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SCIENTIA
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A TREATISE
ON
INFINITESIMAL CALCULUS;
CONTAINING
DIFFERENTIAL AND INTEGRAL CALCULUS, CALCULUS OF VARIA-
TIONS, APPLICATIONS TO ALGEBRA AND GEOMETRY,
AND ANALYTICAL MECHANICS.
BY
BARTHOLOMEW PRICE, M.A., F.R.S., F.R.A.S.,
=
FELLOW OF PEMBROKE COLLEGE, AND
SEDLEIAN PROFESSOR OF NATURAL PHILOSOPHY, Oxford.
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VOL. IV.
THE DYNAMICS
OF
MATERIAL SYSTEMS.
"Les progrès de la science ne sont vraiment fructueux, que quand ils amènent
aussi le progrès des Traités élémentaires."-CH. DUPIN.
OXFORD:
AT THE UNIVERSITY PRESS.
MDCCCLXII.
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PREFACE.
ALTHOUGH I have entitled the present Volume,
"Dynamics of Material Systems;" yet the investiga-
tions contained in it are far from comprising all
which a complete treatise on that subject requires.
They are indeed almost wholly confined to those par-
ticular systems in which the internal forces, brought
into action, either effectively or potentially, by means
of the external forces, enter in equal and opposite
pairs; so that they disappear in the equations of
motion formed on D'Alembert's principle.
I say
almost wholly, because, in the last Chapter but one
of the Volume, the motion of the particles of an
elastic body is to a certain extent discussed: and
herein the elastic forces, which are internal forces,
do not disappear, but enter as effective forces, the
action of which is determined by Hooke's law, or by
an equivalent assumption of a property of such
matter. In all other cases, in a rigid body, in a
rigid system which is maintained in a state of rela-
tive rest by rigid rods and similar modes of con-
straint, in systems wherein every mutual action of
attraction or repulsion is accompanied by an equal
and opposite reaction, the internal forces disappear
from the equations of motion.
The expediency, nay, almost the necessity, of giving
a 2
iv
PREFACE.
a geometrical image to complicated mechanical laws
demanded the insertion of a preliminary Chapter,
which should contain the required geometrical theo-
rems. The importance of familiarity with the sym-
bols of this Chapter and their symmetrical manipu-
lation, with the linear and angular directions, with
the geometrical forms, which being in tridimensional
space are difficult of imagination, cannot be over-
estimated. A system of notation has also been hereby
obtained, and this is preserved uniformly throughout
the Treatise.
The motion of a system is of course more compli-
cated than that of a single particle, and thus greater
prominence has been given to the distinction between
cinematics and dynamics than was necessary in the
preceding Volume. A force surely cannot be perfectly
apprehended as to its effects on the motion of a sys-
tem, unless the effects have been previously exa-
mined, and, I may say, examined in all their gene-
rality. Hence arises the importance of the Chapter
on cinematics, in which it is shewn that the most
general motion of a rigid system is compounded of,
and may be resolved into, a translation of any par-
ticle and a rotation about an axis which passes
through that particle. The applications of this ana-
lysis of motion in the subsequent parts of the Treatise
are many and various. The Analytical Table of Con-
tents will sufficiently indicate the course of inquiry :
it will manifest the logical sequence of the several
parts; the first formation of the equations of motion
by means of D'Alembert's principle: the theorems de-
duced from particular forms of these equations: the
more general theorems and principles which they in-
volve: the transformation of the equations of rotation
into angular velocities: the consequent geometry of
masses, and the theory of principal axes, and their
PREFACE
V
distribution in space: the motion of a body subject
to constraint either of a fixed axis or of a fixed point;
that of a body perfectly free from all constraint; the
theory of relative motion; and the theory of machines
in motion and of work done thereby. I may, too, ob-
serve, that we are herein led to some curious proper-
ties of mechanical units, and to the mode of reducing
all force-action to an uniform standard of mechanical
work.
66
The concluding Chapter is the work of Mr. W. F.
DONKIN, M.A., F.R.S., of University College, and Savi-
lian Professor of Astronomy, Oxford; Theoretical
Dynamics" is the subject of it; and the theory is dis-
cussed which assigns the number and the order of the
differential equations of motion in the most general
problem; the possibility of the solution of some or all
of them; and the forms of the resulting integrals.
The Lagrangian and Hamiltonian equations are in-
vestigated; and theorems, important in reference to
these equations, discovered by Poisson, Jacobi, Pro-
fessor Donkin himself, M. Bour, M. Liouville, are de-
monstrated. Perhaps no one is better able to expound
the difficulties of the theory than the accomplished
mathematician who has contributed the Chapter; he
has studied the subject, and has made real advances
in it.
Investigations of problems of a special character
are introduced more generally than in the preceding
Volume. Thus the subject of the precession and nu-
tation of the earth's axis has been considered at some
length; the apparent effects due to the earth's diurnal
rotation on the action of the pendulum-experiment
devised by Foucault, and of the gyroscope, on the de-
viation of heavy bodies, whether falling freely or pro-
jected with high velocities, are discussed with con-
siderable minuteness. This course has been to a
vi
PREFACE.
certain extent unavoidable; because cosmical phæno-
mena, and machines devised to illustrate them, are the
most simple and most appropriate examples of general
mechanical processes; consequently it has been un-
necessary to devise hard problems for the purpose of
exhibiting the power of the equations, when they are
best illustrated by the movements in which we our-
selves daily take part. It is thought also that the
utility of the work is hereby increased.
The general principle on which the equations of
motion are formed is the same as that which is so
frequently and so prominently stated in the preceding
Volume; viz., the equality of the impressed and the
expressed momentum on a single particle. This prin-
ciple is indeed directly applicable to the determina-
tion of the motion of a material system, only when
the internal forces which act on the several molecules
are taken account of; and as the nature, the laws,
and the action of these forces are generally unknown,
some other mode of estimating the general results is
required. If the system is so organized that the in-
ternal forces enter in equal and opposite pairs, they
disappear in the equations of motion, and the circum-
stances are expressed without difficulty in a sufficient
number of equations.
All the incidents of motion which arise out of con-
tinuous laws are expressed as infinitesimals. This is
indeed the reason why the present and the preceding
Volumes, in which mechanical subjects are treated,
are included in a course of Infinitesimal Calculus. In-
finitesimals are, as heretofore, stated and applied in
their barest forms; and subject to the axiomatic pro-
perties of Art. 9, Vol. I. Infinitesimals and finite quan-
tities are the materies of calculation according to the
same laws. And it is submitted that continuous laws
can only thus be adequately expressed symbolically.
PREFACE.
vii
The object of the Author has been the construction
of an uniform scientific Treatise, pervaded by one idea,
and applying one principle. Thus, at the outset, a
certain form is given to the equations of motion by
D'Alembert's principle; and in that form they are ap-
plied to all subsequent purposes and processes; for
they are directly applicable to all classes of dynamical
problems. In some cases indeed they are conveniently
applied in the transformed state, as they are known
by the name of Euler's Equations; generally however
all special artifices, however ingenious they may be,
and whatever abridgement of work they may intro-
duce, are avoided; the circumstances of a problem
are resolved into their simplest elements, and these
are expressed by the general equations and their in-
tegrals.
The object again is not to make new discoveries or
to open new lines of research; but to use the present
knowledge and the present materials; to digest, to
arrange, to consolidate all into one harmonious Trea-
tise; to make such additions as are necessary for the
process; and to present all to a student on an uniform
plan. The Treatise has arisen out of the want which
the Author himself has frequently experienced in his
professional employment; and the attempt to supply
that want has given to the work its didactic character
and its colloquial style.
The Author is of course under obligations to many
writers on Mechanics and kindred subjects. These
obligations he has attempted to acknowledge from time
to time, as well as to specify the treatises wherein
certain subjects have been originally or fully treated.
It has however been impossible to satisfy the claims
of such writers in all cases. In many cases, the Author
has found, that theorems, to which he was led in
the course of his investigation, had been previously
viii
PREFACE.
discovered, and he is also bound to say, that many
theorems which are attributed to certain authors have
been known and proved long before the time of the
writer to whom the credit is commonly given. This
is a disappointment to which an inquirer in any branch
of science must be liable. He will rejoice however to
find that truth has advanced, although his share in
the work may not be as large as he might expect.
The benefit of the progress will be permanent, his
disappointment will be temporary; and if he will take
heed to use it aright, it will be an inducement to
greater industry and further research.
PEMBROKE COLLEGE, OXFORD,
Nov. 14, 1861.
ANALYTICAL
ANALYTICAL TABLE OF CONTENTS.
Art.
PART I.
DYNAMICS: THE MOTION OF A MATERIAL SYSTEM.
CHAPTER I.
PRELIMINARY GEOMETRICAL INVESTIGATIONS.
1. The necessity of a geometrical investigation..
2. Transformation of rectangular systems of reference in terms of
nine direction-cosines
3. Do. in terms of Euler's three angles, viz. 0, 4, 4
•
Page
1
2 3 4
4. Euler's angles expressed in terms of direction-cosines.
5. Reduction of central surfaces of the second degree to the
centre as origin
5
•
5
6
6. Do. to principal axes as coordinate axes
7. Proof that the latter reduction is possible, and determination
of the position of the principal axes
•
8. Principal axes coincide with singular radii vectores
9. Principal axes are perpendicular to their conjugate planes
10. Degenerate forms of surfaces of the second degree
11. Cones and reciprocal cones of the second degree ..
12. Cyclic planes of cones
9
11
13
15
18
19
13. Cyclic planes of ellipsoids
20
14, 15. Conjugate axes and planes in an ellipsoid
16. On central radii vectores of an ellipsoid
23
26
17. The focal conics of an ellipsoid..
26
18. Confocal surfaces of the second degree.
27
19. On the principal axes of a cone enveloping an ellipsoid
28
20. The sphero-polar reciprocal of an ellipsoid
30
·
•
CHAPTER II.
THE CINEMATICS OF A RIGID BODY.
21. Necessity of a distinct investigation of the cinematics of a rigid
body
PRICE, VOL. IV.
b
32
X
ANALYTICAL TABLE
22. The most general motion consists of a translation, and of two
rotations. Definition and explanation of angular velocity
33
23. Relation of linear velocity to angular velocity
24. Line-representatives of angular velocities..
25. Composition of coaxal angular velocities
36
37
38
26. Composition of angular velocities whose rotation-axes meet..
27. Particular forms, when the rotation-axes are at right angles..
28. Bohnenberger's machine
29. Composition of many angular velocities, whose rotation-axes
pass through one point..
•
✓ 30. Geometrical explanation of Foucault's pendulum experiment..
31. Composition of angular velocities, whose rotation-axes are
parallel .
32. Couples of angular velocities.
33. General case of the composition of angular velocities
34. Particular cases of the preceding Article
35-37. The central axis..
38. Determination of the linear velocity due to given angular ve-
locities
40
41
42
43
44
45
47
48
50
52
56
39. Motion of a body defined by two systems of reference.
40. Relation between angular velocities and the t-differentials of
direction-cosines
58
•
60
41. Analytical proof that all motion consists of a translation and
of a rotation
62
42. Angular velocities expressed in terms of Euler's three angles
and their t-differentials.
64
•
CHAPTER III.
THE DYNAMICS PROPER OF A MATERIAL SYSTEM.
SECTION 1.-D'Alembert's principle: the equations of motion of a ma-
terial system.
43. Various forms of material systems and their definitions
44, 45. Explanation and statement of D'Alembert's principle..
46. Illustrations of its application to particular problems..
V 47. The statement and proof given by D'Alembert
67
69
73
77
48. The equations of motion of a material system, the internal
forces of which are in equilibrium.
78
49, 50. The equations of motion expressed in a single equation by
means of the principle of virtual velocities
80
51. The number of equations necessary for the complete solution
of a problem..
83
•
52. Application to the motion of a flexible string
84
OF CONTENTS.
xi
SECTION 2.-Independence of the motions of translation of centre of gra-
vity, and of rotation about an axis passing through it.
53. Mathematical properties of the centre of gravity or centre of
masses
86
54. Proof of the theorems when the forces are instantaneous. . . .
55. Proof of the theorems when the forces arefinite
56. Illustration of the theorems
88
90
91
SECTION 3.-Conservation of motion of centre of gravity, of moments,
of areas. Laplace's invariable plane.
and
57. Conservation of motion of centre of gravity proved..
58. Proof of conservation of moments..
59. Conservation of areas
60. The invariable plane
61. The invariable plane of the solar system
62. The moments of the momenta relative to the centre of gravity. 102
SECTION 4.-The principle of vis viva. Lagrange's principle of least
action. Carnot's theorem.
92
94
97
98
99
63. Conditions under which the equation of vis viva exists......
64. Conservation of vis viva; critical value of vis viva when the
system is in equilibrium
103
105
65. Circumstances under which .m(xdx+xdy+zdz) is an exact
differential..
107
66. Relation between the vis viva relative to a given origin and
that relative to the centre of gravity
109
V 67. Principle of least action..
110
68. Carnot's theorem
112
SECTION 5.-Gauss' theorem of least constraint.
69. Explanation and proof of the theorem
115
SECTION 6.-Newton's principle of similitude.
70. Explanation, proof, and application of the theorem
CHAPTER IV.
119
EQUATIONS OF MOTION OF A RIGID BODY IN TERMS OF ANGULAR VE-
LOCITIES, PRINCIPAL AXES, AND MOMENTS OF INERTIA.
SECTION 1.—The transformation of the equations of motion.
71. A change of angular velocity is due to a force..
72. Angular forces and angular velocity-increments
124
125
b 2
xii
ANALYTICAL TABLE
73. Transformation of equations; instantaneous forces
74. Moment of inertia, and radius of gyration
75. The equations deduced from first principles
76. The axial components of the resulting angular velocity
77. Transformation of equations; finite forces
78. The resulting angular velocity-increment
79. Axial components of velocity-increment
80. Analysis of the equations..
•
81. Moment of the expressed momentum increments
82. Centrifugal forces
127
128
130
131
•
132
133
134
135
135
137
83. Resulting equations of motion. .
138
84, 85. Simplification of the equations; principal axes.
139
SECTION 2.-Principal axes, and their properties.
86. Proof of the existence of a system of principal axes; and the
position of it at a given point. . .
141
87. Interpretation of the results by means of an ellipsoid.
88. Particular forms of the ellipsoid of principal axes
89. Principal axes determined in a particular problem
143
145
146
90. One principal axis being given, the determination of the two
others
147
•
91. Examples in illustration
148
92. Reduced forms of equations of motion
150
93. Permanent axes
151
94. The central principal axes the only permanent axes
153
95. Foucault's gyroscope
154
SECTION 3.-Moments of inertia, and distribution in space of
principal axes.
96. General value of moment of inertia.
•
97. Example in illustration.
98. The momental ellipsoid
156
157
157
99. Singular values of principal moments
100. The equimomental cone.
•
101. Moments of inertia relative to parallel axes
160
161
162
102. Moments of inertia in relation to the central ellipsoid
163
103. The position of principal axes at any point.
166
104. The central ellipsoid of gyration
167
169
105. The symmetry of a body
106. The cone reciprocal to that of principal axes is equimomental 179
107. The equimomental surface..
108. Professor Maccullagh's construction by Apsidals
109, 110. Particular forms of the equimomental surface
•
171
173
•
174
OF CONTENTS.
xiii
111. Distribution in space of principal axes
178
179
112. Conditions when a line is a principal axis.
113. All lines in a principal plane are principal axes at some point 182
114. Every plane is principal at some point in it
•
SECTION 4.—Examples of moments of inertia.
115. Two general theorems
•
116. Moments of inertia of thin wires
117. Moments of inertia of thin plates and shells..
118-120. Moments of inertia of solid bodies
121. Moments of inertia of shells derived from those of solid
bodies
183
185
186
188
193
197
CHAPTER V.
THE ROTATION OF A BODY ABOUT A FIXED AXIS.
SECTION 1.-The rotation of a rigid body about a fixed axis under the
action of instantaneous forces.
122. Explanation of the motion. .
199
123. The equations of motion when the z-axis is the rotation-axis
200
124. The angular velocity determined in several examples
201
125. Determination of the pressure on the axis. .
204
126. Conditions when there is no pressure on the axis.
206
127. The centre of percussion..
209
128. The axis of percussion
211
SECTION 2.-Rotation of a body about a fixed axis under the action of
finite accelerating forces.
129. The equations of motion when the z-axis is the rotation axis 213
130. Particular case when the lines of action of all the forces are
parallel to the rotation-axis..
215
131. Rotation of a heavy body about a horizontal axis..
132. Small oscillations; the centre of oscillation
216
•
217
133. Examples of the simple isochronous pendulum.
220
134. The length of the seconds' pendulum
224
135. Experimental determination of the radius of gyration
226
136, 137. Isochronal axes, and axes of shortest time...
227
138. Illustrative examples ..
231
139. Capt. Robins' ballistic pendulum
231
140. Motion of machines with fixed axes
233
• •
141.142. Determination of the pressures at the fixed points of
the axis
239
xiv
ANALYTICAL TABLE
CHAPTER VI.
THE ROTATION OF A BODY ABOUT A FIXED POINT.
SECTION 1.-Rotation of a body about a fixed point under the action of
instantaneous forces.
143. Two systems of reference
144. The equations of motion in their general and reduced forms
145. The instantaneous rotation-axis, the instantaneous pole, and
the couple of impulsion
146. Examples in illustration.
•
245
246
247
250
252
147. The pressure at the fixed point
SECTION 2.-Rotation of a rigid body about a fixed point under the
action of finite forces.
148. The general equations of motion; Euler's equations. .
149. The equations and their results when no forces act
150, 151. Explanations of these results; the invariable axis..
152. Position determined by the three angles 0, 4, V. . .
153. The component of the instantaneous angular velocity along
the invariable axis is constant.....
253
255
256
258
259
154. The differential equation in terms of the time and the an-
gular velocity..
260
155. Certain peculiarities of the developed centrifugal force..
260
156. Poinsot's interpretation of the preceding result.
261
157. General and particular properties of the polhode
264
158, 159. Do. of the herpolhode..
266
160. The stability of the rotation-axis
269
•
161. Particular cases of the preceding theorems depending on
particular initial circumstances
271
162, 163. Do. depending on particular constitutions of the body
164. Discussion of the case when the oscillation of the rotation-
axis is small
273
277
165. The cone described by the rotation-axis in the body....
166. Certain properties of the principal axes of the moving body
167. Rotation of a heavy body about a fixed point
278
279
281
168. Particular case when AB, and the initial axis of rotation
is the principal axis of unequal moment..
285
169. Do. when the initial angular velocity is very great
170, 171. Do. when the axis of unequal moment is inclined to
288
the vertical at a constant angle ..
291
172. Bohnenberger's and Fessel's machines
173. Precession and nutation of the earth ..
295
•
297
OF CONTENTS.
XV
174. Simplification of the equations
301
175. Determination of small quantities
302
176. Transformation of certain terms of the equations .
304
177. General integral of the equations
306
178. The effects of the action of the moon
309
179. Determination of the luni-solar precession and nutation
311
ral reasoning
•
180. Poinsot's determination of the preceding results from gene-
181. The pressure on the fixed point..
315
318
CHAPTER VII.
MOTION OF A RIGID BODY OR INVARIABLE MATERIAL SYSTEM
FREE FROM ALL CONSTRAINT.
SECTION 1.-Motion of a free invariable system under the action of
instantaneous forces.
182. Explanation of mode of determining motion
183. The equations of motion
184. The components of velocity resulting from a given force..
185. The locus of points which move with the same velocity
186. The spontaneous axis..
320
322
323
324
•
•
326
187. The vis viva of the system is a maximum when the rotation-
axis is the spontaneous axis
327
188. The motion of a body due to a blow parallel to a central
principal axis and in a central principal plane
329
189. Properties of centre of percussion and spontaneous centre
of rotation
331
190. The centres of greatest percussion.
333
191. The body struck may be equivalently replaced by two mole-
cules of given masses at the ends of an inflexible bar....
192. The position of the point on which the moving body im-
pinges with the greatest velocity....
335
337
193. The centres of greatest reflexion and greatest conversion
194. Motion of a body due to a blow parallel to a central princi-
pal axis
339
•
342
195. Theorems relating to the spontaneous axis and spontaneous
centre corresponding to a given centre of impulsion
344
196. Other incidents of the motion
346
197. Centres of greatest percussion
349
198. Case in which a couple of impulsion initially acts.
199, 200. Case when the body strikes against a moveable mass
201. Points of greatest reflexion and greatest conversion....
350
352
355
xvi
ANALYTICAL TABLE
202. Points of perfect reflexion and perfect conversion.
•
•
203. Effects of an impinging mass on another given mass.
204. The limits of a blow given by a hammer on a fixed prop.
205. The initial motion of a billiard ball
356
359
361
•
362
SECTION 2.-Motion of a free invariable system under the action of
finite forces.
206. General equations of motion ..
364
207. Problems in which the rotation-axes move parallel to them-
selves ...
367
208. The general case of rocking or titubation.
379
209. The general theory of small oscillations. .
210. The coexistence of small oscillations.
382
384
211. Examples of small oscillations
385
212. Small oscillations of a body of which one point is fixed ...
391
213. The motion of a top on a rough horizontal plane.
393
214, 215. Rolling and sliding friction
395
216, 217. Examples of motion of a rigid body
399
218. The motion of a top on a smooth horizontal plane
402
219-222. The motion of a billiard ball on a rough horizontal
table
405
CHAPTER VIII.
RELATIVE MOTION OF A MATERIAL SYSTEM.
SYSTE
SECTION 1.-Investigation of the general equations.
223. Explanation of the term relative motion.
•
224. Components of the relative velocity
•
413
414
225. The components of the expressed relative velocity-increments 416
226. Analysis of the expressions in the preceding Article. Velo-
city-increment of transference and compound centrifugal
force
227. The equations derived from fictitious forces
228. Particular case when the motion of two coordinate axes
takes place in one plane.
1
•
229. Explanation of the resulting expressions
•
418
420
421
423
SECTION 2.-The relative motion of a material particle.
230. Relative constrained motion in one plane
231, 232. Examples in illustration. .
•
424
426
OF CONTENTS.
xvii
233. Motion of a heavy particle in a rotating tube
429
234. Problems in illustration..
430
235, 236. Constrained motion in curved tubes and on curved
surfaces
432
237. The apparent motion of a particle relative to the rotating
earth
434
238. Adaptation of the equations with the omission of small
quantities
436
239. The apparent path of a projectile
437
240. Particular cases of the preceding Article
439
241. The investigation carried to a higher approximation..
242. The effects of the rotation of the earth on a body falling
freely from the top of a tower
243,244. The equations of Foucault's pendulum experiment...
245, 246. Discussion of the equations ..
442
445
446
449
247. Relative motion of a particle on a smooth inclined plane . .
248. Do. on a horizontal plane..
453
457
SECTION 3.-The relative motion of a material system.
249. Formation of the equations of translation and of rotation..
250. All combined into one equation by means of the principle of
457
virtual velocities ..
459
251. Relative vis viva of a material system
459
252. Particular forms of the general equations.
460
253. The relative motion of the centre of gravity of a system
254. Relations between three co-original systems of axes.
255. Relations between absolute and relative angular velocities
due to the rotation of the system of reference
256. The vis viva expressed in terms of angular velocity
257. Adaptation of the preceding equations to the rotation of the
earth
462
464
465
•
466
•
467
258. The gyroscope of M. Foucault when the axis moves in a
given plane..
468
259. Determination of the forces which act on it; and the dis-
cussion of the equations .
469
260. Particular cases of the preceding
474
261. The effects of the gyroscope when the axis is constrained to
move in a right circular cone . . . .
475
262, 263. General case of motion of the gyroscope when the ac-
tion is unconstrained
476
•
264. The results of the gyroscope .
480
PRICE, VOL. IV.
C
xviii
ANALYTICAL TABLE
CHAPTER IX.
THE THEORY OF MACHINES IN MOTION.
265. Explanation of the equation of vis viva. .
481
265. Definition of a machine; its working point, its work and la-
bouring force ..
482
267.
Work" in the ordinary sense, and units of work
483
268. Relation between work and vis viva...
485
269. Moving work and resisting work; useful work and lost
work
487
270. Maximum and minimum values of vis viva
489
271. The efficiency of a machine
272. Uniformity of motion secured by fly-wheels.
273. On mechanical units
489
491
•
493
CHAPTER X.
THE MOTION OF ELASTIC BODIES.
274. The equations of motion of a rigid body
496
275. A rigid body in contrast with an elastic body
276. Two modes of forming the equations of motion of an elastic
body
498
•
499
277. The equations of motion of a fine flexible string
500
278. Motion of an elastic string of which the tension is constant
279. The general motion of an elastic string..
500
501
280. The integrals of the equations of motion expressed in terms
of arbitrary functions ..
504
281. The properties of a vibrating string deduced from the pre-
ceding functions ...
506
282. The oscillations and periodic times of the string
283. The nodes and ventral segments
508
509
284. The relation between the periodic times of the transverse
and longitudinal vibrations.
510
285. The preceding results in reference to the theory of music. .
286. Another mode of expressing the integrals..
511
513
287. The longitudinal vibrations of the molecules of a fine elastic
rod ....
514
288. The motion of the molecules of a thin elastic lamina
289. The general equations of motion of a molecule of an elastic
body
517
519
290. Particular case when the body is a thin elastic membrane..
291. Particular case when the body is a fine elastic string
520
522
•
OF CONTENTS.
xix
CHAPTER XI.
THEORETICAL DYNAMICS.
292. Explanation of notation.
•
524
293. General formula .m("dx + y″dy + z″dz) =Σ. (xdx + ydy + z dz),
and explanations.
525
294-297. Discussion of symbol 8; principle of introduction of
δ
arbitrary constants ..
526
298. Case in which the equation 8A 8A is significant and
530
true..
299. Proof that (du)' = d (u')
531
300-302. Deduction of the Lagrangian formula
T
Σ
αξ
d T
αξ
du)
dé
d S
6ઠ્ઠું =
=
= 0....
531
303. Illustration taken from the transformation to polar coordi-
nates in space
536
•
304. Second example; transformation from fixed to moving rect-
angular axes
537
305. Application to the general case of motion near the earth's
surface when the earth's rotation is considered; deduction
of general formulæ ...
538
306. Transformation of the general Lagrangian form to the
Hamiltonian system
Σ
= ( p² + 1 ) 8 9
17) 8 g = 0,
dн
qi =
dpi
d q
• •
307. Case of n independent coordinates 91, In gives the 2n
“Hamiltonian equations" (i = 1 to i = n)
540
d H
dH
Pi=
Ii =
dqi
dpi
541
308. Considerations on the integrals of such a system .
309. Definition of elements
544
545
310. Illustration; the case of central forces in two dimensions
reduced to Hamiltonian form..
545
•
311, 312. Definition of Poisson's symbol (u, v); its elementary
properties
546
313. Demonstration that (dpi ▲ qi Api & qi) is independent
of t
547
314. Peculiar properties of initial values of p, q, considered as
elements or arbitrary constants
548
315. Definition of canonical elements. The initial values are
canonical, but the number of other systems is infinite
550
XX
ANALYTICAL TABLE OF CONTENTS.
316. If f, g be any two integrals, (f, g) is constant; example
from central forces . . . . .
•
317-320. Sir W. R. Hamilton's discovery of the expressibility of
all the integral equations by means of a single function s.
Discussion of some of its properties
321-323. Jacobi's theorem. The discovery of s is reducible to
that of a complete solution of a partial differential equation
of first order, but not of first degree ..
324. Modification in the case in which principle of vis viva holds
good
325-329. Demonstration and discussion of theorem, that if n out
of the 2 n integrals are known, say, αı, ɑn, and satisfy
the
n (n-1)
2
•
conditions, (a;, aj) = 0, the completion of a
canonical solution is reducible to quadratures
330. Remarks on the practical application of the theorem
331, 332. Application to the case of central forces in three di-
mensions; equations of motion put in Hamiltonian form;
three of the known integrals shewn to be applicable; three
more found by the method...
551
553
557
560
562
567
568
333. Interpretation of the three new elements
570
334. Application to the case of a planet; canonical elements of
an elliptic orbit
571
335. Variation of elements. Demonstration, by means of Hamil-
tonian equations, of Poisson's general formula
}
do
d Q
ci =
(C1, Ci) +
dc1
+
dcz n
(C2n, Ci)
572
336. Simplification in the case of
canonical elements;
d Q
d n
αί
bi
dbi
dai
337, 338. Notice and demonstration of Bour's theorem
339. Conclusion..
574
574
578
ANALYTICAL MECHANICS.
PART III.
DYNAMICS; THE MOTION OF A MATERIAL SYSTEM.
CHAPTER I
PRELIMINARY GEOMETRICAL INVESTIGATIONS.
ARTICLE 1.] In following the course suggested by the nature
of the science of mechanics, the subject next for discussion is
the motion of a material system; that is, of a system of mate-
rial particles which are related to each other by means of cer-
tain forces of attraction, tension, and such like. These will be
explained hereafter. This motion I shall consider in its greatest
generality, and by the light of the best processes which modern
science has discovered: we shall hereby be enabled to apply our
principles to problems of great interest and of practical import-
ance, and to their solution by most elegant methods. I shall
also enuntiate and explain certain very general principles, which
in their mathematical expression include all Dynamical pro-
blems. These will be introduced towards the close of our
treatise; because I think that such and similar general pro-
positions are more adequately apprehended, when they have
been previously applied as it were piecemeal to particular pro-
blems. This is the course which I have taken heretofore, and
which I shall still take, in the conviction that it is that which
is best suited to a didactic treatise.
The general motion of a material system takes place in space;
and is capable of determination only by means of properties of
space; by means, that is, of systems of coordinates, or of some
other equivalent mode of reference. It is necessary therefore for
us to be prepared with a sufficient knowledge of these properties.
Moreover in the course of our treatise we shall often have
occasion to translate mechanical results into analogous geo-
metrical theorems, whereby we shall obtain a fertile interpreta-
PRICE, VOL. IV.
B
4
[3.
TRANSFORMATION OF SYSTEMS OF REFERENCE.
cosines are independent. It is however to be observed that the
transformation is thus effected by means of symmetrical linear
equations; in many cases the advantage of employing such
formulæ is greater than the inconvenience of introducing many
variables which are not independent; but in other cases it is
more convenient to introduce as few variables as possible; and
I proceed therefore to explain Euler's process of transformation,
in which only three new quantities are required.
Let, as heretofore, x, y, z refer to the original system, and
έ, n, to the transformed system.
(1) Let the system of axes be turned about the axis of ≈ in
a positive direction through an angle : see Fig. 1; and let
x', y', z′ be the values of x, y, z, when this transformation has
taken place; so that
x = x' cos ↓ — y' sin √,
Y
xsin y+y' cos y,
(14)
2 = 2.
(2) Let the system of x', y', z' be turned through an angle ℗
about the line ON, which is the axis of x'; and let x", y", z" be the
coordinates when this transformation has taken place; so that
x'
x'',
ý
y" cos — "sin 0,
(15)
y" sin + "cos 0.
(3) Let the system of x", y", z″ be turned about the axis of
" in a positive direction through an angle ; and let έ, n, be
the coordinates when this transformation has taken place; so
that
x"
=
έ cos — ŋ sin ò̟,
y" έ sin on cos 0,
2" = 5.
(16)
Then by these successive transformations the system of axes
will be transformed in the most general manner possible; and
substituting in (14) from (15) and (16), we have
x = (cos + cosy - sino sin cos 0)+(-sino cos-cos sin cos 0)
X
+ (sin sin 0,
(17)
y = (cos + sin + sino cos y cose) +(-sin o sin + cos cosy cose)
N
-cos y sin 0,
έsin & sin 0+ n cos o sin + cos 0,
(18)
(19)
5.]
5
SURFACES OF THE SECOND DEGREE.
whereby the relations between the old and new coordinates are
expressed in terms of three undetermined quantities 0, 4, and y.
4.] The comparison of (17), (18) and (19) with (2) indicates
the following equivalences:
α1
b₁
=
cos o cos y — sin & sin √ cos 0,
sin cos
&
cos o sin √ cos 0,
(20)
C1
C₁ = sin y sin 0,
a2
cos sin + sin o cos y cos §,
b2
sin sin + cos o cos y cos 9,
(21)
-
C₂ = cos y sin 0,
α z =
&
sin sin 0,
b3
= cos sin 0,
(22)
C3
= cos 0;
these equations also satisfy the conditions (4) ... (7); and from
them we have
C1
cos = C3,
tan =
a3
b3
tan
; (23)
C2
so that the nine direction-cosines are expressed by means of
three quantities 0, 0, ¥.
5.] In the course of our work we shall frequently require for
illustration surfaces of the second degree. The most general
form of the equation of which is
Ax²+By²+cz²+2dyz+2Ezx+2Fxy+2Gx+2Hу+2J≈+k=0: (24)
but as we shall need only central surfaces, and these referred
to the centre as the origin, we had better reduce (24) to the
most simple form which the equation of such surfaces admits of.
Let (x', y', z) be the new origin, and be the centre; then sub-
stituting x+x', y+y', z+ severally for x, y, z, (24) becomes
Ax²+By²+cz²+2Dуz+2E≈x + 2 F x y
+2(Ax+ ry' + Ez + G) x + 2 (Fx + By + Dz' + H) y
+2(Ex' + Dy' + C≈' + J) ≈
+ Ax²² + BY'² + Cz²²+2DY'≈ + QEZ'x' + 2 Fx'y'
+2Gx' + 2Hу'+2J+K=0; (25)
as (x', y', ') is the centre, this equation is to be unaltered when
6
[6.
SURFACES OF THE SECOND DEGREE.
for a, y, z we substitute −x, −y, −z; therefore the coefficients
of x, y, ≈ must vanish; so that
AX' + Fу' + EZ + G = 0,
F x′ + BY' + Dz′+H = 0,
Ex' + Dy + cz' + J = 0;
whence we have finite values for x', y', '; unless
(26)
A B C — A D² — BE² - CF² + 2DEF = ▼ (say) = 0,
(27)
in which case the values of x, y, z are infinite. Let us suppose
▾ to be finite: then the equation to the surface becomes
Ax²+By²+C² + 2 Dyz + 2 Ezx + 2гxy + K' = 0; (28)
wherein K' is the constant term, and represents the last two lines.
of (25): and as is evident from (26),
K′ = GX' + Hy' + JŹ + K ;
(29)
and if we substitute the values of x, y, z, which are determined
by (26), we have
K'V
G² (D2-BC) + H2(E2-CA) + J2 (F2-AB)
+2HJ (AD-EF) +2JG (BE — FD) + 2GH (CF - DE) — KV, (30)
= v′ (say).
(31)
In passing I would observe, that v is the determinant of the
three equations (26), when the last terms are omitted; and
that v′ is, omitting a factor, the determinant of the four equa-
tions (26) and (29). This condition has been already deter-
mined in Ex. 3, Art. 355, Vol. I. Ed. 2.
If v =
O, the coordinates of the centre are infinite: the sur-
face in this case is non-central, and is a paraboloid, or one of
its degenerate varieties.
If v = 0, the equation to the surface is
Ax²+By²+Cz²+2Dyz+2Ex+2rxy = 0,
(32)
and the centre is on the surface. The surface is therefore a
cone, or one of its degenerate varieties.
6.] We can further reduce the general equation to central
surfaces by means of another transformation of coordinate axes.
Let the centre still be the origin, and let another system of
rectangular axes originate at it. Let us omit the accent on K'
in (28), and for x, y, z let us substitute the values given in
(2). Then (28) becomes
و
6.]
7
REDUCTION OF SURFACES OF THE SECOND DEGREE.
2
2
§ ² { ▲ α₁² + в α₂²+caz²+2 Dɑ₂α3 +2Еаzа₁+2Ƒа₁α₂}
2
2
D
+ n² { ab₂² + B b₂² + c bz² + 2 » b₂b3+2 £ bžb₁+2 гb₁b₂}
1
2
2
2
2
+ 5² {AC₁² + B C₂² + C Cz² + 2 DC2 C3 + 2 EC3C₂+2 F C₁ C₂}
+ 2 nŚ { Ab₁C₁ + B b 2 C2 + Cbz C3
+D (b₂C3 +b3C2) +E (b3C1 +b1C3) +F (b₁ C2+b₂C1) }
+25§ {Ac₁α₁ + B C 2 Aq + C Cz Az
22
+D(C2A3+C3ɑ2) +E (C3α1 + C₁α3) + F (C₁α₂+C2α₁) }
+2§n {Aα₁b₁+в а₂b₂+с аzbз
1
2
1
2
+D(a2b3+ɑ3b2) +E (аžb₁+а¸b3) +r (a₁b₂+ a2b₁)} + K = 0. (33)
In this equation nine direction-cosines are involved, and these
thus far are subject to only six conditions; viz. (4) and (6), or
(5) and (7) of Art. 2. Three other equations therefore are neces-
sary for their complete determination; assuming the following
conditions to be possible and sufficient, let us suppose the coeffi-
cients of n, SE, and έn in (33) to vanish: so that we have
η
A b₁C₁+B b₂C₂+cb3C3 +D (b2C3+b3C2) + E (b3C1 +b₁C3) + F(b₁C₂+b₂C1) = 0,
AC1α1+B C₂ α2+CCzαz +D (C2A3+C3α2) +E (C3α1 + €1α3) +F (C₁α₂+C2α1) = 0, > (34)
▲ а₁b₁+ва₂b₂+саzbz+D (α₂bz+ɑzb₂) +E (аžb₁ +а₁b¾) +F(α₁b₂+ α₂b₁) = 0.,
Ί
a
2
3
3
2
Also let the new coefficients of 2, 72, 2 in the transformed equa-
tion severally be a', B', c'; so that
2
2
2
sа²² +в а‚² +¤ª²² + 2D A2 A3 + 2 E αz α1 +2Ƒа₁α₂ = A',
1
2
2
a b₁² +в b₂² + cb3²+2 » b₂b3+2 E b¸ b₁ + 2 Ƒ b₁ b₂
F
2
АС
2
2
2
= B
B',
•
A c₁ ² + B C ₂ ² + C C 3² + 2D C2 C3 + 2 E Cz C₁ + 2 FC₁ C₂ = c′;
3 1
(35)
whereby (if these equations are possible) the transformed equa-
tion is
2
A' §² + B'n² + c' §² + K = 0.
(36)
Now the last two equations of (34) may be put into the fol-
lowing forms:
2
2
(Aα₂+Fα₂+Eα3) C1+ (Fα₂+ Bα₂+Dα3) C₂+ (Eα₂+DA₂ + CA3) Cz = 0, |
(Aα₁+ Fɑ₂+Eɑ3) b₁ + (Fα, +Bа₂+Dα3) b₂+ ( E α1 +Dα₂+ Cα3) b3 = 0;
and our hypothesis requires these to coexist with the second
and third of (7), Article 2; viz. with
a₁ C1 + α z Cz + α z Cz = 0,)
2
(38)
(37)
a
1
а₁ b₁ + α₂ b ½ + α z bz
= 0.
But if we have two pairs of equations of the forms
REDUCTION OF SURFACES OF THE SECOND DEGREE.
[6.
lx + my + nz = 0 )
l§ + mn + n = 05
Š
from the first pair we have
LX+My+N≈ =
LE+ Mn+ N = 0 S'
Ś
m
y5—nz
ZĘ – Šx
N
xn-Ey
and from the second,
L
M
N
y s − n z
ZĘ – Śx
x n − ¿y
m
n
L
M
N
(39)
As (37) and (38) are pairs of equations of the same form as
these, we have
Aα1
Aа₁ + Fа₂ + Eа3
Fa₁ + Bа₂+Daz
Eα₁ + Dа₂ + cα3
(40)
а1
Az
аз
2
2
E
(41)
2
(42)
=A', from (35);
(41) being inferred from (40) by operating on the numerators
and denominators of (40) severally with the factors ɑ1, ɑ2, ɑ3, and
by adding numerators and denominators.
Similarly from the third and first, and from the first and
second of (34), we have
2
E
B
2
Ab₁ + F b₂+ Е b3
1
F b₁+в b₂+ D b3
ba
E b₁ + D b₂ + c b 3
ხვ
B',
A C₁ + FC₂ + EC3
FC₁ + BC2 + D C3
C1
C2
EC₁ + DC₂+ CC 3
C3
= c'.
As these last equations are of precisely the same form as
(42), let us take a type-expression of all; and assume x to be
the type of A', B', c'; and t, to be the type of an, bn, cn: so that
we have the following typical form:
(A-x) t₁ + F t₂ + Et3
0,
Ft,
+ (B−X)t₂ + Dtz
0,
(43)
Et₁
+ Dt 2
+(c-x)t₂ = 0;
whence by cross multiplication,
(A-X) (B-X) (c-x)-D² (A-X)-E2 (B-X)-F2(c-x)+2 DEF = 0; (44)
which equation is the condition of the coexistence of the three
equations of (43).
7.] REDUCTION of surfaces of THE SECOND DEGREE. 9
As (44) is a cubic in x it has three roots; and these are the
values of A', B', c' which are the coefficients of 2, n², 2 in equa-
tion (36): so that we have
(A —X) (B —X) (C—X) — D² (A — X)--E² (B-X) - F² (c-x)+2 DEF
M
of which the x-differential is
= (A'-X) (B'- x) (c' - x) = 0, (45)
(B-X) (C-X) + (C−x) (A −x) + (A−X) (B −X) — D² — E2 — F²
X
— (B' — X) (c' — x) + (c' — x) (a' — x) + (A′ — X) (B′ — x). (46)
2
7.] As the roots of the cubic (44) are the coefficients of
¿², n², ¿² in the reduced equation of the surface, and as these
coefficients must be real quantities, the possibility of the pre-
ceding reduction depends on the reality of these roots; and
this is demonstrated by the following process, due to Cauchy.
Let r1 and r2 be the two roots of
(A-X) (B-X) — F² = 0,
so that
r₁ =
A+ B
2
+
{(
A+ B
r2
2
1
2
B
2
+ F2
2
F2
A B
! — { ( ^~ ~ ~ ¹³)² + x² } * ;
2
r₁ and r½ are evidently real quantities. In (44) let us sub-
stitute for x,
(1) +∞; the result is negative;
(2); the result is positive;
(3) r; the result is negative;
2
(4); the result is positive;
1
2
1
therefore the roots of (44) lie respectively between +∞ and r₁;
between r₁ and r
r2; between r, and co; and are all real. Thus
the assumptions made in (34) are demonstrated to be legitimate;
and A', B', c' are real quantities, which are determined by the
equation (44). Henceforth we shall suppose them to be known.
Also from (43) another form of the cubic equation may be
found, which is for many purposes more useful than (44). The
several equations of (43) may be put into the following form:
t₁
to
1
2
+
D
E
+
to
F
PRICE, VOL. IV.
tı
EF
to
FD
{
{
Er
X A +
D
FD
ta
X B+
E
DE
{
X
c+
DE}. (47)
F
C
10
REDUCTION OF SURFACES OF THE SECOND DEGREE. [7.
Whence we have
EF
FD
DE
+
D(X-A) + EF
+
E(X-B) + FD
−1 = 0, (48)
F(XC) + DE
which is a cubic equation in terms of x, and is indeed the same
as (44)*.
And the values of the corresponding direction-cosines may
thus be found: from (42) we have
(A−A')α1 + Faz +
E α3
0,
Fa + (B−A′) α₂ +
Da3
0,
(49)
Ε α
+ Dag + (c-A') αz
0;
and taking these equations two and two together, we have
αι
a2
аз
(B — A′) (C — A′) — D²
2
DE — F (C — A′)
FD-E (B-A)
а1
DE — F (C — A')
а1
a2
аз
(50)
(C — A′) (A — A′) — E²
EF — D (A — A')
a2
a3
FD — E (B — A′)
Also we have
а 1 az
DE - F (C-A)
EF — D (A — A′)
(A-A') (B-A) - F2
from any one of which the direction-cosines corresponding to
A' may be found.
απ
2
(B — A′) (C — A′) — D²
a²
a, 2
а1 аз
FD-E (B-A')
az²
2
(C — A′) (A — A′) — E²
(A — A′) (B — A′) — F²
(51)
1
(B — A'′) (C — A′) + (C — A′) (A — A′) + (A — A′) (B — A′) — D² — E ² — F²
(52)
1
(B′ — A′) (C′ — A')
(53)
the denominators of (52) and (53) being equal, by reason of
(46).
Also from (50) we have
1
1
1
A1 A2 A3
=
EF-D (A — A′)
FD — E (B — A′)
—
DE — F (C — A′)
; (54)
this last system is also evident by reason of (47).
* On the proof that all the roots of (44) and (48) are real, see also a paper
by Kummer in Crelle's Journal, Vol. xxvi. p. 268.
8.]
11
PRINCIPAL AXES AND PRINCIPAL PLANES.
We have thus two systems of symmetrical equations for de-
termining the values of a1, a2, as which correspond to a'. In a
similar way two symmetrical systems may be determined in
terms of B' and c', whereby the corresponding values of b₁, b₂, bз,
C1, C2, C3 will be found; and therefore generally as these values
will be determinate, so will the position of the three lines per-
pendicular to each other to which A', B', c' correspond be also
determinate; and the equation to the surface will be of the
form
(55)
2
2
A´§² + B'n² + c´§² + K = 0.
This is the most simple form to which the equation of a cen-
tral surface of the second degree can be reduced. The three
rectangular axes to which it is referred are called Principal Axes.
These names are specially given to those parts of the coordinate
axes which are intercepted between the origin and the surface.
The three planes passing through the centre, which are per-
pendicular to the principal axes, are called Principal Planes:
they are the three coordinate planes of the equation (55).
8.] As the equation of a central surface of the second degree
will be applied hereafter for the purpose of illustrating certain
mechanical laws, it is necessary also to demonstrate other pro-
perties of principal axes and principal planes. In the first place
I shall shew that the central radii vectores of these surfaces
which coincide with the principal axes have singular values;
that is, are maxima or minima, either totally or partially.
Let us take the equation (28) to be the equation to central
surfaces; and let (x, y, z) on its surface be the extremity of a
central radius vector r; then
p² = x² + y² + ~²;
and as r is to have a singular value,
r Dr = x dx+y dy +≈ dz = 0:
(56)
but the differentials of these variables are connected also by the
differential of (28), whereby we have
(Ax+FY + Ez) dx + (Fx + By + Dz) dy + (Ex + Dy + cz) dz = 0; (57)
and therefore from (56) and (57),
AX+FY + E≈
FX+ BY + D≈
Ex + DY + Cz
(58)
X
Y
Z
C 2
12
[8.
PRINCIPAL AXES AND PRINCIPAL PLANES.
Let l, m, n be the direction-cosines of the singular radius vector
r: so that
and (58) becomes
X
Y
W
7
M
n
Al+FM+ EN
Fl+BM + DN
El+Dm+CN
m
N
(59)
Now these equations are in form identical with (40); and there-
fore the singular radii vectores are coincident with the principal
axes; that is, with those lines for which, when taken as coor-
dinate axes, the terms in the equation to the surface involving
ns, CE, Én disappear.
Let each term of (59) be equal to s; so that we have
S,
Al² + Bm² + cn²+2Dmn +2 Enl+2rlm = 8,
and (A-8)+ Fm +
En
0,
0,
Fl+ (B-8)m + D N
El + Dm +(cs)n = 0;.
whence we have the cubic equation
2
(60)
(A—S) (B — S) (C—s) — D² (A—s) — E² (B-s) - F² (c-s) + 2 DEF = 0, (61)
:
which is identical with (44) and of which therefore the three
real roots are A', B', c', which are given by the equations (35);
and the corresponding values of l, m, n are a₁, b1, C1 ; ɑ2, b2, C2 ;
ɑ3, b3, c3, because the equations for the determination of the three
different values of l, m, n which correspond to the three roots
of (61) are the same as those by which, in the preceding
Article, the direction-cosines of the principal axes have been
determined.
It is also evident from the form of the equation that the
three singular radii vectores are at right angles to each other.
Let us take the cubic which arises from (60) in the form given
in (48) and let us take the equations which correspond to
B′ and c'; whereby we have
:
EF
FD
DE
+
+
− 1 = 0,
D (B' — A) + EF
E (B' — B) + FD
F (B' — C) + DE
EF
FD
DE
+
+
− 1 = 0;
D (C′ — A) + EF E (C′ — B) + FD
and subtracting the latter from the former, we have
F (C' — C) + DE
9.]
13
PRINCIPAL AXES AND PRINCIPAL PLANES.
1
(c' — B') }{ { D (B'— A) + E F } {D (C'— A) + EF}
+
+
1
{E (B'— B) + FD} {E(C'— B)+FD}
1
{F(B'— C)+DE} {F(C'— c)+DE}
}
= 0; (62)
and by reason of the equations which are analogous to (54) this
becomes
(C'— B′) { b₁ C1+b₂ C2 +b3C3} = 0;
and as B' is not generally equal to c', we must have
b₁ C1 + bz C2 + b3 C3 = 0;
2
and therefore the two corresponding singular radii vectores are
perpendicular to each other: in the same manner it may be
shewn that the other singular radius vector is at right angles to
each of these: so that the three form a system of rectangular
axes; and if the equation to the surface is referred to them as
its coordinate axes, its equation is (55).
It is also evident that the normals at the points where these
principal axes meet the surface are coincident with the axes.
9.] The theory of principal axes and planes may also be de-
rived from another property of surfaces of the second degree.
I shall in the first place demonstrate that the locus of the middle
points of a system of parallel chords is a plane.
Let us take (28) to be the equation to the surface; and let
the equations to one of a system of parallel chords be
X x
1
y—y'
m
r,
п
where (x, y, z) is a point on the surface, (x', y', z′) is a point
through which the chord passes, and r is the distance between
these two points. For x, y, z in (28) let us substitute x'+lr,
y' + mr, z' + nr respectively, and let us arrange the result in
terms of r; then (28) becomes
r² {Al²+Bm² + cn² + 2Dmn +2 Enl+2Flm}
+2r{(Al+Fm+En) x' + (Fl+BM+DN) y′+ (El+DM+CN) z' }
+Ax²²+BY²² + cz'2+2 Dyz' + 2EZ'x' + 2 Fx'y' + K = 0.
=
(63)
This is a quadratic equation in terms of r, and has two roots;
and therefore from a point (x', y', ') two radii vectores can be
drawn to a surface of the second degree along the same straight
line.
14
[9.
PRINCIPAL AXES AND PRINCIPAL PLANES.
Let us suppose (x', y', z') to be the middle point of the chord :
then the two values of r are equal and of opposite directions
and signs, and consequently the term involving the first power
of r in (63) must vanish: hence we have the condition
(Al+FM+EN) x' + (Fl+BM+DN) y′+ (El+DM+cn)z′ = 0; (64)
As the chords are all parallel to each other, and to a central
radius vector whose equations are
Y
X
m
n
(65)
l, m, n are constant: therefore (64) is the equation to a plane
passing through the centre, of which the current coordinates
are x', y', z': and therefore the middle points of a system of
parallel chords is a plane passing through the centre of the
surface.
The plane and the line whose equations are (64) and (65) are
called relatively to each other a conjugate plane and a conju-
gate diameter.
Now it is evident that generally a diameter will not be per-
pendicular to its conjugate plane. Let us examine whether this
relation between them is ever possible; and, if so, the circum-
stances under which it may exist.
If (65) is perpendicular to the plane (64),
Al + Fm + En
7
F l + B M + D N E l + Dm + c n
M
n
= Al² + Bm² + cn² + 2Dmn+2 Enl+2 Flm
= 8,
where s is the coefficient of r² in (63).
(66)
(67)
As (66) are identical with (59) they involve similar conclusions.
There are therefore three diameters which are respectively
perpendicular to their conjugate planes; and these are the prin-
cipal diameters, and their conjugate planes are the principal
planes of the surface.
We have thus considered the properties of principal axes
under three different aspects: (1) if the surface is referred to
them as coordinate axes, its equation takes the form (36), and
has no term containing the products of the variables: (2) they
are singular radii vectores: (3) they are diameters which are
perpendicular to their conjugate planes.
Principal axes may be defined by either one of these pro-
10.] VARIETIES OF SURFACES OF THE SECOND DEGREE. 15
perties; and all three mutually involve each other; and are in
fact identical in the geometrical conception of infinitesimals.
10.] Let us next consider those cases in which the roots of
the cubic equation (44) have particular values.
(1) Let two roots be equal: say, let A'B'; then (46) vanishes
when X A'B'; and we have
(B-A') (C-A') + (C—A′) (A—A') + (A—A') (B —A') — D² - E²-F² = 0. (68)
3
Also as α1, ɑ2, α3, b₁, b₂, bз cannot be infinite, by reason of (52) we
must have
(B — A′) (C — A′) — D² — (C — a′ ) (a — A ′) — E² — (A — A′) (B — A′) — F²
=
0; (69)
EF
FD
DE
that is, A-A':
B-A'
C
c— A' =
;
(70)
D
E
F
and consequently,
A = A
EF
D
FD
DE
- B
=
;
(71)
E
F
3
in which case the direction-cosines a₁, α2, α, and similarly the
direction-cosines b₁, ba, b, are indeterminate; but C1, C2, C3, which
correspond to the unequal root c', are determinate as here-
tofore. In this case the equation to the surface is
c'
a′ (§² + n²) + c´¿² + K =
0.
(72)
The principal axis of being determinate, any two axes in
the plane of (§, n) perpendicular to each other are the other prin-
cipal axes. Equation (72), in this case, represents a surface of
revolution, whose axis of revolution is the - axis. If a' were the
unequal root of (44), and B'= c', then the reduced equation is
A'§² + c′(n² +5²) + K = 0,
=
(73)
which also represents a surface of revolution: the axis of έ is
the determined axis, and is the axis of revolution of the surface,
the position of the other two axes being indeterminate.
(2) Let all the roots of (44) be equal: that is, let a′= B′= c':
then differentiating (69),
2A′ = B + C = C + A = A + B ;
C+A=A+B;
and from (69),
A = B = c;
DE = F =
0;
=
(74)
(75)
... A'B' c′ = a;
and then all the direction-cosines are indeterminate, and any
16
VARIETIES OF SURFACES OF THE SECOND DEGREE. [10.
system of rectangular axes originating at the centre is a system
of principal axes. And the equation to the surface is
K
ε² + n² + 5² + ==
2
= 0;
A
(76)
and if Ka'A', this is the equation to a sphere whose radius
is a.
=
(3) Let one root of (44) be zero; say, c'= 0; then we have
ABC-AD²-BE2-CF2+2 DEF =
and the equation to the surface becomes
A'§² + B'n² + K =
0 ;
0 ;
(77)
(78)
and therefore = 0. As (77) is the same expression as (27), the
centre of the surface is at an infinite distance. Also (78) is
the equation of a central conic in the plane of (§, n); therefore
the surface is a cylinder whose trace on the plane of (, ) is.
the conic (78).
(4) Let two roots of (44) be zero; say, B'= c'= 0; then,
besides the condition (77), we have
BC+CA+AB-D2-E2-F² = 0;
and the equation to the surface becomes
A'§²
A' &² + K = 0;
(79)
(80)
and as 7=5=0, it represents two planes parallel to the plane
of (n, S.)
(5) Let all the roots of (44) be zero; so that A'B'C' = 0;
then the equation to the surface becomes
K = 0;
which represents a plane at an infinite distance.
(81)
We may express these several equations in a more convenient
form.
If ▾ does not vanish, and if v′ = 0, in which case the sur-
face is central, and the constant K in the reduced equation
(36) vanishes, then we have
A´§² + B'n² + c´§² = 0;
(82)
and if a′, B', c′ are all positive, the only values of έ, ŋ, Ŝ which
satisfy the equation are έ==(=0; that is, the equation
represents a point at the origin.
10.] VARIETIES OF SURFACES OF THE SECOND DEGREE. 17
If one coefficient, say c', is negative, and a' and в' are posi-
tive, then if
(82) becomes
y.
1
1
a²,
B′
1
b2
c'
C
25
2ع
n²
+
a²
b2
C2
0 ;
(83)
which is the equation to an elliptical cone, and all plane sec-
tions of it perpendicular to the axis of are ellipses; and if
a=b, the surface is a right circular cone, whose axis of rota-
tion is the axis of .
If however v' does not vanish, the equation to the surface is
2
A'§² + B'n² + c'С² + K = 0.
(84)
If A', B', c' and K are all positive, the equation does not admit
of geometrical interpretation. Let us therefore assume к to be
negative: so that with obvious substitutions, and with all the
varieties of sign which the quantities admit of, the equation
may take either of the forms,
2
172
im alam al
+
+
+
1,
b2
c2
(85)
n2
१
1,
62
C2
(86)
n²
a²
1;
b2 c2
(87)
which severally represent an ellipsoid, a hyperboloid of one sheet,
and a hyperboloid of two sheets. Let us assume a>b>c; then
of (85) degenerate species are, (1) an oblate spheroid, when
a = b; (2) a prolate spheroid, when b= c; (3) a sphere, when
a = b = c. And if a = b, the hyperboloids of revolution are par-
ticular species of (86) and (87). We have not however space or
occasion to enter into all these particulars, or into the nature
and forms of the surfaces. This information must be obtained
from treatises wherein the properties of these surfaces are spe-
cially treated of. The omission of this and similar matter is
necessarily incidental to preliminary chapters which must be
incomplete. It is our intention to demonstrate for the most
part only those geometrical theorems which will be required in
the sequel; not because other theorems are in themselves un-
important, but because the interpretation of our mechanical
results will not require them.
PRICE, VOL. IV.
D
18 CONES OF THE SECOND DEGREE. RECIPROCAL CONES. [II.
11.] I will first take the cone whose equation is (83); but
will, for convenience of symbols, use, y, z for the coordinates
of any point on its surface; so that its equation is
x2
+
a2
y2 ≈2
b2
0 ;
(88)
and I shall observe with respect to it that the vertex is its
centre, and that, like other surfaces of the second order which
are represented by (84), it has three principal axes; which are
respectively the axis of the cone, commonly so called, or, as we
may call it, the internal axis; and two lines through the vertex
which are respectively parallel to the major and minor axes
of those elliptic sections whose planes are perpendicular to the
axis of the cone; these are called the external axes.
The plane which contains the internal axis and the major
axis of the principal elliptic sections is the plane of greatest
section of the cone; and that which contains the internal axis
and the minor axis of the elliptic sections is the plane of least
section. The principal axes of a cone of the second degree,
when its equation is given in the most general form, are de-
termined by the process of Art. 6; for this is applicable to
equation (28), whether K' 0 or not.
At the vertex of the cone (88) let straight lines be drawn
perpendicular to the tangent planes, these will all lie in a
second cone which is called the supplementary or reciprocal
cone; its equation may thus be found. The equation to the
tangent plane of (88) is
x E
+
a2 b2
yn 25
C2
=
0 ;
(89)
and the equations to the line through the vertex perpendicular
to it are
2
a² §
b2n
X
Y
c²š
(90)
therefore squaring these, and multiplying the terms of (88) re-
spectively by them, we have
a² §² + b² n² - c²¿² = 0;
(91)
which is the equation required, and represents a cone of the
second degree, which has the same internal axis as (88), but
whose major and minor external axes are respectively the minor
and major external axes of (88).
For the construction of the second cone (91) it is evident that
as every line through its vertex is perpendicular to a tangent
12.] CONES OF THE SECOND DEGREE. CYCLIC PLANES. 19
plane of (88), so is every tangent plane of (91) perpendicular to
a line through the vertex of (88). Thus the cones have cor-
responding or reciprocal properties, and to a tangent plane of the
first cone and to its line of contact correspond a line through
the vertex of the second and the tangent plane along that line.
Again, to the first cone let two tangent planes be drawn, and
let the corresponding lines on the second cone be taken; the
plane containing these lines is perpendicular to the line of
intersection of the two tangent planes of the former cone; and
the tangent planes of the second cone along these lines are per-
pendicular to the lines on the first cone; and their line of
intersection is perpendicular to the plane through the lines of
contact on the first cone, so that to a line and its polar plane on
the first cone correspond a plane and its polar relatively to the
second cone.
It is evident therefore that properties relative to the angles
contained by certain planes and right lines on the first cone will
give rise to properties of corresponding right lines and planes
in the second cone; in other words the properties of cones of
the second degree are double; and the principle of duality is
established.
12.] Assuming that a cone admits of being cut by a plane
such that the section is a circle, I propose to determine the
position of the circular planes of (88).
Let us suppose a² to be greater than b²; and let (88) be put
into the form
x² + y² + ≈² = m² (~² — n² y²),
as we may do by the following substitutions;
(92)
m²
a² + c²
c2
;
n²
c² a² - b²
b² a² + c²
(93)
Then (92) may be resolved into the two factors.
(x² + y² + 2²) = km (≈—ny),
y²+ ì
M
(x² + y² +≈²) ³
(~+ny);
k
and as k is an undetermined quantity, these are satisfied by
(1)
x² + y²+~² = 0,
≈ = ny,
(2) x²+ y²+≈² = 0,
ny;
(94)
(95)
each of which pairs of equations represents a plane section of a
sphere; that is, represents a circle;
the cone made by the two planes,
so that the sections of
ny and ≈ =
t
ny, are
z
D 2
20
[13.
PROPERTIES OF THE ELLIPSOID.
circles; and it is evident that the sections made by all planes
parallel to either of these will also be circles; for this reason
these planes are called the cyclic planes of the cone, and their
equations are
C a² — b² 3
ba²² + c²
≈ = +
y;
(96)
they pass through the axis of a which is parallel to the major
axis of the elliptic sections perpendicular to the axis of the
cone; and they make with the plane of (x,y)
C a² - b² /
± tan-1
ba²
a² +
+ c²
(97)
Now if through the vertex of the cone (88) two lines are
drawn perpendicular to these cyclic planes, as the line of inter-
section of the cyclic planes is perpendicular to the plane of least
section of (88), so will these lines lie in the plane of the greatest
section of (91); and because every plane perpendicular to one
of these right lines cuts the reciprocal one in a conic, one of
whose foci is on this right line, these lines are called focal lines.
The analytical proof of this property of focal lines is contained
in the preceding equations; a geometrical proof will be found
in the Memoir of M. Chasles, entitled, "Sur les propriétés des
cones du second degré," and contained in the VIth volume of the
Memoirs of the Royal Academy of Brussels.
13.] We must also investigate certain properties of the ellip-
soid, as we shall frequently require this surface for the purposes
of illustration and interpretation, but our description of the
properties will be incomplete, because we shall demonstrate
those only which are wanted hereafter.
Let the equation to the ellipsoid be
x2 y2 z2
+ +
a2 b2
૮૭
1,
(98)
where a² > b2>c2; so that a is the greatest, and c is the least
of all central radii vectores; b, however, has also a critical value,
for it is the semi-axis minor of the elliptic trace in the plane
of (x, y), and for that plane is a minimum, but it is the semi-axis
major of the elliptic trace in the plane of (y, z), and for that plane
is a maximum; we shall immediately determine lines on the
surface which will indicate the regions for which b is a maximum,
and for which b is a minimum, and the dividing line of these
regions will indicate certain singular positions of central radii
vectores.
13.]
21
CYCLIC PLANES.
Let us in the first place inquire into the position of the
cyclic planes of the ellipsoid; viz. those planes which cut the
ellipsoid in circles, and we will take the central equation (98),
and suppose the plane to pass through the origin; let (98) be
transformed to the system of axes of έ, n, of Art. 2; and let us
suppose the plane of (§, ŋ) to be the cyclic plane; so that from
(2) we have the following substitutions,
y = a 2 &+ b₂n,
and therefore from (98) we have
x = a₁ & + b₁n,
1
2 =
a 3 §+b3n ;
(99)
a₁2
£2 + +
а22
a²
b2
c2
+ 25n {
a bi
а-2b2
аз вз
+
a²
+
b2
c²
S b₁2
+n²
b₂2 b₂2
la²
+
2 b2
+
= 1.
C2
(100)
Now this is to represent a circle; therefore.
2
+
α1² a22 a32
6,2 b22 b32
+
a2
b2
C2
a²
+ +
b2
c2
(101)
a₁b₁
az bz
a3b3
+
a²
+
b²
· 0.
c²
ૐ
(102)
As the positions of the axes of § and ŋ in the plane of (§, ŋ) are
indeterminate, let us suppose the axis of έ to coincide with the
line of intersection of the two planes of (§, ŋ) and of (x, y), so
that a3 = 0; therefore from (7),
a₁ b₁ + a₂ b₂ = 0;
and therefore from (102)
1
1
1 )
a₁ b₁
0 ;
a²
دنا
=
0, or b₁ = 0.
which is satisfied by either a
1
(103)
(1) Let a₁ = 0; therefore a₂ = 1 and b₂ = 0; so that the
plane of section passes through the axis of y; and if 0 is the
angle between the plane of section and the plane of (x, y), so that
=sin 0, b = 0, from (101) we have
b₁
= cos 0, b3 = sin 0,
1
1
b2
tan 0
аё
+
1
c2
B'— A
+
C' - B
>
(104)
(105)
according to the notation of equation (84); so that there are
་
22
PROPERTIES OF THE ELLIPSOID. CYCLIC PLANES. [13.
two planes of section equally inclined to the plane of (x, y), and
passing through the axis of y. The mean principal axis of the
ellipsoid is evidently the radius of the circle, and the cyclic
planes are related to the ellipsoid in the manner indicated by
the lines in Fig. 2, where UOA = U′oa' ◊ ; ou = ou′ = b.
Thus all the central radii vectores in these two cyclic planes
are equal; and are equal to the mean semi-axis of the ellipsoid.
For all parts of the surface of the ellipsoid contained between
these two planes towards the maximum axis, the radii vectores
are greater than b; and for all parts towards the minimum axis,
the radii vectores are less than b; these cyclic planes therefore
divide the ellipsoid into four parts, corresponding to two of
which at B b is a minimum, and corresponding to the other two
it is a maximum.
Also all planes parallel to the central cyclic planes deter-
mined by (104) are cyclic planes; for the conditions involved
in (101) and (102) depend only on the coefficients of x2, y2, z²
in (98), and these quantities are not changed by a change of
origin, if the new axes are parallel to the old. And the four
points at which planes touch the ellipsoid are the umbilics.
This is also manifest from Ex. I. Art. 408. Vol. I. Ed. 2.
2
3
(2) Let b₁ = 0; therefore ab₂ = 0; and b²² + bå³ = 1; either
A₂ = 0, or b₂ = 0.
аг
Let a₂ = 0; therefore a₁ = 1, so that the cutting plane passes
through the axis of x; and (101) becomes
1 b₂2
+
a²
b2
2
1 - b₂²
c2
;
b₂2
a²bc-b² c²
a² b² — a² c²;
which is greater than unity; and therefore this result is im-
possible.
