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A TREATISE
ON
BY
BARTHOLOMEW PRICE, M.A., F.R.S., F.R.A.S.,
SKDLEIAV PROFESSOR OF NATURAL PHILOSOPHY, OXFORD.
VOL. I.
STATICS, ATTEACTIONS,
DYNAMICS OF A MATERIAL PARTICLE.
SECOND EDITION.
AT THE CLARENDON PRESS.
M.DCCC.LXVIII.
[All riykts reserved.]
A TREATISE
ON
INFINITESIMAL CALCULUS;
CONTAINING
DIFFERENTIAL AND INTEGRAL CALCULUS,
CALCULUS OF VARIATIONS. APPLICATIONS TO ALGEBRA AND GEOMETRY,
AND ANALYTICAL MECHANICS.
BY
BARTHOLOMEW PRICE, M.A., F.R.S., F.R.A.S.,
SBDLEIAN PROFESSOR OF NATURAL PHILOSOPHY, OXFORD.
VOL. III.
STATICS, AND DYNAMICS OF MATERIAL PARTICLES.
SECOND EDITION.
" Les progres de la science ne sont vraiment fructueux, que quand ils amenent
aussi le progres des Traite's elementaires." CH. DUPIN.
AT THE CLAEENDON PRESS.
M.DCCC.LXVIII.
[All right* reserved]
PEEFACE TO THE SECOND EDITION,
JLHIS volume is the third of a Treatise on Infini-
tesimal Calculus and its capital applications. It is
also the first of a Treatise on Mechanics, and may
be considered and studied independently of the two
preceding volumes. In it are contained Statics or-
dinarily so called, Attractions, and the Dynamics
of a Material Particle.
The investigations are for the most part confined
to subjects which are within the range of the general
principles of Mechanics, and are not extended to par-
ticular sciences wherein these principles are specifi-
cally applied. Thus, the principles are discussed
on which the equilibrium and stability of bridges,
arches, and roofs depend ; yet the practical rules
of the engineer's and the builder's arts are not con-
sidered. Also as physical astronomy, the theories
of light, heat, and electricity require the explanation
and discussion of certain experimental laws which
rule their subject-matter, so the inquiry into these
special subjects is beyond the scope of this work
at its present stage.
viii PREFACE.
Chapter I is introductory to the whole of this part
of the Treatise on Infinitesimal Calculus. It seemed
desirable to explain as accurately as possible the
relation between "applied Mathematics," as some
parts of the subject are called, and the sciences of
number and geometrical space ; and so I have en-
tered on a discussion of one or two salient points
of the subject with the object of shewing that an
exact knowledge of Mathematics is necessary for
the complete inquiry into such sciences. I have
also ventured to submit to the common judgment
of Mathematicians the statement, that Mechanics,
enlarged in its idea and principles, as I have at-
tempted to enlarge it, is nothing else than the
science of motion, and ought, as such, to be called
by that name. Thus there are three principal ma-
thematical sciences, those viz. of number, space, and
motion : the last of which it has been my purpose
to develope in the following pages.
A course of inquiry somewhat irregular has been
followed, because it has been found most conve-
nient for a didactic treatise ; and Chapters II V
contain Statics, wherein the laws of pressure as they
produce equilibrium, or neutralize each other's effects,
are considered. In Chapter VI I have considered
the theory of Attractions at some length, and have
also employed the indirect mode of investigation
which the potential-function supplies. In Chapter
VII the principles, incidents, laws, and conditions
of the science of motion are formally drawn out.
The Chapter is thus introductory to Dynamics. The
mode of investigation and the forms of statement
PREFACE. ix
of some of the pregnant principles are different from
those which are commonly given. Only two of the
three ordinary laws of motion (axiomata molds, as
they are called by Newton) are admitted. The truth
of these is made to depend on and to flow from an
intelligible conception of the idea of motion and its
incidents; and on an inductive verification only so
far as the science is applied. This distinction is
important, and appears to solve some questions which
are in dispute between the two schools of writers
on Mechanics. The method which I have taken is
indeed counter to that of most English authorities
on the subject : it is rather in accordance with that
of foreign, and chiefly French, writers. If any one
after reflection should hesitate or refuse to admit
my principles, and the mode of arriving at and of
stating them, I must ask him to consider the subject
from the point of view which the Infinitesimal Cal-
culus and a reasonable conception of Infinitesimals
present to him ; and which, with great respect for
the great names and the sober judgment of those
who take the opposite course, I venture to think to
be the most natural and the most rational.
The first principles of the science are drawn from
an intelligible conception of motion itself. For the
mathematical expression of these, the language and
the symbols of Infinitesimals are peculiarly appro-
priate : effects are produced by causes which act
according to continuous laws: thus the effects be-
come continuously developed, and a peculiar system
of symbols is required to express them. New
ideas necessitate a ncAv language, and new language
i'KICE, VOL. I] I. b
x PREFACE.
requires new characters; and these are supplied by
the Infinitesimal Calculus.
A license has been taken, for which I must crave
some indulgence ; certain words are used which are
either new or are used in a new relation. In the ab-
sence of generally recognised rules for the formation
of scientific language, I have used compounded words ;
and have thereby obtained expressive, though some-
what long, words. This course I found myself obliged
to take. For ideas which are in themselves clear and
distinct have been so much obscured by ambiguity
and indistinctness of language, that there is no source
of error more fertile. Let me cite an instance. In
former books no word occurs more frequently than
the word " force." Indeed Mechanics has been called
the science of forces. But what does " force" mean ?
Will any one give an accurate definition of it ? a defi-
nition, that is, which will be correct, when the word
is applied to " the cause of motion," to " accelerating
forces," to " effective forces," to " forces lost and forces
gained," to "living force," to "labouring force?" In
some of these various meanings it indicates effect, in
others it indicates cause. Surely herein is confusion ;
and herein too, as it seems to me, is the reason why
the principles of mechanical science, or the science
of motion, are so imperfectly understood. Similar is
the ambiguity of the word " motion :" it is frequently
used synonymously with the word velocity: thus
.x'ntuin" has been called "quantity of motion:"
it is <i>int'(ty of velocity; and it is at all events per-
plexing to most minds to have a thing called by a
nami' which moans what it is not. Thus I have
PREFACE. xi
endeavoured in those parts of the treatise where first
principles are expounded, and where clearness of
language no less than clearness of conception is re-
quired, to call things by names which are expressive
vi significationis ; although in the more popular parts
I have used words in their ordinary and less exact
meaning. The subject is not in itself difficult, but
it has been made difficult by the maze of indistinct
nomenclature by which its fundamental notions have
been obscured.
As in the previous volumes, I am under obligation
to many friends, and to many writers on these sub-
jects. It is almost superfluous to mention Euler,
Lagrange, Laplace, Poisson, Poinsot, Jacobi, M. Ber-
trand, Sir W. R. Hamilton of Dublin, and now,
Sir William Thomson and Professor P. G. Tait, the
authors of the treatise on Natural Philosophy, the
first volume of which has lately been published at
the Clarendon Press ; because no one has a right
to form a judgment, and much less to compose a
didactic treatise, on the subject of Mechanics, with-
out a previous and preparatory study of the works
of these eminent men. From the works of Dr.
Whewell, lately the Master of Trinity College, Cam-
bridge, I have derived much aid: I know not how
much : for in the Appendices to the second volume
of his Philosophy of the Inductive Sciences so much
suggestive matter on Mechanical Philosophy is con-
tained, that opinions which appear to be one's own
may perhaps owe their origin to those essays. The
Journals of Crelle and Liouville have given much
assistance. To the editors of those Journals and
6 2
xii PREFACE.
to their contributors, whose names are too many
to be mentioned here, I tender my acknowledg-
ments.
References are made to the second editions of
the Differential and Integral Calculus, which are
the two preceding volumes of this treatise ; and
also to the numbers of the Articles and of the
equation as in these volumes. The colloquial style
has been retained.
11, ST. GILES', OXFORD.
Nov. 3, 1868.
ANALYTICAL TABLE OF CONTENTS.
CHAPTER I.
INTRODUCTORY : METHOD OF THE TREATISE.
Art. I'a-e
1. Importance and object of the Treatise 1
2, 3. The inductive process .. .. 1
4. The deductive process 4
5. Mathematics the most powerful instrument of the deductive
process 5
6. Mathematics : the attention which they require : the normal
sciences which they include 5
7. The science of number 7
8. The principle of homogeneity 8
9. The science of space 8
10. The science of motion 10
11. The method of the pure science of motion 11
12. For didactic purposes it is better to adopt a course of inquiry
not altogether philosophical, and to investigate the laws of
pressures first 12
PART I.
STATICS.
CHAPTER II.
STATICAL FORCES ACTING AT THE SAME POINT.
SECTION 1. Explanation of matter, force, mechanics.
13. Matter; force; statics; dynamics 14
xiv ANALYTICAL TABLE
Art. Pas 6
14. Statical forces: their four incidents; their units and mode
of measurement, and line-representatives 15
15. The resultant, and components; equilibrium .. '.- 17
SECTION 2. Composition of statical forces acting on a particle in
one plane.
1 6. Composition of forces which have one and the same line of
action 17
17. The parallelogram of forces ; two equal pressures .. .. 19
18. Geometrical interpretation of the same 21
19. 20. The parallelogram offerees ; two unequal pressures .. 22
21. The triangle of forces 25
22. The moment of a force defined 25
23. The composition and resolution of many pressures acting at a
point in one plane 26
24. Examples in illustration 27
25. Problems involving tension of strings .. .. 28
26. Problems involving pressures from planes .. 30
27. 28. Further reduction of the resultant of many forces acting
in one plane at a point 31
29. The polygon of forces 32
SECTION 3. Comj)osition and resolution offerees acting in any
directions on a material particle.
30. Composition of three forces acting along rectangular axes .. 33
31. Composition of many forces acting on a particle at the origin 34
32. Geometrical interpretation of the preceding result .. .. 35
33. The generalization of the parallelogram of forces 35
34. The conditions of equilibrium 30
35. The resolution of a force into three forces having action-lines
not in the same plane 36
SECTION 4. Equilibrium of forces acting on a constrained particle.
36. Equilibrium of forces acting on a particle which is in contact
with a surface 37
37. Equilibrium of forces acting on a particle in contact with a
smooth curve 40
38. The three degrees of freedom of a particle 43
OF CONTENTS. xv
CHAPTER III.
COMPOSITION AND RESOLUTION OF STATICAL FORCES ACTING ON
A RIGID BODY.
SECTION 1. Composition of two forces acting on a rigid body in
one plane.
Art. Page
39. Properties of a rigid body : transmissibility of force .. .. 44
40. Composition of two forces acting on a rigid body .. .. 45
41. Rotatory effect of a force : its name and its measure .. .. 46
42. Another form of the result of Art. 40 48
43. Composition of two parallel forces acting on a rigid body .. 49
44. Particular case when the parallel forces are equal, and act in
opposite directions .. .. 50
45. The equation of the line of action 51
SECTION 2. On couples ; their laws and composition.
46. A couple; its axis, arm, and moment .. .. 51
47. Theorems on coaxal couples 53
48. The rotation-axis, and the moment-axis of a couple .. .. 55
49. The resultant couple of many coaxal couples is another coaxal
couple 56
50,51. The composition of two couples not coaxal 57
52. The composition of couples whose rotation -axes have any
position in space 59
SECTION 3. The composition and resolution of forces acting on
a rigid body, the lines of action of ivhich are in one plane.
53. Composition of many parallel forces acting on a rigid body .. 60
54. The resultant of a system of parallel forces, and the equation
of its line of action 62
55. The equilibrium of a system of parallel forces 63
56. The centre of a system of parallel forces 63
57. Composition of forces acting in one plane on a rigid body .. 65
58. R and G are both finite 66
59. Other cases of particular values of R and G 67
60. Problems in illustration 68
61. Form of the preceding equations when the coordinate axes
are oblique * .. 72
62. Theorems on the moment of the resultant couple .. .. 74
63. The radial moment .. 75
xvi ANALYTICAL TABLE
Art. **
64. The centre of the system 78
65. Geometrical determination of the centre 79
66. Theorems on the radial moment 79
67. Amount of rotation necessary for bringing a non-equilibrium-
system into an equilibrium-system 80
SECTION 4. Composition and resolution of forces acting on a rigid
body in any directions.
68. Composition of many forces acting on a rigid body .. .. 81
69. Another interpretation of the result 82
70. Conditions of equilibrium 84
71. Geometrical theorems in interpretation of the conditions of
equilibrium 85
72-75. Theorems concerning the action-lines and points of appli-
cation of an equilibrium-system 87
76. Consideration of the case wherein R = 0, and a is finite .. 90
77. Consideration of the case wherein R is finite and G = .. 91
78. Consideration of the invariant LX + MY + NZ 93
79. Resultant of a system of parallel forces 94
80. The centre of a system of parallel forces 96
81. Consideration of the case wherein R and G are both finite .. 97
82. The central axis; the central plane; and the central principal
moment 98
83. Another demonstration of the theorems 99
84. Certain other theorems concerning the central principal
moment 101
85. Theorems on moment-centres and momental planes .. .. 103
86. A more general investigation 104
87,88. Further theorems on moment-centres 105
SECTION 5. TJie reduction of a system of forces in space to two
forces of translation.
89. The first demonstration of the possibility of the reduction .. 112
90. The second demonstration of the same ..113
91. A third demonstration 114
92. A fourth demonstration by means of the resultant of trans-
lation and of the central principal moment 116
93. Theorems concerning the two forces to which a system may
be reduced 118
SECTION 6. The equilibrium-axis of an equilibrium-system.
94. Definition of an equilibrium-axis ; and condition requisite
for its existence 120
OP CONTENTS. xv'i
Art. Page
95. Interpretation of the condition .. ..123
96. The condition when two lines not parallel are equilibrium-
axes 124
97. The introduction into a system of two equal forces acting in
opposite directions along parallel lines will satisfy the con-
dition of an equilibrium-axis 125
98. Reduction of a system to two forces, which with two other
new forces shall be in equilibrium, and shall have an equi-
librium-axis 127
SECTION 7. Stability and instability of equilibrium.
99. Explanation of stability, neutrality, continuity, instability,
of equilibrium 129
100. The theory of displacement 129
101. Case of two forces 130
102. Case of forces acting in one plane 132
103. Character of equilibrium dependent on the radial moment . 133
104. Examples illustrative of stability of equilibrium 134
105. Character of equilibrium of a body under the action of
many forces in space 135
106. Geometrical interpretation of the condition 137
107. Stability dependent on the radial moment 138
SECTION 8. Tlie principle of virtual velocities.
108. The principle stated) and deduced from the six equations of
equilibrium 140
109. Examples wherein the principle is applied 143
110. Gauss' theorem of least statical constraint 146
SECTION 9. Constrained equilibrium.
111. Firstly, when one point of the body is fixed .. .. ' .. 148
112. Secondly, when two points are fixed: indeterminateness of
the pressures on the points 148
113. Thirdly, when three or more points are fixed 150
114. When the body is in contact with a fixed surface .. .. 151
115. When the body is in contact with many surfaces .. .. 152
116. Equilibrium of many bodies under the action of given forces,
and in contact with each other 153
117. Examples of the preceding 154
SECTION 10. On friction.
118. The rationale of friction : the laws of friction 155
119. Problems involving friction 158
PRICE, VOL. III. C
xviii ANALYTICAL TABLE
CHAPTER IV.
OK GRAVITY, AND CENTRE OF GRAVITY.
SECTION 1. Elementary consideration on mass, gravity, and weight.
Art. Page
120. Further properties of matter : impenetrability, porousness,
density 163
121. Mass; specific density 164
122. Mass-centre: its coordinates .. .. 166
123. Gravity, and weight 167
124. The variation of gravity 169
125. Centre of gravity : its coordinates 170
126. Relations of mass and weight 172
SECTION 2. The centre of gravity of material lines or wires.
127. Investigation of the coordinates in this particular case .. 173
128. Examples in illustration 174
129. Application to curved wires in space 176
130. The curve which a heavy and flexible wire takes when its
centre of gravity is in the lowest position 177
131. The first theorem of Pappus 179
SECTION 3. Centre of gravity of thin plates and shells.
132. Investigation of the coordinates of the centre of gravity
in reference to rectangular coordinates in one plane .. 181
133. The same in reference to polar coordinates 184
134. Centre of gravity of a thin shell of revolution 185
135. Centre of gravity of a thin curved shell 188
136. The second theorem of Pappus 189
SECTION 4. Centre of gravity of heavy bodies.
137. Investigation of the coordinates of the centre of gravity of
a solid body bounded by a surface of revolution .. .. 191
138. The same for a solid body bounded by any curved surface .. 194
139. The same in reference to polar coordinates 195
140. Various examples 196
SECTION 5. Stability and instability of equilibrium of heavy bodies.
141. A position of equilibrium is stable, neutral, or unstable, ac-
cording to the position of the centre of gravity .. .. 198
OF CONTENTS. xix
Art. Pag*
142, 143. The stability of a solid body, resting on a curved
surface 199
144. Examples of the preceding conditions 201
SECTION 6. General theorems of the centre of gravity.
145-148. Theorems I, II, III, IV. .. 203
CHAPTER V.
THE ACTION OF FORCES ON BODIES OF VARIABLE FORM.
SECTION 1 . Flexible and inextensible strings.
149. Investigation of some properties of the funicular polygon .. 207
150. Funicular polygon under the action of normal forces .. .. 208
151. The catenary 209
152. The catenarian curve under the action of many forces in all
directions 210
153. Particular properties of the curve .. .. 212
154. The catenarian curve in one plane 213
155. The equation of the heavy catenary 213
156. The equation deduced from the triangle of forces .. .. 214
157. Integral forms of the equation 215
158. Some geometrical properties of the catenary 217
159. The equation of the curve in which a heavy chain suspended
by its two ends hangs 219
160. The centre of gravity of the catenary 220
161. The heavy catenary of variable thickness and density .. 221
162. The form of the curve when the centre of gravity has the
lowest possible position 223
163. The string-curve under the action of central forces .. .. 225
164. Properties of this string-curve .. 226
165. Examples of the curve 227
166. The catenary on a smooth surface 228
167. The catenary on a smooth plane curve 229
168. The catenary on a rough surface 232
169. Examples in illustration 233
SECTION 2. The equilibrium of elastic strings.
170. Our notions of elasticity, and ignorance of elastic action .. 234
171. An extensible string : Hooke's law 235
C 2
XX ANALYTICAL TABLE
Art. P"* 6
1 72. The form of an extensible string-curve under the action of
given forces 237
173. The extensible catenary 240
174. The heavy extensible catenary 241
SECTION 3. The equilibrium of elastic plates or springs.
175. The bending of an elastic lamina 242
176. The forces brought into action 245
177. The equation to the curves of the fibres 246
178.179. Two particular cases 246
180. The vertical strength of a spring or a beam 248
181. The deflexion of a beam bent by its own weight .. .. 249
182. Examples in illustration .. 250
CHAPTER VI.
ON ATTRACTIONS.
SECTION 1. The direct investigation of the attraction of bodies.
183. Introductory and explanatory 252
184. The mathematical expression for the attraction of two par-
ticles 253
185. The attraction of a straight rod or wire on an external
particle 254
186. A remarkable geometrical construction of the result .. .. 255
187. Illustrative examples 256
188. The attraction of a bent rod or bar on an external particle . 257
189. The attraction of a circular ring on a particle in its plane .. 259
190. The attraction of a cylindrical tube on a particle in its axis 259
191. Problems on the attraction of thin wires .. .. .. ..260
192. The attraction of a circular plate on a particle in the per-
pendicular through its centre 261
193. The attraction of a solid of revolution on a particle in its
axis 262
194. The form of the solid of revolution of greatest attraction .. 265
195. The attraction of a homogeneous spherical shell on an ex-
ternal particle 266
196. The attraction of a sphere, (1) homogeneous, (2) hetero-
geneous 268
197. What are the laws for which the attraction of a shell on an
external particle is the same, as if the shell were condensed
into its centre? .. .270
OP CONTENTS. xxi
Art. Page
198. The attraction of a homogeneous spherical shell on a par-
ticle within it 272
199. What are the laws for which the attraction of a spherical
shell on a particle within it is zero ? 273
200. The attraction of a rectangular plate 274
201. The attraction of thin plates on particles in their planes .. 275
202. Various problems of attractions 276
203. The attraction of a homogeneous ellipsoid 277
204. Jacobi's expression of the components of attraction .. .. 279
The attraction of ellipsoidal shells 280
206. The attraction of spheroids and spheres 281
^207. Attraction of an ellipsoid on an external particle .. .. 283
208. Theorem of concentric and confocal surfaces 284
209. Corresponding points 285
210. Ivory's theorem 286
211. The attraction of spheroids on external particles .. .. 287
212. Attraction of an oblate spheroid of small eccentricity .. 288
213. Attraction of a homogeneous elliptic cylinder 289
214. Maclaurin's theorem in attractions 289
215. Two remarkable theorems in attractions .. .. .290
SECTION 2. Indirect investigation of attractions. Tlte potential.
216. Investigation of a function, the partial derived functions of
which are the axial-components of the attraction .. .. 292
217. The form of the preceding when the law of attraction is the
inverse square of the distance. The potential .. .. 294
218. The physical meaning of the potential 295
219. The attraction along any line deduced from the potential .. 296
220. The potential of a thin straight rod on an external particle 297
221. The potential of a thin spherical shell 298
222. The potential of a sphere 299
223. The potential of a finite body on a particle at a very great
distance 300
224. The axial-components of the attraction of such a body .. 303
"^225. The potential of an ellipsoid 304
v226. Theorems concerning the attraction of an ellipsoid .. .. 307
<227. The potential and attraction of ellipsoidal shells .. .. 309
228. The action -line of such an attraction 311
229. The amount of the attraction 311
I J 2 30. A remarkable theorem concerning the total attraction .. 314
v 231. The attraction of an ellipsoid deduced from the preceding .. 315
xxii ANALYTICAL TABLE
SECTION 3. General theorems in attractions.
Art. Pa ^
232. The equilibrium-surface, or equipotential surface .. ..I
233. Laplace's theorem of the potential .......... 318
234. Another proof of the theorem ............ 320
235. Another form of the theorem ............ 322
236. The potential deduced from the theorem in certain cases .. 322
237. Integral form of the preceding theorem ........ 325
238. The differential form deduced from the preceding .. .. 328
7^239. No maximum or minimum value of a potential ...... 329
240. Laws of attraction deduced from the equipotential surface 330
241. Green's theorem in attractions ............ 331
242. Certain theorems and general remarks ........ 333
PART H.
DYNAMICS ; THE MOTION OF MATERIAL PARTICLES.
CHAPTER VII.
MOTION ; ITS AFFECTIONS, LAWS, AND EQUATIONS.
SECTION 1. Introductory ; motion, matter, time, space.
243. Dynamics, the subject of the following investigations; its
most general form : its symbols and their nature .. .. 335
244. Motion, the fundamental idea ; matter; kinematics, and me-
chanics 337
245. Matter; its mobility, and divisibility; time and space, as
incidents of motion; volume and form, as incidents of
matter 338
SECTION 2. The kinematics ofapartick in a straight line.
246. Velocity, constant .. 340
247. Velocity, varying : acceleration 342
248. The mathematical expressions for acceleration 344
249. Illustrative examples of acceleration 344
OF CONTENTS. xxiii
SECTION 3. TJte dynamics of a particle moving in a straight line.
Art. Page
250. The inertia of matter 346
251. The inertia of terrestrial matter 347
252. Force; the cause of a change of velocity 349
253. Force; its action-line ; its measure 351
254. Force; finite and impulsive .. 352
255. Mass ; the dynamical mode of measuring quantities of
matter 353
256. Momentum, or quantity of velocity 355
257. Equality of momentum expressed to momentum impressed 356
258. The same law true of infinitesimal momenta ; equations of
motion 357
259. The integral equations of motion; theory of equivalence
of work 359
260. Proof of the preceding theorems in the case Of terrestrial
matter. Attwood's machine 362
261. Pressure is momentum virtually developed 363
CHAPTER VIII.
THE RECTILINEAR MOTION OP PARTICLES.
SECTION 1. Direct impact and collision.
262. Impact is direct or oblique. Explanation of the circum-
stances of collision of two particles or spherical balls .. 365
263. Investigation of the velocities after direct impact of two balls 367
264. Modification of the preceding when the elasticity is (1) per-
fect, (2) zero 369
265. The velocity of the centre of gravity of the two balls is not
altered by the collision 370
266. Examples in illustration 371
267. The resistance of a medium on a body passing through it .. 372
SECTION 2. Rectilinear motion of particles under tJie action of an
uniformly accelerating force.
268. The incidents of a particle moving in a straight path under
the action of a constant accelerating force 374
xxiv ANALYTICAL TABLE
Art. Pa?e
269. The relation between the space, and the time, deduced from
first principles 377
270. Examples in illustration 378
SECTION 3. Gravity as an uniformly accelerating force.
271. Gravity : its variation at different places on the earth .. 379
272. Gravity: the velocity -increment due to it 381
273. Experimental evidence by means of Attwood's machine .. 382
274. General results of the action of gravity 383
275. Illustrative examples 386
276. 277. Motion of two particles connected by a string passing
over a pulley 387
SECTION 4. Rectilinear motion of particles under the action of varying
accelerating forces.
278. Accelerating forces are supposed to be explicitly functions
of the distance and not of the time 390
279. The force varies directly as the distance 391
280. Cases of this law of force in Nature 393
281. A different case of the same law 395
282. The force repulsive 396
283. The equation of harmonic motion 396
284. The force varies inversely as the square of the distance .. 397
285. The force varies inversely as the square root of the distance 398
286. The force varies inversely as the nth power of the distance 399
287. The force varies inversely as the distance 400
288. A particle moves under the action of two forces which vary
directly as the distance 400
289. A particle moves under the action of two forces which vary
inversely as the square of the distance 401
290. Motion of two particles under their mutual action .. .. 403
291. Motion of a particle when the centre of force also moves .. 404
292. The same problem solved relatively 405
SECTION 5. Rectilinear motion of particles in resisting media.
293. Motion of a particle, when 'the resistance varies as the
square of the velocity 405
294. Motion of a heavy particle in air ., 406
OF CONTENTS. XXV
Art. Pag
295. Motion of a heavy particle in air, when it moves contrary
to the direction of the action of gravity 408
296. Motion of a particle under the action of a constant force,
when the resistance varies as the velocity 410
297. Motion of a particle in a medium, of which the density
varies . 411
CHAPTER IX.
THE THEORY OF CURVILINEAR MOTION.
SECTION 1. The kinematics of a particle moving in
curvilinear path.
298. Extension of the definitions of velocity and velocity-in-
crements 413
299. Resolution of velocity 413
300. Axial-components of velocity-increments 415
301. Problems of resolved velocities 415
302. Problems of resolved velocity -increments 416
303. Tangential and normal resolution 419
304. The same deduced from axial resolution 420
305. Cases when the process is convenient 421
306. The hodograph 422
307. Angular velocity ; axis of rotation 423
308. The measure, direction, and notation of angular velocities . 424
309. Angular acceleration 425
310. Problems in illustration of angular velocities and angular
acceleration 426
311. Radial or paracentric, and transversal resolution .. .. 428
312. Particular and remarkable forms of these 429
313. Coordinate-resolution in space 430
314. Tangential and normal components derived from general
considerations 431
315. The same derived from the axial accelerations 432
316. Polar resolution in space 432
317. The theory of relative motion of a particle 433
318. The analytical expressions for relative velocities and velocity-
increments 434
319. Particular forms of the preceding 436
PRICE, VOL. III. (I
xxvi ANALYTICAL TABLE
SECTION 2. Tlie dynamics of a particle moving in a
curvilinear path.
Art. Page
320. The simultaneous action of many forces which have different
lines of action 437
321. Extension of the law of inertia 438
322. Experimental illustrations of the law 439
323. Mathematical expressions of the expressed momentum-in-
crements 441
324. The equations of motion when the path is a plane curve .. 442
325. The equation of vis viva, or of work 442
326. Centripetal and centrifugal force 443
327. Problems in centrifugal force .. 444
328. The earth's gravity as affected by centrifugal force .. .. 445
329. Equations of motion of a path in space 447
330. A particle acted on by no forces moves in a straight line .. 447
331. Polar resolution in space 448
332. The equations of relative motion 448
CHAPTER X.
THE PRECEDING PRINCIPLES APPLIED TO THE MOTION OF
PARTICLES IN SPACE.
SECTION 1. Oblique impact and collision of particles and of smooth
splierical balls.
333. Consideration of some circumstances of collision .. .. 449
334. Oblique impact on a smooth plane 450
335. Illustrative examples 452
336. Oblique impact of two particles or balls, m and in .. .. 453
337. Oblique impact of elastic balls 454
338. The line of motion of the centre of gravity is not changed
by the collision 455
339. Illustrative examples 456
340. The oblique effects of a resisting medium 456
341. The resistance of a fluid on a surface of revolution .. .. 458
342. The form of the surface when the resistance is a minimum . 460
SECTION 2. Motion of Jieavy particles on smooth inclined planes.
343. General investigations of the motion of a heavy particle on
a smooth inclined plane .. 461
344. The synchronism of a circle- in a vertical or in an inclined
plane 464
OF CONTENTS. \\vii
Art. l'_
345. The determination of planes of quickest and slowest descent 464
346. Illustrative examples of the motion of a particle on an in-
clined plane 466
347. Motion of two particles connected by a string passing over
a pulley at the common vertex of two inclined planes . . 468
348. Illustrative examples 470
SECTION 3. Determination oftfte paths when the laws of force
are given.
349. The case of a projectile in vacuo 471
350. The path is a parabola : its latus rectum, and the coordi-
nates of its vertex ,. .. 472
351. The velocity at any point of the path is equal to that ac-
quired in falling from the directrix 474
352. The path of the projectile is also found from first principles 474
353. The range on an inclined plane 475
354. Conditions necessary that the projectile may pass through a
given point 476
355. Examples in illustration 477
356. Motion in a parabola when the y -axial component varies
as y 479
357. Determination of a curve when two conditions are given .. 481
358. A particle moves under the action of a force perpendicular
to, and varying inversely as the square of the distance from
a given straight line 482
359. Motion of a particle under the action of a central force
varying directly as the distance 482
360. Motion of a particle under the action of a central force
varying inversely as the square of the distance .. .. 484
361. An example in which the axes of reference are oblique .. 486
362. The theorem of M. Bonnet 486
363. The path of a projectile when the forces are resolved tan-
gentially and normally 488
364. Motion of a particle describing a helix 489
365. The relative motion of two particles attracting each other
inversely as the square of the distance 490
366. The centre of gravity of two such particles either remains
at rest or moves in a right line 493
367. The equations of relative motion of two particles disturbed
by a third particle. The disturbing function 495
il 2
xxviii ANALYTICAL TABLE
SECTION 4. Curvilinear motion in a resisting medium.
Art. Page
368. General equations of motion in a resisting medium . . . . 496
369. The forms assumed by them when the motion is wholly in
one plane 498
370. Determination of the law of resistance, so that a given curve
may be described. Examples in illustration 499
371. The law of resistance when the force is central 500
372. A projectile in a medium the resistance of which varies as
the velocity 502
373. If the resistance is small, the path is parabolic 503
374. A projectile in a medium of which the resistance varies as
the square of the velocity 504
375. Determination of the asymptote to the path 506
376. Particular case of the preceding 507
CHAPTER XL
FREE MOTION OF A PARTICLE UNDER THE ACTION OF
CENTRAL FORCES.
SECTION 1. General investigations. Determination oftlie laws of
force when the equations of the patJts are given.
377. Explanation of a central force : expediency of an inde-
pendent investigation 508
378. The motion takes place in one plane 508
379. The sectorial areas vary as the times in which they are de-
scribed 509
380. The equation of vis viva : mathematical expressions of the
central forces 511
381. Examples in illustration : conies ; centre of force is in the
focus 513
382. Central conies; the centre of force is in the centre .. .. 516
383. Motion in a circle 518
384. Motion in the lemniscata and the cardioid 519
385. Motion in revolving orbits 520
386. Other examples 523
387. Investigation of the gem-nil expression of central force from
first principles . 524
OF CONTENTS. xxix
SECTION 2. The determination of the orbits when the laws offeree
are given.
Art. Page
388. Explanation of the necessary constants 525
389. The orbits, when the force varies as the distance .. .. 525
390. Application of the results to the wave-theory of light .. 528
391. The orbits, when the force varies inversely as the square of
the distance 529
392. The ellipse : its major axis, eccentricity, and periodic time . 531
393. The true, the mean, and the eccentric anomalies .. .. 533
394. The parabola 535
395. The hyperbola 536
396. The orbits, when the force varies inversely as the cube of
the distance 537
397. The orbits, when the force varies inversely as the fifth
power of the distance 540
398. Some general properties of central orbits 541
399. The orbit, when the velocity is that acquired in moving
from an infinite distance under the action of the central
force 542
400. Other cases which admit of integration in finite terms .. 543
401. Problems in illustration 545
SECTION 3. The elements of physical astronomy.
402. Observation is required ere the results of the science of mo-
tion can be applied to physical astronomy 549
403. The laws of Kepler ; their mathematical interpretation .. 552
404. The truth of these laws is approximate : the motion is re-
lative 555
405. The plane of the ecliptic. The equinoxes ; the seasons .. 556
406. Precession: nutation: the first point of Aries 558
407. The elements of a planet's orbit : generally seven .. .. 560
408. The radius-vector and longitude in terms of the time .. 561
409. The corresponding expansions by means of Lagrange's
theorem 563
410. The values determined by successive approximations .. 564
411. The equation of the centre 565
412. The time in the parabolic orbit 567
413. An approximate determination of the masses of the planets 568
SECTION 4. Tlie polar equations of motion of a disturbed planet.
414. The general differential equations of a disturbed body .. 570
415. The first approximate solution of the equation in latitude .. 573
xxx ANALYTICAL TABLE
CHAPTER XII.
THE CONSTRAINED MOTION OF PARTICLES.
SECTION 1. A particle constrained to move on a given cwrved line.
Art. Page
416. A normal pressure always acts on the moving particle in
addition to the impressed momentum -increments .. .. 575
417. The general equations of motion in a curved line or tube .. 575
418. The equations when the motion is in one plane 578
419. The equation of vis viva 579
420. The expression for the normal pressure 580
421. Examples in application of the principles 580
422. The equations of motion in the case of a heavy particle .. 582
423. The motion of a heavy particle on a cycloid 583
424. The cycloidal pendulum 584
425. Another problem of motion on a cycloid 585
426. The motion of a heavy particle on a circle 586
427. The expression for the time of an oscillation 587
428. The circular pendulum : its application to the determination
of (1) gravity, (2) the height of mountains, (3) the depth
of mines 589
429. The tangential equation of a heavy particle on a circle .. 590
430. Examples illustrative of constrained motion .. .. .. 592
431. The general problem of tautochronism 596
432. Examples illustrative of tautochronism 598
433. The tautochronous curve of a heavy particle in vacuo .. 599
434. Synchronous curves 600
435. Brachistochronism ; the general equation of brachisto-
chronous curves 601
436. Some particular cases of brachistochronous curves .. .. 604
437. Motion of particles in moving tubes 605
438. The same solved by the equations of relative motion .. .. 607
SECTION 2. Particles constrained to move on a given surface.
439. General equations of motion : and general properties .. 610
440. The motion of a heavy particle in a sphere 612
441. The motion of a pendulum in a spherical surface .. .. 614
OP CONTENTS. xxxi
Art. !
442. The motion of a particle on a surface of revolution .. .. 617
443. The circumstances under which a parallel of latitude is de-
scribed 618
444. The brachistochron on a given surface 618
445. Lines of easy motion on a surface 620
446. The relations between a brachistochron, a geodesic line,
and a line of easy motion 622
447. The brachistochron of a heavy particle, and on a surface of
revolution *. 623
SECTION 3. Constrained motion in resisting media.
448. Solution of some particular problems ; the cycloidal pendu-
lum in a resisting medium 625
449. The circular pendulum in a resisting medium .. , .. .. 627
450. The same problem when the resistance varies as the square
of the velocity 628
451. The tautochronous curve in a resisting medium .. .. 630
452. An application of the general equation .. 633
CHAPTER XIII.
GENERAL THEOREMS IN THE MOTION OF A PARTICLE.
SECTION 1. The principle of vis viva, or of work.
453. The principle of vis viva is deduced from the equations of
motion 635
454. The principle of vis viva is also the principle of work .. 635
455. The nature of the forces when the principle is applicable .. 637
SECTION 2. The principle of least action.
456. The meaning of least action explained 640
457. The principle of least action is assumed, and the general
equations of (1) free motion, (2) constrained motion are
deduced 641
458. The problem of the path of the projectile in vacuo is solved
by the principle of least action 643
459. The same principle is applied to the path of a particle
under the action of a central force which varies inversely
as the square of the distance 644
xxxii ANALYTICAL TABLE OP CONTENTS.
Art. Page
460. If the velocity is constant, the path due to least action is a
geodesic line 644
SECTION 3. The variation of parameters.
461. General explanation ." 645
462. The method applied to a heavy particle falling in a medium
of which the resistance varies as the square of the velocity 647
463. Also to the problem of the path of a projectile 648
464. Also to the motion of a particle on a cycloid 650
465. Two examples in illustration , 652
CHAPTER XIV.
ON VIETUAL VELOCITIES. -
466. Enunciation and mathematical expression of the principle 653
467. General investigation of the principle 654
468. The equations of (1) statical equilibrium, (2) of motion of a
particle deduced from the principle 655
ANALYTICAL MECHANICS.
CHAPTER I.
INTRODUCTORY ; THE METHOD OF THE TREATISE.
ARTICLE 1.] Of all parts of Infinitesimal Calculus, Analytical
Mechanics, or (as I shall hereafter have reason to call it) the
Science of Motion, is in its results and its applications the most
important; the principles and processes of all mathematical
physics are derived from it ; and as, for reasons which shall be
assigned hereafter, it is in itself the most perfect of physical
sciences, so do the others approach more or less to completeness
according as the laws and methods of mechanics are more or
less satisfied by them ; and the object to be attained in all is,
to make them parts of this principal and normal science. Now
in the process of our application of the science of number to
that of motion, new subject-matter, or new kinds of quantity
measurable by number, will be introduced j and also as the
results of our investigations will be applicable to the phenomena
of the external world, and to the unravelling of complex effects,
it is necessary to premise some few observations on the method
of our inquiry ; and especially to shew how, and how far, the
pure sciences of number, space, and motion may aid us in the
discovery of the proximate causes of such effects ; proximate, I
say, in order that the objects of our search may be definite and
intelligible, and that we may not be lost in the subtleties of
metaphysics.
2.] There are generally two processes, by one or other of
which our knowledge of natural phenomena is obtained, and
with both of which it is in many cases absolutely necessary, and
in all cases desirable, that an inquirer into nature's laws should
be acquainted ; and although in their use one of these processes
frequently runs into the other, and they are alternately applied
PRICE, VOL. III. B
2 THE INDUCTIVE PKOCESS. [2.
for the purposes of discovery and verification, yet they are in
themselves distinct, and for philosophical reasons it is requisite
to keep them so. In one of these processes we take the facts
of nature as they are presented to us in their simple and con-
crete forms ; and animated by a conviction deep-seated in our
nature that they are not isolated, but instances of a grand and
comprehensive law, which has been impressed on them, and by
virtue of which they are, we seek for that law : with this object
in view we study them, analyse them ; and in the analysis we
subject them to trials of various kinds, if they admit of experi-
ment, or observe them in such varying relations as they exhibit
to us, if they do not ; we separate what is extraneous and thus
accidental from that by virtue of which they seem to us to be,
and without which they would not be : and by this process
detect the general law which lies latent in the fact; or, in
perhaps more precise terms, the cause of which the fact is the
effect. Thus we ascend from the fact to the cause ; and when
many facts have been subjected to a similar process, and the
same law has been detected in all, we collect them under a
general formula which expresses this law, and thereby a cause
of which all the examined facts are the effects ; and the human
mind, endowed with a love of continuity, extends this to other
facts similar in kind, and beside those which have been exa-
mined. In this process therefore we interrogate nature as she
offers herself to us in her simple forms and particular develop-
ments ; and so long as any branch of knowledge consists only
of such isolated facts it is little else than mere empiricism;
but when a bond of union has been imported from some other
source, and these facts have been collected into general propo-
sitions ; when on these phenomena has been induced a distinct
idea, and the information obtained from them in their isolated
forms has been studied, arranged, and reasoned upon, then, and
not until then, has it a right to bear the name of Science; it
is then no longer accumulated experience, but it is experience
systematized, digested, assimilated, organized into a whole ; it
has harmony, regularity, and law ; and the physical sciences
thus formed will be found to satisfy another most exact, and
perhaps the most searching, test of their truth; they predict
similar effects from similar causes.
3.] All the physical sciences are progressive, and pass through
the experimental or observational stage which I have described ;
3.] THE INDUCTIVE PROCESS. 3
in their infancy the subjects of them have been in this discon-
nected state. Experience in the way of experiment and observa-
tion has been the chief instrument by which their boundaries
have been advanced, distinct ideas for the colligation of facts
been obtained, and inquirers been led to the discovery and enun-
tiation of their peculiar laws : the discovery of the laws of
motion by Galileo, of the laws of planetary orbits by Kepler, of
the law of refraction by SHC!!, are, amongst many, early and
salient instances. Invariably, so long as any science is in this
imperfect condition, its phenomena must be examined for the
purpose of discovering such normal laws, and it is in the pro-
secution of this work that the most eminent philosophers of the
present age have earned their glory : in short, the analysis of
such facts has been in a great measure the characteristic of the
science of the XlXth century : and no mean work is it : it
demands the highest intellectual and moral qualities that can
adorn human nature ; an eager and honest desire after truth ;
patience and endurance of labour; a courage that never fails
under non-success ; the keenest intellectual acuteness in detect-
ing resemblances ; a mind gifted with a plastic power of framing
an idea distinct and pregnant, which shall collect all into one
general formula; an inventiveness and a never- fail ing command
of resources : and in our days these qualities have not been
wanting, and have not been unrewarded. It is however un-
necessary for me to do more than to indicate the methods of
experimental philosophy, in order that I may contrast with
them, and thus bring into greater prominence, the process of
investigating truth which will be developed in the following
volume ; and the reader desirous of further information on the
methods of inductive philosophy must have recourse to works
wherein such subjects are specially treated of. Let me refer
him to Sir John Herschel's Treatise of Natural Philosophy, a
work which contains in a short compass a masterly exposition
of the methods, and to Dr. Whewell's Philosophy of the In-
ductive Sciences, wherein he will find the subject treated by an
eloquent author, whose knowledge of physical science seems to
be limited only by the limits of science itself*.
* See also an article on ' Whewell on Inductive Sciences ' in the Quarterly
Review, Vol. LXVIII : and subsequently republished in the Collection of Essays
contributed to the Edinburgh and Quarterly Reviews by Sir John F. W. Herschel.
B 2
4 THE DEDUCTIVE PROCESS. [4.
4.] It will be seen then that the first step in experimental
philosophy is to colligate facts by means of a distinct and ap-
propriate idea ; afterwards a consilience of inductions takes
place ; and hereby we arrive at the last step in the construction
of a science, which is the enuntiation of a theory ; the determi-
nation, that is, of a law which rules all the subject facts, and
the discovery of a general cause, of which the facts of the science
are the single and (as they seem at first) isolated or independent
effects; and when such perfection is attained the aggregate of
the knowledge receives the name of a science, having all the
characteristics of arrangement, order, system, completeness,
which are necessary for such perfection.
And now comes in the second process to which allusion has
been made. If the theory is true, not only is it an explanation
of all the facts which it comprises in its formula, but it has
also a prophetic power : when the cause is active, results similar
to the former ones must be produced ; the theory requires verifi-
cation; and the verification consists in the prediction of the
future : and it is only when such future facts have been shewn
to accord with a theory, that it satisfies those stringent rules of
induction which have been constructed in a jealous care of truth.
The theory may also be pregnant with results different from
those out of which it has grown ; these must also be traced
and examined : the theory must be tested in all ways and in all
directions ; and when such tests have been satisfied, it has a
claim on our acceptance, and for this purpose a process, the
reverse of the former, is necessary : facts were in that analysed,
so that their latent cause might be detected ; in this causes are
to be developed into their effects ; the former is the historical
process through which the science has grown from an imperfect
state to perhaps full maturity ; the latter takes the science in
its perfect state, and explores the riches which it contains ; the
former is the process by which the science has been constructed,
and is somewhat analogous to the manner in which we indi-
vidually learn it; the latter is the form wherein the man of
science knows it. Now this distinction is important : for as it
is under the latter and more perfect aspect that I shall have
to consider the science of motion, so the method is dogmatic ;
and the fundamental and axiomatic laws will be enuntiated, and
no formal proof of them will be given ; it may sometimes be
desirable to indicate the steps by which historically they have
6.] WHAT ARE MATHEMATICS? 5
been arrived at, but such an explanation will be only incidental
and that the learner may have adequate knowledge of them ;
and I shall not lose sight of the chief object, which is to trace
into their farthest results those general laws which an inductive
philosophy has supplied.
5.] Mathematics is the most powerful instrument, which we
possess, for this purpose : in many sciences a profound know-
ledge of mathematics is indispensable for a successful investiga-
tion. In the most delicate researches into the theories of light,
heat, and sound it is the only instrument ; they have properties
which no other language can express ; and their argumentative
processes are beyond the reach of other symbols. For other
sciences, for Mechanics, and Astronomy, and for Mechanism
they are almost as necessary; and I am sure that to any one
who has taken the pains to compare the general explanation
of planetary disturbances given in Sir John Herschel's Outlines
of Astronomy with that of the same phenomena as discussed
with the aid of mathematical appliances, there cannot be a doubt
that, however successful Sir John Herschel may have been,
even beyond his expectation, yet for an accurate comprehension
of the circumstances the other method is absolutely necessary.
The following extract from that work * is unimpeachable testi-
mony : ' Admission to its sanctuary' (that is, of astronomy)
' and to the privileges and feelings of a votary is only to be
gained by one means sound and sufficient knowledge of mathe-
matics, the great instrument of all exact inquiry, without which no
man can ever make such advances in this or in any other of the
higher departments of science as can entitle him to form an in-
dependent opinion on any subject of discussion within their range.'
I can truly use the same language as to the necessity of mathe-
matics for the successful study of the other higher branches of
the science of motion.
6.] Here it may be asked, What are mathematics ? Define
them. Do they require and apply reasoning processes different
from those of the ordinary discourse of men ? have they a dif-
ferent logic ? and a different language ? What distinction exists
between pure and mixed mathematics, since they are commonly
divided into these two classes ? and what does the term include ?
Many of these questions may be matter of words only; it is
* See Outlines of Astronomy, 4th edition, p. 5. Longman and Co., London,
1851.
6 WHAT ARE MATHEMATICS? [6.
not necessary for me to define mathematics in a way which
would satisfy a metaphysician, or to inquire how far 'science
of quantity/ or 'science of measuring quantity' may be a
sufficient definition, and whether there is not a large class of
propositions of geometrical position which such definitions will
not include ; it is enough for me to be able to give you such
an account of the means which mathematics afford for pursuing
our present inquiry that I may excite in you good hope of final
success. I would however observe, that the reasoning process
is not different from that of any other branch of knowledge ;
their logic is the same as that of chemistry, of political economy,
or moral philosophy; it is addressed to the same faculties of
man, and does not require any peculiar formation or deformity
of human nature, as some seem to think ; but there is required,
and in a great degree, that attention of mind which is in some
part necessary for the acquisition of all knowledge, and in this
branch is indispensably necessary. This must be given in its
fullest intensity ; this is the excellency which Sir Isaac Newton
claimed % for himself, and thus placed his superiority on moral
rather than on intellectual grounds : the other elements espe-
cially characteristic of a mathematical mind are quickness in
perceiving logical sequence, love of order, methodical arrange-
ment and harmony, distinctness of conception. The language
of mathematics is to a certain extent peculiarly its own ; its
symbols are certainly its own ; but these may generally, if it
is desirable, be translated into ordinary language ; and its
language is peculiar, because the subjects of which it treats are
peculiar. Now mathematics include three normal sciences ;
(1) science of number, (2) science of space, (3) science of motion;
and under one or other of these all sciences which are treated
mathematically may be ranged ; or the several parts of any one
may come under different normal sciences : thus, formal or
geometrical optics is an application of geometry ; physical optics
of the science of motion ; plane astronomy is geometrical, physical
astronomy is mechanical. The division of mathematics into
pure and mixed is arbitrary and useless, because it leads to no
practical result ; and therefore I do not care to retain it. I may
however observe that the first two sciences, those viz. of number
and space, are commonly included under the term pure mathe-
matics, and that the last one and its subordinates are called
mixed ; the reason being that the subject-matter of the last has
7-] THE SCIENCE OF NUMBER. 7
been thought to be terrestrial, or, at all events, cosmical matter ;
and that therefore the science involves considerations of the
properties of this matter, and which must be discovered by
examination and analysis, and that these processes are extra-
neous to pure motion : whereas the other sciences consider sub-
jects only which are proper to them, and therefore they are
called pure.
7.~\ The science of number, or, as the French call it, le calcul,
has for its subject-matter number in its pure and abstract form ;
number, that is, as an abstract quantuplicity ; not this or that
thing taken so many times, but the times which it is taken ;
it does not treat therefore of concrete things; and it is im-
portant to observe this property of the science, because the
truths of number are for this reason so generally, almost uni-
versally, applicable ; time, space, pressure, weight, velocity,
quantity of light, of heat, of electrical action, may be all mea-
sured by it ; and so long as the conditions imposed by the nu-
merical science are observed, the truths of number have their
counterpart in the applied science. The science also includes
number in its twofold division of discontinuous and continuous
number ; the former of which is the subject of arithmetic and
algebra, and the latter of infinitesimal calculus ; these being
distinguished by a difference of species of subject-matter, and
not of process. It is most important to observe that the nume-
rical symbols represent abstract quantuplicities, and that the
results are true, because they are correct developments of the
idea of number, and are independent of the concrete matter to
which they are applied. Yet they may be applied, and by the
following process : the numerical proposition is operated on
by the concrete unit of the matter of the particular science;
whether it be linear length, or area, or cubical content, or
weight, or velocity ; that is, each term of the numerical equa-
tion has the concrete unit affixed to it, and thereby itself be-
comes concrete, and expresses the concrete thing taken a cer-
tain number of times; thus suppose we have a numerical
equation 4 + 3 = 7
and suppose that the operating concrete unit is an inch : then
we have
4 times x one inch + 3 times x one inch = 7 times x one inch ;
an inch being matter of such a kind as to be consistent with the
8 THE PRINCIPLE OF HOMOGENEITY. [8.
fundamental operations of arithmetic; that is, if one inch is
added to one inch, no part of either one is absorbed into the
other, but the matter is continuously additive. Similarly might
the operating unit be a pound, or an unit of velocity, and in
both cases the result would be true because the arithmetical
equality is correct.
8.] Now this process of introducing a concrete factor into an
arithmetical equation is of the greatest importance, and deserves
careful consideration. The effects of it will frequently be dis-
cussed hereafter ; but one above all others requires explanation
at the outset of our work. Although the equations are made
concrete by the process, yet they are still subject to the laws
of algebra. In being made concrete they become also homo-
geneous as to the concrete unit; consequently they are intel-
ligible and interpretable : indeed no meaning can be attached
to an equation which is not homogeneous. Also if an equation
is once homogeneous, it continues homogeneous, whatever are
the algebraical processes to which it is subjected. Hence
homogeneity supplies a test of the correctness of the opera-
tions ; if this character of an equation is lost, error has been
introduced. The principle of expressing homogeneity in refer-
ence to various concrete units will be explained hereafter.
9.] The second mathematical science is that of space, or, as
it is usually called, geometry ; the subject-matter is in general
tridimensional space ; whatever is the origin of our conception
of it, whether it is experience, or whether space is a phenomenal
condition of our knowing things at all, or whether it is an in-
tuitive notion, yet at all events the subject-matter of geometry
is space, abstracted from all consideration of the space which
we occupy, and in which we are : and the science consists in
the development of this idea of space. The axioms contain
enuntiations of constituent parts and properties of it ; the defi-
nitions are explanations of terms arising out of, and necessary
to, the division of space which flows from the fundamental idea ;
thus, for instance, space is such that the whole is greater than
its part; that if equal spaces are added to equal spaces, the
wholes are equal ; spaces are equal which occupy equal parts of
space, the comparison being made on the principle of super-
position. The truths of geometry may be directly deduced
from the axioms and definitions by means of postulates and
more complex constructions, and the science of space thus
9.] THE SCIENCE OF SPACE. 9
treated of is called pure geometry; as such it neither requires
nor involves the properties of number; its additions and sub-
tractions and equalities are made on the principle of superposi-
tion; thus, if an angle is added to an angle, no reference is
made to any unit angle, but one concrete angle is superposed
on the other; and the symbols in pure geometry are symbols
of the concrete quantities and are not the subjects of arithmeti-
cal laws and operations. The old geometricians employed this
process only. But Descartes, perceiving that geometrical space
accords with the fundamental requirements of number, treated
of its properties by means of arithmetic and algebra : in this
view we may operate on any numerical equation with a concrete
geometrical unit whereby it becomes concrete and homogeneous,
and becomes a geometrical proposition ; and whatever numerical
truths are contained in, and deducible from, the numerical equa-
tion, analogous geometrical propositions are also deducible ; and
therefore if the equation is transformed or operated on according
to arithmetical laws, so will the transformation carry with it
the correctness of the corresponding geometrical changes; the
geometrical process is parallel with, and proved by, the nume-
rical process. Thus suppose the following equation to be true
for certain numerical values,
then by operating on each term with the linear unit, and inter-
preting x and y according to the conventional signification of
rectangular axes, we have the geometrical property of the curve
of which it is the equation, viz. (y 2 ) times the linear unit=(2a#)
times the linear unit (# 2 ) times the linear unit; y, x, and a
being numbers. Or otherwise suppose that we operate on the
same equation with the (linear unit) 2 , then the equation be-
coming arithmetically
y*y (2a ar)a?;
and we have the square of the ordinate = the rectangle con-
tained by the segments of the base.
By this process algebraical geometry has been constructed :
the equations in their original forms are numerical ; but as geo-
metrical space satisfies the conditions as to quantity which the
science of number requires, we operate on these numerical equa-
tions with a geometrical unit, and hereby transform them into
geometrical propositions; and we can further employ all the
PRICE, VOL. III. C
10 THE SCIENCE OF MOTION. [lO.
processes of algebra for deducing and proving geometrical truths
which are contained in other given geometrical propositions.
In both these sciences it will be observed that the process
of inference is the same : the deduction from the fundamental
ideas of number and space of the truths with which they are
pregnant.
10.] The third and last of the mathematical sciences is that
of motion ; into the foundation, laws, and processes of which I
shall enter at length in the following pages ; but as my method
is that of a positive deductive science, intended for didactic use,
and therefore to a certain extent dogmatical, it is not neces-
sary formally to discuss the history of the laws of motion, or
the growth of the fundamental idea, and the successive steps
through which it has reached that perfect state in which parts
of it can be expressed in definite axioms, and thus be made the
major premisses of the first syllogisms from which all the other
truths of the science are to be inferred. I shall not relate the
logomachy of mechanics in the days of Aristotle, and the dispu-
tations of the Schoolmen who taught that rest was natural and
motion was unnatural, and that some bodies fall faster than others
because they are heavier ; nor shall I indicate the several steps
by which Galileo first obtained a clear insight into the laws of
motion, and how Stevinus first proved the laws of oblique pressure
by means of a continuous chain resting on two inclined planes :
neither shall I generally detail or explain experiments by which
evidence is given to the truth of the axioms. My work, on the
contrary, is to take the idea of motion as recognized, and its
laws as acknowledged, and to deduce from them their results.
To this end mathematics, and especially the science of continuous
number, will be found most useful instruments of inquiry : a
word or two will shew this. Matter of motion, space, time,
velocity, and combinations of these, such as momentum, work,
vis viva, pressure, weight, will come under consideration. All
these quantities are continuously additive and subtractive, and
satisfy the requirements of the science of number: and they
admit of infinite divisibility ; nay, more than this, some of these
are within the grasp of our minds only when they are resolved
into infinitesimal elements : as, for instance, it is necessary to
know the law of change of velocity of a particle moving with
a varying velocity, before we can determine the actual change
of velocity which takes place in a given finite time; that is,
II.] THE SCIENCE OF MOTION. 11
the infinitesimal increment must be known, and this is deter-
mined by the law, before we can find the finite change, the
latter being determined from the former by means of integra-
tion ; in these respects then the subject-matter of our science
will be found to harmonize with the laws of the science of
number : and these latter may be applied.
11.] Suppose now that the axiomatic laws of mechanics are
deduced from the fundamental idea of motion, and that we know
them : let them be translated into mathematical language and
symbols, and so stated that the propositions take the form of
equations; if the concrete mechanical unit be removed, the
equation will stand as a numerical equation : to it in this state
all the rules of the science of number may be applied, and what-
ever are the results which can be inferred by means of them,
they may be translated by an operating factor into their me-
chanical equivalents, and these again into ordinary language.
As therefore the resources which the science of number supplies
become more numerous, the more fruitful is the deductive pro-
cess ; and hence it is that the progress of the sciences is simul-
taneous ; whatever retards the one is also an obstacle to the
progress of the other.
Consequently the following will be the course of our enquiry.
The idea of motion will be first described together with space
and time which are two incidentals of it. This is the funda-
mental idea of the science ; and pregnant properties of it will
be enuntiated : as matter is the subject of motion, so will certain
properties of matter have to be explained, and especially the
property which is called inertia, as we are hereby led to the
formation of equations of motion, in which the equality of mo-
mentum impressed and momentum expressed will be stated. These
pregnant properties of motion and of matter are called Laws of
Motion, and will be found to be only two ; we shall translate
them into mathematical language and symbols ; and by the pro-
cesses of infinitesimal calculus deduce from them their results,
which we shall in many cases trace in the applications of me-
chanics, and especially in the phenomena of gravitation, whether
in the case of bodies being near to the earth and falling towards
it, or in the case of the approximate motion of the planetary
bodies, herein laying the dynamical foundations of physical astro-
nomy. By this method the foundations of mechanics will be laid
in breadth sufficient to include all kinds of matter; whether
c 2
12 STATICS. THE SCIENCE OP PRESSURES. [l2.
cosmical or of that of light, if there is an ethereal medium ;
and all kinds of motion, whether direct or orbital or oscillatory ;
the basis therefore will be wide enough to comprehend the ma-
thematical theories of hydromechanics, light, heat, electricity,
magnetism; these several sciences, as they advance towards
perfection, satisfy more and more the notes of the science of
motion, but the perfect state will be reached only when they
wholly do so.
12.] Such is the philosophical form of the perfect and exact
science of motion ; and such is the philosophical course of learn-
ing it ; but there are reasons why a different method is more
suitable to a didactic treatise. It is better to begin with what
is apparently more simple and more concrete, than with an ab-
stract verity ; we are not accustomed to analyse cases of motion,
but we are familiar with an effect of the same cause as that
which produces motion, but which in mechanics is actually more
complex ; we have all of us a notion more or less exact of pressure
or of weight ; the tension of a string caused by a weight sus-
pended at the end of it, or a pressure caused by a weight resting
on the hand, gives us a notion more distinct than that of a body
falling under the action of the earth's attraction. Now let me
analyse such a pressure from a dynamical point of view : take
the case of, a weight resting on a table ; the same force which
produces the pressure on the table would cause the body to fall
towards the earth, if the table were removed ; the falling effort
is the same, although the table is there : the earth attracts the
body, impresses velocity on it, and causes it to penetrate the
table ; but the material of the table is elastic, and therefore so
often as the body penetrates the table and causes the particles
of the table which are in contact with or are near the body
to approach each other, an elastic force of recoil is called into
action and causes the body to retire : thus an oscillatory motion
of the body is established, which is however so slight that the
motion of the body is to the senses imperceptible. It may
perhaps be thought that this is an indirect mode of considering
such a simple case as that of a body resting on a table : perhaps
it is ; but it is the mode of applying the principles of the science
of pure motion to the case of a body resting on a table.
Thus although in the order of the pure science other and more
simple cases of motion would be discussed before this, yet as
this case of pressure is so simple, as it seems, and so common,
12.] STATICS. THE SCIENCE OP PRESSURES. 13
it is for didactic purposes desirable, even if it does cost a loss
of order scientifically correct, to consider first those forms of
problems with which a learner is most familiar ; we shall hereby
take advantage of his previous knowledge, and lead him from
that which is to him more simple to that which is more com-
plex. I propose therefore to defer the pure science of motion
to the second part of the treatise ; and to consider at present
pressures only, and these apart from the properties, real or
virtual, of motion. The science of pressures is called statics ;
and in establishing the principles from which I shall begin, I
shall be obliged to appeal to experience, to what we see and
observe : and whatever assumptions or hypotheses I may make,
I shall refer for proof to our observation of such pressures and
to the common sense of mankind. Let me make one other
observation on the difference which exists in the views of the
same effect as presented to us in a statical and a dynamical light.
Suppose that a pound weight rests on the hand, which is at
rest ; a pressure is experienced which the hand bears ; and if
another pound be added a pressure twice as great is experienced;
but are you conscious of or do you think about the cause of that
pressure ? are you aware that it is due to the earth's attraction,
and to a motion which the body would have if your hand were
removed ? I think that you consider it as a pressure- only, and
not in reference to velocity : this is, I say, the common judg-
ment about such pressures : it does not refer them to motion ;
and it is to such common judgment that I shall appeal in laying
the foundation of statics : it may be that I shall now and then
use language appropriate to the conception of a real or virtual
motion, and that I thereby elucidate difficulties ; but it must be
remembered that such conceptions are extraneous to statics thus
considered, and are such as the subject does not of itself require.
ANALYTICAL MECHANICS.
PART I.
STATICS.
CHAPTER II.
STATICAL PRESSURES ACTING AT THE SAME POINT.
SECTION 1. Explanation of matter, force, mechanics.
13.] A formal definition of matter such as would satisfy a
metaphysician or a physicist is not required for this work. It
is sufficient for us to conceive of it, as the subject of pressure :
capable of receiving- and of, as we shall hereafter see, trans-
mitting pressure : and as such, having volume and form ; be-
cause it is in this aspect only that it is of importance to us in
the present treatise*. Matter is rigid or stiff, when its com-
ponent particles are kept in a state of relative rest by the action
of cohesion or attraction, or of similar molecular forces ; and of
these we require at present only to know that the external
pressures acting on matter are in magnitude, in comparison of
these internal forces, infinitesimal. The consideration of other
properties of matter, as the subject of force, will be undertaken
in the sequel.
Matter is assumed to be infinitely divisible ; an infinitesimal
portion of it is called a particle : and the space occupied by a
particle is so small that it is a geometrical point. A finite
portion of matter is called a body. The quantity of matter
contained in a body is called the mass of the body.
* M. Poisson says, ' La matiere eat tout ce qui peut affecter nos sens d'une
mauiere quelconque.' Dr. Whewell, 'Body or matter is anything extended
and possessing the power of resisting the action of force.' Mechanics, gth
edition, Cambridge, 1836.
14.] MATTER AND FORCE. 15
Force is a cause which changes or tends to change matter's
state as to motion or rest. A % particle is at rest when it con-
stantly occupies the same place in space. A particle moves
when the place occupied by it changes its position.
Mechanics is the science which treats of the action and effects
of forces in this respect.
Statics is that part of Mechanics in which the relations of
forces are considered as they produce pressure or a tendency to
motion.
Dynamics, or as they are sometimes termed Kinetics, is that
part of Mechanics in which the relations of forces are considered
as they produce motion. In the first part of this work I
consider Statics, and only so far, for the most part, as the bodies
on which the forces act are rigid. Dynamics and other subjects
will be considered in subsequent parts.
14.] "When force acts definitely on matter, it is subject to the
four following incidents : it acts (1) at a certain point; (2) along
a definite line ; (3) in a given direction along that line ; (4) with
a certain magnitude or intensity. And a force is not said to be
given unless all these four incidents of it are given.
As Statics is that part of Mechanics which considers the
relations of forces as they produce pressure or a tendency to
motion, so are statical forces pressures. Weight is one of the
most common forms of pressure. Whenever in this first part
I speak of forces, the term signifies pressures ; but I employ the
word force in accordance with common usuage.
The point at which a force acts is called its point of application.
The straight line passing through the point of application of a
force, along which the force tends to make the particle at the
point of application of the force move, is called the line of action
or the action-line of the force ; the direction of the line toioards
which the force tends to make the particle move is called the
direction of the force. Thus we take the direction to be that in
which the force pulls or attracts the particle at its point of
application. The magnitudes of forces are measured by com-
paring them with some other force, the magnitude of this latter
force being taken to be an unit-force. The following is the
mode of measuring force.
Two forces are equal, which acting at the same point, along
the same line of action, and in opposite directions, neutralize
each other.
16 FORCE AND ITS INCIDENTS. [14.
Statical forces are continuously additive, and, as such, satisfy
the requirements of the science of number : thus, if one pound
is added to one pound, the sum is two pounds ; no part of
either of the weights is absorbed into the other ; the weight
of a basket of stones is the same, whatever is the arrangement
of the stones. Statical forces also admit of continuous increase
and decrease, and of infinite divisibility : they thus satisfy the
requirements of the science of continuous number.
If two statical forces, thus proved to be equal, act on a particle
at a point along the same line and in the same direction, the
acting force is twice each of the original forces : if three forces
act similarly, the resulting force is thrice each of the original
forces : and so on. Thus it is that forces admit of measure-
ment : an unit of force is chosen, and other forces are compared
with it ; and are expressed as being so many times the unit-
force. Thus forces are expressed by numbers, being referred to
a concrete unit-force. The unit-force is arbitrary, and may be
a finite or an infinitesimal force. If forces are expressed by
numbers which are commonly called incommensurable, they
possess the properties of commensurables, if they are referred to
an infinitesimal unit-force. If the unit-force is changed, the
numbers expressing the forces which are referred to it are also
changed in an inverse ratio. Thus a weight of six pounds is
expressed by 6, if a pound is the unit-force ; by 1 2, if one-half
of a pound is the unit-force ; by 3, if two pounds is the unit-
force. It is manifest that general laws connecting the point of
application, action-line, direction, and magnitude of a force, must
be independent of the conventional unit-force.
Statical forces will hereafter be expressed by symbols, such as
p, Q, R, . . . . These are numbers expressing the number of times
which the concrete unit-force is contained in the given force ;
hence also when we meet with such symbols as p 2 , Q 2 , . . . these
are also numbers. It is plain that if P represents a concrete force,
P* is uninterpretable and unintelligible.
Forces may be represented by geometrical straight lines. As
a force has a definite point of application, a definite action-line,
a definite direction, and is of a definite magnitude, so does a line
starting from the point of application of the force and coincident
with the action-line in its direction, and in length containing
the same number of linear units that the force contains units of
force, adequately and completely represent the force in all its
1 6.] THE COMPOSITION AND RESOLUTION OP FORCES. 17
circumstances. This mode has the advantage not only of sim-
plifying the enuntiation of many theorems, but also of enabling
us to infer mechanical propositions from their geometrical ana-
logues ; and vice vers&. Of this process we shall hereafter have
many instances.
15.] When a material particle is acted on by many forces
simultaneously, there is generally a definite line and a definite
direction along which it experiences a definite pressure, or, in
other words, along which it has a tendency to move. Now the
one force which would produce on this particle a pressure equal,
along the same action-line and in the same direction, is called
the resultant of the acting or impressed forces : and its action-
line is called the action-line of the resultant : and the several
impressed forces are called components in reference to it. The
resultant is evidently unique, definite as to its point of applica-
tion, action-line, direction of action, and magnitude.
If the forces acting on a particle are so related as to produce
a resultant whose magnitude is zero, then the forces are said to
be in equilibrium, and the system of forces is called an equi-
librium-system.
Hence we infer that when many forces act on a particle, if a
new force is introduced equal in magnitude to their resultant,
and acting along the same line and in an opposite direction, it
neutralizes the effects of all the others, the system of forces is
in equilibrium, and the particle is at rest.
The process of combining the effects of many forces, and of
thereby determining one force which would produce an equal
effect, is called the composition of forces. And as the process
evidently admits of inversion, and the effect of one force may be
decomposed into the effects of many forces acting simultaneously
at the same point, so this latter process is called the resolution
of a force. These processes will be very extensively employed
in the sequel.
SECTION 2. TJie composition and resolution of many forces acting
on a material particle, the lines of action of which are in one
plane.
16.] Let us first take the case of many forces acting on a
particle along the same action-line, and in the same direction.
PRICE, VOL. III. D
18 THE COMPOSITION AND RESOLUTION OF FORCES. [l6.
Let o, fig. 1, be the particle, and let OA be the line of action of
all the forces, and let them act from o towards A. Let them
be represented by the symbols p t , P 2 , ... P n ; then, since statical
forces acting at a point along the same line and in the same
direction are continuously additive, the resultant is equal to the
sum of all. So that if R represents the resultant,
R = P 1 +P 2 +...+P,, (1)
= 2.P, (2)
where P is the type-symbol of a force, and 2 is the summation-
symbol.
Again, suppose o to be acted on by two forces, along the same
line, and in opposite directions : let them be P and Q, of which
p is the greater : let P be resolved into two parts, Q and P Q j
then at the point o three forces act, viz. P Q, Q, and Q, of
which the last two act in opposite directions ; therefore they
neutralize each other ; and, if R is the resultant, we have
R = p Q. (3)
And as a similar result is true for any number of forces acting
in either direction, and along the same action-line, the equation
(2) may be extended so as to include the algebraical sum of the
forces acting on a point and along the same line.
Hence we infer that a particle is in equilibrium under the
action of many forces acting along the same line, if the sum of
those acting in one direction is equal to the sum of those acting
in the opposite direction ; and the condition of equilibrium is
2.P = 0. (4)
Let us also take another simple case : that of three equal
forces P, Q, R, see fig. 2, acting at o, all of which are in the same
plane, and the lines of action of which are inclined to each other
at 120. Let the forces be represented, both in direction and
in intensity, by the equal definite lines OP, OQ, OR: then the
particle at o is in equilibrium : for by the principle of sufficient
reason it cannot move out of the plane of the forces, neither can
there be any resultant pressure in the plane ; the particle there-
fore is in equilibrium; and either of the forces may be con-
sidered to be equal in magnitude to the resultant of the other
two, and to act in the same line, but in an opposite direction.
Hence we have the following geometrical construction of the
resultant. Let P and Q be the components ; then R neutralizes
the effects of p and Q on o; produce RO to 11' so that OR' is
1 7.] THE PARALLELOGKAM OF F011CES. 19
equal to OR; then the force of which OR' is the geometrical
representative neutralizes R ; but the resultant of P and Q also
neutralizes u : therefore the force R' is the resultant of P and Q ;
and by the geometry OR' is the diagonal of the parallelogram of
which OP and OQ are the adjacent containing sides.
17.] The more general problem however is the determination
as to action -line, direction, and magnitude, of the resultant of
two forces acting on a particle. This proposition is commonly
called the parallelogram of farces by reason of the geometrical
form of it.
Let the meaning of the problem be clearly understood ; it is
required to determine the line of action, the direction, and the
magnitude of a force which acting at a given point shall produce
the same effect in all respects as two forces acting simultane-
ously at the same point.
It is evident by the principle of sufficient reason that the line
of action of the resultant is in the same plane with the lines of
action of the components.
* Let us first take the case of two equal forces P and P acting
at o, and with their lines of action inclined at an angle 20.
It is manifest that the line of action of the resultant bisects the
angle contained between the lines of action of the components ;
because every reason which can be alleged why it should be on
one side of this line is equally valid to prove that it should be
on the other : and an integral part of the conception of a re-
sultant is that it should be unique both as to line of action
and as to magnitude ; hence by the principle of sufficient reason
we conclude that the line of action of the resultant bisects the
angle between the lines of action of the components.
To determine the magnitude of the resultant. Let OP, Of l
represent, see fig. 3, the two equal forces acting at o ; let the
angle PO?! = 20 ; let OR be the line of action of the resultant R,
so that FOR = FiOR = 6. Now the magnitude of R can depend
on only P and 6 ; so that if f denotes a function which is to be
determined, R _ y ^ ^ . (5)
in this equation R and P are numbers depending on the arbitrarily
chosen unit of force, and varying of course as the unit varies ;
* The following proof of the parallelogram of forces is due to M. Poisson, and
commonly bears his name. A discussion, more or less complete, on 45 other
proofs will be found in ' Praecipuorum inde a Neutono conatuum, compositionem
virium demonstrandi, recensio. Auctore Carolo Jacobi. Gottingse, MDCCCXVIII.'
20 THE PARALLELOGRAM OF FORCES. [17.
but the law of relation between R, P, and Q cannot depend on
this unit ; in other words, the equation must be homogeneous
in terms of P and R ; therefore (5) must be such that the unit
may be divided out, whatever be its magnitude ; and this can
only be the case when the equation is of the form
R = P/(0). (6)
It remains for us to determine the form off.
Suppose P to be the resultant of two equal forces Q and Q t
acting at equal angles on the opposite sides of P'S line of action ;
and let QOP = Q^P = </> ; therefore by (6),
?=Q/(</>); (7)
similarly let P! be the resultant of two forces Q, and Q;, equal to
each other and to the former QS, acting at equal angles <f> on
the opposite sides of P/S line of action ; so that
i f Pi = Q/(0); (8)
consequently from (6),
Now R is the resultant of P and PJ ; and therefore, as P and P,
are the resultants of Q, q, q u and q x , R is the resultant of these
also ; let them be taken in pairs, so that R is the resultant of
Q, q, and of q,, ^ ; but by (6),
the resultant of q, q =
therefore substituting in (9;,
and /(**+)+A'-*) : /C^f).i (11)
that is, the form of f is such as to satisfy the functional equa-
tion (11).
Expanding the left-hand member of (11) by Taylor's series,
we have
2.34 -
but as no relation exists between 6 and <, Q is constant in re
ference to 0: therefore in (12), which is the expansion o
we may put, if a is constant,
/^?._ fl ,. /""(*)
m ' " 7W = '
and so for the other terms ;
1 8.] THE PARALLELOGRAM OF FORCES. 21
= 2 cos a$ ;
.-. f(0) = 2cos0; (13)
and R = 2pcosa0; (14)
a is still undetermined; it must however be some uneven num-
ber, because R = 0, when =. 90, that is, when the two equal
forces act in the same line and in opposite directions : and the
uneven number can be none other than unity, because if it
were 3 or 5, or ... or 2n+ I, R would vanish when = -, = ,
6 10
. and this would be absurd : therefore the func-
'
tional relation between R, p, and 6 is .
R=2Pcos0*. (15)
The form of function given in (13) evidently satisfies (11),
because
2 cos a (6 + 0) + 2 cos a (#</>) = 4 cos ad cos a<f>.
If I had assumed in the preceding/" (9) = a*f(0], then
so that f(0} =e** + -**,
and thus /(0) would increase without limit as increased
without limit; and consequently R would increase indefinitely
with 0. This of course cannot be the case, and the solution is
accordingly excluded, and (15) is the only solution admissible
by the conditions of the problem.
18.] The following is the geometrical interpretation of this
theorem; Let OP and o~p 13 fig. 4, represent the components in
line of action, direction, and magnitude, so that POP! = 20 ; let
OR bisect the angle P! OP; from p draw PD perpendicular to OR,
and produce OD to R, so that DR = OD; then OR = 2 OP cos 0,
and therefore OR by its length and direction represents the re-
sultant of P and P! ; join PR, Rpj : then p t OPR is manifestly a
rhombus, of which OP, o?! are two adjacent sides, and OR is the
diagonal.
If therefore two adjacent sides of a rhombus represent two
forces acting at o, the diagonal of the rhombus abutting on o
* Another mode of solving (n) is given in Ex. 7, Art. 456, Vol. II. (Integral
Calculus).
22 THE PARALLELOGRAM OF FORCES. [19.
represents the resultant both as to line of action and intensity ;
hence also, since
OR 2 = OP* + OP, 2 + 20P.OPiCOSP 1 OP,
R = 2P"-f 2P 2 cos20. (16)
Hence also conversely we infer that a force acting on a
particle may be equivalently replaced by two equal forces acting
at equal angles on either side of its line of action if, R being the
force to be replaced, P being one of the equal components of
it, and being the angle between the lines of action of R and P,
p = -sec0; (17)
T>
p therefore cannot be less than - ; and increases as Q increases,
m
and lastly becomes infinite when = 90 : hence we infer that
the effect of R on o cannot be produced by any force whose line
of action is perpendicular to that of R ; and therefore that two
forces whose lines of action are perpendicular to each other do
not affect each other's effects. As the theorem admits of the
preceding geometrical interpretation, it has received the name
of the parallelogram offerees.
19.J Let us in the next place take the case of two unequal
forces P and Q acting at a point O, fig. 5, and along lines of
action perpendicular to each other. Let p and Q, be represented
by the lines OP and OQ; complete the rectangle OPRQ, and draw
the diagonal OR; let the angle ROP = a; then the force P may
by reason of the preceding Articles be resolved into two forces
p' and P' acting at equal angles a on either side of OP, and by
reason of (17), P
P = -seca; (18)
4
and therefore P' is geometrically and equivalently represented by
half of the diagonal OR. Again, let Q be resolved into two equal
forces Q' and Q,' acting at equal angles 90 a on each side of
OQ, so that by reason of (17)
Q'=|coseca, (19)
m
and therefore p/ is geometrically and equivalently represented
by half of the diagonal of the rectangle. Hence we have two
forces, each of which is represented by half of OR, acting along
OR and in the same direction, and of which therefore OR is the
resultant both as to line of action and as to magnitude; and
20.] THE PARALLELOGRAM OF FORCES. 23
also two forces Q,' and p' acting at o in the same line and in
opposite directions : and as these are equal, both being repre-
sented by half of OR, they neutralize each other; and therefore
the resultant of the two forces p and Q acting at o is represented
by the diagonal of the rectangle of which the containing sides
are the representatives of the components. Hence if R is the
R 2 = P a + Q S ; (20)
R = pseca = Qcoseca. (21)
Hence also conversely, fig. 6 ; if a force P acts at o, and is
represented in line of action, direction, and magnitude by the
line OP; it may be resolved into two forces acting along two
lines originating at o and perpendicular to each other ; so that
if x and Y are the resolved forces, and if the angle between the
lines of action of P and x is 0, then by (21)
x = Pcos0, Y = Psin0; (22)
P 2 = x 2 -f Y 2 . (23)
Hence the resolved part of a force along any line is equal to
the product of the force and the cosine of the angle between
the given line and the action-line of the given force.
This theorem is most important, and is very frequently em-
ployed in subsequent investigations. By virtue of it forces may
be resolved, or projected, according to the same law as lines and
areas are projected. It is for this, with many other reasons,
that the cosine of an angle is called the protective coefficient,
20.] Lastly, let us consider the case of two unequal forces
p and Q acting on a point O, along lines of action inclined to
each other at an angle y ; see fig. 7 ; let OP and OQ be the geo-
metrical representatives of the forces, and let QOP = y ; com-
plete the parallelogram QOPR, and draw the diagonal OR. Now
resolve P into two forces P' and p" along OR and perpendicularly
to OR, and suppose ROP = ; then by (22),
p'=pcos0, p"=Psin0; (24)
so that by the geometry of the figure, OP' is the geometrical
representative of P', and OP" of P". Again, resolve Q into two
forces Q' and Q", in lines along and at right angles to OR ; then,
by (22),
Q' = qcos(y-0), Q" = Qsin(y-0); (25.)
and therefore OQ' is the geometrical representative of Q', and
24 THE PARALLELOGRAM OF FORCES. [20.
OQ" of Q". Now P" and Q" are manifestly equal, and act in
the same line but in opposite directions ; they therefore neutralize
each other; and there remain p' and Q' acting along OK in the
same direction, and therefore the resultant is equal to the sum
of them, and is geometrically represented byop'+oo/, that is,
by on, which is the diagonal of the parallelogram of which OP
and o Q are the containing sides ; and since
OR* = OP 2 -{-PR* 2.0P.PRCOSOP11
= OP 2 + OQ 8 -f2.OP.OQcosPOQ; (26)
therefore replacing the geometrical lines by their statical pro-
portionals, R2 _ p*_j-Q* + 2PQCosy. (27)
Evidently the former two cases are particular instances of this :
for if y - 90, R 2 = p' -f Q ;
if p = Q, R = 2 P cos ^
%
Hence in all cases we may enuntiate the theorem in the fol-
lowing form :
If two forces acting at a point are represented by two lines
meeting at the point, the resultant is represented as to line of
action, direction, and magnitude by the diagonal of the parallel-
ogram of which the two lines are adjacent, sides.
This theorem is, as above mentioned, called the parallelogram
offerees, on account of the geometrical interpretation of it.
Hence, conversely, if any force R acts at a point o, it may be
resolved into any two forces p and Q, whose lines of action are
inclined at an angle y, if P, Q, and y satisfy the condition (27).
And from (24) and (25), if is the angle between the action-
lines of R and P, if we resolve P and Q along, and at right-angles
to, the action-line of R,
R = Pcos0+Qcos(y-0),)
p sin Q Q sin (y 0) = 0. f
Hence, fig. 8, if a force R, equal to R', say, the resultant of p
and Q, acts on a particle at o in the line OR', but in an opposite
direction to R', the three forces p, Q, R are in equilibrium : and
either force is equal to the resultant of the other two ; and there-
fore if qOR = a, ROP = ft, POQ = y,
P 2 = Q 2 + 2QRCOSO + R 2 ,
Q 2 = R 2 -f 2RPCOS/3 + P 2 , \- (29)
..:;,
R = P 2 + 2PQCOSy+Q*. J
22.] THE MOMENTS OF FORCES. 25
Hence also it is plain that a force acting at a given point may
be resolved into two forces whose lines of action pass each
through a given point, if the three points and the action-line of
the given force are in one plane.
21.] Also since the three equilibrating forces P, Q, R are pro-
portional to the three lines OP, oq, OR, or to OP, PR', R'O; and
since the three sides of a triangle are proportional to the sines
of the opposite angles, therefore
P q R .
sinOR'p sinR'op ~~ sinopR 7 '
or _^-= * =^_; (30)
sin a sin ft sin y
that is, if three forces acting at a point are in equilibrium, each
is proportional to the sine of the angle contained between the
lines of action of the other two.
From (30) we infer that three forces acting at a point are
in equilibrium, if they are proportional to the three sides of any
triangle whose sides are parallel to the lines of action of the
forces, and if their directions are those of a point traversing the
perimeter of the triangle. This theorem is known by the name
of the triangle of forces.
22.] Also from the second equation in (28) it appears that if
p and q are the lengths of the perpendiculars drawn from any
point in the line of action of R to the lines of action of P and Q,
then p _ sing _ Q m
q ~ sin (y 0) ~ P '
.'. PJ? = q?. (31)
And thus if PJ and P 2 are forces acting at a given point along
lines of action, the equations to which are
1 b 1 =0,)
, 5 a = 0,)
which we may represent by the abridged notation a, = 0, and
a 2 = ; then attaching the proper signs to c^ and a,, the equa-
tion to the line of action of the resultant is
p iai +P a a 2 = 0. (33)
The product of a force and the perpendicular from a given
point on the action-line of the force is called the moment of the
force with reference to the given point, and denotes a certain
property of the force which will be explained at length here-
after; consequently (33) contains the following theorem ;
PRICE, VOL. III. E
26 THE COMPOSITION OP MANY FORCES. [23.
The moments of the components are equal with reference to
any point in the action-line of the resultant.
23.] Let us next consider the general case of many forces
acting at a given point, the lines of action of all of which are in
one plane.
Let o be the point at which all the forces act : and through
it let two lines, as coordinate axes, be drawn perpendicular to
each other, and in the plane in which the forces act.
Let the force be P,, P a , . . . P W , of which let p be the type-force :
and let the angles between the a?-axis and their action-lines
severally be a,, a 2 , . . . o^ of which let a be the type-angle ; and
let the several forces be resolved along the axes of x and y\
then by equations (22), Art. 19, the resolved parts along the
ar-axis severally are P! cos a l} P 2 cos a a , ... P M cos a u ; and those
along the y-axis are P! sin a l} P 2 sin a 2 , ... P n sin a n ; and there-
fore if x and Y denote the forces along the axes of x and y
respectively,
X = P! cos d! + P 2 cos a 2 + ... +P B cosa B ) ,_ v
= 2. P cos a. )
Y = P! sin dt+Pjj sin a a -f ...+:
= 2. P sin a.
Let E be the resultant of all the forces acting at o, and 6 the
angle which its line of action makes with the axis of x ; then as
E produces at o the same effect as to magnitude, line of action,
and direction as all the impressed pressures taken in com-
bination, so are the resolved parts of E along the axes equal
severally to x and Y : consequently
E cos 6 = x = 2. P cos a, )
c (o)
Esm0 = Y = 2.Psma; }
.'. E' = x'-f Y a ; tantf = -; (37)
!^1 = ^? _ I; (38)
Y X E
and hereby may the magnitude, line of action, and direction of
the resultant of many forces acting in one plane on a given
particle be determined.
If the forces are so related that the particle is at rest, then
the resultant vanishes ; and
)
)
E a = x'+Y 2 = 0; (39)
= 2.Pcosa = 0, Y = 2.rsina = 0. (40)
24-] THE COMPOSITION OP MANY FORCES. 27
As the conditions of equilibrium must be independent of the
particular system of coordinate axes, we infer that, if many
forces acting on a particle in one plane are in equilibrium, the
sum of the resolved parts of the forces along every straight line
is equal to zero.
24.] The following examples are in illustration of the pre-
ceding theorems.
Ex. 1 . Four equal forces whose directions are inclined to the
axis of.r at angles of 15, 75, 135 and 225 act at a point:
determine the magnitude and direction of their resultant.
Let each pressure be equal to P ; then
3* 2.
Y = Psin 1 5 -I- P sin 75 + psin 135 + Psin 225
.-. E = p(5 2.3*)*; tan0 =
3* -2
Ex. 2. Three forces act perpendicularly to the sides of a tri-
angle at their middle points, and are proportional to the sides ;
it is required to prove that they are in equilibrium.
Let ABC, fig. 9, be the triangle, and let the forces be P, Q, R,
and act in the directions indicated by the arrow-heads; their
lines of action meet at the point o; let QOR = a, ROP == /3,
POQ = y ; a, /J, y being manifestly the supplements of A, B, c;
then by the data
p o R
and since the sides are proportional to the sines of the opposite
angles, P Q R
sin A ~~ sin B ~ sin c '
P Q R .
sin a ~ sin /3 " sin y '
and therefore by (30), P, Q, R are in equilibrium.
Or thus resolving along BC ;
The forces along BC = Q sin c R sin B
= k (5 sine csin B}, by (41),
= 0;
E 2
28 THE COMPOSITION OF MANY FORCES. [25.
and similarly will the sum of the resolved parts of the forces
along any other line vanish. And therefore the system is in
equilibrium.
Ex. 3. If R is the resultant of P and Q acting at o, fig. 7, and
A is any point in the plane POQ, from which perpendiculars
A/?, Aq, Ar are drawn to OP, OQ, on respectively, then
(1) P.Aj9 + Q-A<? = R.Afj
(2) P.OJO +Q.o<7 = R.or.
Join AO, and let AOP = 0. Let P, Q, R be resolved along and
perpendicularly to AO ; then as R is in all respects equivalent to
p and Q in combination, the component of R along any line is
equal to the sum of the components of P and Q, : consequently
PCOSAOJO + QCOSAOJ = RcosAor:
and replacing the sines and cosines by their geometrical values,
we have
(1) P.AjO + Q.AJ' = R.A/j
(2) P.OjO+Q.O^ = R.Of.
(1) is the theorem of the equivalence of moments which has
already been proved analytically in Art. 22 ; and (2) is the
theorem of virtual velocities the general investigation of which
will be made hereafter.
Hence also if p, Q, R are three forces which equilibrate at o,
and A is another point in the plane PQRO from which Aj9, A q, Ay-
are drawn perpendicular to the action-lines of P, Q, R respec-
tively, P.Aj0 + Q.A + R.Ar = 0,
p.qp + Q.o^ + R.or = 0.
Hence also generally if many forces P t , P 2 , . . . P n equilibrate at o,
25.]] In the application of the preceding principles, statical
forces often arise from (1) the determinate tension of strings,
(2) reacting pressures. It is worth while to say a few words on
each of these cases.
Suppose in fig. 1 OA to be a string, fastened at o, and pulled
at its other extremity with a certain force = P ; then it is (expe-
rimentally) plain that o is pulled with a force equal to that
exerted on the string at A, and that the tension of the string is
the same throughout ; the line of the string of course expresses
the line in which the pressure acts on o, but the length of it is
25.] THE COMPOSITION OF MANY FORCES. 29
not a measure of the intensity of the pull, although a length
may be taken along it which shall be proportional to that in-
tensity. One or two examples, in which such pressures are
involved, are subjoined.
Ex. 1. A and B, fig. 10, are two fixed points in a horizontal
line ; at A is fastened a string of length c with a smooth ring
at its other extremity c, through which passes another string
fastened at one end at B ; the other end of which is attached to
a given weight w ; it is required to determine the position of c.
Let AB = 2a, AC = c, CAB = 6, ABC = 0. Let the tension
of the string AC = T ; which is undetermined. Now as the ring
at c is smooth, the tension of WCB is the same throughout, and
is of course equal to the weight w ; and therefore c is kept at
rest by three forces, w, w, and T ; let us apply equations (40)
and resolve the forces horizontally and vertically; and equate
those acting towards the right-hand to those acting towards the
left ; and those acting upwards to those acting downwards. Then
the horizontal forces are, w cos < = T cos 6 ;
and the vertical forces are, w sin $ -f T sin Q = w.
Therefore eliminating T,
cos Q = sin (0 + 0) ;
.-. 20 + ^ = 90. (42)
Also from the geometry
sin (e -f 0) _ 2a t
from (42) and (43) and $ may be found : and thence T may be
determined ; and thus all the circumstances of the problem are
determined.
Ex. 2. A and B are two points in a horizontal line; a string
fastened at A, fig. 11, passes over a small pulley at B, and sup-
ports at its other end a weight w ; a small and smooth heavy
ring of weight w' slides on the string between A and B ; deter-
mine the position in which the string rests.
Let c be the point at which the heavy ring rests : as the
pully is smooth, and has no friction, and as the ring is also
smooth, the tension of the string is the same throughout and
is equal to the weight of w ; hence the point c is kept in equi-
librium by three forces, w along CA, w along CB, and w / which
acts vertically downwards: let_CAB = 0, CBA = <; therefore,
taking horizontal and vertical forces, we have
30 THE COMPOSITION OF MANY FORCES. [26.
Horizontal forces ; w cos 6 = w cos < ;
Vertical forces ; w sin + w sin $ = w' ;
V
.. = <f> = sm- 1 --
2w
26.] Again, suppose the particle, on which the statical forces
act, to be on a smooth plane surface, which is capable of bearing
the resultant of the component forces which acts along the
normal and in a direction towards the plane; but by reason
of its smoothness does not offer any resistance to motion in the
direction of its surface ; then, since the actual normal pressure
of such a plane is equal, and in direction opposite, to that im-
pressed on it by the component forces, this normal reaction of
the plane is one of the forces by which such a material particle
is kept at rest, and, as such, will enter into the equations of
equilibrium.
Ex. 1 . A particle of weight w is kept at rest on a smooth
inclined plane by a force P acting at a given angle to the plane ;
determine the pressure on the plane, and the magnitude of P.
Let fig. 12 be a vertical section of the system ; AC the inclined
plane; CAB = a, PQC = Q, R = the reaction of the plane against
the particle Q: then, as the lines along which forces may be
resolved are arbitrary, let us resolve along, and perpendicularly
to, the plane. Then we have
Forces along the plane, P cos (3 = w sin a ;
Forces perpendicular to the plane, E + P sin 8 = w cos a ;
sin a cos (a + 8)
.-. P=w- ; R = w - ^ .
cos 8 cos 8
The force P therefore acts to the greatest advantage, that is, w
is the greatest, when 8 = 0.
Ex. 2. Two forces P and Q acting respectively parallel to the
base and length of an inclined plane will each singly sustain on
it a particle of weight w ; to determine the weight of w.
Let a be the inclination of the plane to the horizon ; then in
each case resolving along the plane, so that the normal pressures
may not enter into the equations,
P cos a = w sin a, Q = w sin a ;
The case of this Article is a particular one of the general theory
of a constrained particle which is fully discussed in Art. 32.
28.] THE MOMENTS OF FORCES. 31
27.] The resultant of forces acting at a point in one plane
must be, as to line of action and intensity, independent of the
particular origin and the particular system of coordinates ; and
we may in the following manner deduce this property from the
preceding results :
.. x = 2. P cos a = PI cos a, + P 2 cosa 2 + ... -f P n cosa n , )
.
Y = 2.P sin a = Pj sin a l +P a sin a a -f . . . -f P n sin a n , )
.-. R 2 = x"+Y a
- Pl * + p 2 * + ... +Pn *
+ 2 (PJ P 2 cos (a, a a ) + P! P 3 cos (a, a 3 )+ ...
- + ?_! P n cos(a n _ 1 -a w )} (45)
= 2.P 2 + 22.PP'cos(a a'), (46)
where P, P' are the symbols for any two of the forces, and a a"
is the angle contained between their lines of action; and the
sign of summation prefixed to pp'cos(a a') indicates the sum
of the products corresponding to the n forces taken two and
two together; and therefore (46) is independent of the system
of coordinate axes. The parallelogram of forces which is given
in equation (27) is a particular case of (46). A further gene-
ralization of this theorem is given in (68), Art. 31.
28.] We have also the following relation between the several
components and their lines of action, and any point in the line
of action of the resultant.
Let the equations to the lines of action of the components be
xcosa l +y sinaj = = a u 1
x cos a a +y sin a 2 = = a 2 , i ,._,.
x cos a n +y sin a n = = a n ; j
the point at which they act being the origin, a being the angle
between the axis of x and the normal to the line of action ;
and the a on the right-hand side of the equation being the
length of the perpendicular from the point (x, y] to the line.
Now if the components are P U P 2 , ... P M , and the resultant is
R, and a is the angle between the normal to R'S direction and
the axis of x, then the equation to R'S line of action is
xcosa-\-y sma = ; (48)
.. xncosa+ynsina = 0;
but R cos a = 2.P cos a, R sin a = 2.P sin a ;
.. #2. POOS a-fys.Psin a = ;
32 THE POLYGON OF FOKCES. [29.
...-fp n {^cosa n +ysina n } = 0; (49)
Pia 1 + P J a,+ ... + P H a f( = 0, (50)
where a,, a a , ... a n are the perpendiculars from (#, y\ any point
in the line of action of R, on the lines of action of the compo-
nents; therefore, bearing in mind the meaning of the word
moment as given in Art. 22, we have the following theorem;
With reference to any point in the line of action of the
resultant, the sum of the moments of the components vanishes.
The theorem given in (33) Art. 22 is a particular case of the
preceding.
The following also is a more general theorem ; if (x, y] is a
point in the plane of the forces but not on the resultant, then
x cos a +y sin a = the perpendicular distance from (x, y] on the
line of action of R: let this = r; then from (49),
p 1 a 1 + P 3 a 2 + ...+P n a n = Rr;
that is, with reference to any point in the plane of the forces the
sum of the moments of the components is equal to the moment
of the resultant.
Hence if two forces only act, as is the case in the parallelogram
of forces, with reference to any point in the plane of the forces,
the moments of the resultant is equal to the sum of the moments
of the components.
As the moment is the product of the line-representative of
the force and of the perpendicular on the action-line of the
force from a given point, it expresses geometrically twice the
area of the triangle of which the given point is the vertex and
the line-representative of the force is the base. Hence, in fig. 7,
if A is any point in the plane POQR, and if AO, AP, AQ, AR are
drawn, the triangle A OR is equal to the sum of the two triangles
AOP and AOQ,. This is easily demonstrated geometrically.
29.] The following is another geometrical interpretation of
the conditions of equilibrium in equations (40).
It is a well-known property of a closed polygon that the sum
of the projections of its sides on any given straight line is zero ;
the projections of the sides being affected with positive or nega-
tive signs according as the angles made by them with the given
straight line are acute or obtuse, and care being taken to esti-
mate the angles between the given line and the sides of the
polygon which are turned all towards the inside or all towards
JO.] COMPOSITION OF FORCES. 33
the outside of the figure. Hence, if l lf l y , ... 1 H are the lengths
of the sides, and a u a a , ... a n are the angles between them and
the given straight line,
2./cosa = 0. (51)
Now if n forces act at a point, the condition of equilibrium is
2.pcosa = 0. (52)
Hence if n forces, having their lines of action parallel to the
successive sides of a closed polygon, their directions the same
as that of a point traversing the sides of the polygon, and their
magnitudes represented by the lengths of those sides, act at a
point, (51) assumes the analogous mechanical form (52), and the
forces are in equilibrium : hence conversely, if many pressures
whose action-lines are in one plane act at a point and are in
equilibrium, their action-lines are parallel to the sides of a closed
polygon, the sides being proportional to the magnitudes of the
forces ; or in other words, the line-representatives of a system of
forces, acting in equilibrium and in one plane at a point, will
form the contour of a closed polygon, the sides of which taken
in order are equal and parallel to these line-representatives taken
in the same order.
This proposition is known by the name of the polygon of
forces, and the triangle of forces proved in Article 21 is a
particular case of it.
SECTION 3. Composition and resolution of forces acting in any
directions on a material particle.
30.] Here and elsewhere we shall refer the effects of forces
acting in space to a system of rectangular coordinates ; because
the results are not more general, and are much more compli-
cated, when they are referred to a system of oblique axes. And
let us in the first place take the case of three forces x, y, z
acting at the origin o, see fig. 1 3, and along the coordinate axes.
Let the resultant of x and Y, which are at right-angles to each
other in the plane of (x, y}, be R'; then, by (20), Art. 17,
R' 2 = X s -fY 2 .
Again, of R' and z, which are at right-angles to each other, let
the resultant be R ; then
R* = R''+Z
= x a + Y a + z 2 ; (53)
PRICE, VOL. III. I
34 COMPOSITION OP FORCES. [31.
and R is the resultant of the three forces. Let the direction-
angles of its line of action be a, b, c; then, by equation (22),
X = RCOSfl, Y = RCOS#, Z = RCOS. (54)
Hence, conversely, any force P, acting at o, the direction-angles
of whose line of action are a, /3, y, may be resolved into three
forces x, Y, z acting along the coordinate axes, such that
x = PCOSO, Y = Pcos/3, z = pcosy. (55)
31] Next let us take the case of many forces acting in any
lines at the point o.
Let the forces be ? 1} P 2 , . . . P n ; and let the direction-angles of
their lines of action be Oj, /?,, y l ; a a , /3 S , y j . . .a B , /3 n , y n ; let these
be resolved severally along the coordinate axes, and let x, Y, z
be the sums of the resolved parts along the axes respectively
of x, y } and z ; then
x = p t cos Oj + P, cos a a + . . . + P cos a n ^
= 2 .Pcosa;
Y = 2.P cos /3 ;
Z = S.PCOSy. J
Let R be the resultant of all the impressed forces ; and let the
direction-angles of its line of action be a, 6, C', then as the
resolved parts of R along the three coordinate axes are equal to
the sum of the resolved parts of the several components along
the same lines,
Rcosa = x, Rcosi = Y, RCOSC = z; (57)
and squaring and adding,
(58)
cos a = -, cos b = -, cose = -: (59)
R R R
and the equations to the line of action of the resultant are
-JL = -J- (60)
2.P COS a 2.P COS /3 2.P COS y
Also from (58), R = X-+Y- +z-
K R xt
= x cos a -\- Y cos I + z cos c,
that is, R is equal to the sum of the forces along the coordinate
axes resolved along the line of action of R.
If the point at which all the forces act is (x, y', /), so that
the equations to the lines of action of the components are
33-] A FREE PARTICLE. 35
x-af zz'
cos/?!
xx' _ yy zz'
cosa 2 ~~ cos/3, "cosy,' (61)
xx' y y z z f
cos a tt cos /3 n " cos y n ' J
then the equations to the line of action of the resultant are
*-* = y-y = *-' (62)
2.P cos a 5.P cos /3 2.P cos y
32.] Now from the point at which the forces act, let straight
lines be drawn, which are in length and direction geometrical
representatives of the forces : and let the extremities of these
lines be (x u y^z^ t (*i,y*,z t \ ... fo,,^,,,*.), and let their lengths
be s l} * a , ...*; then
' 'x 1 = *j cos o
(63)
and 2.P cos a =
S.Pcos/8 = %(j/ y] = ny' (y\.+y*-\- ~{-y n )) \- (64)
2.P cos y =
and therefore (62) become
xx'
n n n
which are the equations to the line of action of the resultant.
The point whose coordinates are
(65)
is that which is known by the name of the geometrical centre of
mean distances of the points which are the extremities of the line-
representatives of the forces : and therefore from (65) it appears
that the line of action of the resultant passes through this point.
33.] Also the magnitude of the resultant of the pressures,
which is of course independent of the particular system of
coordinate axes, may thus be found ; since
x = P! cos a! + P 2 cos a 2 + . . . + P n cos a n , 1
Y = Pj COS0J + P, COS /3 2 +...+? COS , } (66)
Z = P^osyj-fp, cos y 2 + ... + P M cosy,,; J
2
36 COMPOSITION OF FORCES. [34.
R' =
+ 2 PI PS (cos tti cos a 2 -I- cos & cos /8 2 + cos y! cos y,}
(67)
= 2.P" + 22.pp'cos (P, P'), (68)
where P,P' are the symbols for any two of the forces, and COS(P,P')
is the cosine of the angle between their lines of action. And
from the forms, which the resolved parts of R take in equations
(64), it follows that the geometrical representative of it is n
times the length of the line joining the point of application of
the forces and the centre of mean distances of the extremities
of the geometrical representatives of them. This theorem is
due to M. Chasles, and is the true generalization of the paral-
lelogram of forces.
34.] If the forces are in equilibrium, R = ; in which case,
by reason of (58), x = 0, Y = 0, z = ; or,
2.PCOSO = 0, 2.PCOS/J = 0, S.PCOSy = 0; (69)
that is, the sum of the resolved parts of the forces along each
of three coordinate axes is equal to zero.
35.] We have thus far employed rectangular coordinate axes,
and have in reference to them proved that a force may be
resolved into three components whose lines of action are at
right angles to each other, and that these three forces equiva-
lently replace the given force. A force may however be equiva-
lently replaced by three forces whose action-lines meet on a
point in its action-line, provided that the action -lines of these
three forces are not in one and the same plane. To demonstrate
this theorem, let P be the force, and let x, Y, z be its axial
components ; and let (l l} m lt nj (l a , m a) n a ~) (l s , m 3 , 3 ) be the
three straight lines along which the forces p^ P 2 , p s are to act,
and which are equivalently to replace P ; then
x = pj, + P 2 ^ + P S 1 3)
Y = p, nil + P Wt + P 3 MS,
z = Pi ! + ?., ft 2 -4-p s M 3 ;
from which equations, if they are independent of each other,
PU ?*, PS can be determined. If however the action-lines of
PI, P 2 , P, are in the same plane, and the action-line of P does not
lie in this plane, then, employing the symbols of determinants,
2 + / v w 2 w 3 = 0, and P,, p,,, P 3 are infinite, and the proposed
36.] EQUILIBRIUM OF FORCES. 37
equivalent substitution is impossible. The values of p,, p,, p,
are indeterminate if their action -lines and that of P are in the
same plane.
SECTION 4. Conditions of equilibrium of many forces acting on a
particle which is in contact with a smooth surface or a smooth
curve.
36.] Let us first take the case of a smooth surface, and
suppose a particle acted on by many forces to be in contact with
it at a given point. As the surface is smooth, the only direc-
tion along which it can offer any resistance to the particle's
motion is that of its normal ; and as it is conceived to have no
active power of its own, but only a capacity of resisting any
force that acts against it along its normal, so must the resultant
of the impressed forces act along the normal and towards the
surface : these conditions therefore are sufficient for the equi-
librium of the particle.
Let the equation to the surface be
F(*,y, = 0; (70)
and employing the same notation as in Art. 332, Vol. I. (Dif-
ferential Calculus), and Art. 236, Vol. II. (Integral Calculus),
let
so that if A, \L, v are the direction-cosines of the normal at (x,y, z),
u v w
cos A = , cos u = - . cos v = j
Q Q Q
then as this line is to be coincident with the line of action of
the resultant of the acting forces, whose direction-cosines are
proportional to x, Y, z, the conditions of equilibrium are
5 = I = ; (72)
U V W
and if these equations are not, and cannot be, satisfied, equi-
librium on the surface cannot exist. Consequently the point on
a given surface, at which a particle under the action of given
forces will rest in equilibrium, is the point on the surface at
which the preceding equations are satisfied.
The normal pressure of the surface, which arises from the
action of the impressed forces, may thus be determined. Let
38 EQUILIBRIUM OP FORCES. [36.
N represent the normal pressure ; then the resolved parts of it
along the coordinate axes are
u v w
N-, N-, N j
Q Q Q
and these together with the acting forces must be in equili-
brium : therefore
u
2.P cos a = x = N - ,
"
v
2.Pcosy3 = Y = N-, }
TW
W
s.pcosy = z = N :
Q J
whence, squaring and adding,
If the normal resistance of the surface acts in only one direction,
the resultant of the acting forces must act in the direction
opposite to that of the resistance. We subjoin some examples
of the preceding formulae.
Ex. 1. A particle is placed on the surface of an ellipsoid and
is acted on by attracting forces which vary directly as the dis-
tance of the particle from the principal planes of section ; it is
required to determine the position of equilibrium.
Let the equation to the ellipsoid be
2x 2y
TT -IT - _ ;. -117
~^ > TI '
*
2z
_
,,2
a 3 o* c"
let x = JM?, Y = \i t y, z = n
then equations (72) become
Mi Ma M 3 Mi + Mz + Ma .
if these conditions are fulfilled, the particle will rest at all points
of the surface.
Ex. 2. Again, take the same surface, and let the forces vary
inversely as the distances of the point from the principal planes :
it is required to determine the position of equilibrium.
x -- i Y - Ma 7 - ***
' *' 7' ~z~'
36.] A CONSTRAINED PARTICLE.
therefore (72) become
ar* y* z*
~ ~ ~
1 1
= -, (say);
z
x - a-\ , y = d-) , 2 = c
\ / \ /
- i . L
' x* y* z*
Ex. 3. A heavy particle is placed inside a smooth sphere on
the concave surface, and is acted on by a repulsive force varying
inversely as the square of the distance from the lowest point of
the sphere : find the position of rest of the particle.
Let the lowest point of the sphere be taken for the origin,
and let the axis of z be vertical; then the equation of the
sphere, whose radius is a, is
Let w = the weight of the particle, and r = the distance of it
from the lowest point ; then
= 2az.
Also let the repulsive force = -- = ;
. x = ** X Y = * y
2az r ' 2az r' 2az r
Let N = the normal pressure of the curve ; then
_M_* _ N * Ji_^_ N ^
2az r a' 2az r a'
u. z z a
-^ w = N ;
2az r a
from which we have
whence the position of the particle is known for a given weight
of it, and for a given value of jz.
If another force of the same kind, and in which /i is replaced
40 EQUILIBRIUM OF FORCES. [37.
by \L ', makes the particle to rest at a distance / from the lowest
point, then , r ,
1/3 J_ . ' :
/ >. f, '
w pi r 3
that is, the absolute values of the repulsive forces at an unit-
distance vary as the cubes of the distances from the lowest point
of their positions of rest.
37.] Next let us consider the circumstances of pressure of a
particle resting-, or (to fix our thoughts) of a small ring- sliding,
on a given curved line which is smooth and offers no resistance
to motion along itself.
As the curve is smooth, the resultant of the impressed forces
is manifestly perpendicular to the tangent of the curve at the
point of equilibrium ; therefore if the curve is of double curva-
ture, so that the direction-cosines of its tangent are proportional
to dx, dy } dz, the required condition is
Jidx + ^dy + zdz = 0; (73)
and if N is the normal pressure, and A, /z, v are the direction-
angles of its line of action,
N cos X = x, N cos fj. = Y, N cos v = z; (74)
N a = x 2 + Y 2 + z a ; (75)
whence N, A, p, v are known. If the equation (73) cannot be
satisfied at any point of the curve, equilibrium is impossible ;
and if the forces are given, the point, at which equilibrium takes
place, may be determined by means of (73) and the equations
to the curve.
If the curve is a plane curve, (73) becomes
Kdx + idy = 0. (76)
And if F (x, y) = is the equation to the curve, this may be
expressed in the form
X Y
x^J\ " " xfl?Fx '
\~dx) \dy)
Also (75) becomes
N 2 = x 2 + Y 2 . (78)
Ex. 1. A ring is capable of sliding on a smooth helix, and is
acted on by a constant force perpendicular to the axis; shew
that equilibrium is impossible, unless the force parallel to the
axis of is zero.
37-] A CONSTRAINED PARTICLE. 41
The equations to the helix are
y = a sin <, dy = xdQ,
z = ka$\ dz = kad<\>;
and if pa is the constant force which acts towards and perpen-
dicular to the axis,
x=-M-r, Y=-^;
and therefore substituting in (73), we have
nxypxy + zka = 0;
which can be satisfied only when z = 0.
Ex. 2. A small ring, capable of sliding on a smooth ellipse,
whose equation is x * yt
^~ + F = l '
is acted on by forces parallel to the axes of x and y represented
by nx n and py n ; find the position of equilibrium.
In this case (77) becomes
a*z n ~ l fry"-*-,
n+l f n+l n+l-v i
> - I o - 72 - I TS
.'. X = abn-i la n-l+d n-lt ;
and a similar value may be found for p.
Ex. 3. Two weights P and Q are fastened to the ends of a
string, fig. 14, which passes over a pulley o; and Q hangs freely
when P rests on a plane curve AP in a vertical plane ; it is re-
quired to find the position of rest when the curve is given.
The forces which act on P are, (1) the tension of the string
in the line OP, and which is equal to the weight of Q, (2) the
weight of P acting vertically downwards, (3) the normal reaction
of the curve, viz. R.
Let F (x, y] c be the equation to the plane curve, o being
the origin, and the axis of x being vertical. Let OM = #, MP=y,
OP = r, POM = 6, OA = a. Then
dy dx
x = P Qcos0 R-j-} Y = Qsm0 + R-T-;
therefore from (76),
(P Q cos 6} dx Q sin Q dy = 0,
xdx + ydy
vdx Q -- J J = 0;
v
PRICE, VOL. III. G
42 EQUILIBRIUM OP FORCES. [37-
but since ar +y 8 = r" ; . . xdx+ydy = rdr-,
.-. vdx-qdr = 0; (79)
and this condition must be satisfied by P, Q, and the equation to
the curve. Also
R* = P 2 2PQcos0 + Q 2 . (80)
(1) Let the curve AP be a hyperbola of which o is the centre;
then ^. IP
r* = x*+y* = e^x* 6*;
W*xdx = 0;
Jp
(2) Let it be required to find the equation to the curve, on all
points of which P will rest. In this case (79) must be satisfied
at all points of the curve ; therefore
Par Qr = a constant
if the curve passes through A, and OA = a ; therefore
(1 - *-)a
> ^
p
1 -- cos Q
Q
(81)
which is the equation to a conic section, of which the focus is
the pole ; and is an ellipse, parabola, or hyperbola, according as
p is less than, equal to, or greater than, Q.
(3) Let the curve be a circular quadrant, convex downwards,
with a horizontal radius passing through o, which is also a
point on the circle, and let P = 2 Q ; then the equation to the
circle is, if a is the radius,
r = 2 a sin 6 ;
and therefore (79) becomes
4(cos0) 2 cos 2 = 0;
whence Q may be determined.
(4) Another form of the problem is, The length of the string
being given, and Q always resting on a given curve, to find the
curve on which p shall rest in all positions.
Let the tension of the string be equal to T, and let / and ^
38.] A CONSTRAINED PARTICLE. 43
refer to the curve BQ, fig. 15, on which q rests, and of which let
the equation be / _ f(tf\ (82)
where f is the symbol of a known function : then we have
from (79) Q<&'-T<// = 0,
also ndxfdr = 0;
and since r + / = 2c = length of the string; (83)
.-. dr+d/ = 0; .-. qdaf + vdat 0; (84)
and by means of (82), (83), and (84), / and & are to be elimi-
nated, and the resulting equation in terms of r and Q will be
that required.
Let the curve on the left-hand side in the diagram be a
parabola of which o is the focus ; then
1 cosw
and from (84), qaf + Px = 2/fcQ,
where k is an arbitrary constant; therefore from (85),
1 -- COS0
Q
(86)
which is the equation to a conic section of which the focus is o.
