and dividing by a^ , we find
M'2 ^ — 6'«= i
a^^ \x' -f 2w<» V =0 (277).
Putting the coefficient of x' equal to zero, we have
n
6''
this value being introduced into the last term of equation
(277) reduces it to
y-" ■
But the equation (276) gives
substituting this value for the last term of equation (277),
and suppressing the second term, which by hypothesis is
equal to zero, we shall obtain
or by clearing the denominators,
h'^^y^—b'^a^x''^^a^W^Q (278).
270 DYNAMICS.
In this equation the origin of co-ordinates is at the centre
of the curve ; hence, if we make ^=0, and deduce the
corresponding value of x', we shall have
M'
semi-axis major =-^ (279) ;
and by making a similar supposition with respect to x', we
find
semi-axis mmor= w^ — tt-
This value becomes imaginary when b' is positive, agreeing
with the -esult in Art. 510, since the curve described is then
a hyperbola ; but the value is real when b' is negative, the
curve then being an ellipse. In this case^ if we replace b' by
— b', we shall have
semi-axis minor= — — (280).
515. This result corresponds with that which would have
been obtained from the consideration that the minor axis is a
mean proportional between the major axis and the parameter,
the values of which have been already obtained.
516. Having determined the principal elements of the
curve described, it will now be easy to establish the third of
Kepler's laws. Let tt denote the number 3.1416 ; then, the
area of an ellipse whose semi-axes are represented by A and
B will be expressed by ^AB ; and if A and B be replaced by
their values determined in equations (279) and (280), we
shall find
waM'
area of the ellipse described by the planet=7; — jj • • • • (281) ;
or,
area of the ellipse described by the planet =-^^( ~ ) ^*
But it has already been shown that if i represent the time
required by a planet to describe the sector JjAm (Fig. 188),
the equation (262) will give
2 sector LAm
a
When t becomes the time of an entire revolution, which we
GRAVITATION. 271
will represent by T, the sector hAm will become the area of
the ellipse, and we shall then have
2^ /M'\l.
M'
and since — represents the semi-axis major, we shall have,
by representing its value by D,
or, replacing M' by its value (266), we obtain
''-i^^) (^^^>-
In like manner, for a second planet m', which performs its
revolution in the time T', in an ellipse whose semi-axis major
is denoted by D', we shall have, since the mass of the sun
remains invariable,
3
T/=-~?^l—^ (283) :
but the masses of the planets being extremely small when
compared with the mass of the sun, we may neglect the
quantities 7Ji and m' in comparison with M ; and the equa-
tions (282) and (283), being then compared, will give the
proportion
T : T' : : D^ : D'% or T^ : T'^ : : D^ : D'^ ;
the squares of the times of revolution will therefore be pro-
portional to the cubes of the greater axes of the orbits de-
scribed, or to the cubes of the mean distances of the planets
from the sun.
517. The inverse problem may also be resolved, and the
law of gravitation deduced, from the elliptical motions of the
planets. For this purpose, we must adopt the hypothesis
that the equation (260) refers to an ellipse : but the polar
equation of the ellipse being of the form Cr cos ç>=B^ — Ar,
its differential will give
*^^r^[(C2-A^)r»-B* +2AB'r]'
272 DYNAMICS.
The condition of identity between this equation and equation
(260) requires that we should have
— 7/Rrfr= AB2 =a constant, or -fRdr ^^""^tant .
differentiating, and suppressing dr, there remains
Tj _ constant
which proves that the force varies in the inverse ratio of the
square of the distance.
Of the Motions of Projectiles.
518. If an impulse be communicated to a material point in
a direction oblique to the surface of the earth, the point being
at the same time solicited by the force of gravity, it will
describe a trajectory, the nature of which it is proposed to
investigate. To determine the circumstances of this motion,
we will denote by Aa-, Ay, and Az the three co-ordinate axes ,
the axis Ajz being supposed vertical. The force of gravity
will then tend to diminish the co-ordinates z which are
reckoned positive upward, and if its intensity be supposed
constant, we shall have
X=0, Y=0, Z=--.
These values being substituted in the general equations (180)
reduce them to
d^x_f. ^''y_n d^z_^
'dF ' JF ' dF~~^''
the first two of these equations being multiplied by dt, and
integrated, give
dx dy ,
dt ' dt '
the constants a and b represent the velocities of the material
point in the directions of the axes of x and y respectively.
These velocities distinguish the motion under consideration
from that which takes place when the point is projected ver-
tically, their values in the latter case becoming equal to zero.
If tiie preceding equations be multiplied h^ dt, and again
integrated, we shall obtain
PROJECTILES IN VACUO. 273
x=at-\-a\ y=ht + h'\
and eliminating t between these relations, there results
hx . ab' — a'h
a a
This equation appertains to a right line EC {Fig- 191), situ-
ated in the plane of.r, y, and the trajectory ELC will therefore
be contained in a vertical plane.
519. Since the trajectory described is confined to a vertical
plane, it will only be necessary to consider the two co-ordi-
nate axes of X and y, the former being supposed horizontal
and the latter vertical ; we therefore employ the two equations
'dF ' dr-~ '^'
Multiplying by dt^ and integrating, we find
%=a, 'i^-St^r. (284).
If we multiply again by dt^ and integrate, we shall obtain
.r=a^ + a', y— — \gt^ +ht-\-h' (285).
To determine the constants, we suppose the time to be
reckoned from the instant at which the material point leaves
the origin of co-ordinates ; whence,
.r=0, y=0, and ^=0;
this supposition gives
a'=0, 6'=0;
and the equations (285) are thus reduced to
x—at^ y — — \gt'^-\-ht.
Eliminating t between these two equations, we find
y--a^-'^4' ('«")•
The equations (284) indicate that the constants a and h
express the values of the horizontal and vertical compo-
nents of the velocity at the instant from which the time is
reckoned, or when ^=0. If, therefore, V denote the initial
velocity, and a the angle formed by the direction of the initial
impulse with the axis of x^ the componeYits of this velocity
will be
S
274 DYNAMICS.
V . COS « parallel to the eixis of x,
V . sin « parallel to the axis of y ;
whence,
a=V cos «, 6=V sin «.
These values reduce equation (286) to
y=^ tang .-^g^-^ (287).
520. This equation appertains to a parabola, having its
origin at the point A {Ptg. 192), the vertex being situated at
a point C, above AB, and the curve extending indefinitely
below AB ; for. the equation (287) being of the form
i/=mx — iix'^,
by making y =0, we shall obtain for the abscisses of the points
at which the curve intersects the axis of x,
x=0, and x= — .
n
7/1
But every value of :r less than — will give a positive value
for 7/, whilst every value greater than — will give y a nega-
tive value. For, if we multiply by nx both members of the
inequality
m
x<—,
n
we shall obtain nx^^mx, the condition which is obviously
necessary, that the ordinate y may be positive. In like man-
771
ner, it may be shown that when .t>— , the value of y will
n
become negative.
521. If /i denote the height from which a body must fall to
acquire the initial velocity V, we shall have (Art. 401)
V=^(2^A) (288):
by means of this value, the equation (287) is reduced to
x^
y=x . tang » — — . — (289).
^ ^ 4/i cos^' « ^ '
522. The distance from the origin A to the point B, at
which the curve intersects the axis of a-, is called the range.
PROJECTILES IN VACUO. 275
To determine its value, we make y=0, and the corresponding
value of .r, which is not zero, will express the range. Thus
making y=0, in (289), we have
3;=0, and a:=4A. tang «. cos^rtj
the second value of x gives, by reduction,
x=-4Ji . sin a. . cos « ;
and consequently,
range =4/i . sin a . cos u, (290) ;
or, replacing 2 sin « . cos » by its equal sin 2<«, we have
range =2/i . sin 2* (291).
This equation may be employed in the construction of tables
which shall express the ranges corresponding to different
velocities, and different angles of projection.
523. The greatest positive ordinate will express the maxi-
mum elevation of the moveable point above the axis of x.
To determine its value, we make -^=0; or,
ax
-^ =tang «■—— =0 ;
dx ^ 2h cos=^ a
from which we deduce
x=2h . cos2 « . tang a,
or,
x=2h . cos « . sin «6 ;
and consequently, the absciss of the highest point of the tra-
jectory will be equal to one-half the range.
Replacing x by 2/t . cos « . sin « in equation (289), we find
for the maximum elevation of the moveable point,
y=/t . sin^ ct.
524. The projectik may be impelled in two different direc-
tions, so as to produce the same range. For, let «' represent
an angle equal to the complement of « ; the equation (290)
will give the value of the range,
4A . sin « . cos «=4/i . sin « . sin « .
But if the projectile be thrown in a direction forming an
angle *' with the axis of x, the range will be expressed by
Ah . sin «'. cos x=Ah . sin «' . sin «.
S2
276 DYNAMICS.
The identity of these expressions for the ranges corresponding
to the angles » and <*', evidently proves that the ranges will be
equal when the two angles of projection are complements of
each other.
525. To determine the angle of projection which corres-
ponds to the greatest range, we remark that the range is in
general expressed by 2h sin 2«, and that this expression will
become a maximum when the angle 2« is equal to 90° ;
hence it follows that a projectile in vacuo will have the
greatest range upon a horizontal plane when the angle of
projection is equal to 45°.
The supposition of 2<«=90° gives sin 2»=! ; consequently,
the expression for the range then becomes equal to 2h ; or
the range corresponding to the angle of 45° is equal to twice
the height due to the velocity of projection.
Let this range be denoted by P ; we shall have
h = \V (292).
To determine the value of the coefficient h, the projectile
may be thrown in a direction forming an angle of 45° with
the horizontal plane, and the corresponding range may then
be measured. If this range be represented by P, the value
of h will immediately result from equation (292). In fire-
arms, the coefficient h serves £is a measure of the force of the
powder, since the extent of the range evidently depends on
the intensity of the force of projection.
526. The quantity h having been determined by taking
the mean result of a large number of experiments, we substi-
tute its value in equation (289), which will thus become
'y=x tanga-
2P cos- «
If we represent by P' the range corresponding to an angle «',
the equation (291) will give
F=2^sin2«' (293);
or, .replacing h by its value iP (292), we find
P'=Psin2«'.
This relation will determine the range P' corresponding to
the angle «', when the value of the maximum range has
been previously ascertained ; and, in general, we can calcu-
PROJECTILES IN VACUO. 277
late the range P' which corresponds to an angle «', from, a
knowledge of the range P" given by any other angle «" ; for,
since
P'=P sin 2«', P"=P sin 2«",
we obtain, by division,
F _ sin 2cc' ^
P" sin 2<*" '
if, therefore, the range P" corresponding to the angle »" be
determined by measurement, the value of P' corresponding
to X will result immediately from the preceding equation.
527. The value of h (292), being substituted in equation
(288), will give, for the value of the initial velocity,
V=v/(%) = v/(32ift.xP).
If, for example, the range corresponding to an angle of 45"^
were equal to 1000 feet, we should find
V=:^(1000 ft.x32ift.) = 179.3ft., nearly.
528. If, on the contrary, the initial velocity and angle of
projection were given, we might determine the range : for
example, let the initial velocity be supposed equal to 200 feet
per second, and the angle of projection 15" ; we first determine
h from the following formula, deduced from (288),
V2 (200 ft.')*
A= — = ^-— — ^ = 6^1 7 ft •
^ 2g 64ift. ^■^■^"•'
and the range P' will then become, (293),
F=2x621.7ft.xsin30°=261.7ft.
529. The problem may also be presented under the follow-
ing form : — Having given the initial velocity and the co-ordi-
nates a:'=AB, and y'=BC, of a point C {Fig. 193), it is re-
quired to determine the angle of projection such that the
trajectory may pass through a given point C. The equation
V= v^(2g-A) will determine the value of h ; and since the co-
ordinates x' and y' should satisfy the equation (287), we shall
have by substituting i/ and y' for x and y,
y'=x' tang«----^^l— (294).
^ ^ 4A . cos* cc
In this equation the quantity « is alone undetermined : mak-
ing tang ««=z, we have
278 DYNAMICS.
1 1
COS «=-
sec* ^(1+taiig^ct) ^{i^z')'
and by substituting these values in equation (294) we find
j/=x'.z-^{l+z=) (295).
This equation being resolved with reference to z, will give
two values which determine the two angles of projection
corresponding to the directions in which the projectile should
be thrown in order that it may strike the point C ; we select
the greater of these two angles when we wish to crush the
object upon which the projectile falls, £is the vertical velocity
at the point C will then be the greatest.
It may occur, that instead of the line CB, we have given
the angle CAB subtended by the object CB. Let this angle
be denoted by ^ ; we shall have
CB=:r' tang^=y';
this value of y', being introduced into equation (295), trans-
forms it into
• x'
ta.ng<p=z——{l-{-z^)]
from which we deduce
-'^±v/(^^-'-^f^-0-
Of the Motions of Projectiles in a Resisting Medium.
530. The theory of projectiles in vacuo, which has been
examined in the preceding paragraphs, afibrds results which
diifer greatly from those obtained by direct experiments per-
formed in the atmosphere : these discrepances are very con-
siderable when the velocity of projection is great, and are to
be attributed to the resistance opposed by the atmosphere to
the motion of a body. If this resistance, represented by R, be
supposed, as in Art. 412, to vary in the duplicate ratio of the
velocity, we shall have
The resistance R at each point of the trajectory will be
exerted in the direction of the element of the curve, but in an
PROJECTILES IN A RESISTING MEDIUM. 279
opposite direction to that of the motion ; and the force R
will form with the axes of co-ordinates the same angles as the
element ds. Thus, denoting by «, |3, and y the angles included
between the tangent to the curve at any point and the co-
ordinate axes, the components of R will be expressed by
R cos », R cos /3, R cos y.
To obtain expressions for these cosines, let mm' {Pig- 194)
represent an element ds of the curve : the projection of this
element on the axis of z will be equal to m'n. But the tri-
angle 7}i'mn gives the proportion
1 : cos mm'n : : mm/ : m'n ;
or,
1 : cos y :: ds : dz]
hence,
dz
cos y^^-j- '}
as
and the component of R in the direction of the axis of z, will
therefore be expressed by
R—
ds'
We attribute the negative sign to this component, because the
tendency of the force R, while the projectile is moving from
9)1 to m', will be to diminish the co-ordinate z. Fora similar
reason the other components of the resistance R should be
affected with the negative sign.
531. An analogous course of reasoning will give
— R— - for the component of R in the direction of the axis ofx,
ds
— R— for the component in the direction of y.
ds
Thus, the equations expressing the circumstances of the
motion will be
d^x -odx
'di^~~~ ds'
^ = _R^
dt^ ds*
d^z_ j^dz
'dF — ^d^~^'
280
DYNAMICS.
From the first two we obtain, by division,
d^y _dy ^
or,
S='i^ <^^«)^
and by integration,
log dy~\og dx-\-\og a=log adx.
Passing from logarithms to numbers, we find
dy^adx ;
and by a second integration,
y=ax + b ;
hence we conclude, that the projection of the trajectory on
the plane of a:-, y is a right line, and therefore that the trajec-
tory is contained in a vertical plane.
532. If we resume the consideration of the problem with
this restriction, that the curve shall be confined to a vertical
plane, it will only be necessary to employ the two equations
d^x_ Tydx d^y _ -^dy
dF ds' ~dt^ ds~~^'
It has already been remarked, that the vertical component of
the resistance R-r^ should be affected with the negative sign,
since this resistance tends to diminish the co-ordinate ; but
this tendency will only exist whilst the projectile is describing
the ascending branch of the trajectory. If, on the contrary,
the projectile be supposed at a point M" in the descending
branch {Pig- 194), the resistance, being exerted in the direc-
tion from M" to M', would tend to increase the co-ordinate y.
It might, therefore, appear that the component R— should
change its sign ; but since dy becomes negative in the second
branch of the curve, the vertical component will still be ex-
pressed by — R J^,
If the quantity R in the preceding equations be replaced
by its value mv"^ , they will become
PROJECTILES IN A RESISTING MEDIUM. 281
d^x „dx d^y Ay
df^ ds' dt"" ds ^
The quantity v^ may be eliminated by means of the equation
ds^
dp'
and we shall have
IF- "^W^Ts ^'^^^^'
^=_^^xf^-^ (298).
dp dt"" ds ^ ^ '
533. The first of these equations being multiplied by dU
gives
d^x J dx ds
—-—=—mds .—-.-—
dt ds dt
or.
d^x J dx
dt dt '
from this equation, we deduce
d^x
dt
= —mds\
dx
W
and by integration,
log — = — ms-\-C.
534. Let A represent the number whose logarithm is equal
to C, and e the base of tl^e Naperian system ; we shall have
C=log A, log e=l ;
the preceding equation may therefore be transformed into
dor
log -z- = — ms log e+log A,
or^
log^ r^log e-™+log A=log Ae-"-î
dt
passing from logarithms to numbers, we have
$=Ae-^ (299).
dt
282 DYNAMICS.
535. To determine the constant A, let V represent the
initial velocity, and « the angle formed by the direction of the
initial impulse with the axis of w. The component of V in
the direction of this axis will be expressed by V cos «. But
when 5=0, -^ will express the comp 'Uent of the initial velocity
along the axis of a- : hence the preceding equation will, on this
supposition, be reduced to
V cos«=Ae°=A.
This value substituted in (299) converts it into
^= V cos u . e-"" (300).
at
536, Since this equation contains three variables, we must
obtain a second relation between them, in order to render the
integration possible. For this purpose, the equations (297)
and (298) may be written under the form
dx_ ~ 'dP dy_ ~\dP^)
ds ds^^ lis dT^ '
dV" dV
the quantity ds may be eliminated immediately by division ;
and we thus obtain
d^y
dy_dt^ ^
dx d^x
w
From this equation we deduce
_dy d^x _d^y ,
^~di''dF~dF^
or, by reduction,
Jyd^.x-dxd^y 3Q^^
dx
The second member being divided by —dx becomes the ex-
act differential of -^ ; and the equation (301) may there-
at:
fore be written
PROJECTILES IN A RESISTING MEDIUM. 283
(lu
If, for greater simplicity, we make j~ =zp, there will result
gdP^-dx.dp (302);
and eliminating dt, by means of equation (300), we find
^=-V2 cos^^ * . e-^™ . ^ (303).
537. This equation still contains three variables ; but one
of them may be readily eliminated by means of the relation
ds=^{dx^ -\-dy-) ; in which, replacing dy by its value pdx,
we obtain
ds^dx^il+j^"") (304);
and consequently, by eliminating dx between this equation
and (303), there will result
dp^(^+p')=^S^ (305).
Litegrating, we have
i/V(l+7^^)+|log[i^ + x/(l+P^)]=C-^^J^^...(306);
and by making C=iB, and suppressing the common divisor
2, we obtain
2V(l+2^^)+log b+^/a+P=')]=B-^J^^^^ (307).
To determine the value of the constant B, we observe that
— expresses the trigonometrical tangent of the angle formed
dx
by the element of the curve with the axis of x. At the point
A, the origin of the motion, this angle is denoted by * , the
quantity t being at the same time equal to zero ; we shall
therefore have
:r=0, y=0, s=0, jo=tang«.
These values of s and jt being substituted in the preceding
equation give
B=tang »v^(14-tang2*)
+log[tang.+ v/(l+tanr*)]+^^^VWi: '
the value of the constant B in equation (307) may therefore
be regarded as known.
284 DYNAMICS.
538. If we eliminate e'^™ between the equations (303) and
(307), we shall obtain
dx= ^ = (308).
The two members of this equation being multiplied by the
corresponding members of the equation
dy
there will result
dy= = A ^.^^ (309).
m[py/l-]-p'' +log (p + \/l+p^)— B]
539. To determine the time t, we substitute in the equation
dt^ = -±L^,
g
the value of dx, given by equation (308), and we thus obtain
dt'' = — ^:^ (310) ;
m^[jo^/l+p2+log(Jo + ^/l4-232 )— B]
or, by changing the signs of the numerator and denomi-
nator,
dt^z=
dp'-
mg[—pVl+p^—\og{p + V\+p'')-\-B]
In extracting the square root of the two members of this
equation, the second might be affected with the double sign,
but in the present instance we shall attribute to it the nega-
tive sign. For, since every equation between two variables
t and p may be regarded as that of a curve, of which t is the
absciss, and j» the ordinate, Up increases whilst t diminishes,
the elements dt and dp will necessarily be affected with con-
trary signs. But, in the present case, it is obvious that whilst
t augments, the quantity p, which expresses the trigono-
metrical tangent of the angle formed by the element of the
curve with the axis of x^ constantly diminishes in the ascend-
ing branch of the trajectory, which is the one at present under
consideration; hence, we shall have
dt= ^^ __ (311).
y/mg[-2}y/l-\-p^—\og{p+\^l-\-p'')-{-B]
PROJECTILES IN A RESISTING MEDIUM. 2S5
540. The expression for the velocity can now be obtained
in functions of jo ; for, the velocity resulting from the equation
ds ^(dx^ -\-dy-) dx .^ , .
we obtain, after replacing dx and dt by their respective
values,
V:
|xwi+i>^)
\/-/>v/(l+P=')-logb + v^(l+P')]+B
541. We can also express the arc s in functions oip ; for
the equation (307) gives
Taking the logarithms, and reducing, we obtain
'mV-cos^ <*
1 {'
[B— _p%/l+p2— log (p + v/l+jo*)]
2711
542. To obtain the equation of the trajectory, it would be
necessary to integrate equations (308) and (309) : these inte-
grations cannot be effected except by the aid of series. Never-
theless, by employing equations (308) and (309), the curve
may be constructed approximatively by points.
For this purpose, we will write those equations under the
form,
dx=<pj) .dp (312),
dy=^p .dp (313) ;
in which <pp and ■<l'p represent certain known functions of».
The first of these equations gives
dx
dx
the quantity — represents the tangent of the angle included
between the axis of abscisses and the element of a curve
whose co-ordinates are denoted by p and x respectively. We
will first construct this curve, which will serve to determine
points in the trajectory. It is distinguished by the name of
the auxiliary curve.
286 DYNAMICS.
Having drawn two rectangular axes A^ and Ax {Pig- 195),
lay off from A to B a distance AB=tang« ; the point B will
appertain to the auxiliary curve, since the ordinate x=0
corresponds to the absciss p=ta.nga.'
If the line AB be divided into equal parts BB', B'B", âcc.^
each of these parts being represented by dp, it will be easy to
construct approximatively the points M, M', M", <fec. of the
auxiliary curve, corresponding to the points B, B', B", <fcc.
For, if we suppose the points B, B', B", &.c. to be exceedingly
near to each other, we may regard the arcs M'B, M"M',
M"'M", &c. of the curve as coinciding with the tangents
drawn to the points M', M", M'", &c. The ordinates M'B',
M"B", M"'B"', &c. may then be calculated ; for, the trigono-
metrical tangent of the angle formed by the element of the
dv
curve with the axis of p, being represented in general by J~ ,
its value will always be given by means of equation (312)^
whenever we assume a value for p. Thus, if we wish to
determine the trigonometrical tangent of the angle WBp in-
cluded between the tangent at M', and the axis of ab-
scisses, since the absciss of the point M' is AB'=:AB — BB'=
tang cc — dp, it will be necessary to change p into tang d—dp,
dx
in the value <pp=--, given by equation (312) : we thus
obtain
tang WBp=Ç){iQXig a.— dp) ;
whence,
tang M'BB'=— ^(tang ^-dp).
The ordinate M'B' being expressed by BB' X tang M'BB', we
shall have
M'B'=BB'x tang M'BB';
or,
WB!=dp X — ^(tang »—dp).
Thus, the point M', of the auxihary curve BC, may be con-
structed by means of the co-ordinates
AB'=tang a. — dp,
and
BW=dp X — ^(tang a— dp).
PROJECTILES IN A RESISTING MEDIUM. 287
To determine a third point M", we make AB"=tang a.—2dp ;
and by the same course of reasoning prove that the trigo-
nometrical tangent of the angle M"M'0 is expressed by
— ^(tang *— 2rfp), and consequently,
WO=dp X -^(tang ^—2dp)
substituting this value and that of M'B' in the equation
M"B"=M'B' + M"0,
given by an inspection of the figure, we find
]\|"B"= —dp . ^(tang it— dp) —dp . <Z)(tang a—2dp).
To calculate the ordinate M"'B"' which corresponds to the
absciss AB"'=tang *— Srfja, it will only be necessary to add to
the value of M"B" that of the portion M"'0', which, by an
investigation similar to the preceding, may be proved equal
to — dp . (f(tangct— Mp) : thus we have
B"'M"'= —dp . ^(tang <t—dp)—dp . ^(tang a.—2dp)
— dp .<p{t3Lng cc—3dp).
In this manner we may determine a series of points which
will appertain to the auxiliary curve, the co-ordinates of
which are .v . nd p. Connecting these points by right lines,
we form a polygon BM'M"M"', &c., which will coincide more
nearly with the curve, in proportion as dp has a smaller
value assigned to it.
By performing similar operations with reference to the
equation
di/=4^p . dp,
we may construct a second auxiliary curve BD, the co-ordi-
nates of which will represent the quantities p and y. The
co-ordinates mb and /'6, which in these two curves correspond to
the same value of p, will represent the two co-ordinates of a
point in the trajectory ; so that by taking the co-ordinates
B'M', B"M", B"'M"', &c. of the first curve as the abscisses
of the trajectory, its ordinales will be represented by the lines
B'L', B"L", B"'L"', &c.
288 DYNAMICS.
Of the different Methods of measuring the Effects of Forces.
543. It has been remarked (Art. 388), that two forces F
and F' applied to the same body are proportional to the veloci-
ties which they can impress upon that body. Let it now
be supposed that these forces are applied to different maisses.
If two equal forces acting in opposite directions be applied
to equal and spherical masses M and M', they will commu-
nicate to these masses the equal velocities V and V ; and if
these masses be supposed to impinge directly upon each
other, they will mutually destroy each other's motion, and an
equilibrium will ensue, since the circumstances of motion in
each are precisely similar. But if the mass M be supposed
equal to nW, and V greater than V, we may regard M as
composed of n masses m', m", m'", m'"', each equal to
the mass M'. In consequence of the mutual connexion of
the different parts of the system, each of the nicisses ni', m",
wl'\ (fee. must move with the same velocity V, so that if the
body M be supposed to piss over the space of three feet in one
second of time, each of the masses m', ml\ m'", (fee. will like-
wise pass over a distance of three feet in one second ; or, if
V represent the velocity of the mass M, V will likewise ex-
press the velocity of each of the masses m', w", m'", (fee. But
if the mass m', moving with the velocity V, should impinge
against the equal mass M', which moves with the velocity V,
it would destroy a portion of the velocity of the second body
equal to V ; and if, at the same instant, the mass m", acting by
its connexion with the other masses, should impinge against
the body M', it would likewise destroy a portion of the velo-
city V, equal to V : and the same may be said of the other
masses m!'\ rrû% (fee. Thus, the joint effect of the several
masses m', m", m"', m*"\ would be to destroy in the
mass M' a velocity represented by iiN. If we suppose the
velocity V to be entirely destroyed, an equilibrium will ensue,
and it will be necessary that V'=?iV.
By eliminating n between this equation and the relation
M=wM', we obtain the proportion
M:M':: V: V;
MEASURE OF FORCES. 289
from which we concUide that an equilibrium will ensue when
tv)o bodies are caused to impinge directly against each otlier^
with velocities inversely proportional to their masses.
544. It may be readily demonstrated that the same propo-
sition is equally true when the mass M does not contain the
mass M' an exact number of times. For, if the mass M be
supposed to contain 'n masses, each of which is equal to m^
and the mass M' to contain a number of these equal masses,
denoted by n' ; each mass m contained in M, will destroy a
portion V of the velocity V of a mass m contained in M' ; or,
since M' is supposed to contain n' masses, each of whicli is
equal to m, the mass m, moving with the velocity V, will de-
V
stroy in M'=m'/w a velocity expressed by — : and since the
other equal masses contained in the body M will produce
similar effects, the entire velocity destroyed in M' by M will
V
be equal to — repeated as many times as the mass m is con-
it'
V
tained in M, or it will be equal to — Xw: if we suppose the
velocity V to be entirely destroyed, we must have
V— V*il •
or,
V : V ::/«': w : : mn' : tnn ;
and replacing mn, mn', by their values M, M', we obtain the
proportion
V : V : : M' : M :
whence the truth of the proposition is manifest.
545. Since the masses of the bodies are in the inverse ratio
of their velocities when an equilibrium is produced, it follows,
that if the bodies have equal volumes, and unequal densities,
their velocities will be in the inverse ratio of their densities.
546. Let F represent a force which impresses a velocity V
upon a meiss M : if the same force be supposed to act upon a
mass M times less, and which will consequently be repre-
sented bv — =1, this force will communicate to the mass
' M
T 25
290 DYNAMICS.
unity, a velocity M times greater than that communicated to
the mass M : this velocity will therefore be expressed by MV.
For a similar reason, the force F', which communicates to the
mass M' a velocity V, would communicate to the mass
M'
— = 1, a velocity represented by M'V.
The velocities represented by MV and M'V being com-
municated by the forces F and F' to the mass unity, it follows,
from the principles enunciated in Art. 388, that we shall have
the proportion
F : F' : : MV : M'V.
The expressions MV and M'V are called the quantities of
^notion communicated by the forces F and F' ; and it should
be recollected that the characters M, V, F, M', V, and F'
represent abstract numbers, which merely express the number
of times which the quantity under consideration contains the
unit of its own species.
547. The unit of force being arbitrary, we may represent
it by the quantity of motion which it produces. Thus, by
supposing F'to represent this unit, we can replace F'by M'V
in the preceding proportion ; and we thence infer that
F=MV.
548. When the force <p acts incessantly, it has been shown,
Art. 388, that this force will be represented by the velocity
which it would communicate in a unit of time, if the value
of the force should become constant ; hence we obtain, by
substituting for V its value «p,
F=M^.
If the mass M be supposed equal to unity, we shall have
F=^;
consequently, (p represents the force exerted upon the unit of
mass ; the quantity 4> is usually called the acceleratiJig force,
and F is called the waving force. When F is given, the
value of ^ can.be determined by simply dividing by M, the
mass moved.
549. It has been shown. Art. 163, that if g represent the
force of gravity, P the weight of the body, and M its mass,
we shall have
MEASURE OF FORCES. 291
eliminating M between this equation and the preceding, there
results
g
and if the incessant force <p be that of gravity, we have <p =g ;
hence,
F=P;
and in this case the moving force is measured by the weight
of the body upon which the force is exerted.
550. The writers upon Mechanics were long divided in
opinion as to the proper measure of forces. TJiis disagree-
ment, like many others, arose entirely from a misapprehen-
sion of the signification of words.
The nature of forces being known to us only by the effects
which they produce, we may with propriety measure these
eifects in different ways, according to the object which it is
desired to accomplish. If, for example, it be proposed to
determine the load which a man can support for an instant
of lime, it is evident that the force exerted by the man will
be proportional to the weight which he can sustain, and may
therefore be measured by this weight : but if we wish to
measure the force of this man by the work which he can
perform in a given time, we must adopt a measure for the
force entirely different from the preceding : for, it might
happen that a man absolutely weaker, but endued with a
greater capacity of sustaining a continued effort, would give
by his labour a result greater than that given by the first
man, and might therefore be considered as actually possessed
of greater force.