2
Let b₂ = 0; therefore b3 = 1; a₁²+a² = 1; and the cutting
plane passes through the axis of z; then (101) becomes
а12 1-a₁2
1
+
..
a²
b2
c2
a² (b²-c²)
2
a² =
c² (b² — a²)
which is a negative quantity; and therefore a₁ is impossible.
The only cyclic planes then are those determined in the first
case, and these two planes pass through the mean principal axis
of the ellipsoid.
The cyclic planes of a cone, and indeed of all the surfaces of
the second order, may be determined by the method of this
Article.
14.] THE ELLIPSOID.
CONJUGATE AXES AND PLANES. 23
I may observe also that the process which has been applied
to the cone is also applicable to the ellipsoid; for the equation
(98) may be thrown into the form
k² (x² + y²+z² —b²) =
where
k² =
a²
a² - b²,
-
x² — n² z²,
a² (b²- c²)
n²
b² (a² — b²)
and then the equation to the ellipsoid is satisfied by the pairs of
simultaneous equations
(1) x² + y²+z² = b²,
(2) x² + y²+≈² = b²,
x = NZ;
x =
nz.
Each of which pair represents a plane section of the sphere
whose radius is b, and therefore represents a circular section of
the surface.
14.] Let us also inquire somewhat briefly into the relations
which exist between an axis and its conjugate plane relatively
to the ellipsoid (98). We have already investigated the condi-
tion which generally exists between a radius vector and the
plane which bisects all chords of the ellipsoid which are parallel
to that axis; but some further properties of axes in conjugate
relations to each other will be required in the sequel.
Let the equations to a radius vector be
X
Y
2
;
7
M
n
(106)
then, by reason of equation (64), the equation to its conjugate
plane is
lx my nz
+
+
0;
a2
12
c2
and therefore if the equation to a plane is
LX+My+NZ = 0,
the equations to the axis conjugate to it are
(107)
(108)
Z
X
Y
a² L
b² M
C² N
(109)
if the line (106) meets the surface of the ellipsoid at the point
(x', y', z′), then we have
Z
m
ጎ
x'
y
yy
+
+
a²
b2
0;
c²
(110)
and (107) becomes
X X
૧૩.
24
CONJUGATE AXES AND PLANES. [14.
THE ELLIPSOID.
which is evidently the plane parallel to that which touches the
ellipsoid at (x, y, z); so that if a tangent plane be drawn to an
ellipsoid at a given point, the central plane parallel to that plane
is conjugate to the axis drawn to the point of contact.
Now if three axes of an ellipsoid are such that each is the
axis conjugate to the plane which contains the other two, these
lines form a system of conjugate axes. And if three planes are
such that the line of intersection of any two is the conjugate
axis of the third, these planes form a system of conjugate planes.
Of such systems we have already had an instance in the princi-
pal axes and the principal planes. Let us determine the rela-
tions which exist generally between these lines and planes.
Let (x1,y1: 21) (X2, Y2, Z2) (X3, y3, Z3) be the three points on
the ellipsoid to which the system of conjugate axes corresponds;
so that the equations to the three axes are
X
Y
Z
X1
Y1
21
X
Y
Z
X 2
Y2
يم
Y
༧
N
X3 Y3
23
and the equations to the conjugate planes are
X X 1
a²
+
X X z
+
a²
Y Y 1
b2
Y Y 2
b2
+ 221
c²
=
0,
+ 223 = 0,
c²
ZZ3
X X 3
+
a²
YY 3
b2
+ 223 = 0.
شه
1
(111)
(112)
But since the first of (111) coincides with the line of intersection
of the second and third of (112), we have
X1
a² (Y2Z3-Z2Y3)
Y1
21
(113)
b² (Z2 X 3 — X 2 Z3)
c² ( X 2 Y 3+ Y 2 X3)
which equations are equivalent to the two equations,
X1 X2
a²
Y1Y2 2122
+
+
0,
b2
c2
X1 X3
Y1 Y3
2123
+
+
= 0;
a²
b2
c²
(114)
and as the other two lines of (111) must coincide with the lines
15.]
25
THE ELLIPSOID. CONJUGATE AXES AND PLANES.
of intersection of the other planes of (112), we shall, in addition
to (114), have also the equation
X2 X3
a2
Y2Y3 2223
+
b2
C2
+ = 0
0;
(115)
these are three relations between the coordinates of the ex-
tremities of three conjugate axes.
By a similar process it may be shewn, that if
L₁ x + M₁Y + N₁≈ = 0,
1
LqX + MqY + Ng≈ = 0,
LgX+Mgy + Ng≈ = 0,
(116)
are the equations to three planes of a conjugate system, then
a² L2 L3 + b² M₂ M3+c² N2 N3 = 0,
2
a² L3 L₁+b² M3 M₁ + c² N3 N₁ = 0,
Lz 1
1
N1
a² L1 L2+ b² M₁ M2+ c² N₁ N₂ = 0.
(117)
The equations (114) and (115) contain apparently nine un-
known quantities, but as the equations are homogeneous these
are equivalent to only four; and as these are subject to only three
conditions, the system is indeterminate; the number of systems
of conjugate axes is therefore unlimited; if, however, one axis
is given the other two are determinate.
15.] Again, a system of conjugate axes may be defined by
the following equations;
x₁ = al₁,
X1 α
Y₁ = bm₂,
1
:}
(118)
X 2 = al 2,
X3
=
a l 39
Y 2
= bm2,
Y
y 3 =
bm3,
Z 2 = C N 2 ;
спа
≈3 =
c nzi
2
1₁² + m² ² + n₂²
2
2
2
1
1₂² + m² ² + n₂ ²
2
13² + m² ²+n²² = 1. (119)
21 = c n 1 ;
in which cases the equation of the ellipse gives
and from (114) and (115) we have
lq l z + MzMz + Nz Nz = = zl1
Z z lj + m z my + n z n₁ = l₂ ↳½ + m² m² + n₁ n₂ = 0; (120)
3
1
l2 m2
and from these six equations we have the inverse systems
2
2
2
1½² + 1½² + b² ² m₁² + m₂² + m²²
2
2
2
=
n²² + n² + n²² = 1; (121)
l3M3
N..
2
M1 N1 +MzNq+MzNz = N1l1+Nqlq+Nz lz = lym₂ + l₂ M₂+lzMz = 0. (122)
Also we have theorems analogous to (11), (12), and (13) of
Art. 2. Now these relations are useful for the proof of many
properties of conjugate axes; thus, let r1, 72, 73 be three conju-
gate axes; then
2
2
r3
r²² + r²² + r²² = a² (1½² +12² +13²) +b² (m¸² +m²² +m²²)
2
1
2
2
3
2
2
+ c² (n₂ ² + n₂² + n3²)
= a² + b² + c²;
(123)
PRICE, VOL. IV.
E
26
[16.
THE ELLIPSOID.
CENTRAL RADII VECTORES.
that is, the sum of the squares of three conjugate axes is con-
stant.
16.] Let there be three central radii vectores of an ellipsoid
mutually at right angles to each other; then the sum of the
squares of their reciprocals is constant.
1
Let r₁, 72, 73 be the three central radii vectores, of which let
the direction cosines be (l, m, n1), (l2, M2, N2), (l3, M3, ng); then
we have
1,2 m,2
+
1
n, 2
+
2
ན་ གྭ
a²
b2
c2
T
1
72
2
m₂
nz
n₂2
+
+
(124)
2
a²
b2
C2
2
7,કૈ
2
mz
+
+
nz
2
;
2
3
a²
b2
c²
2°
1
and therefore by addition
1
1
1
1 1
1
r.
2
+
2
+
12
r r3²
2
a²
+ +
b2 c2
(125)
17.] The normal and the tangent plane drawn at any point
of a central surface of the second degree meet each of the prin-
cipal planes at a point and along a straight line respectively,
which are such that in each principal plane the point is the pole,
and the straight line is the corresponding polar, relatively to
a certain determinate conic in that principal plane.
Let us take the ellipsoid whose equation is
x2 y2 ≈2
+ + = 1,
a2 b2 c²
(126)
to be the typical case; and consider the normal and the tangent
plane at the point (x, y, z); and let us also take the principal
plane of (x, y), which is that of the greatest and mean principal
axes. Then the normal pierces this plane at
a² - c² b² — c²
a²
X,
b2
y);
and the equation to the line of intersection of the tangent plane
and the plane of (x, y) is
this is evidently the polar of the pole
tively to the conic
αξ yn
a²
+ = 1;
b2
(127)
a²- c²
a²
b² - c²
X,
b2
y) rela-
x²
a² - c²
y2
+
1.
b2 - c²
(128)
18.]
27
THE ELLIPSOID. CONFOCAL SURFACES.
By a similar process we may shew that the conics in the
other principal planes are expressed by the equations
x2
22
+
1.
a² - b²
c²-b2
y2
22
+
1
b² — a² c² — a²
;
(129)
(130)
of these equations (128) is that of an ellipse in the plane of (x, y);
(129) of an hyperbola in the plane of (x, z); and (130) of a curve
which is wholly imaginary in the plane of (y, z). These curves
are called the excentrical or the focal conics of the ellipsoid
(126); and for this reason; the vertices of (128) are the foci of
the elliptic sections of the ellipsoid by the principal planes of (y, ≈)
and (x, z); and the foci of it are the foci of the elliptic section
of the plane of (x, y): also the vertices of (129) are the foci of the
elliptic sections of the ellipsoid in the planes of (x,y) and of (z, y);
and the foci are the foci of the elliptic section made by the
principal plane of (x, z). The third curve is imaginary, although
its foci are, as in the other two cases, real. Also the hyperbola
(129) passes through the umbilics of the ellipsoid.
18.] Now we call those surfaces of the second degree confocal,
the principal sections of which are confocal; hence it appears
that all surfaces of the second degree, which have the same focal
conics, are confocal.
Thus the general equation of all surfaces of the second degree
confocal with (126) is
x2
+
+
a² + 0
y2
b² +0 c² + o
~2
1;
(131)
the equations to whose focal conics are (128), (129), and
(130). And if (131) passes through a given point (x', y', ~), we
have from it the cubic equation in
(0+ a²) (0+b²) (0 + c²) − x² ² (0 + b²) (0 + c²) − y'²² (0 + c²)(0 + a²)
2
—¿² (0+ a²)(0+ b²) = 0 ;
in which if we substitute for ◊ successively +∞, −c², —b², —a²,
the results are severally +,
respectively between + and -c2,
,
to
; so that the roots lie
c2 and -b2, b2 and -a²;
in which cases (131) represents respectively an ellipsoid, a hy-
perboloid of one sheet, and a hyperboloid of two sheets. Thus,
at the point (x, y, z) these three confocal surfaces intersect.
We have also proved (see Vol. I. Art. 411. Ed. 2.) that they
intersect at right angles and along their lines of curvature.
E 2
28
19.]
THE ELLIPSOID.
Thus, at the common points of intersection of these three sur-
faces, their normals are at right angles at each other. It is also
evident that these surfaces intersect in eight points, one in each
octant of space about their centre.
Now if 0
c², (131) requires that z = 0, and we have
x2
+
y²
2
a²- c² b² — c²
1,
which is the equation to the focal conic in the plane of (x, y);
similarly if
b², and if 0 = a², we have the focal conics
in the planes of (x, z) and (y, z) respectively; whence it appears
that the focal conics are only particular cases of the surface of
the second order confocal with (126.)
And therefore surfaces which are confocal may also be de-
scribed as those which have the same focal conics.
19.] If from any point (§, n, Ŝ) an enveloping cone is drawn
to the ellipsoid (126), the principal axes of that cone coincide
with the normals of the three confocal surfaces of the second
degree which intersect at the vertex of the cone.
By Ex. 2. Art. 355. Vol. I. Ed. II. the equation to the cone
whose vertex is (έ, n, ), and which envelopes the ellipsoid
(126), is
2ع
2
-1}³-
a²
2
c2
y2
b2
{ = + + ² - 1 } - ( C + Z + −1) (+ + −1) = 0. (132)
x
a²
yn
b2
n2
2
2
a²
n²
+ +
b2 c2
2
1 = K ;
For the sake of abbreviation let
22
c²
(133)
so that (132) on expansion becomes
कर
(-) + (-)을 ​+(을​)를
​a²
+
K
215 255
b² cz y z + c² az z x +
Y
2 &n
a ² b x x Y +
0; (134)
the other terms being omitted because the position of the prin-
cipal axes of the cone depends on the first six terms only of the
expanded equation.
In this case equation (48) becomes
¿2
a² (a²x + K)
also from (133) we have
2
دع
a² K
n²
2
+
+
b² (b²x+K)
n²
+
+
b² K
2ع
c² (c²x+K)
1
;
2
c² K
=
1 +
K
1; (135)
19.]
29
CONFOCAL SURFACES.
therefore by subtraction
हुर
2
n²
(2
+
+
= 1.
(136)
K
K
K
a² +
b² +
c² +
X
X
X
Now κ and x are functions of the coordinates of the given
vertex (§, n, §) and are therefore known: hence if we describe
the surface of the second degree whose equation is
x2
y²
22
+
+
=
1,
(137)
K
K
a² +
b² +
c² +
X
X
X
K
(136) shews that that surface passes through the vertex of the
enveloping cone; and this surface is evidently confocal with
the original ellipsoid (126); and as x has three values which
are the roots of (135), so, as we have shewn in Art. 18, the
equation (137) represents three surfaces which are severally
an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of
two sheets, all of which are confocal with (126); and which
intersect orthogonally at (έ, n, §).
For the determination of the principal axes of the cone, let us
take the system of direction-cosines of Art. 6, and the forms
of them given in (54); in the case of (134) these take the
following values
α1
ૐ
(a² +
K
+ ===
=
X
a2
η
(ε³ +
K
a3
X
5
(c²+
K
; (138)
X
and to fix our thoughts let us suppose x in this equation to be
that root of (135) which corresponds to the ellipsoid. Now the
direction-cosines of the normal to the ellipsoid (137) at the
point (§, n, Š) are proportional to
η,
を
​a² +
C
η
K
K
b² +
c² +
X
X
K
X
and a comparison of these values with (138) shews that the
principal axis (a1, a2, ag), the position of which is determined by
(138), coincides with the normal of the confocal ellipsoid which
passes through the point. This axis is generally the internal
axis of the cone. By a similar process we may shew that
the two external axes of the cone, which correspond to the
two other roots of the cubic, are normal to the two confocal
hyperboloids which intersect at the given vertex.
The normals to these two hyperboloids are, as we have shewn,
30
[20.
THE ELLIPSOID.
tangents to the lines of curvature on the ellipsoid at the point
(§, n, (); and therefore if a cone envelopes an ellipsoid, and
if through the vertex of the cone an ellipsoid be described
confocal with the former ellipsoid, the normal to the ellipsoid,
and the tangents to the two lines of curvature on it, are the
principal axes of the enveloping cone.
M. Chasles has also proved that the generating lines of the
confocal hyperboloid of one sheet which passes through the
vertex are the focal lines of the cone.
Now these same properties are true if instead of the ellipsoid
(126) we had taken any other surface confocal with it; and
therefore are true if the focal conics are the directors of the
cone, because the focal conics are the limiting forms of the
confocal surfaces.
Hence also it follows that if two surfaces of the second degree
have the same focal conic, and if from any point in space as
vertex two cones are described enveloping these surfaces, these
cones have the same principal axes, and the same focal lines.
20.] Another principle of duality also arises from the theory
of reciprocation which has been explained within the limits
of plane geometry in Section 3, Chapter XIII. Vol. I. 2nd Ed.;
and which I must here extend to the geometry of surfaces, so
far as the sequel requires.
Let the equation to an ellipsoid be
2
हुर
a²
←
n²
+ +
b2 c²
1.
(139)
and from every point of it as a pole let the polar plane be taken
relatively to the sphere
x² + y² + z² = r²;
the general equation of the polar plane is
x & + yn+≈5 = p² :
z
we propose to find the envelope of these planes;
(139) and (141), we have
x d § + y d n + z d$ = 0,
5
² de + "/dn+ & α5 = 0;
d}
c²
whence we have
a² x
b²y
c² z
;
हुं
η
5
(140)
(141)
differentiating
and eliminating έ, n, Š, we have
a²x² + b² y² + c²² = pb;
(142)
20.]
31
THE SPHERO-RECIPROCAL.
which is the equation to another ellipsoid, and is called the
sphero-polar reciprocal of (139).
Now it is evident that a tangent plane of (142) corresponds
to a point of (139); and to the intersection of two tangent
planes of (142) corresponds a line passing through the two cor-
responding points of (139); and to a point on (142) corresponds
a tangent plane of (139). Also to a tangent line of (142) cor-
responds a tangent line of (139). These surfaces therefore have
reciprocal properties, and to a plane a line and a point on either,
a point a line and a plane on the other severally correspond, so
that all properties admit of being doubled. It is manifest that
the theory of the reciprocal cone which has been explained
in Art. 11. is a particular case of this principle.
I may also observe that the sphero-polar reciprocal of any
surface of the second degree is also another surface of the second
degree; but as we shall require only the simple form which has
just been discussed, the more general case may be omitted.
32
[21.
CINEMATICS OF A RIGID BODY.
CHAPTER II.
THE CINEMATICS OF A RIGID BODY. ANGULAR VELOCITIES;
THEIR COMPOSITION AND RESOLUTION, AND RELATION TO
LINEAR VELOCITIES.
21.] SEVERAL times in the course of the treatise on Me-
chanics allusion has been made to a division of the subject into
two parts, Cinematics and Dynamics proper. In the former of
these motion and its incidents are discussed apart from all con-
sideration of the action of forces which produce that motion;
the affections of pure motion are investigated; as, for instance,
it is shewn that motion takes place in time and space; that
a particle moves with a certain velocity, and that velocity de-
pends on time and space. In the latter motion is considered as
the effect of certain producing causes, and the relations between
it as the effect and force as the cause are investigated; thus the
laws of motion and the equations of motion belong to Dynamics
proper. In the exposition of the first principles of Dynamics,
Chapter VII. Vol. III, this division of the subject is made, and
the parts are treated separately; but it was unnecessary to bring
the division into more special prominence, because the Cine-
matics of a moving particle do not present to the mind images
difficult of formation. Every one can form a conception, more
or less perfect, of the motion of a single particle; it describes
a certain line, which is its path, and we can easily imagine
that path; it moves with a certain velocity, and if its velocity
varies, it is not difficult to conceive the rate of variation. But
the motion of a system of particles is more complex; we can
indeed follow the path of any one particle of it; it describes
a line, just as if it were not connected with the other particles;
but what is the motion of all the particles of the system re-
latively to it? Let us however confine our attention to the
motion of a rigid body, which is a system of particles of in-
variable form. The body can, as it were, pirouette about any
one particle in all ways, and it is difficult to imagine and to
trace the motion of any other particle. So if a body rotates
about an axis absolutely fixed, we can easily picture to ourselves
22.]
33
MOTION OF ROTATION.
the path described by every particle; it is a circle, the plane of
which is perpendicular to the fixed axis, and the centre of which
is in that axis. But when the body has the most general motion
of which it is capable, our conception is for the most part very
obscure. Hence arises the necessity of resolving that motion
into its simplest elements; so that when we have an adequate
conception of each separate element, we may combine them,
and thus obtain an adequate conception of the motion of the
whole body. We must therefore first discuss the several mo-
tions of which a rigid body is capable, independently of the
forces which produce these motions; and subsequently consider
the relations which subsist between these effects and their
causes. In the present Chapter we shall confine our attention
to the former part, viz. the Cinematics of a rigid body; and
in the following Chapters we shall consider the Dynamics
proper, the fundamental axioms, and the theorems deducible
from them.
22.] Let us imagine a rigid body or a system of material
particles of invariable form to be in motion. The form of this
system will be definite if (1) the distances of three particles
from each other which are not on the same straight line are
given; (2) the distances of every other particle from each of
these three particles is given; so that, as the system is rigid, if
the positions of the first three particles are determined, the
position of every other particle is also determined, and that
of the whole body is also known. Thus it is sufficient for us to
consider the motion of the first three particles.
Let the three particles of the system which form a triangle,
and relatively to which every other particle is known, be P, Q, R,
and let these be joined by straight lines. Now if the motion is
such that the sides of this triangle are always parallel to their
original positions, it is plain that the line joining any other
point to each of these three points is also parallel to its original
position; such a motion is said to be a motion of translation
of the body or system. In this case the paths of all particles
are equal and parallel lines, and are described with equal and
parallel velocities; and the motion of the whole body will be
easily inferred from that of any one particle. As the incidents
of such a motion have been fully discussed in the previous
volume of our work it is unnecessary to say more on this part of
the subject.
PRICE, VOL. IV.
F
34
[22.
MOTION OF ROTATION.
If however the paths described by the several particles are
not equal and parallel the body has another motion besides that
of translation. Let us consider its nature.
Now let P, Q, R be the positions of the three particles of the
body, to which the position of every other particle is referred at
a given time; and let r', q', R' be their positions after a certain
motion; suppose moreover that the paths described by these
three particles in their motion are not equal and parallel; we
may analyse the motion by the following process: first let all
the three particles be moved so as to describe paths equal and
parallel to that described by P; that is, let us suppose a motion
of translation of the whole body such that every particle of
it moves over a space equal and parallel to PP'; let the positions
taken by Q and R after this motion be q", R"; through P', Q", R"
let a plane be drawn, which is manifestly parallel to the original
plane PQR; and also let a plane be drawn through the three
final positions P', Q', R'; let these two planes intersect along the
line P'N; let the body rotate about the line P'N, until the plane
Q"R"P' is brought into the plane q´R´r'; and, if it is necessary,
let the body again rotate about a line passing through P' and
perpendicular to the plane r'q'n', until q" and " coincide.
respectively with Q' and R'; by these several motions the body
will have passed from its first to its final position. The motions
are three; the first is a motion of translation; the other two
are motions of turning or of rotation about certain axes; and as
the motion has been of the most general kind, so may all motion
be resolved into separate motions of the kinds which we have
mentioned.
R
P
{
This motion of rotation requires careful consideration. It
always takes place about a certain straight line or axis. If
a body rotates all points along the axis are at rest so far as
the motion of rotation is concerned; they may move by reason
of other circumstances, but they do not move by reason of
the rotation of the body about that axis; and the straight
line along which the quiescent points are is called the axis of
rotation.
Also this axis may or may not meet the body. If it meets
the body, the particles of the body along the axis are at rest; if
it does not meet the body, all the particles of the body move by
reason of the rotation,
Many rotations about different axes may co-exist; we must
22.]
35.
ANGULAR VELOCITY.
consider how this is, and investigate laws by which these may
be combined into one or more resultants.
Now the most simple rotation is that of a body rotating about
an axis fixed absolutely; that is, relatively to it and to space.
In this case every particle of the body describes a circle in a
plane perpendicular to the axis; and the body being rigid, the
times in which the circles are described are the same for all the
particles; and their relative position is not changed by or during
the motion.
Imagine a particle m at a distance from the axis of a rotat-
ing body; and through the fixed axis and containing it let a
plane be drawn fixed in space; then the position of the particle
may be determined at any instant by means of r and the angle
at which r is inclined to this fixed plane. Thus in Fig. 3, let
P be the place of m at the time t; let oz be the rotation-axis,
fixed relatively to the body and to space; through it let the
plane cox be drawn, and let it be fixed in space, so that when
the body rotates, this plane as well as the axis remains fixed;
let op be drawn at right angles to o≈; op = r, Pox = 0; then
r and ✪ are sufficient to determine the place of m.
Firstly, let us suppose the body to rotate uniformly about the
axis; that is, let us suppose to receive equal increments in
equal times; let o be the angle by which is increased, that
is through which r revolves, in an unit of time; then if is the
angle through which r has revolved in t units of time,
Ө = wt;
(1)
so that if or coincides with or when t = 0, Pox = 0 = wt.
And from (1) we have
W
Ꮎ
t
(2)
We must enlarge our language; let us take our nomenclature
from that of motion of translation. Since the linear velocity
of a particle moving uniformly is the linear space described
by it in an unit of time; so let the angle through which an
uniformly rotating body rotates in an unit of time be called the
angular velocity of the body. Thus w is the angular velocity of
the body, and is defined mathematically by (2). I must observe
that the angular velocity is independent of r and is the same for
all parts of the body.
Secondly, suppose the body not to rotate uniformly about
the axis, so that the radius vector of any particle does not
F 2
36
[23.
ANGULAR VELOCITY.
describe equal angles in equal times; then the angular velocity
is a function of the time. Let the time be resolved into in-
finitesimal elements; and let us suppose the angular velocity at
the time t to be w, and to be w+do at the time t + dt; and let do
be the angle through which the body has rotated in the time dt.
Then since w is the angular velocity at the beginning of dt, and
w+do is the angular velocity at the end of dt, the mean angular
velocity with which de has been described in w+pdw if is
a proper fraction; and is positive or negative according as the
angular velocity is increasing or decreasing; so that by reason
of (1)
do = (w + & dw) dt;
ω
and omitting the infinitesimal of the second order,
do = wdt;
(3)
thus do is the angle described in dt units of time by the body
rotating with the angular velocity o at the beginning of dt; and
therefore dividing both sides by dt, we have
do
W
;
d t
(4)
and therefore o or
de
d t
is the angle described in an unit of time,
and is the angular velocity of the body.
Thus in both cases, of uniform and of continuously varying
angular velocity, angular velocity is the angle described by the
radius vector of any particle 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 of a body which rotates
through an unit angle in an unit of time; and if the angular
velocity of a body is o, w is a number designating the number of
unit angles through which the body rotates in an unit of time.
23.] Again let a rigid body rotate about a fixed axis; at a
given instant the angular velocity is the same for all particles of
the body; but the linear velocity is evidently not the same for
all; the linear velocity of those at a greater distance from the
axis is greater than of those at a less distance; the relation
between the angular and the linear velocities of a particle is
thus found.
Let us take a particle m at a distance r from the axis. Let
w be the angular velocity, d◊ be the angle described by r in the
24.]
37
ANGULAR VELOCITY.
time dt, and let ds be the space described by m; then ds =rd0;
and
ds
d Ꮎ
= r
dt
dť'
dt
= rw;
(5)
(6)
so that the linear velocity of m is the product of the angular
velocity and the radius of m, and therefore varies directly as the
distance of m from the rotation-axis. If therefore r 1, the
angular velocity is identical with the linear velocity.
24.] Hence is derived the principle on which angular velocities
are measured; if two bodies rotate with angular velocities such
that the particles in each at an unit distance from the axis de-
scribe equal spaces in equal times, the angular velocities of the
bodies being uniform during that time, these angular velocities.
are said to be equal. And this mode of determining equal an-
gular velocities being adopted, it is evident that one angular
velocity may be double, or treble, or n times another. If the
equal spaces are described by each particle in the same direc-
tion, the angular velocities are equal and in the same direction;
but if the equal spaces are described in opposite directions, the
angular velocities are equal and opposite. Angular velocities
may therefore be affected with signs. Thus if w represents the
angular velocity with which a body rotates in a given direction,
- will represent the equal angular velocity of a body 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 are
determined in Statics. 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
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 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 ro-
38
[25.
COMPOSITION AND RESOLUTION
tation be called positive, and let its line-representative be mea-
sured 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. Thus in Fig. 4; let o be the pole and or the rotation-
axis; as we look from o towards let the body rotate as the
hands of a watch which we face; that is, in the direction of the
letters PQRS, then that rotation is positive, and its line-repre-
sentative is to be measured from o towards the right; let oa be
that line, then OA is as to direction and length a representative
of the angular velocity. If, on the contrary, when we look from
o towards a, the rotation of the body is in the opposite direction,
then the line oA is to be measured along or produced back-
wards; that is, oA' is the line-representative of the angular
velocity; this principle of interpretation is in accordance with
that of Art. 251. Vol. I. Ed. 2. I may observe that if we look
from o towards a', that is towards the left instead of the right,
OA is the representative of the rotation in the second case
according to the principle we have adopted, for as we look from
o towards a' the body rotates in the same direction as the hands
of a watch; thus the line-representative is independent of the
direction in which we look from 0. We shall hereafter use
straight lines as adequate representatives of angular velocities.
25.] Thus much as to single angular velocities, and their line.
representatives. We will now investigate the circumstances of a
body which rotates with many simultaneous angular velocities;
that is, we suppose a body to rotate about a determinate axis
with a given angular velocity, and another angular velocity to be
communicated to it; what change of motion is due to the addi-
tion of this new angular velocity, and what is the combined
resultant of the new and the original angular velocities? And
again, we shall suppose other angular velocities to be communi-
cated, and we shall have to determine the resultant of all of
them. At present we say nothing about the source of these
velocities or the mode of communication; we shall consider
only the combined effect of them as expressed angular velocities.
The problem therefore is the composition and resolution of
angular velocities; and we shall in order consider (1) those
which have the same rotation-axes; (2) those whose rotation-
25.]
39
OF ANGULAR VELOCITIES.
axes meet in a point; (3) those whose rotation-axes do not
meet; and, as a special case of (2), those whose rotation-axes
are parallel, that is, meet at an infinite distance.
Let a body rotate about the axis ox, Fig. 5, with two positive
angular velocities wa, w, whose representatives are oA and oв;
let P be the place of any element of the body, which we will
take to be in the plane of the paper. Let PM be drawn from
P at right angles to or, and let PM = r; as both the angular
velocities are positive, all elements of the body lying in the
plane of the paper and above or move from below to above the
plane of the paper; and all elements lying below or move from
above to below; therefore the spaces through which P passes
from below to above the plane of the paper in the time dt are
wɑrdt due to w, and ordt due to w; so that the whole space
over which P passes in dt = (wa + wv)rdt.
Now suppose to be the resultant of w, and w; then if the
body rotates with this velocity, P will pass in the time dt over a
space due to w which is equal to the spaces due to w。 and to wr;
but the space passed over by p due to the angular velocity o in
the time dt is wrdt; therefore
ordt = (wa + wo) rdt
.*. ω
wa + wb;
(7)
that is, the resultant angular velocity is the sum of the two com-
ponent angular velocities.
If one of the components, say wʊ, is negative, the positive
space passed over by P in the time dt will manifestly be
(wa-w)rdt; and we shall have
W
Wit Wb;
(8)
and if wo =
wa, then from (7) w = 0, and two equal and
opposite angular velocities neutralize each other.
Similarly if a body rotates about a given axis with coaxal
angular velocities w1, 2, ,, and if o is the resultant angular
Ω
velocity,
Ω = 1 + ως +
+ One
= Σ.w;
(9)
where z. expresses the algebraical sum of the several component
angular velocities.
In all these cases the line-representative of the resultant is
the algebraical sum of the line-representatives of the com-
ponents.
40
[26.
COMPOSITION AND RESOLUTION
26.] Next let us suppose the body to move with two simul-
taneous angular velocities wa
wa and Wu, whose axes OA and Oв
intersect each other in the point o.
Let us take o to be the pole, and OA, OB, Fig. 6, to be the
line-representatives of the angular velocities waw respectively;
so that due to wa all particles of the body in the plane of the
paper which are above the line oA pass from below to above the
plane of the paper, and all those below OA pass from above
to below; similarly by means of w, all particles to the right
of oв pass from above to below and all those to the left from
below to above. Let us take a point P within the angle BOA,
and from P let PL and PK be drawn perpendicular to oA and OB
respectively; let
let Oмr, MP =
OM =
Y, BOA = y; then PK = sin y,
PL = y sin y. Let us investigate the paths described by p in
the time dt, which are due to these two angular velocities; the
upward path of P due to waway siny dt; and the downward
path of p due to wɩ = wzx sin y dt: so that the resultant upward
path of P = (?! ωα
x w) sin y dt. Now let us suppose P to be
at rest under the effects of the two angular velocities, then
P
wʊ
x
y
;
Θα
Wb
and replacing wa and or by their representatives oA and Oв,
X
O A
y
ов
1
(10)
(11)
but either of these is the equation to a straight line passing
through o; and as all particles along it are at rest, it is the axis
of the resultant angular velocity. From (11) it appears that it
is the diagonal of the parallelogram of which OA and oв are the
containing sides; so that the axis of the resultant angular
velocity lies along the diagonal of the parallelogram of which
the line-representatives of the component angular velocities are
the containing sides.
The intensity of the resultant may be found as follows; let us
suppose it to be we; then as the path which A, or indeed any
particle on the line oa, describes in dt in the case of the
component angular velocities is that due to w only, and is
wɩ o A sin y dt, and as the path described by a in dt in the case
of the resultant angular velocity is o, oa sin coa dt, these paths
are to be equal; and therefore
C
w sin y = we sin coa;
шо
(12)
similarly if we equate to each other the two paths described by
27.]
41
OF ANGULAR VELOCITIES.
B in the cases of the component and of the resultant angular
velocities, we shall have
wa
wɑ sin y
we sin COB;
(13)
from either of which equations it appears that w, is represented
in length by the diagonal oc. Hence it follows that if a body
rotates with two simultaneous velocities, whose line-representa-
tives meet in a point and are the adjacent sides of a parallelo-
gram, the resultant angular velocity is equivalently replaced in
all respects by that diagonal of the parallelogram which abuts
at the point of insersection of the line-representatives of the
component angular velocities.
=
Hence if, as in Fig. 7, co is produced to c', so that c'o co,
then if OA, OB, oc' are the line-representatives of three simul-
taneous angular velocities, the body is at rest; because oc, which
represents the resultant of wa and w, represents an angular
velocity which is neutralized by that of which oc' is the line-
representative; and therefore if AOC' B, BOC' = a, from (12)
and (13) we have
Wa
sin a
We
b
sin ß
sin y
(14)
and therefore if a body rotates with three simultaneous angular
velocities whose axes meet in a point, and whose line representa-
tives are parallel and equal to the three sides of a triangle, the
angular velocities neutralize each other, and the body remains
at rest.
Hence also it follows that if a body has two simultaneous
angular velocities wa and w about two axes which intersect
at an angle y, these are equivalent to a single angular velocity
we, which is given by the equation
2
ω
2
= wa² + 2 wa wr COS y + wr²;
(15)
the rotation-axis of which makes angles a and ẞ with the rota-
tion-axes of w, and we are such that
a
Wa
sin a
Wo
sin ß
sin y
(16)
27.] Now suppose a body to have two simultaneous angular
velocities w, and w, about two axes intersecting each other
at right angles; then, if w is the resultant angular velocity,
Wx
Wy
w² = wx² + w₁₂²;
2
2
y
and, if a is the angle between the axes of ∞ and wx,
(17)
Wx = @Cos a,
(18)
Wy
= ∞ sin a;
G
PRICE, VOL. IV.
42
[28.
BOHNENBERGER'S MACHINE.
so that angular velocities may by means of their line-representa-
tives be resolved and compounded according to the projective
laws of pure geometry, the laws of resolution and composition
of statical pressures, and of dynamical linear velocities.
Similarly if a body has three simultaneous angular velocities
wx, wy, wz about three axes which intersect at right angles; then,
if is the resultant angular velocity,
2
2
w² = w₁₂² + w₁² + w ²;
ω
(19)
and if a, ß, y are the angles which the axis of the resultant angu-
lar velocity makes with those of the component angular velocities,
W.
COS a
W!!
cos B
W:
ω.
COS Y
(20)
28.] The machine represented in Fig. 8 may probably facili-
tate the conception of simultaneous angular velocities, and their
combined effects; the machine is delineated in its primary state
of rest. o is the centre of a sphere, through which a horizontal
axis AA' passes, the ends of which are in a horizontal circular
ring AB A'B', so that the sphere can rotate about this axis. To
this horizontal ring two pivots are attached at в and B', the line
joining which is perpendicular to AA'; these pivots work in a ver-
tical circular ring CB C'B', so that the ring containing the sphere
can rotate about the axis BB'; the ring cc' has also two pivots
at c and c', which are fixed points, the line joining which is
vertical and is at right angles to both AA' and BB'; and the
vertical ring can rotate about cc' as an axis. The three lines
AA', BB', cc' intersect in o the centre of the sphere, and thus
form a system of rectangular axes in space, the origin of which
is at the centre of the sphere. Now this is the state of the
machine at rest, and the problem is this; let an angular
velocity w be given to the sphere about the line AA'; to the
ring AB A'B' carrying the sphere let an angular velocity w, be
given about the axis B B'; and to the ring CB C'B' carrying the
former ring and the sphere let an angular velocity o be given
about the axis cc'; then the sphere has from its connexion with
the rings all these angular velocities simultaneously, and the
question is, what line of particles is at rest? What is the axis
of rotation, and what is the angular velocity of the sphere about
it? Try to follow in your mind the path described by any
particle of the sphere, when it moves with all these simultaneous
angular velocities; and try thence to determine the line of
29.]
43
COMPOSITION OF ANGULAR VELOCITIES.
quiescent particles. Probably in the difficulty of doing so, you
will perceive the necessity of such composition and resolution of
angular velocities as we have just explained. Let us assume
the three angular velocities to be positive; then the resultant
angular velocity will be given by equation (19) and the direction-
cosines of its axis by (20). This may be thus exhibited; let us
suppose w₂ = wyw₂; then a = B = y; let a diameter of
the sphere be drawn making equal angles with AA', BB', cc' in
its original position; and at the poles where this diameter meets
the surface, let the surface be divided into three equal lunes, and
let them be coloured respectively red, yellow, and blue; then it
will be found that the sphere will rotate about this axis which is
equally inclined to the three lines AA', BB', cc', and the rotating
sphere will, if its angular velocity is great enough, appear white;
whereas if the resultant rotation-axis does not pass through the
point where the differently coloured lunes intersect, the colour
of the rotating sphere will be that of the lune in which the
rotation-axis pierces the sphere.
29.] Hence we can deduce the single resultant angular ve-
locity of many angular velocities, whose rotation-axes pass
through a given point: let that point be the origin, and at
it let a system of coordinate axes originate; let the several
angular velocities be w₁, w2, ...... w; and let their rotation-axes
be (a1, B1, 71), (az, B₂, Y2), ...... (an, Bu, Yn); let each angular velo-
city be resolved into three components along the three coor-
dinate axes; so that those, whose rotation-axis is the axis of x,
are w₁ cos α1, W₂ COS A2,
w₁ cos an; and if x.o cos a is the
sum of all these,
Σ.w cos a = w₁ cos α₁ + w₂ COS a₂ +
similarly
1
Σ.ω cos β = w₁ cos ß₁ + w₂ cos B₂+
Σ.@ COS Y = w₁ COS Y1 + w₂ COS Y½ +
Let a be the resultant angular velocity;
direction-angles of its rotation-axis; then
+wn cos an;
.... + wn
(21)
+ wn cos Br
+wn COS Yn•
and let a, b, c be the
♪ cos α = Σ.@ cos a, o cos b = x.w cos B, o cos c = Σ.w cos y ;
(22)
22
(Σ.w cos a)² + (Σ.w cos ẞ)² + (Σ. w cos y)²;
(23)
and
Cos a
Σ. ω COS α Σ. ω cos β
cos b
Σ.w cos y
Ω
(24)
COS C
G 2
44
[30.
THE PENDULUM EXPERIMENT OF FOUCAULT.
which equations give the intensity of the resultant angular ve-
locity, and the direction-cosines of its rotation-axis.
30.] The experiment with the pendulum, devised by Foucault
to exhibit to the eye the rotation of the earth about its axis, is a
simple application of the laws of resolution and composition
of angular velocities which have been investigated. Let us
suppose the earth to be a perfect sphere, of which a plane
section through the poles is drawn in Fig. 9; P and P' being the
north and south poles, c being the centre, and wCE being the
intersection of the plane of the paper and the plane of the
equator. Let a pendulum be suspended at the north pole, and so
that it may vibrate and turn freely in all directions. Now if
the pendulum is at rest, and were suspended from a point
not rotating with the earth, but fixed absolutely in space, the
earth would rotate under the pendulum from west to east in
24 hours; and the apparent effect to a person on the earth
would be a complete rotation of the pendulum through 360°
from east to west in the same time. A point of suspension
however not fixed to the earth cannot be obtained, and of
course if the point of suspension is joined to the earth, all
moves together, and the pendulum has no apparent rotatory
motion. Let however the pendulum be suspended from a point
fixed to the earth; and let it vibrate in a plane; then as the
earth turns, the point of suspension turns, and the pendulum
turns, but the plane of vibration is not affected by these rota-
tions; it is as stationary as if the point of suspension were
absolutely fixed; so that in the course of 24 hours, if the pen-
dulum vibrates so long, the plane of vibration will apparently
pass in succession over all the meridians from east to west,
because the earth in that time performs a complete revolution
from west to east under the pendulum; and the angular velo-
city of the earth will be the apparent angular velocity of the
plane of vibration of the pendulum. A similar phænomenon
will be presented by a pendulum at the south pole, but the
direction of the apparent rotation of the plane of vibration will
be from west to east. At the equator no such effect takes place.
For suppose a pendulum to be suspended at the equator, and its
plane of vibration to be, say, north and south; as the earth
rotates about its axis, it is evident that neither the point of
suspension of the pendulum nor the plane of vibration has any
31.]
45
COMPOSITION OF ANGULAR VELOCITIES.
rotation; the point of suspension of the pendulum is carried
round in a circle, and the plane of vibration continues north
and south. At the equator therefore no effects of the earth's
rotation such as we have described will be exhibited by a pen-
dulum. The full effect is exhibited at either pole; and no
effect at the equator. Now let us take any place a, whose
latitude is λ; and let AC be drawn to the centre of the earth.
Let o be the angular velocity of the earth about its axis; then
sin A is the angular velocity about the line AC, which is the
normal to the earth's surface at the point A; so that the plane
of vibration of the pendulum at A will undergo a displacement
from east to west similar to that which takes place at P, but
more slowly; for whereas the time of a complete revolution at
24 hours, the time of a complete revolution at
2π
P
A =
W
2π
w sin x
; so that
The time of revolution at a
24 hours
sin λ
(25)
This law has been verified by numerous observations made at
various places on the earth; for although the vibration of the
pendulum has not been continued through 24 hours, yet the
arcs described by the plane of vibration in a given time have
been found to vary in different latitudes as the sines of these
latitudes.
wa
31.] We proceed now to consider the resolution and compo-
sition of angular velocities, the rotation-axes of which do not
meet; and we will first consider the particular case of angular
velocities whose axes are parallel, and about which separately
the body rotates in the same direction. Let the angular ve-
locities be wɑ and w; and let their poles be o and o', and their
axes OA, O'B; oo' being perpendicular to each of these lines;
see Fig. 10; let P be the place of any particle in the line oo',
and to fix our thoughts let us take it between o and o'; let
00 C, OP = x, PO' = y; then the downward path of p in the
time at which is due to wa is x wadt, and the upward path in
the same time due to w is y w dt; so that the downward path
of p in the time dt
v
= (X wa — y wr) dt.
P
Now suppose to be a point in oo' which under the effects
of the two angular velocities w, and w remains at rest; then
Θα
46
[31.
COMPOSITION OF ANGULAR VELOCITIES.
if x and y are the respective distances of a from oA and from
O′B, X wa—Y wz = 0; whence
X
y
шъ
;
Θα
(26)
whereby Q is determined; and as every point in the line through
a perpendicular to oo' is at rest, so qc is the axis of the resultant
angular velocity; and (26) shews that it divides the distance
between the axes of the two component angular velocities into.
two parts which are to each other inversely as the angular
velocities.
Let o be the resultant angular velocity; then in the case
of the component angular velocities, the downward path of o' in
the time dt = wa cdt; and in the case of the resultant angular
velocity, the downward path of o' = wydt; these are of course
equal; whence we have
wa C = wy;
α
(27)
similarly, if we equate the two paths of o in the two cases, we
have
whence
Wz C = w X;
310
Θα
Wb
C
Y
X
wa + wo
Y + x
Θα
+ wo
C
W ∞ = wat wo;
two;
(28)
(29)
(30)
(31)
that is, the resultant angular velocity is the sum of the com-
ponent angular velocities.
A similar theorem is true, whatever is the number of the
component angular velocities which have parallel axes.
If one of the component angular velocities, say w, is negative,
the paths of all particles between o and o' due to both wa and
w will be downward; let us then, see Fig. 11, consider the path
taken by a point p in the line oo' produced; now the line-repre-
sentative of w is o'в, which is drawn from o' in a direction
opposite to that in which OA is drawn from o.
Let OPX,
O'ry; then the downward path described by P in dt due
to wa is wax dt; and the upward path due to w is wr y dt;
therefore the whole downward path in the time dt
= (wax — wz y) dt.
wo
32.]
47
COUPLES OF ANGULAR VELOCITIES.
Let a be a point in oo' which remains at rest; then, if oq = X,
O'Q
= y,
Wu X = wz Y ;
(32)
so that the line oo' is divided externally into two parts which
are inversely proportional to the component angular velocities.
Also let be the resultant angular velocity whose rotation-
axis is qc; then equating the downward paths of o which
and to a respectively, we have
are due to w
шъс
Wz C = W X ;
(33)
and equating the downward paths of o' which are due to wɑ and
to o respectively,
(34)
wac = wy;
W
шо
Θα
C
X
Y
Wb
x — Y
ωα
>
Wi
Θα
C
W
ω = ων – ωα;
шо
(35)
that is, the resultant angular velocity is the excess of the larger
component over the less; and has therefore the same sign as
the larger.
Hence if a body moves with many angular velocities w₁, w2, . . . wng
all of which have parallel rotation-axes, and if n is the resultant
angular velocity,
Ω = 1 + wg +
+ w₁
= Σ.w;
(36)
where . is the algebraical sum of the several components;
but more will be said hereafter on this subject.
wɑ
32.] If however the difference between wa and w, is infi-
nitesimal, then w is also infinitesimal; and if wɑ = wv, w = 0,
and the resultant angular velocity vanishes. In this case how-
ever =
x y; which can be only if x = y = ∞. Here then
a paradox presents itself; when two component angular velo-
cities with parallel axes are equal and have opposite signs, the
resultant angular velocity is zero, and its axis is at an infinite
distance. We must return to first principles.
Consider Fig. 12, wherein oa and o'в are the line-representa-
tives of two equal angular velocities which have opposite di-
rections let oo' = c, and take any particle P in the line oo':
48
[33.
COMPOSITION OF ANGULAR VELOCITIES.
let op = x, o'p = y; then the downward path of P in the time
dt due to wa and to
- Wa
= wax dt +way dt,
=wa (x + y) dt,
=wa cdt;
α
(37)
and therefore is the same, whatever is the place of P. Thus all
particles of the body are advanced in the time dt along a distance
equal to wc dt and perpendicular to the plane containing the
two parallel axes of the component angular velocities. The
effect therefore of a body moving with such a pair of equal and
opposite angular velocities is a displacement of translation of
the body over a distance proportional to the product of either
angular velocity and the perpendicular distance between the two
axes. M. Poinsot, to whom we are indebted for the laws of
composition and resolution of angular velocities, calls such a
pair of equal and opposite angular velocities a couple of an-
gular velocities, and the product wc he calls the moment of
the couple. The analogy is evident between these theorems and
those of statical couples.
a
Hence a couple of angular velocities gives a body a displace-
ment of translation equal to w cdt in the time dt, and along a
line perpendicular to the plane of the axes of the couple.
a
Hence also it is evident that a couple may be equivalently
replaced by any other equimomental couple provided that the
planes of the axes of the couples are either parallel or identical.
And the geometrical representation of a couple is a straight
line whose length is proportional to the moment of the couple,
and which is perpendicular to the plane of the axes of the
couple.
33.] Lastly, let us consider the most general case; that in
which a body moves with many simultaneous angular velocities,
the rotation-axes of which do not pass through one and the
same point, and are not parallel.
As the signs of angular velocity are arbitrary, it is convenient
to affect them with those which are best suited to a system of
coordinate axes in space. Let those angular velocities be con-
sidered positive with which, having for their rotation-axes sever-
ally the axes of x, y, z, the body turns from the y-axis to the
z-axis, from the z-axis to the x-axis, from the x-axis to the
y-axis respectively; and let those be negative with which the
33.]
49
COMPOSITION OF ANGULAR VELOCITIES.
body rotates in contrary directions. This system is evidently cy-
clical, and is easily remembered.
Let the angular velocities be w₁, w₂, wn; and let a point
o rigidly connected with the body be the origin; at it let a
system of rectangular coordinates fixed in space originate; and
let the direction-angles of the rotation-axes be a₁, ẞ₁, Y1, ɑ2, B2, V2,
... an, ẞn, Yn; let (x1, Y1, 1), (X2, Y2, Z2), ... (Xu, Yn, Zn) be points
severally on the rotation-axis of each, and let P1, P2, ... Pn be
the perpendicular distances from o on the several rotation-axes.
And of all these quantities let w, (a, B, y), (x, y, z), p be the
types. Let us consider the type velocity w. At the origin let
a pair of equal and opposite angular velocities be introduced, each
of which is equal to o, and the rotation-axis of which is parallel
to that of w; and from o let the perpendicular distance p be
drawn to the rotation-axis of w; so that instead of the original
w, we have w at o equal to the original o and with a rotation-
axis parallel to that of the original o, and a couple of angular
velocities, each of which is w, and the distance between whose
axes is p; so that po is the moment of the couple; and the
effect of which is a displacement of the body in the time dt over
a distance equal to wpdt in a line through the origin perpen-
dicular to the plane which contains the rotation-axis of w. Let
a similar process be performed on all the angular velocities;
then we have a system of angular velocities the rotation-axes of
which pass through o, and also a system of couples of angular
velocities, the effects of which are severally a displacement of
translation of the body.
Let a be the resultant angular velocity of all those which act
at o; let x, y, be its axial components; and let a, b, c be
the direction-angles of its rotation-axis; then
x = 1 COs a₁ + w₂ cos az +
= Σ.ω COs a;
Dy
= Σ.w cos ẞ;
Q₂ = Σ.W cos y ;
+ wn cos an,
(38)
(39)
(40)
and
.*.
2² = 2₂² + 2₂² + Dz²,
= (Σ.w cos a)² + (≥. w cos ẞ)² + (x.w cos y)²; (41)
Σ. ω COS α
cos a
Σ.ω cos
cos b
Σ.ω cos γ
COS C
(42)
whereby the intensity and the direction-cosines of the rotation-
axis of the resultant angular velocity through o are known.
PRICE, VOL. IV.
H
ނ
50
[34.
COMPOSITION OF ANGULAR VELOCITIES.
As to the couple of angular velocities which arises from o, the
moment of the couple is po; and as p is the perpendicular from
the origin on a line passing through (x, y, z), whose direction-
angles are (a, ß, y), we have
p²=(y cosy - zcos B)2 + (z cos a-x cos y)² + (x cosẞ-ycosa)². (43)
y)²+(x
Now the displacement of translation which the body undergoes
by virtue of this couple of angular velocities is along a line per-
pendicular to the rotation-axis of w and to p; so that its di-
rection-cosines are
y cos y-z cos B
Ρ
z cos a-x cos y
COS x cos ẞ-y cos a
Ρ
p
(44)
Let A be the space through which the body is displaced in
the time dt by reason of this couple of angular velocities; then
ΔΟΞ ω wpdt;
(45)
and the direction-cosines of Aσ are given by (44); so that if
a§, aŋ, ağ are the axial projections of a ʊ,
ΔΙ w (y cos y − z cos ß) dt,
An = w(z cos a x cos y) dt,
Δη
Δζ = ∞ (x cos ß — y cos a) dt.
(46)
A result similar to this is true for each component angular
velocity; and therefore if σ is the whole space through which
the origin is transferred, and if §, ŋ, are the axial projections of o,
and
η
έ — Σ.w (y cos y − z cos ß) dt,
n = x.w (z cos a − x cos y) dt,
Š
= x.w (x cos ẞ— y cos a) dt;
2
{² + n² + 8²;
and the direction-cosines of σ are
0²
=
Σ.w(ycosy―zcos ß) dt
x.w(≈ cosa — x cosy) dt
σ
(47)
(48)
σ
σ
Σ.w(x cosẞ — y cos a) dt
; (49)
whereby we have the resultant motions, both of translation of
the origin, and of rotation about an axis through it, of a body
moving with many simultaneous angular velocities.
34.] If the simultaneous angular velocities w1, 2, ... wn are
such that the body is at rest, then î = 0, and σ = 0; so that
we have the six conditions
Σ. ω cos a
B
0,
Σ.w (y cos y —≈ cos ß)
0,
(50)
Σ. (z cos a-— x cos y)
w
0,
(51)
Σ.w cos y = 0;
z.w(x cosẞ—y cos a) = 0;
1.0 cos 8 = 0.
34.]
51
COMPOSITION OF ANGULAR VELOCITIES.
from these equations theorems can be deduced similar to those of
Art. 57, Vol. III. Also if the angular velocities are capable of
composition into a single angular velocity, we must have
§ Q²x² + n a + Šo₂ = 0;
Y
DZ
(52)
which equation shews that the resultant line of displacement is
perpendicular to the rotation-axes of the resultant angular velo-
city; and as the resultant displacement may be replaced by two
equal and opposite angular velocities whose rotation-axes are
perpendicular to the line of displacement, we may take these
rotation-axes to be parallel to that of 2, whereby we shall have
three angular velocities with parallel rotation-axes which may be
compounded into a single angular velocity.
If the axes of all the simultaneous couples are parallel, then
n cos a = cos a Σ.w,
cos b
=
cos ẞx.w,
a cos c = cos y Σ.w;
a = a,
b
B,
(53)
c = y;
(54)
(55)
Ω = Σ.ω ;
that is, the resultant angular velocity is equal to the sum of all
the component angular velocities, and its rotation-axis is parallel
to the rotation-axes of the components. Also
हु
n =
=
(cos y Σ.wy-cos ẞ z. o z) dt,
(56)
5
(cos a Σ.wz-cos y Σ.w x) dt,
(cos β Σ.ω α cos a x.wy) dt.
In this case (52) is satisfied; and the angular velocities are
capable of reduction to a single angular velocity; let o be the
resultant, and let (x, y, z) be a point on its rotation-axis; then
as it produces the same effect as all the components taken in
combination, and as the direction-angles of its axis are a, ß, y,
हु
έ = (cos y nÿ — cos ẞa z) dt,
n =
१
(cos anz cos y ax) dt,
(cos Box
Ω
cos a oÿ) dt;
(57)
which are severally equal to the values given in (56). But
a, ß, y are manifestly indeterminate; so that we have
x =
Σ.ων
Σ.
Y
Σ.ω.
Σ.Φ
Σ.ΘΕ
Σ.ω
(58)
(x, y, z) is called the centre of the parallel angular velocities.
*
H 2
52
[35.
COMPOSITION OF ANGULAR VELOCITIES.
35.] From the preceding Articles it appears that if a body is
moving with many simultaneous angular velocities, the resultant
motion consists (1) of a determinate angular velocity n, the
rotation-axis of which is determinate in direction but not in
position; and (2) of a displacement of translation of an arbitra-
rily chosen point on the axis of the resultant angular velocity 2,
the line of motion and amount of which are given by equations
(47) and (48). Now whatever is the position of this arbitrarily
chosen point, the resultant angular velocity is the same in
intensity and direction, but the amount and the line of the
displacement of translation of the point varies with the point.
In the infinitesimal time dt the displacement is infinitesimal,
but equations (47) shew that it varies with the origin. Let us
therefore inquire what is the position of the origin, when that
displacement is the least; the form of (47) indicates that it does
not admit of a maximum, but it may be a minimum.
Let the directions of the coordinate axes be unchanged; and
let (xo, yo, 。) be the origin for which the displacement of transla-
tion is the least; so that for x, y, ≈ in (47) we must substitute
XC Xo, Y — Yo, ≈≈0; and let §。, o, So be the projections of the
displacement o。 of the new origin; then
0
É。 = x.w {(y—Yo) cos y-(—) cos ẞ} dt;
= x.w(y cosy - zcos ẞ) dt-y, z.wcos y dt +.wcos ßdt;
) dt,
= § — (Yo λ ; — 2
R = - Z)
No = n(zo Qx-xon) dt,
0
Šo = 5 — (Xo Dy — YoQx) dt ;
σό
2
0
{ § — (Y。 ^ ₂ — Z¸N) dt}² + {n−(20Qx — XqQz) dt}²
(59)
+ {5−(x¸ˆ‚—Yox) dt}². (60)
Thus σ2 is a function of xo, Yo, o which are three independent
variables; whence equating to zero the three partial differentials
σo, as in Art. 72, Vol. III, we shall have
of σ
Ś Dy — 11 Dz
η Ωχ
xo
ฝูง
22 dt
§ Dz – Č D x
2² dt
Dy
Dx
n Dx — § Dy
%
20
2.
Q2 dt
; (61)
Ωχ
which are the equations to a straight line; this line therefore is
the locus of points for which the displacement of translation due
to the simultaneous angular velocities is a minimum. The line
is evidently parallel to the rotation-axis of n; and passes through
a point whose coordinates are
1
36.]
IMAGES OF A MOVING BODY.
53
És
5211-naz
η Ω
Ω
2 d t
EQz - CDx
Ω 22dt
η Ωχ
ξΩΝ
(62)
22 dt
which give the following geometrical construction. At any point
o taken arbitrarily, let the displacement of translation oσ and
its axial components έ, n, & be drawn; also let the rotation-
axis of a be drawn. Let & be the angle between these two
lines; through o draw a line perpendicular to both of them;
and along that perpendicular from o take a length oD = p, such
that
(63)
then a line drawn through D parallel to the rotation-axis of a is
that whose equations are (61).
p =
o sin
2 dt
;
Ω
36.] If the origin is taken on the line, p = 0; and therefore
sin = 0; so that the line of displacement lies along the rota-
tion-axis of . In this case therefore the system of simultaneous
angular velocities is reduced to (1) an angular velocity n about a
determinate axis, and (2) a displacement of translation σ along
this axis; this axis is called the central axis of the system. In
the infinitesimal time dt the body rotates through an angle odt
about the axis, and moves along the axis over a distance σ which
is given by equation (48). Thus it has a helical motion, like a
screw: while it rotates with a given angular velocity, it also
advances along the rotation-axis with a determinate linear velo-
city. This is one of the most simple images which the motion
of a rigid body admits of.
I may observe that the equations (61) of the central axis may
be found by investigating the locus of points at which the dis-
placement of translation lies along the rotation-axis of o.
Now when a body has a continuous motion the system of the
central axes forms a ruled surface in space, and another ruled
surface in the body; all the generators of the second succes-
sively coincide with those of the first. In imagining such a
motion let us suppose that of translation or the sliding of the
axis to take place before the rotation about the axis; let two
generators be placed on each other in their corresponding posi-
tions, and let the sliding along them take place over the distance
σ; and then let the body turn about the common generator of
the two surfaces through the angle a dt; by this means the
next two generators will be brought into coincidence; and the
corresponding sliding and rotation will again take place; and
54
[37.
PROPERTIES OF THE CENTRAL AXIS.
so on; whereby the two surfaces will be successively brought into
contact with each other along their generators. If one of the
surfaces is developable, the other is; and if σ = 0 throughout
the motion, in which case there is no sliding, the two surfaces
are evidently cones. This also is the case when one point in the
body is fixed, so that all the axes pass through that point. The
line of contact of the two surfaces is called the instantaneous
sliding axis.
The case however in which the body has a fixed point, and in
which the ruled surfaces become cones, deserves further illustra-
tion. Let o, see Fig. 13, be the fixed point of the body; and
from it let a sphere be described of any radius; and let the cones
which are respectively fixed in space and fixed in the body be
cut by the surface of the sphere in the curves Is and Is'; let 1
be the point where the instantaneous axis meets the spherical
surface; let s and s' be the arcs of the two curves is and is' re-
spectively; of which let the length-elements be ds and ds'; let
OI be the instantaneous line of contact of the two cones, and let
the time be divided into equal infinitesimal elements dt; also
let the curves s and s' be divided into elements corresponding
to successive dt's, and so that the corresponding elements are
equal. Let os be the cone fixed in space, and os' that fixed in
the body; then the motion of the body will be represented by
the rolling, without sliding, of the latter cone on the former;
the line of contact of the two cones being always the instan-
taneous axis of the body. Hereafter we shall find the equations
to these cones, and thus have their relative magnitudes and
position. In the mean time I may observe that the rolling may
take place in many ways. One cone may roll outside the other, as
in Fig. 13; or the moveable cone may roll inside, as in Fig. 14;
or again as in Fig. 15, where the moveable cone is larger than
the fixed cone and rolls on the outside of it. Or again either
of the cones may degenerate into a plane, as in Fig. 16; and
either of the cones may become a straight line, in which case
the axis of rotation is fixed. Or the cones may become identical,
in which case the position of the rotation-axis is indeterminate.
Also the vertex of either cone may be at an infinite distance, in
which case the cone becomes a cylinder; and as the instantane-
ous axes are all parallel, the path of every particle is in a plane
perpendicular to the axis of the cylinder.
37.] Let us consider other properties of the central axis.
37.1
55
PROPERTIES OF THE CENTRAL AXIS.
From (59) we have
0
Eo 2x + no 2, + So Dz
§ Qx + n Qy + b fz;
(64)
Ω
Ω
and by a process similar to that of Art. 71, Vol. III, it may be
shewn that this quantity is constant, through whatever angle
the system of coordinate axes is turned. But if o is the angle
between o corresponding to any origin and the rotation-axis of w,
§ Qx + n 2 + Ś DZ
.*.
cos
ΤΩ
σ cos p = a constant
σ0;
(65)
where σ is the displacement of translation along the central axis;
that is, the projections of the displacements of translation of all
points of the body on the central axis are equal. If therefore a
plane is drawn perpendicular to the central axis, the distances
of all points of the body from this fixed plane are increased or
diminished by the same quantity in the time dt. And the line
of this quantity is, as shewn in Art. 35, that along which the
displacement of translation is the least. Hereby we have the
following construction for the direction of the central axis.
Through any point in space draw straight lines equal and paral-
lel to the actual displacements of all the points of a body due to
the time dt; then the ends of these lines will be all in one plane
which is perpendicular to the central axis.
Again, all the properties of reciprocal lines which are demon-
strated in Vol. III, Art. 70-87, are true, mutatis mutandis, of
angular velocities; indeed the principle of duality is completely
applicable to these theorems. X, Y, Z, R, L, M, N, G are to be
changed into x, y, z, î, §, 11, S, σ respectively, and the enun-
ciations of the theorems are to be changed accordingly; that is,
pressures are to be changed into angular velocities, and moments
are to be changed into displacements of translation. Thus, it is
there proved that any system of pressures may be reduced into
two pressures acting along lines at right angles to each other:
hence it follows that every system of angular velocities may be
reduced into two angular velocities whose axes are perpendicular
to each other.
Again, as all moment-centres of equal moment lie on a cylin-
drical surface whose axis lies along the central axis; so all points
for which the displacement of translation is the same lie on a
cylindrical surface; also all the lines of equal displacement
corresponding to points on a circle whose plane is perpendicular
56
[38.
RELATION BETWEEN ANGULAR
to the central axis lie on the surface of a hyperboloid of revo-
lution.
So for all points on a line perpendicular to the central axis,
the lines of displacement are in the surface of a hyperbolic
paraboloid.
From the theorems of Art. 85, Vol. III, we have the following
deductions. If a body moves with many simultaneous angular
velocities, they may be reduced to two others whose rotation-
axes are such that if one is given, the other is the reciprocal
line; and the perpendicular line to these two axes passes through
and is perpendicular to the central axis of the system.
And from Art. 86, it follows that every system of angular ve-
locities may be replaced by two equal angular velocities, whose
rotation-axes are perpendicular to each other, and each of which
is inclined at 45° to the central axis, and the axes are perpen-
dicular to a line which is bisected at right angles by the central
Ω
axis; each of the angular velocities = ; and the length of
23
the perpendicular distance between the axes =
200
2 ༦༠ .
Ω
38.] We have in the next place to determine the motion of
translation of any particle of a body which rotates about a given
determinate axis; and I will consider the body te have a fixed.
point, through which of course the rotation-axis passes, so that
the displacement of a particle may be due to the rotation only;
and I shall consider the displacements which take place during
the infinitesimal time dt.
Let the fixed point of the body be taken as the origin, and a
system of coordinate axes fixed in space originate at it; let w
be the angular velocity with which the body rotates about the
axis, of which let the direction-angles be (a, B, y); let (x, y, z)
be the place of the particle r, the displacement of which in the
time dt is to be calculated; let ds be the displacement, of which
let dx, dy, d≈ be the axial projections; let wx, wy, w; be the axial
components of w; so that
W 1
Wx
Cos a cos B
W~
ω,
COS Y
(66)
Let p be the perpendicular distance from (a, y, z) on the rota-
tion-axis (a, ẞ, y); then as wdt is the infinitesimal angle through
which the body rotates in the time dt,
ds = pw dt;
pwdt;
(67)
and
38.]
57
AND LINEAR VELOCITIES.
p²= (zcosẞ— y cosy)² + (x cosy−≈cosa)² + (y cosa-x cosß)². (68)
Now ds is perpendicular to the rotation-axis, and to the line
drawn from the origin to (x, y, z); also dx, dy, dz are propor-
tional to its direction-cosines; therefore
cos adx + cos ẞdy + cos y dz
0,
(69)
xdx+
y dy +
zd z =
0; S
0:5
dx
dy
dz
(70)
zcosẞ-y cos y
x cosy z cos a
y cos a-x cosß
ds
(71)
Ρ
(≈ cos ẞy cos y) w dt =
(z wyw:) dt,
(x cos y -
z cos a) w dt =
(x w,
zwx) dt,
(72)
(y cos a
x cos ß) w dt =
(y wx — x w„) dt ;
dx
da
dy
dz
ds²
dt2
=
=
=
(z wy—y w :)² + (x w z − z w¿)² + (y wx-xw₁)². (73)
z
x
In all these expressions there has been an ambiguity of sign,
which I have omitted; that sign has been taken which is in ac-
cordance with the principle of signs of angular velocities deter-
mined in Article 33; for suppose the body to rotate about the
axis of x only; so that w₁ = w = 0; then for a particle in the
w,
first octant of space, dz, the increment of, will be positive, and
dy, the increment of y, will be negative; similarly for single
rotations about the other coordinate axes; and equations (72) are
in accordance with these conditions.
Hence the equations to the tangent line of the path which
the particle m at (x, y, z) is taking at the time t are
ૐ X
n-y
Y wx - Xw y
(74)
≈ wy - Y W z X w = — ~ W x
Hence also if a body has a fixed point at the origin, and
rotates with angular velocities wx, y, z about its three coor-
dinate axes respectively, the rotation-axis of the body is the
locus of the points which are at rest, and its equations couse-
quently are
ૐ η
Wy
ωχ
Č
ω
-
(75)
As an exact comprehension of (72) is of great importance, let
us investigate them by another and a more elementary process.