38.] In review of the preceding results it appears that, (1) if
the particle on which certain forces act is entirely free, so that
three variables are independent, the forces must satisfy three
conditions ; (2) if the particle is constrained to be on a given
surface, there are two equations of equilibrium; and (3) only
one condition is requisite, when the particle is on a given curve.
That is, if a particle is entirely unconstrained it has three
degrees of freedom ; if it is constrained to a given surface it has
only two degrees of freedom, one degree being lost because the
particle cannot move in the line of the normal to the surface ;
and if it is constrained to a given curve, it has only one degree
of freedom, as it can move from an assigned point in the direc-
tion of the tangent of the curve, and along that line only.
o 2
CHAPTER III.
COMPOSITION AND RESOLUTION OF STATICAL FORCES
ACTING ON A RIGID BODY.
SECTION 1. Composition of two forces acting on a rigid body
in one plane.
39.] Before we enter on the formal inquiry into the mode
and results of the composition of forces acting on a rigid body
it is necessary to explain some properties of such bodies, with
the view of obtaining a principle which is necessary to the
discussion.
A rigid body is such that its component particles are in a
state of relative rest by the action of unknown molecular forces,
such as attractions, cohesions, &c. : and the intensity of these
forces is so great, that the relative equilibrium of the particles,
which is due to them, is not disturbed by the forces which act
on the body.
When a force acts at a definite point of a body and along a
definite line it produces a pressure of the particle on which it
acts against the contiguous particle in the line of its action, and
from the contiguous particle in the opposite direction : and this
pressure on these particles, although infinitesimal in comparison
of the molecular forces, is propagated from one particle to an-
other along the whole line of action of the acting force ; and is
the same at all points in this line. Hence we infer that the effect
of a force on a rigid body, acting in a definite line, is unaltered,
whatever is the point in its line of action at which it is applied.
This principle is called that of Transmissibility of Pressure, and
the truth of it depends on the rigidity of the body which in-
volves such a mode of action as that described above.
Now two equal forces acting on a particle in the same line
and in opposite directions neutralize each other ; and this pro-
perty may be extended by means of the preceding principle, so
that, Two equal forces acting in the same line and in opposite
directions at any points of a rigid body in that line neutralize
each other. Hence \ve infer, that when many forces are acting
40.] MOMENT OF A FORCE. 45
on a rigid body, any two which are equal and have the same
line of action and act in opposite directions may be omitted;
and similarly the introduction of two equal forces along the
same line of action and in opposite directions does not change
the circumstances of the system as to resultant pressure.
The effects of the forces which have been considered in the
preceding chapter are a tendency to motion in a given straight
line, and, so far as we have considered them, along that straight
line only : these are called pressures or forces of translation. But
suppose a point o, fig. 1 6, of a rigid body to be fixed, so that
there cannot be any motion of translation of the whole body ;
and suppose a force p to act on the body at a definite point M in
the line MP ; join OM, and resolve P into two parts, one along,
and the other perpendicular to, OM; then the part along OM
produces a pressure at o, which being fixed is capable of bearing
it without the body having thereby any tendency to motion:
but the other component causes a pressure on M in a direction
at right angles to OM ; but as o is fixed, M can only describe a
circle about o as the centre ; the effect therefore of this latter
component is a tendency to circular motion of M, or, as it is
commonly called, to rotation about O; a force producing such
an effect is called a pressure or force of rotation about or in
reference to a given point ; and we have now to consider these,
their measures, and their laws at length, and fully discuss them.
Single particles are subject to forces of translation, but, having
neither magnitude nor parts, not to pressures of rotation.
40.] Composition of two forces acting at definite points on a
rigid body in one plane.
Let the two forces be P and Q, and let them act in the plane
of the paper at the points A and B, fig. 1 7 ; join AB, and let us
assume that the lines of action of P and Q are not parallel ; let
the angles between AB and the lines of action of P and Q be
respectively a and ft ; produce the lines of action to meet in O,
o being supposed to be in the rigid body or to be rigidly con-
nected with it ; then by virtue of the principle of transmissibility,
we may suppose P and Q to be applied at o. Let R be the
resultant of them so transferred, and let the line of action of R
intersect AB in the point c ; then we have to determine the
magnitude of R, its line of action, and a point in that line ; these
last two will be conveniently known, if we find AC, and the
angle between A B and co.
46 MOMENT OF A FORCE. [41.
Let AC = x, CB = y, AB = a; .*. x+y = a;
OAB = a, OBA = /3, OCB = Q \
then, by the parallelogram of forces,
R* = p' 2PQcos(a + /3) + Q 2 ; (1)
whereby the magnitude of the resultant is known. And re-
solving P, Q, R at o along lines through o, parallel, and perpen-
dicular to, AB, we have
RCOS0 = PCOSa QCOS& ) ,<.
R sin0 = Qsin/3 + Psin a; j
Qsin/3 + Psina. ( }
.. tan0 = - -t (3)
p cos a Q cos /3
and by reason of equations (30) Art. 21,
(4)
sin (^ + ^3) sin (6 a) sin(a
Let p and q be the lengths of the perpendiculars on the lines
of action of P and Q from any point in the line of action of R,
say, from c ; then
p = co sin (0- a),) , g)
q = cosin(0 + /3);J
therefore from the first two terms of (4),
pj> = q ? ; (6)
and therefore, since p = x sin a, q = y sin /3,
P#sina = Qysin^S, (7)
Qsin/J Psina Psina + Qsin/8
= --j (8)
Rsm0
whereby x and y are given in terms of known quantities : the
magnitude, line of action, and point of application on the line
AB of the resultant are therefore determined.
41.] The equation (6) requires especial consideration with
reference to the properties of moments which have been men-
tioned in the previous chapters ; two forces, P and Q, act on the
body, each of which alone produces a pressure of translation
along its line of action : but the resultant of the two taken in
combination is a single force R, the position of whose line of
action is given by (3) ; a force therefore equal to R, along the
same line of action, and opposite in direction to R, will with
41.] MOMENT OF A FORCE. 47
p and Q produce equilibrium. Now this force may be applied
at any point in the line of action of n ; let c be the point of
application ; and thus the system is in equilibrium, and is as if
c were a fixed point. Let us consider this in the light of the
remarks of Art. 39 ; P and Q severally produce a pressure of
rotation about c, and manifestly in opposite directions; and
they neutralize each other, for the body is at rest : therefore
their rotatory eifects are equal. But what relation exists between
them ? because we may thence infer a measure of their rotatory
effects with reference to the point or centre c. P and q, balance
when (6) is satisfied ; that is, the rotatory effect due to one force
is equal to, and neutralized by, that due to the other, when the
products of the force and the perpendicular distance from c on
its line of action are equal. This product therefore may be
taken as the measure of the rotatory effect of a force. And as
it is desirable to have a distinctive name for such an effect, it is
called a force's moment ; and therefore we define as follows :
DBF. Moment of a force with reference to a given point is the
rotatory effect of it with reference to that point ; and is measured
by the product of the numbers which represent the force and
the perpendicular distance from the point on the line of action
of the force. This is the algebraical measure of the moment.
Two forces are said to be equimomental with respect to a point
when their moments with respect to that point are equal.
As the forces act in one plane we have spoken of the moments
with respect to a point : it is more correct to say, with respect
to an axis passing through the point and perpendicular to the
plane in which the forces act, because it is about this line that
the forces per se, and all other things neglected, tend to make
the body turn. However, when the body, on which the forces
act, moves, we shall have a modification of this statement.
A force may tend to make a body turn about an axis in either
one or the other of two directions ; it is necessary therefore to
distinguish these, and to affect them with different signs : let
therefore the moment of a force be positive if it tends to turn
a body from right to left, that is, in the direction in which the
hands of a clock revolve, when it is opposite to us; and let
the moment of a force be negative, when it turns a body in the
opposite direction.
As the moment of a force in reference to a point is the product
of the perpendicular from that point on the line-representative
48 MOMENT OF A FORCE. [42.
of the force and that line-representative, its geometrical repre-
sentative, as we have observed in Art. 28, is twice the area of
the triangle, of which the given point is the vertex, and the
line-representative is the base. Hence as properties of forces of
translation have their geometrical analogues in lines, so properties
of moments are translated directly geometrically into theorems
concerning areas. We shall however see hereafter that moments
are also frequently represented by lines whose lengths are pro-
portionals to the moments.
Moments of forces, being quantities measurable by number,
are capable of addition and subtraction. Thus if three forces
are proportional to, and act along, the sides of a plane triangle
in the same direction, as to translation they neutralize each
other, and the result is zero. But as to rotation, the resultant
moment with reference to any point in the plane of the triangle
is equal to twice the area of the triangle.
42.] Let us return to equation (7), and consider c as a point
at rest, by means of the force R acting on it which is in equi-
librium with p and Q : then resolving p and Q, along and per-
pendicular to AB, we have P sin a and Q sin /3 perpendicular to
AB, and pcosa and qcos/3 along AB : these latter forces pro-
duce a pressure on c which is equal to their difference ; but the
former components produce a rotatory pressure about c, and
equilibrate when the moments of the two are equal, that is,
wn en #P sin a = y Q sin /3 ;
and this is equation (7).
Again, suppose that the components are P t and P 4 , and that
the equations to their lines of action are given ; and let it be
required to find that of the line of action of the resultant n.
Let the equations to the lines of action of the components be
x cos a t +y sin a t p l = = a l} ) , g .
x cos a a -\-y sin a 2 /> = = a 2 , )
ttj and a a being symbols of notation for the left-hand members
of the equations : then, if x and y refer to any point in the line
of action of the resultant, by equation (6) we have
P^j + Pjo, = 0;
. ' . (p t cos ttj + P 2 cos a 2 ) x + (P! sin a! + P., sin a^)y
-/iPi-j,P, = 0; (10)
which is the equation to the line of action of R.
43-] MOMENT OF A FORCE. 49
Hence if r is the perpendicular from the origin on the line
of action of R,
J cos dj -f P a cos a 2 ) a + (P! sin a t + P, sin a,) 8
.-. Rr =^ 1 P 1 +j0 2 P i ; (11)
that is, the moment of the resultant is equal to the sum of the
moments of the components.
43.] Let us consider the subject from another point of view,
and take two forces, whose lines of action are parallel, acting in
the same direction on a rigid body.
Let P, Q, be the two parallel forces acting at A and B, fig. 1 8 :
join AB, and let a be the angle between AB and the lines of action
of P and Q ; at A and B introduce two equal forces s and s which
act along AB, and in opposite directions : the circumstances of
pressure are not hereby altered. Let P' be the resultant of P
and s at A, and Q' the resultant of Q, and s at B ; let the lines
of action of P' and Q,' be produced to meet in o, o being sup-
posed to be rigidly connected with the body : at o resolve P'
and Q,' into the forces of which they were compounded; the
components along the line parallel to AB manifestly cancel each
other, and there remains P + Q acting in a line parallel to the
lines of action of P and Q. Let this resultant be R, so that
R = P + Q; (12)
that is, the resultant is the sum of the two parallel forces.
Let AC = x, CB = y, AB = a ; therefore x +y a ; then P' is
the resultant of P and s, and these pressures are parallel to the
sides of the triangle ACO ;
s P . ., . s Q .
.-. - = : similarly - = ;
x co y co
.-. P# = Q^. (13)
Let p and q be the perpendicular distances from c on the
lines of action of P and Q, : then p x sin a, q = y sin a, and
thus (13) becomes
PjO = Q0;
that is, the moments of P and Q about c are equal.
PRICE, VOL. III. H
50 MOMENT OP A FORCE. [44.
Again, from (13),
* = y = x y. == -', (15)
Q P P+Q R
whence x and y are known ; and are reciprocally proportional to
the forces at their extremities. Hence also when three parallel
forces are in equilibrium, each is proportional to the distance
between the action-lines of the other two.
If P = Q, y = x = ~, R=2p;
that is, the resultant is equal to twice one of the forces, and is
applied at the point of bisection of the line joining the points
of application of the forces.
As (14) is independent of the angle between AB and the direc-
tion of the forces, c is the same whatever that angle is ; c is for
this reason called the centre of the two parallel forces.
44.] Suppose one of the parallel forces of the preceding
Article to act in a direction contrary to that of the other : then
fig. 19, introducing as before two equal forces s, s acting along
AB and in opposite directions, and compounding P and s into P',
and Q, and s into Q', let us suppose the lines of action of P' and
Q' to meet at o, o being rigidly connected with the body ; and
at o let P' and Q' be resolved into the forces of which they were
compounded; the forces parallel to the line AB cancel each
other, and there remain P and Q acting in a line parallel to the
original lines of action of P and Q, the resultant of which is equal
to their difference : let us suppose Q to be the greater, then
R = Q p. (16)
Let AB = a, AC = x, BC = y ; therefore x y = a; and let a
be the angle ,between AB and the lines of action of P and Q.
Since P' is the resultant of P and Q,
s P . ., , s Q,
= : similarly = ,
x co J y co
.-. rx = Qjr. (17)
Let p and q be the perpendicular distances from c on the
lines of action of P and q ; then p x sin a, q y sin a ; there-
fore (17) becomes
P^ = Q2; (18)
that is, the moments of p and Q about c, and similarly about
every point in the line of action of R, are equal.
46.] COUPLES, AND THEIR MOMENTS. 51
Again, from (17)
x _y _ x-y _ a m .
= = , \ iy )
Q P Q P R
whence x and y are known, and are reciprocally proportional to
the forces acting- at their extremities.
This theorem of the equality of moments, whether of parallel
forces as I have demonstrated in this and the preceding articles,
or of forces whose lines of action are not parallel, has been called
the principle of the lever, and has been by many writers on
mechanics made fundamental; and other mechanical theorems,
including that of the parallelogram of forces, have been derived
from it. I, on the other hand, have derived the equality of
moments from the parallelogram of forces, in the conviction
that the latter proposition is more simple, and that the former
follows more directly from it. The immediate application of
the theorem is so easy, that it is unnecessary to insert examples
at this stage of the work.
45.] The equation to the line of action of the resultant of two
parallel forces PJ and p a may be determined as follows :
Let the equations to the lines of actions of the components be
arcosa + ysina 8 X = = a l} *
#cosa+y sin a 8., = = a 3 ; 5
therefore by (14) or (18) the equation to the line of action of the
resultant is
(p 1 + p a );rcosa + (PI+P,) y sin a (8 t Fi + 8, P,) = ;
that is, since PJ + p a = E,
tfRcosa+yRsina (StPi-f-SjPa) = 0. (21)
If P! + p a = ; that is, if the forces are equal and act in
opposite directions, then
(8 1 -8 2 )P 1 = 0, (22)
which is the equation to a straight line at an infinite distance ;
consequently the resultant of two equal and opposite forces acts
at an infinite distance.
SECTION 2. On couples their laws and composition.
46.] These results arising from the simultaneous action of
two equal forces, working in opposite directions along two
H 2
52 COUPLES, AND THEIR MOMENTS. [46.
parallel straight lines which are at a finite distance apart, require
closer consideration ; for they open to us a series of theorems
in themselves and in their inferences of very great use in the
simplification of mechanical propositions. It is indeed on these
theorems that a large and distinct part of our subject has been
raised ; and it is consequently necessary to investigate them at
considerable length. I will start from the results of Art. 43
which refer to the composition of two unequal forces p and Q,
which act in opposite directions along parallel straight lines, and
I will suppose Q to be the larger of the two ; let us suppose the
difference between Q and P gradually to diminish, and Q ulti-
mately to become equal to P ; then R becomes less ; and x becomes
greater; and ultimately, when Q=P, R=0, and x-=.y=.<x> ; that
is, there is no single force of translation which will be equivalent
to such a pair of forces ; and therefore there is no one force of
translation which will be in equilibrium with them. It is also
by the principle of sufficient reason manifest that such a system
cannot have a single resultant of translation; because such a
resultant is unique; and whatever is the process of reasoning by
which its line of action is assigned in respect of one of the
forces, by the same will it be assigned in a similar position with
respect to the other force.
Such a pair of forces, equal and acting in parallel lines and in
opposite directions, is called a couple* ; its effect is evidently a
pressure of rotation about a line perpendicular to the plane in
which the forces act, and which line is called the axis of the
couple. Now in statics, as the motion is only virtual and not
actua^ the direction of the axis is fixed, but not the position of it ;
it is some line perpendicular to the plane in which the forces act.
If motion takes place the position of the axis, as well as its di-
rection, becomes fixed, as we shall see hereafter. If the axes of
couples are parallel, that is, if the planes of these forces are
parallel, the couples are coaxal.
The perpendicular distance between the lines of action of the
forces is called the arm of the couple.
The rotatory effect of a couple is called the moment of the
couple. In estimating its measure we must examine all possible
positions of the axis. Let the couple be that indicated in fig. 20 ;
* See Poinsot, " Me"moire sur la composition des Moments et des Aires dans la
Mecanique." The tract is appended to "Elements de Statique" of the same
author, 8n edition, Paris, 1842.
47.] THEOREMS ON THE TRANSFERENCE OF COUPLES. 53
and (1) let us suppose the axis to pierce the plane of the couple
at the point o which lies between the forces ; then
the moment of the couple = PXOA + PXOB
= PXAB. (23)
(2) Suppose the axis to pass through A, one of the extremities of
the arm : then the force which acts at A produces no pressure of
rotation, and we have
the moment of the couple = P x AB. (24)
(3) Suppose the axis to pierce the plane of the couple at a point
o, fig. 21, in the arm produced : then
the moment of the couple = PXOB PXOA
= PXAB. (25)
In all cases therefore the moment of the couple is equal to the
product of the numbers expressing the force and the length of
the arm. Thus if the force contains 6 units of pressure, and the
arm 3 units of linear length, the moment of the couple is ex-
pressed by 1 8 ; that is,
the moment of couple = the force x the length of the arm. (26)
A couple may evidently tend to make a body revolve in either
one or the other of two opposite directions ; that is, in the di-
rection of the hands of a watch, as we face it, or in the opposite
direction ; and it is desirable to affect these different directions
with different signs ; for the present, let the former be positive
or right-handed couples, and the latter, negative or left-handed
couples. In figs. 20 and 21 right-handed couples are repre-
sented.
Two couples whose moments are equal are said to be equi-
momental.
The forces applied in turning the handle of a corkscrew, of
a gimlet and of an auger, are familiar instances of couples.
47.] The following three theorems concern the transference
of couples :
THEOREM I. The effect of a couple on a rigid body is not
altered, if the length of the arm and the force being the same,
the arm is turned about its extremity through any angle in the
plane of the couple.
Let AB, fig. 22, be the arm of the original couple, and P, P its
forces ; through A draw any straight line AB' in the plane of the
couple equal to AB, and at A and B' respectively introduce in the
54 THEOREMS ON THE TRANSFERENCE OP COUPLES. [47.
plane of the couple two forces equal to P, with their lines of
action perpendicular to the arm AB', and opposite in direction to
each other ; then the original circumstances of pressure are not
altered by the introduction of these forces. Let B AB'= 2 6 ; then
the resultant of P acting at B, and of P acting at B', whose lines
of action meet at Q, is 2 p sin Q, and acts along the line A q :
similarly the resultant of P acting at A perpendicularly to AB, and
of P perpendicularly to AB', is 2p sin 6, and acts along the line A Q
in a direction opposite to that of the former resultant : these
two resultants therefore neutralize each other, and there remains
the couple whose arm is AB' and the forces P, P : and this is
equimomental with the original couple and replaces it, and con-
sequently the theorem is true.
THEOEEM II. The effect of a couple on a rigid body is not
altered, if the plane of the forces is transferred to any other
parallel plane, the arm being parallel to its original line, and of
an equal length, and the forces being unaltered in magnitude.
Let AB, fig. 23, be the arm, and P, P the forces of the given
couple : let A'B' be an arm equal and parallel to AB ; at A' and
B' respectively introduce two forces equal to P, acting perpen-
dicularly to A'B', and in opposite directions, and in a plane
parallel to the plane of the original couple : the original circum-
stances of pressure are not altered by the introduction of these
new forces. Join AB', A'B ; these lines evidently intersect and
bisect each other in o ; then P at A and P at B', acting in parallel
lines and in the same direction, are equivalent to a force 2p
acting at o : similarly p at B and P at A', acting in parallel lines
and in the same direction, are equivalent to 2p acting at o in a
line parallel to their original lines of action : at o therefore these
two resultants, being equal and opposite, neutralize each other;
and there remains the couple whose arm is A'B', and whose forces
are p, p, acting ip. the same direction as those of the original
couple, in a parallel plane, and with an equal arm : it is there-
fore coaxal and equimomental, and may equivalently replace the
original couple.
The proof which is here given for a parallel plane is of course
valid for the less general case of the same plane : and there-
fore from this and Theorem I. we infer, that the effect of a
couple on a rigid body is not changed whatever is the position
of its plane, if the direction of the axis is unaltered, and the arm
and the forces are equal.
48.] THE COMPOSITION OP COUPLES. 55
THEOREM III. The effect of a couple on a rigid body is not
altered, whatever is the position of its plane, arm, and force,
provided that its axis and moment are unaltered.
In fig. 24, let AB be the arm, and P, p the forces of the given
couple ; at A and B introduce any equal forces s and s acting
along AB and in opposite directions. Let p' be the resultant of
p and s at A, and let P' also be the resultant of P and s at B : the
lines of action of P' and P' are of course parallel ; produce P'A
backwards, and from B draw BA' perpendicular to A A' : then the
forces p 7 and P" form a couple whose arm is BA', and each of
whose forces is P'; let B A A' = ; then A'B = AB sin ; p' = p cosec 6 ;
s = P'COS Q = P cot Q ; and
the moment of the new couple = P' x A'B
= P cosec 0xAB sin
= PXAB
= the moment of the original couple. (27)
It will be observed that s is arbitrary, and that 6 and con-
sequently the length of the new arm, as also the force of the
new couple, depend on it : consequently they are also arbitrary ;
but they are subject to the condition (27), which requires the
new couple to be equimomental with the original one. And
thus it appears that a couple is equivalent to, and may be re-
placed by, another couple, of which the moment is the same,
the forces are in the same plane, and the arms have a common
extremity.
Combining this theorem with the preceding, we conclude
that a couple is equivalent to, and may be replaced by, any other
equimomental and coaxal couple.
48.] Now in all these transformations, the axis of the couple,
that is, the direction of the line about which the couple tends
to make the body rotate, has not been altered ; the arm and the
force have been altered in position, in length, in magnitude;
and the plane in which the forces act has been changed from
any one into any other parallel plane ; but the normal to the
plane, which is the axis, has continued unaltered ; and the
moment has continued the same ; and these quantities cannot
be changed without changing the effect of the couple ; the
former of these then has a fixed direction, and the latter is a fixed
quantity. It is convenient, as of forces of translation, so of
these forces of rotation, to have geometrical lengths as adequate
56 THE COMPOSITION OF COUPLES. [49.
representatives ; and such we shall obtain, if along the axis we
take lengths containing the same number of linear units as the
moment of the couple contains units of pressure. Thus if the
force of a couple is 4 and the length of the arm is 3, the mo-
ment is represented by the number 1 2 ; and if along the axis
1 2 linear units are measured, this length is a full and adequate
representative of the couple ; and moreover as couples may be
right-handed or left-handed, that is, have positive or negative
signs, so from a fixed point (the origin) on the axis may the
line be taken in one or the other direction, and thus indicate
the sign of the couple. Now this line is technically called the
axis of the couple, the word being used in a sense different to
the former one : there it indicated line of rotation only ; here it
indicates three things, viz. the line of rotation, a finite length
of that line measured from a given point on it, and the direction
in which it is measured. This axis therefore fully determines
all the circumstances of the couple. Some confusion may arise
from the ambiguous use of the word, and therefore I shall
always take care to specify axis as to rotation, and axis as to
rotation and moment, by calling the former rotation-axis, and
the latter moment-axis, bearing in mind however that the latter
is indicative of direction as well as the former ; and when couples
are said to be coaxal, it is with respect to the former meaning
of the word only ; and when two couples are statically equi-
valent they are coaxal and equimomental.
49.] The following theorems concern the composition of
couples :
THEOREM IV. The resultant of many coaxal couples is a coaxal
couple whose moment is equal to the algebraical sum of the mo-
ments of the component couples.
Let the forces of the several couples be ? P 2 , . . . p n ; and the
lengths of the arms jo u p,, . . . j n ; so that their moments are
p, p 1} P a jj, ... P W JO H . Let all, by virtue of Theorem II, be trans-
ferred to the same plane, and let all the arms have a common
extremity ; again, by virtue of Theorem III, let all be trans-
formed into equivalent couples with arms of the same length,
equal to r, and let the forces thereby changed be P/, p/, ...?';
so that
PI> = PI p lt P a V = P a p a , P n V = P^ H ; (28)
and lastly, by virtue of Theorem I, let all the arms be turned
about their common extremity, and become coincident; then
5O.] THE COMPOSITION" OF COUPLES. 57
the length of it is r, and at each extremity there are equal and
opposite forces, of which let the sum be R, where
R = P/+P/+ ...P.'; (29)
so that the moment of the resultant couple is
Rr = p/r -f P/r + . . + P B V
= *ipi+v,p, + ...+vJ> m
= 2.PJ0; (30)
that is, the moment of the resultant couple is equal to the sum
of the moments of the several component couples.
If some of the couples are negative, the forces belonging to
them will in (29) have negative signs, and R will be equal to
the difference of the forces which have positive signs and of
those which have negative signs : and the same result will
appear in (30), so that the right-hand member denotes the alge-
braical sum.
The moment-axis of the resultant is equal to the sum of the
moment-axes of the component couples.
Two equimomental and coaxal couples acting in opposite di-
rections evidently neutralize each other.
A close analogy exists between parallel forces of translation
applied at the same point and coaxal couples : in either case the
effect of the resultant is equal to the algebraical sum of the
effects of the components. We shall trace this analogy further
in the succeeding Article. As to the geometrical representatives
of the effects, in the case of couples the moment-axis may be
transferred parallel to itself in any manner ; in the case of forces
of translation, the representative line can, by the principle of
transmissibility, be transferred only along its own line of action.
50.] THEOREM V. If two lines meeting at a point represent
the moment-axes of two couples, the diagonal of the parallelo-
gram originating at the same point, and of which the two lines
are adjacent sides, will represent the moment-axis of a single
equivalent couple.
Suppose two couples to act in planes which are inclined to
each other at an angle y ; let the couples be transferred in their
own planes so as to have the same arm lying along the line of
intersection of the two planes ; let the forces of the couples thus
transferred be P and Q. And, fig. 25, let AB be the common
arm, and let us suppose it to lie in the plane of the paper : then
PRICE, VOL. III. I
58 THE COMPOSITION OF COUPLES. [51.
compounding p and Q at A into a single force R, and p and Q at
B in the same way, since PAQ = y, we have
R 2 = p a + 2PQCOsy-f Q; (31)
and the R at B is eqiial and parallel to the B, at A. At A draw
A0, A.6 perpendicular respectively to the planes PBAP, QBAQ,
and of lengths equal to the moment-axes of the couples ; com-
plete the parallelogram A.act>, and draw the diagonal AC ; then
AC is the moment-axis of the resultant couple whose arm is AB
and whose force is R. For since A=PXAB, and A = QXAB,
therefore A a and A.6 are proportional to p and Q, that is, to AP and
AQ; and they are also perpendicular to these lines, and are in
the same plane with them ; therefore the diagonal AC is perpen-
dicular, and proportional in the same ratio, to AR; therefore
AC = R x AB, and is the moment-axis of the resultant couple.
Therefore, if A and A.& are the moment-axes of two couples, AC
the diagonal of the parallelogram of which A and A are the
two adjacent sides is the moment-axis of the resultant couple.
Hence if L and M are the moment-axes of two couples, and are
inclined to each other at an angle y, and if G is the moment-
axis of the resultant couple,
G a = L s + 2LMCOSy + M*. (32)
Attention must of course be paid to the direction of the couple ;
thus, if A a is the moment-axis, to an eye placed at A and look-
ing along A0, the couple is right-handed.
Hereby also we are authorized to resolve a couple whose mo-
ment-axis is given into any two couples, such that their moment-
axes are the sides of the parallelogram of which the given mo-
ment-axis is the diagonal. And the number of ways in which
such resolution can be effected is infinite.
51.] If the moment-axes of two couples are perpendicular to
each other, then y = 90 ; and
G* = L S + M*; (33)
if X is the angle between the rotation -axes of G and L, then
L = GcosA, M = osinA, (34)
tan A = -; (35)
L
a couple therefore whose moment-axis is G may be resolved into
any two couples such that their moment-axes are the sides of
the rectangle whose diagonal is the given moment-axis.
52.] THE COMPOSITION OF COUPLES. 59
Hence also a couple, whose moment-axis is equal to o, but is
in an opposite direction, neutralizes L and M, and the whole
system is in equilibrium.
Also from (32) by a process analogous to that of Article 21
we can shew that if, fig. 26, OL, OM, ON represent the moment-
axes of three couples L, M, N ; and if MON = CI, NOL=/3, LOM = y,
and if
L M N
sin a sin /3 sin y '
then the three couples are in equilibrium ; and conversely, if
three couples are in equilibrium, the moment-axis of each is
proportional to the sine of the angle contained between the
rotation-axes of the other two.
Hence also if many couples acting on a rigid body are in
equilibrium, their rotation-axes are parallel to the sides of a
closed polygon, the sides themselves being the moment-axes.
And finally we conclude that couples may by means of their
moment-axes, which are their geometrical representatives, be
resolved and compounded according to the same laws as forces
of translation by means of their equivalent lines of action. And
whatever is true of pressures of translation is also true, mutatis
mutandis, of pressures of rotation as exhibited by the moment-
axes of the couples which are their geometrical representatives.
52.] The analogy which has been traced between the moment-
axes of couples and the line-representatives of the forces of
translation also holds good when there are many couples of
which the moment-axes are not all parallel and are not all in
one plane. And to take the most general case, let us consider
the composition of couples whose rotation-axes have any position
in space.
Take any point o in space for an origin of coordinate-axes,
and at it let three straight lines originate, forming a system of
rectangular axes.
Let the axis of every component couple be shifted, and pass
through o, and let the moment-axis of each component couple
be resolved into two moment-axes, one of which coincides with
the .z-axis, and the other lies in the plane of (x, y] ; also let this
latter moment-axis be resolved into two others which coincide
with the axes of x and y respectively ; then when every com-
ponent couple has been resolved in this way, we have three
series of coaxal couples, whose axes are the coordinate axes of
i 2
60 COMPOSITION OF PARALLEL FORCES. [53.
x, y, z respectively. Let the sum of these coaxal couples be
taken ; and let L, M, N be the moment-axes of the sums which
respectively have their rotation-axes coincident with the axes of
X) y, z. Thus all the component couples are reduced to three
couples whose rotation-axes are perpendicular, each to every
other two, and of which the moment-axes are L, M, N.
Let us further compound these three couples. Let G' be the
resultant moment-axis of L and M; then by (33),
G /a = L 2 -fM 2 .
Also again compounding- G' and N which are perpendicular to
each other, if G is the resultant moment-axis,
G 2 = L 2 +G /a
= L 2 + M 2 +N a . (36)
Let X, \i, v be the direction-angles of the rotation-axis of G :
then L = G cos X, M = G cos JA, N = G cos v ;
L M N. , _.
.'. COSX = -, COS^ = -y cosv = -> ( 37 )
so that if L, M, N are given, we can find G and the line of its
rotation-axis ; and if a moment-axis is given, we can resolve it
into three component moment-axes, which are at right angles
to each other. It is to M. Poinsot that we are indebted for
this great simplification of a problem which it is very difficult
to follow in its complex form.
The analogy which has thus been traced to composition and
resolution between couples as, expressed by their moment-axes
and forces of translation by means of their line-representatives
establishes a real and a large principle of., duality, and of which
we shall hereafter have many illustrations. Every theorem
hereby becomes double. It admits of interpretation with re-
spect to couples, that is, with respect to pressure of rotation, as
well as with respect to pressure of translation ; and the proof of
a theorem of one class authorizes the inference of the analogous
theorem in the other class.
SECTION 3. On the composition and resolution of forces acting on
a rigid body, the lines of action of which are in one plane.
53.] I propose in the first place to investigate the composi-
tion of those forces, the action-lines of which are parallel to each
other, and which are consequently called parallel forces.
53-] COMPOSITION OP PARALLEL FORCES. 61
Let the plane in which the forces act be the plane of (x, y] ;
and let the origin o be, fig. 27, any point which is in, or rigidly
connected with, the body; and let the forces be P,, p a , ... P n , of
which let P be the type : let p l} p t , . . . p n be the perpendiculars
from the origin on their lines of action, of which let p be the
type-perpendicular : let (x, y] be any point in the line of action
of the type-force P, and let a be the angle between the line of
action of P and the axis of x : then the equation to the Hue of
action of P is g gin a _ y CQg a _ p = Q
Let two forces each equal to P, with their lines of action parallel
to that of P, and acting in opposite directions, be introduced at
the origin O; so that instead of the original force P, we have
p acting at o in a parallel line and the same direction, and a
couple whose moment is PJP and whose rotation-axis is perpen-
dicular to the plane of the forces.
Let P at o be resolved into two forces along the coordinate
axes, viz. P cos a, and p sin a ; and let all the forces be similarly
transformed; then, if x and Y are the resultants of the forces
severally along the axes of x and y,
x = P! cos a -f P., cos a + -f P n cos a
= cosa2.P; (38)
Y = P! sin a + P a sin a + . . . + P n sin a
= sinaS.P. (39)
Also the moment of the couple arising from p is equal to p/?, the
tendency of which is to turn the body from the axis of x towards
that of y ; and, as a similar couple and moment will arise from
every one of the forces, if G is the moment of the resultant
couple, by reason of Art. 49,
G = 2.pp
== 2.P (x sin a y cos a)
= sinaS.Ptf cosaS.P^, (40)
placing sin a and cos a outside the signs of summation, because
they are the same for all the forces : and observing that x and y
refer to some point in the line of action of each pressure, which
will generally be different for each. G in (40) consists of two
parts, which are affected with different signs; the resultant
couple therefore is the difference between the resultants of two
systems of coaxal couples acting in contrary directions : sinaS.Ptf
tend to turn the body from the axis of x towards that of y, and
act in the -Apposite-direction. ... ...
62 COMPOSITION OF PARALLEL FORCES. [54.
54.] Suppose now that all the forces are capable of being
reduced to a single force R ; or, in other words, suppose that one
force R will have the same effect on the rigid body as all the
impressed forces taken in combination. Let a be the angle at
which the line of action of R is inclined to the axis of x, and let
(x, ~y] be any point in the line of action of R, and 7 the perpen-
dicular distance from the origin on it. Then introducing at O
two forces, each equal to R, with their lines of action parallel to
that of R, and acting in opposite directions, we have the force
of translation R acting at the origin, and a couple R? ; whence,
resolving R at the origin along the coordinate axes, and equating
the resolved parts to the sum of the resolved parts of the im-
pressed forces, we have
R cos a = 2.P cos a = cos a 2.P, ) , .
R sin a = S.P sin a = sin a 5.P ; )
therefore R = 2.P, a = a; (42)
that is, the resultant is equal to the algebraical sum of the com-
ponents, and its line of action is parallel to those of the several
components.
Also the couple Rr, due to the resultant R, must be equal to
G; so that (43)
2.P
and thus the force R is determined as to magnitude, line of
action, and direction.
The equation to its line of action may thus be found. Re-
placing J in (43) in terms of ~x and y, the current coordinates of
the line of action of R, we have
R(#sina ycosa) = G; I (44)
.. aFsina ycosa = -; (45)
which is the equation required.
We may however employ the abridged form of the equation
to a straight line ; in which case let the equations to the lines
of action of P O P 2 , . . . p n be
a, =0, a, = 0, ... a n = 0, (46)
where o is the length of the perpendicular from any point (x, y)
on the line of action of p. Now since R^ = o, it is plain that in
reference to any point in the line of action of the resultant,
G = ; therefore
. +?(!,, =s 2.?a = 0, (47)
56.] CENTRE OF PARALLEL FORCES. 63
which is the equation to the line of action of R ; and written at
length is
a?cosa2.P-fysina2.P S.PJO = 0; (48)
and therefore the perpendicular distance from the origin on the
line of action of R is ^
2.P
Thus if the equations of the lines of actions of the several
parallel forces are given, that of the line of action of the re-
sultant is given by (45) or (48) : and it is the locus of point in
the plane of the forces with reference to which the sum of the
moments of the component couples vanishes.
55.] If the forces are in equilibrium, that is, if the system is
what we shall call an equilibrium-system, whatever point is
taken as the origin, the particle at that point is at rest, and the
moment of the couple producing rotation about that point
vanishes. If this is the case we must have the two following
conditions; viz.
R=2.p =0; (49)
G = 2.PJ0 = 0; (50)
and these are the conditions of equilibrium of a system of
parallel forces.
If 2.P = 0, and 2.Pjo is a finite quantity, then R = 0, ~r = oo,
and the forces are reducible to a couple whose moment is S.PJO.
If 2.P/> = 0, and 2.P is a finite quantity, the forces are reduced
to a single force of translation, the line of action of which passes
through the origin.
It will be observed that S.P which is equal to R is a quantity
independent of the position of the origin and of the coordinate
axes; and is accordingly an invariant. Not so is 2.P/? or G; it
depends on the position of the origin, although it is independent
of that of the coordinate axes. The law of dependence will be
considered at length in a more general case hereafter.
56.] In the preceding Articles the line of action, the direc-
tion, and the magnitude of the resultant of a system of parallel
forces have been determined, when the lines of action, direction,
and magnitudes of the component forces have been given : that
is, we have considered the forces with reference to only three
out of the four incidents as stated in Art. 14. The problem
which I have now to investigate will require the fourth incident
also, viz. the point of application of each force. The problem is
64 CENTRE OF PARALLEL FORCES. [56.
this. Suppose that an equilibrium-system consists of n parallel
forces, of each of which the four incidents are given ; what
conditions must it fulfil, so that it should be an equilibrium-
system, when, the direction and points of application being
unchanged, the lines of action are all turned in the same direc-
tion in the plane of the forces through the same angle ?
As the action-lines of the forces are all turned through the
same angle, the system after the displacement is also one of
parallel forces. Let P U P 2 , . . . P B be the forces, and let (x lt y^
( x i>y*) ( x *> y^) be their points of application, and let a be the
angle between the new lines of action and the ar-axis. Then
the conditions of equilibrium of the displaced system are (1)
2.P = 0; (2) 2.Pj = 2. P (a? sin a" ycosa') = 0; the former of
which is satisfied because the system was originally in equili-
brium ; and as a in the latter is indeterminate, we must have
2.P"# = 0, 2.Py = ; (51)
and these together with 2.P = are the conditions requisite
that an equilibrium-system of parallel forces should also be an
equilibrium-system when the lines of action of the forces are all
turned through the same angle in the plane of the forces.
From these conditions we have the following results. Let us
suppose the equilibrium-system to consist of n forces p u P 2 , ... p,,
whose points of applications are (x 1} y^), (a? 2 , y^) . . . (a? n y n ) and of
a force R, whose point of application is (x, y}; then R, acting
along the action- line of R, will neutralize R, and is conse-
quently the resultant of the n forces P u P 2 , ... P n ; and the
preceding conditions become
_ 2. Pa? 2. Pa? ~|
~ == _ - >
x =
y =
R 2.P J
which are the coordinates of the point of application of the
resultant of the n components, and are the same whatever is
the angle through which the action-lines of the forces are turned
in the plane of the forces. It is for this reason that the point
(x, y} is called the centre of parallel forces. We shall hereafter
have many applications in which the position of it is of great
importance.
If the centre of parallel forces is at the origin, then in that
system of forces, and in that reference, 2. Pa? = 2.Py = (X.
57-] COMPOSITION OF FORCES IN ONE PLANE. 65
If the system consists of two forces PI and P 2 applied at the
points (.r u y^ (.r 2 , y 2 ) respectively, then
and if p 2 = P O x=y = oo ; consequently, as in this case the sys-
tem is a couple, the centre of a couple is at an infinite distance.
If the forces are all equal, viz. p t = P 2 = . . . = p n , then
X =
,p . I
(53)
nv 11
and the centre of parallel forces is the centre of mean distances
of the points at which the forces are applied.
The following are examples in which the centre of parallel
forces is determined.
Ex. 1. Suppose six parallel pressures proportional to the
numbers 1, 2, ... 6 to act at points whose coordinates are seve-
rally ( 2, 1), (1, 0), (0, 1) ... (3, 4); find the resultant, and
the centre of these parallel forces.
R = 2.p = 1 +2 + . ..+6
= 21;
2.P.T = 2 2 + 4 + 10 + 18
= 28;
2. Py = 1+3 + 8 + 15 + 24
= 49;
28 49
* x -^-'> y
21' 21
Ex. 2. At the three angular points of a triangle parallel
forces are applied severally proportional to the opposite sides
of the triangle ; it is required to find the centre of these forces.
Let (# u y^) (# 2 , y^) (x. t) y z ] be the angular points of the tri-
angle, and let a, b, c be the sides severally opposite to them ;
then ax ,fa , , i ,
a+d+c y a + d + c
57.] Composition of many forces acting in one plane on a
rigid body or a rigid system of material particles.
Let the plane in which the forces act be that of (x, y) ; and
let o, the origin, fig. 27, be a point of the body, or rigidly con-
nected with it : let the forces be P O P 2 , . . . P,, : let a 1} a t , ... a f be
PRICE, VOL. III. K
66 COMPOSITION OF FORCES IN ONE PLANE. [58.
the angles between their lines of action and the axis of x : let
Pi, PI, . . . p n be the lengths of the perpendiculars drawn from the
origin on the lines of action : and of these quantities let P, a,
and p be the types : so that
p = x sin a y cos a. (54)
At o let there be introduced two forces equal to P, with their
lines of action parallel to that of P, and in opposite directions;
so that, in the place of the original force p, we have p acting at
o in a parallel line and the same direction, and a couple whose
moment is PJO, and whose rotation-axis is perpendicular to the
plane of the forces. Let p at o be resolved into parts along the
coordinate axes, so that P cos a acts along the axis of x, and
p sin a along that of y ; and let all the forces be similarly re-
placed. Then if x and Y are the sums of the resolved parts of
the forces along the axes of x and y respectively,
x = P! cos (*! + P a cos a a + . . . + P n cos a n ,
= 2. P cos a, (55)
Y = P! sin a a + P 2 sin a 2 + . . . + P n sin a n ,
= 2. P sin a; (56)
and if R is the resultant of x and Y, and a is the angle between
the action-line of R and the #-axis,
R 2 = x 2 +Y 2 ; (57)
X Y
cos a = -, sin a = (58)
R R
Also the moment of the couple arising from p is PJO ; the ten-
dency of which is to turn the body from the axis of x towards
that ofy ; and as a similar couple will arise from every one of
the forces, and as all these couples are coaxal, the moment of
their resultant is equal to the sum of the moments of the com-
ponents. Let G be the moment of the resultant couple ; then
G = Pi^i+P a jo a + ...+Pj w
= 2.PJ3
= 2.p(#sina ycosa)
= 2. Par sin a 2.Py cos a. (59)
58.] From these results four cases arise: (1) that in which R
and G have both finite values ; (2) that in which R is finite, and
G = 0; (3) that in which R = 0, and G is finite; (4) that in
which R = 0, and G = 0. These cases severally require con-
sideration.
59-] COMPOSITION OF FORCES IN ONE PLANE. 67
The first case in which R and G have both finite values is that
in which these resultants are equivalent to a single force of
translation which acts along a definite line of action. For let
the couple whose moment is G be turned about its rotation-axis
until its arm is perpendicular to the action-line of R ; and let
the length of the arm of G = r, and the force = R, so that
rR = G. Also let the couple be so placed that one of its forces
acts along the action-line of the resultant of translation, and in
a direction opposite to that of that resultant ; and the other acts
along a line parallel to the resultant, and at a distance r from it.
Then one force of the couple is neutralized by the resultant of
translation, but the other force remains, and is the final single
resultant of translation ; and as its action-line is parallel to that
of the original resultant and at a distance r from it, where
R/ = G, if x and y are its current coordinates, r = x sin a y cos a;
and either
_ y R cos a _ G ^
or a?Y yji = G, (61)
is the equation to the action-line of R.
If the equations of the action-lines of the several components
are given in the ordinarily abridged forms of notation ; that is,
if a t = 0, a, = 0, . . . O B = are the equations to the lines along
which Pj, p a , . . . P n act, then the equation to the action-line of
Ris P 1 a 1 +P 3 a a +...+P n a n = 0, (62)
or #2.pcos a+^2.Psina = 2.P/?; (63)
either of which equations states that the action-line of the
resultant is the locus of points in reference to which the moment
of the resultant couple vanishes.
59.] The second case is that in which R is finite, and G = 0.
This is that particular case of the preceding Article, in which the
forces have a resultant of translation, on the action-line of which
the origin has been taken.
In the third case, R = 0, and G is finite. Here the forces are
equivalent to a couple whose moment is G, and the value of
which is independent of the position of the origin in the plane
of the forces.
In the fourth case R = 0, and G = ; that is, no force acts
at the origin, and there is no force of rotation tending to turn
the body about an axis perpendicular to the plane of the forces ;
that is, there is no pressure of translation on the origin, and no
pressure of rotation about it ; in other words the forces are io
K 2
68 PROBLEMS OF STATICAL FORCES. [60.
equilibrium and the body is at rest. And since by reason of
(57), when R=0, x = 0, Y = 0, three conditions must be satisfied
by a system of forces, whose action-lines are in one plane, which
are in equilibrium ; viz.
x = 2. P cos a = 0, ) / 64 x
Y = 2 . P sin a = ; )
G = 2.P/J = 0. (65)
As the origin is arbitrary and the directions of the axes are
also arbitrary, a system of forces acting- in one plane on a body
is in equilibrium, if the sums of the resolved parts of the forces
along- any two straight lines in the plane perpendicular to one
another vanish, and if the sum of the moments of the forces
about an axis perpendicular to the plane also vanishes.
As the three conditions given in (64) and (65) are all that can
in the most general case be required for the equilibrium of a
system of forces in one plane, they show that the body on
which the forces act has at the most three degrees of freedom ;
which have to be severally neutralized. There are two displace-
ments of translation along any two lines which are perpendicular
to each other, and a displacement of rotation about an axis per-
pendicular to the plane of the forces.
If one point of the body in which the forces act is fixed, and
the point is in the plane of the forces, the body can have no
displacement of translation, and this circumstance satisfies the
first two conditions, viz. (64); and this effect is also otherwise
manifest, inasmuch as the determination of a point requires two
conditions, and these may be the first two of (64).
If two points of the body are fixed in the plane in which the
forces act, the body is entirely fixed. These circumstances
indeed give one condition in excess of those which are requisite ;
they give four conditions, whereas three are sufficient to satisfy
(64) and (65).
The four preceding cases show that when a body is acted on
by a system of forces whose action-lines are in one plane, the
system is either one of equilibrium, or is reducible to a single
force of translation, or to a single couple of rotation.
60.] The examples in which the equations of equilibrium (64)
and (65) are applied are extremely numerous ; and a large supply
will be found in any of the ordinary collections ; it is desirable
however to insert a few, that the reader may understand the
mode of application.
60.] PROBLEMS OP STATICAL FORCES. 69
Ex. 1. A heavy uniform beam AB rests in a vertical plane,
fig-. 28, with one end A on a smooth horizontal plane and the
other end B against a smooth vertical wall : the end A is pre-
vented from sliding- by a horizontal string of given length
fastened to the end of the beam and to the wall : determine the
tension of the string and the pressures against the horizontal
plane and the wall.
Let the length of the beam be 2 a, and let w be its weight ;
which, as the beam is uniform, we may suppose to act at its
middle point G; let R be the vertical pressure of the horizontal
plane against the beam ; and R' the horizontal pressure of the
vertical wall, and T the tension of the horizontal string AC ; let
BAC = a, which is a known angle, as the lengths of the beam
and the string are given. Then equations (64) and (65) become,
for horizontal forces, T = R';
for vertical forces, w = R;
for moments about A, Vfa cos a = R'2 sin a ;
w
.'. R = T = cot a.
A
Ex. 2. A heavy uniform beam rests on two given smooth in-
clined planes : it is required to find the position of the beam,
and the pressures on the planes.
Let AB, fig. 29, be the beam, whose length is 2 a, and whose
weight is w acting at the centre of gravity G : let the inclina-
tions of the planes AC and BC to the horizon be respectively a
and /3 ; and let the inclination of the beam be ; let R and R'
be the pressures of the planes on the beam, and the lines of
action of which are perpendicular to the planes by reason of
their smoothness. Then we have
for horizontal forces, R sin a = R'sin /3 ;
for vertical forces, w = R cos a -f R'COS /3 ;
for moments about G, R cos (a 6) = R'# cos (/3 + 0) ;
sin (a /3)
tan0 =
2 sin a sin /3
wsin/3 ,_ wsina
R ^ ~- ~ t ~ ^r i R =
sin (a
Ex. 3. A heavy uniform beam AB, fig. 30, rests with one end
A against a smooth vertical wall, and the other B is fastened by
a string BC of given length to a point c in the wall; the beam
70 PROBLEMS OF STATICAL FORCES. [60.
and the string are in a vertical plane : it is required to determine
the pressure against the wall, the tension of the string, and the
position of the beam and the string.
Let AG == GB = a, AC = x, BC = b,
weight of beam =w, tension of string =T, pressure of wall =R,
BAE = 0, BCA = <;
then for horizontal forces, R = T sin < ;
for vertical forces, w = T cos <j> ;
for moments about A, wa sin = fx sin $ ;
.-. a sin0 = #tan</> ;
and, by the geometry of the figure,
b 2a x
sin ~~ sin< sin (0 <)
sm =
2a
whence R and T are known.
Ex. 4. A system of forces acting on a rigid body in one plane
is represented by the sides of a plane closed polygon taken in
order ; it is required to determine the resultant.
Let some point within the polygon be taken for the origin,
and two lines drawn perpendicularly to each other for coordi-
nate axes. Let the lengths of the sides of the polygon be
*t, # 2 , ... s n ; and let their angles of inclination to the axis of a?
be ciu a 2 , . . . a,,, and the perpendiculars from the origin on the
lines of action be fli,p a , ...p n : at the origin let pairs of equal
and opposite forces be introduced, equal and parallel to those
along the sides of the polygon : so that the system is changed
into (1) a system of forces acting at the origin, which are in
equilibrium by reason of Article 29, and (2) a system of coaxal
couples, the moment of the resultant of which is equal to
*jJ 9 iH-*a^a+ +*J0 B ; that is, to a moment of which the
geometrical representative is twice the area of the polygon.
A particular case is that of a triangle, whose sides are geo-
metrical representatives of three forces : of which the resultant
of translation vanishes, and the moment of the resultant couple
is represented by twice the area of the triangle. See Art. 4 1 .
6o.] PROBLEMS OF STATICAL FORCES. 71
Ex. 5. A heavy and smooth circular ring- rests on two hori-
zontal bars, which are not in the same horizontal plane : deter-
mine the pressure on each bar.
Let fig. 3 1 represent a vertical section of the system ; p and Q
being the two bars, R and R' the pressures of the ring 1 against
them, w the weight of the ring acting at its centre o ; let the
angle POQ = a, which is known ; and let the angles of inclina-
tion to the vertical of the lines of action of R and of R' be ft
and y ; then, as the three forces meet in the centre of the ring,
we have
R R w
sin y sin ft sm a
Ex. 6. A parabolic curve, fig. 32, is placed in a vertical plane
with its axis vertical and vertex downwards, and inside of it and
against a peg in the focus a smooth uniform and heavy beam
rests : required the position of rest.
Let PQ be the beam, of length 2c and of weight w ; let SA = ,
SP = r, PSA = Q ;
2a
r
14- cos B
a
also SPT = STP = 90 -- ; PG = GQ = c,
for forces along PQ, R sin STP = w cos 6;
for moments about s, ur cos SPT = w (r c) sin
")
; J
Suppose that it were required to find the curve AP such that
(16
the beam should rest in all positions : then tan SPT = r -^- ',
dr
therefore from (66),
(19 cos
~r 1 - \ A> * r = c+asecd; (67)
dr (rc)sm6
where a is an arbitrary constant ; and this is the equation to
the conchoid with an arbitrary modulus.
Ex. 7. To discuss the properties and conditions of equilibrium
of a balance ; fig. 33.
Let AB be the arm of the balance ; AC = CB = ; and let the
balance be suspended by a point o in a line perpendicular to AB
at its middle point c, and let o c = c ; let the balance be symme-
trical with respect to the line oc, and let the centre of gravity
72 COMPOSITION OF FORCES IN ONE PLANE. [6 1.
of the beam, scales, &c. be at G ; let OG =' k, and let the weight
of the whole machine, short of the weights in the scales, = w ;
and to consider the general case suppose the weights in the
scales P and Q to be unequal, Q being greater than p ; and let
the arm of the balance be inclined to the horizontal line at an
angle 0. Then the vertical pressure on o = P + Q + w ; and
taking moments about o,
Q (a cos Q c sin 6} = P (a cos + c sin 6} -f v?h sin ;
(68)
Now the conditions required in a balance are (1) horizontality
of the beam, when the arms and weights are equal; (2) sensi-
bility, which is estimated by the angle through which the arm
is turned when the weights are unequal; (3) stability, or the
tendency to return after the cause of displacement is removed.
Condition (1) is fulfilled when Q = p, since, by (68), in that
case, 6=0.
Condition (2) is more or less satisfied according as d is larger
or smaller for a small difference between p and Q ; now in (68),
if Q p is very small, tan 6, and therefore Q, is large,
(1) when a is large, that is, when the arms of the balance
are long ;
(2) when c is small, that is, when the point of suspension
is not far above the beam ;
(3) when p + Q is small, that is, when the weights are small ;
(4) when w is small, that is, when the weight of the whole
balance is small ;
(5) when k is small, that is, when the centre of gravity of
the machine is not far below the beam ;
and either c or Ti or both may be negative ; and then as a limit-
ing case we may have tan = oo, and = 90 ; in which case the
beam becomes vertical when it is displaced at all, and may have
no tendency to return to its horizontal position ; and thus the
sensibility of the balance may be very great, but there may be
no stability, and one of the necessary conditions is not satisfied :
this last condition therefore may be inconsistent with the second,
and the two must be adjusted as is practically most convenient.
61.] Although in all cases it is possible, and in most cases
scarcely less general, to refer forces and conditions of equilibrium
6 1.] COMPOSITION OF FORCES IN ONE PLANE. 73
to rectangular coordinates, yet it is desirable to indicate the
forms which the reduced resultants take, if the coordinate axes
are oblique.
Let the angle of ordination be o> ; let the forces be t lf P,...P W ;
(*i>yi), (*,y*),-(x, y^ their points of application ; Pi,p>,...p n
the perpendiculars from the origin on their lines of action j
ai/3i, a a /3 a , ... a. n f3 n the angles between the perpendiculars to the
lines of action and the axes of x and y respectively ; then, em-
ploying the symbols without any subscripts as the type-symbols,
we have for the line of action of p
ircosa+^cos/3 p 0. (69)
Let two equal and opposite forces, each of which is equal to P
and has its line of action parallel to that of p, be introduced at
the origin ; so that, instead of the one force p applied at (x, y),
there are (1) a parallel and equal force at the origin, (2) a couple
whose arm is p and whose force is p. Let the former be resolved
into parts along the coordinate axes, viz. P sin a, and P sin ft ;
and let all the forces be similarly reduced ; let x and Y be the
sums of the resolved parts along the axes of x and y respect-
ively ; then
x = PI sin a t -f P 3 sin a, + . . . + P,, sin a n
= 2.P sin a ; (70)
Y = PJ sin P! + P 2 sin /3 a + . . . -4- P n sin /?
= 2.psin; (71)
and therefore if a is the resultant of x and of Y,
R 2 = X 2 +2XYCOSft>+Y 2 . (72)
And let G be the moment of the resultant couple : then
G = PX^, +P 2 ^2 + ...+P n ^ n
= 2.PJ0
= 2.p(# cos a+y cos/3). (73)
If the impressed forces are in equilibrium, E = 0, and G = ;
.' . 2.P sin a = 0, 2.P sin y3 = 0, S.PJO = 0.
If the equations to the lines of action of the impressed forces
are given, that to the line of action of the resultant may thus be
found ; let the equation to the lines of action of the forces be
tfcosaj+y cos/3i PI = 0, -
tfCOSaj+ycos/3., j 2 = 0,
x cos a w +y cos /3 M -j M = ; J
PRICE, VOL, III. L
74 COMPOSITION OF FORCES. [62.
then in reference to any point in the line of action of the re-
sultant, 2.PJ0 = ; therefore we have
2.P (x cos a +y cos/3j}) = 0,
#5.p cosa+y 2.P cos ^3 s.p/> = 0. (74)
62.] On referring to Arts. 58 and 59 it will be seen that the
effects of a system of forces acting in one plane as to translation
and as to rotation depend on R and G, since these are respect-
ively the resultant of translation and the moment of the re-
sultant couple with respect to an arbitrarily chosen origin. It
will be observed that R is independent of the origin and of the
coordinate axes, being the same whatever they are ; it is accord-
ingly an invariant. But not so is G, which is equal to 2.P/J, and
consequently depends on the origin, though it is independent of
the coordinate axes ; thus the value of it varies according as the
point varies in reference to which it is estimated. The general
value of it is determined as follows :
Let G be the value of the moment of the resultant couple
with reference to (a? , y ] ; and let (#', y] be a point in the
action-line of P with respect to (# , y ] ; so that x = x +af>
y - #o+f; then from (61),
(75)
The following are theorems deduced from this equation :
(1) On comparing (75) with the equation of the action-line of
the resultant given in (61), it is seen that if the right-hand
member vanishes, that is, if the point (.r , y ] is on the line of
action of the resultant of translation, G O = ; that is, the
moment of the resultant couple vanishes for all points on the
line of action of the resultant, and this is the absolutely least
value of G.
(2) If G is a constant, the locus of (# c , y ) is a straight line
parallel to the action-line of the resultant ; hence for all points
in a straight line parallel to the action-line of the resultant, the
moment of the resultant couple is the same.
(3) If the forces are in equilibrium, so that x = Y = G = 0,
G = 0; so that if a system of forces is in equilibrium, the
moment of the resultant couple vanishes for all points in the
plane of the forces.
(7G)
63.] CENTRE OF FORCES. 75
(4) If the system of forces is reducible to a couple, in which
case R = 0, that is, x = Y = 0, G O = G ; consequently the mo-
ment of that couple is the same for all points in the plane of the
forces.
(5) If the moment of the resultant couple vanishes for three
points in the plane of the forces which are not in the same
straight line, the system is in equilibrium. For if (#,,y,),
(*> y*)) ( x z> $3) ai 'e three points in the plane of the forces, and
with reference to them we have
G-Y^+X^ = 0,
G Y# a + Xy a = 0,
G-Y# 3 + xy 3 = 0;
then eliminating x and Y we have
G{* t y.* t y t +x,y l x l y t +x l y,x,y l } = o :
but the second factor of the left-hand member of this equation is
twice the area of the triangle of which the three given points
are the angular points ; and as they are not in the same straight
line, it does not vanish : consequently G = ; and similarly
x = 0, Y = ; and therefore the system is in equilibrium.
(6) Hence if the moment of the resultant couple of the system
vanishes for three points in the plane which are not in the same
straight line, it also vanishes for all points in the plane.
(7) If the moments of the resultant couples of a system are
given for three points not in the same straight line, the moment
G is given for every other point (#> ^o) of the plane. The given
equations are
G c* J_ v -v -_ Y .*/ > f 7 T\
2 "" v"" "i * w 2 " / %) \ J
from which G, x, Y may be determined ; and consequently G O ,
of which the value is given in (75), may be found.
63.] The preceding investigations on the composition of forces
in one plane have depended on the magnitude, line of action,
and direction of the acting forces ; but, the principle of trans-
missibility having been applied, have been independent of the
points of application of the forces. I come now to the problem
analogous to that of Art. 56, and propose to consider a case in
which the last incidents are required ; viz. to investigate the
circumstances under which an equilibrium-system of foi'ces in a
L 2
76 COMPOSITION OF FORCES. [63.
plane will also be in equilibrium, wben the body is displaced in
the most general manner in the plane ; the magnitudes, points
of application in the body, and directions of the forces being the
same as before the displacement, and the lines of action in the
new position of the body being parallel to those in the former
position ; or, in other words, when the action-lines of the forces
are all turned in the same direction through the same angle in
the plane of the forces.
Let us take two systems of rectangular coordinate axes, one
of (x, $} fixed in the body, and the other of (of, /) fixed in the
plane of the forces; and let these coincide in the original
position of the body. Let the body be shifted through distances
(#, y c ) respectively, parallel to the original fixed axes, so that the
Origin of the axes fixed in the body is brought to the point
(#, y c ) ; and let the body be turned through the angle 6 about
an axis perpendicular to the plane of the forces, and passing
through (# , y ) : then, if (x, y"] is in reference to the axes fixed
in space the same point as (x, y] in reference to the axes fixed in
the body,
x' # 4-#cos0 y sin 0, ~)
y'= y 9 + x sin d +y cos 6. J
Now as the system of forces is in equilibrium in the original
and in the new positions of the body, and as the lines of action
of a force in the new position is parallel to that in the former
position, we have
5.P cos a = 2.P sin a = 2.P (# sin ay cos a) = 0, (79)
5.P (tf'sin a /cos a) = ; (80)
let the values of x f , / which are given in (78) be substituted in
(80): then
# 2.P sin a y 2.P cos a
+ cos02.p(;rsma ycosa) sin s.p (3- cos a +^ sin a) =0. (81)
As the first three terms of this expression vanish by reason of
(79), we must have also
2.p(#cosa+^sina) = ; (82)
and as this is independent of x , y , and 0, it holds good for all
displacements of the body, and gives a fourth relation to be
satisfied by the forces and the points of application, when the
system is in equilibrium, whatever is the displacement of the
body, so long as the plane of the forces is the same and the
63.] CENTRE OF FORCES. 77
displacement of rotation is about an axis perpendicular to the
plane of the forces. Hence four conditions must be satisfied,
three in (79), and one in (82) when the equilibrium-system
satisfies the stated requirement.
The condition (82) admits of the following interpretation.
Let the point of application of each force be referred to polar
coordinates, the original origin being the pole, and the fixed
ar-axis the prime radius. Let (r, 6) be the point of application
of P, and let p be resolved along and perpendicularly to the
radius vector. Let u be the component along the radius vector
and acting from the pole, and let u be called the central com-
ponent ; let v be the component acting perpendicularly to the
radius vector, and tending to increase d, and let it be called the
transversal component; all these being type-symbols, and type-
names. Then
u = pcos(a d)
p^cosa+ysina).
:= ,
r
v = Psin(a 0)
_ p(#sina ycosa)
~r~
.*. 5.P (#cosa+y sin a) = S.ur = H, say: (83)
2.p (x sin a y cos a) = s.vr = a. {84)
Thus H, which represents (82), is the sum of the products of
each central component and the distance from the origin of its
point of application. Let H be called the radial moment* . As
the lines of action of all the central components pass through
the origin, they produce no pressure of rotation about that
point ; consequently the moment of the resultant couple is due
to the transversal components only ; and evidently, as in (84),
G = 2.vr.
Thus if an equilibrium-system of forces in one plane is also
in equilibrium after the displacement of the body, subject to the
stated conditions, the requisite relations of the forces are given
by the four conditions
X = Y = G = H=0. (85)
The first three being requisite so that the system should be an
equilibrium-system in its original position ; and the last being
* German writers on Mechanics call H "Fliehmomente ;" see Dr. Schweius
in Crelle's Journal, Vol. XXXVIII, p. 77.
78 COMPOSITION OF FORCES. [64.
an additional condition so that it should be an equilibrium-
system after displacement.
64.] Suppose now one force to be taken out of this equi-
librium-system, and to be replaced by an equal one acting at the
same point of application and along the same line of action but
in an opposite direction ; then this new force is the resultant of
all the other remaining forces. Let us slightly modify the
system as before conceived, and suppose it to consist of (n+ 1)
forces, viz. n forces, f l} P 2 , ... P,,, of which the points of applica-
tion are (Xi,y^, (x 2 , y *)>(%*) y^)> and of u, of which the point
of application is (x, ~y], and a the angle at which its line of
action is inclined to the #-axis. Let this be an equilibrium-
system, then K, is the resultant of the other n forces ; let it also
be an equilibrium-system after an arbitrary displacement ; then
the four conditions (85) become
x = S.pcosa = Rcosa; Y = 2.P sin a = Rsina; (86)
G = 2.P (x sin a y cos a) = R (x sin a y cos a) ; (87)
H = S.P (x cos a+y sin a) = R (x cos a + ~y sin a). (88)
Now (#, ~y] is the point of application of E, and is the same for
all positions of the body ; that is, the magnitudes of the forces
and their points of applications being unaltered, if these lines
of action are all turned in the same direction through equal
angles in the plane of the forces, the resultant will always be
applied at (.r, j/), its magnitude being unaltered, and its line of
action being turned in the plane of the forces through the same
angle as the lines of action of the other forces. The point (He, ~y)
is for this reason called the centre of the forces, and its position
is determined by means of (87) and (88). Thus let the moment
of the resultant couple of the n forces p,, P 2 , ... p n be G, and let
the radial moment of the same forces be H ; then we have
G = B,(irsin# ^cosa), (89)
H = lificosa+y sin a) : (90)
whence = _ H cos a + Gsina HX + GY
x =
B, R 2
H sin a G cos a HY GX
y = = :
E R
and these assign the position of the centre of the forces.
If the system consists of parallel forces,
H = cos a S.P# -f- sin a Z.vy, G = sin a 2.p;r cos a 2.
66.] CENTRE OP FORCES. 79
and consequently
2.
which are the same values as (52).
65.] The centre of two forces acting- in a plane on two given
points may be determined in the following manner by a geo-
metrical construction. Let the forces be p, Q, and let their
points of application be A and B ; let the lines of action of the
forces meet in o ; describe a circle passing through o, A, B ; and
let oc be the line of action of the resultant R, and let it cut the
circle in c ; then c is the centre of p, q. Whatever is the posi-
tion of o in the circumference of the circle between A and B, and
suppose it to be at o', the angles AO'B, BO'C, CO'A are equal
severally to A OB, BOC, COA; so that the action-lines of all the
forces are turned through equal angles in the plane of the forces,
as long as o is on the circumference of the circle; and as the
equilibrating relation between p, Q, R depends on these angles
only, it is the same whatever is the position of o' : but in all
cases c remains the same ; therefore c is the centre of the forces.
66.] The radial moment of which the value is given in (83)
has the following properties :
(1) Since H = s.ur = 2.p# cosa + s.py sin a, it appears that
the radial moment of the whole system is equal to the sum of
the radial moments of the two systems of the resolved forces
along the axes.
(2) It is evident that the value of the radial moment is not
altered, whatever is the position of the coordinate axes, if the
origin remains the same.
(3) If the origin be moved to the point (# , y ) ; so that, if
x' } y are the coordinates at the new origin,
then 2.?'U = 2.P (#'cos a +/sin a) -f # S.P cos a +y c 2.P sin a
= S.rV+tfo 2.P cos a-f y 5.A sin a; (93)
so that if H is its value at (# > y<>}>
H = H + # x+y Y;
.'. H = H-X;r -Yy ; (94)
and thus the radial moment varies with the position of the origin
to which it is referred.
80 COMPOSITION OF FORCES. [67.
If (#, y ) is a point at which the central moment vanishes,
that is, at which H O = 0, then
x# + Yy = H; (95)
which is the equation to a straight line, of which x , y are the
coordinates; and consequently at any point in this line the
radial moment vanishes. This line is called the line of radial
moments.
On comparing this equation with (88) and (90) it appears that
the centre of the forces lies on this line of radial moments ; and
as it also, as it appears from (87) and (89), lies in the line of
action of the resultant ; the centre of forces is at the intersection
of these two lines, and these two lines intersect, as their equa-
tions shew, at right angles.
From (94) a series of theorems may be inferred similar to
those which have been inferred in Art. 62, from (75).
67.] If the system of forces in its original condition is re-
ducible to a couple, so that 2.P cos a = 0, 2.P sin a = 0, but that
G = S.P (x sin a y cos a) does not vanish ; and if after the dis-
placement the system is an equilibrium-system, then from (81),
5. P (x sin a ycosa) G
tan = - *. ( = - ; (96)
2.P (x cos a +y sin a) H
and thus the angle is assigned through which the sj'stem must
be turned, so as to be brought into an equilibrium-system. This
result is also manifest from the following reasoning.
Let the forces of the couple to which the original system is
equivalent be PJ, P, ; and let their points of applications be
(^u ^i) (&> y*} and let a be the angle between their action-lines
and the #-axis ; and let r be the distance between their points
of application, and the angle between this line and the action-
lines of the forces. Then if the lines of action of the forces are
turned through an angle towards the line r } these lines will
lie along r and the two forces will neutralize each other, and the
system will become an equilibrium-system. Now G = PI r sin 0,
H = P, (#i # a ) cos a + (y v y a )sina} = P,rcos0 :
/. tan0 = -:
H
which is the same result as (96).
68.] COMPOSITION OP FORCES IN SPACE. 81
SECTION 4. Composition and resolution of forces acting on a
rigid body or system of material particles in any directions.
68.] We proceed now to the most general case of statical
forces acting in any directions on a rigid body or system of
material particles in space.
Let any point, either of the body, or rigidly connected with
it, be taken as the origin, and let a system of rectangular co-
ordinate axes originate at it. Let the forces be P,,p 2 ,...p n j the
direction-angles of their lines of action, a,, /3 U y u a 2 , /3 8 , y 2 , ...
a n> P> y n ', a point in the line of application of each (x l} y l} zj,
( x *> y*> Zi})-- (#*> y n t z n ) > the perpendiculars from the origin on
their lines of action, p 1} p t) ...p n ', and of these quantities let
the types be P, a, /3, y, (x, y, z\p. At the origin O, fig. 35, let
there be introduced a pair of equal and opposite forces, each of
which is equal to P, and has its line of action parallel to that of
P; from o let the perpendicular OD (= p] be drawn to the line
of action of p : then, instead of the original p, we have p at o
equal to the former force and acting in the same direction along
a parallel line of action, and a couple each of whose forces is p,
whose arm is OD, and whose rotation-axis is perpendicular to
the plane PODP : and let a similar process be performed on all
the other forces. As to the force of translation at o, let p be
resolved into three components p cos a, P cos /3, P cos y along the
axes of x, y, z respectively ; and let x, y, z be the sums of the
resolved parts of all the forces along these axes ; then
x = PJ COSC^+PJ cos a 2 + ...+P n cos a w
= 2.P cos a; (97)
Y = p x cos/3 1 +p s eos s + ...-f P n cosy3,,
= S.P cos ft ', (98)
z = P x cos /! + P 2 cos y a + . . . + p n cos y n
= 2.P cosy; (99)
and consequently, if R is the resultant of these three forces,
E = x' + Y'+z 1 ; (100)
and if a, I, c are the direction-angles of the line of action of R,
cos# = -, cos 6 =. - , cos c = - : (101)
R R R
so that the magnitude, the line of action, and the direction of R
are known.
PRICE, VOL. III. M
82 COMPOSITION OF FORCES IN SPACE. [69.
As to the couple which arises from p, its moment is PJO : and
as p is the perpendicular distance from the origin on a line pass-
ing- through a point (x, y, z}, and having direction-angles, a, /3, y,
jo* = (ycosy 2cos/3) 2 -f (.zcosa #cosy) s + (#cosj3 ycosa) 2 ;
and as the rotation-axis of the couple is perpendicular to the
plane passing through the origin and containing this line, its
direction-cosines are
y cos y z cos z cos a x cosy #cos/3 y cos a / 1AO
; (10J)
P P P
in accordance with the law of Article (52) let us resolve the
moment-axis of the couple along the three coordinate axes;
then the resolved parts are P (^cosy 2COS/3), P (z cos a x cosy),
p (tfcos y3 y cos a), which are the moment-axes of the three com-
ponent couples, and whose rotation-axes are along the three
coordinate axes. Let the couples corresponding to all the im-
pressed forces be similarly resolved, and let L, M, N be the sums
of the moment-axes of those couples whose rotation-axes are
severally along the three coordinate axes : so that by reason of
(30) Article 49,
L = PI (^i cos y x 1 cos/3 1 )+... + P n (^ n cosy n -,2: n cos0 n ); (103)
L = 2.P (y cos y z cos /3) ; 1
similarly M = 5.P (2 cos a #cosy); <* (104)
N = s.P (x cos/3y cos a) ; j
and if G is the resultant moment-axis of these three couples,
G 3 = L + M* + N; (105)
and if the direction -angles of the resultant rotation-axis are
A > V> v > L M N , .
COS A = -, COSU = , COS V = J ( 106 J
G G G
so that both the moment-axis and the rotation-axis of the re-
sultant couple are determined. Thus the forces are reduced to
a force of translation, viz. R, acting at the origin, and to a
couple G, whose axis is determined by (105) and (106).
69.] The formulae (104) require closer consideration; the
right-hand member of each of the equations consists of two
parts, one of which is aifected with a positive, and the other
with a negative sign. Thus L is composed of two sets of coaxal
couples, viz. s.P^cosy and 2.P0cos/3; the former of which is
the sum of a system of couples, the force in each of which is the
^-component of the impressed force, and the arm is the y-ordi-
nate of its point of application ; and in the latter system, the
69.] COMPOSITION OF FORCES IN SPACE. 83
force of each couple is the y-component of the impressed force,
and the arm is the 2-ordinate of its point of application. Ima-
gine therefore the force P to be, at its point of application,
resolved into thre,e components along lines parallel to the co-
ordinate axes ; and let these be P cos a, P cos j3, P cos y ; and let
couples be considered positive, which having for their rotation-
axes severally the axes of x, y } and z, tend to turn the body
from the ^-axis to the 2-axis, from the 2-axis to the #-axis, from
the ar-axis to the y-axis; and let those couples be negative
which act in a contrary direction : which arrangement, it will
be observed, is cyclical. Now consider pcosy; and, fig. 36,
introduce at M and at o two equal and opposite forces, equal to
it and acting parallel to its line of action ; so that we have a
parallel and equal force acting at o, and two couples, of one of
which the arm is OM, and of the other the arm is MN ; of which
the former has the axis of y for its rotation-axis and is negative,
and the latter has the axis of x for its rotation-axis and is
positive ; hence P cos y acting at P is replaced by
A parallel and equal force, = P cos y, acting at o,
And a couple whose moment is pcosyy, and whose rota-
tion-axis is the axis of x,
And a couple whose moment is P cosy a?, and whose rota-
tion-axis is the axis of y.
By a similar process will P cos a and P cos /3 be replaced : and
the same process having been performed on all the impressed
forces, we have ultimately
S.P cos a acting at o along the axis of x,
5.PCOS/3 y,
S.Pcosy - z;
and the couples whose moments are
2.p(ycosy zcos/3), the rotation-axis of which is the axis of x,
2.P(^cosa #cosy), ------------- y,
2.p(#cos/3 ycosa), - z }
which results are the same as those investigated in the preceding
Article.
The principle on which signs are affixed to couples is of
course arbitrary; we have chosen one depending on the order
of the letters which distinguish the coordinate axes ; the con-
ventionality of the sign and direction is involved in the sign
in (102), which may be either positive or negative.
M 2
84 AN EQUILIBRIUM-SYSTEM. [70.
70.] The system of forces being thus reduced to a force of
translation R, the line of action of which passes through the
origin, an arbitrarily chosen point, and to a couple whose
moment is G, there are four cases separately to be considered :
(1) when R = G = 0, and the body is at rest because there is
neither a force of translation nor a couple acting on it ; in which
case we have an equilibrium- system ; (2) when R = 0, and G
has a finite magnitude, in which case the system is reducible to
a couple the direction of whose rotation-axis is assigned by (106) ;
(3) when G = 0, and B^has a finite magnitude, in which case the
system is reduced to a single force of translation the line of
action of which passes through the origin; (4) when R and G
are both of finite magnitude ; in this last case also if the line of
action of R lies in the plane of the forces of G, R and these two
forces having lines of action in the same plane are reducible to a
single force = R, and we have the third case. All these cases
will be considered in the following pages.
Let us first take the case when R = G = ; that is, when the
particle at the arbitrarily chosen origin is at rest, and when
there is no tendency to rotation about any axis passing through
that point, so that the whole system is in equilibrium : and by
reason of (100) and (105) we have
x=0, Y = 0, z = 0; (107)
L = 0, M = 0, N = ; (18)
or, s.Pcosa = 0, 2.pcosj8 = 0, 2.Pcosy = 0; (109)
2.p(^cosy .zcos/3) = 0, "I
2.p(^cosa #cosy) = 0, j- (HO)
2.P (x cos/3 y cos a) = ; J
which are six independent conditions to be satisfied for an equi-
librium-system ; that is, the sums of the resolved parts of the
forces along any three rectangular axes vanish ; and the sums of
the moments of the couples whose rotation-axes coincide with
the axes of any system of rectangular coordinates also vanish.
The following is an example in which the preceding conditions
are required:
Three planes, whose equations are
= 0,
= 0,
= 0,
71.] AN EQUILIBRIUM-SYSTEM. 85
meet at the origin, and support between them a heavy sphere of
weight w : determine the pressure on each of the planes.
Let the axis of z be taken in a vertical direction ; and let the
pressures on the planes be R U R 2 , R 3 ; the lines of action of which
are of course normal to the planes ; and let the equations of the
planes be such as to satisfy in each the condition, A* -f B a -f c 2 = 1 :
then (109) become
-K| -A-j "T" Jttj -A-2 ~T~ Jttg Ag ^ \J m
i\i j J5 j j~ -IV 2 -D 2 "l -"3^3 """"* ^ J
Rj^+RaCa+RjCs = Wj
from which we have, using the notation of determinants, t
2. + A 2 B 3 S. + AjB! 2. + A,B,
R, = W - = - , R~ = W - = - , R, = W -- = - ;
3. + A 1 B a C 3 5. + A.JB3C! S. + AjBiC.,
and _ Sl = __ ^ - = _ ^ __
Aj .03 ^^ -^2 -"-3 31 """ 3 1 1 li "^ 12
As the six conditions given in (107) and (108) are all that can
be required in the most general case for the equilibrium of a
body under the action of given forces, they shew that such a
body has six degrees of freedom, which they severally neutralize.
These are three displacements of translation along any three
lines which are perpendicular to each other, and three displace-
ments of rotation about three lines which are also perpendicular
to each other. These conditions are also equivalent to three dis-
placements of translation along any three lines which are not all
in the same plane, and to three rotations about any three lines
which are similarly not in the same plane.
71.] These conditions of an equilibrium-system admit of the
following geometrical interpretation. Let (x, y, z) be any point
in space ; then since x = Y = z = 0,
Kx + vy + zz = 0; (HI)
and replacing x, Y, z by those equivalents given in (109), we
nave
+ P 3 (x cos a z +y cos /3 3 + z cos y 2 )
+ ..........
+ p n (arcosa n +ycos)3 n +2cosy n ) = 0. (I 12 )
Now as x, y, z are the coordinates of any point in space,
# cos di+ycosfii+z cosyj is the projection on the line of action
of PJ of the distance of (x, y, z} from the origin ; and therefore,
as the origin also is an arbitrary point, this equation expresses
the following theorem. If the resultant of translation of a
86 AN EQUILIBRIUM-SYSTEM. [71.
system of forces vanishes, the sum of the products of each force
and of the projection on its line of action of a line joining two
given points (fixed arbitrarily) is equal to zero.
Also as one of the forces of this system is, when taken in an
opposite direction along its action-line, the resultant of all the
others, we have the following theorem :
In a system of forces acting on a rigid body, the sum of the
products of each force and of the projection on its line of action
of a line joining two given points fixed arbitrarily, is equal to
the product of the resultant of translation and of the projection
on its line of action of the same straight line.
Also if L = 0, M = 0, N = 0, then multiplying these severally
by x,y y z, we have L # + My + N0=0; (113)
and replacing them by their values given in (110), we have
p i {(y\ cosy x ^ cos /3J # + (,?! cosdi x cosyjy
+ (#! cos /3j y t cos ajz}
+ ...................
+ p { (y cos y n z n cos /3,,) x + (z % cos a n - x n cos y B ) y
+ (ar n cos/3 n y n cosa n )*} = 0. (114)
Now this expression admits of the following interpretation.
The equations to the planes passing through the origin and the
lines of action of the forces are
^f + C^costti a^cos y t ) 77 + (a^cos^ j^cosaj)^ 0, "1
= ;
and if J9 u j9 2 , ... p n are the lengths of the perpendiculars from
the origin on the lines of action of the forces, then
= (y l cosy! Z-L cos/S^ 2 + (^ cos a l x l cos y^ 2 + (x^ cos/3 x y l cos aj) a , (116)
with similar values for p 2 ...p n ; so that, if b l} b 3 ... 8 n are the
lengths of the perpendiculars from (x, y } z] on the planes whose
equations are given in (115),
_ lll l1l1
~~7T~
with similar values for 5 a , 8 3 ... b n ; and thus (114) becomes,
Pi^i8i + P a ^ a 8 a + ... + P n ^,,8 n = 0. (118)
Suppose that along the lines of action of the forces lengths are
taken proportional to the magnitude of the forces, and thus
proportional to P,, P 2 , . . . P n : then PJO is twice the area of the
triangle whose vertex is at the origin, and of which the base is
72.] AN EQUILIBRIUM-SYSTEM. 87
the straight line represented by P-: and as 8 is the perpendicular
distance from (x } y, z) on the plane of the triangle, ?pb is six
times the volume of the tetrahedron whose base is the triangle
and whose vertex is (x, y, z] ; that is, whose four vertices are at
the origin, the point (x, y } z}, and the two extremities of the
line representative of P; and as the first two points, viz. the
origin and (x, y, z], are arbitrary, this equation expresses the
following theorem :
If at any point the resultant couple of a system of forces
vanishes, the sum of the volumes of the tetrahedra which have
for one edge lines along the action-lines of the forces propor-
tional to the forces and for the opposite edge the line joining the
given point and any other fixed point in space, is equal to zero.
This and the former theorem are of course true for any system
of forces in equilibrium ; and in the latter theorem it is to be
observed that the base of each tetrahedron is proportional to the
moment of the couple which corresponds to the force.
72.] When the number of forces of which an equilibrium-
system consists does not exceed six, equations (109) and (110)
contain some remarkable theorems concerning their lines of
action and points of application. The equations of equilibrium
are six in number, and the symbols of the forces enter into them
homogeneously and symmetrically in the first degree, the co-
efficients being functions of the direction-cosines and current
coordinates of the action-lines of the forces. Consequently if
the number of forces does not exceed six, relations exist among
these coefficients ; that is, amongst the elements of their action-
lines ; and these relations express geometrical theorems.
To abridge the notation I shall take /, my n to be the direction-
cosines of the action-line of P, and I shall employ the notation
of determinants. In consequence of the former assumption, the
equations of equilibrium become
2.p = 2.PWZ = s.Pra = ; (119)
2.?(ny mz] = 2.p(z nx] = -2.?(mx ly} = 0. (120)
If the equilibrium-system consists of only two forces, these
equations become
P! /! + ?,, = P 1 W 1 +P 2 W2 2 = P t Mj+PjW, = 0; (121)
PI fai^i iSi) + Pi(i.y. m 2 z 3 ) = 0, "I
?,(/!*! ia?i) + P,(/,2, ,#,) = 0, i. (122)
PI (M?i Jijfi) +v,(m,x, l,y,) = 0;
88 AN EQUILIBRIUM-SYSTEM. [73.
from which groups, by the elimination of P, and v 3 , we have
whence it is evident that the action-lines of the forces are coin-
cident, the forces being equal and acting along them in opposite
directions.
73.] If the equilibrium-system consists of three forces, then
(119) and (120) become
" = 0; (123)
I 3 z 3 n 3
= 0; (124)
!*! lift, m a x t 1 2 y^ m 3 x 3 I 3
(123) shews that the action-lines of the three forces are parallel
to the same plane ; and (124) shews that they meet in a point ;
consequently these lines meet in a point and are in the same
plane. These equations are also satisfied when the action-lines
are parallel and lie in the same plane. Hence three straight
lines can be the action-lines of an equilibrium -system only when
they meet in a point and lie in the same plane.
74.] If the equilibrium-system consists of four forces; then-
we have the following equations :
(125)
Let the ratios of p l : P
denoted by the letters
the several equations of (126) we have
p s : p 4 be determined from (125) and be
, q t , q 3 , q t then substituting these in
J
74-] AN EQUILIBRIUM-SYSTEM. 89
Let us suppose three action-lines to be given, and consider
the fourth as that which is to be determined ; so that # 4 , y 4 , z t
are variables and l t , m t , n t are undetermined in the preceding
equations. Then the product of the left-hand members equated
to that of the right-hand members is, in terms of these variables,
the equation to a hyperboloid of one sheet, the three equations
in (127) being those of three fixed lines on which each of the
lines (1 1} m 1} #,), (l^ m t) n t ) rests; and consequently these
four lines are generators of the surface of the same class ; the
three lines given in (127) being generators of the surface of the
other class. Hence we have the following theorem : If an
equilibrium-system consists of four forces, their lines of action
must be generators of the same class of a hyperboloid of one
sheet.
This is also otherwise evident ; as the system consists of four
forces, and three enter homogeneously into the six equations of
equilibrium, we have three different and independent relations
which contain the elements of the lines of action only. Let us
consider three of the action-lines to be given ; then the action-
line of the fourth must satisfy these three conditions. Now the
equations of a straight line in space contain four independent
constants ; three of these may be satisfied by the three preceding
conditions, but one other is still required for the complete de-
termination of the line. Such a condition might be that the
line should meet a given line. Then this condition leads to the
following result : Let the four action-lines of the forces be called
Pi>P*>Ps>P*> an< i let q be any straight line which meets the
first three ; then as the moments of the forces vanish about any
straight line, and as the moments of the first three vanish about
q which meets their action-lines ; the moment of P 4 also vanishes
about it ; and consequently p t meets q. Let four several posi-
tions of q be taken, and let these be q l} q a , q 3) qi ; then the line p t
lies on all these lines. But this relation between the jt/s and the
q's is that which we know to exist between the generators of the
two classes of the hypei'boloid of one sheet ; viz. every line of
one class of generating lines intersects every line of the other
class of generators. Hence any four lines which are the action-
lines of an equilibrium-system of four forces lie on the surface of
a hyperboloid of one sheet.
As the cone is a degenerate form of a hyperboloid, so does it
give a particular case of the preceding theorem. In it the
PRICE, VOL. III. N
90 THE THEORY OP MOMENTS. [75.
action-lines of the forces pass through the same point, and they
are the generating lines of the cone.
75.] If the equilibrium-system consists of five forces, only
two independent conditions can be derived from the six equa-
tions of equilibrium ; and consequently if the action-lines of
four forces are supposed to be given, we have only two con-
ditions for the determination of that of the fifth force; and
accordingly two others are required ; these may be that the line
should pass through a given point or lie in a given plane.
If six forces constitute an equilibrium-system, then only one
condition can be obtained from the six equations of equilibrium ;
and consequently if the action-lines of five forces are given, that
of the sixth force must satisfy three other conditions; that is, it
may lie on three given straight lines, or it may pass through
a given point and intersect a given straight line.
Six straight lines fulfilling the condition requisite that they
should be the action-lines of forces of an equilibrium-system are
said, by Professor Sylvester *, to be in involution ; and certain
geometrical relations concerning them have been discovered by
him, whereby he has arrived at a geometrical construction of the
sixth, when five are given. M. Chasles has added to Professor
Sylvester's paper some remarks which well deserve attention.
If an equilibrium system consists of seven forces, the ratios of
the forces can be determined from the six equations of equi-
librium in terms of the elements of the action-lines of the forces ;
and if an arbitrary magnitude is given to one of the forces those
of all the other forces will also be given.
76.] We now come to the second case mentioned in Art. 70,
viz. when B, = and G has a finite value. Here it is to be
observed that K is independent of the origin and of the coor-
dinate axes ; and consequently if n = at any one point, this
circumstance holds good for all places of the origin and for all
positions of the coordinate axes ; and accordingly E is an inva-
riant. G, however, generally depends on the position of the
origin; but is an invariant when R = ^because the system of
forces is in this case reducible to a couple of which G is the
moment ; and theorems already demonstrated shew that the effect
of a couple is the same so long as its moment is unaltered and
its rotation-axis is parallel to a given straight line.
* Comptes Rendus, Tome LII. p. 741. 1861.
77-] THE THEORY OF MOMENTS. 91
The following process also proves that if R = 0, G is an in-
variant :
Let the origin be transferred to (x^y*, z ), and let L O , M O , N , G O
be the values of L, M, N, G corresponding to the new origin ; then
L = s.p{(y y c ) cosy (zz,} cos /3}
= S.P ( y cos y z cos /3) y 2.P cos y + z s.P cos /3 ;
.'. LO = L Zy +Y* j'
M = M X2 + z# ; . (128)
N = N Ytfo + X^o^
and since R=0, x = Y = z=0; consequently L O = L, M O = M,
N = N, G = G, and the moment of the resultant couple is the
same for all points in space; and thus the system is always
equivalent to a couple whose moment is G.
77.] The third case is that in which the system is reducible
to a single force of translation. If at the arbitrarily chosen
origin, G=0, and R has a finite value, in reference to that origin
the system has a single resultant of translation acting at that
origin ; but since G depends on the position of the origin, as
(128) shew, some condition or conditions are required so that the
reduction may hold good for all origins,
In reference to any arbitrarily chosen origin let E be the
single force of translation to which the system is reducible ; let
(x, y, z] be its point of application ; a, b, c the direction-angles
of its line of action ; r the perpendicular distance from the
origin on that line ; so that
r 2 = (y cos c z cos 6)* + (z cos a x cos c) 2 + (x cos I y cos a) J .
Let there be introduced at the origin two equal and opposite
forces, each of which is equal to E, and whose line of action is
parallel to that of R : so that we have now R acting at the
origin, and a couple whose moment is nr ; and resolving each
of these along the three coordinate axes, and equating the re-
solved parts to the corresponding parts of the aggregate of the
impressed forces, we have
R cos a = S.P cos a *= x,
E cos b = 2.P cos /3 = Y,
R cos c = S.P cos y = z ;
N 2
92
THE THEORY OF MOMENTS.
[77-
C Z COS 1} = L = ZyXZ, "I
-R.(Z cos a tfcos c) = M = X2 zz, I (129)
R (x cos y cos a) = N = y x'S.y ; J
These equations are not independent, and so do not assign de-
finite values to x,y, z : they are subject to a condition ; for if we
multiply them severally by x, Y, z, we have
LX + MY + NZ = 0; ( 13 )
and this relation is one which the forces must satisfy if they are
reducible to a single resultant of translation.
Now LX + MY + NZ is an invariant; being independent of the
position of the origin, and of any particular system of coordinate
axes. From (128) it is evident that it is independent of the
position of the origin ; for from those equations
L X + M Y + N Z = LX + MY + NZ.
It is also independent of the position of the coordinate axes ;
for let a new system of axes, say of x', y , /, originating at the
same point be connected with the former by the system of di-
rection-cosines given in the following scheme :
y
(131)
Let x', Y', z', if, M', N' be the values of x, Y, z, L, M, N re-
spectively in reference to the new coordinate axes ; so that
x = a l x' + t>! Y / + c l z',
N = <Z S L +
with also corresponding inverse systems ; so that
LX + MY+NZ = (a l x' + i, Y' + C,Z')L + +
= xVf Y'M
78.] THE THEORY OF MOMENT& 93
and thus LX + MY + NZ is an invariant for all positions of the
origin and of the coordinate axes ; and if it vanishes, the system
is reducible to a single force of translation.
78.] Let this invariant be denoted by KR, so that, R being
constant, K is also constant : that is, let
LX+MY + NZ = KRj ( 132 )
then we have the following interpretation of K. Replacing
L, M, N, x, Y, z by their values given in (101) and (106), we have
RG (cos a cos A + cos /3 cos p + cos y cos v\ = KR ;
consequently if <f> is the angle between the action-line of R and
the rotation-axis of G,
K = GCOS$; (133)
that is, K is the component of G along the action-line of R ; and
this is consequently constant for all origins and for all systems
of coordinate axes.
As K = 0, when the system of forces is reducible to a single
resultant, therefore, from (133), </> = 90; that is, the rotation-
axis of the resultant-couple is perpendicular to the action-line
of R, and consequently the action-line of the resultant of trans-
lation lies in the plane of the forces of the resultant couple;
which is the circumstance alluded to in the fourth case in
Art. 70. Thus there are three forces acting in the plane of the
couple : viz. R, and the two forces of the couple. These may
evidently be compounded into a single force. As the arm of
the couple may be turned round in its own plane without alter-
ing the effect of the couple, let it be so arranged that the line of
action of each of its forces may be parallel to that of R ; and thus
if R' is a force and a an arm such that R'O = G, we shall have
three parallel forces R', R', and R acting in one and the same
plane, and these manifestly have a single resultant, whose mag-
nitude is R.
Its position, or the equations which determine the position of
the action -line of this resultant, may be found as follows : As
(x, y, z) in (129) is any point in the line of action of R, (129) are
the equations to that line ; and they may bejput into the follow-
ing form :
M N = X + Z X
M N
__ x+y+z .
X+Y4Z '
94 THE COMPOSITION OF [79.
and therefore from the symmetry of the right-hand member
M N N L L M
v- A.* I *v i/ "v I ~v \ rz
X+Y+Z X+Y + Z X + Y + Z
Or the equations (129) may be put into the following form:
multiplying the second by z and the third by Y, and subtracting,
we have
MZ NY =
= x (x# + \y + zz) #E a ;
MZ NY
E a
X E 2
MZ NY NX LZ LY MX
(134)
and either system is that of the equations to the line of action
of the single resultant, which is plainly parallel to that of the
resultant E acting at the origin.
If L = M s= N = 0, that is, if G = 0, then K = identically,
and the condition requisite for a single resultant of translation
is satisfied ; in this case the resultant passes through the origin.
79.] If the impressed forces are parallel, the condition (130)
is satisfied, and the system admits of a single resultant of trans-
lation. Let the forces be PJ, P 2 , ... P W , and be applied at the
points (x lt y lt z v ), (x t , y a , z a ), . . . (x n , y u , z n ) ; then
X = 2.P cos a = cos a 2.P, "
Y = 5.PCOS/3 = COS/32.P, (^35)
z = 2.P cos y = cos y 2.P; ^
and consequently from (135),
x
cosfl = - = cos a, cos 6 = cos ft, cose = cosy; (136)
E
that is, the resultant of translation at the origin is equal to the
sum of all the impressed forces, and acts along a line which is
parallel to the lines of action of the components. Again,
79-] PARALLEL FORCES. 95
L = COSCS.Py COsis.PZ, "I
M = COS02.P2 COSCS.Ptf, I (137)
N = cos^s.Ptf cosas.P^j J
{cos a s.P# + cos b s.py + cos <? S.P.Z} * ; (138)
and hereby may cos A, cos/*, cos y, equations (106), be found.
From (137) we have
L cos a + M cos b + N cos c = ;
and therefore from (136),
LX + MY-f NZ = 0,
whletris the condition requisite that the system should be redu-
cible to a single force of translation. Let R be this force ; a, d, c
the direction-angles of its line of action ; (x } y, z) its point of
application; then introducing at the origin two equal and op-
posite forces, each of which is equal to R and acts along a line
of action parallel to that of R, we have a force acting at the
origin equal to R, and in a parallel line of action, and a couple
each of whose forces is R, and whose arm is r, where
r 2 = (y cose 5cos$) 2 -f (zcosa a? cos c) 2 -t- (# cos # jFcosa)",
and the direction-cosines of the rotation-axis of which are
ycosc zcosb zcosa a- cose xcosbycosa^ /in Q \
then, as this system is to produce the same effect as the aggre-
gate of all the impressed parallel forces, we have
R cos a = cos a 2.P, ~j
Rcos/5 = COS/32.P, I (140)
R cos c = cos y 2.P ; J
whence squaring and adding,
R* = (2.p)% and therefore R = 2.P. (141)
cos a = cos a, cos b = cos ft, cos c = cos y ;
.-. a = a, b = /3, c y. (142)
Also
L = R(ycosy 0COS/3) = cosy 2.Py cos/3s.P2, 1
M = R(0 cos a ircos y) = cosaS.Pz cosy 2.P#, ^ (143)
N = R (x cos 13 y cos a) = cos /3 2.p# cos a 2.Py. J
Thus (141) and (142) assign the magnitude and direction-angles
96 THE CENTRE OP PARALLEL FORCES. [80.
of the line of action of the single resultant ; and as (a; y, z) is
any point in that line of action, (143) are the equations to it;
and the resultant is defined in all its incidents.
80.] Another property of a system of parallel forces requires
notice. In the preceding Article the magnitude, line of action,
and direction of the resultant have been deduced from the similar
incidents of the acting parallel forces ; and the fourth incident,
viz. the points of application, have not been brought under con-
sideration. In (143), which are the equations to the line of
action of R, (x, y, z) is any point in that line. Suppose, however,
all the forces to act at definite points, so that (#, y, z] in the
right-hand members of (143) have given values; also suppose
the lines of action of all the forces at their points of applications
to be turned through equal angles in the same or in parallel
planes, so that the system consists of parallel forces after the
change of line of action ; and consequently has a single resultant.
Now the magnitude of this resultant is equal to the sum of those
of the given forces, and the line of action is parallel to those of
the acting forces ; and both these quantities are independent
of the particular system of coordinate axes, consequently a, ft, y
are indeterminate, and the point of application of R must be
consistent with this condition. But from (143)
R# s.Ptf ~&y 2.Py RZ 2.P2
= -Z- ~- -~ (144)
cos a cos p cos y
/. R# 2.P# = ~R,y 2.Py = nz 2P2 = 0; (145)
2.P# 2.PW 2.P.Z
x = - -, V=~ -, z=- -; (146)
2.P 2.P 2.P
the point (x, y, z) is the point at which the resultant is applied
in all these cases, and consequently is called the centre of the
parallel forces.
The following are examples in which the centre of parallel
forces is determined :
Ex. 1. Four parallel forces 2, 4, 6, 8 are applied at the angles
of a square the length of the side of which is 2 a : find the centre
of these parallel forces.
Let the plane of the square be the plane of (x, y}, and let the
origin be at the centre of the square. Let (a, a, 0) be the point
of application of 2, ( a, a, 0) of 4, and so on ; then
8 1.] COMPOSITION OP FORCES IN SPACE. 97
5.P = E = 20 ;
2.P.T = ; s.P^ = 8a; 2.P3 = :
.. __ .
81.] The last case mentioned in Art. 70, viz. that in which R
and G have both finite magnitudes, remains for discussion. In
reference to an origin and a system of coordinate axes, both
of which are arbitrarily chosen, the system of forces is reduced
to a force of translation acting at the origin, and to a couple
whose moment is G, the line of action of R and the rotation-axis
of G being given by (101) and (106).
Whatever point is taken as the origin the magnitude of R is
the same ; all its lines of action are parallel, and its direction
is the same.
But the value of G varies as the place of the origin varies,
because L, M, N depend on the coordinates of the points of appli-
cation of the forces ; and if L O , M O , N O are the values of L, M, N
at the new origin (#, y c , z ), then by Art. 76, the new axes
being parallel to the former,
L = L-Zy +Y2 , "j
M = M X2 +Ztf , ^ (147)
N = N Y# + xy ;J
also if the axes are changed, see Art. 78,
LX + MY + NZ = L'X' + M'Y' + N'Z' = KR; ( 148 )
and if < is the angle between the action-line of R and the rota-
tion-axis of G,
GCOS $ = K; (!49)
so that the resolved part of every moment-axis along the line
of action of R is constant. These are properties of G which have
already been investigated.
And further let it be observed that of all axes passing through
a given point, that corresponding to G is the one whose moment
or moment-axis is the greatest ; for the moment of the impressed
couples with respect to a rotation-axis inclined at an angle to
that of G is G cos 0, as is plain from the law of resolution of such
moment-axes ; and as G cos 6 is less than G, it follows that of all
lines passing through a given point, the rotation-axis of the
resultant couple is that with respect to which the moment-axis
is the greatest. For this reason G is called the complete or
PRICE, VOL. III. O
98 THE CENTRAL AXIS. [82.
principal moment-axis with reference to the point which is called
the moment-centre. Hence also we infer that at a given moment-
centre the moment-axis is the same for all axes which are in-
clined at the same angle to the axis of the principal moment ;
that is, all axes of equal moment with reference to a given
moment-centre form a right circular cone which has the axis
of principal moment for its axis of figure.
82.] Since G cos <f> = K = a constant, G has its least value
when cos </> has its greatest ; that is, when < = 0, and when the
rotation-axis coincides with the line of action of R.
Let (# , ^ , ) be the moment-centre at which this circum-
stance takes place ; then
L^_MO_NO__^O __ K
X~Y'Z""R~R'
and replacing L O , M O , N O by their values given in (128),
L Zy -fY0 M X^o + Z^o N YiT +XY K
X Y Z ~ R J
whence we have
NY MZ LZ NX MX LY
(151)
R-
which are the equations to a straight line whose current coor-
dinates are # , ^ , z ; and as no other relation is given between
ar , y 0) z<>, that point may be anywhere on this line ; and con-
sequently this straight line is the locus of those moment-centres
at which the rotation-axis of the principal moment coincides
with the line of action of the resultant of translation. This line
is called the central axis of the system (Hauptdrehlinie) ; and
any plane perpendicular to it is called a central plane. If the
system is reducible to a single force of translation, that force
evidently acts along the central axis; and for this reason (134)
and (152) are identical.
At all points of this line the principal moment is a minimum
and is K ; and K is called the central principal moment : and its
rotation-axis coincides with the line of action of R. Consequently
The central axis is that line along which the system of forces
produces a pressure of translation = R; and which is also the
rotation-axis of the resultant couple whose moment is K. Thus
the forces produce a shifting pressure along the central axis and
a tendency to make the body rotate about the same line. This is
83-] THE CENTRAL AXIS. 99
indeed the most simple form in the nature of the ease to which
the system of forces can be reduced, and from this point of view
the result is most important ; but the complexity of the equations
(152) often precludes us from making that use of them which we
might, were they more simple, and the reduction to a single
force of translation and to a couple whose moment-axis is G is
employed in preference.
These results might have been arrived at from investigating
the locus of those moment-centres at which the principal mo-
ment is a minimum, viz. when x 0) y , z vary so that
is a minimum ; and we should have the following results :
(1) With respect to moment-centres taken at any point in
space, the moment of the rotation-axis coincident with the
central axis is the least. Thus K is the minimum maximorum
moment.
(2) If any point of the central axis is taken as the moment-
centre, of all axes of rotation passing through that point, that
coincident with the central axis has the greatest moment.
83.] The following is another mode of demonstrating the pre-
ceding results. In fig. 42, let o be the original moment-centre ;
OR the line of action of the force of translation acting at it ; OG
the moment-axis of the resultant principal couple at o : let
GOR = ty, so that
LX + MY + NZ
COS (b = - -',
GR
resolve OG into two parts, one along, and the other perpen-
dicular to OR; then the part along OR is G cos <, and that per-
pendicular to OR is G sin 0; the rotation-axis of G cos</> is OR,
and that of G sin < is a line in the plane containing OG and OR :
at o draw OP perpendicular to this plane, and take OP of a
length such that RxOP = Gsin$; at p introduce two equal
and opposite forces, each of which is equal to R, and whose line
of action is parallel to that of R : then the couple whose arm is
OP, and whose force is R, neutralizes the couple whose moment-
axis is ON; and there remain (1) the force R acting at p, and in
a line parallel to the original line of action of R, and (2) a couple
whose moment-axis is G cos <, and whose rotation-axis is along
OR. Let the rotation-axis be transferred parallel to itself so as
to pass through p, and we have finally a force of translation a
O 2
100 THE CENTRAL AXIS. [83.
acting along PR, and a couple whose rotation-axis is along the
line of action of R, and whose moment-axis is G cos </>, which = K.
Thus the line through p, and parallel to OR, is the central-axis ;
and its equation may thus be found. It passes through p, and
its direction-cosines are proportional to x, Y, z. Since OP = - sin <p,
K
and OP is perpendicular to OR and to OG, the coordinates of P are
NY MZ LZ NX MX LY
consequently the equations to PR are
NY MZ LZ NX MX LY
X Y Z
which are the equations to the central axis.
As OP is perpendicular to both OG and PR, it is the shortest
line between the rotation-axes of G and of K.
If OP = r, we have
= G" : (154)
therefore G, the principal moment at a point, is the same at all
points for which r is constant ; that is, at all points equally
distant from the central axis; and therefore the locus of all
moment-centres, at which the principal moments are equal, is a
circular cylindrical surface, of which the central axis is the axis
of figure ; and at all points along the same generating line of
this cylinder, the rotation-axes of the principal moments are
parallel, and all therefore lie in the plane touching the cylinder
along the generating line.
But the directions of the rotation-axes change as we pass from
one generating line to another ; for since <J> is the angle between
the central axis and the rotation-axis of the principal moment
corresponding to a moment-centre at a distance r from the
central axis we have from (153)
T> A*
tan<f> = ; (155)
K
and this is therefore constant for all points of the cylindrical
surface mentioned above; and as the direction-cosines of the
central axis are proportional to x, Y, z, and those of the rotation-
axis of the principal moment G to L, M, N, these lines in general
84-J THEOREMS ON MOMENTS AND MOMENT-CENTRES. 101
do not meet : and therefore if a section is made of the cylin-
drical surface mentioned above by a plane perpendicular to the
central axis, and the principal moment-axes are drawn for the
moment-centres situated in this circular section, they will form
a hyperboloid of revolution of one sheet, having the central axis
for its axis of figure.
84.] These theorems however, and others of a like kind, may
be investigated more easily by the following process :
Let a point in the central axis be taken as the origin, and let
the central axis be the axis of z ; so that the system of forces
consists of a force of translation R acting along the 2-axis, and
a couple whose moment is K and whose rotation-axis is the 2-axis
also. At (# , y , 0) let two equal and opposite forces, and each
equal to R and acting parallel to the 2-axis, be introduced ; and
let G be the moment of the resultant couple, of which let L ,
M , NO be the axial components : then
L O = R^O, M O = R# O , N, = K; (156)
.'. G ' = R'(# 3 +y ! <)+K*. (157)
Let a?o*-f^o a = **> and let < be the ^-direction angle of the
rotation-axis of G O ; then
N = GO cos< = K; (158)
G sin = (L 2 +M 2 )* = Rr; (159)
V -4-^0* = ^ (tan <). ( 16 )
B
From these equations we have the following theorems :
(1) All moment-centres of equal principal moment are on the
surface of a right circular cylinder, of which the central axis is
the axis of revolution.
For from (157) we have
the right-hand member of which is constant, if G O is constant ;
and consequently all the moment-centres, at which G O is con-
stant, lie on the surface of the right circular cylinder whose
equation is (161).
Also the greater G O is, the greater is the radius of the cylinder,
and the farther is the moment- centre from the central axis ; and
the least value of G is K.
(2) At all points of equal principal moments, the rotation-
axis is inclined at the same angle to the central axis.
102 THEOREMS ON MOMENTS [84.
This follows from (158), because cos <J> = ; hence </> is con-
Go
stant when G is constant, and the equation to the cylinder in
(161) becomes
v . 1 .. t . == * a ( fam *)'. (162)
B 2
Also at all points in the same generating line of this cylinder,
the principal rotation-axes are parallel, and lie in the plane
which touches the cylinder along that generating line. Hence
also the larger G becomes, the smaller is cos $, and if G O = oo,
< = 90; and as the tangent of the angle between the rotation-
axis and the central axis is proportional to the distance of the
moment-centre from the central axis, the rotation-axis is per-
pendicular to the central axis only when the moment-centre is
at an infinite distance.
(3) The rotation-axes of the principal moments for the mo-
ment-centres lying in the circle given in (161) are in the surface
of a hyperboloid of revolution of one sheet of which the central
axis is the axis of figure.
By reason of (156) the equations of the rotation-axis corre-
sponding to the moment-centre (x , y Q} 0) are
^H^l=fc^ = -; (163)
from which and (162), eliminating # andy , we have
x* +y*z* (tan 0) 2 = (tan 0) 8 ; (164)
it
which is the equation to a hyperboloid of revolution of one
sheet, of which the .z-axis, that is, the central axis, is the axis of
figure.
This theorem is only a special one of a general class; viz.
given the locus of the moment-centre to find the equation to
the ruled surface generated by the corresponding rotation-axis
of the principal moment. For from (163) we have
_ (
consequently if the moment-centre moves along a given curve
in the plane of (x, y] a relation is given between x and y , and
the substitution of the preceding values of x and y in that
relation will give the equation of the ruled surface which is
generated by the rotation-axis of the principal moment. The
following theorem is an example of such a ruled surface :
85.] AND MOMENT-CENTRES. 103
(4) For all moment-centres lying in a straight line cutting
the central axes at right angles, the corresponding rotation-axes
of the principal moments lie on the surface of a hyperbolic para-
boloid.
Let the straight line on which the moment-centre is be the
axis of #/ so that the moment-centre is (x 0) 0, 0) ; consequently
L O = 0, M O = Ba? , N = K; and the equations to the rotation-
axis of the principal moment are
.'. Ky = KXZ; (167)
which is the equation to a hyperbolic paraboloid.
Also generally if the moment-centre moves along a straight
line which is perpendicular to, but does not cut, the central axis,
the rotation-axis lies on a surface of the second degree.
(5) The line whose equations are (163) is evidently perpen-
dicular to that which passes through the origin and (xo,y a )'>
consequently this latter line is the shortest distance between the
central axis and the principal rotation-axis corresponding to
(*o, #)
(6) The plane which contains the line of action of the re-
sultant and the principal rotation-axis at a given moment-centre
is perpendicular to the line drawn from that centre at right
angles to the central axis.
85.] The preceding theorems supply means for investigating
certain general properties of planes and lines with reference to
moment-centres, and also criteria as to the reduction of systems
of forces to a force of translation, and to a couple whose rotation-
axis may coincide with a given line or be perpendicular to a
given plane.
Whatever is the position of a plane, that plane is always a
momental plane with reference to some point in itself which is
the corresponding moment-centre : that is, the system of forces
may always be reduced to a force of translation acting at a
point in the plane, and to a couple the rotation-axis of which is
normal to the plane.
If the plane is perpendicular to the central axis, it is a central
plane, and the theorem is self-contained.
If the plane is not perpendicular to the central axis, at the
point where the central axis intersects it, let a line be drawn in
104 THEOREMS ON MOMENTS [86.
the plane perpendicular to the central axis ; and along this line
let a distance r be taken of such a length that if < is the angle
between the central axis, and the normal to the plane,
r = 5tan^; (168)
then the point at the extremity of this line is the moment-
centre ; and the normal to the plane at it is the principal rota-
tion-axis ; and the line parallel to the central axis is the line of
action of the resultant.
When the equation to the plane is given, the coordinates of
its moment-centre may be found by the following process :
Let the equation to the plane be
AX + xy + Cz = D; (169)
and let the moment-centre in the plane be (#, y , z ) ; then as
the equations to the corresponding rotation-axis are
# a'o _ yy __ z z
ny ~ R# O K
and as this line is perpendicular to the given plane, we have
A B _ c .
R^o R# K*
BK AK D / 1 i,->\
' * = S' '=- z = c'
which assign the moment-centre of the plane (169).
The value of z shews that the moment-centre lies in the in-
tersection of the given plane by a plane parallel to that of (x, y),
and passing through the point at which the given plane cuts
the central axis ; and the line of intersection of these two planes
is perpendicular to both the central axis and the principal rota-
tion-axis. For a series of parallel planes, the values of .r and y 9
are constant; consequently all the moment-centres lie in a
straight line parallel to the central axis.
Hence also if (#, y , z ) is the moment-centre, the equation
to the corresponding momental plane is
Ry.tf-fRtfoy+K^ 2 o ) = o. (171)
86.] And to consider this problem more generally, let the
system be referred to an origin and coordinate-axes taken arbi-
trarily ; then from the comparison of the direction -cosines of the
normal of the plane (169), and of the axial components of the
principal moment-axis given in (128), we have
87.] AND MOMENT-CENTRES. 105
L Z^o+Y^o M X-Zo + Z-r, N Y# + X^
c
(172)
C
LX + MY + NZ
AX + BY + CZ
_ DX + BN CM 1
vft ^~
DY + CL-AN
AX+BY + CZ
DZ + AM BL
AX + BY + CZ '
Hence the coordinates of the moment-centres of the three
coordinate planes are,
Of the plane (y, z}, x= 0, y= -, z= - ;
(z x} - *=0 z=--' VU74)
\ f Jt Y ' "7 ' w ' I * *
M L
all which points evidently lie in the plane whose equation is
LOT + M^ + NZ = 0,
and which is the momental plane of the origin ; and hence also
we infer that the moment-centres of the three coordinate planes
lie in a plane passing through the origin of coordinates.