In this second method of considering the effects of forces,
we regard them as proportional to the weight raised, and the
height to which it is elevated in a given time ; it being always
understood that the effort necessary to overcome the weight
is not supposed to vary with the elevation.
If, for example, two men raise the same weight, in the same
time, to the heights of 600 and 200 yards respectively, we
would, according to this method of estimating the effects of
T2
292
DYNAMICS.
forces, reo^ard the first as possessed of three times the force of
the second.
Again, if, in the working day, one man can raise a weight
of 50 lbs. through a height of 200 yards, and a second a weight
of 251bs. through a height of 400 yards, we should regard the
two men, according to the present hypothesis, as possessed of
equal strength, although the absolute strengths of the two
might be very different ; the strengths of the two individuals
are here considered only with reference to the work done.
This method of estimating forces was adopted by Descartes.
The difference in the opinions entertained by him and other
geometers rested entirely on the definition of the word /orce.
He contended that a force should be measured by the product
of the mass into the square of the velocity. This conse-
quence may be deduced from the definition of the effect of a
force, adopted by Descartes, in the following manner.
Let P represent a weight, and h the height to which it can
be raised in a given time : the force employed to raise it, ac-
cording to the definition of Descartes, will be measured by
the product
PX/i.
We can replace P in this expression by its value M^ (Art.
163), and we shall have
?h=mgh ;
or, multiplying by 2,
2PA=Mx2^A;
and since the velocity v due to the height h is expressed by
i/{2gh) (Art. 401), the preceding expression becomes
2PA=My^
Having given a definition of the word force difierent, from
that adopted by Descartes, we shall not say that the force is
measured by the product Mv^*, but that it is measured by the
quantity of motion Mv whieli it is capable of producing, as
has been explained in Art. 547 ; and to avoid confusion, we
shall, according to ordinary usage, apply the term living force
to the product M.v^, of the mass by the square of the velocity.
551. The consideration of living forces is of great utility
in estimating the effects produced by a machine. Thus, if
COLLISION OP UNELASTIC BODIES. 293
it were required to calculate the eifect of a given fall of water,
the force necessary to move a carriage on a given piece of
ground, or the eflfort requisite to raise a given mass of coals
from the bottom of a mnie, we might in each case compare
the effect of the moving force to the product of a certain
weight by a given height, or to an expression of the form P/i,
the double of which, as has been before shown, is equivalent
to the product Mv'^.
Of the Direct Impact of Bodies.
552. Bodies are usually distinguished as elastic or unelastic.
An elastic body is that which, when compressed by the appli-
cation of an impulse, will resume its original figure with a
force equal to that of compression, in virtue of a quality pos-
sessed by the body. An unelastic body, on the contrary, is
one whose figure either undergoes no change by the action
of a force applied to it, or which, if compressed, has no tendency
to restore itself to its original form.
All natural bodies are found to partake more or less of these
two qualities ; there being none which are perfectly elastic,
or perfectly unelastic.
Of the Direct Impact of Unelastic Bodies..
553. Let M and M' {Pig. 196) represent two spherical un-
elastic bodies, which move in the direction from A to 0. If
the velocity of M be supposed to exceed that of M', the former
will overtake the latter, and will communicate to it a portion
of its motion, until the velocities of the two bodies become
equal. Let F and F' represent the forces which communicate
to the bodies M and M' their respective velocities V and V ;
since these forces can be represented by the quantities of mo-
tion which they produce (Art. 547), we shall have
F=MV, F'=M'V';
and by compounding these two forces, their resultant will be
expressed by
F + F=MV-fMV.
294 DYNAMICS.
To obtain a second expression for F+F', let v represent the
common velocity of the two bodies after impact : we may
regard the mass M + M' a^ a single body, to which the velocity
V has been imparted by the exertion of a force F+F'. We
shall then have
F+F=(M+M')v.
By equating these two values of F + F', we obtain
(M + M')î;=MV + M'V';
whence, we deduce
_ MY + M'V
'"~ M + M' ■
554. If the bodies move in opposite directions, we regard
one of the velocities V as negative, and we then have
_MY-MT'
^~ M + M' ■
The body M' being supposed at rest, and impinged against
by the body M, V will become equal to zero, and the pre-
ceding formula will reduce to
MV
^~M + M''
If the bodies have equal masses and move in the same direc-
tion, we shall have M=!M' ; and consequently,
^=i(V4-V'),
or, if they move in contrary directions,
and when the body M impinges upon an equal msiss M' at
rest, this expression reduces to
Of the Direct Impact of Elastic Bodies.
555. We will first consider the circumstances of motion
when an elastic spherical body impinges upon an immoveable
plane AB {F^g. 197) in a direction perpendicular to the sur-
face of the plane. At the instant when the body comes in
contact with the plane, it will begin to experience a com-
pression in the direction of the diameter ED, the point D
being caused to approach the centre of the sphere. This
COLLISION OF ELASTIC BODIES. 295
eôect will continue until the velocity of the sphere is entirely-
destroyed ; then, in virtue of the elasticity possessed by the
body, an equal velocity will be generated in an opposite direc-
tion, the body at the same time resuming its original figure.
Hence, the body will recoil with a velocity precisely equal to
that vnth which it impinged upon the plane.
550. Let us next consider the impact of two elastic bodies
M and M' {Fig. 196), which move in the same direction
from A towards C, with velocities represented by Y and V.
Tliat an impact may be possible, it is necessary that the
velocity of M should exceed that of M'. When the body M
overtakes M', a mutual compression will commence, and will
continue until tlie bodies have acquired a common velocity ;
so that a material point D of the body M {Pig. 198), which,
in virtue of the velocity V, would have described the lino DE,
being retarded in its motion by the eftect of the compression,
will, instead of having reached the point E at the instant of
maximum of compression, have only arrived at a point F :
then the force of restitution, beginning to act upon the mate-
rial point, will communicate to it a velocity in a direction
opposite to that of the motion, equal to that which it has lost
by the compression, and which would transfer it to the ex-
tremity G of a line FG=EF, whilst the body is resuming its
original figure.
The velocity of the body being common to all its points,
(Art. 443), if we represent this velocity before impact by DE,
it may be represented after impact by
DE-GE=:DE-2FE.
557. To express these conditions analytically, let u repre-
sent the velocity common to all the particles of the two
bodies at the moment of maximum compression. At this
instant, the bodies may be regarded as unelastic, and the
velocity m will therefore be given by the formula
MV+MT-
''= M + M- (^^^)-
The velocity lost by the body M during the compression,
being equal to the velocity V diminished by that which
remains at the instant of greatest compression, it will be ex-
pressed by V— ?/. Such will be the velocity lost at the
296
DYNAMICS.
moment of greatest compression, but the force of elasticity,
tending to restore the figures of the bodies, will cause the
body M to sustain an additional loss of velocity, represented
by V — u ; tluis, the total loss of velocity experienced by M
will be expressed by 2(V— ?i). Let v denote the velocity of
the mass M after the impact ; we shall have
or, by reduction,
v=2m— V (315);
The body M', at the instant of greatest compression, may
likewise be regarded as unelastic, and will then have gained
a velocity expressed by ii — V : for the velocity gained is evi-
dently equal to the velocity u which the body has at this
instant, diminished by the original velocity V. The force of
restitution, being then exerted, will cause the body to gain the
additional velocity u — V ; whence, the entire gain of velocity
by M' will be equal to 2(z^— V), and the velocity of M', after
collision, will therefore be expressed by
Representing this velocity by v'y we have
v'=2u—T (316).
By substituting in equations (315) and (316) the value of
u given by (314), we find
2(MV+MT0 2(MV + M T0
M + M' V: ^- M+M' '
from which, by reduction, we obtain
_V(M-M0+2MT' ,_ r(M^- M)+2MV ,g,„
"^ M+W' ' '" MTW ^^'^^•
If M=M', we shall have
v=V', v'^V (318).
These equations indicate, that when the masses are equal,
the impact will cause them to exchange velocities.
558. If the bodies move in opposite directions, the velocity
V may be regarded as negative in the preceding formulas,
which then become
V(M-M')-2M'V' , V'(M— M')+2MV .„.„.
""^ M+W ' '" M+W ^^^^^'
COLLISION OF BODIES. 297
559. The bodies being supposed equal in mass, and moving"
in opposite directions, we make M=M' in equations (319),
which are thus reduced to
v=-Y', v'=\ (320).
Hence we conchide that the bodies will recoil, having ex-
changed velocities.
560. When the bodies impinge in opposite directions, with
equal velocities, the masses of the two being unequal, we
make V'=V in equations (319), and thus obtain
_ Y(M-3iVr) , _ V(3M-M')
^ M+ivr ' '"~ M+M '
In this case, the motion of M will be entirely destroyed by
the impact, if its mass be supposed triple that of M' ; for
when M=3M', the first equation reduces to v=0 : the same
supposition gives 'y'=2V.
561. Lastly, the body M' being supposed at rest, and
impinged against by an equal body M, we make M=M', and
V'=.0, in equations (317), and we thus have
v=0, v'=y:
hence, the body M will be brought to rest, and M' will acquire
its entire velocity.
Of the Preservation of the Motion of the Centre of Gravity
in the Impact of Bodies,
562. Let the two bodies M and M' be supposed to have
arrived at the positions B and C {Fig. 199), immediately before
impinging upon each other ; and let S and S' represent their
distances from the point A, and X the distance of their com-
mon centre of gravity from the same point. From the known
property of the centre of gravity, we shall have
(M-1-M')X=MS+M'S';
and since the distances X, S, and S' vary with the time t, we
shall obtain, by differentiating with reference to t,
(M+M')^=M§+M'^'.
at at ai
298 DYNAMICS.
The differential coefficients -^ and -r- represent the velo-
dt dt ^
cities of the bodies M and M' at the instant when they have
arrived at the points B and C, the distances of which from the
point A are represented by S and S' respectively. Let these
velocities be denoted by V and V', and that of the centre of
gravity by W=-7- : we shall obtain, by substitution,
W="+M-V-
M + M' ^ '
Such is the expression for the velocity of the common centre
of gravity before the impact : but immediately after the im-
pact, the bodies, being found ai the points B' and C, will have
experienced a change in their velocities, and it is required to
determine what effect has been produced upon, the velocity
of their centre of gravity. Let w denote the velocity of the
common centre of gravity after impact, and x its distance from
the point A, in the new positions of the bodies ; the distances
of the bodies from A being represented by s and s' respectively,
and their velocities by U and U', we shall have, as above,
(M+M>=M5+MV :
and by differentiating with reference to t, we find
(M + M')^=M^4-M'^'.
^ ' dt dt dt
Replacing — -, -—, and -r- by their respective values w^ U, and
dt dt dt
U', there results
MU+iAfU -
'^~ M+M' ^^^^^'
563. Two different cases may now be presented for exami-
nation ; viz. the bodies may be elastic, or they may be un-
elastic ; when they are unelastic, we have
U=w=U';
whence.
. M+M'
M + M'
COLLISION OF BODIES. 299
But it has been shown (Art. 553), that the velocity lo common
to the two bodies after the impact will be equal to
MV+MV
M+M' '
this velocity being precisely equal to the velocity W, it fol-
lows that we shall have w = W ] or, the velocity of the com-
mon centre of gravity of two unelastic bodies is not affected
by their im,pact.
564. When the bodies are elastic, their velocities after im-
pact will be expressed (Art. 557) by 2w— V, and 2m — V.
Substituting these values' of U and U' in equation (322),
we find
m{2u—Y)-[-M'(2u—Y')
^~ M-fM' '
or, by reduction,
„ MV + M'V
^=^^^- M + M- '■
replacing the second term of the second member by its value
u, there will result
w=7i ;
or,
MV-fM'V
'w:=i :
M+M'
and eliminating the second member of this equation by
means of equation (321), we find
hence we conclude, that in the im,pact of elastic bodies, as in
that of unelastic bodies, the velocity of the centre of gravity
is the same before and after impact.
Of the Preservatioîi of living Forces in the Impact of
Elastic Bodies — Relative Velocity before and after Im-
pact — Loss of living Force in the Collision of Unelastic
Bodies.
565. The principle of the preservation of living forces in
the collision of elastic bodies may be enunciated as follows :
Wlien two elastic bodies impinge on each other, the sunt of
their living forces is the same before and after imj'act.
300 DYNAMICS.
Let V and V represent the velocities of the bodies before
colHsion, and v and v' their velocities after collision ; the sum
of the living forces before the impact will be expressed by
MV^+M'V'2 ; and it is required to prove that this sum is
equal to Mv^ +M'î;'=, the sum of the living forces after the
impact.
It has been shown (Art. 557), that the velocities v and v',
after impact, are given by the equations
v==2u—Y, v'=2u-\",
hence,
mv^-\-M'v'^=M(2ii—Yy+M'{2u--Y'y ;
and by performing the operations mdicated in the second
member, we have
Mi;=^ +MV2 =MV2 +M'Y'2
+4(Mm2+M'm''— MVî^-M'V'm) (323) :
but the terms included within the brackets mutually destroy
each other, in consequence of the relation (314),
_ MV+]vrv ^
'' M+M' '
for, by clearing the denominator, aiid multiplying by u, we
find
M?^^ i-Wu"' ^MY If -{-M'Y '71 ;
consequently, the equation (323) will reduce to
Mv^ +MV2 =MV2 +M'V'2.
This equation may be written under the form
Mv^ -f M'î;'^ —MY' — M'V'2 =0 ;
from which we conclude that when elastic bodies impinge
on each other, the difference between the sums of their living
forces before and after impact, will be equal to zero.
566. The relative velocity of the two bodies is the velocity
with which they approach towards, or recede from, each
other ; and another remarkable property of elastic bodies con-
sists in the equality of their relative velocities before and after
impact. This may be proved by subtracting the equations
v=2u—Y, v'=2u—Y']
from which we obtain
v-v'=-{Y-Y') ;
hence v' exceeds v by the same quantity that V surpasses V' ;
PRINCIPLE OF d'aLEMBERT. 301
and the bodies will therefore separate after impact, with a
velocity precisely equal to that with which they approached.
567. In the collision of unela^tic bodies, the difference be-
tween the sums of the living forces before and after impact
will not be equal to zero ; but it will be equal to the sum of
the living forces of the bodies when moving with the veloci-
ties lost or gained.
This theorem is due to Carnot, and may be demonstrated
in the following manner :
The velocities lost and gained by M and M' respectively,
being equal to V— ?«, m — V, if the masses were moved with
these velocities, their living forces would be expressed by
M{Y-uy, M'(m-V')= ;
performing the operations indicated, we shall have
M(V -uy- +M'(u-Y')"~ =
MV-^+M'V'2 + (M+M')w2-2m(MV+M'V') (324) ;
eliminating MV-fM'V, by means of the equation
_ MV+M'V'
" xM+M' '
the second member of equation (324) will reduce to
MV^' +M'V'2— (M+M')«%
and we shall therefore have
M(V-tiy +M'(^i-V')==MV2+M'V'2-(M+M')M^ ;
hence the truth of the theorem enunciated becomes apparent.
Principle of U Alembert.
568. When the several bodies which compose a system are
connected together in any manner, and subjected to the action
of different forces, this connexion will in general prevent
each body from taking the motion which would have been
communicated to it if the connexion had not existed. For
example, if several material points M, M', M", &c. [Fig. 200)
be attached to an inflexible right line AL, moveable about the
point A, it is evident that these points, being unable to move
except with the line AL, will, when acted on by the force of
gravity, oscillate together about the point A, describing arcs
26
302
DYNAMICS.
proportional to their distances from A, and will at the end of
a certain time be brought into the positions K, K', K", &.C.;
whereas, if the points were unconnected, being merely attached
to the point A, they would, from the principles of the simple
pendulum, explained in Art. 471, oscillate in very unequal
times, depending on their distances from the point A. More-
over, if we resolve each of the several forces which are exerted
in vertical directions upon the points M, M', M", &.C., into
two components, one of which shall act along the line AL,
and the other in a direction perpendicular to this line ; the
latter component will alone tend to communicate motion to
the point ; and since the several perpendicular components,
exerted on the different points, will be equal to each other,
they would communicate in the instant of time c?^ equal veloci-
ties to the points M, M', M" <fcc., if these points were uncon-
nected. But in consequence of their connexion, the veloci-
ties assumed are evidently proportional to their distances from
the point A.
569. It thus appears, that the effective velocities assumed
by the several parts of the system differ from the velocities
impressed, and hence the circumstances of the motion can
only be discovered when we have succeeded in expressing the
effective velocities in functions of the velocities impressed.
This object is readily accomplished with the assistance of a
dynamical principle first employed by D'Alembert.
570. Let v, v\ v", (fee. represent the velocities which Avould
be impressed by certain forces on the bodies M, M', M", &c.,
if they were perfectly free, and u, u', ii", &c, the velocities
assumed by these bodies in consequence of their connexion.
The velocity v being resolved into two components, one of
these components may be assumed arbitrarily, and the second
will then become determinate. Let the effective velocity ii
be assumed as the arbitrary component of the impressed
velocity v, and denote the other component by U. Making a
similar decomposition of the other velocities v',v", &c., we have
u and U for the components of v,
u' and U' for those of v',
u" and U" for those of v",
(fee. &.C. (fee. ;
PRINCIPLE OF d'aLEMBERT. 303
and the quantities of motion impressed upon the system,
which are Mv, M'v', M"v", &c., will become, after the de-
composition,
Mu, MV, M"u", (fee,
MU, M'U', M"U", &c.
But, in consequence of the connexion of the different parts of
the system, these quantities of motion will be reduced to
Mu, M'«', M'V, (fee. ;
hence, it is necessary that the quantities of motion MU, M'U',
M"U", (fee. should destroy each other, or should produce an
equilibrium. For, if it were otherwise, we might combine the
resultant of the quantities of motion MU, M'U', M"U", cfec.
with the quantities of motion Mu, M'u', M"w", (fee. ; thus the
effective velocities of the several parts of the system would
no longer be represented by u, u', u", (fee, which is contrary
to the hypothesis.
571. It may be observed that the products MU, M'U',
M"U", (fee. express the quantities of motion due to the veloci-
ties lost or gained by the several bodies. For the velocity v
may be replaced by its two components u and U ; the former
of which expresses the effective velocity of the body M, and
the latter represents that velocity which, combined with ii,
would produce the impressed velocity. Thus, U is a velocity
introduced or destroyed in the system by the connexion of its
parts.
The general principle may therefore be enunciated in the
following manner : It is necessary that the quantities of
motion due to the velocities lost or gained should he such as
would maintain the system hi equilibrio.
572. It has been remarked that the quantity of motion Mv
may be resolved into the two components Mu and MU ; and
since an equilibrium will always subsist between three forces,
one of which is equal and directly opposed to the resultant
of the other two, it follows that the forces represented by Mu
and MU will sustain in equilibrio a force equal and opposite
to Mv ; and consequently, that the force Mv will sustain in
equilibrio two forces which are respectively equal and oppo-
site to Mu and MU.
304 DYNAMICS.
The same remarks being applicable to the other forces, it
appears that the forces Mr, M'v', M"v", &c. will sustain in
equilibrio two systems offerees which are equal and directly
opposed to the forces
Mu, M'w', M"u", &c.,
MU, MU', M"U", &c.
But the forces MU, M'U', M"U", (fcc. destroy each other ; and
hence we obtain a second enunciation of the principle of
D'Alembert, viz. ; An equilibrium will subsist between the
quantities of motion Mî;, JM'v', M"v", cj'c. impressed upon the
several bodies, and the effective quantities of motion Mtf,
M'?<', M'w", ^'c, tJie latter being applied in dii^ections con-
trary to those of the motions actually asstimed.
573. This principle is equally true, whether the velocities
V, v', v", &yC. are finite velocities, acquired by the masses M,
M', M", &c. during a finite time, or communicated instanta-
neously by forces of impulsion ; or, when these velocities are
infinitely small, being generated by incessant forces ; or,
finally, when some of these velocities are finite, and some of
them infinitely small.
574. To apply this principle, let us consider the impact of
two unelastic bodies M and M', which move in the same
direction. Let v and v' represent their velocities before im-
pact, and 71 the common velo 'ty after impact. The velocity
lost by M being equal to its original velocity diminished by that
which remains after collision, it will be expressed by ?;—?«:
in like manner, the velocity lost by M' will be expressed by
!>' — 11. The quantities of motion due to these velocities being
such, by the principle of D'Alembert, as to produce an equi-
libriinn, we shall have
M{v—2i) + M'{v' -zi)=0 ;
whence we deduce for the velocity after impact,
_Mv + M'v'
''~ M+M' •
When the bodies move in opposite directions, v' will become
negative.
575. As a second example, let it be required to determine
the circumstances of motion of two bodies M and M', which
PRINCIPLE OF d'aLEMBERT. ; 305
rest on two inclined planes AB and AC {Fig. 201) having a
common altitude, and are connected by a thread MEM', pass-
ing over a fixed pulley.
If the vertical line M^, drawn through the centre of gravity
of the body M, be supposed to represent the intensity of the
force of gravity ; the component of the force in the direction
of the plane will be represented by MR ; this component will
alone tend to urge the body down the plane : its value will
be expressed by
AD
§• Xcos RMg-^^ . cos BAD=^— — .
AB
In like manner, the component of gravity, which tends to
cause the descent of the body M' on the plane AC, will be
expressed by g-j-^-
AC
Let the lines AD, AB, and AC be denoted by h, I, and V
respectively ; the incessant forces exerted upon the bodies
will then be
gh J gh
T' ""■* T-
But if we suppose the motion to take place in the direction
M'EM, and the velocities to be reckoned as positive in this
direction, the force j^, which is opposed to the motion, must
be regarded as negative ; and the incessant forces will there-
fore be expressed by
it, and _f .
The general expression for the value of an incessant force
being
dv
<P =
we have
^=^^'
dv=ç>dt :
hence, the velocities imparted to the bodies in the time dU
when they are unconnected, will be expressed by
u
306 DYNAMICS.
and the quantities of motion due to these velocities will be
Ug!^di, -WffydL
But the bodies being supposed connected by a thread of inva-
riable length, if M should descend through any distance on
the plane AB, M' will necessarily ascend through an equal
distance on the plane AC ; or, in other words, the velocities
of the bodies at any instant will be equal to each other. De-
noting by V their common velocity at the end of the time t,
the eifective velocities communicated to them in the succeed-
ing instant dt, will be expressed by dv, and the effective
quantity of motion imparted in the same time, will there-
fore be
(M.-\-M')dv.
By the principle of D'Alembert, this quantity of motion when
applied in a contrary direction, wil produce an equilibrium
with the quantities of motion impressed on the bodies : hence,
the sum of these quantities of motion will be equal to zero, or
-{M+M)dv+Mg^dt-^^dt=-0 (325) :
from which we deduce
and by integration,
■ "=4+^-^'+° (^'^"'^
or, if we denote by G the coefficient of ;, we shall have
v=Gt-\-0 (326).
Let X represent the distance OK of the body M from the
point O, the origin of the spaces, at the end of the time t ;
the general expression for the velocity gives
dx
and therefore,
''' 'dt
at
PRINCIPLE OF d'aLEMBERT. 307
from which, by integration, we obtain
ar=iG^2+o^ + C' (327).
The formulas (326) and (327) indicate that the circumstances
of motion in this system are precisely similar to those which
attend the fall of heavy bodies ; the only difference consisting
in the value of the incessant force, which in the latter case is
denoted by g, and in the former by G.
576. If the planes AB and AC be supposed to become ver-
tical, the case will be reduced to that of two weights con-
nected by a cord which passes over a fixed pulley : the quan-
tities A, Z, and I' are then equal, and the equations (325 a) and
(327) may then be reduced to
M— M' M— M'
''=ffw^^'+''' "=ra''*^''+'''+''' • • • <'''''^-
577. These formulas will serve to explain the principle of
Atwood's machine, which is employed for the verification of
the laws of constant forces.
This machine consists essentially of, 1°, A fixed pulley,
over which passes a very fine flexible thread, having its ex-
tremities attached to two equal brass basins ; 2°. A vertical
graduated scale with a moveabls stage to maxk the space
passed over by the descending basin ; and, 3°. A seconds
pendulum, by means of which the time of descent may be
accurately observed.
When the two basins are loaded with equal weights, they
will sustain each other in equilibrio ; but if an addition be
made to either, it will immediately preponderate, aîid will
produce a motion uniformly varied. Moreover, by rendering
the difference M— M' of the weights M and M' attached to
the extremities of the thread, very small in comparison with
their sum M + M', the space described and the velocity ac-
quired in a given time which result from equations (327 a)
may likewise be rendered small, and the observations will
thus become susceptible of great accuracy.
For the purpose of observing the velocity acquired at the
end of any time, we give to the additional weight placed in
the descending basin the form of a flat bar, and the basin
being allowed to pass through a sliding ring attached to the
U2
308 DYNAMICS.
vertical scale, the bar may be removed at any instant during
the descent. The equality of the weights in the two basins
being restored by the removal of the bar, the motion becomes
uniform with the velocity acquired at the instant when the
bar was removed.
By comparing the spaces described, the velocities acquired,
and the times elapsed, we find that when the basins move
from rest under the influence of a constant force, the velocities
are constantly ])roj)ortional to the times, and that the spaces
are proportional to the squares of the times.
578. For a third example, let it be required to investigate
the circumstances of motion of two weights M and M', which
are attached to cords passing around the respective circum-
ferences of a wheel and of its axle.
If we suppose the body M to prevail, and reckon the veloci-
ties positive in the direction of its motion, the force of grav-
ity will impress upon the bodies M and M', in the instant dt^
which succeeds the time t, the velocities gdt and — gdt ; and
the quantities of motion impressed will therefore be
Mgdt, and —Wgdt.
But if V and v' represent the velocities of M and M' at the ex-
piration of the time t, the effective velocities communicated
in the succeeding instant dt will be expressed by dv and dv'.
Thus, denoting by R and r the radii of the wheel and axle,
we shall have
Masses. Impressed velocities. Effective velocities. Distances from the axis.
M . ... gdt dv R,
M' . . . —gdt dv' r.
The effective quantities of motion, being applied in directions
contrary to those of the motions assumed, will sustain in equi-
librio the quantities of motion impressed ; and since the equi-
librium is maintained through the intervention of the wheel
and axle, it is necessary that the sum of the moments with
reference to the axis should be equal to zero : hence, we
obtain
WRgdt—Wrgdt-MRdv—'m:rdv'=() (328).
This equation containing the two unknown quantities v and
r', it will be necessary to discover a second relation between
UNIFORM MOTION ABOUT AN AXIS. 309
them. For this purpose, we remark that the velocities v
and v' bear to each other the constant ratio of R : r ; thus, we
have
V : v' ::B. : r ]
or,
r
and by differentiating,
V —V-
R
T
dv'=—dv
R
substituting this value in equation (328), we find
MRgdt—M'rgdt—MKdv-M'^dv=0 ;
or, by reduction and transposition,
MR'dv-^-M'r^dv^MR^gdt-M'Rrgdt;
whence,
, MR2-M'Rr ,,
^^^=mrm:mv^^^''
Denoting by K the constant coefficient of df, this equation
becomes
dv=K.dt ;
and by integration,
Replacing v by its value — -, and performing a second integra-
tion, we find
These results indicate that the motion is uniformly varied,
the circumstances of the motion being similar to those of a
body falling under the influence of the force of gravity.
Of the Motion of a Body about a Fixed Axis.
579. When an impulse is applied to a system of material
points connected together in an invariable manner, and sub-
jected to the condition of turning about a fixed axis, which
we will suppose to pass through the point A {Fig. 202), per-
pendicular to the plane of the figure, the several particles tw,
310
DYNAMICS.
m\ m", &c. will describe circles won, rr^o'i}!^ m"o"n", <fcc., the
planes of which will be parallel to each other, and perpen-
dicular to the fixed axis ; and the arcs described by the several
points in the same time will contain the same number of
degrees. These arcs being proportional to their radii, the
velocities of the several particles will be in the same propor-
tion ; so that if we denote by a the velocity of the particle e,
whose distance eA from the axis of rotation is equal to unity,
the velocities of the particles w, m.\ m", <fcc., at the distances
r, r', r", &c. from the fixed axis, will be expressed by r*, r'«,
r% &c. Thus, the efiective quantities of motion of the dif-
ferent particles Avill be represented by
mro}, m'r'oj, 'm"r"u, &c.
Let V, v', v", &c. be the velocities impressed : the correspond-
ing quantities of motion will be expressed by niv, m'v', ni!'v'\
&.C. It will therefore be necessary, according to the second
enunciation of the principle of D'Alembert, that an equilib-
rium should subsist between the forces mv^ oyi'v', 'm"v", &c.,
and — mro)^ — ni'r'u^ — 'm"r"o), &c.
To establish the conditions of equilibrium between these
forces, we will first consider the force niv, and represent it by
w/ja portion of its line of direction : from the point/ let the
perpendicular //i be demitted upon the plane of the section
om/2,and denote by <p the angle /?7iA, formed by /w with this
plane ; by constructing the rectangle hh', the force mv may
be resolved into the two components
mh' = (mv) . sin <p, parallel to the fixed axis,
mh = (mv) . cos <p, situated in the plane om?i.
The first of these components will have no tendency to turn
the system about the fixed axis ; but the second will produce
its entire effect in communicating a motion of rotation.
Tf we represent in like manner by <z>', p", &.c. the angles
formed by the directions of the forces m'v', vi'v", &c. with
the planes o'm'?i', o"m"ii", (fee, the quantities of motion im-
pressed will become
mv cos (p, m!v' cos <p\ m"v" cos <p", (fee.
These quantities of motion, as well as the quantities - ????>,
UNIFORM MOTION ABOUT AN AXIS. 311
—m'r'u, —}7i"r"u, &c. are situated in planes perpendicular to
the fixed axis.
The conditions of equilibrium between these forces will
evidently be the same as those which arise when the forces
are situated in the same plane ; if, therefore, the forces be
regarded as situated in the plane of the figure, the conditions
of equilibrium will require that the sum of the moments of
the forces which tend to turn the system in one direction
about the point A, shall be equal to the sum of the moments
of those which tend to produce rotation in a contrary direc-
tion ; or, that the algebraic sum of the moments shall be
equal to zero.
But the quantities of motion — mr<y, —m'r'a, —m"r"a>, &c.
being derived from the common motion of the system, they
will tend to turn it in the same direction ; and since these
motions take place in the circumferences of the circles mno,
m'u'o', m"n"o", &c., the radii r, r', r", &c. will represent the
perpendiculars demitted from the point A upon their respect-
ive directions ; consequently, the sum of the moments of the
effective quantities of motion, when applied in opposite direc-
tions, will be expressed by
— mr'^61 — mr'^u — rn'r'^u — &Lc. = — «(mr^-j-TOr'^-f-OT V'^-i-&c.).
Let the quantity within the brackets be denoted by :s{mr^) ;
the sum of these quantities of motion will then be repre-
sented by — û)2(mr2).