From (x, y, z), the place of m, let perpendiculars 7, 7, 7 be
drawn to the three rectangular axes of x, y, respectively; and
let be the angle between 7 and the plane of (xy), p the angle
PRICE, VOL. IV.
Ꮖ
I
58
[39.
MOTION OF A BODY DEFINED
between r, and the plane of (yz), y the angle between r, and the
plane of (zx); so that do, do, d↓ are positive according to our
hypothesis of signs for small rotations about the axes of x, y, Z
respectively. Let rotations through infinitesimal angles take
place successively about these axes, and let the changes of the
variables due to these rotations be calculated. For a rotation
about the axis of x through do we have
y = rx cos 0,
dy
rx sin o do,
= − zdo;
z = r, sin 0,
dz
= rx cos e de,
Ꮎ
= yd0;
so that the infinitesimal variations of y and ≈ are respectively
-zde, and yde; similarly for a rotation through an angle do
about the axis of y, the variations of z and x are respectively
-xdo and zd; and for a rotation through dy about the axis
of z, the variations of x and y are respectively -ydy and x d¥;
so that if da, dy, dz are the total variations of x, y, and z due
to these combined rotations,
dx = — z do — y dy,
dy = x dy-z do,
—
dz = y de x do.
(76)
39.] We shall hereafter find it convenient to refer a body and
its motion to two sets of coordinate axes at the same time; one
of which is assumed to be fixed in space, and the other to be
fixed in the body and to be moving with it. At first I will
assume these two systems to originate at the same fixed point of
the body so that the body is capable of only a motion of rota-
tion about an axis passing through this fixed point; and I will
suppose a point P to be (x, y, z) and (§, n, ) in reference re-
spectively to the systems fixed in space and fixed in the body;
and I shall suppose the two systems to be related by the scheme
of direction-cosines given in Art. 2. As the body moves about
an axis, which passes through the origin, x, y, z and the nine
direction-cosines vary, but έ, n, do not vary; so that from (2),
§, Ŝ
Art. 2, we have for the variations in the time dt
da₁ db₁ dc1
dt
dx
ૐ
dt
dt
τη +8
dt
dy
daz
ૐ
dt
dt
dt
db 2
τη +8
dcz
(77)
dt
dz
da3
db3
dc 3
ૐ
τη
+8
- ;
dt
dt
dt
dt
39.]
59
BY TWO SYSTEMS OF AXES.
which values of
dx dy dz
dt' dt' at express at the time t the axial com-
ponents of the velocity of the element which is at (x, y, z).
Let wx, wy, wz be the axial components of the angular velocity
w of the body along the fixed axes of x, y, z respectively; then
from (72), Art. 38,
dx
Z wy — Y w z ;
dt
dx
dt
dy
dt
dz
dt
= wy (ɑ3 § + bzŋ + C3 Ś) − wz (ɑ2 § + b₂n + c₂ Š) ;
§ (αzw, − ɑzWx) + n (bzwy − b₂ wz) + Ŝ (Cz Wy − C2 Wz),
§ (α₁ w z — Az Wx) + n ( b₁ w z − b z wx) + Ś (C₂ W z — Cz Wx),
§ (α₂ wx − α₂wy) + n (b₂ wx − b₁wy) + Ś (C2Wx−C1Wy).
(78)
In (77) and (78) (§, n, Č) is the place of any particle, and there-
fore έ, n, are indeterminate; so that the systems of equations
are identical; hence we may equate coefficients, and we have
= C 1 W z — Cz W x ; > (79)
dc₁
da₁
= Az Wy — A z Wz
d b₁
dt
=
b z wy — bq w z
= Cz Wy - CqWz;
dt
dt
db
da2
d b₂
dcz
2
=
= A₁ W x — A z W x 9
b₁ w z − b z wx,
dt
dt
dbs
dc 3
daz
= A z W x — A z wy,
=
bq
b q w x - by wy, d t
dt
dt
dt
= CqW x — C1@y⋅
These formulæ are important, and it is necessary to understand
their meaning; we have arrived at them indirectly, and there-
fore let us prove them by another process.
The circumstances of the body are these. A point in it is
fixed and is the origin; at it originate (1) a system of coor-
dinate axes (x, y, z) fixed absolutely in space; (2) a system of
coordinate axes (έ, n, ¿) fixed in the moving body: the body
rotates with an angular velocity w about an axis such that the
axial components of w along the axes of x, y, z are wx, wy, wz; at
the time t the direction-cosines of the axis of έ are a1, α½, ɑz;
What are the changes of these quantities due to the angular
velocity w? Take a point (x, y, z) on the axis of έ at a distance
from the origin; then
ρ
X
Y
= p;
α1
az
Az
dx
da
dy
daz
dz
P
1
P
P
dt
dt
dt
dt
dt
daz
d t
I 2
60
[40.
MOTION OF A BODY DEFINED
and replacing t'at' at
dx dy dz
by their values given in (72), we have
dt
da1
1
(zw, — YWz)
dt
ρ
=
Az wy
A z wz
da2
=
A₁ w z — Az Wxi
dt
daz
A z W x — A 1 Wy;
dt
and the other six equivalents of (79) may be found by similar
processes.
da₁ da₂ daz
Thus, if p = 1,
are severally the axial com-
dt dt dt
ponents of the velocity of the particle on the axis of έ at an unit
distance from the origin at the time t.
Also from (79) we have
2
by
2
2
dc₁
dt
2
2
dar)² + ( db; )² + (dą; )² = ∞, ² + w.³,
(day
dt
2
dt
2
dc₂
2
2
x
(80)
(da² )² + ( db; )² + ( dc:)² = ws² + ws²,
dt
2
dt
3
2
das)² + (dbs) 3.
dt
3
dt
+(
dt
dc3
dt
2
2
2
= w ² + w₁²;
40.] Hereby also the axial components wx, wy, w may be de-
termined in terms of the t-differentials of the direction-cosines.
Let us multiply the three equations in the last horizontal row
of (79) by ɑ2, b2, c₂ respectively and add, then
dc3
ωχ
wx = Aq
dt
daz dbz
+ b₂
+ C₂
2
dt
dt
(81)
also let us multiply the three equations in the middle row of
(79) by α3, b3, C3 respectively and add; then
daz d b₂
-Wx az +63 + C3
dt
dt
dc₂
dt
(82)
these two values of ∞ are in accordance with the first equation
of (6) Art. 2.
Hence, and from similar processes, we have
db3
dc3
+ Cz dt
daz + b₂ dt
Wx
аг
dt
da
1
) ၂
Ο
= az dt
da
= a 1 d t
dby
+bs dt
3
+b₂
dba
dt
dc
+ C3 dt
dc2
C1
+ C i dt
{
{
az dt
daz dbz
+b3
dt
daz
a 1 d t
A₂
da₁
dt
dcz
+C3
+ C3 dt
; (83)
ac³ }; (84)
db3
dcz
+ b₁ dt
+ C1
db₁
+ b₂ dt
+ C₂ dt S
; (85)
dc1
40.]
61
BY TWO SYSTEMS OF AXES.
Hence also the equations (75) to the rotation-axis may be ex-
pressed in terms of the t-differentials of the direction-cosines of
the axes fixed in the body.
Also, if we wn, we are the axial components along the axes of
È, n, of the angular velocity of the body at the time t,
A y w x + A z wy + Az
az wz,
W ś =
ωη
b
₁ w x + b z wy + b
@5 =
and conversely, w₁x =
Wx
3
C 1 W x + C 2 Wy + C z W ≈ ;
a₁ w¢ + by wn + cz ws
w₁ =
a₂ w & + b₂ wn + C₂ w5,
ωχ
(86)
(87)
2
2
Az wę + bz wn + C3 @5-
Also, from the second vertical row of (79), we have
db₁
db₂
+ Cz dt
С1
C ₁ d t
db3
+C3 dt
= (b₂C3 — bzC2)wx + (b3C1−b₁C3)wy + (b₁C₂—b₂C1) wz
= A₂ w x + A₂ W y + A z W z
= Wes
ως
using the equalities contained in (11), (12), and (13) of Art. 2. By
a similar process, equivalents of w, and wg in terms of the t-dif-
ferentials of the direction-cosines are determined, and we have
dc.
3
G1
day
1
+b3
dc3
dt
}
da2 da3
+ C₂ dt
ως C1 + C₂
db₁
dt
db.,
dt
db3
+ C 3 dt
dc₁ dc2
b₁ +b,
dt 2 dt
dc₁
Θη = α1
а1
Az
dt
+ α ₂ dt
dca
+a
{ c
da
da
w5 = b₂
+ b₂
dt
dt
da3
+ b 3 dt
db₁
db2
α
dt
dt
2
3 dt
b₂·
dt
+C3
dt
db3
+az dt
And the components of the absolute velocity of the particle m
at the time t along the moving axes which are fixed in the body
may thus be found; let them be v§, vn, vg; then
as
૬ = d
VE
dx
dt
dy
dz
+α3
+ az a t
da
= (
a1
dt
مله
1
-
2
dt
daz
+ as d t
+αz
daz)
dt
+n(a, db₂
1 d t
dc₁
+ 5 (a₂ at
dt
2
d by
+ a 2 d t
dc 2
db3
+ az dt
:)
+ a₂ dc3);
+ uz dt
+az
dt
(88)
whence, and by similar processes for the other components, we
have
VE
ζωη η ως
2₁ = {ws (we
υς = η ως—ξωη.
(89)
These equations enable us to determine the position of the ro-
62
[41.
MOTION OF A BODY DEFINED
tation-axis at the time t relatively to the axes fixed in the moving
body; for if vε = v₁ = vg = 0,
' 0 Vn
ૐ
ωξ
η
१
;
Θη
ως
(90)
and these are the equations to the line of quiescent particles.
Now, in the general motion of a body, the axial components
of the angular velocities are functions of the time, and may be
expressed in terms of t; and therefore the position of the rota-
tion-axis will vary from time to time relatively to both systems
of axes, and will describe a conical surface the vertex of which
is the fixed point. If then we eliminate t from (75), the result-
ing equation will be that of the conical surface fixed in space;
and if we eliminate † from (90), the resultant equation will be
the conical surface fixed in the body; and these two conical
surfaces will always have a generating line common; which
will be the rotation-axis at the time t. These are the cones
referred to in Art. 36.
41.] Let us now suppose the body to be free from all con-
straint: let us as heretofore take a point in or rigidly connected
with the body to be the origin of a system of rectangular coor-
dinate axes fixed in it and moving with it; relatively to a system
of axes fixed in space, let, at the time t, (x, y, z) be the place of
a type-particle m of the body, and let (x。, yo, ≈。) be the place of
the moving origin; and let (§, 7, §) be the place of m relatively
to the origin and to the axes which are fixed in and move with
the body; then, taking the scheme of direction-cosines of Art. 2,
we have
x = x。 + α₁ & + b₁ n + c₁ Ś,
хо
1
y = Yo + a₂ & + b₂ n + c₂ Ś,
2
z = 20 + A3 § + b3 N + C3 Ś ;
where also we have the inverse system
(91)
ૐ
1
α₁ (x − x) + α₂ (Y-Yo) + az (≈-20),
n =
b₁ (x − xoq) + b₂ (y —Yo) + bz (≈ —2g),
(92)
1
2
$ = c₁ (x − xq) + C₂ (Y —Yo) + C3 (≈—Z0)•
20
As the body moves ro, yo, 2o and the nine direction-cosines evi-
dently vary; but §, 7, ‹ are constant; therefore
da₁ db₁
dt
dbq
2
dic dxo
Ο + §
+ n
+8
dc .
dt
dt
dt
dt
dca
(93)
dy
dyo
daz
+ §
dt
dt
dt
dt
dz
dzo
da3
db3
dc3
dt
dt
+ & d t
τη
dt
+5
dt
;
dt
τη +8
dt
41.]
63
BY TWO SYSTEMS OF AXES.
1
which are the components along the fixed axes of the velocity
of any particle m, and admit also of expression in the following
forms;
da
db₁
dc₁
dx dxo
dt
dt
+(x − xo) (α₂
dt
+ b ₁ d t
+ C i dt
1
+(y-Yo) (α 2 dt
(α₂
da₁
db₁
dc1
+ b₂
dt
+ C₂ dt
+ (≈ −20) (α3
da₁
db1
dc
dt.
+ b z dł
dt
+ C3 dt
de);
dx
dxo
+ (≈ − %o) wy
dt
dt
Also similarly,
dy
23
dyo
dt
dz
dzo
dt
dt
dt
Zo) w, — (y — Yo) wz,
+ (x − xo) ∞ z − (Z — Zŋ) wx,
+ (Y-Yo) w x − (X——X。) wy.
(94)
Let vε, vn, vg be the components of the velocity of m, along
ης
the moving axes of §, n, §; so that
— w
dz
dx
dy
+ az a t
da3
+ α z d t
da 2
WE
α1
a1 d t
= α1
Ալ
dxo
dt
+ az dt
dyo + az dt
dzo
+ §
ε (a₁ dt
day + az dt
Az
+dz at
dt
1
dc3
dc2
;
.'.
τη
VE
db₁ + az dt
1 dt
db2
1
+az dt
dzo
dxo
+ az at
dyo + as dt
db 3 ) + 5 (α₁
+ ζωη η ως
dc1
+ ar dt
+α3 dt
dt
dt
(95)
a1
a₁
dt
dzo
dyo + b s dt
+ έως ζω
2
Vn
bi
b₁
dt
dzo
dyo
dxo
05
v5 = C₁ dt
C1
dx o + b₂ dt
+ Cz dt
3
+ C3 d t
+n wε - {wn•
In these expressions the first three terms of each are the re-
solved parts along the axes of §, n, § of the velocity of the moving
origin; and the last two terms are the velocities due to the ro-
tation about an axis passing through the moving origin. But
the point (x, y, z) is arbitrary; so that we have the following
theorem ;
An infinitesimal motion of a body in a time dt is compounded
of a motion of translation of any particle of it at (o, yo, ≈o),
and of a motion of rotation of the body about an axis passing
through that point.
64
[42.
A BODY'S POSITION DEFINED
If @ is the angular velocity about the rotation-axis passing
through (~。, Yo, ≈0),
شه
2
and the direction-cosines of its rotation-axis are
ωξ
@
= wε ² + w₁² + w; ²;
Θη
,
W
ως
ω
(96)
(97)
If the displacement of translation lies along the rotation-axis of
the angular velocity, the motion is compounded of a sliding
along a line and of a rotation about that line: this is indeed the
case which we have considered in Art. 36; the sliding axis of
rotation being the central axis of the body; the motion of the
body is helical; and we may thus find the equations of the cen-
tral axis.
Let (%, yo, 2。) be any point chosen arbitrarily; of the displace-
ment of translation of which let dæ, dy, dzo be the projections
on the fixed axes. Let (x, y, ≈) be a point on the central axis;
this point has therefore only the sliding displacement of transla-
tion, and the line of its motion is along the rotation-axis of the
angular velocity; therefore dx, dy, dz are proportional to w
respectively. Hence from (94),
wx,
wy,
dxo
dyo
+ (~~~。) w₁ — (y — Yo) w ż
+ (x − xo) ∞ ~ — (≈ — Zo) wx
→
dt
dt
Wx
dzo
-
dt
+ (y — Yo) w x − (x − ∞。) wy
; (98)
W z
from which we have
w₁ dz。 — w z dy。
wz d xo
x-xo
xo-
y - Yo-
w² dt
dx。 — wx dzo
w² dt
Wx
Wy
wx dy — w „d x¸
- wy
w² dt
(99)
W~
which are the equations to the central axis; and are identical
with (61), Art. 35, if the origin is taken at the point (xo, Yo, Zo) ;
in which case dx。, dyo, dz。 become respectively §, n, ¿ of that
Article.
42.] The two systems to which the rotation of a rigid body
has been referred in the preceding Articles are related to each
other by means of nine direction-cosines, but as six equations of
condition are given, only three of the direction-cosines are arbi-
trary; that is, three independent variables are sufficient for pass-
42.]
65
BY THREE ANGLES.
ing from one system of rectangular coordinates to another, both
of which originate at the same point. I proceed to explain the
mode of expressing angular velocities referred to one system in
terms of angular velocities referred to another system by means
of the formulæ which are investigated in Art. 3; and to give
clearness to our thoughts let us consider the systems of axes
which are delineated in Fig. 1. o is the common point at which
the two systems originate, and from o as a centre let a sphere of
radius unity be described, on the surface of which the great
circles delineated in the figure are supposed to be. Let w, wy, w
we, wn, we be the angular velocities about the axes of x, y, z, §, n, 8
respectively. Let the planes of (xy) and of (n) intersect in the
line on, and be inclined to each other at the angle ; so that 0
is also the angle between the axes of z and ; ON is technically
called the line of nodes. Let, in Art. 3, xon = Y, SON
ON = 0;
then, as the body moves, 0, 4, and ovary. The angle is tech-
də
dt
༧
nically called the obliquity, and is the angular velocity of the
body about the line on;
αφ
dt
is the angular velocity about the axis
dyi
of <; is the angular velocity about the axis of , and indi-
dt
cates the velocity with which on moves along the plane of (xy);
it is called the velocity of precession, the precession being the
angle xon; and the precession is direct or retrograde according
as the angle is increasing or diminishing. The angular velo-
y
d Ꮎ
city is sometimes called the nutation; of these terms how-
dt
ever and of the origin of them more will be said hereafter.
Let us express the angular velocities we, w, wg in terms of
d0 dp
dv
and
dt
that is, let us resolve the latter along the ro-
dt' dt'
tation axes of the former;
d Ꮎ
аф
d &
ωξ
Q =
COS & ON +
cos ×× +
cos έoz
0
dt
dt
dt
d Ꮎ
αψ
cos o +
sin 0 sin &;
(100)
dt
dt
d Ꮎ
аф
d↓
G
cos noN +
cos nos +
COS OS
dt
dt
dt
d Ꮎ
sin &+
dt
d↓
di
sin cos ;
(101)
K
PRICE, VOL. IV.
F
66
A BODY'S POSITION DEFINED BY THREE ANGLES. [42.
d Ꮎ
do
ως
COS CON +
dt
dt
αψ
cos o+ cos (oz
dt
аф
dy
+
cos 0.
(102)
dt
dt
Therefore by elimination.
do
=
dt
wę cos &— w, sin ø.
(103)
d f
ως
sin + cos &
(104)
dt
sin 0
аф
cos e
dt
ως
(we sin + cos 4);
(105)
sin 0
do dy do
whereby dt' dt' dt
are given in terms of the angular velo-
cities of the body about three axes fixed in and moving with it;
and if by integration or otherwise 0, 4, 4 can be found in terms
of t, these quantities will determine the place of the body at a
given time.
If, to fix our thoughts, the moving body is the earth, of
which of is the polar axis; then the plane of (n) is the equator;
and, if the plane of (xy) is the ecliptic, the axis of z passes
through the pole of the ecliptic, which is fixed in the heavens.
In this case is the obliquity of the ecliptic, on passes through
the vernal and the autumnal equinoxes and is the line of equi-
noxes; and is the longitude of a certain meridian plane, viz.
έo, measured from the line of equinoxes.
Let this much at present suffice for the cinematics of a mov-
ing body; the subject however is far from being exhausted; we
shall return to it hereafter in a more general case, and we shall
arrive at formulæ of which all the preceding are only particular
cases.
43.]
67
D'ALEMBERT'S PRINCIPLE.
CHAPTER III.
*
THE DYNAMICS PROPER OF A MATERIAL SYSTEM.
SECTION I.— D'Alembert's Principle; the equations of motion of
a material system.
43.] We now come to the dynamics proper of a material
system.
A material system is an assemblage of particles dependent on
each other by the action of certain forces which have their origin
in and pervade the system. These are called internal forces and
are generally different in different systems. The system also is
such that if one particle or body of it moves by the action of a
force external to the system, one or more of the other particles
will also move. Thus a material system is always subject to
the action of internal forces; and may also be acted on by ex-
ternal forces, in which case the system will move. Let us first
consider the former forces; and the constitution of the system
of the particles as it depends on the nature of these internal
forces.
(1) The system may be a rigid body; then the internal forces.
are molecular, and of such an intensity that all the particles of
the body are at relative rest during the whole motion; and the
external forces, whatever are the particles they act on, do not
effect any separation of the particles; so that the molecular forces
are infinitely great in comparison of them.
Such a system is
not probably found in nature: all bodies are more or less com-
pressible and elastic, and the particles have a relative motion
under the action of external forces. Nevertheless as such a
system is imaginable, and is the limit towards which rigid bodies
in nature tend, it is necessary to consider it and to discuss its
properties.
(2) The system may consist of particles invariably connected
by rigid and inextensible rods, which are capable of bearing force
of either compression or extension; so that during the motion
the particles are at relative rest. This system has dynamically
K 2
68
[43.
D'ALEMBERT'S PRINCIPLE.
the same properties as the former, and differs only in the nature
of the internal forces. As in this and the former cases the par-
ticles have always the same relative places, these systems are
called geometrical.
(3) The several particles of the system may be connected by
flexible and extensible strings or rods; in which case the in-
ternal forces acting on the particles, by the strings or rods, may
vary from time to time as the system moves, and according to
the nature of the external forces. We have instances of these
systems, when bodies move about pulleys by means of flexible
strings, either extensible or inextensible; when a perfect pen-
dulum vibrates with an extensible rod.
(4) The system may consist of particles and of bodies which.
act on each other by mutual attractions or repulsions; and these
may be functions of the several distances of the particles or
bodies; so that during the motion of the system its form may
not be invariable. The solar system is of this nature, in which
we have a series of planets, secondaries, &c. subject to mutual
attractions. We have also other systems of the same kind in
the motion of liquids; of oscillating flexible cords; of air; of
the ethereal medium, &c.; in all which cases the form of the
system is continually changing.
These two latter are generally called dynamical systems.
Such are some of the material systems, the motion of which
we have now to examine. They consist of material particles,
each of which is acted on by certain forces both internal and
external. Now if all the forces which act on any one particle
are given, the motion of that particle may be determined by the
processes of Vol. III. The internal forces however are generally
not known, and the determination of them is beyond our power.
Let us then consider whether the equations of motion of the
aggregate system cannot be constructed without a knowledge
of these internal forces, and thus without a knowledge of the
motion of the several constituent particles. With this object in
view, we must examine the circumstances of motion more ex-
actly, and this we shall do by contrasting it with that of a single
particle.
When a single particle moves under the action of a certain
external force, the expressed momentum is equal to that im-
pressed on it by this force, whether these are respectively infini-
tesimal, which is the case when a finite force acts for an infini-
44.]
69
D'ALEMBERT'S PRINCIPLE.
tesimal time-element; or whether the momentum-increase is
finite, as is the case when an instantaneous force acts. These
results are consequent on that property of matter which we call
Inertia.
In the motion of a material system, the momentum expressed
in any particle is not necessarily equal to that impressed on it
from an external source, because each particle is influenced or
constrained by one or more of the other particles; that is, is
acted on by internal forces, as we have just now explained,
and thus it is not free for the development of the momentum
communicated to it. The expressed momentum may be either
greater or less than that impressed by the external forces accord-
ing to the internal forces which act on it from the other parti-
cles. Some other principle therefore beyond that of inertia is
necessary for the construction of equations by which the motion.
of the system may be determined. This has been supplied by
D'Alembert. It was first enunciated in a Memoir read before
the Academy of Sciences in Paris at the end of the year 1742;
and it is now always known as "D'Alembert's Principle." I
propose to consider the circumstances which require it in one or
two particular cases; because by these it will be better under-
stood. And although I shall take continuously acting finite
forces by which momentum-increments are impressed, yet, as
what is true of them will also be true of momenta impressed
by instantaneous forces, the explanation will be applicable to
both kinds of force.
44.] In the first place, a difference usually exists between the
momentum-increment impressed on, and that expressed in the
motion of a particle of a material system. To shew that this is
the case, let us suppose a heavy rigid body to be composed, say,
one half of pith, and the other half of gold; and suppose it
to fall towards the earth through the air which is a resisting
medium. Now gravity acts as an accelerating force equally on
the pith and the gold; that is, gravity impresses equal velocity
on both; and in an exhausted receiver of an air pump, as we
well know, both fall through equal vertical spaces in equal times,
and both in equal times acquire equal velocities. The air how-
ever by its resistance is a retarding force, and acts with greater
effect, cæteris paribus, in diminishing velocity on bodies whose
density is more nearly equal to its own density than it does on
those whose density is greater. Hence the velocity of the pith
70
[44.
D'ALEMBERT'S PRINCIPLE.
will be diminished by the action of the air more than the velo-
city of the gold if both were of equal size and shape and were
separate. If the two substances were separated the gold would
fall faster than the pith; that is, in other words, in a given time
greater velocity would be impressed upon and expressed in the
gold than in the pith. Let the two be connected so that the
retarding action of the air is equal on both; by reason of the
connection all falls together; and has therefore a common velo-
city; and in neither the pith nor the gold is the momentum ex-
pressed equal to that impressed, for the expressed momentum
of the gold is less than it would be if it were not attached to the
pith; and the expressed momentum of the pith is greater than
it would be if it were separate from the gold. Some of the mo-
mentum which is impressed on the gold is not expressed in its
motion; and that expressed in the pith is in excess of that which
is impressed on it. The gold therefore loses momentum as ex-
hibited in that expressed, and the pith gains.
Again, let us take an instance which is similar to that for the
solution of which the principle was first devised. Let us sup-
pose a circular horizontal plate to rotate about a vertical axis
passing through its centre; and let us suppose it to rotate in
the exhausted receiver of an air pump, so that no diminution of
velocity takes place by reason of the resistance of the air. At
the ends of the vertical axis let pivots be placed in fixed centres,
so that the plate continues to rotate about the fixed vertical
axis; and let the friction of the pivots be the sole cause of the
diminution of the velocity of the plate. Now to the plate let a
certain angular velocity be imparted; then if the plate thus
rotating were divided into two equal concentric parts, the mo-
mentum of the exterior part would be greater than that of the
interior, in the ratio indeed of 2-1 to 1; and therefore if the
interior part alone rotated it would be brought to rest by the
friction of the pivots much sooner than the exterior part would,
if it were connected with the axis, the interior part having been
removed. However if the whole forms one rigid body all is
simultaneously brought to rest; the exterior in less time than it
would be, if it were separated from the interior; and the in-
terior in longer time than it would be if it were alone. In the
withdrawal therefore of the momentum of the plate by the fric-
tion of the pivots, the diminution of the exterior part is greater,
and that of the interior part is less than it would be if each were
45.]
71
D'ALEMBERT'S PRINCIPLE.
*~
separate from the other. Thus the exterior loses and the in-
terior gains momentum. In neither one part nor the other
is the momentum expressed equal to that impressed. Similarly
if the plate is divided into concentric rings of infinitesimal
breadth, generally the expressed momentum-increment of any
ring (I use the term increment algebraically) is not equal to the
impressed. It is so doubtless in a certain ring, but all rings
external to that lose momentum; and all rings internal to it
gain momentum; and, as we shall presently shew, the aggregate
of the momentum lost throughout the plate is equal to the aggre-
gate of that which is gained.
45.] What has here been said of rigid bodies, is also generally
true of material systems. In the motion of each particle a dif-
ference will exist between the momentum impressed by a given
external force and that expressed in the motion of the particle;
and this difference too exists not only in the intensity of these
momenta but also as to their lines of action; the particle m (say)
will not move along the line of action of the force which impresses
momentum, as it would do if it were free, but it will generally
move along some other line; thus the momentum due to the
acting force is not expressed in the
to intensity or as to line of action.
particle with other particles of the
difference.
particle's motion either as
And the connection of the
system is the cause of this
On what however is this difference spent? the momentum is
impressed; matter is inert and cannot absorb it; it is not ex-
pressed in the motion of the particle on which the force acts; it
must therefore be expressed elsewhere; and must in the first
place produce a strain or an action between that particle and
one or more of the other particles of the system. And what is
the result of this? It must be that these other particles will
gain exactly as much momentum as the original particle m has
lost. A similar result will also be true for every other particle
of the system; so that the sum of the momenta expressed in
all the particles will be equal to the sum of those impressed.
Thus, if the momenta are calculated throughout the system, the
sum of those which are lost is exactly equal to those which are
gained. Hence the differences between the impressed and the
expressed momenta, taken throughout the system, are in equili
brium, and satisfy the equations (94) and (95), Art. 57. Vol. III.
This theorem of the equality of the impressed and expressed
72
[45.
D'ALEMBERT'S PRINCIPLE.
momenta, taken through the whole material system in motion,
was devised first by D'Alembert, and is now called D'Alembert's
Principle. It is enunciated in the following form;
When a material system is in motion, and is acted on by
forces which impress momenta, the momenta lost by all the par-
ticles of the system are in equilibrium.
In this enunciation the term "momentum lost" is equivalent
to the excess of the momentum impressed over that expressed
in any particle of the system. The term also is employed alge-
braically, and includes cases in which the expressed momentum
is in excess of that impressed. It will be perceived that in the
explanation above, we have fixed our thoughts on a particle in
which the impressed momentum is greater than that expressed.
If the system is acted on by finite accelerating forces, so that
infinitesimal momentum-increments are impressed in infinitesi-
mal time-elements, the term momentum in the preceding enun-
ciation must be replaced by "momentum-increment."
M3
The mathematical expression of this principle is as follows;
Let m₁, m2, mz... m, be the particles of which a system is
composed; of which let m be the type, and let (x, y, z) be its
place at the time t; let x, y, z be the axial components of the
momentum-increment impressed by the external forces on m;
and let I cosa, I cos B, I cos y be the axial components of the
momentum-increment arising from the internal forces; then the
equations of motion of translation of m are
d2x
X m
= I cos α,
dt2
d2y
Y กาว = I cos ß,
dt2
d² z
2
m
= ICOS Y;
dt2
let the equations of which these are the types be written for
every particle of the system; and let them be added; then we
have
M
M
(x-
M
d² x
dt2
= Σ.I COs a,
Y
M
dt2
d² y
= Σ. I cos B,
d² ~
Σ Z
M
=Σ. I COS Y;
dt 2
46.]
73
D'ALEMBERT'S PRINCIPLE.
if however the system is such that all the momentum-increments
arising from the internal forces neutralize each other, the right
hand members of the preceding equations vanish; and we have
d² x
Σ.
x. (x
X M
0,
dt2
y
Σ.
(x — m 124) = 0,
dt2
d² z
= 0.
dt2
x. (z – m² 1223) =
Σ.
This is one form of the equations of motion which is involved in
D'Alembert's Principle.
46.] Before we proceed to the complete and purely mathema-
tical expression of the principle, and to the investigation of the
general equations which arise out of it, I will explain two or
three simple problems, so that the mode of application may be
more exactly apprehended. Certain circumstances must be
cmitted, because we have not yet deduced from the principle
theorems which they require. Hereafter the problems will be
treated completely.
Ex. 1. Let m and m', Fig. 17, be two heavy particles attached
to the ends of a perfectly flexible and inextensible string, which
we will suppose to be without weight. The string with the
weights at its ends is suspended over a small pulley which we
will assume to be without inertia and to be perfectly smooth.
It is required to determine the motion of the particles and the
tension of the string.
Let A and A' be the places of m
the time, P and P' at the time t;
O'P'
x';
and m' at the beginning of
OA a, o'A' a'; OP = x,
oa = =
=
+α.
.'. x + x = a + a.
(1)
Let us consider the circumstances at the beginning of the
motion. Let us suppose impulsive forces to act on m and m'
downwards, and to impress on them velocities u and u'; that is,
m and m' would move with velocities u and u by the action of
the impulsive forces, if they were free. Let v and v′ be the ex-
pressed velocities of each in its constrained state; and let 7 and
T' be the tensions of the strings oA and o'a', when t=0. Then,
as the impressed and expressed momenta of m are respectively
mu and mv,
and similarly
T = M U mv;
7' = m'u̸' —m'v'.
PRICE, VOL. IV.
L
T
(2)
(3)
74
[46.
D'ALEMBERT'S PRINCIPLE.
dx'
=0; .'.v+v′= 0, and v′
V.
dx
But since from (1) +
dt dt
As the pulley has no inertia, and neither friction nor roughness,
T = T';
mu— mv — m'u̸' + m'v ;
=
mu m'u'
V
;
m + m²
(4)
which gives the velocity of descent of m, and of ascent of m',
when t = 0. Also
T
mm' (u+u')
m+m'
(5)
Let us next consider the circumstances at the time t; that is,
when the bodies are no longer under the action of impulsive
forces, but of the continuously accelerating force of gravity.
Then the impressed and the expressed momentum-increments
d² x
dt2
d²x'
dt2
of m are respectively mg and m ; and of m', m'g and m'
also let the tension of the string OPT, and of the string o'r'r';
but from (1)
d² x
.'.
T = mg - m
dt2'
d²x'
T' = m'g — m'
dt2
d2 x d2x
+
dt2 dt2
0.
(6)
(7)
And as the pulley has no inertia and is smooth and free from
friction, by D'Alembert's principle, T =
=T',
d² x
d2x
mg
m
dt2
m'g + m'
dt2
d2 x
m m'
d²x'
g
(8)
dt2
m + m'
dt2
dx
m · m'
v =
gt,
dt
m + m
(9)
dx'
m
m'
+ v =
dť
m + m² I t ;
m m' 1
x = a + vt +
m + m² 29t²,
m
x′ = a′ — vt
m' 1
gt2:
m + m² 2
(10)
T = T
2mm'g
m + m'
•.
(11)
so that all the circumstances of motion are determined.
46.]
75
D'ALEMBERT'S PRINCIPLE.
Ex. 2. Let m and m', Fig. 18, be two heavy particles attached
by means of flexible and inextensible strings without weight to
a wheel and axle respectively, which are supposed to be without
inertia; the initial circumstances being given, it is required to
determine the subsequent circumstances of motion.
In Fig. 18, c is the common centre of the wheel and axle;
co = c, co' = c' are the radii; a and a' are the places of m and
m' when t = O, P and P' when t = t
t; oa = a, o's′ = a', op = x,
o'p':
As a increases by dx, and as x' decreases by dx', let the wheel
and axle rotate through an angle de; so that
dx = c do,
dx
d Ꮎ
C
dt
dt
d2 x
d20
C
dt2
dt2
dx'
— c do,
dx'
d Ꮎ
- c
;
(12)
dt
dt
d2x
d20
- c
(13)
dt2
dt2'
Let us consider the circumstances at the beginning of the
motion; and let the symbols be the same as those of the pre-
ceding example; then
(14)
(15)
By D'Alembert's principle these tensions are in equilibrium;
therefore
and from (12)
T = M U ти
7' = m'u' — m'v' ;)
CT = C'T'
V
v
C
210
cmu — c'm'u'
*
mc² + m'c'²
;
(16)
T
T
mm' (c'u + cu')
C
mc² + m'c²
2
;
(17)
whereby the initial velocities of the particles and the initial ten-
sions of the strings are known.
Let us now consider the circumstances at the time t; and let
the symbols be the same as those of the preceding example; then
d2 x
T = mg — M
dt2
d20
= mg mc
;
(18)
dt2
d²x'
=
T' m'g - m'
dt2
d20
= m'g+m'c'
;
(19)
dt2
L 2
76
[46.
D'ALEMBERT'S PRINCIPLE.
and as these tensions are in equilibrium by D'Alembert's prin-
ciple, we have
.'.
CT = c'T';
d20
mc-m'c'
dt2
mc² + m'c'²
g;
(20)
(21)
1 d² x
c dt2
1
2
d²x²
m c
m'c'
c dt2
1
dx'
27
c
dt
1
..
dx
dt
x = a + v t + c
mc² + m² c'² 9 ;
mc
· m'c'
mc² + m'c29t;
mc-m'c' gt2
(22)
mc² + m'c²² 2
mc
m'c' gt2
x' = a + v't-c
(23)
mc² + m'c² 2
cc'mm'g(c+c')
(24)
mc² + m'c²²
2
CT = C'T'
whereby all the circumstances of motion in both the initial and
the general states are known.
Ex. 3. A heavy chain, flexible and inextensible, homogeneous
and smooth, hangs over a small pulley at the common vertex of
two smooth inclined planes; it is required to determine the mo-
tion of the chain.
Let the two inclined planes, the chain and the pulley, be re-
presented in Fig. 19, each of the inclined planes being supposed
to be longer than the length of the chain; so that the chain, as
we consider its motion, is on one or the other of the planes. Let
o be the common vertex of the two planes; A and A' the ends of
the chain when t = O, P and P' the ends when t = t; OA = ɑ,
OP'
OA' a', OP = x, or' = x'; and let a and a' be the angles of in-
= X,
clination of the planes to the horizon;
chain; therefore
the length of the
x + x = a + á = 1.
(25)
We will suppose the chain to be initially at rest. Let w be the
area of a transverse section, p = the density; T =
the density; T = the tension
at the time t
• T = wpx
३०
sin a
d2x)
dt2
w pa′ {g sin a'
d²x'
dt2
}; (26)
d2x
sin a + sin a
α
ga-g sin a':
(27)
dt2
7
d2x'
sin a + sin a'
gx' -g sin a;
(28)
dt2
47.]
77
D'ALEMBERT'S PRINCIPLE.
dx2
g (sin a + sin a')
(x2
dt2
a²) — 2 g sin a' (x
a); (29)
dx' 2
g (sin a + sin a')
2
d12
1
(x¹² — a′²) — 2 g sin a (x′ — a′); (30)
whence the relations between x and t, and between x' and t,
may be found; but the form of the equations is too compli-
cated to be of any use.
Also
T
@pgxx
7
(sin a+ sin a').
(31)
If the chain, instead of resting on two inclined planes, hangs
over a small pulley without inertia, then, all the other circum-
stances being the same, a = a 90°; and the equations of
motion are
dt2
d² x 2g
ī
X
9,
d²x'
dtz
2g
x' — g.
(32)
47.] The following explanation of D'Alembert's principle is
much the same as that which he first gave in the Traité de Dy-
namique; and as it will thus be stated in a mathematical form,
the general equations of motion will be most conveniently de-
duced from it.
Let P, Fig. 20, be the place of a particle m of a material
system. During the infinitesimal time dt let a force act on m
which would impress on it, if it were free, a velocity whose
line-representative is PA; let the impressed velocity be v; so
that my is the impressed momentum along, and proportional to
PA; let v be the velocity of m; that is, let mv be the expressed
momentum; and let its line of action be PB; let PC be the line
which would complete the parallelogram of which PA is the
diagonal, and PB one of the containing sides: then resolving v
into the velocity v along PB, and v' along PC, v', which is repre-
sented by PC, is the velocity lost; and mv', which is proportional
to and acts along PC, is the momentum lost. D'Alembert's
principle asserts that all the lost momenta taken throughout
the system are in equilibrium. His words are;
Décomposez les mouvements a, b, c ...... imprimeés à chaque
corps, chacun en deux autres a, a; b, ß; c, y ;
qui soient
tels que si l'on n'eût imprimeé aux corps que les mouvements
a, b, c,….. ils eussent pu conserver ces mouvements sans se nuire
78 THE EQUATIONS OF MOTION. INSTANTANEOUS FORCES. [48.
réciproquement; et que si on ne leur eût imprimé que les mouve-
ments a, ß, y, …….………., le systême fût demeuré en repos.
Again, produce BP to B', so that PB'
PB; then the momen-
tum represented by PC is evidently the resultant of those repre-
sented by PA and PB'; hence we have D'Alembert's principle in
the following form;
If the expressed momenta of the several particles of a ma-
terial system are estimated in a direction the contrary of that
in which they act, they, together with the impressed momenta
when taken through the whole system, will satisfy the conditions
of statical equilibrium.
48.] Such is D'Alembert's principle, as to its origin and as
to its form of expression; it reduces all the theorems of motion
of material systems to those of statical equilibrium; and so it is
commonly said that D'Alembert reduced dynamics to statics.
The principle does not indeed directly furnish the equations ne-
cessary for the solution of the different problems of dynamics;
but it teaches the mode by which they are to be deduced
from the equations of equilibrium; and thus, if we apply to
the "momenta lost," the conditions of statical equilibrium, the
dynamical equations will be formed. It is evident too that we
may introduce them as pressures into the equation of vertical
velocities, and this will hereafter be done. The equations of
equilibrium of a system of pressures acting on the several points
of a rigid body are investigated in Vol. III. The number of
them is six; of which three are of translation and three are of
rotation: the momenta lost must satisfy these six conditions.
Firstly, let us suppose the acting forces on the system of par-
ticles to be impulsive and instantaneous, so that finite momenta
are impressed instantaneously, and the expressed momenta are
also instantaneously developed.
Let m be the symbol of a type-particle; (x, y, z) its place at
dx dy dz
the time t;
و
>
dt dt' dt
the axial components of its expressed
velocity due to the acting instantaneous forces; V, V, V₂ the
axial components of the velocity impressed on m; so that the
differences between the axial components of the impressed and
expressed momenta are
dx
m (v. — da),
V
dt
m(v
dy
m (v: — dz). (38)
リ
​dt
48.]
THE EQUATIONS OF MOTION. FINITE FORCES. 79
By D'Alembert's principle these and similar quantities for all
the other particles of the system are in equilibrium; therefore
the six following equations must be satisfied by them;
n (v₂ — da)
Σ.mVx
Σ.Μ n
Ꮖ
0,
(v, — dy) = 0,
dt
(v₂ — dz) = 0;
Σ.m v
dz) — z (v, — dy)}}
0,
z.my — = 0
Σ.Μ
{y (v.
—
{ = (v.
Vx
Σ.m
-
dt
dx
dt
X
い
​z. m { x (v, — dy) — y (v.
dt
dt
dz
dt
(34)
=
0,
(35)
di')
dx
— dc)} = 0;
dt
whereof the first three are the equations of translation, and the
last three are the equations of the moments of the couples which
arise from the excess of the expressed over the impressed mo-
menta about the three coordinate axes. The sign of summation
extends to, and includes, all the particles of the system; and
the expressed velocities are those due to the action of the im-
pressed forces.
are the
Hence, if u, v, w are the axial components of the velocity of m
dx dy dz
before the instantaneous forces act, and if
dt' dt' dt
axial components after the velocities Vr, Vy, Vz have been im-
dx dy
pressed, ·U,
dt dt
dz
dt
v, -w are the expressed axial velocities
due to the instantaneous forces; and in equations (34) and (35)
dx dy dz
are to be replaced by these quantities.
dt' dt' dt
Secondly, let us suppose the system of particles to be under
the action of finite accelerating forces, so that in infinitesimal
time-elements, infinitesimal momentum-increments are impressed
upon and expressed in the type-particle m.
Let x, y, z be the axial components of the impressed velocity-
increment on m, which is supposed to be at (x, y, z) at the time t;
d²x d²y d²z
and
dt², dt, dtz are the axial components of the expressed
velocity-increment; so that the differences between the axial
80
[49.
D'ALEMBERT'S PRINCIPLE
components of the impressed and the expressed momentum-
increments of m are
m (3
(x - da,x),
d²
dt2
m (x — day),
2
m (z – d²). (36)
dt2
By D'Alembert's principle these and similar quantities for all
the other particles of the system are in equilibrium; therefore
the six following equations must be satisfied by them;
Σ.m
2
x
(x — daa) = 0,
dt2
d²y
x.m (v - day)
dt2
d2 z
n (z — 122 ) =
Σ.m | Z
dt2
0,
= 0;
(37)
Σ.Μ
{y ( z
−
d22
dt2
1/2zz) — ~ ( x
−
d2y
dt2
124 ) }
0,
Σ.m
{ ≈ ( x
−
d2x
dt2
1²x) − x (
z
−
dt2
1² z ) } = 0,
(38)
X | Y
d2
dt2
૨૦૧
)-3(x
d² x
0 ;
dt2
Σ.Π
{x(
the sign of summation includes all the particles of the system.
In these equations the power of the sign of summation should
be carefully observed; it includes all the particles of the moving
system, whether that system be continuous or discontinuous.
Thus, if the several particles are m₁, m2, ... mn; and their places
at the time t are (x₁, Y₁, ~1), (X2, Y2, Z2), . . . (Xn, Yn, n), and the im-
pressed velocity-increments are (X1, Y1, 21), (X2, V2, Z2), . . . (Xn, Yn, Zn);
then the first of (37) is the abbreviated form of
m1X1
m (x₁ — d²x²)
d² x 1) + m² ( X 2
+ m² (x₂ — d²x²).
dt2
dt2
2 xn
+ ... + Mn
+ mn (xn− d²x₁₂)
Xn
dt2
= 0.
Similarly the other five equations are abbreviated forms of ana-
logous expressions.
49.] The equations of motion of a material system may also
be expressed in a shorter form. For since D'Alembert's prin-
ciple enables us to deduce them from the equilibrium which
subsists among the "lost momenta;" that equilibrium will be
obtained not only from the six equations which correspond to
(37) and (38) of the preceding Article, but also from the equa-
tion of virtual velocities. The truth of this last equation has
been demonstrated in Articles 104 and 395 of Vol. III, but
49.]
81
VIRTUAL VELOCITIES.
since we shall now apply it somewhat extensively, and very gene-
rally, as it will include all dynamics, it is necessary to say a few
words on its form and the conditions of its exactness.
We
imagine a material system to be at rest under the action of many
forces, which may be external as well as internal to the system :
of these forces we take P to be the type, and we suppose it to
act on m, which we take to be the type-particle: so that .P
will be the sum of all the forces which act on all the particles;
many of which may act on one and the same particle; and
others of which will enter in pairs of equal and opposite forces,
when there are mutual tensions or reactions or constraints among
the particles of the system. We imagine the system to receive
an arbitrary infinitesimal displacement, consistently with its geo-
metrical relations, whereby the points of application of the forces
are changed, but neither the intensities nor the directions of the
lines of action are altered. Let the displacements of the points
of application of the forces be estimated along the lines of action
of the forces; and let 8p be the infinitesimal displacement of
(x, y, z), the point of application of P, thus estimated; then the
equation of virtual velocities is
Σ.Pdp = 0;
δρ
and this expresses the condition that the forces are in equili-
brium.
2
As
Let us put into an equation of this form the several quanti-
ties which are active in the motion of a material system.
d²x d²y d²z
dt' d' dt are the axial components of the expressed velo-
city-increments of m, which is at (x, y, z) at the time t, it is evi-
dent that the impressed momentum-increments along these axes,
which would have their full effect in producing pressure if the
d2x d³y d2z
M
dt2' dt?" dt²
system were at rest, must be diminished by m
now that the system moves; and the actual effects will be the
excesses of the former over the latter: in the equation therefore
of virtual velocities these latter quantities must be affected with
negative signs. We shall use the symbol & to express the varia-
tions of the points of application of the forces which are due to
the arbitrary geometrical displacement of the system; and we
shall still indicate by the symbol d the time-variation of the
coordinates and velocities. Let me be the type momentum-
increment acting on the type particle m; and let us suppose dp
PRICE, VOL. IV.
M
82
[49.
D'ALEMBERT'S PRINCIPLE.
to be the infinitesimal displacement of the point of application
of P estimated along its line of action, P tending to remove m
from the origin, and dp being positive when the point of appli-
cation of p is moved in the direction along which P acts; then,
if the line of action of a force is along a coordinate axis, say that
of a, the variation of the point of application is ox. Now, esti-
mating forces according to these conditions, the equation of vir-
tual velocities is
Σ.m p dp — Σ.m
d²x
dt2
d³y dy +
бох + бу
d2
82
dt 2
dt2
}
= 0; (39)
which is the most general equation of motion of a material
system.
Р
x.mpdp includes all the forces which act on the several par-
ticles of the system, both internal and external; if however
two particles are acted on by a force along the line which joins
them, and if the distance between these particles is unchanged in
the geometrical displacement, this force will disappear in the ag-
gregate; because the geometrical displacements of the two parti-
cles estimated along the line of force will be equal and opposite,
and therefore the two effects, as they are measured in the pre-
ceding equation, will neutralize each other, and will disappear.
50.] Let all the impressed momentum-increments, as they
are applied to each particle, be resolved into components parallel
to the three coordinate axes; and let x, y, z be the axial compo-
nents of p as it acts on m at (x, y, z); then the infinitesimal dis-
placements of m along the three axes will be dx, dy, dz; which
are the same as the displacements of the point of application of
the expressed momentum-increments: so that (39) becomes in
this case
d2y
x.m {(x − d²) 8x + ( − 1 )by + (zd²)z} = 0; (40)
Σ.Μ X
d2x
dt2
Y
d² y
dt2
dt2
δ
which is another form of the equation of virtual velocities.
Now no restriction has been made as to the kind of displace-
ment of (x, y, z), of which the axial projections are represented
by dx, dy, dz; it is only to be consistent with the geometrical
relations of the system: let us suppose it therefore to be most
general, and to be compounded of motions of translation and of
rotation of the whole system. Let the system receive a displace-
ment of translation, so that every particle moves over an equal
and parallel space in the direction of the coordinate axes, which
51.]
83
VIRTUAL VELOCITIES.
we will represent severally by dxo, dyo, do; and also let the
system receive three successive displacements of rotation through
the angles 80, dp, dy about the three coordinate axes: then the
total variations of the coordinates of the point (x, y, z) are
бх = dx + zd4ydy,
δι = δυο + αδψ - 28θ,
бу
δε = 80 + 1 δθ – αδφ;
y
and substituting these in (40) we have
d2x
dx, z. m ( x − 1²x²) + 8 y。 z.m
0
dt2
dy
+ ò0 z.m {y (
δ θ {y (z
Y
dx
dz z ) − z ( x − 1 2 3 )
dt2
(41)
(x — day)
+8% Σ.m
dt2
(z - 122)
d2z
dt2
d2y
–
dt2
+ 8 4 z.m { z
(
x
−
d²
12x)
X
(
d2≈
Z
dt2
(x
−
δφ
+d4w.m
+84 2. m { x
-
dt2
d2 y
d2
1 1) — y (x-2)=0; (42)
dt2
d t²
as the several variations on which the displacement depends are
independent of each other, their coefficients must separately
vanish; and hence we have the six equations of motion, viz.
(37) and (38) of Art. 48.
By a similar process we may deduce the six equations (34) and
(35) for instantaneous forces.
51.] When the system of particles on which the forces act is
rigid or invariable in form, that invariability of form will be
secured by means of certain equations which the coordinates of
the several particles must satisfy. Thus, if the number of par-
ticles is ", and if all the distances of these particles from each
other are invariable during the motion, (3n-6) equations must
be satisfied; for if the distances between three are given, and the
distances of every one of the remaining (n−3) from each of these
three are given, whereby we have 3 n-6 given distances. the
system of particles is invariable in form. Now the position of
every particle of the system at any time is determined, when
the coordinates of every particle are expressed in terms of t;
and as each particle has three coordinates, 3n quantities must
be expressed in terms of t: these however are subjected to 3 n 6
conditions of relative position: six other conditions therefore
are necessary, and are sufficient for the complete solution of the
problem; and these are given by the equations (37) and (38) of
Art. 48.
D
M 2
84
[52.
MOTION OF A FLEXIBLE CORD.
And now let us take a more general case; and let us suppose
certain particles of a material system to be constrained to move
on certain curves, so that certain relations will be given which
the coordinates of the particles must satisfy. Suppose the num-
ber of these relations to be k; and let them be F, 0, F₂ = 0,
….. F; = 0; then, taking the most general case, and supposing
each of these functions to contain all the coordinates of all the
points, their total variations are
1
dr
df1
1
0.
dr.
dz
dFi
dy
dy
dr.
dri
dr
F
d F z \ d r i
+
d F e \ 8 9 ₁ =
+
) ¿ µm +(
18=
=
= 0.
(43)
d=
dyn
d1
dai
drs
drk
+
d F F
dr
dyi
1
* ) a + ( 1 ² ² ) ō = n
and from (40) we have an equation of the form
Aq & Xq + B₁ & Ụ₂ → C₁ = 2 + ...... +1 « & X « + B„ dy + C₂d == 0. (44)
Y ¿
à
If then we multiply (43) severally by indeterminate multipliers
Ay, Ay, ….. Ax, and add all and (11); and if we equate to zero the
coefficients of 8.x₁, dy…. ….. d.„. we have
dæ₁,
drž
= 0:
dz.
dyn
dri
dr
λ1
ايد
dr
drt
= 4; =
A 0.
+
dri
dr
dy
!
dr
λε
đụ.
1
dr
11
+
X
()
dz
dz
= 0,
B. =
÷ c, = 0;
(45)
by means of which equations, and of the given equations of con-
dition, A₂. Aş……..A may be eliminated, and the coordinates of the
particles may be determined.
52.] We will now apply the process of the preceding Article
to the motion of a flexible and inextensible string, which has its
two ends fixed, under the action of given forces.
Let ds be the length-element of the string at the point
(z, v. :); let & be the area of the transverse section,
↑ = the
density: so that the mass-element of the string is pods. Let
the points (4. g₁ = • (22, 22, #2) be the two ends of the string.
and let 2. p. PA & be the values of p. & at them respectively.
Let 1. 1. z be the axial components of the impressed velocity-
increment at the point .. and let 1. 12. 21. I, I. Za be
o 01:
1 Wy
*
7, y.
52.]
85
MOTION OF A FLEXIBLE CORD.
the values of these quantities at the two ends of the string
which we suppose to be dynamically fixed, but to admit of geo-
metrical variation. Then (40) becomes
d²
[ " p w d s } ( 1 − d² 2 ) d x + ( 5 dzy ) òy + (z — dzz) òz
1
dt2
I
Σ
dt2
}
2
Y₂ Ò Y 2
Z
2
0; (16)
and since the string is of constant length
2
ds = a constant.
dz
a.dyds
d.dz
[ { dr¿.dr + dy &.dy + d= 8.d= = 8. 1" de
Ids
du
ò. ļ¯ds
ds
0:
(47)
(48)
and multiplying the quantity under the sign of integration by à,
and adding it to (46), we have
ρωάς
[² pods (3 - 11
der
dr
d. c x
di²
ds
x₁ ò x ½ + I₁ Ò Y₂+
- Z₂ = 0; (49)
and integrating by parts the second terms of the several mem-
bers of the upper line, as we have explained in the Calculus of
Variations, Vol. II, we have
[² { pwds (x - d ² 1)
der
di
dr
-d.λ
OF T
ds!
dr
dy
dz
DI+ A
દેy + A
二
​ds
d's
ds
m
÷ I₂ ÒX½ + I¸ ôY, +
1
÷Za dia = 0. (50)
As no other relation is given between dr, dy, and -, we have
A
dr
pods
p w ds ( I
di²
dr
-d.λ = 0,
ds
dy
dy
pode T
- d.A d² = 0,
diz
d²:
d-
Dods Z
-d.λ
=
die
ds
0:
A...
(51)
which are the equations of motion of the cord. The latter terms
in (50) give values at the ends of the string; if the ends are in-
dependent of each other,
dr.
de
dx:
Z-
3 -
— 2-12, = 0.
=
(52
I
入​了
​d=
dv.
+ I
d
86
TRANSLATION OF THE CENTRE OF GRAVITY. [53.
and if the ends are fixed
d = 1 + 21
di asi
d=2+22
+2₁ =
0;
dy
1
dx1
λι
Midsi
+ x₁ = 0,
λι
+ Y 1
0,
·1 ds 1
1
dx 2 + x 2
A s 2
+ X₂ =
= 0,
- dsa
dy2
d $2
+ Y₂ = 0,
一​入口
​+22= = 0.
ds₂
The form of the last terms of (51) shews that A is the tension of
the string at the point (x, y, z), and acts along the length-element
ds: indeed the equations (51) are only the particular form of
(37) when the forces arising from the tension are introduced
into them. If we eliminate λ from (51), we shall have two
equations in terms of x, y, z, t, which will give the position of
the string at any time. I may observe too that A is evidently
the tension, because the mode in which it is introduced shews
that it is an internal force acting along ds; and that the varia-
tions of its points of action are the same as the variations of the
ends of the length-element.
Thus much must at present suffice for this problem; and we
will proceed to the demonstration of various general theorems
which arise out of the equations of motion.
SECTION 2.- Independence of the motions of translation of the cen-
tre of gravity, and of rotation about an axis passing through it.
53.] Many general theorems of dynamics are deducible from
D'Alembert's principle as expressed in the six equations, either
(34) and (35) of instantaneous forces, or (37) and (38) of finite
accelerating forces the investigation of the most general of
these will be reserved to a subsequent Chapter of our Treatise;
but it is convenient at once to discuss certain theorems relative
to the centre of gravity of a material system as well as some
other simple properties of motion which follow directly from them.
I must first observe on the name "centre of gravity;" it is given
to the so called point of a heavy body or system for a reason
much too narrow for the properties of it which we shall presently
develope; and a term of wider application is required. We shall
also apply the term to a system of particles which perhaps may
not gravitate at all. In geometry, as it is well known, a similar
point is called the centre of mean distances; in dynamics the term
"centre of masses" has been proposed; but the inconvenience of
a new name is so great that I propose to retain the old one, "cen-
53.]
87
TRANSLATION OF THE CENTRE OF GRAVITY.
*
tre of gravity." The student however must remember that we do
not employ the term in its restricted sense, that is, in the stati-
cal meaning as applied to heavy bodies under the action of the
earth's attraction; but we mean that point (x, y, z) of a system
of particles which, relatively to all the particles, is defined by
the following equations:
x Σ.m = Σ.mx,
Σ.m = Σ.my,
ZΣ.m = Σ. m 2 ;
dx
dx
d2 x
d2x
so that
Σ.Μ = Σ.
Σ.Μ = Σ.m
dt
dt
dt2
dt 2,
dy
dy
d2y
d2y
Σ.Μ = Σ.
Σ.Μ = Σ.
dt
dt'
dt2
dt2
dz
dz
d2
d2~
Σ. = Σ.
Σ.M = E.M
dt
dt
dt2
dt2
so that if the centre of gravity is the origin
Σ.mx = Σ.my = Σ.MZ =
0;
dx
dy
dz
Σ.
=Σ.m
= Σ.m
= 0;
(53)
dt
dt
dt
Σ.Μ
d² x
dt2
= Σ.000
d2y
dt2
d² z
= Σ.m
=
0.
(54)
dt2
In the preceding Chapter it has been proved that the general
motion of a body may be resolved into a motion of translation
of any one point of it chosen arbitrarily, and into a motion of
rotation about an axis passing through that arbitrarily chosen
point. One simplification indeed of that theorem was pointed
out on principles purely cinematical; viz. that if that point was
taken on the central axis, the motion of the point would lie along
the axis of rotation; and this is true of course, when all the
expressed velocities of the body or system due to the acting
forces are given, whereby the position of the central axis at the
time t may be determined. But here another problem is offered
for consideration. When a body or system is in motion, certain
velocities are developed during the motion by reason of the
inertia of the particles, and hereby certain velocities are im-
pressed on the particles: these vary according to the point
which is assumed to be the moving point of translation, and
through which the instantaneous rotation-axis of the body passes.
We shall shew that these velocities, due to the motion, neutralize
each other, when the centre of gravity is taken to be the moving
88
TRANSLATION OF THE CENTRE OF GRAVITY. [54.
point of translation, so that the instantaneous rotation-axis passes
through it; and that the motion of translation of the centre of
gravity is independent of the motion of rotation of the system
about the instantaneous axis passing through it.
54.] Let (x, y, z) be the place of the centre of gravity at the
time t; (x, y, ≈) the place of the type-particle m; and let us
suppose a system of coordinate axes to originate at the centre
of gravity parallel to the original system of reference; and let
the place of m relatively to the new axes be (a', y', ✅'); so that
~ = ~ + ~ ;
z.my' = x.m² = 0;
E.MZ
y = ÿ + y',
x = x + x',
ɛ.mx' =
..
Σ
dx'
dy
dz
Σ.»
= Σ.m
= Σ.m
= 0:
dt
dt
dt
d²x'
day'
d²
Σ.Μ
= Σ.m
=Σ.m
dt2
dt2
= 0.
dt2
E.MX = NT,
dx
dx
Σ.»
= M
Σ.Μ
dt
dt
dt
Let м = the mass of all the particles of the moving system; then
Σ.my = My,
dy dy
Σ.M≈ = M≈
dz
dz
= M
Σ.Μ
= M
dt
>
dt
dt
Σ.Μ
d² x
dt2
d2x
d2y
d2y
d² z
d2z
M
dt²,
Σ.η
= M
dt2
dt
d12,
Σ.Π
= M
dt2
dt²
Firstly, let us take equations (34) and (35) which refer to in-
stantaneous forces; then
dx dx dx'
dy
dy dy' dz
dz dz
+
+
+
(55)
dt
dt dt
dt
dt dt
dt
dt dť
and (34) become
dx
M
= Σ. M Vx
dt
dy
M
= 2. MV {
(56)
dt
dz
M
= Σ. MV; ;
dt
which equations are of the same form as those of the motion of
a material particle whose mass is м. Whence it appears that
The motion of translation of the centre of gravity of a system.
of particles under the action of instantaneous forces is the same
as if the whole mass were collected into it, and all the impressed
momenta were applied at it, each in a line parallel to its own
line of action.
54.]
89
ROTATION ABOUT THE CENTRE OF GRAVITY.
If
d's
dt
is the velocity of the centre of gravity,
M
ds
dt
{(z.mv¸)² + (z.mv,)² + (x. mv.) 2 } };
and the direction-cosines of the path which the centre of gravity
takes are given by (56).
Again, taking the first of (35), and making similar substitu-
tions, we have
Σ.Μ
{ (J + y´) (v=
dz
dz
dt
dt
da ) — (z + ~') (v, — dy - dy ) }}
dt
dt)
= 0;
which may be expressed
ÿ z.m (v. — dz) — y z.m
dt
as follows;
ď z
dt
+ z.my (v. — dz) -
dź
dz
z.my'
dt
dt
Ξ Σ.
Z z . m ( vy
dy
dy
dy
+ZZ.M
Σ.Π
dt
dt
dt
dý ) +
dy
x.mz' = 0;
dt
and thus we have
Σ.Μ
d:
dy
{y' (v: — d ) — ~' (v, — ( 4 ) }
dt
dt
similarly from the second and third of (35) we have
Σ.Μ
{ ~ ' (v,
Vx
Σ.Π
0 ;
-
d='
(57)
= 0,
dt
y
ズ
​(xx — da') }
= 0.
dt
— da' ) — x' (v =
dt
dý
{ x' (v, — day
dt
) —
dx
These are evidently the equations of the three couples of the
lost momenta relatively to the axes of a system originating at the
centre of gravity; and the impressed momenta are the same as
in the original equations (35): whence we infer that
If the motion of a system of material particles, under the
action of impulsive forces, is resolved into a motion of trans-
lation of the centre of gravity, and of rotation about an axis
passing through that point, the motion of the centre of gravity
is the same as if the masses of all the particles were collected at
it, and all the impressed momenta were applied at it each in a
line parallel to its own line of action; and the motion of rota-
tion about the axis passing through the centre of gravity is the
same as if the centre of gravity were a fixed point, and the sys-
tem rotated about an axis passing through that point under the
PRICE, VOL. IV.
N
90
TRANSLATION OF THE CENTRE OF GRAVITY. [55-
action of the impressed momenta which are actually applied to
the several particles of the system.
And the centre of gravity of the system is the only point
which has this property; for there is no other point for which
dx
dy
dz
Σ.Μ
= Σ.m
= Σ.Μ
0 ;
dt
dt
dt
so that the terms omitted in (57) should disappear.
55.] Secondly, let us take equations (37) and (38), which
D'Alembert's principle gives when the system of particles is
subject to finite accelerating forces; then, differentiating (55),
we have
d2x d2x
d²a'
d² y
d² y
d³y' d2z
d2 z
d2'
dt²
2 dt2 dt2'
+
dt2
+
+
dt2
2
dt², dt²
dt2
dt2
so that (37) become
d2 x
M
Σ.mx,
dt2
d2y
M
= Σ.MY,
(58)
dt2
d2z
M
= Σ . m Z;
Σ.MZ;
dt2
which equations have the same form as those of the motion of a
material particle whose mass is M. Whence it appears that
If the whole mass of a material system is collected into its
centre of gravity, and the several impressed momentum-incre-
ments are applied at it, each in a line parallel to its own line of
action, the expressed momentum-increment, and therefore the
motion of translation, of the mass thus condensed is that due to
all the impressed momentum-increments thereat applied.
Again let us substitute in (38); then we have
Σ.Μ
{T+y') ( z −
d2z
d2z'
dt2 dt2
d2y
day'
-) — (≈ + 2') ( x
dt2
dt2
)} = 0;
and this when expressed at length becomes
d2z
d2z
yx.m
2 (z.
-yΣ.m
dt²
dt2
+ z.my (z — d³z)
d2z
Σ.my'
dt2
dt2
- zz. m (x — 121)
d2y'
dt2
dt2
d² y) + zz.m
+ Σ.mz' = 0;
dt2 dt2
and therefore, omitting terms which vanish by reason of pre-
ceding equations and conditions, we have.
x.mz (
~(x_dzy)
dj
d2y
56.] ROTATION ABOUT THE CENTRE OF GRAVITY.
91
d2z
Σ.Μ
૫
Z
dt2
-) − = (x
day'
dt2
-} }
0 ;
similarly from the other two equations of (38) we have
Σ.Μ
Σ.Μ
d2x'
{ z′ ( x − d²x ) − x (z —
dt2
—
d2z
d12
= 0,
= 0.
{ x' (x — dzy) — y' (x — 12')}
dt2
dt2
(59)
But these are evidently, in reference to the system of coordinate
axes originating at the centre of gravity, the three equations of
the couples which arise from the excess of the impressed over
the expressed momentum-increments; and the impressed mo-
mentum-increments are at each point the same as in the original
equations (38); whence we infer that
If the motion of a system of material particles, under the
action of finite accelerating forces, is resolved into a translation
of the centre of gravity, and a rotation about an axis passing
through that point, the motion of the centre of gravity is the
same as if the mass of all the particles were collected at it, and
all the impressed momentum-increments were applied at it, each
in a line parallel to its own line of action; and the motion of
rotation about an axis passing through the centre of gravity is
the same as if that centre of gravity were a fixed point, and
the system rotated about an axis passing through it under the
action of the impressed momentum-increments which are actually
applied to the several particles of the system.
56.] Thus, when a body is projected in any direction, and
moves under the force of gravity, which acts on all the particles
of the body, the centre of gravity of the body describes a para-
bola in a vertical plane. Again, if a shell is projected and the
shell bursts before it meets the earth, by the action of internal
forces, these latter forces, being related in equal and opposite
pairs, do not appear in the right-hand member of (58), and the
centre of gravity of all the broken parts moves in the same
parabolic path as before the explosion.