Also if G O is the principal moment-axis with reference to the
point (x 9) y Q) z c ) given in equations (173),
G = ^ KR. (175)
AX + BY + CZ
Hence if & x) G tf , G 2 are the principal moment-axes of the planes
of (y, z), (z, x) } and (x } y] respectively,
K Ji, K I! K K / i /> \
*-.. r i *. = :T J ^ = T ;
the moment-centres of these planes to which these moment-axes
correspond are given in (174).
87.] In Article (85) it is demonstrated that if
= D
is the equation to a momental plane, ( , . -) is its mo-
V CR CR c'
ment-centre ; and also that, if (# , y Q) z ) is a moment-centre,
-Ry # + R# 0< y + K(2 Z ) = (178)
is its momental plane. Now from these relations problems of
PRICE, VOL. III. P
106 THEOREMS ON MOMENTS [87.
the following nature arise: (1) Given the locus of the moment-
centres, find the envelope of the corresponding momental-planes ;
this will evidently be generally a developable surface, and the
problem is the discovery of its equation ; and (2) Given the law
according to which momental planes are drawn, to find the
locus of the corresponding moment-centres. The following are
examples of these problems :
Ex. 1. To find the envelope of the momental planes, when
the locus of the corresponding moment-centres is a plane.
Let (x 0) y , z ) be the moment-centre; and let the plane in
which it always is be
A# -f B^ + CZ = 0, (179)
the origin, the position of which on the central axis is arbitrary,
being taken at the point where the central axis intersects this
plane. Consequently making # , y , z 9 to vary, and equating
the ratios of the coefficients of the differentials of # , y oy z in
(178) and (179), we have
ay Ear K
A B C '
.'. x , y = z = 0; (180)
CR' CR*
which assign a point in the plane of (179), and which lies in the
line of its intersection with the plane of (x, y} ; and this point
is, as (170) shew, the moment-centre of the plane (179); conse-
quently all the momental planes, corresponding to the moment-
centres in (179), pass through the moment-centre of that plane,
which is thus the envelope of them.
Let lines drawn in a plane from the moment-centre of the
plane be called rays ; then from tne preceding result the follow-
ing theorems arise :
If the moment-centre of a plane lies in the line of intersection
of it with another plane, the moment-centre of the latter plane
also lies in the same line of intersection.
The momental planes of all moment-centres lying in a ray
intersect in that ray ; or, in other words, a ray is the locus of
the moment-centres of all planes passing through that ray.
The moment-centres of all planes which pass through one and
the same point lie in a plane which is the momental plane of
the point through which all the planes pass.
If the moment-centre is in the plane of .(a-, y), so that in (179)
87.] AND MOMENT-CENTRES. 107
A = B = 0, then from (180) the origin is the moment-centre,
and the origin of rays ; so that all the momental planes corre-
sponding to moment-centres in the plane of (x, y) pass through
the origin.
Since from (180) we have Aar + B^ = 0, and this is inde-
pendent of c, all the moment-centres of the planes intersecting
the plane of (x, y] in the line A.x + '&y = 0, lie in that line : and
as this line passes through the origin which is the moment-
centre of the plane of (x, y), it is a ray of that plane ; conse-
quently the ray is the locus of the moment-centres of all the
planes passing through that ray.
Ex. 2. To find the envelope of the momental planes corre-
sponding to moment-centres, of which the locus is a spherical
surface ; whose centre is on the central axis.
Let the equation to the sphere be
V+y a + V = '; (181)
then the envelope of the plane (178), when x a) y t) z a are subject
which is the equation to a hyperboloid of revolution of one sheet,
the -axis being the axis of figure.
Ex. 3. If the locus of the moment-centres is the ellipse
x a y ~
^- + ^j = I, the envelope of the corresponding momenta!
u.
planes is the elliptic cone
B a (a 8 y + 4*d?)--K a s" = 0.
Ex. 4. To find the envelope of the momental planes, when
the locus of the moment-centres is a straight line.
Let the line which is perpendicular to both the central-axis
and the locus-line of the moment-centres be the axis of x, and
let r be the perpendicular distance between those two lines ;
then the line is parallel to the plane of (y, z) and cuts the axis
of a? at a distance = r from the origin. Let a be the angle at
which it is inclined to the plane of (x, z) ; so that the equations
to the locus of the moment-centre (x , y , z ) are
x ~ r _ y* _ z . (182)
sin a ~ cos a'
then replacing x 9 and y by these values in the equation of the
momental plane, we have
Z 9 ) = 0, (183)
p 2
108 THEOREMS ON MOMENTS [87.
whence, as z* varies, we have
= 0; R#tana+K = 0;
Kcota K f-, ol \
.-. x = --- , y = -- z; (184)
R Rr
which express a straight line cutting the axis of x at right
angles at a distance = - on the negative side of the origin,
Hi
and inclined at an angle tan -1 ( -- ) to the plane of (y, z); and
thus lying on the opposite side of the plane of (y, z) to that on
which (182) is.
Consequently all the momenta! planes whose moment-centres
are on (182) pass through the line (184), which is the envelope
of them ; and conversely, the moment-centres of all momental
planes which pass through the same straight line lie in a
straight line.
Now these two lines have many remarkable relations. If
(184) is the locus of moment-centres, all the corresponding
momental planes intersect along (182). For let (x lf y^ be a
moment-centre on (184), and let -- = r l} -- = tan a, :
K, Rr
so that the equations to (184) become
K cot a K
#= -- - = r 1 m t y -- z = tan c^ z.
R Rr
Consequently the equations to the line of intersection of the
corresponding momental planes are
K K
x = -- cot a x = r : y -- z = tan az,
R Rfj
which are the equations (182). Thus we have the following :
The momental planes of all moment-centres on (182) intersect
in (184), and the momental planes of all moment-centres on
(184) intersect in (182).
As these two lines have reciprocal relations, they are called
reciprocal lines, (gegenlinien.} It is evident that to every line
there is a reciprocal line.
Hence also it appears that the line, viz. the #-axis, which is
perpendicular to both of them is also perpendicular to and inter-
sects the central axis.
If r and i\ are, irrespectively of sign, the perpendicular
distances between the central axis and the two reciprocal lines,
and a n and a, are, also irrespectively of sign, the angles at
87.] AND MOMENT-CENTRES. 109
which these lines are inclined to the central axis, we have the
following relations :
K K
r,=-cota : tana,= - ; (185)
R Rr
j^
.'. /! tana = r tana, = - (!86)
R
If two reciprocal lines are coincident, this line is a ray of all
planes passing through it. The analytical condition is
Rrtana-fK = 0.
If two reciprocal lines are perpendicular to each other,
a + a 1 = 90; ., K 3 + R 2 r r x = 0. (187)
Ex. 5. Find the locus of the moment-centres of a series of
planes, which intersect in one and the same straight line.
Let the equations of the line in which they intersect be
x r = ; yzta.na=Q;
so that the equation to the planes which pass through this line
X(# r}+y zta.na = 0,
where X is an indeterminate quantity; then by (170) the co-
ordinates of the moment-centre are
K XK Xr
/ti __
If
, ,
R tan a R tan a tan a
.'. y=- Z', (188)
Rr '
which shew that all the moment-centres are in the line which is
reciprocal to that in which the planes intersect.
Ex. 6. Find the locus of the moment-centres of all the planes
which touch the sphere # 2 +y 2 + .z 2 = a 2 .
Let the equation of one of the tangent planes be
x cos a -\-y cos /3 -f- z cos y = a ;
so that by (170), if (x a) y , z ) is the moment-centre,
K cos 8 K cos a a
np ^ .. _ 4/ ^ ^_ _
R COS y R COS y COS y
R a?/ R aff a
.'. cosa= -- -^ , cos/3=- , cosy = ;
K - ,, K ZQ Zg
.-. R'a a (tf 2 +y 2 )-K 2 (V-a 2 ) = 0; (189)
which is the equation to a hyperboloid of revolution of two
sheets, the axis of figure of which is the central axis.
110 THEOREMS ON MOMENTS [88.
88.] Although every point in space may be a moment-centre
and have a momental plane and a principal rotation-axis passing
through it, and although every plane may be a momental plane,
and have its moment-centre in it, yet every straight line may
not be a principal rotation-axis, and may not consequently have
a moment-centre corresponding to itself in it. This result is
evident from the properties of principal rotation-axes which are
proved in Art. 84 ; for every principal rotation-axis touches a
cylinder whose axis is the central axis, its corresponding mo-
ment-centre being the point of contact, and it is inclined to the
central axis at an angle <p such that
R f
tand>= , (190)
K
if r is the perpendicular distance between the given line and the
central axis ; and this is a relation between r and < which all
straight lines evidently do not satisfy.
The conditions however to be satisfied when a straight line is
a principal rotation-axis, and also the coordinates of its moment-
centre, may be ascertained in the following manner :
Let the equations to the straight line be
z^ = i* = fzf. (191)
I m n
and let (x , y<>, z ) be the moment-centre on it. Then comparing
(191) with (156) and (157), we have
I m n (I 1 + m^ 1
= = - = ', - = ; (192)
Ry Rtf n K -R(.X ao.^i G
and from (191),
which assign the moment-centre. Also from the two values of
z, we have the condition
K = nu(amdl). (195)
The geometrical meaning of this condition is that if <f> is the
angle at which the line is inclined to the central-axis, tan d>= ;
K
for from the first two members of (192) it appears that the line
drawn from (o- , y ) at right angles to the central axis is also
88.] AND MOMENT-CENTRES. Ill
perpendicular to the given straight line j so that this line is the
shortest distance between them ; let it be equal to r; then
. . tan d> = .
K
Thus (193) and (194) assign the moment-centre ; and if G is
the principal moment at it,
GO* = R 2 r a + K 2 . (196)
If the origin and axes of coordinates are taken in the most
general position, and the equations to the straight line are
x a _ y b z e
7 - == >
I/ in, n
then, if this line is a principal rotation-axis, whose moment-
centre is (#, y , 2 ),
I m n
L Z^ +Y2 M X2 +Z# N
= l ^"?, (197)
KB,
whence # > y 0) z a may be determined; and the values are similar
to those given in (173).
If one of the coordinate axes, say the axis of x, is a principal
rotation-axis, m = n = ; y = z = ; consequently
M N
#o = -- = - >
Z Y
and the condition, when this is the case, is
MY + NZ = 0, (198)
and the moment-axis is L. A similar result is true of the other
axes.
In further illustration of the preceding conditions, we can
hereby shew that if two reciprocal lines are perpendicular to each
other, each is then a principal rotation-axis, the moment-centres
being on the axis of x in the configuration of Art. 87, Ex. 4.
For in this case, by (186) and (187),
K nr a
tan a =
E7>1 (199)
K Er
Rfo K- -
consequently both the reciprocal lines are principal rotation-axes.
112 REDUCTION OF A SYSTEM [89.
Let G and G t be the corresponding principal moment-axes ;
K = G cos a c = G t cos a 1}
= G! sin a ;
.-. _ + _ = __. (200)
G 4 G t 2 K 2
Since the product r r t is a constant, by (187), when the re-
ciprocal lines are perpendicular to each other, r +r l is a mini-
mum, when
/, = /> = 5. (201)
in which case a = a x = 45, and G = G l = K 2* ; thus the two
reciprocal lines are each inclined at 45 to the plane of (y, z).
Hereby also it may be shewn that the principal rotation-axes
at (r , 0, 0) and at ( r l) 0, 0) make equal angles with the re-
ciprocal lines at these points.
SECTION 5. The reduction of a system of forces in space to two
forces of translation acting along lines which are not in the same
plane.
89.] The reduction of a system of forces acting in space to
two forces acting along lines which are not in the same plane,
and consequently do not intersect one another, may be effected
in various ways. Each of course demonstrates the possibility of
the reduction. The following arise out of the processes of com-
position which have been employed in the preceding Articles.
Let us take the most general case of forces acting along lines
in space.
Let P be the type-force, and (x, y, z) a point in its line of
action, which we will suppose to be its point of application.
Let A, B, o be three points taken arbitrarily and fixed ; and let
us assume that the point of application of P is not in the plane
containing A, B, c. Let P at its point of application be equi-
valently replaced by three forces along lines passing through
A, B, c respectively ; and let all the forces be similarly resolved ;
then we shall have three groups of forces, corresponding to the
points A, B, c respectively, each group consisting of forces whose
lines of action have a common point. Let the forces of each
90.] TO TWO FORCES OF TRANSLATION. 113
group be compounded into a single force ; so that the system is
reduced to three forces acting each at an arbitrarily chosen
point: let these forces be respectively Q, R, s acting at A, B, c
respectively. Let D be a point in the line of intersection of the
planes ABR, ACS ; and let R be resolved into two forces, whose
lines of action are BA and BD; and let s be resolved into two
forces whose lines of action are CA and CD: thus the system is
reduced to three forces whose lines of action pass through A
and to two forces whose lines of action pass through D ; let each
of these groups be compounded into a single force; then we
have finally two forces whose lines of action pass through A
and D respectively, and evidently do not generally meet each
other.
The magnitudes and lines of action of these two final re-
sultants depend on the positions of A and D, and indeed of A, B, c ;
and as all these are arbitrary, so is the system of the two final
resultants arbitrary ; the extent to which the arbitrariness ex-
tends, that is, the determination of the conditions to which the
elements of these two resultants must be subject, will be in-
vestigated hereafter : at all events the system is not unique, and
the number of pairs of forces, which are equivalent to a system
of forces in space, is indeterminate.
90.] For a second way of reduction, let the forces and their
lines of action be referred to a system of rectangular coordinates.
Let P, as heretofore, be the type-force, and by virtue of the prin-
ciple of transmissibility let us assume it to act at the point
where its line of action intersects the plane of (x, y]. At that
point let it be resolved into two forces the lines of action of
which are in and perpendicular to the plane of (x, y] respectively.
Then all the forces having been similarly resolved, we shall have
(1) a group of forces the action-lines of which are all in the
plane of (x, y}, and which consequently generally admit of com-
position into a single force of translation; and (2) a group of
forces all the action-lines of which are parallel to the axis of z,
and which can be compounded into a single force of translation,
the magnitude of which is equal to the sum of the magnitudes
of the several components. Thus the system is reduced to two
forces of translation, the lines of action of which do not gene-
rally meet ; which, however, have the special property that the
lines of action are perpendicular to each other.
However, as the origin and the coordinate-axes are arbitrary,
PRICE, VOL. III. Q
114 REDUCTION OF A SYSTEM [91.
and as the choice of the coordinate-axis along and perpendicular
to which the forces are resolved is also arbitrary, so the system
of the two resultants is arbitrary ; and the number of ways in
which a system of forces can be reduced to a pair of forces,
whose action-lines are perpendicular to each other and do not
meet, is indeterminate.
The magnitudes and lines of action of these two resultants
may be determined in the following 1 manner :
Let p be the type-force, and (x, y, 0) its point of application :
also let sin 6 cos <, sin 6 sin </>, cos 6 be the direction-cosines of
its line of action. Consequently if p at its point of application
is resolved into components whose action-lines are in and per-
pendicular to the plane of (x, y), P sin and P cos are these
components respectively ; and they are applied at the point
(x, y } 0). Let all the forces be similarly resolved : and let R t
and R 2 be the two resultants respectively in and perpendicular to
the plane of (x, y}. Then
R a = 2.PCOS0; (202)
and if (x } y, 0) is a point in its line of action
;? 2.P cos = 2.P x cos 6, y 2.P cos 6 = 2.P^ cos 6 ', (203)
and compounding the forces whose lines of action are in the
plane of (x, y),
Rj 2 = (2.P sin B cos <) 2 + (2.P sin 6 sin <) 2 ; (204)
and the equation to its line of action is, see (60), Art. 58,
#2.P sinflsin <j> ^2.P sin 0cos $ = 2.P sin 6 (.rsin < ycos <). (205)
Thus the magnitudes and lines of action of R t and R, are de-
termined.
If the point (.?, Tf) given in (203) lies in (205), the lines of
action of R, and R 2 intersect, and as these may in that case be
compounded into a single resultant, the system of forces is
reducible to a single resultant. The substitution of (203) in
(205) leads to the condition (130), Art. 77.
91.] Again, if all the forces are reduced, as in AiH. 68, to a
single force of translation acting at an arbitrarily chosen origin,
and to a single couple, we may suppose one of the forces of the
couple to act at the origin, the other acting along a determinate
line parallel to the line of action of the former. Now the former
force and the resultant of translation may be compounded into
a single force acting at the arbitrarily chosen origin ; and thus
91.] TO TWO FORCES OF TRANSLATION. 115
the system is reduced to two forces of translation acting along
lines which do not meet.
If the arm of the resultant couple is turned in its own plane,
the point of application of one of its forces, viz. of that at the
origin, being unaltered, the resultant of that and of the original
resultant of translation will vary, and consequently the system
of pair of forces to which all the forces may be reduced is inde-
terminate.
The reduction, however, admits of the following simplification :
Let R be the resultant of translation at the origin, and let G be
the moment of the resultant couple, and let all the other symbols
be employed as in Art. 68 : let the arm of the couple be turned
in its own plane until it is perpendicular to the line of action of
E ; let R' and a be the force and the arm of the couple ; both of
these quantities being arbitrary, but subject to the condition
n'o = G. Then, if < is the angle between the line of action of R
and the rotation-axis of G, so that
LX + MY-fNZ K
cos d> = - = - , (206)
RG G
-j ^ is the angle between the action-lines of n and R', these
m
action-lines meeting at the origin. Let these forces be com-
pounded into a single force R" ; then
R" 2 = R 2 + 2 RR'sin + R' 2 ; (207)
and the system is reduced to the two forces R' and R", the lines
of action of which do not meet, and the shortest distance between
them being a which is perpendicular to both lines of action.
Also this reduction may be so arranged that the lines of action
of the two forces shall be perpendicular to each other. Thus, as
before, let the arm of the couple be perpendicular to the line of
action of R ; and let R be resolved into two parts R sin < and
R cos <f> respectively in and perpendicular to the plane of the
couple : so that there are, (1) three forces R', R', Rsin</> in the
plane of the couple, the lines of action of all of which are parallel
and are perpendicular to the arm of the couple, and the resultant
of which is R sin $, which acts in the plane of the couple, at
right angles to its arm, and at a distance r from the origin
along the arm, such that nr sin <f> = G ; and (2) the force
R cos whose line of action is perpendicular to the plane of the
couple.
Thus the system is reduced to the two forces R sin < and
2
116 REDUCTION OF A SYSTEM [92.
R cos <f> acting 1 along lines perpendicular to each other which do
not meet, and between which the shortest distance is r, where
(208)
As the line of action of R cos $ passes through the origin and
is perpendicular to the plane of the couple, its equations are
* = $=*- (209)
L M N
and as the line of action of R sin </> lies in the plane of the couple
and passes at right angles through the extremity of r which is
perpendicular to both the line of action of the original resultant
of translation and to the rotation-axis of the couple, its equa-
tions are
NY MZ LZ NX MX LY
X -- ; - V -- : - Z
G 2 X LK G a Y MK G*Z NK
Thus the lines of action of the two forces are determined, and
also the shortest distance between them.
As the equations to the line on which r lies are
V I/ V
= * = , (211)
NY MZ LZ NX MX LY
this line is perpendicular to the central axis whose equations are
given in (152), and also intersects it. Consequently we have
the following theorem :
If a system of forces is reduced to two forces of translation,
which act along lines perpendicular to each other, the shortest
distance between their lines of action intersects the central axis
at right angles.
The sole indeterminateness which is involved in this mode of
reduction arises from the arbitrary position of the origin. When
that is h'xed, all the quantities are assigned.
92.] Also if the system of forces is reduced to the force of
translation R acting along the central axis, and to the couple K
whose rotation-axis is the central axis, we may replace K by its
two equal and opposite forces each of which is equal to , if a is
the length of an arbitrary arm. Of these two forces let one
act along a line passing through the central axis, and of course
perpendicular to it ; then it and R may be compounded into a
single force R", such that
*"' = R + ^> ( 212 )
92.] TO TWO FORCES OF TRANSLATION. 117
and there remains the other force of the principal central couple
acting along a line, perpendicular indeed to the central axis but
not meeting it, and not meeting the action-line of R" ; and the
shortest line between the action-line of these two resultants is a,
which is such that, if R' is the force of K, R'a = K.
This reduction may also be effected more generally by the
following process : Let us suppose the central axis to be the
axis of z; and let R be replaced by two forces R, and R,,
the action-lines of which are parallel to the central axis, and
which pass through two points Q t and Q a on the axis of x at
distances r t and r, respectively from the origin, and on opposite
sides of it ; then we have
R =
R a R! + R,
Let the principal central couple be replaced by two equal forces
R' acting in opposite directions along lines passing through Qj
and Q a and parallel to the axis ofy; then
K = R'(r 1 + r 4 ). (214)
Thus there are now four forces, viz. R t and R' at Q,, and R,
and R' at Q a ; each pair acts in a plane perpendicular to the
ar-axis, and the action-lines of the forces in each pair are perpen-
dicular to each other : let Pj be the resultant of RJ and R' which
act at QJ, and let p a be the resultant of R a and R' which act at
Q a ; then
p 1 a = R 1 +R /a ; p a ' = R a ' + R''; (215)
so that the system is now reduced to the two forces PI and p,,
the shortest distance between the action-lines of which is r l + r t .
As to the action-lines of P, and P a ; let O l and 3 be the angles
between them and the central axis ; then
R' = P! sin #1 = P 3 sin a ; (216)
R 1 =P l COS^,j R a =P a COS0, ; ( 217 )
consequently P, cos0! + P 3 cos0 2 = R ; (218)
jr
! sin O l = P, sin 2 = - -; (219)
fi + ra
!/! cos^ = p a r a cos0, ; (220)
K
' '
r l r, Rr, r t
so that if r, and r a are given, the forces and their incidents are
completely determined.
118 REDUCTION OF A SYSTEM [93.
93.] In reference to this system of two forces to which the
general system has been reduced, the following theorems are
noteworthy :
(1) On comparing (221) with (186) it appears that the action-
lines of P! and P a are reciprocal lines ; consequently as the posi-
tion of a line is given when that of its reciprocal line is given,
so if the action-line of one force is given that of the other force
is also given.
(2) Let GJ and G 2 be the principal moments at Q t and Q, ;
then evidently,
Gj COS d 1 = G, COS 6 2 = K j
therefore from (220), -^- = -^-; (222)
T! P! ft ?2
which gives the ratio of the principal moments at Q! and Q 2 .
(3) The volume of the tetrahedron of which the line-repre-
sentatives of P! and P 2 are opposite edges is constant. For let
v be the volume, then
v=
rt _i_ 4
- {P! sinfli p 2 cos 63 + PI cos0, P 2 sin 2 }
KB,
= - (223)
which is constant ; and consequently the volume of the tetra-
hedron is constant whatever is the position of the two forces
which equivalently replace a system of forces.
If the volume of the tetrahedron vanishes, the two forces act
in the same plane, and the system is reducible, either to a single
force of translation, or to a couple : that is, either K = 0, or R=0.
Hence also it is evident that if four forces are in equilibrium,
the volume of the tetrahedron constructed on the line-representa-
tives of any two is equal to that of the tetrahedron constructed
on the line-representatives of the other two.
(4) If the action-lines of p t and P 2 are at right angles to each
other, then t -f 2 = 90 ; sin Q = cos O a ; sin 2 = cos 6^ ; and
consequently tan 0, tan 2 = 1 :
K* R R
r t r t = ; tan 6, r l -; tan 0, = -,-; (224)
it K. K.
whence if any one of the four quantities / r. i} O l} d. 2 is given,
all the others are given : as, however, the number of equations
93-] TO TWO FORCES OF TRANSLATION. 1 1 !)
connecting the unknown quantities is less by one than the
number of unknown quantities, the number of ways is infinite in
which a system of forces may be reduced to two forces acting
along lines at right angles to each other.
When any one of these quantities relating to one of the forces
is assigned, then all the incidents of the other force are also
assigned.
(5) The system of two forces is however unique, when the
forces are equal and act along lines perpendicular to each other.
In this case PJ = P 2 ; and consequently
_ K R
and we have the following theorem :
A given system of forces acting on a rigid body may be re-
placed by two equal forces whose lines of action are perpendicular
to each other, and each of which has a line of action inclined at
45 to the central axis; and the forces act perpendicularly at
the ends of an arm which is bisected at right angles by that
-p
axis ; the magnitude of each force is equal to , and the length
2K 2 *
of the arm is
R
This result may also be arrived at directly in the following
manner :
Let R be resolved into two equal and parallel forces, each of
which = - ; and let them act at two points Q t and Q 2 on the
m
axis of x which are equidistant from the central axis, and at a
distance r from it on either side; also let the forces of K be
TJ
- , and act at the points Q, and Q 2 , so that nr = K. Then we
have at Qt and at Q 2 two equal forces acting along lines which
are perpendicular to each other ; and the resultant at each point
E.
is equal to , and acts along a line inclined to the central axis
2
at an angle of 45; but as these lines are on opposite sides of
that axis, they are at right angles to each other.
This is the only unique system of a pair of forces to which a
system may be reduced.
(6) The distance between the action-lines of the two forces
which equivalently replace a system of forces is a minimum,
when the forces are equal and their action-lines are perpendicular
to each other.
120 THE EQUILIBRIUM-AXIS OF [94.
SECTION 6. The equilibrium-axis of an equilibrium-system.
94.] In this section I propose to investigate for an equili-
brium-system of forces acting in space the conditions requisite
that the system should also be an equilibrium-system, when the
body receives the most general displacement, and the forces act
at the same points of the body, along lines parallel to their
former action-lines, and in the same direction as before dis-
placement.
Whatever the displacement be, it may always be resolved into
a displacement of translation and a displacement of rotation, the
effects of which may be separately considered. Now the dis-
placement of translation will be effected by transferring the
point of the body which coincides with the origin in its original
position to the point (# , y , 2 ), and making all particles of the
body describe equal and parallel paths : then if (#', / ', /) is the
place of the particle which was originally at (x } y, z),
x r =x + x 0} y'=y-\ry , z = z + z . (225)
As the systems, both displaced and original, are equilibrium-
systems, and as the direction-angles of the action-lines of the
forces are unchanged, we have the following conditions ; viz.
5.P cos a = 2.P cos /3 = 2.P cos y = ; (226)
2.p(y cosy zcos/3) = 2.P (2 cos a arcosy)
= 2.p(#cos ycosa) = 0, (227)
2.p(/cosy /cos/3) = 2.p(/cosa x'cosy)
= s.P^cos/J /cos a) = 0; (228)
and substituting from (225) in (228), (228) are identically satis-
fied by reason of (226) and (227); so that whatever is the dis-
placement of translation an equilibrium-system continues an
equilibrium -system.
Let the displacement of rotation be produced by making the
body turn through an angle about an axis passing through the
origin and of which the direction-angles are f,g,h: let (x, y, z}
be the place of any particle of the body in its original position,
and let this point after the rotation be (x-\-*x, y + &y, z + *z) :
let A* be the distance between the two positions of this point,
so that ( A j) _ ( Aar ) _j_ ( A ^) + ( A 2) . (229)
and let p be the perpendicular distance from (x, y, z) to the axis
94-] AN EQUILIBRIUM-SYSTEM. 121
of rotation ; so that A* is the chord of a circular arc, of radius
p and angle d, described by (x, y } z] revolving about the axis of
rotation ; and therefore
a
A* = 2j5sin-. (230)
I
As this point is in both its positions at the same distance from
the origin, and also in the same plane perpendicular to the rota-
tion-axis, we have
= 0, (231)
s/+Ay cos^-f AzcosA =0. (232)
Also from (230),
(A#) 9 + (A^)* + (A*)* = 4/; 3 (sin-) ; and (233)
(ecosg ycos^) 2 -f (#cos^ 2cos/) 3 -f (ycos/ #cos#) a =j9 4 . (234)
Also as z cos g y cos h, x cos k z cos/, y cos/ x cos g are pro-
portional to the direction-cosines of the normal to the plane
which contains the rotation-axis and (x } y, z), and A#, Ay, A z
are proportional to the direction-cosines of the chord A*, and as
Q
- is the angle contained between these lines,
Q
(z cosff y cos Tt] A x + (x cos h z cos/) Ay -f- (y cos/ xcosg)&z = p A cos .
Thus we have the three following linear equations in terms of
A#, Ajf, A.Z,
=- j9 2 sin 9,
cos/ A^ + cos^ Ay+ cos^ A2= 0;
and from them we have
A x = sin (z cos g y cos h}
0\ 2
+ 2 (sin-) {cos/(^cos/+ycos^4-2cos/i) x} ;
Ay = sin 6 (x cos h z cos/)
/} 3 ^23^
+ 2(sin-) { cos ^ (a; cos/4- y cos ^-f 2cos>&) y} ; '
A^ = sin (y cosfx cos ^7)
2
+ 2 (sin-) (cos>5(a?cos/-f ycos^ + ^cos^) z}.
I may by the way observe, that if the angle through which
the body is turned is infinitesimal, say = dO, then omitting
PRICE, VOL. III. R
122 THE EQUILIBRIUM-AXIS OF [94.
powers of it higher than the first, and replacing A.r, Ay, &z,
which are also infinitesimal, by dx, dy, dz,
dx = (z cos g y cos K) dd, (236)
dy = (x cos h z cos/)^0, (237)
dz = (y cosf x cos g) dO. (238)
The signs of the terms in the right-hand members of these
equations, which are ambiguous by reason of the system of
squares in (234), have been taken in such a manner that if the
ar-axis were the axis of rotation, the positive direction of rotation
would be from the y-axis to the 2-axis. And the rotations about
the other axes would have similar positive directions ; so that
the system is cyclical.
In (228) let of, y', z be replaced by ar-f-Atf, y + Ay, z + &z re-
spectively ; and let the following symbols be employed for the
abbreviation of the results ; viz.
s.py cos y = 2.P z cos /3 = D,
2.P z cos a = s.P x cos y = E, \ (239)
2.P#cos = 2.py cos a .= F ;
2.p(y cos/3 + z cos y) = u,
2.P (z cos y + x cos a) = v, \. (240)
2.P (x cos a +y cos /3) = w ;
the first three equalities following from (227); then we have the
following equations, viz.
a
cot - ( u cos/" 4- F cos g -f E cos k} cos h (F cosf \ cos g -f D cos K)
m
= 0; (241)
a
cot - (Fcosy* vcos^ + DCOS^) cosf (Ecosy-f Dcosy wcos^)
+ cos/5( u cosy +F cosy -f E cos K) = 0; (242)
A
cot - (E cos/+ D cos^ w cos k] cosy ( u cos/+ F cosy + E cos h]
m
+ cos h (F cosf v cosy + D cos h] = 0; (243)
but as equilibrium is to subsist for all angles through which the
body is turned about the rotation-axis, Q is indeterminate ; and
consequently from these three equations the following result ;
u cos/+ F cos g -f E cos h = 0,
FCOS/ VCOS^+DCOS^ = 0, \ (244)
E COS/+ D cos g w cos h ;
95-] AN EQUILIBRIUM-SYSTEM. 123
and from these the direction-cosines of the rotation-axis are to be
determined. As, however, they are more than sufficient for the
purpose, a relation exists between them ; and eliminating cos/,
cos^, cos h we have
uvw D S U E J V FW 2DEF = 0; (245)
which expresses a relation between the forces, their action-lines,
and their points of application, when an equilibrium-system is
also an equilibrium-system after rotation through any angle
about a certain axis. As this axis has important properties, it
is convenient for it to have a distinctive name, and so it has
been called the equilibrium-axis. Equation (245) is the condition
that an equilibrium-system should have an equilibrium-axis.
When that condition is satisfied, the direction-cosines of the
equilibrium-axis are given by (244), and we have
(cos/) 3 = (eoeg)' _ (cos*)' = 1 ^
D a VW E J WU F 2 UV D 2 +E 2 +F l (VW+WU + UV) '
As these equations give only the direction-cosines of the equi-
librium-axis, all straight lines parallel to that thus assigned are
also equilibrium-axes.
If D S = vw, E 2 = wu, F a = uv, f, g, h are indeterminate, and
the body is in equilibrium, whatever is the position of the axis
about which it is turned.
If all the forces act in one plane, say in that of (x, y), then
cos y = 0, and consequently D = E = 0, and from the last of
(244), w=0; that is,
2.p(# cos a +y cos /3) = 0,
which is the same condition as (82), Art. 63. Hence also
cos/= cos^ = and cos = 1, so that the equilibrium-axis is
perpendicular to the plane of the forces.
95.] The condition for the existence of an equilibrium-axis
given in (245) will be more easily interpreted, if we take the
most simple case. For this purpose let us assume the system of
coordinate-axes to be so taken that the z-axis is the equilibrium-
axis ; then coef= cos g = ; and consequently D = E = 0, w = ;
that is,
2.pycosy = 0, 2.P#cosy = 0, S.P(# cosa+^cos/3) = 0; (247)
from the first two of which taken in combination with 2.P cos y
= 0, see (226), we infer that, if the forces are at their points of
application resolved in directions parallel to the coordinate axes,
B 2
124 THE EQUILIBRIUM- AXIS OF [96.
those parallel to the axis of z are in equilibrium ; and from the
last, combined with the first two of (226) and the last of (227),
we infer that the forces whose lines of action are parallel to the
plane of (x, y] satisfy the conditions required for a centre, see
Art. 63, and are therefore in equilibrium when the body is
turned through any angle about the axis of z. Hence the
meaning of the condition (245) is,
If the forces acting on a body are resolved along a certain
straight line, and in planes perpendicular to that line; and if
the forces parallel to the straight line are in equilibrium, and
those in the planes perpendicular to the straight line are also in
equilibrium, and satisfy the conditions required for a centre,
then every line parallel to that line is an equilibrium-axis.
Also if the forces are such that the x- and y-axes are both
equilibrium-axes : then from the equations (246)
D = E = F=0, U = V = 0;
and therefore cos h = ; and therefore any line parallel to the
plane of (x, y] will also be an equilibrium-axis.
96.] To investigate generally the conditions requisite that
any two lines inclined at any angle to each other should be
equilibrium-axes ; let the direction-angles of the two lines be
f> 9, b, f, i, V', then from (244),
u cosy + F cos g + E cos h = 0, -|
F cosy -vcos^ + D cos h = 0, r (248)
E cosy +D cos^ w cos h ; J
+F COS ff'+ ECOS^' = 0, "
' vcos/+ DCOS' = 0, (249)
Ecosy'-f DCOS^' wcos^'= 0;-
whence we have the following relations :
DU+EF=0, EV + FD = 0, FW-fDE = 0; (250)
and D = VTT, E a = wu, F S = uv; (251)
which are the conditions necessary that an equilibrium-system
should admit of two equilibrium-axes not parallel to each other.
But by reason of (251), cosy, cos^, cos h, as also cosy, cos^,
cos h' are indeterminate ; they are however subject to the follow-
ing relation; if we substitute from (251) in either of (248), we
have u* cosy+ v* cos^ + w* cos h = ; (252)
and if we substitute in either of (249), we have
u* cosy -f v* cos/+ w* cos h' ; (253)
97-] AN EQUILIBRIUM-SYSTEM. 125
which shews that both these lines are parallel to or lie in the
plane whose equation is
u*#+vfy + w*z = 0; (254)
but that the position of the lines in the plane is indeterminate.
Hence we infer that a body which is in equilibrium for two
equilibrium-axes which meet and are not parallel to each other,
is also in equilibrium for all axes parallel to the plane which
contains these two equilibrium-axes. And hence
If a body has three equilibrium-axes which are not parallel to
one and the same plane, so is any fourth axis an equilibrium-axis.
And as a body has an equilibrium-axis, if it is in equilibrium
in two different and not parallel positions, so if it is in equi-
librium in four different and not parallel positions, it is also in
equilibrium in every fifth position.
And when this last case occurs, D = E = F = 0, u=v=w=0;
so that the position of the plane (254) becomes indeterminate.
97.] Although a system of forces acting- on a rigid body and
being in equilibrium admits of an equilibrium-axis, only when
(245) is satisfied, and therefore not generally; yet if a system is
in equilibrium, two new equal forces acting at certain definite
points, along the same line of action and in opposite directions,
may be introduced in such a manner that the system thus modi-
fied may have an equilibrium-axis in a given direction. The
new forces, it will be observed, as introduced into the first posi-
tion of the body, being equal and opposite, neutralize each other,
and do not disturb equilibrium, and in the other positions form
a couple which equilibrates with the impressed forces of the
system in their new position.
Let, as in the preceding Articles, f, ff, h be the direction-
angles of the given equilibrium-axis; P' and P' the two new
forces, equal and opposite to each other; (x',y', /), (#",./'> 2")
their points of application ; I, m, n the direction-cosines of their
common line of action ; r the distance between their points of
application; let x"x', y"y, /' z be positive quantities;
then yf'_yr y "_y> z _ z >
/ y y .
I m, n
and if the accented letters refer to the system when increased
by the two new forces, and the unaccented letters to the ori-
ginal system, D ' = D +/'P'-/F'
= D-f (/'-/)P';
126 THE EQUILIBRIUM-AXIS OP (97.
.. D' = v + p'rmn, ~]
similarly if = E + -p'rnl, i- (255)
F' = wT'rlmJ
and substituting these values in the conditions (244), which are
requisite for an equilibrium-axis, we have from the first of them
U COS/-f F COSy + E COS k
= p'/ {(m? + ft 2 ) cosf Im cos g In cos h}
= p'r {cos,/ l(lcosf+m cosy + n cos k}}. (257)
Let </> be the angle between the line of action of P' and the
equilibrium-axis; then
cos (f> = I cosf+ m cos g + n cos h ; (258)
and therefore we have
u cos/-f F cosy + E cos h = p'r {cosf I cos 0} = it, -
Fcosy 1 v cosy -fD cos ^ = p'r {cos^ ?# cos </>} = v, -(259)
E cos/4-Dcosy wcos^ = p'r (cos>i n cos 0} = w, -
employing M, v, w as abbreviating symbols for the left-hand
members of the equations, which are known quantities.
Hence we have
u cosy+ v cos g + w cos Ji = p'r { 1 (cos <f>) 2 }
= pV(sin</>) 2 .
Also u 2 -{- f 2 +w 2 = p /2 / 2 (1 (cos </>) 2 },
= p' 2 / 2 (sin </>)*;
P'r=- -.. (260)
u cosy 4- v cos y + w cos ^
; (261)
and therefore from (259) we are able to determine I, m, n; and
thus the direction of the line of action of P' is completely deter-
mined. The intensity -of P' and the position of its point of
application are involved in only (260) ; and therefore we may
take any two points on the line defined by (I, m, n) at a distance
r apart, and at them apply two equal and opposite forces P' and
p' of such magnitude that p'r is equal to the right-hand mem-
ber of (260).
From the preceding it appears that two equal forces, acting
originally in opposite directions along the same line of action,
98.] AN EQUILIBRIUM-SYSTEM. 127
will, when the body is turned about a certain axis, equilibrate
with the forces of the system : but as the two forces in this dis-
placed position form a couple, we infer that
If a rigid body, on which a system of forces in equilibrium
acts, is turned about any axis, and if the forces act on the same
points of the body as before and in the same directions, they
are generally reducible to a couple; but in the particular case
when the condition (245) is fulfilled, the moment of the couple
vanishes.
98.] In Section 5 of the present Chapter it has been shewn
in various ways that it is possible to reduce a system of forces
acting on a rigid body to two forces, and that the two forces
are generally indeterminate in all their elements ; it was shewn,
however, that the pair is unique and determinate, when the
two forces were equal and acted along lines at right angles to
each other. I propose now to shew that it is always possible to
reduce a system of forces to two forces of translation, such that
they with two other new forces shall be in equilibrium, and also
shall have a given equilibrium-axis.
Let the two new forces be P' and P" ; let a ft /, a" ft' y" be
the direction-angles of their lines of action ; (x, y', z'}, (x" ', y" , z"}
their points of application ; then for the condition of equilibrium
of these two new forces, with the former forces of the system,
W6have
= 0, (262)
P'COS / -f P" cos y" + z = 0.
Also let
2.PyCOSy = D, 2.P2COS/3 = D ,
2.P3 COS a = E', S.P# COS y E",
S.ptf cos0 = F', s.py cos a = F",
then, as the three expressions for the moment-axes of the couples
about the coordinate-axes are to vanish, we have
= p'/cos/S'-f p'V'cos/3"+D"= D (say),
P'/COS a + p'Y'cos a" + E'
= pVcos y' + p'V'cos y" + E"= E (say),
P Vcos p + P ' Vcos /3" + F'
= P'/COS O' + P"/'COS a"+ F"= F (say). .
128 EQUILIBRIUM-AXIS OF AN EQUILIBRIUM-SYSTEM. [98.
Also let
p' (/cos^ + /cos /) + P"(/'COS /3" + /'cos /') 4- u = u',
p' (/cos / -|- x'cos a') + P"(/'COS y" + #"cos a") + v = V,
P' (of cos a' +/COB + P"(#"COS a" +/'cos/3") + w = V ;
and therefore, if the direction-angles of the given equilibrium-
axis are/, g, h, the conditions required are, see (244),
Tj'cOSy+FCOS^+ECOS^ = 0, ~]
F cosy* V'COS^ + D cos Ji = 0, > (264)
Ecosy+Dcos^ w'cos^ = ; J
and these are all the conditions which are requisite for the
existence of an equilibrium-axis : viz. the equations severally
of (262), (263), and (264), and' of which the whole number is
nine ; and they contain twelve undetermined quantities : viz.
p'cos of, P'COS ft', ... P"COS y", x', ,/,... /'; of these therefore nine
may be eliminated, and there will remain a condition involving
the other three : the elimination, however, is so long that I shall
only state results. If we eliminate the forces P', P", the direction-
angles of their lines of action, and the coordinates of the point
of application of one of them, say, x", y" , z", it will be found
that the resulting equation is of the second degree in terms of
x 't y f > z '> an< i w iU therefore represent a surface of the second
order : and it will also be found that the point of application of
the other force is also upon the same surface, and also that
every point in the line joining the two points is on the same
surface : the surface is therefore an hyperboloid of one sheet,
the line joining the points of application of the forces being one
of the generating straight lines of the surface ; and 'the equi-
librium-axis is the imaginary axis of the surface. And hence
we conclude that into a system of forces which is not in equili-
brium two forces may be introduced, so that the system thus
modified may be in equilibrium and may also have an equi-
librium-axis ; and the points of application of these two forces
may be at such points on the surface of a certain hyperboloid
of one sheet, that the line joining them lies wholly in the
surface; and when these points of application are given the
lines of action of the forces are also determined.
Although I have applied to the theory of the equilibrium-axis
only the geometrical changes of x, y and z, given in Art. 94,
equations (235), yet they are of much wider application, and will
hereafter be largely used.
100.] STABILITY OF EQUILIBRIUM. 129
SECTION 7. Stability and Instability of Equilibrium.
99.] The investigations of the preceding- section, as also those
of Art. 63, are of great importance in determining a delicate
question, viz. the character of equilibrium of an equilibrium-
system. For if a body is at rest under the action of many
forces, and receives a small displacement of the most general
kind, but of such an infinitesimal amount that the forces, when
applied at the same points as before, act in the same directions
along lines parallel to, and infinitesimally distant from, their
former lines of action; then the body in its new position will
generally not be in equilibrium ; and the acting forces may tend
either to bring it back to its former position or to remove it
farther from it ; if the former is the character of the forces the
equilibrium is said to be stable; and if the latter the equilibrium
is said to be unstable. A heavy homogeneous sphere resting in
a hollow bowl, a heavy oblate spheroid resting on a horizontal
plane with its axis vertical, a heavy weight suspended as a
pendulum and at rest, a loaded wheel with the load in the
lowest possible position, are all cases of stable equilibrium.
On the other hand, a loaded ball with its load as high as
possible and resting on a horizontal plane, an egg balanced on
the smaller end, a heavy beam resting on two inclined planes,
a heavy ball balanced on the highest point of a sphere, are all
instances of unstable equilibrium. If, however, the body in its
displaced state is in a position of equilibrium, it may be so either
for the displacement which it has actually undergone and for
no other near to it, in which case the equilibrium is said to be
neutral ; or it may be in equilibrium for this and all other infi-
nitesimal displacements, and then the equilibrium is said to be
continuous,- A heavy homogeneous cylinder having its ends
perpendicular to the axis resting on a horizontal plane with its
axis horizontal, and a heavy homogeneous circular cone having
its base perpendicular to its axis resting with its slant side on
a horizontal plane, are instances of neutral equilibrium ; a heavy
homogeneous sphere resting on a horizontal plane is an instance
of continuous equilibrium.
100.] Now the most general displacement which a body can
undergo always consists of a displacement of translation, and of
a displacement of rotation about a determinate axis. In Art. 94
it has been shewn that if a body is at rest under the action of
PRICE, VOL. m. s
130 STABILITY AND INSTABILITY [lOI.
given forces, it is also at rest when it has undergone a displace-
ment of translation, the paths described by every particle of the
body being equal and parallel, the forces being applied at the
same points as before, in the same direction, and along action-
lines parallel to, and infinitesimally distant from, the former
action-lines. Thus we have to consider only the effects of an
infinitesimal displacement of rotation about a certain deter-
minate axis. Let the direction-angles of the axis of rotation
be/, g, h; and let dO be the infinitesimal angle through which
the body is turned about that axis; then the changes in the
coordinates of the point (x, y, z), which are due to this infini-
tesimal displacement of rotation, are those which are given in
Art. 94 ; and we have
dx = (z cos g y cos K) dO, "\
dy = (xcoshzcosf)dQ, > (265)
dz = (yco$fxco$g}d6.J
If, however, all the action-lines of the forces are in one plane,
say, in the plane of (x } y), and the rotation-axis is perpendicular
to that plane, then
dx=ydQ, dy = xdd. (266)
In reference to equilibrium-axes it is evident that if a body
in equilibrium under the action of certain forces has no equi-
librium-axis, its equilibrium is either stable or unstable ; if it
has one or two equilibrium-axes which meet, its equilibrium is
neutral, when the displacement of rotation takes place about one
of them ; and if the system of forces is such that every axis is
an equilibrium-axis, then the equilibrium is continuous.
101.] In application of this theory I will first take the most
simple case of a body held in equilibrium under the action of
two forces only: these of course are equal to each other, and
act along the same line, and in opposite directions : but these
conditions may be satisfied in two ways : the forces may act to
draw their points of application either nearer to, or farther from,
each other. Let P U p,, see fig. 66, be the two forces; A t , A,
their respective points of application. Let the body receive an
infinitesimal displacement of rotation about an axis perpendicular
to the line of action of the forces : so that the line A I A a , which
before the displacement was in the same line with the line of
action of the forces, is now in one of the positions, relatively to
them, indicated in the figures (a) and ($) : (a) is evidently the
state in which the forces applied at A, and A, tend to bring the
1 01.] OP EQUILIBRIUM. 131
points nearer to each other; and in which, now that the dis-
placement has taken place, the action of the forces tends to
remove the system farther and farther from its original position,
and in which therefore the original equilibrium was unstable :
(/3) is the state in which the forces act to separate their points of
application, and in which the forces act after the displacement to
bring the body back to its original position ; and in which there-
fore the equilibrium is stable. If the two forces act at the same
point, equilibrium is continuous for every displacement of the body
about an axis perpendicular to the line of action of the forces*
and also because the point at which they act is their centre.
The following analytical investigation supplies a criterion of
these several states of equilibrium. Let (x lf y^ (# 2 , y^) be the
points of application of p t and of P, respectively; then the con-
ditions of equilibrium of these two forces are
P 1 + p 2 = 0,
G = sinars.Par cosaS.Pjf = 0. (267)
Let the body be turned about an axis perpendicular to the
plane of (#, y] through an angle dO ; then the forces, their points
of application in the body, and directions being unchanged, and
their lines of action being parallel and infimtesimally near to
the former action-lines, G varies ; and the change of it which is
due to the displacement is the moment of the couple which
acts on the body in its displaced state. Now the displacement
involves a change of x and y, and we have
dQ = sin a 2.P dx cos a 5.P dy,
= {sinaS.Py + cosaS.Ptf} dQ; (268)
but according as -^ is positive or negative, so does the couple
u/d
brought into action by the displacement tend to remove the
body further from, or to bring it back nearer to, the original
place of equilibrium : that is, so is the equilibrium of the body
unstable or stable. And consequently the equilibrium is stable
or unstable, according as
2.p#cosa-f 2.pysina (269)
is positive or negative.
And because a is the same for both the forces, and is also
generally indeterminate, since the directions of the axes are
arbitrary, the criterion (269) reduces itself to either 2.P# or 2.P,y,
and thus the stability depends on the sign of either of these.
s 2
STABILITY AND INSTABILITY [lO2.
, If (269) = 0, then, since PJ + p 2 = 0, a?, = # 2 = 0, y =^ 2 = ;
that is, the forces are applied at the same point, viz. the origin,
and the equilibrium is continuous.
The rotation has taken place about an axis perpendicular to
ihe line of action of the forces. I would only further observe,
that if it takes place about the line of action of the two forces^
their points of application undergo no displacement, and no
criterion of stability is obtained.
102.] The process of the preceding article is also generally
applicable to the determination of the criterion of the stability
and instability of forces all the action-lines of which are in the
plane of (x, y]. Let the forces and their several incidents be
denoted by the same symbols as heretofore. Then for the equi-
librium of the system we have
x = 2.P cos a = 0, Y = 2.Psina=0; (270)
G = 2.P(#-sin a y cos a). (271)
Let the body on which the forces act undergo an infinitesimal
displacement of rotation through dd about an axis perpendicular
to the plane of the forces ; then
d,G =. 2.P (sin a dx cos a clif]
= 2.P(ysina + #cosa)6?fl; (272)
and consequently the effect of the couple brought into action by
the displacement is to remove the body further from, or to bring
it back into, its former state, according as 2.P (x cos a +y sin a) is
positive or negative; but this quantity is the radial moment,
see Art. 63; consequently the equilibrium is stable or unstable
according as the radial moment is positive or negative. If the
radial moment vanishes, then the system has a centre, and an
equilibrium-axis perpendicular to the plane of the forces, so that
the body is in equilibrium in its displaced state, as also in its
former state, and the equilibrium is neutral or continuous.
Hence we have the following theorem :
Of a system of forces acting jon a rigid body in a plane, and
being in equilibrium, the equilibrium is stable, neutral, or un-
stable, according as 2.P (x cos a +y sin a), that is, the radial
moment, is positive, zero, or negative.
The preceding criterion is true only for a displacement of the
body about an axis perpendicular to the plane in which the
forces act ; for let us suppose four forces to act on a body in one
plane and to be in equilibrium; and supppse them to be such
1 03.] OF EQUILIBRIUM. 133
that a pair of them is in equilibrium ; and that therefore the
other pair also equilibrates; let the body be turned about an
axis coinciding with the line of action of the latter pair, the
equilibrium of the other pair may evidently be either stable or
unstable : and if the rotation takes place about the line of action
of the former pair, the equilibrium of the latter pair may be
either stable or unstable; and evidently there is no necessity
that it should be of the same character as the other ; hence in
this case we are unable to determine a priori the axes of stable
or of unstable equilibrium.
And the preceding test is applicable to the case of forces
whose lines of action are parallel to a given plane when the dis-
placement takes place about a line perpendicular to that plane.
103.] We can also hence derive another remarkable criterion
of the stability and instability of an equilibrium-system. Let
the radial moment, as in Art. 63, be denoted by H, so that
H = 2.P (x cos a+y sin a) ; (273)
. . dv. = 2.P (dx cos a + dy sin a),
= 2.P (x sin a y cos a) (Id, (274)
= G dO = 0,
since the system is in equilibrium and consequently G = 0.
Hence in an equilibrium-system the radial moment has a critical
value, and is a maximum, a minimum, or a constant, zero being
a particular value of the constant. To determine the character
of this critical value, we differentiate again, and we have
rf 2 H dG f dx . dy \
= - v - = 2.P ( -r^ sm a cos a ) ,
(16* d6 \dd dB '
= 2.P (X cos a+y sin a),
= -H; (275)
so that H has a maximum or minimum value according as it is
positive or negative; but according as H, which is equal to
JQ , is positive or negative so is the equilibrium stable or
unstable ; consequently we have the following criteria as to the
character of equilibrium of a system of forces.
The equilibrium is stable or unstable according as H is a maxi-
mum or minimum ; or according as H is positive or negative.
If H = 0, the system has an equilibrium-axis, and the equi-
librium is neutral.
If the action-lines of all the forces are parallel, let us take a
134 STABILITY AND INSTABILITY [ 1 04.
line parallel to them for the axis of y ; so that in this case a = 90,
and H = S.Py, (276)
and equilibrium is stable or unstable according as this quantity-
is a maximum or a minimum. We shall hereafter have many
applications of this equation.
104.] The following are examples in which the preceding
criteria of stability are applied :
Ex. 1. When a heavy uniform beam rests on two inclined
planes, is the equilibrium stable or unstable ?
This is the case which is discussed in Ex. 2, Art. 60 ; and
I will take the notation therein employed, and c for the origin,
and the horizontal line through c for the axis of x. Then if
CB = /, CA = /, and as the forces are R, R', w,
H = 5.P(;rcosa+^sina)
= R'/sin /J cos /3 R/ sin a cos a
4- EY sin /3 cos /3 + Rr sin a cos a w (/sin y3 a sin 6}
sin i
consequently H is a negative quantity, and the equilibrium is
unstable.
Also as the beam is at rest -=-: = 0, and thus
tan0 = ^ ^r-
2 sin a sin j3
Also -5 - is positive, so that the value of H is a minimum.
Ex. 2. If a heavy beam rests against a smooth wall, and has
the other end fastened by a string to a given point in the wall,
as in Ex. 3, Art. 60, what is the character of equilibrium ?
Let us take the symbols which are given in Art. 60, and take
c to be the origin, fig. 30, and the horizontal line drawn through
it to be the avaxis, the y-axis being taken downwards. Then
H = 2.P (x cos a+y sin a)
and substituting in this equation the values given in Ex. 3,
Art. 60, we have
H =
Thus H is a negative quantity, and the equilibrium is unstable.
1 05.] OP EQUILIBRIUM. 135
Ex. 3. What is the character of equilibrium in the problem of
Ex. 6, Art. 60 ?
Let s be the origin ; then
H = urcos- +wcos0(rc)
= W (<?-),
and this is positive or negative according as c is greater or less
than a ; hence the equilibrium is stable or unstable according as
c is greater or less than a.
Ex. 4. Two heavy particles connected by a string support
each other on the circumference of a circle in a vertical plane.
Determine the nature of the equilibrium.
Let the weights of the particles be p and Q, and let the radii
of the circle drawn to the points where P and q rest make angles
6 and </> with the vertical. Let the string subtend an angle = a
at the centre, so that 04-0 = a : then, if the origin is taken
at the centre,
H = P cos d a Q cos < ;
dia. = #P sin0fi?0-f0Qsin4>6?0
= a {rsin Qsin<}^0 = 0,
. , sin sin </>
T- _ _
11
Q P
consequently H is negative, and the equilibrium is unstable.
105.] In the case of a rigid body in equilibrium under the
action of many forces acting along lines of action in space, we
have to consider only the effects of a displacement of rotation, as
to the kind of equilibrium which the body is in.
Let the direction-angles of the axis of rotation be/, g, h ; and
let the moment-axes of the impressed couples along the three
coordinate axes be L, M, N ; then, if G is the moment of the
couple tending to turn the body about the rotation-axis, by
reason of the law of resolution of couples,
G = L cos/+ M cosy + N cos h
cos/s.pfy cos y z cos /3) + ... + ... ', (277)
do, d dz
(cos/)' 2.p( < y cos/3 +z cosy) -f cos g cos ft 2.P (y cos y + z cos/3)
(cos g]* S.P(Z cosy + iF cosa) + cos Ji COS/S.P (zcos a + tf cosy)
(cos K)* 2.?(ar cosa +y cos/3) + cos/ cos y 2.p(arcos/3-f ^ cosa);
136 STABILITY AND INSTABILITY
and employing the abbreviating notation of Art. 94,
-5-r = U (COS/) 2 V (COS #) 2 W (COS h}*
+ 2D cos^ cos h+ 2E cos h cosf+2v cos/cos^ ; (278)
and since the effect of G due to a small variation of 6 is to bring
back the system to its former position, or to remove it farther
therefrom, according as -^ is negative or positive, so is the
Civ
equilibrium stable or unstable according as the right-hand mem-
ber of (278) is negative or positive.
For convenience of reference let us denote this quantity by s ;
so that
s = u(cos/) a v(cos^) 2 w(cos^)*
+ 2Dcos#cos/& + 2E cos hcosf+2?cosg;
then equilibrium is stable or unstable, according as s is negative
or. positive : and the sign evidently depends, not only on the
impressed forces and their incidents, but also on the direction-
angles of the rotation-axis ; and therefore an equilibrium-system
may be stable for one rotation-axis, unstable for another, and
neutral for a third; that is, in the third case the system may
have an equilibrium-axis, and s may be equal to zero.
For suppose that s is arranged in the form
{ u cosf+ F cos g + E cos h} CQS/+ (F cosfv cos g + D cos h} cosg
+ (E cos/"-f D cos^ w cos k} cos h y
and that we have also
u COS/+F COS^ + E cos h 0,
DCOS^ = 0,
wcos h = ;
so that uvw D 2 u E*V F S W 2DEF = ;
then this is the condition requisite for the existence of an equi-
librium-axis; and in this case s = 0, and the equilibrium is
neutral.
If also, according to Art. 96, equations (251),
D a = vw, E a = wu, F 8 = uv,
and if the axis about which the rotation takes place is parallel
to the plane whose equation is
u x-\-\*y+ vf^ z = 0,
then equilibrium is neutral for all such axes ; and is continuous,
I06.] OF EQUILIBRIUM. 137
if the change of axis is from any one line to any other line lying
in the plane.
And if in addition, D = E = F=O, }
u = v = w = 0, (
so that any axis about which the body is turned is an equili-
brium-axis, then the equilibrium is continuous for all axes.
I may also observe that, if the directions of action of all the
forces are reversed, the signs of u, v, w, D, E, F are changed, and
therefore the sign of s is changed ; and thus the nature of the
equilibrium is changed : in the case, however, of neutral equili-
brium no alteration takes place.
106.] And s admits of the following geometrical interpreta-
tion : on the straight line drawn through the origin, and whose
direction-angles are f, g, h, let a point (x, y, z] be taken : then
x, y, z are proportional to cos/^ cos^, cos Ji, and s becomes pro-
portional to
Utf 2 vy 2 wz* + 2 vyz + 2 E^a?+ 2 ?xy, (279)
which, when equated to zero, is the equation to a cone of the
second degree ; and therefore for all lines passing through the
origin, and lying within this cone, and employed as rotation-
axes, the above expression has a different sign to that which it
has for all lines lying outside of the cone j and for all lines on
the surface of the cone it vanishes ; so that for all the generating
lines of the cone, equilibrium is neutral ; and the cone divides
space into two parts such, that for all axes within its surface,
the equilibrium is the opposite to that which it is for axes out-
side the surface.
I may, however, observe that, if lines are drawn through the
vertex of the cone, and if these are called interior or exterior
lines according as from points on them real tangent planes
cannot, or can, be drawn to the cone ; then will interior lines be
axes of stable, and exterior lines axes of unstable, equilibrium, if
uyw D a u E'V F 2 w 2DEF = v (say)
is positive ; and if v is negative, the converse is the case.
If v = 0, we have the following circumstances. If we reduce
the expression (279) so as to deprive it of the terms containing
the products of the variables, we obtain the discriminating cubic,
of which the constant term is v ; and therefore if v = 0, one of the
roots of this cubic is zero, and the reduced equation becomes
of the form v '
PRICE, VOL. III.
138 STABILITY AND INSTABILITY [ l 7-
which, if the upper sign is taken, represents the axis of z ; and,
if the lower sign is taken, two planes perpendicular to the plane
of (#, y]. In the former of these two cases the axis of z is an
axis of neutral equilibrium, and other lines are axes either all
of unstable, or all of stable, equilibrium : in the latter case, any
line in either of the planes is an axis of neutral equilibrium, and
the other lines are either all axes of stable, or all of unstable,
equilibrium.
One or two special forms of (277) require notice : if the 2-axis
is the rotation-axis, the condition requires that
w = 2.P (x cos a-f ycos/3)
should be positive for stable, and negative for unstable, equili-
brium : which is the same result as that of Art. 102.
And if all the forces are parallel to the axis of z, so that
cos a = cos /3 = 0, cos y = 1, then
y} ; (280)
-^
and if the axis about which the infinitesimal rotation takes
place is at right angles to the lines of action of the forces, then
h = 90, and we have
and therefore equilibrium is stable or unstable according as 2.P z
is positive or negative.
Now on referring to Art. 80, (146), it appears that if (x, y, z)
is the centre of a system of parallel forces, 22.P = 2.P2; conse-
quently the equilibrium is stable or unstable according as z is
positive or negative. In the following Chapter we shall have
many illustrations of this theorem.
107-3 The condition for the stability of equilibrium of a
system of forces acting in space may be expressed in a form
similar to that of Art. 103 by the following process :
Let the infinitesimal rotation take place about an axis whose
direction-angles are f, g, h; so that, as the moment-axes of the
couples, whose rotation-axes are the coordinate-axes, are L, M, N,
for equilibrium we have
L cosf+ M cos g -f N cos h = ;
and thus, replacing L, M, N by their values, and introducing d0,
2.p {cos a (z cosffy cos ^) 4 cos /3 (# cos Ti z cos/")
-I- cos y (y cos/ x cos y)} dO = j
and by means of (265),
2.P. {cos a dx + cos fidi/ + cos y dz} = = dn (say) ;
I0;.] OP EQUILIBRIUM. 139
therefore by integration
H = 2.p(#cos a+y cos p + z cosy); (282)
and therefore H is a maximum, a minimum, or a constant. And
since, see equation (277),
ds. _
d's. _ do
= s, (283)
see equation (278); therefore H = 2.p(#cosa+y cos /3 + 2 cosy)
is a maximum or minimum, according as s is negative or posi-
tive, that is, according as equilibrium is stable or unstable.
Now s, as given in (278), admits of being put into the form,
s = 2.P { (x cosf+y cos g + z cos K) (cos a cosy+ cos /3 cos g + cos y cos K) }
2.p(#cos a+ycos/3 + 2Cosy); (284)
and as for a given rotation-axis x cosy+y cos y + 2 cos ^ is the
projection on the axis of rotation of the distance from the origin
of the point of application of the force P, and
P (cos a cosy+ cos /3 cosy + cos y cos k]
is the resolved part of P, along the rotation-axis ; and as both
these quantities are constant for a given-rotation-axis, and inde-
pendent of the rotation ; the value of s can only change by
means of the last term in the right-hand member of (284) : but
this term is H ; hence equilibrium is stable or unstable according
as H is greater than or less than
2.P {(# cosf+y cos g + z cos Ji) (cos a cosy+ cos /3 cos g + cos y cos k}} ;
and if s = 0, equilibrium is either neutral or continuous.
In Art. 60, the forces have been resolved along, and perpen-
dicular to, the radius vector of the point of application ; and
2.P (x cos a +y sin o)
has been called the radial moment of the system, because it is
the product of the radius vector of the point of application, and
of the radial component. Similarly in space, if we resolve P
along the radius vector of its point of application, and call u
its radial component,
p (x cos a+y cos ft + z cos y)
r
where r is the radius vector of the point of application of P :
therefore
H = 2. P(# COS 0+^008/3 + Z COSy) = 2.U/, (285)
T %
140 VIRTUAL VELOCITIES. [lo8.
and H is called the radial moment of the system. Hence we have
the following theorem :
The equilibrium of a system of forces is stable or unstable
according as the radial moment is a maximum or a minimum.
The radial moment also possesses the following two other pro-
perties. Let us suppose the body or system of particles on which
the forces act to receive a small displacement, and all the forces
to act at their points of application, along lines of action parallel
to the former ones, and in the same directions. Then if the
motion of the body is constrained in translation along a given
ds.
line, and ds is the space described along that line, -=- is the sum
U^
of the components of the forces estimated along that line ; and
if the motion is one of rotation about a given axis, and 6 is
dR
the amplitude of rotation, then is, in any position, the mo-
ment-axis of the couple arising out of the system of forces about
that axis.
SECTION 8. The principle of Virtual Velocities.
108.] Let a body, or a system of material particles on which
an equilibrium-system of forces acts, receive the most general
infinitesimal geometrical displacement that is possible, so that
the forces may act at the same points as before the displace-
ment, along lines parallel to, and infinitesimally distant from,
the original action-lines, and in the same directions. Let , 77,
be the infinitesimal distances along the coordinate-axes through
which the body is displaced, and let f, g, h be the direction-
angles of the rotation-axis about which the body is turned
through the angle dd. Then all these quantities being arbitrary,
the total displacement is of the most general kind.
Let us employ the symbol 5 to signify this most general
displacement ; so that d signifies a particular form of it, viz.
that in which the change of value is restricted to given condi-
tions. Then 8.r, by, bz being the variations of x, y, z, which are
the coordinates of any point in the original system, due to these
displacements,
bf = +(zcosg y cos ft) dO, ~j
by = rj + (.? cos h z cos/) dO, [ (286)
5 z = f -f (y cos/ .r cos g) dd. J
108.] VIRTUAL VELOCITIES. 141
As the system of forces is in equilibrium, we have the follow-
ing six conditions :
S.P cos a = 0, 2.P cos /3 = 0, 2.P cos y 0,
2.P (y cos y z cos $) = 0,
2.P (2 cos ax cos y) = 0,
2.P (# cos /3 ^ cos a) = ;
let these be severally multiplied by , 77, {, cosfdd, cos g dd,
cos k dd, and added ; then we have
2.P (cos a-\-y cos/3+Ccos y
(2 cos^ y cos 7i) cos a dO + (x cos h z cosf ) cos ft dd + (y cosf x coag} cosfdd } = ;
and by reason of (286) this becomes
2.P (cos a8tf + cos/36y+cosy52) = 0. (287)
Now as bx, by, bz are the projections on the coordinate -axes of
the displacement of (x, y, z), which is the point of application of
of P, and as a, ft, y are the direction-angles of the action-line of
p, cos a bx -f cos /3 8^ -f cos y bz is the projection of the displace-
ment along the action-line of p. Let this projected displace-
ment = bjj ; then (287) becomes
2.P8jo = 0. (288)
This equation expresses a theorem which is known as the
Principle of Virtual Velocities, and which may be enuntiated as
follows : ,
If a system of forces acting on a rigid body, or on a system \J
of material particles which are at relative rest, is in equilibrium,
and the body receives an infinitesimal displacement of the most
general kind possible, whereby the points of application of the
forces are displaced; but the forces act along lines parallel to,
and infinitesimally distant from, their former lines of action;
then the sum of the products of each force and the projection on
its line of action of the displacement of its point of application,
is equal to zero.
The projection on the line of action of a force of the infini-
tesimal displacement of its point of application is called the
virtual velocity of the force : and as that projection may take
place along the line either in the direction of the force or in the
opposite direction, so it is in these alternative cases to be affected
with a different sign. I shall take the virtual velocity to be
positive when the projection on the action-line of P is in the
direction in which the force acts. Thus in fig. 140, let A P be
142 VIRTUAL VELOCITIES. [108.
the line of action of P, ere the displacement takes place : let the
system be infinitesimally displaced, so that the point of applica-
cation of the force is shifted from A to A'; A A' being of infi-
nitesimal length ; let us suppose the line of action of the force
after the displacement to be parallel to its line of action before
the displacement, so that A'P' is parallel to A p. From A' let a
perpendicular A' M be drawn to the original line of action of the
force, so that AM is the orthogonal projection of AA' on that line:
AM is called the virtual velocity of the force P ; and is the infini-
tesimal distance, over which the point of application of P moves,
in its own line of action. If, as in the first figure of fig. 140,
AM lies along AP in the direction in which p acts, the virtual
velocity is taken to be positive : and if it lies in the direction of
AP produced backwards, as in the second figure, then it is taken
to be negative.
Hence, if the displacement of the point of application takes
place along the line of action of P, the whole displacement be-
comes the virtual velocity : and is positive or negative accord-
ing as it takes place in the direction towards which p acts, or in
the opposite direction.
Hence also, if the point of application of the force is displaced
in a line which is perpendicular to the line of action of the
force, the virtual velocity of the force is zero.
The quantity ~ebp is frequently called the virtual moment of
the force P in any assigned displacement. The importance and
meaning of this quantity in a Dynamical respect will be seen
hereafter.
This principle of virtual velocities is of the greatest import-
ance. It includes all Statics under the single equation (288),
for as bp in its most general form involves six arbitrary quantities
which correspond to the six possible degrees of freedom, so it
comprehends six conditions, which are the six equations of equi-
librium, and which may be deduced from it by a process the
reverse of the preceding. It also includes all Dynamics, as we
shall see hereafter ; and we shall also see that the equation of it
may be deduced from Dynamical principles, and may be inde-
pendent of the parallelogram of forces, by means of which we
have now proved it.
This principle has been made by Lagrange the foundation of
that great work of his on Mechanics, Mecanique Analytique.
Also, if every force at its point of application is resolved into
1 09.] VIRTUAL VELOCITIES. 143
three forces of which the action-lines are parallel to the axes of
x, y, z respectively, and if we call x, Y, z the axial components of
the force P, then the equation of virtual velocities takes the form
2.(x&r + Yty + z8,z) = 0. (289)
In connection with the theory of stability of equilibrium and
of the radial moment, which have been discussed in the preceding
section, it will be observed that as H = 2.P(#cosa-fycos/3-4-2cosy),
so the principle of virtual velocities as given in (287) expresses,
that consistently with the most general variations of x, y, z,
dH = ; and that consequently in an equilibrium-system the
radial moment has a critical value. This is indeed no more than
what is expressed by (288).
109.] The following are various problems which are solved by
the principle of virtual velocities.
Ex. 1 . Three forces p, Q, R act in given lines at the point A,
and are in equilibrium : it is required to determine the relation
between them.
Let the angles severally between the lines of action of Q, and
a, of R and p, of P and Q, be a, /3, y : let the point of application
of the forces be shifted from A to A', see fig. 141; and from A'
let perpendiculars A.'m, A?n, A.'j) be drawn to the lines of action of
p, Q, R respectively; then Am, An, AJO are the virtual velocities
of P, Q, R respectively : so that (288) becomes
PX AW + QX An RX AJP = 0.
Let AA'= bs; A'AP = ; QAR = a, RAP = /3, PAQ = y : so that
this equation becomes
s(/3 0) = 0;
Qsiny)tan0 = 0;
and as the line along which A is displaced is indeterminate, is
indeterminate, and therefore
P + QCOS y + Rcos/3 = 0,
Rsin /3 Q, siny = :
from the latter we have
P Q R .
sin a ~ sin/3 ~ silly/
the first term of the equality being inferred by reason of the
symmetry. Also we have
R cos /3 = p Q cos y, R sin /3 = Q sin y ;
144 VIRTUAL VELOCITIES. [ IO 9-
whence, squaring and adding,
R 2 = p 1 + 2 PQ cos y 4- Q* :
these are respectively the mathematical expressions of the tri-
angle and of the parallelogram of forces.
Ex. 2. To determine the conditions of equilibrium of the
straight lever.
Let ACB be the lever, fig. 142, which turns about a horizontal
axis through c : let the forces p and Q act at the ends A and B
along lines of action which are inclined to ACB at angles a and
/3 respectively : let AC = a, CB = b.
Let the lever be turned about the horizontal axis through an
infinitesimal angle dO, so that AA' = add, BB'= bdd : then the
projections of these quantities on the lines of action of p and Q
respectively are a dd sin a, b dd sin $ ; and as the virtual velocity
of Q is negative, (288) becomes
Tfadd sin a Q dd sin /3 = ;
.. Pflsina = Q#sin/3 :
which is the ordinary equation of moments about c.
Ex. 3. To determine the conditions of equilibrium of the
wheel and axle.
Let a = the radius of the wheel on which P acts : b = the
radius of the axle on which w acts : and let the system be
turned through a small angle dd, so that p (say) descends
through a vertical distance a dd, and w ascends through a ver-
tical space bdd : then (288) becomes
ad9p + bdOw ; .'. pa = w.
Ex. 4. To find the conditions of equilibrium in the screw.
In this mechanical power, as it is called, I shall assume that
there is no friction. Let h be the vertical distance between two
successive winds of the thread : let I be the length of the lever,
measured from the axis of the screw, at the end of which p acts :
let w be the weight on the screw. Then as w descends through
a vertical distance equal to k, the point of application of P moves
round the circumference of a circle whose radius is b : so that k
and 2-nb are evidently proportional to the virtual velocities of w
and P ; and equation (288) becomes
27rp-t-w = 0;
h
.'. p = - j-w.
2-nd
I Op.] VIRTUAL VELOCITIES. 145
Ex. 5. To determine the condition of equilibrium of a heavy
body resting on an inclined plane under the action of given
forces.
In applying the principle of virtual velocities to problems
wherein some of the forces are pressures against lines or sur-
faces, the reactions will not enter into the equation, if the
displacement of the point of application of the reaction is per-
pendicular to its line of action, because in that case the virtual
velocity vanishes. Hence also if one surface rolls on another,
and the resulting displacement is the arbitrary displacement out
of which the virtual velocity arises, the mutual reaction of the
surface does not appear in the equation of virtual velocities.
Several instances of this circumstance will be given in this and
the following examples.
In this example let us take the symbols, &c., of Ex. 1, Art. 26,
fig. 12. Let Q be shifted over a distance bs up the plane; then
the virtual velocity of P is 8 s cos /3, that of w= 8$ sin a, and
that of K = ; so that
p8scos/3 w8sina = ;
.'. Pcos/3 wsina = 0.
Ex. 6. Solve by virtual velocities the problem given in Ex. 1,
Art. 60.
Let the system as described in fig. 28 be shifted so that A and
B may still be in contact with the horizontal and vertical planes
respectively; and let a = BAG be diminished by 8a; then the
virtual velocity of T = b.2a cos a = 2 a sin a 6 a, and that of
w = 8.0 sin a = cosa8a; and those of the reactions vanish :
so that
T2asina8a4 wacosa8a = ;
.*. 2Tsina wcosa = 0.
Ex. 7. In the problem given in Ex. 3, Art. 60, fig. 30, let
the beam be shifted so that A is still in contact with the wall ;
then the principle of virtual velocities gives
w8.(#cos0 a cos 6) = 0;
.. sin$8$ asm 9 dO = 0.
But d sin < = 2 a sin ;
. . # cos c/> 8<J> = 2 a cos Q bO :
.'. tan = 2 tan <;
which leads to the results given in Ex. 3, Art. 60.
PKICE, VOL. III. U
146 VIRTUAL VELOCITIES. [lIO.
Ex. 8. Find the form of the curve in a vertical plane, such
that a heavy rod resting on its concave side, and on a peg at
a given point, say the origin, may be at rest in all positions.
Let the place of the peg be the origin, and let the rod be
inclined to the vertical at the angle ; let r be the radius vector
of the curve which coincides with the rod, and let 2 a be the
length of the rod. Then by the principle of virtual velocities,
w8.(y a)cos& = ;
.-. (r a) cos 6 = a constant = k } say:
. . r = a + k sec 6 ;
which is the equation to the conchoid of Nicomedes.
Ex. 9. In Ex. 3. Art. 37, prove that (79) is the equation of
virtual velocities ; and that in case (4), (84) is also the equation
of virtual velocities.
Ex. 10. A particle is attracted by two centres of force which
vary inversely as the square of the distance ; find the form of
the surface on all points of which the particle will be at rest.
Let ft and p be the absolute attractive forces, and let r and /
be the distances of the particle from the centres ; then by the
principle of virtual velocities we have
fidr n'dr _
-/T- :
r
U U
.'. - + ^ = a constant :
r r
which condition expresses the form of the surface.
110.] A remarkable theorem discovered by Gauss, and pub-
lished for the first time, so far as I know, in the fourth volume
of Crelle's Journal, may be deduced immediately from the equa-
tion of virtual velocities.
For a system of forces in equilibrium we have
2.P {cos a dx + cos ft dy + cos y dz} =0. (290)
Let the forces be replaced by line-representatives, and let (#, y, z)
be the point of application of the type-force P, and (, TJ, the
other extremity of the representative ; then replacing P cos a,
PCOS, P cos y respectively by x, yy, (z, (290) becomes
(say); (291)
and if the displacement of the system is such that the extremity
(6 V) f ^ ne line-representative of the type-force is fixed, while
HO.] VIRTUAL VELOCITIES. 147
the other extremity (.r, y, z) receives an infinitesimal displace-
ment, then integrating (291) we have
2-{(-*) a + Ol-J) f + -)'} = n; (292)
and thus n, which is the sum of the squares of the line-represen-
tatives of the forces, is a maximum, a minimum, or a constant.
Hence we have the following theorem :
If there are n points, at invariable distances apart, the sys-
tem of which is however moveable, and also if there is a system
of n points wholly fixed, each of which corresponds to a point of
the former system, then if the sum of the squares of the dis-
tances between each of the moveable points and its correspond-
ing fixed point has a critical value, the system of forces repre-
sented as to intensity and line of action by these distances, and
acting severally at the moveable points, is in equilibrium ; and
the equilibrium is stable or unstable, according as the sum of
the squares of the distances is a minimum or a maximum, and
is neutral if it is constant.
Also differentiating again (291) we have
D'n = 22.(dz* + dy' i + dz*)
-22. {(- x }d* X +(n-y}d*y + (S-z}d*z} ; (293)
and if the displacement, to which the variations of the coor-
dinates of the points of application of the forces are due, is such
that d*x = d*y = d' l z=0, then D 2 n is necessarily positive, and
n is a minimum ; also if 2.{( x)d a x + (r) y)d*y + (( z}d*z} is
negative, that is, when equilibrium is stable, n is a minimum.
The line-representatives of the forces, however, can always be
taken so small that x, ^y, z shall be infinitesimal;
whereby the second part of (293) being infinitesimal, and of the
third order, must be neglected ; and as the first part is positive,
n is a minimum ; that is, the sum of the squares of the line-
representatives is a minimum.
To this subject, however, we shall return hereafter, and in
a more general way. And in respect of the preceding it is
also to be observed that, in the displaced position of the body
on which the forces act, = ... = ...= - are supposed to
PCOSd P
act along lines parallel to their original lines of action ; whereas,
in the most general case, the new lines of action would be
functions of the original points of application.
u 2
148 CONSTRAINED EQUILIBRIUM. [ill.
SECTION 9. Constrained Equilibrium.
111.] The material body or system of material particles, which
receives the pressures considered in the preceding Articles, has
been supposed to be free from all constraint ; we must now in-
vestigate the modifications required in the general results when
the system is subject to certain given constraints.
Firstly, suppose one point of the body to be fixed ; let this
be taken for the origin : it is evident that, because it is fixed,
it will bear any pressure of translation acting on it, and that the
body will not move owing to that pressure ; but the effects of a
pressure of rotation about a rotation-axis passing through that
point are not affected by the fixedness of the point ; the im-
pressed forces therefore must be so related that, see Art. 70,
G = ; and therefore that,
L=0, M = 0, N = 0; (294)
which three conditions are requisite, so that a body, of which
one point is fixed, should be at rest. These three conditions, it
will be observed, satisfy equation (130), and therefore indicate
that the impressed pressures may be compounded into a single
force of translation : that, viz. which passes through the fixed
point.
And the pressure on the fixed point, and the direction of its
line of action, may thus be found : let R be the pressure, and
, 6, c the direction-angles of its line of action ; let the impressed
forces be p u P 2 , ... P N , and the direction-angles of their lines of
action a lf (3 lf y 1} &c. ; then
R cos a = 2.P cos a, ->
R cos b = s.P cos /8, > (295)
R cos c = s.p cos y -, J
.- . R* = (2.P cos a) z + (S.P cos /3) + (S.P cos y) J ; (296)
and therefore by (295) a, b, c are known.
112.] Secondly, let us suppose two points of the body to be
fixed; and let the axis of z pass through the two points, and
the origin be at the middle point of the line joining them ; and
let the z-ordinates to the points be +z^ and z l ; then it is
manifest that the body cannot have any motion of translation,
and can have motion of rotation about the axis of z only. The
impressed forces therefore must be so related that the rotation-
112.] CONSTRAINED EQUILIBRIUM. 149
pressure about the axis of z should be equal to zero ; therefore
the necessary condition is
N = 0. (297)
And the pressures on the two points may be determined in the
following manner : let them be represented by R, and R 2 , and
let the direction-angles of their lines of action be a lt & c t ;
a 3) $ a , Cy f then
E x cosa l + R 2 cos a, 2.P cos a, -
R! COS^+Rj COS b a = 2.P COS ft, I (298)
R! COS <?! +R 2 COS C 3 = 2.P COS y J J
L + R! Z l COS &i R 2 ?! COS # 2 = 0, "1
> [2991
M R!^ 0080! + RjZj COS # a = 0. J
From the first two of (298), and from (299), we have
Z 2.PCOS/3 L
RI cos 0, = ;
U Z i
Z i 5.PCOS/3 + L
R 2 cos a = ;
2z l
^,2.P cos a + M
R! cos a x = ;
2 z
z, 2.P cos a M
RJ cos 2 = ;
Z Zi
and thus the pressures on the fixed points, which are parallel to
the axes of x and y, are determined : but the pressures along the
axis of z are involved in only the third equation of (298), which
shews that the sum of the pressures is equal to 5.P cos y, and
therefore that each pressure is indeterminate : now this is, at
first sight, a startling fact, and has been urged heretofore as an
argument against the truth of our mechanical results and prin-
ciples ; because it is said that, when a body is supported in the
manner assumed in the problem, say a gate or a door on its two
hinges, the vertical pressures are determinate and may be ex-
perimentally determined at both hinges ; our mechanical formulae
therefore ought to yield a corresponding result. In any actual
case the pressures without doubt are determinate, and may be
determined by mechanical means : but then the bodies which
are the subjects of the experiments are more or less compressible
and extensible : they are not rigid ; and therefore do not satisfy
the conditions required in the preceding theory, however nearly
they may approach to them; thus if to a door, being in a
150 CONSTRAINED EQUILIBRIUM. [1*3-
horizontal position, two ' eyes' are attached, which correspond
to two hooks fixed in a vertical doorpost, and if the distance
between the eyes when the door is horizontal is equal to that
between the hooks in the vertical doorpost ; then doubtless, if
the body were perfectly rigid and inextensible, and were attached
by the eyes to the hooks, either one or the other hook would be
sufficient to bear the vertical pressure ; and we should be unable
to determine whether one or the other carried the whole weight,
and whether it was distributed between them, and in what pro-
portion ; yet as such a door is extensible, both hooks would bear
a part of the weight, and the respective proportions will depend
on the extensibility and the elasticity of the material. Thus if
the distance between the eyes is greater than that between the
hooks, the pressure will for the most part be on the lower hook,
although the compression of the material due to its weight may
cause the eyes so to approach eaoh other, that some of the pres-
sure may be brought upon the upper hook ; and a similar effect
may occur at the lower hook, when the distance between the
hooks is greater than that between the eyes. Thus it appears that
the determinateness of the pressures is due to the extensibility,
compressibility and elasticity of the material which is in nature
the subject of the experiment ; and the truth of the result which
is arrived at in (298) for a rigid body is not affected : for in nature
we have nothing of perfect rigidity. We shall see a further ex-
ample of indeterminateness of the same kind in dynamics.
Again, suppose the circumstances of constraint to be such,
that the body is capable of sliding along, as well as of turning
about, the axis passing through the two fixed points ; then the
points will be able to bear the pressures arising from the forces
which are resolved at right angles to the axis, and parallel to
the axes of x and y ; but will not offer any resistance to those
along the axis of z : if therefore equilibrium exists, the forces
must satisfy the conditions,
S.P cos y = 0, N = 0.
113.] And lastly, if three or more points of the body are fixed,
and if all these are not in the same straight line, it is evident
that the body is fixed ; and therefore whatever are the impressed
forces as to intensity, point of application, line of action, and
direction, the body is in equilibrium, if we suppose the fixed
points of it to be capable of bearing the pressures which are due
to the impressed forces.
II4-] CONSTRAINED EQUILIBRIUM. 151
And it is evident by the following reasoning that, if these
points are fixed, the body is also fixed. For suppose the body
to consist of n particles ; then each of these particles is at rest,
if the forces, including the tensions, mutual reactions, &c., act-
ing on it satisfy the three conditions (69), Art. 34 : and there-
fore if all are at rest, 3 n conditions are required. Now if three
points of a body are fixed, the mutual distances of them are
also fixed, and hereby we have three conditions ; also as the
body is rigid, the distances of each of the remaining n 3 par-
ticles from each of the three fixed points are given, and thus
we have 3n 9 conditions ; and as the equations of equilibrium
of a rigid body are six, we have six more conditions : and thus
altogether we have, as before, 3 n equations. If the three fixed
points are in one and the same straight line, one of the con-
ditions is lost, and the number is insufficient for equilibrium.
114.] Another form in which a body under the action of im-
pressed forces may be in constraint is, when it rests with points
of it on a plane, or against any surface.
Let us consider first the more simple case of a smooth plane :
and let us suppose the plane to be that of (x, y), and n points of
the body to rest on it; let these be (x lt y^ t (v t , y a ),...(x Hf y n ) ',
and let the pressures at these points be RI, R 2 ,...R n ; the lines of
action of which are parallel to the axis of z : thus the equations
of equilibrium become
2.P cos a = 0, 5.P cos /3 = 0, 2.P cos y 2.R = ; (300)
L 2.R^ = 0, M + 5.KO? = 0, N = 0. (301)
Here are six equations, of which only three involve the pres-
sures against the plane and the coordinates of their points of
action ; there are always therefore three independent conditions
to be fulfilled by the impressed forces.
Now if only one point of the body is in contact with the
plane, the pressure at that point will be given by the third equa-
tion, and the impressed forces must be such as to fulfil the other
five.
If two points are in contact, the pressures at them may be
determined by either two of the third, fourth, and fifth equa-
tions, and the forces must satisfy the remaining four conditions.
But if three points are in contact, the pressures at them may
be determined by means of the three equations which involve
the pressures, and the other three equations must be satisfied by
the impressed forces.
152 CONSTRAINED EQUILIBRIUM. [H5-
If more than three points are in contact, the pressures are
indeterminate, because there is not a sufficient number of equa-
tions for their determination.
In all cases the pressure which the plane has to bear is given
by the third equation of (300); and for the existence of equi-
librium, if the body only presses against the plane, it is neces-
sary that the 2.P cos y should act towards, and not from, the
plane ; it is also necessary that the line of action of this pressure
should pierce the plane of (x, y) at some point within the area
determined by straight lines joining the points of contact of
the body and the plane : otherwise the rotation-pressure of the
.z-force will cause the body to turn about one of the bounding
lines of this area.
And of the indeterminateness of the several pressures, which
act at the points of contact, when more than three points are in
contact with the plane, an explanation similar to that of Art. 112
may be given. Suppose a heavy body to rest on a horizontal
table, and to be in contact with it at many points ; the sum
of all the pressures is doubtless equal to the weight of the
body ; but if the points of contact are more than three, each
pressure, so far as the preceding theory enables us to determine
it, is indeterminate; and so it would be in fact, if the table
were accurately plane, and it and the body were perfectly rigid ;
but such a table and such a body do not exist : and so our
results when applied to flexible and compressible matter are not
true. If however we knew the laws of flexibility and elasticity,
and could thus bring into calculation all the conditions of the
problem, the result would be determinate and true ; and thus it
seems that the non-applicability of the mechanical principles is
only apparent, and is due to the omission of certain conditions
which the true solution of the problem requires.
115.] Again, suppose the body to be in contact with surfaces
whose equations are F X = 0, F 2 = 0, . . . F n = ; and the mutual
pressures between the body and the several surfaces to be R,,
R 3 , . . . R n ; the direction-angles of the lines of action of these
to be a l} &i, Cj ; a t) b 2 , c a ; . . .a n , 6 n , c n ; and the points of contact
to be (a?!,^,,^), &>?t> *)*... (**>jr>*) then employing the
ordinary notation, see Art. 36,
u v w
cos a = , cos = , cos c =
and the equations of equilibrium become
1 1 6.] CONSTRAINED EQUILIBRIUM. 15.'J
2.P cos a 4- 2.R cos a = 0, ->
2.P cos ft + 2.R cos b = 0, I- (302)
2.P COS y + S.R COS 6' = ; ^
L + 2.R(yCOS C Z COS #) = 0, -,
M + 2.R(2 cos a x cose) = 0, \ (303)
N-f 2.R(# cos b y cos a) = 0.
To which equations, as to the number of points in contact be-
tween the body and the surfaces, the remarks of the last three
Articles are applicable.
One point however requires further elucidation : suppose that
the surface of the body on which the forces act meets n given
and fixed points ; then the equations (302) and (303) contain
n undetermined pressures which act at these points. Now as
the equations are six in number, if n = 6, the six pressures at
the points may be determined ; and the directions of their lines
of action will be along the normals to the surface of the body
at the points; if n is greater than 6, n 6 of the pressures may
be indeterminate, and when they receive given values, the other
6 will be known : and when n is less than 6, the pressures at
the given points may be eliminated from the preceding equa-
tions, and the remaining 6 n conditions must be fulfilled by
the impressed forces acting on the body. And hence we infer
that generally a body under the action of given forces is in equi-
librium and fixed, if the bounding surface of it passes through
six given and fixed points * ; and that the mobility of it is not
taken away, if the surface has to pass through fixed points of
which the number is less than six.
116.] And hereby I am led to another subject : viz. to the
investigation of the conditions requisite that many bodies subject
to given pressures, and in contact with, or under mutual action
from, each other, should be in equilibrium.
Let the number of bodies be n ; let P U P 2 , ... P n be the types
of the forces which act on the first, second, ... nth body re-
spectively ; let R be the general type of the reacting pressures at
the points of contact, and a, b, c the direction-angles of its line
of action, and (x,y, z) the point of its application ; L U M,, N, ;
L 2 , M 2 , N 2 ; . . . the moment-axes of the component couples which
* For various other properties of this kind let me refer the reader to Mb'biua,
Lehrbuch der Statik, Zweiten Theil, Erstes Kapitel ; Leipzig, 1837.
PRICE, VOL. III. X
154 CONSTRAINED EQUILIBRIUM. [n?-
act on the several bodies ; then the conditions of equilibrium for
the several bodies are
2.P X COS Oj + 2.R t COS <Zj = 0, -j
2.P 1 cos/3 1 + 2R 1 cos<3 1 = 0, > (304)
2.Pi cos y l + 2.R t cos c l = ; -*
.R t (y t COS x ^j COS 3 t ) = 0, -i
2! cos*?! x l cosc 1 ) = 0, ^ (305)
N, + 2.R t (a?i cos b l y^ cos aj = ; ^
(306)
2.P,, cos a n + 2.R W cos a n = 0, -
2.P n COS ^ n + 2.R,, COS ^ B = 0,
2.P n COS y n + 2.R n COS C n = j -*
L B -4- 2.R (y, cos c n - z n cos J = 0, ~j
a?,,cosc n ) = 0, j- (307)
= 0. J
Now if, of all these groups of equations, all the first of the
first sets are added, 2.Rcos# will disappear, because, the reactions
of the several bodies being equal and opposite, the same quantity
will appear twice, and with different signs; so that we shall
finally obtain 2.P cos a ; similarly, by adding all the second
equations of the first set in each group, and by adding all the
third equations of the first set, we shall have
2.P cos /3 = 0, 2.P cos y = 0.
In the same way, by adding the several equations of the second
sets of the groups, we shall obtain equations free from the R'S,
and shall have ultimately
L = 0, M = 0, N = 0;
and thus the equations of condition necessary for the equili-
brium of a system of rigid bodies are of the same form and of
the same number as those required for the equilibrium of a
single rigid body.
117.] Examples illustrative of the preceding Articles.
Ex. 1 . A heavy uniform beam is fixed by a hinge to a given in-
clined plane : between the beam and the plane a heavy sphere is
in equilibrium ; determine its position and the several pressures.
Let fig. 37 represent a vertical section of the system made by
the plane of the paper: POB = a; POQ=20; oo = GA-=a; CP
II 8.] FRICTION. 155
= CQ = c ; w = the weight of the beam ; w = the weight of the
sphere ; R = reaction existing between the beam and the sphere ;
R'= the pressure of the sphere on the inclined plane. And let
us consider separately the conditions of equilibrium of the sphere
and of the beam.
For the equilibrium of the sphere, resolving the forces along
the plane, we have
w sin a = R sin 2 6.
For the equilibrium of the beam, taking moments about o,
we have wa cos (a + 20) = RXOQ,
r= RC COt ;
.'. \va cos (a + 20) sin 20 = we sin a cot 6 :
whence may Q be determined ; and thence R ; and since
R' = w cos a -f R cos 2 Q,
R' may also be found.
Ex. 2. Two heavy beams OA and OA' of equal lengths are
connected, fig. 38, at o by a hinge, and at A A' by a string of
given length ; between them a heavy sphere is placed, and the
string remains horizontal ; determine the tension of the string
and the pressure against the beams.
Let length of each beam be 2 a, weight of each beam = w;
2 c length of string ; T = the tension of the string ; b = the
radius of the sphere ; w = the weight of the sphere ; a = the
y
angle A OB = sin- 1 ; then for the equilibrium of either of the
JU Q/
beams, taking the moments of the forces about o, we have
T2#cosa = wa sina + R^ cot a;
and for the equilibrium of the sphere, taking vertical forces, we
w J b w ,
T = tan a + (coseca) 4 .
SECTION 10. On Friction.
118.] All the surfaces, which we have imagined to be in con-
tact in the preceding Articles, are supposed to be smooth, and,
as such, to offer no resistance to the motion of the points in
contact with them in directions perpendicular to the normal at
X 2
156 FRICTION. [ll8.
the points ; and therefore the reaction arising from the contact
acts along the common normal line only. In nature, however,
we have no surfaces perfectly smooth ; the constitution of all
bodies is such, that on their bounding surfaces are small eleva-
tions and depressions, arising, as it seems, from their constituent
molecules not being continuous and in perfect contact : so that
if the surfaces of two bodies are pressed against each other, the
elevations of one fit, at least in a measure, into the depressions
of the other, and the surfaces interpenetrate each other; and
the mutual penetration is of course greater, if the pressing force
is greater ; much of this roughness may be removed by polishing,
and the effect of much of it may be destroyed by lubrication :
all however cannot be, and there still remains a resistance due
to it, when force is applied so as to cause one body to move or
to have a tendency to move on another with which it is in con-
tact. This resistance is called friction, and is of two kinds ;
either of sliding or of rolling : the first is that of a heavy body
dragged on a plane or other surface ; of an axle turning in a
fixed box ; of a vertical shaft turning on a horizontal plate, or
of a millstone turning upon another concentric stone about a
vertical axis. Friction of the second kind is that of a wheel
rolling along a plane ; the resistance however of which seems to
arise from the necessity of the wheel overcoming small obsta-
cles which are successively in its path. It is of friction of the
first kind only that I shall at present state the laws and give
examples ; and first as to its line of action : it is manifestly
along that tangent line of the surfaces at the point of contact
which is the line of the tendency to motion ; and its direction
is opposite to that of the line of motion. Suppose therefore
many forces to act on a material particle which is in contact
with a rough surface; and, the lines of action of the forces
being unaltered, their magnitudes to change, so that motion is
on the point of taking place (1) in one direction, and (2) in an
opposite direction : the line of action of friction is in both cases
the same ; but the direction of it in the former case is contrary
to that of it in the latter. Also the magnitudes of the forces
may evidently vary within certain limits, and the particle may
still be at rest. Examples of the determination of these limits
are given in the following Article.
In our ignorance of the constitution of bodies, and of their
molecular action, the laws of friction must be deduced from
I 1 8.] FRICTION. 157
experiment ; and therefore I shall enunciate those only which
are necessary for our purpose, and refer the reader to the Trea-
tise by M. Morin*, wherein he will find the subject investigated
in all its completeness.
I. Friction is proportional to the normal pressure, when the
materials of the surfaces in contact are the same.
II. Friction is independent of the extent of the surfaces in
contact.
III. Friction is independent of the velocity of motion.
As to law I ; suppose R to be the normal pressure between
two surfaces, and F to be the friction, then F = /xR, where /u is
a constant quantity for the same materials and is the value of
F when R = 1 ; ft, is called the coefficient of friction. And this
law, it may be observed, appears to arise out of the preceding
theory of friction ; because the greater is the pressure, the
greater is the interpenetration of the molecules at the surface
of the bodies, and the greater is the resistance to be overcome,
when motion is just about to take place.
As to law II ; it signifies that if the pressure remains the
same, and the surface in contact increases, the total resistance
is still the same, whilst the pressure on each element and the
friction corresponding to that element are diminished in the
inverse ratio of the area of the surfaces in contact.
The treatise of M. Morin will be found to contain a complete
account of the modes of determining jtx for different substances ;
but the following manner of considering the subject is suffi-
ciently simple, and sufficiently general for our purpose.
Let a given heavy body rest with a plane face of a finite area
on a horizontal plane; and let the plane be turned about a
horizontal line in it, so that it becomes inclined to the hori-
zontal plane, that is, becomes tilted : the body will begin to
slide when the inclination has reached a certain limit ; and this
inclination will manifestly depend on the friction which exists
between the body and the plane, and may be determined as
follows. See fig. 39.
Let w be the weight of the body ; ju, = the coefficient of
friction ; a = the angle between the inclined and the horizontal
* Nouvelles Experiences sur le frotteiuent faites a Metz, iniprimees par ordre
<lc I'Acaddmie des Sciences ; 3 vols. in 410. 1832-1835.
158 FRICTION. [ JI 9-
planes just as motion is beginning to take place ; R = the pres-
sure on the plane ; so that
F = /UR; (308)
and resolving along, and perpendicular to, the plane,
F = w sin a, R = w cos a ;
.'. tana = /u, a = tan~ > : (309)
a is called the angle of friction, and the angle of rejwse. The
body will rest on the plane when the angle of inclination is less
than the angle of friction, and will slide, if the angle of incli-
nation exceeds that angle.
11 9.] Various problems involving friction.
Ex. 1. A small ring under the action of known pressures is
capable of sliding on a rough curved material line in space ; it
is required to determine the limits of the forces, so that the ring
may be at rest.
Let the resolved parts of the impressed forces along the co-
ordinate axes be x, Y, z, of which let the resultant be R ; so that
if x } y, z are the coordinates to the position of the ring on the
curve, the whole impressed force along the tangent, which we
will call T, is
dx du dz
T = X-T- + Y-f +Z-y . (310)
ds ds ds
Let N = the normal pressure : then
= R 2 ,
/ dx dy dz \*
.'. N 2 = X 2 +Y 2 +Z 2 (X-j- +Y- + Z-y- ) .
^ ds ds ds '
Now in order that motion should not take place,
T 2 < /x 2 N 2 < /^(R 2 T 2 );
T a Ll 8
.*. < - a < (sin a) 2 , see equation (309);
iv 1 ~f~ /-*.
(3U)
, . Jidx+vdy + zdz
and if f - =+smo, (312)
R CIS
the particle will begin to slide ; the + sign assigning the limits
within which the forces are to be confined.
Ex. 2. As an example, let us take the helix whose equations are
1 1 9.] FRICTION. 159
and let the force which acts on the ring be its own weight, and
= ID, and have its line of action parallel to the axis of z : then
z = R = w : and
dz k ,
-=- = - = -v- sin a : .* . k = + tan a :
ds (i + )*
that is, the angle of inclination of the thread of the helix to the
horizontal plane is equal to the angle of friction.
Ex. 3. To determine the limits of the pressures, so that a par-
ticle under the action of them may be at rest on a given rough
surface.
Let F (a?, y, z] be the equation to the surface : then em-
ploying the ordinary symbols, if N = the normal pressure, T =
the tangential force, and R = the resultant of the acting forces,
of which the resolved parts along the coordinate axes are x, Y, z,
XU + YV + ZW
N = - ! - -, T 4 = R 2 -N 2 ;
"
therefore that the particle should be at rest
R 2
T 2 <JU 2 N 2 , R 2 N 2 </X 2 N 2 , <14^ 2 ;
Q 2 R S
' , r- < l+ju 2 <(seca) 2 ; (313)
(XU+YV + ZW) a
and therefore if - = + sec a, (314)
XU + YV-f ZW
the particle will just begin to move ; the + sign assigns the
limits of the impressed pressures. As an example let us take
the following :
Ex. 4. An ellipsoid has its least axis in a vertical direction ;
determine on the surface the curve, on all points within which
a heavy material particle being placed shall remain at rest.
In this case x = 0, Y = 0, z = R ;
^! .^..fl- i.
+ + ~
V^T + -|T
therefore (314) becomes
* ~ + 77-( tan )'-
a* It* ^ ' c*
160 FRICTION. [119.
which is the equation to a cone, whose vertex is at the centre of
the ellipsoid; and the line of intersection of which with the
ellipsoid is the required bounding curve.
Ex. 5. A heavy particle rests on a rough inclined plane, and
is acted on by a given force in a vertical plane which is perpen-
dicular to the inclined plane ; determine the limits of the force,
and the angle at which the least force capable of drawing the
particle up the plane must act.
Let fig. 40 represent a vertical section of the inclined plane,
and containing the force p ; let the inclination of the plane to
the horizontal plane be i ; and let 6 be the angle between the
inclined plane and the line of action of P ; /* = coefficient of
friction : and let us first suppose the tendency to motion to be
down the plane, so that friction is a force acting up the plane :
then resolving along, and perpendicular to, the plane,
F + p cos 9 = w sin i, K + P sin = w cos i, F = /xu ;
sinz fjicosi
- r
..
cosy f
And if P is increased so that motion up the plane is just be-
ginning, F acts in an opposite direction, and therefore the sign
of p. must be changed, and we have
sin i + n cos i ftic\
p=w - - - : -. (dlv)
cosd + psind
Now .to determine Q in this latter case, so that P shall be the
least, ^ P sin0 u cos Q
-j- = w (sm a + LI cos /) ; - - - -. = 0,
dd ; (cos0 + jisin0) 2
if tan 6 = p ;
that is, if 6 is equal to the angle of friction. Hence we infer that
A given power acts to the greatest advantage in dragging a
weight up a hill, if the angle at which its line of action is in-
clined to the hill is equal to the angle of friction of the hill.
And, similarly, a power acts to the greatest advantage in drag-
ging a weight along a horizontal plane, if its line of action is
inclined to the plane at the angle of friction of the plane.
Hereby also may we determine the angle at which the 'traces'
of a drawing horse should be inclined to the plane of traction.
The preceding results are those which are a priori to be ex-
pected, because some part of the power ought to be expended
in lifting the weight from the plane, so that friction may be
diminished.
1 1 9.] FRICTION. 161
Ex. 6. Also let us consider the case of a rough cylindrical
axis, on which given forces act and produce a pressure of rota-
tion, capable of turning within a rough hollow coaxal cylinder.
Let fig. 4 1 be a section perpendicular to the axis of the cylin-
der ; the smaller and interior circle being a section of the cylin-
drical axis, and the larger circle of the hollow cylinder ; let C
be the point of contact of the two cylinders, and at which of
course the resultant of all the impressed forces acts : let this
force = P, and let be the angle between the lines of action of
E and P : then
E = P cos 6, F = P sin 0,
F = ftEj .-. tanfl = JA;
therefore is equal to the angle of friction. If therefore the
angle between B, and p is less than the angle of friction, the
cylinder will continue at rest ; and if it is greater, it will move.
Ex. 7. A heavy circular shaft rests in a vertical position, with
its end, which is a circular section, on a horizontal plate ; deter-
mine the resistance due to friction which is to be overcome,
when the shaft begins to revolve about a vertical axis.
Let a be the radius of the circular section of the shaft ; and
let the plane of (r, 9) be the horizontal one of contact between
the end of the shaft and the plate ; and let the centre of the
circular area of contact be the pole ; now the vertical pressure
on each element of this area manifestly varies as the area ; and
therefore, if r dr dO is the area-element and k is the coefficient
of variation, since, by law III, friction is independent of the
velocity of motion,
the pressure on the element = kr dr dQ ;
.-. the friction of the element = pkrdrdO ;
the moment of friction about the vertical axis through the centre
= phr* drdO-,
. ' . the moment of friction of the circular end
flit fa
= I / i*.kr
JQ JQ
*drdO
3
Now if w = the weight of the shaft ; since k is the pressure
on an unit of area,
w = TT ka? ;
PEICE, VOL. III. Y
162 FRICTION. [ IJ 9-
.*. the moment of the friction of the circular end = ^-- ,
O
and consequently varies as the radius. Hence arises the ad-
vantage of reducing to the smallest possible dimensions the
area of the base of a vertical shaft revolving with its end resting
on a horizontal bed.
Similarly may the friction of the upper millstone moving on
the nether one be calculated.
Ex. 8. If the shaft is a square prism of the weight w, and
rotates about an axis in the centre of the shaft, then the mo-
ment of friction varies as the side of the square section of the
shaft.
Ex. 9. If the shaft is composed of two circular cylinders
placed side by side, and rotates about the line of contact of the
two cylinders, then
the moment of the friction of the surface
in contact with the horizontal plane = .
9ir
Ex. 10. A heavy straight rod rests on a rough horizontal
plane, and at one end of the rod, in a line perpendicular to its
length and in the plane, a force pulls the rod, the magnitude of
which is just sufficient to move the rod in the plane. Shew that
the point, about which the rod begins to turn is at a distance
= a \/2 from the other end of the rod, if the length of the rod
is 2 a.
CHAPTER IV.
ON GRAVITY, AND CENTRE OF GRAVITY.
SECTION 1. Elementary considerations on mass, gravity, and
weight.
120.] Into the investigations of this and of subsequent Chap-
ters there will enter certain elementary conceptions of matter
beyond those which have hitherto been stated. In Chapter II.
matter was defined as the subject of force ; occupying space, and
consequently possessing form : capable of infinite divisibility,
and thus resoluble into particles ; capable of rigidity, in which
state the particles are in relative rest ; and transmitting force in
the line of action of the force only, so that the external forces
acting on the matter are of infinitesimal magnitude in com-
parison of the internal forces which act on the several particles
and keep them in relative rest ; for the relative equilibrium is
not affected by the action of the forces which act on the matter
from without. Now we require other properties of matter.
Matter is impenetrable; that is, two particles of matter cannot
occupy the same place at the same time.
Matter is porous; that is, although matter is composed of
particles or molecules or atoms, yet these are not packed in
close and continued contact; but there are intervals or inter-
stices, which do not contain the matter of the body, whatever
that is by which they are occupied.
According to the greater or less degree of closeness with
which the particles are packed, so is matter more or less dense ;
and density is predicated of it in respect of this quality. If the
density of matter is constant throughout a given body, the body
is said to be homogeneous; but if the density changes, either
continuously or discontinuously, the body is said to be hetero-
geneous ; in the more general case the density varies continu-
ously, and at a given point is a function of the coordinates
of the point. Thus the earth is not homogeneous ; the density
Y 2
164 MASS. [121.
of it increases as we pass from the surface to the centre ; it is
doubtless composed of concentric shells, each of which has sur-
faces of the form of an oblate spheroid and is homogeneous;
and the density of which is a function of the axes of the shell.
The average density of a heterogeneous body is called its mean
density. The mean density of the earth is about five times that
of distilled water.
121.] As the quantity of matter contained in a body is a func-
tion of the volume of the body and of the density of the matter,
it is necessary to have means of measuring the same with
precision.
Quantity of matter is called mass ; so that the mass of a body
is the quantity of matter contained in the body.
Density \& the quantity of matter contained in an unit- volume;
the absolute density or the closeness with which the particles
are packed, being uniform throughout that unit-volume. This
definition is directly applicable if a body is homogeneous ; but
if it is heterogeneous, and the density varies from point to point,
the density at any point is the quantity of matter contained in
an unit-volume, throughout which the density is the same as
that at the point. Density is commonly denoted by the symbol
p, which is constant in homogeneous bodies, and in heterogeneous
bodies is a function of the coordinates.
Thus if v is the volume of a homogeneous body of which p is
the density, the mass = P v; (1)
and if the body is heterogeneous, and is referred to a system of
rectangular coordinate axes ; and if p is the density at (x, y y z],
then the mass = fpdv; (2)
dv being an element of the volume, p being a function of the
coordinates of the place of dv, and the sign of integration
denoting the process of summation, whether that involves one
two, or three integrations, according to the dimensions of the
body, and the integrations extending through the space occupied
by the body.
Density is usually measured by means of comparison with
some substance the density of which is assumed to be the unit-
density. This latter substance is commonly taken to be distilled
water at the temperature 39.4 Fahrenheit, and under a baro-
metric pressure of 2116.4 Ibs. on the square foot; so that by
means of this comparison p is a number ; and the value of it
121.] MASS. 165
for any given substance is called the specific density of that
substance. Thus for platinum, p = 21.5, and this means that,
bulk for bulk, and under the stated conditions, platinum con-
tains 21.5 times more matter than distilled water.
The following are examples in which mass is determined,
when the law of varying density is given.
Ex. 1 . To find the mass of a straight wire or rod, the density
of which varies directly as the distance from one end.
Let the end of the rod be taken as the origin, and let a be
the length of it ; and let the distance of any point of it from
that end = x ; let o> = the area of a transverse section of it ;
then d\ = u>dx; and p = kx; therefore
the mass of the rod = / kvxdx
r a
= /
J
2
Ex. 2. To find the mass of a circular plate of uniform thick-
ness, the density of which varies as the distance from the centre.
Let T be the thickness of the plate and a its radius : let the
centre of the plate be the origin, and let it be referred to polar
coordinates; so that dv = rrdrdd : let p = kr ; then
/*2 TT r Q,
the mass of the plate = / / krr*drdO
rZ
plate = /
JQ
3
If the density is constant, and the thickness varies directly as
the distance from the centre; then T = kr, and we have
Mir ra
the mass of the plate = / / pkr*dr
^0 *^0
dO
3
Ex. 3. The mass of a sphere, the density of which varies in-
versely as the distance from the centre = 27j-p 3 , where p is the
density of the outside stratum.
Ex. 4. The mass of an ellipsoid composed of shells the prin-
cipal sections of which are similar ellipses, and the density of
which varies as the semi-axis major of the largest principal
section of each shell, is equal to Ttpa^bc, where p is the density of
the outside stratum.
166 CENTRE OP MASS. [l22.
Ex. 5. To determine the bounding curve of a thin ribbon of
uniform thickness and density, such that the breadth of it
corresponding to each ordinate may be proportional to the mass
of the ribbon beyond it.
Let the curve be that delineated in fig. 63. Let the axis of x
be vertical, and that of y horizontal. OM = x, MP = y, OA = a.
Let r be the constant thickness of the ribbon, p its density ;
then taking the part of the ribbon on the positive side of the
axis of x, the mass of it below MP
Cy=y
- I rpydx;
Jy=0
fy=y
therefore by the data / rpydx = py,
.. ydx = kdy, dx k',
the equation to the logarithmic curve. Similarly, if OA'=', for
the curve on the other side we shall have
/= <.
122.] The letter m is usually employed to denote mass, and M
to denote the sum of many masses, and consequently the mass
of a body, so that M = 2.m. Now when many particles occupy-
ing points in space are the subjects of our inquiry, there is a
certain point in reference to their masses and to their positions
which is frequently of great importance towards the simplifica-
tion of the investigation. Let there be n particles whose masses
are respectively m lf z 2 , . . . m n , and let the places of them be
(*>i, y Zi}> (*w y*> z*}, " (%*, y n , *) If these particles are all
equal, and each is equal to the unit-particle, the mean, or
average, of their distances from a given plane is
if p l} j)i, ... are the distances of the particles severally from the
plane. But if the mass of a particle is m, that particle contains
m unit-particles, so that in the preceding formula m of the p's
become identical ; and thus if all the particles are of masses
different or not as the case may be, the formula becomes
1 23.] GRAVITY AND WEIGHT. 167
2 'tfl *D
which we denote by - . Hence if x, y, z are the mean dis-
tances of the places of the several particles from the planes
of (y, z), (z, x}, (x, y) respectively,
s..mx -s.my -s..mz
x -, = _ . z - -. (3)
Z.m z.m -s..m
The point (x, y } z) thus defined, and thus determined, is called
the centre of mass, or mass-centre, of the system of particles, and
is a definite point in every system ; for whatever are the values
of the numerators in the preceding expressions, the denominator
is a positive quantity, and cannot vanish, so that the expres-
sions cannot take an indeterminate form.
If the system of masses is a body, and is continuous, and the
density at any point is p, then
_fpydv .
' fpdv ' fpdv ' ~~'
so that the centre of mass of any system of particles is that
point whose distance from any plane is equal to the sum of the
products of each mass into its distance from that plane divided
by the sum of the masses.