To determine the value of the sum of the moments of the
impressed forces,
mv . cos (p, mJv' . cos <p', m"v" . cos <?", &c.,
let Az {Pig. 203) represent the fixed axis, and ml, m'l', m"l",
&c. the forces qîiv . cos (p, m'v' . cos <?', ni"v" . cos <p", <fec. situated
in the planes mno, m'ti'o', m"n"o", &c., perpendicular to the
fixed axis : from the points A, A', A", &c., at which the axis
intersects the perpendicular planes, let the perpendiculars
M=p, A'l'=p', A"l"=p", (fcc. be demitted upon the directions
of the several forces mv cos (p, m'v' cos ç>', m"v" cos <f>", <fcc. ;
the moments of these forces will be expressed by
mv cos ç> . p, m'v' cos <p' . p', m"v" cos p" . p".
The algebraic sum of these moments will be expressed by
312
DYNAMICS.
2(mu cos <p .2^) ] and hence, by the conditions of equilibrium
before enunciated, we shall have
2(mv . cos ç> . p) — al.(mr^ ) =0.
This equation gives the value of the angular velocity
^^Hmv.cosç.p)
and the motion of the body about the fixed axis will there-
fore be uniform.
580. When the forces mv^ m'v\ m"v", (fcc. are exerted in
planes perpendicular to the axis, the angles <p, <p', <p", &c.
become equal to zero, and we have
sin ^=0, cos ^=1,
sin ^'=0, cos (p'=l,
sin ç>"=0, cos <p"~l,
&c. (fee. ;
consequently, the equation (329) reduces to
I.{mvp)
581. If equal velocities be impressed, in parallel directions,
upon the several particles m, m', m", (fee, we shall have
v=v'=v"=(fec.
and the moments of the quantities of motion impressed will
become
mvp + ?n'vp' -{•m"vp" -]-ôcc.=v{mp -^7ii'p' +'m"p" + ôcc.) :
the sum of these moments may be represented by v^{mp)f
and the equation (329) will be transformed into
.=.!^^ (330).
Let a plane AK be now drawn through the axis As {Fig.
204), parallel to the directions of the several forces mv, vi'v',
7n"v", (fee. : the perpendiculars p, p\ jj", (fee. demitted from the
points A, A', A", (fee. upon the directions of these forces, are
evidently equal to the perpendiculars mq, ni'q', 7n"q", (fee, let
fall from the points ???, m,', m'\ (fee. upon the plane AK. Let
q, q', q", (fee, represent these perpendiculars, and Q,the perpen-
dicular demitted from the centre of gravity of the system, upon
the plane AK ; then, denoting by M the sum of the particles
UNIFORM MOTION ABOUT AN AXIS. 313
which compose the system, or the entire mass of the body,
we shall have, by the property of the centre of gravity,
MGi—mq-\-m'q' -\-m,"q" ■\-&LQ,. ;
and since
p=q, p'=q', p" = q", (fec,
the preceding equation may be written
MGi=mp -{-fn'jy -\-'m"p" -\'(Scc. = 7:{mp).
This value being substituted in equation (330), there results
vMGi .^„.>
«=: — (OOi).
582. It may happen that the velocity v has been impressed
iipon only a limited number of the particles ?n, m\ m", &c. :
then, M will no longer represent the entire mass of the system,
but merely the sum of those particles upon which the velocity
has been impressed ; and Q, will express the perpendicular
demitted from the centre of gravity of this part of the system
upon the plane AK.
The quantity ^(mr^) is called the moment of inertia : the
method of determining its value will be explained in the next
section.
583. It is frequently necessary to consider the effects pro-
duced upon the fixed axis by the application of an impulsive
force to any point of the system. For this purpose, let the
axis of rotation Kz {Fig. 205), be assumed as the axis of z,
and resolve the impulsion P, which is supposed to be applied
at a point O, into two components P' and P", which shall be
respectively parallel and perpendicular to the plane of x, y.
liCt the axis of y be then assumed parallel to the direction
of P', and denote the co-ordinates of the point O by a, è, and
c : since the force P may be applied at any point in its line
of direction, we can always suppose the point of application
O to be contained in the plane oi x^z'. this supposition gives
6=0.
Instead of regarding the axis as fixed, let such forces be
introduced as may be necessary to retain it. These forces
will be equal, and directly opposed to the impulsions expe-
rienced by the axis, and may in general be reduced to three
forces respectively parallel to the axes of ^, y, and z. Let X,
27
314 DYNAMICS.
Y, and Z represent the impulses communicated to the axis,
and call AB=«, AC=/3.
The particle m will describe a circle parallel to the plane
of X, y, and its velocity in the direction of the tangent ml will
be expressed by ret (Art. 579) : the cosines of the angles
formed by this direction with the axes of x and y respectively,
will be — and — — ; hence, the effective quantity of motion of
r r
the particle m will be mrai, and its components in the direction
of the axes of x and y will be mi/a and —mxa : the same
remarks apply to the other particles m', m", m"\ &c.
But, by the principle of D'AIembert, an equilibrium will
subsist between the effective forces and the force P, the latter
being applied in a contrary direction ; thus, we shall have
Forces. Components parallel to axes of Co-ordinates of points of application parallel to
X y z X y z
— P Pcos^ — Psin^. ... a c,
X X
Y Y ^,
Z Z 0,
niru myu — mxu x y z,
m'r'a) m'y' a — mx'u x' y' z\
<fcc. (fee. &c.
The general equations (66) and (67), which express the
conditions of equihbrium of forces lying in different planes,
and acting upon various points of a body, may be written
under the form
2(X)=0, 2:(Xy-Y:i')=0,
2(Y)=0, 2(Zt-Xz)=0,
s(Z)=0, s(Y;s-Zy)=0;
and when applied to the system under consideration, will
give
X4-*2;(m2/)=0,
Y + P cos ^ — «2(m:r)=0,
Z— Psin^=0;
«2:(7/w=) — P cos ç>a=0,
X«+«2;(my2;) + P sin ^a=0,
Y/3+P cos <pc —a'z{?nxz)=0.
UNIFORIW MOTION ABOUT AN AXIS. 315
Let M represent the mass of the body, r„ y„ and z, the
co-ordinates of its centre of gravity, and Mv the quantity of
motion which the force P is capable of communicating : these
six equations will be reduced to
Y=cMx,—mv . cos ^ V (331 a) ;
Z=Mv.sin<^ 5
«s(mr2)=My .cos^ . a ^
X<«= — «s(my2;)— Mv . sin p . a > (331 b).
Y^=<u'z(ni.Tz) — Mv . cos p .c j
From the fourth equation we deduce the value of u, which
being substituted in the first and second, the values of X and
Y become known : the third determines the value of Z, and
the fifth and sixth give the co-ordinates « and /3 of the points
B and C, at which the forces X and Y are applied. The
solution of the problem is therefore complete.
When we wish to communicate the impulse P in such a
manner that the axis shall receive no shock, we make X, Y,
and Z equal to zero. This supposition reduces the equations
(331 a) and (331 b) to the following forms :
ax, = V, ^{myz) = 0,
The third equation indicates that the direction of the impulse
must be parallel to the plane of x,y; the first, that the centre
of gravity of the body must lie in the plane of .r, z, perpen-
dicular to which the impulse is applied ; the second deter-
mines the angular velocity a ; and the fourth and sixth make
known the vakies of the co-ordinates a and c of the point O.
The point O is then called the centre of percussion^ which
may be defined to be that point in the plane passing through
the centre of gravity and the axis of rotation, at which an
hnpulse must be applied in a direction perpendicidar to this
plane, in order that the axis may receive no shock.
584. The equation l(myz)=0 expresses a relation which
is evidently dependent on the figure of the body and tlie
position of the axis of rotation. This relation will exist only
in particular cases, and it therefore follows that a body
3lB DYNAMICS.
retained by a fixed axis will not necessarily have a centre of
percussion.
585. The distance of the centre of percussion from the
axis of rotation being equal to the absciss AN=:a, its value
will be
a= — ^ — _: = -> '-.
586. Although the axis will receive no impulse at the in-
stant of impact, yet the motion of rotation will immediately
give rise to centrifugal forces which will exert a pressure
upon the axis.
Of the Moment of Inertia.
587. The momert of inertia being the sum of the products
formed by multiplying each material point of a system by
the square of its distance from a fixed axis, it has been repre-
sented in the preceding section by 2(wir"). In this expres-
sion, we may replace the particle m by dM., the element of the
mass ; and the moment of inertia will then result from the
integration of an expression of the form/r^rfM.
588. For example, let it be required to determine the
moment of inertia of a material right line CB {Pig. 206), with
reference to an a\is AZ perpendicular to the plane CAB.
Let AB=/i represent the perpendicular demitted from the
point A upon the right line, and BP=;r the distance of a
point P assumed arbitrarily on this line, from the point B :
we shall have
PA2=A2^.'r2.
This expression being multiplied by the diflerential of the
mass, the integral of the product will express the moment of
inertia. The volume, in the present case, being a right line,
the element of the volume will be represented by the infinitely
small difference dx between two consecutive abscisses BP=^
and BV=x-\-dx] and the element of the mass rfM will
therefore be expressed by dx multiplied by the density D, or
by Ddx. Thus, by multiplying h- -^-x^ by Ddx^ and inte-
MOMENT OF INERTIA. 317
grating, we obtain for the expression of the moment of inertia
of the right hne,
In the present disposition of the figure, the integral should be
taken between the limits of the point B, where x=0, and the
point C, at which x=a] the moment of inertia thus becomes
('"«+Ï)d-
In effecting this integration, we have regarded the line as
homogeneous, or the density D as constant : but if the differ-
ent parts of the line be supposed unequally dense, the quan-
tity D will be variable, and may in general be regarded as a
function of x. The form of this function will depend on the
law according to which the density is supposed to vary.
589. When the body is homogeneous, it is frequently con-
venient to regard the density as equal to unity ; and the factor
D is then replaced by 1, in the general expression for the
moment of inertia. Having determined the moment of inertia
of a body whose density is equal to unity, we can determine
that of a similar body whose density is equal to D, by simply
multiplying the former moment by the density D. In the
succeeding examples, we shall regard the density as equal to
unity.
590. As a second example, we will determine the moment
of inertia of the area of a circle CBD {Pig. 207), ivith refer-
ence to the axis AZ passing through its centre, and perfen-
dicular to its plane.
Let m represent a point in the plane of the circle, at a dis-
tance mA=x from the fixed axis : the areas of the circles
described with the radii x and x-\-dx will be expressed
respectively by
vx^, and 7r(x-\-dxy ;
and the difference between these areas, by neglecting the infi-
nitely small quantities of the second order, will be 25rx . dx:.
This expression will represent an elementary ring, every
point of which will be at the distance x from the axis : hence,
by multiplying this element by x^, we shall obtain 27rx^dxfoi
the differential of the moment of inertia. Taking the integral
318 DYNAMICS.
from a?=:0 to x='>' we shall find i^rr* as the moment of
inertia of the area of a circle whosk radius is denoted by r.
591. Let it be required to determine the iiioment of inertia
of a sphere with reference to an axis passing through its
centre. If the sphere be cut by a plane EE' perpendicular to
tlie fixed axis AB {Fig. 208), the section will be a circle
whose centre will be fuund at the point D. Denote by x the
absciss AD of this section, and by y the ordinate DE; or the
radius of the section. The moment of inertia of the area of
this circle taken with reference to the axis AB, will be expressed
(Art. 590) by
and if this expression be multiplied by dr=DD', the product,
l^y'dx,
will express the moment of inertia of the elementary volume
EE'F'F bounded by parallel planes drawn through the con-
secutive points D and D'. The integral of this expression,
being taken between the limits :r=0 and a:=AB=2r, will
give the moment of inertia of the entire sphere.
But by the property of the circle, we have
y 2 =2rx — x^ ;
and therefore,
f^7ri/*dx=^^f(2rx—x''ydx
=7rf(2r^x^ —2rx^-\- ^x*)dx ;
or,
f^^y'dx=^x^{"fr^-irx+-f\x^)-\-C.
The constant C will be equal to zero, since the moment is
zero when x=0: and by making x=2r, we obtain for the
moment of the whole sphere,
_*_«■/• ^.
15' •
These examples are sufficient to explain the manner in
which the determination of the moment of inertia is reduced
to a simple problem of the integral calculus.
592. When the moment of inertia of any body with refer-
ence to an axis passing through its centre of gravity has
been determined, its moment with respect to a parallel axis is
readily found.
MOMENT OF INERTIA. 319
For let GF and CK {Pig. 209) represent two parallel axes,
the first of which passes through G, the centre of gravity of
a body : let the origin be assumed at the point G, the line
GF being the axis of z. Through a point m, assumed arbi-
trarily within the limits of the body, let the plane mKF be
drawn, parallel to the plane oi x^y \ this plane wilt cut the
axes GF and CK at two points F and K, and the distances
of the point ni from these axes will be represented respect-
ively, by the right lines mK and mF, which we shall denote
by r and r'. From the point m let the perpendicular mE be
demitted upon the plane of a;, y ; the triangles ECG, mKF will
be equal in all respects, and the sides of the former may there-
fore be substituted for those of the latter. Denote by
a, and j3, the co-ordinates GD and DC of the point C,
X and y, the co-ordinates GP and PE of the point E,
a, the distance between the axes :
we shall have
GC2=GD2.fDC=, GE2=GP2-fPE2,
or,
«2 =^2 4-^2^ ^'2 ^j.2 J^y2 (332).
Again, the right line CE passing through points whose
co-ordinates are x and y, « and /3 ; the value of CE— r will
result from the equation
or, by developing the terms of the second member,
r2 = r2 -f y2 _2«.r— 2/3y+«2 4.^2 j
and reducing by means of equations (33,";), we obtain
multiplying by dM and integrating, we have
fr''dM=fr'^d^l-2ccfx(M—2i2fydM.+a^fdM (333).
The expressions fxdM. and fydM. which enter into this equa-
tion, are equal to zero ; for, let x and y represent the co-
ordinates of the element dM of the mass M ; the moments of
this element with reference to the planes ofx, z and y, z will be
ydM and xdM. : hence, the co-ordinates x, and y, of the centre
of gravity of the mass M will be determined by the equations
M.x=fxdM, My=fydM..
320 DYNAMICS.
But in the present instance, the centre of gravity is situated
in the axis of z ; and the co-ordinates x, and y, are therefore
equal to zero : hence,
/r^M=0, fy(m.=0.
Reducing equation (333) by means of these values and sub-
stituting M for its equal /c/M, we shall obtain
fr-dM^fr'^dM + Ma^ (334).
The expression y?''2rfM being the moment of inertia with re-
ference to the axis passing through the centre of gravity, we
conclude that when the value of this moment has been found,
that of the moment of inertia /r^cZM, taken with reference to
a parallel axis, may be immediately determined, by adding to
the former the product of the mass of the body by the square
of the distance between the two axes.
The equation (334) may be written under the form
y-..<M=M(-qf+„=),
and this expression maybe simplified by putting-^ — :^ =k^.
Adopting this notation, the moment of inertia taken with
reference to any axis will be expressed by the formula
Of the Motion of a Body about a Fixed Axis when acted
upon by Incessant Forces.
593. Let us now suppose that the several material points
of a system which is retained by a fixed axis kz {Fig. 210),
are acted upon by incessant forces : each particle m will
describe about the fixed axis, the arc of a circle mno^ the
plane of which will be perpendicular to this axis, and will
intersect it at a point C. Let <p denote the incessant force
acting upon the particle w, and S" the angle TmP formed by
its direction with the tangent to the circle mno at the point m.
The force 4» may be resolved into three components ; one
parallel to the fixed axis, which will have no tendency to turn
the body about this axis ; a second directed along the radius
VARIED MOTION ABOUT AN AXIS. 321
mC, which will be destroyed by the resistance of the axis ;
and a third coinciding in direction with the element of the
curve described by the particle m : this last component will
be expressed by (p cos ^, and will be the only portion of the
force <p which tends to turn the system about the axis Az.
Let ta represent the angular velocity of the system at the
expiration of the time /, and r the distance Cm of the particle
m from the axis of rotation : the absolute velocity of m, at
the end of the time if, will be expressed by ru (Art. 579), and
in the succeeding instant dt, this velocity will be increased or
diminished by the action of the incessant force.
If the particle m were unconnected with the other parti-
cles, the force 4» cos ^ would communicate to it in the instant
dt, the velocity represented by <p cos ^.dt] consequently, the
velocity of the particle m, at the expiration of the time t + dt,
would be expressed by
ra + 4» cos S-.dt]
but this particle being connected with the other parts of the
system, its effective velocity at the end of the time t-{-dt
will actually be represented by
ro>-\-rda) ;
and the effective quantity of motion of the particle m will
be {7'ù) + rdw)m.
The same remarks being applicable to the other particles
which compose the system, it is necessary that the quantities
of motion impressed, or
2,[[ru-{-(p cos S'.dt)m]
should, by the principle of D'Alembert, sustain in equilibrio
the effective quantities of motion
l,[(ru-\-rd<i))m\,
the latter being applied in directions contrary to those of the
motions assumed.
But, in order that an equiUbriimi may subsist between
these two sets of forces, it is necessary that the sum of the
moments of the several forces taken with reference to the
fixed axis, shall be equal to zero : and since these forces are
exerted in the directions of the elements of the circles described
by the material points, the radii of these circles will represent
X
322
DYNAMICS.
the perpendiculars demitted upon the directions of the several
forces. The equation of the moments will thus become
l.[{r''<v-\-r<p cos ^. dt)m]—i:[{r'' u + r^ dc^)m]=0 ]
or, by reduction,
1( r" do). m)=2(r<p cos S'.di.m) (335).
The quantities dt and du being the same in all the terms
of this equation, they may be placed without the sign 2 ; and
when the number of terms is regarded as infinite, the value
of each being infinitely small, the character 2 may be re-
placed by the integral sign f, and the particle m by dM, the
diiferential of the entire mass : thus we shall have
dtfr .(pcos^.dM. =do,/r^ dM ;
from this equation we deduce
d<u_ fr .ç cos S'.dM. cv^e^
dt JFdM ^ '''
To complete the integrations here indicated, it is necessary
to know the positions of the elements which compose the
body, and the directions and intensities of the incessant forces
exerted upon each particle. These particulars will be exam-
ined in the following section.
Of the Compound Pendidum.
594. The compound pendulum, represented in Fig. 211, is
composed of a body, or a system of material points, connected
together in an invariable manner, and supported by a hori-
zontal axis KL. When the body is turned around this axis,
the points m, m', m", &c. describe arcs of circles 7nn, in'n',
Qn"n", &c. ; the centres of these circles are situated in the axis
KL, and their planes are perpendicular to it.
595, The motion of the pendulum being referred to three
rectangular axes, let the axis of z be supposed to coincide
with the horizontal line Cz {Fig. 212), about which the body
turns, and the axis of x to be vertical ; the plane of 0, y will
then be horizontal. If we suppose the incessant force exerted
upon each particle to be that of gravity, we shall have
^=^'=^"=(fcc.=g-.
COMPOUND PENDULUM. 323
The direction of tlie force which sohcits a particle m, being
parallel to the axis of .r, the intensity of this force may be
represented by a portion nig of a vertical line ; the angle <^
will be equal to ^mg ; and if the perpendicular mD be de-
niitted upon the axis of x, the angles CmD and 'Tmg will be
equal to each other, being each the complement of the angle
TmD: hence, C7nD=J'; and consequently, the equation
wiD=Cm xcos CmD
will become
mD=Cm . cos<^;
or,
y=r . cos <^.
The values of cos ^ and ç> being substituted in equation (336),
we obtain
dç^_/gpdM .
dt fr^dM '
or, since g is constant,
dco^gfi/dM
dt .fr^dM^
The expression ydW represents the moment of the elementary
mass dM taken with reference to the plane oî x, z\ if, there-
fore, we denote by y, the distance of the centre of gravity of
the entire mass M from the same plane, we may replace fydM.
by My,, and the preceding equation will then become
d^_ gWy, ^^
dt-J^l ^'^*^^^-
and since //'^6?M expresses the moment of inertia with refer-
ence to the axis C2;, this moment may be represented (Art.
592) by M(Â:2+a^). Substituting this value in equation
(337), we find
^ = , ^y' (338).
596. It has been shown (Art. 592), that the quantity a in
the expression M(a- + k"^ ) represents the distance CG (Pig.
209) between the axis CK and the parallel axis GF passing
through the- centre of gravity. But, by the motion of the
system, the centre of gravity describes a circle having its
radius CG=a {Fig. 213), and its plane arCL perpendicular to
324
DYNAMICS.
the axis CK ; hence, the ordinate DG will represent the
quantity y,, and we shall have from the property of the circle,
Again, if s denote the arc described by the point G, the velo-
ds
city of this point will be expressed by — - : but this velocity
at
will also be expressed by a» (Art. 579). Hence, we shall
have
ds
and consequently,
ds
«= — —'
adt
The values of a and y^ being substituted in equation (338),
convert it into
ade kr-^w"
597. If we multiply each member of this equation by 2ac?5,
the first member will become an exact differential, and we
shall obtain by integration,
%' ""' ""' ^y^FS^^*^^^"^^'""^''^ ^^^^^'
The integral of the second member can only be obtained
after eliminating one of the two variables which it contains :
this may be effected by means of the equations
ds=^{dx^- -\-dy;-), y=y/{2ax,—x;-) ;
and by proceeding as in Art. 465, we find
, _ — adx,
^(2ax,—x,^) '
substituting this value in equation (339), we have
V2 = — / , ^^ dx, ;
whence, by integration,
^^^_2a2^ (340).
To determine the value of the constant C, let EB=6 repre»
COMPOUND PENDULUM. 325
sent the value of .x\ at the instant when v~0 ; the supposition
of w=0 and x=b gives
C =
2ar-gb .
k^-+a '
and the
equation (340)
will therefore become
v^, or
dp
-'^"/) ;
whence,
dt— ■ 7 ^
ds
dt=-
This equation can be readily integrated when the oscillations
are performed through very small arcs, as usually happens ;
for, by replacing ds by its value — -!_ obtained on the
y/{2ax)
supposition that x, may be neglected as exceedingly small in
comparison with 2a, in the expression
» — CiUiX I
^~ y/(2ax,—x,^)'
th^ equation (341) becomes
jdx^
which may be written under the form
di=-.^(^^l±^) X ,f - , , (342).
'^ \ ag J ^[{b-x)x]
598. By comparing this equation with the equation (228),
it will appear that they differ only by the constant factor,
/ ( k"^ _L^2\
which in the former is \ \/ \ j , and in the latter
-\/ — Hence, the integral of (342) may be immediately
obtained from that of (228), the constants being determined
by the same condition, that when jf =0, x,=b. Consequently,
if we denote by I the length of a simple pendulum, or if we
replace — in equation (228) by - -, and determine I by the con-
dition 2§
326 DYNAMICS.
g «^
the simple pendulum and the compound pendulum will per-
form their oscillations in the same time. The preceding
equation gives
a
Thus, by means of this formula we can always find the length
of the simple pendulum which will perform its oscillations
in the same time as a given compound pendulum.
599. If, at the distance I from the axis of suspension AB, a
line EF {Fig. 214) be drawn parallel to the axis AB, this
parallel will enjoy the property, that all points contained in it
will perform their oscillations in the same time as though
they were unconnected with the other points of the body.
When the line EF is contained in the plane passing through
the axis of suspension AB and the centre of gravity of the
body, this line is called the aads of oscillation^ and its several
points are called centres of oscillation.
600. The axes of suspejision and oscillation are recipro-
cal ; that is to say, if we take the axis of oscillation EF
{Fig. 214) as a new axis of suspension, the corresponding
axis of oscillation will coincide with the original axis of sus-
pension.
To demonstrate this property, we resume the expression for
CDj the distance between the axes of suspension and oscilla-
tion given in Art. 598,
l^ a^+k^ .343)
a
If we then assume the line EF as an axis of suspension,
and represent by V and a' the corresponding distances of the
centres of oscillation and gravity from this axis, we shall have
by the nature of the centre of oscillation,
n'- -4-Z-2
1'-= ^ ^ (344).
a'
And since the equation (343) indicates that the distance I
exceeds a, it follows liiat the centre of gravity will be situated
COMPOUND PENDULUM. 327
between the axes of suspension and oscillation. We shall
therefore have the following relation,
a-{-a'=l,
or,
a'=l — a.
By means of this value, the equation (344) becomes
l,^ (l-a)'-\-fc' .345)^
I— a
Again, from equation (343) we have
7 ^'
l^a= — ;
a
and the value of I' may therefore be changed into
I'
or, by reduction,
l'='L + a=l
a
consequently, when the line EF is taken as the axis of sus-
pension, the axis of oscillation KH is situated at a distance
MX from the line EF, precisely equal to that which separates
the axes AB and EF.
601. The equation (343) gives
a{l—a)=k^ ;
and by replacing I — a by its value a', we have
aa!-=k^ :
but the value of /j^^ which is dependent on the moment of in-
ertia taken with reference to an axis passing through the centre
of gravity, and parallel to the axis AB, will remain constant
so long as the direction of the axis remains unchanged : hence
it appears that if the body be caused to oscillate about any
axis parallel to AB, and at a distance from the centre of
gravity represented by a, the corresponding axis of oscillation
will be found at a distance a' from the centre of gravity ; thus
the value of a-\-a\ or the length of the equivalent simple
pendulum, will be the same as when the oscillations were per-
328 DYNAMICS.
formed about the axis AB. A similar remark is applicable to
all those axes parallel to AB which are situated at a distance
a' from the centre of gravity. If, therefore, the body be sus-
pended successively from any number of axes parallel to AB,
and at a distance from the centre of gravity equal to a or a',
the times of oscillation about such axes will be equal to each
other.
These parallel axes of suspension about which the oscilla-
tions are performed in equal times, will evidently be found
in the surfaces of two cylinders having a common axis pass-
ing through the centre of gravity.
602. The expression for the distance I between the axes of
suspension and oscillation may be put under the form
Wi Ma '
and since this value is precisely equal to that which was
obtained for the distance of the centre of percussion from the
axis of rotation (Art. 585), it appears that the centre of per-
cussion, when it exists, will be found upon the axis of
oscillation.
Of the Motions of a Body in Space when acted upon by
Impidsive Forces.
603. In the preceding sections, the circumstances of
motion of a body retained by a fixed axis have been alone
discussed ; it now becomes necessary to consider the motions
of a body in space when unconnected with fixed objects.
IjCt m, ni', m", (fcc. represent material points composing a
system whose several particles are unconnected, and let v, v',
v", &c. represent the velocities respectively impressed upon
these particles in directions parallel to each other : it is
required to determine the motion of the common centre of
gravity of the system.
If a plane be passed through the primitive position of the
centre of gravity parallel to the common direction in which
the impulses are applied, the sum of the moments of the
particles m, m', rii'% (fcc, taken w\û\ reference to this plane,
Mall be equal to zero at the commencement of tlie motion ;
PERCUSSION. 329
and it is likewise evident that this sum will remain equal to
zero during the motion, since the distances of the bodies from
the assumed plane remain invariable. Hence, the motion of
the centre of gravity will be confined to this plane ; and since
the same may be said of any other plane drawn through the
primitive position of the centre of gravity and parallel to the
direction of the motions, it follows that the centre of gravity
will continue in each of these planes, or in their line of inter-
section ; and we therefore conclude that the motion of the
centre of gravity of such a system is rectilinear , and parallel
to the direction of the m,otions of its several parts.
Let a plane be drawn perpendicular to the direction in
which the bodies move, and represent the distances of the
several bodies from this plane at the commencement of the
motion, by S, S', S", &c. : their distances, at the expiration of
the time t, will be expressed by
S+î)^, S'-f v'/, S"+D'7, (fee.
If a and x, represent the distances of the centre of gravity
of the system from the perpendicular plane, at the commence-
ment of the motion, and at the end of the time t, we shall
have, by the property of the centre of gravity,
wS -{- m'S' -j- m"S" + «fee. = (m 4- ^W'' • I- W 4- <fec. )a,
m{^^-vt)-\-m'{^' -{-v't)-[-m"{^" ■\-v"t)-\-&^c,=
{m-\-m' -\-'in" -\-ÔLc)x , ;
and by subtraction, we obtain
{m-\-m' -\-m" -\-éLC.)[x, — a)={niv-\-m'v'-'rm,"v"-\-&Lc)t:
hence, it appears that the space passed over by the centre of
gravity is proportional to the time, or the motion of the centre
of gravity is uniform.
It is to be understood that those velocities are regarded as
negative, whose directions are opposite to such as we con-
sider positive.
604. The preceding equation may be written under the
form
^x—a
X, — a
{m-{-m! -{-mf'-^-oi^c)— =wu + wV-f- wV4-<fec. ;
the expression — represents the velocity of the centre of
330
DYNAMICS.
gravity, and is independent of the positions of the particles
m, m', m", <fcc., to which the quantities of motion mv, Tti'v',
tn"v"^ &c. are respectively apphed : it follows, therefore, that
if we suppose a mass M equal to the sum of the masses m,
m\ 7Ji'\ (fee. to be concentrated at the centre of gravity, the
quantity of motion of this mass will be equal to the sum of
the quantities of motion in the entire system.
We also conclude, that the centre of gravity will have the
name Tnotion as though the several masses m, m\ m", Sf'c.
were concentrated in this jjoint, and the several forces applied
immediately to it in directions parallel to those along which
they were originally applied.
605. When the forces applied to the different particles are
not parallel, ;hey may be resolved into components parallel to
three rectangular axes, and since the effects produced by each
system of parallel components will be independent of the
other two systems, it may in like manner be shown that the
motion of the centre of gravity parallel to each of the axes
will be uniform, and equal to that which would be produced
by concentrating the masses at the centre of gravity, and
applying the several forces directly to that point.
606. Let the several masses be now supposed connected in
an invariable manner, the same property will be equally true.
For, let mv, mv', m"v", (fee. represent the quantities of motion
impressed upon the particles m, m', m!', (fee, and let each of
these quantities of motion be resolved into components mu
and wU, (fee, the first of which shall be the effective quantity
of motion retained by the particle, the second being 'Ipstroyed
by the mutual connexion of the parts of the systein : then,
since the quantities of motion mxi, m!u', m"u", (fee, commu-
nicated to the masses m, m', m!', (fee, produce their full effects,
these masses will move under their influence, in the same
manner, whether we regard them as free or connected.
Hence, it appears that the centre of gravity of the system
will move in the same manner as though the quantities of
motion mn, mJii!, in"u", (fee. were applied directly to it. The
quantities of motion ii\\}, m'\}', m"U", (fee. being such as to
destroy each other when applied to the different points m, vi'^
PERCUSSION. 331
tn"^ (fcc, they must (Arts. 54 and 130) destroy each other
when appHed to the centre of gravity.
But the two systems mi/, m'u'j m"u", (fcc, WiU, wt'U', m"U",
«fee, may be replaced by the original system mv, m'v', m,"v'\
&.C., and we therefore conclude that the centre of gravity will
have the same motion as though the several masses had been
concentrated at that point, and the original quantities of
motion niv, m'v', m"v", ^-c. impressed immediately upon it.