This theorem is also true of the solar system; because it is a
material system of the nature explained in Art. 43; so that if
the solar system has a proper motion in space by the action of
forces external to it, which have either acted once for all, or
are finite and continuous, that proper motion will be shewn in
the change of place of the centre of gravity of the system; and
N 2
92
[57.
CONSERVATION OF MOTION
if the path, velocity, &c., of the centre of gravity can be deter-
mined by observation, the force to which it is due may also be
determined. Now as the mass of the sun is so very much greater
than that of the other constituent bodies of its system, and as
they are arranged around it, we may, without great error, as-
sume the centre of the sun to be the centre of gravity of its
system; and this being so, the result of the calculations of M. F.
G. W. Struvè, founded on the studies of Argelander, O. Struvè,
and Peters, is that the sun advances annually in space through
154,185,000 miles towards a point in the heavens situated in the
constellation Hercules*. This result is arrived at from an esti-
mation of the proper motion of the stars: but our knowledge of
these motions is at present far too imperfect for us to decide.
how far the assigned velocity and direction of the solar motion
deviates from exactness; and whether it continues uniform, or
whether it shows any symptoms of deflection from rectilinearity.
At present, says sir John Herschel, we require more precise and
extensive knowledge, before we can hold out a prospect of being
one day enabled to trace out an arc of the solar orbit, and to in-
dicate the direction in which the preponderant gravitation of the
sidereal firmament is urging the central body of our system.
SECTION 3.- Principles of the conservation of the motion of the
centre of gravity; and of the conservation of moments and of
areas. Laplace's invariable plane.
57.] I PROPOSE now to consider certain theorems which arise
out of equations (37) and (38), when the impressed momentum-
increments are of certain particular forms; and, firstly, we will
take equations (37).
Suppose a material system to have been put into motion by
the action of instantaneous forces, so that the axial components
of the velocity of its centre of gravity are those given in equa-
ions (56); and let us suppose the forces which subsequently act
on the system to be such that
Σ.MX = Σ.MY = Σ.m z
Σ.mz = 0;
(60)
the meaning of which condition is, that either the system is free
from the action of any forces; or the forces are such that the
* See Études d'Astronomie Stellaire. St. Petersburg, 1847.
57.]
93
OF CENTRE OF GRAVITY.
momentum-increments impressed by them mutually destroy each
other, when all are transferred to the centre of gravity in lines
parallel to their own lines of action; then, from (58), we have
d² x
dt2
d2 y
d2z
0,
0,
0 ;
(61)
dt2
dt 2
dx
Σ.mvx
dy
Σ.mvy
dz
Σ.m vz
;
(62)
dt
M
dt
M
dt
M
Σ.mvx
x — α =
a
t, ÿ-b
Σ.mVy
Σ.m Vz
t, z-c
t; (63)
M
M
M
X
· a
y-b
z-c
;
(64)
Σ.ην
Σ.M Vy
Σ.m v Z
(a, b, c) being the place of the centre of gravity of the system
when t = 0.
As (64) are the equations to a straight line, it follows that the
centre of gravity moves along a straight line, of which the di-
rection-cosines are proportional to the axial components of its
initial velocity; and its velocity is given in Art. 54.
If the centre of gravity is initially at rest, so that
Σ.MVx = Σ.mv₁ = Σ.mV½ = 0,
z
it remains at rest during the whole motion of the system.
This theorem is called the principle of the conservation of the
motion of the centre of gravity; and by virtue of it, in all cases
of motion of a free system of particles, and of a system which
is subject to forces which mutually destroy each other, the cen-
tre of gravity of the system either remains at rest or moves
with a constant velocity along a determinate straight line.
Thus the motion of the centre of gravity of a system of par-
ticles is not altered by their mutual collision, whatever is their
degree of elasticity, because a reaction always exists equal and
opposite to the action. If an explosion takes place in a moving
body, whereby it is broken into pieces, the line of motion and
the velocity of the centre of gravity of the body are not changed
by the explosion; thus the motion of the centre of gravity of
the earth is unaltered by earthquakes; volcanic explosions in the
moon will not change its motion in space. The motion of the
centre of gravity of the solar system is not affected by the mu-
tual and reciprocal action of its several members; it is only
changed by the action of forces external to the system.
94
[58.
CONSERVATION OF MOMENTS.
58.] Next let us take equations (35), and put them into the
form
x.m (y
dz
dy
༧
= Σ.m (y vz— Z Vy),
dt
dt
dx
dz
Σ.m Ξ
dt
x = x.m (≈Vx— XV z),
dt
(65)
m(
(x dy
dy_yda
dx
= x.m (xvy-YVx);
dt
dt
Σ.η χ
the left-hand members of which equations are the axial com-
ponents of the moments of the couples of the expressed momenta
of all the particles of the system. And as the right-hand mem-
bers are the similar quantities for the impressed momenta, the
equality of the two is asserted in the equations. If therefore
the system of particles moves at any time t with such momenta
that the left-hand members of (65) express the axial components
of the moments of the couples of the expressed momenta of all
the particles, then z.mvx, Σ.mv, .mv, are the momenta which
impressed in a direction contrary to that of the motion will
destroy the rotation of the system; and moreover if z.m vx, Σ.mVy,
z.mv₂ are subject to the relations (34), the system will be brought
to rest.
ولا
Now let us take equations (38), and express them in the fol-
lowing form
d2z
Σ.Μ
(ญ
d2y
Z
= x.m (y z―z Y),
dt2
dt2
Σ.Μ
(~
d² x
d² z
X
= Σ.m (z x − x z),
(66)
dt2
dt2
Σ.mx
m(
(x
d2y
d² x
-y
= Σ.m (xx — yx);
dt2
dt2
and let us suppose the system of particles to have been put into
motion.
And now let us suppose the acting forces to be such that the
impressed momentum-increments satisfy the equations,
Σ.m (y z − z Y) = Σ.m (≈ x − x z) = z.m (x Y — yx) = 0. (67)
This is the case;
(1) When, for every particle of the system, x = y = z = 0;
that is, when the system is free of the action of all continuous
forces.
Hence also it is true when the origin is a fixed point, because
the pressures which it bears satisfy this equation.
1
58.]
95
CONSERVATION OF MOMENTS.
(2) When the members of the system are subject to forces, to
each of which an equal and opposite one corresponds. Thus,
for example, suppose m' and m", situated at (x, y, z) and
(x", y", z") respectively, to be attracted towards each other by a
certain force P, dependent on their distance (r) from each other;
then
m'x'=*=*}, wv=P=V1, mt=
y' — y'
m"x"
r
m'
x′
P, m'Y'
P,
γ
r
(68)
20
P, m"x"
y" — Y′p, m″z″=
Z"
Z
P,
γ
r
.. m' (y' z' — z' x') + m" (y" z″ — "Y")
Р
— { y' (z'' — z'′) — z′ (y″ — y′) — y″ (z″ — z′ ) + z″ (y″ — y')}
r
0;
and similar results are true for every other pair of equal and
opposite actions and reactions; and also for the other couples.
(3) When the lines of action of the forces acting on the se-
veral particles of the system pass through the origin; because
in this case
X
Y
Z
X
y
2
(69)
(4) When the forces would be in equilibrium, were the system
on which they act brought suddenly to rest; because in that
case (67) are the conditions of statical equilibrium.
In all these cases (67) are satisfied, and we have
Σ.m y
(3
d2 z
dt2
d2y
2
= 0,
dt2
Σ.MZ
m(:
d2 x
dt2
d2z
X
0,
(70)
dt2
Σ.mx
(a
d2y
d² x
-y
= 0.
dt2
dt2
d
But since
dt dt
(น
dz dy
d2 z
d³y
= y
Z
;
(71)
dt
dt2
dt2
therefore, observing the effect of the summatory symbols, we
have
dt
Σ.Μ
(3
dz dy
=
hv
dt
Σ.mz
(≈
dx
X
dt
dz
dt
dx
hë,
2. m (xdy -y 2) = h₂ ;
dt
dt
(72)
96
[58.
CONSERVATION OF MOMENTS.
where h₁, ha, ha are certain constants of integration; and, as
is evident from (65), are the axial components of the moments
of the couples of momenta of all the particles due to the in-
stantaneous forces by which the system of particles is origin-
ally put into motion; or, as we may say, they are the axial
components of the sum of the moments of all the momenta at
any time. Hence
If a system of material particles is put into motion by the
action of instantaneous or other forces, and, when the action
of these forces ceases, is acted on by forces which satisfy the
conditions (67), then, notwithstanding the alteration in the ex-
pressed momenta of individual particles, the axial components
of the moments of the couples of the expressed momenta of all
the particles, at any time t, are constant. Also the moment-axis
of this resultant couple of all the expressed momenta is constant,
and the direction of the rotation-axis is fixed; that is, these are
independent of the time, and remain the same throughout the
motion. Thus, if h is the moment-axis of the resultant couple,
and a, ẞ, y are the direction-angles of the rotation-axis,
2
2
h² = h₂² + h₂² + h₂²;
2
3
COS a cos B
cos y
1
h₁
h₂
h3
h
(73)
(74)
This theorem is called the principle of the conservation of mo-
ments, and holds good when the equations (67) are satisfied; and
the cases where this occurs are numerous enough to make the
theorem of great importance. Thus, it is true when a collision
takes place between two or more members of the system, because
equal and opposite actions are generated thereby, whatever is the
degree of elasticity. It is also true when two or more members
become suddenly united; when parts of the system pass from
the gaseous to the fluid state, or from the fluid to the solid state;
provided that the causes by which such a transmutation takes
place produce equal and opposite actions. This is a remarkable
case, because the forces may be functions of the time explicitly,
but, as they disappear, the principle is true. Thus the moment
of the couple of all the momenta of the earth, as well as the
direction of its rotation-axis, would remain the same, supposing
the earth to be cooled down, without loss of gravitating matter.
And the principle is also true when the magnetic or electrical
state of two particles or of two members of the system is altered,
59.]
97
CONSERVATION OF AREAS.
if the change is accompanied by an equal and opposite action.
Thus, no alteration is caused either in the length of the day, or
in the position of the earth's axis, that is, in the place of the
polar star, by earthquakes, volcanic explosions, the rolling of the
sea on the shore, the fall of avalanches, the continual friction
of the wind against the surface of the earth, &c.; because all
these actions are accompanied by equal and opposite reactions,
and therefore satisfy the equations (67).
59.] Equations (72) also admit of further interpretation. From
the origin let radii vectores be drawn to each of the particles of
the system; as the system moves, then, if the origin is assumed
not to move during the time dt, each radius describes an infi-
nitesimal sectorial area, which is part of a conical surface; let
these sectorial areas be projected on the coordinate planes; from
(72) we shall infer that the aggregate of the products of each
particle, and the projection of the sectorial area described by its
radius vector relatively to each of the coordinates planes, varies
as the time.
Let be the radius vector, drawn from the origin to the place
of m, at the time t; let da be twice the infinitesimal sectorial
area over which r passes in the time dt; and let d.ax, d.Aŋ,
Aŋ, d.Az
be the projections of da on the planes of (y, z), (z, x), (x, y) re-
spectively; then, as in Art. 307, Vol. III,
-
d. Ax ydz z dy,
d.sy =zdx x dz,
d.s₂ = x dyy dx;
Az
(75)
so that (72) become
z.md.s,
h₁dt,
z.md.sy
h₂ dt,
Σ.Π
z.md.sz
A
h₂ dt.
(76)
Now as these equations are true for an infinitesimal time dt,
and for any point as centre of areas, provided that in case (3) of
Art. 58 that point is the source of the central forces; so will
they be true for a finite time, if the centre is fixed, or if the
centre moves in a straight line; under either of these circum-
stances we may integrate (76), and we have
Σ.m Ax = h₁t,
2.ms, hot,
Ay =
Σ.Μ Λ.
hzt;
(77)
the limits of integration being such that the areas and the time
PRICE, VOL. IV.
0
98
[60.
THE INVARIABLE PLANE.
begin simultaneously. Thus, the sum of the products of the mass
of every particle, and the projection of the sectorial area described
by its radius vector on each coordinate plane, varies as the time;
and for an unit of time is constant throughout the motion.
This theorem is called the principle of the conservation of
areas, and is true whenever equations (67) are satisfied.
The signs of the areas are thus far determined by the signs of
the right-hand members of (75); they are therefore to be con-
sidered positive when the direction of rotation is positive ac-
cording to the principle of Art. 33. Thus, for instance, for rota-
tion about the axis of z, if the projection of the radius vector on
the plane of (x, y) moves from the axis of x towards the axis of y,
d.Az xdy-y da; in which case d.A is positive; and the
signs of the other projections are to be taken on an analogous
principle. If therefore the motion is retrograde, the areas will
have negative signs.
=
60.] As the sum of the products of the mass of each particle,
and its projected sectorial area, varies as the time, or is constant
for an unit of time, for each of the coordinate planes, so will it
be also for every plane; the sum however of these products
varies as the position of the plane on which they are projected
varies: it is evident that there is an infinite number of planes,
for which the sum vanishes; viz. all those planes, the direction-
cosines of whose normals are l, m, n, and which satisfy the con-
dition
lh₁ + mh₂+nhz = 0:
hence it is evident that all these planes may intersect along the
same straight line. And the plane which is perpendicular to
this straight line has the peculiarity that the sum of the pro-
ducts of the masses, and of the projected areas, vanishes for all
planes perpendicular to it; and, for a given plane, varies as the
cosine of the angle at which the planes are inclined to each
other. It is evident then that for this plane the sum is a maxi-
mum; and the position of it may thus be found :
Let l, m, n be the direction-cosines of the normal of the re-
quired plane; and let u be the sum of the products of each mass
and its sectorial area projected on the required plane; then the
theory of projections of areas gives us
u = 1Σ.m Ax + M Σ.M Ay + N Σ. M A z
(l h₂+mh2 + nh3)t;
and
1 = 1² + m² + n²;
(78)
(79)
61.]
99
THE INVARIABLE PLANE.
therefore
Du = 0 =
1
h₂ dl + h₂ dm + h dn,
0 = l dl + m dm + n dn ;
n
hz
2
1
h
{ h₂² + h₂² + h²² } ž t
M
h₂
h₂
u = ht =
..
h1
1 =
m =
h'
hq
h'
W
(80)
h2t
2
(81)
h
n =
(82)
h
whereby the maximum value of the products is determined, and
also the direction-cosines of the plane for which the sum of the
products of the masses and the projected sectorial areas is a
maximum. And, if that plane passes through the origin, its
equation is
h₂ x + h₂ y + hz z = 0.
(83)
Hence it appears that the position of it is independent of t; and
is therefore the same throughout the motion. For this reason
it is called the invariable plane. The preceding equation shews
that if at any time the masses of the moving particles, their
places with reference to a centre fulfilling the conditions (67),
and their velocities are known, then h,, ha, h, may be calculated,
and the position of the invariable plane will be completely deter-
mined.
This plane is evidently that also of the maximum couple of
the momenta of all the particles at any time t; and the posi-
tion therefore of the plane of the maximum couple is constant
throughout the motion. And whatever is true of the invariable
plane and its normal, is true also, mutatis mutandis, of the plane
of the maximum couple of moments and its normal.
Hereafter we shall meet with particular cases of this theorem;
(1) when one point of a body or of a system of particles is fixed ;
(2) when two points are fixed so that the system rotates about a
fixed axis passing through them.
61.] In the determination of the places and motion of the hea-
venly bodies astronomers are always subject to the difficulty that
they have no fixed planes and no fixed lines to which they can re-
fer them. It is true that they generally take the sun as a fixed
centre and the plane of the ecliptic, that is, the plane in which
the centre of the earth always is in its motion around the sun,
to be a fixed plane. The proper motion of the stars however
renders it almost more than probable, that their motions are in
a great measure only apparent, and are due to a true proper
motion of the sun and the position of the plane of the ecliptic
เ
0.2
100
[61.
THE INVARIABLE PLANE.
is subject to small variations by the disturbing effects of the moon,
planets, and perhaps other members of the solar system.
Astronomers therefore are referring the places and motions
of the planets to the sun, which is not a fixed centre, and to the
ecliptic, which is a moving plane: herein lies what may be a
fruitful source of uncertainty and inaccuracy; inferences from
observations, and theory built upon them, are carried over long
ages; and it would be of advantage to astronomy if a fixed point.
and a fixed plane could be determined, to which all observa-
tions and calculations could be conveniently referred; or if, the
position of the latter being given in direction, the motion, recti-
lineal or other, of the former were known. Now it has been
before observed that probably the sun has a proper motion in
space; and that this is rectilineal, so far as our observations at
present indicate, and with a known velocity. Thus far then, if
the sun is taken as the centre of areas, the principle of areas
may be true for the solar system. The forces which act on the
solar system are (1) chiefly internal forces of attraction which
will disappear in the aggregate of the moving masses of the
system; and (2) the external forces acting on the sun and other
members of the system from stars and other bodies, some of
which are perhaps not visible to us. As the mass of the sun
however is so much larger than the masses of all the other bodies
of the solar system, we may assume the sun's centre to be the
centre of gravity of all the bodies of the system, and the ex-
ternal forces which act on the several members of the system to
be applied at it, in accordance with the principle of Art. 55.
We may reasonably suppose that these forces produce the sun's
proper motion in space, and do not produce any sensible effect
on the rotation of the bodies about it; that is, we shall assume
these external forces, approximately and sensibly, to be such as
satisfy the equations (67).
These forces therefore are such that the theory of the in-
variable plane is applicable to the solar system; and as its
position is the same during the whole motion, being independent
of the time, it is a plane to which the places and motions of
the members of the system may be advantageously referred.
The determination of its place however requires a knowledge
of the masses of all the members of the system, and of the ele-
ments of their orbits. Approximate values of these are known
for the planets and their satellites, but of the masses of the
61.]
101
THE INVARIABLE PLANE.
comets we are in total ignorance. As the mutual attractions and
perturbations of the several planets however are sufficient for
the explanation of all these inequalities, it is manifest that hi-
therto at least the action of the comets on the planetary system
is insensible. The comet of 1770 approached so near to the
earth, that the periodic time of the comet is calculated to have
been increased by 2.046 days; and, if its mass had been equal
to that of the earth, it would, according to Laplace, have in-
creased the length of our year by nearly one hour and fifty-six
minutes; but Laplace adds, that if an increase of only two se-
conds had taken place in the length of the year, it would have
been detected; and as such an increase has not been detected,
it follows that the mass of the comet must be less than
1
5000
dth
part of the mass of the earth. The same comet passed through
the satellites of Jupiter in the years 1767 and 1779, without
producing any effect. Thus, though comets are greatly dis-
turbed by the action of the planets, it does not appear that they
produce any sensible effect on the planets by their action. In
the determination therefore of the position of the invariable plane
of the solar system, their effect is insensible.
If therefore h₁, h2, h3 have been determined for the plane of
the ecliptic, as that of (x, y), by observation, and 0 is the inclina-
tion of the invariable plane to that of the ecliptic, and y is the
longitude of its ascending node, from Art. 4 we have
cos y sin 0, h3 = cos 0; (81)
h₁ =sin sin 0,
h₂
..
cos 0 =
h3, tan y
h₁
ha
(85)
and thus the position of the invariable plane would be known.
It will be observed that h₁, ha, ha are in (72) the axial com-
ponents of the moments of the couples of the expressed momenta
of all the particles due to an unit of time; and, in (77), are the
sums of the products of every particle and the projected sectorial
area of its radius vector about the origin in an unit of time. In
calculating therefore these quantities for the determination of
the position of the invariable plane of the solar system, since the
planets and satellites rotate about their own axes, and the satel-
lites revolve about their primaries, we cannot estimate their mo-
ments or their sectorial areas, as if they were single particles; but
the required quantities must be calculated separately for each
102
[62.
THE INVARIABLE PLANE.
individual particle. Thus, as the sun rotates, the sectorial area
corresponding to each of its elements has to be estimated. As
satellites revolve about their primaries, and also rotate about
their own axes, these have to be estimated. It would be out
of place here to enter on these calculations, although they are
of extreme interest, and of great importance in the calculations
of accurate astronomy; I can do no more than refer the stu-
dent to places where the mode of calculation is explained:
(1) Laplace, Exposition du Système du Monde, 5me Ed. Paris,
1824, p. 199, lib. IV, ch. II. (2) Poinsot, Équateur du Sys-
tème Solaire; appended to the Éléments de Statique; 8me Ed.
1842. (3) Poisson, Traité de Mécanique, 2nde Ed. 1833, Vol. II,
p. 469. (4) A note, "Du plan invariable du Système du
Moude," appended to the 3rd Vol. of Pontecoulant, "Système
du Monde," Paris, 1834. The real dynamical things which are
invariable, and on which the position of the plane depends, are
the momentum-moments; the products of the masses and the
sectorial areas are geometrical representatives of them; and the
theorem has been stated in the latter form probably because
Kepler's Law of Areas becomes hereby generalized.
62.] In the calculation of the moments of the momenta of
these several bodies, and systems relative to particular axes, it is
convenient to calculate the moments of an individual body or
system relative to parallel axes passing through the centre of
gravity of that body; and then to increase that quantity by the
moment of the momentum of its whole mass condensed into its
centre of gravity. The following process proves the theorem:
Let the moment of the momentum, relative to the axis of x,
be taken as the type of that relative to each of the other coor-
dinate axes, and indeed to any other axis. Then, taking the
equations of transformation given in Art. 54, we have
dz
h1
= Σ.Μ Y
Z
dt
dy
dt S
= Σ.m
{
dz
dz
(ÿ + y')
+
- ( + x') (dy
dt
dt
dy
z) +
dt dt
}}
M
(T
dz
dy
Z
+ &.m
dt
dt
zy dz
dy'
Ź
;
(86)
dt
dt
where м is the mass of the body or system.
M
So that h₁, which
is the sum of the moments of the momenta of all the particles
of м relative to the axis of x, is the sum of the moment of the
63.]
103
THE PRINCIPLE OF VIS VIVA.
momentum of м condensed into a particle at its centre of gravity,
and of the sum of the moments of the momenta of all the parti-
cles, relative to an axis passing through the centre of gravity,
and parallel to that of x. Similar values are of course true for
ha and hg, and consequently for every axis.
SECTION 4.-The principle of vis viva; Lagrange's principle of
least action; Carnot's theorem.
63.] THE theorems which have been proved in the preceding
section of the motion of a material system are only true when
certain relations exist between the acting forces, whether internal
or external. I proceed now to a theorem which is much more
generally true, and which gives one integral of the equations of
motion. Let us return to the equation (40) of Art. 50. What-
ever is the relative disposition of the several particles of a mate-
rial system, provided that the relation is independent of the
time, that is, provided that the equations of condition F, = 0,
F2 0, Fk = O, do not contain explicitly the variable t, it is
evident that we can always replace the variations dæ, dy, dz by
the actual spaces dx, dy, dz, which the point of application of
a force describes in the time dt. For suppose F
For suppose F = 0 to be an
equation of condition; then, for the geometrical displacement,
the variation of this is
...
(dr. ) d x₁ + (dv) öy₂ +
dx1
dr ) d = 1
+ ... +
(dr ) d = n
-) d≈n = 0; (87)
n
dzi
and if F contained t explicitly, there would be no variation of it
in this equation, because the shifting of the system is a virtual
geometrical displacement.
But if the changes in the coordinates of the points of applica-
tion of the forces are due to the dynamical forces, then t varies,
and we have for the total variation of F = 0, if it contains t ex-
plicitly,
(dr) dt + (dr) dx, + (dr) dy, +
dt
+(
(dr) d2n =
) d = 0; (88)
= 0; that
which equation cannot consist with (87), unless (7)
is, unless F does not contain the time explicitly.
dt
We will assume that in the general equation of motion, viz.
104
[63.
THE PRINCIPLE OF VIS VIVA.
(40) of Art. 50, we can replace da, dy, dz by dx, dy, dz; then
the equation becomes.
m {(x — d²x)
Σ.Μ.
dt2
y
dx + (x − 124) dy + ( z − d) dz} = 0;
1) d x
which may be put into the form
Σ.Π
{
d² x
dt2
dx +
d2y
dt2
d2~
dy +
dz
dt2
}
= ɛ.m (x dx + x dy + z dz) ;
and if v is the velocity of the particle m at the time t, this gives
d.z.mv² = 2 x .m (x dx + y dy + z dz) ;
... z.mv²-z.m v2 = 2 x.m (x dx + x dy + z dz);
22. m f
(89)
the limits of the integral in the right-hand member correspond-
ing to those in the left-hand member.
Let us consider the meaning of the right-hand member: let r
be the velocity-increment impressed on m in an unit of time
along the line ds in which m moves at the time t, so that mr is
the impressed momentum-increment in an unit of time. Let
x, y, z be the axial components of r; then
and (89) becomes
x dx + y dy + z dz = rds;
d.z.mv² = 2 x.m Fds;
(90)
that is, the increment of the vires vivæ of all the particles of the
system, or, as we call it, of the vis viva of the system, in the time
dt, is equal to twice the sum of the products of each particle, its
impressed velocity-increment, and the space through which it
moves.
This latter is a cumbrous form of expression, and it has been
found convenient to introduce a new term; mrds is called the
work or the labouring force of m*, or the impressed quantity of
work of m, and, as it is taken for an infinitesimal time only, it
may be called the impressed increment of work, or the element
of impressed work due to the labouring force; so that from
(90) we have the following theorem ;
In the motion of a system of particles of invariable form the
infinitesimal increment of the vis viva of the system is equal to
twice the increment of the impressed work. If the labouring
force acts to increase the vis viva of the system, it is called a
motive force; if it acts to diminish the vis viva, it is called a re-
* See Traité de la Mécanique des corps Solides et du Calcul de l'effet des
Machines, par G. Coriolis, 2nd Ed., Paris, 1844, page 37.
64.]
105
THE PRINCIPLE OF VIS VIVA.
tarding force. As we intend to enter on this subject at greater
length hereafter, it is unnecessary now to say more upon it.
It must be observed, that in the motion of a system of parti-
cles, which are subject to the action of internal forces as well as
of external forces, the internal forces do not necessarily disap-
pear in the right-hand member of (89), even if they enter in
pairs of equal and opposite forces; for suppose an internal force,
say T, to act mutually on m and m' in equal intensity and in op-
posite directions along the same line of action; the virtual ve-
locity corresponding to r will not disappear in (89), unless the
displacement of m' is exactly equal and parallel to that of m:
and as the geometrical displacements are the actual dynamical
displacements, the displacements of these two particles are not
necessarily equal and opposite. To exemplify this fact, in any
displacement of the solar system, the theorems of the last two
sections are true; the vis viva however of that system is not
always the same, even if we neglect the external forces, because
a change of it arises from the internal forces producing an al-
teration in the form of the system. In the motion of a rigid
body, and of a system of particles of invariable form, the inter-
nal forces will cancel each other. But if there are elastic con-
nections, or springs, and if expansions or contractions arise from
such forces, these may not disappear in the equation of vis viva.
64.] Let us suppose .m (x dx + y dy + z dz) to be an exact
differential of a function of x1, Y1, Z1, X2, Y2, Z2 Xn, Yn, Zn; so that
z.m (x dx + x dy + z dz) = D. f (x, y, z) ;
(91)
where ƒ (x, y, z) is a function of some or all of the coordinates of
the several particles of the system: then, taking the definite
integral of (89), we have
z.m v² — z.mv² = 2 f(x, y, z) - 2 ƒ (xo, Yo, ≈o); (92)
where the quantities with the subscript o are the initial values
of the similar quantities without the subscript; and therefore
The increase of the vires vivæ of all the particles of the system,
that is, of the vis viva of the whole system, in passing from one
position to another, depends on only the two positions of the
system, and is independent of the path described by each par-
ticle of the system. This is called the principle of vis viva of a
system.
Hence, if all the particles resume the positions which they ori-
ginally had, the sum of the vires vivæ is the same in both cases;
PRICE, VOL. IV.
Р
106
[64.
THE PRINCIPLE OF VIS VIVA.
and consequently, whenever a system in motion resumes a posi-
tion which it formerly had, the vis viva of the system is the same
in both cases.
If, for every particle of the system,
X = Y = 2 =
= 0,
Σ.mv² = z.m v²;
and the sum of the vires vivæ is the same throughout the mo-
tion; this theorem is called the principle of conservation of vires
vivæ,
If all the particles of the system are subject to the action of
gravity only, then, taking the plane of (x, y) to be horizontal,
X = Y = 0, z = g, and therefore
0
Σ.mv²
z. m v² — z.m v² = 2 z.mg (2-0).
(z
(93)
But if z, and Z are the distances from the plane of (x, y) to the
centre of gravity of the system at the times 0 and t respectively;
and if м is the mass of the particles,
M ZO = Σ.m %0,
... Σ.m v2
2
M2 = Σ.MZ;
Σ.m vo = 2мg (z-Zo);
(94)
that is, the increase of vis viva of a heavy system depends only
on the vertical distance over which the centre of gravity passes;
and therefore the vis viva is the same whenever the centre of
gravity passes through a given horizontal plane.
Whenever the system passes through a position in which it
would be in equilibrium under the action of the impressed la-
bouring force, if it had no velocity at the time, then, in that
position, by the principle of virtual velocities,
Σ.m (x dx + y dy + z dz) = 0.
(95)
And therefore from (89), d.z.mv² = 0; so that the vis viva is
either a maximum or a minimum, or is constant; the last being
the case when no forces are impressed, and when therefore every
position of the body would be a position of equilibrium, if there
was no velocity. And therefore conversely the sum of all the
vires vivæ is a maximum or a minimum when the system passes
through what would be a position of equilibrium if the particles
had no velocity. And the vis viva of the system is a maximum
or a minimum according as the position of equilibrium is of
stable or of unstable equilibrium.
For on referring to the notation and the process of Art. 103
and 104, Vol. III, it appears that half the vis viva of the system
65.]
107
THE PRINCIPLE OF VIS VIVA.
is the quantity which is therein expressed by u, and is called the
central moment of the system; for, from (373) Art. 103, we have
du = Σ.P(cos a dx + cos ẞ dy + cos y dz)
= x.m (x dx + y dy + z dz)
1
d.z.m v2
2
(96)
and therefore, as the equilibrium of a system of forces is stable
or unstable, according as the central moment is a maximum or
a minimum, so the vis viva of the system is a maximum or a
minimum, according as the equilibrium-state through which the
system passes is a state of stable or of unstable equilibrium.
Thus, it appears that in a heavy system of particles, such as in
machines &c., the vis viva of the system is the greatest when
the centre of gravity has its lowest position, because the equili-
brium in that case is stable: and the vis viva is the least when
the centre of gravity has the highest position compatible with
the constraints of the machine. This also may be deduced di-
rectly from (94).
65.] Let us now consider under what circumstances
Σ.m (x dx + x dy + z dz)
is an exact differential.
It is to be observed, that this cannot be the case if x, y, or z
contains t explicitly. Of course the general conditions for any
one particle m are those given in (15), Art. 383, Vol. III. It is
however an exact differential so far as any particle m is acted
on by a central force whose centre is fixed at (a, b, c); and which
is a function of the distance r between the centre and (x, y, z)
the place of m. Thus, let p be the central force = f(r), say;
then
X
a
X
·ƒ (r), Y =
y - b
if (r), Z
= ² = 0 f(r) ;
C
γ
γ
2.2
p2 = (x − a)² + (y — b)² + (≈ — c)²;
r dr = (x-a) dx + (y - b) dy + (z — c) dz;
m (x dx + x dy + z dz) = m f(r) dr ;
(97)
which is an exact differential, and the change of vis viva due to it
= 2 m
S" ƒ (r) dr.
S.
70
So far therefore as the labouring forces, which act on the par-
ticles of the system, are central forces with fixed centres, the
impressed quantity of work is an exact differential. And if
f(r) is positive, so that the force is repulsive, and if dr is posi-
tive, then the vis viva is increased; and if the particles approach,
P 2
108
[65.
THE PRINCIPLE OF VIS VIVA.
in which case dr is negative, the vis viva is diminished; simi-
larly, if the central force is attractive, the vis viva is increased
or diminished according as the attracted particle approaches to
or recedes from the centre of attraction.
The impressed quantity of work is also an exact differential,
so far as any two particles of the system are attracted towards
or repelled from each other by a force which varies as the mass
of each, and is a function of the distance between them. Let
the two particles be m and m', and let their places at the time t be
(x, y, z) (x', y', '), and let the distance between them ber; let
P = f(r) be the attractive or repulsive force of an unit particle
acting upon them from one to the other; then
Xx
x'
y - y'
X
x = m'
·ƒ(r),
Y = m'
f(r),
r
Υ
z = m' f(r);
2°
x-x'
f(r),
y' =
M
r
y—y_f(r),
(98)
x' = − m² = 2 ƒ (r).
2
m
x =
Also
rdr =
p2 = (x − x')² + (y — y' )² + (≈ — ~')²;
(x — x') (dx — dx') + (y —y') (dy — dy') + (z—z'′) (dz−dz'); (99)
therefore thus far
m (x dx + x dy + z dz) + m′ (x' dx' + 'dy' + źdź)
m m'
2
{(x − x') (dx — dix') + (y—y') (dy — dy') + (≈—z') (dz— dz')}f(r)
= mm'f(r) dr;
which is an exact differential. And the change in the vis viva
due to this impressed labouring force
2°
2 mm'
["
f(r) dr.
то
(100)
And here, as in the former case, if the force is repulsive, the vis
viva of the particles is increased or diminished according as the
distance between them is increased or diminished; and if the
force is attractive, the vis viva is increased or diminished accord-
ing as the distance is diminished or increased.
Thus, if a system of particles, gaseous or solid, receives an in-
crease of heat, whereby repulsive forces are brought into action,
the particles are two and two repelled further from each other,
and there is an increase of vis viva. If, on the other hand, heat
is withdrawn, the particles are drawn nearer together, and a di-
minution takes place. Hence also if in a system of moving parti-
cles an explosion takes place, so that some of the particles are re-
moved farther from each other, an increase of vis viva takes place.
66.]
109
THE PRINCIPLE OF VIS VIVA.
=
If a particle is constrained to move on a smooth surface whose
equation is F 0, no change of vis viva is due to the reaction
of the surface. For let R be the reaction of the surface; and
let u, v, w, be the x , y—, ≈—, partial differential coefficients
of F;
and let q² = u² + v² + w²; then, so far as the reaction
enters into the impressed quantity of work,
R
m (x dx + x dy + z dz)
U
{u dx + v dy + w dz}
Q
= 0;
so that the vis viva is not changed by this reaction. If however
the surface on which m is constrained to move is rough, so that
there is friction, the above condition is not satisfied, and a loss
of vis viva takes place. Hence, if a collision or a sliding occurs
amongst the particles of a moving system, a loss of vis viva
ensues.
Also, if a particle m moves freely on a given line, and x, y, 2
are functions of the coordinates of its place (x, y, z) at the time t;
so far x dx + y dy + z dz is an exact differential. Because x, y, z
may be all expressed in terms of one variable ø, as in the equa-
tions to the helix, Art. 347, Vol. I, 2nd Ed., and thus, if F ex-
presses a function of 4,
x dx + y dy + z dz = Fdp;
which is integrable.
Also, if a particle m moves freely on a given surface, and
X, Y, Z are functions of the coordinates of its place (x, y, z), then
x dx + y dy + z d≈ will be an exact differential if one condition
is satisfied, viz. that which is given in (20), Art. 383, Vol. III.
This however may more shortly be obtained as follows: Let
x, y, z be expressed in terms of two variables 0 and ; then, if
F₁ and F₂ are functions of 0 and ø,
1
x d x + x dy + z dz
which is an exact differential, if
dF1
do
F1
F₁ do + F₂ do;
(dr₂).
66.] The vires vivæ of a system of particles may also be con-
veniently expressed by the vis viva of the whole mass collected
at its centre of gravity, and the sum of the vires vivæ of all
the particles relatively to the centre of gravity.
Let the centre of gravity be (x, y, z), and let the place of m,
relatively to a system of coordinates originating at the centre of
110
[67.
PRINCIPLE OF LEAST ACTION.
gravity and parallel to the original system, be (x, y, z). So that
x = x + x',
Y
ÿ + y',
z = z + z
dx
dx
dx'
dy dy dy'
dz
dz
dz
dt
+
dt dt
dt
+
dt dt
+
dt
dt dt; (101)
...
v2
(
da 2
+
dt
(dy)
2
dz
2
+
dt
dt
+ 2(
dx dx'
+
+
dt dt dt dt dt dt
2
2
dy dy
dz dz
+
dt
(da')² + (dy')³ + ( de )³; (102)
dt
dt
therefore, if м the mass of all the particles, by reason of (53),
Art. 53,
Σ.m v² = x.m v² + Σ. m v² 2,
= M v² + z.m v'2,
(103)
where is the velocity of the centre of gravity; that is, the vis
viva of a system, relatively to a given origin, is equal to the sum
of the vis viva of the whole mass collected at its centre of gravity,
and the vis viva of the system relatively to the centre of gravity.
Hence also, if the element of the quantity of work can be ex-
pressed in the form of an exact differential, say D. f(x, y, z); then
z.mv'² — z.mv '² +м (v² — v²) = 2f(x, y, z)-2 f (xo, Yo, Zo). (104)
67.] If the principle of vis viva is true of a system of parti-
cles, so that
12
2
x.m (x dx + y dy + z dz) = D. f (x, y, z) ;
.
then the equations of motion (37), Art. 48, are such that z.mfvds
is a minimum for the system as it passes from one position to
another; in other words, the principle of least action, which has
been proved in Art. 384, Vol. III, to be true of a single particle
m, is also true of a system of particles.
ds.
Since v ds = r² dt, z.m n f v d s
Jm
Sm v² at ;
m v² dt; thus, the prin-
ciple asserts that the sum of the vires vivæ of all the particles of
the system which accumulates during the time occupied by the
system in passing from one position to another, is under the law
of connexion of the impressed and expressed momentum-incre-
ments, given in (37) Art. 48, less than it would be for any other
v² dt is called the action of the system; and the
theorem is called the principle of least action, and in this ex-
tended form is due to Lagrange.
law: Σ.
:.fmv²
Let the limits of integration, which correspond to the two
67.]
111
PRINCIPLE OF LEAST ACTION.
U = Σ.M
positions of the system, be expressed as in Vol. II; then, if u is
the action of all the particles,
(105)
Let & be the symbol of variation; then, by the calculus of va-
riations,
1
S
v ds.
Su = Σ.m
.mf' (ds òv + vò.ds).
(106)
X
Now
dv= -8x+ бу + 8%9
Y
Z
(107)
V
V
ย
dx
dy
dz
and
8.ds
8.dx +
d.dy +
8.dz;
(108)
ds
ds
ds
so that
V
X
(x8+rô+zôz) +
ds
V
8x +
ds
dy sy +
dz
d=8=)]
1
d.
v x
ds
δκ = Σ.m
1
m [ v
= Σ.m
1
dx
ds
2. m [ { (x -d
{(xdds
+ E.m
ds
"da") &x + (d³-d."dy) by + (d-d.")}; (110)
(vds o
δι
v
ds
ds
since the limits of the integral are given and are fixed, the first
part vanishes of itself.
And the second also vanishes identically;
(dxd.dx+dyd.dy+dz d.dz) }; (109)
z ds
v dz
because
dt
Σ. Π
xds_d. v dx\
=
V
ds
z.m(xdt—d.
dx
d²x
2
dt
Similarly, each of the other
= Σ.η 2
= 0.
=
(x — dze) d
dt2
terms in the latter part of (110)
and u is either a maximum or a
minimum, or is constant. Although it may be a maximum and
vanishes; therefore du = 0;
a constant in certain cases, yet it will generally be a minimum.
1
mf
Since z.m
v ds = Σ.
1
mv² dt, and this quantity is a mini-
mum, the principle will be more correctly called, "the principle
of least vis viva :" and we may then say, that under the existing
laws of motion as expressed by the equations of motion, the vis
viva acquired by the system during the time of its passage from
one position to another is less than it would be under any other
law of connexion between momenta impressed and momenta ex-
pressed. It is necessary that the first and last positions of the
system should be given, because we have assumed the variations
of the coordinates which correspond to them to vanish.
This principle of least action is useless as a method of so-
112
[68.
THE VIS VIVA OF A SYSTEM.
lution of dynamical problems; because, assuming it to be true
from a priori or other reasoning, it gives only the equations of
motion (37), Art. 48, which are derived more satisfactorily from
D'Alembert's principle; and if the variations of x, y, and ≈ had
been taken in their most general forms, which are (41), Art. 50,
being due to not only a motion of translation, but also to that of
rotation, we should from the principle infer the equations (38)
as well as (37) of Art. 48. It is merely then a formula which
includes them. The other principles, however, which we have
proved in the preceding Articles, are more useful; under certain
circumstances, they give us actual integrals of the equations of
motion: thus, if the impressed element of work is an exact dif-
ferential, the equation of vis viva is a first integral, and that
from which the time may be found by a single integration. So,
if no forces act on the system, or only internal forces which have
equal and opposite ones, the principles of conservation of the
centre of gravity and of the conservation of moments give in-
tegrals of the equations of motion.
I may however observe, that we shall generally find it the most.
convenient, as it is the most philosophical method, to state the
equations of motion in their general forms, and to integrate
them directly with the introduction of those limiting values, which
are given by the conditions of the problem.
68] Let us next consider what changes the vis viva of a sys-
tem undergoes under the action of impulsive forces: and here.
we are arrested by a difficulty. D'Alembert's principle is ap-
plicable to such forces, which act for an infinitesimal time, and
in that time impress very great velocities, because the points of
the system on which they act are not sensibly displaced rela-
tively to each other; and thus the principle of virtual velocities.
is applicable also to the momenta lost; whence we have
dx
dz
x.m {} (v, dr) 8x + (v, dy) by + (v. - d=) 3 = {}
Σ.Μ
Ꮖ
dt
dt
= 0. (111)
8z } = 0.
If the type-particle m is moving with a velocity whose axial
components are u, v, w before the instantaneous force acted on
dx dy dz
it; and with a velocity whose axial components are dt' dt' dt
dx dy dz
after the force has acted, at d
dt' dt' dt
in (111) must be replaced by
dx
dy
dz
u,
v,
w.
dt
dt
dt
68.]
113
INSTANTANEOUS FORCES.
If however a collision takes place between bodies which are more
or less elastic, a change of figure takes place; and the relative
position of the particles is changed; so that the conditions under
which the preceding equations of motion have been found are
not fulfilled. The space however through which a particle is
displaced is, it may be thought, so small that we may neglect it
when taken absolutely; but, as the time in which the displace-
ment takes place is also infinitesimal, the velocity continuously
varies, and the change of it cannot be neglected: thus, there
will be a difference of velocities of two contiguous particles; and
this is inconsistent with the immediate application of D'Alem-
bert's principle. If however the bodies are hard and inelastic,
so that no compression takes place, the particles move with the
same velocity, and are at relative rest, and the principle applies:
also, in the case of elastic or imperfectly elastic bodies, the bodies
move with the same velocity when the compression is a maxi-
mum, and before the restitution of the figure has begun to take
place. At this instant the principle of D'Alembert supplies the
equations of motion. Again, when an explosion takes place, the
particles in contact move with the same velocity when the ex-
plosive forces begin to act, and at that instant again the princi-
ple of D'Alembert gives the equations of motion.
I propose then to investigate the following problem: A sys-
tem of material particles is in motion, and is subject to impul-
sive forces, either amongst each other or against fixed obstacles,
of the nature of collision or explosion; what change of vis viva
of the system takes place by virtue of the action of these in-
stantaneous forces?
Firstly, let us take the case of impact and collision of the mem-
bers of the system against each other or against fixed obstacles.
And let us take the equation of motion (111) at the instant
when the compression is a maximum, because then the particles
in contact are at relative rest, and in the case of impact against
a fixed obstacle are at absolute rest. In these cases Vx, Vy, Vz
disappear from the equation, because any momentum arising from
the mutual action of the particles of the system is always accom-
dx dy dz
panied by an equal and opposite reaction. Now let
be the axial components of the velocity v of m before collision
dx dy dz
takes place; and
dt' dt' dt
dt' dt dt
the axial components of the
PRICE, VOL. IV.
114
[68.
THE VIS VIVA OF A SYSTEM.
velocity v' of m, when compression is a maximum; also let the
actual spaces described by m along the coordinate axes, at the
instant when the compression is a maximum, be the particular
values of the virtual velocities da, dy, dz; so that (111) becomes
Σ.Μ
dx
dx'
{ (dr_da) dx + (dy - dy) dy + (dz
dt dt
dt dt
Σ.Μ
{
dx dx'
dy dy
dz dz')
+
+
= Σ.Μ
dt dt
dt dt
dt dt
{
Also we have identically
{(dx
dx' 12
-
dz
– de ) de'} = 0; (112)
dt dt
(dar' )² + (dy )²
= Σ.m v2.
dt
x. m { (dr - da )² + (dy - d ) + (de - dz)"}
dt dt
dt
dz
2
+
dt
(113)
dz
12
dt
dt
dt dť
= Σ.m v² — 2ɛ.m
{ da da
dx dx dy dy'
+
dz dz
+
dz dz
+x.mv2; (114)
dt dt
dt dt dt dt
therefore (113) becomes
2
dx
dx' 12
dt
2
dz2)
Zmv² — z.mv² = x.m {(dr — dr)² + (d-dy) + (dz - dx)"}
—
dt
= x.m w² (say).
dt dt
dt dt
(115)
Thus the loss of vis viva during compression is the vis viva of all
the particles, were each to move with a velocity equal to the
excess of that at the beginning of collision over that at the end
of compression: this quantity we have expressed by z.mw²;
and as it is a positive quantity the compression causes a loss of
vis viva.
If the particles of the system are either inelastic or perfectly
hard, there is no force of restitution of figure, and the velocity,
at the instant when the compression is a maximum, is the per-
manent velocity of the system: we have
z.mv² = x.mv² — z.mw².
This is Carnot's theorem, having been given by him first in his
"Essai sur les Machines en général," Basle, 1797. See also
"Principes fondamentaux de l'Equilibre et du Mouvement,"
Paris, 1803.
Secondly, let us suppose the instantaneous forces, by which
the change of vis viva of the system takes place, to be forces of
explosion; then, using the same notation as before, the virtual
velocities may be replaced by dx, dy, dz, the actual displace-
ments along the axes of the place of m, during the instant dt,
before the explosion took place; then, (112) becomes in this case
69.]
115
THE PRINCIPLE OF LEAST CONSTRAINT.
Σ.Μ
{(da
dx
dx'
dx
dt dt
dy_dy dy +
) d x + (dy - dy)
(dz
dz
dt
dź ) dz} = 0;
(116)
and following the same process, we have
Σ.m v′2 Σ.m v² + Σ.m w²;
that is, vis viva is gained by the explosion.
It is indeed evident, that if the particles of the system are at
rest before the explosion, the separation of the particles must be
accompanied with an increase of vis viva.
In the case now of the collision of bodies, the former of the
two cases is that which occurs until the compression is a maxi-
mum, and the latter is that which occurs during the restitution
of the figure. Vis viva is lost during compression, and is gained
during restitution. If the bodies are perfectly elastic, so that
they recover the same form that they had before collision, the
increase of vis viva during the restitution will be doubtless equal
to that lost during compression; and thus the vis viva of the
system is the same before and after collision. If they are per-
fectly inelastic there is a loss during compression, and there is
no gain, because there is no restitution. And if the bodies are
only partially elastic, as is the case with all substances we are
acquainted with, the recovery of vis viva during the restitution
of the figure is not as large as the loss during compression; and
consequently there is a loss of vis viva.
We have supposed the bodies to be smooth, so that the vis
viva is not affected by friction. If however they are rough, and
the impact is oblique in the cases of collision, terms will enter
into equation (111) corresponding to friction, and our results
will require modification accordingly.
SECTION 5.— Gauss' theorem of least constraint.
69.] In Art. 105, Vol. III, a statical theorem is given which
is there deduced from the principle of virtual velocities. It is
however only a particular form of a very general theorem which
includes all dynamics. It is useless for the direct solution of
dynamical problems, but in the same way as the principle of
least action is useless: yet, as it comprises the equations of mo-
tion given in (37) and (38) of Art. 48, and gives a new meaning
to them, it deserves attention. If we could either assume its
Q 2
116
[69.
THE PRINCIPLE OF LEAST CONSTRAINT.
truth, or prove it to be true by general reasoning, we might
deduce from it all the equations of motion: but it is better to
take an opposite course; to state the principle, and then to shew
that it is deducible from the equations of motion.
The theorem is due to Gauss, and is now called "Gauss' prin-
ciple of least constraint:" it was first given in Crelle's Journal,
Vol, IV, 1829; and a French translation of the memoir is in-
serted as a note to the 2nd volume of the Mécanique Analytique
of Lagrange, edited by M. Bertrand, Paris, 1855. A full expla-
nation of it with examples is given in "Zeitschrift für Mathe-
matik und Physik von Schlömilch und Witzschel," III. Band,
Leipzig, 1858, by Dr. Hermann Scheffler. The following is the
enunciation of the theorem: If a system of material particles is
in motion, under the action of given finite accelerating forces,
the sum of the products of each particle and of the square of the
distance between its place at the end of dt, and the place which
it would have under the action of the given forces, and in the
same initial circumstances, if it were free, is a minimum.
If therefore we measure constraint by the square of the dis-
tance between the actual place of m, and the place which it
would have if it were under the action of the same forces and
were a single unconstrained particle, then the theorem is, that
the sum of the products of each particle and its constraint is a
minimum.
Let the particles of the system be, as heretofore, m₁, M2, ... Mn ;
of which let m be the type; at the time t let (x, y, z) be its place;
let u, v, w be the axial components of the velocity of m, and let
x, y, z be the axial components of the impressed velocity-incre-
ments. Then, at the time t + dt, the coordinates of m are re-
spectively
1 du
2 dt
x+udt + dť², y+vdt+
1 dv
2 dt
1 dw
dt², z+wdt+ dt²; (117)
2 dt
but if m were unconstrained, and moved with the same initial
circumstances, the coordinates of its place would be
1
x + udt +
x dt²,
2
1
y + v dt +
Y dt2,
2
(118)
1
z + w dt +
z
dt²;
2
69.]
117
THE PRINCIPLE OF LEAST CONSTRAINT.
Let us suppose (x+ §, y +n, z + 5) to be any other place which
it were possible for m to take; and let u be the sum of the pro-
ducts of every particle, and the square of the distance between
the possible place and the place which the particle would have
if it were unconstrained 1; so that
σ = Σ.m } (§ — udt
{ (ε — udt
X
Y
dt²)²+(5—wdt
dt²)2}.
— — dt²)² + (n−vdt — — { di²)² + ( 5 — w d t − 2 { di²)² } . (119)
2
2
Then Gauss' theorem consists in the assertion that u is a mini-
mum when the possible place is that given by the coordinates
(117).
Let us differentiate (119); then, if the total differential va-
nishes,
X
-
2
Y
DU—0—2z.m(§—udt — 1, dt²) d§ + 2 z.m (n − v dt — — dť²) dn
-
2
Z
+ 2 z.m (8 — w dt — — dt²)d§; (120)
2
and as d§, dŋ, dy are independent of each other, the coefficient
of each must separately vanish; therefore
X
z.m (§ — u dt — — dť²) = 0,
Y
x.m (n − v dt — 1, dt²) = 0,
Z
dt²) = 0.
.m (―w dt- - 1/2 dt²)
But by the equations (37), Art. 48,
(121)
+
d² x
du
Σ.mx = Σ.m
= Σ.Μ
dt2
dt
d2y
dv
Σ.MY = E.M
= Σ.m
(122)
dt2
dt
Σ.ΜΖ = Σ.m
d2%
dt2
dw
= Σ.m
;
dt
so that (121) become
1 du
Σ.m (§ — udt
dt²) = 0,
2 dt
1 dv
x.m (n − v dt
dt²) = 0,
(123)
2
dt
Σ.m (8 — wdt
1 dw dt²) = 0;
2 dt
and therefore it is necessary that the possible place should coin-
118
[69.
THE PRINCIPLE OF LEAST CONSTRAINT.
cide with the actual place of which the coordinates are given
in (117).
Since u, in (119), is the sum of a series of positive quantities,
it is evident that it does not admit of a maximum; neither
generally is it a constant: it must therefore be a minimum; so
that the sum of the products of each particle and its constraint
is a minimum.
And that it is a minimum, we may also thus demonstrate.
Differentiating again (120) we have,
2
D³v = 2 z.md§² + 2 z.m (§ — u dt — — dt²) d² § + ...
= 2 x.m (d§² + dn² + d¿²),
+...
by reason of (121); and this is necessarily a positive quantity;
so that u is a minimum for the particular values of έ, n, given
in (123).
The minimum value of u is evidently
1
4
U = Σ.Μ
2
{ (x — du)² + ( x − dv)² + ( z − du )*} dt. (124)
dt
dt
In statics m has no velocity, so that u = v = w = 0; and the
theorem takes the following form:
If a system of particles of invariable form is in equilibrium
under the action of given pressures, and is disturbed; the sum
of the products of each particle and the square of the distance
between the original place, and the place which it would have in
the displacement, if it had been free, is a minimum.
Let us apply this principle to the following example. Two
inelastic particles, m and m', impinge on each other; it is re-
quired to find their common velocity after impact.
Let v and v' be the velocities of m and m' respectively before
impact; and let v be the common velocity after impact: then,
if u is the sum of the products of each mass and the square
of
the distance between the place which it has actually, and the
place which it would have if it were unconstrained at the end
of dt,
v = m (v—v)² dt² + m² (v —v′)² dt²;
if
2 m (v − v) dť² + 2 m' (v — v′) d t²
m (v—v) = m' (v — v′ ) ;
.*.
V =
mv + m' v
m+m'
0,
It is hardly necessary to observe the close analogy which exists.
between the principle of least constraint and the method of least
70.]
119
THE PRINCIPLE OF SIMILITUDE.
squares. As that method gives the most likely value to quanti-
ties determined by observation, which are subject to small acci-
dental errors, so, if we apply to dynamics similar considerations,
this theorem of least constraint would assign the most plausible
position to a system of constrained particles. We have how-
ever deduced this theorem from the ordinary laws of motion,
and from D'Alembert's principle; from laws that are incon-
testably and otherwise true; so that the places which would be
assigned to the particles as the most plausible by the method of
least squares are proved to be their actual places.
Notwithstanding the disparaging remarks which are not un-
frequently made on the principle of least action, and which La-
grange thought it worth while to reply to; and although similar
remarks may be made on this theorem, inasmuch as it is, like
that of least action, the expression of a metaphysical idea, yet it
commends itself to every mathematician on account of its ele-
gance and its comprehensiveness. The end of all science is a
knowledge of laws which govern phænomena: and therefore a
theorem which includes all mechanical phænomena, and gives a
new meaning to them, and indicates that there is no waste of
power, is not to be discarded as useless. In the solution of
particular problems we may indeed apply less general laws: but
the more general law cannot fail of exciting curiosity and of
creating a desire to know the details which it contains.
SECTION 6.— Newton's principle of similitude.
70.] As in the present chapter we are investigating those
theorems and principles which follow immediately from the equa-
tious of motion of Art. 48, and which will be, each in its own
degree, applied in the following Chapters; so shall we here intro-
duce a principle of similitude which is given by Newton in Book
II, Section VII, Prop. xxxII, of the Principia; and of which a
proof and some applications are given by M. Bertrand in Cahier
XXXII of the Journal de l'Ecole Polytechnique, Paris, 1848.
The problem which the principle of similitude solves is this:
A system of material particles of a certain form is in motion
under the action of certain forces; we have a new system exactly
similar, and either larger or smaller in what proportions are
:
120
[70.
THE PRINCIPLE OF SIMILITUDE.
the masses, their expressed velocities, the time of motion, and
the impressed momenta or the acting forces to be changed, that
is, either increased or diminished, so that the motion of the new
system may be similar to that of the old system? In other words,
a machine "works" on a given scale; in what proportion are its
parts to be changed, that it should "work" on another given
scale? A model succeeds; a machine made after the model
breaks into pieces: what is the cause of this? Here is a dis-
tinguishing point between geometry and mechanics; whatever
in geometry is true of a triangle on a small scale, is equally true
of a triangle on a large scale: in mechanics it is not so; If
a large machine is made with all its parts geometrically propor-
tional to all the parts of a small one, the "working" of the
large one cannot be inferred from that of the small one. Now
the proportion in which the parts are to be changed is rightly
given by Newton in the proposition above mentioned, and of
which the enunciation is;
Si corporum systemata duo similia ex æquali particularum
numero constent et particulæ correspondentes similes sint et
proportionales, singulæ in uno systemate singulis in altero, et
similiter sitæ inter se, ac datam habeant rationem densitatis ad
invicem, et inter se temporibus proportionalibus similiter moveri
incipiant (ea inter se quæ in uno sunt systemate, et ea inter se
quæ sunt in altero), et si non tangant se mutuo quæ in eodem
sunt systemate, nisi in momentis reflexionum, neque attrahant,
vel fugent se mutuo nisi viribus acceleratricibus quæ sint ut par-
ticularum correspondentium diametri inversi et quadrata velo-
citatum directi, dico quod systematum particulæ illæ pergent
inter se temporibus proportionalibus similiter moveri.
According to this theorem, to any system of particles of a
certain form, the number of similar systems is infinite. A pro-
portionality however must exist between five elements of the
two systems, each to each; the lengths, the times, the velo-
cities, the acting forces, and the masses; instead of between
the lengths only, as the principle of similitude in geometry re-
quires.
Let us take the general equation of motion (40), Art. 50, as
given by the principle of virtual velocities; and modify it so far as to
make x, y, z the impressed momentum-increments; then we have.
Σ
z. {(x — m 12x)
dt2
d2
y
d x + (x − m ¹² 2 ) ò y + (z
dt2
m
d2z
dt2
) 8≈ } = 0; (12
70.]
121
THE PRINCIPLE OF SIMILITUDE.
and this equation, as we have shewn in that Article, includes all
possible cases of motion. Suppose now that all the circum-
stances of motion of the system are deduced from this equation
and that the place of every particle at the time t is given in terms
of t. Also, let us suppose that we have a second system of par-
ticles of a form similar to the first; so that when the motion
begins, the position of the particles in one system is similar and
similarly situated to that of those in the other; and let both
systems be subject to similar constraints, so that both have
similar equations of condition; and let the equation of motion
which applies to this latter system be
Σ.
x. {( x' — m
{(x' – m²'
d²x'
dť2
X - dt z ) d x² + (x — m²
— m'
dť'2
Let
d2z
8
dt'2
-) dx'} = 0; (126)
k
p
ρ Ξ
the ratio of the linear similitude in the two systems,
the ratio of the masses of corresponding particles,
the ratio of the corresponding impressed momenta,
n = the ratio of the times in each,
σ = the ratio of the velocities of corresponding particles in each;
(127)
so that
2?
y
X
Y
δι'
dy'
8 =
M
бх
бу
|_
z
= pi
(128)
X
Y
Z
ds'
ds
m' =
km,
ť
=
nt,
σ
dť
dt
then it is evident that, if these quantities are substituted in
(126), the result is identical with (125) if
P =
22
kr
;
(129)
in which equation any three quantities being given, we can de-
termine the fourth. Thus, for instance, if all the elements of a
system are changed in the ratios given in (127), the time varies
directly as the square root of the linear distances, directly as
the square root of the masses, and inversely as the square root
of the acting force. Newton's principle of similitude consists in
equation (129).
Also
0
ין
n
ds' dt
dt ds
-13
(27) * .
(130)
PRICE, VOL. IV.
غم
122
[70.
THE PRINCIPLE OF SIMILITUDE.
Thus, in all questions of Dynamics, if the motion or "working"
of any system is to be inferred from that of a similar system: if
the linear distances, the times, and the masses are increased
in the ratios: 1, n: 1, k: 1 respectively; then the velocities,
and the impressed momenta or acting forces, must be increased
in the ratios respectively of rn and of kr: n². Let us take
the following case of this: A large locomotive works; what
conditions are necessary that a small similar locomotive should
also work?
Let the ratio of the linear dimensions of the parts in the small
machine be to that in the larger machine as r 1, where r is a
proper fraction; then the ratio of the masses is as r³: 1; so that
k = r³; also, since gravity acts in both cases, and since the
ratio of the weights of the several parts are as the masses, there-
fore the ratio of the forces is as 3; 1; and this is to be constant
with all the forces, so that p 3: thus, from (129), n = r²;
and, from (130), σ = rẻ: thus the times and the velocities will
:
=³:
both be in the ratio r: 1. Let us consider other forces: the
pressure of steam on the piston will be diminished in the ratio
of 23: 1. As the resistance of the air varies as the square of
the velocity and as the area of the surface, it is diminished in
the ratio of r³ : 1. Sliding frictions, being proportional to the
pressures, are diminished in the same ratio. Rolling frictions,
which are found to vary directly as the pressures, and inversely
as the radii of the rolling wheels, will be diminished in the ratio
of 21. Thus, in a small machine, this ratio may be very
great.
I subjoin two simple examples of this principle of similitude;
others will be found in the Memoir of M. Bertrand, which has
already been alluded to.
Ex. 1. Two equal particles at rest are attracted towards a
centre of force, which varies directly as the distance; to prove
that, whatever are their initial distances, they arrive at the centre
simultaneously, and that their distances at any instant from the
centre are proportional to their initial distances.
Ο
0
0
0
Let x and x'。
XO and x be the initial distances of the particles from the
centre of force; so that xo =rx: also, if μx is the attractive
force, then pappa; therefore pr; also k = 1; so that
(129) gives n = 1; therefore the times are equal in the two
cases and both arrive simultaneously at the centre of force.
And as the systems are similar at first, they are similar through-
70.]
123
THE PRINCIPLE OF SIMILITUDE.
out the motion; and thus, if x and x are the distances of the
particles from the centre of force at the time t,
Х
Xo
x'
;
Хо
which is the second part of the theorem.
Ex. 2. Two simple pendulums, with equal weights, whose
lengths are l and l' respectively, are moved from their vertical
position through equal angles, and are under the action of the
gravities g and g' respectively; it is required to compare the
times of oscillation.
These are evidently two similar systems: l'rl, gpg, k=1.
r
l' g
n² =
ρ
lg
therefore, if T and T' are the times of oscillation,
• : ~ :: ()* : (-)*:
T: T
which is the well known result.
;
R. 24
124
[71.
ANGULAR VELOCITY-INCREMENTS.
CHAPTER IV.
THE EQUATIONS OF MOTION OF A RIGID BODY EXPRESSED
IN TERMS OF ANGULAR VELOCITIES AND THEIR INCRE-
MENTS. PRINCIPAL AXES AND MOMENTS OF INERTIA.
SECTION 1.-The transformation of the equations of motion.
71.] THE most general motion of a system of material parti-
cles of invariable form may be, as we have already proved, re-
solved into a motion of translation of any point, and a motion.
of rotation about an axis passing through that point. Generally
the position and direction of that axis undergoes a continual
change, and the axis may be considered to be constant during
only an infinitesimal time-element dt; for it is only in a few
cases that the axis is fixed during the whole motion.
:
From the nature of angular velocities, which have been ex-
plained in Chapter II, it is evident that they admit of increase
and decrease, either continuously or discontinuously; and, in
the general motion of a body, there will generally be a continu-
ous variation of angular velocity, whether the rotation-axis is
permanently fixed, or has the same position for only an infinitesi-
mal time-element; and the angular velocity may either increase
or decrease. Now this change of angular velocity cannot take
place without the action of some force of which it is the effect :
a rotating body can no more increase or diminish its angular
velocity, than a material particle increase or diminish its linear
velocity this fact is involved in the law of inertia of matter:
whenever a change takes place, some force acts to produce that
change; and the relation between the change of angular velo-
city and the producing forces will be the subject of our inquiry
in the present Chapter. We shall demonstrate the relation in-
directly at first, by a transformation of the preceding equations
of motion: but we shall introduce direct proofs as occasion
arises in the course of our inquiry. Thus, while the process of
transformation will enable us to conduct our treatise in a syste-
matic form, the direct proofs will remove all intermediate opera-
tions, shew the close dependence of our results on first principles,
and thus enable us to view the relations as they are in them-
selves. Thus, as I conceive, we have the advantage of both the
72.]
125
ANGULAR FORCES.
analytical and the synthetical processes, of which such admirable
examples are given respectively in the Mécanique Analytique of
Lagrange, and the Nouvelle Théorie de Rotation of Poinsot.
72.] As, for the most part, we consider changes of angular
velocities during an infinitesimal time only, the position of the
rotation-axis, whether in the body or in space, will be unchanged
during that time; and, if the rotation-axis is unchanged during
a finite time, the change in angular velocity will be found by
integration. In all cases therefore, during the change of the
angular velocity, we shall suppose the position of the rotation-
axis to be fixed.
If, by the action of an impulsive force, the angular velocity of
a body is abruptly changed, or if a body at rest receives a finite
angular velocity, we consider only the whole velocity which is
communicated to be the effect of the force: we do not inquire
into the law of communication, which would assign the rate at
which, in successive time-elements, the communication took
place, but as the whole process is completed in an infinitesimal
time, we take the whole at once.
If however a finite force acts, whereby the angular velocity of
the body about the given axis continuously varies, then there
are two cases to be considered, according as equal or unequal
angular velocities are communicated (or abstracted) in equal
time-elements; these two cases corresponding to those of a con-
stant and of a variable force in the linear motion of material
particles respectively. Let us first take the case in which equal
angular velocities are impressed in equal times. Let be the
angular velocity impressed in an unit of time; and let o be the
angular velocity impressed and also expressed in t units of time:
therefore
W = $t.
(1)
As equal angular velocities are impressed in equal times, and as
is the measure of them, is called a constant angular force.
If the body moves with an angular velocity, before the force
• acts, and if ∞ is its angular velocity, when has acted for t
units of time,
and if acts in a direction contrary to that of 2,
(2)
W = 2 + $t;
Ω
ω Ξ Ω Ω - Φί.
pt.
(3)
do
Now from Art. 22, equation (4),
dt
= ; so that, if 0 is the
126
[72.
ANGULAR FORCES.
angle through which the body rotates in the time t, and if 0=0,
when t = 0, then generally from (2)
d Ꮎ
Ω
= 2 + $t;
dt
1
Ꮎ
0 = at +
$ 12;
2
(4)
which gives the angle through which the body rotates in the
time t under the action of a constant angular force .