Hence, if the centre of mass of a system of material particles
is taken as the origin,
Z.mx z.my = z.mz = ; (5)
and if the system of particles is a continuous body
J*pxd\ =.fpyd\ = fpzdv = 0. (6)
And here I might proceed to consider the various forms which
(3) and (4) take according to the continuous or other distribution
of matter, and according to the bounding forms of bodies, and
to apply them largely to special cases, and there would be a
theoretical advantage in such a method, as it would preserve the
generality of the expressions, and this point is of great import-
ance in many subsequent investigations. But as the preceding
expressions have been almost universally considered and applied
from another point of view, and as there is no practical incon-
venience in following that course, I will take it ; the number of
applications of (3) and (4) will not thereby be lessened ; and these
remarks will prevent the student from limiting his view of the
subject to the restricted aspect which this latter conception of
it presents to him.
123.] Of all terrestrial, and indeed of all cosmical matter, as
168 GRAVITY AND WEIGHT.
far as our knowledge extends, every particle attracts towards
itself every other particle; and all would come into close contact,
did not some forces act to hinder them. This property is in-
herent in cosmical matter, but we know neither the cause of it
nor its mode of operation. It is called gravity, and its action-
line is the line which joins the two particles, and its intensity
varies inversely as the square of the distance between the par-
ticles, so that if the distance is increased, say, twofold, the
attraction is diminished, and is only one-fourth of what it was
before. We shall enter on the inquiry into these and kindred
subjects hereafter. By reason of this power of attraction the
earth attracts all other matter towards itself, and we shall
shew hereafter that the resultant attractive force of all the par-
ticles of the earth on a particle outside of it varies approximately
inversely as the square of the distance of the particle from the
earth's centre.
Now of bodies which are the subject of investigation to us,
and are near to the earth's surface, the dimensions in all direc-
tions are usually so small in comparison of the distance of the
body from the centre of the earth that we may, without sensible
error, suppose the earth to exert an equal force on all particles of
the body which are of equal mass ; and as gravity is a force
which penetrates matter, and acts with equal effect, whether the
particle on which it acts is within a body, or on its bounding
surface, or separate, so the effect of it on a body varies as the
mass of the body ; the amount of this attraction of the earth on
a body is called its weight; and is thus measured. Let the
mass-element of the body be m, and let g be the weight of an
unit-mass ; that is, g is the amount of the earth's attraction on
an unit-mass at the place ; then
the weight of m = mg ;
and if the mass of the body is M,
the weight of the body = M# :
M having been determined by the processes indicated in Art. 121.
So that of a body of the most general form, and heterogeneous
in structure, the weight = yyy^ dx dy dz (7)
If the volume of a homogeneous body is v and its density is p,
then its weight = pg\; consequently if v = 1,
the weight of an unit-volume = pg ; (8)
this weight is sometimes called the specific gravity, but some-
I 24.] GRAVITY AND WEIGHT. I C'j
times and more correctly called the specific weight of a substance.
It is evidently the product of the specific density, and the weight
of the unit-mass at the place.
124.] I have been obliged to limit g to the weight of an
unit-mass at a given place: for although mass is the same where-
ever the body may be, yet the weight of it varies from place
to place; gravity is not the same at all places of the earth's
surface : it increases as we go from the equator, where it has its
least value, towards the poles, where it has its greatest value :
and this increase is according to the following law given by
Clairaut. Let G and g be gravity at the equator, and a place
whose latitude is A, respectively ; then
g = G{1 +. 005 133 (sin A)*}.
This increase is due to two causes : (1) the statical attraction of
the earth, and (2) the dynamical action of centrifugal force :
to the consideration of both these causes we shall return here-
after. And it also changes, as we pass further from the centre
of the earth: for bodies external to the earth's gravity decreases
in the ratio of the inverse square of the distance from the centre
of the earth; also as we pass from the surface of the earth
towards the centre, as e. g. down a mine, its intensity decreases,
and varies directly as the distance from the centre of the earth.
A proof of these propositions will be given hereafter. Gravity
also varies according to the nature of the materials of the earth
in the neighbourhood of the place where it is considered : its
value on an island is different to that on a continent : it is also
affected by neighbouring mountains, and in line of action as
well as in intensity.
The line of action of it is vertical, that is, is perpendicular to
the surface of still water. Now although the earth is not quite
spherical, so that all verticals do not meet at the centre ; yet its
radius, about 4000 miles, is so large, compared with the dimen-
sions of any bodies which we shall at present consider to be
subject to gravity, that all vertical lines corresponding to mole-
cules of the same body may be reckoned parallel ; and therefore
all the particles of material bodies may be considered to be acted
on by forces whose lines of action are parallel.
Another point also requires some remarks. In these Articles
different concrete units are involved. Now the symbols p, dv, g
are symbols of numbers ; and therefore their product is a
number ; but the quantity which we commence with is volume-
PRICE, VOL. III. Z
170 CENTRE OF GRAVITY.
element, and that which we end with is weight-element : it
remains therefore to seek the source whence this change arises ;
it is true, as it is convenient, that dv expresses the number of
the volume-units, p the number of mass-units in a volume-unit,
and g the number of earth's attraction-units in a mass-unit : but
how does the result of all this imply weight ? In the first place,
the process ( multiplication ' must be used in a sense wider
than its numerical one, so as to include within its subjects of
operation quantities of different kinds ; and so that the product
may be of a kind different to that of either of the multiplicands :
and thus the product of two concrete units is a concrete unit
of a different kind ; the product of the volume-unit and of the
density-unit is mass-unit ; and the product of the mass-unit and
of the earth's attraction-unit is weight-unit ; the change of con-
crete unit therefore arises from the product of the different
concrete units ; and weight-unit is the product of three different
concrete units. The units are of course ai-bitrary, and therefore
we choose those which are most convenient ; and thus we take
a cubic inch to be the volume-unit ; the density of distilled
water, at a certain temperature and under certain atmospheric
pressure, to be the density unit ; and the earth's attraction at
a given place on a mass-unit to be the gravity-unit ; and by
means of these we obtain the weight of a cubic inch of distilled
water at a certain place, and compare all other weights with it.
125.] Thus by reason of the earth's attraction every mass-
element of the body becomes the source and point of application
of a force which varies as the mass of the element; and the
action-lines of all these forces are vertical and parallel. Conse-
quently they are subject to the laws of composition of such
forces which are investigated in Arts. 79, 80. The resultant is
equal to the sum of the components -, that is, the weight of the
body or system of particles is equal to the sum of the weights of
the component particles. Its action-line is vertical. It has
also a definite point of application the coordinates of which are
assigned by (146) Art. 80. This point is called the centre of
gravity, being the centre of the parallel forces ; and if it is fixed
the body will rest in all positions, and every line passing through
it is an equilibrium-axis, the equilibrium of the body thus sup-
ported being continuous.
Firstly, let the system consist of many material particles sepa-
rate from each other; let their masses be m l} m t , ...m n , and let
1 25.] CENTRE OP GRAVITY. 171
the positions of them be (x l} y lt z,), . . . (x n ,y n) .?) ; let the centre
of gravity be (x, y, z) ; then as the weights are m l g, m 2 g,...in n g,
R = t.mg = g ?.m ; (9)
x 'Z.mg = s.mgx ; .. x z.m = 'S.MX j -j
y-s..mg *.mgy\ y-Z.m = ^my\ \ (10)
zi.mg = s.mgz', zz.m = -s,.mz;J
whereby both the resultant and the position of its point of ap-
plication are known. And from the form of these equations it
follows that, in the investigation of the centre of gravity of a
system of material particles or bodies, we may, if it is conve-
nient, divide the system into groups, and calculate separately
the centre of gravity of each group ; and by a similar process
deduce from them the centre of gravity of the whole system.
Secondly, let us take the case of many material particles
aggregated into a continuous body, so that the symbol of sum-
mation becomes that of integration ; and let the coordinates to
the type volume-element of the body be x, y, z : then the type-
force is pg dv ; let (x, y, z) be the centre of gravity ; then from
(146) Art. 80,
xj pgd\ =
_ r r
y] pgdv = ]pgydv,
(11)
V V
z I pg dv = I p
r .
I is used on both sides of the equations as a general symbol
of summation ; and is to be replaced by the symbols of single,
double, or triple integration according to the different values
of dv, and the integration is to extend through the space
occupied by the body.
In reference to these values it is to be observed that the
centre of gravity is the point of application of the resultant of
all the weights of the several component particles of a body,
which resultant is equal to the sum of the separate weights ; it
is therefore that point at which, if the weight of the whole body
acts, an effect is produced the same as that of all the particles of
the body taken in combination ; or, in other and equivalent
words, the centre of gravity is that point at which, if the body is
collected into a material particle, the circumstances of pressure are
the same as those of the body in its actual state.
There are of course many cases where the centre of gravity
z 2
172 CENTRE OF GRAVITY. [l2,6.
is known at once, by reason of the symmetry of the figure;
thus the centre of gravity of a straight wire or rod, of the same
density and thickness throughout, is at the middle point of the
rod ; the centre of gravity of a circular wire of the same density
and thickness throughout is at the centre of the circle : that of
a circular or of an elliptical plate of constant thickness and den-
sity is at the centre : that of a homogeneous sphere and of a
homogeneous ellipsoid is at the centre : and in a similar manner
we shall frequently conclude from the symmetry of the figure,
that the centre of gravity of a body is in a particular line which
can be at once determined.
126.] Since g in (10) and (11) denotes a constant quantity,
it may be divided out from both sides of the equations ; and if
this is done, the results are then identical with (3) and (4),
Art. 122 ; and thus it appears that the centre of gravity always
coincides with centre of mass. These points however arise from
two different and distinct conceptions ; the latter depends on
the constitution of the body only, and its position is geome-
trically derivable from that constitution without any relation to
external circumstances ; it is independent of the place of the
body and of any forces acting on it. The former, on the other
hand, involves the conception of the earth's attraction, assumes
that the action -lines of the force of gravity which acts on each
particle are parallel for all particles, and that these forces are
proportional to the masses of the particles. These assumptions
are only approximately true ; and consequently the point is more
truly conceived of as the centre of mass than as the centre of
gravity. Although in deference to usage I shall call the point
the centre of gravity, yet the place of it will always be deter-
mined by the formulae which were investigated by means of its
conception as a centre of mass ; and I may say that the most
important applications of it involve the conception of centre of
mass and not that of centre of gravity.
It is also to be observed that as gravity is not the same at
different places on the earth, the weight of a given mass is not
the same at all places. Mass however is the same at all places ;
and consequently a certain mass and not a certain weight must
Jbe taken as the measure of comparison of other masses. Thus
.standards of weight, as they are called, are masses and not
weights. As the weight however at a given place varies as the
jnass, two masses may at a given place be compared by means
1 27.] THIN WIRES. 173
of their weights at that place. Two masses are equal if their
weights at the same place are equal, and thus one mass is n
times another if the weight of the former is n times that of the
latter. Weights are easily compared by means of the balance
and its varied forms. Thus these instruments indirectly compare
masses : and herein their great value consists ; and hence arises
the necessity of their perfection. In commerce too, no less than
in experimental physics, the comparison of mass and not the
comparison of weight is required. Mass is absolute ; weight is
relative. We shall return to the subject of the comparison of
masses at a future stage of the treatise.
SECTION 2. The centre of gravity of material lines or wires,
straight and curved.
127.] Let us first consider the centre of gravity of a curved
material line or wire, of which the thickness is infinitesimal in
comparison of the length.
Let a> = the area of a transverse section of the wire, and
da = a length-element, so that dv = (ads; let p be the density
at the point (x,y} } and g = the earth's attraction; and let (x,y,z}
be the coordinates of the centre of gravity ; then
x I pgads = ipgvxds,
yjpgads =
z I pgu>ds = Ipgazds.
(12)
The integrals are of course definite, and the limits are fixed by
the conditions of the problem. If the curve of the wire lies
approximately wholly in one plane, we may take that to be the
plane of (x, y], or of (r, ff), and in that case, the first two of (12)
are sufficient to determine the centre of gravity, since 5=0.
If the curve of the wire is of double curvature all three equations
are required.
It will be found that in many cases the centre of gravity of a
material line is outside of the line ; and it is necessary therefore
that it should be rigidly connected with it if the wire or rod is
to have physical support ; but such connection is not necessary
for the centre of mass.
174 CENTRE OF GRAVITY. [128.
128.] Ex. 1 . To find the centre of gravity of a wire of uniform
thickness and density, bent into the form of a quadrant of a circle.
Let the radius of the circle be a; fig. 43; then as po> and#
are constant, they may be divided out, and (12) become
x \ ds I xds, y I ds = I yds ;
also x* +g* = a* ;
dx dy ds f
y " x ~ a
_ C a adx f a axdx
.'. XI - : = / - - '>
Jo (a* x*y* Jo ( a *x*y
x rsin- 1 -l a = f (a *}*Tj
L a J L 'Jo
adx r a _r . #T r T.
7 = I adx, y\ sm- 1 - = \x \ ',
-t -'o J o L J
Or thus by means of polar coordinates ; r a ;
2a
rf rf
.. x dd = I acos0d8,
JQ Jo
/IF /*ar
y / dO = / a sin dd,
Jo JQ
x =
77
2a
y =
7T
Ex. 2. To find the centre of gravity of a wire of constant thick-
ness and density, and bent into the form of a complete cycloid.
Let the starting point of the cycloid be the origin, and let
the equation to the curve be
x = a versin- 1 - (2 ay y^ ;
a
dx dy ds
y \ " (2a-y) ~ (2^*'
it is evident that the centre of gravity will be in the line per-
pendicular to the base at its point of bisection ; therefore x=.-na\
and as p, g, o> are constant,
2a za ydy _ _ 4a
' '
4 & 4 a
For a wire in the form of a semicycloid, x , y .
3 o
128.] THIN WIRES. 175
Ex. 3. To find the centre of gravity of a wire of constant
thickness and density, bent into the form of an arc of a circle.
Let the radius of the circle be a ; and let the line passing
through the middle point (the vertex) of the circular arc and
the centre of the circle be the axis of x; then as the arc, fig. 44,
is symmetrical with respect to this line, y = 0. Let the arc
BOB'= 2s, and let the chord BB'= 2c, OD = d; then
y* 2axx" ;
dy da; ds
a x y ' ' a
_ C b dx [ b x dx
and x I - r = / - ;
Jo (2ax #')* .'o (2ax-x*}*
ac
x = a
Ex. 4. To find the centre of gravity of a wire in the form of
a half of one loop of a lemniscate.
Let the equation be r 2 = a* cos 2 d ; and let I be the length of
the half loop ; then
dr _ r dd _ ds m
a* sin 26 ~ a 3 cos 2 ~ a 7 '
T5 a'
. . x I = / r cos ds = ',
J n 0*
2*
7 f f ' * 7 2*- 1
yl = I r sin as = a* - - .
*'o 2*
Ex. 5. To find the centre of gravity of a straight rod, the
thickness of which varies directly as the distance from one end.
Let the end of the rod whence the variation of the thickness
is reckoned be taken as the origin, and the line as the axis of x :
then b> = kx ; let a = the length of the rod ; and we have
/;/> 2 a
xl pgtixdx = / pgtix^ax', .f =
-'o -'o
Ex. 6. To find the centre of gravity of a straight rod, the
density of which varies as the wth power of the distance of each
point from a given point in the line of the rod produced.
Let o be the point from which the variation of the density
takes place; fig. 45; OA = a, OB = , OP = #, VQ=dx; p = jcar*;
then _ rb /*&
/' / K W ^7 T (IJT / K mfjX (X/3C 'y
J a J<*
n + 2
176 CENTRE OF GRAVITY. [ 12 9-
If n = 2, then
b dx b dx ab . b
Ex. 7. To find the centre of gravity of a wire bent into the
form of a cycloid, the thickness of which varies directly as the
distance from the middle point of the wire.
The middle point of the wire is the highest point of the
cycloid ; let it be taken as the origin ; and let the axis of the
cycloid be the axis of x ; then y = ; let the length of the wire
be 8a; then, see Integral Calculus, Art. 155, Ex. 3, the radius
of the base-circle is a ; and the equation to the cycloid is
s* = Sax;
and since p = KS, we have
_ C r /* 4a 1 /* 4a
x\gu>K.sds = tgaiK.sxds, xl sds = / s 3 ds;
J J JQ ott JQ
x = a.
Ex. 8. Find the curve whose extreme points are (#,y ), (x 3 y) t
such that mx = x #, ny = yy -
129.] If the wire is in space, having all its elements either
in or not in one plane, we must determine all the coordinates of
the mass-centre which are given in (12).
Ex. 1. A wire of constant thickness and density is bent into
the form of a helix ; find its centre of gravity.
Let a = the radius of the base-cylinder; and let the wire
commence at the axis of x, that is, at the point (a, 0, 0), see
fig. 125, Differential Calculus ; and let its end be at (x, y, z}; then
x a cos 0, y a sin 0, z = ka<j>;
/*
xl
'
V
x = ka-\
= a (I cos</>); y =
z
ax
z
Ex. 2. To find the centre of gravity of the perimeter of a
triangle in space, the three sides of which are thin rods of con-
stant thickness and density.
Let the lengths of the sides be / I 3i 1 3 ; and the angular
130.] THIN WIRES. 177
points be (x l ,y l) 2,) ... (x t >y*, z a ) ' p = the constant density,
<o = the area of a transverse section of the rods : then the centres
of gravity of l lt l a , l s are manifestly at the points
2 ' 2 ' 2
and therefore by the formula? (10),
x (^ + l t + 1 3 } = - {/! (a?, + a?,) + 1* (or, + a?0 + , (x, + a?,) },
&
y (li + 1, -f /,) = K (y +^3) + a (y, -f y,) + 1 3 (y, +y a )},
By a similar process the centre of gravity of the perimeter of a
polygon formed by heavy rods in space may be determined.
130.] The determination of the centres of gravity of material
lines or wires also suggests the following problem, which is
solved by the Calculus of Variations :
To find the equation to the curve into which a thin heavy
rod or string of uniform thickness and density and of given
length is to be bent, so that its ends being fixed at two given
points, the centre of gravity may be in the lowest possible
position.
Let the axis of z be parallel to the direction of gravity ; and
let *lc be the length of the rod; and (x 1} y 1} z^) and (x 9) y 9) z^)
the ends of the line ; then
ds = 2c, (13)
z2c = I zds; (14)
*A)
and z will be a maximum or a minimum according as the plane
of (x } y) is above or below the centre of gravity of the suspended
wire; in either case, 5.z = ; therefore from (14),
= = / \i
Jo
= = / (zb.
Jo
2cbz = = / 8.2 ds
f:
2cbz
PRICE, VOL. III. A a
178 CENTRE OF GRAVITY.
ds
Of this quantity the first part vanishes by reason of the limits
being fixed; also from (13),
8.2c = = 8 ds
.2c = = 8 /
-Jo
/ 1 ( dx 7 ^ 7 dz , ^
{ -j- d.bx + -f d.by + -7- d.bz
( ds ds ds
rdx . dy . ffe, I 1
= &+-f- 8y+ 5^
L* f* J 09 JH
/*M 7 ^# ..
Id.-^-bx.- .-j-
J ( ds ds " ds
and of this quantity the first part vanishes by reason of the
limits being fixed; and as the second part is to consist with
the second part of (15), we have
dx , dy dz
d.z -T=- d.z -j- d.z -j- ds
ds ds ds
- = - = - = A (say); (16)
7 J 7 \*f/' \ J
, dx -.ay dz
d. --^ d. -j- d. -=~
ds ds ds
from the first two members of which equality we have
, dx , dy
d.-j- d.-f- , ,
ds ds dx dy
dx dy x^x, y^y
ds ds
the constants being introduced consistently with the curve
passing through (x lf y lf z,) and (a? ,y c , z );
whence it follows that the curve is a plane cui-ve, and is in a
plane perpendicular to that of (x,y). Let the plane of (x, z) be
taken so as to contain the curve ; then y = ; and from the
first of ( 1 6) we have
dx , dx
d.z-r- \.-r->
as ds ,
dx
jdx . dx ,dx dz ' ds
.*. zd.-j- +dz-,- = Arf.^-, -| -- = 0;
ds ds ds z\ dx
ds
I3I-] THIN WIRES. 179
.-. log (z- A) + log ^r = log a; or, ( z \}^ =a ,
where a is an arbitrary constant of integration ; and since
ds> = <fe + <fe, fo dz .
" {( 2 _x) _*}*'
xb xb
i~ +e~~\; (17)
where is another arbitrary constant of integration ; and the
three undetermined constants a, 6, \ may be determined by the
conditions of the curve passing through two given points, and
of the length of the curve between those points being given.
The equation (17) is that of the catenary, the properties of
which will be investigated hereafter; and the result is im-
portant, inasmuch as it shews that the curve in which a per-
fectly flexible and inextensible heavy string will hang when
suspended from two fixed points is also that of which the centre
of gravity has the lowest possible position.
The form of the problem as stated in equation (14) shews that
it is identical with the determination of the form of the curve
of given length, which passes through two given points, and
revolving about a line in the same plane with the two points
generates a surface whose area is a maximum. This problem is
solved in Art. 326, Vol. II.
131.] The formulae given in (12) lead also to the following
theorem. If the wire or line is of constant thickness and den-
sity, and is infimtesimally then, then
= jyds;
= Iz-nyds. (18)
x s
Now if the plane curve whose length is s revolves about the
axis of x, and generates thereby a thin shell (or surface) of revo-
lution, the right-hand member of (18) is the area of the surface
generated; see Art. 232, Vol. II; and the left-hand member of
(18) is the product of the length of the generating line and of
the path described during an entire revolution by the centre of
gravity of it ; hence we conclude that,
If a plane curve lies wholly on one side of a line in its own
plane, and revolving about that line generates thereby a surface
of revolution, the area of the surface is equal to the (geometrical)
A a a
180 CENTRE OF GRAVITY.
product of the length of the revolving line, and of the path
described by its centre of gravity.
This theorem is one of those known by the name of the
Theorems of Pappus or of Guldinus ; it is a geometrical relation
existing between a curve, the surface which it generates by
revolving about a line in its own plane, and the distance of its
centre of gravity from that line ; the curve must not intersect
the axis of a?; for if it does, y will change its sign; and (18)
may be an inexact expression ; the generating curve may however
be a closed figure. Also as (18) expresses the equality of the
two sides of the equation for a whole revolution, so will a similar
theorem be true for any part of a revolution. Two or three
examples are subjoined.
Ex. 1 . A circle of radius a, revolves about an axis in its own
plane at a distance c from its centre ; it is required to find the
area of the surface of the ring thereby generated.
The circumference of the generating curve is 2?ra; and as
the centre of gravity of it is at its centre, the path described by
the centre of gravity during a complete revolution is 2 TIC;
.. the area of the surface of the ring = 4 -n*ac.
Ex. 2. A right-angled triangle revolves about its hypothe-
nuse, and its sides thereby describe a surface ; it is required to
find the area of the surface described.
Let a, b be the sides of the triangle, and k the length of the
perpendicular from the right angle to the hypothennse, so that
_L JL J_
A' == a* + F J
then the area of the surface = TT (a -f b] h
n (a -\-fyab
(' + ')* *
Also if the area of a surface is known, and the length of the
generating line is known, the distance of the centre of gravity
of the line from the axis of revolution may be determined. Thus,
the surface of a sphere of radius a l-no,*, the length of a semi-
circle = va; therefore from (18),
1-ny x ii a = 47ra* ;
2a
132.] THIN PLATES AND SHELLS. 181
SECTION 3. Centre of gravity of thin plates and curved shells,
bounded ly lines straight or curved.
132.] In the next place let us consider a plane plate of infini-
tesimal thickness, bounded by curved or straight lines, and refer
it to rectangular coordinates. Let the plane of the plate be
that of (x, y] and let the coordinates of any element in the plane
surface of the plate be x, y ; so that the area of the element E is
dxdy-y see fig. 46. Let the thickness of the plate at E=T; then
and the first two of equations (11) become
* I IpffTdy&e = \\ pg-rxdy dx,
rr
y] J p
the integrations extending over the area assigned by the problem.
Ex. 1 . It is required to find the centre of gravity of a thin
plate of uniform thickness and density, bounded by a parabola,
its axis, and an ordinate ; fig. 46.
Let OA = , AB = b ; T = the thickness of the plate, p = the
density : then the equation to the parabola is ay 2 = 6' x ; let
Y 4 = 6*x; so that we have
fa fv fa fv
y\ \ dydx - I / xdydx,
JQ JQ JQ Jo
_ Ta fa a
x\ x* dx I x^dx\
JQ JQ
3
x = -a :
5
fa /*Y fa fv
y] I dydx = / / ydydx;
JQ JQ J() /o
36
Ex. 2. To find the centre of gravity of a thin plate of uni-
form thickness and density in the form of an elliptic quadrant.
Let Y = -(-_ a 1
- f a /* Y /* a f v
then x\ I dydx = / / xdydx,
JQ JQ JQ JQ
182 CENTRE OF GRAVITY. [ I 3 2 -
4a
fa TY fa /*Y
y I I dy dx \ I y dydx;
JQ JQ Jo JQ
"" ~~ BIT'
Hence for a thin plate in the form of a quadrant, the position
of the mass-centre in reference to the centre of the circle is
given by 4 a
*=y = sv'
Ex. 3. To find the centre of gravity of a thin triangular plate
of constant thickness and density.
Let T be the thickness of the plate, and p = the density. Take
the angle o, fig. 47, for the origin, and the sides OA, OB for the
coordinate axes; OA = a, OB = b, so that the equation to A B is
x y
~+ fi= l -
a b ,
Let the angle at o = &> ; then the area of the surface at E
= dx dy sin o> ; dv = r dx dy sin o>. Then if
y = -(a-x],
the equations of moments about the axes are
[a TY fa fv
x sin to / l dy dx sin &> = / / x (sin to) 2 dy dx }
JQ JQ Jo JQ
fa /*Y fa TY
y sm &>/ / dydxsmn = I I y (si
Jo JQ Jo JQ
sn v
a b
x = , y = ;
3' y 3
the centre of gravity therefore is situated on the line passing
through o and bisecting AB, at a distance from o equal to two-
thirds of the bisecting line; and as the result is independent
of the particular angle, it is equally true for all the angles ; and
therefore the centre of gravity of a triangular thin plate is at
the point of intersection of the three bisectors of the sides drawn
from the opposite angles. This is also manifest from the follow-
ing reason : let OAB be a triangular plate, fig. 48 ; and let oc be
drawn from o to c, the middle point of the opposite side AB;
132.] THIN PLATES AND SHELLS. 183
let us imagine the plate to be divided into a series of thin slices
by lines parallel to AB ; then the centre of gravity of each of
these slices will be at its middle point, that is, at its intersection
with oc. Imagine therefore each slice to be condensed into its
centre of gravity ; there is then a series of particles of increasing
weight arranged along the line oc, the law of increase being
that of the distance directly, because PP' varies as OM; if there-
fore OM = x, and oc = h, we have from (19)
_/* r*
x I xdx \ x 3
JQ JQ
dx;
Hereby also we conclude that if the coordinates to the angles
of a triangular plate in space are x l} y l} z^ ; x t) y t) z, ; # 3 , y 3) z 3 ;
x =
z =
3
Ex. 4. If a thin plate is in the form of a complete cycloid,
the distance of the centre of gravity from the vertex is .
6
Ex. 5. Of a thin plate bounded by a cissoid and its asymptote,
the distance of the centre of gravity from the cusp is five-sixths
of the diameter of the base-circle.
Ex. 6. The centre of gravity of a thin plate bounded by the
witch of Agnesi is at a distance from the asymptote equal to
the eighth part of the diameter of the base circle.
Ex. 7. To find the centre of gravity of a cycloidal plate, the
thickness of which varies as the nth power of the distance from
the base, and of which the density is constant.
In this case taking the starting point as the origin, and the
base as the axis of x,
x = aversin" 1 ^ Za**
<b- =
Let T = liy" thickness, p = density ; it is plain that ,?= -na ;
184 CENTRE OF GRAVITY. [l33-
/*2ira fy CZxa fy
ana y\ I y*dydx\ I y n+l dydx;
JQ JQ J Q JQ
/*2a y n+i /*2ira y n+2
.. y\ dx = / dx,
y J Q n+1 JQ n+2
/ 2d 9/**^" 2 /7-j/ tn i 1 / 2c( 2/**~l~ 3 ///
I y ^7 ~i I .7 ^y
' ""a a ^ n + 2Jn (^n.1111^
n + 2
n+1 2n+5
a.
.
n+2 n + 3
Ex. 8. Find the centre of gravity of a thin plate contained
by an ellipse, and the chord joining the extremities of the two
principal axes.
Ex. 9. Find the centre of gravity of a thin plate contained by
a parabola and a straight line through the vertex.
Ex. 10. If x = mx, where x is the abscissa to the bounding
ordinate of a thin plate contained between the axis of x } the
origin and the bounding curve, the equation to the bounding
curve is x ,- m m - t
() -<5)
133.] If the plane surface of the plate is referred to polar
coordinates, and rectangular coordinates are retained for the
centre of gravity, then the area of the surface-element of the
plate is r dr dQ, and x r cos 6, y = r sin 0, so that the equa-
tions (19) become
x \ I pgrrdr dQ = / / pgrr* cos 6 dr dQ, ~j
rr rr (20)
y I I pffrrdr dQ = / / pgrr* sin 6 dr dd.J
Ex. 1 . To find the centre of gravity of a plate in the form of
a sector of a circle, the thickness of which varies directly as the
distance from the centre of the circle.
Let a = radius of circle, 2 a = the angle which the sector
subtends at the centre ; and let the axis of x be the line bisect-
ing the angle 2 a, so that y = ; then T = kr, and we have
x] I r*drdd = r 3 cos6drd6;
J-aJQ J-a.Jto
3a sin a
^ =
4 a
134- ] TH1N PLATES AND SHELLS. 185
Ex. 2. To find the centre of gravity of a thin plate of uni-
form thickness and density in the form of the loop of the
lemniscata.
The equation to the bounding curve is
r* = a 2 cos 2 e ;
and as the loop is symmetrical with respect to the axis of x t
y = 0. Let r = a (cos 20)* ; then from (20),
x\ I rdrdd l I r'cosddrdO,
J_5 JQ J_* JQ
x ^ - J* (cos 2 0)2 cos S dd
= y 2* jf*{ i -(sin
let k* = -, and sin0 = a?; then
2
3 J_fc
2|a 3 w
~3~8 4
na
~~ w
Ex. 3. The centre of gravity of a thin plate bounded by the
curve whose equation is r = a(l+cos0) is at a distance from
the origin equal to
Ex. 4. A thin plate in the form of a circular sector is gene-
rated by the motion of one of its bounding radii ; if a is the
radius, prove that the locus of the centre of gravity is
2a sin0
r =
3
134.] Centre of gravity of a thin shell of revolution.
Let the axis of revolution be the axis of x ; and let the
equation to the curve, by the revolution of which the exterior
surface of the shell is generated, be y =f(x): let T = the
thickness of the shell ; p = the density ; g = the earth's attrac-
tion; and imagine the shell, see fig. 49, to be divided into
a series of circular rings or annuli of breadth dx by means
of planes perpendicular to the axis of revolution, and at an
PRICE, VOL. III. B b
CENTRE OF GRAVITY. [*34-
infinitesimal distance apart : then, if ds is a length-element of the
generating curve, the volume of any one of these rings corre-
sponding to a point (x, y] on the generating curve is 2,-nyrds;
and therefore the weight of it is lirpgry ds: now imagine this
weight to be condensed into a point at the centre of gravity of
the ring, which is at M on the axis of x : the circumstances of
pressure are not hereby changed : and let us imagine the weight
of each ring to be similarly collected at its centre of gravity ;
then we have a series of weights arranged along the line ox, of
variable magnitude, the law of variation depending on the equa-
tion of the generating curve : but such that the weight at the
distance x is equal to Zvypgrds: hence we have to find the
centre of gravity of this rod of variable density ; and therefore,
by virtue of equations (12),
xl 2 -npg Tyds = / 2 Ttpgrxy ds,
and cancelling 2irff, _ r r
xl pryds = / prxyds. (21)
Ex. 1 . To find the centre of gravity of a thin shell of uniform
thickness and density, the exterior surface of which is generated
by the revolution of a quadrant of a circle about one of its
bounding radii.
Let T = thickness of shell; p = density; then, fig. 50, the
equation to the generating curve is
z*+y* = a 2 ;
dx dy ds f
y ' x 'a'
- C a C a
.'. xl adx=t axdx;
This result is also manifest by the method of infinitesimals : in
Vol. I (Differential Calculus), Art. 24, Ex. 7, it is shewn that
each zone of the shell is equal to the corresponding zone of the
cylinder of the same thickness circumscribing the spherical
shell ; and therefore as these zones are equal and equivalent as
to the position of their centres of gravity, the latter may replace
the former, and the centre of gravity of the hemispherical shell
is the same as that of the cylindrical shell; and this Jatter is
evidently on OA in the middle point of OA.
1 34-] THIN PLATES AND SHELLS. 187
Ex. 2. To find the centre of gravity of a thin right conical
shell of uniform thickness and density.
Let T = the thickness of the shell ; p = the density ; and let
the equation to the generating straight line be
y = ax;
let the altitude of the shell = a : then els' 1 = (1-f a^dx*; and
from (21) we have
_ ["a ra
xl x dx = / x* dx,
/0 J
2a
* = T-
This is also manifest by the following reasoning : the conical
shell may be imagined to be resolved into a series of triangular
plates all the vertices of which meet at the vertex of the cone,
and the bases of which form the circular base of the conical
shell : now the centre of gravity of a triangular plate is on the
line which is drawn from the vertex to the middle point of the
base, and is at a distance from the vertex equal to two-thirds of
that line ; and therefore the centre of gravity of the shell is on
the axis at a distance from the vertex equal to two-thirds of the
axis.
And suppose the thickness of the conical shell to vary as the
distance from the vertex : then p = k ( 1 + a*)* x ;
_f , f a ,
I I f - III' I (' 1 1 I-
IV f W '?'' f ' '<<.')
/0 * I)
3a
x =
4
Ex. 3. To find the centre of gravity of a thin shell of uniform
thickness and density formed by the revolution about its base
of a wire bent into a semi-cycloid.
The equation to the generating curve is
x = flversin" 1 (lay y*}^ ;
dx dy ds
(2*-,
260
B b
188 CENTRE OF GRAVITY. [l35-
Ex. 4. The centre of gravity of a thin shell formed by the
revolution of a semi-cycloidal wire about its axis is at a distance
from the vertex ^ 2a 15w _ 8
= Fs STT 4 '
Ex. 5. If x determines the place of the centre of gravity of a
thin shell formed by the revolution about the #-axis of a thin
wire, of which the limiting abscissae are and x, and if mx nx,
shew that the differential equation of the wire-curve is
4n-2m
_
ydy = | l^x m ~ n y* [ dx.
What curves are expressed (1) when m 2n; (2) when
2m = 3n?
135.] Centre of gravity of a thin curved shell.
Lastly, let us investigate the coordinates of the centre of
gravity of a thin curved shell ; of which let the thickness = T,
the density = p ; and let the equation to the bounding surface
of the shell be F (x, y, z] = 0. Then using the ordinary symbols,
if fh is the surface-element at (x, y, z), civ = T?A; and
dA = 5 dydz = ^ dzdx = -^- dxdy, (22)
u " v w
so that, taking for dA the last value of (22), (11) become,
Q j j /Y Q
JJ pffT w" dxdy = Jj pffTZ w"
(23)
If the surface of the shell is more conveniently referred to
that system of polar coordinates in space which is explained in
Art. 165, Vol. II (Integral Calculus), the general equations (11)
instead of taking the form (23) will be modified according to it.
Ex. 1 . To find the centre of gravity of the octant of a thin
spherical shell of uniform thickness and density.
x*+y 2 + z* = a* ;
u = 2x, v = 2y, w = 2z;
so that if ( 2 ..*) = Y, we have
136.] THIN PLATES AND SHELLS. 189
- f" f v adydx C a f v ax dy dx t
J J (a*x*y*}l Jo Jo (a^x^y^'
fa ^ fa^
x\ -dxl -xdx-,
Jn & Jo &
'0 * ^0
a
2
_ / /* Y adydx _ /* C v aydydx .
^ o Jo (a*x*y*}* *M> Jo (a*x>y*)*'
a
2'
a Jy <fo? f a
'o
Suppose the thickness of the shell to vary as the z-ordinate to
any point of it j then r = kz, and
_ Ta TY fa /*y
^/ / akdydx = / / aTcxdydx;
- o ^o *^o *^o
Ta TY Ta TY
3?/ / akdydx = / / akydydx;
JQ JQ JQ JQ
- f a /* Y T a /* Y
^/ / akdydx =11 ak(a*x*
JQ JQ JQ JQ
2a
136-3 The following theorem, due to Pappus, expresses a
relation between a plane area, the volume of the solid gene-
rated by it as it revolves about a, line on its own plane, and the
distance of the centre of gravity of the area from the axis,
whereby, when any two of these quantities are given, we are
able to discover the third.
Let the revolving area be of constant density and thickness,
and be so thin as to be conceived to be a geometrical surface ;
then, if y is the distance of the centre of gravity of this area
from the axis of #, we have,
l/j jdydv = I jydydx;
190 CENTRE OF GRAVITY.
,-. 2 Try x / Idydx = / llTtydydx, (24)
Now these integrals being definite, the second factor of the left-
hand member of the equation expresses the area in the plane
(x, y) } and the first factor is the length of the path described by
the centre of gravity of that area, as it revolves through four
right angles about the axis of x: and because dydx is the area-
element, and 2 Try is the path described by the area-element
during a complete revolution of the area about the axis of x } the
right-hand member is the product of all the area-elements of
the given area and of their paths, and is therefore the volume
described by the area during a complete revolution : if therefore
the curve lies wholly on the same side of the axis of x, so that y
does not change sign, the above equation expresses the following
theorem :
If a plane area, lying wholly on the same side of a line in its
own plane, revolves about that line, and thereby generates a
solid of revolution, the volume of the solid thus generated is
equal to the (geometrical) product of the revolving area and of
the path described by its centre of gravity during the revolution.
As (24) is true for the whole revolution, a similar theorem is
also true for any part of the revolution : and if the generating
area is such as that described in fig. 46, where the axis of a? is
one of the bounding lines, then the limits of the ^-integration
in (24) are the ordinate to the curve and zero : therefore
2 ny \y dx = / Try 2 dx,
and the right-hand member is the ordinary expression for the
volume of a solid of revolution. In other cases the limits ofy
are given by the geometrical conditions of the problem.
Ex. 1 . An ellipse revolves about a line in its own plane, the
perpendicular distance of which from the centre is equal to c ;
it is required to find the volume of the ring generated during a
complete revolution.
Let a and b be the semi-axes of the generating ellipse ; then
the generating area = ?ra#; and as 2 -no is the path described
by the centre of gravity,
the volume = 2*n*abc.
It will be observed that the volume is the same, whatever direc-
tion the axis of revolution has with respect to the axes of the
1 3 7.] HEAVY BODIES. 191
ellipse, provided that the perpendicular distance from the centre
to the axis of revolution is the same.
Ex. 2. The volume of a sphere of radius a is - ; and the
area of a semicircle is - : it is required to deduce from these
m
data the position of the centre of gravity of the semicircle.
Let y be the distance of the centre of gravity of the semi-
circle from the diameter ; then considering it as the generating
area of the sphere, we have
irrt 2 4ir 3 40
and by reason of the symmetry, the centre of gravity is on the
line which is perpendicular to the diameter through the centre
of the circle.
SECTION 4. Centre of gravity of heavy bodies bounded by plane
and curved surfaces.
137.] Before I proceed to the general case, I will consider
that of a solid bounded by a surface of revolution, and refer the
body to the axis of revolution as the axis of x : let the equation
to the generating curve of the bounding surface be y -=.f(x].
Imagine the solid, (see fig. 51,) to be divided into thin circular
slices by planes at an infinitesimal distance apart and perpen-
dicular to the axis of revolution : of these let the circular slice
PP'Q'Q be the type, and let OM = x, MN = dx, so that dx is the
thickness of it. Of this slice take a particle at a distance r from
the axis, and so that the plane passing through ox and that
particle may be inclined at an angle to the plane passing
through ox and oy ; then the volume of the element is equal to
rdBdrdx. Let p = the density of the body at the particle,
then the mass-element = prdrdddx, and the weight-element
= pgrdrdddx.
Now if the constitution of the body as to density is symme-
trical with respect to the axis of revolution, the centre of gravity
is plainly on the axis of x, and therefore we have to find only
x ; and we have from (11)
x I I I pgrdddrdx II I pgxrdQ drdx; (25)
and performing the ^-integration through a whole revolution,
192 CENTRE OF GRAVITY. [ T 37-
so as to obtain the required result for a ring of radius r, and
observing that the symmetry of the body renders p independent
of 6, we have, dividing out 2 Tig,
x I I prdrdx = pxrdrdx. (26)
And if the density is uniform throughout a complete slice, we
may perform the r-integration between r = 0, and r = y, where
y is the ordinate to the generating curve : and (26) becomes
xlpy^dx = ipy^xdx', (27)
J J
the limits of integration depending on the circumstances of the
problem.
Ex. 1. To find the centre of gravity of a paraboloid of revolu-
tion of uniform density, the length of whose axis is c.
Let the equation to the generating parabola be y 1 = 4ax ;
therefore from (27), as p is constant,
_ /* /* 2
'JT' j i ' ' < ! U'vT ^^ I i QvC (A/X * X ^ ~~* C*
Jo Jo 3
Ex. 2. To find the centre of gravity of a portion of a prolate
spheroid of uniform density, the length of whose axis measured
from the vertex is c.
Let the equation to the generating curve of the bounding
surface be i*
then, as p is constant, (27) becomes
xl (Zaxx^dx = / (2ax x*)xdx\
J -0
4 3a c
Thus for a hemi-spheroid, c = a, and we have
5a
S "T-
As b does not enter into either of the last two values, they are
the same for a spherical segment and for a hemisphere.
Ex. 3. To find the centre of gravity of a double convex lens
of uniform density.
Let the equations to the generating circles of the two inter-
secting spheres be, fig. 52,
#+ = a* jcc+* = b*,
1 37.] HEAVY BODIES. 193
where OA = a, BC = b, oc = c ; then the equation to the plane
of intersection of the spheres is
then from (27),
(a?
fa
= / ( 2 x*)xdx +
/*
whence may x be determined.
Ex. 4. To find the centre of gravity of a cone, the density of
each circular slice of which varies as the th power of its dis-
tance from a parallel plane through the vertex.
Let the vertex be the origin, and the equation to the gene-
rating line of the cone be y = ax ; and let a be the altitude ;
then p = kx n : and (27) becomes
fa
x x n
'o
= / x n+3 dx; .'. x = - a.
Ex. 5. To find the centre of gravity of a cone, the density of
every particle of which increases as its distance from the axis.
Let the vertex be the origin, #=the altitude, and let the equa-
tion of the generating line of the bounding surface be y = ax ;
then in equation (26) p = kr, so that
_ ra rax r a rax 4
xl I r*drdx = I r*xdrdx', x -a.
^ Q JQ JQ JQ 5
Ex. 6. To find the centre of gravity of the volume of uniform
density contained between a hemisphere and a cone whose vertex
is the vertex of the hemisphere and base is the base of the hemi-
sphere.
Let the common vertex, see fig. 53, be the origin; and let
the equations to the bounding surfaces be
y* = 2 ax x* = Y 2 , y* x* ;
so that Y and x are the limits of the r-integration in equation
(26) : then, as p is constant,
-
xl I rdrdx I I rxdrdx,
Jo J x JQ J x
r a r a
xl (2axx*x*)dx = / (2ax x* x*)xdx,
JQ JQ
PRICE. VOL. III. C C
194 CENTRE OF GRAVITY.
Ex. 7. If x = mx, shew that the equation to the generating
2-t
curve of the solid of revolution is ky* = x n ~ l .
138.] Now let us take the most general case of a body in
space; and first let it be referred to three rectangular axes
originating at o : let (x, y, z) be the position of any particle of
it, so that the volume-element abutting at it is dx dy dz ; then
dv = dxdydz:
let the density = p; so that equations (11) become
x I I I pdx dy dz == / / / pxdxdydz, -
yjjjpdxdydz=jjjpy dxdydz, - (28)
zl 1 1 pdxdydz III pzdx dydz. J
The integrals are of course definite and the extent of integration
is assigned by the conditions of the problem.
Ex. 1 . To find the centre of gravity of a homogeneous body
in the form of the octant of an ellipsoid.
Let the equation to the ellipsoid be
x* 11* z
/ i
and let z = c(l ^-\ ,
\ Q* b*'
~ a
ra r\ fz ra r\ rz
then xl I I dzdydx = l I I xdzdydx;
_ 3a
3d 3c
similarly, y = , z = .
The integrals required in the preceding example have already
been determined by Dirichlet's process of evaluation in Ex. 2,
Art. 280, Vol. II (Integral Calculus).
Ex. 2. To find the centre of gravity of a body of uniform
density bounded by the Cono-Cuneus of Wallis and by the
planes z = 0, y = c.
The equation to the Cono-Cuneus is, equation (89), Art. 367,
Vol. I. /. z * 7/ ( a t _ r *\ .
y i/ & t/ lit- ^~ *c i .
1 39.] HEAVY BODIES. 195
and performing the z- y y-, ^-integrations in order, the limits are
| (a 2 #*)* and 0, c and 0, a and ; so that if
Z = f('-*>)4,
m^ [a Ce /*z
dzdydx = / / / xdxdydz,
~ J$ JQ J%
xl I y(a t x t }^dydx = I I xyitfx
JQ JQ JQ JQ
xl (a*xrfdx = I x(a? x^dx,
Jo Jo
4a
mra re rz
dzdydx = I y dzdydx,
'O *^0 'O
ny ( a * a? 2 ) * dy dx = / / y 2 ( 2 a? 2 )* dk,
^0 '0
m z f a f c C z
dzdydx III z dzdydx,
. JQ JQ JQ
Sa
139-3 Again, let the curved bounding surface be referred to
a system of polar coordinates of the construction of Art. 165,
Vol. II; then
x 1 1 I pr* sin 6 dr dd d<f> = pr 3 (sin 0) 2 cos $dr dQ d<j>, ^
y I 1 1 pr* sin dr d6 d<j> = / / / pr 3 (sin 0) 2 sin ^ dr dd d$, (29)
z I 1 1 pr* sin dr dQ d(f> = / / / pr 3 sin Q cos QdrdQdQ-, J
the integrals of course being definite, and the limits being
assigned by the geometrical conditions of the problem.
Ex. 1 . To find the centre of gravity of an octant of a sphere,
the density of which varies as the nth power of the distance of
any particle from the centre.
c c 2
196 CENTRE OF GRAVITY. [l4-
Let a = the radius of the sphere; and let p = kr n ; then
equations (29) become
ay r3 fa r? /*?
/ r n+3 sm8(l6d(j>dr / / / ?- n+3 (sin 0) 2 cos<^0^tfr;
Jo Jo Jo *A)
n + 3 a
a? = =.v =. z\
n + 4 2 9
the last two values being inferred from the symmetry of the body.
Ex. 2. The vertex of a right circular cone is at the centre of
a sphere ; it is required to find the centre of gravity of a body
of uniform density contained within the cone and the sphere.
Let the axis of z be the axis of the cone : and let a be the
semi- vertical angle of the cone ; a = the* radius of the sphere ;
p = the constant density : then x and y are evidently equal to
zero ; and we have
f2ir fa fa fZir fa. fa
z\ I I r^ sin dr dO d(p = I I r 3 sin 6 cos 6 dr eld d^-,
M) *M) *M) *% *9 JQ
a 3 . a* (sin a)*
z (l-cosa)27r = ^-2:r,
3a
z = (1+cosa).
o
Ex. 3. The vertex of a right circular cone is on the surface of
a sphere, and the axis of the cone passes through the centre of
the sphere ; if 2 a is the vertical angle of the cone, and z is the
distance of the centre of gravity from the vertex, shew that
1 (cos a)'
z = a ~, 7 T7 '
1 (cos a)*
Ex. 4. If the equation to the cardioid is r = a (1 +cos &}, the
distance from the origin of the centre of gravity of the solid
formed by the revolution of the curve about the prime radius is
4a
equal to .
5
140.] I shall conclude this section with a few examples of
determining the centres of gravity of bodies which do not come
under any of the former methods, but to which the principles
are equally applicable.
Ex. I. To find the centre of gravity of a right pyramid of
uniform density, whose base is any regular plane figure.
Let the vertex of the pyramid be the origin, and the axis of
the pyramid the axis of x ; divide the pyramid into slices of the
thickness dx by planes perpendicular to the axis : then as the
140.] HEAVY BODIES. 197
areas of the sections thus formed will vary as the squares of
their homologous sides, and as these sides will vary as the dis-
tances from the vertex, so will the areas of the sections vary as
the squares of the distances from the vertex; and therefore if
the axis of the pyramid is divided into equal infinitesimal ele-
ments, the masses of the several slices will vary as the squares
of the distance from the vertex. Now imagine each slice to be
condensed into its centre of gravity, which point is on the axis
of x ; then if a = the altitude of the pyramid, we shall have
_ /* /* 3
x I x 3 dx = / x s dx ; .*. x = -a.
A) Jo
Ex. 2. On the base of a hemisphere a right circular cone is
constructed, the whole body being of uniform density ; determine
the altitude of the cone, so that the centre of gravity of the
whole may be at the centre of the circular base of the hemisphere.
Let a = the radius of the hemisphere, <? = the altitude of the
cone : then if we imagine the hemisphere and the cone to be
condensed into their centres of gravity, the moments of these
weights must be equal about the centre of the circular base of
the hemisphere : that is,
C a C c a 2
I (a <t x*}xdx=l ~(cx
JQ JQ C*
.-. c 2 = 3a 2 ;
and therefore the vertical angle of the cone is 60.
Ex. 3. "When a heavy body with a convex surface rests on a
horizontal plane, the vertical line through the centre of gravity
also passes through the point of contact : because as the body is
acted on by only two forces, viz. the weight acting downwards
at the centre of gravity, and the reaction of the plane upwards
at the point of contact, these forces cannot be in equilibrium
unless they are equal, and act along the same line in opposite
directions.
Hence it appears that the compound body of the last example
will rest in any position on its convex spherical surface.
Hence also it follows that if a body is suspended from any
point, the point of suspension and the centre of gravity are in
the same vertical line.
A body in the form of a paraboloid of revolution of given
altitude and uniform density is suspended from a point in the
edge of its circular base ; it is required to find the inclination of
its axis to the vertical.
198 STABILITY AND INSTABILITY
Let a = the altitude of the paraboloid ; I = the radius of its
circular base ; 6 = the angle between the axis of the paraboloid
and the vertical : then, since the distance of the centre of gravity
from the centre of the circular base = -, see Ex. 1, Art. 137,
3
3d
tan 6 =
a
Ex. 4. If a heavy body is placed on a rough inclined plane,
the friction of which is sufficient to prevent sliding, the body
will be at rest so long as the vertical line through the centre of
gravity passes within the part of the body which is in contact
with the inclined plane; and if it falls beyond that part, the
body will fall over ; and if it passes through the edge of it, the
body is just in its limiting position of rest.
A given cone rests with its base on an inclined plane : it is
required to determine the inclination of the plane, when the
cone is just on the point of falling over.
Let a = the altitude of the cone, and I = the radius of the
base : then CG = -, see fig. 54 : let cox = a :
.-. tana = tancoor,
= tanCGB,
and when the angle of inclination of the plane exceeds this
angle, the cone will fall over.
SECTION 5. Stability and instability of the equilibrium of
heavy bodies.
141.] The character of the equilibrium of heavy bodies, in
respect of the stability or instability of the same, requires
especial notice, although the discriminating conditions have
already been investigated in the general case in Section 7 of
the preceding Chapter. Let us refer at first to (280), Article 106,
as in this case the action-lines of all the forces are parallel,
and the axis of z may be taken parallel to these action-lines ;
and consequently, as a horizontal line may be taken for the axis
142.] OF HEAVY BODIES. 199
of infinitesimal displacement of rotation, the equilibrium will
be stable or unstable according as 2.P z is positive or negative ;
that is, by Art. 107, according as S.PZ is a maximum or a
minimum. Hence in the case of a heavy body the equilibrium is
stable or unstable for infinitesimal displacement about a hori-
zontal axis according as ~s,.pgzdv is a maximum or a minimum :
but z.pgzdv = zz.pffdv ; consequently the equilibrium is stable
or unstable according as z is a maximum or a minimum.
The theorem, however, may be demonstrated as follows by
means of virtual velocities. Suppose a heavy body to be at
rest on a horizontal plane, and no forces to act upon it, except
gravity and the resistance of the plane ; and suppose the body
to have such an infinitesimal motion of displacement that it
remains in contact with the plane ; then as the virtual velocity
of the reaction of the plane vanishes, the single condition of
equilibrium is *. pff dvd,= 0. (30)
But if z is the distance of the centre of gravity from the hori-
zontal plane, zz.pgdv = s.pgzclv -, (31)
so that from (30) 8z = 0; consequently z is a maximum or a
minimum ; and as equilibrium is stable or unstable according as
the radial moment is a maximum or a minimum, so observing
that the action of all the weights is towards the plane of (x, y)>
the equilibrium is stable or unstable according as the position
of the centre of gravity is the lowest or the highest.
This problem is that which is presented to us by rocking
stones, and by many children's toys. We shall hereafter investi-
gate the rocking motion of bodies thus placed.
142.] And to take a more general case. Let us consider
that of a heavy body bounded by a convex surface resting on
another body also with a convex surface. And let fig. 55 re-
present the bodies : the continuous lines indicating the position
of the bodies when they are at rest at first, and the dotted lines
the position of displacement. Let CAO be the vertical line pass-
ing through A the point of contact of the two surfaces when
they are at rest, and through the centre of gravity of the upper
body : let c be the centre of curvature of the lower body cor-
responding to the point A, and o that of the upper body ; let G
be the centre of gravity of the upper body : now suppose a small
displacement of the upper body to take place by means of
rolling on the lower one, so that there is no virtual velocity of
200 STABILITY AND INSTABILITY [*42.
the normal reactions of the surfaces : then if p is the new point
of contact, and A' is the point which was originally in contact
with A, A'P=AP, the axis about which the rolling takes place
being perpendicular to the plane of the paper. Let the curva-
ture of the two surfaces be continuous about the points A and P ;
and by reason of the small displacement let o and G respectively
be moved to o' and G'; let CA = CP = pj ; OA = O'A' = O'P = p 2 ;
ACF = ; OG = O'G'=C ; therefore since the arc AP = the arc A'P ;
.-. p t = p 2 A'o'p; .-. A'O'P = 0.
Pi
Let h G'K = vertical height of G' above the horizontal line
through c ; therefore
k = (pi+p 2 ) cos 6 c cos (l -|- -} 6 ;
P*'
and replacing the cosines by the first two terms of their equiva-
lent series, because is small, we have
1.2
= o, if e = o,
and changes sign from + to , if c is less than
Pi+Pa
P 2
to + , if c is greater than -
and therefore h is a maximum or a minimum according as
AG = p 2 c is greater or less than LrJ_. that is, as
Pi + p 2
is less than or greater than -\ ;
AG Pl T p,
and therefore the equilibrium is stable or unstable according as
is greater than or less than I - . (32)
AG Pl T p a
If the equilibrium is neutral,
- = -+-> (33)
AG p t p a
and in this case, for a small displacement, the centre of gravity
of the upper body neither ascends nor descends.
If the lower surface is plane, p, = oo, and the equilibrium is
stable or unstable, according as AG is less or greater than p a ;
1 44.] OF HEAVY BODIES. 201
that is, according as the centre of gravity is below or above the
centre of curvature corresponding to the point A.
If the lower surface is concave, PI is negative, and the equili-
brium is stable or unstable according as
is greater or less than (34)
AG p, Pl
143.] The values of p, and p 2 will of course depend on the
position of the normal planes of the greatest and least curva-
ture of the two surfaces, and therefore the stability will be
different for the different rotation-axes which are perpendicular
to the normal planes through A ; the stability therefore will be
greatest or least according as
1 J_
Pi Pa
is a minimum or a maximum.
If therefore in this latter case, which is the most unfavour-
able, the equilibrium is stable, it is also stable for every normal
section passing through A, and therefore the position of the
body is one of complete stability.
Suppose however that the upper and lower surfaces are so
arranged, that a is the angle between the normal section of
greatest curvature in the lowest, and that of the greatest cur-
vature in the upper ; and suppose that it is required to find the
nature of the stability of any particular normal plane.
Let 6 be the angle between the normal plane of displacement,
and that of maximum curvature in the lowest surface : then if
Kj and r l are the principal radii of curvature of the lower surface,
by Euler's theorem, Art. 403, Vol. I (Differential Calculus),
1 (cos 0) 2 (sin 0) 2 e
-\- f
PI f\ RI
and if R a and r 3 are the principal radii of curvature of the upper
surface,
1 {cos(0+a)} !
p a r, a,
therefore
_1_ 1 __ (cosfl 2 ) (cos(0-fq)} a (sin 6)*
Pi Pa fi f a RI
whereby the normal plane of least stability may be determined.
144.] The following are problems in which the stability of
equilibrium is determined by the position of the centre of gravity ;
PKICE. VOL. III. D d
202 STABILITY AND INSTABILITY OF HEAVY BODIES. [144.
the equilibrium being stable, neutral, or unstable according as
the centre of gravity is in its lowest position, moves in a hori-
zontal line, or is in its highest position.
Ex. 1. A heavy uniform beam rests against a smooth curve,
and against a vertical wall, all of which are in the same vertical
plane ; it is required to find the nature of the curve so that the
beam may be at rest in all positions.
Let the beam be QP, fig. 56, of which let G be the middle
point and the centre of gravity ; and let the horizontal line, in
which the centre of gravity is in all positions of the beam, be
the axis of x, and let it meet the vertical wall in the point o ;
let o be the origin, let the length of the beam be 2, so that
the curve required meets the wall at a distance OA(= a) below
o ; let OA be the axis of y ; OM = x, MP = y, QGO = ;
x y
.1. = cos 6, - = sm ;
2a a
therefore squaring and adding,
v* , y' _ , .
40* "*" a 2 "
the equation to an ellipse, whose centre is o, horizontal semi-
axis is 2 a, and vertical semi-axis is a.
The property of the curve required in the problem is evi-
dently the same as that of the elliptic compasses.
Ex. 2. A heavy uniform beam rests against a smooth vertical
wall, and on a smooth curve ; determine the nature of the curve
so that the beam may rest in all positions.
Let EQ be the beam of length 2 a, whose centre of gravity is G,
fig. 57 ; p the point in the curve at which the beam touches it;
let the horizontal line OMG, in which in all positions of the
beam its centre of gravity is, be the axis of x; and let it meet
the wall at o, and let o be the origin, OM = #, MP=^, QG =
GR = a. Then, as the line RQ, is a tangent to the required
curve at P,
dy
tanoGQ = -.
(vtX>
Therefore a = QP + PG,
.rds yds _
f/.r dy
1 45-] GENERAL THEOREMS ON CENTRE OF GRAVITY. 203
which is a differential equation of Clairaut's form : and of which
the singular solution is, y$ + x$ = a*.
Ex. 3. To determine whether the position of the beam resting
on two planes, as investigated in Ex. 2, Art. 60, is of stable or
of unstable equilibrium.
In fig. 29 let GK = h; therefore
h = AC sin a a sin 0,
sn a sn
= 20 - ' asmd,
sm(a
a
= -r f - (sin (a /3) sin + 2 sin a sin /3 cos 6} ;
sin {a -j- p^
.-. tan e = a - ( S ee Ex. 2, Art. 60) ;
2 sm asmp
and -^ changes sign from + to ; therefore h is a maximum,
(10
and the equilibrium is unstable.
SECTION 6. General properties of the centre of gravity.
145.] THEOREM I. Of all points in space the centre of gravity
is, with reference to a system of material particles, such that
the sum of the products of the mass of each particle and the
square of its distance from the point is a minimum.
Let (x,y, z) be the required point; m lt m tf ,.,m n the masses of
the particles ; (x u y lt zj, (x 3 , y u z a ), . . . (#, y n) *) their positions ;
then if
and if u* is to be a minimum,
unu = m l {(x x^dx + ty y l }dy-\-(z z^
+ m a { (x - x n ) dx + (y -y n ) dy + (z z n ) dz}
D d 2
204 GENERAL THEOREMS ON
and equating to zero the coefficients of dz, dy, dz, we have
Z.mx s.my z.mz ,,,.,
* = -=-, y = - , z --; (35)
2.m -s..m -S.M
and as the function by the form of the expression admits of
infinite increase, it evidently cannot be a maximum; (35) there-
fore render u a minimum ; and these are the coordinates of the
centre of gravity.
146.] THEOREM II. If a system of material particles is inva-
riable in form, and its centre of gravity is at a constant distance
from a fixed point, the sum of the products of the mass of each
particle and the square of its distance from the fixed point is
constant.
Let the fixed point be the origin, and let (x, y, z] be the centre
of gravity, and (a?,, y l} z,}, (a?,, y,, *,), ... (a?., y n , z n ) the positions
of the particles in a given position of the system, these co-
ordinates being measured from the centre of gravity ; also let
ac*+y* +z* =*;
and let r l} r?,.. .r n be the distances of the particles from the fixed
point: then
if pi, p 2) . . .p n are the distances of m lt m i} ...m n from the centre of
gravity. But -S.MX 0, -z.my 0, s.mz = 0, because the centre
of gravity is the origin ; therefore
and as the right-hand member is constant, so is the left-hand
member, and the proposition is proved.
] 47.] THEOREM III. If there is a system of heavy material
particles, the product of the sum of the masses and of the sum
of the products of each mass and the square of its distance from
the centre of gravity is equal to the sum of the product of every
two masses and of the square of the distance between them.
Let the centre of gravity be the origin : then
n = 0, ~\
. . . 4- m n z n =. 0.
148.] CENTRE OP GRAVITY. 205
Let pi,p t ,...pH be the distances of m lt m t , . . . m n from the origin ;
then squaring and adding the above, we have
-f- 2 m l m t (^ # 2 +y, y, + z l z t )
+ .........
l m n (x n _ l x n +y n _ 1 y n + z tt _ l z n ) = 0;
p'cos(p,p') = Q, (36)
if m, m are the symbols for every two of the material particles,
and (p, p) is the angle contained between p and p f . Now sup-
pose u to be the distance between the positions of the two par-
ticles m and m, then
a = P 3 + p' a - 2 pp' cos (p, p'} ;
.-. 2ppcos(p,p') = p* + p'*-u*.
Therefore (36) becomes
2.w 8 p 2 + 2.m'(p 2 +p'* w2 ) = 0:
and when written at length
n i }2.mm'u t = 0;
and if M = ~s,.m = m^ + m t -f . . . -f m n ; we have
M2.zp a = z.mm'M*, (37)
which is the proposition required*.
148.] THEOREM IV. If a material particle is in equilibrium
under the action of many pressures which are represented as to
intensity and line of action by straight lines meeting at the
particle ; and if at the extremities of each of these lines heavy
particles equal in weight are placed, the centre of gravity of
these is at the point which is at rest under the action of the
impressed pressures.
By reason of equations (69), Art. 34, we have
S.pcosa = 0, 5.PCOS/3 = 0, 2.Pcosy = : (38)
let * * 3 , ... s n be the line-representatives of the impressed forces
acting on the material particle, the place of which we will take
to be the origin : so that the equations (38) become
2.* cos a = 0, 5.* cos/3 = 0, 2.* cosy = 0. (39)
* In the " M^canique Analytique " of Lagrange, Premiere partie, Section III,
Art. 20, an extension of this Theorem is given.
206 GENERAL THEOREMS ON CENTRE OF GRAVITY. [148.
Let (x ls y lt zj, (x z ,y t) z 2 ), ... (x n) y n) z n ] be the extremities
of * s )} ... s n ; so that
x l = s l cosai, y^ = s l
ar a = g t cos a 2 , y a = * a cos /3 2 , ^ 2 = * 2 cos y 2
^ = * cos a n ; y = * cos^ B ; * = * n cos y n ;
whereby (39) become
2.a? = 0, s.y = 0, 2.0 = ;
and if the mass of the particle at the extremity of every line-
representative is m, we have
?,.mx = 0, 'S.my = 0, t.mz = ;
and therefore the origin is the centre of gravity of all the
particles.
CHAPTER V.
SECTION 1. The action of forces on flexible and inextensible
strings or cords.
149.] Thus far the bodies or systems of material particles, on
which the statical forces act, have been assumed to be rigid, and
their forms, or the relative position of the particles, have been
supposed not to change on account of the acting forces. We
shall now extend the inquiry to the case of bodies whose form
varies by the action of the pressures, but becomes permanent,
and may be considered rigid, under the action of the impressed
forces. I shall first shortly investigate the case of the Funi-
cular Polygon.
Suppose a string or cord, fig. 58, AB to be fastened at the two
points A, B ; the cord being without weight, perfectly flexible,
and perfectly inextensible ; and suppose at Q u Q 2 , Q 3 , Q 4 , definite
points of it, pressures P n P a , P S , P 4 to act with definite intensities
and along definite lines of action, so that the cord assumes the
permanent position indicated in the figure ; the object is the
determination of the form of the polygonal figure which the
cord of given length assumes under the action of these forces,
and of the tensions of each of its component straight elements.
It is manifest that the tension is the same throughout each
element ; and that as each point Q u Q S , . . . Q 4 is at rest, the forces
acting at each are in equilibrium. Let the tensions along
A Q n QI Q a , ... Q 4 B, be respectively T^ T 3 , ... T 5 , so that the pres-
sures at the fixed points A and B are respectively T t and T 5 ; and
let the angles between the successive straight parts of the cord
be Oj, a a , . . . a t ; then as the point Q, is kept at rest by the three
forces TJ, PJ, and T 2 , the lines of action of all are in the same
plane, and we have
T _ = _2i_ = _ _!__ (i)
sin PI qx A
208 THE FUNICULAR POLYGON.
In the same way for the point Q 2 we have
T, P, T a
; (2)
sin a 2 sin P 2 Q 2 QJ
and so on for the other points ; and therefore when the form of
the polygon and the magnitudes and lines of action of the forces
PI, p 2 , are given, the tensions of the several connecting
strings may be determined.
150.] Suppose that the lines of action of the forces p x , P 2V . .P 4
bisect the angles a 1} a 2 , . . . a t ; then TJ = T 2 = . . . = T S ; and
cos cos cos
22 2
and this condition maybe secured in two ways; (1) by fixing
smooth pins at the points Qj . . . Q 4 , and passing the string round
them, so that the tension of the string is the same on both sides
of the pin, and the pressure on the pin is the resultant of these
two equal forces, and therefore its line of action bisects the
angle between their lines of action : and (2) by making the im-
pressed forces act on the cord at the points Q t ... by means of
smooth rings which slide on the cord, and are at rest at these
points ; and the line of action of P t will manifestly under this
arrangement bisect the angle A^Q-J, because considering A and
Q 2 to be fixed, and the cord to be also fastened at them, if the
ring Qj slides, it can move only on the surface of a prolate
spheroid, of the generating ellipse of which A and Q 2 are the
foci, and the length AQ^-J of the cord is the major axis, and
therefore the normal at QJ which is the line of action of P,
bisects the angle between the focal distances.
If we suppose that the two sides of the polygon abutting at
(say) QJ are equal; then if AQj = Q^ = # and the radius of
the circle passing through AQ X Q 2 is p lf we have
a i *i . /4\
s y : = "277'
and therefore if all the sides are equal, from (3) it follows that
each impressed force is inversely as the radius of the circle pass-
ing through its point of application and the two angular points
of the polygon adjacent on each side.
Now of such a polygon with equal sides a curve is a particular
case, when the length of the curve is the equicrescent variable ;
and the circle just mentioned is the circle lying in the oscu-
lating plane at the point, and its radius is the radius of absolute
151.] THE FUNICULAR POLYGON. 209
curvature of the curve at the point; and therefore when a funi-
cular curve fastened at its two ends is acted on in all its equal
elements by normal forces, the tension is the same throughout,
and each normal force varies as the absolute curvature of the
curve at the point where it is applied.
Thus suppose a cord to be stretched by given forces at its
ends on a curved surface, then the pressure caused by the sur-
face is at every point in the direction of the normal of the
surface, and is therefore proportional to the absolute curvature
of the curve which the cord assumes on the surface ; and as the
normal-line of the reaction is in the same plane with two con-
secutive elements of the funicular curve, the osculating plane of
the curve is a normal plane to the surface at the common point;
and therefore, see Art. 336, Vol. II (Integral Calculus), the curve
is the geodesic line joining the two points : and this geodesic
line may evidently be either the maximum or the minimum ;
thus, a cord stretched between two given points on a sphere
will arrange itself along the geodesic line, which is a great
circle; and as one great circle-arc abutting at the points will
be a minimum, so will the remainder of the same great circle be
the maximum.
151.] If the lines of action of all the forces acting on the
funicular polygon are parallel, the cord is evidently in one
plane. Let the lines of action of the forces be vertical; then
sin p, Q, q 2 = sin P 2 Q a Q t , sin P 2 Q 2 Q 3 = sin P 3 Q 3 Q 2 , ; so that
if Pa ft*, are the angles of inclination of the successive lengths
to the horizontal line, \^
T, cos 0! = T 2 cos /3 2 = T S cos 3 = . . . ; (5)
and therefore the successive tensions are inversely as the cosines
of the angles of inclination to the horizon of the sides along
which they act; and therefore if T O is the tension of a side
which is horizontal, and if T is the tension along any side whose
inclination to the horizontal line is /3,
T = T cos 0. (6)
Suppose however that the vertical forces are the weights of the
several parts of the cord, so that P,, P 2 , ... are proportional to
the lengths AQ^Q^,,... ; and moreover suppose that the lengths
of the elements are infinitesimal, so that the polygon becomes a
plane curve, then if the density and thickness, that is, the area
of a transverse section, of the cord are constant throughout, and
PRICE, VOL. in. E e
210 THE CATENARY.
if p == the density, and o> = the area of a transverse section,
p = pwffds, dx = ds cos /3, dy = ds sin /3 ; and if T and T 7 are
the tensions at the beginning and end of an element respec-
tively,
TCOs/3 = Tcos/3 + #.Tcos/3, ) ,-.
T'sin ft=. T sin /3 -f d.t e0a-/3 f
therefore taking vertical forces,
p -f T sin /3 = T? sin /3',
and replacing p, T and T'sin ft by their values,
= 6?.Tsin/3
and if we consider T O to be known, and to be equal to the weight
of a length = c of the string of the string-curve, so that
T = pu>cg } then from (6) we have
--
and placing the origin at the lowest point of the curve,
which expresses the property of the curve, that the length of it
reckoned from the lowest point varies as the tangent of the
angle at which the tangent of the string at the upper end is
inclined to the horizon. This is a characteristic property of the
curve, and from it all the other properties may be deduced.
The equation in terms of x and y has been determined by means
of (8) in Ex. 7, Art. 166, Vol. II (Integral Calculus). The
curve which a heavy flexible and inextensible string thus takes
is called the catenary. I propose however to investigate the
form of string-curves under the action of given forces in a more
general way, and in the course of the inquiry to return to the
special form of the heavy catenary.
152.] Suppose a perfectly flexible and inextensible string to
be in space, and to be at all its parts subject to the action of
certain given forces ; let it be referred to a system of coordinate
axes, and at the point (x, y, z), let p be the density, co the area
of a transverse section of the cord, these quantities being in the
general case functions of x, y, and z ; and let ds be the length-
element; and thus patds is the mass-element of the cord. Let
x, Y, z be the axial components of the impressed forces acting
152.] THE CATENARY. 211
on an unit of mass at that point ; so that the pressures acting
on the mass-element at the point are
patxds, pvYds, pvzds. (9)
Let T be the tension of the cord at the point (x, y, z) ; then as
it acts along the curve, its resolved parts are
dx dy dz
and therefore at the point (x-\- dx, y-\-dy, z-\-dz) the resolved
parts of the tension are
dx , dx dy , dy dz . dz
the tension varying continuously as we pass along the curve ;
let us suppose x, y, z, and s to increase simultaneously ; then
the element of the curve being in equilibrium under the action
of the forces (9) (10) and (11), the conditions of equilibrium are
. dx
fit-;- +p(axas = 0,
as
it it
fj m " | f, /.* "V ft Q - O > ( 1 O \
U/,\. j -f- pCO I IvS \J , f I 1 i\
ds
dz
d.T-j- -\-pGtzas 0;-
as
and from these equations all the properties of the curve are to
be deduced.
First, integrating the equations, we have
J'pM'x.ds J*pu>Yds CpuiZds T
dx dy dz ds '
and therefore the numerators are proportional to the direction-
cosines of the arc-element on which the forces act.
Also expressing at length the first terms of (12), and taking s
to be equicrescent, we have
7 dx dx ,
Td '^ +-*+/M"W = >
as as
id.-j- + -jr-dT + ptovds = 0,
as as
. dz dz 7
+ -- ffa + patzds = ;
(13)
' ds
Multiplying these equations severally by dx, dy, dz, and adding,
we have dT+pufadx + Yffy + zdz} = 0, (14)
where d? is the total differential of T. This equation is evidently
E e 2
212 THE CATENARY. [ J 53-
that of the tangential components of the forces. Let the inte-
gral of it be taken between the limits which carry the subscripts
n and ; and we have
r
T tt T O + / p(ti{-s.(Ix + Yffy + zdz} = 0. (15)
-'o
If therefore p, o>, x, Y, z are functions of the coordinates of the
point to which they apply, and are such that the quantity under
the sign of integration is a complete differential, then the dif-
ference between the tensions at the limits is a function of the
coordinates of those points only, and is independent of the form
of the curve which joins them.
The analytical conditions which are satisfied when the second
part of (15) is an exact differential have been investigated in
Articles 373, 397, Vol. II (Integral Calculus), and the corre-
sponding geometrical theorems have also been worked out.
Many mechanical properties which satisfy the conditions will
be exhibited hereafter ; and it will be more convenient to con-
sider the character of the preceding equations when they are
under discussion. The tension of the string-curve is constant
throughout its length, that is,
T M = T O , (16)
whenever x^+Y<^ + z^=0; (17)
and this occurs (1) when x = Y = z = 0; that is, when the
string is under the action of no force; (2) when the resultant
force acts at every point along a line normal to the curve at the
point.
153.] Also let us transfer the last term in each of (13) to the
right-hand side of the equation, and take the squares of these
equations, and add them : then if * is equicrescent, p' = the
absolute curvature of the curve at the point (x, y, z], and P is
the impressed force on an unit-mass at (#, y, z) ; so that
/
^+(j s ) = "' '"'*'>
and consequently, if the tension is constant throughout the curve,
and thus the impressed force varies inversely as the radius of
1 5 5.] THE CATENARY. 213
absolute curvature at each point of the string, see Art. 150.
Moreover, if the force is also constant, p is constant, and the
curvature is the same at all points ; and if the string-curve is a
plane-curve, it is also an arc of a circle.
Also from (13) eliminating T and df } we have
(dzd. f~dyd. -j-\x.+ (datd. ~dzd. -^)Y + (dyd.-j-dxd.-%-)z =
V ds y d%' ^ ds ds> ^ y ds ds'
.'. (dzd*y dyd t z)x + (dxd lt z dzd t z)v + (dyd*xdxd*y)z=:0', (20)
and as the former factors of each term are proportional to the
direction-cosines of the binomial, we conclude that the impressed
force lies in the osculating plane of the string-curve.
Also if is the angle between the line of action of P and the
arc-clement, jidx + xdy + zdz = ndscosQ;
therefore from (14),
dT + p<a-pdscos<j> ; (21)
and substituting this value for dT in (18) we have
T = pp'o>psin$; (22)
these are the equations of the tangential and normal components.
^ 154.] If the impressed forces all act in one plane, we may take
that plane to be the plane of (x, y], and equations (12) become
, dx
d.t -^- +pti>-x.ds = 0,
(23)
d.f - +p wds = :
as
and taking the tangential and normal components, we have
= 0; (24)
so that if T is constant,
T = pp'wP. (26)
Of these general formulae the following are particular ex-
amples.
155.] Let us suppose gravity, or the earth's attraction, to be
the only acting force, in which case the curve is the free cate-
nary ; and let the axis of x be horizontal, and that of y vertical ;
then x = 0, Y = g ; so that the equations (23) become
rf.T~ = 0, d.i:--g<,>pds=0; (27)
dx
214 THE CATENARY.
when T is the horizontal tension of the catenary ; that is, it is
dx
the value of the tension, when -=- = 1 . Thus the horizontal
ds
component of the tension is constant. It may be expressed more
conveniently in the following form. Let o- = the density and
a = the area of a transverse section of the string at the point
where the string is horizontal; and let c = the length of a
string of that density and thickness whose weight = T O ; so that
-- (29)
ds
Also from (27), T-^- = gpa>ds ;
us J
' *^!l "/'"** (30)
and if the string is of the same thickness and density through-
out, so that p = or, o> = o, then
if s 0, when - - = ; that is, if s begins at the point at
tut
which the curve is horizontal. All the properties of the curve
may be inferred from (31).
As the heavy catenary however has many remarkable geo-
metrical properties, and has important applications to the theory
of Suspension Bridges, I will also deduce its equation from first
principles, so that it may be presented to the student in the
clearest possible form.
156.] Suppose the curve, see fig. 59, to be suspended from
two fixed points, A and B, in the plane of the paper, which is
supposed to be vertical ; let c be the lowest point of the catenary,
and let a vertical line through it be taken for the axis of y, and
let the horizontal line, which will also touch the curve at c, be
the axis of x. Let CM = x } MP = y, the arc CP = *, p = density
at P, o> = the area of the transverse section of the cord. Then
the arc CP, after it has assumed its permanent form of equili-
brium, may be considered as a rigid body kept at rest by three
forces, (1) T the tension acting at p in the direction of the tan-
gent, (2) the weight of the cord CP acting vertically downwards
and which is equal to / yputds, and (3) the horizontal tension at
J* ,
the lowest point c ; as to the last force, let us suppose, as in the
1 57.] THE CATENARY. 215
preceding Article, o- to be the density of the cord at c, a to be
the area of a transverse section at the same point, and c to be
the length of cord such that gaac is equal to the tension at c ;
then by the triangle of forces, these forces are proportional to
the three lines PT^ T'N, NP, which their lines of action are re-
spectively parallel to ; and therefore we have
/*
/
J
PT" ' T'N NP
but '
(/puds
(32)
/'
t'O
atrc-?- = / puds; (34)
-'o
ds ' ' dy dx '
(33)
dy dx
so that the equation to the curve is given by
dy
dx
and the tension at any point by the equation
T = ya<rc -=--; (35)
dx
which are the same equations as those found in the preceding
Article.
157.] Now let us take a particular case, and suppose to and p
to be constant throughout the cord ; so that p = cr, o> = a, and
the curve to become that of a cord of constant thickness and
density, suspended from two given points A and B : therefore
from (34), dy s , .
= - ; (3b)
dx c
which is the same equation as (31); then differentiating, and
making x equicrescent,
d*y f du*^
a if /ii y \
dx dx
and integrating, and taking the limits such that -/- = 0, when
dx
x = 0, we have
216 THE CATENARY. [l57-
dy x dy\t
'
dx V 6fo a '
. '-'
... 2 = e'-e's (37)
#
and integrating again, and observing that y = 0, when x = 0,
we have
(38)
(39)
Also equating the values of -jj- in (36) and (37) we have
and either (38) or (39) is the equation to the catenary of con-
stant thickness and density, when the lowest point of the curve
is the origin, and the horizontal tangent at it is the axis of x.
To simplify the equation, let the origin be moved to a point
o, see fig. 60, at a distance c below c and on the axis of y, so
that (38) becomes
f-|{|.~?}; (40)
and (39) is unaltered. The horizontal line through o is called
the directrix of the catenary. Thus the ordinate of the catenary
measured from the directrix is the sum of the ordinates of two
logarithmic curves.
Now c oc is the length of a cord of the same thickness and
density as the cord of the curve, the weight of which is equal
to the tension of the curve at its lowest point : if therefore
a smooth small pulley were placed at c, and if over it a cord of
length c, and of the same thickness and density as the cord of
the curve, and joined to the arc CP, were suspended, the curve
would be in equilibrium.
j X X
Since from (39) J* = -{e~ c + e~} = $-\ (41)
therefore from (35), T = gaay, (42)
that is, the tension at every point of the curve is equal to the 1
weight of a cord of the same thickness and density, the length
158.] THE CATENARY. 217
of which is equal to the ordinate of the point. The tension
therefore is the least at the lowest point of the catenary, and
varies directly as the ordinate : it is consequently the same for
the two points in the same horizontal line. And therefore if,
see fig. 61, a cord of constant thickness and density is suspended
over two small pulleys A and B, and is at rest by means of certain
lengths hanging over the pulleys, the two ends H and K are
in the same horizontal line, and the tension at the lowest point
c is equal to the weight of a cord similar in all respects, and
whose length is CO.
158.] Let us investigate some of the more prominent geome-
trical properties of the catenary. From (40) and (39) we have
<">
ds
y = c^=-'
dx
Now as (40) is unaltered when x is replaced by x, it follows
that the catenary is symmetrical with respect to the axis of y.
Also squaring (39) and (40), and subtracting, we have
y* $* = c\ (45)
From the preceding equation it will be found that the radius
y'
of curvature of the catenary = > and is equal to the normal ;
C
and that these lines are drawn from the curve in opposite direc-
tions ; hence the radius of curvature at c is equal to c. Also
from (42),
T 2 = "*Q?a**
= (tension of curve at lowest point) 2
-{-(weight of cord of length = s)*.
Also let a tangent Pn, fig. 60, be drawn to the catenary at
the point P, and from M, the foot of the ordinate, let a perpen-
dx
dicular to pn be drawn ; then since -=- is the sine of IIPM,
as
PRICE, VOL. in. F f
218 THE CATENARY.
dx
nM = y-^-
y ds
= c; (46)
and therefore from (44) or (36) pn = * = the arc CP. Therefore
the point n is on the involute of the catenary which originates
from the curve at c, and nit is a tangent to this involute ; and
as nil is the tangent to this last curve, and is equal to the con-
stant quantity c, the involute is the equitangential curve or
tractrix, the asymptote of which is the axis of x. Let therefore
77 and be the current coordinates to this curve; ON = f,
N n = 77 ; then
= tan
= __E! = -- *, (47)
NM {C 2 -7J 2 }*
which is the differential equation to the equitangential curve.
And producing pn, so that it cuts the axis of x in T, pn is the
radius of curvature of the tractrix at the point n, and HT is the
normal ; and therefore as PMT is a right angle, pnxnT = nM J ;
therefore in the tractrix,
the radius of curvature x the normal = c 2 . (48)
The intrinsic equation of the catenary is
s = ccot^r. (49)
This may be derived analytically from the preceding equations by
the process developed in Art. 168, Vol. II (Integral Calculus),
see Ex. 5 ; or it may be proved geometrically : for pn = *,
FIM = c } FIMT = \/r; therefore pn = riMcotnMT. Also the ca-
tenary at its lowest point approximately coincides with a conical
parabola. For taking the equation (38), the origin of which is
at the lowest point,
c t -
X X 3 X s
I I I I
/. ' 1 9 />2 "*1 o 9 x.s ' ' ' '
+ i-2
f 1 9 />"* 1 9 *? /> 3
C J..A>C 1 . a.d.C
,}
ar' J a: 4 i
+ 1.2.c a H " L2.3.4.C 4 + J '
1 59.] THE CATENARY. 219
and omitting terms which involve powers of x higher than the
second.
*
the equation to a parabola, whose vertex is c, whose principal
axis is cy, and whose latus rectum is 2c.
159.] The constant c which enters into the equations of the
curve may be determined experimentally by means of the tension
at the lowest point c. Suppose however that the data of the
problem are different to those which we have taken. Suppose,
for instance, that a homogeneous heavy cord of the length 2 1
is suspended from two points in the same horizontal line at
a distance 26 apart, and that it is required to determine the
equation of the catenary, the position of the lowest point, and
the tension at every point.
Let the origin be taken at the point of bisection of the hori-
zontal line which joints the two given points ; see fig. 62 ; the
horizontal line being the axis of x } and the vertical line reckoned
positive downwards being the axis of y, OB = OB' = $; let
oc = h ; so that the equations become
- -- - --
h + c-y = -{e~ c + e '}; s=-{e c -e c }', (50)
and in these we have to determine h and c in terms of I and b.
Let a be the angle at which the curve is inclined to OB at the
- I
point B ; then we have sec a -f tan a = e c , and from (4 3) tan a = - ;
C
.'. j = cot a log (sec a -f tan a)
= cot a log tan (45 + -) ;
whence a may be determined ; and consequently c may be found.
And from (50), if y = 0, we have
6
= ce c \
h = I (coseca cot a}
= /tan--
Ff 2
220 THE CATENARY OF [l6o.
therefore the tension at the lowest point =
and the tension at B and at B'= pg <D I cosec a ;
thus all the circumstances of the curve are determined.
Another problem of the same kind is, To determine the form
and circumstances of the catenary when a heavy homogeneous
string 1 of given length is suspended over two smooth pulleys in
the same horizontal line, and the ends of the string hang freely
so that the string supports itself.
160.] To determine the position of the centre of gravity of
the cord of the catenary of uniform thickness and density, be-
ginning at the lowest point c; fig. 60.
lgpads ffpatxds; .-. xlds^i
/. X X
C i - \ ,
-{e e c }dx
, 2< f
= xs
(si)
yjds=jyds',
C* c , x - -V
ys f - \e c -f e c \ dx
sy cx
__ _i __
' 2 2 '
And by geometrical construction in fig. 60,
y =
In Art. 130 it has been proved that of all curves which a
heavy wire or a flexible string of uniform thickness and density
and of given length with its ends at fixed points can assume,
the catenary is that of which the centre of gravity has the
lowest position. The form therefore which a heavy flexible
cord of uniform thickness and density assumes when suspended
from two fixed points is that of stable equilibrium.
l6l.] VARIABLE DENSITY AND THICKNESS. 221
161.] Next let us consider the circumstances of a heavy
string of varying thickness and density, under the action of
gravity only.
From (33) we have
_
da dy dx '
.-. ga.ac~- =. I ffpuds; (53)
ttX J
Q
and differentiating,
d* ds
from which the variation of the density or of the thickness may
be determined, when the catenarian curve is given; and the
1 curve may be found, when the law of the thickness or of the/ i
I density is given : also ^
whereby the tension at any point of the curve may be found.
Some examples are subjoined.
Ex. 1. It is required to determine the law of variation of the
thickness of a heavy homogeneous string, that it may be in
equilibrium in the form of a parabola with its vertex downwards
and its axis vertical.
Let the equation be z* = lay;
dx _ dy ds d*y 1
2a ~" x (40 s + # 2 )* ' dx 2 ~~ 2a'
and therefore from (53), as p is constant and equal to <r,
so that ft> varies inversely as, and T varies directly as, the square
root of the distance of any element from the directrix : therefore
when x is small, o> is constant, which fact has already been
proved in Art. 158.
Ex. 2. It is required to find the law of variation of the den-
sity of a heavy string of uniform thickness that it may hang
in the form of a semicircle with its diameter horizontal under
the action of gravity.
dx _ dy _ ds
ay
Therefore from (53), p = ^ ;
222 THE CATENARY OF [l6l.
that is, the density varies inversely as the square of the depth
below the horizontal diameter of the semicircle.
gaaca
Also T =
ay
If therefore y a } p = <*> T = OO: that is, the density and the
tension are both infinite; and rightly so, because the string is
vertical at the points of its support at the extremities of the
horizontal diameter of the circle, and there is at them no counter-
acting horizontal force to balance the horizontal tension at the
lowest point.
Ex. 3. To find the form of a heavy string, the thickness of
which varies inversely as the square root of its length from the
lowest point, when it is acted on by gravity.
In this case o> = fi*~* ;
therefore from (53),
gaac-f = /
ax JQ
d ( d l\
\> _
because the origin is at the lowest point, where the curve is
horizontal ; and making obvious substitutions,
a-- = x 9
whence the equation to the curve will be found without diffi-
culty. Also a + x
T = gave --
Ex. 4. To find the equation to the catenarian curve, when the
weight of each element of the curve varies as the horizontal
projection of it.
This case is approximately that of suspension bridges, in which
the weight of the chain and of the vertical suspending rods is
neglected, and each element of the chain has to bear that part of
the roadway which corresponds to the horizontal projection of it.
In this case ptagds =
therefore from (53),
dy T
-j- = /
I'd . u
1
2
1 6 2.] VARIABLE DENSITY AND THICKNESS. 223
the equation of a parabola with its axis vertical, and vertex
downwards.
Ex. 5. To determine the equation to the catenarian curve of
uniform density, and the law of variation of the thickness, so
that the thickness may be at all points proportional to the
tension.
In this case G> = JU,T; (55)
therefore (33) becomes
/
Jo
dy dx
du , dy ds*
' rf._
dx
(56)
.-. log sec pyjbur =
secffppx = e^w, (57)
which is the equation to the required curve. This curve is
called the catenary of uniform strength. If we substitute for
a
y
gpn, we have e = sec- ; if a?=0, y=0; and if #= + -z-, y = 00 ;
a LI
so that the curve has two vertical asymptotes, equally distant
from the origin, which are at a distance = ira apart. Also
T = garrc sec gppx,
o> = fj,ga<rc secgpiAX. (58)
162.] In Art. 130 it is shewn that of all uniform and heavy
curved lines of given length joining two given points in the
same vertical plane, the catenary is that of which the centre
of gravity has the lowest position ; I propose to extend the
problem to the case of heavy flexible strings of varying density
and thickness, and to find the form of the curve so that the
place of the centre of gravity of it may be the lowest possible.
Let the axis of z be vertical, and let a point on the curve be
(x, y, z), and let the element ds begin at this point ; let fj. ds
= the mass-element of the string-curve, where p. is a function
of x, y } z ; then z is to be a minimum, where
z] (j.fo = I fj.zds. (59)
^0 M)
224 THE CATENAKY. [162.
Now / y.ds is the mass of the string, and this evidently is
Ja
constant, so that the variation of the right-hand member of
(59) is to vanish consistently with this condition;
0. b.l nzds = Q, and 8./|^=0; (60)
JQ J o
from the former we have
rt
= / b.
Jo
a
; (61)
and from the latter of (60),
= / 8./A<&
^o
t/dx fc <??/ . f&
n(-j-8a?+ -v-8y+ -s-
^\ds, ds ds
Now for (61) and (62) to consist, it is necessary that
u.\ dx , /-du.^ 7 dy
f\d.u,z^r zds(^-)d.u.z~
x' ds ^d' ds
, ,du.^ , dx 7 ,du\ 7 dy
ds(-j-\- dp. -r- ds(-f)- d.p -f-
^dx' ds ^d' ds
/W-*A\ - UiX
\-j- \-d.lJLZ-j-
W *' * = A, (63)
dx
where X is an undetermined constant; and from these equa-
tions, when jx is given, the equation to the catenary is to be
deduced. If /x = 1, the equations (63) become (16), Art. 130.
163.] CENTRAL FORCES. 225
163.] When the catenary is at rest under the action of forces,
the action-lines of which pass all through a fixed point, and
when that point is the source of the action of the force, so that
the intensity of the force depends on the distance from that
point of the particle on which the force acts, the equation and
the properties of the catenary may be more conveniently in-
vestigated by the following process :
Let the point at which the forces originate, and which is
called the centre of force, be taken for the origin, and let the
central force acting on an unit of mass of the string be p ; let
the force be repulsive, so that its tendency is to remove the
molecules of the string further from the origin, and therefore
the string will be concave towards it ; if the force is attractive
p will be affected with a negative sign and the string-curve will
be convex towards the origin. The components along the co-
ordinate-axes of P acting on an unit-mass of the curve at the
point (x, y, z] and at a distance r from the centre are
so that the equations (12) become
dx
f = > 1
fl.T-/- + poxfo = 0,
ds r
j dz , PZ
.T -r- + pads = : J
ds r
(64)
multiplying the second of these equations by z, and the third
by y, and subtracting,
dy j dz
ds ds
du dz
.. integrating. z^-~- yT-=- = li,
ffn if 9
dz dx
and similarly #T-= 21-=- = # 2 , > (60)
ds ds
dx dy
ds ds
and therefore multiplying these last equations severally by
x, y, z, and adding,
h^x-\-h^y -\-h z z = 0; (66)
which is the equation to a plane passing through the origin,
which is the centre of force: whence we infer that the curve
PRICE, VOL. in. G g
226 THE CATENARY.
and the centre of force are in one and the same plane, and thus
the catenary under the action of a central force is a plane curve.
164.] Let the plane in which the catenarian curve is be
taken as the plane of reference ; and let the curve be referred to
a system of polar coordinates in it. Let (r, &} be the place of
the mass-element whose length is ds } and of which p and o> are
respectively the density and the area of a transverse section.
Also let P be the repulsive force and T the tension at this point.
Then resolving along the tangent
dr
pox&P-j- + #T = 0;
.-. dT+pa>vdr = 0; (67)
which equation is also that of the virtual velocities, when the
arbitrary displacement of the point of application of P and T
takes place along the tangent. And resolving along the normal,
if d\\r is the angle contained between two consecutive normals,
so that ds == pdty, where p' is the radius of curvature and is
equal to r -7- >
d/p o
= 0;
.-. pwPjo + T^ = 0; (68)
and if P is eliminated between (67) and (68),
dT dp
+ = ;
T p
.'. tp == T j5 = a constant, (69)
if T andjo, are simultaneous given values of T and. p.
Hence we conclude that the tension at any point of the curve
varies inversely as the perpendicular from the centre of force on
the tangent of the curve at that point.
The equation (69) is the equation of moments, with reference
to the centre of force, of the forces acting on the element of the
curve, and might have been deduced directly from (50), Art. 55.
If we eliminate T from (68) and (69) we have
dp upvdr _
~~
(70)
P
the limits of the integral being given by the conditions of the
problem. From (70), when p is given, the equation to the
curve may be found; and if the curve is given, P may be
165.] CENTRAL FORCES. 227
found; also from (69) the tension at any point of the curve
may be found.
165.] In illustration of the preceding theorems let us take
the following examples :
Ex. 1 . If the central force is constant and is attractive, find
the equation to the catenarian curve of constant thickness and
density.
Let the force = f\ so that (70) becomes
P P*
the curve being such that r = oo , when p = ; making an
obvious substitution, we have
jar = k* ;
whence we have 2 = r 1 cos 2 0, which is the equation of, the
equilateral hyperbola.
Also from (69), T = o>pfr.
Ex. 2. Find the equation to the curve of constant thickness
and density when the central force is repulsive and varies as the
distance.
Let P = fxr ; so that from (70), if p = 0, when r = oo ,
p ~ 2p T
r 9
" e'
whence by integration we have
Ex. 3. Find the equation of the catenarian curve of constant
thickness and density, when the central force is attractive and
varies inversely as the square of the distance.
Let P = ; so that from (70),
r 3
and making obvious substitutions, and replacing - by n, we have
c(uK] = -:
P
Gg 2
228 THE CATENARY [l66.
therefore -j = (o* l)u 2 2c*ku + c*fr; (71)
and the integral of this equation will be of three different forms,
according as c is greater than, equal to, or less than, unity.
(1) Let c 2 be greater than unity ; then, if c 2 1 =%% the in-
tegral of (71) is of the form
u-a = | {"' + *-"*}.
2
(2) Let c* = 1, then the integral is of the form
c
~~
(3) Let c 2 be less than unity; then, if 1 c 2 = a ,
u a = bcos 116.
Ex. 4. If the catenarian curve of uniform thickness and
density is a parabola under the action of a central force in the
focus, that force varies as r~%.
Ex. 5. Prove that a parabola is the catenarian curve of con-
stant density when the force varies inversely as the distance, and
the thickness varies inversely as the square root of the distance
from the centre of force.
Ex. 6. If the catenarian curve of uniform thickness and
density is a circle, and has the centre of force in the circum-
ference, shew that the force varies inversely as the cube of the
distance.
166.] The catenary thus far has been considered a free curve.
If however the string is stretched on a curved surface, and is
also under the action of given forces by which it is kept on the
surface, the equations of equilibrium may be investigated in the
following manner:
Let us in the first place consider the surface to be smooth.
Let the equation to it be F (x, y, z] = ; and let its partial
derived functions be IT, v, w ; and let Q 2 = u* + v a + w 2 : let nds
be the pressure of the surface against the mass-element whose
length is ds, so that the equations of equilibrium are
dx
(72)
R ds = 0. J
u , dx v T du w , dz) , (XU + YV + ZW)
-d.- r + -d.-?- + -d.-j-\+p<*ds \ - - ( + -Rd# = Q; (74)
q * Q 4& - 4 4r ) 7 ( Q I
167.] ON A SMOOTH SURFACE. 229
Multiply these equations severally by dx, dy } dz, and add, and
let * be equicrescent ; then because
we have df + pw {x.dx + tdy + zdz} = ; (73)
which assigns the tension in terms of the impressed forces, and
shews that it is independent of the reaction of the surface ; and
if x, Y, z are functions of the coordinates of ds, and such that
p(t)(xdx+Ydy + zdz) is an exact differential, then T depends on
the coordinates of the extreme points of the string, and is inde-
pendent of the form of the surface.
If Jidx + vdy + zdz = 0, T is constant throughout the length
of the string, whatever is the form of the surface.
Again, differentiating the first terras of (72), and multiplying
U V W
the equations severally by - > - > - > and adding, we have
J . du w . dz
T \-d.-j- -f -(
^ ds o
and therefore if = the angle between the normal to the surface
and the principal normal to the curve at a cqmmon point, and
if tj> = the angle between the normal to the surface and the line
of action of the resultant of the impressed forces, viz. p, and if
p'= the radius of absolute curvature of the curve, we have
TCOS0 /_ eN
hpo>pcos<j> + R = 0; (75)
P
so that from (73) and (75) R may be determined. And since
R ds is the pressure of an element of the curve against the surface,
the whole pressure = / nds. (76)
Again, suppose that x = Y = z = 0, and that we differentiate
the first terms of each of the equations (72), and eliminate T and
dT by cross-multiplication, then
(dzd*ydyd*z}\i + (dxd*zdzd*x}v + (dyd^xdxd^y)^^ ; (77)
and therefore the binormal of the curve is perpendicular to the
normal of the surface ; the curve therefore along which the
string is laid is a geodesic line on the surface.
167.] If the string rests on a smooth plane curve, we may
take the plane of the curve to be that of (x, y], and F (x, y) =
to l>e the equation to the curve ; in which case the equations are
230 THE CATENARY
dx
!T '^ (?8)
x = 0;
u
whence we have
<?T-f pta(x.dx+ vdy) 0; (79)
?v dir\ ,_,
_ Y _ =E;
whereby T and R may be found.
If gravity is the only acting force, we may take the plane of
(x, y) to be vertical, and take the horizontal line to be the ar-axis,
and the y-axis to be positive upwards : then, if the string is of
uniform thickness and density,
T-T O = p<*g(y y,)', (81)
' . (82)
The following are examples in which the pressure of strings
on smooth surfaces and curves is calculated :
Ex. 1. On the smooth surface of a circular cylinder whose
radius = a, and whose axis is horizontal, a heavy homogeneous
string of given length rests in a vertical plane : determine the
tension at any point and the whole pressure on the cylinder.
Let the section of the cylinder be represented in fig. 64. Let
0, and be the angles corresponding to the ends of the string,
6 being measured from the horizontal line through the centre of
the circle. Let the place of ds be (a, 6) ; then, if is the angle
corresponding to the lower end of the string, T O = ; and the
tension at any point is equal to the sum of the weights of the
successive elements of the string resolved along the curve ; so
that re
T = / a pat g cos Odd
*>e
sin0 ); (83)
ne o ). (84)
Hence if the string reaches from the highest point to the hori-
zontal line, = 0, #1 = - > and the tension at the highest point
2
= a pug } but the weight of the string = - = w, say ;
_ 2w .
A "" ~ *
167.] ON A SMOOTH SURFACE. 231
so that if a weight = w 7 is suspended to the string at the lowest
point where it touches the cylinder,
2w
T =
TT
The pressure on the surface may thus be found. It is due (1) to
the weight of the element of the string which corresponds to it,
and this = apa>g sin 6 dd ; (2) to the tension ; let the tension at
ds = T, and let ds subtend an angle = dQ at the centre of the
circle ; the action-lines of T at both ends of ds coincide with the
tangents at these points, and E acts along the line which joins
the centre of the circle to the point of intersection of these two
tangents; consequently
uad6 = 2Tsm = idO-, .. R = -J
2 a
T
and we have R = - -f pw^sinfl; (85)
which result is the same as (82). Hence
/**!
the whole pressure = /
J
(0 1 ). (86)
Hence if the string reaches from the highest point to the hori-
zontal line the whole pressure = 2 a pug; that is, the whole
pressure is equal to twice the tension at the highest point.
The preceding investigation shews that the part of the pressure
due to the tension varies inversely as the radius of the cylinder ;
and as the investigation involves only the infinitesimal angles at
which two consecutive normals are inclined to each other, the
result is true for any cylinder of continuous curvature ; so that,
if p is the radius of curvature,
m-f; (87)
this being that part of the normal pressure which is due to the
tension of the string.
Hence also for a given pressure the tension varies inversely
as the curvature of the cylinder.
Ex. 2. If a string, whose mass is so small that it may be
neglected in comparison of the tension which acts on it, rests
on a smooth surface, what are the circumstances of pressure and
tension ?
THE CATENAE Y [l68.
In this case, all the terms involving o>p are to be omitted ; so
that from (73) C?T= ; and T is constant throughout the length
of the strin.
Also from (75), R = (88)
If the string lies in a plane curve, cos = 1 ; and we have, as
also from (82). T
m-4- ( 89 )
p
Let d\l/ be the angle of contingence at the point (x, y) ; so that
the whole pressure = / R ds
JQ
= T(^-V,). (90)
Thus the whole pressure along the curve between the given
limits varies as the angle between the normals at the ends of the
curve.
Thus, if over a smooth horizontal cylinder a fine string is
suspended, which has at its ends weights, each of which = w,
and these hang vertically downwards,
the whole pressure = TTW.
168.] Suppose however the surface on which the string rests
to be rough, and the string to be on the point of motion along
its length, so that friction arises from the roughness ; then this
friction is a force which acts along the string in the direction
contrary to that of the motion : and if E ds is the pressure on
the surface of a length-element of the string, and ?ds is the
friction corresponding to ds, and p is the coefficient of friction,
see Art. 118, *<fo = /*<&;
and as F acts in the direction of the string along which motion
is about to take place, the components of F ds are
F dx t F dy, F dz ;
or p.ndx, p^dy,
so that the equations of pressure are
. dx
-- (91)
Q
. dz z ,
a.T-j- + pa>z</* + /ARdl2 + R-<& = 0; J
US Q
and from these equations general properties may be deduced.
169.] ON A ROUGH SURFACE. 233
As the investigation, however, presents no difficulties, and is
similar to those of the preceding Articles, we need not occupy
our space with it ; and I will take a particular form which gives
some practical results of considerable interest.
Over the surface of a rough circular cylinder, whose axis is
horizontal, a fine inextensible string, whose mass may be neg-
lected, is placed in a vertical plane, and given forces act at the
ends of the string. What are the circumstances of pressure
and tension ?
Let fig. 64 represent the string resting on the cylinder, of
which the plane of the paper is a section perpendicular to the
axis of the cylinder: let the string be in contact with the
cylinder over an arc which subtends at the centre the angle
ACB = a ; and let the forces at the ends of the string be T, and
T' ; and these are also the tensions at A and B. Let AC = a,
ACP = 0, PCQ = dd; then resolving normally and tangentially,
we have T = an . dT = F< & _ ^adO: (92)
T = T e^, (93)
as T is the tension when = 0', hence as increases in arith-
metical progression, T increases in geometrical progression. The
value of T is the greatest just as the rope begins to slip ; let T,
be the value of T at B just as the slipping begins ; then
T, = T O ^; (94)
so that if the force at B is less than the value of T, thus de-
termined, the rope will not move. Thus, if a rope were wound
twice round the cyclinder,
T! = T! e tir >>;
and if p = 4, which is an usual value of /z, we have approxi-
mately T! = 165 T O , which shews how great is the force which
one man may exert by merely coiling a rope round a post.
T T
From the first of (92) we have R = - = e* 9 ; consequently
the normal pressure on the cylinder = / T O e* 6 d6
JQ
= Ii(eM_i). (95)
169.] Ex. 1. A string passes over three rough cylindrical
horizontal bars which are at equal distances apart, and the
lower two of which are in the same horizontal plane; and at
PRICE, VOL. in. H h
234 ON ELASTICITY. [l?O.
the ends of the string weights are suspended : find the differ-
ence between them just as motion begins to take place.
As the cord is in contact with the surfaces through an angle
_ 2 IT
- at each of the lower bars, and through an angle at the
upper bar, TJ = T e**. (96)
Ex. 2. A string passes over a rough horizontal cylinder ; and
two weights p and Q are suspended at its ends so that p is just
beginning to descend : what weight must be added to Q, so
that Q may be beginning to descend ?
Let Q' be the additional weight required ; then we have
P = Qg^,
. ,_
Ex. 3. A heavy uniform chain is hung over a rough hori-
zontal cylinder ; how much lower will one end of the chain be
than the other, just when the chain begins to move ?
Let c be the length of chain which hangs down on one side,
and c + x the length of that which hangs on the other, just when
the chain begins to move, so that the pressures at the ends of
the horizontal diameters are c&pg and (c + x)wpg respectively :
then, taking account of the weight of the chain, and resolving
tangentially and normally, we have
dT = pvgdy + iiiads ', (97)
R = - + pco#sin0; (98)
.*. ch nfdQ =. pa>gadd(cosd + nsm0)', (99)
and integrating, and introducing the values at the given limits,
we have o a/z
W ' Mr -l). (100)
If c = 0, no string hangs on one side of the cylinder ; and x
then determines the force which must be applied at the other
end to make the string move round the cylinder.
SECTION 2. The equilibrium of elastic strings.
170.] Our knowledge of the internal constitution of bodies is
doubtless very imperfect ; but so far as it goes, there is no ma-
terial substance in nature, the relative positions of the particles
170.] ON ELASTICITY. 235
of which are not changed when the matter is acted on by ex-
ternal pressures : if a force acts on a body at a certain point,
and in the way of pressure against it, the particles of the body
at, or about the point of application, approach to each other ;
and if the force is a pulling force, the distances between the
constituent molecules of the body, at and about the point of
application, are increased. It seems indeed that a body is made
up of a system of molecules, infinitesimal in volume, and at an
infinitesimal distance apart, and that these are held in a state
of relative rest by forces acting reciprocally from one to another;
and that these forces are functions of the distances between the
molecules ; and that when an external force acts on the system,
the molecules are either separated farther from, or are brought
nearer to, each other, by reason of the action of the force ;
so that either a compression or a dilatation of the system takes
place ; all bodies, that is, are compressible and extensible to a
certain degree : the relative position of the molecules is not the
same when the body is free from, and when it is subject to,
external pressures. Into the particular mode of action of such
forces on the constitution of a body, or the change of molecular
action of the internal forces under the influence of such external
force, I shall enter only briefly, and generally, and reserve the
special study of the subject to a subsequent portion of this
course, where I hope fully to enter into it; and also now we
have not data sufficient for the full solution of the problem.
But I would observe, that our previous results of forces acting
on rigid bodies, that is, on bodies the constituent molecules
of which are in a state of relative rest, are not hereby falsified,
because the molecules of the body though disturbed at first
are ultimately in relative rest. It is the amount of this dis-
turbance which we shall generally calculate : and upon the
hypothesis of the truth of certain laws, which are for the most
part empirical, and will not be deduced from more remote prin-
ciples of the structural constitution of bodies.
The disturbances or displacements which the molecules un-
dergo are of three kinds : there may be (1) a longitudinal com-
pression or dilatation ; I shall calculate the effects of this on a
bar or a string : (2) a flexure or a bending, as of a thin flexible
membrane, or plate or spring ; this I shall also consider : (3) a
twisting or a torsion, as of a twisted bar. Now in all these,
as in all similar displacements, one result is the same; no
H h 2
236 THE ELASTIC STRING.
disturbance or disarrangement, at least within certain limits,
takes place, unless there is also called into action a force of
restitution, whereby the body tends to recover its former state ;
the molecular forces are such that, so long as temperature, &c.,
remain the same, they tend to bring the body back again into
that state which it had before the disturbance due to the external
force : this energy of restitution is called Elasticity ; " La force
elastique," says D'Alembert, "est une propriete ou puissance
des corps, au moyen de laquelle ils se retablissent dans la figure
et Tetendue, qu'une cause exterieure leur avait fait perdre."
Thus elasticity in the first of the three cases mentioned above,
is the tendency which a stretched string has to return to its
former and unstretched length : in the second case it is the force
of a spring, as that of a coil which is the motive power of a
watch : in the third case it is the force of return which a twisted
wire exhibits, as in Coulomb's Torsion Balance, or in Cavendish's
experiment with leaden balls. Let this term then be plainly
distinguished from expansibility, extensibility, compressibility,
and so on : it is consequent upon these last, but expresses a pro-
perty quite distinct from them ; and the greater or less perfect-
ness of elasticity of a given substance depends on the degree with
which it recovers the state, as to the arrangement of its mole-
cules, whence it has been displaced : if the state is altogether
recovered, elasticity is perfect : if the body remains in the state
into which it has been put by the disturbing force, it is said to
be wholly inelastic : neither of these conditions is ever fully
satisfied in nature. Thus much as to elasticity is sufficient for
our present purpose.
171.] I will in the first place take the most simple case of an
extensible string, which is stretched by the action of certain
forces in the direction of its length.
The law to which the extension is subject, and which is com-
monly called Hooke's law, is, The extension is as the tension :
that is, the length added to an extensible string by means of a
stretching force varies as the force. Also the same law may
be supposed to be applicable to compression, that is, the com-
pression varies as the compressing force. Suppose the length
of an extensible string of an unit-length, and the area of whose
transverse section is an unit-area, to be by the action of an unit-
force increased by a length e, so that 1 becomes 1 + e ; then, by
reason of the preceding law, under the action of a force T, the
172.] THE ELASTIC STK1NG. 237
length is increased by ef, so that 1 becomes 1 +ex; and there-
fore, the circumstances as to thickness, density, &c., of the string
being the same throughout, the length of a string of length a
becomes a (1 + ex); e is called the coefficient of elasticity. If the
stretching force is not the same throughout the length of the
string, this formula is inapplicable as it stands; but we may
resolve the string into infinitesimal parts, and apply the law to
each of these.
It is sometimes convenient to express e in another form. Let
a' be the length of a when stretched by the constant force T
throughout ; so that
' = (!+ ex); (101)
and let E be the value of T, when a is stretched so that its
length is doubled :
then 2 = a(l+*E); .-. e=-; (102)
and (101) becomes a = a(l + -): (103)
E is called the modulus of elasticity.
172.] Ex. 1. A heavy extensible string of constant thickness
and density is suspended by one end, and hangs vertically ; it is
required to find the length of it thus stretched.
Let o, fig. 68, be the end by which it is suspended : a = the
length of it when unstretched : OA = a'= the length when
stretched : p=the density : <a=the area of a transverse section :
g = earth's attraction on an unit-mass : OP = #', pQ=d!/: and
suppose x to be the distance of P from o, when the string is not
stretched: .then the weight of PA = pgu>(ax}'. and this is the
stretching force on PQ: therefore
dx' = dx (1 -epgtw (a x}} ;
\x'\ = / {1 4 epff<a(a x}}dx;
J
o
,
a = a +
If w is the weight of the chain, w = pvga, and if E is the
modulus of elasticity,
238 THE ELASTIC STRING.
If the whole weight of the string had been collected at the
lowest point, then
a'=. a {I +evf} ;
and therefore by its own weight the string is stretched only half
as much as it would be, if that weight were collected at its lowest
point.
If p or a) varies, the corresponding alteration must be made in
the preceding integral.
Ex. 2. A heavy extensible string of constant thickness and
density is suspended by one end, and hangs vertically; at a
given point in it a weight is fixed : it is required to find the
length of the string thus stretched.
Let o be the end by which it is suspended : let A be the point
at which the weight, say w, is placed, OA = a, AB = b } a and b
referring to the string unstretched : then, using the same symbols
as in the preceding example, we have
fa p
a' + b'= I {l+pg<oe(a + d x)-\-evr}dx+ / (1 +pg&e(b x}}dx
*>o JQ
= a 4- b +
Ex. 3. Two weights P and q resting on two inclined planes,
fig. 69, are connected by an elastic string PQ; it is required to
find the position of equilibrium.
Let CP = x, CQ = y ; let the inclinations to the horizon of CA,
CB, PQ be a, 8 } 9 ; let the tension of PQ = T, and the unstretched
length = a: ^ P q = (i _j. eT }.