607. If an impulse P be communicated to any point of a
body in a direction not passing through the centre of gravity,
this centre will assume a motion precisely equal to that which
would have been produced by the direct application of the
force to it. But a motion of rotation will also be commu-
nicated to the body ; for, if an equal force Q, [Fig. 215) be
applied to the centre of gravity in a parallel and opposite
direction, the joint action of the two forces P and Q, will
maintain the centre of gravity at rest. From the centre of
gravity G demit the perpendicular GA upon the direction of
the force P, and lay off on the opposite side of the point G a
distance GB=AG. liCt the force d be then resolved into
two components, each equal to iQ, or |P, applied at the
points A and B. The forces P and \Q, applied at the point
A, and acting in contrary directions, will have a resultant
equal to ^P ; thus the body will be acted on by two forces
each equal to |P, acting at the distance AG=BG from the
centre of gravity, and tending to turn the body about that
point. And since the point G may be regarded as fixed, the
two forces will have the same effect to turn the body about
that point as the single force P acting at A. The effect of
the force d will be simply to destroy the motion of transla-
tion, without affecting the motion of rotation.
Hence we conclude^ that when a body receives^ an impulse
in a direction which does not pass through the centre of gra-
vity, that centime icill assume a motion of translation as though
the impidse were applied i^nmediately to it ; and the body
will likeioise have a ynotion of rotation about the centre of
gravity, as though that point were immoveable.
608. The circumstances of motion of a body which is
divided symmetrically by a plane passing through the direc-
332 DYNAMICS.
tion of the impulse can now be readily determined. For, the
motion of translation of the centre of gravity will be similar
to that of a material point to which an impulse is applied ;
and the motion of rotation being precisely the same as that
which would take place if a fixed axis passed through the
centre of gravity, perpendicular to the dividing plane, it will
merely be necessary to apply the results obtained in Arts. 581
and 582.
Let Mv represent the quantity of motion impressed upon a
body whose mass is represented by M (Fig. 216), a.ndp the
perpendicular distance from the centre of gravity G to the
line of direction of the impulse. The centre of gravity will
assume a uniform motion with the velocity v, in a direction
parallel to that of the impulsive force. The angular velocity
will result immediately from equation (331), and will be
expressed by
Mvp vp
609. The absolute velocity of each point of the body will
be compounded of the two velocities of translation and rota-
tion. Thus, the point O, for example, to which the force is
applied, has two velocities ; a velocity of translation Oi equal
to that of the centre of gravity, and a velocity of rotation ih
about that point ; so that if we assume any point on the line
OGC, at a distance a from the centre of gravity, its velocity
will be expressed by v±a6>: the superior sign applies to
those points which are situated upon the same side of the
centre of gravity as the point O ; and the inferior sign to
points situated on the opposite side.
610. If we consider the motion of the point O for an ex-
ceedingly short interval of time, the path Oih described by
this point, whilst the centre of gravity describes the line GG',
may be regarded as a right line : thus, the line OGC will
assume the position AG'C, the point C remaining at rest
during this interval. This point is called the centre of spon-
taneous Isolation : its position may be determined by the con-
dition that its velocity of rotation shall be equal to that of
translation : indeed, whilst the point C would be carried for-
ward over the line CC by the motion of translation, it would
FREE MOTION OF A SYSTEM, 333
be moved backward through the same distance by the motion
of rotation : this condition will give the absolute velocity of
the point C
V — aa>=0 ;
whence,
and we therefore have
OC=OG + GC=»+a-=» + — ;
P
from which we conclude, that the centre of spontaneous rota-
tion will coincide with the centre of percussion, if the axis of
rotation he supposed to pass through the point O.
611. When the plane passing through the direction of the
impulse and the centre of gravity divides the body into two
portions which are not symmetrically situated with respect
to this plane, it will usually occur that the axis about which
the body revolves will not retain an invariable position.
For, the rotatory motion of the body will develop in each
particle a centrifugal force, producing a pressure upon the
axis ; and unless these pressures are such as to destroy each
other, the direction of the axis will necessarily be changed.
Of the Motions of a System in Space when acted upon by
Incessant Forces.
G12, We will next investigate the circumstances of motion
in a system whose different particles are acted upon by inces-
sant forces. Let the force acting on a particle m be resolved
into three components X, Y, Z, respectively parallel to three
rectangular axes ; that acting on m' into the three X', Y', Z',
(fee. Let a, 6, and c represent the variable co-ordinates of the
centre of gravity referred to the fixed axes, and let three axes
be drawn through the centre of gravity, parallel to the fixed
axes, and moveable with the system in space. Then, if x, y,
z, x', y\ z\ &c. denote the co-ordinates of the points m, m\
m", &c. referred to the moveable axes; a+a:, 6-fy, c + 5?,
a-\-x', h-\-y\ c-\-z', &c. will express the co-ordinates of the
same points when referred to the fixed axes.
334 DYNAMICS.
613. The velocity of the particle m in the direction of the
axis of X, at the expiration of the time t^ will be expressed by
dla-\-x) da-\-dx
v=—^ = :
dt dt '
and in the succeeding instant dt, this velocity would receive
the increment Xdt, by the action of the incessant force X, if
the praticle m were entirely free ; but in consequence of the
connexion existing between the diiferent parts of the system,
the effective velocity communicated to the particle m in the
time dt, will be expressed by
, , da-\-dx
dv—d ; .
dt
and the velocity destroyed in the particle m, by the connexion
of the parts of the system, will therefore be
Xdt-d ^^.
dt
The same remarks being applicable to the velocities parallel
to the axes of y and z, we shall have for the quantities of
motion destroyed in the particle m, parallel to the three axes,
m\
m
Similar expressions may in like manner be obtained for the
quantities of motion lost by the other particles ; and we shall
therefore obtain, for the sum of the quantities of motion lost
parallel to the axis of x,
2[m(xdt-d ^^'^)] (346);
or, by completing the differentiation indicated, regarding dt
as constant, we have
In like manner, the sums of the quantities of motion lost in
directions parallel to the axes of y and z, will be expressed by
.[„(y..-'^/L^)] (34r),
FREE MOTION OF A SYSTEM. 335
4.(z..-*£±*f)] (348)
The quantities of motion (346), (347), (348), or the forces
capable of producing them, being such as to destroy each
otlier, they must satisfy the general equations of equilibrium
(66) and (67), which appertain to a system of forces having
various directions and applied to different points of a body.
The equations (66) indicate that the sum of the components
parallel to each of the axes will be equal to zero ; we shall
therefore have for those components parallel to the axis of a:
;Kx<._^ii^^)]=0;
(349 a) :
(349 b).
or, by multiplying by dt, and changing the form of the expres-
sion, we have
Q={mX-{-m'X'-\-m"X"-\-iSLC.)di^ —d^ a(m+m'+m"+&c.)
—{md^x+m'd'x'+m"d^x"-\-&c.) (349).
But, by the nature of the centre of gravity,
mx+m'x'-\-m"x"-{-ôcc.=0 )
my-\-m'i/'-\-m"y"-\-ôùC.=0 ) ' '
and by differentiating twice, we find
md'x+m'd^x'-{-m"d^x"-\-ôùC.=0
md'i/+m'd''y'-\-m"d' y"+&c. =
The first of these values being substituted in (349), and the
mass of the system being denoted by M, there will result
Md^a=(mX+m'X'-\-m"X"+&c.)dt',
or,
M^=2(mX):
dt^ ^ '
the same being true with respect to the components parallel
to the axes of y and z^ we shall obtain, for the three first
equations expressing the circumstances of motion of the
system,
M^=.(»X) 1
i
M^-2(mY) \ (350).
MÇ^=2(mZ)
336 DYNAMICS.
These equations serve to determine the motion of the centre
of gravity of the mass M; for when integrated, they will ex-
press the velocities -r > ;r j ;t- ^^f the centre of gravity, par-
allel to the three axes.
614. The equations (350) make known a remarkable
property of the centre of gravity. For, let the particles m,
m\ m'\ (fee be supposed concentrated at their common centre
of gravity, and let the forces wiX, ?/iY, mZ, w'X', m'Y', m'7i\
&c. be applied directly to that point, parallel to their original
directions. These forces may be reduced to three, MX,, MY,,
MZ„ the values of which will result from the equations
MX=2(mX), MY=2(mY), MZ=2(mZ.)
Eliminating the second members of these equations by means
of equations (350), we have
^=^» 1^'=^» '^-^ (^^'>-
But when the forces MX,, MY,, MZ, are applied to the centre
of gravity regarded as a material point whose mass is M, the
circumstances of its motion are expressed by the equations
(180), which are precisely similar to the equations (351) ;
hence, we conclude that the centre of gravity of the system
has the same motion as though the forces were applied directly
to that point.
615. To determine the circumstances of motion of the
several particles m, ra\ m", cfec. with respect to the centre of
gravity, we resume the equations (67), which express the
conditions that the forces have no tendency to turn the sys-
tem about either axis : that this may be the case, it is neces-
sary that the sum of the differences of the moments of the
components parallel to any two of the axes, as x and y, taken
with reference to the corresponding planes of y, z and a-, z,
should be equal to zero. But if we consider the particle m,
the distance of the component X, which acts upon it, from
the plane of a-, z will be equal to y-f6, the co-ordinate of the
point m, parallel to the axis of y : in like manner, the distance
of the force Y from the plane of y, z will be expressed by
x+a: we shall therefore have, for the difference of the
moments,
TU
FREE MOTION OF A SYSTEM. 337
The same remarks being applicable to the particles m\ m",
&c., we shall obtain a similar expression for each. By-
placing the sum of these expressions equal to zero, as in
equation (67), performing the multiplications, and reducing
by means of equations (349 a) and (349 6), we shall obtain
62(mX) -M6|^+2(myX) -2 (^y^)
— a2(mY)+Ma^-2(ma:Y)+2^w^:^•^^ =0.
This equation admits of simplification ; for, if we multiply
the first of equations (350) by 6, and the second by a, and
take their difference, we shall have
62(mX)-a2(mY)— Mô^ + M«— =0.
This relation reduces the previous equation to
2(myX)-2(m;2;Y) -2 (^y^) +2 {jnx^^ =0 ;
whence,
1 (m î^!ty^) =x[«(Yx-Xy) A].
The integral of the first member, taken with reference to the
time L is
;(m-!^^^^):
and by adopting the same process with reference to the other
two axes, putting, for brevity,
l.[mf(Yx-^Xy)dt]='L,
j\mf{Zx—Xz)dt\='^,
^mf{Zy-Yz)dt\=^,
we shall obtain the three equations of motion
^/ ri^-^\ ^ |. (351a).
Y 29
J
338 DYNAMICS.
The equations (351 a) are independent of the co-ordinates of
the centre of gravity, and would undergo no change if forces
were appUed at that point sufficient to destroy its motion of
translation, since such forces would not enter into the ex-
pressions L, M, and N ; thus, the motion of rotation about
the centre of gravity, determined by these equations, is pre-
cisely similar to that which would take place if the centre of
gravity were immoveable.
Hence we conclude, that whe?i arvy body is acted upon hy
incessant forces applied to its several particles, the body will
receive two motioiis : one of translation, in virtue of which its
centre of gravity 2vill be transported in space as though the
forces were apjjlied directly to that point ; and a seco?id, of
rotation about the centre of gravity, as though that point were
absolutely at rest.
General Equations of the Motions of a System of Bodies.
616. Let Z, /', l", &.C. represent the velocities lost or gained
by the several material points which compose a system, in
consequence of the mutual connexions of its parts ; the cor-
responding quantities of motion lost or gained will be ml,
m'l', m'l", «fcc, and, by the principle of D'Alembert, these
quantities of motion, when impressed upon the particles
m, m', m", &c. are such as will produce an equilibrium : hence,
they must fulfil the conditions of equilibrium expressed in
equations (66) and (67).
The components of these quantities of motion, or the
forces capable of producing them, estimated in the directions
of three rectangular axes, will be
ml cos», 7w/cos/3,' ml cos y components of ml.
m'l' cos », m'l' cos /3', m'l' cosy' components of m'l'.
m"l" cos »", m"r cos /3", m"l" cos y" components oîm"l".
(fee. (fee. &c. (fee.
We shall therefore have for the equations of equilibrium,
2(mZ. cos «)=0 i
2(mZ.cos(s)=:0 V (352).
1{m.l . cosy)=0 J
FREE MOTION OF A SYSTEM. 339
^[ml{x COS /3 — y cos a)] =0 i
^ml(z COS »—x COS y)] = > (353).
2[m% cos y — 2; cos /3)] =0 ^
617. If the system is retained by a fixed point, the three
equations (352) cease to be necessary ; the equations (353)
being alone sufficient, provided the origin be placed at the
fixed point.
618. When there are two fixed points within the system,
we connect them by a right line, and assume this line as one
of the co-ordinate axes, z for example ; the first of equations
(353) will then be sufficient to ensure the equilibrium (Arts.
132 and 133).
619. The velocities lost or gained are here indicated by the
letters /, l', I", &.c. ; but to express these quantities in functions
of the incessant forces which solicit the several material
points, we shall first consider the particle m, and suppose that
the forces acting upon this point have been reduced to three,
X, Y, and Z, respectively parallel to the co-ordinate axes.
The velocity of the particle m, parallel to the axis of x, at the
expiration of the time t, will be expressed by -j- (Art. 430) ;
and at the end of the time t-{-di, this velocity will become
-^J^d — ; this will be the expression for the effective
dt dt' ^
velocity of the particle m.
But if the particle m were perfectly free, the incessant force
X would communicate to it in the time dt, a velocity repre-
sented by ILdt (Art. 391), and the velocity of m at the expira-
tion of the time ^4-<^^j would be expressed by — +Xd/;
(XZ
hence, the velocity lost or gained by the particle m will be
equal to
dx , ■^j. (dx jdx\
and by reduction, we shall find that Xc?/— rf^— will express
the velocity lost or gained by the particle m, in the direction
of the axis of x. This velocity being multiplied by the mass
m, gives
Y2
840
m
DYNAMICS.
(x..-.^),
for the quantity of motion lost or gained by m, in the direc-
tion of the axis of x : we shall therefore have
ml . cos ct=m { Xdt r— Ï
V dt /
(354).
In like manner, by considering the velocities lost by m, in
directions parallel to the axis of y and z^ we shall find
ml. cos ^=m I Y dt — -^j (355).
7nl. cosy=m(Zdt—-—\ (356).
Similar expressions may be obtained for the quantities of
motion lost or gained by the particles m', m", «fcc. ; and by
including their sums under the sign 2, the equations (352)
and (353) may be reduced to
.(.|f)=.(.X)'
2(m^) =2(mY) [ (357).
2(.|£)=.(.z)_
de ^ ^
iK5^!^:=^^^=2[m(X;2-Z:r)] .
dt^
^^^(y^y-^^'^J^2[m(Zy-Yz)]
CLZ
(358).
Such are the most general forms of the equations expressing
the circumstances of motion of a system.
620. The expressions Ya:— Xy, X2; — Zx, 7,y—Yz, &c.
become equal to zero under the following circumstances : 1°.
when the incessant forces acting on the particles m, m', m",
(fee. are equal to zero ; 2°. when all the forces are directed
towards the origin of co-ordinates : 3°. when the forces are
such as arise from the mutual attractions of the différent parts
of the system.
FREE MOTION OP A SYSTEM. 341
In the first case, the incessant forces being equal to zero,
their components must likewise be equal to zero ; and hence
X=0, Y=0, Z=0, X'=0, &c.:
the second members of equations (358) will therefore dis-
appear.
621. The second members will likewise disappear, when
the forces are directed towards the origin of co-ordinates.
For, it has been shown (Art. 436), that when the fixed point
towards which the forces are directed does not coincide with
the origin of co-ordinates, if we represent by a, 6, and c the
co-ordinates of this point, and by jo, p', /?", &c. the distances
of the several particles from the fixed point, the components
of the forces P, P', P", (fcc, in the directions of the co-ordi-
nate axes, will be expressed by
p p p
p p p
P^ P'^-=^, P"f^,&c.;
p p p
but, by hypothesis, the origin coincides with the fixed point
towards which the forces are directed, and we therefore have
a=C, h=% c=0:
hence, the preceding expressions are reduced to
Vx
V'x'
V"x"
P"'-
, &c..
Py
py
P"y"
&c.,
Vz
Vz'
F"z"
&p
v'
p'
P"'
And by substituting these values of the components for X,
X', X", Y, Y', Y", Z, Z', Z", &c. in the expressions
Yx—Xy, Xz-Zx, Zy-Yz, YV-Xy, &c (359),
we shall find each of these expressions equal to zero. Con-
sequently, when the incessant forces which act upon the
several particles are constantly directed towards the origin,
342 DYNAMICS.
the expressions (359) become equal to zero, and the second
members of equations (358) will therefore disappear.
622. The same consequences may be deduced when the
material particles are subjected only to their mutual attrac-
tions. For, by putting the second members of the equations
(358) under the following forms :
m(Y.-r— X2/) + m'(YV-Xy)+(fcc. ^
m(X2;-Za:)+m'(X';s'-ZV)+&c. > (360),
m(Zy — Yz) + m'{Z'y'-Y'z')-[-&.c. )
and considering the material points two by two, it is evident
that the moving force exerted by the point m upon m' is equal
to that exerted by m' upon m. Hence, if X, Y, Z, X', Y', Z',
&c. represent the components of the incessant forces P, P,
P", (fee, we shall have
w'X'= — wîX, m'Y'= — mY, m'7/= — wiZ, &c. :
eliminating X' and Y' by means of these values, the first of
the expressions (360) becomes
mYix-x')-mlL{y—y') (361) :
but the force whose components are X, Y, and Z being
denoted by P, and the distance between the points m and m!
by /?, the cosines of the angles formed by the direction of the
force P with the co-ordinate axes, will be represented respect-
ively, by
x~x' y—y' z—z\
p P p
and we shall have
,x-—x -^ j^y y „ -ç^z z
X=pl— ^, Y=P^— ^, Z=P
P p P
Substituting these values in the expressions (361), we obtain
mV .- — —{x—x')—mV. iy—y) ]
a quantity evidently equal to zero.
In like manner, it may be proved that the other terms of
the expressions (360) destroy each other ; it therefore follows,
that when the material particles m, rn', m'\ &.c. are subjected
only to their mutual attractions, the second members of the
equations (358) will disappear ; and since this result is inde-
FREE MOTION OF A SYSTEM.
343
pendent of the position of the origin, that point may be
selected arbitrarily.
623. When either of the three cases just considered
presents itself, the equations (358) will reduce to
dt' '
I,['m(zd^x—xd^z)]_f.
df^ '
2[m{yd^z-zd^p)] _^
dp
The quantities included within the brackets being exact dif-
ferentials, these equations may be written under the form
2[*w . d{xdy — ydx)\_ç.
~~dP " '
2[m . d(zdx — xdz)] _ „
^[m.d{ydz—zdy)] _
dt^
And by multiplying by dt, and integrating with respect to the
time, denoting the arbitrary constants by a, a', and a", we shall
have
'S\m{xdy—ydx)\=adt ^
'l[m,{zdx—xdz)\=a'dt \ (362).
'2[m{ydz —zdy)'\ = a"dt ^
624. To understand the signification of these integrals,
draw the three rectangular axes Ax, Ay, and kz {Fig. 217),
and call AP=:r, VQi—y : let AQ, the projection of the radius
vector Am on the plane of x, y, be denoted by r, and the
angle formed by AQ with the axis of a; by tf ; the infinitely
small arc Q,Q,' described with the radius r will be expressed
by rdè ; the right-angled triangle APQ, gives
a;=r.costf, y=r .svni;
and, by differentiating, we obtain
dx = —r . sin 6 .d6 + cos ê . dr,
dy=r . cos ê . dê+sia ê . dr.
Substituting these values in the expression xdy —ydx, we find
xdy - ydx =r^d6=2x^r'Xrd6=2.a.Te3L Q AQ,' ;
and therefore,
344 DYNAMICS.
m{xdi/—ydx)=2m{a.re2iCiAQ,').
By forming similar products for the other masses m', m", <kc.,
we shall find that the quantity ^[m{xdi/ —yds)] is composed
of the sum of the products formed by multiplying each mass
m, m', m", <fcc. by twice the area of the elementary surface
described by the projection of its radius vector Am on the
plane of a;, y, in the time dt.
625. If we integrate again with respect to the time, the
equations (362) will give
/l [m {xdy — ydx)\ = at + b ^
fi[m{zdx-xdz)] = a't + b' \ (363) ;
/2[m{ydz-zdy)] = a"t + b" ^
and if the areas described be supposed to commence from the
instant when ^=0, the constants 6, 6', and 6" will be equal to
zero, and the preceding equations will reduce to
fl^ini^xdy—ydxy^—atj
y 2 [m {zdx — xdz)\ = «7,
f^^,n{ydz-zdy)\=a"t.
These equations express that the sums of the products formed
by multiplying each mass by the projection of the area de-
scribed by its radius vector, are constantly proportional to
the tirn^es em/ployed in describing these areas.
This enunciation contains the principle of the preservation
of areas in its most general form.
626. The system here considered has been supposed free ;
but if it were retained by a fixed point, the equations (358)
would only be applicable when the origin was taken at this
point : the same may be said of equations (363), which result
from (358). Thus the principle of areas then becomes less,
general, the origin being no longer arbitrary.
627. It has been shown (Arts. 132 and 133) that when the
system contains two fixed points, it will be necessary to sat-
isfy but one of the general equations of equilibrium (67).
The same is true with respect to equations (358) ; and there-
fore but one of the equations (362) will be satisfied : thus, the
principle of areas is only true in this case with respect to one
of the co-ordinate planes.
628. By comparing the results obtained in Art. 155 with
FREE MOTION OF A SYSTEM. S4S
those of Art. 153, we shall find that the quantities A, B, and C
represent, in Art. 155, the sums of the moments of the pro-
jections of the forces on the co-ordinate planes, these mo-
ments being taken with reference to the origin. Thus these
sums will be the same as those denoted by a, a', a" in equa-
tions (362). Hence, the sum of the projections on the prin-
cipal plane given by equation (79), will, in the present instance,
be expressed by
/ / {i:[m{xdy—ydx)'\) " (^[m{zdx—xdz)f) ^ (^[m{ydz—zdy)]) ^ \
^ \ dT^ ' dv' ' dt^ /
This expression may be simplified by putting it under the
form
v'(a" +«"+«'");
and replacing the functions A, B, and C in equations (81) by
their values a, a', and a", we obtain the following expressions
for the cosines of the angles formed by the principal plane
with the co-ordinate planes :
a ^ a'
cos a= -, ; — ;— - — r-r-> COS /3
a"
cos y= .
The angles «, /s, y are constant ; and hence we conclude that
the position of the principal plane remains invariable during
the motions of the several particles of which the system is
composed.
General Principle of the Preservation of the Motion of the
Centre of Gravity.
629. In discussing the circumstances of motion of a sys-
tem of material particles, acted upon by incessant forces, it
was proved that the centre of gravity of the entire system
has the same motion as though the several forces were
applied directly to that point. Thus, denoting by x„ y„ and
z, the variable co-ordinates of the centre of gravity, we shall
have, as in Art. 614,
MX,=s(mX), MY,=2(mY), MZ=2(mZ) (366).
346 DYNAMICS.
and,
'^=^" ^■=^" ^'=^ (^«'^-
630. If the material points which compose the system be
subjected only to the action of forces arising from their mutual
attractions, the equations (367) will reduce to
^'^/^n d^y.-a ^'^/_n.
these equations being integrated give
dx, dy, , dz,
—!-=a, -~=h, -r^=Cj
dt dt dt
and by a second integration we find
x, = at-\-a\ y,~bt + b', z=ct-{-c/.
eliminating t, we have
x—a'=-{z—c'), y—h'=-{z—d).
c c
These equations appertain to a right line in space, and the
motion of the centre of gravity will therefore be rectilinear.
This motion will also be uniform ; for we have the velocity
of the centre of gravity expressed by
y(dxldyldzl\
which is evidently a constant quantity.
631. If the masses m, m', m", (fee. be subjected to the action
of constant forces whose directions are parallel to a given
line, we may adopt this line as one of the co-ordinate axes, z
for example, and the equations expressing the circumstances
of motion of the centre of gravity, then become
^=0, ±^1=0, t^=Z;
df" ' dt^ ' dt"" '
and it may then be proved, as in Arts. 518 and 519, that the
trajectory described by the centre of gravity is a parabola.
632. Finally, it may be shown that if two or more of the
bodies composing the system impinge against each other
during the motion, the velocity of the centre of gravity will
remain unchanged. For, by the nature of the centre of grav-
ity, we have
FREE MOTION OP A SYSTEM. 347
Mx^—l{mx), My=-S.{my), M.z,=:L{mz):
differentiating with respect to the time t^ we obtain
M^-l;=^(J-^\ M^l^=^(Jy\ M^-§^=4m^-pi.
dt \ dt)' dt \ dt )' dt \ dt)
And if we denote by a, a', a", &c. the velocities of the parti-
cles m, m', m", (fee. before the colhsion, and by A, A', A", <fcc.
the corresponding velocities after collision, these values, sub-
stituted in the first of the preceding equations will give
dx
1(ma)— the value of M—- ' before collision,
dt
dx
i:(mA)= the value of M — ^after collision.
^ ^ dt
Thus the sum of the quantities of motion lost by the impact,
in the direction of the axis of x, will be 2(ma) — S(mA). In
like manner, the sums of the quantities of motion lost in the
direction of the axes of 1/ and z respectively, will be
2(m6)— 2(mB), and 2(?nc)— 2(mC) :
but, by the principle of D'Alembert, these quantities should
maintain the system in equilibrio ; and we therefore have
2(?/ia)— 2(wA)=0, 2(7n6) — 2(mB)=0, 2(mc)— 2(wC)=0;
hence, the expressions — -', -p, ~, which represent the velo-
dt dt dt
cities of the centre of gravity parallel to the co-ordinate axes,
remain unchanged by the act of collision.
633. This property of the centre of gravity, in virtue of
which its motion is independent of the mutual actions of the
parts of the system, constitutes the principle of the preserva-
tion of the motion of the centre of gravity.
PART THIRD.
HYDROSTATICS.
OF THE PRESSURE OF FLUIDS.
634. A fluid is a collection of material particles, which
yield to the slightest eiFort, and which move freely among
each other in all directions.
When the material particles adhere to each other in any
degree, the fluid is said to be imperfect ; in the following"
pages the particles will be supposed entirely destitute of any
adhesion.
635. Fluids are divided into incompressible and compressi-
ble or elastic fluids.
Incompressible fluids are such as always occupy the same
volume at the same temperature ; such are water, mercury,
wine, &c.
Elastic fluids are those whose volumes admit of change by
the application of pressure ; such are the vapour of water,
atmospheric air, and the different gases.
636. Let ABCD {Fig. 218) represent a vessel entirely closed,
and filled with a fluid destitute of weight : if two apertures
EF and HI, having equal surfaces, be pierced in this vessel,
and if pistons K and L be applied to these apertures, and urged
by forces RK and SL, equal in intensity, and directed per-
pendicularly to the surfaces HI and EF, these forces will re-
main in equilibrio. Hence, it is necessary that the pressure
exerted upon the surface EF should be communicated to the
surface HI, through the intervention of the fluid medium ]
and this can only happen provided the particles of the fluid
experience the same pressure at every point of the fluid mass.
Adopting the result of this experiment as a basis, we can
establish the following principle :
30
350 HYDROSTATICS.
The characteristic •property ofjiuids is that they transmit
a pressure applied to them, equally in all directions.
637. To express analytically this property, which is
termed the principle of equal pressure, we shall consider a
fluid mass enclosed in a vessel AL {Pig. 219) having the
form of a rectangular parallelopiped, the base of which is
horizontal. Let a piston be applied to the upper surface
EH of the fluid, and let it be urged downward by a force P,
acting: in the vertical direction : the base of the vessel will
experience the same pressure as though the force were ap-
plied directly to it ; and each portion of the base will support
a pressure proportional to its extent ; so that if A denote the
area ABCD, and a the area Abed, of a portion of this base ;
and if p denote the pressure sustained by a, the value of p
will result from the following proportion,
A : a : : V : p.
Let a represent the unit of surface ; we shall then have
P
hence, if a represent the ratio between the surface Abed',
and the surface Abed assumed as the unit, the pressure P'
supported by the surface Ab'c'd, will be expressed by
V'=p» (381) ;
and since all portions of the fluid mass must sustain equal
pressures for the same extent of surface, it follows that if the
surface containing « units were situated in any other portion
of the vessel, on the sides for example, it would still sustain
the same pressure peo.
638. When the surface pressed is indefinitely small, it may
be represented by the elementary rectangle dxdy ; and the
pressure exerted by the piston on this elementary portion of
the surface of the vessel, will be expressed by pdxdy : this
expression will be equally applicable in whatever portion of
the vessel the element may be situated, and whether the sur-
face be plane or curved.
639. In the preceding paragraphs, the fluid has been sup-
posed subjected merely to the action of the pressure applied
at its surface ; but when the particles of the» fluid are acted
EQUILIBRIUM OF FLUIDS. 361
upon by incessant forces, the pressure will cease to be con-
stant throughout the mass. In this case, the pressure sus-
tained by the fluid arises from two distinct causes : 1°. a
pressure resuhing from the force P apphed to the surface,
and equally distributed throughout the mass ; and, 2°. the
pressure arising from the action of the incessant forces. The
latter pressure is usually different in different parts of the
fluid mass, since each particle may be acted on by a force
having any intensity.
640. To offer an example of this second kind of pressure,
let the fluid contahied in the vessel ABCD {Fig. 218) be con-
sidered heavy : then we must regard each particle as acted on
by the force of gravity.
We shall find in the sequel, in discussing the properties of
heavy fluids, that the principle of equal pressure is greatly
modified by this circumstance. It follows from the preceding
remarks, that the pressure p should in general be regarded as
variable, in passing from one point to another of a fluid mass,
when the particles are acted upon by incessant forces. In
this case, the pressure p exerted at any point whose co-ordi-
nates are ar, y, z, when referred to the unit of surface, must
be understood to denote the pressure which would be exerted
upon a unit of surface, if every point in this unit should sus-
tain a pressure equal to that exerted at the point x, y, z.
General Equations of the Equilihriinn of Fluids.
641. Let a fluid particle solicited by incessant forces be
supposed to rest in equilibrio in a fluid mass, and let it be
required to determine the equations necessary to establish the
state of equilibrium.
For this purpose, let the co-ordinate plane of x, y be
assumed horizontal and above the fluid mass, which we will
suppose divided into infinitely small rectangular parallelo-
pipeds by planes parallel to the co-ordinate planes. Let dM.
represent one of these elements whose co-ordinates are x^ y, and
z : the volume of this element will be expressed by dxdydz ;
and by multiplying this volume by the density D, supposed
constant throughout the element, we shall have D . dxdydz
352 HYDROSTATICS.
for the expression of the elementary mass of the fluid : hence,
we derive the equation
cm=B . d.Td}/dz (382).
If X, Y, and Z represent the incessant forces which act upon
the element dM, and which are supposed constant throughout
the extent of this element, X^^M, YdM, and ZrfM will express
the moving forces exerted upon the elementary parallelopiped,
and these forces, acting conjointly with the pressure sus-
tained by the several faces of the element, should maintain
this element in equilifcrio. Let the superior surface d.idi/ of
the parallelopiped be extended (Pig. 220) until its area
becomes equal to the assumed unit represented by BG ; and
let the pressure p sustained throughout this unit be con-
ceived constant, and equal to that exerted at each point of the
face axdy. When the ordi'^ate BT)=z is changed into
D^—z + dZj the pressure p, which varies with z, will become
and will express the pressure exerted on the unit of surface,
each point of which sustains a pressure equal to that sup-
ported by the points in the base EF of the parallelopiped.