Next let us suppose unequal angular velocities to be impressed
in equal time-elements; then the force is called a variable an-
gular force. Let us however suppose it to be such at the time t,
that an angular velocity would be impressed by it in an unit
of time, if the force were constant during that unit of time;
and to be such at the time t+dt, that an angular velocity + do
would be impressed by it under the same supposition as to con-
stancy; let o be the angular velocity at the time t, and w + dw
at the time t + dt; then, if e is the symbol for a proper fraction,
+e dø will express the mean or average value of the impressed
angular velocity due to an unit of time during dt; that is, dur-
ing dt, + edo is the mean constant angular force; and as do is
the angular velocity expressed in dt, we have from (2)
dw = ($ + e dø) dt;
and neglecting do x dt, which is an infinitesimal of the second
order,
(5)
dw
& dt;
dw
d do
.'.
dt
dt dt
d² 0
dt2
(6)
ift is equicrescent; and this supposition we shall make through-
out the treatise, unless it is expressly stated that t is not equi-
crescent.
Hence, if is given in terms of either 0 or t, we can deduce
from (6) by means of two integrations the relation between 0
and t, and thus determine the angle through which the body
rotates in the time t.
As do is an angular velocity, although it is infinitesimal, it is
capable of resolution and composition according to the laws
which have been investigated in Chapter II. This observation
is important.
Let thus much suffice for angular velocity-increments; we re-
turn to the equations for rotatory motion which have been found
73.] ANGULAR VELOCITIES. INSTANTANEOUS FORCES.
127
in Art. 48, with the purpose of expressing them in terms of an-
gular velocities, and angular velocity-increments.
73.] Let us first take equations (35), Art. 48; and replace the
linear velocities in them in terms of the angular velocities about
the three coordinate axes, these angular velocities being due to
the acting forces.
Let us take any point of the body for the origin; and let three
rectangular axes fixed in space originate at it: the origin we
may consider to be fixed, while we calculate the rotation about
an axis through it. We will assume the body to be initially at
rest. Let a, ẞ, y be the direction-angles of the rotation-axis;
let be the angular velocity due to the acting forces; let
♫x, y, be the axial components of ; then
B,
Ω
= n cosa,
Dy Ω cos B,
2₂ = 0 COS
a cos y ;
(7)
2² = 2x² + 2,² + 2₂².
Ωχ
Let L, M, N be the moments of the axial components of the
couple of the impressed momenta; so that
L = x.m (y V₂ — ≈ Vy),
z
M = Σ.m (z Vx Ꮨ Ꮩ . ),
N = 2.m (ť Vy — Y Vx) ;
(8)
then from (35), Art. 48, we have
dz
Σ.Μ ?/
m (
át
dt
dy)
= 1,
dx
dz
Σ.Μ.
X
M,
dt
dt
Σ.mx
m (a
( x dy
dx
-y
N.
dt
(9)
Now, by (72) Art. 38, we have the following values for the axial
components of the linear velocity of m, which is due to the
angular velocity o,
dx
= Z Qy — Y DE
= (z cosß-— y cos y),
dt
dy
= X Dz Z Ax = (x cos y − z cos a),
(10)
dt
dz
dt
Y D x — X D y
= n (y cos a
x cos ẞß);
2{
then substituting in (9), we have
{ cosa.m (y²+2)-cosẞz.mxy-cos y mzx} = L,
î {-cosaz.mxy+cosẞz.m (2+2)-cos y z.myz} = M,
{-cosaz.mzx-cosẞz.my z + cosy z.m (x²+ y²)} = N ;
2
cosß≥.mxy
(11)
128
[74.
ANGULAR VELOCITIES.
a, cos a, cos ẞ, cos y having being placed outside the summatory
symbols, because they are the same for all particles of the body.
And these three equations are, in terms of the resultant angular
velocity and the direction-angles of the rotation-axis, the equi-
valents of (35), Art. 48; and by these the angular velocity o
and the position of the instantaneous rotation-axis are to be
determined.
Let us multiply them severally by cos a, cos B, cos y, and add ;
then
î x.m {(y² + z²) (cos a)² + (≈² + x²) (cos ß)² + (x² + y²) (cos y)²
-2yz cos ẞ cos y -2 zx cos y cos a -2 xy cos a cos ß}
= L COS a + M COS ẞ+ N cos y; (12)
which may be expressed as follows;
ox.m{(zcosß—y cos y)² + (x cos y − z cos a)² + (y cos a − x cos ẞ)2}
= L COS a + M cos ẞ + N cos y ; (13)
and if r is the perpendicular distance from (x, y, z), the place of
m, to the rotation-axis,
2.2
p2 = (cosẞ-ycosy)2 + (xcosy-zcosa)2 + (y cos a-x cosß)2; (14)
so that (13) becomes
n.Σ.mr² = L COS a + M COs ẞ + N cos y ;
(15)
Ω Ξ
L COS a + M cos ẞ+ N cos y
Σ.m r2
(16)
which gives the angular velocity about the instantaneous axis.
74.] The right-hand member of this equation requires ac-
curate and close examination. The numerator of the fraction
is the moment of the couple of the impressed momenta of all the
particles about the rotation-axis; for L, M. N are the moments
of the axial components of the couples of the impressed mo-
menta, and the numerator is the sum of the parts of those axial
components which are effective about the rotation-axis. The
denominator is the sum of the products of every moving particle
and the square of its distance from the rotation-axis: and in
the case of a continuous body it becomes the integral of r² dm,
the integration extending over and including all the mass-ele-
ments of the body. This quantity is called the moment of inertia
of the body or of the moving system, relatively to the particular
rotation-axis, and the geometrical definition of it is that just
given. It appears also from (15) that it is the factor by which
the angular velocity n is multiplied, and thus equated to the
74.]
129
INSTANTANEOUS FORCES.
***
moment of the couple of the impressed momenta about the
rotation-axis. This last is the dynamical definition of it.
The name
"moment of inertia" has been given for the follow-
ing reason. Let us compare (15) with the fundamental theory
of Art. 210, Vol. III, of the motion of translation of a material
particle m, which is acted on by an impulsive force. It appears
that if v is the expressed velocity of m, and if x is the momentum
impressed by the instantaneous force, then
MV = x;
(17)
so that m, which symbolises the mass, is the factor by which v is
multiplied, and so equated to the impressed momentum; and as
in (15) x.mr2 is the factor by which a is multiplied, and thus
equated to the moment of the impressed momentum, so the old
mechanicians compared the m in (17) with the z.mr² in (15); and
as they were wont to say that a body's inertia was proportional
to or identical with its mass, so, by an analogy somewhat rough,
did they call .mr2 the moment of inertia. It seems difficult to
demonstrate the correctness of the term; but as it is undesirable
to introduce a new name, except by urgent necessity, I shall
retain the old one, and call z.mr2 the moment of inertia of the
body or system of particles relatively to the rotation-axis. The
determination of this quantity is evidently the first step in the
solution of a problem which depends on the equation (16); and
is otherwise of great importance. Hereafter many properties of
moments of inertia will be investigated, and I shall calculate the
moments of inertia of bodies and moving systems in many par-
ticular cases.
Sometimes the moment of inertia is expressed in the following
manner: Let м be the mass of the moving system, and let us
suppose the whole system to be condensed into a particle of mass
M, at a distance k from the rotation-axis, so that the moment of
inertia of the system thus condensed may be the same relatively
to the axis as that of the moving system: then, as the moment
of inertia of м in this imaginary and condensed state is м k², so
by our assumption,
M
M м k² = x.r² dm;
(18)
k is called the radius of gyration of the body relatively to the
particular rotation-axis.
Hence, if a continuous body is referred to three rectangular
axes in space, and if p is the density of the particle at (x, y, z),
dm = pdx dy dz,
(19)
PRICE, VOL. IV.
S
130
[75.
ANGULAR VELOCITIES.
and the moments of inertia of the body relatively to the three
coordinate axes of x, y, and are severally
[[[p(y²+2°) dx dydz, [[[p(2² + x²)dxdydz, [[] p(x²+ y²) dx dy dz;
P(~2
the integrals being definite, and including all the elements of the
body.
75.] As (16) is the fundamental equation of rotation of a
body under the action of instantaneous forces, it is worth while
to deduce it immediately from the first principles of motion.
For the sake of simplicity, let us take the rotation-axis to be
the coordinate axis of ≈, and suppose the line of action of the
impressed momentum to be in a plane perpendicular to this
axis. Let my be the momentum impressed on m at the place
(x, y, z); of which let N be the moment of the couple about the
axis of z; let r be the distance of m from the axis of ≈; and let
o be the expressed angular velocity, and v the expressed velocity
due to the instantaneous force; so that
ds
ย
= rn;
dt
ds
di
hence the expressed momentum is m
(20)
=mro, of which the
moment, relatively to the rotation-axis, is mr²; so that the
excess of the moment of the couple of the impressed momentum
over that of the expressed momentum in the case of the particle
m is
Năm Q;
r²
and as by D'Alembert's principle these taken throughout the
moving mass are in equilibrium, we have
Σ.Ν Σ.m r² a = 0.
(21)
As is the same for all mass-elements, it may be placed outside
the sign of summation; also let & be the moment of the couple
of the impressed momenta of all the particles, then we have
Q.E.Mr² = G ;
G
Ω.Σ.
ΩΞ
..
Σ.η7.3
(22)
(23)
Z
which is the same equation as (16); for if the axis of ≈ is the ro-
tation axis, then in (16) cos a = cos ẞ= 0, cos y = 1; and we have
N
Ω
Σ.mr2
(24)
where N is the moment of the couple of the impressed momenta
about the rotation-axis, and is the same as G in (23).
76.]
131
INSTANTANEOUS FORCES.
Equations (11) are so close on the first principles of motion,
as explained in Art. 38, and of the measure of couples, that fur-
ther explanation is unnecessary.
76.] The direction-cosines of the instantaneous-axis are pro-
portional to x, y, z; and these latter quantities may be thus
found:
Let us once for all make the following abbreviations; let
Σ.m (y² + z²) = A, Σ.m (x²+x²) = B, E.m(x²+ y²) = c; (25)
A',
Σ. M x² = A
Σ.myz D,
Σ.my² = B′,
Σ.mzx = E,
z.mz² = c'; (26)
Σ.mxy = F. (27)
These nine quantities are of great importance in the following
investigations, and the substitutions which are here made will
be continued throughout the treatise.
A, B, C are the sums of the products of each particle of the
moving mass and the square of its distance from the axes of
x, y, z respectively in other words, A, B, C are the moments of
inertia of the moving system relatively to the axes of x, y,≈ re-
spectively.
:
A′, B, c´ are severally the sum of the products of each particle
and the square of the x-, y-, - coordinate of its place.
D, E, F will have full explanation in the following section, al-
though (27) evidently exhibit their meaning.
I may observe, that
A = B' + C',
B = c'+ A',
С C = A' + B';
(28)
A
B+C A
2
B'
C + A
2
B
A+ B C
2
; (29)
whereby a, b, c are severally determined in terms of A', B', c';
and A', B', c' in terms of A, B, C. Also
c′.
A + A′ = B+ B′ = c + C′ = A' + B'+C'.
(30)
Now, using in (11) these abbreviating symbols, we have, by
means of (7),
whence
2་
nx =
Sy
-
ΑΩ
F
Ly
E 2 = L,
F Q x + B ny D Q₂ = M,
E Qx — D Qy + C L x = N ;
L (BC-D2) M (DE+CF) N (DF+BE)
ABC
- A D² 2
—
BE2-CF2
L(ED+CF) + M (CA - E²)
ABC — AD² BE2 CF
2 DEF
N (EF+AD)
2 DEF
L (FD+BE) — M (FE + AD) + N (A B − F²)
(31)
(32)
Ως
A B C — AD²
AD² — BE² — CF2
;
CF² — 2 DEF
S 2
132
[77.
ANGULAR VELOCITIES..
from which, and from (7), a, B, y may be determined. The
equations to the instantaneous-axis are
X
y
Пос
Ων
Ω Dz
(33)
We shall hereafter have geometrical interpretations of these
results.
77.] Next let us consider equations (38), Art. 48. Let any
point of the body be the origin; and at it let three rectangular
coordinate axes fixed in space originate; and let us consider the
body at the time t, and during dt, so that the rotation-axis may
be considered fixed during that time. Let a, ß, y be the direc-
tion-angles of the rotation-axis, and let o be the angular velocity
about it at the time t; of which let wx, wy, we be the axial-com-
ponents; so that
W
w² = w
w₁₂ ² + w₁ ² + w 2;
Wx
= @ cos a,
w cos B, W = = @ cos y.
(34)
(35)
Let the moments of the axial-components of the couples of the
impressed momentum-increments at the time t be L, M, N; so
that from (38), Art. 48, we have
Σ.m (y z − z Y)
= L,
Σ.m (z x − x z)
Σ. m (x x − y
= M,
(36)
x)
= N ;
and
Σ.Μ
d² z
dt2
d² y
dt2
2
= L,
Σ.mz
(
d2x
d2z
20
= M,
(37)
dt2
dt2
Σ.Μ
미
​(a dzy
2
2
d² x
dt2
Y
= N.
dt2
Now, by (72), Art. 38, for the linear velocity of m at (x, y, z),
which is due to the angular velocity w, we have
dx
= w(z cos ẞ y cos y),
dt
dy
∞ (x cos y − z cos a),
(38)
dt
dz
w (y cos a x cos ẞ);
dt
and therefore for the increment of the linear velocity, which is
due to the increment of angular velocity, we have
d² x
dt 2
dw
dt
(≈ cos B―y cos y) + w (cos B
dz
dt
dy
cos y at); (39)
78.]
133
FINITE FORCES.
d2x dw
(zcosẞ-ycosy) + w² cosa (x cosa + y cosẞ+zcosy) - w²x.
dt2
dt
Similarly,
d2y
dw
(40):
(xcosy-zcosa)+w2cosß (x cosa + y cosẞ+zcosy) — w²y,
dt2
dt
d2z
dw
(ycosa-x cosẞ) + w² cosy (x cosa +ycosẞ+zcosy) — w²z.
dt2
dt
dw
Therefore, substituting in (37), and placing w and outside
dt
the sign of summation, as they, as well as the direction-cosines
of the rotation-axis, are the same for all the particles, we have
cos a Σ.m (x² + y² + z²)
+w²z.m {(x cosa+ycosẞ+zcosy) (y cosy-zcos ẞ)} = L; (41)
dw
dw
Σmx(x cosa + y cosẞ+zcosy)
dt
dt
dw
dt
do
cosßΣ.m (x² + y²+~²) Σ.my (x cosa+y cosẞ+zcosy)
dt
+w²x.m{(x cosa + y cosẞ+zcos y) (≈ cosa — x cos y)}
dw
dt
= M; (42)
do
cos y ≤.m (x² + y² + z²) z.mz (x cosa + y cosẞ+zcos y)
dt
+w² z.m {(x cosa + y cosß+cosy) (x cosẞ-y cosa)} = N. (43)
The complete solution of the problem requires that w, cos a,
cos B, cos y should be expressed in terms of t; L, M, and N being
functions of these five quantities: now, as a relation exists be-
tween a, ß, y, the preceding equations contain only three inde-
pendent quantities which are to be expressed in terms of t; for
this purpose the number of the equations is sufficient; but as
they generally do not admit of integration, we can apply them
only to particular cases, and have recourse to such artifices as
a particular problem suggests.
78.] Let us multiply (41), (12), and (43), severally, by cos a,
cos B, cos y, and let us add them; then
dw
dt
Σ.m {x² + y² + ≈² — (x cos a + y cos ẞ + cos y)²}
= L COS a + M COS ẞ+ N cos y; (44);
but if r is the perpendicular distance from (x, y, z), the place of
m at the time t, on the rotation-axis,
2.2
r² = x² + y² + ≈² — (x cos a + y cos ẞ+z cos y)²;
(45)
so that (44) becomes
do
Σ.m² = L COS a + M COS ẞ+ N cos y;
(46)
dt
134
[79.
ANGULAR VELOCITIES.
FINITE FORCES.
do
dt
Σ.η γ2
;
(47)
L COS α + M Cos ẞ+ N cos y
which gives the angular velocity-increment about the rotation-
axis which is due to the impressed momentum-increments.
Now this equation, like (16), requires careful attention; it is
that from which, by integration, the increase or diminution of
the angular velocity in a finite time is to be found. The nu-
merator of the right-hand member is the moment of the couple
of the impressed momentum-increments of all the particles re-
latively to the rotation-axis; for L, M, N are the axial components
of the moments of these couples; and L cos a + M COS ẞ+N COS Y
is the sum of the parts of these axial components which are
effective about the rotation-axis. The denominator is the mo-
ment of inertia of the body or moving mass relatively to the
rotation-axis; and the remarks made in Art. 74 are applicable
equally to that and this case.
dwx
A
dt
B
dwy
dt
79.] If the rotation-axis of the body has the same position
during the whole motion, either because two or more points in
it are fixed, or because it bears a certain relation to the parti-
cles of the moving mass, then a, ß, y are constant, and are
known, and the numerator of (47) is given at the time t; and,
if the integration can be performed, the angular velocity will
be determined. If, however, the position of the rotation-axis
changes continuously from time to time, so that it can be con-
sidered fixed only for an infinitesimal time-element dt, then
a, ß, y are functions of t, and equation (47) cannot generally be
integrated as it stands. In this case we must return to equations
(41), (42), (43); in them let us replace cos a, w cos ẞ, w cos y,
severally, by wx, w, w, and use the abbreviating symbols of
Art. 76; then,
d w z
C
dt
+ (C−B) w, W₂ — D (w,² — w₂²) — E
W≈
+ (A−c) w₂ w¸x — E (w² — w²) — F
C) W
X
wy)-
dwz
+ w x wy
dt
Wx
= M; (49)
F
dw
dt
W x W
wz) =
= L; (48)
d w z
wy w x
dt
d w x
E
dt
(dwx +w, w :) — D (
dt
doy
wx) -
F (w, ² — u), ²) — D ( du " + w;Wx
2
— —
+(B—A) W, w,, -F (w,² — w,²) -
x®,
dt
z
dwx
رهاد
= N; (50)
from which three equations wx, wy, we are to be determined in
wz
terms of t: the integration however is beyond our power, except
in a very few special cases, which we shall consider hereafter.
睿
​81.]
135
ANALYSIS OF THE EQUATIONS.
80.] As these last are the fundamental equations of rotation
of a solid body, or of a material system of invariable form, and
will be employed in all our subsequent investigations, they re-
quire close examination. We have arrived at them by trans-
formations from expressions involving velocities of translation
into those involving angular velocities. I will now shew that
they may be found more directly by D'Alembert's principle:
and, in the course of the inquiry, we shall dissect the equations,
and shew the independent origin of their several terms; and I
shall also exhibit other properties of these equations of rotating
rigid systems besides those of the preceding pages.
As the particles of the moving system are in a state of rela-
tive rest, the moments of the forces acting on them, relatively to
every and any axial line, satisfy the conditions of statical equi-
librium; and thus, by D'Alembert's principle, the moments of
the tensions or strains which arise from the excess of the im-
pressed over the expressed momentum-increments must satisfy
the laws of equilibrium when they are taken throughout the
whole system. In reference to any axis for any one particle m
we have the following moments: (1) the moment of the im-
pressed momentum-increments; (2) the moment of the expressed
momentum-increment; (3) the moment of the centrifugal force
which is due to the motion of the body about the instantaneous
axis at the time t; and the moment of the tension, which is
effective at m, is the excess of the moment of the impressed
momentum-increment and of the centrifugal force over the mo-
ment of the expressed momentum-increment; and the moment
of all these tensions vanishes for every axis.
Let us employ the same notation as heretofore; m is the type-
particle of the system; (x, y, z) is its place at the time t; wis
the angular velocity at the time t about the instantaneous axis,
of which wx, w, we are the axial components; a, ß, y are the
direction-angles of the instantaneous axis; r is the perpendicular
distance from (x, y, z) on the instantaneous axis; L, M, N are the
axial components of the moments of the couples of the impressed
momentum-increments on all the particles of the system.
81.] Let x', y', z' be the axial components of the expressed mo-
mentum-increments of all the particles of the system due to the
increments of the angular velocities at the time t; and let p' be
the resultant of these; then, from Art. 38, we have
**
136
[81.
EXPRESSED MOMENTUM-INCREMENTS.
Σ.Μ
(
dwy
dwz
Z
Y
= x',
dt
dt
Σ.m
(
X
do:
dt
2
dwx)
Y',
(51)
dt
Σ.m
(y dwx
dwy
X
= z
z' ;
dt
dt
P′2
2
(52)
and
p'² = x²²+Y'²+z²².
If the origin moves, p' is proportional to the expressed velocity-
increment of it, which is due to the increments of the angular
velocities at the time t; and if the origin is absolutely fixed, it
is the increment of pressure on it during the time dt.
Let L', m', N' be the axial components of the moments of the
couples which arise from these expressed momentum-increments;
and let G' be the resultant moment of these; then, as the axial
components of the expressed momentum-increment of m are
severally,
dwx), m (ydwr - xdwy), (58)
m ( 2
dwy
Y
dw z )
dt
dt
·), m(.
d w z
dt
dt
dt
dt
so the axial components of the moment of the couple of this
expressed-momentum of m are respectively
d w x
m (y²+z²)
dt
dwy
d wz
m x Y
dt
dt
dwy
d wz
dwx
(54)
m (z² + x²)
my z
myx,
dt
dt
dt
doz
dwx
dwy
m (x² + y²)
M Z X
m zy;
dt
dt
dt
and taking the aggregates of these for all the particles of the
system, and using the abridging symbols of Art. 76, we have
dwx dwy
dwz
F
E
L',
A
dt
dt
dt
B
dwy
dt
dwz
dwx
D
F
= M',
(55)
dt
dt
C
d w x
dt
d w x
dwy
E
D
N';
dt
dt
(56)
G′2 = L'2
2
L'² + M²² + N'2 2;
which are indeed the same expressions as (31), Art. 76, and give
the moment of the couple of the expressed momentum-incre-
ments of the system.
1
82.]
137
CENTRIFUGAL FORCES.
82.] Let x", y", z" be the axial components of the momentum-
increments of all the particles which arise from the centrifugal
forces; and let p" be their resultant. As w is the angular velo-
city of the system about the instantaneous axis at the time t,
and as r is the perpendicular distance from (x, y, z) the place of
m on that axis, m war is the centrifugal force of m, the line of
action of which lies along r. Now as r is drawn from (x, y, z)
at right angles to (a, ß, y) which passes through the origin, the
direction-cosines of r are
x-cosa (xcosa+ycosß +zcosy) y-cosẞ(xcosa+ycosẞ+zcosy)
до
z-cosy(xcosa + ycosẞ+zcosy)
g
; (57)
and therefore the axial components of the momentum-increments
of m, due to the centrifugal force, are respectively
2
mw² {x-cos a (x cosa + y cosẞ+cosy)},
mw² {y — cosß (x cos a+y cosß+≈cosy)},
(58)
mw² {z-cos y (x cosa + y cosẞ+zcos y)},
the tendency of these forces being to increase x, y, z as t in-
creases: and, taking the aggregate of these for all the particles
of the system,
and
Σ.mw² {x — cosa (xcosa+ycosẞ+zcosy)} = x″,
Σ.mw² {y — cosẞ (x cos a+y cosß +z cos y)}
Σ.mw² {z — cos y (x cosa + y cosẞ+zcosy)}
—
12
p"2 = X"²+Y"² + z″².
(59)
z"
(60)
If the origin moves, p" is proportional to the impressed velocity-
increment of it, which is due to the centrifugal forces of all the
particles; and if the origin is absolutely fixed, p" produces a
pressure on it.
Let L", M", N″ be the axial components of the moments of the
couples which arise from these centrifugal forces; and let G" be
the moment of the resultant of them; then the axial components
of the moment of the couple which arises from the centrifugal
force of m are, by reason of (58),
mw² {(xcosa+ycos 6+zcosy) (cosẞ-ycosy)},
mw² {(xcosa+ycosẞ+zcosy) (xcosy-zcosa)},
mw² {(xcosa+ycosẞ+zcosy) (y cos a-xcosẞ)};
}
(61)
then taking the aggregates of these for all the particles of the
system, and using the abridging symbols of Art. 76, we have
PRICE, VOL. IV.
T
138
[83.
ANALYSIS OF THE EQUATIONS OF MOTION.
2
D (wy² — wz²) + Ex Wy - FW x W z + (B — C) wyw₂ = L″,
WyWz
ω
E (w₂²-w₂²) M",
(W 2 — w x ² ) + F wy w z − D wy w x + (C — A) W z Wx
z
W
x W y
F(w₂²-w,²)+Dw z wy - Ew z wy + (A--B) w, w₁ = N";
2
x
and
W x
2
Wy
G″2 = L"² + M"2 + N'2;
G"
(62)
(63)
Thus (60) gives the pressure at the origin which is due to the
centrifugal forces, and (63) gives the moment of the couple which
arises from them.
In reference to (59) and to (62) I must observe that
x" cos a+y" cosẞ+z" cosy = 0, )
α
L'"cosa+M"cosẞ+ N'cosy 0; S
so that the instantaneous axis of rotation is perpendicular to
both the line of action of the resultant of translation, and to the
axis of the couple, which arise from the centrifugal forces of all
the particles.
83.] Now, as we have already observed, the moment of the
couple of the expressed momentum-increment is equal to the
moment of the couple of the impressed momentum-increment
together with that of the couple which arises from the centri-
fugal forces and this equality is true relatively to any rotation-
axis, so that for the coordinate-axes of x, y, z we have respec-
tively L'=L+L", M' = M + M", N = N+N";
:
(
(
dwy _ wxwx ) =
dt
dwz
d w z
and therefore we have
dw₂
d w x
A
dt
+ (C — B) w, w; — D (w‚² — w₂ ²) —
— w₂²) —
E (
(·
ωχ
dt
+wy wy) - F
dox
2
B
W
— —
(
dt
dt
d w z
C
dt
+ (B — A) w‚¿‚— F (w‚¸² — w‚¸²?) — D (dw,
2
— —
+
w₂Wx) —
do x
E
dt
dwy
dt
+ (A−C)w₂ Wx − E (w₂² — w‚³) — F
wxWy
dt
w₂) – D
+ Wy Wz
(
(64)
L,
— wywx) = M,
Wx
;w,) = N ;
W z Wy
which are the same equations as (48), (49), and (50). By the
preceding process therefore they are, as it were, dissected, and
the meaning and origin of the several terms are traced out; and
those which arise from the expressed momentum-increments are
distinguished from those which arise from the centrifugal forces.
If the system is at rest when the forces begin to act, then
∞ = 0, and L″ = M" N″ = 0; and the equations are reduced
dwx dwy dwz
to forms identical with (31), Art. 76, where however dt' dt' dt
take the places of x, y, z.
N"
In the course also of this inquiry we have arrived at a more
(65)
84.] SIMPLIFICATION OF THE EQUATIONS OF MOTION. 139
complete explanation of those parts of the equations of transla-
tion of the origin which arise from the angular velocities. The
origin, through which the rotation-axis passes, is considered not
to change its place during the time dt, so that the velocity and
the velocity-increment of m are due to the rotation only. Now,
if we recall to mind the process by which the original equations
of statical equilibrium are formed, it will be found that terms
are introduced at the origin equal to and having the same line
of action with those which are effective at m; and that these
produce a pressure of translation on the origin. In this case
they produce of course a motion of translation of the origin;
the axial components of the expressed velocity-increment of
which, as due to these angular velocities, are given in (40). Here
we have these expressions dissected into three, viz. x', y', z' which
correspond to the expressed angular velocity-increments; and
three, viz. x', y", z", which correspond to the centrifugal forces :
thus
z-z"
(66)
are the axial components of the momentum-increments of the
whole mass collected at the origin, due to the angular velocities
of all the particles and replacing x', x' &c. by their values,
we have
d w z
dwy
Σ.Μ
Z
-y
dt
dt
(
d w z
dwx
x
dt
dwr
Σ.m
x.m (y
dt
dt
x-x',
Y-Y",
+w2.m {cosa (x cosa + y cosẞ+zcosy)-x},
+w²x.m {cos ẞ (x cosa+ycosẞ+zcosy)—y},
dwy) cosa+ycosẞ+zcosy)—z};
X
dt
+w² z.m {cos y (x cosa+y cosß+zcos y) −z} ;
and these are the same expressions as are deducible from (40).
They, if the origin is fixed, are the axial components of the in-
crease of the pressure on the origin due to the angular velocity
which is effective during the time dt.
84.] In all cases the equations (31) and (65) admit of great
simplification. It will have been observed that the equations of
translation of a system of material particles, viz. (34) and (37),
Art. 48, are much simplified if the centre of gravity is taken for
the origin, as we have shewn in section 2 of the preceding
chapter; equations (56) and (58) in Arts. 54 and 55 are more
simple than (34) and (37) of Art. 48. It does not however ap-
pear thus far that any simplification is hereby introduced into
the forms of the equations of motion of rotation; (57) and (59),
> (67)
T 2
140
SIMPLIFICATION OF THE EQUATIONS OF MOTION. [85.
in Arts. 54 and 55, are exactly the same in form as (35) and (38)
of Art. 48. Neither does it appear that any change of axes will
generally introduce a further simplification into the equations of
motion of translation; it may do so in a particular case, because
2.MX, Z.MY, Σ.mz may then take simple forms. In the equa-
tions of rotation however it is otherwise. Consider the equations
(31), which are equivalent to (35), and equations (48), (49), and
(50), which are equivalent to (38), of Art. 48; they contain the
quantities .mx², z.my², z.mz², ɛ.myz, z.mzx, z.mxy; and
these are dependent on the position of the coordinate axes rela-
tive to and in the body. They will be determined by the or-
dinary processes of summation, and of integration if the moving
mass is a continuous body. Now thus far the position of the
coordinate axes, to which the moving system is referred, has not
been determined; it is fixed neither in the body nor in space.
Henceforward we shall suppose a system to be fixed in the body
and to move with it, and to have a particular position relatively
to the body, which we shall determine with the view of simpli-
fying the preceding equations (31), (32), and (65). By this
method we shall investigate the angular velocities of the body
about three axes fixed in the moving body; and we can thence
determine the angular velocities about three axes fixed in space
do do dy
by means of the equations (87), Art. 40; and
dt' dt' dt
may
be determined by means of (103), (104), (105) of Art. 42. By
either process the position of the body in space at the time t will
be determined.
85.] Let us examine the coefficients in (31) and in (65) of the
angular velocities and of their t-differentials; and let us suppose
the moving mass to have volume of three dimensions. What-
ever is the system of coordinate axes, it is evident that they
cannot be such that generally either z.mx2 = 0, or z.my² = 0,
Σ.mx²
or z.mz² = 0; because each of these expressions is the sum of
the products of the mass-element and of a quantity which is
necessarily positive. Thus, A, B, C, A', B', c', defined as they are
in (25) and (26) of Art. 76, are always positive quantities for
masses whose volume is of three dimensions: in plates of infini-
tesimal thickness, if the surface of the plate is taken for the plane
of (x, y), z=0 for every element; and therefore .mz20:
and in straight wires or rods, of which the transverse section is
an infinitesimal area, if the axis of a lies along the rod, y== 0
X
86.]
141
PRINCIPAL AXES.
for every mass-element, and consequently z.my² = x.mz² = 0;
in all other cases ▲, B, C and a′, B', c' are positive quantities. In
reference however to D, E, F, which are the symbols for x.myz,
z.mzx, z.may respectively, the coordinate-axes may have a po-
sition such that
Σ.myz = z.mzx = Σ.mx y = = 0,
(68)
or that one or two of them may be zero; because in the series,
the sums of which are represented by these abridging symbols,
some of the terms may be negative and others may be positive,
so that the result of the whole may be zero.
Thus, for instance, let us suppose an elliptical plate of infini-
tesimal thickness to be referred to the centre as origin, and to
a system of rectangular coordinate-axes, of which the axes of x
and y are coincident with the major and minor axes of the
ellipse, and that of ≈ is perpendicular to the plane of the elliptic
plate. Then, as z = 0 for all the mass-elements of the plate, it
is evident that
Σ.my z = x.mzx = 0;
and since for an element of the plate at (x, y) there is always
an equal element at (−x, y), it is plain that .mxy = 0.
b
This last result may also thus be found. Let y = (a² — x²) ³,
and let the density, r = the thickness of the plate; then
p
a
a
Σ.mxy = pT
= = 0.
[º [ˇ x y d y d x
- Ü
Y
We will now prove that a system of rectangular coordinate-
axes, fulfilling the conditions (68) exists at every point of a body
or system of particles. Such a system is called a system of
principal axes relatively to or at that point. The geometrical
definition of them is, that they satisfy the conditions (68): in
the following sections however several mechanical properties of
them will be demonstrated.
SECTION 2.— Principal axes, and their properties.
86.] Let us consider a body or a moving mass in reference to
a point of it which we take as the origin; and at it let two sys-
tems of rectangular coordinates originate; one of which (x, y, z)
shall be fixed absolutely; and the other of (§, n, () is fixed in the
body; and the position of which is to be determined, if it is
possible, so that
Σ.ηζέ
Σ.mn = 0; z.m¢§ = 0; x.mεn = 0.
Σ.Μηζ
Σ.ηξη
(69)
142
[86.
PRINCIPAL AXES.
Let these two systems be related by the direction-cosines of the
scheme (1), Art. 2. Then, as the systems are rectangular, the
nine direction-cosines are subject to six conditions (4) and (6),
or (5) and (7), of Art. 2: and as three other conditions are given
in (69), we have sufficient data for the determination of the nine
direction-cosines.
Substituting in (69) the values of §, n, $, given in (3), Art. 2,
and replacing .mx², z.my², z.mz², z.myz, z.mzx, z.mxy by
their symbols, we have
A b₁ C₁ + B' b₂ C 2 + C′ b3 C3
2
+ D (b₂ C3 + b3 C2) + E (b3 C1 + b₁ C3) + F (b₁ C2 + b₂ C₁)
A' C₁α₁ + B′ C₂ α₂ + C′ C3 A3
Аа
0,
+D(Czαz + Czαg) +E (Czα₂ + C₁αz) + F (C₁ α₂+ C₂α₁) = 0,
2 3
A' α₁ b₁ + B' α₂ b₂ + c αz bz
2
3
3
2
0 ; j
+D(ɑ2b3 + ɑz b₂) + E (α3b1 + α₁b₂) +Ƒ(α₁b₂+ɑ¿b₁)
Now these equations are in form identical with (34), Art. 6,
and are subject to the same conditions, viz. (5) and (7), Art. 2,
so that we have
(70)
a2
A'α₁ + Fa₂+ Eag
Fα₁ + B´α₂ + DX3
Eα₁ + Dɑ₂+ c'α3.
C
(71)
az
а1
2
аг
2
2
s'α₁² + B'α₂² + c'а² + 2D α₂α, +2Еа₂α₁ +2Ƒα¸α₂
2
2
2
3
2
a₁² + α₂² + α¸²
as
2
az²
1
= s'α₁² + в'α₂² + c'az² + 2 Dа₂α3 +2ɛαzα₁+2Ƒα¸α₂½ ; (72)
A
2
a
= x.m (a₁x + α₂Y + Az≈)²
= x.mε²
=A" (say).
By a similar process we may obtain the following:
a' b₁ + F b₂ + Е b3
1
b₁
2
A' C₁ + FC₂ + EC3
C1
2
(73)
Fb₁₂+ B′ b₂+ D b₂
b₂
E b₁ + D b₂ + c´b3
= x.mn² = B" (say); (74)
= Σ.m (² = c'' (say). (75)
2
2
FC₁ + B′ C₂+ DC3
C2
1
b3
2
EC₁ + DC2 + C´C3
C3
As these last three equations are of precisely the same form, let
us, as in Art. 6, take a type-expression of all; and assume к to
be the type of A", B", c"; so that the discriminating cubic will.
take the forms
2
(A' — K) (B′ — K) (c' — K) — D² (A' —K) - E² (B'-K) - F² (c'-K) + 2DEF = 0; (76)
and
EF
FD
DE
+
+
-10; (77)
D (K — A') + E F
E (K-B') + FD
F(K—C') + DE
87.]
143
THE ELLIPSOID OF PRINCIPAL AXES.
B
each of which equations has three real roots, as we have shewn
in Art. 7, which are A", в", c″; and these quantities are there-
fore known functions of A', B', c', D, E, F, and will henceforth be
treated as such.
Also the direction-cosines of the three principal axes of §, n, 8
are given by either of the following formula: Let us take the
axis of έ, say; and let a₁, α, α, A" correspond to it; then equa-
tions (51) and (54), Art. 7, give
a, 2
૯૭
аз
2
2
a2
(B′ — A″) (C′ — A″) — D²
(c′ — A″)(a′ — A″) — E²
(A' — A″) (B′ — A″) — F²
; (78)
and
α₁ { E F — D (A′ — A″)} = α₂ {FD — E (B′ — A″)} =ag {DE-F(C'—a")}; (79)
and similar forms are true for b₁, b₂, bз, and for C1, C2, C3, in terms
of B" and c" respectively.
It appears then that at every point of a body, and of a system
of material particles, a system of coordinate-axes in terms of
§, ŋ, Č exists, so that, if the body is referred to it,
z.mns = Σ.ηζξ = Σ.ηξη = 0;
and this system at any point is generally unique, and is called
the system of principal axes at that point. Thus the term Prin-
cipal Axis properly belongs to an axis which is one of a system of
three axes.
But we find it convenient to apply the term to an
axis fulfilling any two of the three conditions (69). Thus, if
the axis of x is such that Σ.mxy = Σ.mxz =
X
Σ.mxz = 0, the axis of x is
called a principal axis; and if x.mzx = x.mzy = 0, the axis of
z is a principal axis. Hence, if any two of the rectangular coor-
dinate-axes are principal, the third is also principal. Also the
three planes which are perpendicular to the three principal axes
are called principal planes.
87.] Now all these results admit of a geometrical interpreta-
tion by means of the ellipsoid. From (73), which we will take
to be the type of (74) and (75) also, since the form is the same
in all, we have
2
B
2
2
=A". (80)
s'a₁² + в'a₂² + c'az² + 2 vα₂αz+2Еаzα₁+2Ƒα₁α₂ = A″.
1
23
31
Along the axis (α₁, α2, α3), which is the axis of έ, say, take a
length r, and let its extremity be (x, y, z); so that, in reference
to the system of axes fixed in space,
20
y
๕
a
Az
(81)
144
[87.
THE ELLIPSOID OF PRINCIPAL AXES.
and therefore, from (81), we have
A²x² + B'Y² + C² z² +2Dyz+2Ezx+2Fxy-A″r² = 0.
(82)
As is indeterminate thus far, let it be such that A'r² = µ';
where μ is an undetermined constant, which we may assume to
be unity, if such an assumption is convenient; theu
2.2
A
;
and (82) becomes
(83)
A²x² + B'y² + c'z²+2Dуz+2EZx+2xxy-µ = 0;
which is the equation to an ellipsoid, because A', B', c' are all
positive quantities, of which the origin is the centre, and the
radius vector, corresponding to the direction-cosines (α, α, α),
is that along which the axis of έ lies; and the length of the
corresponding radius vector is
(
By a similar process we may obtain the same equation of an
ellipsoid from (74) and (75), if we take the lengths of the central
radii vectores, which lie along the axes of ŋ, respectively, equal
to
B
and
14
; so that (83) represents an ellipsoid, the
lengths of three of whose central radii vectores, which are at
right angles to each other, are given: this ellipsoid is called the
ellipsoid of principal axes.
And these three radii are the geometrical principal axes of the
ellipsoid; for if we apply to (83) the processes for determining
the lengths and the position of the principal axes of an ellipsoid,
which have been developed in Arts. 6 and 7, the equations for
the direction-cosines, given in (51) and (54) of Art. 7, are the
same as (78) and (79), by which the position of our principal
axes is determined; and the coefficients of έ2, 2, and ² in the
reduced equation are A", в", and c", which are determined by
(73), (74), (75); so that the equation to the ellipsoid, referred to
the axes of έ, n, Š, is
A” §² + B″ n² + c″ §² — µ′ = 0.
2
(84)
Hence it appears that at every point of a body or system of par-
ticles as a centre, an ellipsoid, whose equation is (83), may be
described, the principal axes of which are the principal axes of
the body relatively to that point. And if for the body, relatively
to the centre and the principal axes of the ellipsoid,
د تاریخ
A" =
= Σ.m.§²,
2
Σ.ην
B" = x.mn², c" = Σ.m(²,
88.]
145
THE ELLIPSOID OF PRINCIPAL AXES.
the principal axes of the ellipsoid are inversely proportional to
the square roots of A", B", c" respectively. Thus the form of
the ellipsoid, as well as the position of its principal axes, depends
on the configuration of the system of particles relatively to the
point which is the centre of the ellipsoid.
In our subsequent investigations in this subject we shall assume
Σ.m ¿² > Σ.m n² > x.m (²,
that is,
2
A" > B" > c";
(85)
so that the §- axis, and the §- axis, of the ellipsoid are respect-
ively the least and the greatest of all the axes.
88.] Let us shortly examine the particular forms which the
ellipsoid (84) and the position of the principal axes take, cor-
responding to singular values of the roots of the cubic (76) or
(77). The analytical criteria of the conditions it is unnecessary
to specify, as they are precisely the same as those which have
been determined in Art. 10.
ૐ
a
(1) Let two roots be equal; say A″= B″; then the equation
(84) represents a prolate spheroid whose axis of revolution is
the coordinate-axis of ; as c" is definite, c₁, C2, C3 are also de-
terminate, and the axis of is a principal axis; the direction-
cosines of the axes of έ and ŋ, which are the other two principal
axes, are indeterminate, and any pair of rectangular axes in the
plane of (έ, ŋ) is a pair of principal axes, and with the axis of
completes the system. If в" c", the ellipsoid becomes an oblate
spheroid; the axis of revolution of which is the axis of έ, and is
the determinate principal axis; and any two axes in the plane
of (n, (), which are perpendicular to each other, will complete the
system of principal axes.
(2) Let all the roots of the cubic be equal; then
A″= B″= C"=
c" = K, say;
and equation (84) represents a sphere whose equation is
2
ε² + n² +8² = ;
(86)
K
and every three axes passing through the given point at right
angles to each other will form a system of principal axes. In
this case
n
Σ.mx² = x.my² = x.mz² = x.m§² = z.mŋ² = x.m5²; (87)
and all the nine direction-cosines are indeterminate.
Hence, for every point of a system of particles, there is always
PRICE, VOL. IV.
U
146
[89.
THE ELLIPSOID OF PRINCIPAL AXES.
one set of principal axes; and, for certain points of certain bodies,
every system of rectangular axes originating thereat may be a
principal system.
The ellipsoid (84) will also assume particular forms if one or
two of the quantities A", B", c" vanish, that is, if the body is a
plate or a straight wire; but these cases are so evident that it is
unnecessary to explain them at length.
89.] Let us apply the preceding processes to a particular ex-
ample. Let us take a cube, each of whose sides = a; and let
the origin be at one of the angles; and let the axes of x, y, z lie
along the edges of the cube: it is required to find the position
of the principal axes. Let p be the density of the cube; then
x2
A = Σ.m x²
ɛ.m
a
α
ραδ
pa ja
= [ " [ * ] " p x² dz dy d x = Pa²
0
= B' = C',
as the symmetry indicates.
D = Σ.MY Z
α
α
a
[ª Sª Sª p y z dz d y d x
= } = F F;
so that the cubic equation (76) becomes
3
ραδ
4
7
5
K³ — ρ а5 K² +
2
p² al0 K
48
864
p³ α15 = 0;
5 pa5
the roots of which are
pa5
pa5
6
12 12
; two therefore are equal;
let these be B", c"; so that
>
A"
5 pa5
6
ραδ
B" = c"
12
Hence from (78),
a2
= a
a₂2
a32
2
b₂2
b32
010010001-
2
c₁² = c₁₂²
2
2
so that the axis of έ is the diagonal of the cube; and the position
of the other principal axes is indeterminate; and therefore any
two lines perpendicular to each other, and in the plane passing
through the angle of the cube and perpendicular to the diagonal,
will complete a system of principal axes.
90.]
147
THE DETERMINATION OF PRINCIPAL AXES.
Thus the equation to the ellipsoid (83) is
ραδ
3
(x² + y² + z²) +
ραδ
2
(y z + zx + xy) —µ' = 0,
and the equation to the reduced ellipsoid (84) is
5 ραδ
6
§2 +
ραδ
12
(n² + (2)
= flig
which represents an oblate spheroid, whose axis of revolution is
the axis of §.
If it is required to determine the position of the principal axes
at the centre of the cube; then, if 2a = the length of a side of
the cube,
A′ = B′ = C′
8 pa5
3
DE
= E = F = 0;
A"B" d=
8 pa
3
;
therefore the position of each axis is indeterminate; and any
rectangular system originating at the centre of the cube is a
system of principal axes; and the ellipsoid (83) becomes a sphere.
90.] It is evident from the preceding general investigation
that the position of the principal axes of a body, relatively to a
given point, depends on the values of the definite integrals which
are expressed by the symbols a', B', c', D, E, F; and therefore on
the symmetry of the body relatively to the origin and to the
axes of x, y, z: thus, for a solid of revolution bounded by a
plane perpendicular to the axis of revolution, for any point on
the axis of revolution that axis is evidently one of the principal
axes, and the other two are indeterminate in the plane perpen-
dicular to the axis of revolution. Of a sphere, relatively to the
centre, every system of rectangular axes is principal. Of an
ellipsoid, relatively to the centre, the principal system is unique;
and the principal axes coincide with the geometrical principal
axes of the solid. Similarly the principal system can often be
inferred by general reasoning from the symmetry of the body or
system of particles.
If two principal axes relatively to a given point are given, the
third is also given. If however one principal axis is given, the
other two are at right angles to each other in a plane perpendi-
cular to the given principal axis, and may be determined by the
following process.
T 2
148
[91.
THE DETERMINATION OF
Let the given principal axis be the axis of z; so that
Σ.mxz = Σ.myz
Σ.myz = 0.
(88)
Let the new axis of έ, which is to be principal, be inclined at an
angle
therefore
to the axis of r; so that
x
ૐ = y sin &+ x cos ò̟,
n = y cos p x sin ;
(89)
z.mn = {(cosp)2 — (sin p)2} z.mxy+ sino cos pz.m (y² — x²); (90)
and, as this is to vanish, we have,
and the equations to the two principal axes are
(§² — n²) z. m xy +§n ≥.m (y² — x²) = 0.
2z.mxy
2 F
(91)
tan 2 p
Σ.m (x² — y²) A
B
(92)
As by this process z is unchanged,
(93)
= 0
Σ.m§§ Σ.mz (y sin 4+x sin 4) = 0,
Σ.mŋ5 = x.mz (y cos 4−x cos p)
which, by reason of (88), are true for all values of : if there-
fore (88) are true for any pair of rectangular axes in the plane
of (x, y), they are true for every pair of rectangular axes in that
plane; and indeed for any pair of lines in that plane.
91.] The following are examples of the process for determin-
ing the position of two principal axes at a given point when the
other principal axis is given.
Ex. 1. To determine the principal axes of a rectangular plate
of infinitesimal thickness relatively to the point of intersection
of the two diagonals of the plate.
In this case, as in all problems of plane plates of infinitesima]
thickness, an axis which passes through the origin and is per-
pendicular to the plate is one of the principal axes: and if it is
taken to be the axis of z, and the plate to be the plane of (x, y),
Σ.mxz = Σ.my z = 0, because ≈ = 0 for all the elements of
the plate.
T
Let 2a and 26 be the sides of the plate, r = the thickness,
p = the density;
P
a
Σ.mxy =
L
pτ xy dy dx =
0 ;
a
-b
α
b
Σ.m (x² — y²)
=
- a
та
4рTab
3
рT (x² — у²) dy dx
(a² — b²) ;
91.]
149
PRINCIPAL AXES.
=
therefore tan 24=0; and p=0, 90°; so that the other
two principal axes are parallel to the sides of the plate. If the
0
rectangle is a square, b = a; in which case tan 24 ;
and
is indeterminate, so that every pair of rectangular axes in the
plane of the plate, together with the given axis, constitutes a
principal system.
Ex. 2. To determine the principal axes of a triangular plate
through one of its angles.
Let o, an angle of the triangle, be the origin; oa = α, 0в= b;
and let the axes of x and y lie along these sides; then the equa-
tion to the base is
a
b
let Y =
then
+- 1/2 = 1;
Ꮖ Y
a
b
(a-x); and let w be the angle of the triangle at o;
F=
=5° 5°
PT (X + Y COS W) y (sin w)² dy dx
pr (sin w)² ab²
(a + 2b cos w);
24
a
Y
A'B' =
PT {(x + y cos ∞)2- (y sin w)2} dy dr sin w
Ο
10
PT sin w ab
12
tan 2
(a² + ab cos w + b² cos 2 w);
b sin w (a + 2 b cos w)
a² + ab cos ∞ + b² cos 2 w
;
if b = a, 2p=w; in which case the triangle is isosceles, and of
the principal axes in its plane one bisects the vertical angle, and
the other is perpendicular to the bisecting line.
Ex. 3. To determine the position of the principal axes in the
plane of a thin elliptic plate relatively to a point whose place,
relatively to the centre and principal axes of the ellipse, is (a, ß).
x2 y2
a2 62
Let the equation to be ellipse be +
1; and let
Y (a² — x²): then the ellipse is referred to its centre of
b
a
gravity as origin, and to its principal axes as coordinate-axes;
so that
Then
Σ.m x = 0, x.my = 0 ; x.mxy = 0.
α
Y
F
=[[pr
C
a
α
Y
PT (x-α) (y-B) dy dx
prady dr
= πρτα αβ.
150
[92.
THE EQUATIONS OF MOTION.
A' - B'
=j
Y
= πρταό
PT {(x — α)² — (y — ß)²} dy dx
pr {x² — y²+ a² — ß²} dy dx
a² — b²
+ a² - B²
4
8aß
.'.
tan 2
=
a² — b² + 4 (a² — ß²)
whereby we have the position of the two principal axes in the
plane of the plate, and these, with the axis perpendicular to the
plate, form the complete system of principal axes.
92.] Many other properties of principal axes will arise inci-
dentally in the following section, and will be there demonstrated.
We may therefore now return to the equations of motion, and
make those simplifications in them with which the theory of
principal axes supplies us.
Let the rotation of the body or system of particles be referred
to a system of axes fixed in the body and moving with it; and
let this system be that of principal axes, so that
Σ.mys = Σ.mzx = Σ.mxy = 0, or D = E = F = 0.
Firstly, let us take the case of a body, originally at rest, acted
on by instantaneous forces; and, for the sake of distinctness, let
the axial components of the expressed angular velocities, due to
the instantaneous forces about the principal axes, be symbolized
by 1, 22, 23; and let us reserve î, y, z for the axial compo-
nents of the expressed angular velocities when the system of
axes is not principal; then equations (32), Art. 76, become
L
M
N
21
2
£3
A
B
C
(9-1)
where A, B, C are the moments of inertia of the body about the
principal axes; and L, M, N are the moments of the axial com-
ponents of the couples of the impressed momenta about the cor-
responding axes.
Secondly, let us take the case of a body rotating under the
action of continuous forces; and here again let w₁, w, w, repre-
sent the axial components of the expressed angular velocity
relatively to the principal axes; and let wx, w, w. be the axial
components of the expressed angular velocity when the axes are
not principal; then, since D E F = 0, equations (65), Art.
83, become
93.]
PRINCIPAL AXES.
151
dwy
A
dt
+ (C-B) W₂ W3 = L,
dwz
B
+ (A − C) W3 W1 = M,
(95)
dt
dw3
C
+ (B−A) W₂ W₂ = N ;
dt
which are evidently much simpler than (65), and are equally
general; they were investigated first by Euler, and are now
commonly called Euler's Equations of Rotation. No way of in-
tegrating them in the form in which they stand is known at
present. Particular forms of them will be discussed in the fol-
lowing Chapters; and we shall then employ such artifices of
abbreviation and of interpretation as the particular problem
suggests.
As (95) are of great importance, let us examine the origin of
the several terms: let us take the first of the three; for what is
true of that is also true, mutatis mutandis, of the other two.
On referring to Art. 81 it will be seen that a is the moment
dwi
dt
of the axial component of the couple which arises from the ex-
pressed momentum-increments of all the particles, the other
terms in (55) disappearing because the coordinate axes are prin-
cipal. And from (62) it appears that (B-C) w₂ w, is the moment
of the axial component of all the couples which arise from the
centrifugal forces of the system; hence, by D'Alembert's prin-
ciple, the three equations (95) are formed.
If (95) could be integrated, w₁, wa, w, would be expressed in
terms of t; and thence by equations (103), (104), (105), Art. 42,
✪, 4, and √ could be determined; and the position of the prin-
cipal axes, and therefore the position of the body, as the principal
axes are fixed in it, would also be determined.
93.] Heretofore the point at which the coordinate axes, whe-
ther principal or other, originate, has been arbitrary: let us
consider whether any simplification will be introduced into the
results if we take the origin to be the centre of gravity of the
moving system or body; that is, we may suppose the point
which has motion of translation to be the centre of gravity, and
the axis of rotation to pass through it; and we shall also sup-
pose the coordinate axes fixed in the body, and originating at
the centre of gravity, to be principal axes; and these axes we
152
[93.
THE EQUATIONS OF MOTION.
shall henceforth call central principal axes.
definition of such an origin and such axes is,
Σ.mx = x.my = x.m z = 0,
z.my z = z.mzx = x.mxy = 0.
The geometrical
Let us first consider the axial components of the expressed
momentum-increments of all the particles which arise from the
angular velocity-increments. Then, on referring to Art. 81
wherein the effects have been investigated, from (51) and (55)
we have
z'
X' = Y' = 7′ = 0;
(96)
p′ = 0.
..
L = A
dw1
dt
d w z
dwz
N′ = C
(97)
M' = B
dt
dt
so that the momentum-increments are as to translation in equi-
librium at the centre of gravity, neither accelerating its motion
if it is moving, nor producing a pressure if it is fixed; and the
moments of the axial components of the couple arising from
these expressed angular velocity-increments are given in (97),
and are respectively the first terms of (95).
Next let us take the axial components which arise from the
centrifugal forces which are discussed in Art. 82. From (59)
we have
x"
Y"
= z″ = 0;
P" =
0;
(98)
so that all the centrifugal forces as to translation balance at the
centre of gravity; and from (62)
L″= (B—C) W2W3, M"= (C-A) Wz W1, N"= (A—B)w₁W2. (99)
Thus it appears that relatively to the centre of gravity as the
origin, the forces of translation which are due to the angular velo-
city-increments and to the centrifugal forces are each separately
in equilibrium; but that the equations of rotation are the same
as before.
Suppose however that the axis of rotation is a principal axis,
say the axis of x; then w₂ = w3 = 0; and
L' A
dw,
dt
M' = 0,
N' = 0;
L″ = 0,
M" =
0,
N"= = 0;
so that the centrifugal forces balance themselves as to rotatory
effects on the body; producing no change either of the angular
velocity of the body or of the position of the axis of rotation;
94.]
153
PERMANENT AXES.
and, as (98) shew, they balance each other as to effect of trans-
lation.
Hence if a body rotates about a central principal axis, the
centrifugal forces which are thereby generated are in equilibrium,
and thus do not cause any change of rotation or of the position
of the rotation-axis.
Hence also, if no force acts on the body, its equation of mo-
tion is
dwi
Á = 0;
dt
;
.. wy a constant.
Thus the angular velocity is constant, and the position of the
rotation-axis is the same throughout the motion. For this rea-
son the central principal axes are called the permanent axes of
the body or of the system; they are also sometimes called the
natural axes.
It is evident from what has been said that a permanent axis
of a body at a certain point may be defined as that line about
which if the body revolves, the centrifugal forces generated by
the rotation are either in equilibrium, or have a simple resultant
passing through the point.
94.] The central principal axes are the only axes which possess
this property, that the centrifugal forces balance each other
about them both as to translation and as to rotation; for this
object it is necessary that, see Art. 82,
رچ
0,
x"= 1″ = 2″
Y" =
L″=M"=N"= 0;
(100)
(101)
from these last three we have, from (62), Art. 82, remembering
that
A+ A B+ B' = c + C';
(E cosa + Dcosẞ+c'cosy) cosẞ- (Fcosa + B' cosẞ+D cosy) cosy = 0,
(A'cosa +Fcosẞ+ Ecosy) cos y-(Ecosa + D Cosẞ+c'cosy) cosa = 0,
·0, ↓ (102)
(F COSα + B´COSẞ+ D Cosy) cos a— (A'cosa + F cosẞ+ E cos y) cosß = 0;
so that
A'cosa +Fcosẞ+ Ecosy
COS a
FCosa + B'cosẞ+DCosy
cos B
Ecosa + Dcosẞ+c'cosy
COS Y
; (103)
which are the same equations as (71) by which the principal axes
are determined; the principal axes therefore are permanent axes
so far as the conditions (101) indicate; that is, corresponding to
principal axes the centrifugal forces either are in equilibrium, or
have a single resultant passing through the origin.
PRICE, VOL. IV.
X
1
154
[95.
M. FOUCAULT'S EXPERIMENT.
As to (100) let us replace x, y, z in (59) by ☎ +x', ÿ+ y',ï +ź,
wherein the centre of gravity is (x, y, z); then, since
Σ.mx' x.my' = x.m² = 0,
(100) become
a-cos a (a cosa + y cosẞ+zcosy) = 0,
y-cosẞ (cosa + y cosẞ+zcos y) = 0,
z-cos y (a cosa + y cosß+ cos y) = 0;
(104)
and therefore
X
Z
(105)
cos y
Y
cos a cos B
so that the rotation-axis, that is, the principal axis at the point,
must pass through the centre of gravity of the body. Thus
(105) are to be true for each principal axis at the point; that is,
for each of three different values of (a, ß, y); and this is possible
only when y = 0; only when the point is the centre
≈ = Z
of gravity, in which case either of the three central principal
axes is a permanent axis. If however a principal axis at a point
passes through the centre of gravity, that axis is a permanent
axis for that point.
95.] A remarkable application of this theory of permanent
axes has been made by M. Foucault to the proof of the rotation
of the earth about its polar axis. He presented it to the Academy
of Sciences in Paris in the month of September, 1852. He has
devised a machine which he calls a Gyroscope, and of which a
drawing is given in Fig. 21. I will describe it as it is originally in
its position of rest. ABA'B' is a metallic ring suspended by a wire
SA from a point s which is fixed to the earth; and at a' is a small
pivot working in a small hole, by which the motion of the ring
about the vertical line SAA' is kept steady but is not retarded: BB'
is the horizontal diameter of the vertical ring; and at в and B'
are small holes capable of receiving small pivots or axles; BC B' is
a horizontal metallic ring, of which BB' is the diameter, and o is
the centre, at û and в' are pivots which work in the beforemen-
tioned small holes, so that BCB' is capable of rotating about the
horizontal diameter BB'. Across the horizontal ring BCB' an-
other axis coc' is placed, at right angles to, and bisecting, BOB',
and capable of rotating about pivots at c and c'; on this axis is
fixed a heavy metallic disc DD', whose centre is at o, and which
is consequently capable of rotation about the axis coc', and the
greater part of the matter of the disc is arranged in a ring as near
to the circumference as possible, so as to increase the centrifugal
95.]
155
M. FOUCAULT'S EXPERIMENT.
force of the disc. This is the arrangement of the several parts of
the Gyroscope; and it is evident from the arrangement that the
disc is capable of rotation about any axis, so that whatever are
the forces which act upon it, it can take the axis which they
require, and is indeed in construction identical with Bohnen-
berger's machine, except that the central mass is a disc instead
of a sphere. It is evident also that the centre of gravity of the
whole machine is at o, and that all axes of rotation pass through
o, and thus gravity does not produce any change of position in
the rotation-axis or in the velocity of the disc. Adjustment-
screws are placed in various parts of the machine, so that the
conditions required may be fulfilled as nearly as possible.
Now the disc D D' is taken out of the ring BCB', and has a very
rapid rotation given to it by a machine properly contrived for
that purpose. While it has this rapid rotation it is replaced in
the horizontal ring BCB'; and as the axis coc' is manifestly one
of the principal axes of the ring through its centre of gravity, it
is a permanent axis, and as the disc is not under the action of
any forces, whether external or centrifugal, whereby its velocity
or the position of its rotation-axis may be changed, its axis coc'
keeps an invariable position in space. But what apparent effect
is produced by this invariable position? Let us suppose the
Gyroscope to be at the north pole: then the earth rotates about
the axis sAs', and to an observer the axis of coc' will retain its
horizontal position, and will have a motion in azimuth, in the
same direction as the fixed stars appear to have in passing
through 360° in 24 hours, if the rotation of the disc can be kept
up as long if the Gyroscope is at the equator, no such apparent
effect will take place, because the axis will have only a parallel
displacement of itself in space. In any other latitude the polar
axis of the earth will be inclined to the principal axis coc' of the
disc; and as this, being a permanent axis, retains its direction in
space, it has an apparent motion about the polar axis; the
ring BCB' will revolve slowly about the axis BB', and the ring AA'
will also revolve slowly about the vertical axis AA'; and these
rotations may be observed by means of microscopes properly
placed for the purpose. Indeed the earth truly rotates about the
line coc' which has an invariable position, and that rotation is
shewn by the apparent motion of the line. We shall hereafter
come to the mathematical calculation of these quantities.
X 2
156
[96.
MOMENTS OF INERTIA.
SECTION 5.-Moments of inertia, and the distribution in space of
principal axes.
96.] In the present section I propose to examine more closely
the theory and properties of moments of inertia, of which defini-
tions have been given in Art. 74.
Moment of inertia is (geometrically) the sum of the products
of every particle of a body or of a system of particles, and the
square of its distance from the rotation-axis. Thus, if m is a
particle of a moving system, and r is the perpendicular distance
from the place of m on the rotation-axis, z.mr2 is the moment
of inertia, the summation including all the moving particles, and
becoming integration if the moving system is a continuous body.
We shall however find it convenient to use the symbol z.mr²
in all cases, and it is to be observed that this symbol includes
integration in the cases wherein the moving system is a continu-
ous mass.
Let us in the first place investigate the moment of inertia in
its most general form. Let the origin be taken on the rotation-
axis, of which relatively to a system of axes fixed in the body
let the direction-angles be (a, ẞ, y); let (x, y, z) be the place of
m; and let r be the perpendicular distance from m on the rota-
tion-axis; so that
r² = (y cosy —zcos,3)² + (zcosa - x cos y)² + (xcosß-ycosa)2; (106)
Σ.mr² = x.m (y²+22) (cosa)² + z.m (z²+x²) (cos ẞ)² + z.m (x²+ y²) (cos y)2
-2z.myzcosẞcosy-2z.mzx cos y cosa-2z.mxy cosa cosẞ. (107)
Let н be the general symbol for the moment of inertia; then,
using the symbols of Art. 76,
H = A (COsa)² + B (cos ß)²+c (cosy)²
-2D Cosẞcosy-2 Ecos y cosa-2 F cosa cosß; (108)
and thus if A, B, C, D, E, F are determined for any body and for
a particular system of rectangular axes, the moment of inertia
of the body for any axis may be found by means of this equation.
If the system of axes to which the body is referred is a prin-
cipal system at the point, then D E F = 0; and (108) be-
comes
= = =
H = A (COS a)² + B (cos ẞ)2+ c (cos y)2;
(109)
where A, B, C are the moments of inertia of the body relatively
to the three principal axes of x, y, z respectively, and are for that
reason called the principal moments of inertia. *
In terms of A', B', c', (109) becomes
H = A' (sin a)² + B' (sin ẞ)² + c' (sin y)².
(110)
}
157
98.]
THE MOMENTAL ELLIPSOID.
97.] As an example of (108) let us investigate the moment of
inertia of a rectangular parallelepipedon about an axis passing
through one of its angles.
Let the sides of the parallelepipedon, which meet at the angle
through which the rotation-axis passes, be a, b, c; and let the
coordinate axes lie along these sides respectively; let p be the
density of the volume element at (x, y, z); so that the mass-
element = pdx dy dz. Then
A
Q
b
[ª ↓ ↓˚ p (y² + z² ) d z dy dx =
/
0
B
pabc (c² + a²)
3
p a b c ( b² + c² );
3
C
p a b c (a²+b²).
3
;
a
b
C
D
· = ["[]" pyzdzdyde =
dx
pab2c2
;
4
pbc2a2
E
4
pc a2 b2
F =
;
4
and the moment of inertia about the line (a, ß, y)
= pabc {
I b² + c²
3
c² + a²
a² + b²
(cos a)² +
(cos ẞ)² +
(cos y)2
3
3
b c
2
ca
ab
cos ẞ cosy
cos y cos a
cos a cos B
2
2
and if k is the corresponding radius of gyration,
k² =
b²+c²
3
(cos a)2 +
c² + a²
3
a² + b²
(cos B)² +
(cos y)2
3
b c
ca
ab
cos ẞ cos y
cos y cos a
cos a cos B.
2
2
2
Similarly the moment of inertia of a cube about a diagonal
ραδ
;
6
and about one of its edges
2 pa5
3
Other examples of the determination of moments of inertia
will be given in the following section of this chapter.
98.] The following process gives a geometrical interpretation
of (103), and consequently of (109).
Along the rotation-axis from the origin let a length p be taken,
of which let the end be (x, y, ≈); then
X
Y
2
p;
COS a
cos B
cos y
and (107) becomes
(111)
Ax² + BY² + Cz² -2Dyz-2Ezx-2Fxy = p²x.mr²; (112)
Let
p²z.mr² = μ;
so that z.mr2
μ
(113)
p2
158
[98.
THE MOMENTAL ELLIPSOID.
μ
where is a constant quantity at present undetermined, and
may be unity if such a value is convenient; then (112) becomes
Ax²+By²+Cz2-2Dyz-2Ezx-2гxy-μ = 0.
F
(114)
By the assumption made in (113) it will be observed that the
moment of inertia about any axis p varies as the square of the
reciprocal of p.
As A, B, C are quantities necessarily positive, (114) is the equa-
tion to an ellipsoid, whose centre is at the origin, and of which
p is a central radius vector; and which is such that whatever
radius vector is the rotation-axis, the moment of inertia of the
body relatively to that axis varies as the square of the reciprocal
of the central radius vector. For this reason the ellipsoid is
called the momental ellipsoid. As μ is at present undetermined
the actual size of the ellipsoid is not fixed: however, according
as μ varies, all the corresponding ellipsoids are concentric and
are similar, and that corresponding to any particular value of u
will suffice for our present purpose. Let us imagine therefore
the ellipsoid, whose equation is (114), to be described with the
given point as centre; then that ellipsoid, by means of its cen-
tral radii vectores, indicates the law of variation of the several
moments of inertia of the moving system which correspond to
the radii vectores as rotation-axes, the moment of inertia rela-
tively to any one being proportional to the square of the reci-
procal of that radius vector.