Then resolving along the planes, and eliminating T, we have
O sin 8 cos a P sin a cos 8
-- r-. -- :
(p -t- Q) sm a sin 8
tan 6 =
Ex. 4. A heavy string whose density varies as the distance
from one end is suspended by that end and stretched by its own
weight : find the extension.
Employing the same notation as before, and replacing p by kx,
see fig. 68,
/"a Jkobf/
the weight of AP = / kttgxdx = -(a 3 x*);
J x 2
and this is the stretching weight of dx ;
T72.] THE ELASTIC STRING. 239
a =
3
If w is the weight of the string,
/a
w = / k<agxdx
/Q
kaga* ^
( 3
Ex. 5. A heavy elastic ring is placed round a smooth vertical
cone, and descends by its own weight ; it is required to find the
position of equilibrium.
Consider the cone to be the limiting form of a regular pyramid
of n sides, of which two adjacent ones are the triangles APQ, AQR
in fig. 70 : and let PQ and QR be two adjoining elements of the
string which rest on these sides : let the triangles APQ and AQR
be bisected by the lines AJJ and A# drawn to the middle points of
their bases ; and so that the string contained between p and q
is the nth part of the whole ring. Let w = the weight of the
ring, a = the radius of it unstretched ; r the radius of it
stretched ; 2 a = the vertical angle of the cone ; then the weight
wr -ry
of no a = , and this resolved along AQ = cos a: now the
n n
other forces acting on JOQ^ are the two tensions along JOQ and
Q^, and these are equal to each other and to T (say) \ let AQP =
AQR = /3 ; and resolving along AQ, we have
cos a
w
cos a = 2 T cos /3 ;
n
now 2irr = 2ita{l+eT};
cos (3) '
J9Q 1TOQ .
but cos = - - = - - sin a ;
AQ ttOQ
f
( , ew
a 1 I + -
( 2n
and this determines the position of the ring.
240 THE ELASTIC CATENARY. [^73-
173.] The last example of the preceding Article differs from
the others, in that the string, by reason of its increased length,
also undergoes a change of curvature ; and this change of form
is doubtless to a certain extent resisted, or favoured, as the case
may be, by the elastic forces of the matter of the string : that
is, by those forces of elasticity which affect the curvature of the
string. And of these forces no account has been taken; the
ring is supposed to be perfectly flexible, and yet extensible.
We may however consider, in a more general form, the curve
which is taken by a string, perfectly flexible, and extensible
according to Hooke's Law, under the action of given forces.
Let ds' be a length-element of the curve before it is stretched,
and ds the corresponding length -element in its stretched state :
then, if T is the tension,
ds .= d/(l 4 <?T) ; ,( 1Q 4)
also let the thickness and density of the curve when stretched
be the same throughout the length ; this supposition is of course
in applications generally only approximately true ; and let x, Y, z
be the impressed forces acting on an unit-mass of the string
before it is stretched ; then the equations (12), Art. 152, become
dx
.T-j- = 0,
p(ayds-\-(l+eT)d.T-jr = 0,
(tS
dz
.T!-- = j
(105)
from which the general properties of the curve are to be deduced,
and the properties of any particular curve when the impressed
forces are given.
Let & be equicrescent ; then, expanding the last terms of each
of the equations (105), and multiplying the equations severally
by dx, dy, dz, and adding, we have
p<i> {x.dx + vdy + zdz} + (l +eT)dT ="D. ( 10 6)
And from (104), edi = d.-^\
1 d-s , ds
- j-,d-fo, = 0;
d-(^) = 0; (107)
1 74.] THE ELASTIC CATENAKY. 241
ds
whence by integration -^, and thence the extension of the
string, may be determined.
174.] Suppose however the string to be heavy, and gravity
to be the sole acting force : the string-curve will manifestly be
wholly in one vertical plane. Let the plane be that of (x, y) } and
let the axes of x and y be respectively horizontal and vertical :
and let the curve be above the axis of x : then
x = z = 0, Y = g ;
therefore from the first of (105),
fa^ Q. (108)
and therefore the horizontal tension* is constant throughout the
curve : let it be equal to the weight of a string of length <?, the
thickness and density of which are the same as those of the
string-curve : then integrating (108) we have
T-j-=pa)Cff. (109)
as
Again, from the second of (105),
-H^- (no)
fi&J /7"7*
To integrate this, let -j- = tan r : therefore
dx
ds = c(l+epa>cgsecT)d.ta,nT; (HI)
.. dx = dscosr = c (COST + epa)Cff)d. tan r, )
dy = dssmr = c (sin r + ep<0 extant) ?.tanr; j
and integrating,
x = c \ log tan ( h -) + <?po)C^tanr > , (H3)
y = c \ sec r + ~^~- (tan r) a , (114)
the limits of integration being such that x = 0, y = c, when
r = ; so that the axis of x is the directrix of the curve, and at
a distance c below the lowest point of the curve ; and the axis
of y passes through the lowest point. Also from (111) and
(109) we have,
PRICE, VOL. III. I i
242 ELASTIC PLATES. [175.
s = c j tan T + - (tan r seer + log (tan r + seer)) > , (1 1 5)
T = p&cgsecT. (116)
If T is eliminated by means of (113) and (114), the resulting
equation is that of the string-curve : the expression however is
so complicated that it is not worth while to write it at length.
But in the case wherein <? is small, and the second and higher
powers of it may be neglected without appreciable error, from
(113) we have,
and therefore from (114),
y =
which is the equation to the catenary of slight extensibility.
Also to determine the increase of the length of the arc, in this
case we have from (104),
ds' - - = ds(l ex);
1+6T
therefore from (116), and neglecting terms in (114) involving e,
ds'^=. ds epatcffsecrds
= dsepa>gyds,
dsds'= epagyds,
.. ss'=epa>ffyds. (118)
Now if y is the distance from the directrix of the centre of
gravity of the arc s,
ys = jy ds ;
.-. s /= epvgsy-,
that is, the increase of the arc * due to the tension varies as the
distance from the directrix of the centre of gravity of the arc.
SECTION 3. The equilibrium of the elastic plates or springs.
175.] In this section I propose to take only a few cases of a
simple character, and to select those which not only exemplify
1 75-] ELASTIC PLATES. 243
the general mode of investigating problems of elastic plates, but
are also useful as establishing the principles on which the strength
of materials is estimated by civil engineers. And as the first
example I will consider the effects of forces applied to the
bending of a flexible and elastic thin plate whose bounding
outline in its plane and original form is a rectangle.
Imagine a rectangular plate of an uniform elastic action and
of constant density, of a finite length and breadth, a, and b;
and of infinitesimal thickness 2r, which however is such as to
develop forces of elasticity when the lamina is bent in the
direction of its length by the action of certain external forces.
Also imagine the plate to be resolved into a series of rods, all
of which are parallel to the length a of the plate, and are of in-
finitesimal depth dz; so that of each of these the thickness is
2 T and the length is a. When the flexure takes place each of
these rods may undergo three different kinds of change : (1) the
length may be contracted or increased; (2) the absolute curva-
ture may be altered ; (3) one element of a rod may be twisted
upon the consecutive element of the same rod : the first two
effects I shall consider : the latter will not enter into the inves-
tigation, as the material is supposed to be of a non-crystalline
texture, and, as such, to be incapable of developing forces which
would cause the twisting.
Suppose the rectangular plate, fig. 73, to be that whose length
is a and breadth is b ; and suppose it to be perpendicular to the
plane of the paper, and in its original unbent form to pierce the
plane of the paper along the axis of x : also suppose it to be
fixed throughout its breadth at the extremity passing through o,
so that when the plate is bent, that end of it which is inter-
sected by the plane of the paper at o may be unchanged as to
position; and suppose the end of the plate at A to be stiff
throughout its breadth, so that the plate may be bent by a
single force applied at that extremity; and thus that its sur-
faces, which were originally plane and parallel and at a distance
2 T apart, may be the two surfaces of a cylinder : and thus all the
rods, into which we have imagined the plate to be divided, will
be rods, equal and similarly bent, of the form delineated in the
figure ; and where x and Y are the pressures parallel to the axes,
applied at the extremity A and causing the flexure of the plate.
Let us consider the bent rod of infinitesimal depth dz, and
whose under-surface in the figure is OAB ; and let us assume that
i i 2
244 ELASTIC PLATES. [*75-
the molecules, which in the bent state are along the normal com-
mon to both surfaces, were originally in a line normal to the
two plane faces; so that np'Vis the common normal to the two
curves OP' and BP"; let another consecutive normal be drawn to
these curves, and let it meet the former normal in n, so that n
is the centre of curvature. Again, let the rod be resolved into
other smaller rods or fibres, the depth of each of which is the
same as that of the rod, and the sum of the several breadths of
which is 2 r : then each of those contained within the space
P'Q'Q"P" i g of course parallel to P'Q' and to P"Q"; and of these
let us consider p Q. n is the common centre of curvature of all :
let p be the radius of curvature of that one which is equally
distant from P' and if", and which I shall call the mean fibre ;
and let a be the length of this mean fibre contained between
P'P" and Q'Q", and let the angle at n = dty ; so that
o- = pd\fr. (119)
Let or' be the length of p Q, and let p Q be at a distance u from
the mean fibre, . *' = (p + u) d+ ; (120)
u being positive or negative according as PQ is nearer to or
farther from the convex, side of the plate than the mean fibre,
and the limits of u being T and T.
Now in the process of bending, the fibres on the side towards
the convexity of the bent plate will undergo dilatation, and
those towards the concavity will undergo contraction. For
assuming the coefficients of dilatation (or contraction, as the
case may be,) to be different for different fibres, if * is the ori-
ginal length of the fibre contained between P'P" and Q'Q", we
have by Hooke's law,
o- = *(!+<?), </= #(l+e'), (121)
and which correspond to dilatation or contraction according as
e is positive or negative. Therefore from the last three equations
1+ef
whence, as e and e' are infinitesimal,
S=e+-. (122)
P
Whence it appears that if the length of the mean fibre is not
changed, that is, if e= 0, then e' and u have the same sign ; and
therefore the fibres undergo dilatation or contraction according
as they are on the side towards P'Q' or P"Q" ; and in either case
176.] ELASTIC PLATES. 245
the change of length is proportional to the distance from the
mean fibre.
176.] And imagining the bent lamina to be in a rigid state
under the action of the several forces, let us investigate the
elastic forces which act on the part AQ'Q" by means of the sec-
tion Q"QQ'. Now as any fibre PQ has undergone expansion or
contraction, so does it tend to contract or expand ; let us sup-
pose that this elastic force, corresponding to an unit of surface,
varies as the extent of displacement ; that is, as the coefficient
of elasticity; so that the force acting on an unit of surface
= ke'; and let us suppose the thickness of the plate to be,
with the exception of a variation infinitesimal in comparison
with the thickness, the same as before the flexure, so that its
thickness is 2r; and its depth is dz, then if T = the whole
force, and this acts in a line normal to Q"QQ',
(123)
and if we replace e' by its value from (122), and integrate,
T = 2kerdz. (124)
Also let L be the moment of these elastic forces about an axis
perpendicular to the plane of the paper and passing through
the mean fibre; then
[*T
L = / ke'dzudu
= kdzi (e-\ \udu
= * kr * dz . (125)
3p
Hence it appears (1) that T varies as the contraction or expan-
sion of the mean fibre, and is independent of its curvature;
(2) that L is independent of the extension, and varies directly
as the curvature of the mean fibre ; (3) that T varies directly as
the thickness, and L varies as the cube of the thickness.
Also when the length of the mean fibre is not changed by the
bending, e = 0, T = 0, and L remains the same.
And because similar results are true for each rod into which
the plate is divided, so for a section parallel to the side whose
length is b and through the whole breadth of the plate,
(126)
3/3
246 ELASTIC PLATES,
and if o> is the area of the section, o> = 2&r ; therefore
T = keta* L =
177.] The preceding investigations also enable us to find the
equation to the curves which the fibres take. Since the force
T acts on the element P'Q,'Q"P" at the side Q'Q,''J an equal and
opposite force acts on the side P'P", because the mass-element is
at rest, and no other force acts. And as the same result is true
for all the elements of the lamina, T is constant throughout,
and is therefore equal to the parts of x and Y which are normal
to the end of the plate at A ; e is also constant, and by virtue of
equation (124) is proportional to this force, and is positive or
negative according as the impressed forces act to dilate or
contract the mean fibre.
Let (x , y^) be the point A at which is applied the force which
causes the bending of the plate ; let x and Y be the axial com-
ponents of this force ; then these forces, together with those
applied on that section of the plate whose intersection with the
plane of the paper is Q'Q", keep in equilibrium the part of the
plate between A and Q'Q". Now if (x, y] is the point P, x and y
will also, neglecting infinitesimals, be the coordinates to the
point of intersection of the mean fibre and Q'Q"; and therefore,
taking moments about that point, we have
L + x(y -y)-T(*.-ar) = 0; (127)
and substituting for L from (126), and replacing p by its equiva-
lent expression, we have for the differential equation of the curve,
the integral of which will contain two arbitrary constants : and
on
these will be determined by the condition that -j- = 0, when
dx
x = y = 0, and by the length of the curve between the origin
and (a? , y fl ) which is given.
If the lamina is not fixed at o, a force equal and opposite to
the resultant of x and Y must be supplied at it.
178.] Let us consider the two particular cases of (128), in
which the forces act, (1) wholly perpendicular to, (2) along, the
plate in its original unbent state. In the first case x = ; so
that (128) becomes
-'*"+* (129)
1 79.] ELASTIC PLATES. 247
therefore integrating, and observing that at the inferior limit,
dy
x = 0, when -- = 0,
neither of which expressions can be integrated further.
If however the elastic force of the lamina is very great com-
pared with the deflecting force the bending is slight, and thus
j- is very small throughout, and neglecting the second and
ftx
higher powers of it we have from (130),
2cdy = (2x 9 x x^dx,
.-. 6cy = 3x a x*-x 3 ', (131)
which is the equation to the curve taken by the lamina : that is,
the lamina is bent into a cylindrical surface, the trace of which
on the plane of (x } y) is given by (131). This equation however
jr/xy
expresses the form of the lamina only as long as ->- is small.
Let y be replaced by y in (131) ; then
and therefore replacing c by its equivalent,
(132)
that is, the distance through which the end of the lamina has
been moved varies as the deflecting force, as the cube of the
length (approximately), inversely as the breadth of the plate,
and inversely as the cube of the thickness.
179.] Another form of the problem of the preceding Article
is that of a heavy thin flexible rectangular plate fixed to a
horizontal edge along one of its edges, which is placed in a
horizontal position, and is then bent by its own weight.
Let a be the length of the plate, b its breadth, and 2r its
thickness ; and let it be placed in a horizontal position, with the
248 ELASTIC PLATES. [l8o.
side 2r vertical : then as the deflexion is very small, we may, in
fjfj
equation (127), consider x = a, x = 0, and -^- to be very small,
so that its second and higher powers may be neglected. Taking
the moments of the section Q'Q,", in fig. 73,
_ 2&&T 3 d^
~^~ dx~*'
and as the weight of that part of the plate which lies beyond
Q'Q" is in equilibrium with this force of elasticity, we have,
dx
because -/- 0, when x : therefore
dx
and therefore the whole deflexion at the extremity is
< 13S >
that is, the deflexion of the extremity of the plate varies as the
fourth power of the length of the beam, and inversely as the
square of the depth of the beam.
180.] Let us now take the second case of Art. 180 ; viz. that
in which Y = ; and let us suppose x to act, like a crushing
pressure, towards o : then if the lamina under the action of such
a force is bent at all, its deflexion from a straight line is very
slight, and thus -j- is very small : I shall neglect therefore the
second and higher powers of it : also y = 0, since the force x
acts still along the axis of x : therefore (128) becomes
3x
and if we put = '*' ( ! 34 )
fry
we have -^- + c* y = ;
dx'
and supposing the end of the plate to press against a rough
fixed plane at o, but not to be fixed as heretofore, we have
dv*
+c'G''-a') = 0; (135)
1 8 1.] ELASTIC PLATES. 249
so that r ca } when y = 0, and where a is undetermined :
ax
and integrating again,
y = asincx f (136)
because y = 0, when x = 0. The plate therefore takes a corru-
gated form, the section of which, by a plane perpendicular to it,
and parallel to its length, is the curve of sines. And from (136)
a is the greatest amplitude of this curve, and therefore is very
small in comparison of the length of the curve, because the de-
flection of the plate from a plane is supposed to be very small.
If a = 0, the plate will continue plane, and its length will,
by reason of Art. 173, be slightly diminished, and become
a(\ ^ T-) If a is not equal to zero, the plate, which is like
a rectangular piece of watch-spring, takes the corrugated form,
and the number of undulations on the cylindrical surface will
depend on c; let h be the distance OA, then since y = 0, when
x = h, c and h must be related by the equation, ch = iti f where
i is any whole number ;
.. y = a sin -j- x ; (I 37 )
Mfl
and therefore also y 0, when x = , where n is any number
P
from up to a, so that the curve cuts the axis of x in i+1
points, and therefore the surface has i elevations or depressions.
Also, if I is the length of the curve,
,
-j-ff) ( dx-, (138)
and omitting the fourth and higher powers of a, we have
7r 2 i*a 2
JT* + HJ-; (139)
(140)
whereby a is given in terms of h and I.
181-3 The greatest value of the compressing force x which
can be applied at the end of a spring, and not bend it, is called
the vertical strength of the spring ; in this case I = h, i = 1 :
therefore from (134),
_
3 3
PRICE, VOL. III. K k
250 ELASTIC PLATES. [182.
and using for I its approximate value a,
so that, other incidents being the same, the vertical strength of
the spring varies inversely as the square of the length.
Hereby also are we enabled to calculate the greatest weight
that a vertical pillar of a given form and height can bear with-
out being bent by the weight.
Suppose the pillar to be of a height h and its transverse sec-
tion to be rectangular, the sides of the rectangle being a and b ;
then the greatest weight which it will bear, without being bent
perpendicularly to the side b, is
kit*ab 3
and, without being bent perpendicularly to the side a, the
greatest weight is kTi*a*b
and if the transverse section is a square, a = b, and the strength
of the beam perpendicularly to either of the sides is
^, (142)
12/fc'
and varies therefore as the fourth power of the side.
Suppose a transverse section to be square and to be hollow,
so that the side of the external square is a and of the internal
square b ; then
the vertical strength of the beam = -- j- '
1 t ft
182.] We may also approximately investigate the vertical
strength of beams, the transverse sections of which are of forms
other than rectangles ; and let us assume, as the most probable
hypothesis, that the mean fibre is that which passes through the
centres of gravity of all similar transverse sections ; then L must
be calculated in each case, as in Art. 176, so that we may sub-
stitute in equation (127) ; let and 77 be the coordinates to any
element of the area of the transverse section ; and let us con-
sider the following examples :
Ex. 1 . The section of the beam is a circle, of which the radius
is a.
Suppose the mean fibre of the cylindrical beam originally to
be coincident with the axis of x ; and ultimately, if bent, to be
1 82.] ELASTIC PLATES. 251
in the plane of (x, y), so that the bending takes place about an
axis perpendicular to the plane of (x, y] : let be taken in, and T\
perpendicular to, the plane of (x, y] ; then
L =
so that (127) becomes
a comparison of this result with (142) shews, that if the areas
of the transverse sections are equal in the two cases, the vertical
strengths of the square and the circular beams are as TT : 3 ;
there is therefore a small advantage in favour of the square
beam.
Ex. 2. Let the beam be circular and hollow : let a be the
radius of the external, b the radius of the internal surface : then
by the last result, if x is the vertical strength,
Ex. 3. Let the transverse section of the beam be an isosceles
triangle, of which the base is a and the altitude c : then if the
altitude lies in the plane of (x, y} and the base of the triangle
becomes convex,
kac* k-n'ac*
L = ; . . x = ;
OJ~ * w QC/i2
OOp OQ/l
Such are the principles on which is founded the mathematical
theory of the strength of materials : for a more complete in-
vestigation I must refer the reader to treatises wherein the
subject is specially discussed ; because the constants, which are
left undetermined in the preceding expressions, are to be found
by experiment ; and particular and very delicate apparatus, the
construction of which requires minute explanation, is needed for
their determination.
K k a
CHAPTER VI.
ON ATTRACTIONS.
SECTION 1 . The direct investigation of the attraction of bodies.
183.] In the following chapter, amongst many properties of
matter which will be formally stated as axiomatic principles of
the science of motion, will occur one which is called the law of
inertia, and which declares that matter has no power to change
the state in which itself is ; and experiment amply verifies it in
the phaenomena of nature : it is not however hence to be in-
ferred that matter has no power of acting on, or of influencing,
other matter : on the contrary, matter does act on other matter
in the way of either attraction or repulsion, and according to
certain laws : and this action is not impeded by the presence
or the intervention of other matter ; every particle of matter
attracts or repels every other particle in the same way as if the
two existed alone. Nature presents to us many phaenomena in
evidence of this active power of matter. There is, in the first
place, that universal law of gravitation, by reason of which every
material particle of the celestial system exercises on every other
particle a force which varies as the product of the masses of the
particles, and inversely as the square of the distance between
them ; and which acts along the line joining the two particles,
and tends to draw them nearer together. So again in the ex-
planation of magnetic and electrical phaenomena, there are
doubtless two states in which particles active with the influence
may be : and the attraction or repulsion which mutually acts
between them varies as the product of the intensities of the two
particles, and inversely as the square of the distance between
them ; and the force is attractive or repulsive according as the
particles are in opposite or in the same magnetic states ; and
the line of action is that which joins the two particles. There
are also other phaenomena where the attraction varies inversely
184.] ATTRACTIONS. 253
as the square of the distance, but where the line of action is
not that which joins the two particles. These and similar cases
require investigation, and for this reason : when two single
material particles attract or repel each other, it is easy to con-
ceive the force which mutually acts from one to the other ; we
can easily imagine the tendency of the one to move towards or
from the other in the straight line which joins the two. But
when one material particle is attracted simultaneously by many
others, aggregated into a finite body of a given form and den-
sity, the determination of the intensity and of the line of action
of the resultant force requires investigation; and perhaps also
the density of the attracting body may vary, in which case the
difficulty is increased. The following inquiry will be for the
most part confined to the cases where the law of attraction is
that of the product of the two attracting particles, and of the
inverse square of the distance between them, because this is the
law of gravitation, and generally rules in cosmical phaenomena :
but it will also embrace other laws ; so that by operating
with general laws we may determine the results which they
necessitate, and by a comparison of these with the works of
Nature, may obtain a knowledge of the special laws which rule
therein.
184.] As to the attraction varying as the product of the
masses of the attracting and the attracted particles : let there
be two particles m and m' at a distance r apart ; and let the law
of attraction, which is a function of the distance between them,
\>ef(r) ; so that the attraction of an unit-particle in the position
of m on an unit-particle in the position of m is f(r) : now m'
contains m' unit-particles; and each one of these attracts the
unit-particle in the position of m with a force f(r) ; therefore
the whole force of m' on the unit-particle in the position of m is
m>'f(r) : but m also contains m unit-particles, and each of these is
attracted with equal force by m' ; therefore the whole attractive
(or repulsive) force of m on m is
mm'f(r}. (1)
If the attraction varies inversely as the square of the distance,
the attractive force = ; - ; (2)
254 ATTRACTIONS. [185.
and in all cases which we shall investigate, the line of action of
the force lies along r.
Suppose now m to be the mass of an attracted particle, and
dv to be a volume-element of the attracting body, and p to be
the density of dv, and r the distance between m and dv, then
the attraction of dv on m is
mpdvf(r); (3)
and the attraction of the whole body on m will be found by
means of the Integral Calculus.
In this section I propose to investigate the attraction of bodies
on particles, and in some cases on other bodies, directly by in-
tegration. An indirect method for the investigation of these
attractions will be given in a following section.
Whenever the law of attraction is not expressly stated, it is
assumed to be that of gravitation.
185.] The attraction of a straight rod or wire of uniform
thickness and density on an external particle; fig. 74.
Let o be the attracted particle whose mass is m, and let AB
be the attracting bar : of which let the density be p, and let the
area of a transverse section be o> ; from o draw oc perpendicular
to AB; let CA=0, CB=# : oc = c. Let PQ be a volume-element
of the bar, CP = y, PQ = dy : therefore the mass-element at
p = po> dy ; and let the attractions be calculated along, and per-
pendicular to, oc ; let the attraction of the bar on o along oc
and towards c = x, and let the attraction at right angles to oc
and towards A = Y. Then
the attraction of P on o along o p = - ^- :
c'+y*
therefore the attraction of P on o in the direction o c
c'+y*
motpcdy
"(+*')*"
The attraction of P on o at right angles to oc
mvpdy .
= - S111POC,
1 86.] STRAIGHT BARS. 255
/" mupcdy
y
C
( a # )
((<?" + a')* (<?+)*>
I CA CB)
\ OA OB )
{sinAOC-j-smBoc} ; (4)
Y =
C
mutpydy
S -- [- (5)
(OB OA)
If a = #, so that c is the middle point of the bar, Y = 0, and
x= - sin AOC ; that is, the attraction of the bar acts only
C
in a direction at right angles to its length.
186.] By the following geometrical construction we obtain a
remarkable equivalent for the attraction of a rod on a particle
outside of it, as in the last Article.
From centre o, fig. 75, and radius oc, describe an arc of a
circle meeting OA, OB, OP, OQ in the points a, 6, p } q- 3 and sup-
pose a bar of the same material, density, and thickness as A B to
be bent into an arc of a circle, and to coincide with the arc ab ;
then the attraction of this bent bar on o is the same as that of
the straight bar AB.
From o as a centre, and with the radius OP, describe a small
arc PR ; then
PO PRSeCOPR OP
- = - = sec POC,
pq pq op
_ opa .
= op*'
OP 2 OJ5 2
Now the attraction of PO on o in the direction OP = - - >
OP"
256 ATTRACTIONS.
and therefore is equal to ----', that is, is the same as that of
the element pq of the circular arc : and as a similar result is
true for all the elements of the circular arc, so the total attrac-
tion of the bar AB on o is the same as that of the circular bar ab.
If the angle AOB is bisected by the line OD, the line of action of
the whole attraction of the bar ab manifestly is OD : OD is there-
fore also the line of action of the whole attraction of the bar
AB on o.
Hence it follows that if o is capable of moving towards AB,
each element of its path will bisect the angle AOB, and the path
will be a hyperbola of which A and B are the two foci ; and the
particle will ultimately meet the bar at a point, the difference
between whose distances from A and B is equal to OA OB.
Thus if o is a particle of iron filings and AB is a magnetized
bar, the path which o will take in moving towards AB is a
hyperbola.
Hence also if from A and B as foci, an ellipse is described
passing through o, OD will bisect the focal distances, and is
evidently a normal to the ellipse at o ; thus the action-line of
the force on the particle at o will be perpendicular to the ellipse,
and the particle will rest in equilibrium on the ellipse. We shall
speak on this subject more at length in the following section.
Hence also if three bars of the same thickness and density,
and attracting with a force varying inversely as the square of
the distance, are arranged as a triangle, a particle placed in the
centre of the circle inscribed in the triangle is equally attracted
in all directions.
The preceding process of integration is also applicable when
the density of the attracting bar is variable.
187.] Also let the following results be proved :
(1) The attraction of a bar of uniform thickness and density,
when the attraction varies directly as the distance, on a particle
in contact with it at distances a and b respectively from the
ends of the bar is
and therefore if the attracted particle is placed at the end of
a bar whose length is a, so that 3 = 0,
the attraction = - >
M
1 88.] CURVED BARS. 257
and is the same as if the whole bar were condensed into a
particle at its centre of gravity in the middle point of a.
(2) The attraction of a bar of uniform thickness and density
on a particle in the same straight line with it, and at distances
a and b severally from the ends of the bar, is
a b
mpta j
ab
(3) The attraction of two straight bars, each of which is of
uniform thickness and density, on each other, in the same straight
line, of the lengths a and b, and at a distance c apart, is
,
OpOOCii log:
3 c(a
and this is of course the force which is required to keep the
bars asunder.
Since the result involves the anharmonic ratio of the four
points which are the ends of the bars, it follows that if AB
and CD are the bars, and if through any point v lines VA, VB,
vc, VD are drawn of any length, and any line A'B'C'D' is drawn
cutting them, the mutual attraction of A'B' and C'D' is the same
as that of AB and CD.
(4) Two straight bars of lengths 2 a and 2 b and of constant
thickness and density, and each particle of which attracts with
a force varying inversely as the square of the distance, are
placed parallel to each other at a distance c apart, and so that
the line joining their middle points is perpendicular to each of
them : it is required to shew that the force necessary to keep
them apart is
188.] The attraction of a bent rod of uniform thickess and
density on a given particle.
Let us first investigate the attraction of a bar bent into the
form of a circular arc on a particle at the centre.
Let p = the density, o> = the area of a transverse section of
the bar : a the radius of the circle, 2 a = the angle subtended
at the centre by the bar; fig. 76. Now it is manifest that the
resultant attraction acts along the line oc bisecting the sub-
tended angle, for the resultant attraction which is perpendicular
to that line vanishes. Let POC = 0, AOC = BOC = a : then
PRICE, VOL. III. L 1
258 ATTRACTIONS. [l88.
the attraction of the bar in the direction oc
#2 COp / a -7x.
= I cos v ad
a J_ a
2 #z cop sin a
(6)
a
and therefore varies directly as the sine of half the subtended
angle and inversely as the radius of the arc.
Hence the whole attraction of the bar AB, in Art. 186, on o,
and along the line OD, see fig. 75, is
2/Wcop . AOB
sin
OC 2
Hereby we are enabled to solve the following problems :
Ex. 1 . Three bars, each of which is of uniform density and
thickness, form a triangle ; find the position of a particle placed
within the triangle which is equally attracted in all directions.
Let the densities of the bars be respectively p, cr, T } and let
the transverse sections of all three be the same ; let the perpen-
diculars from the attracted particle on the sides be p, q, r; and
let the sides subtend at the attracted particle angles 2 a, 2/3, 2y ;
then the particle is kept at rest by the three forces
p q r
the angles between the lines of action of which are /3 -f y, y + o,
o + /3; or 180 a, 180 /3, 180 y, because a -f/3 + y = 180;
and therefore by the triangle of forces, Art. 21, the forces are
proportional to the sines of these angles ; therefore
And if p <r = r, then p q=.r y and the attracted particle is at
the centre of the circle inscribed in the triangle.
Ex. 2. Two bars CA and CB of the same constant thickness
and density meet at right angles and attract a particle placed
at the foot of the perpendicular from c on AB ; it is required to
find the magnitude and the line of action of the resultant at-
traction.
Let CA = a, CB = b, a 3 + 6' = c* ; and let P be the position of
the attracted particle. Then the attraction of c A on p in the
line bisecting the angle APC is - ~ sin 45; and, similarly,
a o
the attraction of CB on P in the line bisecting the angle BPC is
1 90.] CURVED BARS. 259
O 2 j) ff\ f\ /^
j- sin 45; and as these two lines of action are perpen-
dicular to each other,
the resultant attraction = T- ;
a 3 o*
and the line of action of it is inclined at 45 to each of the lines
CA and CB.
From (6) it appears, that the attraction of a circular rod on a
particle at its centre is the greatest when a = 90, that is, when
the arc is a semicircle ; and if a = 1 80, that is, if the circle is
complete, the attraction vanishes.
Suppose however the ring to be complete, and the attracted
particle to be in the plane of the ring, and at a small distance
x from the centre ; then we have the following problem.
189.] To find the attraction of a circular ring on a particle in
its plane, and near to its centre.
Let p be the density, and o> the area of a transverse section
of the ring : a = the radius, fig. 77, CO = x, which is very small,
and such that we shall neglect the third and higher powers of
it; PCA = 6, QCP = dd', m = the mass of o, OM = acosd #,
OP' = a 3 2o?cos0 + # J . It is manifest that the ring attracts
o along the line COA alone ; and the attraction
= 2m'
cos POM
OP*
(acos0 x)dQ
'
o a*
2mpo>
a a?
2mpa> f w ( 3cos20-fl 9 cos0+ 15 cos 30
- - - -a?+- -
2 8a
and therefore the attraction varies directly as the distance of
the particle from the centre of the ring.
190.] To find the attraction of a circular ring of uniform
thickness and density on a particle m at a given distance from
its plane, and in the line perpendicular to the plane and passing
through the centre of the ring.
Let a be the radius, p the density, o> the area of a transverse
section of the ring, c the distance of the attracted particle from
260
ATTRACTIONS.
the plane of the ring ; see fig. 78 ; suppose the plane of the ring
to be perpendicular to that of the paper; let p be an element
of it, and let the line PC drawn from p to c, the centre of the
ring, make an angle 6 with the plane of the paper; then the
mass of the element at p = pwadd; and the attraction of p on
o along the line oc
= P ^, ^ COSPOC;
therefore the attraction of the ring = 2 TT.
(c* + a")!
Hereby we can solve the following problems :
Ex. 1. To determine the attraction of a hollow cylindrical
tube on a particle at a given point in its axis.
Let r = the radius of the interior surface of the tube, T = the
thickness, p=the density; and let the distances of the attracted
particle m from the ends of the axis of the tube be a and 6;
and let the tube be resolved into a series of rings of infinitesimal
depth by means of planes perpendicular to the axis of the tube :
then if x is the distance from m of any ring whose thickness is dx,
the attraction of the whole tube
xdx
= 2-nmpTri
r
= 2-nmpTr
Ex. 2. To prove that the attraction of a thin paraboloidal shell
or cup, limited by a plane through the focus perpendicular to
the axis of the shell, on a particle at the focus is equal to
'Sirmpr . i
_ (2* 1).
3
191.] The following are other problems on the attraction of
thin wires.
Ex. 1 . The attraction of a thin wire in the form of a parabola
on a particle in its focus = > where 4# is the latus rectum
34
of the curve.
Ex. 2. The attraction of a semicircular ring, on a particle
at the extremity of the diameter which bisects the ring
= - - log tan , where a is the radius of the ring.
a \5
1 9 2.] THIN PLATES. 261
192.] The attraction of a thin circular plate on a particle
m in the line passing- through the centre of the plate and per-
pendicular to it.
Let the attracted particle and the centre of the plate be in
the plane of the paper, fig. 79 ; and let the plane of the plate be
perpendicular to it. Let p = the density, T = the thickness of
the plate: a = the radius, and oc = c, the distance of the at-
tracted particle from the plate. Resolve the plate into concen-
tric circular rings, of which let the radius of that containing the
element p be r, and the depth be dr ; then if ACP = 0, PCQ=6?0,
the mass of the element at p = prrdrdd;
therefore the attraction of the plate on o in the direction oc
I"* /* 2jr rdOdr
= mprc I
^o J o (c' + r")*
r i -\ r =
= 2irmprc -
L ( c -f /-3)5J,.=o
= 2-nmpr\l C ,\. ' (7)
(a* + crf)
The attraction of the plate in a direction at right angles to oc
is evidently zero.
Since in (7) = cos AOC, it follows that the attraction
' (+*)*
of all circular plates of the same thickness and density on a
particle in the line passing through their centres and perpen-
dicular to their planes is the same, if their diameters subtend
the same angle at the attracted particle. Hence if a right cone
is divided into a series of circular plates, all of which are of the
same thickness, by means of planes perpendicular to the axis
of the cone, the attraction of each of these on a particle at the
vertex is the same.
In (7) if the radius of the plate is infinite, that is, if a = oo,
the attraction = 2-nmpT,
which is independent of the distance of the attracted particle
from the attracting plate : therefore the attraction of a plate of
infinite extent on a particle outside of it is the same, whatever
is the distance from the plate at which the particle is placed.
Hence for particles near to the surface of the earth, the earth's
attraction is constant ; because the earth may be conceived to
be divided into a series of thin plates by planes perpendicular
262 ATTRACTION [ I 93-
to the vertical line passing through the attracted particle, the
radius of each of which is very large in comparison of the dis-
tance of the particle.
If the law of attraction varies as the nth power of the dis-
tance, the attraction of the circular plate on a particle outside
of it in the line passing through its centre, and perpendicular
to its plane,
(8)
which is the same as (7), if n 2.
193.] To determine the attraction of a solid of revolution on
a particle in its axis.
Let the solid, fig. 80, be resolved into circular slices of infi-
nitesimal thickness by means of planes perpendicular to the
axis of revolution. Let o be the attracted particle, of which the
mass is m ; and let y =.f(x] be the equation to the generating
curve of the bounding surface of the solid.
Let OM=#, MP=y, OA=, OB = ; and let the thickness of
the circular slice PMP' be dx; then by (7),
the attraction on o of the differential circular slice
Idx;
J
(9)
therefore the attraction of the solid on m
f b ( x )
= 2 imp / -si 1C >
y having been replaced by its equivalent value in terms of x
by means of the equation to the generating curve.
Similarly may the whole attraction be found from (8), when
the attraction varies as the nth power of the distance.
Ex. 1 . To find the attraction of a homogeneous circular cylinder
of length a and radius b on a particle in its axis at a distance c
from one end.
/"+ ( x )
The attraction = 2 imp / \ 1 r \ dx
so that if the particle m is in contact with the cylinder, c = 0,
and the attraction = 2irmp {a + b (a* + *)*}
If the cylinder is of infinite length in the direction from the
attracted particle, a = oo, and the attraction
5 :i +e 9 )* c};
1 93-] OF SOLIDS OF REVOLUTION. 263
and if the particle is in contact with the end of the cylinder of
infinite length c = 0, and the attraction = 2 -nm pb, and varies
as the radius of the cylinder.
Ex. 2. To find the attraction of a homogeneous right cone on
a particle at its vertex.
Let the vertical angle = a ; so that the equation to the gene-
rating line is y _ #tana;
and let a be the altitude of the cone : then from (10),
the attraction = 2irmp I (1 cosd)dx
= 2Trmp(lcosa}a.
If the attraction varies as the wth power of the distance, the
attraction of the cone on a particle at its vertex is
(seca) n+t l}a n+s .
Ex. 3. The attraction of a circular cylinder of length a and
radius c, whose density is constant, on a particle in the centre
of its circular end, is, if the attraction varies as the nih power
of the distance,
Ex. 4. To find the attraction of a homogeneous sphere on- a
particle external to it.
Let a = the radius, p = the density of the sphere ; m = the
mass of the attracted particle ; c = the distance of the particle
from the centre of the sphere, so that the equation to the
generating circle of the sphere is
The attraction of the sphere on m
C c+a ( x )
= 2 TT mp / si [ dx
Jc-a ' (a 3 C' + 2 CX)* *
__^nmpa^
O /2 * '
= " < 12 >
if M = the mass of the sphere : but as c is the distance of the
attracted particle from the centre of the sphere, this result ex-
presses the attraction on each other of two particles m and M
264 ATTRACTIONS.- [ r 93'
at the distance c apart : consequently the attraction of a sphere
on a particle external to it is the same as if the mass of the
sphere were condensed into its centre.
This result is physically of great importance ; because in the
investigation of the circumstances of a particle moving under
the attraction of a sphere, every particle of which attracts it
with a force varying inversely as the square of the distance, the
attracting sphere may be supposed to be condensed into its
centre ; and the problem becomes reduced to that of the mutual
attraction of two particles.
Also if two spheres attract each other, the action is the same
as that of two particles whose masses are equal to those of the
spheres, and placed at the centres of the spheres : and therefore
the force which acts mutually on them is equal to the product of
their masses divided by the square of the distance between their
centres.
Ex. 5. To find the attraction of a homogeneous sphere on a
particle on its surface.
In this case, y" 1 = 2azx 3 ; therefore
/2a C x ^
\ 1 r^ I
( V 2a'
2a
= Zitmp
O
4irmpa _
~~3 '
and therefore the attraction varies directly as the radius of the
sphere.
Ex. 6. To find the attraction of a homogeneous sphere on a
particle within it.
If c is the distance of the particle from the centre of the
sphere, and a is the radius of the sphere, the attraction of the
larger segment of the sphere whose base is the plane through
the attracted particle and perpendicular to the line joining it
and the centre is
and the attraction of the lesser segment is
1 94-] SOLID OP GREATEST ATTKACT10N. 2(55
and the attraction of the whole sphere, being the excess of the
former of these over the latter, is
3
and varies therefore as the distance of the attracted particle
from the centre of the sphere. But by the last example this
would be the case if the particle were on the surface of a sphere
whose radius is c ; therefore the spherical shell, of the thickness
ac, exerts no attraction on the particle.
Ex. 7. By similar processes let it be proved that the attrac-
tion of a homogeneous oblate spheroid on a particle m at its
pole is 1-npml , _ (1 -e*}* . _ t ,
e a e 1
where b and e are respectively the semi-minor axis and the ec-
centricity of the generating ellipse. And that the attraction of
a prolate spheroid on a particle m at its pole is
where a and e are respectively the semi-major axis and the ec-
centricity of the generating ellipse.
194.] The Calculus of Variations enables us to solve the fol-
lowing problem :
To determine the form of the bounding surface of revolution
of a homogeneous mass of given volume, so that the attraction
of it on a particle in its axis may be a maximum.
Let u be the attraction, and let -nc 3 be the volume of the
given mass, which is to be contained between x l and # ; and
thus, if r* = x i +^ t ,
u = 2-nmp j | \--\dx; (13)
1TC 3 =
(16)
i r\
b.c 3 - = U'&r] -/ {lydyKx-lydxly}; (16)
o ^o
therefore, if k* is an arbitrary constant,
^ = a?* (**_;F*). (17)
PRICE, VOL. III. Mm
266 ATTRACTION OP SPHERICAL SHELLS. [l95-
Hence it appears that the curve, which by its revolution about
the axis of x generates the solid, cuts the axis of x at the origin,
and also when x = k ; thus x = 0, x k, and the integrated
part of (15) shews that it cuts it in both points at right angles.
Also substituting from (17) in (14) we have
n
= I (x%k*x*}dx
Jn
' s
= e'
4
substituting from which in (17), the equation to the curve is
completely determined. And the attraction of it on m
f k f x")
= 2 irmp I \ 1 > dx
/e > &)
Timpk
o
Now the attraction of a sphere whose mass is -npc 3 on a particle
m at its surface is
Every particle on the surface of the solid of given mass and of
greatest attraction attracts m with equal force in the direction
of the axis ; for if r is the distance of any particle on the sur-
face from m, and if 6 is the angle between r and the axis, from
the preceding equations we have,
COS0 _ 1
~^~ ~"F'
which is constant, and therefore is the same for all particles on
the surface.
195.] To find the attraction of a spherical shell of infinitesimal
thickness, and of constant density on an external particle, when
the law of attraction is represented byy (distance).
Let the centre of the shell be the origin ; and let the shell
be referred to that system of polar coordinates in space, which is
explained in Art. 165, Vol. II, and let the attracted particle be
on the axis of z at a distance c from the centre of the shell ; let
1 95.] ATTRACTION OF SPHERICAL SHELLS. 267
r the radius, dr = the thickness, p = the density of the spheri-
cal shell : so that
the mass-element of the shell = p/ a Bin6drd9d<f) :
let u = the distance of this mass-element from the attracted
particle m; then the attraction of the mass-element on the
attracted particle along the line joining m and the centre of
the shell is
drded<j>; (18)
u
and therefore the attraction of the shell on m
r*
r /
Jo JQ
f
r t dr
J
rcosfl)
J
w sin 6(cr cos 6) ,.,.j a
(19)
U
But since u' 1 = r* 2 r c cos + c 2 ;
itdu = rcsinddd,
and 2c(c rcosd) = ^ 2 + c 2 r 2 ,
and when = TT, u = c + r, when = 0, u cr; therefore
substituting in (19), the attraction of the shell on m
V + o--r VW *. (20)
Ex. 1 . Let the law of attraction be that of the inverse square of
the distance : 1
/M = 5r;
irpmrdrr c 3 r'~\ e+r
.. the attraction = - \u --
C 2 L U J c -r
^W*., (21)
C
and the mass of the shell = 4 Ttpr*dr = M (say) ;
. . the attraction of the shell = - ;
c 2
and is therefore the same as if the mass of the shell were con-
densed into its centre.
Ex. 2. Let the attraction vary directly as the distance : then
f(u) = u, and
*.- c AT. u u
the attraction of the shell =
r 3
T- 'J
= 4iipmr 2 cdr (22)
m 2
268 ATTRACTION OF SPHERICAL SHELLS.
and therefore the attraction is the same as if the mass of the
shell were condensed into its centre.
Similarly may the attraction be determined for any other
law of attraction.
] 96.] Hereby the attraction of a sphere on an external particle
can be determined by considering it as resolved into a series of
concentric spherical shells of infinitesimal thickness.
Ex. 1 . Let the sphere be homogeneous, and let a be its radius :
then if the law of attraction is that of the inverse square of the
distance, from (21),
the attraction on m = - / r*dr
* '
(
and is the same as if the sphere were condensed into its centre :
and if the particle m is on the surface, c = a, and
the attraction = 4- a, (24)
I
and varies directly as the radius of the sphere.
Now, under certain conditions, this gives a value to g, which
in Art. 123 has been defined to be the weight of a mass-unit ;
that is, g is the earth's attraction on a mass-unit placed at its
surface. Consequently if the mass of the earth is homogeneous,
and its figure is a sphere of radius a,
4
ff = -irpa,
which gives g in terms of the radius and mean density of the
earth.
Let the law of attraction be that of the direct distance : then
from (22),
fa
the attraction on m = 4^ up me r* dr
^o
= *'P**'. (25)
3
and therefore is the same as if the sphere were condensed into
its centre.
Ex. 2. Let us assume the density of a particle of the sphere
to vary as some power of the distance of the particle from the
centre : so that the sphere is composed of a series of homo-
geneous concentric shells, the density of which is different for
different shells.
196.] ATTRACTION OF SPHERICAL SHELLS. 269
Thus suppose the density to vary inversely as the distance
from the centre, and the law of attraction to be that of the
inverse square of the distance ; then by reason of (2 1 ), since
k
the attraction on m = / rdr
c a J
= < ^mk^_ (26)
And if the attraction varies directly as the distance,
the attraction on m = 2irpmcfca t .
Ex. 3. Let us suppose the density of the concentric spherical
shells to decrease in arithmetic progression from the centre of
the sphere : so that if p is the density at the centre, p t kr is
the density of the shell whose radius is r, where k is a constant.
Then, if the attracted particle is within the sphere and at a
distance c from the centre, the matter of the sphere which lies
outside of the spherical surface passing through the attracted
particle exercises no attraction on it : and of that within this
spherical surface
4 77 ttl C C
the attraction = I (p<,kr}r*dr
C Jn
and consequently varies partly as the distance and partly as the
square of the distance of the particle from the centre of the
sphere.
In each of the cases, (21) and (22), as the attraction of the shell
on m is the same as if the shell were condensed into its centre,
so will the attraction of the whole full sphere be the same as if
it were condensed into its centre.
Now in celestial mechanics this fact is of great importance :
for the planetary bodies are nearly spherical, and the density of
each of them is variable; and they are probably composed of
concentric shells, each of which is of uniform density, and the
density of which decreases as we pass from the centre to the
surface. Thus by this property we can avoid the difficulty of
investigating the attracting properties of them as solid bodies,
and we can treat them as single attracting material particles.
From the preceding results also it follows that supposing the
270 ATTRACTION OF SPHERICAL SHELLS. [l97-
earth, to be a sphere, the attraction of it on particles external to
it varies inversely as the square of their distance from the centre
of the earth. Thus if g and g' are the attractions of the earth
on the same particle respectively at the mean surface which
corresponds to the radius r, and on the top of a mountain whose
height is k, then
g_ (r+K)*
9 ' r*
197.] And the preceding results suggest another important
question : Are there any other laws of attraction, besides those
of the inverse square of the distance, and of the direct distance,
for which the attraction of a spherical shell on a particle with-
out it is the same as if the shell were condensed into its centre ?
If p = the density of the shell, and c is the distance of its
centre from m, the attraction of the shell, condensed into its
centre, is ^Tipmr^drf^}; and as this is equal to its attraction in
its actual form, we have from (20),
\ u > + c * -r*}f(u)du, (27)
whence the form of f is to be determined. Omitting common
factors, and integrating by parts the right-hand member of (27),
r c+r rc+r /
4 rc*f(e} = [(M S + c 2 -r 2 ) //()<**] 2 / {u lf(u}du} du. (28)
J cr J c-r J
Let ]f(u}du = (u), and let / u < (u} du ^f (u) ; (29)
.-. $'() =/(), - ^() = $() (30)
and therefore from (28),
4 r <?'<? =
_ d
which 'is a functional equation to be satisfied by the form of f.
Now expanding by Taylor's series, we have
a
1 97-] ATTRACTION OP SPHERICAL SHELLS. 271
and as no relation exists between r and c, the coefficients of the
several powers of r must vanish separately ; therefore
/) = *); 03)
= 4- *-&; and so on; (34)
dc c
but from (30), ^'(e) = c<f>(c); therefore from (33),
which is only an identity. Also since from (30)
), from (30);
+ e/'W
therefore from (34), if 3 A is an arbitrary constant,
C'/(c) = AC 3 + B,
where B is another arbitrary constant : therefore
; (35)
and this value off(c) also makes to vanish the coefficients of all
the other powers of r in (32) : it is therefore the complete solu-
tion of the equation (31).
Thus the only laws which satisfy the requirements of the
problem are (1) that of the inverse square of the distance, when
A=0 and B is finite; (2) that of the direct distance, when B =
and A is finite ; and (3) that of these laws in combination, when
A and B are both finite.
These are of course the only laws of attraction for which a
sphere can attract an external particle with the same force as if
it is condensed into its centre ; because the sphere may be re-
solved into a series of concentric homogeneous shells, each of
which will attract with the same force as if it is condensed into
its centre.
272 ATTRACTION OF SPHERICAL SHELLS.
Hence also two homogeneous shells external to each other will
attract each other with the same force as if each is condensed
into its centre.
198.] To investigate the attraction of a homogeneous spherical
shell of infinitesimal thickness on a particle m placed within it,
when the law of attraction is represented by f (distance).
Let all the quantities and symbols be the same as in Art. 195 :
in this case however c is less than r, and the limits of integration
in the expression corresponding to (20) are r + c and rc; so
that the attraction of the shell on m,
Ex. 1 . Let the law of attraction be that of the inverse square
of the distance ; so that f(u) = ;
irpmrdr f r*c*~] r+c
the attraction = - - u -\ --
c* L u J r _ e
= ; (37)
therefore the attraction of the shell on an interior particle is
zero, and the particle is equally attracted in all directions.
The geometrical proof of this proposition is so simple that it
is desirable to insert it. In fig. 81, let the centre of the shell
and the attracted particle be in the plane of the paper, and let
the circular ring APBP' be the section of the shell by the same
plane. At o, the place of m, let solid angles be formed, which
occupy all space about it : and let each be considered with
reference to an equal opposite and vertical one; let &> be the
area of a spherical surface, described from o at the radius =
unity, which is intercepted by one of these solid angles : then
the area of the spherical surface intercepted at the distance r
is r 2 < : thus a mass-element of PJO^Q at the distance r from o
= pr a u>dr; and as the attraction of this on m
flft
= p r* to dr = mp o> dr,
so will the attraction of all the mass at vpqQ be
similarly, the attraction at P'J//Q' is mpux'p'j}': but by the
geometry of the circle PJP = P'JO'; therefore the attractions of
these masses are equal : and acting on o in opposite directions,
199'] ATTRACTION OF SPHERICAL SHELLS. 273
they neutralize each other. And because the same result is
true of every pair of such opposite small masses into which the
whole shell may be divided, the attraction exercised by it on
the particle m at o is zero.
The shell has been considered to be of finite thickness, but it
is obvious that the same result is true for a shell of infinitesimal
thickness.
Hence it follows that the attraction of a full homogeneous
sphere on a particle within it varies as the distance of the par-
ticle from the centre : for if a concentric spherical surface is
described passing through the attracted particle, the shell lying
outside of that sphere has no attraction on the particle ; and it
is attracted only by the mass lying within that sphere ; and that
varies directly as its radius : see equation (24).
If the sphere is composed of a series of concentric homo-
geneous shells, the density of which however varies, then the
attraction of all those lying outside of the concentric sphere
passing through the attracted particle is zero : and as the at-
traction of each of the others is the same as if it were condensed
into its centre, so if c = the distance of m from the centre
and M = the mass of all those shells lying nearer than m to
the centre,
the attraction of the sphere on m = - ;
C
and if the matter of M is homogeneous, so that M varies as c 3 ,
the attraction varies directly as the distance from the centre.
Ex. 2. Let the attraction vary directly as the distance ; then
f(u) = ;
and from (36) we have
TrpmrdrTu* (r* c*)u*~] r+e
the attraction = - - -- s - I
c> L4 2 J r ^c
and this is the same as if the shell were condensed into its
centre.
199-3 ^ ne resu ^ * n ( 37 ) leads us to inquire whether there
are any other laws besides that of the inverse square of the dis-
tance, for which the attraction of a homogeneous spherical shell
on a particle within it is zero.
In this case from (36),
-
^r-c
PRICE, VOL. III.
274
ATTRACTION OF THIN PLATES. [2OO.
f _-+c fr+c f
c 2 r*)\f(u}du\ 2 {u //() du}du\
J rc Jrc "
now making the substitutions of (29), we have
r+c
=
d
dc \
) ^
) '
and integrating,
c)
1 ---- = a constant = A j (38)
1 u O
and as no relation exists between r and c,
V(r) = A, (39)
fy'"(r) = 0, and so on.
And since from (30), i\r'(r} =
therefore the law of the inverse square of the distance is the
only one for which the attraction of a homogeneous spherical
shell on an internal particle is zero.
200.] To determine the attraction of a rectangular plate on a
particle at a given distance from the plate in the line passing
through the centre and perpendicular to the plane of the plate.
Let 2 a and 26 be the sides of the plate, p its density, and r
its thickness ; let c = the distance of the particle from the
plate ; then the attraction of the plate on the particle
f a C b dydx
4mprc I *
/*a
= 4mprdc I
Jo
Hence, if the plate is a square plate, 6 = a ; and the attraction
201.] ATTRACTION OP THIN PLATES. 275
If a = oo, the extent of the plate is infinite, and as the right-
hand member = 2TimpT, the attraction is constant; and thus
the attraction on an external particle of a square plate of infinite
extent is the same, whatever is the distance of the particle from
the plate. This result is the same as that already found for a
circular plate in Art. 192.
Also, if c = 0, that is, if the particle is on the plate, the
attraction is equal to l-nmpr, and is the same as that of an
infinite square plate.
Hereby also we can solve the following problems :
Ex. 1 . To find the attraction of a homogeneous prism, whose
transverse section is a square, on a particle in its axis at a given
distance from one end.
Let 2 a = the side of the square transverse section of the prism ;
I = the length of the prism ; c = the distance of m from one
end : then the attraction of the prism on m
' T . a* . a + (x* +
= 4 mp \x sin- 1 -- 2 a log -
"* + ** ' 1
Ex. 2. The attraction of a pyramid on a square base, of which
the altitude is a and 2d is the side of the base, on a particle at
its vertex
201 .] The attraction of thin plates on particles in the plane
of the plate.
Ex. 1 . To find the attraction of a thin rectangular plate on a
particle external to it and in its own plane.
Let a of, and b I' be the sides of the plate; then, if the
axes are parallel to the sides of the plate, and (a', ft} is the angle
nearest to the origin, and x, Y are the axial-components of the
attraction,
n xdxdy
~ ' (x 3 + y 8 )'
X =r
f
= m P T I
Jtr
= mpT
hog *
*
= mpT log _ log
5 ''' 'i *
N n 2
276 ATTRACTION OF SOLID BODIES. [2O2.
If a'= I' ; that is, if the attracted particle is at an angle
of the rectangle x = Y = oo, and the attraction is infinite. If
one, however, of these quantities is zero, the attraction is still
finite.
Ex. 2. The attraction of an elliptical plate on a particle at
the focus, when the attraction varies directly as the distance is
equal to mts.ae } if M = the mass of the plate.
202.] The attraction of various bodies on particles.
Ex. 1 . The vertex of a right circular cone is at the centre of a
sphere : find the attraction of the part of the sphere intercepted
by the cone on a particle at the vertex.
If we refer to the system of polar coordinates in space, and
take the axis of the cone to be the 2-axis, then
fZyf ra fa <
the attraction = mp I / / sin0cosddrd6d<f>
/0 ''O /<
If a = 90, the portion becomes a hemisphere, and consequently
the attraction of a hemisphere on a particle at its centre = -nmpa.
Ex. 2. The vertex of a cone is on the surface of a sphere, and
the axis of the cone passes through the centre of the sphere :
find the attraction of the intercepted mass on a particle at the
vertex of the cone.
Let the vertex of the cone be the origin, and the axis of the
cone the axis of z ; so that according to the notation of Art. 165,
Vol. II (Integral Calculus), the equation to the sphere is r =
2 a cos 0', then, if z is the attraction along the axis of z } and
2 a cos Q = r,
/*3ir [a /*r
z = mp I I sm6cos0drdOfl<t>
Jo JQ ^o
4irpma .
= y (1 -(cos 0)'}.
Ex. 3. To find the attraction of a hemisphere on a particle at
its edge.
Let the place of the particle be the origin, and let the line
passing through it and the centre be the s-axis, the plane cir-
cular base of the hemisphere being in the plane of (y, z) ; then,
taking the system of coordinates given in Art. 165, Vol. II
(Integral Calculus), if 2 a cos = r, and x, Y, z are the axial-
components of the attraction,
203- ] ATTRACTION OF HOMOGENEOUS ELLIPSOIDS. 277
\ = pml I I (sm6yc
*L 'o A)
5
Ipma
3
Y
= 0.
t/_S JQ /o
= 0.
/*? ri r*
z = pw; / / smOcosOdrddd<f>
3
203.] The attraction of a homogeneous ellipsoid.
Let the place of the attracted particle m be the origin ; and
let the coordinate axes be parallel to the principal axes of the
ellipsoid, the centre of the ellipsoid being at (a, /3, y), so that
the equation to its surface is
and let us refer it to the system of polar coordinates explained
in Art. 165, Vol. II; so that, if
a sin 6 cos /3 sin 9 sin y cos d
j^- - + -75 = B > r
! + :
(40) becomes Ar 2 2Br+c = 0; (42)
then if r, and r, are the roots of this equation, r t being the
greater, and r a the less,
B+(B 2 AC)* B
/P v :_ ^
A
If p is the density of the ellipsoid,
its mass-element = pr*sm0drd6d<p',
and as the direction-cosines of r are sin cos 0, sin 6 sin </>, and
cos 0, so the resolved attractions on m of the mass-element are
mp (sin 6}* cos <#r ?# d<J>, mp (sin 0) 2 sin r/r 6?0 </</>,
z p sin cos 0drd0d<p',
the integrals of which for limits assigned by the geometrical
conditions of the problem are the axial-components of the total
attraction.
278 ATTRACTION OF HOMOGENEOUS ELLIPSOIDS. [203.
Let x, Y, z be the axial-components of the attraction ; and let
us first consider the attracted particle to be within the ellipsoid ;
so that the limits of the /-integration are r and r t ; then
1 mp(sm6} 3 cos<pdrded({) } (44)
) ' r a
mp(sin.d)*sm<}>drdOd<t>, (45)
1 = 1 I I
JQ A) '
T = I / I
Jo -A) '
z=/ / / mpsin.6cosOdrd0d<t>. (46)
/0 *^-r 2
Of these three I shall consider the last, because it is the most
simple ; and results which are derived from it may be extended
to the other two by an exchange of letters only.
Performing the r-integration, and replacing r l and r a by their
values in (43),
n* 2s
mpsinOcosd dOd(p. (47)
~ *
Now
sin0eos0B
A
and observing that the denominator is a rational function of
sin and (cos 0) 2 , and that the first two of the three terms con-
tained in the numerator are rational functions of the same
quantities ; and observing that the limits of the ^-integration are
TT and ; by virtue of (40), Art. 88, Vol. II (Integral Calculus),
the integrals of the quantities corresponding to these two terms
vanish; and we have
,,, r* f w sm0 (cos eydedd)
z=:2mpa*o*yl I - - - * - - (48)
Jo JQ (be sin 6 cos </>) 2 + (ca sin 6 sin 0) 2 + (ab cos 0) 2
and performing first the ^-integration, we have
6 (cos & d0
, f w sin 6 (
z=2Ttmpaoy I -
^o {^(siney + a^cosey
and therefore by virtue of Art. 88, Vol. II (Integral Calculus),
sin0(cos0) 2 fi?0
s - '- - .(49)
Let cosd = t; .. sinddO =dl', and since t = 0, when
6 = - , and t=l, when = 0, we have, substituting in (49),
ri p. fa
y - - - - . (50)
J 0*fee*'t**.-0**i
204-] ATTRACTION OF ELLIPSOIDAL SHELLS. 279
Similarly,
I
/
1 - :J (51)
dt
; (52)
which definite integrals represent the axial-components of the
attraction of a homogeneous ellipsoid on an internal particle.
Now these three expressions involve elliptic integrals which
cannot be expressed in circular or logarithmic functions. The
problem however is reduced to simple quadrature ; and the
required integration involves the summation of the attractions
of a series of conical shells, whose common vertex is the at-
tracted point, and the thickness of which increases directly
as the distance from the vertex, because the r- and the ^-inte-
grations have been taken between limits which give a double
conical shell; and therefore the element-function in (51) &c. is
the attraction of such a double shell.
t, it will be observed, is the cosine of the semi-vertical angle
of the cone : the axis of the cone being parallel to that prin-
cipal axis of the ellipsoid parallel to which the attraction is
resolved.
204.] Jacobi has put the three preceding expressions for x, Y, z
under an elegant form by means of the following substitution :
a 3 adu
Let t* = - -; .-. dt = -- :
and when t = 0, u = oo ; when t = 1, u = : therefore
du
x =
r
a /
'o
3Mfl?g /** ?w /_ 3 x
if M = the mass of the ellipsoid. Similarly,
Y = "" f " H / ^ ; (54)
z= 3 -^r + '** + '^ . (55)
2 J f7^4.c 2 HC?<-i-fl 2 H?f4-i 2 H?<-i-e 2 H*
280 ATTRACTION OP ELLIPSOIDAL SHELLS. [205.
These values may also be expressed in the following form :
u = /
>
then
And they also give the following remarkable relation :
dv dz _ Bum /* (u + &') ( + c) + (it + c 1
~
dy~ 2
= 4irpm. (56)
We shall see hereafter that this is a general theorem of at-
tractions.
205.] Of the values of x, Y, z it is to be observed that each
is proportional to the distance of the attracted particle from the
principal plane which is perpendicular to the principal axis of
the ellipsoid parallel to which it is the component of attraction.
Consequently the particle is attracted by three components
along the principal axes, each of which varies as the corre-
sponding coordinate of the attracted particle.
It is also to be observed that the values of x, Y, z in (51) &c.
are not changed, if the quantities a, 6, c are replaced by ka, kb,
TcCy where k is any number : the attractions therefore are not
changed by the addition or subtraction of a shell contained
between two ellipsoidal surfaces concentric and similar, provided
that the attracted particle is within the interior. Hence we
infer that a homogeneous shell contained between two similar
and concentric ellipsoids attracts a particle within it equally in
all directions. This theorem is generally known by the name
of Newton's theorem on attractions, and is proved synthetically
in the Principia. To it also the geometrical method of Art. 198
is immediately applicable.
In fig. 82 let o be the attracted particle, and let the shell, of
which the section through o and the centre by the plane of the
paper is drawn in the figure, be contained between two similar
ellipsoidal surfaces concentric and similarly placed, and let us
suppose the shell to be homogeneous. Consider o to be the
vertex of solid angles which fill up the space around it ; and to
206.] ATTRACTION OF SPHEROIDS. 281
each one of these angles let the opposite and vertical angle be
drawn as in the figure : let QPOP'Q' be one of the lines of such
angles made by the paper. Now of similar and similarly situated
ellipsoids it is a property that PQ = P'Q'; let o> be the area of the
spherical surface, described about o as a centre, with unity as
the radius, which the cone intercepts : so that the volume, of
which PQ^JO is the section made by the paper, consists of ele-
ments, each of which is equal to u>r*dr } and the attraction of
each of which on m placed at o is
and of which the sum is zpo>xPQ; similarly the attraction of
p'q'^jt/ is mpat x P'Q', which is equal to the preceding ; therefore
the two attractions acting in opposite directions neutralize each
other : and as the same result holds true for all the solid angles
at o, so the resultant attraction of the shell on o vanishes.
As this proposition is independent of the thickness of the
shell, it is also true for a shell of infinitesimal thickness ; and
therefore it is true also for a shell of any thickness, composed of
homogeneous concentric, similar, and similarly-placed shells, the
density of each of which varies according to any given law.
Hence also if an ellipsoid attracts a particle of its own mass,
and a concentric and similar ellipsoidal surface is drawn through
the place of the attracted particle, the ellipsoidal shell lying
outside this latter surface will have no action on the particle;
and the attraction of the ellipsoid will be reduced to that of
the body lying within the latter surface.
Hence also it appears that if, as in fig. 83, m is without the
shell, and from o a cone, intercepting a spherical area (o>) with
a radius unity, is drawn, the attraction of the intercepted part
of the shell at P is equal to that at Q ; and as the same result is
true for all similar cones, it follows that if o is considered to be
a pole with reference to the exterior ellipsoid, the polar plane
will divide the shell into two parts, the attractions of which on
m at the pole are equal.
206.] In certain cases however the values given for x, Y, z in
Art. 203 can be integrated again.
Ex. 1. Let the bounding surface be an oblate spheroid; that
is, let a = b ; and let e be the eccentricity of the generating
ellipse of the spheroid, so that
a> c 2 a' 1 e' 2 ; c 1 = a' 2 (I e' 2 ) ;
PRICE, VOL. III. 00
282 ATTRACTION OF SPHEROIDS. [206.
_ 4irpmca f l
a
similarly Y = PP ^ } {ar*e-e(\ - ; (58)
" J Pdt
in _, )
J
Hence it appears that the attraction depends solely on the
eccentricity of the bounding spheroid, and is independent of its
magnitude.
Thus if through the attracted particle a spheroidal surface is
drawn similar to the given one, it will attract m with the same
force as the given spheroid, and as any other similar concentric
spheroid which includes m within its mass. Hence a spheroidal
shell, the surfaces of which are similar and concentric, attracts
a particle within it equally in all directions.
Ex. 2. Let the bounding surface be a prolate spheroid, so
that b = c; also let the eccentricity of the generating ellipse
be e, so that
a a = a*<? 2 j 2 =(! e 3 );
then we have
dt
4*nmp& 3 a f l Pi
x = - -/
a 2 J 1 (
(a,)
As these expressions involve the eccentricity only, it follows
that a spheroidal shell, the bounding surfaces of which are
similar and concentric prolate spheroids, attracts a particle
within it equally in all directions.
207.] ATTRACTION OF ELLIPSOIDS. 283
Ex. 3. If the bounding surface is a sphere : then a = b = c ;
47npa 4vmpP > 4irmpy t , .
~3 ' ~3 ' ~~T~' ( }
where a, /3, y are the coordinates of the attracted particle from
the centre of the sphere as origin.
207.] Returning now to the expressions (44), (45) and (46),
with the object of applying them to the case of the ellipsoid
attracting an external particle, the limits of the /-integration
will be /! and r 2 , so that
mri
mpsinO cos 6 dr dd d(f>
. , -i
r 2 (B" _ AC)^
mp sine cos0 - - dOdQ. (64)
A
Let A, B, c be replaced by their values, which are given in
(41): then the element-function contains circular functions in
an irrational form, and does not admit of further direct integra-
tion. We are therefore obliged to have recourse to an indirect
process ; and with a view to it, I will return to the symmetrical
equation of the bounding ellipsoid.
Let the centre of the ellipsoid be the origin, and (a, /3, y) the
place of m the attracted particle. Let a, b, c be the semi-axes
of the ellipsoid, of which the equation is
then the axial-components of the attraction of this -ellipsoid
on m are the following :
x = fff- P m (a-x)d*fydz . / 66)
JJJ {(a-^+Gfr-jO'-Ky-*)'}*'
Y = fff pm(fr-y)dxdydz _ ,
JJJ {(a-^ + ^-^ + fr-^jS'
z = fff _ pn(y-z}d*dydz _
JJJ {(a-x)* + (p-yy + (y- z y}$
the range of integration in each case being the space included
within the surface of the ellipsoid.
From the symmetry of the formulae it is evident that the
results of the others may be inferred from the integral of one
of these expressions. We need therefore only consider one of
them, and I will take the first, viz. (66).
o o a
284 ATTRACTION OF ELLIPSOIDS. [208.
Let a ji____J =x; ^ (<?-*)* = Y; (69)
then
(a.x}dxdydz
x =
=,./ r r
J_ c J_ Y L a _ ar
Now when x is replaced by its limiting values, the element-
function has a form which does not admit of further direct inte-
gration, and we are obliged to have recourse to an indirect
method. This has been discovered by Mr. Ivory, who has
shewn that (70) expresses the corresponding axial-component of
a certain ellipsoid on an internal particle ; and as the latter has
in the preceding Article been investigated and reduced to the
form of a single integral, so may this also be expressed in the
terms of a single integral ; and consequently by Ivory's theorem
we can effect one integration in the right-hand member of (70),
and thus reduce it from a double integral to a single integral.
208.] It will be observed that when x is replaced by x, the
point (x,y } z] in (70) is on the surface of the ellipsoid (65).
Through the point (a, /3, y) let there be described an ellipsoid
concentric with (65), and having moreover the foci of its prin-
cipal sections coincident with the foci of the principal sections of
the given ellipsoid. In this case the latter ellipsoid is said to
be concentric and confocal with the former. Its equation is
consequently #* y* z *
where w is a quantity to be determined by the fact that (a, ft, y)
lies in the surface of (71) ; so that we have
1 . (72)
Now when the fractions are removed from this equation, and
it is reduced to an integral form, it is a cubic in o>, and has
three real roots.
Let us suppose a' > b* > c a : then if in (72) we replace o> succes-
sively by oo, c 2 , #% % the results are -f-, , -f , ; so
that of the roots one is greater than c a ; another lies between
c a and b 3 ; and the other lies between b* and a" ; and
accordingly in the first case all the coefficients in (71) are
positive, and the surface is an ellipsoid ; in the second case the
2og.] IVORY'S THEOREM. 285
coefficients of the first two terms are positive, and that of the
third term is negative, and the surface is a hyperboloid of one
sheet; in the third case the coefficient of the first term is
positive, and those of the other two terms are negative, and the
surface is a hyperboloid of two sheets. Thus through a given
point there can be drawn three surfaces of the second order
concentric and confocal with a given ellipsoid ; and indeed with
any central surface of the -second order. As however an ellip-
soid is the surface which is required for the present problem, we
must take the largest root of (72) as the corresponding value
of o> ; and this is a positive quantity, because the result is + oo
when < = oo, and is when o> = 0.
209.] Let a', b', c' be the semi-principal axes of the concentric
and confocal ellipsoid which has been thus described : so that
a /J = <z a + o>, b'* = 6 2 + o>, c' 3 = c 2 + o>; (73)
and the equation to the ellipsoid is
and let x', Y 7 , z 7 be the axial-components of the attraction of this
ellipsoid on a particle m, equal to the other particle, placed at a
point (a, tf, /) within it. Then if
r*' /* r dtfdaf Y
x' = pm I / : (76)
J-c- J_ Y - Lj( a '_^. + (0'_y'). + ( y '_/).jiJ_ J /
with similar values for Y 7 and z 7 . It will be observed that
when x' is replaced by x', the point (x' } y f ) /) is on the surface
of (74).
Let us determine the points (#', y', z) (a, {, /), which are
thus far arbitrary, as follows : let (a, ft', y') be taken on the
surface of the first ellipsoid ; so that we have
and let its place be related to that of (a, /3, y), so that
a a p /3 y' y
T : ''~tf' T :: y T :: ~7 ;
also let (x, y', /) be so related to (x, y } z} that
*=*; sL-i, iW2-
a' - a V " b c ' ~ c'
286 ATTRACTION OP AN ELLIPSOID. [2IO.
these relations being consistent, as it is evident, with the equa-
tions to the surfaces. Let (x } y, z) and (a", ft, y'} which are on
the first ellipsoid be P and Q'; and let (x' } if, /) and (a, /3, y)
which are on the second ellipsoid be P' and Q; so that the
denominators of the element-functions in (70) and (76) are re-
spectively PQ and P'Q'.
Now P''= P f r
fa' ,N 8 / a' /x
= ( x-a,') -(x a')+"- +
\a ' v a '
= 0;
.-. PV=PQ. (80)
Two points, as P and P', on two concentric and confocal ellip-
soids, the coordinates of which are related by the conditions (79),
are called corresponding points. Similarly Q and Q' are also cor-
responding points.
210.] The preceding relations enable us to express the ele-
ment-function of the double integral (70) in terms of that of the
integral (76). Let us replace y, z, x, Y, in terms of ^, /, x', Y'
by means of (79), and the denominator of the element-function
by means of (80), and we have
be C<' /**' r Wdsf -i*
X=pfl& 1 ^/ / ^ r
Vc' J_ c . J_ y > L {(a'-tfO' + OS'-/) 8 + (/-/)'}*-! -x-
=*' _ ^ < 81 >
Similarly Y = -^,Y / , z = ~z f . (82)
And we have thus determined the axial-components of the
attraction of an ellipsoid on an external particle in terms of the
similar components of a concentric and confocal homogeneous
ellipsoid on an equal particle within it, whose place is that on
the interior ellipsoid which corresponds to that of the originally
attracted particle.
211.] ATTRACTION OF A SPHEROID. 287
Let x', Y', z' be replaced by" their values given in (51), (52),
and (50); and a', tf, c' by their values given in (73), and we
have
a f 1 t*dt < . .
Jo a + a>-f s -a*^*a a + o> + c a a a rf 2 *'
X =
and also similar values for Y and z.
Thus the attraction depends on the evaluation of a single
definite integral.
From (82) we have the following theorem :
The attractions with which two confocal homogeneous ellipsoids
attract, parallel to each axis, equal particles placed at corre-
sponding points on their surfaces, are as the products of the axes
perpendicular to each component.
This theorem has also been extended by Poisson to the case
in which the law of attraction is any function of the distance.
211.] By the preceding theorem we can determine the attrac-
tion of a spheroid on an external particle. Thus, for example,
let us take the oblate spheroid ; in which case a = d ; and let
e be the eccentricity of the generating ellipse, so that a* c* =
tfe* =a f *(f t , if (f is the eccentricity of the ellipse which gene-
rates the concentric and confocal spheroid which passes through
the attracted particle. Then from (57),
ac
X = -r-, X
ac , t
~ e
= 2itmp -{sin-V (f(\ (f*)*}, (84)
0,6
where d = - - , and co is the positive root of the quadratic
equation a 2 + 5 2 v*
Similarly
e'(\ -/)*}; (85)
k{i r &^*r?4 <)
Similarly may we determine the attraction of a prolate spheroid
on an external particle.
288 ATTRACTIONS. [212.
212.] If the oblate spheroid is of small eccentricity, so that it
differs little from a sphere, the right-hand members of (84), (85),
(86) may be expanded in ascending- powers of /, and approximate
results may be obtained in an algebraical form. We shall neglect
powers of / above the second, and if we suppose the attracted
particle to be close to the spheroid, /= e } if e is the eccentricity
of the generating ellipse of the spheroid. I will also suppose
the attracted particle to lie in the plane of (x, z), so that /3 = 0,
and Y = ; for no loss of generality arises from this assumption.
Thus 2-nmpa , ., .
x=- - {(1 e*)*Bin~ l e e(\ e 2 )}
(88)
Let R be the resultant attraction ; and X be the latitude of the
place of m ; that is, the angle at which the normal to the spheroid
at the place of m is inclined to the equator. Let M = the mass
of the spheroid ; then
M =
a 2 =
3
a* (cos
y 2 = c 2 (sinA) 2 (l-<? 2 ){l+e 2 (sinA) 2 };
3 6*
" To"
Now as E is the resultant attraction of the spheroid on m, it
is what in Art. 123 has been called the weight of m ; in reference
of course to a homogeneous oblate spheroid of small eccentricity ;
and consequently R is identical with mg ; hence
* = ^( 1+ lFM 1+ T5< sinX >'}- (89)
Let G be the least value of g ; that is, G is the value of g at
the equator, when A = ; then
2 1 4.] MACLAUBIN'S THEOREM. 289
so that o varies directly as the mass and inversely as the square
of the equatorial radius. Also
sr=G {i + ( sin A )'};
that is, the increase of gravity of a homogeneous spheroid of
small eccentricity varies as the square of the sine of the latitude
as we pass from the equator to the pole.
In reference to this result, however, I would observe that the
latest reductions of geodetical observations seem to shew that
the figure of the earth is that of an ellipsoid with three unequal
axes ; and that the greatest and least equatorial axes are in the
longitudes 15 34' E and 105 34' E respectively from Greenwich.
In an ellipse of small eccentricity, the ratio of the excess ot
the equatorial radius over the polar radius to the equatorial
radius is called the ellipticity, and is denoted by e ; so that
CL __ C S^ &*
ac 0e; but - = 1 (1 a ) = ; therefore e = - ; and
Or 22
^=G{l + |(sinX) 2 }. (92)
5
213.] The preceding formulae will also give the attraction of
a homogeneous elliptical cylinder of an infinite length on an
external particle on its surface.
In (83) and in the two similar values for Y and z, let a = oo,
while b and c remain finite ; then as the particle is on the sur-
face, w = 0, and ,
_ 2Timp6ca f 1 fidt
~~a* /.I l-t> ~
as we may foresee. Let e be the eccentricity of the elliptic
section ; then
tdt
_ g3\ _ j -\-6'* C
(93)
_
__
Ess 4*fpy - ^ - - -=litmpy-j -- (94)
C ~\~ C
214.] Hence also we may deduce the following theorem,
which was stated first by Maclaurin in a particular form, and
was demonstrated by him in that form :
Two confocal homogeneous ellipsoids attract an external particle
PRICE, VOL. in. p p
290 MACLAURIN'S THEOREM. [215.
along the same action-line, and with forces proportional to their
masses.
Let the two confocal ellipsoids be E and E', of which let the
semi-axes be respectively a, d, c ; a', b', c'; and the masses be M
and M 7 . Let (a, /3, y) be the place of the attracted particle, and
through it let a concentric and confocal ellipsoid E O be described.
Let x, Y, z, x', Y', z' be the axial-components of the attraction of
E and E' on m at (a, /3, y) respectively. Let a , 6 , c be the semi-
axes of E : also let x t , Y X , z t be the components of the attraction
of E on a particle at the point on the surface of E which cor-
responds to (a, ft, y) ; then
x be Y ca z ab
Also let x , Y , z be the components of the attraction of E O on m
at (a, /3, y) which is in its surface; then we have from (53), if
MI, = the mass of E O ,
3M z a a /"* du
du
x n =
x xx, abc M
x c ~" x t X ~ a 6 c ~ MO '
X Y Z M
JXQ JL Q JQ <&*()
the last two members of the equality following from the sym-
metry of the formulae. Similarly for the attraction of E' we
have, x ' ^ #,. M '
Xo ~ Y," " "z7 ~ M^ '
X Y Z M
"F == Y 7 == V '" F'
which equality is the statement of the theorem.
215.] At the conclusion of this direct process of investigating
attractions the two following observations are to be made :
(1) In the preceding investigations, whenever the law of at-
traction has been, as to distance, that of the distance directly,
the attraction of the body on the attracted particle has been
the same as if the body were condensed into its centre of gravity.
Now this property admits of generalization, so that in all cases,
2 1 5.] THEORY OP ATTRACTION. 291
whatever is the form of the attracting body, the total attraction
of it on a material particle is the same as if the body were con-
densed into its centre of gravity.
Let the centre of gravity be taken as the origin ; (a, 13, y) the
position of m the attracted particle ; (x, y> z) a point of the
attracting body, at which let the element dxdydz of its volume
abut : let p be the density, and r = the distance between m and
(x, y, z) : then if x, Y, z are the axial-components of the at-
traction, rrr . a-x
x = m I I I prdxdydz -
= m I 1 1 p (ax) dx dy dz ;
Y = mjj j p (j3 -y) dx dy dz ;
z = m I I p (yz) dx dy dz ;
but because the origin is the centre of gravity,
/ / / p x dx dy dz j I I py dx dy dz = / / / p z dx dy dz ;
and if M = the mass of the attracting body,
M = / / / p dx dy dz ;
and these are the axial-components of the attraction of M, which
is at the origin, on m placed at the point (a, /3, y) ; and therefore
the proposition is proved.
(2) Some remarkable results arise, when a spherical shell or
a sphere attracts a material particle, if the law of attraction is
that of the inverse square of the distance; (1) the attraction
of a spherical shell or of a sphere on an external particle is the
same as if they were respectively condensed into their centres ;
and (2) the attraction of a spherical shell on an internal par-
ticle is zero. Also we shall hereafter shew that, Kepler's laws
being assumed to be true, this law of attraction holds good
in the motion of the celestial bodies ; and also in electrical and
magnetic phaenomena, results can be explained and accounted
for by the same law. Now is there of this law any a priori
probability? Can we assign any reasons why the attraction
should vary directly as the product of the attracting masses,
and inversely as the square of the distance? Suppose m to be
P p 2
292 THE POTENTIAL. [216.
the mass of an attracting- particle, the influence of the attrac-
tion of which on other matter radiates from it in all directions;
and which is such that none of its quantity is lost by the pro-
cess of propagation ; let m be the vertex of a cone ; and let us
consider the parts of spherical surfaces which are described from
m as a centre with different radii, and are intercepted by the
cone. The areas of these spherical segments vary as the squares
of their radii, and the same amount of attracting influence is
spread over each one ; therefore the intensity of the amount of
attraction on an unit-surface varies inversely as the area of the
spherical surface intercepted by the cone ; and therefore varies
inversely as the square of the distance of the unit-surface from
the centre of the cone, that is, from the position of m ; and as
the same result is true for each unit which is at the same distance
from m, therefore, if there are m' units, and r is the distance,
. mm'
the attraction = ;
and thus, on the hypothesis that none of the attracting influence
is lost by means of or during the propagation, the law of attrac-
tion is that of the inverse square of the distance.
SECTION 2. Indirect investigation of the attraction of bodies.
The potential.
216.] When one particle m attracts or repels another particle
m', the two particles being at a distance r apart, the amount of
force which is in action between them varies directly as the pro-
duct of the masses, and is also a function of r ; and the action-
line of this force is the straight line which joins the particles.
Now from these circumstances arise very remarkable relations
between the components of the force of attraction. These com-
ponents are expressed by quantities which are partial differential
coefficients of the same function, so that when this function is
determined, the components of attraction may be found by differ-
entiation. Thus, let m' be the attracting and m the attracted
particle ; and let the law of attraction be represented by/(r),
where r is the distance between m' and m, so that the attraction
or repulsion of m' on m is mm'f(r). Let m be at (a, /3, y), and
m' at (of, /, /) ; then
2 1 6.] THE POTENTIAL. 293
Let x, Y, z be the axial-components of the attraction of m'
on m ; then
x = mnff(r) ~^, Y = w'/(r) =^ , z. = mm'f(r) y -^-
Letf(r) be the derived function of F (r), so that
d.v(r)=f(r)dr; (96)
also rdr = (a a') da; and consequently
/^ p ( r ) ft7\
X = m j (97)
, . (r) , .
Similarly Y = mm j~> z = mm , v . (98)
all these being partial differential coefficients with respect to the
coordinates of the attracted point ; and these can be found when
F(r) is known.
Now let us suppose m to be attracted by many particles m l)
m t ,...m n , whose places are respectively (#u j'u *i); (*s> Jfo *t)j
( x *>y*>Zu)) an< i whose distances from m are r l} r 3 ,...r n re-
spectively; then
d , .
= m-r n ?.mj?(r}',
A i \ (">
Ml ^ / / \ ^//\
OlYim lO fl V "V -Ml ^ -W3 1? I -} 1 ^ ^ *>M "^ >*M T? I /J* I
olllllIciFly X ^ lib ~T~Z ***ffv JP I/ J ^ //* ^ 2^,1/1 J? I/ )
dp dy
Let 2.m'F(r) = v; (100)
+Ti /^ V \ /^ V \ (dv\.
that is, the axial -components of the attraction are partial dif-
ferential coefficients of the function v, which is defined by (100).
In the formula (100) 2 expresses a sum; and becomes the
symbol of a definite integral, when the attracting mass is a
continuous body. In this latter case, if p is the density of the
body at (x, y, z}, the mass-element at that point is p dx dy dz, and
(102)
Thus, if this function can be determined, the axial-components
294 THE POTENTIAL. [ 21 7-
are expressed very simply as the partial derived functions of it
with respect to the coordinates of the attracted particle.
v, as defined by (100) or by (102), is called the potential of the
mass z.m', in reference to m at (a, /3, y), which is attracted ac-
cording to the law given by /"(?), where F(r) = lf(r) dr.
217.] Let us now take the law of attraction to be that of the
inverse square of the distance ; so that
consequently
(104)
the latter or the former value of v being taken according as the
attracting mass is continuous or discontinuous.
The value of r being that which is given in (95), we have
,dv^ /Y/*p(a x]dxdydz
x = (3;) = m JJJ - -V ' < 105 >
p(/3 y)dxdydz
r> ' < 106 >
y z)dxdydz
-;- - ;
and thus the axial-components of the resultant attraction de-
pend on the definite integral which is given in the right-hand
member of (104). Now this integral is evidently finite so long
as the attracted particle is outside of, and not a part of, the
attracting mass, even if the attracting mass is a closed shell
having m within it : because in this case r would never vanish,
and no value of the element-function would be infinite. If,
however, the attracted particle were upon or were a particle of
the attracting mass, it is not so evident but that the definite
integral might be infinite, because for attracting particles con-
tiguous to m, r = 0, and the corresponding value of the element-
function is infinite. But if we take the origin at the place of
the attracted particle, and refer to polar coordinates in space, we
have rrr
v= / / prsm6drdOd<t>; (108)
and this evidently does not take an infinite value when r = 0.
Similarly, if the attracted particle is at the origin, and x, Y, z
are expressed in terms of polar coordinates, it will be seen that
2 1 8.] ITS PHYSICAL MEANING. 295
these do not take infinite values, notwithstanding the attracted
particle is contiguous to particles of the attracting mass.
Although the potential is a term by which the general in-
tegral given in (102) is called, yet the name is specially applied
to the particular form in (104) and (108) where the law of attrac-
tion varies inversely as the square of the distance : for this is
the law of gravitation, as also the law according to which elec-
trical and magnetic forces act.
Thus the potential is the sum of every mass-element of the
attracting mass divided by its distance from the attracted par-
ticle.
As the potential is expressed by a sum or by an integral, it is
evident that the potential of a mass is the sum of the potentials
of the parts of which that mass is composed.
The name of " Potential-function" was given to v by George
Green, in his most remarkable " Essay on the Application of Ma-
thematical Analysis to the Theories of Electricity and Mag-
netism," which was published at Nottingham in 1828. But the
abridged name " Potential" was given by Gauss in his memoir
" On General Propositions relating to Attractive and Repulsive
Forces varying inversely as the Square of the Distance," which is
printed in " Resultate aus den Beobachtungen des Magnetischen
Vereins im Jahre 1839."
218.] The potential is of so great importance not only in the
theory of attractions but also in dynamics, hydromechanics,
electricity, magnetism, and heat, that before we determine its
value in particular cases it is desirable to consider its meaning
from two points of view.
As it is the sum of every mass-element of the attracting mass
divided by its distance from the attracted particle, it is a func-
tion independent of any particular system of coordinates to
which the particle and the attracting mass may be referred.
Its value depends indeed on the place of the attracted particle,
and varies as that place varies ; but it is independent of the
mode of determining that place.
It has however the following important physical meaning :
"We estimate work, as we shall hereafter explain at length, by the
product of pressure and the distance through which the pressure
has acted estimated along the action-line of the pressure. Thus,
for instance, if a weight, = mg, is raised through a vertical
distance equal to h, the work which has been spent on the lifting
296 THE POTENTIAL [ 2I 9-
that body, and is due, as we say, to its change of place,
is mgh; and the work which would be recovered, if the body
were replaced in its former position, is also mgh. Now potential
is the particular form which the work of an unit-particle takes,
when it is under the attraction of one or more other par-
ticles. For suppose an unit-particle to have come from an in-
finite distance under the attraction of a particle whose mass is m',
and where the law of attraction is that of the inverse square of
the distance, so that the attraction on the unit-particle at that
distance is zero ; then if w is the work when the particle is at
the distance r, and dw is the increment of work due to, or ob-
tained by its passage over, the space dr,
m .
dw=--dr;
.'. w = ; (109)
r
so that the mass of the attracting particle divided by the dis-
tance of the unit-particle from the place of the attracting particle
is the work of the unit-particle acquired in moving from an
infinite distance to a point at a distance r from the attracting
particle. And as a similar result is true for an unit-particle
under the attraction of many particles, and as the whole work
of the unit particle is the sum of all the separate works, so
the whole work = ,. = ffff**., (110)
the latter or former being taken according as the attracting
mass is a continuous body, or is composed of separate particles.
But as the right-hand member of ( 1 1 0) is the potential as defined
by (104), the potential is the whole work obtained by an unit-
particle in its passage from infinity to a point at a distance r,
under the attraction of the several particles of the attracting
body. This is the mechanical interpretation of the potential.
219.] The component along any line of the resultant attrac-
tion of the attracting mass on m at (a, /3, y) may thus be
found.
Suppose (a, /3, y) to be a point on a curve or other line, of
which ds is a length-element, so that the direction-cosines of ds
are -=-i -=-> -y-: then the component along d* of the resultant
ds ds ds
attraction is
220.] OP A STRAIGHT BAR. 297
da dB dy
X- r 4- Y-y- +Z-r
ds (h as
I
so that the force of attraction which acts in the line ds varies
dv
as
Hence if R is the resultant attraction, and Q is the angle be-
tween its action-line and ds,
dv
Rcosfl = m -j- ;
L
ds
= -0z(v 1 -v ).
'0
And if the integration is carried through a closed ring, then
v, = v , and r
I R cos 6 ds = 0.
The axial-components are evidently particular cases of (111).
If the place of the attracted particle, in reference to a system of
polar coordinates in space, is (r, Q, <), then the components of
attraction perpendicular to the meridian plane, along the radius
vector, and perpendicular to the radius vector, the action-lines of
both these last components being in the meridian plane, are re-
spectively
m ,dv\ /dv^ m
If the attracting mass is a plane wire or a plane plate, and the
attracted particle is in that plane, and the place of it is (r, 6} in
reference to a system of polar coordinates, then the radial and
transversal components of the attraction are respectively
,dv\ m /-dv^. ., , rt .
m(-j-), (-n;)- (113)
^dr'' r V0'
Before I enter on the investigation of other general properties
of the potential, and especially of the geometrical interpretation
of it, I will determine its value in some particular cases, and
derive from them the corresponding components of attraction.
220.] The potential of a thin straight rod on an external
particle.
Let to and p be respectively the area of the transverse section
PRICE, VOL. III. Q q
298 THE POTENTIAL [22 E.
and the density of the rod. Let the place of the attracted particle
m be (a, /3) ; the extremity of the rod being the origin, and the
axis of x lying along the rod, whose length = a. Then
pa> dx
/ a
. {* + (#_ a) 2 }*
. a-a*
= p &) log -
x = m(-j-\ = mp<a
/dv\ moot ( a a a
Y = mi i = - > __ t-
V ^/3 ; ft I {/?+(*-a)}* (/*+*
which are the same results as those found in Art. 185.
221 .] The potential of a thin homogeneous spherical shell.
Let p = the density, T = the thickness, a = the radius of the
shell ; let (0, 0, y) be the place of the attracted particle. Then
if the centre of the shell is the origin,
sin6ded(b
?*)*] (114)
y o
Now this radical takes different forms according as the attracted
particle is within, or is external to, the shell.
If the particle is within the shell, y is less than a ; and
2-npra
v= -
= 4irpr.
If the particle is outside the shell, y is greater than a ; and
2 itpra
v= -
In the former case -=- = : and the shell exerts no attraction
dy
on a particle within it.
T ,, , ,
In the latter case z =
if the mass of the shell = M ; and the attraction of the shell on
222.] OP A SPHERICAL SHELL. 299
the external particle is the same as if the mass of the shell were
condensed into a particle at its centre.
If the shell is not thin, but contained between two concentric
spherical shells whose radii are a and a' respectively, the poten-
tials are thus found.
If the attracted particle is in the shell, and at a distance y
from the centre, v consists of two integrals ; one of which cor-
responds to the matter outside the concentric sphere which passes
through m, and the other to the matter within that sphere.
Consequently, since v = 0, when a = y = a, that is, when there
is no attracting matter,
C a 4-irp fy
= 47TO ada + - - \ a
J y y J a -
And if the particle is outside the shell,
4wp f a ,
v = / <2 a <z
y Ja
47Tp , 3 / 3 .. M (118)
: ~7^ ; -y
since v = 0, when a = a', because in that case there is no at-
tracting matter.
Consequently if the attracting body be a full homogeneous
sphere, then #'= ; and for an internal particle
and for an external particle,
M
3y ' y
As both these expressions give the same value for v when
y = a, that is, when m is on the external surface of the shell,
v undergoes no discontinuity at that point.
222.] The potential of a sphere composed of concentric spheri-
cal shells, the density of each being a function of its radius.
Of the shell of which the radius is r, and the thickness is dr,
let the density bef(r) : then we have two cases according as the
particle is outside the sphere, or is within it.
If the particle is outside the sphere, and at a distance from
the centre = y, and if a is the radius of the sphere,
Q 2
300 THE POTENTIAL [223.
m aw r" f(r) sin 6 dd> dd dr
J } (119)
r 2rcos0-f j *
y
whereby the potential may be found, when f(r) is given. If
M = the mass of the sphere,
fa
47T / r*f(r}dr = M; so
^
that
and consequently the potential is the same as that of a particle,
whose mass is M, placed at the centre of the sphere.
If the particle is inside the sphere, so that y is less than a,
then the integral given in (119), and which expresses the poten-
tial, must be divided into two parts, the former of which will
correspond to the shell on the interior surface of which the
attracted particle is, and the latter to the sphere on the surface
of which the attracted particle is : so that
ra f* r *f( r \ s i n Q M d r [y f* r
= 2ir / ^ ,+2-n I J
J y ~'o a 2rcos0 + r 2 * ^o ^o
,
(y a 2yrcos0 + r 2 )* ^o ^o (y 2yrcos0+ r a )*
47r r>
rf(r)dr+ / r*/(r)dr; (120)
y y JQ
and this is the required potential.
If we take the y-differential, according to the principles ex-
plained in Art. 96, Vol. II (Integral Calculus), we have
fy
and because 4w / r*f(r}dr is the mass of the sphere whose
/o
radius is y, it follows that the shell lying outside that sphere
exercises no effect on the attracted particle.
223.] The potential of a body of finite dimensions on a particle
at a very great distance.
Let us suppose the attracting mass to be a continuous body,
and M to be ite mass. Let the origin be taken at its centre of
mass ; so that
= I I pydxdydz = / / / pzdxdydz 0. (122)
223.] OP A BODY. 301
Also let us so far anticipate what will be proved hereafter as to
assume the possibility of a system of coordinate-axes such that
with reference to them
II I py*datdyds= lpzxdxdydz = \\\ pxy dxdydz =, (123)
and let the attracting body be referred to this system ; also let
(a, p, y) be the place of m the attracted particle. Let
so that <r is very large in comparison of the distance of any mass-
element of the attracting body from the origin. Then
C f f p dx du dz
v = I I I - Z- _,
where r* = (a a?) + (&yY + (yz)*
-M'. (124)
Therefore expanding by the binomial theorem,
11 ax + fiy + yz 3 (as - -
r a cr 8 2(r 5
omitting terms involving in the denominator higher powers of a ;
so that
i ,
2<r
dxdydz. (126)
Now of the several definite integrals which this expression
contains, the first = > because // I p dxdydz = MJ those
corresponding to the second term in the expression vanish by
reason of (122); of those in the third term some vanish by
reason of (123), and we have ultimately,
(_a 2 p* + 2y*)z t }datdydz. (127)
To abridge this expression let
A =jjjp(y*+z*}dxdydz,
B* = / / / p(z*+ x*)dxdydz,
c =ljjp(x*+y t ')dxdydz;
(128)
302 THE POTENTIAL [223.
where A, B, c express the sums of the products of each mass-
element of the attracting body and the square of its distance
from the axes of #, y, z respectively ; and substituting these in
(127) we have
V = + -i-{(B + C-2A)a'+(c + A-2B)/3' + (A + B-2c)y}; (129)
<r 2<r
and this is the required value to the stated degree of approxi-
mation.
If we neglect the whole second term,
v = ; (130)
<T
and the potential is the same as it would be if the whole attract-
ing body were condensed into a particle at its centre of mass ;
and thus if R is the resultant attraction, by (1 1 1),
dv
R =m^-
aar
Hence we have the following very important theorem :
The attraction of a body or of a material system on a particle
at a distance from its centre of mass, which is very great in com-
parison of the linear dimensions of the body, is the same as it
would be if the whole attracting mass were condensed into a
particle at its mass-centre.
Also as m may be a mass-element of a finite body M', and as
this theorem will be true for every mass-element, so we have the
following theorem :
Two bodies or material systems, of which the linear dimen-
sions are very small in comparison of the distance between their
mass-centres, attract each other in the same manner as if the
mass of each were condensed into a particle at its mass-centre.
Thus as each heavenly body consists of attracting particles,
and the linear dimensions of these bodies are small in comparison
of their distances from each other, they attract each other ap-
proximately as if the mass of each were condensed into a particle
at its mass-centre. Consequently in the investigation of the
motion of any one of these bodies, we may consider it to be
under the attraction of many particles, each attracting as its
mass and inversely as the square of its distance from the mass-
centre of the attracted body.
224-] OF A BODY. 303
Theorems similar to these have already been demonstrated of
spheres, whatever is the distance between their centres ; so that
what is approximately true of bodies of any form, the mass-centres
of which are at a very great distance apart, is true of two spheres,
whatever is the distance between their centres. This result may
also be inferred from the preceding process of investigation ; for
in spheres, by reason of the symmetrical distribution of their
matter, A = B = c, and all the terms in the expansion of v,
except the first, vanish.
224.] Returning to the general value in (129), and taking in
all the terms, we may, with the object of determining the com-
ponents of attraction, express it as follows :
which are the axial-components of the attraction of the mass M
on m. If we are investigating the action of M on m, these must
be taken with negative signs, as M tends to draw m towards the
origin.
In reference to these expressions, the first terms are evidently
the axial-components of a central force, whose centre, that is, the
point in which the force originates, is the origin ; and con-
sequently, as regards rotatory motion, this force has no effect :
but this is not the case as to the other terms of the expressions,
inasmuch as the corresponding coefficients are generally not
equal : and if L, M, N are the axial-components of the resulting
couple, then by (104), Art. 68,
(136)
.
L = z/3-Yy = ^ (B-c) ;
3mya .
M = xy za = f (c A) ;
3wa/3 ,
N = Ya x/3 = (A B).
rt-9 < ' J
304 THE POTENTIAL [225.
These results will be of considerable use hereafter, when we come
to the investigation of the motion of a body about an axis pass-
ing through its mass-centre.
225.] The potential of a homogeneous ellipsoid.
Let the centre of the ellipsoid be the origin, and (a, /3, y) the
place of m the attracted particle. Then
=/*//- pd *
JJJ {(_) +p-
(137)
,)*}4
the range of integration being the space included within the
ellipsoid whose equation is
For the evaluation of this triple integral, I propose to take
the method explained in Art. 283, Vol. II (Integral Calculus),
and to determine it, as far as it is possible, by Dirichlet's process
of a discontinuous function.
Taking the discontinuous function given in (68) Art. 283,
Vol. II (Integral Calculus), we have
2 /"" sin tcoskt . . ~\
-I ~dt\ } when k is less than 1 ; , .
TfjQ I r {lay)
= 0, when k is greater than 1 . j
x* 1/* z*
Also since + ~- -\ - is less than, or greater than 1, ac-
cording as (x, y, z) is within or without the ellipsoid,
t
according as (x, y } z] is within or without the ellipsoid which is
given in (138).
To abridge the expression let
(a #) s + (/3 y)* + (y *)' = u* ; (140)
then we may express v in the following form :
2 P /- r r r smt _ ,& , y *
=-- /
n J_ J-
^ 8 0* C* ' U
the limits of integration having been extended to oo and to oo,
and thus being constant, so that the order, in which the several
integrations may be effected, is indifferent.
And v can be expressed in another form which is more con-
venient for our present purpose ;
225-] OF A HOMOGENEOUS ELLIPSOID. 305
v = the real part
O f ^ r r r r ^(S+s+jD*******. (142)
tf^O ^-oo^-co^-oo ^ W
We must however replace - by a definite integral, which will
give an expression of a more convenient form.
By (280) Art. 129, Vol. II (Integral Calculus),
Let a- 2 be replaced by Vltfs; then we have
v = the real part
!(_!)*/"" rr r r^s^^^
77? *^0 ''O J -<a J -<x> J -v> t
where i = the triple integral which includes the #- y- ^-inte-
grations.
Now for the ar-integral we have by Art. 138, Vol. II (Integral
Calculus),
.
,-V-i
and similar values are true for the y- and the ^-integrals; so
, a *g? ny c y >
v 8 ' +< 6S ' + ' ca ' + ' /
"
Consequently v is the real part of
, (<?*? b*ffl
-u^^i . (1
The variables * and are thus far unconnected ; let us suppose
s u = t, where o> is a new variable dependent on u and inde-
pendent of t : then
7 td(d
as =. -- and
PRICE, VOL. III. R r
306 THE POTENTIAL [225.
f <z 2 a s 3 2 /3 s e'y 1 ) t ( 3 a 2 2 * C 2 y 2 )
_ I I v. ^^ j _|_
e \ ~|~ "j "t" ( ^ = \ i T~ i 4
if *= + + ; (144)
so that v is the real part
2pabc r r sin* . ^
' '
/ - sinrf (cos^-f \/ lsin^)^<?a) ,, ._
ofv I 2 pate I I - ; (146)
f
smktdtda
v= 2padc/ / - -- rj (147)
*
where k is a function of a, /3, y and o)^ by reason of (144).
This is the potential of a homogeneous ellipsoid on a particle
at (a, ft, y) whether the place of it is within or without the
ellipsoid.
For the components of attraction, as k is a function of a, we have
^Vx , r o /* co sin cosktdtdu
) = Ipaabc / ; (148)
da' J 9 J 9 t (*+) {(0*+)(*'+X**+*)i*
with similar values of (-^\ and C-j V
>a/5' v y^
Now if (a, & y) is within the ellipsoid, is evidently less
than 1 ; and consequently,
/"" sinrf
/
JQ t
TT
-,
t 2
therefore
: (149)
Y= m- i7t
V <#*' (d+i*){(*+
dta
r do
r ; (150)
J (d+i*){(*+.(iJ + )(+)}*
(<x
-) =
dy j
which are the values already found by the direct process.
226.] OF A HOMOGENEOUS ELLIPSOID. 307
a a /3* y"
If (a, /3, y) is without the ellipsoid, so that + -^- + -^-- is
ft C
greater than 1 , let n be the greatest root of the cubic
s -fo>
then as the left-hand member of this equation decreases as o
increases, so is the quantity less than 1 for all values of CD from
ao to n; and consequently for all these values of o>, k is less than
1, and for other values k is greater than 1. Hence if we
confine oo within the limits for which k is less than 1, we
have, as before,
sin t cos let ,. TT
/
.
. 2
and accordingly for an external particle,
dv \ i r
-7-1 = 2-npa.abc I
fi? a' ^o a'
X = Zirpmaabc
and if in this quantity we replace &> by o> + n, and put
:C /a , (153)
X == 2irpmaabc I (154)
Similarly,
/ao j
-^- -; (155)
/CO 7
r. (156)
. (<f' + a)t(af* + aWi' + Utf'+wn*
These quantities which contain only single definite integrals
give the axial-components of the attraction of a homogeneous
ellipsoid on an external particle ; they are of the same form as
(53), (54), (55) which assign the components of attraction of a
homogeneous ellipsoid on an internal particle.
From (152) and (153) it is evident that a, V, cf are the
principal semi-axes of the surface concentric and confocal with
the given ellipsoid, and the value of o> which we have chosen
shews that they are the principal semi-axes of a concentric and
confocal ellipsoid which passes through (a, /3, y).
226.] From these values of the components of attraction the
following theorems are deduced :
R r 2
308 THE POTENTIAL [226.
I. If in (149), (150), (151) a, b, c are replaced by &a, &&, kc
respectively, and o> is replaced by *&>, x, Y, z are unchanged.
Thus the homogeneous shell contained between two similar
concentric and similarly -placed ellipsoids has no attraction on
a particle placed within the smaller surface. This is Newton's
theorem.
For such a shell (-T-) = (^5) = (;/~) = > consequently
the potential has the same value for all points within the inner
surface of the shell.
II. Let x', Y', z' be the components of attraction of a homo-
geneous ellipsoid whose semi-axes are a' ', V , c' on a particle m
situated at (a, ft, y] within the ellipsoid; then by (149),
/oo
x' = 2-npmda'b'c' I
./n (n'*
(of* + o>) { (a'* + o>) (i /a + w) (c'* + w) } T
but by (154) if x is the ^-component of attraction of a homo-
geneous concentric and confocal ellipsoid on an equal particle at
/ <ttt)
x.= 2TTpmaaoc t ;
^^ , , (Io7)
X O.OOC
Let (a', ft', /) on the ellipsoid (a, 6, c) be the point corre-
sponding to (a, ft, y) on the ellipsoid (a', V,(T)', so that
ct ct
x' 4V
then = -z ; (158)
x oc
... Y- tfV z' a'V
similarly = ; = T-; (159)
Y ca z ad
and these three equations constitute Ivory's theorem.
III. Maclaurin's theorem follows immediately from the ex-
pressions given in (154), (155), (156).
Let E and E' be the two homogeneous concentric and confocal
ellipsoids which attract m at (a, ft, y} ; and let x, Y, z, x^, Y', z'
be the respective components of attraction; leta t ,^ 1} c l be the
principal semi-axes of the concentric and confocal ellipsoid
passing through (a, ft, y) : then
, r da>
x = 2-npmaaoc
227-] OF ELLIPSOIDAL SHELLS. 309
,,, , T 00 dot
X = 2-npmaa be I - ;
*^o (*+){(!' + )(*!* +)(' -M)} 4
x abc M Y z
' x' = aW = ^ = -? =Y'
if M and M' are the masses of the two ellipsoids. Thus the
ellipsoids attract the external particle with forces proportional
to their masses, and along the same line of action.
JiV. Hence also the attractions on an external particle of
/ two homogeneous ellipsoidal shells, the external bounding sur-
/ faces of which are concentric and confocal ellipsoids, and the
internal surfaces of which are ellipsoids concentric, similar and
similarly-placed to the external surfaces respectively, and the
thicknesses along the same axis are as the axes of the external
surfaces, are as the masses of the shells, and have the same
action-line.
! . As this theorem is true for shells of any thickness, it is also
true when the shells are infinitesimally thin.
V. If v and v' are the potentials of E and E 7 with respect to
an external particle,
v v 7
227-3 ^ ne potential and attraction of ellipsoidal shells.
By an ellipsoidal shell we mean a shell of which the thickness
is infinitesimally small, and the bounding surfaces are two
similar, similarly placed, and concentric ellipsoidal surfaces.
Consequently if a, b, c are the principal semi-axes of the
interior surface, and ka, kb, kc, of the exterior surface, the
thickness of the shell at the extremities of the principal axes
are severally (k 1) a, (k 1) b, ( 1) c, and are proportional to
the corresponding axes respectively ;
also the volume of the shell = (k 3 l}abc. (162)
O
Now the potentials of two thin ellipsoidal shells on an external
particle are to one another as the masses.
In proof of this it is to be observed that we may deduce it
from theorem IV. of the preceding Article ; for as the attrac-
tions of such shells on an external particle in all directions are
as the masses of the shells, so must also the potentials of the
two be equal.
The following however is a proof, independent of the preceding
310 THE POTENTIAL [227.
calculation of the potential of the homogeneous ellipsoid in an
external particle.
Let (a, /3, y) be the place of the attracted particle m ; and
through it let an ellipsoidal surface E O be described concentric
and confocal with the exterior surface E of the shell : and also
let a similar, similarly situated and concentric ellipsoid be de-
scribed within the former, and infinitesimally near to it, so that
a shell is formed on the exterior surface of which m is.
Let , #, c be the principal semi-axes of E O ; and let P O
(# , y , 2 ) be any point on E O . To (# ,y , z ) let the corre-
sponding point P (x, y, z) on E be taken ; so that
_ ^ ?L = l?-- - = . fi
a ~~ a ' b ~ b ' c ~~ c '
dxdydz dx n dydz
' z - == - Z - '
abc a^O^Co
To the place Q (a, ft, y) of the attracted particle let the corre-
sponding point Q'(a', &, /) be taken on E; then by reason of
(80) Art. 209, PQ = P O Q';
1 dx dy dz 1 dx dy dz >
*~~J j / 5 11O4)
abc PQ a 6 c P O Q
and summing these expressions so as to include the whole shells,
for these are corresponding spaces,
Cr f p dxdydz _ abc C C C pdx.dy^dz^
JJJ PQ a 6 c JJJ p Q'
and as the volumes of the shells vary as the product of the
principal semi-axes, this equation shews that the potential of
the inner shell on the particle m at Q which is exterior to it,
has to the potential of the exterior shell on the particle m
at Q' which is interior to it, the ratio of the masses of the
shells.
Now the potential of an ellipsoidal shell on a particle within
it is the same for all positions of the particle, and is consequently
constant. Let v c be the potential of E O on an internal particle
m ; and let v be the potential of E on the external particle m at
(a, ft, y): then from (165) if M and M O are the masses of the
shells,
=^- (166)
M M
Also let there be another ellipsoidal shell E', of which the
outer surface is concentric and confocal with the outer surface
22Q.] OF ELLIPSOIDAL SHELLS. 311
of E, and to which m at (a, p, y) is also external ; let M' be its
mass, and v' its potential ; then
v 7 v
-, = ; (167)
M MO
.-. -1-^; 068)
that is, the potentials of two concentric and confocal ellipsoidal
shells on an external particle are to one another as their masses.
Hence also if two concentric and confocal ellipsoidal shells
attract a particle m, which is external to both, the components
of the attraction along any line vary as the masses of the
shells; and the shells also attract the particle along the same
action-line.
228.] The action-line of the attraction may thus be deter-
mined. Since v and M O are the same for all positions of m on
the surface of E O , v by reason of (166) is also constant for all
these positions of m ; consequently as m is shifted from one
place to another on this surface there is no change in the value
of v ; that is, for such a displacement = ; and therefore
there is no action of attraction along the surface, and the action
is wholly normal to it. And therefore if through the place of
the attracted particle an ellipsoid is described concentric and
confocal with the exterior surface of the attracting shell, the
action-line of the attraction is normal to this ellipsoid.
Hence also if a conical surface is described having its vertex
at the place of the attracted particle and enveloping the ellip-
soid, the internal axis of the cone is the action-line of the
resultant attraction of the shell. Steiner has given a geome-
trical proof of this theorem. See Crelle, Vol. XII. p. 141.
229.] The amount of attraction of an ellipsoidal shell on an
external particle m may thus be fcund.
Let the ellipsoid E' in Art. 227 be that on the external surface
of which m lies. Then if we can determine the attraction of the
shell corresponding to E' on m, we can by means of (168) deduce
that of the shell corresponding to E.
Let o be the position of m on the exterior surface of the
shell, a section of which by a plane through the normal OG and
the centre c is delineated in fig. 84 : then the line of action
of the resultant attraction of the shell is, as just now shewn, the
line OG. Let a series of very small solid angles originate at o :
312 THE ATTRACTION [229.
and let that one of which the section is opjt/^'q intercept an
area <o of a spherical surface described from o as a centre with
the radius = unity : so that the area intercepted at a distance
r = a>r 3 : now the volume of each of the mass-elements of the
shell thus intercepted and at a distance r from o = p o> r 2 dr ;
and therefore its attraction on m = mpoxlr; and therefore the
attraction of OPQ on m = mp<axof, and the attraction of pp'tfq
on o = mp<i> x pp' : but by reason of the similarity of the surfaces
OP = pp r : therefore the attraction on m of the part of the shell
intercepted by the cone o> in the direction OP = 2mp<a x OP ; and
therefore the attraction in the direction OG
= 2mptoOn = 2mpa>T, (169)
if T is the normal thickness of the shell. And since the surface
of a hemisphere, whose radius = unity, is 2 TT, therefore
the attraction of the whole shell = l-nmpr, (170)
and thus varies as the thickness of the shell.