Consequently, to obtain the total pressures on the superior
base BG and on the inferior base EF of the element, we
must multiply the surfaces BG and EF, each of which is
equal to dxdy, by the respective pressures exerted upon their
unit of surface : thus, we shall obtain for the pressures sup-
ported by BG and EF,
pdxdy^ and (ji-\--~dz)dxdy ]
the first of these pressures is exerted downwards, and the
latter upwards. Their difference will be a pressure exerted
upwards, if we suppose the pressure to increase with the
co-ordinate z, and it will be expressed by
-^dzdxdy :
and since this difference should sustain in equilibrio the ver-
tical force Z^iM, we shall have
EaUILIBRIUM OF FLUIDS. 363
^dzdxdy=ZdM. :
dz ^
substituting for dM its value given by equation (382), and
reducing, we find
dz
In like manner, by denoting the lateral pressures on a unit
of surface exerted against the faces dxdz, dydz, by q and r,
we shall obtain
— =:DY =DX.
dy ^ dx
It has been shown (Art. 640) that the pressures exerted upon
any one of the faces is composed of the pressure uniformly
distributed throughout the fluid, and of the pressure due to
the incessant forces. Thus, to estimate the pressure qdxdz,
exerted upon the face dxdz, it is obvious that this pressure
• may be considered as resulting from, 1°. The pressure exerted
upon the superior base, which is transmitted equally through-
out the parallelopiped ; and, 2°. The pressure due to the
incessant forces exerted upon the particles which compose
the parallelopiped. But the pressure exerted upon the upper
base being pdxdy, it will be transmitted to the face dxdz,
exerting a pressure pdxdz proportional to the area of this
face.
The incessant forces being Xé?M, Yc?M, and XdM. respect-
ively, the pressure arising from their joint action will be a
function of their intensities, which we shall represent by
F(X^M, Ydm, ZdM) ;
and we shall thus obtain
qdxdz=pdxdz + F(X.dm, Ydm, Zdm) (383).
The function represented by •
F(Xrfw, Ydm, Zdm)
must be such that it will disappear when the forces are sup-
posed equal to zero : hence it is necessary that every term of
the function should contain at least one of the factors X^M,
YdM, or ZdM. : and by arranging the terms with reference
to the powers of dM, commencing with the least, we may
suppose
Z
354 HYDROSTATICS.
F(X^M, YdM, ZdM)^hX<m + J<iY(M + VZdM+&,c.
Substituting this value in equation (383), we shall have
qdsdz=pd3;dz + hX(M+l^Y(m-^VZ(m-\-&.c. ■
and replacing dM by its equal Ddxd^dz, this equation will
become
qdxdz =pdxdz + T)hKdxdydz + jy^Ydxdydz
-{-BFZdxdydz + âcc:
<iividing by dxdz, there results
q=p-\-'DhXdy+BNYd7/ + T)PZdi/-]-6cc (384).
The terms BLXdy, DNYrfy, BPZdy, &c. being infinitely
small with respect to p, it follows that the equation (384)
may be reduced to
q=p.
In like manner, it may be demonstrated that r=p ; and
hence, the equations of equilibrium will become
$=DZ, i^=DY, i^=DX (385).
dz dy dx
If we multiply these equations by dz, dy, and dx respectively,
and take their sum. we shall obtain an expression for the
diâërential of the pressure, when the co-ordinates x, y, and
z are supposed to vary together ; thus,
dp=^dx^'^-^dy+^dz=B(Xdx+Ydy+Zdz) (386).
dx dy dz
Such is the equation which, when integrated, will determine
the pressure upon the unit of surface at any point of the
fluid.
Application of the General Equations of Equilibrium to
Incompressible Fluids.
642. Let us suppose an incompressible homogeneous fluid
to be in equilibrio in a vessel capable of opposing an indefi-
nite resistance to pressure : the pressure p exerted upon the
unit of surface, at a point whose co-ordinates are x=a, y=b,
z=c, will be determined by substituting the values a, b, and c
for X, y, and z, in the integral of equation (386) : and if the
density D be supposed constant, the determination of the
INCOMPRESSIBLE FLUIDS. 355
value of p will depend on the possibility of integrating the
formula
Xdx + Ydy + Zdz (387).
This integration will always be possible, when the pre-
ceding expression is an exact differential of the variables
X, y, and z.
Let it be supposed that this condition is fulfilled, and that
the pressure at any point on the sides or bottom of the vessel
has been determined ; this pressure will be destroyed by the
resistance of the vessel. But if we consider a point in the free
surface of the fluid, and suppose that no exterior pressure is
applied to the fluid by means of a piston or otherwise, it is
obvious that the pressure at such point will be equal to zero.
The same being true for every point in the free surface of the
fluid, it follows that in passing from any point in the surface
of the fluid to a consecutive point in the same surface, the
pressure j9 will remain invariable, being equal to zero at each
of these points ; hence dp=0, and the equation (38G) con-
dered as applicable to points situated in this surface, will
reduce to
Xda; + Ydi/+Zdz=0 (388).
This equation will likewise appertain to the surface of the
fluid when this surface experiences a constant pressure, that of
the atmosphere for example, since we shall still have dp=0.
It will also subsist for those points within the fluid mass
which su&tain equal pressures.
643. When the expression (387) is an exact diflèrential,
and the equation (388) is satisfied, we shall have ijo=0, and
the pressure, if it exist, must be constant. But, in this case,
in order that the equilibrium may be preserved, it is neces-
sary that the resultant of the forces exerted upon each par-
ticle in the surface, and directed towards the interior of the
fluid, should be normal to the surface of the fluid : for if it
were not, we might decompose this resultant into two forces,
one normal and the other tangent to the surface ; and it is
evident that the latter would impart a motion to the fluid par-
ticle.
644. This condition is likewise indicated by the equation
Z 2
358 HYDROSTATICS.
(388) ; for, let x\ y\ and z' represent the co-ordinates of a par-
ticle in the surface of the fluid, and X, Y, and Z the incessant
forces applied to this particle. The general equations of the
normal to a curved surface at the point x\ y\ z, are
x—x — ■j—{z—z')
ax
y—v = — ; — {z—z)
^ ^ dy' '
(389);
dz'
and if we substitute in these equations the values of -— ^and
CJL3j
dz'
-— determined by equation (388), the equations (389) will
become those of the normal to the surface of which (388) is
the equation. But by regarding X, Y, and Z as functions
of the co-ordinates x^ y, and '2:, and employing the usual nota-
tion, the equation (388) will give
~ dx'~~7l' ~ dy'~ Z'
Substituting these values in (389), we find, for the equations
of the normal at the point x\ y\ z\
x-x'=^{z-z'), y-y'=^iz-z').
These equations are precisely similar to those of the result-
ant cf the forces X, Y, and Z, found in Art 57.
645. The equation (388), when susceptible of being inte-
grated, leads to several remarkable consequences. For, if we
represent the integral of this equation by F{x, y, z)-{-C, and
make C = — A, we shall have
F{x,y, z)=A.
If we assign to A arbitrary values successively increasing,
such as 0, a, a', a", (fee, we shall obtain the equations
F(x,y,z)=0,
F{x,y,z)=a,
F{x,y, z)=a',
F{x, y, z)=a",
F{x, y, z)=a^"\
<fec. (fee.
INCOMPRESSIBLE FLUIDS. 357
Each of these equations being diflferentiated will produce
equation (388), and among them will be found that apper-
taining to the surface of the fluid, which is supposed to have
produced equation (388) by differentiation.
Let this equation be represented by F{x, y, 2;)=a^"' : then
the other equations will appertain to different surfaces, each
of which will possess the property, that the resultant R of the
forces X, Y, Z, exerted upon any particle situated in such sur-
face, will be perpendicular to the surface.
The directions of the forces being cut perpendicularly by
the surfaces of constant pressure, such surfaces are said to be
level. If we suppose the arbitrary constants 0, a, a', of', &c.
to differ by indefinitely small increments, the fluid mass will
be divided by these level surfaces into a series of extremely
thin layers, which are denominated level strata.
646. It follows, from the preceding remarks, that when the
particles of the fluid are solicited by forces constantly di-
rected towards a fixed point, its exterior will assume the
spherical form. The same consequence may be deduced ana-
lytically. For, let the origin be taken at the centre of attrac-
tion, and denote by .v, y, z the co-ordinates of a particle dM. in
the surface of the fluid: the distance of the point t, i/,z from the
origin will be expressed by ^{x^ +f/^ -\-z^). If this distance
be denoted by r, and the force of attraction exerted upon the
particle dM by a, the cosines of the angles formed by the direc-
tion of this force with the co-ordinate axes will be expressed
by—) -> ^^^ - j 3.nd the components of the force a will be
r r
X=-A-, Y=-x^, Z=-A-;
r r r
the negative signs are prefixed to these components because
they tend to diminish the co-ordinates of the particle dM..
By substituting these values in equation (388), we shall
obtain for the diîîerential equation of the surface of the fluid
t(xdx+ydy + zdz)=0 (390).
r
Suppressing the common factor , and integrating, we find
358 HYDROSTATICS.
an equation appertaining to a spherical surface ; hence the
surface of the fluid will be spherical.
647. If the radius of the sphere be very great in compari-
son with the extent of the surface, as is the case when we
consider a small portion of the earth's surface, the curvature
will be insensible, and the surface may therefore be regarded
as a plane.
648. The integration of equation (390) was effected imme-
diately in consequence of equation (388) becoming, in that
example, a particular case of the theorem demonstrated in
Art. 436, relative to forces directed to fixed centres. It is by
virtue of this theorem that equation (388) will always be
integrable in such cases as refer to the equilibrium of fluids
resting upon fixed surfaces.
649. If, in equation (386), we replace the quantity within
the brackets by its equal d[F{x, y, z)], we shall obtain
dp=T>xd[F{x,y,z)]]
or, by division,
^=4F(:r, i/,z)] (391).
But d[F{x, y, z)] being by hypothesis an exact differential,
-^ must likewise be an exact differential ; hence, D will con-
tain no variable except/? ; this condition may be expressed by
the equation
Ty^fp (392).
If the pressure ^ be supposed constant, the density D will
be likewise constant, and (391) will reduce to
d[F{x,7/,z)]=0.
The integration of this equation will reproduce that already
found in Art. 645, the properties of whic^ have been dis-
cussed.
650. The fluid being still supposed incompressible, but
heterogeneous, the density D will be variable ; and in order
that the pressure p may be determinate, the quantity
DQidx+Ydy+Zdz) must be an exact differential : but if
ELASTIC FLUIDS. 359
X-dx -\-Y dy ^'Zadz be likewise supposed an exact differential,
it will appear, as in equation (392), that the density will be
always a function of the pressure. Thus the pressure and
density will become constant together, and will remain in-
variable for all points situated in a level stratum.
We conclude, therefore, that a heterogeneous fluid mass
cannot remain in equilibrio, unless it be disposed in such man-
ner that each of the level strata shall be of equal density
throughout. The law of variation in the density in passing
from one stratum to another, will depend on the manner in
which D is expressed in functions of x^ y, and z : and since
the nature of the function is entirely arbitrary, the law of
the density will likewise be arbitrary.
Application of the General Equations of Equilibrium to
Elastic Fluids.
651. The characteristic property of an elastic fluid is its
power of sustaining compression, and subsequently regaining
its original volume and elasticity, when the compressing force
is removed.
Thus, a fluid which is elastic exerts in addition to the
pressure due to the forces which act upon it, an eflîbrt arising
from the elasticity of its particles.
It has been ascertained experimentally, that this effort,
which is called the elastic force of the fluid, is proportional to
its density, so long as the temperature remains invariable.
Thus, if .we suppose the temperature to remain constant,
and represent by P that pressure exerted upon the unit of
surface which is necessary to produce a certain density
assumed as the unit, this density will be doubled when the
pressure becomes 2P ; trebled when the pressure becomes
3P, (fee. ; and hence, if the density be expressed by D, the
corresponding pressure will be PD. This pressure being
denoted by p, we shall have
p^VJy (393) ;
the quantity p represents, as heretofore, the pressure exerted
upon the unit of surface.
360 HYDROSTATICS.
653. By combining equation (393) with the equation
dp=\y{lidx^-Ydy+Zdz\
there results
dp _K.dx-\-Yd'y-\-'Ldz tOQA\
7 p ^"^^^^ '
and by integration, we have
_ PlLdx + Ydy-\-Zdz ç,
653. The temperature being supposed constant throughout
the mass, and the nature of the fluid particles everywhere
the same, the quantity P will be constant, and may therefore
be placed without the integral sign : thus, by representing
the constant C by log C, we shall have
or, if we denote by e the base of the Naperian system, this
equation will reduce to
fÇ%.dx+Ydy+Zdz)
log^=loge ^ +log C :
reducing, and passing from logarithms to numbers, we find
/(Xrfj+Yrfy+Zdz)
p=C'e P
This value being substituted in equation (393), we obtain
/(Xrfj+Yrfy+Zdz)
Ce ^
P
The pressure and density being both functions of the quan-
tity /(Xc^:r+Yc?y+Z^2;), they will become constant at the
same time ; and hence, the density of the fluid throughout
each level stratum will remain invariable. The value of the
density in any stratum results immediately from the pre-
ceding equation.
654. It should be remarked, that in the case of elastic
fluids, the equation
:Ldx+Ydy+Zdz=Q
cannot be deduced from the hypothesis of ^=0: for, if we
suppose ^=0, the equation will give D=0 ; and hence, we
perceive that it would be necessary that the density of the
PRESSURE OF HEAVY FLUIDS. 361
fluid should be likewise equal to zero ; a supposition which
would destroy the existence of the fluid.
We conclude, therefore, that in an elastic fluid, the pressure
cannot be equal to zero at the surface ')f the fluid, as is the
case with incompressible fluids. Thus, a mass of elastic
fluid cannot be in equilibrio unless contained in a close
vessel, or extended indefinitely in space.
Of the Pressure of Heavy Fluids,
655. It is now proposed to examine the circumstances of
equilibrium in fluids whose particles are acted on by the force
of gravity. For this purpose, let it be supposed that a vessel
is placed upon a horizontal plane, and filled with water, or
other heavy fluid, to a certain height. The surface of the
fluid, as has been demonstrated, will assume a horizontal posi-
tion ; let this surface be assumed as the plane of a-, y, and let
the co-ordinates z be reckoned positive downwards ; the force
of gravity being the only force exerted upon the fluid parti-
cles, we shall have
X=0, Y=0, Z=^;
and the equation (386) will become
dp^Dgdz.
The density of the fluid and the intensity of gravity being
supposed constant, the integration of this equation will ^ive
p='Dgz-\-C (395).
To determine the value of the constant C, we make z—0,
and since the pressure p is equal to zero at ihe same time, we
deduce C=0 : thus the equation (395) is reduced to
p=T>gz (396).
656. If a horizontal plane be drawn below the surface of
the fluid, every point in such plane will have a common ordi-
nate z ; and the pressure p—Dgz will therefore be constant
throughout this plane.
657. Let h represent the distance between the surface of
the fluid and the horizontal base of the vessel ; the pressure
supported by the unit of surface of the base will be determined
362
HYDROSTATICS.
by equation (396), in which we replace z by A, and thus
obtain
p=T)gh (397).
Let J)' represent the pressure supported by the entire base,
which is supposed to contain h units of surface : the quantity
p will be contained b times in p' : we therefore have
p'=bp (398),
and by substituting for p its value given in equation (397),
we find
p'=Tighb (399).
But bh represents the volume of a prism whose base is 6,
and height h ; and by multiplying this vokime by the density
D, we obtain bliD for the mass of the prism : therefore bg/iD
will express the weight of such prism ; and hence, it appears
that the base b supports a pressure equal to the weight of the
column of fluid which rests immediately upon it.
658. The pressure p', exerted by the same fluid, being
dependent only on the base b and height h, it follows that the
pressures supported by the bases of diflerent vessels will be
equal, whatever may be the forms of the vessels, provided
their bases, and the heights of the fluid above them, be
respectively equal.
659. To determine the lateral pressure exerted against the
sides of the vessel, let da represent the element of this sur-
face, and z the distance of the element from the surface of the
fluid ; the pressure j) (referred to the unit of surface), which
is supported by the element da, will be determined by equa-
tion (396) : this value being substituted in equation (399),
and the area b being replaced by da, we obtain T) . gz . da> for
the expression of the entire pressure on the element da. A
similar expression may be obtained for the pressure upon
each element ; and since the pressures will be exerted in par-
allel directions when the side of the vessel is supposed plane,
we shall have, for the total pressure exerted against the side,
p'=fDgzda.
The second member of this equation contains two variables,
one of which must be eliminated before the integration can
PRESSURE OF HEAVY FLUIDS. 363
be efiected. This elimination is readily accomplished when
the figure of the surface <y is known.
660. Let it be required, for example, to determine the pres-
sure exerted against the inclined rectangle ACDB {Fig. 221),
whose sides AB and CD are parallel to the horizon. Denote
by h and I the base AB and length BD of the rectangle, and
conceive its surface to be divided into an infinite number of
elements, by lines parallel to AB or CD ; the pressure will be
the same upon every point of the same element. Let v denote
the distance D/ of any one element af from the base CD ;
the height of this element will be expressed by dv=ae^ and
the surface of the element by
abXae=bdv ;
substituting this value for do in the expression /Dg-zda, we
obtain
/Dgzdu^/Dgzbdv :
such will be the expression for the pressure exerted upon the
surface ABDC. The integral should be taken between the
limits v—0 and v=l, the variable z being previously elimi-
nated. To effect this elimination, let (p represent the angle
BDL included between the plane of the rectangle and the
vertical line NL, and a the distance DN of the superior base
CD from the surface of the fluid ; we shall have
K/orLN=NDH-DL;
or,
z=a + v . cos <J) :
hence, the pressure exerted upon the surface will be ex-
pressed by
p'=f'Dg{a-{-v . cos <p)bdv ;
and by performing the integration indicated, we find
p'='Dgb{av-\-^v'' cos(|>)4-C.
The integral being taken between the limits v=0, and v=l,
we obtain
p'=T>gb(al-{-ll^ cos (p).
661. To determine the point of application of the resultant
of all the pressures exerted upon the rectangle, we remark,
in the first place, that this point must be situated upon the line
364 HYDROSTATICS.
EH, which bisects the sides AB and CD. We next regard
the pressures exerted upon the different points of the surface
ABDC as parallel forces, and determine their moments with
reference to a vertical plane passing tlirough the horizontal
line CD : the pressure sustained by the element abfe being
Dgzbdv, its moment will be expressed by DgzbdvXv . sin (p ;
and by denoting the distance EG of the point of application
of the resultant from the line CD by v,, the principle of mo-
ments will give
p'v, sin 4)= sin <pfDgzhvdv ;
or,
*" p'v,—fT)gzhvdv.
If, in this expression, we replace z by its value determined in
the preceding Art., we shall obtain
v'v,—Y)ghf{avdv-\-cos <p . v^dv) ;
whence^ by integration.
— +cos^yJ+C.
The integral being taken between the limits v==0 and v=^,
there results
p'v=Dgh ^^-^+008 Ç-J ;
and by substituting for p' its value, we find, after reduction,
al , l^
2+cos<.-
v=- J-
a + cosç> -
Having found the pressures exerted upon the base and
upon each of the sides of the vessel, we combine these pres-
sures, and determine their resultant : such resultant will
express the entire pressure produced by ihe fluid.
662. We will next consider a body immersed in a homo-
geneous heavy fluid : the pressure exerted by this fluid
against any portion of the surface of the body may be deter-
mined by the method for finding the pressure against the
sides of a vessel ; but when it is required to consider the
total pressure exerted against tiie surface of a body immersed
PRESSURE OF HEAVY FLUIDS. 365
in a fluid, we commonly employ the following theorems, the
truth of which will be demonstrated.
1°. The pressures exerted upon the surface of a body en-
tirely immersed in a fluid have a single resultant ^ which is
vertical and directed ujjwards.
2°. The resultant of all the pressures is equal in intensity
to the weight of the fluid displaced.
3°. The line of direction of this resultant passes through
the centre of gravity of the displaced fluid.
4°. The horizontal pressures destroy each other.
To establish the truth of these propositions, let us suppose
a vessel ADE {Fig. 222) to be filled with a heavy fluid in
equilibrio, and let a portion of this fluid KL be conceived to
become solid, its density remaining unchanged : the state of
equilibrium will not be disturbed by this change. But this
solid is urged downwards by a force equal to its weight,
applied at its centre of gravity. This force can only be
destroyed by the resultant of all the pressures exerted by the
fluid against the solid ; hence, it follows that these pressures
must have a single resultant equal in intensity to the weight
of the displaced fluid, and that this resultant must be applied
at the centre of gravity of the displaced fluid, and be directed
vertically upwards. Moreover, as the direction of the result-
ant is vertical, the horizontal pressures will mutually destroy
each other.
When a body is partially immersed in a fluid, an equilibrium
cannot subsist unless the centres of gravity of the body and
of the fluid displaced be situated upon the same vertical line :
this condition will necessarily be fulfilled when the body is
entirely immersed, provided it be homogeneous ; since its
centre of gravity will then coincide with that of the fluid
displaced.
The buoyant eflbrt exerted by the fluid being directed along
a line which passes through the centre of gravity of the dis-
placed fluid, that point is called the centre of buoyancy.
663. Let V represent the volume of fluid displaced, and v'
that of the body ; D the density of the fluid, and D' that of
the body : the weights of the volume of displaced fluid, and
366
HYDROSTATICS.
of the body will be respectively Dgv and D'gv'. If the body
be supposed to rest in equilibrio, we shall have
Dgv=:D'g-v' ]
and if we suppose it to be entirely immersed, the volumes v
and v' will be equal, and the densities D and D' must likewise
be equal, in order that tft«^ equilibrium may be preserved.
But if the weight of the body bc.lelsg than that of the fluid
displaced, w^e shall have ■*>.. ..
and the body will be urged upwards by a force equal to the
difference Dgp—'D'gi'.
If, on the contrary, we should have
T>gv<D'gv',
the body would tend downwards with a force equal to
Wgv' — T>gv.
Of the Equilihrium, Stability, and Oscillations of Floating
Bodies.
664. The propositions demonstrated in Arts. 662 and 663
establish two principles which serve as tlie basis of the theory
of floating bodies ; these principles are,
l'^. When a body is partially or totally miTnersed in a
fluid, an equilibrium cannot subsist unless the centre of
gravity and centre of buoyancy be situated upon the same
vertical line.
2°. If an equilibrimn be maintained, the weight of the
body will be equal to that of the fluid displacef.
The latter principle is frequently employed for the purpose
of estimating the weight of a ship either with or without her
cargo. For this purpose, we measure the capacity of the
part immersed, and allow a weight of one ton for every 35
cubic feet which it contains. By taking the difference of the
weights of the vessel with and without the cargo, the weight
of the latter may be obtained. We can also arrive at the
same result, by simply measuring the additional portion of the
vessel immersed, when the cargo is introduced.
EQUILIBRIUM OF FLOATING BODIES. 367
665. The horizontal surface of the fluid is called the flane
of floatation.
666. If V denote the volume of fluid displaced, D its den-
sity, and g the intensity of the force of gravity, the weight P
of the body ABC {Fig. 223), which floats upon the surface
of the fluid, and is partially immersed, will be equal to Dgv.
667. When the floating body and fluid are both homogene-
ous, the centre of gravity of the part immersed will coincide
with the centre of buoyancy.
668. The fluid and body being homogeneous, the centre of
gravity G {Pig. 223) will be situated above the point O, the
centre of buoyancy. For let g be the centre of gravity of
that portion of the body which lies without the fluid : then,
the centre of gravity G of the entire body will necessarily
be situated upon the line ^O, and between the points g and O ;
hence, it will be found above the point O.
669. But if the floating body be heterogeneous, it may
happen that the centre of gravity of the entire body will lie
below the centre of buoyancy. For by supposing the density
of the lower part of the body to be very much greater than
that of the upper portion, the centre of gravity of the entire
body may be situated extremely near the lower surface : but
the position of the centre of buoyancy depends only on the
figure of the part immersed, since the density of the fluid is
supposed uniform, and it may therefore be situated at a
greater distance from the lower surface of the body than
the centre of gravity of the entire mass.
Hence we conclude, that the centre of gravity of the float-
ing body is sometimes situated above, and sometimes below,
the centre of buoyancy.
670. When the body is but partially immersed, the weight
of the immersed portion is less than that of the fluid dis-
placed, and the equilibrium is maintained by the weight of
that portion of the body which lies without the fluid : this
weight is equal to the difiîsrence of the weights of the fluid
displaced and of the part of the body immersed. If the weight
of the body be increased, it will sink to a greater depth, until
the weight of the additional quantity of fluid displaced shall
be equal to the weight added.
368 HYDROSTATICS.
G71, Let us now suppose that a body floating upon the
surface of a fluid {Fig. 224) is deranged in a very sUght de-
gree from its position of equilibrium, by the application of
any force, and let us examine whether the body will tend to
return to its original position, or, on the contrary, to deviate
farther from it. Let ADB represent the immersed part of the
body before derangement, and a6D that immersed after de-
rangement : we suppose the new position of the body to be
such, that the weight of the fluid displaced shall still be equal
to the weight of the body, or that ABD = a6D. The centre
of gravity G may be regarded as fixed during the rotation^
since the forces will tend to turn the system about that point,
as though it were immoveable. The centre of buoyancy
will not retain its position O, but will be found nearer to the
portion CB6, which, by the rotation, has become immersed in
the fluid : and if we suppose, for the sake of symplifying the
question, that the body is divided symmetrically by the plane
ABD, the centre of buoyancy will obviously be found in this
plane after the derangement. Let o represent the centre of
buoyancy in the deranged position, and through o and G let
perpendiculars oi and Gk be demitted upon the line ab. If
an equilibrium subsist, the weight of the body and the up-
ward pressure of the fluid will be equal and directly opposed.
The first condition will necessarily be satisfied, since we
have supposed the volume of fluid displaced to remain un-
changed; the second condition will be fulfilled when the
points i and k coincide with each other : but if this coinci-
dence should not take place, the point i may fall either to the
right or to the left of the point k. In the first case, the pres-
sure of the fluid applied at o and acting upwards, will evi-
dently tend to restore the body to its primitive position,, or to
render the line DG vertical. But if the point i should fall to^
the left of k, this pressure would tend to turn the body in a
contrary direction about the point G, and would thus cause
it to deviate farther from its original position.
If the body, when deranged in a very slight degree from its
position of equilibrium, should tend to resume its former posi-
tion, the equilibrium is said to be stable ; but if, on the con-
trary, it should tend to depart still farther from this position,
EaUILIBRIUM OF FLOATING BODIES. 369
the equilibrium is called unstable ; when the body neither
tends to return to its original position, nor to deviate farther
from it, the equilibrium is said to be one of indifference.
672. By examining the directions of the pressures before
and after derangement, we shall find that the lines OG and
oi perpendicular to AB and ab respectively, are inclined to
each other, and being contained in the same plane, they will
intersect in some point m, {Fig. 225).
This point is called the metacentre ; and it appears from
Art. 671, that when the point G is situated below w, the
extremity k of the perpendicular GA; will fall to the left of the
point i, and the equilibrium will be stable ; but if the point G
be situated above the point w, the extremity k of the perpen-
dicular Ok will fall to the right of the point ?', and the equi-
librium will become unstable. If the points i and k coincide,
the equilibrium becomes one of indifference.
673. Let it now be required to determine the position of
the metacentre. This point being found upon the line con-
necting the centre of gravity and centre of buoyancy in the
primitive position of the body, it will be sufficient to determine
its distance from the point G, or the point O.
For this purpose we remark, that when the body is slightly
inclined, the line AB {Fig. 226) which represents the profile
of the plane of floatation in the primitive position, assumes a
position inclined to the new plane of floatation ab in a certain
angle «, the portion ACa being at the same time withdrawn
from the fluid, and the portion BC6 being immersed. Hence,
the immersed portions of the body in the two positions will be,
aCBD+ACa in the primitive position,
aCBD + BC6 ..... after the derangement.
But if y, g, and g' represent the respective centres of gravity
of the volumes aCBD, aCA, and èCB, the centre of gravity
O of the volume ABRD will be found by dividing the line gy
in the inverse ratio of the volumes aCBD and ACa ; and in
like manner, we may find the centre of gravity o of the
volume aèBD : thus, we shall obtain the proportions
vol aCBD : vol aCA ; : O^ : Oy (400),
vol aCBD : vol 6CB ::og' :oy (401) ;
Aa
370 HYDROSTATICS.
but the second terms of these proportions are equal to each
other : for, the floating body being supposed to displace the
same quantity of fluid after it has been deranged as it did in
its primitive position, the volumes ABRD and abRD will be
equal to each other ; and if from these equals we subtract
the common part aCED, there will remain the volumes aCA
and BC6 equal to each other. Hence, we deduce from the
proportions (400) and (401),
Og- : og' : : Oy : oy ;
which proves that the lines gy and g'y are cut proportionally
by the right line Oo, which line is therefore parallel to gg'.
But, the derangement of the body being, by hypothesis,
extremely slight, the line gg' may be considered as nearly
coincident with the primitive plane of floatation ; and since
Oo is parallel to gg', this line may be regarded as parallel to
the same plane,
674. To determine the value of Oo, we deduce, from the
proportion (400),
vol aCBD+vol aCA : vol aCA : : O^+Oy : Oy,
or,
vol ABRD : vol aCA : : gy : Oy.
But the similar triangles gg'y and Ooy give
gy.Oy:: gg' : Oo ;
and by comparing this proportion with the preceding, we
obtain
vol ABRD : vol aCA :: gg':Oo (402) ;
whence,
vol_aCAx^' .^„.
^'- vol ABRD ^^^^^'
675. Having determined the value of Oo, we can readily
obtain that of Om (Fig. 227) ; for, the lines Om and om
being respectively perpendicular to CA and Ccr, the angles at
C and m will be equal ; and since these angles are exceed-
ingly small, we may regard the triangles ACa and Oîno as
similar and isosceles : hence, we shall obtain the proportion
Aa : Oo : : Ca : viO ;
and therefore,
^Q^OoxCa
Aa
ËQ,lîlLlBRltJM OP t^LOATING BODIES. 371
6r6i To obtain the analytical expressions for Oo and mO,
we remark, that the plane of floatation AB {Pig. 228), which
limits the immersed part of the body in its primitive position,
is replaced by the plane ab after the derangement : these two
planes, being intersected by a vertical plane perpendicular to
their common intersection, will exhibit the section ACa rep-
resented in Fig. 226 ; and if we continue to draw other par-
allel vertical planes, we shall divide the solid included
between the planes KAL, KaL {Pig: 228) into an infinite
number of elementary laminae parallel to the plane ACa.
But it is evident, that when the plane KAL, which in the
primitive position of the body coincided with the surface of
the fluid, shall have been detached from the surface, revolving
around the line KL, each right line in this plane, as CA, will
have described the sector of a circle ; so that the sections of
the solid included between the planes ALB and oLb {Fig.
228) by the system of parallel vertical planes, will be repre-
sented by the sectors ACa, A'C'a', A"C"a", &c. {Fig. 229).