μ
The momental ellipsoid is evidently concentric with the ellipsoid
of principal axes, equation (83); it is also coaxal with it. For,
by Art. 6, the equations for determining the principal axes of
(114) are
Ad - Fag
Edg
а1
Fa₂+Bα₂-DA3
аг
- E α₁ — Dа₂ + c α 3
az
= Ha,
(115)
if Ha
is the moment of inertia about the axis (a1, a2, α3) of έ, so
that Ha.m (n² + 2). Now replacing A, B, C severally by
B' + C', c' + A', A+ B', and subtracting each term of the equa-
lities (115) from a′+ B' + c', we have
c'α
A
▲′α₂+ Fɑ2 + Eα3
FA₁ + B' Aq + D Ag
а1
аг
Eα₁ + Dа₂ + c α z
az
= A'+ B'+ C'- Ha
= Σ.m²;
(116)
which are identical with (73), whereby the principal axes of the
body are determined. By similar processes we might find equa-
98.]
159
THE MOMENTAL ELLIPSOID.
η
tions identical with (74) and (75) in terms of H, and He, which
are the moments of inertia about the axes of n and respectively.
Thus the geometrical principal axes of the momental ellipsoid
lie along the principal axes of the body at the origin; let it be
referred to these as axes; it is manifest from (115) that Ha, Hồ, Hc
are the coefficients of x², y², z² in the reduced equation. Let,
however, henceforth A, B, C represent the moments of inertia
about the principal axes; then the equation to the momental
ellipsoid, referred to the principal axes as coordinate axes of
(x, y, z), is
A x² + B y² + C 2² = µ.
(117)
This result might have been inferred directly from (114).
the position of the coordinate axes is undetermined, let the sys-
tem be the principal system; then D = E = F = 0, and (114)
becomes (117).
This is another instance of the simplification which is intro-
duced into the equations of dynamics by the use of principal
axes. The form of the equations of motion, and the mathema-
tical values of the moments of inertia which enter into those
equations, are much simplified.
In Art. 87 we assumed x.ma² > ɛ.my² > Σ.mz²; therefore
Σ.m(y²+z²) < Σ.m (≈² + x²) < x.m(x² + y²) ;
A < B < C ;
so that the moments of inertia are respectively the greatest and
the least about the axes of z and x; and the maximum, mean,
and minimum axes of the momental ellipsoid lie along the axes
of x, y, z respectively, and correspond to the minimum, mean,
and maximum axes of the ellipsoid of principal axes.
One word as to the meaning of μ; let us give it a value which
will make the equations homogeneous; let м be the mass of the
moving body, and let a, b, c be the radii of gyration about the
axes of x, y, z respectively; so that
A = Ma²,
let therefore
B = м b²,
μ = Mgt;
C = Mc²;
(118)
(119)
we shall hereafter determine the meaning of g; then (117) be-
comes
a²x² + b² y² + c² z² = gª ;
(120)
so that the maximum, mean, and minimum axes of the momen-
a
g² g² g²
b' c
and (120) is homoge-
tal ellipsoid are respectively
neous. Notwithstanding, however, we shall still find it conve-
nient to employ μ.
160
[99.
THE MOMENTAL ELLIPSOID.
99.] Since the moment of inertia of the body about any axis
is proportional to the square of the reciprocal of the radius vec-
tor of the momental ellipsoid which coincides with that axis, it
follows that the moments of inertia and the radii vectores have
simultaneously critical values. Thus, as the x-, the y-, the z-
axes of the ellipsoid (117) are respectively the greatest, the
mean, and the least of all axes of the ellipsoid, so of all ro-
tation-axes passing through the centre of the ellipsoid, those
of x, y, z are the rotation-axes relatively to which the moments
of inertia are the least, the mean, and the greatest. That is,
according to our assumptions, A is the least, B is the mean, and c
is the greatest of all moments of inertia relatively to the given
origin. And these are the principal moments.
Whereas then we have defined principal axes as those in refer-
ence to which ≥.myz = z.mzx = x.mxy = 0, they might have
been defined as those axes for which the moments of inertia have
critical values. The former conception of them arose first, in
the simplification of the equations of the motion, and therefore
we pursued it. It is however to be observed that whatever is
true of the axes of principal moments of inertia is also true of
the principal axes and of the principal planes; and several pro-
perties which are true of principal planes as defined in the last
section, and which might have been there demonstrated, will be
proved in the course of the present section. Henceforth then
we shall treat principal axes, and the geometrical principal axes
of the momental ellipsoid, as identical.
And all other moments of inertia relatively to the given point
are evidently intermediate to c and a; that is, are less than c,
and are greater than a. From (109)
H = A(cos a)² + B (cos B)2 + c (cos y)2;
which may be expressed in either of the following forms;
H = A + (BA) (cos B)2 + (c-A) (COS y)²;
H=C(CA) (COS a)2- (CB) (cos ẞ)²;
(121)
(122)
and as B-A, CA, C-B, are positive quantities, c is the greatest
and a is the least of all moments of inertia.
If two principal moments are equal, the momental ellipsoid
becomes a spheroid; if в = c, the spheroid is prolate and the
moments for all axes lying in the plane of (y, z) are equal to one
another and to c; and other moments are less than c: if A=
the momental ellipsoid becomes an oblate spheroid, and the mo-
AB,
100.]
161
THE EQUIMOMENTAL CONE.
ments for all axes lying in the plane of (x, y) are equal to one an-
other and to a, and the moments for all other axes are greater
than A.
In the former case the ellipsoid of principal axes be-
comes an oblate spheroid; and in the latter case a prolate
spheroid.
If the three principal moments are equal, a = B = C, and the
momental ellipsoid becomes a sphere, and the moments of inertia
for all axes are equal to one another. In this case also the
ellipsoid of principal axes becomes a sphere.
100.] All the rotation-axes passing through a given point, for
which the moments of inertia are equal to each other, lie on a
cone of the second degree, whose vertex is the origin.
Let п be the moment of inertia to which all are to be equal;
then, since p²x² + y²+2, from (112) we have.
= x²+y²+≈²,
A x² + B у² + cz² −2Dyz-2Ezx-2Fxy = н(x² + y² + z²) ;
.'. (H−A) x²+(H—B) y² + (H — C)≈²+2Dy≈+2Ezx+2Fxy = 0; (123)
which is the equation to a cone of the second degree whose
vertex is at the origin; and the principal axes are evidently co-
incident with those of the momental ellipsoid. All rotation-axes
therefore lying on the surface of this cone are axes of equal mo-
ment, and the cone is consequently called equimomental.
If the coordinate axes are principal axes at the origin,
DE = F = 0,
and the equation to the equimomental cone is
2
(H — A) x² + (H −в) y²+(H−c) ≈² = 0,
B
(124)
where a, b, c are the principal moments of inertia. As we have
proved that c is the greatest and A is the least of the moments
of inertia for axes passing through the origin, н must be in-
termediate to c and A; so that necessarily one of the coeffi-
cients in (124) is negative; and not more than two can be
negative.
H
Let н be greater than в; then н—А and н—в are positive,
and н-c is negative; in which case the axis of ≈ is the internal
principal axis, and the axes of x and y are the external principal
axes. See Art. 11. And all plane sections parallel to the plane
of (x, y) are ellipses.
Let HB; then
(B−A)*x = ± (C—B)*~;
(125
which are the equations to two planes; these are indeed, see
PRICE, VOL. v.
Y
162 MOMENTS OF INERTIA ABOUT CENTRE OF GRAVITY. [101.
(105), Art. 13, the cyclic planes of the momental ellipsoid. Thus
all the rotation-axes at any point for which the moments of inertia
are equal to the mean moment of inertia lie in two planes equally
inclined to the axis of greatest moment.
Let н be less than в; then н
H в and H-c are negative, and
H- A is positive, so that the axis of x is the internal principal
axis of the cone, and the axes of y and z are the external prin-
cipal axes; and all plane sections perpendicular to the axis of a
are ellipses.
Thus, according to the configuration which we have chosen,
all axes lying within the planes (125) towards the axes of z are
rotation-axes of greater moment than the mean; all those lying
in the planes (125) are rotation-axes of moment equal to the
mean; and all those lying without the planes towards the axis
of x are rotation-axes of moment less than the mean. See Fig. 2,
in which the cyclic planes are delineated; all axes within the
angles uou' and vov' are of the former kind, and all those within
the angles uov and u'ov' are of the latter kind.
Also the cyclic planes of the equimomental cone (124) are the
cyclic planes of the momental ellipsoid; for, by reason of (96),
Art. 12, the equations to the cyclic planes of (124) are
± {H—C—(H−B) } ³z ;
{H — B — (H — A)}³x = ± {H−C
(BA) x = ± (c — B) ≈ ;
which are the same equations as (125).
(126)
If two principal moments are equal, so that the momental
ellipsoid becomes a spheroid, the equimomental cones become
cones of revolution.
If the three principal moments are equal, the equimomental
cone degenerates into a rotation axis.
101.] I propose now to consider the moments of inertia and
the momental ellipsoid relatively to any point of a body, with
reference to the moments of inertia and the momental ellipsoid
relatively to the centre of gravity. We shall hereby be led to
general theorems which will clear up many obscurities as to the
distribution of principal axes in space, and will indicate remark-
able symmetry as to their arrangement.
The following theorem must be demonstrated in the first place.
The moment of inertia of a body, or of a system of particles,
about any axis is equal to the sum of the moment of inertia
about a parallel axis passing through the centre of gravity, and of
102.]
163
THE CENTRAL ELLIPSOID.
یا
the product of the mass and the square of the distance between
the axes.
Let н be the moment of inertia about the given rotation-axis,
and H' the moment of inertia about a parallel axis passing
through the centre of gravity; let h be the perpendicular distance
between these two axes. Let r and be the distances of m
from the axes of н and н' respectively, and let & be the angle at
which is inclined to h; so that
p2 =
2
p² - 2 r'h cos
+h²;
Σ.mr² = Σ.mr2
≥ .mr' 2 — 2 hz.mr'′ cosp+h² ɛ.m;
(127)
but 'cos is the perpendicular distance from m on the plane
which passes through the centre of gravity and is perpendicular
to h; so that z.mrcosp 0. Let the mass of the body or
system of particles м; then, since z.mr2 = B, z.mr² = H',
(127) becomes
=
H = H' + M h²;
H,
which is the mathematical expression of the theorem.
(128)
Hence, if k is the radius of gyration relatively to the rotation-
axis, and k' is the radius of gyration relatively to the parallel
axis through the centre of gravity,
h2
2
k'² + h².
(129)
Hence also it follows, that if a line is drawn through the centre
of gravity of a body or system, the moments of inertia are equal
for all parallel rotation-axes at equal distances from this line;
or, in other words, all axes lying on the surface of a right cir-
cular cylinder whose axis passes through the centre of gravity of
a body or system are rotation-axes of equal moment.
Hence too of all parallel rotation-axes the moment of inertia
is the least for that which passes through the centre of gravity
of the body.
102.] As we shall often have occasion to refer to the momen-
tal ellipsoid at the centre of gravity, it is convenient to give it a
distinctive name; I shall call it the Central Ellipsoid; and the
principal axes and the principal planes which refer to the centre
of gravity will be called the central principal axes and the cen-
tral principal planes and the principal moments of inertia at
the centre of gravity will be called the principal central mo-
ments.
Let the centre of gravity be the origin; and let the equation
to the central ellipsoid be
Ax² + Bу² + Cz² = μ;
(130)
Y 2
164
[102.
THE CENTRAL ELLIPSOID.
μ being an arbitrary constant which we shall hereafter determine;
and where a, b, c are the principal central moments of inertia,
arranged in order of magnitude as heretofore; viz. A < B < c.
Let also A', B', c' refer to the central ellipsoid; where
A' = x.mx², B' = z.my², c' = x.mz².
Let (§, ~, §) be the point at which the principal moments and
the position of the principal axes are to be determined; and let
(x', y', ') be the place of m relatively to that origin, the coor-
dinate axes being parallel to the central principal axes; so that
= x − &, y' = y―n, 2 = 2−8; (131)
X'
and let м be the mass of the body or system of particles. Now
the equation to the momental ellipsoid whose centre is at (§, 7, 5)
is that given in (114), Art. 98, and in this case is
2
Σ.m (y'² + ≈′²) x² +Σ.m (z'² +x′²) y² +ɛ.m (x²² + y'²) ≈2
— 2ɛ.my'z' y z — 2ɛ.mz'x' zx—2z.mx'y' xy—µ = 0. (132)
Let us calculate the coefficients of this equation relatively to the
centre of gravity as origin. Here, by (131),
Σ.m (y'²+22) = x.m {(y—n)² + (≈ — $)² }
= x.m (y²+~²) — 2ŋ ɛ.my — 2 (≤.mz+ (n² + (²) Σ.m
= A + M (n² + 8²),
as is evident also by reason of the last Article.
Let
then
similarly
/2
z.m (≈′² + x²²) = B+ Mp² - Mn2,
2
§² + n² +8² = p² ;
(133)
2
z.m (y'² + 2′²) = A + M p² — M §²;
(134)
z.m (x²² + y'²) = c + Mp² - M².
Also
z.my'z' = z.m (y — n) (≈ — 8)
= z.myz - ŋz.mz - (z.my+nSz.m;
... Σ z.my' z' = Mn $;"
Μηζ;
similarly
Z.mz'x'= M ČE,
Ꮗ MCE,
Μξη.
z.mx'y' = MEŋ.
(135)
Thus the equation to the momental ellipsoid, whose centre is at
(§, n, 5), which is the origin of x, y, z, is
(A + M p² — M §²) x² + (B+M p² — M n²) y² + (c + M p² — M (²) ~²
-2мny-2M (§ zx-2м §ŋ xy-μ = 0. (136)
The equations for determining the position of the principal
axes of this ellipsoid are given in Art. 6; and for that principal
axis which corresponds to a, a, a3 we have
102.]
165
THE CENTRAL ELLIPSOID.
(A+Mp2-M2) a, -мEnα-мŠČα,
аг
M
− м§ŋα₁ + (B+ M p² — м n²) α₂ — м ngaz
1
a2
− м § Śа¸ — м ŋ Ša₂ + (C + Mp²—MŠ²)α3; (137)
1
2
аз
2
(A + M p² — M §²) a₁² + (B+M p² — M ŋ²) α₂² + (C + M p² — M (²) α,
-2 Μηζα,α3-2 ζέα,α2ξηαια,
M
= Ha,
3
1
(138)
by reason of (108), if H, is the moment of inertia of the body
about the axis (a, d, as) at the given point. We have also
analogous equations in terms of b₁, b₂, bз, нь, and C1, C2, C3, Hc;
and from these equations the direction-cosines of the principal
axes may be determined as in Arts. 6 and 7. Let t₁, të, të be
the direction-cosines of a principal axis, and let н be the type of
Hạ, Hồ, H¿; so that from (137) we have
A+Mp²
м § (t₂ § + t₂n + t35)
t₁
2
Mn(t₁§+t₂n+t25)
to
2
м ¿ (t₁§+t‚ŋ+t35)
= B+M p²
= c + Mp² -
t3
(A+Mp²-H) t₁
1
(B+Mp²-H) t₂
M (t₁§+t₂n+t38) =
= н; (139)
(c+Mp² —H)¹3; (140)
لله
ทุ
h8+69+h's
t₁ § + t₂n + tą Ś
n²
3
+
B+Mp² - H HC+MP-H
Š

;
(141)
دع
+
A+ Mp² - H
whence we have
2
n²
2ع
1
+
+
;
(142)
A+ Mp² — H
B+ Mp² — H
C + M p² — H
M
t₁
to
and
(143)
ૐ
η
5
A+ Mp² — H
B+Mp².
B+M p² — H H
C+Mp² - H
(142) is a cubic equation in H, whose roots are н, Hы, H.; that
is, are the three principal moments at the point (§, 7, 8). And
these three roots are respectively less than A+Mp², greater than
A+Mp² and less than B+Mp², and greater than в + Mp² and less
than c+Mp²; for if we arrange (142) in descending powers of н,
the result is +ve, if ¤ = c + M p² ;
—ve, if н = B+ M p² ;
ve, if H = A + Mp²;
+ve,
-ve, if H =
∞
(144)
thus the roots of (142) are real, and limits of them are assigned;
166
[103.
PRINCIPAL AXES DETERMINED.
and as all the other quantities in (142) are given in terms of the
principal central moments, the mass of the moving system, and
the coordinates of the given point, we shall henceforth consider
Ha, Hʊ, H₂ to be known quantities.
If these several values of H are substituted successively in
(143), we have three different sets of direction-cosines, which
correspond to the three principal axes at (§, n, Š).
103.] These equations (142) and (143) admit of the following
interpretation. The equation
x²
+
y2
A+M p² — H B+M p² — H
~2
1
+
(145)
C+Mp² - H
2
M
represents three confocal surfaces of the second degree; which
are an ellipsoid, an hyperboloid of one sheet, and an hyperboloid
of two sheets; because, according to (144), the coefficients of
x², y², z² (1) are all positive; (2) that of a² is negative and the
other two are positive; (3) those of a² and y² are negative, and
that of 2 is positive; and according to our assumption of the
order of magnitude of A, B, C, in the ellipsoid the x-, y-, and
z- axes, are respectively the greatest, the mean, and the least.
Thus (142) shews that the point (§, 1, 3) is at the point of in-
tersection of these three confocal surfaces of the second degree.
And since the direction-cosines of the normal of either of these
surfaces at the common point are proportional to
§
η
,
B+ Mp² H
5
c+Mp2-H'
(146)
A+ M p² — H
(143) shew that the principal axes of the body at the point
(έ, 7, ) lie along the normals to the three confocal surfaces of
the second degree which intersect orthogonally at it. Or if we
take one surface only, the three principal axes are respectively
normal to it, and touch the lines of curvature of the surface at
the point.
As we pass from one point of a body or system of particles to
another, p² and н vary; so that in the denominators of the left-
hand member of (145) Mp²-H varies according as the point
changes at which the principal moments are to be determined;
and thus the ellipsoid, with which all the surfaces of the second
degree represented by (145) are confocal, is that whose equation is
y2
1
x2
+
22
+
A
B
C
Μ
(147)
104.]
167
THE CENTRAL ELLIPSOID OF GYRATION.
Let a, b, c be the principal radii of gyration, so that
A = M a²,
B = Mb²,
C = Mc²;
then (147) becomes
x2
a²
+ +
62 C2
y2 22
1;
(148)
(149)
which ellipsoid is called the central ellipsoid of gyration, as its
principal axes are the radii of the body relative severally to
them. Hence we have the following construction for the position
of the principal axes at any point in space. Through the given
point let three surfaces of the second degree be drawn which are
confocal with the central ellipsoid of gyration, the tangents to
the three lines of intersection of these surfaces are the principal
axes at the point; and the principal moments at the point are
the three roots of the cubic equation (142).
Again, let us multiply the numerators and denominators of
the three last terms of (140) severally by
(B — C) t₂ tз, (C — A) tз t₁,
3 1
(A — B) t₁tą ;
and let us add the numerators and the denominators respect-
ively; as the sum of the numerators vanishes, so must also the
sum of the denominators; and therefore
(A−B)
(B-C) į tą tз + (C−A)ηtzt₁ + (A—в)¿t₁t₂ = 0.
3
}
1
Now as t₁, ta, ta are the direction-cosines of one of the principal
axes at the point (§, ŋ, §) referred to central principal axes, let
us replace them by a, y, ; whence we have
(B−C) ¿Y≈+(C—− a) n≈x+(1—B) έxy = 0;
(150)
which is the equation to a cone of the second degree; on the
surface of which therefore are all the principal axes at the point
(§, n, (), this being the vertex of the cone. Since (150) is satis-
fied when x, y, z are proportional to έ, 7, it follows that the line
drawn from (έ, n, Č) to the centre of gravity lies on the cone. It
is evident also that the three axes of x, y, z lie in the surface of
the cone.
Hence we have the following geometrical theorem :
The three principal axes at any point of a body, the three
lines drawn through that point parallel to the central principal
axes, and the line drawn from the point to the centre of gravity
all lie in the surface of a cone of the second degree.
104.] The ellipsoid of gyration is the sphero-polar reciprocal
of the central ellipsoid with reference to the sphere
x² + y² + z² = g².
(151)
168
[104.
THE CENTRAL ELLIPSOID OF GYRATION.
The equation to the central ellipsoid (120) is
a² &² + b² n² + c² 5² = g¹;
(152)
and the polar plane of (151), relatively to (έ, n, ) as the pole, is
x § + yn + z § = g²;
therefore we have
x d¿+ydn + zd$ = 0,
a²¿d § + b²nd n + c² ¿d § = 0; S
a² §
b2n
c² Ć
a² §² + b² n² + c² č²
g2;
X
Y
z
x § + y n + z Ś
and, substituting in (152), we have
x2 y2 22
a2
+
+
b2 c2
1
;
(153)
which is the central ellipsoid of gyration; and therefore the
polar reciprocal of the central ellipsoid with respect to the sphere
(151) is an ellipsoid confocal with which are those surfaces of
the second degree which intersect orthogonally at any point of
a rigid body along the principal axes at that point. Of this
ellipsoid of gyration, see Art. 20, the momental ellipsoid is the
sphero-reciprocal polar.
If p is the perpendicular from the origin on the tangent plane
of (153),
p²
a4
1 x2 y2
+ +
22
b4
(154)
and the direction-cosines of the perpendicular are severally
px
рх ру
pz
a²
b2
p, by (109),
૮૭
; so that the moment of inertia of the body about
= A
p²x²
a4
p2 y2
+ B
p² 22
74
+ c
C4
= Mp²
X2
y2
a2
+
b2
+
C2
= M p²;
(155)
so that p is the radius of gyration of the body about the axis
which coincides with p. Hence we have the following theorem ;
If the central ellipsoid of gyration is described, and on a tan-
gent plane drawn to any point of it a perpendicular is let fall
from the centre, the length of that perpendicular is the radius
of gyration of the body relatively to it as the rotation-axis.
The locus-surface of the extremity of p is
2
(x² + y²+z²)² = a²x² + b² y² + c² z² ;
(156)
of this surface any radius vector is the radius of gyration of
the body rotating about it.
105.]
169
THE SYMMETRY OF A BODY.
This result also follows directly from (109).
For if r is the
radius of gyration about the rotation-axis (a, ß, y), then its mo-
ment of inertia is м2, and we have
Mr² = Ma² (cos a)² + м l² (cos ẞ)² + M c² (cos y)² ;
..
(x² + y²+±²)² = a²x² + b² y² + c² ~²;
and thus, if we choose to begin with this equation, the theorems
already proved may be deduced by an inversion of the processes
of the preceding Article. The advantage of the method would
be that we should be rid of the undetermined constant g until
it was introduced as the radius of the sphere, relatively to which
the sphero-polar reciprocal of the central ellipsoid of gyration
would be the central momental ellipsoid.
105.] Now the three surfaces of the second degree, which
equation (145) represents, and which are confocal with the cen-
tral ellipsoid of gyration, intersect orthogonally at not only
(§, n, §), but at seven other points which are situated symmetri-
cally in the other octants, and which correspond to the several
combinations of the double signs of έ, 7, and . The equation
(142) is the same whatever are the signs of έ, n, and (; so that
the principal moments are the same at each of the eight points,
which are the angles of a rectangular parallelepipedon whose
centre is at the centre of gravity, and whose sides are parallel to
the central principal axes: and as the equations for determining
the position of the principal axes (143) are the same when the
signs of §, ŋ, ¿ are all changed, so the principal axes at (§, 77, Š)
are parallel to those at (—§, -n, -(); and similarly the prin-
cipal axes at the other six points of symmetry are arranged in
pairs corresponding to the ends of a diameter.
—n,
Thus the body or system of particles is symmetrically arranged
as to principal axes, principal moments, and all moments of in-
ertia, relative to the centre of gravity, the central axes, and the
central principal planes. And as space is divided by the principal
central planes into eight portions, so to a point in any octant a
point of symmetry corresponds in each of the other seven octants,
at which the principal moments are equal, and the momental
ellipsoid is similarly situated with respect to the centre and the
principal central axes. Therefore whatever is the form of the mov-
ing system, be it a continuous body or a system of disconnected
particles, however various the distribution of its parts, however
unsymmetrical its bounding surface, yet it has a centre of gravity,
PRICE, VOL. IV.
Z
170
[106.
THE SYMMETRY OF A BODY.
central axes, and a central momental ellipsoid; and the arrange-
ment of all other moments and axes is symmetrical relatively
to that point. In discussing therefore the rotation of an ir-
regular mass about an axis passing through a fixed point, we
may dismiss from our minds all the irregularities of the mass,
and consider in its stead either the regular and symmetrical
central ellipsoid, or, as the late Professor Maccullagh taught in
his lectures, the central ellipsoid of gyration; for the properties
of either of these surfaces will express all the possible circum-
stances of motion of the system.
106.] I must also mention another construction whereby the
position of the principal axes at a given point may be deter-
mined.
From the point as a vertex let a cone be described enveloping
the central ellipsoid of gyration (149); then, as we have shewn
in Art. 19, the principal axes of the cone at its vertex are the
normals to the three surfaces of the second degree which inter-
sect at it and are confocal with the enveloped ellipsoid; and as
the principal axes lie along these three normals, so they also
coincide with the principal axes of the cone at the point.
The cone which is reciprocal to this enveloping cone is an
equimomental cone: this might be demonstrated directly from
the equation to the enveloping cone which is given in (134),
Art. 19; for if we determined the equation to its reciprocal cone,
it would be identical with (123), Art. 100. The following proof
however is more concise. Through the given point let a tangent
plane be drawn to the ellipsoid of gyration; this plane being
evidently a tangent plane to the enveloping cone. To it let per-
pendiculars be drawn from the centre of the ellipsoid and from
the given point: let the distance between these perpendiculars
be q, say; then, since by (155) the moment of inertia of the
body about the former line = Mp², therefore the moment of
inertia about the latter line, by reason of Art. 101, is
Mp² + Mq² = Mr²,
if r is the distance of the given point from the centre of gravity:
but this latter line is a generating line of the reciprocal cone,
and r² is the same for all generating lines of the cone; and
therefore the moment of inertia is the same for all generating
lines of the reciprocal cone; and consequently the reciprocal
cone is equimomental.
107.]
171
THE EQUIMOMENTAL SURFACE.
Thus also may a series of equimomental cones be described;
for the number of surfaces of the second degree confocal with
the central ellipsoid of gyration is unlimited and as to each of
these an enveloping cone may be drawn, so may the reciprocal
cone of each be described, and thus will there be a series of equi-
momental cones.
The focal conics are particular and degenerate forms of sur-
faces of the second degree which are confocal with the central
ellipsoid of gyration. Their equations have been found in Art. 17;
and are
y2
z2
+
1,
b² — a²
c² - a²
22
x2
+
1,
c² — f²
a² -f2
(157)
x02
y2
+
a² - c²
b² — c²
1;
and as by our hypothesis c > b > a, so are these curves re-
spectively an ellipse in the plane of y, an hyperbola in the
plane of zx, and an imaginary conic in the plane of xy. These
curves then in their several planes may be taken to be directors
of the cones whose geometrical principal axes are the principal
axes of the body at the vertex of the cone.
If the surface of the second order, confocal with the ellipse of
gyration, passes through the point at which the principal axes
are to be constructed, the enveloping cones degenerate into tan-
gent planes; and the reciprocal cones become normals to these
planes: thus the two constructions, by means of the three con-
focal conics, and by the axes of the enveloping cone, become
identical.
107.] I propose in the next place to inquire into the locus-
surface of those points (§, 1, §) at which one of the principal mo-
ments has a constant value. Let be the radius of gyration
corresponding to this given value of the principal moment of
inertia; so that if H is the principal moment, н = мk²: also
let a, b, c be the principal central radii of gyration; so that
A м a²,
and thus (145) becomes
X2
+
r² ÷ a² — k²
where
B = Mb²,
C = MC²
32
+-
x² + b² k2 r² + c² - k²
=== x² + y² + ±²
1 ;
(158)
Z 2
172
[107.
THE EQUIMOMENTAL SURFACE.
and if we replace 1 in the right-hand member by
have
x² + y² + x²
we
x² (a² — k²)
+
2.2 + a² - k²
y² (b² - k²)
j•² + b² - k²
+
≈² (c² — k²)
j²+c² - k²
0.
(159)
The surfaces which these equations represent have been named
by Professor William Thomson Equimomental Surfaces. As ge-
nerally at every point k has three different values, so will three
equimomental surfaces pass through every point.
By giving different values to k we have different equimomental
surfaces. According to our hypothesis a is the least radius of
gyration for all axes passing through the centre of gravity; it is
therefore absolutely the least of all radii of gyration, but they
have no superior limit; so that k may have all values from a to
+∞. Now (159) will express surfaces different in form ac-
cording as k is greater than c; lies between c and b ; is equal to
b; and lies between b and a. If k is greater than c, the equi-
momental surface is the same as the wave surface in biaxal
crystals*.
The principal axis at a given point (x, y, z) of the equimo-
mental surface lies in the tangent plane at that point; and
passes through the point where a perpendicular from the origin
on the tangent plane meets it.
M
Let us replace A, B, C, н in (143), severally by мa2, м b², Mc²,
Mh2; then the direction-cosines at the point (x, y, z) are re-
spectively proportional to
X
y
(160)
r² + a² - k² ' r² + b² - k²' x² + c² — k² ·
Now let l, m, n be the direction-cosines of the line drawn through
(x, y, z) to the point of intersection of the tangent plane of
(158) with the perpendicular on it from the centre. Then, if
F(x, y, z) = 0 is the equation to the surface, the direction-cosines
of this line are easily shewn to be proportional to
2
F
2
X
F
∞ { (dry)² + (dr)² + (d2)" } − (d') {∞ (dr) + 3 (dy) += (de) },
X
Y
{
dx dy
dr 2 dr 2
dx
dr
F
2
+
F
dy
F
2
+
dz
(dr)²
dx
y
F
d F
d F
X
+y
(dr)
+2
d
dz
F
= {(~1)² + (1)² + (d)" } -
dx
dy
z
dy
d
F
(C) {~ (4) +
(de) y
dz
dx
dz
dr
dz
(4) + = (CZ)};
dy dz
* A full discussion of this surface will be found in a Thèse de Mécanique,
by M. Peslin. Mallet-Bachelier, Paris, 1858.
108.]
173
THE EQUIMOMENTAL SURFACE.
x2
y2
Let
s;
(2ª + 2ª_k¥j² + (r²+ b²_k²)² + (~+2=gj;=*; (161)
(r²
(r² c² k²)²
22
then from (158)
(dr)
2x
r² + a² — k²
2xs,
dr
dy
2y
r² + b² - k²
-2ys,
dr)
2z
-2zs;
r² + c² - k²
dz
so that l, m, n are proportional to
(84-20 to 100-ptor-to
(162)
which are the same as (160): and thus the theorem as enunciated
is proved.
108.] I must also explain another process by which the equi-
momental surface is found. It occurred to Professor Maccullagh
to draw through a fixed point a series of planes intersecting a
given surface, and at the point to draw perpendiculars to the
several planes of lengths equal to the apsidal radii vectores of the
section of the surface by the plane; one or more surfaces would
be the locus of the extremities of these perpendiculars, and these
he called the apsidals or apsidal surfaces.
Let this process be performed on a central surface of the
second degree at its centre; and let the equation to the surface
be
22 y² 2
+ +
a² B2 y²
1;
let any plane of section be
lx + my + nz = 0;
and let the apsidal radius vector in this plane ber; so that
r² = x² + y² + ~²;
y²+
(163)
(164)
(165)
let (§, n, Ŝ) be the end of the perpendicular; then, as the per-
pendicular is to be equal to r, and to be perpendicular to the
plane (164), we have
हुं
77
१
7
M
n
(166)
Differentiating (163), (164), and (165), and putting Dr = 0, as
r is an apsidal radius vector of the section of (163), we have
x
2
a
dx+dy+d=0;
I dx + m dy + ndz = 0;
x d x + y dy +zda = 0:
174
[108.
THE EQUIMOMENTAL SURFACE.
multiplying the first of these by λ, and the second by μ, adding,
and equating to zero the coefficients of dx, dy, dz, we have
λ
X
a2
+ µl + x = 0,
y
λ
B2
+µm + y = 0,
λ
z
22
+μn + z = 0;
multiplying these severally by x, y, z, and adding, we have
λ + r² = 0;
2.2
x ( 1 −
y (1 −
a² 2
α
2.2
B2
2 23 ) + µ l
αμί
+ µ l = 0,
.. x ---
0,
α
a² — p2
:)
12/2) + µm = 0,
B2 μm
y +
0,
(167)
B² — p²
= (1 −
2.2
2
)
)+un
γέμη
= 0;
2+
0 ;
z² — p²
and multiplying these last severally by l, m, n, and adding,
a212
B2 m²
y2n2
+
2
α
a² — p2
B² — r2
+
= 0;
and from (166)
a² €2
B2 n2
y² 52
2-2-a2
+
a²
+
p2
B2
p² — y²
= 0;
(168)
and adding to both sides r², which is equal to §² + n² +-¿²,
دع
n²
(2
+
+
1;
p² — z²
r² — a² 2-B2
which is the same equation as (158) if
a² = k² - a²,
B² = k²- b²,
y² = k² - c²,
(169)
(170)
and therefore the equimomental surface is the apsidal of the sur-
face of the second order (163); which is an ellipsoid, an hyper-
boloid of one sheet, or an hyperboloid of two sheets, according as
k² is > c², is > b² < c², is > a² < b².
109.] We must not continue the discussion of the properties
of the equimomental surface; let us however investigate the
positions of those points at which (1) two of the three principal
moments, (2) all three principal moments, are equal.
If (§, ŋ, Č) is a point at which two principal moments of inertia
are equal, two of the three values of ƒ in (142) are equal to one
109.]
175
TWO PRINCIPAL MOMENTS EQUAL.
another; and consequently (142) and its н-differential are simul-
taneously true. Now the H-differential may be put into the
form
2
+
(A + M p²-H)2
n²
+
2ع
(B + Mp² — H)² (C + M p² — H)²
= 0; (171)
or ¿² (B+Mp² — H)2 (c + Mp² - H)² + n² (c + M p² — H)² (a + M p² — H)²
+ S²(A + Mp² — H)² (B + M p² — H)² = 0. (172)
A
==
All the roots of this equation as it stands are imaginary; and
as the reality of the roots of (142) has been demonstrated, (172)
must be satisfied identically: this may be done as follows:
(1) Let έ = 0, and H = A + мp²; in which case (142) becomes
§
n²
B- A
وع
+
C-A
1
;
M
and since a = мa², B = м b², c = м c², this becomes
n²
+
1
;
b² - a² c² — a²
(173)
(174)
and, according to our assumption, c > b > a; thus (174) repre-
sents an ellipse in the plane of (7, §), and is in that plane the focal
conic of the ellipsoid of gyration. Two of the principal moments
of inertia = A + Mp²; and the third = B+C−A.
2
B+Mp²; in which case (142) becomes
(2) Let n = 0, and H= B+M
₤2
(2
+
= 1
1;
a² — f²
c² — b²
(175)
which represents an hyperbola in the plane of (§, §), whose real
and imaginary axes lie respectively along the axes of § and §,
and is another focal conic of the ellipsoid of gyration. Two of
the principal moments of inertia B+Mp2; and the third prin-
cipal moment = C + A − B.
-
(3) Let (= 0; and H = c + Mp²; then (142) becomes
دع
n²
2
+
a² - c² b² — c²
= 1;
(176)
which is the other focal conic of the ellipsoid of gyration, and is
imaginary. At all points therefore of the real focal conics of the
ellipsoid of gyration, two roots of (142) are equal; and two
principal moments are equal: the tangent line to the focal conic
is the axis of the unequal principal moment, and the normal plane
to the focal conic is the plane which contains the axes of equal
moments. All axes therefore in this plane which pass through
the point of contact are axes of equal moment: so that the num-
ber of axes of equal moment is infinite. Indeed the other two
176
[110.
ALL PRINCIPAL MOMENTS EQUAL.
surfaces of the second degree, which with the focal conic are
confocal with the ellipsoid of gyration, become flat, and infi-
nitesimally thin; so that any plane which passes through the
tangent of the focal conic is a tangent plane to one of these sur-
faces, and the perpendiculars to these planes at the point of
contact are principal axes.
This result is also evident from the construction of principal
axes which is given by the enveloping cone of the ellipsoid of
gyration: the enveloping conic is a cone of revolution if its ver-
tex is on a focal conic; the tangent of the conic is the internal
axis of the cone; and any two lines in the plane through the
vertex of the cone, which is perpendicular to the internal axis,
are external axes. In this construction however it is to be ob-
served that the enveloping cones may be imaginary.
Hence we have two distinct curves of the second degree in the
planes of (n,) and of (Ś, έ), which are respectively an ellipse and
hyperbola, at every point of which the position of one principal
axis of the body is determinate; but as the moments correspond-
ing to the other two principal axes are equal, the position of such
axes is indeterminate.
At every point on a focal conic, the momental ellipsoid be-
comes a spheroid, whose axis of revolution is the tangent to the
focal conic.
110.] If (§, n, Ŝ) is a point at which all the principal moments
are equal, the three roots of (142) are equal; and (136) repre-
sents a sphere; so that
n5 = $§ = εn = 0;
M
(177)
(178)
A+ M (n² + Š²) = B + M (§² + §²) = c + м (§² + n²)
From (177) it follows that of the three quantities έ, n, & two
must be equal to zero; and as c is > B > A, the only possible
supposition is
therefore
と
​ε = n = 0;
A + M (2
= B + M (² = c ;
2
.'.
A = B;
A
+
(179)
(180)
M
therefore two of the central principal moments, viz. the mean
and the least, must be equal to each other; and thus the cen-
tral ellipsoid must be a prolate spheroid; in which case, on the
axis of greatest central moment there are two points, viz. the foci
of the spheroid of gyration, equally distant from the centre of
110.]
177
ALL PRINCIPAL MOMENTS EQUAL.
gravity, at which all the principal moments are equal, and there-
fore all axes are principal axes. At each of these points the
momental ellipsoid becomes a sphere.
If c = a, that is, if all the central principal moments are
equal, from (180), = 0; and at no other point in the body, but
the centre of gravity, are all the principal moments equal.
Now these results might have been arrived at from considera-
tions founded on the properties of the focal conics: the three
principal moments can be equal only when the focal conics of
the ellipsoid of gyration have a common point; and as (174)
and (175) can have a common point only on the central axis
of ¿, § = n = 0; in which case
= ± (c² — a²) ½
= ± (c² — b²) ±
..
a = b,
or A = B;
and
A
S =+ (c = 4)².
(C
5
M
The results of this and the preceding Articles are important in
reference to principal axes, and to their properties as permanent
axes: the couple of the centrifugal forces vanishes whenever the
rotation-axis is a principal axis; and if it is a central principal
axis, having its origin at the centre of gravity, the pressure at
the origin, which is due to the centrifugal forces, also vanishes:
at the centre of gravity therefore there are three, and generally
only three, permanent axes, which are the principal central
axes; unless two of the three quantities A, B, C are equal, in
which case all axes perpendicular to the unequal principal axis
are permanent axes. And if a = B = c, every axis through the
centre of gravity is a permanent axis.
For all points on the focal conics the position of one principal
axis is determinate, and every axis which is perpendicular to
that axis is a permanent axis; so that the number of permanent
axes is infinite; in this case however the resultant pressure of
the centrifugal forces does not vanish; and the axes do not
generally pass through the centre of gravity; in fact a perma-
nent axis passes through the centre of gravity only when the
origin is at a vertex of a focal conic.
At the two points on the central axis of greatest moment,
given by (180), all axes are principal, and therefore permanent
axes; and of course only one axis at each point passes through
the centre of gravity.
PRICE, VOL. IV.
A a
178 THE DISTRIBUTION IN SPACE OF PRINCIPAL AXES. [III.
111.] The investigations of the preceding Articles, and the
methods given for the construction of principal axes, shew that
an axis taken arbitrarily on a body may not be a principal axis
at any point on it; because those axes alone are principal which
are normal to some surface of the second degree which is con-
focal with the ellipsoid of gyration.
If an axis of a body is a principal axis, let us call that point
at which it is principal, its principal point; and let us call the
plane which is perpendicular to it at its principal point, and
which contains the other two principal axes, its principal plane;
so that a principal plane of an axis is a plane tangent to a surface
of the second degree, confocal with the central ellipsoid of gyra-
tion at the point where its axis cuts that surface.
In considering therefore the axes of a body, we may dis-
tinguish (1) those which are principal at every point along them;
(2) those which are principal at one point; (3) those which are
not principal at all. Let us consider them in order.
The three principal central axes cut at right angles all the
surfaces of the second degree which are confocal with the central
ellipsoid of gyration; and as the number of such surfaces is in-
finite, so every point on a central principal axis is principal ;
and as the other two confocal surfaces at these points degenerate
into the coordinate planes, the other two principal axes are always
parallel to the central principal axes.
The three central principal axes are the only lines which have
the property of being principal at every point on them in this
respect then, as in others, they form an unique system.
:
Some special cases of axes which are principal at a particular
point deserve consideration.
The sphere whose centre is at the centre of gravity, and whose
radius is infinitely great, is a surface confocal with the central
ellipsoid of gyration; and as all lines drawn through the centre
of gravity are normal to this sphere at the infinity point, so is
every line drawn through the centre of gravity a principal axis
at a point which is at an infinite distance along it.
If three confocal surfaces of the second order pass through a
point in one of the central principal planes, one of the confocal
surfaces becomes flat, and the normal to this surface is a per-
pendicular to the central plane; so that one of the principal
axes at that point is always normal to the principal central
plane; and thus all axes parallel to a central principal axis are
112.]
179
CRITERIA OF PRINCIPAL AXES.
principal at the points where they intersect a central principal
plane: the other two principal axes are the tangent and the
normal respectively to a line passing through the point, and
which is confocal with the trace of the central ellipsoid of gyra-
tion in that plane.
112.] Let us however determine the general conditions which
are to be satisfied when a line is a principal axis at one of its
points; and let us find its principal point, and the equation to
its principal plane.
Let the centre of gravity be the origin, and let the central prin-
cipal axes be the coordinate axes; let the equations to a certain
line be
Ꮳ
0
y - Yo z
20
= 8, (say);
m
n
= 0.
and let the equation to a plane perpendicular to it be
lx + my + nz — p
p =
(181)
(182)
Now if (181) is a principal axis, and (182) is its principal plane,
(181) is the normal to, and (182) is the tangent plane at the
same point to, a surface of the second degree which is confocal
with the central ellipsoid of gyration.
Let the equation to this confocal surface be
002
y 2
≈2
+
+
1;
(183)
c² + o
a² + 0
as (181) is to be normal to this surface we have
X
1 (a² + 0)
Y
m(b² + 0)
2
(184)
n (c² + 0)
b² — c²
c² - a²
a² - b²
x +
Y +
2= 0;
(185)
M
ԴՆ
which is a condition to be satisfied by the coordinates of the
point on (183), at which the given line pierces the surface.
Also, since from (181),
X
Y
Yo
+ s,
+ 8,
+ S,
(186)
ī
m
m
N
N
(185) becomes
b2 - c²
c²- a²
a² - b²
xo+
Yo+
0,
M
N
or mn(b² — c²) xo+nl (c² — a²) Y。 + lm ( a² — b²) ≈。 = 0;
0
(187)
which is a relation between the elements of the line (181) and
the central principal radii of gyration, which must be satisfied,
when (181) is a principal axis at one of its points.
A a 2
180
[112.
CRITERIA OF PRINCIPAL AXES.
Now (187) deserves consideration, as it expresses geometrical
theorems of great importance. It is the equation to a plane in
terms of xo, yo, 20, if l, m, n are constant; therefore all parallel
straight lines, which are principal axes at some one point on
them, are in the same plane.
Also this plane contains the central radius vector of the cen-
tral ellipsoid which is parallel to the system of lines, and the
normal to the ellipsoid at the point where the radius vector
meets the vector; because (187) is the condition of the coexist-
ence of the three equations
Lxo+Myo+Nz。 = 0,
0 MY
La²l+Mb²m + N c²n = 0,
L l + M M + N N
0 ;
(188)
which express a plane fulfilling the above stated conditions.
Hence also if a straight line is a principal axis, it is parallel
to that central radius vector of the central ellipsoid which is
drawn to the point at which the normal to the ellipsoid meets
the given straight line.
Again, let us consider (187) when x。, Yo, z。 are constant, and
l, m, n vary; let us take the point (xo, Yo, zo) to be an origin,
and on the line (l, m, n) passing through it let us take a point
(x,y,z); so that we may replace l, m, n in (187) generally by x,y,z;
then (187) becomes
0
(b² — c²) x y z + (c² — a²) y。zx + (a² — b²) zxy = 0; (189)
which is the equation of a cone of the second degree. This
shews that all principal axes which pass through the point
(x。, Yo, z。) lie on a cone of the second degree. It is also evi-
dent from the equation (189) that the three lines parallel to the
central principal axes, and the line drawn from the centre of
gravity to the given point, lie on the surface of this cone. This
is an extension of the theorem given by (150).
It is hence evident that all straight lines passing through the
centre of gravity are principal axes at some point on them.
Now the principal point of (181) is its point of section with
the surface (183), at which point also (182) touches the surface;
but, if (182) is the equation to a tangent plane of (183), we have
Z
X
y
1
1 (a² + 0)
m (b² + 0) n (c²+0)
P
lx + my + nz
a² 1² + b² m² + c² n² +0
p
a² 1² + b² m² + c² n² +0
; (190)
112.]
181
PRINCIPAL POINTS.
therefore
and
a² 1² + b² m² + c²n² + 0 = p²;
0 = p² - (a² 1² + b² m² + c² n²).
Also substituting in (190) the values of x, y, z, given in (186),
and eliminating s and 0, we have
mn (b² — c²)
Ρ
ny。-mzo
nl (c² — a²) Im (a² — b²)
(191)
lzo-nxo
mxo-lyo
whereby p is given; and thus is determined; and therefore
the confocal surface of the second degree is determined.
And from (190) we have
X
x =
l (a² + 0)
Ρ
m(b² + 0)
Ρ
m
7
{ p² + a² — a² 1² — b² m² — c² n² },
p
Y
{ p² + b² — u² 1² — b² m² — c² n² },
(192)
Ρ
z =
n (c² + 0)
p
n
{ p² + c² — a² l² — b² m² — c² n² } ;
p
which are the coordinates of the principal point of the line (181).
As p is given in (191), the equation to the plane which is
principal to (181) is
lx+my+nz
mn (b² — c²) nl (c² — a²) Im (a² — b²)
myo-nzo 1z0-nxo
mx-lyo
(193)
If a = bc, this principal plane passes through the centre of
gravity of the system; and the principal point on the axis is
that where the perpendicular from the origin falls on it.
Now of all principal axes passing through the point (xo, Yo, Zo)
and lying on the cone (189), the principal points are the points
of intersection of (193) with the axis (181), so that if we re-
place l, m, n, in (193) by their proportionals x-xo, Y-Yo, Z-Zα,
which are given in (181), we have the equation of a surface on
which these principal points are: this process gives
x(x-x)+y(y-yo)+z(z-z。) =
(y—y。) (z —z。) (b² — c²)
Zyo-yzo
(z—z。)(x − x。)(c² — a²)
XZ0-2x0
(x — x。)(y — Y。)(a² — b²).
yxo-xYo
which are three surfaces of the third degree; any two of which
along their line of intersection give the principal points of all
axes passing through (xo, Yo, 0); or any one of which, together
with the cone (189), determines these points.
182
[113.
PRINCIPAL POINTS.
The lines of the principal points of the system of parallel
principal axes, which lie in the plane (187), is an equilateral
hyperbola, which may thus be found: Let the axis of a lie along
the radius vector of the central ellipsoid so that p = x; and let
the centre of gravity be the origin; then
y² (my。—nzo)² + (lz−nx)² + (mx。—ly)²);
then from (191) we have
r4
xy=
{ (b² — c²)² m² n² + (c² — a²)² n² 1² + ( a² — b²) ² 1² m² } } ;
which is the equation to an equilateral hyperbola, of which axes
of x and y are the asymptotes. Hence it appears that the prin-
cipal points of lines passing through the centre of gravity of a
body are at an infinite distance.
As the relation (187) must be satisfied by the elements of
a line capable of being a principal axis at some one of its points,
it is evident that a line taken arbitrarily may not be a principal
axis at any point.
113.] All lines which lie in a central principal plane are prin-
cipal axes at some point. This is evident, because the condi-
tions (187) will be satisfied in these cases. Thus, let the line be
in the plane of (x, y), n = 0, and z。= 0; and similarly for the
other central principal planes.
It may however be proved, independently of the preceding
Article, by the following process. Whatever is the position of
the line in the plane (say) of (x, y), a conic, confocal with the
conic
x2
a2
y2
+ =
b2
1,
can always be drawn to which this line shall be a normal.
Thus, let the equation to the line be
l x + my
my =
P,
(194)
where 12+ m² = 1. And let the equation to the conic, confocal
with the conic of gyration, be
x2
y2
+
a² + 0 b2+0
1;
(195)
where is to be determined so that (194) may be a normal to
(195); whence we have
lx
my
p
a² + 0
b²+0
a² — b²
a² + 0
+
12
b² + 0) =
m²
-K
;
(196)
114.]
183
PRINCIPAL POINTS.
12 m²
therefore
Ꮎ ;
(a²-b²)² - a² m² - b² 12;
(197)
p2
and therefore from (196)
X =
{ a² m² — b² m² + p²};
(198)
p
M
Y
{ a² 1² — b² 1² — p²};
(199)
p
which determine the principal point in the line (194); and shew
that whatever that line is, it is always a principal axis, and has
consequently a principal point.
0
If p = 0, x = y; that is, if a line in a principal central
plane passes through the centre of gravity, the principal point of
that line is at an infinite distance; a theorem which has been
stated before. If p=0, and 7 = 0, x = Ō, y = 0; thus on the
central axis of x every point is a principal point. A similar
theorem is true of the axis of y.
Also, since through the point (x, y), as defined by (198) and
(199), two confocal conics can be drawn which intersect it at
right angles, the given line (194) will be a normal to one conic, and
a tangent to the other confocal conic. The other principal axis is
perpendicular to the central plane. It is easy therefore to con-
struct the principal axes at a point in a central principal plane ;
through it describe the two conics which are confocal with the
focal conic in that plane, the tangents and the normals to these
two confocal cones are the principal axes.
It is unnecessary to say more as to lines in space which may
not be principal axes at all; the criterion of such lines is, that
their equations do not satisfy the condition (187). I will how-
ever again observe that this fact is evident from this considera-
tion. Let the given straight line be produced to meet one of the
principal central planes, and let the polar of that point be drawn
relatively to the focal conic in that plane; it is evident that the
trace of a plane perpendicular to the given line need not be
parallel with that line. If it is parallel, the original line has a
principal point, and is a principal axis. The condition of pa-
rallelism is expressed by (187).
114.] Although a line in space may not be a principal axis at
all, yet every plane is a principal plane for some point in it, be-
cause, whatever is the position of the plane, it is a tangent plane
to some surface of the second degree which is confocal with the
184
[114.
PRINCIPAL POINTS.
central ellipsoid of gyration. And the principal axis may be
found in the following way. Let us consider the trace of the
plane on one of these central principal planes in which the focal
conic is real; and let this trace be considered a polar relatively
to that conic; let the corresponding pole be then determined, and
from it let a perpendicular be drawn to the given plane; that
perpendicular is evidently the principal axis of the plane, and
the point of intersection of it with the plane is the principal
point of the plane. But we may investigate these results ma-
thematically, by the process which we have followed in Art. 112.
Let the equation to the plane be
lx + my + nz = p,
(200)
where 12+ m² + n² = 1; and let (§, n, ) be the point where this
plane touches the surface
5)
£2
n²
2ع
+
+
1;
(201)
a² +0
b² + o
c² + 0
of which the tangent plane at (§, n, () is
1;
(202)
ξυ
ny
+
+
Cz
a² + 0 b2+0 c² + o
and as (200) and (202) are identical, we have
l(a² +0)
&
m(b²+0)
n
ميد
n(c² + 0)
p
१
(203)
= { 1² (a² + 0) + m²(b² + 0) + n² (c² + 0)}*; (204)
0 =
p² — (a² 1² + b² m² + c² n²) ;
Ρ
m
{ p² + a² — (a² 1² + b² m² + c²n²) },
(205)
η
{p² + b² - (a² 1² + b² m² + c² n²)},
(206)
p
१
n
p
{ p² + c² — (a² 1² + b² m² + c² n²) } ;
so that (205) gives the particular ellipsoid surface which is con-
focal with the ellipsoid of gyration; and (206) assigns the prin-
cipal point in the plane, and is that point at which the plane
touches the surface (201).
Hence it appears that if a plane is given, a confocal surface
can be assigned which shall be touched by that plane; and also
the point of contact can be determined, and this is the principal
point of the plane.
Let the trace of (200) be taken on the plane of (x, y); then, if
115.]
185
MOMENTS OF INERTIA.
it is considered a polar relatively to the focal conic in that
—
plane, the pole is (a² — c²
p
b² - c²
1, m); and therefore the equa-
p
tions to the axis, which is principal to the plane (200), are
غ
a² - c²
b² - c²
し
​η
m
p
p
M
!!!
n
or, as they may be expressed,
&
a²l
p
b² m
c² n
η
p
P
n
m
(207)
Hence every plane is a principal plane at some one point
of it. A central principal plane is a principal plane for every
point of it, because every axis which is perpendicular to a central
principal plane is a principal axis with its principal point in the
central principal plane.
Also the plane at an infinite distance is a principal plane at
every point of it; and all the corresponding principal axes pass
through the centre of gravity. The three central principal planes
and the plane at infinity alone have this property, that every
point in them is a principal point.
SECTION 4.-Examples of moments of inertia.
115.] In this section I propose to apply the general formulæ
of the preceding section to the calculation of moments of inertia
and radii of gyration, relatively to certain given axes, of material
lines or wires, of thin plates and curved shells, and of solid
bodies. It will be found most convenient to make the calcula-
tions with reference to certain axes to which the bodies are
geometrically related, and which yield the most simple forms of
integration. And by means of them, and the theorems of the
preceding section, to investigate the moments of inertia about
the given axes. The following theorems are most useful for the
purpose.
(1) If at any point of a body A, B, C are the principal moments
of inertia, and H is the moment of inertia about the axis (a, ß, y)
passing through that point, then
and if
H = A (COS α)² + B (COS ẞ)² + C (cos y)² ;
A' = Σ.mx², B' = x.my², c = z.mz²,
H = A′(sin a)² + B′ (sin ß)² + c´ (sin y)².
B b
PRICE, VOL. IV.
(208)
(209)
娠
​186
[116.
MOMENTS OF INERTIA.
(2) If н and н' are the moments of inertia of the mass м about
two parallel axes, one of which passes through a given point, and
the other passes through the centre of gravity; and if h is the
distance between these axes, then
H = H' + м h²;
(210)
and therefore if k and k' are the radii of gyration about the
axis through the given point, and the parallel axis through the
centre of gravity respectively, then
H
M k²,
H' = M/'2;
k² = k'² + h².
..
(211)
116.] The moments of inertia of material lines or wires.
Ex. 1. The moment of inertia of a straight wire of uniform
thickness and density.
o
Let the length of the wire be 2a, p its density, ∞ = the
area of a transverse section; and let it lie along the axis of x.
(1) Let the rotation-axis be perpendicular to its length, and
pass through its middle point; then
α
the moment of inertia
=
[° pwx² dx
2
a
pwa³.
3
(2) Let the rotation-axis be perpendicular to the wire, and at
a distance c from the middle point which is its centre of gravity;
then, by (210), since the mass of the wire = 2pwa,
2
the moment of inertia
pwa³+2pwac².
3
Hence, if an equilateral triangle is formed of a wire whose
length is 6a, the moment of inertia relatively to an axis passing
through the centre of gravity of the triangle and perpendicular
to its plane is 4pwa³.
(3) Let the rotation-axis be perpendicular to the wire and
pass through one of its ends; then
2a
the moment of inertia
[² pw x² dx
8pw a³
3
(4) Let the rotation-axis intersect the wire in its middle point
at an angle a; then
the moment of inertia =
3
૫ ૭
2
2 8
pwa³ (sin a)².
116.]
187
MATERIAL LINES.
Ex. 2. The moment of inertia of a wire of uniform thickness
and density whose form is a circular arc.
Let p = w
= the density, ∞ = the area of a transverse section,
a = the radius of the circle, 2a = the angle subtended by the
arc at the centre of the circle.
(1) Let the rotation-axis pass through the centre and be per-
pendicular to the plane of the arc; then
a
the moment of inertia = (2ª pwa³d0
= 2pwa³ a;
and therefore the moment of inertia of a complete circular wire
about an axis which passes through its centre and is perpen-
dicular to its plane is 2πpwa³.
(2) Let the rotation-axis be perpendicular to the plane of the
wire and pass through its middle point; then
2ax-x²;
dx
y²
dy
ds
α X y
a
..
⚫ the moment of inertia
=
= √ pw
w(x² + y²) ds
£0
= 4pwas {a-sina};
Απρωα.
and the moment of inertia of a complete circle 4πpwa³.
(3) Let the rotation-axis be in the plane of the wire and pass
through the centre and its middle point; then
α
the moment of inertia
[ª
α
pwa³ (sin 0)² do
= pwa³ {a—sina cosa};
and therefore the moment of inertia of a complete circular wire
about its diameter is πρωα.
(4) Let the rotation-axis pass through the centre of a com-
plete circular ring, and be inclined at an angle y to the plane of
the circle; then, by (208),
the moment of inertia pwa³ (cos y)² +2πрw a³ (sin y)2
π
= πρωα {1+ (sin γ)}.
Ex. 3. A wire of uniform thickness and density, whose length
is a, is bent into the form of a complete cycloidal arc: the mo-
ment of inertia of it about a rotation-axis which joins its two
pwa³
ends is
30
In each of the preceding examples the mass of the wire can
be easily found: and as the square of the radius of gyration is
B b 2
188
[117.
MOMENTS OF INERTIA.
the moment of inertia divided by the mass, so the radius of
gyration can be found without difficulty.
If the wire lies wholly in one plane, say in the plane of (x, y),
that plane is a principal plane of it; because in this case z = 0
for all elements of it; and therefore z.mxz = Σ.my z
0, and
the axis of≈ is a principal axis. The other two principal axes
must be found by the process of Art. 90.
=
117.] The moment of inertia of thin plates and of curved
shells.
T
In all cases we shall assume the thickness of the plates and
shells to be infinitesimal, and to be represented by the symbol 7 ;
and thus, if it is convenient, we shall take the plate-plane to be
the plane of (x, y); in this case, as z = 0 for all elements of the
plate, z.mxz = z.myz = 0, and the plane of (x,y) is a principal
plane and the axis of z is a principal axis. The other principal axes
will be found by the method of Art. 90; and the principal mo-
ments of inertia having been determined, the moment of inertia
about any other axis may be determined by means of the theo-
rems given in (208) and (210).
༧
Also, since z = O, the moments of inertia about the axes of x
and y are respectively z.my² and z.mx2; and as x.m (x² + y²) is
the moment of inertia about the axis of z, it follows that the
moment of inertia about an axis perpendicular to the plate is
equal to the sum of the moments of inertia about any two axes
at right angles to each other in the plate.
If the axes of coordinates are principal axes, from (208) we
have
H = A (COS α)² + B (COS ẞ)² + (A + B) (COS Y)²
a
2
= A{(cosa)²+(cos y)2} + B {(cos B)2 + (cos y)2}
= A (sin ẞ)² + B (sin a)²;
(212)
and if the rotation-axis is in the plane of (x, y), sin ẞ = cos a; and
2
H = A (COS a)² + B (sin a)².
α
Ex. 1. The moment of inertia of a square plate.
(213)
Let a = the side of the plate, p = the density at the point
(x, y).
(1) Let the rotation-axis pass through the centre of the plate
and be perpendicular to its plane; then
a
2
the moment of inertia
PT (x²+ y²) dy dx
-7
ρτας
6
a
117.]
189
THIN PLATES AND SHELLS.
(2) Let the rotation-axis be the line joining the middle points
of two opposite sides; then
α
the moment of inertia
=L₁₂
2
Sr. Spry² dy dx
ρτας
12
-17
(3) Let the rotation-axis pass through an angular point of the
plate, and be perpendicular to its plane; then
the moment of inertia
2ρτας
3
(4) Let the rotation-axis pass through the centre of the plate;
and let its direction-angles, with reference to two lines bisecting
the opposite sides of the plate and the perpendicular through its
centre, be a, ß, y; then, as these lines are the principal axes of
the plate,
the mom. of in.
ρτας
12
ρτας
{(cosa)² + (cos ẞ)2} +
6
(cos y)2
ρτας
12
ρτας
6
Also, if the rotation-axis is the diagonal of the plate, y = 90°,
and
(sin y)2 + (cos y)2.
the moment of inertia
ρτα
12
As this is a case in which the two principal moments of inertia
in the plane of (x, y) are equal, and the third is greater than each
of them, two points on the axis of, which are at distances
from the origin, which is the centre of gravity of the plate, (see
Art. 110), equal to
A2
M
+234
α
6
are such that at them the principal moments of inertia, and
therefore all the moments of inertia, are equal. At these points.
the momental ellipsoid becomes a sphere.
Ex. 2. The moment of inertia of a triangular plate.
(1) Let the triangular plate be isosceles; and let the rotation-
axis pass through its vertex and be perpendicular to its plane;
let a = the altitude, 26 the base; then
the moment of inertia = 2 ["
bx
α
a
pт (x² + y²) dy dx
0
ρταό
6
(3 a² + b²).
190
117.
MOMENTS OF INERTIA.
(2) Let the triangular plate be isosceles; and let the rotation-
axis be the line which passes through the vertex and bisects the
base; then
bx
α
α
the moment of inertia
2
pту2 dy dx
0
ρτα
6
(3) Let the triangular plate be that whose sides and angles
are a, b, c, A, B, C; and let the rotation-axis pass through c and
be perpendicular to the plane of the plate; let c be the origin,
and let the lines lying along the sides a and b respectively be
the axes of x and y; so that the equation to the side c is
X y
+
1;
and let
a
Y =
b
b
(a− x);
a
a
the mom. of in.
[ª [* p r (x² + 2 x y cos c + y²) dy da sin c
τα
24
(3a²+3b² — c²) sin c.
(4) Let the triangular plate be that of the preceding case;
and let the rotation-axis pass through the centre of gravity of
the plate and be perpendicular to its plane; then, if & is the
2a²+262-c²
; and the mass of the
9
; therefore, by reason of (210),
centre of gravity, (CG)² =
prabsinc
2
ρταό
24
plate
mom. of in.
(3a²+3b2-c²) sinc-
ρταό
18
(2 a²+2b²-c²) sin c
τα
(a²+b²+ c²) sinc;
72
and therefore, if k is the radius of gyration relative to a rota-
tion-axis passing through the centre of gravity of a triangular
plate and perpendicular to its plane,
a² + b² + c²
k² =
36
k2
Ex. 3. The moment of inertia of a circular plate, and of a cir-
cular annulus.
Let the radius of the plate = a; and let p and r express the
same quantities as heretofore.
117.]
191
THIN PLATES AND SHELLS.
(1) Let the rotation-axis pass through the centre and be per-
pendicular to the plane of the plate; then
2π α
the moment of inertia
= [" ["prv³ drde
πρτα
2
(2) Let the rotation-axis pass through the circumference and
be perpendicular to the plate;
then, by (210),
πρτας
the moment of inertia
+ πρτα
2
3πρτα
2
(3) Let the rotation-axis be the diameter of the plate; then
the moment of inertia
*2π α
[** [ª prr³ (sine)² dr de
πρτα
4
(4) Let the rotation-axis be a tangent to the plate; then,
by (210),
the moment of inertia =
5 ποτα
4
(5) Let the interior of the circular plate be removed, so that
the remainder is a circular annulus, the radii of the exterior and
interior bounding circles of which are a and b: then the moment
of inertia relative to a rotation-axis passing through the centre
of the annulus and perpendicular to its plane is
πρτα - 64)
2
Also the moment of inertia of the annulus relative to its diameter
is
a
πρτ (α* — 64)
4
Ex. 4. The moment of inertia of an elliptical plate.
Let the equation to the bounding ellipse be
and let
then
x2
y/2
+
a2
b2
1 ;
b
Y =
(a² — x²) è̟.
a
(1) Let the rotation-axis be the major axis of the ellipse;
<<
術
​192
MOMENTS OF INERTIA
[117.
the moment of inertia
=
k
4 pт [" [ˇ y³ d y d
Αρτ I dy
T
0
a
x
4 p + b³ [ " (a² - x² ) + dx
—
3 a³
πρτα 3
4
(2) Let the rotation-axis be the minor axis of the ellipse; then
the moment of inertia = 4pr[" ["
πρτα
4
Y
x² dy dx
(3) Let the rotation-axis be a line perpendicular to the plane
of the plate and passing through its centre; then
r
Y
the moment of inertia = 4pr[ " [" (x² + y²) dy dx
ποτα
4
(a² + b²).
(4) Let the rotation-axis pass through the centre of the plate
and make angles a, ß, y severally with the major axis, the minor
axis, and the perpendicular to the plate through its centre; then,
as these are the principal axes of the plate, we have, by (208),
the mom. of in.
πρταό
4
{b² (cosa)² + a² (cos ẞ)² + (a² + b²) (cos y)²}
πρτα
4
{a² (sin a)2+ b² (sin 3)2}.
(5) Let the rotation-axis be a central radius vector r of the
plate, making an angle a with the major-axis; then, from the
last result, as a + 6 = 90°, we have
ß
the moment of inertia
but by the equation to the ellipse
ποτα
4
{a² (sin a)2+b² (cos a)2};
α
a² (sin a)2+b² (cosa)2 =
2.2
a2b2
...
the moment of inertia
πρτα33
42.2
Ex. 5. The moment of inertia of a spherical shell of radius a
8
T
and thickness about its diameter ==πpгα¹.
3 πρτα.
118.]
193
SOLID BODIES.
118.] The moment of inertia of a solid body bounded by a
surface of revolution relative to its geometrical axis as its rota-
tion-axis.
Let the axis be that of x; and let the equation to the curve,
by the revolution of which about the axis of x the bounding
surface is formed, be y f(x).
Let the solid be divided into a series of circular plates by
planes at an infinitesimal distance apart and perpendicular to
the axis of revolution; let the density be uniform and be p;
then, at the distance x from the origin, y is the radius of a cir-
cular plate whose thickness is da; and therefore, by Ex. 3,
Art. 117, the moment of inertia of this circular slice, relative to
an axis passing through its centre and perpendicular to its plane,
πρydr
2
is
πρ
2
T½ {f(x)}+ dx;
and therefore, if x and x, are the limits of x,
n
Пр
the moment of inertia = πP ** {ƒ (x)}* dx.