This theorem deserves a passing remark. If m is inside the
shell, the resultant attraction is zero ; if it is outside, and on or
very near to the surface of the shell, the attraction = 4 irmpT ; so
that on passing through the attracting matter of the shell, the
attraction suddenly and discontinuously changes its value. It
will be observed that this value of the attraction is the same
as that of a plate of infinite area, whether that area be square
or circular, see Arts. 192, 200, and whatever is the distance of
the particle from the plate.
If in fig. 84 cm = r, mo = dr; and p is the perpendicular
from the centre on the tangent plane at o,
T &
~* } I 1 i 1 )
ar r
but by reason of the similarity of the bounding surfaces,
dr da db dc
= = -j- = ; (172)
r a o c
therefore the attraction of the shell, in the direction of the
normal OG, 4mp
= - da. (173)
a
Now if the equation to the exterior surface is
x 3 y* z*
2 2 p.] OF ELLIPSOIDAL SHELLS. 313
17*2 2/2 >i
tC it 6
then, - = + - + ; --,
/>* a* #* c*
and the direction-cosines of the normal are
px py pz f
Ht' ~fi' ~e fl
therefore if x', Y', z' are the components of the attraction of
the shell on m, placed at the point (z, y, z], along the three
coordinate axes,
, , _ ,_
~* ; ~^~ ~e~
and if E is the resultant attraction,
E = 4 'W* (175)
Thus the attraction of an ellipsoidal shell on a particle on its
surface varies as the perpendicular from the centre of the shell
on its tangent plane at the place of the attracted particle. And
the attraction is the greatest when the particle is at the ex-
tremity of the longest principal semi-axis, and is least when the
particle is at the extremity of the shortest principal semi-axis.
Also the shell equally attracts all equal particles placed at
points on the exterior surface the tangent planes at which are
equally distant from the centre. Thus at all points on the
curve of double curvature which is the intersection of the two
ellipsoids t , t
and F + F + 7T = IT'-
m is equally attracted by the ellipsoidal shell.
Hereby we are led to the determination of the attraction of
an ellipsoidal shell on an external particle m.
Through (a, /3, y] the place of m let there be described an
ellipsoid E O , whose principal semi-axes are a Q) & , c 0) concentric
and confocal with the exterior surface E of the ellipsoidal shell ;
and let there be described within this ellipsoid a concentric,
similar, and similarly situated ellipsoid, of such a thickness da
at the extremity of the semi-axis a 0) that
^.= ^ ; (176)
#0 *
then m is on the surface of an ellipsoidal shell. And by the
PRICE, VOL. in. s s
314 THE ATTRACTION [ 2 3-
concluding paragraph of Art. 227, if X and x are the x-com-
ponents of the attractions on m of the shells whose external
surfaces are respectively E O and E,
x= ^* . (,77)
Consequently replacing x by its value which is given in (174),
dbc
x = j. ;
a<>o c a s
= ^;7^r' (178)
, . , lirpadc fda db dc\ . . . ,,
where M, which = ^ ( 1- -y- -\ j = Ivpocda, is the
mass of the attracting shell. Hence also
where the letters with the subscript refer to the ellipsoid
passing through (a, (5, y] which is the place of the attracted
particle m.
Hence also if E is the resultant of this attraction,
R = . (,80)
a 9 6 c g
230.] This result leads to a remarkable theorem.
Let R be the attraction of M on an unit-particle on E O corre-
sponding to the perpendicular p v ; then if ds is a surface-element
of E at that point, R ds is the attraction which acts on ds ; and
consequently the attraction of M which acts on the whole surface
ofE r M .
= IE^S = j / p ds:
J a b c J *
but Ip ds= three times the volume of the ellipsoid E O , and con-
sequently = 4ira 6 c ;
.:. / K?s = 4wM; 081)
and as the right-hand member is independent of the position of
the attracted particle, it follows that the sum of the actions
with which an ellipsoidal shell attracts all the elements of a
concentric and confocal ellipsoidal shell is constant.
Hereafter we shall have a general theorem which includes this.
231.] OP ELLIPSOIDAL SHELLS. 315
231.] The results which have been demonstrated for the
attraction of a homogeneous ellipsoidal shell on an external
particle are of course the differentials of the attraction of a full
ellipsoid, either homogeneous or heterogeneous according to
certain laws of varying density; and the attraction of these
latter can be inferred from the preceding values by means of
integration in the following manner :
Let a, b } c be the principal semi-axes of an ellipsoid attracting
a particle m at (a, /3, y] which is a point external to the ellipsoid.
Let the ellipsoid be resolved into a series of similar, similarly
placed, and concentric ellipsoidal shells, the density of all being
the same in a homogeneous ellipsoid, and in the heterogeneous
ellipsoid the law of density being such that it is uniform
throughout each shell. Let a0, bQ, c6 be the principal semi-
axes of the exterior surface of one of these shells, 6 being a
proper fraction ; then the thickness is such that
da _ db dc _ dQ
a b ' c ' 6
Through (a, , y) let an ellipsoid be described concentric and
confocal with the exterior surface of the elementary ellipsoidal
shell whose semi-axes are aO, bO, cO: then the equation of the
ellipsoid thus described will be of the form
of which cubic equation we take that root which makes all the
denominators positive; and let this ellipsoid be the exterior
surface of a thin shell, the interior surface of which is a similar
and similarly placed ellipsoid, and of which the thickness is such
that da^__ db^ __ dc^ __ da _ _d0
a 6 c<> " a '
Now by Article 229 the a?-component of the attraction of the
elementary shell on m is
> (183)
where a , 6 , c are the principal semi-axes of the ellipsoid
(182), andj is the perpendicular from the centre on its tangent
plane at (a, /3, y) ; and consequently the attraction of the whole
S S 2
31 GENERAL THEOREMS [232.
ellipsoid is the ^-integral of (183) as varies from to 1. So
that if x is the ^-component of the attraction,
t*2-i (184)
ao 3 o c
o>)0% c 2 = (c 2 + o>)0 2 j and
/3 2 y 2
2 " 22 '
The definite integral will take a simpler form if we make &>
the variable; for differentiating (182) we have
6*
-- dot 2 6 dQ,
m 2
J*0
and corresponding to = 0, and 6=1, o> = oo and co = n,
where n is the positive root of
And substituting these we have
x = ZTrmaabc -; (187)
^o (a 2 + &)){(a 2 + a))(^ 2 + &))(c 2 + a))}^
with similar expressions for Y and z.
If p is constant, that is, if the ellipsoid is homogeneous, these
expressions are the same as those already determined in Art. 225.
If the ellipsoid is heterogeneous, and the variation of the
density is such that when the ellipsoid is resolved into concen-
tric, similar, and similarly placed shells, p is a function of 0, that
is, of the ratio of any one of its central radii to the coincident
radius of the bounding ellipsoid, then the element-function in
(184) is a function of 6 only, and the problem requires the evalua-
tion of a single definite integral.
If p varies inversely as 6, that is, if the density of each shell
varies inversely as the length of its principal semi-axes, which
is a hypothesis made by some investigators of the figure of the
earth, the definite integral can be expressed in finite terms.
SECTION 3. General theorems in attractions.
232.] Our investigations in attractions have thus far been
limited to the action of matter contained within and filling
232,.] IN ATTRACTIONS. 317
closed surfaces of very special forms ; we proceed now to certain
general theorems which are applicable to matter distributed in a
much more general manner.
The equipotential surface is the locus-surface of all those
points (a, ft y) at which the potential of a given mass has a
given value. Thus it is the surface whose equation is
v =f(a, ft y) = C, (188)
v being otherwise defined as in (104) or (108).
Since (-7-)' (^5)' (y) are proportional to the direction-
cosines of the normal of (188) at the point (a, ft y), and since
these are also severally proportional to the axial-components of
the attraction on m, it follows that the action-line of that at-
traction is normal to the equipotential surfaces ; and as a similar
result is true for all points on the surface, the surface cuts
orthogonally the action-lines of the attraction on all particles
at its surface; and consequently if the surface were a smooth
shell capable of resisting pressure in its normal direction only,
the attracted particle would be at rest at every point on the
surface. For this reason the equipotential surface is called
a surface of equilibrium, or, as the French mathematicians term
it, surface de niveau (a level surface.) In reference to the ma-
thematical theory of heat, it is called an isothermal surface.
As c in the right-hand member of (188) is arbitrary, so as it
varies, a series of equipotential surfaces is formed ; and these all
cut at right angles the lines of action of the resultant attraction.
We thus obtain a system of lines which are cut orthogonally by
the system of equipotential surfaces, and the tangent at every
point of each of these lines is the action-line of the resultant
attraction at that point. For this reason these lines are called
lines of force. We have already had instances of them and of
equipotential surfaces. In Art. 187, all ellipses of which A and B
are the foci are equipotential lines or lines of equilibrium, and
confocal and concentric hyperbolae which intersect these ellipses
at right angles are the lines of force in the plane of the paper ;
and in space, prolate spheroids, of which A and B are the foci,
are the surfaces of equilibrium, the lines of force being a series
of confocal hyperbolae. Also from Art. 228 the equilibrium
surfaces of an ellipsoidal shell are concentric and confocal
ellipsoids, and the lines of force are the intersections of the
concentric and confocal hyperboloids of one and two sheets
318 GENERAL THEOREMS [ 2 33-
respectively which pass through the place of an attracted par-
ticle.
The corresponding points on a series of concentric and con-
focal ellipsoids are all on the same line of force.
If x, Y, z are the axial-components of the attraction, and
R, 2 = x 2 +Y 2 -f z a , the direction-cosines of the normal of the
X Y Z
equipotential surface are severally - > - > -
If two equipotential surfaces with reference to the same
attracting mass have a common point, they are coincident in
all their points ; for v = c and v = c' cannot be satisfied by the
same values of (a, /3, y), unless c = c', in which case they are
identical ; and thus they coincide in all their points.
An equipotential surface, v = c, is a closed surface ; for it is
evidently continuous ; and it cannot go off to infinity j for in
that case v = 0, and this result is inconsistent with v = c.
Of two equipotential surfaces the interior is that to which the
greater attraction corresponds.
For two successive equipotential surfaces the force of attraction
on any point varies inversely as the distance between the surfaces.
This is evident from the theorem given in (111) Art. 219 ; for if
dn = the distance between their surfaces, and R is the resultant
attraction, ,7 V
that is, R varies inversely as dn ; and the magnitude of the at-
traction is given by (189).
233.] The potential is subject to the following theorem,
which is largely applied in subsequent physical investigations.
Since
and r 2 = (a #) 2 -f(/3 y)* + (y zY ; (191)
/dv\ C C C p(aaf)dxdydz
lJ5 " ~JJJ ~^~
(192)
233-] IN ATTRACTIONS. 319
therefore by addition
This theorem was discovered by Laplace. It does not however
hold true when the place of the attracted particle lies within the
space occupied by the attracting matter, this matter being con-
tinuous ; that is, for the attraction of a body acting on one of
its own particles ; because for particles immediately contiguous
to the attracted particle r is infinite, so that v takes the form of
infinity, and its first partial differentials take indeterminate
forms; thus the preceding process is apparently incorrect, and
we must consider the subject with greater exactness. Let us
consider it first in reference to the case of a homogeneous
sphere ; let the centre of the sphere be the origin ; and let
a*+P*+y* = <T*. Then if v and V are the potentials of the
sphere according as m is internal or external to it, we have
from Art. 221,
2irp(r s
T -I.,.' -- f-;
/ dv \ 4 IT pa % f^\
" -- =
3<7 S dp 3<r' ' V' 3 a'
3 a 5
y 2 / 3<r 5
If the particle m is on the surface of the sphere, <r = a ; and
v = V ; also the first partial differential coefficients of v and v 7
become identical, so that all these quantities are continuous,
although the law of variation changes abruptly at the surface.
But the second partial differential coefficients of v and V are not
the same at the surface of the sphere ; thus,
= -3-
with similar values for the other second partial differential co-
efficients. Thus these vary discontinuously ; and the bounding
320 GENEKAL THEOREMS [ 2 34-
surface of the sphere is the locus of the points of discontinuity.
Hence we have
(3?) + (jp) + \2?) = '
We need not be surprised at these cases of discontinuity ; they
are of frequent occurrence in the application of analysis to phy-
sical enquiries, and evidently arise from the discontinuous distri-
bution of matter; we shall demonstrate hereafter, see Art. 237,
the integral equivalent of (193) by another process; and the
cause of the discontinuity will at once become evident ; and so
also will the meaning of the right-hand members of the two
preceding expressions.
This illustration also enables us to determine the value of
the left-hand member of (193) when m is a particle of the
attracting body. For let us suppose a small sphere inclosing
the attracted particle to be taken out of the attracting mass,
and the radius of it to be so small, that the density within the
sphere may be considered constant : let V be the potential of
this small sphere, and Y the potential of the whole excess of the
attracting mass over the sphere : then by reason of (193),
And for the small sphere, as just now proved,
And thus if the attracted particle is a part of the attracting
mass, since v = v + /,
This correction of Laplace's theorem was made by M. Poisson.
(193) and (194) are evidently invariants, whatever is the
system of rectangular axes to which the bodies are referred.
This theorem can be proved easily also from the formulae for
transformation of axes.
234.] The following is another proof of this theorem.*
Let the place of the attracted particle be the origin, and let
* See Cours de Me'canique, par M. Sturm ; No. 127. Paris, 1861.
234-] IN ATTRACTIONS. 321
the system of reference be that of polar coordinates in space;
so that ///
v = / / / pr sin 6 dr dd d<f> ;
and if f } g } h are the direction-angles of r,
-- = (-?-) = I 1 1 pewd cosfdr dd d<p ;
the limits of integration being in all these integrals such as to
include the entire mass of the attracting body.
When a, /3, and y vary, the variation of them causes displace-
ment of the origin ; but the only quantities which are dependent
on this displacement are p and the limits of the /-integration.
Consequently
where r t is the superior limit of r, and ^ is the corresponding
density.
Also
=]]] -f-sinddr dd d<j> jj ^ sin 6 dd d$.
rrrdp . re
III -j- sin 6 df dd d<p = / / (PI /) ) sin $ dd d<f>
J J J or J J
rr rr
= 11 piSindddddip,, I I smdddd<f>
J J J J
=//<
PRICE, VOL. III. T t
322 GENERAL THEOREMS [ 2 35-
when p is the density at the origin. Consequently
235.] Now the equivalents of the expressions (193) and (194)
are often more convenient when expressed in terms of polar
coordinates. Let the positions of the attracting and the at-
tracted particles respectively be (/, tf, <}/), (r, 0, <) : then
=///
JJJ !r> '.
-; (195)
2r/(sin0sin0'cos(< </>'; + cos0cos0 / ) + / 2 }*
and transforming (193) and (194) into their equivalents in terms
of partial differential coefficients of v with respect to r, 0, and </>,
as in Ex. 2, Art. 108, Vol. I (Differential Calculus), we have
= 0, or = 4 TT />, (196)
according as the attracted particle is not or is part of the attract-
ing mass.
The left-hand member of (196) may be put into another form
which is in many cases more convenient. Let cos 6 = f* ; then,
as in the last part of Ex. 2, Art. 108, Vol. I (Differential Cal-
culus), the expression becomes
or = 4w/>, (197)
according as the attracted particle is not or is part of the at-
tracting mass.
236.] Before I present these theorems in another form, and
from another point of view, I will shew their use in determining
the potentials of certain given masses. Hereby we shall verify
results already obtained, and exhibit them in reference to a
different process.
Ex. 1 . To determine the potential of a shell composed of a series
of concentric spherical shells, each of which is homogeneous, and
the density is a function of the radius of the shell.
Let the origin be at the common centre of all the shells ; let
/ be the radius of any shell, and p =/"(/), the density of that
shell; r j and r 9 being the radii of the external and internal
bounding spheres. Let r be the distance from the origin of
236.] IN ATTEACTIONS. 323
the attracted particle ; then v is evidently a function of r only,
and is independent of and <f>.
(1) Let m be external to the shell ; then (196) becomes
d*v 2 dv
where c is an arbitrary constant ; but c evidently is equal to the
mass of the shells, because the attraction of a system of con-
centric shells on an external particle varies as the mass of the
shells. Hence fa M
dr~ ~~r*'
.'. v= , (199)
T
no constant having been introduced, because v = when r = oo .
If the sphere is not full, but there is a hollow cavity within
the interior shell, and m is in that cavity, then in (198) c = 0,
because the resultant attraction vanishes for all positions of m
within that cavity ; consequently for all such positions v is a
constant.
(2) Let m be within the shell occupied by the attracting
mass; then (196) becomes
d a v 2 dv
0; (200)
__
whence we have r-j-+4ir/ rp^r = 0; (201)
"* Jtt
the lower limit of integration having been brought in, because
/]?V
-T- = 0, when r = r . Thus
--'* (202)
but 4 TT / r 1 p dr = the mass of the shell, the external and in-
J r
ternal radii of which are respectively r and r ; consequently
and thus the attraction of the shell varies as the mass contained
within the concentric shell bounded by the internal bounding
surface and the concentric spherical surface which passes through
the attracted particle, and inversely as the square of the radius
T t 2
324 GENERAL THEOREMS [236.
of the latter shell. Consequently the matter lying outside that
surface produces no attractive effect on the particle.
Again, from (202),
dv 4ir f r . ,
-r = -- r / r*f*r
dr r a J ro
A. r*
lirpr+lTtpr -- - I r*pdr;
r *Vo
Tr'pdr,
r Jr
(204)
the lower limits having been introduced in accordance with the
remark made at the end of Art. 221.
Ex. 2. To find the potential of a system of concentric cylin-
drical shells of infinite length, and coaxal, of each of which the
density is uniform.
If the origin is taken at a definite point on the axis, and the
axis of the cylinder is taken to be the z-axis, it is evident that
j- = 0, and that consequently the resultant attraction is per-
d'y
pendicular to the z-axis. Hence also -T-J = ; whence, as v is
evidently a function of r only, we have
* 1 dv
(2 5)
according as the attracted particle is without or within the space
occupied by the attracting matter.
If m is outside the cylinder, by integration of (205) we have
dv
r^ = c, (206)
where c is an arbitrary constant. If the cylinder is hollow, for
all places of m within that cavity the attraction vanishes, and
consequently c = ; for all points external to the cylinder, the
attraction varies inversely as the distance from the axis of the
cylinder, and r
v = clog , (207)
r o
where r is the radius of the internal surface.
If m is in the space occupied by the attracting matter,
d\ C r
r-j- = 4iT prdr, (208)
Or J ro
dv
where r is the radius of the interior surface, and -y- = when
dr
r- r,.
237-] IN ATTBACTIONS. 325
Ex. 3. The potential of matter arranged in parallel plates, each
of which is of uniform density and of infinite extent, may thus
be found :
Let the #-axis be perpendicular to the plates ; so that the
resultant attraction must be parallel to it, the attractions parallel
to the plates vanishing; consequently (-7-5)= (-?-) = ; and
N a/3' x ay x
therefore , a
-=0, or = -47rp, (209)
according as the attracted particle m is without or within the
space occupied by the attracting matter.
For positions of m outside the plates
where c is an arbitrary constant ; and consequently the attraction
of the plates is the same, whatever is the distance of m from the
surface of the plates.
If the particle is within the plate,
= 4wpa, (211)
if the density of the plates is constant, and the origin is placed
midway between the bounding surfaces of the plates, so that the
attraction vanishes when o = 0. Thus if the whole thickness of
the system of plates is t, when a = - > =- = 2-npt', and when
t dv 2
a = -> -=- = 27rp; and the whole attraction of the plates on
2 do,
a particle at the surface = 2,-npmt.
237.] The theorems concerning the partial-differentials of the
potential which have been demonstrated in Arts. 233 and 234,
and have been applied in Art. 236, admit of statement in an
integral form: and in that form have been demonstrated by
Gauss, Sir William Thomson, aod Professor Stokes. The follow-
ing is the most elementary proof :
Let s be any closed surface, and let o be a point either within
it or without it, at which is a particle m' of the attracting matter.
Then if ds is a surface -element of s, and dn is an element of the
normal to the surface drawn outwards at the element ds, so that
326 GENERAL THEOREMS [ 2 37-
dv .
-5- is the force which the attracting matter exerts at ds in a line
normal to s,
dv
-j-ds= 47TM , or =0, (212)
where M O is the whole attracting matter within s, according as
o is within or without s.
Firstly, let o be within s. From o draw a straight line OL
cutting the surface, and produce it in one direction so far that it
cannot cut the surface again. OL will cut s in one point p x at
least, and may cut it in more points if s is a re-entrant surface :
but the number of points of section must be uneven. Let these
points be r lt P 2 , P S , ... ; and let opj = r u OP 2 = r 2 , ... ; about
the line OL and including it, from o as a vertex let a cone be
described, at the vertex of which is the small solid angle o>.
Then if from o as a centre, spheres are described with radii
r 1} r i} . . . , the areas of the spherical surfaces intercepted within
the cone are severally w/x*, a>r a 2 , ...; let ds^ds,,... be the
surface-elements of s at p^ p a , . . . intercepted by this cone, and
let d lt da,... be the angles at which dn } the normal-element, drawn
outwards, at each of these points is inclined to the line OL : then
o>r 1 *=ds 1 cosO l ; o>r 2 3 = ?s a cos0 a ; <or s a = ds 3 cos0 s ;... (213)
Let N!, N 2 , ... be the components along the normals to a at
p n p a , ..., of the attraction of these surface-elements on m' ; so
that m ' m '
ifv lit/ f .
N! = cos0 x ; N a = cos0 a ; (214)
consequently
the number of terms of which is uneven : and
.*. S.N<?S = m'd).
Let this process be repeated for all angular space about o ;
this is effected by the integration of the left-hand member
through the whole of s, and by the corresponding integral of w,
which is 4 77 ; and thus for m }
yds =4irm'.
But a similar result will be true for every particle of the
attracting matter M O which is contained within s ; in which case
237-] IN ATTRACTIONS. 327
N must be replaced by its value -5- given in (189), where v is the
potential of M O ; so that
rfs=-47TM . (216)
Secondly, let o be without s. From draw a straight line OL
as in the former case, cutting the surface in p,, p a , ...; the
number of these points of intersection is at least two, and may
be greater ; but the number is necessarily even. The same con-
struction and the same symbols being used as in the former
case,
u>r 3 * = ds 3 cos0,;.., (217)
in m M
= -COS0,; N, = - COS0 a ; N, = - COS 3 ',. . . (218)
" *
= o,
as the number of terms in the series is even ;
..-. 2.Nd?S = 0.
Let the process be repeated for all that angular space about o,
which is necessary to include the whole surface of s ; then for all
that surface, and for m', we have
= 0.
Also a similar result will be true for every other particle of
the attracting mass which is outside s ; and if v is the potential
of all that matter, then replacing N by its equivalent,
^ds = 0. (219)
dn
Hence we have the theorem :
If v is the potential of any mass M, part of which, viz. M O , is
within, and the rest is without a closed surface s, of which dn is
an element of the normal drawn outwards, then, the whole
surface of s being the range of integration,
(220)
Or, in other words, If we consider the attraction of a mass of
matter on the surface-elements of a closed surface, the sum of the
attractions estimated along the normal to the surface at its
several points drawn outwards is equal to 4TTM,,, where M O is
the attracting matter within the surface. Equation (181) in
Art. 230 contains a particular case of this last theorem.
328 GENERAL THEOREMS [238.
The proof here given of course includes the case in which the
closed surface is an equipotential surface, and in which the re-
sultant attraction on the surface-element acts along the normal.
238.] Laplace's equation (193) and Poisson's extension of it
(194) may be deduced from (220) by the following process, due
to Professor Stokes* :
Let us take the more general case of the attracting particle
being within the surface s, and let us take this surface to be
the surface of the small elementary parallelepipedon da dfi dy.
Then for the face dydz the value of / / -y- ds is (-,-) dp dy ;
and for the opposite face it is {(-7-) + (~r~i) ^ a } dfi dy ; and
7
therefore for this pair, the value of the integral is (-j^
similar results are true for the other two pairs of faces ; so that
the left-hand member of (220) becomes
Now the density in the elementary parallelepipedon being
constant, and being p, say, M O = pdadfidy; therefore from (220)
. .
If the particle is outside the parallelepipedon, the right-hand
member vanishes.
Also (220) may be deduced from (222) by integration in the
following way :
Let (222) be multiplied by da dp dy, and be the subject of
integration through a given space within a closed surface s,
which contains the attracting matter M O ; then
) + () + 0} <"> = - 4 *///' **""
Now // j (-r^dadpdy = /7 [( 3-)] *0 <*7i where a t and
a are the abscissae to the points where a line parallel to the
axis of a, at distances /3 and y from the planes of (y, a) and
(a, /3) respectively, cuts the surface ; and if ds t is the surface-
element at (a, j3, y) and X is the angle between the normal at
* Cambridge and Dublin Mathematical Journal, Vol. IV, p. 215. In the
same memoir the reader will find references to the works of Gauss and Sir William
Thomson.
239-] IN ATTRACTIONS. 329
that point drawn outwards and the a-axis, df} dy = ds l cos A,
= dSo cos A ; and if a cylinder is described with the generating
lines parallel to the axis of x, and circumscribing s, the line of
contact divides s into two parts; and the range of
is the part of the surface which is farthest from the origin ; and
the range of / / (-j )cos A ds 9 is the part nearest to the origin.
Hence if the range is the whole surface of s,
/ / /
and the other two parts of (223) will take similar forms; also
p dx dy dz = M O ; therefore (223) becomes
(225)
but since A, /U, v are the direction-angles of dn, which is an
element of the normal of s drawn outwards, by (111), Art. 219,
, / dv -^ f dv \ , dv N dv
COSA +cos " +cos = --
' (227)
and this is the integral equation of the normal attraction
through a closed surface.
239.] The potential does not admit of a maximum or mini-
mum value at any point in free space, where there is no attracting
matter. For in this problem v is a function of a, /3, y ; and is
subject to the condition given in (193); and thus in (39), Art.
163, Vol. I (Differential Calculus), the coefficient of 6* vanishes,
and consequently the three roots of that cubic equation cannot
be all of the same sign ; but this condition is necessary when v
has a critical value, and accordingly v does not admit of a maxi-
mum or minimum value.
This fact may also be inferred from the statements of Art. 237.
For did the potential admit of a maximum or minimum value
at a point in free space, a closed surface could be described
about that point, and so near to it, that at every point within
it the potential would be less or greater than that at the
point ; so that -y- would be negative or positive all through the
d/n r r ti
surface, and consequently / / -j- ds would be finite ; and this
PRICE, VOL, in. n u
330 GENERAL THEOREMS [240.
is impossible, as such a surface contains none of the attracting
matter.
Hence if the potential is constant at all points on the closed
surface, s, which includes none of the attracting matter, it has also
the same value for all points within s. Because if this is not so,
there must be one or more critical values within it, and this has
been shewn to be impossible.
As v does not admit of a maximum or minimum value in free
space it increases in some directions and decreases in others,
remaining constant for all points on an equipotential surface;
consequently a material particle under the action of attracting
bodies cannot be in a position of stable equilibrium. This theorem
is due to Mr. Earnshaw, and is given by him in the " Philo-
sophical Transactions of Cambridge/ ' Vol. VII, March, 1839.
240.] The theorem given in Art. 237 leads to the following
very remarkable results in reference to equipotential surfaces :
Let an infinitesimal area be taken on an equipotential surface,
v = c ; and let the normals to the surface be drawn all round
the contour of the infinitesimal area; these evidently form a
tubular surface of small section. Also let another equipotential
surface, v = c ', be drawn intersecting the tubular surface : let o>
and 00" be the areas of the sections of the tubular surface made
by these equipotential surfaces. Let R and B,' be the forces of
attraction on o> and a/ respectively, the lines of action of them
being normal to the areas of o> and <o' ; then as there is no force
of attraction perpendicular to the sides of the tube, as the gene-
rating lines are lines of force, and as no attracting matter is
supposed to be within the tube, by (2 1 9) we have
Eco nV= ; (228)
that is, the attracting force is inversely proportional to the
section of the tube. This theorem gives the variation of the
attraction along a line of force so long as it does not pass
through attracting matter.
If the tube contains attracting matter and the volume between
o> and G>' is full of matter, then the difference of the whole at-
tractive forces on the two ends of the tube varies as the quantity
of matter contained in the tube between these two areas.
The following are particular cases of this theorem* :
When the attracting body is symmetrical about a point, the
* See Thomson and Tait, Natural Philosophy < Vol. I, p. 365. Oxford, 1867.
241.] IN ATTRACTIONS. 331
lines of force are obviously straight lines drawn from this point.
So that in this case the tube becomes a cone, and o> is propor-
tional to the square of the distance from the vertex ; conse-
quently the attraction varies inversely as the square of the
distance.
If the attracting matter is distributed symmetrically about an
axis in cylindrical shells of infinite length, the lines of force are
perpendicular to this axis, and the tube becomes a wedge, the
section of which is proportional to the distance from the axis j
and the attraction therefore varies inversely as the distance from
the axis.
If the attracting matter is composed of a system of parallel
plates, the lines of force are all parallel, and the tube becomes a
cylinder, the area of the section of which is the same at all
distances, and consequently the attraction is the same at all
distances.
241.] The following theorem in the Integral Calculus, dis-
covered by George Green, and contained in his " Essay on the
Theories of Electricity and Magnetism," is more general than
the preceding; and is fundamental in many subsequent in-
vestigations.
If s is a closed surface containing a given quantity of matter,
and ds is a surface-element of it, and dn is the element of the
normal drawn outwards ; and if u and v are two functions of
a, )3, y which do not become infinite at any point within s, then
dv 7
a*
f/Y (X' v \ fd*v
JJJ D d + <
in which the triple integrals comprise all the matter contained
within the surface s, and the range of the double integral is the
whole closed surface s.
Let us take the integrals in the left-hand member of the
equality ; then
rd*v^
U U 2
332 GENERAL THEOREMS [241.
replacing the first definite integral by its value as given in
Art. 238 ; so that the whole left-hand member of (229) becomes
/*/* dv -
V/ D * *"
but the third of these integrals is equal to the first, as we have
explained in Art. 238 ; so that omitting these terms, the quan-
tity becomes
M<?UV /^Vv
^)(^
Now as this expression is symmetrical with regard to u and v,
it is likewise the value of the second member of (229) : so that
the theorem is hereby established, whatever is the form of the
functions u and v.
If, however, one of the functions u and v becomes infinite at
any value of a, /3, y within s, certain corrections must be made.
Let us suppose u to be infinite at a point P O (a , /3 , y ) within s;
and let us suppose v to become - at that point. Let a sphere be
described from that point as centre with an infinitesimal radius
= a ; then the preceding theorem is manifestly true for all the
attracting matter external to this sphere. And with regard to
/ x ,^ /^
the sphere, since v = ->() + () + ( ) =
same vaiue
whether the district of integration is diminished by the volume
of the sphere or not. Also
is by the principles of definite integration equal to / / idadfidy,
which is the volume of the sphere, multiplied by some mean
1 j/rf'Uv , /^ 2 iK /^ 2 u>J , . , . . ., ,
value ot -iij ) + (-^ ) + (-; }> which is in magnitude of
r l^da*' W 2 ' \d*'
the same order as -, since (-^ W-.. is finite for all points
r \da*'
within the sphere; and consequently when the radius of the
242.] IN ATTRACTIONS. 333
sphere is infinitesimal this must be omitted. Also / Iv-j-ds
= a mean value of -5- multiplied by / / vds -lit a* 4ira;
and this vanishes when a = 0. Also / / u -=- ds = U / / -j- ds,
where U is the value of u at (a , /3 , y ) ; and since v = - ,
dv dv 1 rfdv , I
-j- = -j- = - ; consequently / / -,- as = -
rf dr r* J JJ dn a*
= 4ir ; so
that in the limit, for the sphere,
dv
-rds = 47ru .
dn
Hence when the whole ranges of integration are considered,
//A *+
Similarly, if v becomes infinite at any point within the range
of integration, an analogous correction must be made for it ; and
also similar corrections for any other points, whatever be their
number, at which such values take place.
242.] One or two remarks have to be made in conclusion :
(1) Throughout this Chapter I have spoken of attracting
masses, and have denoted mass by the symbol m, and mass-
element of the attracting body by the value given in Art. 121;
viz. pdxdydz ; and I have retained this conception, to give con-
sistency to the imagined action. But the preceding theorems
are of much wider application than to gravitating matter in the
ordinary meaning of the word : they apply to electrical and
magnetical action ; and thus the meaning of m must be en-
larged ; and must be taken to denote quantities of attracting
action or of influence, whether of free electricity or of magnetism,
whatever these may be. It is indeed with reference to these
latter subjects that the theorems are so important.
(2) We have spoken always of attraction. But the theorems
are also true for repellent action when the repulsion varies
directly as the products of the repelling masses and inversely as
the square of the distance between them. This extension will
334 GENERAL THEOREMS IN ATTRACTIONS. [242.
be made in the mathematical expressions of the theorems, if the
masses are affected with negative signs ; and such a change is
necessary in applications to electricity, where two influences,
positive and negative electricity, are in operation.
(3) If a good conducting body is charged with electricity, and
then placed in a good non-conducting medium, such as dry
air, there is equilibrium in the interior, and the remaining free
electricity passes to the surface of the body, and there forms
a shell of varying density or power, which is kept at rest by the
pressure of the external air. For since the interior is at rest,
the potential throughout is constant : and consequently
and thus, by (194), 4:rp = ; and p = ; whence it follows
that there is no electricity in the interior, and the free electricity
is carried to the surface.
(4) And the free electricity forms a shell which is in equi-
librium, of which the thickness may be considered constant, and
the density variable. This shell has no action on the interior
parts of the body, and consequently its interior surface is at
rest. The exterior surface is open to the pressure of the air, and
as this acts only normally to the surface, the exterior surface
of the shell is an equipotential surface.
(5) If the electrised body is an ellipsoid, the shell of electricity
on its surface will be ellipsoidal, and we may consider it to be of
constant density, and of variable thickness, and to be contained
between two similar and concentric ellipsoids, so that the thick-
ness at any point varies as the central radius vector to that
point.
(6) The repulsive action on any particle in its external surface
is normal to the surface at the point, and proportional to the
thickness at the point.
(7) The repulsive action on a particle at different points on
the external surface is proportional to the perpendicular distance
from the centre on the tangent plane to the ellipsoid at the point ;
so that at the extremities of the principal axes the repulsive
action varies as the length of the axis.
ANALYTICAL MECHANICS.
PART II.
DYNAMICS; THE MOTION OF MATERIAL PARTICLES.
CHAPTER VIL
MOTION, ITS AFFECTIONS, ITS LAWS, AND ITS EQUATIONS.
SECTION 1. Introductory; on motion, matter, time, space.
243.] On resuming the course of our treatise of the science
of motion which was interrupted at the end of Article 11, it is
necessary to make some preliminary observations.
Mechanics is the science which treats of the action and effects
of forces on material particles and bodies at rest and in motion ;
that part of it which relates to bodies at rest, that is, under the
action of many forces in equilibrium, is called Statics, and has
been discussed in the preceding part : and that part of it which
relates to motion is called Dynamics, and has lately been termed
Kinetics, and will be developed in the following parts of the
work : the passage from the latter to the former, and the process
by which the principles of the latter include those of the former,
as the general science includes its particular subordinate, will be
investigated hereafter.
Dynamics, as it is intended to unfold the subject in the fol-
lowing pages, will be presented to the student in a twofold
aspect : primarily and chiefly it will be considered as a positive
and exact science, such as I have attempted to sketch it in the
first Chapter ; and of that nature of which the pure sciences of
number and geometrical space are supposed to be. Motion is
the fundamental idea of it ; that, viz., out of which spring all
336 DYNAMICS. ITS TWOFOLD ASPECT. [243.
the truths of the science, and from axiomatic statements of
which they are deductively inferred. Dynamics, as such, is a
science of speculation and thought ; doubtless in the construc-
tion of it experience may have suggested much, but the so-
called necessity of its principles is derived from another source.
Secondarily, it is my purpose to shew that the science is useful
to explain phaenomena of the world external to us : hence arises
the necessity of proving that the axioms and the first statements
of the pure science are true in the subject-matter of cosmical
observation, and that the laws of natural phaenomena are in-
cluded within the range of the pure science. Now for this end
large experience, in the way of observation and experiment, is
frequently required. The operations of nature are complex, and
it is only with deep searching that they allow themselves to be
so far unravelled as to exhibit the laws they are subject to. In
this respect then it is necessary to apply a limit to our inquiry ;
and I propose only to shew, and that concisely, that the axioms
of the pure science, or the laws of motion, are true in cosmical
matter ; so that, thus far at least, it is likely that we are on the
right road of natural philosophy. And it will also be desirable,
here and there, to point out certain salient laws, such as the
law of gravitation and Kepler's laws of planetary motion, that
our attention may be directed to them rather than to others.
As in the preceding Chapter it was beside our object to enter on
the complete discussion of attractions as applied to the determi-
nation of the figures of the earth and of the planets, to the theory
of heat, and to magnetism and electricity, because such appli-
cations can be made only on certain hypotheses, and with the
development of functions in series involving infinitesimal terms,
the knowledge of which belongs to the special subject : so in the
following treatise I shall not enter on the planetary or lunar
theories, because such subjects require special knowledge, and
belong to physical astronomy : but the general equations of
dynamics will be investigated in all their breadth, and will be
brought down to that stage where' these special sciences com-
mence ; and will not, except in very simple instances, be applied
to cases or under circumstances wherein such special knowledge
is required. Thus our science is a principal and normal one ;
normal, I say, because it is that to the rules of which each
special subordinate science conforms : and the greater or less
that conformity is, the more or less complete is that special
244-] MOTION AND MATTER. 337
science ; and principal) because the laws of dynamics are those
which the special science takes and applies, each in its form and
degree ; and they are so large, that many forms of them are
included, which observation has not yet shewn to exist in the
material universe. The applied part also serves a moral purpose,
insomuch that it enables man to fathom the depths of the laws
of Cosmos, to express them in a concise form, and thus to study
the works of God. It is for these reasons that the science of
motion is the most perfect of the physical sciences.
Although philosophically perhaps it might be more correct
separately to investigate these two branches of the subject, yet,
as the treatise is didactic, it is more convenient to consider parts
of one or the other, as they arise in the course of it.
The nature of the symbols which will be employed requires a
remark ; we shall have to speak of time, space, velocity, matter;
these are heterogeneous quantities, and cannot be operated on
so as to multiply time into space or mass into velocity ; this is
self-evident. But these quantities will be represented by sym-
bols such as t, dt, s, ds, v, dv, m, dm; and these are numbers,
and not the concrete things. Thus t expresses the t times an
unit of time is taken ; dv the dv times an unit of velocity is
taken ; and the numbers, of course, can be multiplied together,
and the resultant of the operation is number of that kind which
the symbols express before the operation. The unit of con-
cretion however, which is to be introduced after the operation,
may be different to that previous to the operation : see Art. 124.
The concrete units are of course arbitrary, but remain unaltered
during the whole of an operation. Sometimes a second, some-
times a year is taken as the unit of time ; sometimes a foot,
sometimes the earth's radius, sometimes the mean distance of
the earth from the sun, is taken as the line-unit ; these units
vary according to the problem ; and the circumstances of it will
generally guide us to a judicious choice.
244.] Motion is the fundamental idea of mechanics ; motion,
that is, either real or virtual, either in act or in power ; and
therefore the science is more correctly termed the science of
motion. Motion need not be defined : it is too general to be
capable of useful expression by means of a more general term ;
it is a quality or a state : one result of it is change of position of
the thing moving : I say, thing moving ; for a necessary element
in an adequate conception of motion as the fundamental idea of
PRICE, VOL. III. XX
338 MOTION AND 1IATTEB. [245.
mechanics is that something 1 moves : motion exists in, and is of,
something ; and that something, in which it is, and of which it
is a state, is matter.
Now although we do not know matter as free from force
and consequently from motion of which force is the cause, and
we do not know force except as affecting matter, yet the con-
ceptions of motion and matter are distinctly separate, and will
be advantageously considered separate from each other. The
body of doctrine concerning abstract motion, that is, concerning
motion in itself and free from all consideration of its subject and
of its causes, is called Kinematics, and will be treated of in the
following section, and in other parts of the work as the necessity
for it arises. It is a geometrical subject, and the limits of it
are those which geometry imposes. When complete it embraces
the whole theory of pure mechanism, for it teaches all possible
kinds of motion, and the modes of transmitting and converting
them*. We shall give only those elements which are required
for the purposes of this treatise.
When however motion is treated of in connection with matter
as its subject, and as the result of force acting on matter, the
body of doctrine concerning it is called Mechanics; and Dynamics
is that part of mechanics where the force produces active motion
in matter.
It will obviously be convenient to treat these two subjects
separately from each other ; and the former of course is ante-
cedent to the latter. I shall consider them in their most simple
forms in the two following sections of this Chapter ; but I
must first make some other general observations on motion and
matter.
245.] When I speak of matter as the subject of mechanics,
the term is not limited to the matter of the members of the solar
system ; to that which has sensible properties, and which gravi-
tates; but it embraces everything that moves or is capable of
motion ; the particles of air of course are included ; and they
gravitate, and they are the subject-matter by the vibrations of
which sound is propagated; the particles of light which the
emission-theory of light assumes, and the aethereal molecules of
* Many excellent treatises exist on this special subject. Let me mention but
two of marked excellence : (i) Willis' Principles of Mechanism ; London, 1841 :
(a) Cour de Me"canique et Machines, par Edm. Bour (premiere fascicule, Cine'-
matique) ; Paris, 1865.
245-] SPACE AND TIME. 339
the undulatory theory, are also included. Matter is treated as
the subject of motion ; and when it is spoken of, it is supposed
to have one essential property, and that is mobility.
Matter also admits of divisibility without limit : a very large
quantity of it may have motion, or a very small, nay, an infi-
nitesimal part of it ; and this is called a particle ; such as is
analogous to a geometrical point : and its other properties,
mobility and such like, are independent of the quantity of it.
This remark is important ; because it will be necessary to divide
the subject, according as we consider the motion of a finite
quantity of matter, which is supposed to consist of an infinite
number of particles, and which is called a material system or a
material body: or according as we consider that of a material
particle. The quantity of matter which a body or a particle
contains is, as already stated, called its mass.
Of motion, and consequently of matter with reference to its
property of mobility, there are two other affections, which, by
reason of their abstract nature, need not be defined : viz. time
and space : it is sufficient for us to be able to form a notion of
them, and to enunciate of them such properties as are required
for our purpose. Space and time, like matter, are continuous
and divisible ; and these aifections are without limit. Space
may be very large, nay, infinite ; we cannot fix the boundaries
of that space in which the heavenly bodies are ; and it may be
very small, such as that occupied by a chemical atom or a
material particle. Time also admits of degrees as to quantity ;
it may be an instant ; such an infinitesimal, that the aggregate
of an infinity will make only finite time : or it may reach through
the present moment from ages bygone to ages to come. Motion,
matter, time and space, stand to each other in the following
relations. Matter exists in space and time; all matter, even
the minutest particle, occupies space. No two particles of matter
and also no two bodies can occupy the same space at the same
time ; this property of matter is called its impenetrability. The
same matter cannot be in two different places at the same time :
hence a particle of matter or a body cannot pass from one posi-
tion to another without lapse of time : time is consumed in the
passage ; and therefore a change of place requires time. And
as a longer or a shorter time may be spent in the passage, so do
we conceive of the rate or speed at which a particle or a body
moves. And hence arises the quality of matter which is called
x x 2
340 KINEMATICS OF A PARTICLE. [246.
velocity ; velocity being the degree of swiftness or slowness with
which matter moves. From these relations arises the necessity
of measuring space and time, and of determining equal spaces
and equal times. As material bodies exist in space, they have
volume and form; volume depending on the quantity of space
which they occupy A and form on the bounding terms of that
space ; but the knowledge of equal spaces must be found in an
adequate knowledge of space. The method of measuring volume
is founded on the geometrical principle of superposition, and
two volumes are equal which occupy the same or equal spaces.
The notion of equal times and also the measure of equal times
arises out of the idea of time, and an idea of time is not ade-
quate unless it has these notions ; it is true that the passage of
time is marked by events which take place in it ; and equal
times are marked by the regular recurrence of similar events ;
that is, by uniform motion ; but equal times * are in themselves
altogether independent of any particular kind of motion ; they
exist before it and they enable us to apprehend and to measure
such a motion : equal times therefore must be deduced from
the notion of time.
SECTION 2. The Kinematics of a particle in a straight path.
246.] The most simple motion of a material particle is that in
which it describes along a straight line equal linear spaces in
equal times ; the motion of it is then said to be uniform, and
the velocity to be constant ; these two expressions in fact being
equivalent. But when equal spaces are not described in equal
times, the velocity is said to be variable ; such a velocity may
vary continuously or discontinuously ; but it will be necessary
for us to consider only a continuously- varying velocity ; because
a discontinuous variation will be a succession of constant velo-
cities, changing abruptly, and, as it were, by impulses.
* M. Poisson, Trait de Me'canique, 2^e Ed. Tome I, p. 205, writes : " La notion
des temps e'gaux, et la mesure du temps ne sont fondles ne*cessairement sur aucune
loi particuliere de mouvement, et Ton peut, en consequence, les supposer dans la
definition du mouvement uniforme et de toute autre sorte de mouvemens."
Dr. Whewell, in his Treatise on Mechanics, Ed. 5, Art. 102, says : " Those
intervals of time, in which there is no discoverable reason why they should be
unequal, are supposed equal."
246.] VARYING VELOCITY. 341
In the case of constant velocity, equal linear spaces are de-
scribed in equal times ; now although the velocity of a moving
material particle is a quality or state of the particle itself and
resides in it, and is that by which it differs from a particle at
rest, and although no account more exact can be given of it,
yet the velocity can be measured ; and the measure is taken to
be the number of units of linear space passed through in an unit of
time. If therefore a material particle describes uniformly v units
of linear space in one unit of time, v is the measure of the velo-
city : and if s represents the space passed through by the particle
in t units of time, then, bearing in mind the last clause in
Art. 243, 8 = vt; (1)
and v = - - (2)
t
Thus velocity is linear space, and is the linear space described in
one unit of time.
I may observe that, if a particle describes a path with uniform
velocity, this result is true whatever is the form of the path, be
it straight or curved, a curve plane or of double curvature.
If the velocity continuously changes, equal spaces are not
described in equal times, and the velocity becomes a function of
the time. Let the time be resolved into infinitesimal elements ;
and let us suppose the particle at the end of the time t to be at a
distance s from an origin fixed on the line, and to be at that
time moving with a velocity v : that is, if the particle were to
move for one unit of time with the velocity which it has at #,
it would describe v units of space in that unit of time ; and
suppose ds to be the space described in dt, the next element of t ;
then, if v is the velocity at the beginning, and v + dv is the
velocity at the end, of dt, the mean velocity with which ds has
been described may be expressed by v + 0dv, where is a proper
fraction, and is positive or negative according as the velocity is
increasing or decreasing : therefore by reason of (1),
ds = (v + Qdv}dt;
and neglecting the infinitesimal of the second order, as by the
principles of infinitesimal calculus we are obliged to do, we have
ds = vdt; (3)
that is, ds units of space are described in dt units of time by
the particle moving with the velocity v at the beginning of dt ;
ds
and therefore dividing through by dt, we have -^7 equal to the
clt
342 KINEMATICS OF A PARTICLE.
space described in one unit of time ; and this is velocity ; and
thus we have
In the cases therefore, both of constant and of varying 1 velocity,
velocity is the space described in an unit of time ; and is, by
reason of (2) and (4), the ratio of the space described to the
time during 1 which it is described; and is in the latter case the
ratio of two infinitesimals.
The unit of velocity is evidently the velocity with which a
particle describes uniformly an unit of space in an unit of time.
It will be observed that Q has disappeared : now as it is upon
the sign of 6 that an increasing or decreasing velocity depends,
so are the results (3) and (4) true in both cases.
The following are examples of the preceding theory :
Ex. 1. If a particle describes uniformly 100 feet in 10 seconds,
and a foot and a second are respectively the line-unit and the
time-unit, the velocity of the particle is 10. But if the time-
unit is half a second the velocity is 5.
Ex. 2. Find the position of a particle at a given time when
the velocity varies as the distance from a given point on the
rectilinear path.
ds 7 ds
Here ks; .'. = kdt\
at s
o
.. log M; s = s ^ f ',
s o
if is the distance of the particle from the origin, when t = 0.
Ex. 3. Find the position of the particle when the velocity
varies as the time.
Here
dt
Kt
where s is the value of*, when t = 0.
247.] Let us now suppose the particle to be moving along
its path with an increasing (or decreasing) velocity ; and to fix
our thoughts let us suppose the velocity to be increasing ; then
this increase may take place either uniformly, or at varying
rate.
Firstly, let us suppose the increase of velocity to take place at
an uniform rate. Let us suppose /"to be the increment of velocity
247-] VELOCITY-INCREMENT. 343
in an unit of time ; then if the velocity is zero at the beginning
of t, and the velocity is v at the end of f,
v =ft; (5)
but if v is the velocity at the beginning of t t and v is the velo-
city at the end of /, v _ v<t= f l . (6)
so that the velocity-increment in the time t varies as/* and also
varies as t.
f, which is the velocity-increment in an unit of time, is often
called the acceleration; but as the former term is more sug-
gestive, I shall generally employ it ; it evidently takes a negative
sign if the velocity decreases as the time increases. When the
velocity increases uniformly, f is constant, and the acceleration
is constant ; these two expressions being equivalent.
If the particle at the beginning of t is moving with a velocity
v , and the velocity-increment is negative, then if v is the velo-
city at t, v = v 9 -ft. (7)
Also since the particle is at rest when v = 0, that rest takes
place when ~
-f w
Secondly, let us suppose the increase of velocity to take place
at a varying rate, so that there are not equal increments of
velocity in equal times ; then the increase of velocity is a func-
tion, either explicit or implicit, of the time.
Let the time be resolved into equal elements ; and let us sup-
pose the particle at the time t to be moving with a velocity v,
and at the time t + dt to be with a velocity v + ds', then if f is
the velocity-increment at the time t, andy + df at the time t + dt,
f+Odf, where Q is a proper fraction, is the mean velocity-incre-
ment during the time dt ; and consequently by reason of (5),
.-. dv =fdt, (9)
if we omit the infinitesimal of the second order ; that is, dv units
of velocity are added in the time dt. Hence, dividing by dt,
*-/ <
In this latter case the velocity-increment or acceleration is said
to vary. And thus whether it is uniform or varying, it is the
increase of velocity in an unit of time ; and is also the ratio of
the increase of the velocity to the time in which that increase
344 KINEMATICS OF A PARTICLE. [248.
takes place, and is in the latter case the ratio of two infini-
tesimals.
Thus the unit of acceleration or the unit of velocity-increment
is, when the increase of velocity is an unit in an unit of time.
If the velocity decreases,/" is negative ; and from (10) we have
-=-/. (ID
dt J ^ }
These expressions shew that an unit of acceleration is that which
corresponds to an uniform increase of an unit of velocity in an
unit of time.
248.] Taking these results in combination with those of the
preceding Article, we have the following values :
In the general case of varying velocity and of varying positive
acceleration, from (4) and (10),
dv d ds
(12)
dt 3
and therefore if * is equicrescent,
and if t is equicrescent,
.
We shall suppose t to be an equicrescent-variable throughout
the whole treatise, unless it is stated expressly that it is not so.
These values suggest the following remarks :
Let a particle be moving, and let it describe the space * in
the time t :
(1) Let us suppose the space and the time to be resolved into
corresponding infinitesimal increments, so that neither all the
dfs nor all the ds's are equal : in which case neither t nor * is
equicrescent; and thus (12) correctly represents the velocity-
increment due to one unit of time : but the expression is un-
necessarily complicated, and is therefore of little practical use.
(2) Let the time be resolved into equal elements, that is, let t
be equicrescent: then d*t = Q, and (14) expresses the velocity-
increment. Now if the velocity is constant, all the corresponding
elements of space will be equal : that is, all the ds's will be equal
and d's = : there will, in this case, be no velocity-increment.
249-] VELOCITY-INCREMENT. 345
If the velocity is not constant, the d^s corresponding to equal
dfs will not be equal ; there will be an excess of one ds over the
preceding- or succeeding ds, and thus there will be a d's : as dt
is constant, let us assume it to be the unit of time : then ds is
the velocity ; and d*s is the velocity-increment ; and therefore
measures the acceleration force. It is also to be observed, that
if the velocity-increment is constant, d*s is constant, and there-
fore d 3 s : but if on the other hand the velocity-increment is
variable, the d's's vary, and d*s is not equal to zero : similarly
we might proceed, and shew under what circumstances d 3 s would
be constant, and therefore d's = 0.
(3) Let the space be the equicrescent variable ; in which
case, if the velocity is constant, the dfs corresponding to the
ds's are equal, and d*t = ; but if the velocity is not constant,
equal dfs do not correspond to equal ^s, and therefore (fit
will not be equal to zero : in this case (13) is the expression for
the velocity-increment, f being affected with a negative sign
because the velocity-increment becomes greater, as the time to
which it is due becomes less ; and therefore the dfs, to which
equal successive ds's are due, are decreasing, and therefore d*t is
negative.
249.] The following are simple illustrations of the preceding
formulae ; it is unnecessary to add others as the subject will be
amply applied in the succeeding Chapter.
d 3 s
Ex. 1. If there is no velocity-increment, -= = ; so that if
Cvv
v 9 is the constant velocity,
ds
Ex. 2. If the acceleration or velocity-increment is constant,
* = *o+M + 2,/*'J
where s and v are the values of the space and the velocity
respectively, when t 0.
Hence if a particle moves from rest from the origin with a
constant velocity-increment,
* = i/Z' ;
and thus the space described varies as the square of the time.
PRICE, VOL. in. Y y
346 MATTER. MOBILITY. INERTIA. [250.
Ex. 3. If the velocity-increment varies as the time from rest,
dt*
ds__ 1.
v t -t>
6
Ex. 4. If the velocity-increment varies as the distance from a
given point in the line of motion, and is negative,
d's
if a is the value of *, when the particle is at rest ;
[the negative sign being taken, as I will suppose the particle to
I move towards the origin ;
COS" 1 = k*t.
a
if t = 0, when s = a, and the particle is at rest ;
.*. s = a cosk*t.
SECTION 3. The dynamics of a particle moving in a straight line.
250.] The preceding observations on the kinematics of a
particle are all that we require at present. We shall return
to the subject in Chapter IX; and must now enter on the
further consideration of matter as the subject of motion; and
we shall state explicitly certain properties of matter, beyond
those which have been stated, in Part I, of it as the subject
of pressure or statical force. An important question meets us
at the outset; according to our conception of matter, as the
subject of motion, has it any power of changing its state;
251.] MATTER. MOBILITY. INERTIA. 347
has it when at rest a power of putting itself into motion ? has
it when in motion a power of itself either of increasing or of
diminishing its velocity ? An adequate conception of matter
involves a reply to these questions in the negative. Matter is
inert ; it has no power of acting on itself or of changing its own
state as to rest or motion. If it is at rest, it will remain at
rest : if it is moving with a given velocity along a rectilineal
path, it will continue to move with that velocity along that
path : there is no more reason why it should change its course
towards one side of that line than towards the other : this is
equivalent to saying that lapse of time does not affect matter's
state as to rest or motion. And not only does matter remain as
it is, unless acted on by some source of velocity external to itself,
but it also passively submits to external influence : whatever
effect is communicated to it, that is also developed in it. Now
I am not saying that matter does not act on other matter, for
the matter of our physical system does so act : thus leaden balls
attract each other : particles of air repel each other : but it does
not change its own state. Whenever therefore and this is
most important matter's state is changed either from rest to
motion, or vice versa, or when its velocity is increased or di-
minished, that change is due to some adequate cause, and velo-
city is communicated to it from some source external to itself.
This source is called force; and force is either accelerating or
retarding according as the velocity of matter is, by its action,
increased or diminished : a more exact definition by means of
its measure will be given hereafter. From the fact that matter
is inert, or, in other words, from the principle of inertia, will
be inferred the first equations, or propositions, of the science.
The principle may be stated in the following form, and is then
commonly called the first Law of Motion :
Matter at rest remains at rest, and matter in motion continues
to move in the same line and direction, and with unvaried velocity,
unless acted on by some force external to itself.
This principle of inertia is axiomatic, and is the first axiom
in the construction of the science ; it rules that when a change
of state takes place in matter, that change is due to the action
of some cause external to the matter.
251.] As we shall apply our theoretical investigations largely
to the matter of the earth, and of other bodies of the solar sys-
tem, it is worth while shortly to inquire how far the properties of
Y y 2
348 MATTER. MOBILITY. INERTIA.
matter which have been axiomatically stated are fulfilled in that
particular matter of which they consist.
As to Mobility ; the fact is shewn by daily observation :
bodies falling towards the earth, particles of matter constantly
moving in the air and as seen in a sunbeam, the waters of
the sea never at rest, the motion of the moon and of the planets,
the motion of particles of air in the wind, all bear evidence to
this property : nothing is seen quiescent ; everything is in
motion.
As to Inertia : terrestrial matter seldom changes its state
without our being able to assign the cause ; and hence we
inductively infer, that the cause could always be assigned, if
our knowledge of the moving matter and its circumstances was
perfect. Consider a particle of iron, placed on a smooth table ;
relatively to the table it is at rest : but let a magnet be placed
so that the particle of iron is within its influence ; the particle
will begin immediately to move towards it ; and the longer the
space is through which the particle moves, the greater will be
its velocity ; thus the magnet is the cause of the motion of the
particle at first, and also of its subsequently increasing velocity.
Now let another magnet be introduced of the same power as the
former, and acting along the same line of action, and in an
opposite direction, so that the action of the former magnet on
the particle of iron is neutralized : then it is found that the
iron-particle will continue to move with the velocity which it
has at the time when the neutralizing magnet is introduced :
that is, the velocity which it has at that instant is a quality
residing in it, and which it has of itself no power to annihilate :
its velocity will, it is true, during the subsequent motion become
less and less ; yet it appears that such a loss of velocity is caused
by the friction against the table, the resistance of the air, and so
on : for if these impediments are diminished, the particle Con-
tinues to move with a velocity less rapidly decreasing : and
hence we infer that if they were entirely removed, there would
be no diminution of the iron's velocity.
So again if a ball is projected along a level surface, such as
a bowling-green, the rougher the surface is the more impediment
does it offer to the ball's motion, and the sooner is the ball
reduced to rest : but if the surface is smooth, as a pavement, or
smoother still, as a plate of glass, or as ice, the longer will the
ball continue to move ; eventually, however, it will be reduced
252.] FORCE AND VELOCITY I ITS MEASUEE. 349
to rest, because it is impossible to remove all the impediments
which are continually acting on it as retarding forces, and are
thereby withdrawing velocity from it.
Again, if a suspended pendulum oscillates, the time ere its
motion ceases will be longer if it vibrates on a knife-edge than
if it is suspended by a spring, because the resistance of the
former is less than that of the latter ; and if it oscillates in the
exhausted receiver of an air-pump, the time ere its motion
ceases will be longer than if the oscillations take place in air.
From experiments such as these, it is inductively inferred, that
if all the hinderances are removed, and if the moving matter
does not receive velocity from any other source, it has in itself
no power either to increase or to diminish its own velocity.
The nearly uniform periods of the planets, and the almost
constant length of the mean sidereal day, in a similar manner
tend to shew that the same law is true in the matter of which
the bodies of the solar system consist.
The principle of inertia was first recognised by Galileo : me-
chanicians had before his time failed to give a correct exposition
of the principles of mechanics because they knew not this fact.
252.] Matter therefore can neither generate velocity for itself
out of its own resources, neither can it absorb into itself velo-
city which it has, or velocity which is communicated to it : it is
alike " natural " to it to be at rest and in motion ; whenever
therefore its state changes, some cause external to itself is the
origin of the change ; if the velocity is increased, some velocity
has been communicated to it ; if it is diminished, velocit}' has
been abstracted from it : whatever causes a change of velocity
is called force, and the word " force " will be used in Dynamics
in this meaning only.
The word " force," as thus stated, has not the exactness which
an exact science requires. Such terms are not precise enough
unless the quantities which they express are measurable : and as
force is an active cause, it will be measured by its effects. Now
the effect of a force is velocity, and consequently the velocity
communicated to or impressed upon the moving matter in a given
time, say, in an unit of time, is the measure of the force. But the
velocity impressed on a particle is equal to the velocity expressed
in its actual motion, inasmuch as matter has no power to absorb or
to produce any of such impressed velocity ; and consequently the
velocity-increment, or the acceleration, in an unit of time is the
350 FORCE AND VELOCITY: ITS MEASURE. [252.
measure of the force. Hence if a force causes in a moving ma-
terial particle an increase of velocity f in an unit of time, the
force may be correctly denoted by/*, because that signifies its
effect, viz. the velocity-increment of which the force is the cause.
Hence the unit of force is that which impresses an unit of
velocity in an unit of time.
Also the varieties of force are in this respect as many as are
the velocity-increments which they produce. Thus if a force
communicates equal velocities in equal successive time-elements,
the force is said to be constant ; and according as it increases or
diminishes the velocity, it is called an accelerating or a retarding
force. If a force, on the other hand, communicates unequal
velocities in equal successive elements of time, it is called a
variable force, and an accelerating or a retarding variable force
according as the velocity is increased or diminished by its action.
The law according to which the velocity is communicated is
called the law of the force. The velocity which a force transfers
to a body is called the impressed velocity; and the velocity
which is developed by the action of the force in the moving
body is called the expressed velocity. In the case of a single
particle the velocity expressed in its motion is equal to that
impressed by the force on it ; but if that particle is a member of
a material system or of a body, for reasons which will be given
hereafter, it will appear that this is not the case. Thus in a
moving particle the impressed and expressed velocities are indeed
the same thing viewed from different points.
From these explanations of force and its varieties the follow-
ing results arise. Firstly, let us suppose a force to be constant
and to act on a material particle, which is moving in a straight
path, along the line of its motion ; and let f be the velocity
which is communicated by this force in an unit of time ; then if
the force acts for t units of time, the velocity communicated is
ft; and if the particle was moving with a velocity u when the
force began to act, and with a velocity v at the time t } and the
force is accelerating,
v = u+ft', (15)
and if the force is retarding,
v = u-fL (16)
Secondly, let us suppose the force to be variable, and suppose
it to be such that at the time t a velocity/ 1 would be impressed
by it in an unit of time, if it were constant during that unit :
253-] FINITE ACCELERATING FORCE. 351
and to be such that at the time t + dt a velocity f+df would be
impressed by it in an unit of time if it were constant during
that unit. Then, if is a proper fraction, f+Oclf would be the
average or mean velocity impressed in an unit of time during the
time dt : and consequently if dv is the velocity actually impressed
and omitting the infinitesimal of the second order,
dv =fdt>, (17)
and this assigns the increase of velocity which takes place in the
time dt by the action of the force/". This force is the increase
of velocity in an unit of time.
If we require the amount of the velocity which is impressed
by a finite accelerating force in a finite time, this must be
deduced from (17) by integration; and the process can be
effected immediately \if is constant or is a function of t, since
in this case
but it must be done indirectly iff is a function of v or of s.
We have however brought our investigation of the effects of
force to this point : viz. that its effect and its measure is the
acceleration or velocity-increment which has been discussed in
the previous section on Kinematics, and we have
f _ dv _ d ds
'" ~dt == ~dt~di
. .
and consequently all the results of this equation, its various
forms, and the remarks which have been hitherto made on it,
are applicable to it, when/ 1 is the accelerating force. We shall
have so many applications of this equation hereafter that it is
unnecessary now to insert any.
253.] When a force acts on a particle at rest, the action-line
of the force is of course the line of motion of the particle ; but
when a force acts on a particle in motion the action-line of the
force may be, or may not be, the line of motion of the particle.
In the preceding article we have supposed the former case. The
latter case, which is more general and also more important, will
be fully discussed in following Chapters.
If two or more forces act simultaneously on a particle in the
352 FINITE ACCELERATING FORCE. [254.
line of its motion, the resultant effect will evidently be the sum
of their separate effects. Thus suppose a material particle to be
moving with a constant velocity v, and two constant forces
f and f to act on it, the effects of which are severally to
produce velocities f and f in one unit of time ; and suppose
each of these forces to act for t units of time : then the velocity
of the particle at the end of t units of time will be v +ft +f't.
If one of the forces, sayf, act in a direction contrary to that of
the particle's motion, it will abstract velocity, and the velocity
of the particle will, at the end of t units of time, be u+ftf't.
A similar result is of course true when the forces are variable.
Hence if two forces are capable of communicating equal ve-
locities to the same body in equal infinitesimal elements of time,
the two forces are said to be equal, and are such, that when
applied to the same body in opposite directions along the same
line of action, they neutralize each other, and do not change
the body's velocity. This is the definition of equal forces. Simi-
larly, forces which in equal infinitesimal elements of time will
produce in a given body, twice, thrice, &c. the velocity which
another force will, are estimated as double, triple, &c. of this
latter force.
254.] Force, such as we have considered it, impresses finite
velocity in a finite time ; and the effects of it have been resolved
into elements corresponding to infinitesimal elements of time.
Thus if a force acts for a finite time, and if the law of the force
is given, the total velocity impressed by it during the whole
time may be found by integration, and the whole velocity will
be the measure of the force's action. A force of this kind is
commonly called a finite accelerating, or retarding, force. But
suppose a force to act, and to communicate a very great velo-
city in a very short time, such as the explosive force of gun-
powder, which will impress a very great velocity on a cannon-
ball in the very short time during which the ball is passing
along the bore of the gun, then doubtless if the law of the
communication of the velocity is known, the whole velocity
may be found as in the former case, and will be the measure of
the action of the force ; but if the law of the force is not known,
and the force acts for a short time and then ceases, the whole
velocity which is impressed by it may be taken as the measure
of its action. A force of this kind is called an impulsive or in-
stantaneous force. This force does not, it is to be observed,
255-] MASS AS MOVING MATTER. 353
*
differ in kind from finite accelerating force ; the communication
and the development is as gradual in one case as in the other ;
the difference consists in the mode of measurement of its effect :
in the former case the law of force is given, and the total action
of the force is determined by integration : in the latter case,
whether the law of force is known or not, the action of the
force is measured by the whole velocity which has been com-
municated by it.
255.] Hitherto motion and velocity have been considered
independently of the quantity of matter of which they are.
Velocity and its properties have been discussed as being of a
mass, and, to fix our thoughts, we have assumed a material
particle to be the matter moving ; but it has been unnecessary
to introduce any reference to the quantity of matter, because
the velocity of a material particle and of a mass of large dimen-
sions may be the same : and inertia as a property of matter
does not require any conditions as to the quantity of matter : it
is true equally of a particle and of a large body. But now it is
necessary to consider velocity in reference to quantity of matter
or mass : because the equations of motion of moving matter,
from which all the theorems of dynamics will be deduced, are
formed by comparing the velocity impressed with the velocity ex-
pressed ; and thus a question arises, whether two bodies having
equal velocities impressed on them will move with equal veloci-
ties, whatever are their masses ? No doubt, by the principle of
sufficient reason, if their masses are equal, the expressed veloci-
ties will also be equal : but what will be their expressed velo-
cities, if the masses are unequal ? In reply to this question we
must strictly define equality of mass; and be on our guard
against an argument in a circle : equal masses must not be de-
fined to be those on which, when equal forces act, equal velocities
are impressed; when equal forces are defined to be those which
impress equal velocities on equal masses. We have already spoken
of mass, and of its mode of measurement by means of weight,
in Section 1, Chapter IV; but the following process of deter-
mining it is that which is most appropriate to our present
purpose. If two masses having the form of spheres, and moving
with their centres along a straight line, and in opposite direc-
tions, impinge on each other, and if each is by the collision
brought to rest, these masses are said to be equal : so that
equal masses are defined in the following terms :
TRICE, VOL. in. z z
354 MOMENTUM [255-
Two masses are equal which moving with equal velocities along
the same straight line, in opposite directions, and impinging on each
other, are reduced to rest by the collision.
When many masses have by this process been determined to
be equal to each other, we may collect two or more into one
mass, and thus obtain masses which shall be any multiple of a
given mass : and by a reverse process we may obtain masses
which are submultiples of another mass ; and thus we may ob-
tain masses which bear any ratio to each other. Thus if m equal
masses are collected into one mass, and m' into another, the ratio
of these collected masses will be to each other as m to m'.
In the present volume I propose to consider the motion of a
material particle only. It is much more simple than that of
a body, and for this reason : if a body moves, its particles may
all of them describe equal and parallel paths, in which case the
body is said to have only motion of translation ; or the particles
of the body may revolve one about another, without the relative
positions of them being changed, in which case the body has
the motion of rotation : a full investigation of these kinematical
circumstances will be found in Part III of the Treatise : whereas
as a particle occupies space equal to an infinitesimal geometrical
point, the motion of rotation may be neglected, and we have to
consider motion of translation only.
Experience teaches us, in the case of terrestrial matter, that
if two particles are at rest, and if it is required to make them
move with equal velocities, a greater force may be required to
act on one than on the other; and the reason assigned is, that
one has much greater force of inertia than the other. Now this
is an inaccurate expression. Inertia* has no force : it neither
destroys nor generates velocity ; motion and rest are equally
natural to matter. The true reason is, one mass is greater than
another, and therefore has a greater quantity of matter for
velocity to be communicated to. These circumstances however
must be thoroughly examined, as they lie at the very foundation
of our subject.
* Many writers on Mechanics use the expression "force of inertia ;" and
(Ps
lately -^ has been called "force of inertia ;" the expression is surely inaccurate
and unphilosophical, if the words are used in the senses which I have assigned to
them : and I cannot but refer to M. Poisson, who is no mean authority on such
questions, for a corroboration of the view of the subject here taken. Traite 1 de
Me*canique, 2<le Ed. Tome I, Art. 1 20.
256.] AND MOMENTUM-INCREMENT. 355
256.] Let two material particles be in motion ; of which I will
suppose one to be the unit-mass, for the unit is arbitrary, and
the other to contain m unit-masses; and let them move with
equal constant velocities v : if the m unit-masses of the larger
mass are separate, each would move with the same velocity v,
and therefore the sum of the velocities of all the particles
moving separately would be mv ; and by the principle of inertia
the sum of the velocities is not changed when all the particles
are collected into one common mass, and therefore the quantity
of velocity which is expressed in the moving mass m is mv :
that is, is in quantity m times the velocity of the unit-mass.
Although therefore the velocity of both the masses is the same
as to intensity, yet in quantity or amount of velocity, that of the
mass m is m times that of the unit-mass.
As we shall frequently speak of this quantity of velocity, it
is convenient to assign to it a distinctive name : it is, as ex-
plained above, the product of the numbers expressing the mass
and the velocity, and has been ordinarily called momentum or
quantity of motion ; although the term is somewhat inaccurate,
yet, to avoid the inconvenience of new nomenclature, I shall
use it, and shall signify by it the quantity of velocity which
exists in moving matter: and shall henceforth signify by the
term " velocity," velocity as to intensity.
The increments of these will be called respectively the mo-
mentum-increment and the velocity -increment; and of these, when
expressed, the mathematical equivalents will be m ^ and -j~ > if
t is an equicrescent variable.
Thus the momentum-unit is the product of the mass-unit into
the velocity-unit. It is evident also that the momentum of a
body is equal to the sum of the momenta of its several parts.
The distinction which is drawn between velocity as to inten-
sity and velocity as to quantity or momentum, may be illus-
trated by the following analogies. Suppose two masses of the
same substance, one of which is ten times as large as the other,
to be in the same state of temperature, and suppose both to be
heated so as to be of the same higher temperature : then to these
masses heat has been transferred from some external source ; and
to the larger mass ten times as much as that to the smaller one;
and thus, although both are of the same heat as to intensity,
yet the quantity of transferred heat in one is ten times as great
z z a
356 EQUALITY OF IMPRESSED AND [257.
as that in the other. A thermometer measures temperature,
that is, heat as to intensity ; in the same manner does the space
described in one unit of time measure velocity as to intensity.
Again, suppose a certain quantity of light from a given source
to be received by a table or a given area ; the smaller the area
is, if it receives the whole light, so much more intense will the
illumination be. But if a given area is illuminated equally
throughout its surface, the greater the surface is, the greater
also will be the quantity of light received by it. Thus light
does as to intensity vary inversely as the area over which it is
spread : but the quantity of light received by a surface illumi-
nated equally throughout varies directly as the surface.
257.] It appears then that when force acts on matter, and
communicates velocity to it, the effect is momentum, and not
only velocity as to intensity. And this subject has to be con-
sidered both with reference to velocity and to velocity-increment ;
that is, with reference to impulsive and to finite accelerating
force. Now the law of inertia rules that in both these cases,
the momentum expressed is equal to the momentum impressed.
Firstly then if the force is impulsive ; let m be the mass of a
particle, which I will assume to be at rest : and let a force act
upon it in such a way that it instantly moves with a velocity v,
then the expressed momentum is mv. Let Q, be the momentum
impressed by the action of the force, which is like a blow ; then
by the preceding principle,
Q, = mv, (19)
.-. t> = ^: (20)
m
which assigns the velocity communicated to the particle by the blow .
If the particle was previously moving with the velocity u,
and the force acted on it in the line of its motion, then if v is
the velocity after the action of the force,
V = U+ OL. (21)
m
The following are illustrative of this theorem :
If a particle of m mass-units moves with a velocity v its
momentum is mv ; and if all its momentum is transferred to a
particle m', and if is the consequent velocity of m',
m'tf = mv,
mv
.-.. v'- ;
m
which determines the velocity of m' '.
258.] EXPRESSED MOMENTA. 357
Thus if a particle, as a small ball, of mass = 3, moves with a
velocity = 4, its momentum is 12; and if it impinges directly
on another particle of mass = 2, and is reduced to rest by the
impact, the whole of the momentum will have been transferred
to m' y and mf will move with a velocity =6. 6 therefore will
be the expressed velocity, and 12 will be the expressed mo-
mentum, of this latter particle.
Hence also momentum is the measure of the pressure of per-
cussion of a moving mass.
So if two particles, moving along the same line and in oppo-
site directions with velocities which are inversely proportional
to the masses, impinge directly on each other, they will be re-
duced to rest by the collision.
Again, if a cannon-ball of mass = 10 is fired from a gun,
and emerges from the bore with a velocity = 250, the momen-
tum of the ball is 2500, and this will be the measure of the
explosive force of the gunpowder.
Hence also it appears that whenever momentum is impressed
on a mass by means of matter acting upon it, it is withdrawn
from some other source, or an equivalent momentum is simul-
taneously produced in an opposite direction. Hence also we
infer that the whole amount of momentum is always the same.
Momentum cannot be created : it can only be transferred. It
may perhaps be thought that momentum can be generated by
muscular action, say, that a stone may be thrown, and thus
receive momentum, by the muscular action of the arm : we
must not however be deceived by appearances : let a person
stand in a frame suspended as the scale of a balance, and which
is capable of moving freely : if he impresses momentum on any
body, as, for instance, if he throws a stone, it will be found that
he moves in a direction directly opposite : and the product of
his mass and the velocity with which he moves in the scale will
be equal and opposite to that which he has given to the stone :
the apparent creation then of momentum in one direction is ac-
companied by the creation of an equal quantity in the opposite
direction. A similar effect takes place when momentum is im-
parted to a mass by means of a pressure against the earth.
258.] The same principles apply, and lead to similar results
when the force is finite accelerating. If a particle of mass m
d*s
receives a velocity-increment = -= > by the action of a force, the
dt
358 EQUALITY OF IMPRESSED AND [258.
d' 2 s
expressed momentum-increment due to the force is m -j ; and
do
consequently if the force is such as to impress a velocity-incre-
mentyon an unit-mass, then
m f= m ~^'> (22)
and /=. (23)
And if S is the whole impressed momentum, which may or may
not vary with m, j,
s = m nf < 24 >
The source of the impressed-momentum has been usually called
moving force; and as it is equal to the expressed momentum-
increment, we take this latter to be its measure ; and as the
accelerating force is measured by the velocity-increment, that is,
d* s
by j > so the measure of the accelerating force is that of the
moving force acting on a mass-unit. Hence also the moving
force-unit is thai which impresses an unit of velocity on a mass-unit
in an unit of time.
Equations (19) and (24) are called equations of motion; they
define all possible kinds of motion of a particle in a rectilinear
path.
Let us exemplify this result : suppose a particle m to be
falling towards the earth : it is found by experiment that the
earth's attraction is an uniformly accelerating force, which im-
presses on the falling particle a velocity-increment of 32 feet
(approximately) in one second of time ; let a second therefore
be the time-unit, and let us represent the number 32 by g
(= gravity) : then mg is the impressed momentum-increment,
d*s
and m = is the expressed momentum-increment : therefore, by
(tt
reason of (22), d"g
mg = m ; (25)
The illustration may be more correctly represented when we
take account of the mass of the earth. For since the attraction
between the earth and the particle is mutual, the particle at-
tracts the earth while the earth attracts the particle : if there-
259-] EXPRESSED MOMENTA. 359
fore m and M are the masses of the particle and of the earth
respectively, the velocity-increments of the particle and earth in
an infinitesimal element of time are inversely as the masses.
Similar too is the mutual attraction of the earth and moon :
the expressed velocity-increments with which they move towards
each other are inversely as their masses. Hence it follows that
their centre of mass would remain at rest, if the earth and moon
had no other motion than that which is due to their mutual
attraction : but owing to the action of the sun, and the motion
of each in space, the centre of gravity describes an ellipse with
the sun in one of the foci.
Hence then it follows that (1) moving forces do not impress
equal velocities on different masses, unless they are proportional
to the masses ; (2) the velocities expressed in equal masses are
proportional to the moving forces ; (3) the velocities expressed
in unequal masses by equal moving forces are inversely propor-
tional to the masses. Hence also we infer that when a moving
force impresses velocity on a mass, the velocity expressed varies
directly as the moving force and inversely as the mass. This
last proposition has been commonly called the third Law of
Motion, and is enuntiated in a form such as,
When moving force produces velocity in a given mass, the velocity
produced is inversely proportional to the mass.
Sir Isaac Newton calls the following proposition the third
law of motion : " Action and reaction are equal and opposite."
This however is no more than a statement in plain language of
(19) and (24); and it is necessary to explain the meaning of the
terms action and reaction, and how they are measured.
And here we have come to the second axiomatic principle
which is necessary to the construction of our science : when the
state of matter, as to motion, changes, a measure of the change
is hereby given : the product of the mass and of the velocity
which is expressed in a given time is the measure of the force
which has caused the change, and is by the principle of inertia
equal to the impressed momentum. From this equation all the
results of dynamics will be deduced.
259.] As equation (22) is a differential expression of the
second order in terms of * and t, and as in the complete solution of
a dynamical problem it is required that * should be expressed in
integral terms of t, it is evident that this differential equation
must undergo two integrations before the required solution is
360 EQUALITY OF IMPRESSED AND [259.
obtained. Now this process will take different intermediate
forms according to the form of/.
If f is constant, or if it is a function of t, say/" = <t>(t), then
(24) becomes /7z,
m(^-v ) = I m<j>(t)dt, (26)
when v is the velocity, when t = t ;
ds C t
.'. m-j 7 =mv + m<j>(t)dt; (27)
at J to
and thus the result gives momentum; (26) giving the mo-
mentum which accrues in the time t 1 0) and (27) giving the
momentum at the time t. In both cases however the result is
momentum. The space may be found in terms of the time from
(27) by another ^-integration. Both these are cases of time-
integration.
If/ is a function of * ; sayf= <f> (*), then
Let us multiply both sides by ds ; then
m ds -j = m$ (s) ds ;
mv 9 mv * /**
.'. = / m(j>(s)ds', (28)
the left-hand member of this equation is called the vis viva* of
the particle m ; that is, vis viva is a quantity which varies as
the product of the mass of a particle and the square of its
velocity, and the form of the left-hand member shews that it is
convenient to take ^ as the coefficient of variation. The right-
hand member is the sum of the products of the mass of the
particle into the product of the impressed velocity-increment
and the distance over which it acts. This is called the work
done by the force through the distance s # ; consequently the
work done by a force exerting action through a given distance
is equal to the increase of vis viva which has accrued to the
particle in its motion through that distance.
Thus the unit of work is that which is done by an unit of
accelerating force acting on an unit of matter through an unit
of space. And if the earth's attraction at a given place is the
* It is called by Sir W. Thomson and Professor Tait " Kinetic Energy."
259-] EXPRESSED MOMENTA. 361
unit-accelerating force, and the mass of a pound is the unit-mass,
and a foot is the unit-space, the unit of work is that which is
required to raise the weight of one pound through a vertical
space of one foot. This is called a foot-pound; and is the unit
of work generally adopted by British engineers.
Thus the work done in raising a weight through a vertical
distance is proportional to the weight raised and to the vertical
distance through which it is raised.
Generally for any force, the work, as thus defined, done
during an infinitesimal displacement of the particle on which it
acts, is the virtual moment of the force, which has been de-
scribed in Article 108.
If in the motion vis viva is lost, negative work is done by the
force ; that is, the work is stored up as potential work in the
particle or mass on which the force has acted. Thus if work is
spent on winding up a watch, that work is stored in the coiled
spring, and is thus potential and ready to be restored under
adapted circumstances. Similarly, if a weight is raised through
a vertical distance, work is spent in raising it, and that work
may be recovered by lowering the weight through the same
vertical distance.
This theorem, stated in the most general form, is the modern
principle of conservation of energy or of work ; and is made the
fundamental theorem of abstract dynamics as applied to natural
philosophy.
In this case we have an instance of space-integrals. And as
forces in nature are functions of the distance, this is the form
which dynamical problems take in physics ; we shall hereafter
have many examples in the solution of problems which are
capable of such application.
There is still another form which f may have : it may be a
ds
function of the velocity ; that is, f may be of the form <f> (-^-) ;
in which case 7l 7
O / I 1/iS \
and of this equation we may generally take either the time-
integral or the space-integral. Thus if we take the time-in-
tegral, replacing -j- by v } we have
dv
W) = ' (30)
whence, when $(y) is given, we have v in terms of t.
PRICE, VOL. in. 3 A
362 EXPERIMENTAL EVIDENCE OF THE THEOREM. [260.
.
And if we require the space-integral, since
d*s _ dv _ ds dv dv
~dt* "" Tt ~ 'Tt Ts = V Ts'
rdv
wehave W) = '
whence we have v in terms of s.
If f is given in terms of two or more of the three quantities
ds
t, s, and -j- ) it is only in certain cases that the differential equa-
dt
tion admits of integration.
Of all these several forms of f we shall have many examples
in the following Chapter.
260.] The truth of the preceding theorems connecting moving
force, mass, and expressed velocity, in the case of terrestrial
matter is proved by Attwood's machine, for a full description
of which I must refer the reader to Attwood's original treatise
on rectilineal motion, and to other treatises on experimental
mechanics, but of which a concise account is given in Section 3
of the succeeding Chapter. It is shewn by numerous experi-
ments made with it that the expressed velocity-increment in a
second of time varies directly as the moving force and inversely
as the whole mass moved; and therefore the product of the
mass and the velocity-increment varies as the moving force, and
may be taken to be a measure of it. The same theorem is
also proved by the following experiment : it has been shewn
in the preceding Chapter that the earth's attraction on bodies
near to its surface is constant ; and it will be shewn in Section 3
of the following Chapter that, when bodies move under the ac-
tion of the constant accelerating force of gravity, the expressed
velocities, due to given vertical distances through which the
force acts, vary as the square roots of those distances. Suppose
two spherical balls m and m' to be suspended from two points in
the same horizontal line, and by strings of lengths such that
the balls when at rest may just touch, and also have their
centres in the same horizontal plane : let these balls be moved
in circular arcs ; then the velocities acquired by them as they
fall from rest to the lowest point vary as the square roots of the
versed-sines of the arcs through which they descend; and as
the versed-sine varies as the square of the corresponding chord,
so the velocities acquired in the descent vary as the chords. If
therefore the two balls are raised through arcs, the chords of
26 1.] STATICS, A SPECIES OF DYNAMICS. 363
which are inversely as the masses of the balls, the velocities at
the lowest points will also be inversely as the masses ; and it is
found by experiment that balls which have fallen through arcs,
the chords of which are inversely as their masses, and which
impinge on each other at the lowest point, are by the collision
brought to rest : and therefore the momenta of them must have
been equal, and thus being in opposite directions along the
same line of action have neutralized each other. This then is
an experimental proof that the momentum is equal to the pro-
duct of the mass and of the velocity. It is also found that, if
the arc through which one of the balls moves is greater than
that determined above, when the balls come into contact, they
are not reduced to rest, but move in the direction of the motion
of that which has fallen through the proportionally greater arc.
It is also found by experiment that if two balls of unequal
masses are placed in contact, and have a spring so arranged
that when the spring is set free it exerts an equal action against
both of them, the velocities which are expressed in them are
respectively inversely as their masses.
261.] When the matter on which moving force acts rests on
a surface, the normal to which is along the line of action of the
moving force, the effect is not velocity but pressure : for the in-
finitesimal element of velocity, which the moving force would
impress in an infinitesimal element of time, is destroyed by the
resistance of the surface. But if the surface were removed it
would be expressed in the moving matter, and the elements of
velocity being added to each other, a finite velocity would be
expressed. When therefore a moving force impresses velocity,
and the velocity is expressed, the elements of it are added to
each other, and the resultant is the whole expressed velocity :
but when the elements of velocity are destroyed as soon as they
are communicated, the result is pressure. Hence it follows that
two pressures are to each other as the product of their masses
and the infinitesimal elements of velocity which would be ex-
pressed in them in an infinitesimal element of time if they were
free. At this point therefore statics becomes a branch of dy-
namics, and on this principle, which is the principle of virtual
velocities, the theorems of the latter science are applicable to
and become those of the former.
We are hereby supplied with a method, which is in practice
most convenient, for determining the mass of terrestrial matter.
3 A 2
364 STATICS, A SPECIES OF DYNAMICS. [261.
Observation shews that at the same place all bodies, whatever
are their substances, acquire the same velocity in falling in
vacuo in the same time. The earth's attraction therefore is an
accelerating force which acts independently of the particular
kind and quantity of the matter which moves, and is therefore
the same for all matter. Consequently the pressures of bodies
under the attraction of the earth vary as their masses : these
pressures are the weights of the bodies, and therefore the weights
at the same place vary as the masses of the bodies; and as the
balance affords an easy mode of comparing weights, we can
hereby deduce the relative proportions of the masses.
It may probably be thought that this method of determining
mass is more simple than that chosen in Art. 255 ; and prac-
tically for the matter of the earth it is : but there are objections
to it, so far as the principles of the pure science of motion are
concerned: (1) it experimentally assumes the relation between
mass, moving force or its measure momentum-increment, and
accelerating force or its measure velocity-increment ; and this
it is adduced to prove : (2) only terrestrial matter can be com-
pared by it, whereas the principles of the science of motion
should be laid in breadth sufficient to include matter of all
kinds : (3) M. Poisson writes in the Traite de Mecanique,
Art. 62, Ed. 2 de : "Toutefois, on doit avoir un idee prealable de
Tegalite et du rapport des masses, inde'pendamment de la pe-
santeur, qui n'est qu'une propriete secondaire des corps, puis-
qu'elle deviendrait tout-a-fait insensible, sans que les masses
eussent change, en les transportant a une distance suffisamment
grande de la terre." Thus M. Poisson thinks that such a mode
of determining mass would not be sufficiently general for even
terrestrial matter,
262.] DIRECT IMPACT AND COLLISION. 365
CHAPTER VIII.
THE RECTILINEAR MOTION OF PARTICLES.
SECTION 1. Direct impact and collision.
262.] We proceed now to the application of the principles
and equations which have been investigated in the preceding-
Chapter : and we shall begin with the most simple case, that
of the direct impact and collision of two material particles. To
fix our thoughts, however, I shall consider these particles to be
spherical homogeneous balls, which move so that all the parti-
cles describe equal and parallel paths, and the balls have there-
fore no motion of rotation ; the velocities also of the balls will
be supposed to be uniform both before and after collision, and
the paths along which they move are supposed to be rectilineal ;
also the line of action of the mutual pressure of the balls during
the collision is supposed to pass through their centres ; and
if this line is that in which the balls are moving the impact is
said to be direct; but if either of the balls moves in a line not
coincident with this line of action the impact is called oblique.
We shall now investigate the former case : the latter will be
considered in Section 1, Chapter X.
Let the masses of the two material particles be m and m'\ and,
to fix our thoughts, let us suppose them to be moving with
uniform velocities in the same direction along the straight line
OA, fig. 85, say, from left to right: let v and v' be their re-
spective velocities, and let us suppose v to be greater than v',
so that m overtakes and impinges on m' : the momenta of the
two balls are respectively mv and m'v'.
Now no matter is perfectly rigid ; all is more or less ex-
tensible, compressible, and also elastic. Thus when m impinges on
m', a compression of the particles of the two balls at, and about,
the point of contact takes place : a change of form of the balls
thus takes place, and the molecules of them move one relatively
to another : velocity therefore has been impressed on them.
The disturbance of the relative positions of the elements of the
366 DIRECT IMPACT AND COLLISION. [262.
bodies also brings elastic forces of restitution into action : for
the effects of the impact are supposed not to be such that the
balls are broken or crushed by them : and the greater the dis-
turbance of the particles is, the greater is this elastic force :
now although according to the configuration of the balls which
we have imagined, the velocity of m is greater than that of m',
yet during the collision momentum is being withdrawn from m
and is transferred (1) to m' by the means of all its particles,
whereby the velocity of m' is increased ; and (2) to the particles
which are disturbed in and about the place of contact : the
limit of this latter transferred momentum is the elastic force ;
this transference of momentum continues until m and mf move
with the same velocity ; which circumstance eventually occurs :
for so long as the velocity of m is greater than that of m', the
change of the forms of the balls is increased, whereby the elastic
force is also increased; and as this increases in a greater pro-
portion than the compressing force, the two balls must ultimately
move with the same velocity : at this stage of the process, the
compression is, it is to be observed, a maximum.
As soon however as the balls move with the same velocity,
there is no mutual pressure between them : there is then no
force to counteract the elastic forces which have been brought
into action by the compression, and these therefore begin to
produce their effects. Now the common velocity of m and m'
at the instant, when the compression is a maximum, is from o
towards A ; and the effect of the elastic forces in the restitution
of the figure is to increase the velocity of mf and to diminish that
of m in that direction. In other words, during the compression,
momentum of the balls is changed into elastic moving force : and
in the restitution, this elastic force again produces momentum :
and in both processes momentum is abstracted from m and is
given to m'.
A question however arises : What relation does the momentum
impressed by the elastic force during the restitution of the forms
bear to that which was lost by m during the compression ? Here,
in our ignorance of the constitution of bodies and of their
molecular action, we are obliged to have recourse to experi-
ment; and it is found that in two balls of given substances
there is always a certain definite ratio between the momentum
spent in producing a certain compression, and that acquired
during the restitution; the latter quantity being always less
263.] DIRECT IMPACT AND COLLISION. 367
than the former : the ratio is called the measure of the restitu-
tion of the bodies, and is symbolised by e ; the limiting values
of e are and 1 : the former being its value for substances per-
fectly inelastic, and perfectly hard, because if a body is perfectly
hard there is no compression, and therefore there is no elastic
force of restitution : and the latter being the value of e when the
bodies are perfectly elastic, and when the momentum recovered
during the restitution is equal to that spent in producing the
compression. Mr. Hodgkinson has not found in the course of
his experiments (see British Association Reports, Vol. Ill, p. 534)
any matter perfectly fulfilling these conditions. Hence the
value of e for all known substances is a positive proper fraction.
If therefore P represents the momentum impressed during the
compression, e? is that acquired during the restitution.
263.] Let m and mf be the masses of the two balls, which
move in the same direction along the straight line OA, see
fig. 85, with uniform velocities v and v' : and let us suppose v to
be greater than v' ', so that m overtakes and impinges on m' : let
u be the common velocity of the two balls when the compression
is a maximum : let P represent the momentum spent in pro-
ducing the compression, and P that acquired in the restitution
of the form of the bodies. Let v and V be the velocities of m
and m, when the collision ceases ; and which are their uniform
velocities after the collision has taken place. We shall consider
the circumstances of the balls as they are (1) at the instant
when collision begins, (2) at the instant when the compression
is a maximum, (3) when the collision has ceased. Now
mv the momentum of m at the beginning of the collision,
p = the momentum spent in producing compression,
mu = the momentum of m when the compression is a maximum ;
therefore, by reason of (19), Art. 257,
mv = mu + p; (1)
m'tf=. the momentum of m' at the beginning of the collision,
m'u'= the momentum of m', when the compression is a maximum;
.*. m'tf m'u p : (2)
and at the instant when the collision ceases, we have by a
similar process, mv = mu _^ (3)
m'v'= m'u + ep : (4)
and therefore adding (1) and (2), and (3) and (4),
368 DIRECT IMPACT AND COLLISION.
mv + m'v' mv 4 w/V
1)
-
m + m in + m
/. mv + m'tf=z mv + m'v': (6)
therefore the sums of the momenta before and after impact are
equal.
From (1) and (5) we have
mm ^
p = - -,(vv^; (7)
'^
therefore the momentum spent in producing 1 the compression
vanes as the difference between the velocities before impact.
Substituting- in (3) and (4) from (5) and (7), we have
mw + m'v' em ,
v=- --- - f (v 0j (8)
m + m' '^
mv 4- mv' em
v / =- -T- + - (-*0; ( 9
m + m m-\-m
and thus the velocities of. the balls after collision are expressed
in terms of their masses, the coefficient of restitution, and their
velocities before impact.
The momentum which is impressed on m in a direction oppo-
site to that of its motion, by the elastic force ev during the
restitution of the form of the balls, may be such as either wholly
to neutralize the velocity of m and thus to bring it to rest, or to
impress on it a velocity in the opposite direction. In the latter
case, v will have a negative sign, and we shall have
v ,, (l+e}m'
-. greater than -- -. --
v em M
If m e before impact moves in a direction opposite to that which
we have imagined, and so as to meet m, v' must be affected with
a negative sign in all the preceding formulae ; in which case if
mv=m'v' } that is, if the momenta of the impinging balls are
equal, u = 0, and the balls are at rest at the instant when the
compression is a maximum ; and after the restitution has taken
place, v= ev, and v / = ev' ', and thus the balls move in opposite
directions.
Defining vis viva, as in Art. 259, by one half of the product
of the mass of the moving particle or ball and the square of the
velocity, and noting that this is the equivalent of work, the sum
of the vires vivae of the balls before collision is - ; and
m v* -4- m' v 7 *
after collision is - - ; so that by (8) and (9) we have
264.] DIRECT IMPACT AND COLLISION. 369
*v + V J _ mv' + m'v'* (le')mm'
~2~ ~2~ ~2&+m') (
and therefore in the case of imperfectly elastic balls, when e is
less than unity, vis viva, and consequently work, is lost by
collision.
Also since the balls after impact move with constant veloci-
ties v and v', they in t units of time severally pass over v and
Vt units of distance : and therefore the distance between them
= (v--v)*
= e(v-v'}t.
Also V v = e(v i/),
that is, the relative separation after impact is to the relative
separation before impact as e is to 1 .
264.] Let us consider some special cases of the preceding
results.
Ex. 1 . Let the elasticity be perfect : e I : then
-v'), (11)
O iyn
(v-v')-, (12)
and also if m = m', v = v', V v; that is, when a perfectly
elastic ball impinges on another equal and perfectly elastic
ball, each after impact moves with the velocity of the other
before impact; if therefore one is at rest before impact, the
impinging ball remains at rest after impact, and the other will
move with the velocity of the impinging ball. Hence if there
is a row of equal and perfectly elastic balls in a straight line ;
and if the first ball moves in that line with a velocity ?;, and
impinges on the second, the first will be brought to rest, and
the second will move on with the velocity v : similarly will it,
after impact on the third ball, be brought to rest, and the third
ball will move with the velocity v ; and so on through all the
balls, until finally the last ball moves with a velocity v and all
the others are reduced to rest. Now as this result does not
depend on the distances between the balls, it will be true if the
balls touch each other; and thus if there is a row of equal
and perfectly elastic balls in a straight line, which touch each
other, if one of the extreme balls moves with the velocity v, and
impinges on the next ball with a velocity v in the direction of
PRICE, VOL. in. 3 B
370 DIRECT IMPACT AND COLLISION. [265.
the row of balls, the intermediate balls will not be disturbed,
and the last will move with the velocity of the impinging ball.
Also if e \, m v* + m 'v" _ mv* + m'v' a .
2 2
that is, the sum of the vires vivae is the same before and after
impact. This is an instance of the general law of dynamics, viz.
the conservation of work.
Ex. 2. If the bodies are wholly inelastic, e=Q; also if they
are perfectly hard, so that no change of form is caused by the
impact, then no elastic force is brought into action, and e = 0.
In these cases m v + m'v'
v = v / = ; ; (14)
m+m'
that is, the balls after impact move together, and of course with
the same velocity.
Ex. 3. If m' is infinitely greater than m, and if v' 0, or, which
is the same thing, if m impinges on a fixed obstacle, as on a
fixed plane, then v = ev ; (15)
that is, the ball rebounds with a velocity which is e times that
of impact, and in an opposite direction.
And if the elasticity is perfect, e 1, and
v=-v; (16)
that is, the velocity of rebound is equal and opposite to that of
impact.
And if e = 0, v = 0, and the ball remains in contact with the
plane.
265.] The velocity of the centre of gravity or mass-centre is
not changed by the alteration which the velocities of the balls
undergo by reason of the impact.
Let x and of be the distances of the centres of m and m' from
o, fig. 85, at the time t : so that their velocities along OA at that
dx , dx' ,
time are -=- and -=- : and thus
at at
dx , dx'
v = 7(' '=*
Let x be the distance from o of their mass-centre ; then by
(110) Art. 125, (m + m^x = mx + trix'',
dx dx , dx'
_ =*_+._
mv + m'v' j (17)
and is therefore equal to the sum of the momenta of the balls :
but by (6) the sum of the momenta is the same before and after
266.] DIRECT IMPACT AND COLLISION. 371
impact : and therefore the velocity of the centre of gravity is
the same before and after impact. The same property is also
true of any number of balls directly impinging on each other in
a straight row.
266.] Examples illustrative of the preceding equations :
Ex. 1. Determine the velocity of a given ball m which im-
pinges on another equal ball moving with a given velocity,
when the impinging ball remains at rest after the collision.
Here m =. m, and v = : therefore from (8),
1+e ,
v -- v .
le
Ex. 2. To determine the mass of a ball m t) which, interposed
between m l and m 3 is such that the velocity of m 3) which is
originally at rest, may after impact from m t through the inter-
vention of m, be a maximum.
Let v be the velocity of m t at first : then
the vel. of m 3 after impact from m, = - - - :
i f A. 4. _
.. the vel. of h after impact from m, =
, _ s- _
' f
if m t == (/! m^, and changes sign from + to : therefore the
value of f(m t ) is a maximum, if m a is a mean proportional be-
tween the two extreme balls.
Ex. 3. n balls m lf m a , m s , . . . m n , perfectly elastic, are placed
in a row : find the ratio of their masses, when a momentum m^ v
impressed on the first is after impact equally divided amongst
the n balls.
vel. of m, after impact on m, -v; (18)
m.. .0- m..
m l
2m l v
O " ; 3 *
vel. of m, after impact from m l = - - ;
m 1 -f m a
i n n, ul n *^" 77^ A Uv \ C/
vel. of m a after impact on m s = - ; (19)
i f, . ,, 2m 2m, v
vel. of MI after impact from m 2 = - * ^ ;
in n, (tn, Mi t ) 4 m, m t v
vel. of m s after impact on z 4 = v * ^ ; (20)
(m, -4- m t } (m* -4- m 3 ) (ni\ -\- m t )
\ * */ \ */ \ /
3 B 2
372 DIRECT IMPACT AND COLLISION.
and so on: therefore from (18), (19), and (20),
m l v m m t 2m 1 m 2 m 3 m 3 v
r~/
) (m^ -f m^
(m 3
n m a + m 3 n\ m 3 + m t n + 2
m, m, m m, m. m t
n + 1 ~~ nl' n ' n2* nl ft 3
n r + 2 n r
which gives the ratio of every two successive balls.
267.] The theoiy of impact may also be applied to the deter-
mination of the momentum lost by a body as it moves through
a resisting medium.
By the law of inertia, a material particle or body which has
a certain momentum continues to move in a rectilinear path,
and with a constant velocity, unless it is acted on by some
force ; that is, unless momentum is abstracted from it or is com-
municated to it. Now if a particle or body moves in a vacuum,
its velocity is not affected by any action of the medium through
which it passes, because there are no material particles to be
displaced by the body in its passage : but if the particle moves
in a resisting medium, such as air or water, or in any other
medium gaseous or liquid, whose density is finite, the particles
of the medium are to be displaced, to allow the particle to pass
through the medium ; that is, the particles must move, and
must therefore have momentum communicated to them; and
this will be abstracted from the moving body ; hence it loses
momentum, the amount of which it is our object now to deter-
mine. And a loss of momentum will arise not only from the
displacement of the particles which the passage of the body
through the medium requires, but also from the action of the
particles on each other, and from their friction against the
surface of the moving body, whatever the nature of that friction
is : the latter causes of loss of momentum, involving data extra-
neous to the present subject, we cannot now determine; but of
the former cause, and which is the principal one, we can deter-
mine, at least approximately, the effects.
267.] THE ACTION OF A RESISTING MEDIUM. 373
Let the moving mass present to the resisting medium a plane
face, whose area is o>, and the plane of which is perpendicular
to the line of motion of the body : let m be the mass of the
moving body, p = the density of the resisting medium, and let
the plane face o> be at the time i at a distance * from a fixed
point in the line of its motion : let ds be the distance through
which CD moves in the time dt, and let v be the velocity of the
body : so that ds = v dt. In the time dt, the plane face co will
have passed over a space equal to ds, and will have impinged
upon, and communicated momentum to, all the particles of the
medium contained within a small cylindrical surface, of which
o> is the base and ds = v dt is the altitude ; then as p is the den-
sity, and as the particles move with a velocity v so as to allow
the body to pass through, a velocity v will have been commu-
nicated to the mass pa>ds, that is, to patvdt; and as this moves
with a velocity v, its momentum is pnv^dt', and this has been
abstracted from the moving mass; therefore by reason of (22),
the resistance of the medium therefore will have caused to the
moving body a loss of momentum which varies as the density
of the medium, as the plane area of the body on which the
medium acts, and as the square of the velocity with which the
body moves.
It will be seen hereafter that this result gives a loss of mo-
mentum due to the resistance just double of that which is given
by the principles of fluid motion as estimated in hydro-me-
chanics. One source of the discrepancy doubtless arises from
the fact that, as the body moves and displaces particles of the
medium in front of it, it leaves an empty space behind, into
which other particles at once move, and impinging against the
body give momentum to it ; and thus the loss of momentum,
which is given in the preceding expression, is greater than
what actually occurs.
From (21) the following results may be derived : since
dv
v poo 7 ,
-- r = dt ;
v 3 m
^ (22)
m
374 RECTILINEAR MOTION OF PARTICLES. [268.
if v is the value of v when t = ; and this equation gives the
velocity at any time t, if the body moves initially with the velo-
city v . Also t = oo when v = ; so that the body never comes
to rest.
Also since v = -57 > if v = v 0) and * = 0, when t = 0,
(it
(23)
which gives the relation between s and t; and consequently
* = oo, when t = oo, and v = 0.
SECTION 2. Rectilinear motion of particles under the action of
an uniformly accelerating force.
268.] Let m the mass of the moving particle ; and let a
point o, fig. 86, in its line of motion be taken as the origin :
let P be its position at the time t, let OP = x, and let pq = dx
be the space-element described in the time dt : so that if v is the
velocity of m at the time t,
dx
-jj- = v; dx vdt: (24)
let f be the impressed velocity-increment due to, and the mea-
sure of, the accelerating (or retarding) force : then mf is the
impressed momentum-increment of m in an unit of time.
Let dv be the expressed velocity-increment due to the time
dt; therefore mdv is the expressed momentum-increment due
to the same time ; and m -jj is the expressed momentum-incre-
clt
ment due to one unit of time : therefore by reason of (23),
Art. 258, ~_dv d_ dx_ _ d*x_ , }
f == dt == dt' dt == dt*' ( ;
if t is the equicrescent variable : f also is to be affected with a
positive or negative sign according as from (25) the action of
the force makes the velocity increase or decrease as the time
increases. To fix our thoughts, let/" be positive, therefore
Now suppose the circumstances of motion to be such that the
268.] UNIFORMLY ACCELERATING FORCES. 375
velocity ofm = u, when t = ; then, integrating between limits
thus assigned, we have
<*=/*
' d ir u =f t > (27 >
that is, the increase of velocity in the time t \sft\ u is called
the initial velocity.
Again, integrating, and supposing the particle to be at
A (OA = a), when t = 0, we have from (27),
dx udt+ftdt;
ft*
.'. x a = ut + '*-; (28)
j|
/2 2
.'. x a + ut + - (29)
I
If w is at the origin, when t = ; a = 0, and
x = ut+t^\ < (30)
and also if the particle starts from rest, then u = 0, and we have
*=1T ; (31)
in this last equation x is called the space due to f during the
time t : and t is called the time to which x is due under the
action of/!
Again, multiplying both sides of (26) by 2dx, we have
and supposing the velocity of the particle to be u when x = 0,
so that u and respectively are the inferior limits of the defi-
nite integrals of the sides of the equation, we have
dx" 1
^- = V*; (32)
and if the velocity of the particle = 0, when x = 0, then u = 0,
and we have .a
.:. the velocity =
Thus, if the particle m stands from rest and moves through
the distance x,
., . . wv*
the vis viva = - = mfx', (33)
376 RECTILINEAR MOTION OF PARTICLES. [268.
that is, is equal to the product of the mass, the accelerating
force, and the distance.
As (27) gives the relation between the velocity and the time,
(33) that between the velocity and the space, and (30) or (31)
that between the space and the time, it appears that when a
particle moves under the action of a finite accelerating force,
(a) The velocity acquired during a given time varies as the
time.
(/3) The velocity acquired by the particle during its motion
through a certain space varies as the square root of the space.
(y) The space through which the particle passes varies as the
square of the time.
If the force is retarding, f must be affected with a negative
sign, and we have
//'T*
- = velocity = u- ft, (34)
J -- (36)
i
Also if the initial velocity is in a direction the opposite of that
in which the force acts, then u is negative, and the necessary
changes must be made in the preceding formulae.
And if the particle is projected with a velocity u from o in a
direction contrary to that in which the accelerating force acts,
dx
it comes to rest when -=- = ; that is, when
(i/t
t=j,, and * = f^ (37)
It will be observed that two different modes of integration
have been adopted in this Article, the subject of both modes
being the equation (26). One mode has been the time-inte-
gration, and the other the space-integration. (27) is the result
of the former, and (32) of the latter. Thus if we introduce m
into both sides of (26), momentum is given by the time-integral
and vis viva by the space-integral. In the case of a constant
accelerating force, we are able to effect both integrations ; here-
after we shall see that the choice is but seldom offered to us ;
and that the space-integral is the only one that we can effect.
The distinction is of great importance, and will come out more
prominently than at present in a future section.
269.] UNIFORMLY ACCELERATING FORCES. 377
If a particle moves from rest, the space described in t units of
time is given by (31), and we have
~2~ '
Let x l} x^y ... x n be the spaces described in the first, second, . . .
nth units of time : then we have
= g > " * = 2 x
_./ ---^x-,
(38)
/ _/
that is, the spaces described in the first, second, . . . nth units of
time are as the numbers 1, 3, 5, ... (2n 1), and are therefore in
an arithmetical progression, the common difference of which is/.
269.] As a full understanding of the results of a constant
accelerating force is of great importance for future subjects, let
us consider it in its most elementary form, and from first prin-
ciples.
Let m start from rest at o, fig. 86 : and let the time of its
motion be resolved into equal infinitesimal elements, each of
which we shall represent by T : and let x lt x t> # ...# be the
spaces which it describes in the first, second, . . . nth time-ele-
ments ; then since f is the velocity which the accelerating force
impresses in an unit of time, the velocities of the particle at the
end of the first, second, . . . nth time-elements will be
/r, 2/r, 3/r, ... nfr.
Now imagine each successive space-element to be described in
the same time r, and with an uniform velocity through that
space-element : then if Q is a symbol for a positive proper frac-
tion, these successive uniform velocities will be
*A, (/+./) I", (2/+0,/)r, ... {(-!)/+ y} r;
and because the space is equal to the product of the time and
the velocity, #j = 01/V*,
PRICE, VOL. III. 3 C
378 RECTILINEAR MOTION OF PARTICLES. [270.
.-. the whole space = x^ + # 2 + ... + # n
= (l+2 + ... + (-l)}/T
6> + ...+e n )fT* (39)
. + - + <UA'. (40)
Let the whole space described by the particle = x, and let the
whole time = t : then t = nr : and since T is an infinitesimal
time-element, n is an infinity of that order of which T is an
infinitesimal, and we have
t
therefore from (40),
n = -
and omitting the infinitesimals, viz. the terms involving r and T*,
ft*
r t _ .
2 '
which is the same result as (31).
270.] Some examples are added illustrative of the principles
contained in the preceding articles.
Ex. 1 . It is required to divide a straight line whose length is
a into four parts, such that a particle under the action of a con-
stant accelerating force which acts along the line may, starting
from rest, describe each part in an equal time.
Let # #, x 3 , # 4 be the four parts : then, by equations (38),
1357 1+3+5+7
a
a 3a Sa la
16 16 16 16
Ex. 2. A particle moves in a straight line, under the action of
an uniformly accelerating force, and describes spaces P and Q in
the j)th and th units of time respectively ; determine the velo-
city of projection, and the magnitude of the accelerating force.
Let u = the velocity of projection, and let/* be the accelerating
force ; then
The space described in one unit of time on account of the
velocity of projection is u : and that due to the accelerating
force in the wth unit of time = - (2n 1> :
2 '
271.] ON GRAVITY. 379
P-Q
/ - ) U
P-q.
Ex. 3. A particle is projected with a given velocity u in a line
along which an accelerating force acts, and in a direction oppo-
site to that of the force's action : and the time is given between
its leaving a given point and its return to it : it is required to
determine the velocity of projection and the whole time of
motion.
Let u = the velocity with which the particle leaves the origin
o : and let the time between the particle's passage through A,
at a distance a from o, and its return to it be 2 x : let B be the
extreme point which the particle reaches : then, by Art. 268,
,7/2 A/
OB = ^1 and time from o to B = -^\
2 / J
.. the distance AB = ; a;
*/
and the time due to this distance = T : therefore by (31),
u 2 fi a
-- a = - ;
2/ 2
O *i O ft A
and the whole time of motion = = 2 fx 2 + )
J J
SECTION 3. On gravity as an uniformly accelerating force.
271.] In the Chapter on Attractions it is shewn that the at-
traction, on an external particle m, of a sphere consisting of
homogeneous concentric shells, the density of each one of which
may be different, is the same as if the whole sphere were con-
densed into its centre, and therefore the attraction of such a
sphere on an external particle varies as the square of the distance
of the particle from the centre of the sphere. Hence if a particle
moves in vacuo towards such a sphere, and under the influence
of its attraction, the law of force is that of the inverse square
of the distance from the centre of the sphere. But when the
attracted particle is nearly on the surface of such a sphere, and
3 c 2
380 ON GRAVITY. [ 2 7 l -
moves only over distances which are small in comparison of the
radius of the sphere, the variation of the attraction is so small
that it may be neglected, and the accelerating force may be
considered constant. The same result also follows from the in-
vestigation of the attraction of a plate of infinite extent on an
external particle which is given in Art. 192. If the attracted
particle lies within the surface of the sphere, the law of attrac-
tion depends on the densities of those concentric shells of which
the sphere is composed, and which are within that concentric
spherical surface on which the attracted particle is; for the
resultant attraction of all the shells external to that one
vanishes.
Now these results are approximately applicable to the attrac-
tion of the earth on particles and on bodies ; only approximately ',
I say : because the mean bounding surface of the earth is not a
sphere, but approximately an oblate spheroid, of which the equa-
torial diameter is 7925 miles, and the polar diameter is 7899
miles ; and thus the ratio of these diameters is nearly that of
the numbers 299 to 298. Now the effect of this oblateness (1)
is an increase in the earth's attraction, and thus in weight and
in the accelerating force of gravity, on particles at or near to
the earth's surface as we pass from the equator to the poles ;
and the amount of this increase is in weight about the 590th
part of the weight of a body at the equator : (2) is a change
of the line of action of the earth's attraction. If the earth were
a sphere consisting of homogeneous concentric shells, the line of
action on a given particle would be the line joining the position
of the particle and the earth's centre : as the case now is, the
line of action is, by the principle of fluid-equilibrium, perpen-
dicular to the surface of still water at the place : and is therefore
along the normal to the spheroid : all these lines of action there-
fore touch the evolute-surface of the spheroid, but do not pass
through the centre. Laplace has calculated the effect of the
oblateness of the earth on the motion of the moon ; and observa-
tion verifies his results.