But if we assume the line of intersection KL as the axis of
x, and place the axis of y in the plane KAL, the ordinates y
will be the perpendiculars AC, A'C, A"C", (fcc. The infinitely
small angle formed by the planes KAL and KaL being every-
where the same, let the arc described by a point at the distance
unity from the line KL be expressed by <y : the arc described
by the point A will then be determined by the proportion
1 : 41 : : AC or y : arc Aa ;
whence,
arc ka=uy (405).
This arc being multiplied by the half of the radius y, we
shall obtain ^ay^^ for the area of the sector ACa ; and this
area being multiplied by CC'=dxj the portion of the line KL
intercepted between two consecutive sectors, we shall have
for the volume of the solid
Kaa'A!=\wy^dx,
which will express the element of the solid included between
the planes KAL and KaL. Hence, we shall have {Fig. 226)
vol A.Ca = \é>fy^'dx (406).
Such will be the analytical expression for the second term of
the proportion (402).
372 HYDROSTATICS.
To determine the value of the third term, we remark that
the Une C^ {f'ff- 22(3) being the distance of the axis KL
{Pig, 229) from the centre of gravity of the sohd KaLA, we
shall determine this distance, by dividing the sum of the mo-
ments of the elementary solids by the volume KaLA.
If we consider the elementary sector ACa {Fig. 229), the
centre of gravity g of this sector will be found upon the
radius CR=CA {Fig. 230), at a distance from the point C
(Art. 184) expressed by
chordAa
arc Aa
but the angle C being supposed extremely small, the arc Aa
may be regarded as equal to the chord ; and since CR is equal
to CA or y {Fig. 228), the preceding expression will give |y
for the distance of the centre of gravity from the axis KL.
Multiplying the elementary solid \ay^dx by this distance, the
moment of this solid with reference to this axis KL will
become \uy^dx : thus, we shall have
f^uy'^dx=i\\.ç. sum of the elementary solids,
f^iiy^dx=X\ve sum of the moments of the elementary solids :
and from the property of the moments, the distance Cg of the
centre of gravity of the small solid CAa {Fig. 226), or
KALa {Fig. 229), will be expressed by
^ f^^^y'dx '
the quantity « being constant, this expression may be re-
duced to
^ 2jrdx
^ ^fy'dx
Ç)77. The value of C^ will result from the integrations here
indicated ; and that of C^' {Fig. 226) may be obtained in a
similar manner ; but, if the floating body be symmetrical with
respect to a vertical plane passing through the axis KL, as
will always happen in the case of a ship, we shall have
and therefore,
KQtJILlBRIUM OF FLOATING BODIES. 373
The volume of the part immersed, which hkewise enters
into the equation (403), can be calculated directly, when the
figure of the vessel is supposed known. Let this volume be
denoted by V, and let its value and those of the volume ACa
and gg', given in equations (406) and (407), be substituted in
equation (403) : we shall thus obtain
and lastly, by substituting in equation (404) this value, and
that of the arc Aa, given by equation (405), replacing Ca by
y, we find
3V
Such is the formula expressive of the distance of the meta-
cenire from the centre of buoyancy.
678. When the floating body is homogeneous, and of such
figure that its parallel sections will be similar, we may rea-
dily determine the position of the metacentre, without the
necessity of performing an integration. For let a^ represent
the area of the section AEB (Fig. 231), which is supposed to
have been determined by direct measurement, and let b repre-
sent the half-breadth CA of this section : the half-breadths of
the sections A'E'B', A"E"B", «fee. will be represented by C'A',
C'A", &.C. or by the ordinates y of the curve KAL. These
sections being by hypothesis similar figures, they will be
proportional to the squares of their homologous sides ; and
hence, we shall have
section AEB : section A'E'B' : : AC^ : A'C'^,
or,
«2 : section A'E'B' : : b^ x y^\
whence,
section A'E'B'=^.
b^
The distance CC between two consecutive sections being
denoted by dx, we shall have
a^y'^dx
~~b^
for the expression of the elementary solid.
32
374
HYDROSTATICS.
679. Let g represent the centre of gravity of the section
AEB, which, in consequence of the symmetry of the figure,
will be found on the vertical CE, The centre of gravity of
this section having been determined, let its distance from the
surface of the fluid be denoted by n : we shall then have,
from the similarity of figures,
, ( the distance of the centre of ffravitv of the ;
' } section A'E'B' from the surface of the fluid ) '
whence,
Multiplying this distance by the elementary solid, we shall
obtain for the moment of this solid, taken with reference to
the surface of the fluid,
ny a^y^ ,
and therefore the expression— -/ y ^^f a,- will represent the
sum of the moments of the elementary solids taken with refer-
ence to the surface of the fluid. This sum being equal to
the product of the volume V of the solid immersed by the
depth HG of its centre of gravity, if this depth be denoted
by Gj we shall have
whence,
G = ;
-jy^'dx.
Y.b-
But it has been shown (Art. 677) that the distance mO of
the metacentre from the centre of buoyancy is given by the
formula
mu ^^ ,
and if we compare these two expressions, we shall find
G:mO::3na-.2b':
na^ . r ^ J 2 ^^ ly^dx
or
EQUILIBRIUM OF FLOATING BODIES. 375
whence,
™"=l^ w-
680. For the purpose of applying this formula, let it be
required to find the metacentre of a rectangular parallelo-
piped ML. Let AF represent the intersection of the body-
by the surface of the fluid {Fig. 232), supposed parallel to
the base NL. The depth AN, to which the body must be im-
mersed in order that it may be sustained in equilibrio, will
depend on the weight of the parallelopiped and the density of
the fluid (Art. 664) : this depth may be considered as deter-
mined by experiment : the quantity a", which represents the
section BN, and which will be constant for all parallel sec-
tions, will be determined immediately ; for we have
a^=ABxCE.
Again, the semi-breadth of the section being equal to |AB,
there results
6 = iAB=AC;
and since the centres of gravity of all the sections are equally
distant from the surface of the fluid, the centre of gravity
of the fluid displaced will be situated at the same distance ;
so that we shall have
W = G=:iCE.
By substituting these values in formula (408), the distance
of the metacentre m, from the centre of buoyancy will be
found equal to
,^ SAC^»
^^=âÂB^E
or, by reduction,
3CE
For example, if the semi-breadth of the parallelopiped be
supposed equal to 9 feet, ^ud the depth of the part immersed
4 feet, we shall find the height of the metacentre above the
centre of buoyancy equal to 6| feet ; if, therefore, we subtract
from this height, 2 feet, the depth of the centre of buoyancy,
there will remain 4| feet, for the height of the metacentre
376 HYDROSTATICS.
above the surface of the fluid. Hence, the centre of gravity
of the parallelopiped should not be more than 4| feet above
the surface of the fluid, if we wish the equiUbrium to be of
the stable kind.
681. As a second example, let us consider a vessel whose
vertical sections below the surface of the fluid are equal right-
angled isosceles triangles, such as AEB {Fig. 233).
If the perpendicular EC be demitted upon the base, the
triangle ABC will likewise be isosceles, and the height EC
will therefore be equal to one-half the base AB : thus, the
quantities which enter into the formula (408) will be, in the
present case,
a2=area of the triangle AEB^AC^,
^=G=iCE,
h=AC=CE;
consequently, by substituting these values in formula (408), it
will reduce to
and if from this value we subtract that ot the distance of the
point O below the surface of the fluid which is equal to ^CE,
there will remain iCE for the distance of the metacentre
above the surface of the fluid. Hence, in a prismatic vessel
whose vertical sections are right-angled isosceles triangles,
the metacentre will be found at a distance above the surface
of the fluid equal to the distance of the centre of buoyancy
below the surface.
682. If we suppose the body to be slightly deranged from
a position of stable equilibrium, and conceive the resultant of
all the upward pressures of the fluid to be applied on its line
of direction, at the metacentre, we can determine the circum-
stances of oscillation of this body about the centre of gravity,
by a method entirely analogous to that employed in consider-
ing the motion of the compound pendulum. For this pur-
pose, let the origin of co-ordinates be placed at the centre of
gravity, and let the proper value of y, be substituted in for-
mula (337), which may be put under the form (338)
(It k^'+a'"'
OSCILLATIONS OF FLOATING BODIES. 377
This formula admits of simplification in the present case,
from the consideration that the oscillations are performed
about the centre of gravity ; and the general expression of
the moment of inertia M.{k^-\-a'') is therefore reduced to
Mk^ : hence, we obtain
^=^ (409).
dt k^ ^ ^
This equation, when integrated, will serve to determine the
angular velocity, and the time of performing a complete oscil-
lation.
683. To determine the value of y,, which represents the
perpendicular distance from the axis passing through the cen-
tre of gravity, about which the oscillations are performed, to
the line of direction of the upward pressure, we remark, that
the distance of the metacentre from the centre of buoyancy
O is expressed by
2fy=dx
3V *
Let this distance be denoted by A, and the distance GO {Fig.
234) by B ; we shall then have
or, since the point G may fall above O, we may likewise have
Gm=A— B ;
hence we may comprise the two cases under the double sign,
by writing
Gm=A±B.
If the angle LmG {Pig. 234), formed by the vertical mh with
the new direction of the line GO, be represented by 6, we shall
have the relation
GL=Gm sinfl;
or, replacing the sine by the arc, since the arc is supposed
extremely small, and substituting the value of Gm, this equa-
tion will become
GL=(A±B)<»;
and by introducing this value of y, in formula (409), we
shall obtain
dcj_ g{A±B)ê
di ¥^
376
HYDROSTATICS.
684. But the angular velocity a being that which corres-
ponds to the arc i described with a radius unity, this velocity
will be expressed by — ; and since the arc « {Fig. 234) is a
decreasing function of the time t^ cU should be affected with
the negative sign ; hence,
"=^' («o>-
Multiplying the corresponding terms of these equati(Mis toge^
ther dt will disappear : and there will result
Putting, for brevity,
^^^=E (411),
and multiplying by 2, we obtain
2'Eêd6-^2a>dco=Q.
Integrating, we have
whence,
Substituting this value in equation (410), we obtain
or, by reduction,
d6
dt-
and, by integration,
t=—= arc ( cos = -'^ ) +C':
^E V ^G/
from which we deduce
'-^=cos[(^-C')v/E]:
and, finally.
,, ^C.cos[(^-CVE]
; v^
685. When E is negative, tlie value of o becomes imagin-
SPECIFIC GRAVITY. 379
ary, and the oscillatory motion cannot take place ; but in
order that E may be negative, the first member of equation
(411) must likewise be negative ; and consequently,
A ± B=a negative quantity :
this case occurs when B exceeds A, and is affected with the
negative sign ; and since A± B represents the distance of the
centre of gravity from the metacentre, it follows that the meta-
centre will then be situated below the centre of gravity, and-
the equilibrium will be unstable. On th& contrary, if A±B
be positive, the metacentre will be found above the centre of
gravity, the value of E will be positive, and the values of 6
and u will be real : thus, the oscillations can be performed,
and the equilibrium will be of the stable kind.
686. The time of oscillation being determined by a method
entirely similar to that employed in investigating the circum-
stances of motion of the compound pendulum, we may con-
clude that this time will be independent of the extent of the
arc through which the oscillations are performed, provided
the arcs be extremely small.
Specific Gravity — Hydrostatic Balance — Hydrometer.
687. Let P represent the weight of a body M : if this body be
immersed in a fluid, the buoyant effort exerted by the fluid will
tend to support the body, and the force P' necessary to sustain
it will be less than P, that required previous to the immer-
sion, by a quantity equal to the weight of the fluid displaced.
For example, if M be supposed a sphere of lead whose
weight is equal to eleven pounds, and if it be found to weigh
but ten pounds when immersed in water, we should conclude
that the weight of an equal volume of water would be one
pound ; and therefore tfiat the weight of lead was to that of
water as eleven to one.
688. The specific gravity of any substance is the ratio
between its weight and the weight of an equal volume of
some other substance assumed as the standard.
Thus, in the preceding example, if water be adopted as the
standard of comparison, the weight of the sphere of lead
380 HYDROSTATICS.
being eleven times greater than that of an equal volume of
water, the specific gravity of lead will be represented by the
number 11.
The density of a body has been defined (Art. 161) to'be the
ratio between the quantity of matter contained in the body and
that contained in an equal volume of some other substance
assumed as the standard ; and since the weights of bodies are
proportional to the quantities of matter which they con-
'iain, it follows that the ratio of the weights of two bodies
will be equal to the ratio of their quantities of matter.
Hence, the number expressing the specific gravity of a body
will be the same as that which expresses its density, provided
we refer the density and specific gravity to the same sub-
stance as a standard.
In practice, it is usual to adopt water as the standard in
determining the specific gravities of solids and incompressible
fluids ; and for the purpose of rendering the comparison more
exact, the water is first deprived, by distillation, of any im-
purities which it may contain. The specific gravities of
gases and vapours are generally referred to that of atmo-
spheric air.
689. The dimensions of all bodies being more or less
aiFected by changes of temperature, it becomes necessary to
adopt a standard temperature, at which experiments for the
determination of specific gravities may be performed. A
convenient temperature for this purpose is that corresponding
to 60° of Fahrenheit's thermometer, it being easily obtained
at all times : and the tables of specific gravities are usually
calculated for this temperature. When circumstances will
not permit the experiments to be performed at the standard
temperature, the results obtained must be reduced to this tem-
perature, by introducing a correction for the change of vol-
ume which the substance would undergo if reduced to the
standard temperature. This correction is readily applied
when the law of dilatation has been previously ascertained.
690. If we wish to determine the specific gravity of a fluid,
as olive-oil, we may immerse successively the same solid in
water and in this fluid ; we shall thus be enabled to deter-
mine the weights of equal volumes of the two fluids ; and a
SPECIFIC GRAVITY. 381
comparison of these weights will give the specific gravity
of the oil. For example, if the sphere of lead weighing
eleven pounds have its weight reduced to 10.085 lb. when
immersed in oil, the weight of the fluid displaced would be
equal to 0.915 lb. ; and since the weight of an equal bulk of
0.915
water was found equal to 1 lb., we shall obtain -^ — = 0.915,
for the ratio of the weights of equal bulks of the two fluids :
this number will therefore represent the specific gravity of oil.
From the preceding remarks, we may infer that if two
bodies of unequal volumes, suspended from the arms of a
balance, sustain each other in vacuo, the equilibrii^m will not
be maintained when the bodies are similarly suspended in the
atmosphere ; the weight of the larger body being most sup-
ported by the buoyant eflbrt of the atmosphere.
691. The instrument usually employed for determining
with accuracy the specific gravities of bodies, is the hydro-
static balance. This consists merely of a delicate balance,
having a small hook attached to one of its scales, by means
of which the body can be suspended, for the purpose of deter-
mining its weight when immersed in a fluid. The body is
connected with the hook by a hair or slender thread, whose
weight is inconsiderable.
When we wish to determine the specific gravity of a solid,
we place it in the scale to which the hook is attached, and add
weights in the opposite scale until an equilibrium is produced.
The weights thus added will represent the weight of the body
in air. The body is then attached to the hook and im-
mersed in water ; and the weight necessary to be placed in
the opposite scale to produce an equilibrium will give its
weight in wçiter : the difllerence between the weights in air
and water will be equal to the weight of an equal volume
of water, and by comparing this difference with the weight
in air, we shall obtain the specific gravity of the substance
under consideration.
This process is slightly inaccurate ; since the buoyant
efforts exerted by the atmosphere upon the body when im-
mersed in it, and upon the weights introduced into the
opposite scale, have been neglected. But as the density of
382
HYDROSTATICS.
the atmosphere is very small, this omission will not affect the
results materially.
When the given substance is soluble in water, we deter-
mine its specific gravity with reference to some fluid in which
it is insoluble, and then compare the specific gravities of the
two fluids. If the body be lighter than water, we can con-
nect it with a heavier body, which will cause it to sink.
Then, having the weights of the heavier and lighter bodies,
and that of the compound in air, and having ascertained the
loss of weight sustained by the heavier body and the com-
pound when immersed, we can readily deduce the weight of
the fluid displaced by the lighter.
The specific gravity of a fluid may be determined by
weighing successively the same body in this fluid and in
water, and comparing the weights of the equal volumes
displaced. Or it may be ascertained by weighing the same
vessel when filled with water, and with the fluid under con-
sideration ; these weights, being diminished by that of the
vessel when empty, will give the relation between the specific
gravity of the fluid and that of water.
692. The hydrometer is an instrument usually designed to
determine approximatively the specific gravities of fluids. It
is composed of a cylinder of glass or metal, to the lower
extremity of which a cup is attached loaded with shot or
mercury, and terminated at top by a slender graduated wire.
When the hydrometer is plunged into a fluid, the weight
with which its lower extremity is loaded causes it to assume
a vertical position, and it sinks to a greater or less depth,
according to the specific gravity of the fluid. Hence, that
division on the graduated stem which corresponds to the
surface of the fluid will serve to indicate the specific gravity
of the fluid.
For example, if the hydrometer be immersed in distilled
water whose temperature corresponds to 60° Fahrenheit, the
surface of the water will intersect the stem at a certain
division, which we shall suppose to be that marked 10 : if
plunged in wine, it will sink deeper, say to the 11th, 12th,
or 13th division ; and if in brandy, to a still greater depth,
SPECIFIC GRAVITY. 383
the division indicated being dependent on the quantity of
alcohol which the brandy contains.
The use of this instrument evidently depends upon the
principle, that when a body is immersed in a fluid, a portion
of its weight equal to that of the fluid displaced will be sup-
ported by the buoyant effort of the fluid : thus, the heavier
the fluid, the less the depth to which the hydrometer will
sink.
693. The hydrometer, as improved by Nicholson, will
serve to determine the specific gravities of solids or liquids.
The instrument consists of a hollow copper ball A {Fig. 235),
to the lower part of which is attached a brass cup of sufficient
weight to maintain the hydrometer in a vertical position
when immersed in a fluid. The upper part of the ball
carries a slender wire D, which supports a small dish C des-
tined to receive the weights. The weight of the hydrometer
is such that the addition of 500 grains in the dish C will
just sink the instrument in distilled water, at the temperature
60°, luitil the svirface of the water intersects the stem at its
middle point D. If, therefore, a body be placed in the dish
C, and weights be added until the point D shall correspond to
the surface of the water, the difference between 500 grains
and the weights added will express the weight of the body.
The body being then transferred to the lower dish B, it will
be found necessary to place additional weights in the dish C,
in order to sink the hydrometer to the same depth: these
additional weights will be equal to the loss of weight sus.-
tained by the body when immersed. Hence, the specific
gravity of the solid may be readily determined.
When we wish to determine the specific gravity of a fluid
with this hydrometer, we immerse the instrument succes-
sively in distilled water and in the given fluid, and ascertain
the weights necessary to be added in each case to the dish C,
in order to sink it to the same level. Then, the known
weight of the instrument added to the weights introduced
into the upper dish will give the weight of the fluid dis-
placed. Thus, we can compare the weights of equal volumes
of the two fluids.
384 HYDROSTATICS.
Of the Pressure a?id Elasticity of Atmospheric Air.
694. The weight of the atmosphere was first recognised by
Gahieo. Torricelh, his pupil, demonstrated the existence of
this weight by the following experiment. Let AB (Fig. 236)
represent a glass tube, 3 feet in length, filled with mercury,
closed at the lower extremity and open at the upper : let the
finger be applied to the open extremity, and let the tube be
inverted, and its open extremity plunged in the basin of mer-
cury : on withdrawing the finger, the mercury will be found
to descend in the tube, leaving a certain portion of it BE {Pig.
237) unoccupied. If the experiment be tried with tubes of
different lengths or diflîerent diameters, the height of the
column of mercury sustained in the tube will be found, in
each case, to be about 29 or 30 inches above the level of the
fluid in the basin. This column of mercury is sustained by
the pressure of the atmosphere, arising from its weight;
which pressure, being exerted upon the surface CD, is suf-
ficient to counterbalance the weight of the column.
If the experiment be performed with fluids of different
densities, the heights at which they will be supported will be
found to differ : thus, if the fluid be water, whose density is
to that of mercury as 1 to 13i, the height of the column will
be found equal to 30in. Xl3|=34 feet, nearly; the weight
of such column being equal to the weight of the column of
mercury.
695. The operation of the common siphon is also to be
referred to the pressure of the atmosphere.
The siphon is a bent tube having its two branches of
unequal lengths. The shorter branch EF {Pig. 238) being
plunged into the fluid contained in the vessel ABCD, and the
air being withdrawn from the siphon, the pressure of the
atmosphere exerted upon the surface BC will cause the fluid
to rise in the siphon ; and if the height of the point F be less
than that at which the atmospheric pressure can sustain the
given fluid, it will pass into the longer branch, and will be
delivered at the point C. The current having commenced
in the siphon, it is maintained in consequence of the superior
PRESSURE AND ELASTIClTV OF AIR. 385
M'-eight of the fluid in the longer arm overcoming, in part, the
pressure of the atmosphere at the point C, and thus permitting
the equal pressure of the atmosphere exerted upon the surface
BC to force the fluid up the shorter branch. Hence, it is
obvious that the point C nmst always be below the surface
of the fluid in the reservoir ABCD, in order that the siphon
may be effective.
696. Air is an elastic fluid, which is susceptible of being
compressed into spaces which bear to each other the inverse
ratio of the forces applied.
This may be established experimentally as follows : Let
A'BCE {Fig. 239) represent a curved tube closed at E and
open at A' : let mercury be introduced into the tube until it
shall stand at the same level CC in the two branches : the
air contained in the space CE will then be of the same density
as the exterior air. If mercury be now poured into the tube
until the part ABCD be entirely filled, the length AB being
equal to 30 inches, the column of air DE will be found reduced
to one-half its original bulk CE : if mercury be again intro-
duced until it extend from A' to d, the length A'h being equal
to 60 inches, the volume of air will be found reduced to a
space Et^ = iCE.
This experiment establishes the law of compressibility ;
for, before the introduction of the mercury, the air contained
in the space CE, being pressed by the weight of the atmo-
sphere, must support a pressure equivalent to 30 inches of
mercury. When the same volume of air is caused to sustain
the additional pressure of a column of mercury AB=30
inches, it is reduced to one-half its original bulk ; and by the
further addition of 30 inches, the air is reduced to one-third
of this bulk. Thus, it appears, that the spaces occupied by
the same mass of air are inversely proportional to the pressures
applied ; and since the densities of the air are inveisely pro-
portional to the spaces occupied by the same mass, it follows
that the densities will be in the direct ratio of the pressures.
If the mercury be withdrawn from the tube, the air will
expand and occupy the same space as it did previous to
compression.
33
386 HYDROSTATICS.
Of Pumps for raising Water.
697. The pump is a machine employed for the purpose of
raising water. There are three principal kinds of pumps,
viz. the sucking pump, the lifting pump, and the forcing
pump.
The sucking pump, represented in Fig. 240, consists of
two tubes ABDC and DCHL, of unequal diameters, connected
together ; the first of these is called the sucking pipe, and the
second the body of the pump. Within the body of the pump,
an air-tight piston MN, having a valve opening upwards, is
moved through the space MH, which is called the play of the
piston. At the lower extremity of the body of the pump, a
second valve ^-, called the sleeping valve, is placed, which
likewise opens upwards.
The lower extremity AB of the sucking pipe being im-
mersed in a reservoir containing water, and the piston MN
being raised from the position MN to HL, the air contained
in the space CN will expand and fill the space CL, its density
and elastic force being both diminished : at the same time,
the air contained in the pipe AD, having a density equal to
that of the exterior air, will, in virtue of its elasticity, exert
upon the valve A:, a stronger pressure than that arising from
the elasticity of the rarefied air contained in the space CL :
hence, the valve k will be forced open, and the air contained
in the interior of the pump will acquire a density that is uni-
form throughout, but less than that of the exterior air : then
the pressure exerted upon the surface of the water AB being
less than that exerted by the atmosphere upon the surface at
other points of the reservoir, the water will rise in the suck-
ing pipe to the level A'B', such that the weight of the column
A'B , together with the pressure of the rarefied air contained
in the pump, shall be equal to the pressure of the exterior
air. The densities of the air in the body of the pump and in
the sucking pipe having become equal, the valve k closes by
its own weight.
The piston being then depressed from the position HL to
MN, the air contained in the space CL will be compressed
PUMPS. 387
into the space CN, and its density and elastic force will
become greater than those of the air contained in the sucking
pipe : the pressure on the upper surface of the valve k being
now greatest, this valve will continue closed during the de-
scent of the piston, and will intercept the communication be-
tween the sucking pipe and body of the pump : hence, the
density of the air in the sucking pipe will remain unchanged,
and the water will retain the level A'B'. When the piston
shall have regained the position MN, it will have compressed
into the space CN, not only the quantity of air originally con-
tained in CN, but likewise that portion which was introduced
into the body of the pump from the sucking pipe. The den-
sity of the air contained in the space CN will therefore exceed
that of the exterior air, and its elastic force will open the
valve I : the air contained in CN will thus be restored to its
original density. The piston being raised a second time, the
air in MD will be again rarefied, a portion of that con-
tained in A'D will pass into the body of the pump, and the
equilibrium will be restored by the water rising to a new
level A"B".
The same operation being repeated, the water will rise
through the valve k into the body of the pump, will pass
through the valve I in the piston, and will finally be delivered
by the spout Q,R.
698. We will next examine the mechanism of the lifting
pump. In this pump, the piston MN {Fig. 241) is situated
below the fixed valve /r, and being depressed from the posi-
tion MN to HL, is supposed to pass below the surface a'h' of
the water contained in the reservoir : the piston contains a
valve opening upwards, through which the water passes,
regaining its level a'h'. The piston being then elevated, the
column of water a'L, which rests upon its superior base, being
prevented from returning through the valve, will be raised
through a height equal to the play of the piston, and will oc-
cupy the space «N : at the same time, a vacuum being formed
below the piston, the water will be compelled to follow the
piston in its motion by the pressure of the atmosphere on the
surface of the water in the reservoir. But the air contained
in the space a'D being comprpssed by the elevation of tlie
Bb2
388 HYDROSTATICS.
piston, its elastic force will become greater than that of the
exterior air, and the valve k will open, restoring the air below
k to its original density. The circumstances will then be
the same as before the first stroke of the piston, with the ex-
ception that a portion of water has passed above the piston.
When the piston is again depressed, the column of water aN,
which rests upon it, will also descend, and the air contained
in the space Co will therefore be rarefied. The descent of
the water will continue until the elastic force of the rarefied
air contained between the valve k and the surface of the
water, together with the weight of the column of water raised,
shall be equal to the pressure of the atmosphere : the valve
in the piston will then open, and an additional quantity of
water will pass above the piston. By repeating the process,
a certain portion of water will pass above the piston at each
stroke ; and reaching the valve k, will pass into the body of
the pump, and may be delivered at any height.
699. The forcing pump is a combination of the sucking and
lifting pumps. In this pump, the piston MN {Pig. 242) is
without a valve, but the lateral pipe HE is provided with one
at I, opening upwards ; and there is a sleeping valve at L, as
in the sucking pump. The piston being raised, the water rises
into the space MCDEF, for the reasons assigned in describing
the sucking pump ; when the piston is depressed, the water is
forced through the valve I into the tube HG ; and by con-
tinuing the process, it may be delivered at any height.
700. If the dimensions of the sucking pump be improperly
chosen, it may happen that the water will rise only to a cer-
tain height. For the purpose of discovering in what cases
this will occur, we shall simplify the question, by supposing
the pump to be of uniform bore throughout. Let the water
be supposed to have been raised to the level ZX {Pig, 243),
and the piston to move through the space ML : call
a=LN, the play of the piston,
6= LB, the height of the piston at its greatest elevation
above the surface of the water contained in the
reservoir,
.T=the distance LX.
When the piston is raised from the position MN to HL, the
PUMPS. 389
air which was previously contained in the space ZN will
occupy the space ZL, and its elasticity will therefore be
diminished in the ratio of LX to NX ; so that if R represent
the elastic force of the air contained in the space NZ, and R'
the elastic force of the rarefied air contained in LZ, we shall
have
LX : NX : : R : R' ;
or,
X : X — a ; : R : R' :
whence,
,r — a
R'=R:
X
But the air contained iw. the space NZ being of the same
density with the exterior air, its elastic force will be properly
measured by the weight of a column of water whose base c is
equal to the surface MN, and whose height is equal to 34 feet.
Let this height be denoted by h ; the density of water being
supposed equal to unity, and the force of gravity being denoted
by g^ we shall have
R=cA^.
This value, substituted in the preceding equation, gives
T,, x—a ,
R'= dig.
But it is evident that when an equilibrium subsists, the elastic
force of this rarefied air, together with the weight of the
column of water BZ, must be just sufficient to counterbalance
the pressure of the atmosphere, which tends to produce the
ascent of the water. The weight of the column of water
ABXZ will be expressed by ^c X BX, or gc X ^—x) ; and the
pressure exerted by the atmosphere will be expressed by the
column gch ; hence, we shall have, in case of an equilibrium,
~ gch-\-{h—x)gc=gchy
X
or, by suppressing the common factor gc^
X — a
-h + b — x=h.
But, if it were required that the water should rise above the
level ZX, it would then be necessary that the atmospheric
390
HYDROSTATICS.
pressure should exceed that arising from the weight of the
column ZB, and the elastic force of the air contained in the
space ZL : we shall consequently have
-h-\-h — x<h.
X
Let z represent the excess of the second member of this in-
equality ; then
-h-\-h — x-\-z=h :
X
or, by reduction,
— ah-\-hx—x^-\-zx=^ :
whence^
If we make 2;=0, the water will cease to rise, and we shall
then have
These two values of x will be real so long as — exceeds ah :
if, therefore, this condition be fulfilled, there will be two points
at which the water will stop : but if, on the contrary, ah
should exceed —, the values of x will become imaginary, and
there can be no point at which the water will cease to rise.
Such is the condition requisite to ensure the effective per-
formance of the sucking pump.
701. With the lifting pump, the water can be raised to any
lieight, provided sufficient force be applied to the piston.
For, let the water be supposed to have risen to the level EF
{Fig. 241), the water in the reservoir standing at the level aè,
above the piston. Then, the column included between the
surfaces ah and MN being supported by the pressure of the
contiguous fluid, the piston MNP will be loaded only with
the weight of the column extending from ah to EF.
702. But if the level of the fluid in the reservoir be sup-
posed at a'6' below the piston, the weight P of the column of
water included between MN and a'6' must be supported by
AIR-PUMP. 391
the pressure of the atmosphere exerted upon the surface of
the water in the reservoir. Hence, the pressure of the atmo-
sphere exerted upon the upper base of the piston, through the
column EN, will exceed that which is exerted upon the lower
base through a'N, by the weight of the column a'N ; for this
weight counteracts in part the pressure exerted by the atmo-
sphere upon the water in the reservoir. Thus, the piston MN
being urged downwai'ds by the weight of the column MF
situated above it, and hkewise by the difference of the atmo-
spheric pressures, which is equal to the weight of the column
a'N, the effect will be the same as though the piston sup-
ported a column of water whose base is MN, and whose
altitude is equal to the distance between the levels a'h'
and EF.
It thus appears, that with a sufficient effort, the water may
be raised to any height by the lifting pump, the fixed valve
k being supposed near the surface of the water.
The same principles will serve to estimate the force neces-
.sary to raise the water in the sucking pump.