2 xo
Ex. 1. The moment of inertia of a cylinder.
Let the altitude of the cylinder a, and the radius of the
base =
b; therefore
the moment of inertia =
πρασ
2
Ex. 2. The moment of inertia of a cone; let the altitude
and the radius of the base = b; then
=
= α,
the moment of inertia =
про ра
201
2 a
x¹ dx
πραό
10
Ex. 3. If a = the altitude, and b = the radius of the base of
a paraboloid, then
the moment of inertia = πραγ
6
Ex. 4. If.a = the radius of a sphere, then relatively to a
diameter as the rotation-axis,
the moment of inertia =
8 πρα
15
Hence the moment of inertia of a spherical shell contained
PRICE, VOL. IV.
сс
194
[119.
MOMENTS OF INERTIA.
between two concentric spheres whose radii are a and b respect-
ively, relatively to the diameter as the rotation-axis,
8 πρ(α –65)
15
Ex. 5. The moment of inertia of a prolate spheroid relatively
to its axis as the rotation-axis =
πρασ
15
Ex. 6. The moment of inertia of an oblate spheroid, whose axis
is the rotation-axis,
πρα
15
Ex. 7. If the radius of each surface of an equiconvex lens is a,
and the thickness of the lens is 2t, then the moment of inertia
of the lens relative to its axis as the rotation-axis
# p [ ' ( 2 a x − x²)² d x
#pt3
15
-
(20a2-15 at +3t²).
119.] The moment of inertia of a solid body bounded by a
surface of revolution relative to an axis perpendicular to its geo-
metrical axis.
Let the point in which the rotation-axis intersects the axis of
revolution be the origin; and let y = f(x) be the equation of
the generating curve of the bounding surface; then, using the
notation of the preceding Article, and applying the result of
Ex. 3, Art. 117, the moment of inertia of the type-slice relative
to its own diameter
πpy+dx
4
;
and therefore by (210) the moment of inertia of this slice about
the actual rotation-axis is
πру+ dx
4
+ πру²x² dx;
and if x and x are the limits of the x-integration,
n
the moment of inertia = πР
xn
[** ( 2² + y²x²) dx.
200
4
Ex. 1. The moment of inertia of a cone relative to a rotation-
axis passing through its vertex and perpendicular to its own
axis.
Let the altitude of the cone a; let the radius of the base
b; then
120.]
195
SOLID BODIES.
the moment of inertia
πρ
of
a b4
b2
+
4a4
a2
:) x* dx
πραγ
(4a²+b²).
20
It is evident that relative to the vertex of a cone the principal
axes are the axis of the cone and any two lines perpendicular to
each other and to the axis of the cone. So that the moment of
inertia relative to a rotation-axis passing through the vertex of
the cone and inclined at an angle a to the axis
πραγ
20
(4a² + b²) (sin a)² +
πραγ
10
(cosa)².
Ex. 2. The moment of inertia of a cone of which the altitude
= a, and the radius of whose base = b, relative to a rotation-
axis passing through its centre of gravity and perpendicular to
its own axis,
πραγ
80
(a² +46²).
Ex. 3. If the altitude of a paraboloid of revolution is a, and
the radius of the base b, the moment of inertia relative to a
rotation-axis passing through its vertex and perpendicular to its
own axis
πραγ
8
(b²+2a²).
Ex. 4. If the altitude of a cylinder is a, and the radius of its
base = b; and if the rotation-axis is perpendicular to the axis,
and at a distance c from its end, then
the moment of inertia
= ["
fate (
a+c
προ
4
+ προ222) da
πραγ
πραγ
+
4
(a²+3ac+3c²).
3
Hence, if the rotation-axis passes through the end of the axis,
the moment of inertia =
πραγ
12
(3b²+4a²);
and if the rotation-axis passes through the middle point of the
axis of the cylinder,
ર
the moment of inertia
πρό
4
£
+ πp b²x²) dx
-
πραγ
(a²+3b²).
12
120.] The moment of inertia of various solid bodies.
Ex. 1. The moment of inertia of a rectangular parallelepipedon
about an edge.
CC 2
196
[120.
MOMENTS OF INERTIA.
Let the edges be a, b, c; and let the lines which coincide with
the edges be the axes of x, y, z respectively; let the density = p;
then the moment of inertia relative to the edge a
a b C
= S"S" S
pabc
3
p (y² + z²) dz dy dx
(b² + c²);
and symmetrical values are of course true for the moments of
inertia relative to the edges b and c.
Thus the moment of inertia of a cube whose side is a,
to one of its edges as a rotation-axis,
2 pa5
3
relative
Ex. 2. The moment of inertia of a cube relative to a diagonal.
Let the side of the cube be a; and let the centre of the cube
be the origin, and let the three lines which pass through the
centres of the opposite sides be the coordinate axes; these lines
are evidently principal axes; and relatively to either of them
the moment of inertia
p (y²+ z²) dz dy dx
ραδ
6
;
and as the moment of inertia is the same for each of these prin-
cipal axes, it is the same for every axis passing through this
point; thus, the central ellipsoid is a sphere, and all its radii
vectores are equal; and therefore relative to the diagonal of the
cube,
ραδ
the moment of inertia
6
Ex. 3. The moment of inertia of an ellipsoid.
Let the equation to the ellipsoid be
x2 y2 22
+ +
a2 b2 c²
1.
The axes of the ellipsoid are evidently the principal axes of the
body; so that when the moments of inertia relative to these
axes are determined, that about any other axis may be found
from (208).
Let
Now
{1-
x2
a²
y²
b2 S
2.ma² = 8 [ "
11
8/
= 2,
x2
-Y.
a2
[" ["pa² d= dy dœæ
S
S
Απρ
a³ b c
15
dz dx
121.]
197
SOLID BODIES.
4π pab³ c
Απρα
В с ³
similarly
Σ.my 2
Σ.m z² =
15
15
4πραbc
· A = Σ.m (y² + z²)
(b² + c²),
15
4 πp a b c
B = Σ.m(z² + x²)
(c² + a²),
15
4πρα b c
c = x.m(x² + y²) =
(a² + b²);
15
and therefore the moment of inertia relative to the axis (a, ß, y)
4 π ρ a b с
15
4π р a b c
15
{(b² + c²) (cosa)² + (c² + a²) (cosß)² + (a² + b²) (cos y)2}
{a² (sin a)² + b² (sin ß)² + c² (sin y)²}.
>
Ex. 4. If in the preceding example a = b, and a is c, the
ellipsoid becomes an oblate spheroid, and
A B =
4πρας
15
(a² + c²),
πραξε
C =
15
Therefore, by Art. 110, at two points on the axis of all the
moments of inertia are equal, and at them the momental ellipsoid
becomes a sphere: the distances of them from the centre
C
(c-A) /
જે
= +
M
a² - c²
5
W[-
= ±
and if these points are at the poles of the spheroid, a² = 6c2.
121.] From the preceding results the moments of inertia of
many curved shells and of systems of thin plates may be de-
duced.
For if the equation of the bounding surface of the solid con-
tains a single parameter, by the infinitesimal variation of that
parameter, the content of the solid will receive an infinitesimal
variation in the form of a thin shell, the thickness of which will
be the variation of the parameter. Thus, if the radius of a
sphere is increased by an infinitesimal variation, say dr, the con-
tent will be increased by a spherical shell of thickness dr. Simi-
larly, if a solid is increased by the variation of the parameter on
which the bounding surface depends, the moment of inertia of
198
[121.
MOMENTS OF INERTIA.
SOLID BODIES.
that increase is the increase of the moment of inertia of the
solid; and the former is generally a thin shell or a system of
thin plates, so that the moment of inertia of these may be deter-
mined by the variation of the moment of inertia of the solid.
Thus by the preceding Article the moment of inertia of a
ραδ
6
cube about a diagonal is ; let the edge of the cube be in-
creased by da; then all the sides of the cube receive increments
in the form of thin plates, the thickness of which
da = 7, say;
and therefore the moment of inertia of the hollow box, formed
by these six plates relative to a diagonal
5ρατ
6
10 par
3
Similarly, by reason of Ex. 1 in the preceding Article, the
moment of inertia of the box relative to an edge
As the moment of inertia of a sphere relative to a diameter is
so that if a spherical shell of thickness 7, relatively to
15
8 πρτα
the same rotation-axis, =
3
πρασ
As the moment of inertia of a cylinder, relative to its own
πραγ
2
axis as rotation-axis, is
so the moment of inertia of a
cylindrical shell whose thickness is db r, is, relatively to its
own axis, 2π ρταό3.
=
In all the preceding examples we have calculated moments of
inertia; and as the masses of the rotating bodies may be found
in all the cases, the corresponding radii of gyration can be deter-
mined without difficulty.
122.] A FIXED ROTATION-AXIS. INSTANTANEOUS FORCES. 199
CHAPTER V.
THE ROTATION OF A BODY ABOUT A FIXED AXIS.
SECTION 1.-The rotation of a rigid body about a fixed axis under
the action of instantaneous forces.
122.] In the last two sections of the preceding Chapter we
have considered that part of our subject which has been called
the Geometry of Masses: it has indeed nothing directly me-
chanical in it, but the theorems which have been proved are
useful and necessary on account of the form which the process
of transformation into angular velocities has given to the equa-
tions of rotatory motion. We come now to the consideration of
the most simple case of dynamics proper; that, namely, in which
a rigid body under the action of given forces revolves about an
axis fixed in it and in space. Every particle of the body thus
moves in a circle, the plane of which is perpendicular to the ro-
tation-axis, and the centre of which is in that axis.
We shall suppose the form, matter, and density of every part
of the moving body or system to be given; and we shall sup-
pose the body to be capable of an unfettered rotation about the
axis. This axis may be fixed at many points, or, in the language
of machinery, may have many bearings; we shall however sup-
pose that it has only two fixed points; because these are suffi-
cient to fix the axis; and if there are more, the pressures become
indeterminate at them both in intensity and in line of action.
We shall indeed find that even in the case of two points, the
components of the pressures on the fixed points along the rota-
tion-axis are indeterminate. We have already had a similar in-
stance in Art. 62, Vol. III.
Let us in the first place consider the circumstances of rotation
of the body, when it is acted on by instantaneous or impulsive
forces; that is, we shall investigate the resulting angular velo-
city of the body, the pressures on the fixed points, and their
incidents, which are due to one or more blows impressed at given
points of the body. To simplify the formulæ, we shall generally
assume the body to be at rest when the impulsive force acts,
although the results will be equally applicable if the body is
moving with a given angular velocity.
200
[123.
A FIXED ROTATION-AXIS.
?
123.] Let the rotation-axis, on which are the two fixed points,
be the axis of 2; and let the two fixed points be at distances 21,
22 from the origin; let the pressure at these two points be P1, P2,
and let the direction-angles of the lines along which they re-
spectively act be (a1, B1, 1), (a2, B2, V2). Let m be the type-par-
ticle, and let (x, y, z) be its place at the time t, when the instan-
taneous force acts on it; let mv be the momentum impressed by
this force, of which let the axial components be mvx, mvy, MVz;
dx dy dz
let
be the components of the actual velocity (or in-
dt dt dt
crease of velocity) with which m moves in consequence of this
instantaneous force; all these being type-expressions, and there-
fore applicable to each particle on which forces act. Thus the
equations of motion, (34) and (35), Art. 48, become
Σ.Μ
dx
Σ.mVx
(Vx — da)
m ( v
dt
) — Þ¸ cosa¸ — P₂ cos a₂
P2
=
0,
(1)
0;
z.m (v, — dy) — P, cos ³¸ — ¹, cos³½ = 0,
Σ.Μ
dt
x.m(v₂ — dz)
–
1
—
2
- P1 COS Y1 - P₂ COS Y₂ =
{y (v. — dz) — z (v, — dy) }
dt
dx
{z (v - de)
Vx
12
+≈₁ P₁ cosß₁ + Z2 P₂ Cos ß2
1
=
0,
Σ.Μ
x ( v z
dz
-1 P1 Cos a₁-Z2 P₂ COS A2
=
2
= 0,
(2)
dt
Σ.Μ
ই
z.m {a (v, dy) —y
Y
dt
(v,da)}
dx
0 ;
dt
dt
Let us express these equations, as in Art. 73, in terms of angular
velocities. Let o be the angular velocity which results from the
instantaneous forces; then, as its rotation-axis is the axis of z,
and as there is no motion parallel to the axis of z,
dx
dt
DY,
dy
dt
Qx,
dz
= 0.
dt
(3)
Let the moments of the axial components of the couple of the
impressed momenta be L, M, N; then (1) and (2) become
E.mVx+QE.my - P₁ cos a1-P2 COS a2 = 0,
Σ.mvy — Qz.mx-P₁ cosẞ₁-P₂ cos B₂ = 0,
Vy
E.IN V
z
-P₁ COS Y₁- P₂ COS Y2
COS
0
L+QZ. MZ X + P, cos ẞ11+ P₂ cos B22 = 0,
L+QE.MZX+ P1
(4)
(5)
= 0 ;
M+QZ.MYZ — P, cos a, 2-P₂ cos a2z = 0,
Z― α11 2
N-NΣ.m (x² + y²)
124.]
201
INSTANTANEOUS FORCES.
which six equations assign the incidents of motion, and the pres-
sures on the two fixed points.
These equations admit of dissection by means of first prin-
ciples, in a manner similar to that which has been employed in
Art. 81 and 82. As 2 is the expressed angular velocity about
the axis of z, or is the expressed velocity of m at a distance r
from that axis; and mor is the expressed momentum; the x-
and y- axial components of which are may and max. Let
us introduce pairs of momenta equal and opposite to these at
the origin and in the plane of (x, y) at the foot of the z-ordinate of
m; then the momentum mor of m at the point (x, y, z) is equi-
valent to (1) a momentum may acting at the origin and along
the axis of x; (2) a momentum mox also acting at the origin along
the axis of y; (3) three couples -mozx, -mayz, mQ(x² + y²)
whose axes are respectively the coordinate axes of x, y, and z;
and a similar result is true for every element of the body. Now,
by D'Alembert's principle, the sum of all these expressed mo-
menta, together with the pressures at the fixed points, are in
equilibrium with the impressed momenta; and the conditions
requisite for the equilibrium are evidently the six equations (4)
and (5). We have hereby an intelligible meaning of their several
terms. We proceed to deduce from them the value of the an-
gular velocity which results from the impressed forces, and the
pressures on the fixed points.
124.] The angular velocity is given by the last equation of
(5), and we have
Ω
N
Σ.m (x² + y²)
N
Σ.mr2
The moment of the impressed momenta
The moment of inertia
;
(6)
which is the same result as (16), Art. 73. It appears therefore
that the resulting angular velocity does not depend on the pres-
sures at the fixed points, or on the distance between them, but
only on the moment of the impressed momenta, and on the mo-
ment of inertia of the body or system. It is also the same
whatever is the number of the bearings. And if no force ex-
ternal to the system acts, the system continues to rotate uni-
formly with this angular velocity.
PRICE, VOL. IV.
D d
202
[124.
A FIXED ROTATION-AXIS.
Now let us suppose a body capable of rotating about a fixed
axis to be at rest, and let us suppose it to be struck by a blow
of given momentum at a given point and in a determinate line :
we must first resolve the blow into two parts, of one of which the
line of action shall be parallel to the rotation-axis, so that the
angular velocity will not be affected thereby, for it will only pro-
duce pressures at the fixed points along the rotation-axis; of
the other, let the line of action be in the plane of (x, y) which is
perpendicular to the rotation-axis; let the momentum of this
latter be q, and let a be the perpendicular distance from the
axis on its line of action; then (6) becomes
Ω
Q a
The moment of inertia
The following are examples of this equation.
(7)
Ex. 1. A body м at rest, and capable of moving about a fixed
rotation-axis, is simultaneously struck by several masses m, m2,
mn, moving with velocities v₁, v2, vn in planes perpen-
dicular to the fixed axis; the masses adhere to the body: it is
required to find the angular velocity of the body.
Let the distances of the points of impact of the masses sever-
ally from the rotation-axis be 4, 42, ; and let P1, P2, P₂
..Pn
be the perpendiculars from the rotation-axis on the lines of the
velocities v1, v2, ...... vn; then, if k is the radius of gyration of
the body relative to the rotation-axis,
Ω
MĮ V 1 P1 + M 2V2 P2 +
м k² + m₁ l₂² + m₂ 12² +
1
Σ.mvp
2
м k² + x. m l² °
+ Mn Vn Pn
+ Mn ln²
"n
2
Ex. 2. A body м revolving about a fixed axis with an angular
velocity 2, is struck by a particle m, moving with a velocity v in
a line perpendicular to the plane containing the rotation-axis
and the point of impact; it is required to determine the result-
ing angular velocity of the rotating body, the velocity of rebound
of the striking particle, and the place of percussion when the
velocity of rebound is a maximum, the elasticity of the body
and particle being e.
Let мk² be the moment of inertia of the body relative to the
rotation-axis; p = the distance of the point of impact from the
axis; the angular velocity of the body after collision; v =
n'
the velocity of m after rebound; and let us suppose 2 and v to
=
124.]
203
INSTANTANEOUS FORCES.
be such that the motion of m and of the point of impact may be
in the same direction at the instant of collision.
Let v' be the velocity of that point of м at which the impact
takes place; so that
v
ΩΡ;
and let m' be the mass of a particle which, moving with the
velocity v', would produce the same circumstances of velocity
&c. in m after impact on m', as the rotating body м; so that
m'v' is the momentum with which м would strike a body at the
point of impact of m, and in the line of m's motion: therefore
by (6)
M k² Q
m'v'
m' =
P
M k²
p2
Let v' = po' be the velocity of the point of impact after colli-
sion has ceased; then, by (8) and (9), Art. 215, Vol. III,
mp²v + Qpмk² — e м k² (v — po)
M
eм
m p² + M k²
V
V
mp²v + Qpмk² + mep² (v — pQ)
;
mp² + Mk2
mpv+uke+mep (v−pe)
Ω'
mp² + Mk²
Ω-Ω
mp(1+e) (v − p Q)
mp² + Mk2
;
V-V
Mk2 (1+e) (v-po)
mp² + Mk²
;
whereby we know the velocity of m after collision and the an-
gular velocity of м.
Thus, let м be a cricket bat, and m a ball; let us suppose the
ball to meet the bat; then the sign of v must be changed; and
if v = the velocity of rebound of the ball,
V
м k²ap —mp²v + eмk² (v + Qp)
mp² + мk²
and to determine the point of impact so that v may be a maxi-
mum, the p-differential of v must be equated to zero; whereby
we have
ย
v2
Mk2)
Ρ
+
+
Ω
Ω
m
If m is at rest when it is struck by м, v =
0, and
p = k (
M
M
鸞
​D d 2
204
[125.
FIXED ROTATION-AXIS.
Again, let м be a rectangular plate whose sides are a and b,
and let the rotation-axis lie along the side a: let us suppose it
to be at rest and to be struck by m at a point on the side op-
posite to the rotation-axis; then мk² =
Q'
3mv (1 + e)
(3m + M) b
M b2
3
; and
125.] In the next place let us consider the pressures on the
two fixed points of the axis; the x- and y- components of P₁
and P₂ can be determined from the first two of (4), and from
the first two of (5); and we have
M + Z₂Σ.MVx+Q (z₂Σ.my — Σ.myz)
P1 cos α1
(8)
Z2-21
L+ ≈₂ Σ . MV y + Q ( − Z₂ Σ . M x+x.mxz)
P₁ cos B₁
(9)
22-21
P₂ COS α2
(10)
P₂ COS B₂ =
;
(11)
M−%₁E.MVx+Q ( − z₁Σ.my +Σ.myz)
22-21
—L — 2₁E.M Vy + Q (z, z. m x — Σ.mxz)
L-Σ.mv+ ΩΣ.ma
Z9-21
ལ
whereby the components of the pressures which are parallel to
the plane of (x, y) may be determined.
P1
P2
The x-components of P₁ and P₂ enter into only the third equa-
tion of (4), and we have
P₁ COS Y1 + P₂ COS Y₂ = .MV z ;
1
(12)
therefore the sum of these z-components of the pressures is equal
to the sum of the z-components of the impressed momenta; but
as the sum only is given, each is indeterminate. An explanation
of this indeterminateness has been already made in Art. 62,
Vol. III: this is the dynamical case, which is therein alluded
to. And we are unable to determine the pressure which acts
at each fixed point.
To give greater clearness to our ideas, let us suppose the
impressed momenta to arise from a single blow, whose mo-
mentum is Q, say, and whose line of action is in a plane perpen-
dicular to the rotation-axis. Let us take the rotation-axis to be
the z-axis, and the plane perpendicular to it, and containing the
line of the blow, to be the plane of (x, y); let the axis of y be
parallel to the line of the blow; and let a be the distance be-
tween these two lines: then the equations of motion are
125.]
205
PRESSURE ON THE AXIS.
Qx.my-P₁ cos a₁-P₂ cos a₂ = 0,
Q-Qz.mx-P₁ cos B₁-P₂ cos B₂ = 0,
1
2
-P₁ COS Y1-P2
P2 COS Y2
(13)
0;
(14)
Qx.mzx+P₁ cos ß, 21+ P₂ cos B₂%20,
Qz.mzy - P1 COS α1 21-P2 COS A2Z2
Qa − Qɛ.m (x² + y²)
=
0,
= 0.
Let м the mass of the body or system of particles; and let k
M =
be the radius of gyration relative to the rotation-axis; then
from the last of (14)
Ω Ξ
Qa
Mk2
(15)
Let the centre of gravity be (x, y, z) when the blow is given; then
Ω Σ.my =
Qay
k2
ΩΣ.ΜΧ =
Q X X
k2
and the values of the axial components of the pressures may be
determined.
Now these pressures will compound into a single resultant
when
-QΜуΣ.mxz + (Q — QMπ) Σ.myz = 0,
that is, when · ayɛ.mx z + (k² ax)ɛ.myz = 0;
(16)
and this condition is satisfied when the rotation-axis is a prin-
cipal axis, and the line of action of the blow is in its principal
plane; and if R is the single pressure,
R =
k2
{
k¹ — 2 añ k² + a² (x² + y²) }
;
when (16) is satisfied, one point is sufficient to fix the axis.
Hence, if the axis of rotation is a central principal axis, R = Q ;
and evidently acts at the centre of gravity.
Let us apply these results to one or two examples.
Ex. 1. A thin rod of length a revolves with an angular velo-
city about an axis passing through its end and perpendicular
to its length; it is suddenly stopped by a fixed obstacle at its
other end; determine the blow which the obstacle receives, and
the pressure thereby caused on the fixed points of the axis, these
being supposed to be near to each other at the fixed end of the
rod.
the momentum of the blow with which the obstacle
Let Q
is struck; M =
ertia = M
α
3
the mass of the rod; then the moment of in-
α ΜΩ
;
; and
3
206
[126.
FIXED ROTATION-AXIS.
therefore
ΟΜΩ
P1 + P₂ =
3
and P₂ = P₁, since both act close together and perpendicular to
P2
the rod;
ΟΜΩ
P₁ = P₂ =
P2
6
Ex. 2. A circular plate of radius a revolving with an angular
velocity about an axis passing through its centre and fixed at
the extremities of the diameter, is struck with a blow a at right
angles to its plane, at a point in the diameter perpendicular to
the rotation-axis at a distance c from the centre; find the pres-
sures on the fixed points of the rotation-axis.
Let the rotation-axis be the z-axis, and the plane of the plate
be the plane of (z, x); let м be the mass of the plate; then
Ω Ξ
4Qc
Ma²
P₁ = P2
2
126.] Let us however further consider certain particular values
which the pressures at the fixed points may have. It is evident
that they will not generally vanish, whatever is the origin;
whatever are the axes; whether they are central principal axes;
whether the rotation-axis is a principal axis, and whether the
origin is its principal point: yet it may be that they will vanish if
the momenta are impressed under certain conditions, and in a
certain reference to the constitution of the body. Now we will
suppose a single force to act, and to impress a certain momentum
in a given direction at a certain point; and we will inquire the
point at which and the line along which this force must act, if no
pressure is thereby produced on the fixed points.
Let us suppose the momentum impressed by this acting force
or blow to be q; and to be impressed on the body at (§, n, Š),
and along the line (a, ß, y): then, since P₁ = P₂ = 0, (4) become
Qcos a +
Q cosß - nw.mx =
Q cos y
.my — 0,
0,
= 0;
1
2
(17)
the last of which shews that cos y = 0; and therefore the line
of the impressed momentum must lie in a plane which is per-
pendicular to the rotation-axis. Thus (5) become
- CQ cosß+2x.mzx = 0,
CQ cosa + oz.my z = 0,
Q(cosẞ-n cos a) - z.m (x² + y²) = 0
(18)
126.]
207
AXIS OF PERCUSSION.
From the first two of (17) and of (18) we have
Σ.myz
Z.mzx
Ś
;
Σ.my
Σ.ΜΧ
whence we have the condition
(19)
*
Σ.mx Σ.myz Σ.my Σ.mzx = 0;
(20)
and this must be satisfied if the fixed points are free from pres-
sure. Now this expresses a particular constitution of the body
relative to the axis of 2, and is independent of both the impressed
momentum and of the position of the fixed points. It evidently
indicates that the axis of z is a principal axis, and (19) gives the
distance of its principal point from the origin. Hence we have
this first condition. If the fixed points of the rotation-axis are
free from pressure, that axis must be a principal axis of the
body, and the line of action of the force, or the line of the blow,
must be in its principal plane.
Also from the last of (18), in combination with the first two
of (17), we have
§z.mx +nz.my — Σ.mr² = 0;
(21)
so that if (x, y, z) is the centre of gravity, and if k is the radius
of gyration relative to the rotation-axis, (21) becomes
x § + ÿn − k² = 0;
&
(22)
which is the equation to the line of action of the blow, in the plane
parallel to, and at a distance from, the plane of (x, y). (22) is
evidently perpendicular to the line joining the centre of gravity
and the rotation-axis; and if h is the distance of the centre of
gravity from the axis, and is the perpendicular distance from
the axis on the line of the blow, or the line of percussion, from
(22) we have
k2
1 =
h
(23)
hence the line of the blow must be at right angles to the per-
pendicular from the centre of gravity on the rotation-axis, and
at that distance / from the rotation-axis which is given in (23).
Also in this case we have from (6), if м = the whole moving
mass,
Ω
al
Mk2
;
Mh
(24)
certain special forms of the preceding equations deserve remark.
If the plane of (x, y) is the principal plane of the axis of z,
which is the rotation-axis, (20) is satisfied identically; and (=0.
Equation (20) is also satisfied identically if z.mx = x.my = 0;
208
[126.
AXIS OF PERCUSSION.
that is, if the rotation-axis passes through the centre of gravity;
but in this case (= ∞ and therefore Q = 0. So that if a body
capable of rotation about an axis, passing through the centre of
gravity, is struck by a blow, whatever is the direction and the
intensity of the blow, certain pressures are always produced at the
fixed points of the axis. This result obviously depends on the
fact that generally the principal point of an axis passing through
the centre of gravity of a body is at an infinite distance.
If at the time when the blow is given the coordinate planes
are so chosen that that of (x, z) contains the centre of gravity;
then .my = 0; but as z.myz evidently vanishes also, has a
determinate value.
It appears then that if a body capable of rotation about a fixed
axis is struck by a blow and rotates thereby, so that no pressure
is produced on those points at which the axis is fixed, it is ne-
cessary that (1) the rotation-axis should be a principal axis of
the body; (2) the line of the blow should be in the principal
plane of this axis, and perpendicular to the plane containing the
rotation-axis and the centre of gravity, and at a distance from
the axis equal to 7, which is defined by (23).
2
A representation of these circumstances is given in Fig. 22;
OP₁ P₂ is the fixed rotation-axis, and is the z-axis; P₁, P, are the
two fixed points which determine it; o is its principal point, and
is the origin, so that in this figure (= 0; and the plane (x, y) is
the principal plane. G is the centre of gravity of the body which
is taken to be in the plane of (x, z), so that the line of the blow is
parallel to the y-axis. Okk, the radius of gyration of the
system relatively to the rotation-axis. OL; NG OM = h;
so that by (23) OL is a third proportional to ом and ок.
=
If k' is the radius of gyration of the body relatively to мG, by
(129), Art. 101,
k² = h² + k'²;
h² + k'²
so that
;
h
k'2
= h +
ht
1
..
hi
h(l-h) = k'2;
OM X ML = a constant.
(25)
Now be it observed that all lines in the plane (x, z), which are
parallel to op₁ P2, are principal axes at some point on them, by
reason of Art. 113; whatever therefore the point L is to op₁ P2,
1
127.]
209
CENTRE OF PERCUSSION.
P2
so is some point on or, P₂ to a line through L parallel to OP₁₂:
these two parallel lines are therefore to some extent reciprocal
to each other.
127.] The point L, which has been determined in the preced-
ing Article, is called the Centre of Percussion of the body rela-
tive to the given rotation-axis. It determines the line along
which a blow must be impressed on a body capable of rotation
about a principal axis, when the axis receives no strain thereby ;
and conversely, if a body rotates about an axis free from all con-
straint, or if constrained, free from pressure at its bearings, the
centre of percussion determines the line in which a blow must be
given to the body to reduce it to rest without causing pressure on
the bearings; or, in another sense, it determines the positions in
which a fixed obstacle may be placed, on which if the body im-
pinges and is brought to rest, the bearings of the axis will suffer
no pressure.
1 2
It is also evident that as the axis OP, P₂ is free from pressure at
its bearings, it is that axis about which the body continues to
rotate; it is therefore a permanent axis. We have hereby then
arrived at another property of a permanent axis, and have shewn
it to be identical with a principal axis.
It is also evident that if the body is free from all constraint,
so that it is capable of translation as well as of rotation, the
effect of a blow at L along LQ will cause a rotation about op₁ P₂ ;
for this reason the axis OP, P₂ is called the Spontaneous Axis of
the body relative to the point L. This subject however we shall
consider at length in Chapter VII.
2
I propose now to apply the preceding theory to certain ex-
amples, and to exhibit the practical meaning of the results. For
this purpose it is often more convenient to express (23) in the
following form;
Mk2
Mмh
The moment of inertia
Mh
(26)
Ex. 1. Find the centre of percussion of a circular plate, capable
of rotation about an axis which touches it.
Let the rotation-axis which touches the plate, and is in its
plane, be the axis of z; let the plate be the plane of (z, x); and
let the plane passing through the centre of the plate and per-
pendicular to the rotation-axis be the plane of (x, y); then it is
evident that .myz = Σ.mzX = 0; and thus that the rotation-
PRICE, VOL. IV.
E e
210
[127.
CENTRE OF PERCUSSION.
axis is a principal axis, and that the point of contact is its prin-
cipal point. In this case
5 ποτα
the moment of inertia =
;
4
5 a
therefore from (26),
4
5πρτα
4πρτα
Also the line of the blow a must be perpendicular to the plane
of the plate; therefore, by (24),
ΩΞ
Q
πρτα
Ex. 2. Find the centre of percussion of a rectangular cube
whose rotation-axis is parallel to four parallel edges of the cube,
and which is equidistant from the two nearer, as well as from
the two farther edges.
Let the rotation-axis be the z-axis; and let the plane passing
through it and bisecting the cube be the plane of (x, z); it is evi-
dent that .myz = Σ.mzxX 0; so that the rotation-axis is a
principal axis, and the line drawn through the centre of the
cube perpendicular to it cuts it in its principal point. Let 2a be
a side of the cube, and let c be the distance of the rotation-axis
from its centre of gravity; then
2 a2
k² = c² +
3'
and
h = c;
2 a2
...
= c +
;
.
3 c
Q
8p a³ c
3
Ω
Ex. 3. A cylinder is capable of revolving about the diameter
of one of its circular ends: find the centre of percussion.
Let a = the length of the cylinder, b = the radius of its cir-
cular transverse section. It is evident that the rotation-axis is
a principal axis; and that the centre of the circular end is its
principal point.
3b2 + 4a2
6 a
;
Ω Ξ
2Q
πρα 12
Hence the centre of percussion will be at the end of the cylinder
2α
;
3
if 36² = 2a². If b is very small in comparison of a, l =
thus, if a straight rod of small transverse section is held by one
128.]
211
CENTRE OF PERCUSSION.
end in the hand, I gives the point at which it may be struck
when the hand will perceive no jar.
Ex. 4. Find the centre of percussion of a sphere revolving
about an axis, which touches its surface.
This axis is evidently a principal axis, and the point of con-
tact is its principal point; and we find
7 a
5
Ω Ξ
3Q
4πρα
128.] Let us now suppose a single blow a to be applied to
the body at a point (§, n, ), in a line whose direction-angles are
λ, µ, v, relatively to a system of coordinates thus chosen: let
the rotation-axis be the z-axis, and let a line perpendicular to it,
and passing through the centre of gravity, be the x-axis; so that
when the blow is struck the centre of gravity is (h, 0, 0), where
h is the distance from the centre of gravity to the rotation-axis.
And let us suppose the effect of a blow on the axis to be a
single pressure along it, and no pressure at right angles to it;
so that the axis may slide in its own direction, if such a motion
is possible. In this case we have
0;
P₁ Cosα₁ = P₁ cosẞ₁ = 0
and the equations of motion are
P₂ cosa, P₂ cos B₂ = 0;
Q cos λ
0,
h
0,
Q cos μ - Ω ΜΑ
Q cos v — P₁ — P₂ = 0.
(27)
Q (n cos v —
cosµ) + QΣ.mz X
0,
Q ((cosλ
cosv) +
.myz
= 0,
(28)
Q (§ cosμn cosλ) — oz.m (x² + y²) = 0.
From the first of (27), cos λ = 0; so that the line of blow
must be in a plane perpendicular to the line drawn through the
centre of gravity at right angles to the rotation-axis.
And if k is the radius of gyration of the body, relatively
to the rotation-axis, from the last of (28) and the second of
(27) we have
k2
1.2
§
;
h
(29)
which gives the perpendicular distance from the rotation-axis on
the plane which is parallel to it and contains the line of the
blow.
Also if, as heretofore, D = x.myz, E = Σ.mzx, from the first
two of (28), since cos v = sin μ, we have
E e 2
212
[128.
AXIS OF PERCUSSION.
м k² — Dn
Ek2
h
= 0;
(30)
D
Mk2 ;
which is the equation to the line of the blow in the plane given
by (29); this makes with the plane of (x, y) an angle tan-
so that (29) and (30) are the equations to the line of the blow.
The line just determined is called the Axis of Percussion. If
D = E = 0, that is, if the rotation-axis is a principal axis, of
which the origin is the principal point, = 0, and the axis of
percussion lies in the principal plane of the rotation-axis; and
its intersection with the plane containing the rotation-axis and
the centre of gravity is the Centre of Percussion.
Now this axis of percussion may also be arrived at by the
following process. At all points on the rotation-axis let the
momental ellipsoids be described, and let the planes be drawn
which are conjugate to the rotation-axis; these planes shall all
intersect in the same straight line; and that line is the axis of
percussion.
The equation to the momental ellipsoid at the origin is
2
A§² + Bŋ² + c§² — 2Dn5—2E §§—2F§ŋ−1 =
and the plane conjugate to the z-axis is
c-Dn-E = 0;
0;
(31)
so that for the momental ellipsoid, whose centre is at a distance
μ from the origin, the equation to the plane conjugate to the
axis of z is
or
C
c (5—µ) — (D—µz.my)n — (E — µz.mx) § = 0;
c— Dn—E—µ{c—z.myn — x.mx {} = 0.
x.mx{}
(32)
If we take the notation and coordinate-system of the present
Article, C = Mk2, Σ.my = 0,
so that (32) becomes
x.mx = мh ;
(33)
м k² — Dŋ — E § — µ {м k² — мh§} = 0;
M (
which is the equation to a plane, and contains the indeterminate
quantity; it therefore represents a series of planes, all of which
pass through the straight line which is the intersection of the
two planes,
M k² (— D ŋ — E Ć
- E § = 0,
k² — h§ = 0:5
;
(34)
the latter of which is a plane parallel to the plane of (y, z); and
by substitution from the latter in the former we have
E k2
Mk2 - Dn-
0 ;
h
129.]
213
FINITE FORCES.
which is a plane perpendicular to the plane of (n, 5), and inclined
to the plane of (§, n), at an angle whose tangent is
D
Mk2
As the z-axis and the origin are, relatively to the body, arbi-
trary, this theorem is true for all lines which traverse the body;
and therefore,
If at all points of a straight line which traverses a body the
momental ellipsoids are described, the planes of these ellipsoids,
which are conjugate to the given line, all pass through one and
the same straight line.
Hence also we have this theorem:
If a body is capable of rotation about a certain fixed axis, and
at all points of the axis the momental ellipsoids are described,
and the planes of them, conjugate to the axis, are drawn ; then
all these pass through the same straight line; and that straight
line is the direction of a blow which will produce no strain on the
axis. If the axis is principal at one of its points, this line of
blow lies in the corresponding principal plane, and is perpen-
dicular to the plane containing the rotation-axis and the centre
of gravity, and there will be no pressure at all on the axis. But
if the rotation-axis is not principal at any one of its points, the
direction of the blow will be oblique to the plane containing the
axis and the centre of gravity, and there will be a pressure act-
ing on the axis in the direction of its length.
In the preceding process we have supposed the body to be
initially at rest, and motion to be communicated to it under
certain states of pressure on the axis, &c.: the process however
may be reversed; we may suppose the body to be moving about
a fixed axis with the stated conditions of pressure on it; and the
problem which would then have to be solved is, to determine
the point of application, &c. of a force, as, for instance, a fixed
obstacle, which shall withdraw all the momentum from the
body.
SECTION 2.—Rotation of a body about a fixed axis under the
action of finite accelerating forces.
129.] I PROCEED now to the case of a rigid body rotating
about a fixed axis under the action of forces, whereby momenta
are continuously impressed. To this case equations (37) and
(38), Art. 48, are to be applied.
214
[129.
FIXED ROTATION-AXIS.
1
2
Let us take, as in the preceding Articles, the rotation-axis to
be the z-axis; and to be fixed at two points whose distances
from the origin are respectively, and 2; let the pressures at
these points at the time t be respectively P, and P₂; and let the
lines of action of these pressures be (a1, B1, 71), (α2, B2, V2).
Let .P cos a
Z.PZ COS α ......
a be abridging symbols of
the axial components of these pressures, and of their moments
relative to the axes; and let L, M, N, as in Art. 77, be the mo-
ments of the axial components of the couples of the impressed
momentum-increments at the time t; then the equations of
motion are
(x-12x)
Σ.mx-
m (x
x.m(x — day)
이
​dt2
Σ.P COS α = 0,
dt2
Σ.P COSẞ = = 0,
(35)
Σ.Μ Ζ
d2 z
dt2
Σ.P COSY =
= 0;
Σ.Μ
(3
Y
dt2
d²y
dt2
- Σ.PZ Cosß = L,
Σ.Μ
(2
Z
d² x
dt2
d² z
2
X
+E. PZ COS a = M,
(36)
dt2
d2y
d2x
Σ.Μ
X
y
= N.
dt2
dt2
d2z
As the z-axis is the fixed axis, it is more convenient to transform
the last of these into its equivalent in terms of angular velocity
by an independent process, than to take the general equations
given in (48), (49), (50), Art. 79.
Let r be the distance from the rotation-axis of m, whose place
at the time t is (x, y, z); and let o be the angle between r and
the plane of (x, z), which plane is assumed to be fixed in space;
let o be the angular velocity about the fixed z-axis; so that
de
dw
d20
@=
dt,
dt
dt2.
Hence we have,
x = r cos 0,
y = r sin 0;
d² x
dt2
dw
w²r cos ◊ — r sin
dt'
d² y
2
dw
- w²r sine+r cose
dt2
dt
so that the last of (36) becomes
dw
Σ.mr.2
= N ;
dt
(37)
130.]
215
FINITE FORCES.
and as
do
dt
is the same for all the particles of the system, it may
be placed outside the sign of summation, and we have
N
dw
d20
dt
dt2
Σ.mr2
The moment of the impressed momentum-increments
The moment of inertia
; (38)
each of these quantities being estimated relatively to the fixed
rotation-axis. The form in which this equation is put shews
that it is independent of the particular system of coordinate axes
which has been taken. It is indeed identical with (47), Art. 78.
By it the angular velocity-increment about the rotation-axis is
given; and therefore by integration the angular velocity, and by
a subsequent integration the angle described in a given time
may be found. Thus the motion of the body about a fixed axis
will be determined.
Before however we proceed to examples of this motion, let us
shew that (38) may be derived immediately from first principles;
for this process will remove any obscurity which may attach to
its meaning.
Let m be a type-particle of the body or system; let r be its
distance from the rotation-axis of z, so that if 0 is the angle be-
tween r and the fixed plane of (x, z), the linear velocity of m is
d20
d Ꮎ
dt
>
and the linear velocity-increment is r
and therefore
dt2
d20
the moment of the expressed momentum of m is mr2
dt2
; so
that relatively to the axis of z the moment of the whole ex-
pressed momentum-increment is x.mr2 ;
d2 Ꮎ
dt2
and therefore if N
is the moment relatively to the same axis of the whole impressed
momentum-increment in an unit of time, by D'Alembert's prin-
ciple we have
d20
Σ.mr2
dt2
N;
d20
do
N
dt2
dt
Σ.mr2 *
130.] With respect to this equation, I would in the first place
observe, that if the lines of action of all the impressed forces are
216
[131.
FIXED ROTATION-AXIS.
parallel to the axis of 2, which is the rotation-axis, N = 0; and
that
d20
0 ;
dt2
do
S,
dt
if is the angular velocity at the time under consideration; so
that the system moves with a constant angular velocity. Hence
also
st,
Ө - а
if a is the value of 0, when t = 0; so that equal angles are de-
scribed in equal times. This is a particular case of the principle
of conservation of areas; see Art. 59. Thus, if a heavy body
rotates about a vertical axis, the force of gravity has no effect
on the angular velocity.
Z
131.] But one of the most important applications of this the-
orem is the motion of a heavy body rotating about a fixed hori-
zontal axis. Let us take the system of axes delineated in fig. 23;
let the axis of ≈ be vertical downwards; let the y-axis be the rota-
tion-axis, and let o be the angle at which the line from m(x,y, =)
to the y-axis is inclined to the vertical plane of (y, z); thus the
line of action of gravity, which is the only force acting on m, is
parallel to the z-axis. Let & be the centre of gravity of the
body, and let the plane passing through it and perpendicular to
the axis of y be the plane of (x, z); so that as the body rotates
about the axis of y, the line oG moves in the plane of (x,z).
Let м be the mass of the body; OG = h the distance of the
centre of gravity from the rotation-axis; and let mk2 be the
moment of inertia about the rotation-axis; let Go≈ =
as increases the body rotates about the axis of y from the
z-axis to the x-axis: also = is the angular velocity; and
is the same for all particles of the body. Now the moment of
the impressed momentum on m at (x, y, z) at the time t relatively
to the rotation-axis is mgx, and tends to diminish ; so that
the moment of the momenta impressed on all the particles at the
time t
d Ꮎ
dt
Σ.mg x
=-мghsin0;
0, so that
and the moment of the impressed momentum is the same as
if the whole mass were collected at its centre of gravity. And
thus, from (38), we have
132.]
217
ANGULAR VELOCITY.
d20
Mgh sin 0
dt
Mk2
gh
sin 0;
k2
(39)
which equation gives the expressed angular velocity-increment
about the rotation-axis.
Let us multiply both sides of (39) by do; and let us suppose
the body to be at rest when a; then, integrating (39), we
have
d02
dt2
2gh
k2
(cosecos a);
do
dt
(40)
which assigns the angular velocity in terms of e. From this
equation it appears that = 0; that is, that the angular velo-
city vanishes, and the body is at rest, when
Ꮎ
0 = α, 0 = 2π + α, Ө
=
Ө
=
2 mπ ± а;
so that if (40) expresses the circumstances of the body, the mo-
tion of it is oscillatory, the arc of vibration being double of that
between the vertical line and the initial position of the line
through the axis and the centre of gravity; this latter being the
vertical line when the body is at rest. Hence we have the fol-
lowing circumstances of motion of a heavy body capable of oscil-
lation about a horizontal axis. When the body is at rest, the
perpendicular from its centre of gravity to the rotation-axis is
vertical; let this line be moved through an angle a, and let the
body be left to itself; it will oscillate through an angle 2a, the
centre of gravity ascending to equal heights on both sides of the
lowest point. Such an oscillating body is called a "compound
pendulum."
If the body is moving with an angular velocity a when = a,
the equation of the angular velocity becomes
d02
dt2
22
2gh
7.2
(cos — cosa);
(41)
but as this is of the same form as (40), so far as integration is
concerned, we shall inquire into the properties of only (40).
Equations (40) and (41) are evidently those of vis viva.
132.] From (40) we have
dt
k
d Ꮎ
(2gh) (cos 0 — cos a)
(42)
whence, by integration, the time may be found in terms of e,
and the whole time of an oscillation may be determined.
PRICE, VOL. IV.
F f
218
[132.
THE COMPOUND PENDULUM.
This equation however in its present form is an elliptic
transcendent, and therefore cannot be (as it is said) integrated. If
however the displacement of the body is slight, so that a and
are both small, then we may expand cose and cosa, and neglect
powers of and a above the second; whereby we have
02
cos 0 = 1
Q2
2
cosa = 1
and (42) becomes
2
k
dt =
do
(gh) $ (a² — 0²) ↓ '
;
t =
k 20
cos-1 ;
(gh)*
α
(43)
(44)
if t =
0, when
0:
= a; and therefore if T is the time of a small
Τ
oscillation of a heavy body about a horizontal axis, t = r when
= a, and
T=
πλ
(gh) *
(45)
Now if we consider a heavy particle of infinitesimal dimensions
attached to the end of a rigid imponderable rod of length 1, and
without weight, and vibrating about a horizontal axis perpen-
dicular to its length, to be a perfect pendulum, then, as we have
shewn in Art. 357, Vol. III, if r is the time of small oscillation
of such a pendulum,
T
T
π
(2)
and the time of the heavy oscillating body is identical with this, if
1
k2
h
(46)
Thus, the compound pendulum is isochronous with a perfect pen-
dulum of the length 7, which is given in (46); and 7 is called the
length of the simple isochronous pendulum.
The agreement however in motion between the compound and
the simple isochronous pendulum is greater than the preceding
investigations lead to. For the general equation of a heavy parti-
cle attached to the end of a rigid and imponderable rod of length
1, and rotating in a circle, is, see Art. 359, Vol. III,
d20
dt2
9
sin 0 ;
and this equation is identical with (39), which determines the
rotation of the heavy body, if
1 =
k2
;
h
132.]
219
THE COMPOUND PENDULUM.
and hence we conclude that if the whole mass of the rotating
body is condensed into a particle at a distance 7 from the rota-
tion-axis along the line which passes through the centre of gra-
vity, the circumstances of equilibrium and of motion of this par-
ticle would be identical with the similar circumstances of the
body. And if the body is slightly displaced from its position of
stable equilibrium, and oscillates through a small angle, the
time of an oscillation
where
八
​;
k2
h'
k being the radius of gyration of the body about the rotation-
axis, and h being the distance of the centre of gravity from the
rotation-axis.
The point o, Fig. 23, in which the horizontal rotation-axis
pierces the vertical plane containing the centre of gravity, is
called the Centre of Suspension; and if og is produced to c, so
that oc =
1, c is called the Centre of Oscillation, and oc or lis
the length of the simple pendulum isochronous with the body;
that is, if the whole mass is collected into a particle at c, the
circumstances of rotation of the particle thus condensed will be
the same as those of the body.
Let k' be the radius of gyration of the body relative to an axis
through G, and parallel to the rotation-axis; then, by (129),
Art. 101,
k² = k'² + h²;
2
k'² + h²
h
几十
​h
.. (l—h)h = k²;
and replacing these by the geometrical quantities.
(47)
(48)
CG X GO = K2
a constant.
(49)
Now this equation would be unaltered if the places of o and c
were interchanged; whence we infer that if c is the centre of
oscillation for an axis oy through o, o would be the centre of
oscillation for a parallel axis through c. This theorem, as it is
commonly stated, asserts the convertibility of the centres of sus-
pension and oscillation. As the length of the simple isochronous
F f 2
220
[133.
THE ISOCHRONOUS PENDULUM.
pendulum is the same whether c or o is the centre of oscillation,
so the time of oscillation is the same for both parallel axes.
It will be observed that we have the same expressions for the
determination of the centre of oscillation and the centre of per-
cussion relative to a given rotation-axis, see (23), Art. 126: in the
latter case, however, it is necessary that the rotation-axis should
be a principal axis at some point on it, and the centre of percus-
sion should be in its principal plane; here no such restriction as
to the nature of the axis is necessary. We have hereby a method
by which the centre of percussion may be practically determined.
Let the body of which the centre of percussion is to be found be
suspended by, and made to vibrate about, the relative rotation-
axis; let the number of vibrations in a given time be noted; let,
say, n vibrations take place in t; then
..
t
n
π
912
1=
;
π² n²
2
thus, if t and n are carefully observed, as the other quantities
are known, l is also known; and this is the distance of the cen-
tre of percussion from the rotation-axis.
133.] Before we enter on other investigations connected with
the times, &c. of oscillation of bodies, let us determine l in certain
cases; and for this purpose we shall generally find the second
of the following forms the more convenient;
k2
M k²
7
h
Mh
The moment of inertia relative to the rotation-axis
•
(50)
The mass x the distance of the c. of Gr. from the axis
Ex. 1. A straight heavy wire, of length 2a, vibrates about an
axis passing through its end, and perpendicular to its length:
find the length of the simple isochronous pendulum.
4a
1 = ;
3
that is, the length of the simple isochronous pendulum is two
thirds of the length of the wire.
Ex. 2. A wire, in the form of the arc of a circle, vibrates about
an axis passing through its middle point and perpendicular to
its plane; prove that the length of the simple isochronous pen-
133.]
THE ISOCHRONOUS PENDULUM.
221
dulum is that of the diameter of the circle, whatever is the
length of the wire.
Let p and w be the density and the area of the transverse sec-
tion of the wire; let a be the radius of the circle; then the origin
being the middle point of the wire, the equation to the wire is
y² + x² = 2 ax;
Also
the moment of inertia = pof(x² + y²)ds
pw
2apwfxds.
w fxds;
Mh = pw
the limits of integration being the same in both integrals; so that
Į
2a.
Ex. 3. Compare the times of vibration of a thin circular plate
about axes passing through the circumference, and (1) touching
the circle and in its plane; (2) at right angles to the plane of
the circle.
The moment of inertia relative to a tangent
5 ποτα
4
πρτα
2
the moment of inertia relative to a perpendicular axis
Μ = πρτα;
and in each case
therefore, if, and I are the lengths of the corresponding iso-
chronous pendulums,
1-
5 a
4
3 a
12
2
and if t₁ and t₂ are the corresponding times of small vibration,
5
4 = (4) * = (8)*
t₁
t₂
6
Ex. 4. A right cone oscillates about an axis passing through
its vertex and perpendicular to its own axis; it is required to
find the length of the simple isochronous pendulum.
Let a the altitude of the cone; b = the radius of the cir-
cular base; then
the moment of inertia =
πραγ
a
20
(4a² + b²);
and
πραγ
Mh =.
;
4
therefore
4a² + b²
5 a
222
[133.
THE METRONOME.
If ab, that is, if the cone is right-angled, la; and the alti-
tude of the cone is the length of the simple isochronous pen-
dulum thus the centre of oscillation is in the centre of the
base; so that the times of oscillation of a right-angled circular
cone are equal for axes through the vertex and the centre of the
base which are perpendicular to the axis of the cone.
Ex. 5. The mass of the particle at the end of a perfect pen-
dulum of length a is м: to determine the position of another
particle m on the rod, so that the time of oscillation of the whole
pendulum may be a minimum.
Let the distance of m from the centre of suspension; then
mx² + Ma²
dl
dx
mx + Ma
;
M
m²x²+2mмa x — m м a²
(mx + Ma)²
if mx = − мa ±
which gives two values for x,
other is negative.
Let
a {M(M +m) } * ;
0
of which one is positive and the
be the distance from the centre of suspension of the
centre of gravity of м and m, when m is in its required position ;
then (м+m)π = mx + мa
x = + a
M
M + m
that is, the centre of gravity of м and m is equally distant from
the centre of suspension in the two directions along the rod. Also
1 = 2a ± {M(M +m)} *—m
m
M
Ex. 6. A metronome is formed of a heavy rod of given length,
having at one end a heavy sphere of radius r and mass m, at a
distance a from the rotation-axis, which is perpendicular to the
rod; another sphere of radius r' and mass m' slides along the rod :
to find the point at which the centre of this latter sphere must
be fixed, so that the whole system may oscillate n times in a
minute.
Let the metronome be represented in Fig. 24, wherein the
rod, which in the position of equilibrium is vertical, is slightly
inclined to the vertical.
Let the plane of the paper be the plane of vibration, and let o
be the point where the rotation-axis pierces the plane. Let a be
the centre of the fixed sphere, OA = a; let P be the centre of
133.]
223
THE METRONOME.
the sliding sphere, or = x; let OB = b; let м the mass of
the rod. Then, relatively to the rotation-axis,
the moment of inertia of m = m
(27² + a²)
5
27/2
of m' = m' {
5
+x02)
of M = m
(az
a² — ab+b²
ab + b²
3
and the denominator of (50) in this case
= ma M
b- a · m' x ;
2
so that
Į
15
2 m (6r2+15a²) + m² (6r2 +15x2)+5м (a²-ab+b²)
2ma-м(ba) — 2m'x
; (51)
As the metronome is to oscillate n times in a minute, we have
60
П
N
(
Let L = the length of the second's pendulum; then
1 = π
(-)
L
;
3600
and therefore
Z
L;
n²
and if we substitute this in the left hand member of (51), the
equation contains a and all known quantities; whence a may be
determined; and the rod of the metronome may be graduated so
that the system will oscillate in any required time.
If the rod is very thin, as is the case with the ordinary me-
tronomes, м may be neglected; and we have
3600
n²
L
m (2r²+5a²) + m² (2r²² + 5 x²)
5 (ma-m'x)
Ex. 7. A pendulum consists of a rod of length a and mass m;
at the end of which is a circular plate, Fig. 25, of radius r and
mass M, so arranged that the plate is capable of sliding on the
rod, and rests on a nut fixed at the end of the rod; the plane of
the plate is always in the plane of vibration; find the length of
the simple isochronous pendulum ;
The moment of inertia of the plate = M
2.2
2
{ + (a−1) 3 }
The moment of inertia of the rod = m
a2
;
3
224
[134.
THE LENGTH OF THE SECONDS' PENDULUM.
and
Mh
m a
2
+ M (α-r);
...
1 =
3M (3r² - 4ar + 2 a²) + 2ma²
3 {ma + 2M (a− r)}
(52)
Let us suppose the temperature to vary so that a and r are
increased by da and dr respectively; м and m being unaltered;
and let us suppose the pendulum to be compensating; so that
l remains the same, whatever is the temperature; then, since
D70, we have from (52)
{6 m² (2 a² — 4ar+r²)+mm (10a²—8ar−9r²)+2m² a²} da
M
= мdr {6м (37² −6ar+2 a²)+2ma(9r−4a)}. (53)
In the most common form of compensating pendulums the
straight rod is made of steel, and the weight consists of a cylin-
der of mercury which is fixed at the end of the rod, the axis of
the cylinder coinciding with the rod, and the base of the cylinder
resting on a nut at the end of the rod. The amount of expan-
sion of the rod and the mercury having been determined by ex-
periment for an increase of one degree of temperature, and the
length of the seconds' pendulum being also known, the quantity
of mercury may be determined by a process similar to that which
we have just explained.
134.] The convertibility or the reciprocality of the centres of
suspension and oscillation of a pendulum has been applied by
Capt. Kater to the determination of its length; and he has
hereby obtained means for determining the length of a seconds'
pendulum at a given place.
Let the pendulum consist of an ordinary thin straight rod,
and a heavy disc, as in Fig. 26. At the points o and c, at the
distance / apart, let two knife edges be placed parallel to each
other, and at right angles to the rod of the pendulum; so that
the pendulum may vibrate on either of them, as in the diagrams
of the figure, where it rests on two horizontal and parallel plates.
Let a small weight m be capable of sliding on the bar, and of
being clamped to it by means of a screw. It is evident that
* For a full account of this pendulum I must refer the reader to a Memoir
by Mr. Francis Bailey in the Eighth Volume of the Memoirs of the Royal Astro-
nomical Society of London, for the year 1824: and for a description of various
other kinds of compensating pendulums to "Mechanics," by Capt. Kater and
Dr. Lardner; Longman and Co., London, 1830.
134.] THE LENGTH OF THE SECONDS' PENDULUM.
225
whether o or c is the centre of suspension the length of the
simple isochronous pendulum will vary according to the place of
m; let the place of m be so adjusted that the times of oscillation
may be the same, whether the pendulum is suspended by the
knife edge at c or by that at o; so that oc (= /) is the length
of the simple isochronous pendulum; if then this distance oc is
carefully measured, the length of a simple pendulum is accurately
known and by means of it the lengths of all other pendulums
may be determined.
:
Thus, suppose the pendulum above described to make n oscil-
lations in a given time, say in t; these quantities can be found
by means of an astronomical or any other correct clock, by the
method of coincidences: then
t
n
π
* (1) *
Let L be the length of a seconds' pendulum; then
1 = π
(54)
.*.
L =
N2
1 ;
t2
and therefore the length of L is also known.
(55)
Now, as we have before remarked, g, which is the velocity-
increment of a falling particle due to the earth's attraction in a
second of time, varies for different places on the earth's surface,
and for several reasons, as we have explained in Arts. 224 and
258, Vol. III. It is important to have means of determining the
value of g by observation; as well for the purpose of verifying
the theoretical law, as for determining the constants which enter
into Clairaut's expression, Art. 258, Vol. III; and the preceding
theory of pendulum-motion supplies a method. From (54), we
have
g
n² 21
T2
t2
;
(56)
and thus when n and t have been determined by observation,
and 7 by direct measurement, all the quantities in the right hand
member of this equation are known. So that from (55) and
(56) the length of the seconds' pendulum, and the velocity-incre-
ment due to the earth's attraction, which is usually termed “the
force of gravity," may be found at any given place. A table
containing the values of L and g for a few places, with their lati-
tudes N or s, is subjoined; the observations are reduced to the
PRICE, VOL. IV.
G g
226
[135.
DETERMINATION OF RADII OF GYRATION.
level of the sea, and to a pendulum vibrating in vacuo, at a tem-
perature 62° of Fahrenheit*.
Name of Place.
Latitude.
Length of
Pendulum in
Inches.
Gravity in
Feet.
Name of Observer.
Spitzbergen
•
79°49′58″N 39.2146 32.25294 Sabine.
|
63° 25′54″N 39.1745 32.2198
32.2198
51°31′ 8″N 39.13929| 32.1910
48°50′14″N 39.1308 32.1838
32.1838
Drontheim
London
Paris
New York
Jamaica
40° 42′ 43″N 39.101632.1598
Sierra Leone
Cape of Good Hope
Sabine.
Kater, Sabine.
Biot, Borda, &c.
32.1598 Sabine.
17°56′ 7″N 39.0351 32.1052 Sabine.
8°29′ 28″N 39.0199 32.0933
|
32.0933 Sabine.
33°55′ 15″ s 39.0787 32.1409 Freycinet.
These results shew that gravity continually increases as we go
from the Equator to the Poles. And it is found that the differ-
ences between the observed results and the values calculated
according to theory are extremely small.
135.] By means of the preceding value for the length of a
pendulum which vibrates isochronously with a body relative to
a given rotation-axis, we are able to deduce experimentally the
radius of gyration of a body relative to an axis; and consequently
the central principal radii of gyration, and thus the central ellip-
soid of gyration.
About the axis relative to which the radius of gyration of the
body is to be determined, let the body make small oscillations:
let T be the time of an oscillation, which can be observed by
means of a clock; then
Tπ
g
π2 k²
π² =
g
h
* For accounts of the process by which General Sabine determined the
lengths of the pendulum at those places in the following table to which his
name is attached, see "An Account of Experiments to determine the Figure
of the Earth by means of Pendulums vibrating seconds in different latitudes,
as well as on various other subjects of Philosophical Inquiry," by Edward
Sabine, F. R. S., &c., &c.; John Murray, London, 1825; at the expense of
the Board of Longitude. See also three other papers by General Sabine in
the Philosophical Transactions of 1827.
136.]
227
ISOCHRONAL AXES.
Now h, which is the distance of the centre of gravity from the
rotation-axis, must be measured; and we have
k2
T2
T² hg;
72
... M
м k² =
T2 h
if w is the weight of the body.
72
W,
(57)
(58)
Thus, (57) gives the radius of
gyration, and (58) gives the moment of inertia of a body relative
to a given axis.
=
If the axis passes through the centre of gravity the method
fails, because h = 0, and therefore T∞ in this case let an-
other rotation-axis be taken parallel to the given one through
the centre of gravity, and at a distance h from it; then, if k' is
the radius of gyration for the axis through the centre of gravity,
2
π² h² + k²²
..
T2
;
g
h
k'2
T² hg
h2,
П2
T2 h
M k'2
=
W
M h²;
.2
π
(59)
(60)
thus, (59) gives the radius of gyration, and (60) gives the mo-
ment of inertia about an axis passing through the centre of gra-
vity. When these have been found for a sufficient number of
axes, the central ellipsoid of a body may be constructed. When-
ever therefore a body is given, however irregular its bounding
surface is, whatever is the law according to which its density or
the distribution of its elements varies, its central ellipsoid can
always be determined by the preceding method; and conse-
quently every curve or surface connected with it, or which may
be derived from it, may always be assumed as known.
136.] Certain general properties of axes of vibration of a body
also require investigation. Let us refer the body to the centre
of gravity as origin, and to its central principal axes as coor-
dinate axes. Let A, B, C be the three central principal moments
of inertia, in the same order of magnitude and about the same
axes as we have assumed them to be in the preceding Chapter.
So that if Mk2 is the moment of inertia about an axis (a, ß, y)
passing through the centre of gravity,
Mk'2 = A (COS a)2 + B (cos ẞ)2 + c (cos y)2.
(61)
G g 2
228
[136.
ISOCHRONAL AXES.
Let all axes of a body, relative to which the times of vibration
are equal, be called isochronal; then for an axis parallel to a line
(a, ß, y), which passes through the centre of gravity, and at a
distance from it equal to h,
k'2
1 = h +
h
= h +
A (COS a)² + B (cos B)2 + c (cos y)2
Mh
;
(62)
(63)
and since this is true for all axes parallel to (a, ß, y), and equi-
distant from it, it follows that all axes lying on the surface of a
right circular cylinder whose axis passes through the centre of
gravity are isochronal.
Let l-hh'; so that hh'k'2; then as an axis at a distance
h' from the centre of gravity is isochronal with a parallel axis at
a distance h, so all axes lying on the surface of a right circular
cylinder whose radius is h', and whose axis passes through the
centre of gravity, are isochronal; and are isochronal with those
which lie on the surface of the coaxal circular cylinder whose
radius is h.
From (62) it appears that = ∞ when h=0, and when h=∞;
so that there is some value of h between these limits which makes
7 a minimum. Let us equate to zero the h-differential of (62);
then
dl
dh
=1
if h = k';
1 = 2 k':
k' 2
h2
0,
(64)
it appears then that for all parallel axes the time of oscillation
is the least for those which are at a distance k' from the centre
of gravity of the body, where k' is the radius of gyration of the
body relative to a parallel axis through the centre of gravity;
and that the length of the corresponding simple isochronous
pendulum is 2 k'. In this case, the two coaxal cylinders of iso-
chronal rotation-axes become identical; and from (62) we have
(A (COS a)² + B (cos B)2 + c (cos y)2
Since 7
2
{
>
2
α
M
(Cosy)³}
(65)
2k', the time of oscillation depends on the cen-
tral radii of gyration, and is least for an axis parallel to the
least radius of gyration; therefore, from (65), I is the least when
cosß = cosy = 0: that is, when
137.]
229
ISOCHRONAL AXES.
1 = 2
M
(A)*
= 2a,
if a is the least central radius of gyration. And this absolutely
gives the least time of oscillation of all axes about which a body
can oscillate. And as of all parallel axes that which is at a
distance equal to 2k' from the parallel central radius of gyration
yields the least time of oscillation; so of all, that which is pa-
rallel to the axis of the greatest moment of inertia is the maxi-
mum minimorum, and that which is parallel to the axis of the
least moment of inertia is the minimum minimorum, and the
other minima are intermediate to these.
Some examples are subjoined.
Ex. 1. Of all axes passing through and perpendicular to a thin
rod of length 2a, that at a distance a 3- from the middle point
of the rod is the axis for which the time of oscillation of the rod is
the least; and the length of the simple isochronous pendulum is
2 a
·(3)*·.
35
Ex. 2. For a sphere of radius r, all the radii of gyration passing
through the centre are equal, and = r
; so that the axes for
which the time of oscillation is the least are at a distance from
the centre equal to this quantity; and
23r
1 =
5+
Ex. 3. The axis for which an ellipsoid vibrates in the shortest
possible time is parallel to its greatest principal axis, and at a
distance from it =
( 62 + 0²) * .
5
137.] Again, since all central equimomental axes lie on the
surface of the right cone
(H — A) x² + (H — B) y² + (H −c)≈² = 0,
(66)
where H is the moment of inertia relative to any axis on the
cone, this is the locus-surface of all axes of the circular cylinders
of equal radius h, all lines lying on the surface of which are iso-
chronal axes; and for which l = h +
H
Mh
It is similarly the locus surface of all axes of the circular
230
[137.
ISOCHRONAL AXES.
cylinders of equal radius h', all lines lying on the surfaces of which
are isochronal axes; and for which
l' = h' +
H
Mh
where hh' = k'², and the axes lying on the surfaces of all the
cylinders are isochronal.
Thus, if two spheres of radii h and h' are described from the
vertex of the cone (66) as centre, and cones are described touch-
ing them, coaxal with and similar to the given cone, all generat-
ing lines of these two cones are isochronal.
وه
Lastly, let us determine the conical surface on which lie all
isochronal axes passing through the given point (~。, Yo, 。).