Again, as the earth rotates about its polar diameter, the cen-
trifugal force, which diminishes the weight of particles and the
earth's accelerating force on particles near to the surface, is
greatest at the equator, and is zero at the poles : of this cause
of diminution and its measure we shall speak hereafter : I may
observe, however, that at the equator the weight of a body is
2 7 2.] ON GRAVITY. 381
diminished by about the 289th of its true weight, and that the
effect of centrifugal force in passing from the equator to the
poles varies as the square of the cosine of the latitude.
Also the effect of the earth's accelerating force varies by
reason of local causes : it is affected by neighbouring mountains
both as to intensity and as to line of action : it is different on
an island which is surrounded by a large mass of water, and on a
continent : it even varies, as delicate observations with the time-
measuring pendulum shew, with the materials of the earth at
the place of observation : thus may the pendulum, as M. Poisson
observes, and as we shall shew hereafter, become an indicator
of geological conditions.
Gravity also manifestly varies with the altitude of a place
above the level of the sea : experiments however, by which its
value has been determined, are supposed to be made at the
level of highwater-mark.
272.] And notwithstanding all these variations of the earth's
attraction, for bodies near to the surface the accelerating force
due to it is nearly constant at any given place, and increases as
we pass from the equator to the pole ; and decreases as we
remove farther from the centre of the earth. The exact mea-
sure of it as an accelerating force, that is, the velocity-incre-
ment which it impresses on an unit-particle in one unit of time,
for a given place is, of course, to be determined by experiment :
and at Greenwich, if one second is the unit of time, at the level
of highwater-mark, and in vacuo, the most exact pendulum
experiments exhibit a velocity-increment of 386.28 inches, that
is, of 32.19 feet. That is, if a particle falls in vacuo towards
the earth, the excess of the velocity at the end of any second of
time over that at the beginning of the second of time is 32.19
feet.
The velocity-increment is the measure of the accelerating
force called gravity ; and it is independent of the matter, form,
and magnitude of bodies. Thus in the common experiments
under the exhausted receiver of an air-pump, the heaviest metals
and the lightest pith fall from rest through the same distance
in the same time, and acquire equal velocities. And also the
time of oscillation of a pendulum is independent of the matter of
which the pendulum is made : gravity therefore as an acceler-
ating force is independent of the particular kind of matter which
it communicates velocity to.
382 ON GRAVITY. [ a 73-
273.] As to the experimental proof that gravity is an uni-
formly accelerating force : when a heavy particle or body falls
freely by itself in vacuo, the velocity of it quickly becomes so
great and increases so rapidly that the law of the increase can-
not be observed with accuracy ; hence arises the need of some
contrivance which may diminish the velocity, and not change
the law. There are chiefly two contrivances for this purpose :
firstly, Attwood's machine ; in which two unequal masses, dif-
fering slightly from each other in weight, are connected by a
very fine, and, as nearly so as may be, flexible and inextensible
string : this is suspended over a pulley, see fig. 87, ABC ; which,
by means of friction- wheels and other appliances whereby friction
is diminished, moves as easily as possible. Of course the greater
mass descends; and as both the masses move with the same
velocities, their expressed momentum-increment is the product
of the sum of their masses and their common expressed velocity-
increment; and their impressed momentum-increment is that
due to the excess of the momentum-increment of the larger mass
over that of the smaller ; that is, is due to the difference of their
weights. And these momentum-increments are equal, except
that some small part of the impressed momentum is spent in
producing the velocity of the pulley and of the string, which we
at present neglect. Now as the difference of the weights of the
two masses may be as small as it is convenient, so may the ex-
pressed velocity-increment of the masses be diminished as much
as we please, and we are thereby enabled to measure the rate of
increase of the velocity, and also the whole velocity which is ex-
pressed in a given time : and after very careful and numerous
observations it is found that,
(1) The velocity of the descending mass varies as the time
during which it has been in motion from rest.
(2) The spaces described by the descending mass vary as the
squares of the times during which they are described.
(3) The spaces described in successive units of time vary as
the odd numbers 1, 3, 5, ... (2n 1).
And as these results are in accordance with those which have
been deduced in Art. 268, when the accelerating force is con-
stant ; and as the processes by which these results were proved
may be inverted ; it follows that the moving force by which, in
Attwood's machine, moment-increment is impressed, is uniform ;
274-] ON GRAVITY. 383
and therefore the earth's attraction, or gravity, is an uniformly
accelerating force *.
Secondly, the oscillating pendulum is a contrivance by which
great accuracy is attained, in which the velocity-increment is
easily measured, and which is actually employed for the pur-
pose. To the lower end of a fine straight rigid rod a body is
attached, the mass of which is so large in comparison of that of
the rod, that the mass of the latter may approximately be
neglected : the upper end of the rod is fixed to a horizontal
axis, about which the whole rod and body vibrates freely. Now
if the rod is moved out of its position of rest, and turns about
this axis, the rod and body will vibrate ; let the vibrations
be small, and let the motion take place wholly in one plane :
then it is observed that the oscillations are isochronous, that is,
are performed in equal times. In a future Chapter it will be
shewn that such isochronism of bodies moving in small circular
arcs can be true only when the accelerating force is constant;
and therefore we infer that the force of gravity under the action
of which these isochronous oscillations are performed is a con-
stant accelerating force.
274.] In the following examples of the action of gravity, the
time-unit is taken to be one second, the space-unit one foot ;
and the velocity-increment is supposed to be 32.2 feet (rather
greater than its correct value 32.19 feet in the latitude of
Greenwich) for facility of calculation, and is symbolized by g ;
and m, is the mass of the moving particle.
We will consider the case (1) of a falling body : (2) of a body
projected vertically upwards with a certain velocity : and in
both cases I would observe that if the time results with a nega-
tive sign, it expresses an epoch anterior to that at which we
suppose our time to commence.
(1) The motion of a heavy particle m falling towards the
earth.
Let a certain point o, (a) fig. 88, in the line of the particle's
motion be taken as the origin : and let #=OP be its distance
from o at the time t : then, if dx = p Q is the space described in
dt, that is, in dt units of time,
* From this Article, and from Art. 260, it appears that two principal results
are established by Attwood's machine : (i) the matter of the earth is such, that
the expressed momentum-increment is equal to the product of the mass and the
expressed velocity-increment : (a) gravity is an uniformly accelerating force.
384 ON GRAVITY. [^74-
dx
-=7 = the space described in one unit of time,
dt
the velocity at the point P, and at the time t,
= v (say) ;
dx
and therefore m -=- = mv is the momentum of the particle at
{it
/7i} d^y
that time. Hence -77 = -^ is the expressed velocity-increment
U't (tt
d*x
in an unit of time, and m -j- is the expressed momentum-mere-
st*
ment in an unit of time. This last expression is to be equated
to mg, which is the earth's impressed momentum-increment on m
due to a second of time ; so that we have
Now of this equation let us first take the time-integral ; then if
u = the velocity of m, when t = 0, the definite integral of (41),
the superior and inferior limits on both sides corresponding to
t = t and to t = 0, is
dx
Tf -" = fft; (42)
dx
-j- = the velocity of m = u+gt\ (43)
dt
that is, the velocity is equal to the sum of the initial velocity,
and of that which gravity has impressed in t"; and if the particle
is projected upwards from^ o in a direction contrary to that in
which x is measured and g acts, then
%=*-* ()
Again integrating (43), and supposing a to be the distance of m
from o when t = 0, let us take the definite integrals with limits
corresponding to t == t and to t ; and since
ot*
dx (u-\-gf}dt- t .-. xa = ut+ ~;
m
x = a + ut+?' (45)
i
Next let us take the space-integral of (41), and multiplying both
sides by 2 dx } we have
274-] RECTILINEAR MOTION OF HEAVY PARTICLES. 385
and taking the same limits of integration as before,
rit
-); (46)
..
Jmt
.-. = (vel.) a = ' + 2g(x-a). (47)
from which the equation of vis viva and of work may be de-
duced ; for from it, if v = the velocity at the time t, we have
fflt
y(v'-O = mg(x-a), (48)
of which the left-hand member expresses vis viva as defined in
Art. 259, and the right-hand is the work taken from the mass m
as it moves through the vertical distance x a.
If when t=Q, #=0 and the particle is at rest, then a = 0, and
u 0, and
dx at* dx 3
^^ *--'' =
Whence in a particle falling from rest,
(1) the expressed velocity varies as the time;
(2) the expressed velocity varies as the square root of the space ;
(3) the space varies as the square of the time.
And generally, (43) gives the velocity in terms of the time :
(45) gives the space in terms of the time : and (46) gives the
velocity in terms of the space : (45) is of course identical with
the equation which would result from the elimination of the
velocity by means of (43) and (46).
(2) Suppose the particle m to be projected vertically upwards
from o, see (/3), fig. 88; OP = #, pq = d#; and let OH =% where
H is the highest point which m reaches ; and let u = the velocity
of projection from o : then since g in this case causes both the
velocity and the distance to decrease as t increases,
d*x
and taking definite integrals with limits the same as heretofore,
dx
*-*=-*
dx
-jf*-9t, (51)
* = ^-f; (52)
PRICE, VOL. III. 3 D
386 RECTILINEAR MOTION OF HEAVY PARTICLES. [275
dx*
also from (50), -j- n* = 2gx,
JjL-=u*-2gx. (53)
at
fid*
When -=7 = 0, the particle comes to rest; therefore from
dt
(51) and (53),
the time, when m comes to rest, = -; (54)
9
the distance from o to H = ; (55)
after m has come to rest at H, it begins to descend : and from
the preceding formula it is manifest that the time of the descent
from H to o is equal to that of the ascent from o to H ; also that
the velocity acquired in the descent is equal to that lost in the
ascent. These results are also evident from first principles.
275.] Ex. 1. A particle falls from rest; determine its velo-
city, and the space which it has described at the end of 6".
fj^'v* fltf f/f^
.*. the velocity at the end of 6"= 6 x 32.2 feet ;
32 2 x 36
the space described during 6" = - - feet.
i
Ex. 2. A particle is projected vertically upwards with a velo-
city of 100 feet in one second: find the height to which it
ascends, and the time of its ascent.
CL dt/
~dt* ~~ 9 '
dx
dt
dx
, dx u 1000 u* 10000
wnen -77 = 0, t =. = - > x = = -
dt g 322 2g 64.4
Ex. 3. A particle is projected upwards with a velocity u ; find
the time which intervenes between its leaving, and returning
to, a given point in its path.
Let a=the distance of the given point A from o, the point of
projection : then if t is the time from o to A,
276.] RECTILINEAR MOTION OP HEAVY PARTICLES. 387
... , = ^ + K-2^)* (56)
9~ 9
/tt
and the time to the highest point = -
/
In (56) the upper sign refers to the passage of the particle
through the given point in its descent, and when it has left the
highest point; and the lower sign refers to the passage of m
through the given point in its first ascent : therefore
,1 . -. 2(u*-2ag)*
the intervening time = -
Ex. 4. With what velocity must a particle be projected down-
wards, that it may in n" overtake another particle which has
already fallen through a feet.
Let u = the required velocity : therefore the space which the
first particle will pass through in n" is
and the velocity which the second particle has, when the former
starts from rest, is (lag}* : therefore at the end of n", its dis-
tance from the origin, is a -f- (2ag}%n+- ; which, being equated
to the preceding distance, gives
Ex. 5. A particle whose elasticity is e falls through a given
vertical distance a and strikes a horizontal plane, whence it re-
bounds, and falls again ; and so on continually : find the whole
space which it passes through before it comes to rest.
By (49), the velocity of impact on the plane = (2ag}%;
.-. by (1 5), Art. 264, the velocity of rebound == e(2ag)% :
.. the height to which the particle ascends, by (55), = e'a;
similarly after the second impact, the height = e* a ; and so on :
therefore the whole space = a+2 {e*a + e*a + ...}
= a
1-e 1
1+e*
1e*
276.] Let us also investigate, and apply to certain examples,
the equations of motion of two given masses connected by
a fine inextensible and flexible string, which is suspended over
30 a
388 KECTILINEAK MOTION OF HEAVY PARTICLES. [276.
a pulley, as in Attwood's machine, see fig. 87 : we shall suppose
the pulley and the string to be without inertia, so that no part
of the impressed momentum is spent in giving velocity to them.
Let m and m be the masses respectively at p and Q at the
time t : of these let us suppose m to be the greater, so that P
descends: let AP = x, BQ = #': then, since the string is in-
extensible, x + x > _ a constant ;
dx dx' , d*x d*x'
' Tt + ltt= ' and ^ + ^ = 0; < 57 '
whence we conclude that the velocity and the velocity-incre-
ments of the two particles are equal and have opposite signs.
Now the whole mass which receives and develops velocity-
d*x
increment is m + m' ; and as the whole of this has -=-- for its
at 1
velocity-increment, the expressed momentum-increment
d*x
and the impressed momentum-increment is the excess of that of
m over that of m' ; that is, is mg m'g ; therefore
d*x
(m + mf)-^ = mg m'g; (58)
(Px _ mm' /__v
~di? ~~ = ^ri 9 '
from which equation all the circumstances of a motion such as we
have supposed are to be deduced. The corresponding equation
of motion of m' is
**_ _ m^-m , }
dt> " m' + m 9 '
As to the circumstances of the initial velocity : suppose m to be
projected vertically downwards, so that if it were free it would
have the velocity a ; and similarly let m' be projected vertically
downwards, so that if it were free it would have the velocity a':
and let the velocity with which, by virtue of these two separate
velocities, in and m' move when connected by the string, be u :
then (m + taf) u = ma- ma ; (61)
ma m'd
.-. u = -: (62)
'
which gives the initial velocity with which m begins to descend,
if ma is greater than m' a' ; and with which m begins to descend
if ma is greater than ma.
277-] RECTILINEAR MOTION OF HEAVY PARTICLES. 389
Let the initial value of x be a : therefore from (59),
dx mm'
w = - -,gt\ (63)
dt m + m' J
dx* mm' .
(x a); (64)
^
m m' gt* ,,.,,
x = a + ut+- , \ (65)
m + m 2
u being given by (62).
And if a is the initial value of x' , we have
m m' at*
x* = a ut ; , y - (66)
2
277.] Examples illustrative of the preceding formulae :
Ex. 1. m = 16.6 oz. : m'=: 15.6 oz., and they start from rest :
required the space through which m passes in 5", and the velo-
city which it has at the end of the time. From (63) and from
(65)> the velocity of m = 5 feet;
the space =12.5 feet.
Ex. 2. A mass of 10 Ibs. is distributed at the ends of a thin
cord passing over a fixed pulley, so that the heavier weight
descends through 3g feet in 10": it is required to find the
weights at each end of the cord.
Let ? =one weight; therefore 10 ? = the other : then from
(65), _mm'gt* 2w 10
' ' ~ ''
2 10 2 '
.-. w = 5.3, 10 w = 4.7.
Ex. 3. A heavy mass m draws another m' by means of a
flexible and inextensible string over a pulley : at the starting of
the weights, m is thrown downwards through a feet, and m'
through a' feet : it is required to determine the distance through
which m descends in if'.
Using the notation of equation (62),
mafr
u =
m + m'
therefore from (65),
ma? m'a'* tn ... mm' at*
x = - (2 flO* t+ - -,
m + m' m + m' 2
Ex. 4. It is required to determine the velocity-increment of
the centre of gravity of two heavy masses m and m' which are
connected by a string passing over a fixed pulley.
390 RECTILINEAR MOTION OF PARTICLES. [278.
Let x and x' be the vertical distances of m, and m' at the time
t below the horizontal line passing through the centre of the
fixed pulley : and let x be the vertical distance from the same
line of their centre of gravity. Then
and substituting from (59) and (60),
mm' ,m'm
= m - -,g-\-m , -- 9 ',
y y
m
SECTION 4. Rectilinear motion of particles in vacuo under the
action of varying accelerating forces.
278.] The varying accelerating forces whose effects will be
considered in this section are supposed to be explicit functions
of the distance between the moving particle m and the point
wherein the force resides, and whence its influence emanates ;
and the motion of the particle is supposed to be along this line.
Thus the force is only implicitly a function of the time : that is,
only so far as the passage of the particle through a certain dis-
tance requires time, and the distance may thus become a func-
tion of that time; and the equation of motion will be of the
form ~.
and not of the form ~ = f(t). (69)
We limit our considerations to the former form chiefly, be-
cause it expresses the laws of communication of velocity which
present themselves in the salient phaenomena of nature : although
in some problems the latter law will also occur.
The point whence the influence of a force emanates is called
the centre of the force ; and according as the force attracts or
repels, so is it called an attractive or a repulsive force.
Let us consider briefly the general case. Let m = the mass
279-] VARYING ACCELERATING FORCES. 391
of the moving particle, and let the centre of the force be the
origin o, fig. 86 : let P be the position of m at the time t : let
o P = x : let the force vary as the ni\\ power of the distance, and
let p., which is called the absolute * force, be the value of it
when x = 1 and m = 1 ; so that the impressed momentum-incre-
ment is ii.mx n , which is to be affected with a positive or nega-
tive sign according as the force is repulsive or attractive. Now
the expressed momentum-increment is m ^ : therefore by
Art. 258, if the force is repulsive,
because both x and the velocity increase as t increases; and
//, /7 1 J // ^/7*
thus both -ji v and = ^ are positive ; and consequently
(it (it Clt
dividing (70) through by m,
=** <">
If the force varies inversely as the nth power of the distance,
and is repulsive, (71) becomes
*?-JL- (72)
fc' ~ x" '
and (71) and (72) must have negative signs if the force is at-
tractive, because in that case x decreases as t increases, when
the force makes m move towards its centre.
Instead however of deducing from these general values the
circumstances of the corresponding rectilinear motion, it will be
more convenient to consider the results for particular laws of
force : and we shall choose such examples as will either elucidate
natural phaenomena or will suggest general methods for solving
problems in rectilinear motion.
279.] A particle m moves towards a centre of force which
attracts directly as the distance : it is required to determine the
circumstances of motion.
Let o, the centre of force, be the origin : and let P, fig. 89, be
the position of m at the time t : let OP = x and OA = a, where A
* In the preceding investigations on attraction, Chap. VI, I have taken the
mass of the attracting body to be the absolute force, so that at an unit of distance
and on an unit-mass the attraction is equal to the attracting mass : and thus the
attraction of one unit-mass on another unit-mass at an unit-distance apart is made
the attraction-unit.
392 RECTILINEAR MOTION OP PARTICLES. [279.
is the position of the particle when t = : let /ix = the absolute
force : then the equation of motion is
and if the limits of integration are those values which corre-
spond to t == t and to t = 0, then if the particle is at rest, when
!~;=-^-')
/ M n -.t*\ /^A\
= p(a* -a*)-, (74)
dx
the negative sign of the root being taken, because, according to
our configuration, x decreases as t increases : therefore integrating
between the limits corresponding to t = t and to t 0,
x \
cos" 1 - = u?t,
.'. a? = cos /X T tf ; (75)
Cl*K 1 . 1 . /mm*+\
-jr=a^sm^t. (76)
From (74) it appears that the velocity of the particle is zero
when #=<z, and when a?= a ; and is a maximum, viz. a^, when
x ; the particle therefore moves from rest at A ; its velocity
increases until it reaches o, where it becomes a maximum, and
where the force is zero : so that the particle passes through that
point, and its velocity decreases, and at A', at a distance = a,
becomes zero : whence the particle under the action of the force
returns, and continually oscillates over the distance 2 a, of which
o is the middle point. The distance a of A from o is called the
amplitude of the vibration.
7T
Also from (75) it appears that when #=0, t -, and when
j?=0, =0 ; so that the time of passing from A to o = - ; and
the time from o to A' is the same, so that the time of the oscillation
from A to A' is This result may also be more generally
inferred by the following method. The relation between y and t
280.] VARYING ACCELERATING FORCES. 393
is given by the periodic function (75), viz. x a cos/** t. Now
as the greatest value of a cosine is + 1 , and the least value is
1, the greatest and least values of x are +a and a: x also
will have passed once through all its values when \&t is in-
n-
creased by 2 77 ; that is, when t is increased by - ; this therefore
V?
is the time of a complete double oscillation ; and consequently
7T
the time of one oscillation, viz. from A to A', is
/*
And hence we have the remarkable fact that the time of an
oscillation is independent of the distance from the centre of the
point from which the particle began to move, and only depends
on the absolute force, and is the greater the less that is.
280.] The two following cases in nature, wherein an attrac-
tion, the law of which is that of the direct distance, presents
itself, deserve mention.
(1) A homogeneous sphere attracts a particle within its
bounding surface with a force varying directly as the distance
from the centre of the sphere; see Ex. 1, Art. 196. Let us
therefore consider the earth to be such a homogeneous sphere,
and let us suppose a particle to move under the action of the
earth's attraction within the shaft of a mine the direction of
which is vertical. Thus if c is the centre of the earth, fig. 90,
and P is the position of m at the time t, the force acting on m
varies as CP ; and thus if the shaft were continued straight
through the earth, such as that represented by A'CPA in the
figure, and if the particle were free at A, it would move to c,
where its velocity would be a maximum, and thence on to A' on
the opposite side, where it would come to rest : and thence it
would return through c to A again ; and its motion would con-
tinue to be oscillatory, and the time of the oscillation would be
independent of CA, the earth's radius.
(2) In the undulatory or wave theory of light, all space is
supposed to be pervaded in a greater or less degree by the par-
ticles of a fluid excessively elastic and jelly-like ; in the motion
of these particles light is supposed to consist, and when they
are at rest, there is darkness. It is also supposed that these
particles exercise mutual attractions on each other : that the
possible relative displacements of them are very small, and that
when displacements occur elastic forces are brought into action,
PRICE, VOL. in. 3 E
394 RECTILINEAR MOTION OF PARTICLES. [280.
by virtue of which, in conjunction with their mutual attractions,
the motion of them continues : the lines of action, as well as
the intensities of such elastic forces, of course vary from one
medium to another ; and in this variety consists the optical
character of the medium. Doubtless the arrangement of the
particles of a crystallised substance is different to that of one
which is non-crystallised. The sun, the flame of a candle, and
the electric spark, are, together with many others, exciting
causes of the motion of the particles of ether ; and the displace-
ment of each particle is very small. It seems, too, that the force
which acts on a particle in its displaced position varies directly
as the distance of it from its original position of rest ; this force
being the resultant of the elastic forces which arise from the
disturbance of the medium and of the attracting forces of the
particles. Now a ray of light consists in the motion of a series
of ethereal molecules which when at rest are in a straight line
emanating from the source of motion. The mode of propaga-
tion of the motion of the particles it is not my purpose now to
inquire into : I shall consider the motion of only a single mole-
cule of a single ray. The displacement of a molecule may be
in any direction with reference to the line of propagation of the
ray : it might be along that line, or it might, after its first dis-
placement, describe any curve with reference to that line ; it is,
however, in the theory of light supposed, and not without evi-
dence, that the motion of the molecule takes place in a plane
which is perpendicular to the line of propagation of the ray :
that is, the displacement of the particle is transversal to the
line of propagation. Generally the force acting on the molecule,
varying directly as the distance, will have its line of action in-
clined to the line joining the displaced and the original position
of the molecule ; and, as we shall shew hereafter, the molecule
will move in an ellipse, the centre of which is the original place
of rest of the molecule : but in particular constitutions of the
ethereal medium, the line of action of the force may be that
joining the original and the displaced positions of the molecule :
in which case the molecule moves along that line, and is under
the action of a force varying directly as the distance from its
original position of rest : we have then the case of a particle
under the action of a force such as we have supposed that in
Art. 279 to be, and the results of that Article are applicable.
The particle therefore has an oscillatory motion, and the ampli-
28 1.] VARYING ACCELERATING FORCES. 395
tudes of its vibrations are equal on both sides of its original
position ; and the time of the oscillation is independent of the
amplitude, and depends only on the absolute force at the centre.
Now the intensity of light is supposed to depend on the ampli-
tude of the vibration, and the colour of it on the time of vibra-
tion, that is, on the value of /* : it follows therefore that, with
such incidents of motion as we have imagined, the intensity and
the colour may vary independently of each other : the former
will depend on the original exciting cause of the motion ; the
latter on the nature of the medium : and this independence of
these properties of light is amply verified by experiment.
If the motion of all the molecules of a ray is in straight lines,
and is such as that described above, and if all the lines of motion
are parallel to each other, the ray is said to be plane-polarised ;
and as a beam of light consists of an infinite number of rays, if
the molecules of all the rays move in lines parallel to each
other, the beam is said to be plane-polarised. And although
there has not been uniformity on the subject, yet the plane,
perpendicular to which the motion takes place, may be called
the plane of polarisation.
281.] If in Art. 279 m is projected from A with a velocity u
along the line OA, and towards o, then (74) becomes
(77)
and taking the definite integrals with limits corresponding to
t = t and to t = 0, we have
x a
COS" 1 , . COS
,
f *\ / *M
( a + ) ( a + )
f* M
.-. x = acQQ^t -- -smfjfit; ( 78 )
dx **
and j- = a^sm^t ncos^t. (79)
ut
From (77) it appears that the greatest and least distances of
m from o are
/ w\i , \i
( 1 H -- ) > and (a* -\ -- ) ;
I* P- '
TS
and from (78) the time of an oscillation is, as before, r-
M*
3 E *
396 RECTILINEAR MOTION OF PARTICLES. [282.
282.] If the central force varies directly as the distance and
is repulsive, the equation of motion is
Let us suppose m to be projected from the centre of force
with the velocity u ; then we have
^a __ vj/pa (80)
7. _ _L>/*^ an*t\. (811
. . x -igi" e f {oi)
Thus as t increases x also increases, and the particle recedes
further and further from the centre of force ; and the velocity
also increases and ultimately = oo, when x = t = oo. Thus in
this case we have no oscillatory motion.
283.] From this and the preceding equation we have the
following remarkable result, which is of large application and
deserves careful consideration; we shall also frequently appeal
to it in future parts of our treatise.
The equations of motion (73) and (80) are of the same form;
d z x
viz. replacing /u, by n*, ^ = ri*x, but in the former n" is ne-
Civ
gative, and in the latter it is positive.
Now in the former case the motion is oscillatory, and the
particle never recedes from the centre of force beyond points
equally distant from the centre, the position of which is deter-
mined by the initial distance of the particle at rest from the
centre of force, or by the velocity with which it is projected
from the centre of force or from any other given point. Also
7T
the time of an oscillation is - ; and the complete periodic time is
p
> during which the particle has passed through all its possible
places, and has undergone all the different circumstances of its
motion as to position and velocity, and at the end of which the
particle is in precisely the same phase, as it is called, as it was
at the beginning. This motion is called harmonic motion, and
the equation ^ x
is called the equation of harmonic motion. Its most general in-
tegral is x = a cos (nt + a) or x = a sin (nt a), where a and a are
either arbitrary constants introduced in the course of integration,
284.] VARYING ACCELERATING FORCES. 397
or are constants determined by the limits of integration, a being
the amplitude, and - in the latter form the epoch at which the
it
particle is at the centre of force.
In the latter case, where the equation of motion is
as the time increases, the particle recedes further and further
from the centre of force, and never returns. If it is originally
at rest at a distance a from the centre of force, it never comes
nearer to it ; and if it is originally at rest at the centre of force,
it never moves from that centre.
284.] A particle m moves towards a centre of force which
attracts inversely as the square of the distance ; it is required
to determine the circumstances of motion.
Let the centre of force be the origin; and let p, fig. 91, be
the position of m at the time t ; let A be the position of m at
rest, when t = 0, so that the particle is moving towards o : let
OP = #, OA = ; let jx = the absolute force: and let the limits
of the definite integrals correspond to t = t and to t = 0. Then
the equation of motion is
2dxd*x
dx* _ 2/x 2f*
dt* ' ' x ~ ~ a
the negative sign being taken, because x decreases as the time
increases, according to the arrangement of our figure. Therefore
integrating again, and taking the limits corresponding to t = t
and to =0, we have
.i a . 2x Tia /2u\*
(ax-*)*- -versm-' - + - = (-?) t;
. (85)
398 RECTILINEAR MOTION OF PARTICLES. [285.
From (84) it appears that the velocity =0, when a? = a; and
= oo, when x = : thus the velocity increases as the particle
approaches the centre of force, and ultimately, when it arrives
at the centre, becomes infinite; and from (85) it appears that
the time of passing from A to o is
t
If m moves from an infinite distance towards o, then a = oo,
and the velocity at a distance x from o = f )
If m is projected from A with a velocity u } then we have
dx* ^ _ 2f* 2j^
dp ' x a
and the process of integration is the same as the preceding.
This problem is that of a particle moving in vacuo from a
given place above the surface of the earth towards the earth's
centre, the distance through which it moves being so great that
the variation of the earth's attraction due to the distance must
be taken account of. In this case if a is the radius of the earth,
and g is the earth's impressed velocity-increment at the surface,
and is such as we have taken g to be in the preceding section,
and x is the distance from the centre of the earth of the moving
particle at the time t, then the equation of motion is
/7-T P 8
U/ JC K, e .
&=-'+' (86)
and if the particle is projected upwards from the surface of the
earth with the velocity u, we have
dz*
and the particle comes to rest, when
- , (87)
If however the particle falls towards the earth, and also passes
from above to below its surface, as, for instance, down a mine,
the law of force changes at the surface : and having varied in-
versely as the square of the distance, then varies (approximately)
directly as the distance.
285.] Again, let the force vary inversely as the square root
of the distance and be attractive ; and suppose the particle to
be at rest at a distance a from the centre of force ; it is required
to determine the circumstances of motion.
286.] VARYING ACCELERATING FORCES. 399
dx
and thus the velocity at any point of the path and the time occu-
pied in arriving at that point are known : and when the particle
arrives at the centre, a? = 0, and the velocity = 2jx* a*, and the
time =
286.] Let us briefly consider the case in which the force
varies inversely as the nih power of the distance, and investigate
the laws under which the time of moving over a finite distance
can be found.
_ (88)
dt* - 1 -
if the particle is at rest at a distance a from the centre of force.
This equation, which is that of vis viva, assigns the velocity in
terms of the distance. To find the time; from (88), if T is the
time from x = a to x = x,
n i i w1 n ~
~
1 f x n
a*~ I x 3 (a*-* x*~ l )-ldx. (89)
'a
As the element-function under the sign of integration is of
the form (86), Art. 43 (Integral Calculus), the expression is in-
tegrate by rationalization, (1) when = - > (2) when
Wt ^~ 1
n = - > m in both cases being an integer. The series of
values of n in the two cases are
531 35
"7 1 5' 3' *' I' I f
- - - 3 4
\/ 1 Q O ^^OO
400 RECTILINEAR MOTION OF PARTICLES. [287.
287.] If the force varies inversely as the distance, and the
particle moves to the centre from a given finite distance, the
time will be expressed by means of the gamma-function. Thus
dx*
if the particle is at rest, when x = a. If T is the time of passage
from x = a to x = 0,
Let (log -) = y ; then
T=(-) i y x e-^^
that is, the time varies directly as the distance.
288.] Let us now take the case of a particle moving along the
straight line joining two centres of force of equal absolute in-
tensity which vary directly as the distance.
Let A and A', fig. 92, be the centres of the forces, at a distance
2 a apart : let o, the middle point of AA', be the origin : let \i. be
the absolute force of each centre : let B be the position of m at
rest, P its position at the time t : OA = o A'= a : OB = b : OP = x.
Then the equation of motion is
x = b cos (2 fi)* t. (93)
Thus it appears that the velocity of the particle is zero when
x= b : the particle therefore moves from rest at B, and comes
to rest again at a point B' on the opposite side of o, and at a
distance from it equal to that of B : also the velocity is a maxi-
mum at o : and the particle returns from B' to o, and again to B,
289.] VARYING ACCELERATING FORCES. 401
and thus oscillates continually : and from (93) it appears that
7T
the time of an oscillation is -
(2**)*
289.] A particle m is placed at rest at a certain point in the
line joining the centres of two forces, which vary inversely as
the square of the distance : it is required to determine the cir-
cumstances of motion of m.
Let A and A' be the centres of force, fig-. 94, of which let the
absolute forces be p and p : let the point o, which is the neutral
point of attraction between them, be the origin; OA = a,
OA'= a'', let AA'= c : then
+ /*'
whereby a, and a are known. Let B and P be respectively the
places of m when tf=o, and when t=t: let OB = 3, OP = #:
then the equation of motion is
* *' (94)
therefore multiplying by 2dx and integrating, and taking the
limits corresponding to t = t and to t 0,
dx* 2/x 2ju' 2fj. 2 1/
dP ~~ a x a'+x a 6 a' +6'
which equation involves an elliptic transcendent, and does not
generally admit of further integration.
Suppose however the circumstances to be such that the par-
ticle is projected from B with a velocity w, and comes to rest at
o : then from (94) by integration we have generally
dx* 2u 2u,' 2u. 2u
u* = h
dt* ~a xa'+x ad a'+ &'
dx
and since -7- = 0, when x = 0,
at
2ut> 2u'&
,t ^
~
a(a-b] a'(a'+b]
If the velocity of projection is less than that thus determined,
m will not reach o, but will come to rest at some point short of
it, and will, as the force at A is greater than that at A', return
to A : similarly if the velocity of projection is greater, the par-
ticle will pass beyond o, and will eventually fall into A'.
PRICE, VOL. in. 3 F
402 RECTILINEAR MOTION OF PARTICLES. [2,89.
Now if A and A' are the centres of two spheres, each of which
is composed of concentric homogeneous shells, every particle of
which attracts with a force varying directly as the mass, and
inversely as the square of the distance, then each sphere will
attract an external particle with a force which varies directly as
its mass, and inversely as the square of the distance of the par-
ticle from its centre ; see Ex. 4, Art. 193. Now suppose A' and
A to be the centres of the earth and the moon, which are assumed
to be spheres, and to be at rest : and suppose P to be the position
at the time t of a particle in the line joining their centres, and
acted on by their attractions ; then we have the circumstances
of the preceding problem : and since the mass of the earth is
about 75 times that of the moon, as determined by tidal obser-
vation and by the phaenomenon of nutation, we have
and also since the mean distance of the moon's centre from
that of the earth is about 60 (actually 59.9643) of the earth's
equatorial radii, or about 237000 miles, we have a + a'= 237000
miles: therefore a'= 212466 miles, a = 24534 miles; such are
the distances from the centres of the earth and the moon of the
neutral point of attraction of the two bodies,
Suppose now a particle to be projected from the surface of
the moon towards the earth, and with such a velocity as just to
arrive at the neutral point, and to remain at rest there. Then
since the mean radius of the moon is 1080 miles,
b (24534 1080) miles = 23454 miles : and therefore
a b = 1080 miles, of + b = 235920 miles.
32 2
Also if g is gravity at the earth's surface, g = - miles ;
1 7oO X 3
and if r is the earth's mean radius, r = 4000 miles,
therefore substituting in (95), and reducing, we have ultimately,
u = 7852 feet in 1";
and therefore if the moon were not moving, and if there is no
atmosphere, so that the projectile does not meet with a resisting
medium, a particle thrown from its surface with a velocity
greater than 7582 feet in 1" towards the earth, will pass
beyond the neutral point of attraction, and will move towards
the earth.
290.] VARYING ACCELERATING FORCES. 403
290.] Suppose two particles m and m' , which attract each
other directly as their masses, and as the distance between
them, to be placed at rest at two given points, and then to
be left to their mutual action : it is required to determine their
positions at a given time, and the other circumstances of their
motion.
Let a and a' be the distances of m and m' from a certain point
o, fig. 93, in the line joining them, when they are at rest, and
when t = : and let x and x' be the distances of them from the
same point o, when t = t: let OA = , OA'=a': OP = #, OP'= x':
then the equations of motion of m and m' respectively are
Cv y/ / \ CL x ..
= m'(af-x)' f = m ( x - x '). } (96)
d*x ,d*x
.; . m -j -|- m T = :
dt* dt 3
dx ,dx f
= ' < 97 >
///*i n w
the initial values of -=7 and of -^- being simultaneously zero ;
u/t d/t
and taking the definite integral, with limits corresponding to
t = t and to t = 0, we have
m(xa}+m'(x'a') = 0. (98)
If x refers to the centre of gravity of m and m', then
(m + m')x = mx + m'x'
ma + m'a'-, (99)
and therefore the centre of gravity remains at rest. Again
from (96),
d*x d*z , ,
-W~-d = - (m+m](x - x} ''
let x' x z ', and let m' + m = p ;
d*z dz* ,
.'. =-pz; and = n{( a -.a)*-z*},
dz dx' dx
because when t = 0, -=7 = -=- -^- = 0: therefore
dt dt dt
z ,
.. cos- 1 -, = u?t,
a a
because when t = 0, z = a' a. Therefore substituting
x' x = (a' a} cos (m + mf t ;
F a
404 RECTILINEAR MOTION OF PARTICLES. [29 1.
m(a' a) . . .
.'. % = x 4 -- i -- -r- cos (m + m)? t.
m + m
''
m'(a'a) Nl ,
x = x -- T- cos (m + m\* t.
m+m
Thus the motion of the particles is oscillatory, the periodic
rrr
time being - ; this result may be inferred from the pre-
(m + m'y*
ceding equation in terms of z, which is the equation of harmonic
motion.
291.] A centre of force, which varies as the distance, moves
with an uniform velocity along a straight line, and attracts a
particle, which is projected with a given velocity from a given
point in the line of motion of the centre of the force and along
that line : it is required to determine the circumstances of
motion of the particle.
Let o, fig. 95, the position of the centre of force when t = 0,
be taken as the origin ; let o = the constant velocity with which
the centre of force moves along OA, and let c be its position at
the time t, so that oc = at: let A and p be respectively the
positions of m when t = 0, and when t = t: OA = a, OP = # :
and let m be projected from A along the line, and in the direc-
tion AP, with a velocity u. The equation of motion is
= ^(x af); (100)
d*x
.'. -jp + \t.x pat ;
whence by integration we have
x =
where Cj and c a are arbitrary constants introduced in integra-
tion, and which are determined by the following conditions :
when t = 0, x~a } therefore C a = a ; also when t = 0, -=- = u;
_ at
therefore c t = : and thus
M a
x = at H -- sin [jfi t + a cos }ji* t -,
/*
whence it appears that the mean value of x varies directly as
the time : that the particle is sometimes before and sometimes
293.] RELATIVE MOTION. 405
behind the centre of force ; and that while it has on the whole
a progressive motion, it oscillates from one side to the other of
the centre of force ; and that the period of an oscillation is
f**
292-3 Thus far we have referred the place and the velocity of
a moving particle to a fixed origin and to a particle at rest
respectively. It is however frequently convenient to refer to an
origin moving either uniformly or with a varying velocity. In
the former case the motion is called absolute) and in the latter
it is said to be relative. The problem of the preceding Article
affords so good and so simple an instance of rectilineal relative
motion that it is worth while to consider it from that point of
view, although we shall somewhat anticipate the complete dis-
cussion which will be made in the following Chapter.
Let the moving centre of force be the point relative to which
the motion of the particle is to be estimated ; and let the distance
of the particle from it at the time t be z ; then employing the
notation of the preceding Article, z xa.t. Consequently
dz _ dx d*z d*x
~dt ~ dt~ a ' W ~~~' ~d'
and the equation of motion is
^+M* = 0; 002)
which is the equation of harmonic motion ; and thus the motion
is oscillatory about the centre of force, the particle being some-
times in advance and at other times in the rear of the moving
centre, and the periodic time being If the particle is pro-
M*
jected from the moving centre with a relative velocity /3, that is,
with an absolute velocity /3 + a, the integral of (102) is
z
SECTION 5. Rectilinear motion of particles in resisting media.
293.] A particle is projected from a given point with a given
velocity, in a medium of which the density is constant, and of
which the resistance varies as the square of the velocity, and
where no other force acts on the particle : it is required to
determine the circumstances of motion.
406 BECTILINEAR MOTION [ 2 94-
Let the point from which the particle is projected be taken
as the origin : and the line in which it is projected as the axis
of x : let u be the velocity of projection, and let x be the dis-
tance of the particle from the origin at the time t : let the par-
ticle be of the mass m, and be of such a form as to present a
plane surface o> to the medium in the direction in which it
moves : then by (21), Art. 267, the equation of motion is
d*x ,dx\*
m W =<*(*) !
and as p is constant, let p&> = mk ;
k is commonly called the coefficient of resistance. Now putting
(103) in the following form,
~dt
and taking the definite integral at limits corresponding to t = t
and to t = ; we have
log ^ -log =&?;
,- " .-" " 1 = *^ . ' < 104 )
e**dx = udt,
e**-\ = kut. (105)
(104) gives the relation between the velocity and the distance
through which the particle has passed: and (105) gives the
relation between the distance and the time. From (104) it
dx
appears that -5- = 0, or that the particle comes to rest, when
dt
x = oo : in which case also t = oo, as appears by (105).
294.] A heavy particle m acted on by gravity (a constant
accelerating force) moves in the air, which is supposed to be a
resisting medium, whose density is uniform, and the resistance
of which (according to Art. 267) varies as the square of the
velocity: it is required to determine the circumstances of
motion.
Firstly, let us suppose the particle to descend towards the
earth and to start from rest : then if p is the constant density
294-] 1N RESISTING MEDIA. 407
of the air, and if o> is the area of the face, which m presents to
the medium, transversal to the direction of its line of motion,
d' x fd*\* / 1ftc \
m W = mff - p( *(di)' (106)
affecting the resistance with a negative sign, because it tends to
diminish the velocity :
d*x pot fdx^
*' ~dP~ 9 ~~m{w '
Let = k, the coefficient of resistance : so that we have
,dx
~dt
(108)
whence integrating, and taking the definite integrals corre-
sponding to t=t and to ^=0, we have
i 7 i dx
, i g* + k* ^7
^1 dt 14
-log - -. = kt;
dt ~
Also again from (107),
"
= Ikdx;
therefore integrating, and taking the limits as before,
7 /<&\ 2
'
,
log =
y
= | { l-^ } . (1,0)
(109) gives the velocity in terms of the time, and (1 10) in terms
of the distance through which m has passed. Also from (109),
-_ ,
kdx - (%)* -- , -- - dt;
'
408 BECTILINEAR MOTION [ 2 95-
therefore integrating-, and taking the same limits as before,
fa = log {(**>* + <?-(**>*} -log 2 ;
.-. 2e** = (*)*'+-(*)*'; (111)
which gives the relation between the distance and the time to
which it is due. This equation might have been found by
eliminating the velocity between (109) and (110).
When t = oo, x = oo ; that is, an infinite time is required
for an infinite space : but when x oo, and = oo, the velo-
city = (^\ , that is, becomes uniform ; in which case, as appears
d^x
from (107), -= = 0; and although this state is never attained
Cvv
to, yet it is that to which the circumstances of motion approach ;
also this limiting velocity is greater, the less Jc is ; but k varies
directly as the density of the medium, directly as the surface
which the particle presents, and inversely as the mass of the
particle : therefore the terminal velocity is greater, the greater
the mass of the particle is, and the less the density of the
medium is, and the less the area of the face is which the particle
presents to it in its motion. These results are in accordance
with experience. From the form of (107) it appears that the
equation is satisfied if
because in that case -^- = : this therefore is a solution of
Cut
the equation, and is a singular one, because it does not arise
by giving any particular values to the arbitrary constants, to
which the limits of the integrals are equivalent, and which are
therefore dependent on the initial circumstances of motion. It
appears therefore that the general integral represents the cir-
cumstances until the velocity attains its constant value; and
that then the singular solution represents the motion. Other
and similar peculiar properties of singular solutions will be
exhibited hereafter.
295.] Secondly, let us suppose m to be projected upwards,
that is, in a direction contrary to that of the action of gravity,
with a given velocity u : it is required to determine the circum-
stances of motion.
Let us moreover suppose m to be of such a form, that it
presents to the medium an equal area transversal to the line of
295-] IN RESISTING MEDIA. 409
motion, whether it falls, or whether it moves upwards : then if
x is measured upwards, gravity and the resistance of the medium
both tend to diminish the velocity as t increases : so that the
equation of motion is
and if po> = mk, we have
d*x . f dx^
v j-Jtj).; ("3)
7 dx
Tt
therefore integrating, and taking the limits which correspond
to t=t and to t=Q,
dt "
which gives the velocity in terms of the time.
Again, from (113), if we multiply both sides by 2dx,
*(>'
therefore integrating, and taking limits the same as before,
\*
,.
~ = ~ ;
which gives the velocity in terms of the distance.
Also, from (115),
dx - (^ U ^ C
"^ ^*s
and therefore integrating, and taking the limits the same as
* = ilo *** "*&)** +***(&)**. (117)
* y*
PEICE, VOL. III. 3 G
410 RECTILINEAR MOTION" [296.
which gives the space described by the particle in terms of the
time to which it is due.
From (115) and (116), when -77 = 0, that is, when m has
dt
reached the highest point,
f.^A-in-'tfrf)*, (us)
v
which give the distance of the highest point, and the time of
reaching it. After which the particle begins to fall, and the
investigations of the preceding Article are applicable.
Since k is the same in this and the preceding Article, that is,
since m presents an equal area o> in the ascent and the descent,
by (110) the velocity acquired by m in descending to the point
whence it was projected with u is
_, (120)
which is less than u : hence the velocity acquired in the descent
is less than that lost in the ascent, the reason being that mo-
mentum is withdrawn from m both in the ascent and in the
descent, and is transferred to the molecules of the medium.
Again, substituting (119) in (ill), the time occupied in the
descent is
which is different to that required for the ascent, as given in
(118).
296.] Let us also consider the motion of a particle under the
action of a constant force in the line of its motion, and moving
in a medium, the resistance of which varies as the velocity ; and
let us suppose the particle to be projected with a velocity u,
when t = and x = 0. The equation of motion is, in terms of
velocity -increment,
-/-'& <>
wherein / expresses the constant force, and k is the coefficient
of resistance. Therefore integrating, and taking the limits which
correspond to t = t and to t = 0,
297.] IN RESISTING MEDIA. 411
(123)
= T +( ^'- 1) - (124)
Thus from (123) we have the velocity in terms of x and t ; and
in (124) the relation is given between x and t : hence also
dx f
And if t oo, x =. <x>, and -=- = -, ; that is, the velocity has
Clb K
this finite limiting value, which it attains only when = oo. This
7 -/*
result also follows from the equation of motion : -; -- *4 = is
dt k
a singular solution of it : and thus the particular integrals
(123) and (124) express the circumstances of the motion, so
long as the time is finite ; but when t = oo, the singular solu-
tion expresses them.
297.] Lastly, let us consider the case of a particle moving in a
resisting medium, where the density of the medium varies ; and
let us suppose the resistance to vary as the square of the velo-
city, and the density to vary inversely as the square of the dis-
tance from a given point ; and the particle also to move under
the action of an attracting force which varies inversely as the
cube of the distance from the same point.
Let a and x be the distances of m from the given point when
t = and when t = t. Let u = the velocity of m when t 0,
and let /x be the absolute force of the central force : then the
equation of motion is
Let k'u> = mk : so that we have
*=-*+*-(*?{. (126)
dt* x 3 * x* \&'
302
412 RECTILINEAR MOTION IN RESISTING MEDIA. [2.97.
Multiplying by 2 dx, we have
, dx* 2
dx 3 2k /<fo\ 2jx
a linear differential equation, of which the integrating factor is
i
e* ; therefore integrating, and taking the limits which corre-
spond to t = t and to t = 0, we have
2k x
_ ax _
a
which gives the velocity in terms of the distance ; but does not
admit of further integration.
CHAPTER IX.
THE THEORY OF CURVILINEAR MOTION.
SECTION 1. The Kinematics of a particle moving in a curvilinear
path.
298.] The motion, whose incidents we have thus far con-
sidered, has been that of a particle describing a rectilineal path ;
but there is a much more general case, viz. that in which the
path is curvilinear; and I propose to consider the kinematics
of such motion with the object of applying the results dyna-
mically. Here as elsewhere some system of reference is needed,
to which the path of the particle may be referred, and whereby
its position at any time may be determined ; the systems usually
taken are the Cartesian, whether of plane geometry or of geo-
metry in space; the two corresponding systems of polar co-
ordinates; and sometimes peculiar facilities for the solution of
a problem are offered by the intrinsic equation of a curve. We
shall hereafter have examples of all these.
The conception and the definition of velocity and of velocity-
increment acceleration which are given in Arts. 246, 247 are
evidently just as applicable to a particle describing a curvilinear
path as to one moving along a straight line ; and consequently
what has been said on these subjects need not be repeated. In
reference however to a curvilinear path, if * is the length of an
arc measured along the curve from any fixed point in it, and
dt is the time during which an infinitesimal arc-element ds is
ds tl*s
described, then -=- and -j are respectively the velocity and
the velocity-increment of a particle moving along the curve.
299.] Let us first suppose the path of the particle to be a
plane curve, and refer its place to a system of rectangular axes
in that plane, and let us take (x, y} to be its place at the time t,
so that x and y are functions of t\ and consequently if t is
eliminated by means of these two equations, the resulting equa-
tion in terms of x and y is that to the path described. This
.path is technically called the trajectory of the particle.
414 THEORY OF CURVILINEAR MOTION. [299.
Let (x, y] be the place of the particle at the time t, and
(x + dx, y + dy] at the time (t + dt}, so that ds, which is equal to
(fix 3 + dy*}? } is the path described in the time dt ; and dx and dy
are the increments of x and y in that time ; and consequently,
//* ////
according to the definition of velocity, -j- and -jj- are the velo-
M dt
cities of the particle relatively to the axes of x and y respec-
tively ; these velocities being estimated positively or negatively
according as the coordinates are increased or diminished as the
time increases.
Also since ds* = dx* + dy 2 ; (1)
and accordingly the square of the expressed velocity is equal to
the sum of the squares of the expressed velocities relative to the
coordinate axes of x and y.
Also if T is the angle at which the tangent to the path at
(x, y} is inclined to the axis of x,
dx = ds cos T ; dy = ds sin T ;
dx ds dy ds .
.'. = -=- COST, -5- = -y- sin r, (3)
dt dt dt dt
that is, the velocities relative to, or along, the coordinate axes
are severally the product of the expressed velocity and the
cosine of the angle contained between the two lines of esti-
mation. Consequently as the lines of the axes are arbitrary,
this law holds universally; and the cosine is the projective
factor of velocities, as it is also of lines, areas, and statical
pressures.
This kinematical theorem is of the greatest importance in
the treatment of complicated problems of Mechanics. It is
called the parallelogram of velocities, and yields results of com-
position and resolution of velocities which enable us to analyse
and solve questions otherwise beyond our powers. Thus if ds
is the diagonal of a rectangle whose sides are dx and dy, all
these three lines meeting in one point, the velocity along ds
ds
which is expressed by -=- may be resolved into two velocities
, -, dt
-j- and -j- which are effective along the sides dx and dy respec-
dt dt
tively ; and as one velocity may be resolved into two, so may
also two or more be compounded into a single one. We shall
301.] KINEMATICS OF A PARTICLE. 415
have illustrations of this theorem from a dynamical point of
view in the following section.
300.] In the most general case of the motion of a particle,
the velocity of it will vary so that , will not be constant ; and
dt
the resolved velocities along the coordinate axes will also vary ;
and thus -^ as also - will not be constant. Hence during
dt dt
equal and successive dfs, the ds's, dx's, dy's will not all be
equal, and we shall have increments of them expressed by d*s,
d*x, dy. And consequently, there will be velocity-increments
or accelerations along the curve, and along the axes of x and y
d^s d"*x d^v
respectively ; these will be represented by -^-- > -= ^ re-
(* I- Cvv '' '
spectively if t is equicrescent; and if t is not equicrescent
severally by
//2 o /// //2/ /Jo //2/y fj-f //2y //'> /j^ti /"// /"/2/ //j/
tfr o lib tv b tfo il Ju U/V iff b (A/JU Iv U U/b w (/ lltJ * . v
dt* dt* dt*
Before I apply these expressions to the solution of particular
problems I would observe that as they express velocity-incre-
ments, they are subject to the same laws of composition and
resolution as velocities; that is, to the law given in the pre-
ceding Article; and consequently the velocity or velocity-in-
crement of the particle along any line is the sum of the resolved
parts of the axial velocities or velocity-increments along that
line. The following are examples in which the preceding ex-
pressions are applied to cases in which the laws of velocity and
of acceleration are given :
301.] Ex. 1. A particle moves so that the axial-components
of its velocity vary as the corresponding coordinates ; it is re-
quired to find the equation of its path.
dx , dy ,
dt dt
y
if (a, 1} is the initial place of the particle ;
.-. x = ae u , y =
- I,
a b
and this last is the equation to the path.
416 THEORY OP CURVILINEAR MOTION. [3 2 -
In this case the axial velocity-increments are
d*x fry
-3 = frx, -rr- = fry.
dt 3 dt* y
(J T fl*)]
Ex. 2. If -5- = by, -j- = kx, the path is an equilateral
hyperbola, and the axial accelerations are
d*x d*y
- r - = frx, ?- = fry.
clt* di* y
Ex. 3. A wheel rolls along the straight line at a constant
velocity; compare the velocity of a given point in the wheel
with that of the centre of the wheel.
Let the line along which the wheel rolls be the axis of x, and
let u be the velocity of its centre : then a point in the circum-
ference of the wheel describes a cycloid, of which, the origin
being conveniently taken, the equation is
x = a versn
dx dy
y* (2a-yf (2)*
,, d y a dy
JNow u -j- a versm~ l - = -- - ~ ;
dt a (Zayy*]* dt
ds /2?/\i
' ; -(*"'
and this gives the velocity of the point in the circumference of
the wheel. Thus the highest point of the wheel moves with a
a,
velocity twice as great as that of the point at which y
2
This is a problem in which a curve is given, and one axial-
component of the velocity is given. From these data the other
axial-component and the velocity can of course be found.
302.] Let us now take some cases in which two out of the
three quantities, viz. the path and the two axial accelerations,
being given, the third is required.
Ex. 1. A particle describes an ellipse with a constant velocity
= a parallel to the axis of x : find the velocity and velocity-
increment parallel to the axis of y, and the time of describing
the ellipse.
Let the equation to the ellipse be
dx
and let (x, y) be the place of m at the time t : so that --=- = a;
dt
302.] KINEMATICS OF A PARTICLE. 417
dy b*x dx
dt a*y dt
ab* x
which gives the velocity parallel to the axis of y.
dx dy
at 3 a 3 y*
6* a 3
~<^ ;
thus the acceleration parallel to the axis of y varies inversely as
the cube of the ordinate of the ellipse, and acts towards the axis
of x, as is shewn by the negative sign.
dx
Since -5- = a, x = a t, if we assume the position of the par-
ticle to be at the extremity of the minor axis when t = 0.
Hence the time of passing from the extremity of the minor
axis to that of the major axis is - > and the time of describing
the whole ellipse is
a
If the orbit is a circle b = a, and the acceleration parallel to
the axis perpendicular to that along which the velocity is con-
stant is
If the velocity parallel to the y-axis is constant and is equal
to ft, then fa a*
~dt* ~ >#' '
46
and the periodic time = -
P
Ex. 2. A particle describing a parabola moves with a constant
velocity a in a direction perpendicular to the axis. Find the
velocity and the velocity-increment parallel to the axis.
Let the equation to the parabola be y* = 4 ax ; then ~ = a ;
dx ay , d*x a?
-=- = > and r = > and is constant : and as it has a
dt 2a dt 3 2a
positive sign, it shews that the particle moves away from the
tangent to the curve at the vertex with a constant acceleration.
PRICE, VOL. in. 3 H
418 THEORY OF CURVILINEAR MOTION. [3O2.
Hence as the earth acts on particles near to its surface with a
constant acceleration in vertical lines, if a particle m is projected
with a velocity a in a horizontal line, and is attracted towards
the earth in a vertical line, m will move in a parabolic path.
By a similar process, if Y represents the acceleration parallel
to the axis of y, when the velocity parallel to the axis of x is
constant; and if x represents the acceleration parallel to the
axis of #, when the velocity parallel to the axis of y is constant;
and if a and ft are the constant velocities parallel to the axes of
x and y respectively in each case ; then, if a particle m describes
2/32
A Hyperbola, xy = k* } x = -- x 3 ,
K
2 a 9
Y = T - y 3 .
6* y
4 a 2 a*
A Parabola, y' = 4 ax. y =
3
A Hyperbola, -- - - = 1, x =
a*/3'
Op
The Logarithmic Curve, y = a*, x =
log a x a 2:
Y = a* (log a)* y.
The Cycloid, starting point being origin, x = - -
JT
The Catenary, y = -$e a +e a l ,
a 2 a
Y =
x=
a j
Y =
Ex. 3. To determine the laws of acceleration parallel to the
axes of x and y, so that a particle m may describe the parabola
with a constant velocity.
Let the equation to the parabola be
dx _ dy
"
303.] KINEMATICS OF A PARTICLE. 419
And let the constant velocity along the curve = c; so that
ds = c dt ;
dx a c*x ili/" 1 c*y
~dF " x+y' ~di* "
and differentiating,
d*x _ c*(ay)l , d\y
~ ~
As very many examples of a similar kind will arise hereafter
when we treat the subject from a dynamical point of view, it
is unnecessary to insert others in this place.
303.] And we will now consider the motion of a particle in
a plane curve from another point of view and in relation to
another system of reference. The method of resolving velocities
and velocity-increments along two rectangular axes chosen arbi-
trarily is artificial, and has not been deduced from considerations
of the actual motion of the particle ; but it is convenient, and
adapts itself to the Cartesian system of algebraical geometry,
and to the ordinary equations of curves. Let us however consider
whether the actual motion of m does not lead us to another and
more natural method.
d 3 s
-57- is the velocity-increment in the line of motion of m at any
dt
time t ; and therefore if there is velocity -increment only in this
line, the path will be rectilinear : generally however the particle
moves in a curvilinear path, and there is therefore a deflexion
from the rectilinear path, and a corresponding velocity-incre-
ment : the question is, What is the mathematical representative
of this velocity-increment? In fig. 98, let PQ (= ds] be the
element of the curvilinear path described in the time dt; let t
be equicrescent ; then if the particle is not deflected from its
rectilinear path, it will in the next dt describe QR ; but suppose
QT (= ds 4- d*s) to be the element of the curve succeeding PQ,
and to be the path taken by the particle in the second dt; then
at the point Q and along the line QS the particle has received a
velocity with which it moves over QS, in the time dt, so that at
the end ofdf, m is at the point T; our object is to determine the
value of the acceleration which acts along QS.
p, Q, and T being three consecutive points in the curve, the
angle RQT is the angle of contingence; see Art. 284, Vol. I.
Let p be the radius of curvature of the path at p ; that is, p is
3 H a
420 THEORY OF CURVILINEAR MOTION. [34-
the radius of the circle passing- through p, Q, and T : and there-
fore from the geometry,
QT a (ds-\-d*s} 2 ds*
^ = -2j=--2^- = -^'
Now whatever is the law of acceleration with which the particle
m describes QS in the time dt, we may consider it to be constant
for that infinitesimal element of time ; and therefore iff is the
velocity-increment, by Ex. 2, Art. 249,
therefore from (5) and (6),
, d*_
p dt a
2
= -> (7)
P
if v is the velocity of m at P : and the line of action of it is
along the radius of curvature, that is, along the normal.
Hence at any point P of the trajectory, if the velocity-incre-
ment is resolved along the tangent to the curve at P and along
the normal, the velocity-increments along these two lines are
x- i ^ s ^ v *
respectively -r- and --
at 2 p
304,] These results may be deduced from the expressions for
the axial accelerations; for as accelerations are velocities they
may be compounded and resolved along any line according to
the laws of compositions and resolution of velocities. Hence if
T and N are the velocity-increments along the tangent to the
curve of which the direction cosines are -=- and -jj- > and along
the normal respectively, we have
d*x dx d*y dy
~~~dP ds + Wds'
w - fy. d - - ~y (^
~ dp 'ds dt"- <ti' { }
Since however ds* = dx* + dy*, ds d*s = dx d*x -f dy d*y : and
if p is the radius of curvature at p, by (5), Art. 282, Differential
Calculus, fas
dx d*ydy d*x = ;
d*s 1 ds 3 v 2
-w-' N = ^ = 7'
if v is the velocity of the particle at the point (x,y).
305.] NORMAL AND TANGENTIAL RESOLUTION. 421
These results may also be found by the following short
process. Let \fr be the angle between the normal to the path
at (x,y) and the a?-axis. Then
dx d* .
Tt = Tt s *>
d*x el's . . ds d\l/
But when ^ = 90, the path is parallel to the #-axis and -j
is the tangential acceleration ; and when ^ = 0, the path is per-
d*x
pendicular to the #-axis, and -= is the normal acceleration.
ds
Also dty = ; consequently employing the same notation as
P
heretofore. d*s v*
T = -7. N = --
dp p
fl's . d'*x
It will have been observed that -. r is not the resultant of -=
, dt* du*
fj% /*j
and of r- because it is not the resultant acceleration, there
c a
being also the normal component. To shew this still further
let E be the resultant acceleration, then
_
=
since by (15), Art. 285, Vol. I. (Differential Calculus),
^ = (d*xy+(d*yy-(d*sy', (12)
and the form of (11) shews that the resultant maybe resolved
into two components whose action-lines are at right angles to
d^s
each other ; and of which one, viz. -jr > is evidently the tan-
dt
v*
gential component, and consequently the other, viz. > is the
normal component.
305.] This mode of resolution is convenient when the tan-
gential velocity-increment is constant or is given as a function
of s; and also generally when a condition is given in terms
of the quantities of which these components of the velocity-
increments are functions. Thus if a particle describes a curve
with a constant velocity, the velocity-increment along the curve
422 THEORY OF CURVILINEAR MOTION.
vanishes, and the normal acceleration at any point varies inversely
as the radius of curvature of the curve at that point. Con-
sequently if a particle describes a circle with uniform velocity,
the normal acceleration varies as the square of the velocity and
inversely as the radius of the circle.
As we shall have several illustrations of this mode of resolu-
tion of velocity-increments in the dynamical portion of the
work, it is unnecessary to insert others in this place.
306.] The direction of the resultant acceleration may always
be found by means of the following construction due to Sir
W. R. Hamilton :
From a given point o let radii -vectores OP, OQ, ... be drawn,
representing by their length the intensity of the velocity and by
their direction the line of motion of a moving particle at each
successive point of its continuous path. In such a path there
will be no abrupt changes of velocity, and no abrupt deviations
of lines of motion; because directions of the tangents of the
curvilinear path of the particle vary continuously; and the
locus of the extremities of all the radii- vectores thus drawn will
form a continuous curve. Now suppose OP to represent the
velocity and the line of motion of the particle at the time t,
and OQ, which is infinitesimally near to it, to be a similar repre-
sentative at the time t + dt ; then the line p Q represents the
resultant of the velocity-increment, since OQ is the resultant of
OP and PQ. But as dt is infinitesimal, PQ is an arc-element
of the curve-locus of P, Q, . . . ; and consequently the arc-element
of this curve is the line-representative both in magnitude and
direction of the acceleration of the moving particle. Sir "W. R.
Hamilton has named this curve the hodograph,
The preceding properties of this curve may also thus be found.
Let (, TJ) be the point on it which corresponds to (x, y) on the
path of the particle ; then by the definition of the hodograph,
. n p
dt dt dt
if p is the radius-vector of the hodograph, and k is a constant.
Consequently d dl] dp
d*x'~ d*y ~ d*s ~
~W ~dP ~di?
And thus the tangent to the hodograph at the point (, 77) is
parallel to the action-line of the resultant acceleration at (r,y)',
307.] ANGULAR VELOCITY. 423
and the differential of the radius-vector of the hodograph is
proportional to the tangential acceleration.
307.] Before we consider velocities and velocity-increments
in reference to the system of polar coordinates, it is neces-
sary to inquire into a mode of estimating the rate or velocity
with which a radius-vector revolves in one plane about a fixed
point and generates angular quantity at that point.
Let be the angle between a fixed straight line passing
through the fixed point, and the position of the revolving radius-
vector at the time t, and let us suppose the line to revolve
uniformly, that is, to pass through, or to generate, equal angles
in equal times ; let CD be the angle generated in an unit of time ;
then if the generating line coincided with the fixed originating
line when =0, and is the angle generated in t units of time,
e = ut: (15)
.'. 0) = ^. (16)
co is called angular velocity : for as the linear velocity of a par-
ticle moving uniformly is the linear space described by it in an
unit of time, so the angle generated in an unit of time by an
uniformly revolving straight line is called the angular velocity of
the line, and the velocity of rotation. The line passing through
the fixed point and perpendicular to the plane of the rotating
line is called the axis of rotation. If however the radius-vector
does not revolve uniformly, and consequently does not generate
equal angles in equal times, then the angular velocity is a func-
tion of the time. Let the time be resolved into infinitesimal
elements, and let us suppose the angular velocity at the time t
to be , and at the time t + dt to be &> + ^co, and dQ be the angle
generated in the time dt. Then since co is the angular velocity
at the time t and co 4- d<a at the time t -}- dt, the mean uniform
angular velocity with which eld is generated is &> + </> da>, where </>
is a proper fraction, and is positive or negative according as
the velocity is increasing or decreasing; consequently by reason
of ( 15 )> d9 = (co + c/>^co)/& = o><#, (17)
omitting the infinitesimal of the second order. Thus clO is the
angle generated in dt units of time by the radius-vector re-
volving with the angular velocity o> at the beginning of dt; and
consequently dividing both sides by dt,
424 THEORY OF CURVILINEAR MOTION. [308.
d6
and co or -j- is the angle generated in an unit of time, and is
dt
the angular velocity of the revolving radius-vector.
Thus in both cases, of uniform and of continuously varying
angular velocity, angular velocity is the angle described by the
radius-vector in an unit of time ; and is the ratio of the angle
described in a given time to the time in which it is described ;
in the case of varying velocity this ratio is the ratio of two
infinitesimals.
The unit angular velocity is that with which the radius vector
rotates through an unit angle in an unit of time ; and if the
angular velocity is w, CD is a number designating the number of
unit-angles through which the radius rotates in an unit of time.
308.] Hence is derived the principle on which angular velo-
cities are measured ; if two radii rotate with angular velocities
such that each generates equal angles in equal times, the angular
velocities being uniform during that time, these angular velo-
cities are said to be equal. And this mode of determining equal
angular velocities being adopted, it is evident that one angular
velocity may be double, or treble, or n times another. If the
equal angles are described by both radii in the same direction,
the angular velocities are equal and in the same direction;
but if the equal angles are described in opposite directions, the
angular velocities are equal and opposite. Angular velocities
may therefore be affected with signs. Thus if o> represents the
angular velocity with which a line rotates in a given direction,
&) will represent the equal angular velocity of a line rotating
in the opposite direction. As angular velocities have rotation-
axes, intensities, and directions, it is evidently desirable to have
some geometrical representative of them, as of linear velocities.
This is supplied by a straight line on a principle similar to
that by which the line-representatives of couples have been
determined in Art. 46. Along the rotation-axis let a length be
taken containing the same number of linear-units as <o contains
angle-units ; then this line by its position and its length re-
presents the axis of rotation and the intensity of the angular
velocity. Let a point on this rotation-axis be taken as a fixed
pole; as the body may rotate about this axis in either of two
directions, so may the line-representative of the angular velocity
be measured in either of two opposite directions, and therefore
we must choose a principle by which direction of rotation may
309.] ANGULAR VELOCITY-INCREMENT. 425
be determined. Let it be this ; if, as we look along the axis
from the pole, the body rotates from left to right, like the hands
of a watch when we face it, let that rotation be called positive,
and let its line-representative be measured from the pole in the
direction in which we look ; but if the body rotates from right
to left, that is, in the direction opposite to that of the motion of
the hands of a watch, let that rotation be negative, and let the
line-representative be measured from the pole in a direction
opposite to that along which we look. Hereafter, however, it
will be convenient to affect angular velocities with signs in
reference to certain systems of coordinate axes in the manner
explained in Art. 69.
309.] If the angular velocity varies, this variation may take
place either uniformly or at a variable rate. In this case we
have angular velocity-increments, or, as they are often called,
angular accelerations. These changes also may take place when
the angular velocity either increases or decreases : we will at
present at least assume the angular velocity to increase, as such
an assumption will give solidity to our ideas.
Firstly, let us suppose this increase to take place at an uniform
pace; and let * be the angular velocity added in an unit of
time ; so that if the angular velocity is zero at the beginning of
t, and to is the angular velocity at the end of t,
(o = *rf: (19)
and if o> is the angular velocity at the commencement of t, and
<o the angular velocity at the end of t,
o> o> = *t; (20)
so that the increment of angular velocity varies as t, and also
as *.
If the angular velocity decreases uniformly, and * is the an-
gular velocity taken from it in an unit of time, then if w and o>
are respectively the angular velocities at t and when t = 0,
o> = to 4ft-, (21)
and the generating line will come to rest, when t =
$
Secondly, let us suppose the increase of angular velocity to
take place at a varying rate, so that the increments of angular
velocity are not equal in equal times ; and accordingly the in-
crease of angular velocity is a function either explicit or implicit
of the time.
PRICE, VOL. in. 31
426 THEORY OF CURVILINEAR MOTION. [31.
Let the time be resolved into equal elements ; that is, let t be
equicrescent; let us suppose the radius which generates the
angles to be revolving at the time t with an angular velocity (a,
and at the time t + dt with an angular velocity o> -f dw, then if
* is the angular acceleration at the time t, and * + 6?* at the
time t + dt, * -f < d* } where </> is a proper fraction, is the mean
angular velocity-increment during the time dt; and conse-
quently, by reason of (19),
dot = (* -f $ d*} dt
= *dt, (22)
if we omit the infinitesimal of the second order; that is, d<a
units of angular velocity are added in the time dt ; and dividing
through lydt, d M
__
dt ~~ dt dt
(23}
( *
if t is equicrescent. And thus whether the increase of angular
velocity is uniform or is variable, in each case the angular velo-
city-increment is the increase of angular velocity in an unit of
time.
Thus we have the unit of angular velocity-increment or of
angular acceleration, when the increase of angular velocity is an
unit in an unit of time.
310.] The following examples are illustrations of the preceding
mode of estimating velocity and velocity-increments :
Ex. 1. If a particle is placed on the revolving line which
generates angle at a distance r from the origin, and the line
revolves with an uniform angular velocity o>, the relation be-
tween the linear velocity of the particle and the angular velocity
may thus be found.
Let d9 be the angle through which the radius revolves in dt,
and let ds be the path described by the particle, so that ds = rd0;
then ds dO
*-'*'"" * (24)
so that the linear velocity varies as the angular velocity and as
the length of the radius conjointly.
Hence if a particle revolves in a circular orbit with an uniform
velocity o>, the normal component of the velocity-increment is
equal to w*r.
310.] ANGULAR VELOCITY-INCREMENT. 427
Ex. 2. If the angular acceleration is constant, and equal to *,
say ; then from (23), g^Q
= *,
dO
-***
0-0 -
2 '
where and o> are the initial values of and of co.
Hence, if a line revolves from rest with a constant angular
acceleration, we have * 2 4
and the angle described by it varies as the square of the time.
Ex. 3. If the angular velocity-increment varies as the angle
generated from a given fixed line, then the equation which ex-
presses the motion is of the form
~/n* = ^ e '
and is a harmonic equation or not according as k* is negative or
positive. If k* is negative, and a is the initial value of 0,
.. = acos&t.
Ex. 4. If a particle revolves in a circle with uniform velocity,
its angular velocity about any point in the circumference is also
uniform ; and is one-half of what it is about the centre.
Ex. 5. If a particle revolves uniformly in a circle, and its
place is continually projected on a given diameter, the linear
acceleration along that diameter varies directly as the distance
of the projected place from the centre.
Let a) be the constant angular velocity, the angle between
the fixed diameter and the radius drawn from the centre to
its place at the time t, x the distance of this projected place from
the centre, so that x = a cos 0, a being the radius of the circle ;
dx H dQ
.*. -77 = # sin 0-^7= 0o> sin 0,
dt dt
dO
which proves the theorem.
Let this suffice at present for the general explanation of
angular velocity and angular velocity-increment; we shall
3 l 2
428 THEORY OF CURVILINEAR MOTION. [3H-
hereafter return to the subject when we treat of the motion of
rigid bodies; we have now to consider these expressions in
another relation.
311.] As the curvilinear paths of particles are frequently
expressed conveniently in terms of polar coordinates, it is neces-
sary to investigate the mathematical values of the components
of velocity and of velocity-increment along and perpendicular to
the radius-vector of the particle at any point of its path. The
former are termed the radial or the paracentric velocity anil
velocity-increment, the latter the transversal velocity and velocity-
increment respectively. The required values are thus found :
Let (r, ff) be the place of the moving particle at the time t,
(x,y) being its place referred to a system of rectangular axes
having the same origin, and the ar-axis coincident with the
prime radius. Then
x r cos 6 ; (25)
dx dr . d0 . .
> (26)
d*x n d*r n dr dO n fdO^ . n d*Q
-j- COS0-T- 2 sm0 -=--;- r cos Of-Tr) rsmfl-j-. (27)
dt 1 dP dt dt ^dt' dP
In all these expressions if = 0, and -5 are respectively
dt (it
the radial velocity and radial velocity -increment; and if 0= 90,
we have the transversal velocity and velocity-increment. Hence
dr
the radial velocity = -^ ; (28)
at
the transversal velocity = r -=7 ; (29)
dt
d*r /dd\*
the radial velocity-increment ~j7i~ r \j) ( 30 )
ill' (I [
n dr d0 d*0
the transversal velocity-increment = 2 r- -=- + r ~^-
J dt dt dt 9
r dt ^ dt'
The values given in (28) and (29) are evidently from first
principles the radial and transversal components of the velocity
of the particle. The expressions for the radial and transversal
velocity-increments may also be deduced from similar principles.
All these expressions may also be deduced by differentiation
from y r sin 6.
312.] RADIAL AND TRANSVERSAL RESOLUTION. 429
312.] Two particular forms of (30) and (31) deserve notice.
If the acceleration is only radial, so that the transversal accelera-
tion is zero, then
r* -jj = a constant = h, say ;
dd _ h dr _ h dr
'"' Tt~^ ~dt~'r*~dd t
*? (}
*"' dp ~ 7* IIP ~'~f*~{dd' '
and the radial acceleration
_k* d*r 2fr ,dr^ b*
= ^^"W7'
and thus is expressed independently of I.
This expression however may be put into a more convenient
form. Let r = - ; then
u
dr 1 du
M) ~ u^Jd'
d 3 r 1 d*u 2 fdu^
~dfr~ " ** JF "*" * ' 2i' !
substituting which in (32) we have
C d a u 1
the radial acceleration =h 2 u' t < -jj- +u^' (33)
From these formulae the law of radial acceleration may be
deduced when the curve is given; and the curve may be deduced
when the law of radial acceleration is given. But as very many
examples of these processes will be given in a subsequent
chapter, it is unnecessary to insert them in this place.
If the angular velocity is constant, so that = a constant,
= o> (say), then
d?f
the radial acceleration = o>V, (34)
(It
the transversal acceleration = 2 w -=j ; (35)
(It
and these are independent of 0.
In illustration of these formulae let us take the following
example :
A particle describes a path with a constant angular velocity,
and without radial acceleration ; find the equation to the path,
and the transversal acceleration.
Let Q be the required transversal acceleration so that the
required equations are
430 THEORY OF CURVILINEAR MOTION.
dr
Q = 2 ^'
Consequently if r = a when -57- = 0, = 0, and t = 0, we have
by integration j 3
g; -(!--) = 0;
dr
log
a
Also, as -TJ = co, therefore d = o>t, because = when t = ;
and this is the equation to the path described by m ; also
dr aa> ,
_ \ aiat _ au>t\ .
dt - 2 {g
.-. Q = 0&> 2 {e at e- ut }
= aco 2 (e fl e~ 9 }
= 2o) 2 (r 2 a 2 )^;
which is the transversal acceleration.
\ 313.] It remains for us still to investigate the kinematics of
a particle describing a curvilinear path in space ; and we will at
first refer its motion to a system of rectangular axes, and suppose
(x, y, z] to be its place at the time t. If three relations can be
found between x, y, z and t, the position of the particle at the
time t will be completely determined; and if t is eliminated,
two equations in terms of x, y, z will result, which will represent
two surfaces, the line of intersection of which will be the trajec-
tory of the particle.
Now if (x, y, z) is the place of the particle at the time t, and
(x + dx, y + dy, z + dz) at the time t + dt, and if
dx* +<fy* + dz 3 = ds 3 , (36)
then ds is the space described in dt, and the velocity of the
d#
particle in its path is -=- > and the components of this along the
dt
three axes are respectively -77-1 -j- > -^- : these results following
dt dt dt
from the law of resolution of velocities which has just now been
investigated.
3H-] NORMAL AND TANGENTIAL RESOLUTION. 431
In the most general case the velocity of the moving particle
ds
will be variable along its path, that is, -57- will be variable ; more-
U'v ,7j *
over the velocity-increment along the path, viz. -j- y if t is equi-
crescent, will be also variable; and consequently, generally, dx,
dy, dz will vary, and d*x, d*y, d*z will have values ; and thus the
velocity-increments or accelerations along the coordinate axes
. . d*x d*y d*z
will be -77- ^- > -7
dt* dt* dt*
If t is not an equicrescent variable, these quantities severally
are ,7ox7/_/72/,7
-; (37)
dt 3
d*x dtdH dx d*y dtdH dy d*z dt-dH dz
(38)
dt 3 dt 3 dt 3
314.] This process of axial resolution of velocity and of velo-
city-increment is, as we have before observed, artificial ; whereas
the motion itself suggests tangential and normal resolution ; the
tangential acceleration being the velocity-increment along the
tangent at the point (x,y, z\ and the normal acceleration being
that with which the particle is deflected from its rectilineal
tangential path.
The tangential velocity-increment is evidently j-j ; the normal
velocity-increment is thus found.
Let P, q, T, fig. 98, be three consecutive points in the curve,
when PQ = ds and is the element of the path described in dl;
let t be equicrescent, so that QT, which is equal to ds+d*s, is
the path described in the next dt. Consequently the particle
describes QT with two effective accelerations ; one, viz. along QR,
d^s
of which the mathematical expression is -= > and another along
at
QS, the mathematical expression for which is^ to be determined.
As P, Q, T are three consecutive points in the curve, the plane
PQT is the osculating plane of the curve at P, and the angle RQT
is the angle of contingence ; and QS is the action-line of the
normal acceleration, and is the distance through which the
particle is displaced. Since however, whatever is the law of
acceleration, the acceleration may in the beginning of the mo-
tion be taken to be constant, the relation between it the dis-
tance and the time is given in Ex. 2, Art. 249 j and we have
2.QS
the normal acceleration =
dt*
432 THEORY OP CURVILINEAR MOTION.
But if p is the radius of absolute curvature at P ; that is, is the
OT 2 ds*
radius of the circle passing through P, Q, T, QS = = - ;
2p 2 p
1 ds z v*
.*. the normal acceleration = = > (39)
if v is the velocity at the point P.
Hence if the velocity-increment at any point of the trajec-
tory is resolved along and perpendicular to the tangent, these
components are severally expressed by and > the action-line
dt p
of the latter being in the osculating plane. Thus these quan-
tities do not express torsion ; torsion is due to their variations
only.
315.] These values may also be deduced from the expressions
for the axial accelerations : for resolving these latter along the
tangent to the path,
d*x dx d*u du d*z dz
the tangenbal component = w - + --JL-J- + -
- *'*
because from (36), ds d*s = dx d*x + dy d*y + dz d*z ;
and the normal acceleration, by (22) Art. 377, Differential Cal-
culus, j^ ^Li^L d * z j dz I
t* ds + dt* Ts + WdsS
~ ds dsdt*
= ~
316.] If the path of the particle is referred to the system of
polar coordinates in space, which is explained in Art. 165, Inte-
gral Calculus, -=7- > r -JT > r sin 6 -37 are the components of the re-
1 dt dt dt
sultant velocity along the radius-vector, perpendicular to r,
in the plane containing r and the pole and perpendicular to
this latter plane respectively; the line of the last being- also
tangential to the parallel of latitude which passes through the
place of the particle at the time t ; and thus these velocities
are along lines perpendicular to each other. These values are
evident from the explanation given in Art. 165, Integral Cal-
culus. Consequently
ds 9 dr* de* (?d>*
317.] RELATIVE MOTION. 433
The components of the velocity-increment along- these three
lines may be deduced from the axial-components by the ordinary
process of resolution. The work however is long and cumbrous,
and it is unnecessary to insert it : because hereafter, in the
Chapter on Theoretical Dynamics in Vol. IV, a general method
will be given by which these results may be immediately ob-
tained from (42). If the reader wishes to exercise himself in
the longer process, and p, Q, R are the velocity-increments
respectively along the radius- vector, in the plane of r and the
pole perpendicular to the radius-vector, and perpendicular to
this plane, he will find the following results; viz.
d*r de* ,. d$*
'= '-'^'' (43)
d , d6 (1&*
'
m0)' (45)
rsm0 dt ( ' dt }
If the motion of the particle is confined to one plane, viz. that
of (r,6), j- = 0, and we have the results already given in (30)
and (31).
317.] In the preceding cases we have considered the motion
of the particle in reference to a fixed origin, and in the cases of
rectangular coordinate axes in reference to fixed axes; and the
velocity and other incidents of motion are thus determined
absolutely. In many cases however, and especially in astrono-
mical inquiries, it is convenient to refer the motion of the
moving particle to the place of another moving particle, taking
this latter to be the origin of coordinates, the system of co-
ordinates originating at it also moving with it. The motion
of the moving particle is said to be relative to this latter particle,
and the case is called relative motion. I propose at present to
take only some simple cases of it.
Let us in the first place suppose the moving axes to be always
in the same plane, and to have the same directions. Let the
place of the moving particle at the time t be (x, y] in reference
to a fixed system, and (, 77) in reference to the moving system
of axes ; (# , ^ ) being the place of the origin at the same time
in reference to the fixed system : then
= #-#, ri=y-y Q ; (46)
PIUCE, VOL. III. 3 K
434 THEORY OF CURVILINEAR MOTION.
d dx dx 9 dr) dy dy n
~dt ~ ~dt dT' ~dt~~dt dT'
(47)
- (48)
dt* ' ' dt* dt* ' dt 3 " dt* dt* '
and these equations assign the relative velocities and velocity-
increments in terms of the absolute velocities and velocity-in-
crements of the moving particle and of the origin.
These expressions may also be found by the following process :
In the case of velocity as given by (47), let us suppose a velocity,
of which the axial-components are r and - , to be communi-
dt dt
cated to both the moving particle and to the moving origin
in directions towards the origin ; then it is evident that the
relative positions and velocities of the particle and origin are not
hereby changed ; but the effect is to bring to rest the moving
origin, and to leave the moving particle with a velocity relatively
f v i, 4-v i dx dx dy dy*
to it, of which the axial-components are -y- -- ^- > -~- -- ~ >
dt dt dt dt
which are the same as those given in (47). The system may be
subjected to a similar process with reference to velocity-incre-
ments, and we shall procure the results given in (48).
Similar results are also true for the relative motion of a
particle in space referred to three rectangular axes.
318.] Let us now take a more general case in which the
origin describes a curve in the plane of (x, y}, and the moving
axes rotate in that plane through an angle in the time t : let
us suppose the fixed and the moving axes to have coincided at
the time t = 0. Then we have
x = # + cos0 77 sin 6,
y = y + sin0 + Tjcos0; (49)
therefore
dx dx^ dt dn . dd ~]
-ji -ji + -n cos 6 -^ sm 6 ( sin 6 + rt cos 6} -=-
dt dt dt dt ' dt
Let v$ and v^ be the components of the absolute velocity parallel
to the axes of and 77 : then
dx dy
= j cos0+ --
dt dt
dd dg
-
319-] KELATIVE MOTION. 435
r] , . .
+ Tt> (52)
in whicli values of v$ and v^ all the terms except the last of each
are due to the motion of the coordinate system of reference,
and the last expresses the axial-component of the velocity of m
relatively to the moving axes.
Let us next take the ^-differentials of (50) : then
-(sin + cos 6} ^ + ^cos 6- g sin 6;
drj
cosfl.
Let v'% and ?', be the axial-components of the absolute velocity-
increment parallel to the axes of and ?; : then
< 53 >
d a r
d'
of which expressions for v'g and t?^ all the terms except the last
in each are due to the motion of the moving coordinate system
of reference ; and the last expresses the axial-component of the
expressed relative velocity -increment.
319. On examining equations (53) and (54), it appears that
the first two terms in the right-hand member of each express
those parts of the velocity-increment which arise from the mo-
tion of translation of the moving origin, and that the last three
3K 2
436 THEORY OF CURVILINEAR MOTION.
in each arise from the angular motion of the coordinate system
of reference and from the relative motion of m in reference to
that moving- system. Now these last are in accordance with
the results of radial and transversal resolution which have been
discussed in Art. 311. The axial-components of the velocity-in-
crement of m at P are the sums of the components of the velocity-
increments of equal particles at L and N, which are the projec-
tions of P on the axes of and 7; ; for as -=- is the angular
clt
velocity of these axes about the moving origin,
1-1 & f/dO\*
the radial component oi L = -^ ( -=- }
(XrV CvC/
the transversal component of L = -> -r- (* -^ :
f dt ^ at'
the radial component of N = -~ r/ (-=-\ >
the transversal component of N = =- (V -^\ ;
TJ at ^ at'
and consequently of the velocity-increment,
,, , d* ,de^ 1 A t dQ^
the Component == _ -f (_)___ (, 2 _);
d*r} ,dO^ 1 d , > d&
the ,-component = ^-^Q + ~^ (P ^-
Thus the axial-components along the moving axes of and TJ
respectively of the velocity-increment at the time t are
If the origin of the moving axes does not move, and the axes
revolve with an uniform angular velocity a>, then (53) and (54)
become
d* dv d*ri df
&- *-*"&' and W-^+ 2 Tt'
These equations however refer to a very special case of the
general motion.
Kinematics of other and more complicated cases of relative
motion will be discussed hereafter : the preceding is sufficient
for our present purpose.
320.] DYNAMICS OF CURVILINEAR MOTION. 437
SECTION 2. The dynamics of a particle moving in a
curvilinear path.
320.] The incidents of motion of a particle moving in a curvi-
linear path having been considered in the preceding section, we
have to investigate the equations connecting these results with
the forces of which they are the effects.
When two or more forces act simultaneously on a material
particle in motion, the effects are different, and require separate
consideration, according as their lines of action are in the line of
motion of the particle or make angles with that line ; in the
former case the effect is an acceleration or retardation of the
particle in the line along which it is moving: and the total
effect of many such forces is the sum of their several effects ;
in the latter case, the effect of a force acting along a line which
is inclined at a given angle to the line of motion of a particle is
partly to produce a deflexion from the rectilinear path in which
by the law of inertia the particle would move, and partly to
produce an acceleration or a retardation along that line. Such
forces therefore will generally cause a particle to describe a
curvilinear path : for it is to be observed that although a par-
ticle may have a certain velocity, yet that velocity is not an
impediment to the independent action of another force on the
particle : the material particle, whether in motion or at rest, has
the same property of inertia. And another force will produce
its own effect on it, and precisely in the same manner as if the
particle was not moving. Thus the expressed velocity of the
particle will be the resultant of these two several velocities, and
its line of motion will depend on the lines of action and the in-
tensities of the two component velocities, and according to a law
which has already been investigated. The law of inertia how-
ever becomes extended, and we have the following proposition :
When two or more forces impress velocity on a particle, the
change in velocity and line of motion of the particle due to each
is the same as if the others did not act.
This proposition arises partly from the inertia of matter, and
partly from the fact that an adequate and intelligible concep-
tion of force requires that it acts on matter and causes it to
move along a definite line, and impresses a definite velocity ;
and consequently by the laws of composition and resolution of
velocities, which have been investigated in the preceding section,
438 DYNAMICS OF CURVILINEAR MOTION. [321.
the resultant velocity will be represented by the diagonal of the
parallelogram, of which the two adjacent sides meeting at the
position of the particle are the representatives of the separately
impressed velocities. This proposition is commonly called the
Second Law of Motion.
321.] This theorem may be worked out by the following
process :
Let o, fig. 7, be the place of the particle m at rest at the be-
ginning of the time : let two impulsive forces P and Q act on it,
of which the lines of action are OP and OQ; and let us suppose
the force P to impress a velocity on m so that it would describe
the space OP uniformly in t units of time: similarly let the
force Q impress on m a velocity such that it would describe uni-
formly the space OQ in t units of time. Let the figure be con-
structed as in the diagram ; where o R is the diagonal of the
parallelogram of which OP and OQ are two containing and adja-
cent sides; where /'OP" is perpendicular to OR, and OQ"=
OP"= QQ'= PP', and where these four lines are all parallel to
each other. Now the velocity of which OP is the line-represent-
ative may be resolved into two velocities, one of which will
be represented by OP' and the other by OP'' ; similarly may the
velocity of which OQ is the line-representative be resolved into
two, of which OQ' and OQ" are the line-representatives. Then
OQ" and OP", being equal and in opposite directions, destroy
each other ; and OP' and OQ' acting along the same line and in
the same direction must be added, and of their resultant the
line-representative is the sum of OP' and OQ', that is, is on;
OR therefore is the line-representative of the velocity which the
particle has, and therefore of the resultant of the two component
velocities of which OP and OQ are the line-representatives.
Thus if on a particle m two impulsive forces act, the lines of
action of which are inclined at an angle y, and if these acting
singly would impress on m velocities u and v along their lines
of action, then, if w is the velocity which one force acting would
impress on m and produce the same effect as the other two
acting in combination,
w* = tt a + 2w0cosy+ZJ 2 ; (55)
and if a and /3 are the angles between the lines of action of v
and w } and of w and u respectively, then
u v w
sin a ~ sin /3 ~~ sin y '
322.] EXPERIMENTAL EVIDENCE. 439
Similarly if three forces, whose lines of action are mutually
inclined at angles a, ft, y, act on a material particle, and are
such that each acting singly would impress on it velocities
u, v, w along their lines of action, then the one force which
would impress on m the same velocity as these three acting
simultaneously is that which would impress a velocity a-, where
<r 3 = u*+v* + w 1 + 2viccosa+2wucos ft+2uvcosy ; (57)
and its line of action would be parallel to the line whose equa-
tions are at y
- = ?- = . (58)
U V W
If a=/3=y = 90 these results are identical with these deter-
mined kinematically in the preceding section.
322.] This result may be illustrated by the following experi-
ment : Let ABC, fig. 96, be the horizontal deck of a boat which
is moving with a constant velocity in the direction indicated by
the arrow, so that in the time t the point A moves into the
position A', and all the other points on the deck describe straight
lines equal and parallel to AA'; and suppose at A a particle m to
be placed, and from a force to receive a velocity in the direction
AQ, so that if the boat is at rest, in the time t it describes the
line A Q : now if the boat is moving, this latter velocity will be
combined with that of the boat, and the result is the effect of
the two ; but neither of them alters the effect of the other ; and
thus at the end of the time t the particle is found at the point R,
having described the diagonal AE, and which is therefore the
line-representative of its velocity, because it is described uni-
formly and in the time t.
Experiments and observations such as the following shew the
law to be true in the matter of the earth.
A small heavy particle let fall from the top of a mast of a
ship sailing uniformly, falls at the foot of the mast, although
the force under the action of which it falls acts vertically and
is uniformly accelerating. Thus the particle retains the hori-
zontal velocity which it had at the top of the mast, and this
is combined with the vertical impressed velocity.
If a carriage moves evenly along a railway, and if an impulse
is given to a ball in it, the effect of the impulse is the same,
whatever is the direction in which it is given : the motion of
the carriage does not alter the effect of the impulse, and the
path and absolute velocity of the ball are of course compounded
of the two velocities.
440 DYNAMICS OF CURVILINEAR MOTION. [322.
The earth revolves on its polar axis from west to east, and
therefore all points on its surface move with a velocity due to
this rotation. If therefore the law is not true, a body struck in
a direction north or south, would deviate to the west, and this
is not found to be the case. And this fact admits of most
accurate examination : for suppose a pendulum to be suspended
and to oscillate, the time and the extent of oscillation would be
diiferent for the different directions of the plane of oscillation :
no difference however is observed, whatever is the azimuth of
the plane : and the smallest difference, as Laplace has shewn in
the Mecanique Celeste, cannot fail of being appreciable.
Again : of a lofty and vertical tower the top is of course
farther from the centre of the earth than the bottom, and there-
fore as the earth rotates from west to east, the horizontal
velocity of the top is greater than that of the bottom. Let a
heavy ball fall from the top : it will have the horizontal velocity
of the top, and this is greater than that of the bottom : if there-
fore the ball falls on the west side of the tower, it will strike
the tower before it reaches the earth : but if it falls on the east
side of the tower, it will strike the earth at a small distance
from the tower towards the east. These results have been
actually observed ; and from them we infer the law of which they
are the effects.
Also the phaenomena of the aberration of light, and the
accordance with observation of the results arrived at, yield
another proof of the truth of the law of composition of velocities
which we have here investigated. Suppose, see fig. 97, s to be
the place of a star, and E to be the place of the earth in its path
at the same time : now light travels with a velocity of 186,000
miles in one second of time, and the earth moves in its elliptic
path through 19 miles in a second : and let us suppose that in
the time during which the light of the star has travelled from
s to E, the earth has moved from E to E', where EE' is to SE as
19 to 186,000 ; then the effect to us is the same as if the earth
had been at rest, and light had a velocity represented by BE'
from E' to E along EE', in combination with its velocity along SE,
so that the ray of light would come in the direction S'E, where
S'E is the diagonal of the parallelogram of which SE and EE' are
two adjacent containing sides : the star therefore appears to us
to be before its real place in the direction of the line of motion
323.] EQUATIONS OF MOTION. 441
of the earth. See also HerschePs " Outlines of Astronomy,"
Arts. 328-335. Ed. I. 1849.
And, omitting other experiments and observations, I may
remark that the most conclusive evidence of the truth of the
law of composition of velocities is the a posteriori proof arising
out of the results of physical astronomy. The expressed velo-
cities and velocity-increments of the planets are resolved and
estimated according to it, and their places calculated at par-
ticular times ; when these are compared with the observed places,
no discrepancies are discovered; and thus one of the severest
tests of the truth of such a law is applied and is satisfied.
323.] Thus much being all that is necessary to be said as to
the effects of force on matter, when two or more forces act
simultaneously on a particle along different lines of action and
cause it to move in a definite curvilinear path, it remains for us
to investigate equations by which that path may be determined
when the forces are given.
ds
Let m be the mass of the moving particle ; then m -=- is its
fit
expressed momentum in the line of its motion ; m -=- , m-j- >
7 dt dt
m -j2 are its expressed momenta along the axes of x } y, z re-
dt
spectively if the path is referred to coordinate axes in space;
dzr nit
and 0&-rT m ~jn are ^ ne axial-components of the resultant mo-
mentum if the motion takes place in the plane of (x } y] : mo-
menta are resolved and compounded according to the law of
geometrical projection.
d*s
Hence also m -j is the expressed momentum-increment
of m in an unit of time along the line of its motion; and
d'x fry d*z
m ~j , m -jf- , m j are the several expressed momentum-
dt (it/ (to
increments of m in an unit of time along the coordinate axes
of x, y, z.
Also the expressed momentum-increment in the line of motion
of a particle at a given time is the sum of the resolved parts
along that line of the several expressed momentum-increments
along the coordinate axes.
Hence also impressed momenta and momentum-increments
and their causes, accelerating forces and moving forces respec-
PRICE, VOL. in. 3 L
442 DYNAMICS OP CURVILINEAR MOTION. [324.
tively, are resolved and compounded according to the law of
geometrical projection.
And as statical pressures, see Art. 261, are virtual dynamical
momenta, it follows that statical pressures are resolved and
compounded according to the same law : hence we have a proof
of the parallelogram of statical forces.
324.] If the motion of the particle m takes place wholly in
one plane, the equations which determine that motion are thus
found.
Let the motion be referred to a system of rectangular axes ;
and let x', Y be the axial-components of the impressed momentum-
increment on m at the point (x, y] at the time t ; then equating
the impressed and the expressed momentum-increments by reason
of the law explained in Art. 258, we have
d*x , d*y
m - = *> m =*> {59)
and if x', Y' are proportional to the mass of m,* so that
x'= mx, Y= m\ }
the equations of motion are
in which case x, Y are the impressed velocity-increments which
are the effects of the accelerating forces.
If the resultant velocity-increment is resolved tangentially
and normally, and T and N are the corresponding components
of the impressed velocity-increment, then by (10)
^ = T - 7 = "- ()
If the motion is referred to a system of polar coordinates,
and P and Q are the radial and the transversal components of
the impressed velocity-increment, then by (30) and (31)
i - r < 62 >
325.] The three following Chapters will contain so many
illustrations of these equations, that it is unnecessary to shew
* The impressed velocity-increments are not always proportional to m : in-
stances of the non-proportionality have already occurred in the equations of
motion in Articles 293 and 294.
326.] EQUATIONS OF MOTION. 443
their application immediately ; but the attention of the reader
must be at once called to the manner in which they are con-
sistent with the principle of the conservation of work.
Let the two equations (60) be multiplied by mdx and mdy
respectively, where dx and dy are the actual axial displacements
of m in the time dt : so that according to the definition of work
given in Art. 259, m*x.dx and rnxdy are the elements of the
work done by the accelerating forces x and Y in the time dt,
and the integrals of these expressions are the works done by
those forces through the assigned limits, whatever they are.
Let the equations after multiplication by mdx and mdy respec-
tively be added ; then we have
-j - -
m /**'
m
**
(63)
the left-hand member of which is the vis viva of m acquired in
its motion from (x 0) y ) to (x, y) under the action of the given
forces ; and the right-hand member is the work done by the
forces through the spaces assigned by the limits. This equa-
tion is called the equation of vis viva and of work. Also from
(61) and from (62) we have
w, C*
-(v* V) = / **<frj (64)
2 J *
fWt /*/"
_-(f,_t> ) / 'm(pdr + Q,rd6); (65)
Jr , e
each of these being the equation of work in its own system
of reference. The circumstances under which these equations
are possible and are interpretable will be explained in a future
Chapter.
326.] As the normal component of the impressed momentum-
increment does not appear in the equation of vis viva, viz. (64),
the force, which by its action impresses it, does no work ; the
reason being that the path of m is always at right angles to the
action-line of this force. Since however m would, if left to
itself or if left to the action of a force which acted along the
tangent, by the law of inertia, continue to move along that
tnv^
tangent, so - is the effect of the force which deflects m from
p
its otherwise rectilinear path, and draws it towards the centre of
absolute curvature. This force is called the centripetal force ;
3 L2
444 DYNAMICS OF CURVILINEAR -MOTION.
and the expressed normal momentum-increment is called the
centrifugal force.* Hence
4ft'
the centrifugal force of m = -- (66)
Centripetal and centrifugal forces are therefore the same quan-
tity under different aspects. Centripetal force is the force which
impresses the momentum, and is spoken of in reference to that
force : centrifugal force is spoken of with reference to the par-
v*
ticle m, and is that force of which m is the expressed mo-
p
mentum.
327.] As an exact knowledge of the nature of centrifugal
force is necessary for a complete comprehension of the theory
of curvilinear motion, let us examine it in two or three appli-
cations.
Let m move in a circle with a constant velocity v; let #=the
radius of the circle, and T = the time of a complete revolution :
then 2-na = VT;
.*. the centrifugal force of m = m ; (67)
thus the centrifugal force in a circle varies directly as the radius
of the circle, and inversely as the square of the periodic time.
Again, if m moves in the circle with a constant angular velo-
city o>, then by (24), Art. 310, v = ao> ;
.. the centrifugal force of m = ma* a; (68)
and therefore varies directly as the radius of the circle.
Thus if a particle of mass m is fastened by a string of length
a to a point in a horizontal plane, and describes a circle in the
plane about the given point as centre, the centrifugal force
produces a tension of the string, and if o> is the constant an-
gular velocity, the tension = ma* a.
A railway-carriage of given weight and dimensions moves at
a given velocity along a curved railway of which the radius of
curvature is known. It is required to find the height to which
the outer rail should be raised above the inner one so as to
counteract the effects of centrifugal force.
Let 2 a be the horizontal breadth of the railway, and h the
height of the centre of gravity of the carriage above the rails
when they are horizontal ; let z be the elevation of the outer
* This term is inaccurate ; is the expressed effect of an accelerating force.
P
328.] EQUATIONS OF MOTION. 445
rail, and 6 the angle of inclination to the horizon of the trans-
verse section of the road when the outer rail is raised ; so that
z = 2 a tan ; let v = the given velocity of the carriage, p the radius
of the curve ; then as the outer rail is to be raised so that at the
given maximum velocity there shall be no pressure on the inner
rail, we may take about the outer rail moments of the centri-
fugal force and of the weight of the carriage and may equate
them : whereby if m is the mass of the carriage, we have
flM?'
- (k cos 6 a sin 0) = mg (a cos 6 + h sin 6) ;
P
.. z = 2atan0 = 2a ~~ pa
328.] When a solid body rotates about an axis, all its par-
ticles describe in equal times circles, the planes of which are
perpendicular to the axis of rotation, the centres of which are
in this axis, and the radii of which are the perpendiculars from
each point on the axis : therefore from (68), as <o is the same
for all the points, the centrifugal forces vary as these perpen-
diculars. Now as the earth revolves about its polar axis, the
centrifugal forces of particles on its surface vary as the perpen-
dicular distances from the particle on the polar axis ; and there-
fore the centrifugal force of each particle varies as the radius of
the parallel of latitude which the particle describes : and the
line of action, being the radius of the circular path in which the
particle moves, is perpendicular to the polar axis. As the radius
of the circle of the parallel of latitude decreases from the equator
to the pole, so does the centrifugal force which varies as this
radius by reason of (68); the centrifugal force therefore is great-
est at the equator and least at the poles, where it vanishes.
Let us consider the earth to be a perfect sphere, and to be
revolving with an angular velocity such that the time of the re-
volution is 24 hours; and let us consider a place on it of which
the latitude is X ; then, if r is the radius of the earth, the radius
of the circle of the parallel of latitude is r cos X ; and therefore
if a) is the angular velocity, the centrifugal force of m in this
line, which is perpendicular to the polar axis, is 7#&>VcosX;
and if this is resolved into two parts, one of which is horizontal,
and the other is vertical, at the place, the vertical component of
the centrifugal force is zco 2 r(cosX) a ; by this quantity therefore
will the attraction of the earth on a particle m on its surface at
the latitude X be diminished : so that if mg is the weight of m
4:46 DYNAMICS OP CURVILINEAR MOTION. [328.
when diminished by the centrifugal force, and m& were its
weight if the earth were at rest, then
mg == mGmtfr (cos X) a ; (69)
.-. g = G o> 2 r(cosA) a ; (70)
therefore o> 2 r(cos A) a is the diminution of the earth's gravity
due to the centrifugal force.
The diminution of gravity is the greatest when A. = 0, that is,
at the equator ; in which case
-^, (71)
9 I
since the difference between g and G is very small. Let
T = the time of rotation of the earth about its axis ; therefore
T = 24 x 60 x 60 seconds. Also 2ir = o>x; and g 32.2 feet ;
TT= 3.14159; therefore r = 4000 x 1760 x 3 feet,
= nearly;
(72)
that is, the diminution of gravity at the earth's equator due
to the centrifugal force is the 289th part of that which the
earth's attraction at the equator would be if the earth did not
rotate.
Thus also the weight of a body m at the equator is diminished
by its 289th part, and the diminution of its weight at the lati-
tude \ is CT0r(cosX)'
289
The preceding calculation is made on the hypothesis that the
earth is a perfect sphere, whereas it is an oblate spheroid : and
the attraction of the earth on particles at its surface decreases
as we pass from the poles to the equator according to the law
investigated in Art. 212, and given in (91) of that Article. The
present inquiry gives the law of diminution of gravity on
account of centrifugal force ; the combination of these two
effects produces the result given in Art. 124, viz.
<7 = G{l+.005133(sinA) 2 }; (73)
and the whole diminution is nearly a 200th part of the original
330.] EQUATIONS OF MOTION. 447
gravity. Hence also the weight of a body at the poles is one
200th more that its weight at the equator.
Since 289 is the square of 17, it follows that if the earth
completed a revolution about its polar axis in the 1 7th part of
a day, the centrifugal force at the equator would be equal to,
and would neutralize, the earth's attraction ; in which case
matter at the equator would have no weight.
329.] If the path of the moving particle m is a curvilinear
path in space and is referred to three fixed rectangular axes,
then if x, Y, z are the axial-components of the impressed velocity-
increment, the equations of motion are
d t x d* d*z
and these are applicable to the solution of every problem in-
volving such a motion and such forces.
Thus, if the laws of the impressed velocity-increments are
given, and if the problem is the deduction from them of the
equations of the trajectory, (74) must be integrated, whereby
three relations will be given between x, y, z and t ; whence t
may be eliminated, and two equations will result in terms of
x, y } z } which will represent two surfaces, the line of intersection
of which will be the trajectoiy. In the course of integration,
twelve limiting values will be required, viz. the six components
of the velocities corresponding to t = t and to t = ; and the six
coordinates of m corresponding to the same values of t : of these,
six, viz. those corresponding to t = t, will be left in the general
equations in their general form : the other six, which correspond
to tf=0, will enter into the final equations as arbitrary constants,
because the beginning of the time t is arbitrary.
330.] If the particle m is not acted on by any forces, so that
x = Y = z = 0, then
y f\ f\ fr*f\
di*~' ~d&~' dfi = '
Let (a, b, c) be the place of the particle when t = 0, and
(x, y, z), when t = t : also let a, ft, y be the components of its
velocity when t = ; then integrating (75) between the limits
corresponding to these values of t,
dx dy dz
xa yb zc
448 DYNAMICS OF CURVILINEAR MOTION. [331-
which are the equations to a straight line, whose direction-
cosines are proportional to the components of the velocity when
t = 0, and which passes through the point (a, b, c}. If there is
no initial velocity, a = /3 = y = 0; in which case x = a, y = b,
z = c; that is, the particle remains at rest and in its original
position. The result is of course in accordance with the law of
inertia.
331.] If the place and motion of. the particle are referred to
a system of polar coordinates in space, and p, Q, R are the
impressed velocity-increments along the radius-vector, along a
line in the plane of r and the pole and perpendicular to r, and
along a line perpendicular to this plane, then the equations of
motion are
d 3 r de* ,. dd>*
d
(76)
naw\ r ^ mv wr
332.] The equations of relative motion, of which the kine-
matics have been investigated in Arts. 317319, are formed on
the same principle of inertia. Thus as to equations (48), let x, Y
be the axial-components of the impressed velocity-increment
acting on m, and let x , Y O be the axial-components of the im-
pressed velocity-increment acting on a particle of the mass m at
the origin or the axial-components of the acceleration of the
origin ; then the equations of motion are
= x x a = Y Y O . (77)
If the path described by m is referred to three axes in space, a
third equation in terms of has to be added.
If the motion of m is referred to a system of moving axes as
also to a moving origin, then from (53) and (54) the equations of
motion are
(78)
3F -* + Tt(?Tt )= (*-Yo)cos0-(x-x )sin0. (79)
CHAPTER X.
APPLICATION OP THE EQUATIONS OF THE PRECEDING
CHAPTER TO PARTICLES MOVING FREELY IN SPACE.
SECTION 1. Oblique impact and collision of particles and of
smooth spherical balls.
333.] The laws of resolution of velocities and momenta, and
the principles of the theory of curvilinear motion have been
investigated in the preceding Chapter ; our object now is fur-
ther to elucidate these, and to apply them to the special case of
particles moving- freely in space, reserving to subsequent Chap-
ters the cases of particles moving on curves or on surfaces or
in tubes ; cases, that is, of constrained motion.
The most simple case of resolution of momenta is that of a
material particle or of a smooth homogeneous spherical ball,
which is supposed to have motion of translation only, which
moves in a rectilinear path with a constant velocity, and im-
pinges at an oblique angle on a given plane. As the velocity of
the ball is constant before impact, so will the velocity be also
constant after impact : and thus the inquiry is limited to the cir-
cumstances of change during the collision, and to the determina-
tion of the line of motion and of the velocity of the ball after
impact. The elasticity of the ball is supposed to be the same
as that which was explained in Art. 262 ; that is, the momentum
acquired during the restitution of the figure of the ball is sup-
posed to bear a constant ratio to that lost during the compres-
sion of the figure : the limiting values of the ratio being 1 and
0, according as the ball and plane are perfectly elastic or are
perfectly inelastic.
One observation however is to be made on the circumstances
of oblique impact, which was not applicable in that of direct
impact. In oblique impact we assume that the mutual action of
the balls during collision is along the line joining their centres
at the instant when compression is a maximum, and along that
PRICE, VOL. III. 3 M
450 OBLIQUE IMPACT AND COLLISION. [334-
line only; that is, in other words, we assume the balls to be
.perfectly smooth. For suppose a ball of mass m } see fig. 102,
to move uniformly along the line <ZA. and to impinge on the ball
of mass m' } which is moving at an uniform velocity along the
line I$B : and suppose the line OAB to be that which passes
through their centres A, B at the instant at which compression is
a maximum : we assume the action of the two balls on each
other for the time during which the collision takes place to be
wholly along this line ; along this line has momentum been
lost during the compression, along this line will momentum
be acquired during the restitution : the momenta in a line
perpendicular to AB have not been altered by the collision :
thus by virtue of the statements made in Art. 262, if P repre-
sents the momentum along the line AB lost during the compres-
sion, e~p represents that acquired during the restitution along
the same line. Thus although for the time during which the
balls are in collision, they, by reason of their velocity which
is perpendicular to AB, slide on each other, and thereby the
momentum in that line would be changed if the balls are not
perfectly smooth, yet for the sake of simplicity we assume the
roughness of the balls to be so far diminished, that it is not
necessary to take account of the change of momentum along
the line perpendicular to AB.
Hence if a smooth ball impinges obliquely on a smooth plane,
the line of reaction of the plane will be perpendicular to its
surface, and the momentum of the impinging ball will be
affected along that line only, and not along the plane.
334.] The results of the direct impact of a ball on a plane
are given in equations (15), (16), Art. 264, so that if v is the
velocity of impact, ev is the velocity of rebound.
But suppose a perfectly smooth and spherical ball, whose mass
is m, and whose elasticity is e, to move with an uniform velocity
v, and to impinge at P on a smooth plane in the line a A, making
an angle a with the normal to the plane at the point P, fig. 103,
so that a AN = a; let u be the velocity of m when the compres-
sion is a maximum : at which instant the motion of the ball is
wholly along the plane; and suppose v to be the velocity of
rebound, and ft = NA to be the angle which the line of motion
of the ball after rebound at P makes with the normal : let P be
the momentum in the line AN which is lost by the ball during
the compression, and let ev be that recovered during the resti-
334'] OBLIQUE IMPACT AND COLLISION. 451
tution, the line of action of both these momenta being the normal
AN. Let us resolve the momenta along and perpendicular to
the plane : then at the instant when the compression is a maxi-
mum, we have (1) along the plane,
mv sin a = the impressed momentum of m,
mu = the expressed momentum of m ;
(2) perpendicular to the plane,
mv cos a = the impressed momentum of m,
p = momentum lost by m during compression ;
.'. mv sin a = mu ; (1)
mv cos a = p : (2)
by a similar process when restitution ends, we have (1) along
the plane,
mu = the impressed momentum of m,
mv sin ft = the expressed momentum of m;
(2) perpendicular to the plane,
ep = the impressed momentum of m,
mv cos ft = the expressed momentum of m ;
.'. mu = mvsinft, (3)
P = mvcosfi', (4)
.*. v sin a = v sin ft, etan/3 = tana;
tana
... tan/Sss ; (5)
sin a
v = - - t> ; (6)
sm ft
whereby ft and v are known in terms of given quantities ; a and
ft are called respectively the angles of incidence and reflexion.
Generally as e is less than unity, ft is greater than a. If the
ball is perfectly elastic, e = 1 : in which case a = ft, that is, the
angles of incidence and reflexion are equal ; and v = v, that is,
the velocities of incidence and reflexion are equal to each
other.
If the ball is perfectly inelastic, e = ; in which case, /3 = 90,
and v = v sin a ; the ball, that is, after impact moves along the
plane with the component in that line of its velocity of impact.
These results are equally true, when the ball impinges on a
curved surface, the plane of impact being in this case the tan-
gent plane to the surface at the point of impact.
3 M 2
452 OBLIQUE IMPACT AND COLLISION. [335
335. ] Examples in illustration of the preceding principles :
Ex. 1 . To find the line along which a ball of given elasticity e
must be projected from a given point A, so that after reflexion
at a given plane it may strike another given ball at B.
Let CD, fig. 104, be the given plane, A and B the given posi-
tions of the balls. From A and B draw AC and BD at right
angles to the plane CD ; produce AC to A', making CA'= e.CA.