Of the Air-pump.
703. In examining the properties of various substances, it
is frequently necessary to withdraw them from the action of
the atmosphere, and it therefore becomes desirable that we
should have it in our power to exhaust the air from a vessel
in which the substance has been deposited. This vessel is
called the receiver, and is usually constructed of a transparent
substance, such as glass, in order that we may have an opportu-
nity of observing the effects produced on the substance under
consideration by the withdrawal of the atmospheric air.
704. The machine employed to exhaust the air is called
an air-pump, and the term vacuimi is applied to the space
from which the air has been extracted.
705. The general principles upon which the operation of
this machine depends, will be readily understood by a refer-
ence to Fig. 244. A represents a section of the glass receiver
which rests upon the plate BC, the lower edge of the receiver
and the plate being ground exactly plane, so that their con-
392 HYDROSTATICS.
tact may be as perfect as possible. The edge of the receiver
being previously smeared with a little sweet oil, the air will
be effectually prevented from penetrating between the
receiver and plate.
The plate BC is perforated by a cavity DE, which commu-
nicates with the cylindrical barrel CF, in which an air-tight
piston, having a valve opening upwards, is worked by means
of a handle H. At the bottom of the barrel is placed a second
valve E, likewise opening upwards.
706. Let it now be supposed that the piston has been
depressed until it has reached the valve E ; the air in the
receiver, barrel, and communicating pipe being of the same
density as the exterior air, and the valves being closed by
their own weight. Then, if a force be applied to raise the
piston, the valve P will remain closed, and a vacuum would
be left between the piston and the valve E, provided the
weight of the valve E were sufficient to overcome the pres-
sure exerted upon its under surface by the elastic force of
the air contained in the receiver and communicating pipe :
this, however, not being the case, the valve E will be forced
open, and a portion of the air contained in the pipe and
receiver will pass into the barrel, until the density of the air
becomes uniform throughout. This effect will continue until
the piston has reached its highest position, and the valve E
will then close by its own weight. The piston being then
depressed, the valve E will remain closed, and the air con-
tained in the barrel being compressed into a smaller space, its
elastic force will be increased, will become greater than that
of the exterior air, and will finally overcome the weight of
the valve P, causing it to open, and thus reducing the density
of the air contained in the barrel to an equality with that of
the exterior air : this effect will only cease when the piston
has been forced to the bottom of the barrel.
It thus appears that by a single ascent and descent of the.
piston, a portion of air has been withdrawn from the receiver
and pipe of communication. The portion withdrawn will
obviously bear the same ratio to the quantity originally con-
tained in the receiver and pipe that the capacity of the barrel
bears to the sum of the capacities of the barrel, pipe, and
AIR-PUMP. 393
receiver. By a repetition of the same process, a second
quantity can be withdrawn, and the operation may be con-
tinued until the exhaustion has been carried to the desired
extent.
707. Since the quantity of air withdrawn at each ascent
and descent of the piston forms but a part of that previously
contained in the receiver and pipe, it is obvious that a perfect
vacuum can never be produced by the operation of the pump.
The weight of the lower valve likewise opposes an obstacle
to the entire exhaustion ; for, whenever the air contained in
the receiver and pipe shall have had its elastic force so far
reduced as to be incapable of raising the valve E, the pump
will necessarily cease to exhaust. This difficulty may, how-
ever, be obviated, by causing the valve E to open by means
of a mechanical connexion with the piston.
708. As it is frequently necessary to produce a very high
degree of exhaustion, it becomes interesting to ascertain the
density of the air remaining in the receiver after any given
number of strokes of the piston ; and since the portion with-
drawn at each double stroke bears a constant relation to that
remaining, this density may be readily estimated. Thus, if
we denote by 6, />, and r the respective capacities of the
barrel, pipe, and receiver, and by d the original density of the
air, we shall have the proportion
h-\-v-{-i^ : p-^-r :: d'. d-^ =density after the first double
o-f-^-fr '
stroke.
In like manner,
b-{-v-\-r:'p-\-r::d~ : di-^ p=densitv after the
^ ^ b+p+r \b+p + r/ ^
second double stroke.
And generally,
,+p+r :f+r : : d (j|±^,)""' : d (j^J' = density
after the nih double stroke.
For the purpose of illustrating the rate of exhaustion, we will
suppose that the capacity of the barrel is one-fourth of the sum
of the capacities of the receiver and pipe ; then, we shall have
394 HYDROSTATICS.
fc=i(p+r)=i(6+;'+r);
and the density after the first double stroke will be
d-^ =d—=^d.
b-\-p+r 5b '
Thus, by the first double stroke of the piston, one-fifth of the
air contained in the receiver and pipe will be withdrawn, and
the quantity remaining will be four-fifths of the original
quantity. The density after the second stroke will, in like
manner, be four-fifths of that after the first, or || of the ori-
ginal density ; and after the third, the density will be reduced
to -fYjj or nearly one-half It thus appears that every three
strokes will reduce the density nearly one-half; and conse-
quently, that after twenty-seven strokes, the air would be
reduced to about one-five-hundredth of its original density.
709. The preceding calculation is based upon the suppo-
sition that the relative capacities of the barrel, pipe, and
receiver have been accurately ascertained, and that the
mechanical construction of the pump is perfect, neither of
which conditions is strictly fulfilled : and as it is frequently
necessary to know the precise degree of exhaustion that has
been attained, it becomes important to have a gauge, or index,
by the aid of which we may ascertain the density of the
remaining air at any moment. The instruments commonly
employed for this purpose are,
1°. The barometer gauge, which consists of a straight
glass tube about thirty-two inches in length, and open at
both extremities. The tube is placed in a vertical position,
its upper extremity communicating with the receiver of the
pump, and its lower being immersed in a basin of mercury.
When the process of exhaustion has been commenced, the
air in the tube being rarefied, the pressure of the atmosphere
upon the surface of the mercury in the basin will cause the
mercury to rise in the tube, and the height at which it stands
will indicate the difference between the exterior and interior
pressures. These pressures are in the direct ratio of the
densities of the air. The principal inconvenience of this
gauge arises from the necessity of having a barometer with
which to ascertain the pressure of the exterior air at the same
time.
AIR-PUMP. 395
2". The short barometer gauge is formed of a tube eight
or ten inches in length, open at one extremity, and filled with
mercury. This tube being inverted, and immersed at its
open extremity in a basin of mercury, the pressure of the
atmosphere upon the surface of the mercury in the basin will
retain the tube entirely full. This apparatus being placed
under a receiver which communicates with that of the pump,
and the rarefaction being commenced, the short tube will
remain full until the density of the air in the receiver has
been so far reduced that its elastic force is insufficient to sup-
port a column of mercury of a length equal to that of the tube.
The mercury in the tube will then fall, and its height at
any moment will indicate the pressure of the air within.
This gauge is evidently unfit for use when only a moderate
degree of exhaustion is required.
3°. The siphon gauge is composed of a short bent tube,
having two parallel branches, one of which is closed, and the
other open. The closed branch being filled with mercury,
and the tube being placed with the bend downwards, the
mercury will be supported in that branch by the pressure of
the exterior air. The tube is then placed beneath a receiver,
and acts upon the same principle as the short barometer
gauge, the bend in the tube serving as a substitute for the
basin of mercury. This, also, is only applicable when a con-
siderable degree of rarefaction is required.
710. The working of the piston being opposed by the
pressure of the atmosphere on its superior surface, and this
difficulty constantly increasing as the rarefaction proceeds, it
has been found advantageous to adapt a second barrel to the
pump, whose piston shall descend whilst that of the first
barrel ascends, — and the reverse. The rods of the pistons
have the form of a rack whose teeth engage in those of a
wheel which is turned by a winch. The pressures on the
pistons are thus caused to oppose each other, and the pump
works with much greater ease. The rapidity of the exhaus-
tion is likewise doubled by this arrangement.
711. If the construction of the pump be such as to require
the lower valves to be opened by the elasticity of the air
remaining in the receiver, the operation of the pump will evi-
396 HYDROSTATICS.
dently cease whenever the rarefaction has been carried so far
that the weight of the lower valve is sufficient to overcome
the elastic force of the air within. To obviate this inconve-
nience, the- lower valves are opened and closed by the motions
of the piston, as shown in Fig. 245, which represents a sec-
tional view of one of the most approved pumps. The dis-
position of the several parts has been somewhat altered, for
the purpose of exhibiting them more clearly.
A represents the glass receiver resting upon the ground
glass plate BC, and communicating by the cavity DFG with
the tvvo pump barrels VR and V'R'. The receiver likewise
communicates by the cavity svy with the barometer gauge
yz, immersed in the vessel of mercury M, and with the siphon
gauge vx. E is a stopcock for cutting off the communication
between the receiver and the barrels when the exhaustion has
been effected, and E' a second stopcock for re-admitting the
external air. In the best pumps, the barrels are made of
glass, to prevent the corrosion which would take place by the
action of the oil with which the pistons are lubricated to
render them air-tight : for similar reasons, the pistons are
sometimes made of steel. The racks L and L' of the pistons
are worked by the wheel W, which is turned alternately to
the right and left by the winch H. The lower valves Y and
V are metallic, and have the form of a conic frustrum. To
the back of the valve is attached a slender rod VR, which
passes through an air-tight hole in the piston P, and carries
near its upper extremity a small projection or shoulder.
When the piston is raised, the friction of the valve-rod which
passes through it causes the rod likewise to rise, opening the
lower valve V: but this upward motion is soon checked by
the shoulder coming into contact with the top of the barrel,
and the rod then slides through the hole in the piston.
Again, when the piston is depressed, it carries with it the
valve-rod RV, closing the valve at the bottom of the pump,
and the descent of the piston is then continued by sliding
along the rod.
712. The valves of the pistons are variously constructed.
In some instances they are metallic, resting upon a metallic
bed ; and in others, they are composed of strips of oiled silk,
AIR-PUMP. 397
bladder, or parchment, stretched across an opening in the pis-
ton, and ahernately allowing and preventing the communica-
tion between the air beneath the piston and the exterior air.
During the ascent of the piston, the valve remains closed by
the stronger pressure of the atmosphere on its upper surface,
and when the piston descends, the compressed air beneath it
will force open the valve. This latter condition will always be
fulfilled, whatever may be the degree of exhaustion, provided
the piston can be forced into actual contact with the bottom
of the barrel.
713, The pistons are usually composed of two metallic
plates, which carry between them a packing of leather soaked
in oil. The distance between these plates can be varied by
means of a powerful screw ; and by the application of a proper
degree of pressure, the packing is caused to fit the barrel
with accuracy.
714. By the aid of the air-pump we are enabled to exhibit
many of the most important properties of atmospheric air :
1°. The weight of the air may be shown by screwing a
vessel provided with a stopcock to the air-pump, and ex-
hausting the air from within it. The weight of the vessel
will be diminished by about ^\ of a grain for every cubic
inch of air that has been withdrawn.
2^. The pressure of the atmoi^nhere is rendered evident by
the difficulty with which the receiver is removed from the
plate of the pump after the air within it has been withdrawn.
A small strip of bladder being stretched across the moutJj
of an open receiver, and the air exhausted from beneath, the
bladder will be ruptured by the pressure of the exterior air.
Two brass hemispheres, being ground so as to fit accurately
to each other, and attached to the pump, cannot be separated
without great difficulty after the air has been exhausted from
the space enclosed by them. The pressu e of the atmo-
sphere is found to be equivalent to about 15 lb. for each
square inch of surface exposed to its action.
3°. The elasticity of the air may likewise be shown by
various experiments. If, for example, a bladder containino-a
small quantity of air be enclosed in a receiver, from which
the air can be extracted, the elasticity of the air contained iii
34
398 HYDROSTATICS.
the bladder will cause it to distend when the exterior pres-
sure is removed ; and on the re-admission of the air into the
receiver, the bladder will again collapse.
If a light glass bulb, having an opening in its lower surface,
b« loaded with weights so that it will just sink in a vessel of
water when the bulb is partially filled with water ; upon
withdrawing the air from the receiver in which the vessel of
water has been deposited, the portion of air contained in the
bulb will expand, expelling a portion of the water through
the orifice in the bottom of the bulb. The bulb and weight
will thus be rendered specifically lighter than water, and will
consequently rise to the surface of the fluid in the vessel :
upon re-admitting the air into the receiver, a portion of water
will be forced into the bulb, and it will again sink.
4°. The resistance of the air to the motion of bodies may
be exhibited by allowing two bodies of very unequal den-
sities to fall in the exhausted receiver of the air-pump, and
in the same receiver after the re-admission of the air. When
the bodies fall in vacuo, they will reach the bottom of the
receiver at the same instant ; but when the receiver contains
air, the denser body being least retarded by the resistance
which the air offers, it will fall through the height of the
receiver in much less time than that required by the rarer
body.
Many other experiments may be contrived to illustrate the
properties of air, but it is unnecessary to notice them in this
place.
Of the Barometer.
715. The barometer is composed essentially of a bent tube
ABC [Fig. 246), closed at A, and open at C, and filled with
mercury throughout the portion NMBEF. The air is sup-
posed to have been exhausted from the space AMN, and the
column of mercury included between the planes MN and
DFE is supported by the pressure of the atmosphere upon
the surface FE. This column is usually about thirty inches
in length, when the barometer is placed at the level of the
ocean.
BAROMETER.
716. This instrument serves to indicate the changes which
are constantly taking place in the pressure of the atmosphere ;
for, when the pressure becomes greater, the length of the
column of mercury which it can sustain is necessarily
increased, and the mercury therefore rises in the tube AD :
but if, on the contrary, the pressure of the air should dimin-
ish, the length of the column will undergo a corresponding
diminution.
The pressure of the atmosphere at any point being that
due to the weight of a column of air extending from that
point to the top of the atmosphere, it follows that this
pressure will decrease as we ascend above the earth's surface,
and consequently, that the height of the mercurial column
will diminish.
717. This principle has been employed to determine the
difference of level of two places situated at unequal distances
above the surface of the earth. For the purpose of investi-
gating a formula which shall be applicable to this object, we
shall denote by
//.' the height of the mercurial column at the lower station,
h the height of the mercurial column at the upper station,
D' and D the corresponding densities of the atmosphere at
the two stations.
Then, if we suppose the axis of z to be vertical, the general
equation of equilibrium of heavy fluids as obtained in Art.
655, will be
d2i=T)gdz*
Let the origin be assumed at the lower station, and let the
co-ordinates z be reckoned positive upwards ; then, as we
ascend in the atmosphere, the pressure arising from the
weight of the superincumbent strata will diminish, and the
* This result may be obtained directly by considering a column of the atmo-
sphere, whose base AB {Fig. 247) is the unit of surface : the pressure sus-
tained by this base is measured by the \\'eight of the column of air ABDC
extending to the top of the atmosphere ; and the elementary pressure dp will be
represented by the weight of a column having the same base, and a height equal
to dz. The base of this elementary column being equal to unity, its volume
will be expressed by 1 Xdz, or dz, and its mass by Ddz : thus, gDdz represents
the weight which will measure the elementary pressure dp. This result will
obviously be independent of the particular form given to the base AB which
has been assumed as the superficial unit.
400 HYDROSTATICS.
density of the air will undergo a corresponding decrease.
Thus, the pressure p being a decreasing function of the alti-
tude z, dp and dz will be affected with contrary signs :
hence, the preceding equation should be written
dp=—Dgdz (412).
If the difference of level of the two places be but slight, the
force of gravity g- may be regarded as constant : and hence
we shall obtain, by integration,
^—IM («^>-
But it has been shown (Arts. 651 and 696) that when the
temperature is supposed constant, the pressure and density
are proportional to each other ; hence, if P denote the pressure
capable of producing a density represented by unity, we shall
have
p=PD;
and therefore,
dp=VdT):
this value substituted in equation (41 3) gives
__P /*dD
^- gJ D-'
and by effecting the integration indicated, there results
g-
To determine the constant, we remark, that \vhen z=0, the
density becomes that which we have supposed to exist at the
lower station, and which has been denoted by D'. Thus, the
preceding equation becomes
0=-?logD'+C;
g-
eliminating C between this equation and the preceding, we
find
z=^ (log D'-log D),
or.
p, jy
But the densities being proportional to the pressures, they
BAROMETER. 401
will likewise be proportional to the observed altitudes of the
mercurial column : hence,
h: h' ::D : D', or^=5.';
h U
D'
this value of -- being substituted in that of z, we obtam
P, h'
g ^A
h'
The logarithm of —, which appears in this expression, apper-
tains to the Naperian system : if therefore, we represent by
h'
Log—, the tabular logarithm
h
the modulus, we shall have
and, by substitution.
h' h!
Log—, the tabular logarithm of —, and by M the reciprocal of
h h
MLog|'=log^;
MP h!
g h
718. To determine the value of the constant P, which
represents the pressure exerted upon the unit of surface, and
capable of producing a density of air represented by unity,
we remark, that the density D' at the lower station corres-
ponds to the pressure exerted by the atmosphere at that
point : this pressure is measured by the weight of a column
of air whose base is the superficial unit, and whose altitude
is equal to that of the atmosphere : but this column of air is
equal in weight to the mercurial column whose height is h! ;
if therefore D" denote the density of mercury, the mass of
the column will be expressed by 1 X A'D", or /i'D" : and by
multiplying this product by g^ we shall obtain the expression
h'D"g, for the weight of the column supported at the lower
station. Such will be the pressure capable of producing the
density D'. To obtain the pressure P corresponding to the
unit of density, we make the proportion
D' : 1 : : h'jy'g : P ;
whence,
402 HYDROSTATICS.
substituting this value in the formula (414), there results
^=-ËF-Log- (415).
719. The intensity of the force of gravity being different
at di^rent places on the surface of the earth, the weight of
the same column of mercury will likewise vary when it is
transported from one place to another : thus, if the force of
gravity be denoted by g at on« station, and by (1 —^)g at a
second, the mercurial column whose height is h' will become
heavier or lighter at the second station than it was at the
first, according as ^ is negative or positive.
Let the quantity ^ be considered positive : then 1 — ^ will be
positive, and less than unity, since the variations of gravity
are exceedingly small. But a column of mercury whose
height is h' becoming lighter at the point whose gravity is
denoted by (1 — J)^, it will correspond to a less pressure of
the atmosphere, and hence, the density of the air correspond-
ing to this pressure will be less.
The densities of the air being proportional to the pressures
exerted, and these pressures being measured by the weights
of the column of mercury whose height is h'. it follows that
the intensities of gravity, which are represented respectively
by g- and (1 — <J)^ at the two places, will be proportional to
the densities corresponding to the same height h' of the mer-
curial column : thus, if we denote by d the density of the
air at the place where the intensity of gravity is represented
by {\—^)g, we shall have
^:^(l-^)::D':c^;
whence,
This value of the density must be substituted in the for-
mula (415), in order that it may become applicable to the
place at which the gravity is represented by {\—S)g: the
formula will thus become
-=DXW)Log^ (416).
From a comparison of the results obtained by causing pen-
dulums to oscillate in different latitudes, it has been ascer-
BAROMETER. 403
tained that if the intensity of gravity be denoted by g at the
latitude of 45°, the quantity ^ will be expressed by 0.002837 X
cos 2^, when the latitude is supposed to become equal to ■^.
Hence, by substitution in the preceding formula, we obtain
D"M/i'Log|-'
D'( 1-0.002837 cos 2%/^)
The quantity «5" being always extremely small, we may
replace t— r in equation (416) by its development l+^S'+.J'^
i —
4-<5''+(fcc. and neglect the terms ^^^ J 3, dec. as extremely
minute with reference to ^ : we thus obtain- — - = 1 + «J' ; hence,
1 —
the value of z will become
;s= ^5^(1 +0.002837 cos 2^) Log ^ (417).
720. This formula has been obtained upon the supposition
that the temperature remains constant in passing from the
lower to the higher station. To adapt the formula to the
case in which the temperature is variable, it will be necessary to
know the law according to which air expands when subjected
to a change of temperature. The experiments of Gay Lussac
and other philosophers demonstrate conclusively that atmo-
spheric air when perfectly dry, and when subjected to a con-
stant pressure, expands for each degree of Fahrenheit's
thermometer, between the temperatures of 32° and 212°, ^^-^
of its volume at the temperature of 32°. Thus a volume of
air represented by unity at the temperature of 32°, will be-
come l + jôn ^^''^^n its temperature has been raised to
32°+/i° : and since the densities are in the inverse ratio of
the spaces occupied by the same mass, it follows that the
density d! of the air, at the temperature 32° -h n°, will be ex-
pressed by
^480
D' being the density at the temperature 32°.
Cc2
404
HYDROSTATICS.
The coefficient of the number n being- very small, the error
which will be introduced by assigning- to n a value which
shall not differ greatly from its real value will always be ex-
tremely small ; and since the variations in temperature which
occur in passing from a lower to a higher station are nearly
uniform, we may, without sensible error, regard the temper-
ature as constant, provided we assign to it a value equal to
the arithmetical mean between the temperatures t and t' at
the higher and lower stations ; we shall thus have
and the density d' of the air, which was previously represented
by D', will become
d'^ --J^ = ^ . (418)
^480V 2 '^^) ^+ 960
But the density of mercury bein^ increased by a diminution
in the temperature, the height of the mercurial column at the
colder station will be less, for the same pressure, than it would
have been if the temperature had remained constant, and
equal to that at the warmer station ; and since mercury is
found to expand about jy j, P^^^ ^^ ^^^ h\i\k for every degree
of Fahrenheit's thermometer, it will be necessary to increase
the height h, which is supposed to correspond to the colder
station, by the quantity — — taken as many times as there
are degrees of difference between the temperatures of the
mercury at th« two stations, in order to reduce this height h
to what it would have been, if the temperature had remained
constant, and equal to that at the warmer station. Let T
and T' represent the temperatures of the mercury at the two
stations as indicated by thermometers in contact with the
barometers ; then the quantity h in equation (417) should be
replaced by
^ ^(T--T) .
9742 '
hence, by substituting this value for h, and that of d' (418)
for D', in formula (417), we find
BAROMETER. 466
_D"M/i' / , i + i'-6'^ \
-—fy-y-^ 960 /
h'
X (1+0.002837 cos 2^) Log — ^
K^+W")
721. Let it be supposed that the observations which deter-
mine the height h' are made in the latitude of 45°, and at the
level of the ocean ; we shall have
cos 2\^=0 ;
and the preceding formula will give
D'Mh' z
D' /, , ^-f r-64\- h'
/, . <+r-64\-
(419).
a(i+
9742/
If the height z be measured trigonometrically, and the quan-
tities h, h', t, i', T, T' be determined by taking a mean result
of a great number of observations, the second member of this
equation will become entirely known, and therefore the con-
M/i'D"
stant will likewise be known. This constant has been
thus found to be equal to 60345 feet : if its value be substi-
tuted in that ol z, we shall obtain the following formula :
z=60345 ft. (l+ttll^\ (1+0.002837 cos 2^)
XLog —, ^,_ (419 a).
h(l+i — I)
\ 9742 /
722. The second member of this equation may be put
under a more convenient form ; for, we have
^±^=.001042(^+^-64) ;
and if we denote by 6 the difference between the temperatures
T and T', and change the form of the last factor in equation
(419 a), there will result
Log
/ T--T\ =^"g^'-^"g^-^"gC + 9^) '
A 9742/
406
HYDROSTATICS.
or, by developing the last term of the second member, retain-
ing only the first term of the development, we shall have
in which M' represents the modulus of the system. The
numerical value of the coefficient oîê is .000044. Hence, the
equation (419 a) may be reduced to
5;=60345 ft. [1 + .001042(^+r-64)](l +.002837 cos 2^)
X (Log A' -Log /i— .0000440).
723. To apply this formula to the determination of the dif-
ference of level of two stations, it will be necessary to observe
carefully the altitude of the mercurial columns at each station,
and the temperature of the atmosphere as indicated by a
thermometer placed in the shade, and at some distance from
the barometer. The temperature of the mercury as shown
by a thermometer in contact with the tube of the barometer
should likewise be noted. These observations should be
made at the same instant, by different observers, at the two
stations, in order to avoid the errors which might arise from
a change of pressure or temperature during the interval be-
tween the observations. When the condition of simultaneous
observations becomes impracticable, it will be advisable to
make observations at one of the stations, the lower for ex-
ample, at equal intervals before and after the time of observa-
tion at the other station. Then, an arithmetical mean
between the first and last results may be considered as nearly
equivalent to an observation made at the instant corresponding
to the mean between the two times, provided the interval be
but short, and the difference between the results of the two
observations inconsiderable.
724. The general formula for the difference of level of
two stations having been obtained upon the supposition that
the atmosphere is in equilibrio, the results given by it are to
be relied on most confidently when the observations have
been made in calm weather.
PART FOURTH,
HYDRODYNAMICS.
OF THE DISCHARGE OF FLUIDS THROUGH HORIZONTAL ORIFICES,
725. Experience has shown, that when a fluid issues
from a small orifice in the bottom of a vessel, the superior
surface of the fluid maintains itself in a position sensibly hori-
zontal, during the discharge of the fluid. Hence, if we con-
ceive the fluid divided into horizontal strata, these strata may-
be regarded as preserving their parallelism during their
descent, and the particles may be considered as descending in
vertical lines. This hypothesis, however, can only be re-
garded as approximating to the truth ; for, if the form of the
vessel be not prismatic, it will be impossible for any one
stratum to occupy the place of that immediately beneath it
without undergoing some change in its dimensions ; its par-
ticles will therefore be subjected to horizontal motions. More-
over, the particles situated in the immediate vicinity of the
orifice, being without support, yield to the pressure exerted
against them by the adjacent particles, and thereby tend to
deflect the latter from their vertical directions : but, in what
follows, we shall omit the consideration of these circum-
stances, which would greatly complicate the question, and
which are found to produce but a slight effect when the form
of the vessel is nearly prismatic. The accuracy of the
hypothesis may be rendered evident by mixing with the fluid
an insoluble powder of nearly the same density : the panicles
of this powder will be carried along with the fluid, and the
paths which they describe may be readily observed. In this
manner it will be found that the particles descend nearly in
vertical lines until they approach very near to the orifice.
408 HYDRODYNAMICS.
726. Let it be supposed that the ordinate z measures the
distance mo [Fig. 248) of one of the fluid strata from a hori-
zontal plane AB, which will be assumed as coinciding with
the surface of the fluid. The form of the vessel being deter-
mined by the equation of its interior surface, /(^z-, y, 2;,)=0,
"we can deduce from this equation the area 5 of the section
which corresponds to the ordinate z^ and by multiplying this
section by dz^ the thickness of a stratum, we shall obtain sdz
for the volume of the stratum. This being premised, it is
obvious that all the particles composing a single stratum will
have a common velocity ; but the particles of different strata
will have different velocities ; for, the fluid being supposed
incompressible, any one stratum in descending through the
height dz^ in the time dt^ will cause a volume of fluid equal
in volume to the stratum to issue through the orifice. But
if we denote by u the velocity of the fluid at the orifice EF,
and by k the area of the orifice, the space described by a par-
ticle issuing from the orifice, in the time dt^ will be expressed
by udt, and the quantity of fluid discharged in the same time
will be represented by kiidt. Equating this value with that
of the stratum, we shall obtain
sdz^kudt (420) ;
whence,
ku=s^ (421).
dt ^
727. At the expiration of the time t, the velocity of the
dz
stratum whose section is s will be equal to —, and if this
CtL
velocity be represented by v, the equation (421) will reduce to
ku=sv (422) ;
whence we conclude, that the velocities v and u are in the
inverse ratio of the sections 5 and k. This result might
have been anticipated ; for, the velocity must evidently in-
crease in the same ratio that the area of the section is dimin-
ished, in order that the quantity of fluid passed through the
section may remain constant.
728. At the expiration of the time t-\-dt, the velocity v will
become v-\--j-dt: but if the motions of the particles were
dt
DISCHARGE OF FLUIDS.
409
independent of their action upon each other, the incessant
force g^ which sohcits them, would communicate, in the
instant dt^ the velocity gdt ; hence, the velocity lost by the
stratum whose section is s, and velocity o, in the time dl,
will be expressed by gdt——dt, and consequently, the inces-
dt
sant force due to this velocity will be represented by
do
^~dt'
But by the principle of D'Alembert, an equilibrium would sub-
sist in the system if each fluid stratum were acted upon by
the force lost (g j which corresponds to it. This sup-
position will convert the equation dp=J)gdz (Art. 655) into
dp=B(^g-^j)dz (423).
The quantity dp represents the differential of the pressure
at that stratum of the fluid which corresponds to the ordi-
nate z, whilst the fluid is in motion. For, the force g, which
acts on each stratum, being resolved into two components,
one of which j- is just capable of producing the motion
assumed by the stratum, the other component^— -5- will
obviously be alone eflective in producing a pressure on the
other strata ; and the expression for dp has been obtained
upon the supposition that these second components v/ere
alone applied to the fluid particles.
The diflerential dv which enters into the preceding equa-
tion must be replaced by its value deduced from formula
(422) ; from that formula we obtain
v=— (424).
s
729. The second member of this equation contains the
two variables ti and s : the quantity u, which expresses the
velocity at the orifice, is a function of the time, and the sec-
tion 5 is a function of the ordinate z. .~,^
410
HYDRODYNAMICS.
The differential of v regarded as a function of z expresses
the difference between the velocities of two consecutive sec-
tions, these velocities being considered at the same instant.
But if the differential be taken with reference to ^ as a varia-
ble, we shall obtain the difference between the velocities of
two consecutive strata which pass in succession through the
same section of the vessel. And, lastly, if we wish to obtain
the difference between the consecutive velocities of the same
stratum, we must differentiate v with reference to the two
variables t and z, regarding the latter as a function of the
former.
This last supposition should be adopted in finding the value
of dv as employed in equation (423). We shall therefore
differentiate the second member of (424), regarding w as a
function of t, and 5 as a function of z, which is itself a func-
tion of t. But the differential of (424) being in general
dv—-du-\-kud-,,
s s
or,
, kdu , ds
dv— ku . — ,
it will become, when modified according to the hypothesis
assumed,
, k du ,, ku ^ds ^dz
s dt s^ dz dt
If we deduce the value of -7- from this expression, and sub-
dt
stitute it in equation (423), we shall obtain
T-v / ' k du , , ku ds dz J \
dp = \J\gaz — . -^dz-\ .-—.-r-dz I :
^ V® s dt s^ dz dt /
dt
.,=D(,.._.f4>.-,4:).
730. This equation must be integrated with reference to z.
We remark, however, that s will necessarily vary with z, but
that the quantities w and -^, which represent the particular
dt
eliminating -^ by means of equation (420), there results
DISCHARGE OF FLUIDS. 411
values of v and -— corresponding to the orifice, not being
functions of the quantity xr, they may be regarded as con-
stant in effecting this integration.
731. If we regard u and — as constant, it is obvious that
at
all the integrals will be taken with reference to z, and there-
fore apply merely to the dimensions of the vessel. But, when
these integrals have been obtained, we may regard ii and —
(JLt
as variables, and functions of t.
732. By effecting the integration, we obtain
^=K-^-4'/t-S^)+« (^^«)-
The velocity u which enters into this equation is equal to
dz
the value — corresponding to the orifice, and will obviously
be a function of the time. Consequently, as the quantity u
has been supposed constant in the preceding integration, the
time t must be constant likewise. Hence, the constant C
will in general be a function of the time.