Join BA' cutting CD in p, and join AP; APB is the required path.
At P draw the line PN normal to the plane : then APN is the
angle of incidence, and NPB is the angle of reflexion : and since
tan APN = cot APC = ;
AC
CP CP
and tanBPN = cotBPD = cot A PC = -7- = ;
A c e.AC
.'. tanBPN = -tan APN:
e
and therefore the path APB satisfies the condition (5), and is
such that a ball projected from A along AP will strike another
ball at B.
Ex. 2. To determine the path which a ball of elasticity e
must take with reference to two given inclined planes, so that
when projected from a given point A it may after reflexion suc-
cessively at the two planes strike another ball at B, A and B
being in the plane which is perpendicular to the line of inter-
section of the two planes of reflexion.
Let the plane of the paper, fig. 105, be that in which the
two points A and B are, and thus the planes of reflexion are per-
pendicular to the plane of the paper.
From A draw ADA' perpendicular to the plane DO, and make
A'D = .AD: also from B draw BCB' perpendicular to oc, and
such that BC = e.B'c: draw A'B' cutting DO in p and oc in Q,
join AP and BQ ; APQB shall be the path required.
It is evident by the construction in the former example that
the lines AP and PQ satisfy at p the condition required in (5) :
and also that PQ and QB satisfy the same condition at Q : there-
fore APQB is the required path.
Also by a similar process may the path be determined, by
which a ball of given elasticity may after projection from a
given point and reflexion at given planes strike a ball placed at
another given point.
Ex. 3. A ball of given elasticity e is projected from a given
336.] OBLIQUE IMPACT AND COLLISION. 453
point in the circumference of a circle, and after two reflexions
from the circle returns to the same point : determine the angle
at which it was projected.
Let A, fig. 106, be the point in the circle whence the ball is
projected, and let AP, PQ, QA be the paths which it successively
describes: let CAP = 0=CPA, CPQ=: d> = CQP, CQA = ^
Then by (5),
tan d> = - tan 0, tan ty = - tan rf> = tan ;
C
1 tand> tan\lr
.. tan = cot(d> + ^) = -
tan o> + tan y
_ <? 3 -(tan0) a m
~ (e' + e} tan '
.-. tan0 = (
336.] Two smooth spheres of given elasticity and of masses
m and m', moving in given lines, and with given velocities, and
with their centres in the same plane, impinge on each other ; it
is required to determine their velocities and lines of motion
after impact.
Let us suppose the two spheres to be moving in the directions
indicated by the arrows, fig. 102, and along the lines a A. and &E,
and let OAB # be the line passing through their centres at the
instant when the compression is a maximum : and let m, whose
centre is A, impinge on m' t whose centre is B. Now by Art. 333,
the elastic action of the two spheres takes place along the line
OAB only : let e = the elasticity, and p = the momentum lost
during compression, so that ep is that acquired during the resti-
tution of the figure of the balls. The momenta will be resolved
along, and at right angles to, the line OAB.
Let v, v' be the velocities of m, m' before collision begins,
u,u' ----- - when compression is a maximum,
v, v' ----- - when collision ends ;
a, a' be the angles between OAB, and the lines of motion of m, m'
when collision begins,
Q,tf when compression is a maximum,
ft, ft when collision ends.
Then at the instant when the compression is a maximum, the
equations of resolved momenta are,
454 OBLIQUE IMPACT AND COLLISION. [337-
( mv cos a = mu cos 6 + P, (7)
for m < . J ; '
( mv sin a = mu sin 0, (8)
/ CwVcosa / = m'-u'costfF, (9)
n \ wzVsin a" = wVsin & ; (10)
and at the instant when collision ends,
(mucosQ = mvcos8 + ep. (11)
for m -
.
sm = OTV sm ft, (1 2)
,(m,'u'costf= m'v'cos ft ev, (13)
(wVsin0 / = zV sin/S 7 . (14)
But when the compression is a maximum, both balls are moving
with the same velocity along OAB : therefore
u cos0 = w'cos0'. (15)
From these nine equations, v, Y, ft, ft' are to be determined.
From (7), (9) and (15),
, ., mv cos a + wVcos a'
u cos 6 = cos 6 = - - -, - ; (16)
m+m
from which and from (7) and (1 1), we have
mv cos a + tfVoOB a' ew'
v cos 3 = - ^ --- > (y cos a v cos a). (17)
m+m m+m^
Similarly,
. mv cos a + m'v'cos a em
vcosp= - -, -- 1 -- ^(flcosa veosa). (18)
m+m ^
Vcosa r ,i
iVsina';)
Also from (8) and (12), and from (10) and (14),
v sin ft = v sin a, (19)
Vswp'= t/sina'; (20)
so that v, v', /3, /^ are completely determined.
Also from the preceding we have
m v cos /3 + wVcos ft = m v cos a + m'v'cos a', ]
v sin ft + flfcVsin ft' = mv sin a + wV
that is, the sum of the momenta both along, and at right angles
to, OAB is the same before and after impact.
337.] Let the balls be perfectly elastic, that is, let e= 1 : then
mv cos a + m'v'cos a m'(v cos a v'cos of) ,_ ON
v cos ft = - --- - -- - f - ' > (22)
m+m m+m
mv cos a + m'v'cos a m (v cos a 0'cos <0
- 7 - + -* ? - '-> (23)
m+m m+m
v sin ft = v sin a, (24)
/sin & = 0'sin a' ; (25)
(26)
338-] OBLIQUE IMPACT AND COLLISION. 455
that is, the sum of the vires vivae is the same before and after
impact, and no work is lost by the collision.
Again, let the balls be perfectly inelastic, that is, let e = : then
_, , mv cos a -f wVcos a'
vcos/3 = -/cos/3 = - (27)
m + m'
vsin/3 = vsina, (28)
v'sin^'z; fl'sina'; (29)
that is, the balls after impact have the same velocity along the
line OAB, but unequal velocities at right angles to that line; also
tan/3 vsma
tan p ~ t;'sin a' '
338.] The velocity and the line of motion of the centre of
gravity of two smooth balls which impinge on each other is the
same before and after impact.
Let e be the elasticity of the balls : and let us take a line
parallel to that which joins the centres of the balls at the in-
stant when the compression is a maximum to be the axis of x ;
and a line perpendicular to it to be that of^ : let (x } y), (#',;/)
be the places of the centres of the balls at the time t } either
before or after impact : and let (x, y] be the place of the centre
of gravity at the same time : then
(ni + m ) x =. mx -f- tn'x } )
(m + m')y = my + m'y'; )
dx dx , dx'
^ = m ^ m ^ (M)
n j* fix
but -77 and -^- are before impact respectively v cos a and
v'cos a ; and are after impact respectively v cos /3 and Vcos ft :
{it I fill
and -j- and --,- are before impact respectively v sin a and
v' sin a, and are after impact respectively v sin /3 and v' sin /3' :
therefore by virtue of equations (21), the right-hand members
of (32) are the same before and after impact; therefore also the
left-hand members are : and therefore - n and -^ are the same
dt dt
before and after impact : and therefore the velocity and the
line of motion of the centre of gravity of m and m f are the same
before and after impact.
456 OBLIQUE IMPACT AND COLLISION. [339-
339.]] Examples illustrative of the preceding equations :
Ex. 1. A smooth and homogeneous sphere of mass m and
elasticity e moves with a velocity v and impinges directly on
another of the mass m', which is moving with a velocity v' in a
line of motion at right angles to that of m; it is required to
find the velocities and the lines of motion of both balls after
the collision.
In this case a = 0, a'= 90 ; therefore from (17), &c.,
m em'
v cos 8 = v T >
m + m
v sin ft = 0,
- (l +e )_^_ :
' m+m
V sin. ft' = t/;
.-. /3 = 0, tan/3'=
v
m em' , 2 ((\ + e)mv\*
m + m' ' \ m + m' 3
Hence the ball m will continue to move in the same line of
motion, but with a velocity diminished in the ratio of mem'
to m + m' to its former velocity ; the velocity of m' will be in-
creased, and its line of motion will make an angle ft with that
of m's motion.
2
Ex. 2. Two balls m and 2m, whose elasticity is -> move with
O
velocities 2v and v, and impinge, so that the line of motion of
each one makes an angle of 60 with the line joining their
centres at the instant when compression is a maximum; it is
required to determine their motion after collision.
v 2
Here m'= 2m, v'= -, e= - , a= a'= 60 ; therefore from
(17), &c.,
2v , , 70
v cos ft = > v cos/3= ;
whence v, v', ft, and ft' are easily determined.
340.] In Article 267 we estimated the loss of momentum
of a body in its passage through a resisting medium, when the
body presented to the medium a plane surface of area o>, which
340.] A RESISTING MEDIUM. 457
is perpendicular to the line of motion of the body. The inves-
tigation can now be extended to the case in which the surface
on which the medium acts is inclined at any angle to the line
of motion. The velocity of the body will be resolved into two
components, of which one is perpendicular to the plane of the
surface, and the other is along the surface : the former alone
causes the resistance by reason of momentum being transferred
to the elements of the resisting medium ; the latter produces
only a friction along the surface, the calculation of which does
not belong to the present part of our work.
Let there then be a plane surface, of which the area is o,
perpendicular to the plane of the paper, and of which the sec-
tion by the plane of the paper is the line OP, fig. 107 ; and sup-
pose it to be moving in the line MO, and its normal to be in-
clined to MO at the angle i ; let v be the velocity of the body
along the line of motion; then v cos i is the velocity of OP in
the line of its normal ; and therefore by a process similar to
that of Art. 267 it follows, that the momentum which is im-
pressed by to on the particles of the resisting medium during
the time dt, and which has therefore been withdrawn from the
movingbody,is
but the line of action of this resistance is in the normal to OP ;
therefore its component in the line of motion of the moving
bodyis pt; (cos *)<#; (33)
and therefore if m is the mass of the moving body, and dv its
loss of velocity in the line of its motion during the time dt
owing to the resistance of the medium,
mdv =
.'. m-j- = p &> v 2 (cose) 3 . (34)
(it
Hence it appears that the resistance of a plane rudder passing
through the water varies as the cube of the sine of the angle at
which it is inclined to the keel of the vessel.
A few examples illustrative of (33) are subjoined.
Ex. 1. An isosceles triangular wedge, of which the vertical
angle is 2 a, the depth is b, and the altitude is a, moves in a
resisting medium, firstly with its edge forward, secondly with
its top forward : compare the resistances in the two cases.
PRICE, VOL. III. 3 N
458 THE OBLIQUE EFFECTS OF [341-
Let RH and R 2 be the resistances in the first and second cases
respectively, then
Ex. 2. A semicircular lamina of given thickness T moves in a
fluid, firstly with its convex edge forwards, secondly with its
base forwards : compare the resistances in the "two cases.
Let the resistances be R! and R 2 : let a = the radius of the
semicircle : then, fig. 108, if PCA = 0, QCP d6, PQ = add,
f
/
'
R! _ 2
R7~ 3*
341.] By means of (33) also can be determined the resistance
which a solid of revolution meets with in its passage through a
resisting medium, such as water or air.
Let OPB, fig. 109, be the generating curve of the bounding
surface of the solid, and let its equation be y =f(x); OM = x t
MP = y\ and let PG be the normal to the curve at the point P,
so that (l>/
COSPGO = --
as
Let an element of the curve at p = ds ; so that of a surface-
element at P, ds is the section by the plane of the paper : also
let the surface-element subtend at an angle d6 at M ; and thus,
if o> = the surface-element,
co = dsydQ',
and therefore the loss of momentum corresponding to o> in the
line AO and in the time dt is
pVy(-j-\ dsdQdt;
CvO
and as the loss of momentum corresponding to every equal ele-
ment of the ring generated by the revolution of ds about AO is
the same, therefore the loss of momentum due to the ring is
(--' dsdt\
34I-] A RESISTING MEDIUM. 459
and the loss of momentum of the whole surface in the time dt is
2trpv* dtj y (Ir/^; (35)
and the loss of momentum to the moving body in an unit of
time, or the resistance of the medium, as it is called, is
'*, (36)
the limits of the integral being quantities assigned by the con-
ditions of the problem.
Ex. 1. Let the surface COB be a hemisphere; it is required
to compare the resistance of the hemisphere with the resistance
of the base.
Let a be the radius ; then if the line of motion is the axis of
x, and o is the origin,
y* = 2axx'- t
dy _dx _ds
ax ~' y " a '
fa fa _ ,g\s
.*. the resistance = 2-npv 1 I - - ' dx
Jo a 3
and the resistance of the hemisphere moving with the base for-
wards = Trpv 3 a*.
Therefore the resistance of a hemisphere moving with its convex
surface forwards is one-half of its resistance when it moves with
its base forwards.
Ex. 2. A right cone passes through a resisting medium, firstly
with its vertex forwards, secondly with its base forwards : it is
required to compare the resistances in the two cases.
Let the resistances be E t and E 2 : let a = the altitude of the
cone, b = the radius of its base ; so that in the first case
y _ x d dx ds
also E 2 = itpv*b*;
3N 2
460 SURFACE OP LEAST RESISTANCE. [342.
In these investigations no account has been taken of the
action of the particles of the fluid on each other, nor of the
friction of the particles against the surface of the moving body :
also as the body moves forward it leaves a space behind it, which
the particles of the resisting medium rush into and occupy :
doubtless some momentum is imparted by these to the moving
body : it is not therefore to be expected that the preceding
results will be entirely accordant with experiment; and they
are not ; and in fact it appears that the law of the resistance is
not, caeteris paribus, that of the square of the velocity. It is
however worth while even to approximate to a solution of a
problem of some difficulty, and therefore I have inserted the
preceding theory of resistance, springing as it does out of that
of impact and collision. We shall discuss it hereafter from a
hydrodynamical point of view, and it will appear that on that
theory the coefficient of resistance is only one-half of what it is
on the present aspect of the case. There is also one other
problem in the subject which deserves insertion, requiring as
it does the Calculus of Variations, and of which the solution was
first given by Sir Isaac Newton.
342.] To determine the form of a surface of revolution COB,
so that the resistance of a fluid, through which it moves in the
line of its axis, may be the least.
Let u represent the resistance : then
and taking the variation, and equating it to zero, we have
but since ds a = dx*
dx dy
- --
b.ds = -=-. --
ds ds
(37)
To determine the function which represents the curve, we have,
by the Calculus of Variations, the two following equations, viz.
343-] MOTION OP PARTICLES ON PLANES. 461
. .
cfo* ds* di*
either of which gives the equation to the curve ; from the former,
2ydy 3 dx
T-. = a constant = c (say) ;
(Is
.'. c(dy* + dx*)* = 2ydy*dx,
- (' + *>"' (S9 >
and replacing y in (38) by its value from (39), we have
dy 3 7 dy , dx
2 -/ -- c?.-f- 3cc?.-- = 0.
ds* dx dy
7 dy f dx dx\ dx / dx*\ , dx
cd.f( + \_ 3c (i + W = o,
aw? v dy </^ s ' <?y * c^ 8 '' dy
. dy c dx* 3 c dx* 3 c da?*
.'. 2a? clog H --- -, ----- , ---- -=- = cf; (40)
5 dx 2 ^ 2 2 d^ 4 <fy
where c' is also an arbitrary constant. And thus (40) becomes
dy dx* 3 c dx*
2xc\o&^-c-j- --- -=- = e ', (41)
& dx dy* 4 dy*
and either (39) or (40) is the equation to the required curve.
The properties of the curve at the limits would be given by the
integrated part of (37).
SECTION 2. Motion of particles on smooth inclined planes, under
the action of the constant accelerating force of gravity.
343.] As the problem of particles moving on smooth inclined
planes under the action of a constant force, and which, to fix
our thoughts, I will take to be the resolved part of gravity, is
the most simple in which a constant force is resolved, it is con-
venient to treat of it in this part of our work : yet as it properly
belongs to the theory of constrained motion, we are unable to
give a complete solution of it, until the principles of such
motion have been explained in a future Chapter.
Let the smooth plane be inclined to the horizon at the angle
a: and let OA, AB, fig. 110, be the sections of the inclined and
462 MOTION OF PARTICLES [343-
horizontal planes made by the plane of the paper, which is sup-
posed to be vertical and perpendicular to the line of intersection
of the two planes.
Let P be the place of the particle m at the time t, and suppose
m to be under the action of gravity : let g, as in Section 3,
Chapter VIII, represent the velocity-increment impressed by the
earth in one second of time, so that mg represents the earth's
impressed momentum on m due to a second of time in its own
vertical line of action : therefore the component of it along the
plane OA is mg sin a ; let OP = x, and suppose m to be moving
down the plane, then the expressed momentum-increment of m
along the plane in an unit of time is m -= ; and as the plane
clt
and m are smooth, there is no friction, and the impressed mo-
mentum-increment along the plane is equal to the expressed
momentum-increment : therefore
5?=flna, (43)
g sin a being positive, because both x and the velocity of m in-
crease as t increases. Let the velocity of m be u when t 0,
therefore dx
-XT u = gt sin a;
dx
-57 = u + msma: (44)
clt
whereby the velocity due to the time t is known.
Also let x = a, when t = 0, therefore
gt* sin a
x a = ut+ y -; (45)
2
at 1 sin a
x = a + ut+ y ; (46)
2
whereby the distance due to the time t is given.
If m moves from rest, when t = 0, and from 0, where x = 0,
then (44) and (46) become
dx
__ ^ a sina ( 47 >
2 " -
Again, multiplying both sides of (43) by 2dx, and integrating
for the limits corresponding to t t and to t = 0, we have
343-] ON INCLINED PLANES. 463
dx*
and thus the velocity is given in terms of the space described.
If m is at rest when t = 0, and also at o, which is the origin
of distance, then dx*
-^ - 2ffx sin a. (49)
Thus if OA, the length of the plane, is equal to I, and OB, the
vertical projection of I, = h, then
(the velocity due to the plane)* = 2 gl sin a
= 2^; (50)
but 2g&, see Art. 274, equation (49), is equal to the square of
the velocity acquired by m in falling down the altitude o B ;
therefore the velocity acquired by m in falling down the plane
depends only on the vertical projection of the length of the plane,
and not separately on its length or its angle of inclination ; that
is, depends only on the distance through which, the force has
acted in its own line of action. Therefore the velocity acquired
by m in falling down a plane is the same for all planes, the ver-
tical heights of which are equal.
This is a particular instance of the law of work, see Art. 259 ;
gravity acts through the same distance in its own direction,
whether the particle falls freely through the vertical height or
7/2 7"*
down the length of the plane, and the work done = mgk =
If m is projected up the plane, and x is measured from the
bottom of the plane, and thus in the direction contrary to that
in which the resolved part of gravity acts, so that, in fig. 110,
AP = x, then d*x
and if tQ, when m is at A, and if the velocity of projection =,
then dx
dx*
= *
sn a
dx
so that m ascends until -=- 0, in which case,
Clt
u n 3
t = : x = - -. --
464 MOTION OP PARTICLES [344.
344.] If a circle is placed in a vertical plane, the times of
descent down all chords drawn from the highest point are the
same.
Let o, fig. ill, be the highest point of the circle OQA, which
is supposed to be in a vertical plane : let <z=the radius, AOQ=0,
therefore OQ = 2a cos 6 ; OP = r : then
d*r dr
= ffC ose; ^
at* cos d . /ff\i
.*. r = 2<zcos0 = - - - ; .'. t=2(-);
which is independent of d, and is therefore the same, whatever
is the inclination of o Q to the vertical line OCA. Therefore the
times of descent down all chords drawn from o, the highest point,
are the same.
By reason of this property the circle is called the synchronous
curve of all straight lines in a vertical plane passing through o.
Similarly it jnay be shewn that the times of descent down all
chords drawn to A, the lowest point, are equal ; that is, the
time down QA is equal to that down OA.
If the plane of the circle is inclined to the horizon at an
angle i, a similar property is true ; for the resolved part of
gravity along the diameter o A becomes g sin i, of which the re-
solved part along OG is g sin icos 0. Therefore using the same
notation as in the preceding problem,
d*r
-3- = g sin i cos Q ;
at*
.. r = 2acos0 = sinicosO;
which is independent of 0, and is therefore the same for all
chords drawn from o, the highest point of the circle, to the cir-
cumference.
Similarly it may be shewn that the times down all chords
from any point Q on the circle to the lowest point A are equal :
the circle therefore is the synchronous curves for a pencil of
lines drawn, (1) from a given point o, (2) to a given point A, on
an inclined plane.
345-3 By help of the preceding property of the circle, whe-
ther in a vertical, or on an inclined plane, may many problems
345-] ox INCLINED PLANES. 465
be solved, which involve the determination of planes drawn from
given points and lines to other points and lines, and which are
such that the times of descent down them may be maxima or
minima. Some examples are subjoined, and the principle con-
tained in them is equally applicable to all similar problems.
Ex. 1. To determine the plane of quickest descent from (1) a
given point to a given straight line, (2) a given straight line to
a given point.
(1) Let A, fig. 112, be the given point, and BC the given
straight line : the solution of the problem depends on the con-
struction of a circle which passes through A, the highest point of
the vertical diameter, and which touches the given straight line.
Through A draw the horizontal line AB : bisect the angle ABC
by BO, which intersects in o the vertical line drawn through A :
from o draw OP perpendicular to BC : then OP is manifestly
equal to OA, and therefore the circle described from the centre
O and with the radius OA or OP will touch the line BC at P j join
AP : AP is the required line of quickest descent.
For since the time is the same down all chords of the circle
drawn from A, it is manifest that the time down any line other
than AP from A to the line BC is longer than that down A p.
(2) Let A be the given point, fig. 113, and BC the given
straight line : through A draw the vertical line AO, and the
horizontal line AB; bisect the angle ABC by BO, meeting AO in
o; from o draw OP at right angles to BC, and describe a circle
from o as a centre with the radius equal to either OA or OP,
which are evidently equal to each other : join PA : AP is mani-
festly the line of quickest descent from any point in BC to the
point A.
Ex. 2. To determine the line of quickest descent (1) from a
point within a circle to the circle : (2) from a circle to a point
without it.
(1) Let BPD, fig. 114, be the given circle, c its centre, and A
the given point within it; through A draw the vertical line AO,
and draw the vertical diameter BCD : join BA, and produce it to
meet the circle in P: join CP, which intersects AO in O: then
OA is manifestly equal to OP, and therefore the circle described
from o as a centre with the radius o A or OP will touch the given
circle at P : and AP is manifestly the line of quickest descent.
(2) Let BPD, fig. 1 15, be the given circle, c its centre, and A
the point without it : draw the vertical diameter BCD of the
PRICE, VOL. III. 3 O
466 MOTION OF PARTICLES [34-6-
circle ; and join BA cutting the circle in the point P : through A
draw the vertical line AO, and draw the line CPO. From the
geometry it is plain that OP = OA ; and therefore the circle de-
scribed from o as a centre, with the radius OP or OA, will touch
the given circle in the point P : and thus PA is manifestly the
straight line of quickest descent.
Ex. 3. To find the straight line of longest descent from a
circle to a point without it, and which lies below the circle.
Let BPD be the circle, c its centre, fig. 116, D the lowest point
of its vertical diameter BCD, and A the given point; join ADP,
PC ; and produce PC so as to intersect a vertical line through A
in the point o : then the circle described from the centre o with
the radius o A or OP manifestly touches the given circle at P, and
the line AP is evidently that of the longest descent.
346.] Illustrative examples of the motion of a particle on an
inclined plane.
Ex. 1 . Of a parabola, whose axis is vertical and vertex down-
wards, to find that focal radius- vector the time of descent down
which is a minimum.
Let 4 a = the latus rectum : and let be the angle between r
and the shortest focal distance : so that
2a
r =
1 + COS
at 3 cos
Now by (47), r = - ;
at* 1
40 cos0 (l+cos0) '
gt sin0(l + 2cos0)
{cos0(l+cos0)}
'
therefore sin = 0, and the sign of -j- changes from to + ;
eld
therefore t is a minimum : so that the line from the focus to
the vertex is that of quickest descent : also - - = when
.. dd
cos = -> that is, when = 120: the radius-vector corre-
ifi
spending to which is the line of quickest descent from the para-
bola to the vertex.
With reference to these and similar problems it may be ob-
served, that we have here determined the position of that plane
down which, of all drawn from a given point or line to another
346.] ON INCLINED PLANES. 467
given line or point, the time of descent is the least or greatest.
It must not however be hence inferred that a straight line is
that for which of all lines, straight or curved, joining two given
points or two given curved lines, the time of descent is the
least : we shall hereafter shew that the cycloid is the curve
which, in vacuo and under the action of gravity, possesses this
property of Brachistochronism, as it is called ; and that the
cycloid required cuts each of the given curves at right angles.
Ex. 2. To determine the inclination to the horizon of a
smooth inclined plane, so that the time of descent of a particle
m down the length may be n times that down the height of the
plane.
Let = the inclination of the plane to the horizon,
c = the length of the plane,
b = the height of the upper end of the plane.
Therefore b = c sin 0. Now from (47),
the time down the length of the plane = (
V
, ,, ,. ,, , . ,, slb^k /
and the time down the height = ( ) = (
^' ^
9
2c Jb
. n
sm0 = -
n
Ex. 3. It is required to shew that the times of descent down
all the radii of curvature of the cycloid, (fig. 105, Differential
Calculus,) are equal; that is, the time down pn is equal to that
down BC.
Employing our usual notation,
x a versin- 1 - (2 ay y*}^ ;
dx _ dy da
J* (2a-jr)* (20)* 1
.-. p = Pn =
dx s ?/
sinPGo = -- = -
.-. (the time down Pn) 8 = ^ ^ (2 a
-- = (the time down Be) 2 .
30 z
468 MOTION OF PARTICLES [347*
Ex. 4. To determine the form of a surface so that the times
of descent to any point in it from two given points in the same
vertical line may be equal.
It is evident that the surface is one of revolution about the
given vertical line ; we may therefore determine the curve by
the revolution of which the surface is generated : and let us
suppose the curve to be in the plane of (x, z) : let the given ver-
tical line be the axis of z ; and let the two given points A and
A', fig. 117, on it be at a distance 2 a apart : let o, the middle
point of AA', be the origin, and p be any point so that the time
down AP is equal to that down A'P : then
OA = OA' = a ; OM = z, MP = x,
... , 2.AP 2.AP a
(time down AP) 2 = -. - = :
^sm APM g.Kts.
2.A'p 2.AV
(time down A p) 2 = - - = :
^.SinAPM #.AM
therefore by the conditions of the problem,
AP 2 A'P 2
AM AM
/. z* x* a*;
which is the equation to the equilateral hyperbola. And there-
fore the surface required is that which is generated by the
revolution of an equilateral hyperbola about its transverse axis.
The lower sheet is that to any point in which all straight lines
drawn from A and A' are lines down which the times of descent
are equal : and the upper sheet is that from any point in which
the lines drawn to A and A' are those down which the times of
descent are equal.
347.] Two smooth inclined planes, the inclinations of which
to the horizon are respectively a and a, have a common vertex :
on these are placed two smooth particles m and m', connected
by a perfectly flexible and inextensible string which passes
over a small pulley placed at the common vertex of the planes :
it is required to determine the motion of m and m'.
Let the section of the two planes by the plane of the paper,
which is supposed to be vertical and to pass through the pulley
and to be perpendicular to the line of intersection of the two
planes,, be represented in fig. 118: let us suppose the pulley at
347-] ON INCLINED PLANES. 469
c to be so small that we may consider it to be (approximately)
a point, and so that the strings CP and CP' are parallel to the
respective planes. Let CAA'= a, CA'A = (I', CP = x, CP'= #',
p and P' being the places of m and m' at the time t : and to fix
our thoughts let us suppose m to be descending. Because the
string is inextensible,
# + #'= a constant ;
dx dx' d*x d*x'
.-. 1 = o, 1 = 0;
that is, j = : which result is also manifest by gene-
ral reasoning. Now m -f m' is the whole mass moved : and
-j is the velocity-increment expressed in the motion of each :
(m + mf) -777 = the momentum-increment expressed.
clt
And mg sin a, and mg sin a are the respective impressed mo-
mentum-increments along the planes : but as these act in
opposite directions,
mg sin am'g sin a'= the momentum-increment impressed ;
(m + m') -j- - = (msina m'sin a) ff; (51)
d*x m sin am' sin a
dP m + m
Similarly for the equation of motion of mf, we have
d*x' m' sin a' m sin a
dp ' m'+m
If when t = 0, m and m' are at rest,
dx m sin a m'sin a'
(52)
(53)
whereby the velocity acquired during the time t is known.
Also multiplying both sides of (52) by 2dx, and supposing
the limits of the integral to be such that the velocity = 0, when
x = a > dx 3 m sin a m' sin a
And integrating (54) again, and taking the limits of integration
such that x = a, when t = 0, we have
m sin a m'sin a' at*
xa - ; - _ . (56)
m + 1,1 2
470 MOTION OF PARTICLES. [348.
If the velocity, with which m and m' begin to move, is u, then
if x = a, when t = 0,
<fo sin a /sin a'
9^ (57)
/ sn a #' sn a atf *
a; a = *H -- ; - > (58)
m + m 2
dx* m sin a &' sin a
-j- -- = - 7 - 2 or. (59)
dt* m + m'
As to the initial velocity u ; suppose m and m' respectively to
have the velocities v and v' down the corresponding planes ; then
if u is the common velocity with which the two particles by
reason of their connexion by means of the string begin to move,
we have from the equality of the expressed and the impressed
momenta, { m + m')u = mv- mv',
mv mv'
.-. u - - . (so)
The preceding formulae are also applicable, whatever are the
inclinations of the planes. Thus suppose the plane c A' to be
horizontal, then a'== 0, and
(Px mg sin a
(U* ~~ m + m' '
that is, m' has no impressed momentum-increment; and if
a = 90, m is then moving vertically downwards : this case is
that of m hanging by a string over the edge of a horizontal
table, and drawing another body m' which is on the table at the
other end of the string.
If a = a'= 90, we have the same formulae as those which
were investigated in Art. 276.
348.] Examples in illustration :
Ex. 1. A small ball m descending vertically draws an equal
ball 25 feet in 2.5 seconds up a plane inclined at 30 to the
horizon, by means of a string passing over a pulley at the top :
it is required to determine the force of gravity.
Here a = 90, a'= 30, m = m'- y
d r 'x __g . dx _ g
dP~V Tt~'
*=|>, 25 = | (2.5)*, g = 32 feet.
349-] THE MOTION OP A PROJECTILE. 471
Ex. 2. Two smooth inclined planes are placed as in fig. 119 :
BC is inclined at 30, and CA at 60 to the horizontal line OA : it
is required to determine the distance CP through which m moves
in If', m' being equal to m.
The equation of motion becomes
d*x
2m-j- = mg (sin 30 + sin 60);
SECTION 3. The determination of the curvilinear paths described
&y particles moving in vacuo under the action of given accele-
rating forces.
349.] In the preceding section the effects of resolved forces
have been considered, when the path taken by the particle in
consequence of them is straight; we have now to investigate
the effects of resolved forces in a more general way : and I shall
take first the simple case of a particle moving in vacuo under the
action of gravity which is a constant accelerating force, and the
line of action of which is always vertical. The projected body
or particle is called a projectile, and the problem is in this case
that of the motion of a projectile in vacuo.
Let m be the mass of the projectile ; and let us prove, in the
first place, that the particle during the motion is always in one
and the same plane.
Let the horizontal plane passing through the point, whence
the particle is projected, be the plane of (x y y\ so that the axis of
z is parallel to the line of action of gravity : also let the velocity
of projection of the particle be u ; and let it be projected in a
line which makes an angle /3 with the axis of z, and of which
the projection on the plane of (x, y) makes an angle a with the
axis of x ; so that the three components of u along the three
rectangular axes are
u sin /3 cos a, u sin /3 sin a, u cos /3.
Now the components of the impressed accelerating force give
the following equations :
472 THE MOTION OF A PROJECTILE. [35-
therefore integrating the first two, and taking for the limits of
integration the values which correspond to t = t and to t = 0,
we have from (61),
-j- u sin (3 cos a = ; -'~- v. sin ft sin a = 0; (62)
dt (It
.'. x ut sin ^3 cos a, y = ut sin/3 sin a ; (63)
cos a sn a
which is the equation to a plane perpendicular to the plane of
(x, y), and containing the axis of z ; therefore the moving particle
is always in the vertical plane, which is inclined at the angle a
to the plane of (x, z).
350.] Let us take the plane in which the particle moves to
be that of (x, y] : let the point of projection, fig. 121, be the
origin : let the axis of x be horizontal, that of y vertical : let
the velocity of projection = u, and let the line of projection be
inclined at an angle a to the axis of x, so that u cos a and
#sina are the resolved parts of the velocity of projection along
the coordinate axes of x and y.
Let P be the position of m at the time t, OH = #, MP =^;
g = the accelerating force of gravity which acts parallel to the
axis of y ; therefore mg is the impressed momentum-increment;
so that the equations of motion are
d*x d*y
m M=> m &=-**',
the latter being affected with a negative sign, because the ten-
dency of gravity is to make the velocity increase, and y de-
crease, as t increases. Therefore
d s x fry
w=> a=-'- (64)
Now integrating these, and taking the limits corresponding to
I = t and to t = 0, we have
dx dy
jr ucosa = Q; -j- u sin a = gt; (65)
(it ut
of*
x = utcosa; y = utsina (66)
2
whence eliminating t, we have
y = xizna -- ?^ r ( 67 )
2 u 1 (cos a) 4
And this equation is that of a parabola ; whence it follows that
350.] THE MOTION OP A PROJECTILE. 473
a parabola is the trajectory of the particle. And (67) may be
put into the form
f u* cos a sin CK a 2(coso) a f ft* (sina) a
so that we have
it* sin a cos a
(1) the abscissa to the vertex = - ; (69)
(2) the ordinate to the vertex = - } ; (70)
2 #
(3) the latus rectum = 2 ^( cosa ) 2 . (71)
9
Also the form of the equation (68) shews that the parabola is
placed with its axis vertical, as in the figure, and that the vertex
is the highest point of the curve.
The distance OB between the point of projection and the
point where the projectile strikes the horizontal plane is called
the Range on the horizontal plane, and is the value of x when
y=0; that is, putting ^ = in (67),
in? sin 2 a
the range = OB = - ; (72)
u
also, as is geometrically manifest, OB = 2oc; that is, the range
is equal to twice the abscissa to the vertex.
From (72) it appears that for a given velocity of projection,
the range is the greatest when a = 45, in which case the range
2
= ; and the focus of the parabola in this case lies in the
9
horizontal line drawn through the point of projection.
Also from the value of the range (72) it appears, that if u is
the same, the range is unaltered when a is replaced by its com-
plementary angle : so that the range is the same for two dif-
ferent angles which are complementary of each other, if the
velocity of projection is the same : hence if a = 45, these two
angles become identical, and the range is a maximum.
c A is called the altitude or the greatest height of the projectile,
and is the value of y when -~- = ; therefore from (67),
L \, - i-j. ^ 2 (sin a) 2
the greatest height = - - ' (73)
2#
Also from (66), x = utcosa; (74)
that is, the abscissa uniformly increases along OM. Hence if we
substitute the range for x, we shall have an expression for the
TRICE, VOL. in. 3 p
474 THE MOTION OF A PROJECTILE. [35 1 -
time which a particle takes in passing from o to B, and which
is called the time of flight : and thus
1.1. x- DO- i-i. 2^ sin a
the time of flight = - - (75)
c/
351.] Again multiplying (64) respectively by 2dx and 2dy,
and integrating, and taking the limits corresponding to t = t
and to t = 0, we have
dx 2 dy' i
^ = u* (cos a) 2 ; -j^ = u* (sin a)' - 2gy ; (76)
ds*
.-. = (the velocity)' = '- 2^. (77)
Now this result deserves notice. Let the directrix of the
parabolic trajectory be drawn as in the figure : then
1 1.1, r .a. i i. ^ 2 ( cos a ) 2
AD = - th of the latus rectum = - -- ' :
4 2^
i a (sina) 2 ,. u* ,, , .
and since by (70), CA = * - > therefore CD = ; that is,
see equation (49), Art. 274, CD is the vertical height through
which a particle falling in vacuo will acquire the velocity with
which the particle m moves at its projection from o.
Let CD = h; therefore w 2 = 2gh' } and substituting in (77),
/72
j- = 2g ( k-y Y) (78)
that is, the velocity at any point P on the curve is that which
would be acquired by a heavy particle falling freely in vacuo
down a vertical height equal to h y, that is, to SP. Hence it
follows that the velocity of in at any point P in its path is that
which would be acquired by a particle falling freely from the
directrix to the curve. The directrix of the parabola is there-
fore determined by the velocity of projection, and is at a vertical
distance above the point of projection equal to that down which
a particle falling would have the velocity of projection. Hence
also the vis viva of the projectile is at every point of the path
the same as that of an equal particle acquired in falling from
the directrix to the point of the curve.
352.] The equation to the path of the projectile may also be
found by the following process : and as the result of simulta-
neous velocities taking place in combination is well exemplified
by it, I do not hesitate to insert it.
Let the particle m be projected from o, fig. 121, with a velo-
city ?/, in the line OQ making an angle a with the horizontal
353-] THE MOTION OP A PROJECTILE. 475
line ; then if no force acted to impress velocity on m, it would
in the time t describe a space OQ = ut, and its coordinates at
the time t would be
x=utcosa, y = ut sin a. (79)
But as gravity is a constant force, and acts in a line parallel to
the axis of y, and tends to diminish y according to the arrange-
ment which we have assumed in the figure, y will by it be
Qt*
diminished in the jfcime t by a quantity equal to ', see equa-
tion (49), Art. 274 ; so that at the end of the time t we have
at*
x = iitcosa, y = ut %\n. a - ; (80)
z
which values are the same as (66); and therefore it appears,
that if P is the place of m in the parabolic path at the time t,
353.] A particle m is projected from a given point on an in-
clined plane in a given line ; it is required to determine the
point at which it will strike the plane.
Let the angle of inclination of the plane to the horizon be i :
let a = the angle between the line of projection and the hori-
zontal line : u = the velocity of projection : then the equations
to the inclined plane and to the path of the projectile respec-
tively are
y=#tan&; y = a?tano - 7- ;
2w 8 (cosa) a
whence eliminating y, we have
x = - ' (cos a) 8 (tan a tan i)
y
_ 2u* cos a sin (a i} <
"~ y
2 w 2 cos a tan i sin (a i\
.'. y = ^^ ,
which give the point on the plane at which the projectile strikes
it : and the range on the plane is equal to x sec i, that is,
2 w 8 cos a sin (a i)
the range =- ; -^ '. (81)
g (cos i)*
Also the range is the greatest when
and fl -* = s*"'' ; (82)
which latter value gives the angle between the plane and the
3? 2
476 THE MOTION OP A PROJECTILE. [354-
line of projection for which the range is the greatest : and in
this case u *
the greatest range = - -. ^ (83)
Hence it follows that if from a given point a system of straight
lines is drawn in the same vertical plane, and particles are pro-
jected with a given velocity u, and in such fines that the ranges
on the planes are the greatest, the locus of the extreme points
of these ranges is given by the equation (83); and therefore if
u is the velocity of projection, r = the range, = the angle
between the plane and the vertical line through the point of
projection, then from (83),
*
and if h is the vertical distance to which u is due
,- '*,; (85)
i + cos e '
which is the equation to a parabola, the focus of which is at the
origin of coordinates, whose axis is vertical, and of which 4 k is
the latus rectum.
354.] It is required to determine the angle of projection for
a given velocity so that a projectile may pass through a given
point.
Let the given point be (x ly y^) : then these coordinates satisfy
the equation of the path of the projectile, and we have
u* 1
' tan a = - + --{w-2Vyi-y a *!*}*; (86)
y*i y*\
therefore two different values of a satisfy the condition, if
u* is greater than 2u^gy l +ff t x l a ;
only one value of a satisfies it, if
u* = 2aVyi+ 2 tfi a ;
that is, if acS = { - -y, I ; (87)
u ' *9
and the projectile cannot reach the point, if
u* is less than 2u' i ff^ 1 +ff t x 1 9 .
Now (87) is the equation to a parabola of which o, fig. 121,
) //- jiv
is the focus, - - is the latus rectum, and or h, see Art. 351,
355-] THE MOTION OP A PROJECTILE. 477
is the distance to the highest point ; all points therefore on this
parabola are the farthest which the projectile can reach ; all
points without it are beyond the reach ; and all points within it
may be reached by two different angles of projection. The
same result may also evidently be proved by the following
process :
It is required to find the envelope of all parabolas described
by particles projected with a given velocity u from a given
point in the same vertical plane.
The equation to the path of one is
)}; (88)
therefore differentiating by making tan a to vary, we have
= (x - tana") d.tan a :
^ it* '
u*
. . tan a = ;
gx
so that (88) becomes
2w
x
ff
which result is of course the same as (87).
355.] Problems in illustration of the preceding equations :
Ex. 1 . To determine the angle of projection so that the area
contained between the path of the projectile and the horizontal
line may be a maximum.
Since the area of a parabola is two-thirds of that of the cir-
cumscribed rectangle, if A represents the required area,
2
A = - the range x the greatest altitude
o
2 u*
= j~i (sin a) 3 cos a ;
db. 2lt*
if tan a = 3*, and changes sign from + to : therefore the
u* 3^
area is a maximum and = - > if a = 60.
%g*
Ex. 2. It is required to compare the areas of the two para-
bolas described by projectiles, of which the horizontal ranges
are the same, and the angles of projection are therefore comple-
mentary to each other.
478 THE MOTION OF A PROJECTILE. [355-
Let AJ and A 2 be the areas : then as the ranges are equal,
these are to each other as the greatest altitudes : therefore
A, (sin a) a
,
A 2 (cos a) a
Ex. 3. From the top of a tower two particles are projected at
angles a and /3 to the horizon with the same velocity u, and
both strike the horizontal plane passing through the bottom of
the tower at the same point ; it is required to find the height
of the tower.
Let k = the height of the tower : u = the velocity of projec-
tion : then if the particles are projected from the edge of the
top of the tower, and x is the distance from the bottom of the
tower to the point where they strike the horizontal plane,
-Ti = x tan a-|^ { 1 + (tan a) 8 }, (89)
-h = * tan/3-f^ (1 +(tan/8)*> ; (90)
(',
therefore by subtraction,
2u* 2 U* cos a cos B
>
g (tan a + tan /3)
substituting which in either (89) or (90), we have
, _ 2 u* cos a cos /3 cos (a + /3)
Ex. 4. Particles are projected with a given velocity in all
lines in a vertical plane from the point : it is required to find
the locus of them at a given time t.
From (66) we have
9V
x = ut cos a, y = ut sin a ;
.'. lit <x>s a = x, utBina
m
therefore squaring and adding, we have
the equation to a circle of which the radius is ut } and the
at*
centre is on the axis of y at a distance below the origin.
2
Ex. 5. Particles are projected from o with a given velocity in
all lines in a vertical plane : it is required to find the locus of
their highest points.
356.] CURVILINEAR MOTION OF PARTICLES. 479
Let x and y be the coordinates to the highest point : then
from (69) and (70),
u* sin a cos a u* (sin a)*
x = > y = ^r J
O 2/9 qi
therefore adding, 4y* + x* = -\ (92)
fcf
which is the equation to an ellipse, of which the major axis
/l'i nil
= > and the minor axis = ; and the origin is at the
*/ y
extremity of the minor axis.
The preceding investigations into the motion of projectiles
would explain the theory of gunnery, if it were allowable to
neglect the resistance of the air ; but as the velocity with which
balls and shells traverse their paths is very great, much of their
momentum is lost by the resistance of the medium ; and the
ratio of the vertical and horizontal velocities is so much altered,
that the form of the trajectory is very different from that of a
parabolic path. In the last Section of the present Chapter we
shall investigate, as far as it is possible, this path, and shall take
account of the loss of momentum which is due to the resistance
of the medium.
We proceed now to other cases of curvilinear motion in
vacuo ; and I would observe, once for all, that if a particle is
projected with a given velocity in a plane, and if the lines of
action of the forces, which act on the particle, are in that
plane, the particle is during its motion in that plane : this is
evident by the principle of sufficient reason.
356.] From the vertex of a parabola a particle m is projected
with a velocity u at right angles to the principal axis, and is
acted on by an attractive force which is perpendicular to that
axis and varies directly as the distance of the particle from it.
It is required to determine the law of force acting parallel to
the axis of x so that the particle may move in the parabola, and
the other circumstances of motion.
Let the vertex be the origin, and the principal axis and the
tangent at the vertex be the axes of x and y ; and let p, (x, y],
fig. 120, be the position of m at the time t, so that the equation
to the parabola is y* = 4 oar;
dy _ 2 a _ ,a
dx
_ _ ,
y \F'
480 CURVILINEAR MOTION OF PARTICLES.
By the conditions of the problem,
-= = x, which is to be determined ;
at*
d*y _ e / g3 \
therefore multiplying (93) by 2dy, integrating, and taking the
limits corresponding to t = t and to t = 0,
dy*
ty* (QA\
= u 1 4 jz ax ;
O/X It tC ~~ 4 Lt (tX /.. \
' -^r = r^ J ( 95 )
therefore the required force which is parallel to the axis of x
partly is constant and repulsive, and partly varies as the abscissa
and is attractive towards the axis of x.
fj'ti ff.^r
Also from (94) and (95) - and -=- both vanish, when
At Ctt
U U* /r >_s
y = -\ -- r and x = - ; (.97)
-
so that at this point, say B in the figure, m comes to rest ; and
afterwards under the action of the forces returns to the vertex of
the parabola, through which it passes with the original velocity
u, and comes to rest again at B', which is equidistant with B
from the vertex : thus the motion is oscillatory. Also from (94)
taking the positive sign, as we will consider the original motion
from o to B. Therefore integrating, and taking the limits cor-
responding to t = t and to t = 0,
1/i
* y = sin p* t -, (98)
M 5
therefore the time from o to B is - : and the time of an
357-] CURVILINEAR MOTION OF A PARTICLE. 481
oscillation, viz. from B to B', = > which is independent of the
velocity of projection from o, and depends on only the absolute
force parallel to the axis of y.
I have, for the sake of simplicity, taken the parabola for the
example whereby the process may be illustrated, but the method
is the same in all cases. Thus in the ellipse, if m is projected
from the extremity of the major axis with the velocity u,
and the coordinates of the point B to which m passes are
u a ,, w\i
, * = -(* --); (100)
r*
therefore m comes to rest at the extremity of the minor axis,
if u = fj.^6.
357.] From a given point in the axis of y a particle is pro-
jected with a given velocity in a line parallel to the axis of x,
and is acted on by an attractive force parallel to the axis of y
and which varies as the distance from the axis of x : it is re-
quired to determine the circumstances of motion.
Let b = the distance from the origin of the point of projection,
u = the velocity of projection ;
so that the equations of motion are
cPy _
dx dy* .. . ^
dt dt*
x = ut ; -^f = JA* dt j
cos -1 ^ = fjfit;
y = b cos $ t
!
= b cos ; (101)
u
which is the equation to the companion to the cycloid.
If the force is repulsive, we have x = ut, and
TRICE, VOL. in. 3 Q
482 CURVILINEAR MOTION OF A PARTICLE. [358.
(102)
(103)
and consequently the path is the catenary.
358.] From a given point in the axis of y a particle is pro-
iected with a given velocity u along a line parallel to the axis of
x, and is under the action of an attractive force parallel to the
axis of y which varies inve'rsely as the cube of the ordinate :
determine the other circumstances of motion.
du
Hi
^ _ **
~T~ **
bu
which is the equation of an ellipse, whose centre is the origin.
If the force parallel to the axis of y is repulsive, the path is a
hyperbola.
If the initial circumstances had been the same, and the force
had been attractive and varied inversely as the square of the
ordinate, then the equation to the path would be
('-*)* + 0-versin-> ) = -x. (105)
359.]] From a given point in the axis of y a particle is pro-
jected with a velocity u in a line parallel to the (rectangular)
axis of y, and is attracted by a force the intensity of which
varies directly as the distance, and which has its source in the
origin of coordinates : it is required to find the equation of
the path of the particle, and to define the circumstances of
motion.
Let b be the distance on the axis of y from the origin of the
point whence the particle is projected with the velocity u : let p,
359-] CURVILINEAR MOTION OP A PARTICLE. 483
fig. 122, be the place of m at the time t, B its place when t = ;
OM = x, MP =y, OP = r, OB = I -, and let fx be the absolute force
of the attraction at o. Then the equations of motion are
(106)
y
di* ~
Now multiplying these equations respectively by 2dx and 2dy,
and integrating, and taking the limits corresponding to t = t
and to t = 0, we have
*_= _,**; (107)
// > * * *
(108)
therefore adding,
%+%-*=*:-*+*'-)> " 0!
which gives the velocity of m at any point of its path. Also
from (107) and (108),
(no)
i i
( ^_ ^ (*._,.)*'
V '
.'. a?= rsin^^; y = 5 cos \& t ;
fi*
U.T 2 7/ 2
whence we have -- h T~ = 1 > ( 112 )
w o a
2?^
which is the equation to an ellipse whose ar-axis is i and
V?
whose y-axis is 2 b, and whose centre is at the origin, that is,
at the source of the force.
From the preceding values we have
ds*
-j- = (velocity) 2 = u 3 (cos p* t)* +pt>* (sin /z* )* ; (11 3)
and from (111), the time from B to A = -; therefore the
whole periodic time = ; and is independent of the velocity
/*
and distance of projection, and involves only the absolute force
of the impressed force.
If the force at o had been repulsive, then the sign of /x would
be changed, and the equations of motion would be
d*x d'y
484 CURVILINEAR MOTION OP A PARTICLE. [360.
(116)
(118)
whence squaring, and subtracting the former from the latter, we
have . 2
which is the equation to a hyperbola with its centre at the
origin; and which might have been deduced from (112) by
affecting p. with a negative sign.
If, in the case of the force being attractive, the velocity and
distance of projection are such that u = b^ } the path which m
describes is a circle, and the velocity in it is constant and equal
to that of projection.
360.] A particle m is projected from a given point with a
given velocity, and is acted on by a force which varies inversely
as the square of the distance from a given point which is its
source : it is required to determine the path of the particle, and
the other circumstances of motion.
Let u be the velocity of projection, a = the angle between the
line of projection and the axis of x, so that the components of
the velocity of projection along the axes are u cos a and u sin a ;
let (a, 1} be the point of projection, p = the absolute force, and
let a 3 + b* = c 8 : let r be the distance of m at the time t from
the centre of force, which we will take to be the origin, and let
(x, y) be the place of m at the time t ; then the equations of
motion are ,
r r
which are simultaneous differential equations, and from which
the solution of the problem is to be obtained. Multiply (120)
respectively by y and x } and subtract ; then
360.] CURVILINEAR MOTION OP A PARTICLE. 485
/77* /7//
and adding and subtracting -jr -jr > and integrating, and taking
definite integrals with limits corresponding to t t and to t = 0,
we have # dx
x -f y-rr = u(a sin a ocosa) (122)
clt dt
= *(say). (123)
Again,, from (123) and the first of (120), we have
, d'x _ \t.x xdyydx
W ~~^r* dt~
/x x* dyxydx ^
~ ~~dt~
but ar'-f/ 1 = r* ; therefore xdx+ydy = rdr;
d*x _ jx x*dyy(rdr-ydy)
dt*~~ ~ r 3 ' dt
\L r^dyrydr
~~~r* dt~
d y
= -*dt-r'>
-^ + ^; (124)
similarly, from the second of (120),
flit It O* It ft
(125)
at re
Multiplying (125) by #, and (124) by y, and subtracting, we have
x dy dx^ . p
\. fi.f fjf s 31 p
therefore by (123),
h t hu(xsma ycosa) = pr -(ax+by), (126)
C
which is of the form r = A# + By + c; (127)
and as r is a rational function of the coordinates x and y, the
equation is that of a conic of which the focus is the origin.
A conic therefore is the trajectory, with the source of the im-
pressed force in the focus ; and the constants A, B, c are given
in terms of the velocity, the direction of the line of action, and
the coordinates of the point, of projection.
Also from (120), multiplying respectively by 2dx and by 2dy,
and adding and integrating with the limits assigned above,
^_ B . = ^_ 2 _M (
dt % r c
whereby the velocity is given at any point of the curve.
486 CUEVILINEAK MOTION OP A PARTICLE. [361.
361.] In some cases oblique coordinates may be used with
advantage. Thus suppose, as in Art. 359, a particle m to be
projected with a given velocity u in a given line from a given
point, and to be attracted by a force the intensity of which varies
directly as the distance, and which has its source in a given
point : it is required to determine the path which it describes.
Let the given source of the force be the origin ; and let the
line passing through it and the point of projection be the axis
of y : and let the axis of x be drawn parallel to the line of pro-
jection ; let the distance from the origin to the point of projec-
tion be b l : then the equations of motion are
d*x _ d*v _
dp = ~^ f 2F = ~W'
and by a process similar to that of Art. 359, we shall have
which is the equation to an ellipse, referred to oblique coordi-
nates, whose centre is at the origin, and of which the angle of
ordination is (say) &>, where o> is the angle between the line of
projection and the line joining the point of projection and the
centre of force. If a and b are the principal semi-axes, then
by the properties of such axes we have
+a= -+v,
, Ult .
do = sin CD ;
362.] Lagrange, in the " Meeanique Analytique," second part,
Sect. VII, Chap. Ill, Art. 83, remarks that a conic, say, an
ellipse, which would be described by a particle under the action
of a force varying inversely as the square of the distance and
tending to the focus of the ellipse, or under the action of a force
varying directly as the distance and having its source in the
centre of the ellipse, may also be described under the action of
three similar forces which have their sources in the two foci and
in the centre of the ellipse respectively ; and he makes this re-
mark, after he has proved that such forces yield a particular
integral of the differential equation which expresses the motion
362.] THEOREM OF M. BONNET. 487
of a particle under the action of two central forces, which vary
inversely as the square of the distance, and whose centres are
in two given points. This fact however is only a special appli-
cation of the following more general law :
If many particles m lt m t , ... m n which are respectively under
the action of the force P U P a , ... p n are projected from a given
point with the velocities respectively V 1} V 2 , . . . v n along the same
line and in the same direction; and if each of these particles
describes the same path ; then one particle M, projected with
the velocity v from the same point, along the same line, and in
the same direction as the m's, will describe the same path, if the
initial vis viva of M is equal to the sum of the initial vires vivae
of the m'a ; that is, if
MV* = m^ v^ + m, Vj, 2 + . . . + m n v w *.
Let (x, y, z] be the position of M at the time t ; so that its ex-
pressed momentum-increments along the coordinate axes are
d'x fry d*z
*dT*' M M' M ^ ;
and let x n Y M z u x 2 , Y a , z 2 , . . . x n , Y n , z n be the components of the
impressed momentum-increments of the several forces p t , P 2 , . . .
p n ; and let N be a certain normal force, the direction angles of
the line of action of which are a, /3, y, and which is such that M
under the action of it and the P'S describes the required path.
Then the equations of motion of M are
-j = x,-f Xj-f ...+x,,+ NCOS a = 2. x+ NCOS a, -]
= 2.Y + NCOS/3,
= 2.Z+NCOSy. J
(129)
T ,, ...
Multiplying these respectively by Idx, 2,dy, 2dz, and supposing
the velocities of M, m l} m a , ...m n a.t the time t to be v, v ly v^ . . .v n ,
wehave ?. M w a = 2^2.x+2^2.Y+2^5.z, (130)
because dx cos a + dy cos /3 + dz cos y = 0.
But the equations of motion of / m,, ...m n under the action
of their respective forces yield the following equations :
d.m,v t * = 2(x a <r + Yjrt^-f z, dz),
d.m n r n * = 2 (x n dx + Y B dy + z n ^fe) j -J
488 CURVILINEAR MOTION OF A PARTICLE. [363.
so that (130) becomes
= s.d.mv* = d.s.mv*;
and therefore taking definite integrals, with limits corresponding
to t=t and to =0, we have
M*? 2 = s.flzt? 2 ;
and therefore at all points of the path of M, its vis viva is equal
to the sum of the vires vivae of the m's in their separate motions.
Hence it follows that the normal force N assumed in equa-
tions (238) is zero, and consequently M, under the action of the
several impressed momenta which act on m l) m*, ... m n , will de-
scribe the same path as each of the m's. This general propo-
sition is due to M. Ossian Bonnet, and is given in the notes
appended by M. Bertrand to the edition of the "Mecanique
Analytique" of M. Lagrange, Vol. II, 1855.
363.] In illustration of the process of tangential and normal
resolution we will consider the simple case of the motion of a
projectile in vacuo under the action of gravity.
A particle of mass m is projected from a given point, in a
given line, and with a given velocity ; and moves subject to the
action of gravity: it is required to determine the curvilinear
path.
Let the point of projection be taken as the origin of coordi-
nates ', and let the vertical and horizontal lines drawn through it
be the axes of y and x respectively : let u be the velocity of pro-
jection, and let a be the angle between the line of projection
and the axis of x : and let (x, y] be the position of m at the time
t : then because the vertical line, in which gravity acts, makes
with the tangent to the curve at the point (x, y] tan- 1 -T- > we
have the following equations of motion :
d*s dy V dx
W=-^' 7 = *2T
From (131) we have
ds*
.'. --u*=
|1 = --2#, (132)
which result is the same as (77) Art. 351. Also from (131), since
364.] CURVILINEAR MOTION OF PARTICLES. 489
p -
d*y
dx*
d *y,
~ (u* 2ffy
dx*
dx s
ds\
dx*'
Igdy
~
dy*
1 -4-
dx*
log -. r = log
6 (sec a) 2
dy dx
{w 2 (sina) 2
1 . . . ,i wsina
g v g wcosa
(133)
2 ft 2 (cos a) 8
which result is the same as that obtained by the method of
coordinate resolution in Art. 350.
364.] A particle m describes a helix with a constant velocity :
it is required to determine the laws of the accelerating forces
which act on it parallel to the three coordinate axes.
Let the equations to the curve be
'. dx = asin<j>d<f>, ~]
dy acostydfa f (134)
z = lea j J dz = Tea d$ ; J
.-. ds* = a 8 (l+ 2 )^</> 2 . (135)
But since the velocity along the curve is constant, ds cdt;
dx _ c sin <f) dy _ c cos </> dz Jcc
" ~
Hence we infer that the accelerating forces parallel to the axes
PRICE, VOL. in. 3 R
490
RELATIVE MOTION.
[365-
of x and y vary directly as x and y respectively, and are at-
tractive, and have the same absolute force ; that is, the resultant
force of these two forces will have a line of action always pass-
ing through the axis of z, and will be constant. Also the acce-
lerating force parallel to the axis of z vanishes, the velocity of
m parallel to that axis being constant.
365.] The following are problems in relative motion, wherein
the place, velocity and velocity-increment of a particle is referred
to the moving place of another particle, the directions of the co-
ordinate axes being parallel throughout the motion. We shall
have other examples hereafter in which the directions of the axes
move. The following are applications of (77), Art. 332 :
Two material particles m and m' attract each other with a
force varying directly as their masses and inversely as the
square of the distance : it is required to determine the motion
of m relatively to m'.
Let, in reference to m' placed at the origin o, which is move-
able, a?, y, z be the coordinates to m at the time t, and let / be
the distance of m from o ; then since the impressed velocity-
increment which acts on m' and attracts it towards m along
MA
the. line r is > the components of the impressed velocity-
increments on mf at the origin are
mx my mz
and the components of the impressed velocity-increments on
(137)
my
i ^ >
if* y>3
Therefore by (77) in Art. 332,
d*x (m + m'} x
dt* H r 3
^_L
dt* ~"
= 0, -,
(138)
and the equations of motion of m' relatively to m will be similar
in form : and therefore m will describe relatively to m' a curve
similar to that which m' will describe relatively to m.
And to determine the path of m relatively to m' : multiplying
the second of (138) by z, and the third byy, and subtracting,
we have
365.] TWO ATTRACTING PARTICLES.
d"y d*z _
therefore integrating,
491
dy
~rr
dt
dz
dz
-n
dt
dx
similarly, x - z =
dx dy
y ~dt~ X ~dt
= 3 ;J
(139)
where h iy h* } h t are arbitrary constants : now multiplying these
severally by x, y, z, and adding,
htX + hty + hiZ = 0; (140)
which is the equation to a plane passing through the origin,
that is, through m' ; and therefore the motion of the particles is
wholly in one plane.
We may if we choose take this plane to be that of (x, y} } and
thus reduce the inquiry to one of two dimensions. We will
however proceed with the most general case.
From (138) and (139), and writing /x for m + m', we have
^~dP~' l *lkt*
dt
dt
dt dt
= /*
rdxxdr
x
. >
r
since x* +y 3 + z* r*, and x dx -f y dy + z dz rdf,
similarly,
dz , dy fj.x
* ~dt ~ 3 dt ~ ~ + ^ ' n
dx
dt
dy
dz
(141)
where f\,Jt } fs are three undetermined constants: now multi-
plying these severally by x, y, z, and adding, and observing
(139), we have
Hr+f 1 x+f a y+f 3 z = hS + hS+h 3 *; (142)
and if r is replaced by (# a +y* + z*^, and the equation is cleared
of radical quantities, it is that to a surface of the second order ;
and as the intersection of it by the plane (140) is a conic, it
follows that the path of m relatively to m' is a conic : and simi-
larly m' describes a conic of the same species relatively to m.
3R2
492 RELATIVE MOTION. [365.
In reference to (142) I would observe, that if (x,y,z) is a
point in the orbit of m, fiX+f 3 y+f 3 z (hS + h^ + hS} is pro-
portional to the perpendicular from (x, y, z) to the plane
Ax+f*y+f*z-(hS + hS+W = 0; (143)
and r is the distance from the origin to the same point : there-
fore from (142) it appears that the distance from the origin to
any point on the surface has a constant ratio to the perpendi-
cular distance from the point on a fixed plane : the surface
therefore is one of revolution of a conic about its axis, which is
perpendicular to the given plane (143), the origin being the
focus of the conic, and the given plane being generated by the
revolution of the directrix. Hence also the direction-cosines of
the axis of the surface are proportional to fi,f 9 ,f* ' and since
by reason of (14 ),
*!/ + *,/+*/. = <>* (144)
it appears that the plane (140) passes through the axis : there-
fore the conic in which m moves is a principal section of the
surface (142); and m' is placed in the focus: m therefore de-
scribes relatively to m' a conic with m' in the focus : and simi-
larly m' describes relatively to m a conic about m in its focus.
Also multiplying the equations (138) severally by 2dx,
and 2 dz } and adding and integrating, we have
where c is another undetermined constant ; but it is to be
observed that all the undetermined constants may be found in
terms of the initial velocity, the direction-angles of its line of
motion, and the coordinates of the point of projection.
The eccentricity of the conic may thus be found. Since the
ratio of the focal radius of a conic to the perpendicular from a
point on it to the directrix is that of e to 1 : and since from
(143) the perpendicular from the point (x,y, z) on the directrix is
../.*+/*+/.*-(*,' + *.' + *.') _ , , .
(/,+/.'+/.*)*
_ i
p -y
= '(/'+/.'+/.')*
fir
_(/'+/.'+/.')*.
f*
366.] TWO ATTRACTING PARTICLES. 493
., ,(/.+/.'+/.)*. (146 )
If the path of m is an ellipse, the equations of the major axis are
x y z + r
== = '
'
and substituting for these values in (142), and taking r, and r a
to be the greatest and least values of r, and substituting
h* = V + V + V, /' =/i'W+/S we have
-
' rj + r a = 20 =
M'-/'
(Jf zr^ " ' i
TTVlTy-i-x 5 ( 14g )
f* \J i +/ +/s J
and if i is the inclination to the plane of (x, y] of the plane of
motion, then from (140),
cosz = -.; (149)
(i* + >5* + s*)*
and if n is the angle between the axis of x and the line of inter-
section of the plane of m's motion with the plane of (x, y), then
tann= j-; (150)
and thus the plane of the motion is completely determined.
And thus (149) and (150) give the position of the plane in
which m moves : (147) give the direction-angles of the major
axis of the ellipse, and therefore assign the position of the
ellipse; and (146) and (148) give the dimensions of the elliptic
path.
This problem is manifestly the astronomical one of two bodies
m and m' moving relatively to each other, and under the action
of their mutual attractions, and on this account I have con-
sidered it at greater length than would otherwise be necessary.
The determination of the other incidents of the motion requires
data which it would be out of place here to enter upon.
366.] If two particles m and m' move subject to their mutual
attractions, the centre of gravity of them either remains at rest
or moves in a straight line.
494 RELATIVE MOTION. [366.
Let the positions of the particles m and m' at the time t be
respectively (x, y } z), and (x', y' ', /) : and let r be the distance
between them, and, to fix our thoughts, let us suppose m farther
from the origin than m' ; then the equations of motion of the
two relatively to the fixed origin are
d*x_-m'(x-x') d*y _-m'(y-y') d*z _- m \z-z f } . , ieiN
~dP~ ~^~ dP~ r 3 ' dP~ r 3
// 2 />* AM [ /* __ 4*A // 2 ^/ AM ( -7/ _^_ ^/ \ fj. 2 A> AM f & __ fl^|
tv t/ fib \J} ^^ JL } tv u III/ \U U J Us & lib I x/ ^~ 6 J
dP " r 3 dP r 3 dp
(152)
/7"2 /> /7* />' /72 ^/ /72-j/' //2 ^ //2 ^'
iv 3u f \4/Jj Us U mlv U U/ Z . tv Z . .
m-rr + m - T =0, m-ff + m -^=0, t>i-j- + m -^=0. (153)
^2 /7/2 ' /7/2 /7/2 ' tit? fit.'' ^ '
dP
Let (x } y, z) be the position of the centre of gravity of m and
m' at the time t : therefore
(m-\-m')x = mx+m'x', ~\
(m + mT)y = my+m'y'y \ (154)
(m + mT) z mz + m'z' ; J
and differentiating these twice we have, by reason of (153),
(155)
Suppose a, ^3, y to be the components of the velocity of the
centre of gravity, and (a, b, c) to be its position when t = ;
then integrating (156),
dx _ dp _ dz
~di =a ' Tt =fi> ~dt = y '>
xa at, yb = j3t, z c = yt\ (157)
x a y b z c
which are the equations to the rectilinear path of the centre of
gravity of m and m' ; if a = /3 = y = 0, so that the centre of
gravity is at rest when t 0, then for all values of t we have
x = a, y b, z = c; (159)
and the centre of gravity remains in the same position.
The equations of motion of m and m' relatively to their centre
of gravity may be calculated as follows :
(, ri, C) (', V, ') be the positions of m and m' relatively
367.] TWO ATTRACTING PAKTICLES. 495
to the centre of gravity as origin ; let r be the distance between
m and m'- } p and p' the distances of m and m' from their centre
of gravity, so that
m, + m' _ m' _ m
' ' P ~7'
Now the ^-component of the expressed velocity -increment of m
is as follows: ~
m + m' p 3 '
. ., . rj_
lrly ' " = "
By a similar process we find
dt* ' m + m' p' 3 ' dt a " m+m' p' 3 dt* ~ m+m' p' 3
The identity of form of these equations shews that the paths
which m and m' describe about their centre of gravity are
similar; and as the form is the same as that of the equations
(138), it follows that the paths are conies, of which the foci are
in the centre of gravity of the particles.
367.] There is another important problem of the same kind,
the differential equations of which it is desirable to insert.
It is required to calculate the motion of m relatively to M,
when M and m are acted on by another particle m' } the law of
attraction of all three being that of gravitation.
Let (x } y, z) } (x' } y', zf] be the positions of m and of m' relatively
to M at the time t ; and let r and / be the distances of m and
of m' from M at the same time : and, to fix our thoughts, let us
suppose m' to be farther from M than m : then by virtue of the
principle of Article 332, the velocity-increment of M due to the
attraction of m and of m' is to be impressed on M in a direction
the opposite of that along which the attractions of m and m' act ;
therefore, for the ^-component of the velocity-increment of m,
we have
d'x _ mx m'x' Ma? m'(x'x}
dt* ~~~ r s ~~ /* r 3
(m + M)x m'x' m'(x'-
496 CURVILINEAR MOTION [368.
similarly,
fry _ __ (m + *i}y _ </ ^-
**" '' ' *'* {(^-
m'z 1
= R
.
'
Let
m'(xx'+yy'+zz'}
and with similar values for (-^-) and (^-)*
So that the equations become
d*x
' -4_ i JJ _L i i n
dt* + r s " + \X) ~ '
(162)
R is called the disturbing function, because it alone involves m' y
which is the mass of the body which disturbs the otherwise
conic path of m relatively to M.
The problem is manifestly that of the moon moving about
the earth, the sun being 1 the disturbing body : or of a planet
moving about the sun, another planet being the disturbing
body.
SECTION 4. The curvilinear motion of a particle in a
resisting medium.
368.] When a particle describes a curvilinear path in a re-
sisting medium, the momentum of the molecules of the medium,
which they have on account of their displacement and on
account of the particle passing amongst them, is withdrawn
from the moving particle, and from it in the line in which it
moves : the medium therefore has no effect in diminishing the
velocity of the particle in the line which is normal to its path ;
and the loss takes place along the tangent to the curvilinear
368.] IN A RESISTING MEDIUM. 497
path : we proceed to consider the circumstances of a particle
moving in such a resisting medium.
Let us consider the moving particle to be spherical in form,
so that an equal surface is presented to the medium, whatever
is the line in which the particle moves : and to take the general
case, whatever is the law of the resistance of the medium, let R
represent the velocity-increment (or decrement) which the re-
sisting medium withdraws from the velocity of the particle m in
the line of its motion, that is, along the tangent of its curvi-
linear path in an unit of time ; then if x, Y, z are the three
impressed velocity-increments, the equations of motion, referred
to three rectangular axes, of a particle moving in space, are
dx
_
~
dy
**
R *' >
dz
since -r-> -|-> -y- are the direction-cosines of the tangent of
ds as as
the curvilinear path, that is, of the line of action of R : now
multiplying these severally by dx, dy, and dz, and adding, we
have
dx d*x + dyd*y + dz d*z
y y s=x.dx+idy+zdz Rdk; (164)
ctt
let v = the velocity, when i = ; and using the symbols indi-
cative of definite integration according to the following form,
so that the limits maybe those corresponding to t=t and to
tf=0, we have
v 2 v a /"* /**
---!- = / (Kdx + xdy + zdz)- I nds. (165)
2 . - V| JQ
Also because
d*8 __ dx d*x dy d*y dz d*z f
~di*~'ds Ttt* + ~ds Hi** ds~di*'
d*s dx dy dz
.'. -rr = X^ +Y-f +Z^- E; (166)
dt* ds ds ds
that is, the expressed velocity-increment along the curve is that
due to the impressed forces less that due to the resistance.
Also if P is the resultant .of the impressed velocity-incre-
ments; then
PRICE, VOL. in. 3 s
498 CURVILINEAR MOTION [369-
, _
W~ dsdP
- ; -()+<+>. ' (167)
if p is the radius of absolute curvature of the path ; that is, the
resultant of the impressed velocity-increments consists of two
components, the lines of action of which are at right angles to
each other ; and of which one acts along the principal normal
V*
to the curvilinear path, and is equal to > and the other acts
along the tangent and is the sum of the tangential expressed
velocity-increment and the resistance. This resolution is there-
fore the tangential and the normal one j and if T = the tan-
gential impressed velocity-increment, and N = the normal im-
pressed velocity-increment,
d*s &
369.] If the motion is wholly in one plane we may take that
plane to be the plane of (x } y] ; and if we resolve along the rect-
angular axes of x and y, the equations of motion are
d*x dx d*y dy
3F = X - E S' S^ = Y - E S' < 169 >
And if we take the tangential and normal components, we have
d*s v*
^ = T-E, _ = N ; (170)
or, ^ = X^+Y^-B, ^1 = 3 4_ Y ^ ; (in)
dt* ds ds p ds ds
and IF ~ IF = J (* dx + Y ^)~/ * ds ( 172 )
I _ -I) \ /7n / -t ry n\
&)as. (I'd)
If the motion is referred to a system of polar coordinates, and
if P and Q are the radial and transversal components of the
370.] IN A RESISTING MEDIUM. 499
impressed velocity-increments, then because -r- and 5 are re-
spectively the sine and cosine of the angle between r and the
normal to the curve,
d*r d6* dr
_ A* _ lm T> ^^ T> _,.. _ ,
dt* dl* ' ds 1
dr^dei d*0 rdO '
dt~dl +r dt* " Q *~ds"
Similarly by tangential and normal resolution,
d's dr rdO
dr rdO
= Q. P - .
p as ds
The first of these last two being multiplied by ds and inte-
grated gives
y-=jf(p*+qW-a*)j (176)
and these general formulae are sufficient for the solution of all
problems relating to motion in a resisting medium.
370.] A particle moves in a resisting medium under the action
of forces parallel to the axes of x and y : it is required to de-
termine the law of resistance so that a given plane curve may be
described.
Let the impressed forces parallel to the axes of x and y be
x and Y ; and let R be the resistance ; then by (171),
dx dy d*s
R X = *4- Y 5 , >
ds ds dP
dt* ~
d 2 s 1 d
2ds ds
1 d .
~ds~ ~2Ts' ~dT
into which expression t, which has been equicrescent, does not
enter ; we are therefore free to make any other variable equi-
crescent : and the expression will hereby become somewhat less
complicated.
If, according to the law investigated in Art. 267, the resist-
ance varies as the density and as the square of the velocity, and
if the density also varies, and the law of its variation is to be
35 a
500 CURVILINEAR MOTION [37*.
discovered so that a given curve may be described, then if s
represents the varying density, B, = s x s ;
. * . s = = ? ? r~ lofif T (178)
a p -^.dy^dx 2 ds ds
Ex. 1. A particle describes a parabola under the action of a
constant force parallel to its principal axis : it is required to
determine the law of resistance.
Let the equation to the parabola be y" 1 = 4 ax ; so that
dy _dx _ ds _ ds
la" y ~~ (y a +4a)*~ (4# + 4a s )*'
2 (x + a)?
p =
In this case Y = 0, x = a constant = (say) ; therefore by (177),
^ dx , d
that is, in vacuo only does a particle moving under the action of
a constant force parallel to the axis of x describe a parabola.
Ex. 2. A particle moves in a circle under the action of a con-
stant force in parallel lines, and the resistance of the medium
varies as the density and the square of the velocity : it is re-
quired to determine the law of variation of the density of the
medium.
Let the equation to the circle be a? a +y* = a* ; therefore
dx _ dy _ ds
y x a '
and let the line of action of the constant force be parallel to the
axis of y : so that x = 0, Y = a constant = k (say) ;
3#
therefore from (178), s =
2ay
371.] A particle moves in a resisting medium under the action
of a central force P : it is required to find the law of resistance,
so that a given curve may be described.
From (175), if Q = 0, we have
d's _ dr ds* rdd _ dr
dp-*~fa~ ~dt^ ' Pp ~^" : ~ P ^ ;
rdr , rdO p
since p = r- > and r = - ;
dp ds r
d*s Id/ <&\ ,. -^
' dr>=-*n*(**T^ (179)
dr \ d f dr\
.'. R = P -j- + = j~ ( P P 3- )
ds 2 d* \ * dp'
37 1 -] IN A RESISTING MEDIUM. 501
E * '(pp.*); (m)
2jD a ds > * dp'
< 182 >
whereby, if the law of resistance and the equation of the curve
are given, the central force P may be found.
Again, if the resistance varies as the density and as the
square of the velocity, and if the density also varies, and the
law of its variation is to be investigated, so that a given curve
may be described, then if s represents the density, R = s x V ;
B, 1 d
and if the density of the medium is given, and the central force
is to be discovered, then from this last equation we have
**F*fT* J
where Ti* is a constant introduced in integration.
And (184) may be put under the following form :
1 1 du>
r' " p* ~ *" ~dd^ >
2 $> ,/ , d *
p n , 3 WN
r = 2 ( u + -sari
3 * '
.: p*^ (+._)*-/**. (185)
Ex. 1 . A particle moves in the circumference of a circle under
the attraction of a central force whose origin is a point in the
circumference, and the law of which varies as the nth power of
the distance : it is required to determine the law of the density
and the resistance of the medium.
Let the radius of the circle be a, and let the pole be at the
centre of force, then
r j = 2 op, and P = pr n ; also rdr = (r 2
therefore from (181),
+ * dr
-
Sa
502
MOTION OF A PROJECTILE
[372.
and from (183), . = -
(187)
v '
It appears therefore that if n = 5, that is, if the central force
varies inversely as the fifth power of the distance, R, = and
s = ; that is, the particle must move in vacuo ; also if r == 2 a,
whatever n is, the resistance and the density vanish.
372.] To determine the motion of a projectile under the
action of gravity in a medium of which the resistance varies
directly as the velocity and of which the density is uniform.
Let the velocity of projection be u, and let the point of pro-
jection be taken as the origin : let the horizontal plane through
it be that of (x,y) and let the axis of z be measured in a direction
contrary to that of the action of gravity : let the resistance be
ds
Tc -^r > of which the components along the coordinate axes are
dx , dy - dz
3 , k
^> k
dt
dt
so that the equations of motion are
dz
(188)
from the first and second of which we have
dyd^xdxd^y = 0;
d*x _ d*y m
dx " dy '
dx . dy
log = log -f;
tv \f
y
V
(189)
which is the equation to a plane passing through the axis of z,
in which therefore the motion of the particle takes place.
Since then the motion takes place in one plane, let us assume
that to be the plane of (x, y) ; and let the point of projection be
the origin, and let the horizontal line through the origin be the
axis of x ; and let the angle between the line of projection and the
axis ofa? = a: and let the axis ofy be taken in a direction the
opposite of that of gravity : so that the equations of motion are
dx
**^ fli
dt
(190)
373-] IN A RESISTING MEDIUM. 503
from the former of which we have by definite integration
rr wcosa = &r;
at
dx , ucosafcx
= dt. log
j tfC/f AV/Ci 'f J
u cos aKX u cos a
... # _ ^ cos a p_ g _ t< ^ ^ 192 )
the limits of integration being the values corresponding to t = t
and to =0 : and from the latter of (190) we have
du
-~ u sin a = gtky,
at
-~ +ky = wsino gt; (193)
/ d , \"~* r
= e~ M I e M (u sin agf) dt
( 194 )
and eliminating ^ by means of (192) and (194) we have
\- -^-logfl -- "): (195)
* & '
j- -- - -- --
POQB a . J* & v u COS a
which is the equation to the path of the projectile.
373.] If we expand the logarithmic term in the preceding
expression, the equation becomes
gx a f kx k*x* k 3 x*
y - \JQfo JJ _1_ " _ _ " 1 __ L. _ I _ I
ku cos a k* I u cos a 2 w 2 (cos a) 2 3 u 3 (cos a) 3
of which series all terms, except the first two, contain k (the co-
efficient of resistance) ; and if k = 0, the equation is that to the
parabolic path which is the trajectory of a projectile in vacuo,
see Article 350 ; and thus the terms on the right-hand side of
(196), after the first two, express the excess of the ordinate of
the parabola described in vacuo over the ordinate of the curve
whose equation is (196) : the trajectory (196) therefore is of a
form somewhat parabolic, but the curve recedes from the direc-
trix further than a parabola.
504 MOTION OP A PROJECTILE [374-
Also by reason of (191), -3- = if x = r ', for this value
of x therefore the horizontal velocity vanishes, and the projectile
moves in a vertical path ; and therefore a vertical line, at a
distance = r from the point of projection, is an asymptote
to the curve. ,
Also from (195) -jj- == 0, that is, the projectile comes to its
wiK
highest point when
= tana+ 7 -^ -j- f 9 - =-, (197)
KU cos a K (u cos a kx)
u 3 sin a cos a u sin a a ,
viz. when x , -. > y = j H -7- log
g + Kusma K A 1
Also from (197), if x = <x>, ~- = tan a+ ^ - : that is,
ax fcucosa
the curve being produced backwards through the origin con-
tinually approaches to a certain definite angle with the axis of x.
374.] But the most important application of this theory is
that of gunnery, in which the motion takes place in air, the
resistance of which (at least approximately) varies as the square
of the velocity; and thus the following problem offers itself:
A particle, or a spherical homogeneous ball, acted on by gra-
vity, is projected with a given velocity u in a given line, in a
medium of which the resistance varies as the square of the velo-
city and the density is uniform : it is required to determine the
circumstances of motion.
It may be shewn, by a process similar to that at the com-
mencement of Art. 372, that the motion takes place in one
plane ; and therefore we may assume that plane to be the plane
of (*,*)
Let # = the velocity of projection; and let a horizontal and
a vertical line through the point of projection be the axes re-
spectively of x and y : let the axis of y be taken upwards : let a
be the angle between the axis of x and the line of projection :
ds"*
and let the resistance of the medium be k -^ ; then, as the line
0V
of action of this is the tangent of the curve, its components are
7 ds dx , , ds dy
* ~j7 -j7 ' and * 37 -57 '>
dt dt dt dt
and as k } see Art. 294, varies directly as the density of the me-
dium, as the surface which the moving ball presents to the
medium, and inversely as the mass of the ball, then, as the ball
374-] IN A RESISTING MEDIUM. 505
is spherical and the density is uniform, Jc is constant. Thus the
equations of motion are
d*x ,dsdx
' TtTt '
d*y , da dy
~W '' ~ 9 ~ TtlTt'
Integrating the first of these, and taking the limits correspond-
ing to t t and to t = 0, we have
d*x dx
dt* . ds dt 7
^^^^^ _ / ___ li .1 1* ,^^^____ - ,- /<?
/V -L > . , AUti ^-^^"^^ /vo
dx dt u cos a
dt dx
.'. -rr = u cos aer* 1 . (199)
ctt
Again from (198), and transforming the equations so that t
is not equicrescent, we have
x d*xdy
-i ?.-
therefore from (199),
-i = ____
dx 1 u* (cos a)*
(201)
..
d# \ dr*' u 3 (cos a) 2
and integrating, and taking the limits corresponding to t =
and to t = 0, we have
tan a sec a log (tana + sec a) = , ( 9 r- 2 (e tkt 1); (202)
tcu ^cos ctj
and for convenience let us substitute
tanaseca + log(tana + seca)+^|^ = c; (203)
so that (202) becomes
1 d fdy\
(205)
PRICE, VOL. III. 3 T
506 MOTION OF A PROJECTILE [375-
ty j dy
dx ** -. = Jcdy; (206)
from which equations, were it possible to integrate them, x and
y might be found in terms of -/- ; and if -j- were eliminated
dx dx
from the two integrals, the resulting equation in terms of x and
y would be that of the required trajectory.
But as the equations are not integrable, we must deduce
from them in their present forms such results as are possible.
Equating the values of e** which are given in equations (199)
and (204), we have ^ y
d.f-
dt= -- ! __ 2f - : (207)
itafi dy , dy\* . f dy
\ c -( l + 3- lo z(r*
whence might the time be found in terms of -j- : also squaring
(tec
(205) and (206), and dividing by the square of (207), we have
4!
+ *
which equation gives the velocity in terms of -~
(204) is the intrinsic equation to the path of the projectile.
375.] When =-,
becomes -;
dx > dx*' *\dx ^ dx*' ( dx*
so that (205) becomes ^
d "ir
kdx= -- ; (209)
d-y*
dx*
dx>
.-. k(x a) = -r- 1 where a is an arbitrary constant:
dv
therefore if ~=x>, x = a = a constant; that is, the line,
whose equation is x a, is an asymptote to the curve. Also
under the same supposition from (208), we have
IN A RESISTING MEDIUM. 507
(vel.)' = j (210)
that is, the velocity of the projectile, as it falls down the
descending branch of the trajectory, approaches to the constant
limit (f)*.
And at the highest point of the path, when -j- = 0,
ax
Thus the path of the projectile is a curve of the form deli-
neated in fig. 124, where OA = a.
376.] The case however frequently occurs in practice wherein
the angle of projection is very small ; and where the ball rises
very little above the horizontal line; and therefore -j- being
very small, we may throughout the path on the upper side of
the axis of x neglect powers of -j- higher than the first. In
this case then
ds = dx; .'. s = x :
so that (200) becomes
_
dx u* (cos a) a
ate. fro..)-
and expanding in a series the last term, we have
y = .Etana- f ' . - Q ***' ..-...; (212)
2 u* (cos a) 2 3 u 2 (cos a) s
which equation, if the terms involving k are omitted, is that of
a parabola, which is the path of the projectile in vacuo. It
appears therefore that the ordinate of the actual curve is that
of the parabola diminished by a quantity which is the sum of
all the terms of the right-hand member of the last equation
except the first two.
dx
Also from (199), ^7 =
dt
= e kx 1; (213)
which gives the time in terms of the abscissa.
CHAPTER XL
THE FREE MOTION OP PARTICLES, UNDER THE ACTION OF
CENTRAL FORCES.
SECTION 1. General investigations ; determination of laws of force
and other circumstances of motion in given orbits.
377.] A central force is that of which the source of influence
is at a certain point, towards which it attracts or from which it
repels any particle of matter within reach of its action ; and
according as the action of it is attraction or repulsion, so is it
called an attractive or a repulsive force. The forces, whose
effects will be considered, are supposed to be functions of the
distance between their centres and the particle on which they
act, and not to be functions explicitly of either the time or the
velocity ; the case in which the line of motion of the particle is
coincident with that of the action of the force has been con-
sidered in Chapter VIII ; and it remains for us now to discuss
the case in which the line of motion of the particle is inclined
at any angle to the line of action of the central force. The
principles and equations of Chapter IX are sufficient for the
inquiry, and have indeed been applied to the subject in Art.
359, and others, in the form of rectangular coordinate reso-
lution : but as the method of resolution into radial and trans-
versal components is more convenient, and as formulae different
to any heretofore employed are deducible from them ; and
moreover as nature presents to us more instances of this than
of any other kind of dynamical action, it is desirable to devote
a separate Chapter to the inquiry : and in the course of it we
shall take occasion to exhibit the first elements of celestial
mechanics in the form of the simple elliptic orbit which a
planet undisturbed would describe about its primary.
378.] Let m be the mass of the moving particle, and let
(z, y, z) be its position at the time t ; let the centre of force,
which we suppose to be fixed, be the origin of coordinates : and
let P represent the central force ; that is, the impressed velocity-
increment in an unit of time : let r be the distance of m from
379'] CENTRAL FORCES. 509
the centre of force at the time i : then we suppose P to be a
function of r ; let P be attractive, so that the equations of motion
in terms of velocity-increments are
Now in the first place I shall shew that the moving particle is
always in one and the same plane, and that a plane which passes
through the centre of force.
From the second and third of (1) we have
d*z d*y
9 &-*&=*''
. . dz dy
.-. integrating J^-fJ
. ., , dx dz ,
similarly jr^-,^**,; (2)
dy dx
*3i *#=**''*
therefore multiplying these equations severally by x t y, z, and
adding, fa+hy+hz = ; (3)
which is the equation to a plane passing through the origin,
which is the centre of force, and in which therefore m always
is. The orbit therefore of m, as the trajectory is called, is a
plane curve.
This fact too is evident by the principle of sufficient reason :
because every reason which can be urged why m should move
out of the plane, which contains the centre and two consecutive
points of the path, on one side may be shewn to be equally
valid why it should leave the plane towards the other side.
We may therefore, without loss of generality, suppose the
plane in which m moves to be that of (x, y}.
379.] Let P represent the central force and be attractive ; let
the centre of force be the origin, (x, y) the position of m at the
time t; r = the distance of m from the origin : then the equa-
tions of motion in terms of velocity-increments are
d*x _ P# d*y py
H& ~~ "7' dp = ~T'
multiplying the former by y, and the latter by x, and subtract-
ing, we have d*u d'x
510 CENTRAL FORCES. [379-
whence, adding and subtracting -=- -^ > and integrating, we have
dt clt
dy dx
*-'*=*> (6)
where h is an undetermined constant.
Now by Differential Calculus, Art. 219, (44), if p is the per-
pendicular from the origin on the tangent,
xdy ydx pds',
therefore from (6), -=- = -: (7)
dt p
that is, the velocity at any point of the orbit varies inversely as
the perpendicular on the tangent at that point from the centre
of force.
Also since
x = rcosd, dx = drcosQ r sin Qdd,
, .
y = r sin Q ; dy = dr sin 0+ r cos 6 dd ;
r*dQ; (9)
therefore from (6), r a d6 hdt. (10)
Now r a dO is twice the sectorial area which the radius-vector
of m describes in the time dt, and as it is proportional to
dt by (10), we infer that the sectorial areas described by the
radius-vector of m are proportional to the times of describing
them : or in other and equivalent words, equal sectorial areas
are described in equal times. Hence also it appears that
Ti = twice the sectorial area described in one unit of time. (11)
Let us give a geometrical proof and interpretation of the
theorems (7) and (10). Suppose s to be the centre of force,
fig. 125, P to be the position of m at the time t, and the element
PQ to be its path-element in the time dt ; and let t be equicres-
cent; and let SY =. p be the perpendicular from s on PQ pro-
duced. Now let PQ be produced to n' so that QR'= PQ ; then if
no force acted, m would at the end of the second dt be at E'J
but suppose, when m is at Q, the central force to act impulsively
and to draw m over a distance Qr in the time dt; then at the
end of dt, if R'R is equal and parallel to Qr, and the parallelo-
gram rn' is completed, m is at R ; by a similar process and con-
struction it may be shewn that m is at T at the end of the third
dt, and so on : now since PQ = QR', therefore the triangles SPQ,
SQR' are equal, and because SRQ, SR'Q are on the same base
380.] CENTRAL FORCES. 511
SQ and between the same parallels, SRQ = SR'Q; therefore
SPQ = SRQ: similarly it may be shewn that SRQ = STR = SUT,
= . . . ; and thus the sectorial triangles which correspond to
equal dt's are equal. And the same result is true in the limit,
when the polygon which is drawn in the figure becomes a con-
tinuous curve, and the central force acts continuously ; and thus
under the action of a central force equal sectorial areas are
described in equal times.
Also let h = twice the sectorial area described in an unit of
time : let PQ = ds, SP = r, TSQ = d0, SY p : then the triangle
described in dt units of time is PSQ ; and the area of PSQ= -- >
see Integral Calculus, Art. 226, (12) ; and also is equal to
1 pds
-PQ x SY = ^-;
,-. l = f*~*:p~;
(It clt
and therefore as equal sectorial areas are described in equal
times, so does the velocity vary inversely as the perpendicular
from the centre of force on the tangent.
Also since -5- = , it appears that the angular velocity
varies inversely as the square of the radius-vector at the point.
Also if t is the time during which the particle passes from a
point in its orbit corresponding to 6 to another point correspond-
ing to 6, then since
, .. (12)
ft n, Jg
whereby in a given orbit the time may be found in terms of the
angle through which the radius-vector of the particle has moved.
The means of determining h in a given orbit and under a given
absolute force will be shewn hereafter.
Thus also if the orbit is a closed curve whose area is A, and if
T is the whole time, or the periodic time, as it is called, in which
m has described it, o .
380.] Again, multiplying the first of (4) by 2dx, and the
second by 2 dy, and adding, we have
' , .
512 CENTRAL FORCES.
but since s'+y = r" ; .*. xdx+ydy = rdr ;
Let v be the velocity at a given point, at which r = R (say) ;
then integrating (15), we have
(16)
. . (velocity) 2 = v 2 2 / r <fr. (17)
^R
Let v be the velocity at the point to which r corresponds :
and let m be the mass of the moving particle : then
= m I vdr-, (18)
2
which is the equation of vis viva and of work, and is the form
which (65), Art. 325, takes when there is no transversal force.
Thus if P is a function of the distance of m from the centre of
force, so that the right-hand member admits of integration,
this equation shews that the increase of vis viva of the particle
in passing from one point to another depends on the coordinates
of the two points, and not on the path which the particle has
described in the passage : that is, the change in vis viva, or the
increase of work, depends only on the distance through which
the force has acted in its own line of action.
From (17) it appears that the velocity is the same at all
points which are equally distant from the centre ; for if r = R,
the velocity = v : and thus if the orbit is a re-entering curve,
the particle always in its successive revolutions passes through
the same point with the same velocity.
Now equating the values of the velocity in (7) and (16), we
therefore differentiating, we have
_ 2/fc 2
- dp =
.. -
p 3 dr
A j T 1 j du
And if r = - > dr -- :
u U*
i J 'i
and Differential Calculus, Art. 270, (19), = a + ^-- ;
381.] CENTRAL FORCES, 513
( d"u )
and (20) becomes P = fru* 1 -j-- + ut ; (21)
which is the result already found by a different process in
Art. 312.
Thus from either (20) or (21) the law of central force may
be determined, under the action of which a particle moves in a
given curve. And from (12) or (13) the time, which is occupied
by its passage through a given arc, or through the whole curve,
if the curve is closed, may be found. And from (7) or (17) the
velocity at any point in the orbit may be determined.
Also because that part of the radius-vector at any point of a
curve referred to polar coordinates which is intercepted by the
circle of curvature, or the chord of the circle of curvature, as it is
called, see Differential Calculus, Art. 301, is equal to 2p-j-,
equation (20) gives
h* dr
- = (vel.) 2 = JP -=-
j a dp
chord of circle of curvature
= 2Px -- -; (22)
and comparing this with (32), Art. 268, it appears that a par-
ticle at rest on the curve, and moving from it towards the centre
of force under the action of the force continuing constant,
acquires the velocity which the particle has in its curvilinear
course, when it has moved through one-fourth of the chord of
the circle of curvature.
Hence if the orbit is a circle, having the centre of force in the
centre, and R, P, v are respectively the radius, central force, and
velocity, v 2 = P x R.
A point in an orbit at which the curve is at right angles to
the radius-vector is called an apse ; the radius-vector at an apse
is called an aspidal distance ; and the angle between two con-
secutive apsidal distances is called an apsidal angle of the orbit.
The analytical character of an apse is manifestly
^ = 0, or =00. (23)
381.] Examples illustrative of the preceding equations :
Ex. 1. It is required to find the law of force, the velocity,
PRICE, VOL. in. 3 u
514 CENTRAL FORCES. [381.
and the periodic time, in an elliptic orbit, when the centre of
force is in the focus.
Let the equation to the ellipse, the focus being the pole, be
a(\e*} l+ecos0
f x *_ 11 __- f94^
" l + <?cos0' " a(l-e 2 ) '
du _ esinO d*u _ ecosd
p == 7i,*u a \
^(IV* ' 0(1 e 2 )
= *irb)F ; ': < 25 >
and the force varies inversely as the square of the distance, and
is attractive > as appears by its sign and by the convention as to
signs which was assumed in Art. 377. Let \i. be the absolute
force of the central force, then
(26)
so that h is given in terms of the absolute force, which is the
mass of the attracting body, or the sum of the masses of the
attracting and attracted bodies, if the motion is relative, and of
the quantities which determine the magnitude of the orbit.
1 du* 2aul
Also since - = u s + -^ = -. :
h* u(2aul) ._,
.-. (the velocity) 2 = = ^ t- (27)
p a
Hence if s is the focus in which the force is, and H is the
TT p
other focus, (the velocity) 2 varies as
If T is the periodic time in the elliptic orbit, then, as the area
of the ellipse = -nab = ira a (lerf, by (13),
27m 2 (l-e 2 )* 2 IT
T = - *- = -a?; (28)
{ M a(l-O}* ^
thus the periodic time varies as the square root of the cube
(as the sesquiplicate power) of the major axis. As these results
will come under consideration hereafter, it is unnecessary now to
comment on them.
Ex. 2. To find the law of force and the velocity in a parabola,
the focus of which is the centre of force.
381.] CENTRAL FORCES. 515
Let the equation to the parabola be
20
T
_ , (29)
1+COS0'
where the line joining the focus and the vertex is the prime
radius. *
-* <>
thus the central force varies inversely as the square of the dis-
tance, and is attractive. Let p the absolute force ;
(the velocity)' = -1 =
(31)
(32)
Also let t be the time during which the particle moves from a
point corresponding to to another point corresponding to O n ;
then by (12),
_(2)* r* dO
~7uf A>
( S....
(sec-)^
And this value for t may be expressed in terms of r n and r ,
the focal radii vectores corresponding to O n and to , and of
the chord c (say) which joins their extremities.
a
For the sake of more convenient symbols, let tan = t n ,
Q
tan = t 9 ; so that (33) becomes
^ P- ' ( *
9 /r 3 Tt J 2 i J J i t -i
t& a \ 2 // . \/, , t-tt "T^H^o + g o \
" \-|4 / V ^V 3 A
By a substitution due to Gauss, let
1 4- -^-r- 2 - = ?" ;
516 CENTKAL FORCES. [382.
(">
But if c is the chord joining the extremities of r n and r ,
c a r n a -2r n r cos(0 n ) + r 2
= (r n cos O n r cos ) 2 + (r n sin n r sin ) 2 .
Also r n = a . (1 + ^), r = a (1 + *),
1 V
and cos # = - - ^- j sin 0., =
and similar values are true for cos and sin ; therefore
similarly, r n + r c = 2 a | ?; M g
therefore substituting in (34),
Cr. + r.-c)!}. (35)
.
This theorem is generally known by the name of Lambert's
Theorem.
Hence the time through an arc of a parabolic orbit bounded
by a focal chord varies as (the chord).
Ex. 3. If the equation of a hyperbola, of which the focus is
the pole, is a(e* 1)
,, I* 1 , , ., ., fi(2a+l)
then P = . ; (velocity) 8 = - =
a(e t 1) r a a
382.] -^ x - 4< -^ particle moves in an ellipse about a centre
of force in the centre : it is required to find the law of force,
the velocity at any point of the orbit, and the periodic time.
The equation to the ellipse is
(V=' ; (36)
382.] CENTRAL FORCES. 517
du* , 1 IN,
K ^ + ^ = (F-?)f (cosfl) - (sln ^'- < 38 >
But from (36), i 1
u * ~ Ta u * -- 1
(cos 0) = - - j- , (sin 0) = - - i- ;
therefore substituting these values in (38), and also from (37),
we shall find h* h*
m ~*'
thus the force varies directly as the distance, and is attractive :
and i{ n = the absolute force, h* pa'd*. Also
(the velocity) 2 = n(a*+6* r*) (40)
if / is the radius conjugate to r.
And if T = the periodic time, by (13),
2 irad 2 TT
/..,\
that is, the periodic time is independent of the magnitude of the
ellipse, and depends only on the absolute central force.
And the time in which the particle passes through an arc
which subtends a given angle at the centre may thus be found.
Let the arc begin at the extremity of the major axis ; then if
t = the time required, from (12) we have
06 e dO
06 f e
= J
. .
_ _
o a 2 (sin 0) 4 + 6" (cos 6}*
1
and thus if Q - > t = r ; where t is one-fourth of the peri-
2
odic time : and thus the whole time is the same as that given
in (41).
Ex. 5. In the hyperbola described by a particle under the
action of a central force in its centre, and of which the equa-
tion is (cos0) a (sin 0) 1
~^~~ -- ~ *'
r = ~r = -nr; (43)
518 CENTRAL FORCES. [383.
that is, the central force varies directly as the distance, and is
repulsive : also
(velocity) 8 = /u, (r 2 a 1 + b*} ; (44)
and the time from the extremity of the transverse axis
1 . 6+atsm.d ,. K .
383.] Ex. 6. Let the particle move in a circle ; and
(1) Let the centre of force be in the centre : let a = the
radius : then the equation to the circle is
1 du _ d'u _ _ h*
= a' T0~d&- = ^ J
/t 2
(the velocity) 2 = = v*(say);
H
therefore the central force is constant, but varies inversely as
the cube of the radius as we pass from one circle to another :
also the velocity is constant, and varies inversely as the radius :
and if T = the periodic time,
_ _
h
v a
Also P = ; now as v 2 is the velocity and a is the radius of
Ot
V 2
the circle, , see Art. 326, is the centrifugal force : the cen-
Q/
tral force is therefore equal to the centrifugal force in the circle ;
the central force, that is, draws the particle towards the centre
over a space equal to that by which the centrifugal force (so to
speak) removes it from the centre : and as the velocity is con-
stant, no part of the central force acts either to increase or to
diminish the velocity in the circular path.
(2) Let the centre of force be in the circumference of the
circle : and let the equation be
r=2acosd; .-. 2 aw = sec 0;
du d*u
2<*jQ= sec0tan0; 2a jfi = sec (tan 0) a -f (sec 0) 3 ;
' w = Sa * n3 ~ u '>
^-^; (46)
384-] CENTRAL FORCES. 519
that is, the force varies inversely as the fifth power of the dis-
tance, and is attractive ; and if /i=the absolute force, /x= 8 3 A* ;
.*. h* = - ; and (the velocity)* = -^~
And if T = the periodic time,
T= --^T' ( 4? )
And if t is the time of the motion of the particle from the ex-
tremity of the diameter to the point corresponding to 6, then
(3) Let the centre of force be at any point within or without
the circle : and suppose it to be at s, see fig. 126 ; and let c be
the centre of the circle, sc = c y CA = a, SP = r, SY = p : then
because sc 2 = SP 2 2SP.CP cosspc + cp 2 ;
.-. c 3 = r 2 2ap + a*; (48)
dr _ a
dp ~ r'
therefore, from (20), p = ( / + ^ g , )3 (49)
384.]| Ex. 7. It is required to find the law offeree, the velo-
city, and the periodic time in (1) the Lemniscata of Bernoulli,
(2) the Cardioid.
(1) r 2 = 2 cos20;
.-. a' U* = sec 2 ;
du
a u -j2 = sec 2 6 tan 2 6 = a* u* tan 2 ;
aO
du
' -j^ = wtan 20;
ttU
-^ = ~ tan 2 0+ 2u (sec 2 0) 2 ;
, P = ^ ; (50)
and thus the force varies inversely as the seventh power of
the distance, and is attractive : and if p = the absolute force,
H = 3^ 2 a* ; and thus
(the velocity) 2 = -^--
If t= the time from an apse, then from (12),
520 CENTRAL FORCES. [385.
I = (-)** / cos 2 dO
= (!!)* T sin2 ^ < 51 )
a*
~2~
(2) The equation to the Cardioid is
f*
o 1 4
and therefore the time to the node = (\ ; and the time of
3 i '
describing one loop = (-) *.
2
r = a (1 -f cos 0) = 2 (cos -) ;
2au ^= (sec-) ;
du f 0y
' a 'd0~^ seC 2^ Ln 2 ~~ a> in 2*
du
2=T = (tanf) f + |(Becf)' = 3 2 -^;
(52)
r*
and thus the force is attractive and varies inversely as the
fourth power of the distance ; and if /x is the absolute force,
fx = 3 ah* ; and 2 u
(the velocity) 2 = ^
o T
If =the time from the apse to the point corresponding to
3*
i*fl*30 sin
,__.
and thus if = 7r, the time from the apse to the pole= - TT ;
and if T = the periodic time,
T = (3/i<Z 5 )*7T.
385.] It is required to find the law of force in (1) the curve
whose equation is r = 2acosn0, (2) the curve whose equation
20
is r =
1 ecos0
(1) The equation of the first curve in terms of u is
2au = sec0;
385.] CENTBAL FORCES. 521
du
u
2a -f = n<l sec n (*& n^Y + n<t ( scc n d Y
( 54 )
and thus the force varies partly as the inverse fifth power, and
partly as the inverse cube, of the distance.
And if t = the time in which the particle moves from the
point corresponding to = 0, to that corresponding to 0,
It
2 a* , sin
As to (54) it is to be observed that the second term, of the
equivalent of P disappears, if n = 1 ; and that in this case the
central force varies as the inverse fifth power of the distance.
Now if = l, the equation to the orbit is/ = 2cos0; that
is, the orbit is a circle, of which the pole is on the circum-
ference and the prime radius passes through the centre: and
when 0=0, r = 2 a in both the circle and the given curve. A
process of tracing such a curve, and of representing the motion
of a particle on it, is hereby suggested to us. In fig. 127
take a line sx for a prime radius; on it take SCA = 2 a; and
on SA as a diameter describe a semicircle SQA: then since
/ = 2 a, when = 0, in the equation to the orbit, the point A is
common to the circle and to the orbit : let us suppose n to be
less than unity : and let AP be the curve of the orbit : on it
take any point P ; join SP : then as PSX = 0, SP = 2a cos W.PSX ;
let the angle PSA'= W.PSX, and make SA'= SA = 2a; on SA'
as a diameter describe a semicircle; then the semicircle will
pass through P, because by the property of the semicircle
SP = SA'COSPSA'= 2acosn0, and thus SP is the same for both
the semicircle and the curve of the orbit. In the same manner
may every point P be shewn to be on a semicircle, the diameter
of which has a varying position, and as PSA = 0, PSA'= nO,
therefore ASA'= (1 ri)0 : therefore while P has moved over an
arc of the orbit subtending an angle at s, SA' has moved
through an angle (1 n)0: and therefore the ratio of the an-
gular velocity of P, to that of the revolving diameter, is as
PRICE, VOL. III. 3 X
522 CENTRAL FORCES. [385.
1 : l; and, according to our system, SA' revolves in the
same direction as p moves. If however n is greater than 1, SA'
revolves in a direction the opposite of that in which SP revolves.
In either case the moving particle m may be represented as
moving in a circle, the diameter of which revolves about the
pole s with an angular velocity bearing a constant ratio to
the angular velocity of m ; and the orbit of m is for this reason
called a revolving circle.
(2) Again in the second curve which is given,
2au = 1 ecosnO ',
d*u
To* = cos '
fan* 1 (!)
.. P = h ', (55)
2a r* r*
therefore the central force is compounded of two different forces,
of which one varies inversely as the square of the distance, and
the other varies inversely as the cube of the distance.
Now the second term of the right-hand member of (55) dis-
appears if n = 1, and in that case the central force varies
inversely as the square of the distance ; but if n = 1, the equa-
tion to the orbit becomes
r- 2 " , (56)
1 ecos9'
the equation to a conic, of which the focus is the pole, and the
prime radius is the principal axis : and to fix our thoughts I
will suppose e less than unity, so that the conic is an ellipse :
now if = 0, r = > in both the conic and the orbit curve :
\e
so that if sx, fig. 128, is a prime radius, and the line SA is taken
2a
on it such that SA = > then A is a point common to the
1 e
conic and the orbit. And to fix our thoughts, let us also sup-
pose n to be less than unity; and let us suppose the curve APQ
to be that of the orbit, of which P is any point, and PSX = 6:
take an angle PSA'= nQ, and let SA'= SA, and produce A'S to B',
so that SB'= ; and on A'B' as a major axis, with s as the
focus, describe an ellipse; then the radius- vector of the ellipse
corresponding to the angle PSA'
2a 2a
1 ecosPSA'" 1 ecosn$'
386.] CENTRAL FORCES. 523
therefore the radius- vector of the ellipse is equal to that of the
orbit, and therefore the point P which is on the orbit is also on
the ellipse. In a similar way it may be shewn that every point
on the orbit is on an ellipse, the major axis of which has a
varying position: and as PSA = 0, PSA'= nO, therefore A'SA =
(1 ri)d; that is, as m has passed over an arc subtending an
angle at s, so has the major axis of the ellipse passed through
an angle (1 ri) 0; the angular velocity therefore of the former
is to that of the latter as 1 : 1 n : and SA' revolves in a direc-
tion the same as that of m if n is less than 1, but in an opposite
direction if n is greater than 1. In either case the moving
particle may be represented as moving in an ellipse, the major
axis of which revolves about the centre of force with an angular
velocity bearing a constant ratio to that of the moving particle :
the orbit of m is for this reason called a revolving ellipse.
Also since in the orbit
1 ecosn0 t . du __ nesmnO ^
iV M - f f _ Hill "
2a ad 2 a
now if -JQ = 0, the corresponding point in the orbit is an apse,
and the line drawn from the pole to an apse is the apsidal dis-
tance : therefore the orbit has an apse, whenever sin #0 = 0:
that is, when
~ n ~ n
therefore the angle between two successive apsidal distances
TT
n
386.] By processes similar to those employed above let it be
shewn that in the orbits whose equations are the following, viz.
m r- 2a (2\ r a-
I 1 / - e *o + e -*o ' W
(3) r = -; (4) r = asecn0;
the central force varies inversely as the cube of the distance :
that in the lituus, whose equation is # a u* = 0, the force varies
partly directly as the distance, and partly inversely as the cube
of the distance : and that in the involute of the circle, of which
the equation is r 3 a* +/> 3 ,
p t
3x2
524 CENTRAL FORCES.
387.] As an accurate comprehension of the expressions for P,
the central force, is of great importance for the complete under-
standing of the phaenomena of the action of a central force, it
is desirable to insert the following proof which is founded on
first principles :
Let m be the mass of a particle moving in a certain orbit
under the action of a central force, the impressed velocity-
increment of which along the radius- vector is represented by p :
let P, fig. 129, be the place of m at the time t, and let PQ = ds
be the length-element described by m in the time dt ; and let
QS be the length-element described in the succeeding dt. Now
if no central force acted, m would in dt pass through QB, in
the line PQ produced, instead of passing through QS : but if a
force, whose source is in o, acts on m at Q, and causes it to
pass through QV in the line QO in the time dt; then at the
end of the second dt, m is at s ; QS being the diagonal of the
parallelogram of which QV and QR are two containing and
adjacent sides ; it is our purpose to estimate the effect of the
central force as expressed in the deflexion of m from its recti-
linear path.
We may consider the central force P to be constant during
the time dt of its action on m, whereby it draws m over the
space QV; and therefore by (31), Art. 268,
2.QV = ?dt\ (57)
Let Q n be the radius of curvature of the curve at Q, and let
QN be the projection of QV on it : therefore, as it has been
proved in Art. 303, if p = Qn,
ds 3 = 2pxQN.
Now QV = QNsecoQn
ds* r _ ds 3 dp
~ Tp p ~~
therefore from (57), p = ~ ^L = ^ ^ , (58)
HP pdr p 3 dr
because by reason of (7), -= = : and this expression is the
dt* p 3
same as that before deduced analytically in Art. 380. The pre-
ceding process is nearly identical with that employed by Newton
in Prop. VI, Section 2, Book I, of the Principia.
389.] CENTKAL FORCES. 525
SECTION 2. The determination of orbits, and of their dimensions
and position, when the laws of central force and other circum-
stances of motion are given.
388.] In the previous Section the law of force and other
circumstances of motion have been determined, when the equa-
tion to the orbit and the position of the centre of force have
been given : it is our purpose now to inquire into the converse
problem ; and let it be observed, that for a complete determina-
tion of the orbit, when the law and centre of force are given,
four constants, or what are, by means of the limits of the integral
or otherwise, equivalent to four constants, are required : this
is evident from the form of the differential equations (4), which
are two simultaneous differential equations of the second order,
and the complete integral of each of which requires two con-
stants : or again the equation (21) contains an undetermined
constant h', and being of the second order, two more unde-
termined constants will be introduced during the process of
integration: and one other constant will be required in the
integral of (12), by means of which the time at which m is at
any point of the orbit may be found. The conditions which
will for the most part be given in the following examples are,
(1) the distance from the centre of force of the point where m
is at a given time ; (2) the line in which m is moving at the
time, and the inclination of that line to the corresponding
radius-vector ; and (3) the velocity with which m is moving at
the given time : the time at which all these circumstances are
given is called the epoch ; and in terms of them, the constants,
or the limits of the integrals, can always be expressed.
389.] A particle m is projected with a given velocity, in a
given line, from a given point, and moves under the action of a
central force, which varies directly as the distance and is attrac-
tive : it is required to determine the equation to the orbit, and
the circumstances of motion.
The plane in which m moves is manifestly that passing through
the centre of force and the point of projection, and which con-
tains the line of projection.
Let the centre of force be the origin ; let E = the distance
of the point of projection from the centre of force ; v = the
velocity of projection ; and let us suppose m to be projected from
526 CENTRAL FORCES.
an apse, so that the line of v is perpendicular to R : then
since generally the velocity = - >
v = -; /. h = VR. (59)
B
Let =0, when m is projected with the velocity v: let /n=the
absolute force of the central force : then since the force varies
directly as the distance and is attractive,
p = ,ir = ^. (60)
%
Now from (21) we have
ft ^7/ IL
therefore in this case -j - + u=-; (61)
ad* A*u 3
multiplying by 2 du, and integrating, and taking the limits cor-
responding to t = t and to t = 0, and observing that
(the velocity)' = k* ( + ^!),
. du 3 V a fJi JAR a
wehave _ + ._ Jf= _^ + ^.;
(the velocity) 2 = v a -f /xR 2 -^.
-r'); (63)
and therefore the velocity is the greatest and least, according as
r is the least or the greatest.
And replacing h in (62) by its value from (59), we have
du* 1 n u.
__ I ^/a __ _ _ _ ~ [ ~ .
* " 823 T V 2 '
^ f iv_ /
* 55 s + V* 2VR* ) == V
2udu
therefore integrating, and taking the limits corresponding to
t = t and to t = 0, and assuming that the prime radius coincides
with R, we have
2V a R a W a
sm
__
v 1 /xR 8
(64)
389.] CENTRAL FORCES. 527
and changing to rectangular coordinates, 