733. To determine this constant, let P represent the
pressure sustained by the superior surface CD of the fluid
{Fig. 249), the area of this surface being denoted by s'. If
/dz
— be taken in such manner that it shall be
equal to zero when s becomes equal to s\ this section s' will
correspond to an ordinate 2;'=0L, and the equation (425)
will give, upon this hypothesis,
C=V-Ti(gz'-^^\.
This value being substituted in (425), we obtain
,=P + d[,(._V)-..^/^^+,«.C^-|;)J....(426).
734. This pressure is exerted at every point of the stratum
whose distance from the plane AB is equal to z. If we wish
to obtain the pressure Q, at the orifice, we denote by z" the
412
HYDRODYNAMICS.
corresponding value of the ordinate z which will be equal to
Ow, and observe that the section 6- will, at that point, be equal
/dz
— being then taken between the hmits
z=z' and z=z'\ we shall obtain, by representing this inte-
gral by N, and substituting these values in equation (426),
735. This equation makes known the pressure at the ori-
fice : the first member expresses the difference between the
pressures at the orifice and at the surface. Let these pressures
be supposed equal, as is the case when they arise from the
weight of the atmosphere: then, d — P will reduce to zero,
the common factor D will disappear, and there Avill remain
gi,z"-z')-m^^^-\-\^i^ (^-l) =0;
but the area k of the orifice being always supposed less than
the area s' of the superior surface, the fraction -^ will be less
than unity ; if therefore, we wish to render the coefficient of
u"^ positive, we may write this equation under the form
gi^^"^^')-m^^-\u^ {}-Ç) =0 (427).
736. If in this eqiiation we introduce the vertical distance
of the orifice below the surface of the fluid, making
z"-z'=h (428),
we shall have
The quantity h, which represents the distance EP (Pig. 250),
will be constant if the surface of the fluid be supposed to be
maintained at the same height ; but it will be variable if the
vessel be supposed to discharge its contents without being
replenished.
737. In the latter case, if we make EO=a, P0=5r, and
EP=r/j, we shall have the relation
h = a-z (430) ;
DISCHARGE OF FLUIDS. 413
and the equation (429) will become
g{a-z)-m'^-^-iu^ (l-^) =0 (431).
738. If the surface of the fluid be constantly maintained
at the same height, the quantity h will have a constant value,
and the integral N, which will then be a function of constant
quantities, will likewise be invariable. Thus, equation (429),
containing no other variables than t and m, may be put under
the form
a — b~ cu^ =0
dt '
from which we deduce
hdu
dt—-
a — ca^
This equation can be readily integrated by the method of
rational fractions; for, if we make h=h'c^ and a=a'^c, the
quantity c will become a factor of the numerator and de-
nominator, and may be stricken out ; whence we obtain
,^ h'du
dt= .
The second member of this equation being resolved into
factors, we shall have
,^ 2-^^r2^'^"
a'-\-u a' — u
which, being integrated, gives
or,
^=^ log {a'+u)-^, log {a'-u)-hC',
. b' , a'+u , ^
t=—- log — ■ hC.
2a' ^a'—u
The constant C is determined by the condition that the ve-
locity u is equal to zero at the same instant as the time t ;
thus, the supposition of w=0, and i=0, reduces the preceding
equation to
^'og 1+0=0;
414 HYDRODYNAMICS.
or,
C=0:
whence,
6' , a'-\-u
this equation will determine u, if we suppose the time Mo be
given.
739. If we denote by e the base of the Naperian system,
and pass from logarithms to numbers, we shall obtain
2a't
a'-\-u -^r.
-, =e
a — u
and by resolving the equation with reference to w, there
results
2a't
_ «te '''-!) .
or replacing a' and h' by their values (Arts. 737 and 738), the
expression for the velocity will become
But, if the area of the orifice, which is denoted by A-, be sup-
posed extremely small, the exponent of e, increasing with the
time ^, will become exceedingly great after the expiration of a
very short time. Hence, we may neglect unity in the nume-
rator and denominator of the last factor, as very small with
reference to the term which precedes it, and the value of u
will then be reduced to
vte)
by neglecting k^ with reference to s'^.
Thus, it appears that the expression */(^gh) is a limit
which the velocity of the fluid at the orifice never attains, but
to which this velocity becomes very nearly equal after the
expiration of an exceedingly short time.
DISCHARGE OF FLUIDS. 415
The value of the velocity being thus determined, we sub-
stitute it in equation (425), and thence deduce the pressure on
the unit of surface.
740. If the vessel be supposed to empty itself, the upper
surface will be depressed as the fluid is discharged, and the
quantity A, or (a — z) must therefore be regarded as variable
in equations (429) and (431).
The equation (431) will thus contain the three variables
t, u, and z, and will consequently be insufficient for the solu-
tion of the problem : but a second relation may be obtained
by means of equation (420) in which we replace * by s', and
thus obtain
ku=s'^ (432).
741. This equation likewise contains three variables, and
we are therefore unable to integrate it ; but it will serve to
eliminate z. For this purpose, we differentiate equation
(431), which gives
—g- A;N- u-r\ 1 r- I =0>
^dt dV- dt\ 5'V '
dz
and by eliminating — ^, we obtain
(jLZ
~g—, — ^^N-- — ?«-—( 1—— ) =0.
s' di'' dt \ s'V
This equation, which can only be integrated by approxima-
tion, makes known the relation between the time and the
velocity.
742. When the orifice is supposed extremely small, the
terms containing k may be neglected, and the equation (426)
will be reduced to
p=V-\.-Dg{z-z');
but z~z' is represented by On — OL {Fig. 249) ; and it will
therefore express the distance of the point w, whose ordinate
is equal to z, beneath the surface of the fluid. Hence, the
pressure p exerted upon the unit of surface at the point n is
equal to the pressure P at the surface of the fluid, plus the
pressure arising from a column of a fluid whose height is
equal to the distance of this point below the surface.
416 HYDRODYNAMICS.
It should be remarked, that this pressure is precisely that
which would be exerted at the point n if the fluid were sup-
posed at rest.
743. If the terms containing k in equation (429) be neg-
lected as infinitely small, it will reduce to
whence,
u=^{2gh) (433):
and we therefore conclude, that when a fluid escapes from an
infinitely small orifice in the bottom of a vessel, the velocity
will be the same as that acquired by a heavy body in falling
through a distance equal to the height of the surface of the
fluid in the reservoir above the orifice ; and since it has been
shown (Art. 405) that a body projected vertically upwards
will rise to a height equal to that through which it must fall
to acquire the velocity of projection, it follows, that if by
means of a curved tube, the jet of fluid be directed upwards,
it will rise to the level of the surface of the fluid in the
reservoir.
744. The expression for the velocity with which a fluid
will issue from an extremely small orifice in the bottom of a
vessel may be investigated in a more elementary manner, as
follows. Let EF {Fig. 251) represent a very small orifice
in the bottom of a vessel ABCD, which is filled with a fluid
to the level AB, and let GF represent an infinitely thin stra-
tum of the fluid directly above the orifice EF. Denote the
height of this stratum by dh, the entire height of the fluid
FI being represented by h. Then if the stratum of fluid GF
be supposed to fall through the height HF under the influ-
ence of the force of gravity, it will acquire a velocity v, ex-
pressed by
v = ^{2gy.YYL)^^{2gXdh).
But if the stratum be supposed to descend through the same
height, being urged by its weight and the pressure arising
from the column of fluid GI, which is directly over it, the
incessant force g', which is then exerted upon it, will be to
the force of gravity, as FI to FH. Hence, we shall have
^~FH dfi
DISCHAKGK OF FLUIDS. 417
Again, if u' denote the velocity acquired by tne stratum in
descending through the space FH, when urged by the force
g\ we shall have
and by comparing this value with that of v^ we find
V ^(2gxdh)'
or, by substituting the value of — , and reducing, there results
This expression is precisely the same as that which would
be obtained for the velocity of a body falling freely through
the height FI.
745. When the orifice, which is still supposed exceed-
ingly small, is pierced in the vertical face of a vessel, the
fluid will issue in a horizontal direction, and will describe the
arc of a parabola, if the resistance of the air be neglected. The
angle of projection denoted by a in equation (289), being in
the present case equal to zero, we shall have tang «=0,
cos ct=l: these suppositions reduce the formula (289) to
an equation of a parabola whose axis is vertical, and whose
vertex coincides with the origin of co-ordinates.
746. The distance to which the fluid will spout upon a
horizontal plane situated at any distance below the orifice
may be readily determined. For, let O {Fig. 252) represent
an orifice in the vertical side of a vessel which is filled with a
fluid to the level EF ; and let AB represent the horizontal
plane upon which the jet is allowed to fall. Then, the quan-
tity h will represent the distance OF, and the ordinate CD of
the parabola OD will be determined by making y=OC : we
thus obtain
CD=:r = v'(4%)=V(OFxOC).
But the expression y'(OFxOC) is equal to the ordinate OG
of a semicircle described upon CF as a diameter. Hence,
we derive the following rule : The horizontal distance to
wliicli ajluid will spout from an orifice in the vertical side of
Dd
418 HYDRODYNAMICS.
a vessel, is equal to double the ordinate of a semicircle de-
scribed upon the distance intercepted between the upper sur-
face of the fluid and the horizontal plane iipon which the
fluid falls ; this ordinate being drawn through the point
lohich corresponds to the orifice.
When the orifice is pierced at the middle of the hne CF,
the ordinate OG will be a maximum, and the distance to
which the fluid will spout will therefore be the greatest.
747. The velocity u having been determined, we can
readily ascertain the quantity of fluid discharged in the time t.
For this purpose, we remark, that whilst the stratum of fluid
CD {Pig. 250) sinks to the level MN, a volume of fluid
equal to that contained between the planes CD and MN must
pass through the orifice. But if we represent by 5 a section
of the vessel, and by dz the thickness of an elementary
stratum, the integral fsdz taken between limits CD and MN
will express the volume of fluid discharged. If this volume
be denoted by Q, we shall have
Q,=fsdz (434) :
but the equation (420) gives
sdz=kudt ;
whence, by substitution, we obtain
Q, =fkudt.
The value of the quantity discharged may be deduced imme-
diately from that of the velocity. For, if de represent the
space passed over by the fluid filament in the time dt, upon
leaving the orifice, we shall have
udt=de:
and if this expression be multiplied by k, the area of the ori-
fice, we shall obtain kudt for the volume discharged in the
time dt. Taking the integral fkudt, we shall find the quan-
tity discharged in the time t.
To effect the integration, we replace u by its value y/{2gh)
given in equation (433) : we thus find
Qi=k^{2g)f^h.dt (435).
748. Two distinct cases may now be presented, viz. when
h is constant, and when h is variable. The first occurs
DISCHARGE OP FLUIDS. 419
when the fluid in the reservoir is constantly maintained at
the same height, and the preceding equation can then be
integrated without difliculty, since the quantity h may be
replaced by a constant a.
Thus, we shall have
Q,=kt^{2ga) + C.
The constant C may be determined by the condition that the
quantity Q, is equal to zero at the commencement of the time,
or Q,=Oj and ^=0 ; hence,
and the equation therefore reduces to
a=A-V(2^«) (436).
749. If the orifice k be supposed circular, its radius being
represented by r, we shall have
k—vr^\
and the formula will become
a=^^(2^)^rV« (437).
The quantity '^^/{2g) will be the same for all problems which
may be proposed, and its value may be immediately deduced,
since we have
^=3.14159, ^=32.1.598.
The quantity g being expressed in feet, the values of r and a
must be expressed in units of the same kind, and the quan-
tity discharged will then be expressed in cubic feet.
750. The time t must be expressed in seconds, since the
second has been adopted as the unit of time in determining
the value of g.
7o\. If the fluid be water, the weight of the quantity dis-
charged may be determined by allowing 62ilbs for every
cubic foot.
752. The formula (437) likewise serves to determine the
time necessary for a given quantity of fluid to be discharged
from an orifice in a vessel, when the fluid is maintained at a
constant height ; for the formula gives
t= ^_ (438).
420
HYDRODYNAMICS.
753. As an example, let the vessel be supposed cylindrical,
the radius of its base being denoted by b ; and let it be re-
quired to determine the time necessaiy to discharge a volume
of fluid equal to that of the cylinder.
In this case, the horizontal sections being all equal to xb^,
the equation (434) will give
Çi^firb^ dz ;
and consequently,
a^Trb^z+C.
Taking the integral between the limits «=0 and Zz=a, there
results
(X=vb^a.
This value substituted in formula (438) gives
Trab^
t=
or, by reduction.
b^^a
t=
754. If we suppose the fluid to be maintained at a height
a' in a second vessel, and denote by Q,' the quantity dis-
charged from an orifice k' in the time t, the equation (436),
when applied to the present case, will give
Ci'=k'^{2g). t^a' ;
and by comparing this equation with (436), we can estabhsh
the proportion
Gl: a' :: k^{2g).t^a: k'^{2g).t^a' ;
or, by suppressing the common factor t-^{2g), this proportion
becomes
d : Q,' :: k^a: k'^/a'.
Hence it appears, that the quantities discharged in the same
time, from orifices of different sizes, and situated at different
depths, are directly proportional to the areas of those orifices
and the square roots of their deptJis.
755. From the formula (436) we can deduce another con-
venient theorem relative to the quantity of fluid discharged.
For, let s represent the space through which a body would
fall in the time t ; we shall have
DISCHARGE OF FLUIDS. 421
or,
Substituting this value for t in equation (436), we obtain
and since ■^/{as) is equal to a mean proportional between tlie
distances a and s, we deduce the following rule : The volume
of fluid discharged from an oriflce k,in the time t,is equal
to tioice the volume of a cylinder whose base is the area of
the oriflce, and whose height is a mean projjortional between
the depth of the oriflce beloiv the surface and the distance
through which a body u'ouldfall in the time t.
756. Let the vessel be now supposed to discharge itself,
without receiving an additional supply of fluid : the quantity
h in equation (433) must then be regarded as variable, and
being replaced by [a — s), that equation will become
This value of u substituted in (432) gives
fif — *'^^
~k^[2^^^a-z)y
or,
dt^ ^'^^ (439).
The quantity s' represents the section of the vessel which
corresponds to the upper surface of the fluid. This section
will be a function of the variable z, and may be eliminated by
means of the equation of the interior surface of the vessel.
Thus, the value of s' in terms of z being introduced into
equation (439) will render that equation susceptible of inte-
gration, and the relation between z and t will therefore
become known. If we subtract the value of z thus obtained
from the constant a, we shall obtain an expression for h in
terms of t, which substituted in (435) will give, after integra-
tion, a relation between the time t and the quantity dis-
charged Q,.
757. Let us take, as an example, a vessel whose interior
surface has the form of a paraboloid of revolution. T! is
36
422 HYDRODYNAMICS.
surface being generated by the revolution of the parabohc
arc AD {Mg. 253) about the vertical axis AB ; if we denote
by a the distance AB between the orifice and the surface of
the fluid in its primitive position, by z the distance PB,
and by y the ordinate PM, we shall have the relation,
y^=1){a—z) the equation of a parabola referred to its
vertex A.
Hence, if ?r represent the ratio of the circumference to the
diameter, the area of the circle described with the radius PM
will be expressed by jry^ =7rp(a — z) ; and consequently,
s'=^p{a-z) (440).
Let this value be substituted in (439), and we shall obtain
dt=-, — ^— — X — -, ^dz ;
or, by reduction,
dt = :r^^'^---(a-zydz.
758. For the purpose of integrating this equation, we make
a—z=x] whence,
f{a—zYdz=—fx^dx=—%x^ + C'.
replacing x by its value, we have
f{a-zYdz = -%{a-z)^ + C',
and consequently.
The constant C is determined by making 2;=0andi=0;
this supposition gives
and the equation (441) can therefore be reduced to
vp
■[a^—{a-zY].
'k^{2gy
To determine the quantity discharged in a given time, we
find in this equation the value of
■.-.^{J-^>)
t=-^^y.^ ''^
DISCHARGE OF FLUIDS. 423
and substitute it for h in formula (435) : we thus obtain the
relation
This equation may be integrated by a process entirely similar
to that adopted in finding the relation between z and t.
759. Let it be required to determine the time in which the
water contained in a vessel having the form of a right cyl-
inder will be discharged through an orifice in the bottom of
the vessel. Let h represent the radius of a section of the
cylinder by a plane perpendicular to its axis : then, s'=?rb'',
and the equation (439), when applied to the present case, wil.
give
Making a—z=.T, then integrating the transformed equation,
and replacing a: by its value, we find
The constant is determined as in the last example, by making
z=0 and t=0 : whence we deduce
The integral being taken between the limits z=0 and z=a,
we find, for the time of emptying the vessel,
'-iBm^" <^*^'-
If we suppose, as in Art. 749, that the orifice is a circle whose
radius is equal tor, we shall have k=7rr^ : this value reduces
(443) to
By comparing this result with that obtained in Art. 753, it
will appear that the time necessary for the entire discharge
of the fluid when the vessel empties itself, is double that in
which an equal quantity of fluid would flow through the
same orifice if the vessel were kept constantly full.
760. The formulas (442) and (443) will serve as a guide in
424 HYDRODYNAMICS.
the construction of a clepsydra, or water-clock. This instru-
ment consists merely of a vessel from which the water is
allowed to escape through an orifice in the bottom, and the
intervals of time are measured by the depressions of the
upper surface. Thus, if we wish the clock to run 12 hours,
we reduce the time o seconds, which g-ives 12 x (60) 2, or
12 X 3600 ; and by substituting this value of t in formula
(443), we can then assume arbitrarily two of the three quan-
tities A-, 6, and a. Let the values of k and h be assumed ;
that of a, the height of the clepsydra, will then result from
formula (443).
To discover the manner in which this height should be
divided in order that the superior surface of the fluid may
be depressed through the several divisions of the scale in
equal intervals of time, we deduce from equation (442) the
value of {a—z), which is
.A•V(2^)^^
-H^'^-'W)
and by making t successively equal to 1 hour, 2 hours, 3
hours, &c., we can determine the corresponding values of
a—z, which should be laid off from the bottom of the vessel.
We can, however, readily discover the general law according
to which the scale must be divided : for, since the vessel is
supposed to discharge itself in 12 hours,if we make ^=12 hrs.,
we shall have a—z=0; and consequently,
/• Â:(12hrs.)v^(2,^)_^.
or,
(12hrs.)^^'(2o-)
2^6=* ' ^
When ^=11 hrs,, we have
t . V(2^) (llhrs.)V(2g-) _,,
and therefore,
a-z = {^/a-\i^ar- =^(j\y Xa.
In like manner, when ^=10 hrs, we shall find
a—z=:{j%y- xa.
DISCHARGE OF FLUIDS. 425
Thus the successive values of a—z^ which correspond to the
several hours, will bear to each other the same relations as the
terms in the series
(TV)^(T^3)^(^)^&c.
These terms are to each other in the same ratio as the
squares of the natural numbers 1, 2, 3, &.C. Hence, if we
divide the whole height a into 144 equal parts, and lay off
from the bottom of the vessel distances which shall be equal
respectively to 1, 4, 9, &.c. of these parts, we shall obtain the
points of division in the scale which will correspond to the
upper surface of the fluid at the expiration of the several
hours. The form of the vessel being prismatic, the figure
of its base may be assumed arbitrarily.
761. When the surface of the fluid shall have arrived
nearly at the bottom of the orifice, the quantity discharged
will be influenced by the formation of a hollow tunnel, which
is then found to be produced directly above the orifice : it is
therefore advisable to employ only the first eleven divisions
of the scale.
762. It usually occurs that the condition of the particles
descending in vertical lines, and with velocities which are
equal at every point of the same stratum, ceases to be ful-
filled when the surface of the fluid has arrived within 4 or 5
inches of a horizontal orifice. The fluid particles then
assume directions which are more or less inclined to the hori-
zon, and the tunnel spoken of in the last article is then
formed. When the orifice is found at a considerable depth,
the upper surface of the fluid remains sensibly horizontal,
and the tunnel above the orifice is no longer formed, in con-
sequence of the greater velocity with which the fluid par-
ticles near the orifice are compelled to flow into the vacancy
which has been left by those immediately preceding them.
763. This tunnel becomes much less perceptible when the
orifice is formed in the side of a vessel. But when the upper
surface of the fluid has nearly attained the level of the orifice,
a slight depression on the side of the orifice begins to be
observed.
764. This tendency of the fluid particles towards the ori-
fice, occasioned by their sustaining less pressure in that direc-
426 HYDRODYNAMICS.
tion, gives rise to a contraction in the jet of fluid, which, in
issuing from the orifice, assumes the form of a truncated
pyramid or cone, whose greater base corresponds to the ori-
fice. This diminution in the size of the jet is called the con^
traction of tJie vein.
With a circular orifice, the smallest section of the fluid vein
is found at a distance from the orifice equal to the radius of
the orifice. Beyond this point the diameter of the section
again increases, so that the entire jet has the form of two
truncated cones which are united by their smaller bases.
765. The contraction of the vein likewise takes place
when the orifice is pierced in the side of a vessel ; but if the
orifice be large, and be placed at a short distance below the
surface of the fluid in the reservoir, the jet will be found to
be more contracted in the vertical than in the horizontal
direction.
766. When a conical tube whose interior surface corres-
ponds to the form of the contracted vein is adapted to an
orifice pierced in a thin plate, the quantity discharged is found
to be very nearly the same as though the fluid issued directly
through the orifice. Hence, we may regard the vessel as
coiitinued to the point at which the greatest contraction of
tiie stream takes place, and consider the least section as
forming the real orifice.
It is proved by experience, that the quantity actually dis-
charged may be deduced from that calculated according to
the theory, by sinjply changing the value of the constant k.
Thus, if we represent by MA: the area of the orifice which has
been calculated from a knowledge of the quantity actually
discharged, tne theoretic formula
must be modified by substituting Wt for k : we shall thus
obtain, for the actual discharge,
767. When the orifices are pierced in thin plates, the ratio
M is found to be independent of the size of the orifice, and of
its depth below the surface, provided that depth be not very
small. Hence, if we represent by Q,' the quantity discharged
DISCHARGE OF FLUIDS. 427
from an orifice k' at the depth a\ we shall have the pro-
portion
a : a' : : Wc^{2g) . V« : MAV(2^) • V«' i
and we therefore conclude, that the quantities discharged from
two such orifices are to each other as the products of the areas
of those orifices, and the square roots of their depths.
768. The number M has been found by Bossut to be about
0.62, and the orifice k must therefore be multiplied by this
fraction, in order that the quantity given by the formula may
correspond with the results of experiment. Thus, the cor-
rected expression for the quantity discharged will be
Q = (0.62)AV(2^).V«-
This formula is alike applicable, whether the orifice be
pierced in the side or bottom of a vessel.
769. When the vessel is allowed to empty itself, the cir-
cumstances of the discharge become very complicated after
the upper surface of the fluid has fallen to within a short dis-
tance of the orifice. If, however, we only consider the ex-
penditure previous to the arrival of the upper surface within
a few inches of the orifice, the same correction may be
applied to formula (439), which will thus become
-(0.62)Av(2^)v/(«-^)'
and will serve to determine the time necessary for a given
quantity of fluid to be discharged.
770. In applying the preceding correction to the theoreti-
cal discharge, it has been supposed that the orifice was
pierced in a thin plate : when a similar orifice is pierced in a
thick plate, the quantity discharged is found to be consider-
ably greater. Hence it occurs, that when the fluid is dis-
charged through a thick plate, or through a cylindrical tube
applied to the orifice, the coefiicient 0.62, which has been em-
ployed in calculating the discharge through a thin plate, is
no longer applicable. In this case the fluid adheres to the
sides of the tube, and the contraction of the stream is in a
great measure avoided. The lengths of such tubes, accord-
ing to Bossut, should be at least twice the diameter of the
orifice, in order that the contraction of the vein may be pre-
428 HYDRODYNAMICS.
vented. There will however be a limit to the length, proper
to be given to such tubes, since the friction of the fluid
against the sides of the tube will necessarily increase with
its length.
771. The quantities discharged by cylindrical tubes are
proportional to the products of the orifices by the square roots
of their depths, as in the case of apertures pierced in a thin
plate ; but the coefficient M, by which the area of the orifice
must be multiplied for the purpose of reducing the theoretical
discharge to that given by experiment, has been found by
Bossut to be about ||, or, more accurately, 0.81, when a short
cylindrical tube is applied to the orifice. Thus, the formula
(436), which serves to determine the quantity discharged from
a reservoir in which the fluid is maintained at a constant
height, will become, when corrected for the case of a cylin-
drical tube,
a=(0.Sl)^(2^).AV«;
or, if we replace k by its value «-/-s, r denoting the radius of
its section, the formula may be written
a=(0.81)^(2o-).^r=V«-
772. When the vessel is supposed to empty itself by an
orifice to which a cylindrical tube has been adapted, we can
still employ the coefficient (0.81), provided we only consider
the circumstances of discharge previous to the arrival of the
upper surface of the fluid at such a level that the tunnel
begins to be formed above the orifice.
773. By adapting tubes of diflferent forms to an orifice
pierced in the side or bottom of a vessel, the quantity of fluid
discharged is generally found to be more or less increased.
The following table presents a view of the relative quan-
tities discharged in some of the simplest cases.
1°. Theoretical discharge in a given time through
an orifice pierced in a thin plate - - - - 1.00
2°. Actual discharge in the same time through the
same orifice 0.62
3°. Discharge through a cylindrical tube, whose
length is equal to two diameters of the
orifice 0.81
MOTION OF WATER IN PIPES. 429
4°. Discharge through a conical tube having the
form of the contracted vein, the larger base
being regarded as the orifice 0.62
5°. Discharge through the same tube, regarding the
smaller base as the orifice 1.00
Of the Motion of Water in Pipes.
774. Let AB {Fig. 254) represent a cylindrical pipe, by-
means of which the water contained in the reservoir R is
transferred to the reservoir R', and let it be supposed that
the current has assumed a uniform motion : it is proposed to
investigate a formula by means of which the quantity of
water delivered at the point B, in a given time, may be
estimated.
Let CC'D'D represent an elementary stratum of the fluid
included between two consecutive transverse sections. Then,
since the motion is supposed to have become uniform, the
forces which tend to accelerate the motion of the element CD'
must be precisely equal to those which are exerted upon the
element in a contrary direction. The force exerted upon
CD', urging it in a direction from A towards B, is the com-
ponent of the weight of this element, in a direction parallel
to the axis of the pipe : and the forces which urge it in an
opposite direction are, 1^. the difference of the pressures
exerted upon the faces CC and DD' ; and, 2". the resist-
ance arising from the friction of the fluid against the sides of
the pipe.
775. lip denote the mean pressure, referred to the unit of
surface, in the section CC, the corresponding pressure in the
section DD' will be expressed by p-\-dp, and if we denote by
a the area of the transverse section of the pipe, the entire
pressures upon the sections CC and DD' will be respectively
adp, a{p + dp).
These pressures being exerted in contrary directions, the ele-
mentary stratum CD' will be acted upon by a force equal to
their difference adp.
The resistance arising from the friction against the sides
of the pipe will be directly proportional to the surface of the
430 HYDRODYNAMICS.
fluid in contact with the pipe, and will likewise be dependent
upon the velocity of the current. Hence, if v denote the
velocity, c the circumference of the section, and s the distance
of the section CC from the extremity A, the distance CD will
be expressed by ds, and the resistance experienced by the
element CD', in consequence of friction, will be
cds .<p{v) ;
in which ç(v) represents a certain function of v, to be ascer-
tained by experiment.
776. To obtain an expression for the force which acts in
the direction from A towards B, we shall suppose the density
of water to be equal to unity, and resolve the weight of the
element which is expressed by g . ads into two components,
respectively parallel and perpendicular to the axis of the pipe.
Then denoting by ê the angle included between the axis and
the horizon, the component of the weight parallel to the axis
will become ^ . sin ^ . ads. But if z represent the vertical
co-ordinate of the point C referred to A as an origin, dz will
represent the difference of level of the points C and D ; and
we shall have
— =sin Ô, g . sin ô . ads = g adz
ds
And since an equilibrium must subsist between this force and
the forces exerted in an opposite direction, we have
gadz = ad J) + cds . <p{v) ;
and by integration,
gaz — ap-^cs. <p{v) + C.
To determine the value of the constant C, we suppose the
pressure at the origin A to be equal to a known quantity P :
we shall then have p = F, z=0, s=0; and therefore
C=—aF.
Eliminating C between these two equations, we obtain
gaz=a{p—T)-{-cs.<p{v).
And by taking the integral between the limits 5=0, and
s=AB=l, the entire length of the tube, denoting by F the
pressure at the lower extremity, and by z' the co-ordinate of
the point B, there results
g.az'=a(P'—F)-\-cl.<p{v) (444).
MOTION OF WATER IN PIPES. 431
777. It has been found by experiment, that the function
ç>{v) may be expressed by two terms which are respectively
proportional to the first and second powers of the velocity :
thus, we shall have
ç>{v)=bv-^b'v'' ;
b and b' representing constant quantities.
This value of <p{v), being substituted in equation (444),
gives
cl
but if the diameter of the pipe be denoted by D, we shall
have
and therefore,
-=iD-
bv + b'v^ =iD ^^' ^f ^\
Ù
778. The pressure P at the upper extremity of the pipe
may be regarded without material error as that due to the
depth E A of the point A below the surface of the fluid in the
reservoir R. Strictly speaking, the pressure P is somewhat
less than that due to the depth EA, since these pressures be-
come equal only when the orifice is infinitely small (Art. 742) ;
but the difference is inconsiderable when the velocity of the
fluid is not great. In like manner, the pressure at the point
B may be supposed due to the depth E'B of the point B below
the surface of the fluid in the reservoir R' : hence, if h and k'
represent the respective depths EA and E'B, we shall have
(Art. 655) ^=gh, V=-gh' ; and by substitution we obtain
bv + 6'i;2 r= iDg _2_ .
If we divide each member of this equation by g^ and put,
for brevity,
b b' z'—h'+h J
g g «
we shall obtain
uv-\-^v*={T>k (445).
The values of « and ^ may be regarded as known, since
432 HYDRODYNAMICS.
they result immediately from those of a and 6, which are sup-
posed to be determined by observation ; and the value of k
will likewise be given when the length of the pipe, the differ-
ence of level of its two extremities, and the difference of the
pressures at those points are previously given. Hence, the
velocity v in a pipe of a given diameter can be readily cal-
culated.
779. The numerical values of « and /3 have been found by
Prony to be
«=0.00017, /3 =0.000106;
and the preceding equation therefore becomes
0.00017t? + 0.000106^2 ^ ^DA.
If we neglect the first term, which is generally admissible
when the velocity v is not extremely small, the formula will
reduce to
v=48.56^(DA-).
780. Let Q, denote the quantity delivered at the point B
in a second of time, and t the number 3.1416 ; we shall have
and by substituting this value of v in equation (445), there
results
4a , i6a= ,T.,
or, if we neglect the term containing the first power of v, and
make — -=^'^, we shall obtain
The numerical value of — is 38.12 ; and the formula there-
9>'
fore reduces to
a-38.12^(D^A).
In this investigation the dimensions are supposed to be
expressed in English feet.
THE END.
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