Let the equations to one of these isochronal axes referred to
the centre of gravity as origin be
X
0
y - Yo
and
M
=
2-20
n
{(x −x。)²+(y−Yo)² + (≈ — z.)²} * = s (say); (67)
h² = (ny—mz)²+(lz−nx)² + (mx−ly)²;
k'2
A l² + Bm² + cn²
M
= a² 1² + b² m² + c² n²,
if a, b, c are the principal central radii of gyration. In these equa-
tions, replacing l, m, n by their values from (67), we have
s2 h2
=
2
(zy。−y z。)²+(x z。− zx。)² + (y x。—xy。)²,
2
0
(x² + y²+z²) (x²+ y²+z²) − (xx。+YY。+zz。)²;
s² k¹² = a² (x — x。)²+b² (y—y。)²+c² (z—zo)².
If therefore 7 is the length of the simple pendulum, isochronous
with the body about each of the axes passing through (~。, Yo, 。),
lh = h² + k'²;
1 {(x − x )²+(y—y。)² + (≈ — ≈。 )² } ½
2
2
{(zy。—yzo)² + (xz。−zxo)²+(yx。—xy。)² } ì
0
2
(zY。−Yz0)² + (xz。−z xo)²+(yx。—xy。)² +
a² (x − x₁)²+b² (y-yo)² + c² (z — zo)²; (68)
which is evidently the equation to a cone of the fourth degree.
Let the origin be transferred to the point (xo, Yo, zo) which is its
vertex; and let
x² + y² + z² = p² |
then (68) becomes
2
x² + y²+z² = r²
lr {r²r ² — (xx。+Y Yo+zzo)² } }
$
;
r² r² — (xx。+yYo+zzo)²+a²x² + b² y²+ c²x². (69)
139.]
231
ILLUSTRATIVE EXAMPLES.
138.] The following problems are in further illustration of the
principles contained in the preceding Articles.
Ex. 1. A vertical rod of length 2a and of mass M, which turns
about a horizontal axis passing through the upper end, is struck
by a blow at its centre of percussion, and ascends into its posi-
tion of unstable equilibrium; determine the force of the blow.
Let p = the momentum of the blow, and let o be the angular
velocity which is thereby given to the rod about its rotation-axis.
4a
3
Now the centre of percussion is at a distance from the rota-
tion-axis; so that by (6), Art. 124,
P
Ω
ам
d20
3g
sin 0;
dt2
4a
d 02
3g
22
2² =
(cos 0-1),
dt2
2 a
do
since = 2, when 0=0;
dt
unstable equilibrium, 0=, and
and when the rod is in its place of
do
= 0; therefore
dt
3g
2² =
and
P = M (3ag).
α
Ex. 2. A circular plate of radius a and mass м, capable of
rotation about a horizontal axis which is in its plane and touches
it, is struck by a blow at its centre of percussion, and ascends
into its position of unstable equilibrium; determine the blow.
P
Ɑ M
;
Ω Ξ
d20
4g
sin 0;
dt2
5 a
16g
4 M
22
;
and
P =
(5 ag).
5 a
5
139.] Another application of these principles has been made
by Capt. Robins, to the determination of the velocity of a cannon
ball.
A large thick heavy board is suspended by a fixed horizontal
axis; a cannon is so placed that a ball projected horizontally
from the cannon strikes this board at rest at a certain point;
and the board revolves through an angle, which is observed. It
232
[139.
BALLISTIC PENDULUM.
is required to determine the velocity of the ball. The swinging
board with its axis is called a Ballistic Pendulum.
section is given in Fig. 27.
A vertical
We shall suppose the ball to strike the board at right angles
to its plane, and to remain in the board after impact. Let
M the sum of the masses of the pendulum and ball.
m = the mass of the ball.
Mk2
the moment of inertia of the pendulum and ball.
v = the velocity of the ball at the instant of impact.
S = the angular velocity due to the blow of the ball.
a = the distance of the point of impact from the rotation-axis.
h = the distance of the centre of gravity of the masses of the
pendulum and ball from the rotation-axis.
2=
mav
Mk2
d 02
2hg
2² =
(cose-1).
dt2
2
do
Let 0=
then
= a,
when
O, that is, when the body comes to rest.
dt
(max)
2 2hg
k²
(1-cos a)
2мk
S
.*. V=
(hg) sin
(70)
ma
as all the quantities in the right hand member of this equation
may be observed, or are known, v is also known.
We may determine a in the following manner. At a point in
the board at a distance h from the rotation-axis, let the end of
a ribbon be fastened, and let the rest of the ribbon be wound
tightly round a reel; as the pendulum ascends, let a length c be
unwound from the reel; then c is the chord of the angle a to
the radius h, so that
a
c = 2 h sin 2;
.. . V
V =
мkc
(2)*:
ma h
(71)
k is determined by the process explained in Art. 135; and if we
replace k by the value given in (57) we have
дем
V =
Ꭲ ;
παγ
(72)
140.]
233
MOTION OF MACHINES WITH FIXED AXES.
and if м and m are replaced by their weights, say w and w, which
are proportional to them, we have
gcw
V
Ꭲ .
паш
(73)
If the mouth of the cannon is placed near to the pendulum, the
value of v, given by this formula, must be nearly the velocity of
projection. And if the distance of the pendulum from the mouth
of the gun be large, so that the velocity of impact on the pen-
dulum is less than that of projection, if the coefficient of resist-
ance of the air is given, we may by the process of Art. 302, Vol. III,
estimate the diminution of velocity due to the resistance of the
air, and thus determine the velocity of projection.
The velocity however may be determined in the following
manner. Let the gun itself be suspended by a horizontal axis,
and thus form a pendulum; when the gun is discharged, it will
oscillate by reason of the recoil; and by observing the times of
these oscillations, and making the required alterations in (73),
the velocity of projection will be determined: w will represent
in this case the excess of the weight of the gun over that of the
ball, and c and a must be similarly altered.
140.] In further illustration of the motion of bodies about
fixed axes, I will consider the motion of the parts of certain
machines, in which certain rotation-axes are fixed.
Ex. 1. Two weights mg and m'g are connected by a flexible
and inextensible string without weight, which passes over a given
pulley with a fixed axis and a rough surface; it is required to
determine the circumstances of motion of each weight and of the
pulley.
The pulley is supposed to be rough, so that as the string
moves the pulley moves with it.
Let the weights, &c. be arranged as in Fig. 17; and let the
symbols be those of Ex. 1, Art. 46; and let us suppose m and m
to have the initial velocities, &c. of that example. Let м the
mass of the pulley, and a = the radius; then
the moment of inertia of the pulley = M
a²
2
M =
Let a be the initial angular velocity of the pulley due to the in-
stantaneous initial tensions of the string; then, by (6),
M a²
a
2
Q = α (1 ~ T');
PRICE, VOL. IV.
H h
234
[140.
MOTION OF MACHINES
also van; so that
T = MU man,
(74)
. Ω
(75)
+' = m'u' + m'an;
2 (mu — m'u')
a (M + 2m+ 2m')
whence the initial angular-velocity of the pulley, and the initial
tensions of the strings are known.
Ma² d² 0
Again,
= a (T-T');
T′);
2 dt2
also
d²x
dt2
d20
d²x'
d 20
α
dt²
dt2,
α
(76)
dt2
dt²
d 20
. T = mg
ma
dt2
(77)
d20
r':
m'g + m'a
dt2
d20
2 (m-m')g
(78)
dt 2
a (м+ 2 m + 2 m'′ ) '
do
2(m-m')gt
Ω
(79)
dt
a (M + 2 m + 2 m')'
d02
4(m-m') go
(80)
dt2
a (M + 2 m + 2 m')
whence by a further integration & can be determined in terms of
t; and thus the space will be known through which mor m'
will move in a given time.
d20
dt2
If we replace in (77) by its value, given in (78), we shall
find the tensions of the strings at any time t.
If the weights of the strings are taken into account, the equa-
tion of angular motion assumes the following form: Let p: the
density, w = the area of a transverse section of the string;
c = the whole length; then, if м k² is the moment of inertia of
the pulley,
(м k² +ma² + m'a² + pw ca²)
d20
dt2
=
a (m—m') g+ap wg x-— a ρ wg x'
αρωγή
— a {m— m' + pw(x − x')} g.
Ex. 2. To investigate the circumstances of motion of a wheel
and axle, the weights of the strings being neglected, and мk²
being the moment of inertia of the machine relative to its axis.
Let us use the same symbols as in Ex. 2, Art. 46, and those of
140.]
235
WITH FIXED AXES.
the last example; and let Fig. 18 represent the plan of the wheel
and axle when projected on the plane of the paper.
.'.
T = MU mca, I
7' = m'u + m'c's ;)
cmu — c'm'u'
M k² + mc² + m'c ²
2;
(81)
(82)
whereby the initial angular-velocity of the machine, and the
initial tensions of the strings are known.
Again,
d20
T = mg m c
dt²
d20
T:
=
m'g + m'c'
dt2
do
Mk2
CT — C'T';
d t²
(cm-c'm') g
;
(83)
d20
dt 2 Μ k²+mc² + m'c²²
whence all the circumstances of motion may be determined.
(81)
If p = the pressure on the axis at the time t, it is equal to the
weight of the wheel and axle together with the tensions of the
strings; therefore
P = Mg+F+T
= (M +m+m')g
Mg+
(mc — m'c')²
мk²+mc²+m'c'29:
mm²(c + c')² + (m + m²) x k²
Mk² +mc² + m²c²²
g;
(85)
(86)
that is, the pressure on the axis is less than it would be if the
machine were at rest; but it can never vanish.
Ex. 3. It is required to determine the motion of a system of
wheels and pinions, such as a crane, or the like, the power at-
tached to the first wheel being P, and the weight attached to the
last pinion or axle being w.
Whatever is the form of the system, it may always be arranged
as in Fig. 28; where we have taken four wheels and pinions:
C1, C2, C3, C₁ are the centres of the successive wheels and pinions,
c, being that of the axle to which the weight w is attached. Let
the pressures between the successive wheels and pinions, whether
due to the action of teeth or to friction, be T1, T2, T3; let a₁, b₁,
b2, ɑ3, b3, ɑ4, b₁, be the radii of the several pinions and wheels in
order; and let H₁, H2, H3, H4 be their moments of inertia; let r
be the tension of the string to which the weight is attached, and,
A2
Hh 2
236
[140.
MOTION OF MACHINES
t the tension of that by which P acts; let м be the mass of the
weight w, and let p = mg; let us suppose w to descend in the
time at through a space da, and P to ascend through a space
dx'; and let d☺1, d02, d03, d☺ be the angles through which the
wheels rotate in that time; then
4
dx=α₁d☺₁; b₁də₁ = a₂dė½; b₂də½=a3d0z; b3d03 = αşdə; b4d04— — dx'. (8′
båd☺z=aşdos;
For the translation of w and P, we have
d2 x
T = W
M
dt2
d20,
W
as
a₁ M
;
(88)
dt2
d2x
t = P
M
dt2
d20s
= p + b ş m
(89)
dt2
And for the rotation of the pulleys, we have
d20,
1
H1
dt2
= a Ꭲ - b Ꭲ,
T
d202
-
H2 = α₂ T1 — b₂ T2,
dt2
d203
H3 dt2
d204
H4
dt2
2
= A3 T2 — b3 T3,
— αş Tz — b₁t ;
=
whence, by a simple elimination, we have
d2x
dt2
{H1 (α2 α3 α4)² + H½ (α¸ª¸b₁)² + н¸(ɑb¸ b₂)² + н (b¸ b½b3)²
H3
2
1
(90)
+м (α¸ª½ ɑzα¹)² + m (b₁ b₂ b3b4)²}
3 4
2 3
= а₁ª½α3α¹ {α1аžаžα§ w — b¸ b₂ь¸b₁º}; (91)
and, by integration, the space described by w in the time t may
be found.
d²x'
Also, from (87),
dt2
b₁ b₂ b ş bş d² x
1 2 4
a
а1α2α zɑş dt²
зад
;
and thus the motion of P may be determined.
(92)
A similar process may of course be applied, whatever is the
number of the wheels and pinions.
If in the preceding example the wheels are all equal, and all
the pinions are equal,
d2x
dt2
{H (a® + a¹b² + a² bª + b²) + м a² +mb8} = a¹ {wa¹ — pb¹}. (93)
M
P
140.]
237
WITH FIXED AXES.
Ex. 4. A heavy flexible and inextensible string of given length
a, is wound round a solid cylinder of mass м and radius c, which
is capable of rotation about its axis, which is horizontal; a piece
of the string of length b hangs down, so that the cylinder begins
to rotate; it is required to determine the motion of the string
and of the cylinder.
Let the circumstances at the time t be represented in Fig. 29;
and let мk² be the moment of inertia of the cylinder. In the
time t let a chain of length = ∞ = co be unwound from the
cylinder; let w =
∞ = the area of a transverse section, p = the den-
sity of the string; then the weight of the string which hangs
vertically at the time t=pwg (b+co). Let r the tension of
the string at the point P, CP = X.
d20
T
x = pw (b + c 0 ) { g − c
pw(b c 0) −
dt2
1 1 2 }
and the moment of inertia of the cylinder, and the chain wound
round it at the time t,
= Mk²+pw(a−b—c9) c² ;
so that the equation of rotation of the cylinder is
{мx²+pw (a-b-c0) c²}
d20
dt2
= CT
pwc (b+c 0) { g −
d² or
C
dt2
;
d20
pc (b+c0)g
dt2
м k² + pw ac²
d02
dt2
pocg
Mk²+pwac²
(260+c02),
(94)
do
since we have assumed
==
0, when = 0; therefore when
dt
co = a―b, that is, when the whole chain is unwound,
d02
dt2
pwg (a²-b²)
Mk2+pwac²
Again, from (94), for the whole time spent in unwinding the
string, we have
a-b
pwc g
do
C
Mk² + pw ac² (
t
(230+c02)
1
a + ( a² — b²) §
log
(95)
b
which gives the time.
By a similar process we may determine the length of string
238
[140.
MOTION OF MACHINES WITH FIXED AXES.
which a cylinder, rotating with a given angular velocity, would
wind up before it is brought to rest.
Ex. 5. A balance has equal weights in the scales, and oscillates
through small angles, the beam and scales moving in a plane
which is perpendicular to the axis of vibration; it is required to
determine the circumstances of motion.
Let the balance, &c. be represented in Fig. 30, in which the
plane of the paper is the plane of motion of the beam and scales,
and the axis of vibration is perpendicular to the plane of the
paper. Let o be the point where this axis pierces the paper; let
G be the centre of gravity of the balance without the weights; let
M the mass of the balance.
m = the mass of each weight in the scales.
Mk2
a
ов
the moment of inertia of the balance relative to the
rotation-axis.
the length of each arm = AB = BA'.
b.
OG = h.
Let the angle between Oв and the vertical line; which
0
angle, as well as its t-differential, we shall assume to be infini-
tesimal, so that the squares and higher powers may be neglected.
dx dx'
be the vertical velocities of the weights in the scales
dt' dt
Let
at P and P' respectively; let r and r' be the tensions of the
strings at a and a' respectively. We shall neglect the oscilla-
tions of the scales about the points a and A'.
The perpendicular distances from o on AP and A'P' respectively
are a cose-b sine, and a cose+bsine; which quantities, as 0 is
infinitesimal, are a b0 and a +60. So that the equation of
rotation is
Mk2
— T (a — b0) — T′ (a + b0) — мgh0.
d20
dt2
Now
T = m
(9 — dza),
2
but dx = d(asino + bcose),
M
r' = m (g — d'a');
(96)
dad(asine+bcose);
dx
d Ꮎ
dx'
do
(a cos 0-bsin 0)
(—a cos 0—bsin 0)
;
dt
dt'
dt
dt
d2x
d20
d²x'
d20
= (a cose-b sin 0)
(-a cos-bsin 0)
dt2
dt2
dt2
dt2
41.] fixed rotatioN-AXIS. PRESSURE ON THE AXIS. 239
so that (96) becomes
d20
dt2
2mb + Mh
2ma² + Mk2
go.
do
Let a be the value of 0 when
0; then
dt
d02
(2mb + мh)g
dt2
2ma² + Mk2
(a² — 0²);
t =
(97)
(98)
2 ma² +мk² ) *
(2mb +мh) g
M
and therefore the time of an oscillation
=
π
2ma²+Mk2
(2mb+мh)g ་
1
COS
α
(99)
and therefore if I the length of the simple isochronous pen-
dulum
2 ma²+Mk2
l
=
2mb +мh
(100)
141.] We must now return to equations (35) and (36), which
determine the pressures borne by the fixed ponts of the axis
during motion of this kind.
As the z-axis is fixed, the particles of the system have no mo-
tion in a direction parallel to that axis, so that for all particles
d2z
dt2
0; and therefore from the last of (35),
Σ.P COSY = Σ. MZ;
(101)
and as neither P1 COS Y1 nor P₂ cos
P2 cos y₂, enters into the other equa-
tions, this shews that the sum of the components of the pressures
along the z-axis is equal to the sum of the similar axial compo-
nents of the impressed momentum-increments on all the parti-
cles; but as the sum only is given each pressure is indetermi-
This case is similar to that of Art. 125, and admits of
a similar explanation.
nate.
I may observe in passing, that if the axis is capable of sliding
in the direction of its length, then the motion of all the particles
of the body along that line will be derived from the equation
d2z
and as
Σ.Μ
(z – d²) = 0;
dtz)
will be the same for all the particles, if м = the mass
dt2
of the body,
d2z
dt2
Σ.ΥΖ
M
(102)
240
[141.
FIXED ROTATION-AXIS.
whereby the longitudinal displacement of the axis may be de-
termined.
The components of the pressures at the fixed points, which
are perpendicular to the axis, enter into the first two equations
d² y
d2x
of both (35) and (36). In these let us replace and by
dt2
dt2
their equivalents, given in Art. 129, in terms of w; then these
four equations become
do
1
2
P₁ cosa₁ + P₂ cosa₂ = w²x.mx +
Σ.my +Σ.mx,
dt
(103)
dw
P₁ cos B₁+ P₂ cos B₂ = w²x.my
Σ.mx + Σ.MY ;
dt
dw
P₁ 1 CO§ ß1 + P₂%2c0s ß₂ = w²z.myz
Z.MXZ
- L,
dt
(104)
dw
P121 Cos α1 + P₂ ≈2 Cos a₂ = w²x.mxz +
Σ.my z + M ;
dt
from which equations the components of the pressures perpen-
dicular to the rotation-axis may be determined. It is worth while
to consider the forms which the preceding equations take rela-
tively to certain axes of the body.
(1) Let us suppose the rotation-axis to be a principal axis of
the body; and let us moreover take the origin at its principal
point; then .my z = x.mzx = 0; and (104) become
P121 COSB₂+ P₂ COS B₂
2 Z2 2
L, )
(105)
P121 cos a₁ + P₂ 2 COS a2 = M ;
from which, with (103), the pressures may be determined.
(2) Let us suppose the rotation-axis to be a central principal
axis, and the centre of gravity to be the origin; then
Σ.my z = x.mzx 0 ; Σ.mx = Σ.my = 0;
then (104) and (103) become respectively (105), and
P₁ cos a₁ + P₂ COS a₂ =
аг
.MX,
P₁ COS B₁+ P₂ Cos B2
cos
= Σ.MY;
(106)
whence we have
P₁ cosα₁ =
22.mx-M
Z2-21
-2₁.mx+M
P₁ cosẞ₁
P₂ COS α2
8α₂ =
2
P₂ cos B₂ =
Z2-21
Z₂ Z.MY+L;
Z2-Z1
—2₁E.MY-L
22-21
(107)
; (108)
If the points of support of the axis are equally distant from the
centre of gravity, so that Z2
-1
≈1, then
142.]
241
PRESSURE ON THE AXIS.
1
P₁ COS α1
2
P₂ cos az
2 %1
21.mx — M
271
,
1
2₁E.mx+M
Z₁ E. MY — L
2Z1
P₁ Cosẞ₁ =
; (109)
P₂cos B₂
21 E. MY + L
2%1
(110)
Y
(3) If no forces act on the system, so that x = y = z = 0 for
all particles; and L M N = 0; then the body rotates about
the fixed axis with the constant initial velocity n.
And if moreover the rotation-axis passes through the centre of
gravity, so that z.mx = x.my = 0; then, from (103) and from
(101), we have
>
P₁ cos a₁ + P2
cos a₂
1
0,
P₁ cosẞ₁ + P₂ cos B₂ = 0,
2
P₁ COS Y1 + P₂ COS Y2
0
(111)
whence we have
P₁ =
P2;
αγ
a2,
B₁ = B2,
Y₁
= 12;
(112)
so that the pressures at the fixed points are equal and opposite,
and act along parallel straight lines; they therefore form a
couple, the effect of which would be to alter the rotation-axis
of the body, were two points on the axis not fixed.
(4) Moreover, if the rotation-axis is a central principal axis
and no forces act on the system, in addition to (111) we have,
from (104),
PCOS B1+PZ2 COS B₂ = 0, )
2-2
P₁₁cos α1 + P₂ 2 COS α₂ = 0;
(113)
and therefore P₁ = P2 0; and no pressure exists at the fixed
points in the rotation-axis. This result agrees with that of
Art. 93, wherein it is proved that the couple of the centrifugal
forces vanishes for all points on a central principal axis. Hence
it is that such axes are called permanent axes, and are˜axes of
no pressure; they are therefore those axes about which a body
will rotate freely, and without fixed points in them, when no
forces act.
142.] The following are various applications of the preceding
results to certain particular Examples.
Ex. 1. A heavy sphere revolves uniformly about a vertical
chord, which is fixed at the two points where it meets the sphere.
Determine the pressures at the points.
Let the radius of the sphere = a; the mass of the sphere
M; 2c the length of the chord; so that the distance of the
chord from the centre = (a² — c²)³.
PRICE, VOL. IV.
the angular velo-
Let
I i
242
[142.
FIXED ROTATION-AXIS.
city of the sphere.
the plane of (x, y)
moves; then
Let the given chord be the z-axis, and let
be that in which the centre of the sphere
X.MX = Σ.MY = 0; x.mz = Mg; Σ.myz = Σ.mzx = : 0;
and at the time t let us suppose the centre of gravity to be in
the x-axis; so that x.mx = м (a² — c²)³, z.my = 0.
from (103) and (105) we have
Also
21 = C, Z₂ =
72
c; and L = 0, M = Mg (a²
—
~ c²) ✯ ;
1
M
P1 COS α₁ =
cos
2
(w² + 2) (a² — c²) + ;
M
P₂ COS α₂ =
2
(w² — 2) (a² — c²) * ;
P₁ cosẞ₁ =
1
P₁ COS Y ₁ + P₂ COS Y₂ =
Mg.
C
P₂ cos ẞ₂ = 0;
—
Ex. 2. A heavy cube makes a complete revolution about one
of its edges, which is horizontal and is fixed at the two angles of
the cube: find the pressures on these points when the centre of
gravity of the cube is in its lowest position.
M =
Let a = the length of the edge of the cube; м the mass;
and let the system be referred to a system of coordinate axes
which is delineated in Fig. 29; where the edge of the cube,
which is the rotation-axis, is the z-axis; A and A' its extremities
a
are the fixed points, each of which is at a distance from the
2
origin, the origin being taken at the middle point of the edge,
and the plane of (x, y) being that in which &, the centre of gravity
of the cube, moves; at the time t let the plane passing through
the rotation-axis and the centre of gravity be inclined at an
angle ✪ to the plane of (x, z), so that Gox = 0: now, if h is the
distance of the centre of gravity from the rotation-axis, and k is
the radius of gyration relative to the rotation-axis,
h =
also
Σ.mx = Mg,
a
2
L = 0, M
x.mx = мh cos 0,
Z.myz
2a2
k² =
;
3
Σ.ΜΥ
M.MY =
0, z.mz = 0;
Nмgh sine;
0,
x.my
мh sin 0,
Σ.mz = 0;
0,
Σ.mxz = 0 ;
d20
dt2
Mgh sin e
Mk2
;
142.]
243
PRESSURE ON THE AXIS.
d02 2gh
Ө
(cos 0 + 1);
dt2
k2
do
because
0, when 0 = ; and therefore at the lowest point,
dt
when ◊ = 0,
d02
4gh
شه
w² =
dt2
k2
Also, by (103), we have at
the lowest point, when 0 = 0,
d2x
2(x - 12m)
.m(s
P₁ cos α1 + P2 COS a₂ = .mx
..
a2
= Mg+Mhw²;
dt2
4Mg,
P₁ cos ẞ₁ + P₂ cos ß2
0.
• P₁ Cos α₁ + P2 COS α2
1
2
B2
And from (75), P₁ cos B₁ - P₂ cos B₂ = 0,
also
2
az
P1 COS α1- P₂ COS α2 =
0 ;
P₁ cos y₁ + P₂ cosy½ = 0;
Y1 P2
so that the whole pressure on the axis is equal to four times the
weight of the cube.
Ex. 3. A heavy sphere revolves about a horizontal tangent
and makes a complete revolution; the rotation-axis is fixed at
points cqually distant from the point of contact: find the pres-
sure on the axis when the sphere is in the lowest position.
Let us adopt the notation and arrangement of the last ex-
ample; then
h = a,
=
k² =
k2
7 a²
;
5
d02
2gh
(cos@ + 1)
dt2
k2
10g
(cos + 1);
7 a
20g
therefore at the lowest position, w²
;
τα
27
1
.'. P₁ COS α₁ + P₂ COS α₂ =
M 9,
7
0;
}
P₁ cosẞ₁+ P₂ cos Ba
P₁ cosẞP₂ cos B₂ = 0,
COS
az 0,
P1 COS α1 - P2 COS α₂ =
a
Y1
P₁ cos y₁ + P₂ cos y½ = 0;
so that the whole pressure on the axis, when the sphere is in its
27
27
lowest position, is
7
Mg
w, if w= the weight of the sphere.
7
I i2
244
FIXED ROTATION-AXIS. PRESSURE ON THE AXIS. [142.
Ex. 4. A heavy rod of length a and of mass м is fixed at its
two ends in a horizontal position; one support is removed, and
the rod turns about the other end find the pressure at this
latter end when the motion begins.
:
In this case, for the angular motion of the rod at the time t
we have
d20
dt2
3g
sin 0;
2 a
do
and since
0, when 0 =
90°,
dt
d02
3g
w2
cos 0.
dt2
a
0
Thus, from (103) and (104), when = 90°, we have
P1 COS α1 + P2 COS α2
3мд
+ Mg
4
Mg
;
4
= 0,
=
0,
az
0 ;
P₁ COS B₁+ P₂ cos B₂
2
P₁ cosẞ₁- P₂ cos B₂
P₁ COS α1 - P2 COS α2
1
so that while the rod is at rest on its two ends, each support
bears one-half of the weight; and if the support is removed, the
pressure on the other is immediately diminished to one fourth
of the weight of the rod.
143.]
245
ROTATION ABOUT A FIXED POINT.
CHAPTER VI.
THE ROTATION OF A RIGID BODY, OR OF AN INVARIABLE
SYSTEM, ABOUT A FIXED POINT.
SECTION 1.— The rotation of a rigid body about a fixed point
under the action of instantaneous forces.
143.] WHEN a rigid body, or any system of particles of in-
variable form, moves with one point of it fixed, it is evident that
it admits only of rotation about an axis passing through that
fixed point; generally, the position of this axis will continuously
vary, and will describe one cone fixed in the moving body, and
another cone fixed in space, which two cones touch each other,
and the line of contact of which is the instantaneous axis: it is
also evident that any given particle of the system will move on
the surface of a sphere whose centre is the fixed point.
We shall suppose the form, matter, and density of every part
of the moving system to be given; and therefore the position of
the principal axes, and the principal moments of inertia relative
to the fixed point, will also be assumed to be known: these latter
we shall take to be A, B, C, as in Chap. IV; and we shall assume
the order of magnitude to be the same as that of Art. 98; viz.,
A < B < C:
(1)
we shall also assume the position of the principal axes, as well as
the values of the principal moments of inertia, to be given at
every point of the system.
Let the fixed point be the origin; and at it let two systems of
coordinate axes originate; one of which we assume to be fixed
absolutely in space, and the other to be fixed in the body and
to move with it: this latter system we will take to be the system
of principal axes which originates at the point, because our ex-
pressions will be much simplified thereby. The motion of the
body will be in the first place referred to this latter system in
terms of the angular velocities about the principal axes; and the
incidents of its motion in space will be thence inferred by means
246
[144.
ROTATION ABOUT A FIXED POINT.
of the connecting equations (103), (101), (105) of Art. 42, or
some equivalents of them.
The general investigation will consist of two parts, according
as the system is under the action of instantaneous forces or of
finite accelerating forces. We shall consider the effects of in-
stantaneous forces in the present section, and in the succeeding
section those of finite accelerating forces; and in each case I
shall inquire into the resulting angular velocity, the position of
the rotation-axis, the pressure on the fixed point, and the other
incidents of motion.
144.] For the sake of simplicity, we will suppose the body to
be at rest at the time when the instantaneous forces act on it.
Let us first refer all the elements to the system of axes fixed in
space; and let the axial components of the impressed momenta
be Σ.mvx, Σ.mv, z.mv.; and if the force is a single blow which
impresses a momentum Q, let QË, Qy, Q- be its components. Let
(x, y, z) be the initial place of m; let è be the pressure at the
origin due to the forces, and let λ, u, v be the direction-angles
of its line of action; then the equations of motion are
P
Σ.Μ
(V.e-day)
dx
P COSλ = 0,
dt
Σ.Μ
(v, - dy)
P COSμ = = 0,
(2)
P COS v = 0);
Σ.Μ
Σ.Μ
(v₂ — dz)
い
​dt
{y (v₂ — dz) — z (v, — dy) }
dt
dx
dt
dz
= 0,
2.18 { = (v, 1) — x (v. — 1/2) = 0,
Σ.Μ
V
dt
dt
z. — = 0.
{ x (v, — dy) — y (vx — da) }
Σ.Μ
dt
– Y
dt
(3)
Let these equations be transformed into their equivalents in
terms of angular velocities, as in Art. 73. Let o be the angular
velocity which results from the instantaneous forces, and let
Ax, y, be its axial components; then (2) become
Z.Mvx
Q₂ E. M Z + QzE.my-PCOSλ = 0,
Σ.MVy—QzE.mx + QE . m z — PCOSµ =
Σ.mz-Рcosµ 0,
M -
Σ. MV; —Q₁E. my+QΣ.mx — P cos v = 0;
(4)
from which, or from (2), the pressure at the fixed point, and the
direction-cosines of its line of action, may be determined.
145.]
247
INSTANTANEOUS FORCES.
Let us replace (3) by their equivalents in terms of angular
velocities, about the principal axes fixed in the body and moving
with it; let G be the moment of the couple of the impressed
momenta, or of the blow, if this motion is due to a blow; and
let L1, M1, N1, be the axial components of the moment of this
couple relative to the three principal axes; and let, be
the axial components of the instantaneous angular velocity.
Then, by the reduction which has already been made in Art. 92,
the equations (3) become
C² = = N₁:
1
(5)
B₂ = M1,
these equations determine the axial components of the angular
velocity relative to the principal axes fixed in the body and
moving with it. From these the angular velocities relative to
the three axes fixed in space may be found, by means of the
equations given in (87), Art. 40; and thus the position of the
initial rotation-axis is absolutely determined.
The preceding equations admit of dissection, and of deduction
from first principles, in a manner similar to that which has been
employed in Art. 81 and 82. It is consequently unnecessary to
repeat it.
145.] From (5) we have
£1
2₁ = 11
L1
√ =
A
B
2
2
2
LG
M₁
+
A
42
B2
2
+
2
23 =
ཞི་
(6)
C
1
C2
A
(7)
and if a, ß, y are the direction-angles of the instantaneous rota-
tion-axis,
21
23
COS α =
a
cos B
cos y =
(8)
Ω
Ω
Ω
L1
M₁
N1
;
(9)
ΔΩ
ΒΩ
ΣΩ
Ω
whence n and the direction-angles of its rotation-axis are known.
Hence the equations to the instantaneous and initial rotation-
axis are
A X ву
L1 M1
C
N1
(10)
As L1, M1, N₁ are the axial components of the moment of the
impressed couple, or, as Poinsot calls it, of "the couple of im-
pulsion," the equations to its axis are
x
L1
y
;
N
1
(11)
248
}
ROTATION ABOUT A FIXED POINT.
[145.
and the equation to its plane is
L₁ ∞ + M₁Y + N₁ ≈ = 0.
(12)
Now these expressions admit of the following interpretation :
In reference to the principal axes fixed in the body, the equation
to the momental ellipsoid is
Ax² + By² + c≈² = µ ;
(13)
relatively to which the equations to the axis conjugate to the
plane (12) are, see Art. 14,
AX
BY
CZ
;
L1
M1
N1
(14)
but these are the equations to the instantaneous initial axis.
Hence we have the following theorem :
The instantaneous axis of rotation, due to a given impressed
couple, is the axis of the momental ellipsoid which is conjugate
to the plane of the couple. Hence, if the momental ellipsoid of
the moving system is constructed, and a plane is drawn touching
it, and parallel to the plane of the couple of impulsion, the cen-
tral radius vector of the ellipsoid drawn to the point of contact
is the instantaneous axis.
Let the point where the instantaneous axis meets the ellipsoid
be called the Instantaneous Pole; then the tangent plane of the
ellipsoid at the instantaneous pole is parallel to the plane of the
couple of impulsion.
Also, the initial angular velocity varies as the square of the
central radius vector of the momental ellipsoid, which coincides
with the initial rotation-axis. For if (x, y, z) is the initial in-
stantaneous pole, and R is its distance from the centre of the
ellipsoid,
X
y
Z
21
22 £3
||
Ꭱ
Ω
&
A x²² + Bу² + C÷²
2
2
AN² + B22² + C Qz²
με
2
{L₁ ~ + M₂ Q₂+ N₂ ~, } ±
μέ
{GOCOS (} \
if is the angle between the initial rotation-axis and the axis of
the couple of impulsion; and therefore
R2G COS
Ω
μ
(15)
145.]
249
INSTANTANEOUS FORCES.
$
that is, the initial angular velocity varies as the square of the
central radius vector of the ellipsoid which coincides with the
initial rotation-axis, and as the component relative to that axis
μ
of the moment of the couple of impulsion. Also, as is the
R2
moment of inertia relative to that axis, the preceding equation
is the form which the equation (16), Art. 73, takes in this parti-
cular case.
Hence, if a body rotates about an axis passing through a
fixed point, the plane of the momental ellipsoid conjugate to that
axis is the plane of the couple which instantaneously impressed
in an opposite direction will bring the body to rest; and if the
axial components of the angular velocity of the body at that
instant are w₁, 2, 3, and G' is the moment of the couple which
reduces the body to rest,
2
2
G A² w₁² + B² w₂² + c² wz² ;
(16)
and the equation to the plane in which the couple must be im-
pressed is
A W₁ X + B W₂ Y +Cwz≈ = 0;
(17)
we shall hereafter shew that &'= &, and that the position of this
plane in space is invariable.
If the plane of the couple of impulsion is a principal plane of
the momental ellipsoid, the instantaneous rotation-axis is the
axis of the couple; but for no other plane of impulsion will the
axis of the couple be also the instantaneous rotation-axis. Hence,
if a body is rotating about a principal axis, it may be brought to
rest by a couple whose axis is that rotation-axis; but in no other
case will the axis of the couple which brings the body to rest
coincide with the rotation-axis.
Hence also, if a body is rotating about a given axis, and a
blow is given to the body whence a couple of impulsion arises, if
the plane of this couple is conjugate to the original axis of rota-
tion, no change of rotation-axis is caused; but if the plane of
the couple of the blow is not conjugate to the previous rotation-
axis, a change of axis takes place. And therefore if a body is
rotating about a principal axis, and a blow is given to the body
which produces a couple whose axis is that principal axis, the
position of the rotation-axis will be unaltered, and there will only
be a change of angular velocity.
I may observe, that if the axes fixed in the body are not
principal axes, equations (32), Art. 76, will take the place of
PRICE, VOL. IV.
кк
250
[146.
ROTATION ABOUT A FIXED POINT.
(5), and the equation of the momental ellipsoid would be (114),
Art. 98; it is therefore unnecessary to repeat them here. From
these equations however the same geometrical interpretation as
that which we have just arrived at may be deduced.
It is to be observed, that cos a, cos ß, cos y are independent
of G, the moment of the momentum of the couple of impulsion;
so that if the body is put into motion by a blow, the position of
the instantaneous rotation-axis is the same, whatever is the in-
tensity of the blow, provided that its line of action is the same.
Not so however the initial angular velocity.
146.] The following examples are in illustration of the pre-
ceding.
Ex. 1. A right angled triangular plate, fixed at its centre of
gravity, is struck at its right angle by a blow perpendicularly to
its plane it is required to find the position of its initial instan-
taneous axis.
Let 3a, 36, see Fig. 32, be respectively the side CA, CB of the
triangle, p = the density, the thickness of the plate. Let
the origin be taken at the centre of gravity; and let the coor-
dinate axes be parallel to the sides. As these axes are not prin-
cipal, we must recur to (32), Art. 76, for x, y, z. In reference
to the origin and axes which we have chosen, the equations to
the sides BC, CA, AB respectively are x=
1;
X y
a, y = —b, + = 1
α b
and consequently
A = x.m (y² + 2²)
9ρτα63
4
D = E.MY Z
0,
B = Σ.m (z² + x²)
9ρτα
4
EZ.MZ x =
0,
c = Σ.m(x²+ y²)
9 prab (a²+b²)
9 pra2b2
=
> F = x.mxy =
4
8
Let a be the momentum impressed by the blow at c, which is
(—a, —b), in a line perpendicular to the plane of the plate, and
parallel to the axis of z in a positive direction; so that we have
L1
- bq,
8Q
nx =
27 ρταb2
8Q
№1
=
0;
M₁ = αQ,
Ly =
27 pтa²b
ρτ
Ωχ
n₂ = 0;
X
+
20
= 0;
and therefore the equation to the instantaneous initial rotation-
axis is
α
consequently the initial rotation-axis, which of course passes
146.]
251
INSTANTANEOUS FORCES.
through the fixed point, is parallel to the hypothenuse of the
triangle.
Ex. 2. Let us consider the general case of a plate of infinite-
simal thickness, which has one point fixed, and which is struck
at a given point (xo, yo) by a blow q, in a line perpendicular to
the plane of the plate.
The plane of the couple of impulsion is evidently that passing
through the point of the blow and the fixed point, and perpen-
dicular to the plane of the plate. Thus its equation is
X Y
= 0.
Хо
Yo
Now the axis of the momental ellipsoid (13), which is conjugate
to this plane, is
AX。X + BY Y = 0;
and this therefore is the initial rotation-axis.
(18)
Also, if q = the momentum impressed by the blow at the
point (o, yo) in a line perpendicular to the plane of the plate,
and in a direction parallel to the positive direction of the axis of z,
..
L1
YoQ,
YoQ
M1
xoQ,
N1 = 0;
XOQ
A
↓₂ = 0;
B
2
(30
2
xo 0
2
+
Q.
A
B2
Thus, if the plate is elliptical and fixed at its centre,
A =
πρτο
4
B =
πρτα
4
;
and consequently the equation of the initial rotation-axis is
X XO
a²
Y yo
+
0;
b2
that is, the initial axis is conjugate to the axis passing through
the place of the blow.
If the plate is parabolic, and fixed at its vertex, and if a and b
are severally the length and the extreme ordinate of the plate,
A
4ρτα 3
15
B=
Αρτα b
7
;
and the equation to the initial instantaneous axis is
0
7 x x
3a2
5yyo
Ο
+
b2
0.
Ex. 3. A cube fixed at its centre of gravity is struck by a
blow, whose momentum is q, along an edge: it is required to
determine the initial instantaneous axis.
K k 2
252
[147.
ROTATION ABOUT A FIXED POINT.
Let 2a = the edge of the cube; let the origin be taken at the
centre of gravity of the cube, and let the coordinate axes be
parallel to the edges; let the line of the blow a be parallel to
the axis of ≈; and let its point of application be (a, a, 0); so that
L₁ = a Q,
M₁ =
aQ,
N₁ = 0;
16ρτα
A = B = C
;
3
2₁ =
21
3Q
16ρτα
3Q
Q2
0 ;
16ρτα
4
and therefore the equation to the initial rotation-axis ist
x + y = 0:
this result is evident from the theorem, that the instantaneous
initial axis is conjugate to the plane of the impulsive couple ;
for the momental ellipsoid is in this case a sphere, and therefore
the instantaneous rotation-axis coincides with the axis of the
couple of impulsion.
147.] Let us now consider the equations (2), or their equi-
valents (4), by means of which the pressure at the fixed point,
which is due to the impulsive forces, is to be determined.
Let м be the mass of the body; and let its centre of gravity
be (x, y, z); then
Σ.mx = Mx,
Σ.my = My,
Σ.mz = MZ;
and (4) become
P COS λ = Σ.MV x + м (QzŸ — Q2T),
P COS µ = Z.MV, + M (î₁Z — Qzπ),
P COS v = Σ.MV½ + м (ν× îÿ).
M (Q2 X
(20)
Now the last terms of these three equations are evidently the
axial components of the momentum of the whole moving mass
condensed into its centre of gravity; so that the pressure which
acts at the fixed point is the sum of the impressed momentum,
and the momentum of the whole mass condensed at its centre of
gravity, which is due to the initial angular velocity.
To apply these formulæ, let us take Ex. 1 of the preceding
Article in that case
x = ÿ
y
z
=≈ = 0;
Σ.mv
=
X
Σ.MVy
Σ. MV y =
.'.
P COSA
.*. · P = Q;
PCOS μ = = 0;
0, z.mv₂ = Q ;
P COS V Q;
and the line of pressure is perpendicular to the plate.
If there is no pressure at the fixed point, then
148.]
253
INSTANTANEOUS FORCES.
whence we have
23Y)
Σ. MV x — M (QqZ — y) = 0,
Σ. MV₁ — M (3 x — î₁z) = 0,
vy
Σ. MV₂ — м (îÿ — ↓₂π) = 0 ;
xz.mvx + y z . m v₁ + Z z . m v₂ = 0;
y
QE. MV x + Q₁₂ E. M Vy + QzE. M V z
Z. Qq = 0;
(21)
(22)
(23)
and it appears that the line of action of the resultant of the
impressed momenta, or of the blow, if the motion is due to a
single blow, is perpendicular to the plane containing the fixed
point, the centre of gravity, and the rotation-axis of the initial
angular velocity.
If the fixed point is the centre of gravity of the body, then
x = y = z = 0; and
P COS λ = Σ.MV
P COS μ.MVy,
P COS V Σ. m Vz ;
and the pressure is in intensity, direction, and line of action, equal
to the resultant of the impressed momenta.
SECTION 2.-The rotation of a rigid body about a fixed point
under the action of finite accelerating forces.
148.] LET us, as in the preceding section, refer the motion of
the body, or material system, to two sets of coordinate axes,
originating at the fixed point: one of which is fixed in space,
and the other is fixed in the body, and moves with it: let this
latter system be the principal system relative to the fixed point.
Let p be the pressure at the fixed point at the time t, and let
λ, μ, v be the direction-angles of its line of action relatively to
the axes fixed in space: relatively to the same axes let (x, y, z)
be the place of m, and let x, y, z be the axial components of the
impressed velocity-increment; then the equations of motion are
Σ.Μ
d²
m (x
n(x-
(x - 12x) -
Σ.mx
dt2
d2
(x — day)
Σ.Μ Υ
dt2
d2 z
dt2
(2 – 192)
Σ. Ζ
x
−
d2z
dt2
d2 x
P COSλ = 0,
P COSμ =
= 0,
(24)
—P COS v = 0;
- = (x
2 Y
dt2
d2z
(x -
d²y
0,
0,
(25)
= 0.
d²,¤ ) − x ( z
x.m {y (z
m { z (
x
dt2
Σ.Μ
m { x (
Σ.Μ
dt2
2
−
dy) — v (x —
:).
d² z
dt2
2
d²x
dra)
dt2
)
}
}
=
254
[148.
ROTATION ABOUT A FIXED POINT.
Let equations (24) be transformed into their equivalents in terms
of angular velocities, as in Art. 77; let o be the angular velocity
about the instantaneous axis at the time t, of which let wx, wy, wz
be the axial components; then (24) become
dwz
dt
Σ.mx
dwy
dt
·Σ.mz+
do
Σ.ΜΥ
-Σ.mx +
dt
do
dt
Σ.ΜΖ
dox.
dt
z.my — wxz.m{wxx+w»Y+w₂~} +w²z.mx — P cosλ=(),
·z.mz — wyz.m {wxX+w₁Y+wzz}+w²z.my-PCosμ=0,
dwy
·Σ.my + Z.MX
dt
wzZ.M {wxX+w»Y+wzz} + w²z.mz— pcosv=0;
from which, or from (24), the pressure at the fixed point, and
the direction-cosines of its line of action are to be determined.
Let us replace (25) by what they become in terms of angular
velocities about the principal axes fixed in the body and moving
with it this reduction has been made in Art. 92; and for the
equivalents of (25) we have
:
dwi
A
+ (C−B) W2 W3 = L,
dt
dwz
B
+ (A−C) WzW1 = M,
dt
dwz
C
+ (B−A) W1 W₂ = N
dt
(27)
where L, M, N are the axial components of the moment of the
couple of the impressed momentum-increments relative to the
principal axes.
As these equations have been already analysed and deduced
from first principles in Art. 80-83, it is unnecessary to give
further explanation of them. I may observe that the instantane-
ous angular velocity, and its axial components relative to the
moving principal axes, may be (theoretically at least) derived
from (27); and thence the axial components relative to the axes
fixed in space by means of the equations given in Art. 40; or, as
we shall find more convenient, we may determine 0, 4, and √ by
means of Art. 42, and thus determine the position of the body
in space, as well as the incidents of its motion at the time t.
As the pressure at the fixed point, which is to be determined
by means of (26), depends on the instantaneous angular velocity
and its axial components, these must be first determined; and
consequently I proceed to consider (27) and (25).
The general solution is beyond the present range of Mathe-
matical Analysis; and we can investigate only those more simple
cases which can be solved either partially or totally.
149.]
255
FINITE FORCES.
149.] Let us first take the case where L = M = N = 0; that
is, when the conditions (67), Art. 58, are satisfied; viz., when
the forces which act on the body are of the nature explained in
the four cases of that Article.
And this problem too is of wider application. It is that of
a heavy body rotating about an axis which always passes through
its centre of gravity; for as the resultant force of gravity acts
through this point, it creates no moment of rotation about an
axis passing through it, and therefore so far as gravity affects
the motion L = M = N = 0. Thus, if a heavy body has motion
of both translation and rotation; the principle proved in Section
2, Chapter III, shews that the centre of gravity will most con-
veniently be taken as the point whose motion of translation is
estimated; and the motion of rotation of the body about an axis
passing through that point will be estimated by the following
process.
In all these cases the equations of motion are
Σ.Μ
Σ.Μ
d2z
(y d² z
(น
(=
Z
dt2
d2x
d t²
2
d2y
2
=
= 0,
dt2
d2z
X ·) = 0,
d t2
d2 x
(x day — y 12 a) = 0;
Σ.η α dt2
and their equivalents relative to principal axes are
dw₁
A + (CB) W₂ W₂ = 0,
dt
W2 W3
(28)
dw2
B
+(A−Ċ) w₂W₁ = 0,
WzW1
(29)
dt
dwz
C
dt
+ (B−A) W₂ W₁₂ = 0;
equations (28), it will be remembered, are relative to the axes
fixed in space; and (29) refer to the principal axes fixed in the
body; but it is desirable to retain both groups of equations as
each will give theorems of importance.
Let us consider (29), and let 1, 2, 3 be the axial components
of the initial angular velocity o about the principal axes when
t = 0; the incidents of such an initial velocity have been con-
sidered in the preceding Section.
Let equations (29) be severally multiplied (1) by w1, W2, W3 ;
and (2) by Aw1, B2, Cwg; and let them be added: then we have
256
THE VIS VIVA OF THE SYSTEM.
[150.
dw2
dwz
0,
do;
AW1 d t
+ BW2 d t
+ c @ 3 d t
(30)
dw1
dwz
d w z
ΑωΙ
2w1 dt
+B² W₂
+ B = W 2 d t
+c² w z dt
= 0;
then integrating, and taking the limits corresponding to t=t,
and to t = 0, we have
2
2
A (W₁²—₁₂²) + B(w₂² —N2²) + C(wz² — Nz²)
2
₪3²) = 0 ;
(31)
2
2
A² (w₂² — î₂²) + B² (w₂² —₂²) + C² (wz² — £3³) = 0 ;
2
(32)
3
2
which are two integrals of (29). Let us interpret these results.
150.] From (31) we have
.@
2
A W₂² + B W₂² + C wz²
2
2
A Q² + B Q‚² + C Q².
2
2
(33)
Let a, ß, y be the direction-angles of the instantaneous axis
at the time t relative to the principal axis; so that, if w is the
instantaneous angular velocity, and z.mr2 is the moment of in-
ertia relative to that axis,
2
2
2
A w₁²
1² + B w₂² + c wz² =
w² {A (cosa)² + B (cos ẞ)²+c (cosy)2}, (34)
= w²x.mr²
= x.m w² p2
= x.mv²
Σ.m
= the vis viva of the body;
C
(35)
so that from (33) it appears that the vis viva of the body is con-
stant throughout the motion. Let this constant vis viva be
symbolised by k²; then we have
2
2
2
2
2
2
Aw₁²+Bw₂²+Cwg² = A² + B₂² + Cz²;
k².
(36)
(37)
This result is at once evident from the principle of vis viva; see
Art. 64; and also may be deduced immediately from (24). For
if we apply to the equations the process of virtual velocities,
which has been already explained in Art. 63, inasmuch as there
is no virtual velocity of the fixed point at which P acts, we have
Σ.mv2 = a constant.
Σ.Μ
{
S d² x
2
dx +
dt2
dt2
d²y dy +
d2z
dz = 0;
(38)
dt2
(39)
2
2
2
(40)
151.] Again, from (32) we have
2
2
A²w,² + B² w₂²+C² w¸² = A²n²+B²N₂²+C² Nz².
Now, suppose L, M, N, to be the axial components of the mo-
ment of the couple G₁, which impressed on the body at the
time t would reduce it to rest; then, by (6), Art. 144, we have
L₁ = AW], M₁ = BW₂,
1
N₁ = CW3;
(41)
151.]
257
THE INVARIABLE AXIS.
G1
is called the effective couple at the time t; and suppose L, M, N
to be the axial components of the moment of the couple of im-
pulsion G; so that
L = AQ1,
M = BQ2,
N = C^3;
then, from (40), we have
2
2
2
(42)
L₁² + M₁² + N₂² = L² + M² + N²;
1
..
G₁ = G;
(43)
that is, the moment of the effective couple at the time t is equal
to the moment of the couple of impulsion; and therefore, gene-
rally, the moment of the effective couple is constant. This is a
particular case of the principle of the conservation of the mo-
ments of the momenta, which has been proved in Art. 58. Thus
(32) becomes
2
2
A² w₁² + B² w₂² + c² wz²
W
= G².
(44)
Also the position of the rotation-axis of the effective couple is
invariable during the whole motion, being that of the axis of the
couple of impulsion. To prove this theorem we must have re-
course to the equations (28), which refer to the axes fixed in
space. These admit of integration; and we have
Σ.m
(2)
dz
dt
dx
dy
2
dt
=
Σ.Μ
y at
h₁,
dz
( z Cze
Ꮖ
h₂,
(x dy
-
hz;
J
Σ.Μ
dt dt
dx
Y
dt dt
(45)
Now the left-hand members of (45) are the axial components of
the moment of the effective couple at the time t; consequently
each of these is constant; and the moment of the effective couple
is also constant; which is the theorem just now proved by means
of the equations (29); and we have
2
h₂² + h₂² + h²²
2
2
G2 =
G₁²;
2
(46)
and relatively to the axes fixed in space, the direction-cosines of
the axis of this constant effective couple are
h₁
h₂
h3
;
G
G
G
(47)
this line is called the Invariable Axis.
As the position of the coordinate axes fixed in space is arbi-
trary, let us assume it to be that which coincides with the prin-
cipal axes of the body when t = 0, and when the angular velo-
cities about the principal axes are 1, 2, 3; then A1, B2, C♫g
PRICE, VOL. IV.
Ll
258
[152.
THE INVARIABLE AXIS.
are the axial components of the moments of the effective couple
at that time; and we have
hy
= ΑΩ,
H₂
B 22,
hç = соз;
(48)
so that under this supposition the direction-cosines of the fixed
axis of the effective couple are
A 21
G
BQ2
соз
G
G
Hence also the plane of the effective couple is
h₁ x + h₂ y + hz z = 0;
2
(49)
(50)
and this is the invariable plane of Art. 60; and is that in which
the sum of the products of each particle and its projected sectorial
area is a maximum.
In reference to the axes fixed in the body the direction-cosines
of the axis of the effective couple are, see (41),
AW1
G
BW 2
G
C W3
G
and the equation of the invariable plane is
A W₁ z w
X + B W₂Y + C Wz≈ = 0.
(51)
(52)
These results give the following equations of condition between
the nine direction-cosines of the two systems of reference.
h1
Since is the cosine of the angle between the axis of the
G
C@3)
G
and the fixed x-axis (α₁, b₁, c₁), those
direction-cosines referring to the principal axes in the body, we
G
couple (401 BW2 CW3
Ꮐ
have
h1
similarly,
and
B
A ɑ2w1 +в b₂ w₂ + CC2 W3
h₂,
▲ α₁ w₁ + B b₁ w₂ + C C ₂ W z
2
· A Az Wy + B b z W 2 + C Cz wz = hz;
1
(53)
and as six other equations of condition have already been given,
the nine direction-cosines may be determined. Thus, if w₁, w2, W3
are given, we may determine the values of the direction-cosines
which fix the principal axes of the body relative to the axes
fixed in space, and thus the position of the body will be com-
pletely determined.
I may in passing observe, that (53) may be deduced from
(45) by transforming the coordinates in (45) to those of the
principal axes.
152.] It is however frequently more convenient to determine
the position of the body by means of the three angles 0, 4, Y, of
Articles 3, and 42.
153.]
259
THE INVARIABLE AXIS.
Let us take the invariable plane for that of (x, y) fixed in
space, the invariable axis being the axis of z; so that h₁ = h₂ = 0;
G: thus, from (53), we have
h3
A α ₁ w ₁ + B b ₁ w₂ + C C₁ wz = 0,
A
α ₂ w z + B b ₂ w z + C C₂ wz = 0,
A
ɑz w₁ + B bz wz +CCzWz = G.
(54)
W1
Let us multiply these equations severally by a₁, ɑ2, α3 ; by b₁, b2, b3 ;
by C1, C2, C3; and add in each case; then
Αω1 = Gaz,
and replacing a, b, c
BW 2
=
Gb3,
Cwg = GC3;
(55)
have
AW1
by their values given in (22), Art. 4, we
= G sin & sin 0, Bw₂ = G cos & sine, cw, G cos &; (56)
AW1
CW3
tan & =
cos =
BW 2
Ꮐ
(57)
does not enter into these equations, because the axis of x in
the invariable plane from which y is measured is quite arbitrary,
and and are sufficient to determine the position of the prin-
cipal axes at the time t in terms of w₁, w2, w, which are functions
of t, and are to be determined from the equations (29).
The t-variation of y however, that is, the angular velocity of
ON, see Fig. 1, about the invariable axis may thus be found.
Employing (104), Art. 42, we have
dy
@1
sin + w₂ cos &
dt
2
sin
2
Ꮐ
2
A w₂² + B w₂² G;
A² w₂² + B² w₂ ²
W1
2
(58)
in which, if w, and we are replaced by their values in terms of t,
and the equation is integrated with limits corresponding to t=t,
and to t = 0, the precessional angle due to that time will be
determined. As the right hand member of (58) is necessarily
positive, it follows that increases with t; so that the preces-
sional motion in the plane of (x, y) is always from x towards y,
that is, is direct according to our arrangement of signs and axes.
153.] The instantaneous angular velocity being ∞, the direc-
tion-cosines of the instantaneous rotation-axis relative to the
@1 W2 @3
་
principal axes are
;
W
ω
Ο
and the direction-cosines of the
AW1 BW2 CW3
invariable axis are, relatively to the same axes,
Ꮐ
G
G
L 1 2
260
[154.
FINITE FORCES.
thus, if is the angle between the instantaneous axis and the
invariable axis,
cos o
11
2
A w₁² + B w₂² + cwz?
k2
GW
Ꮐ
G W
2
by reason of (37) ;
.. w cos & =
k2
G
(59)
that is, the component of the instantaneous angular velocity
about the invariable axis is constant.
154.] The complete solution of the problem requires w₁, w2, w3
to be expressed in terms of t; the following is the differential
equation in terms of do and dt.
From (37) and (44) we have
W₂ 2
w₂² =
¤ k² — G² — A ( C — A) w₂²
B(C-B)
C (C—B)
A(B-A), 2-B k² + G²
3
2
and therefore from the first of (29)
(B —— A) (C — A)
dw₁
dt
+
{(BA)
BC
(60)
(61)
(w₂ ²
Bk2-G2\ /ck-G2
ck²
2
2
ω =0; (62)
A(B-A
A (C — A)
dw,
= ±
2
G
k² — G²
BC
(B—A) (C—A) } * di; (63)
th
2
A (C — A)
{(w, ²
2
Bk2
A (BA)
the left hand member of this equation is generally an elliptic
transcendent, and so does not admit of integration in finite
terms. In certain particular cases, as we shall see hereafter, it
may be integrated; and in these cases the problem admits of
dwz
complete solution: similar equations may also be found for dt
and dwa; or w, having been determined in terms of t, we may
dt
substitute its value in (60) and (61), and we shall thus obtain
W2
and @3 in terms of t. Hereby (57) and (58) will assign the
position of the body and the precessional velocity.
155.] The centrifugal forces which are generated by the mo-
tion possess some peculiarities which it is desirable to indicate.
Let L″, M″, N″ be the axial components of the moment c" of
the couple of the centrifugal forces; then from (62), Art. 83, or
from the equations (29),
156.]
261
POINSOT'S INTERPRETATION.
L" (B-C) w₂ Wz,
M'= (C—A)WzW1, N"= (A-B) w₂w₂; (64)
L″ w₂ + M" w₂ + N″ w3 = 0,
(65)
L'" Aw₂+M"BW₂+N″Cw₂ = 0;
(66)
so that the axis of the couple of the centrifugal forces is at once
perpendicular to the instantaneous rotation-axis, and to the in-
variable axis; and the plane of the couple of the centrifugal
forces contains the instantaneous and the invariable axes.
And for the magnitude of G" we have
G″2
112
Ꮐ
=
2 2
2
2
2
(B — C)² w₂² w₂² + (C — A)² wz² w₁² + (A — B)² w₂² w₂²
2
2
2
w² (A² w₂ ² + B² w₂ ² + C² wz²) − ( A w₂² + B w ₂ ² + c w z 2 ) 2
= G² w
G2w²-k¹
k4
= G² w² (sin )²,
= 4 (tan p)²;
(67)
(68)
that is, the moment of the couple due to the centrifugal forces
varies as the tangent of the angle contained between the instan-
taneous and the invariable axes; and is represented in magni-
tude by the area of the parallelogram whose sides are the line-
representatives of the instantaneous angular velocity and of the
moment of the couple of impulsion.
156.] Now of all these results a perfect geometrical interpre-
tation has been given by M. Poinsot in his admirable "Théorie
nouvelle de la Rotation des corps." Liouville's Journal, Vol.
XVI; and Paris, Bachelier, 1851. The following Articles are
for the most part extracted from that work: he has extended
his investigations to a point beyond the scope of our volume;
and the student desirous of farther research must have recourse
to the original memoir. Indeed the whole of it deserves a most
careful study; the reasoning of it is, for the most part, synthe-
tical; and it is a model of such a treatment of a mechanical
question; the difficulties with which the problem abounded are
cleared away, and the motion of the body from one position to
another can be traced in the mind's eye without any interrup-
tion.
In the equation to the momental ellipsoid in Art. 98, μ is
arbitrary; here however we shall suppose it to be equal to 1, as
hereby the formulæ will be simplified. Then the equation to it
relatively to the fixed point, and to the principal axes fixed in
the body and originating at it, is
A x² + B у² + C≈² = 1.
Let the point where the instantaneous axis meets this ellipsoid
262
[156.
POINSOT'S INTERPRETATION.
be called the Instantaneous Pole; then, if (x, y, z) is this point,
and if R is its distance from the fixed point,
X
y
2
R
| 3
W2
@:3
W
(69)
2
(A x²+By²+C≈²) §
2
2
(A W₁² + B W₂ ² + C w₂ ²) &
1
k
(70)
so that the instantaneous pole is (1,2,3); and as r is the
k
kk
R
central radius vector of the ellipsoid coinciding with the instan-
taneous axis,
W Rk;
(71)
so that the instantaneous angular velocity varies as the radius
vector of the momental ellipsoid which coincides with the in-
stantaneous axis; and consequently its singular values coincide
with the singular values of this radius vector. Of all possible
values therefore of this angular velocity the greatest will be
when the instantaneous axis is that of least moment, and the
least will be when the instantaneous axis is that of greatest
moment.
;
The equation to the tangent plane at the instantaneous pole is
A W₁ X + B W₂Y + c wz z = k ;
(72)
and this is parallel to the invariable plane (52). The perpendi-
cular distance from the origin on (72)
2
k
2
(A² w₁² + B² w₂² + c² wz²) ³
k
G
(73)
which is constant; and therefore the plane which touches the
momental ellipsoid at the instantaneous pole is absolutely fixed
in space, being parallel to the invariable plane, and at a distance
k
G
from it. Now this fact supplies us with the following
image of the body's motion.
:
Let us suppose the momental ellipsoid of the body relatively
to the fixed point to be described as the preceding equations
involve only the principal moments of inertia of the body, we
may imagine the body to be replaced by its momental ellipsoid,
and the motion of the latter will be a correct representation of
that of the former.
Let the invariable plane be drawn at the fixed point, and let
156.]
263
POINSOT'S INTERPRETATION.
k
a perpendicular be drawn to it equal to (73); through the
G
>
extremity of this perpendicular let a plane be drawn parallel to
the invariable plane; this plane is that whose equation is (72),
and the point where the momental ellipsoid touches this plane
at any time is the instantaneous pole at that time. If then we
imagine the momental ellipsoid, whose centre is fixed at the
given point, to roll without sliding on this fixed plane which it
always touches, we shall have a true image of the body's motion.
Moreover, the point of the contact is the instantaneous pole,
and the radius vector to it is the instantaneous rotation-axis ;
the angular velocity varies as the length of this radius vector,
by reason of (71); and the resolved part of the instantaneous
angular velocity about the invariable axis is constant. These
circumstances are delineated in Fig. 33. o is the fixed point,
and the centre of the momental ellipsoid ABC; OG is the in-
variable axis, and is the axis of the couple of impulsion; along
it is taken a distance oG, equal to ; through & a plane is
Ι
k
G
drawn parallel to the invariable plane, and on it the momental
ellipsoid rolls; let I be the point of contact at the time t; 1 is
the instantaneous pole, or is the instantaneous rotation-axis,
and the angular velocity k.o1, and is consequently propor-
tional to the length of OI. GOI is the angle & of Art. 153; and
if o is resolved into two parts, one of which is in the invariable
plane, and the other along the invariable axis, the latter cos
= &
and is always constant.
It will be observed that the elements which determine the
geometrical position of these lines and planes depend on the
original circumstances of motion; and these latter being given,
the positions of the invariable axis and the invariable plane are
determined from Section 1 of the present Chapter. The con-
struction mentioned in Art. 145 is only the initial case of that
motion of which the present construction gives an interpretation
at all times.
As the ellipsoid rolls on the plane GEF, successive points on
it come continually into contact with the plane; and these points
lie in two curves, one EIF in the plane GEF, and the other IQP
on the surface of the ellipsoid; the former is evidently generally
a plane curve of an undulating character, as we have indicated
in the figure; the latter is a closed curve on the ellipsoid. These
264
[157.
POINSOT'S INTERPRETATION.
curves have been named by M. Poinsot the Herpolhode and the
Polhode respectively.
157.] The polhode will generally be a curve of double curva-
ture, inasmuch as it is the locus of those points on the ellipsoid,
k
the tangent planes at which are all at the same distance, viz., G
from the centre; so that the curve is the line of intersection of
the two surfaces,
A x² + BY² + Cz² = 1,
G2
A²x² +в² y² + c² 2² =
k2'
(74)
(75)
which are the equations to two ellipsoids. For the projections
on the principal planes of the momental ellipsoid we have
B (B—A) y²+C (C — A) ≈²
Q2
A;
k2
(76)
G2
C (C — B) ≈² — A (B — A) x²
k2
B;
(77)
2
A (CA) x²+B (c-B) y2
G
= c
k2
(78)
Now in the general case of an ellipsoid with three unequal
axes, the perpendicular from the centre on a tangent plane is
greater than the least central radius vector, and is less than the
G2
greatest; so that is greater than A, and is less than c; and
k2
consequently (76) and (78) are the equations to ellipses in the
planes of (y, z), and of (x, y) respectively; and (77) represents
a hyperbola in the plane of (z, x), of which the real axis coin-
G2
cides with the axis of z or x, according as is greater than or
less than B.
k2
Figure 33 represents the case in which
G2
k2
is less than B; and
A
the polhode is a closed and symmetrical curve situated symmetri-
cally relatively to the vertex a of the ellipsoid; and the projection
of it on the plane of (y, z) is a complete ellipse: the projection
on the plane of (z, x) is an arc of a hyperbola whose real axis
coincides with the axis of x; and the projection on the plane of
(x, y) is an arc of an ellipse.
G2
If is greater than в, the polhode is a closed and symmetri-
k2
cal curve situated symmetrically relatively to the vertex c of the
157.]
265
POINSOT'S INTERPRETATION.
ellipsoid; and the projection on the plane of (∞, y) is a complete
ellipse; those on the planes of (y, z) and (z, x) being arcs of an
ellipse and a hyperbola respectively.
In each of these cases the curve has evidently four vertices
situated in its points of section with the principal planes of the
ellipsoid; these planes divide the curve into four equal parts.
The radii vectores of the ellipsoid at these vertices are maxima
and minima, and the instantaneous angular velocity has maxima
and minima values when it coincides with the maxima and minima
radii vectores.
G2
If =B, that is, when the distance between the invariable
k2
plane, and the plane on which the ellipsoid rolls, is equal to the
mean axis of the ellipsoid, (77) represents two planes; and these
are equally inclined to the plane of (x, y) at
+ tan-1
SA (B — A)
(BA)
(c(c-B) S
ra
(79)
These planes cut the ellipsoid in two equal ellipses, of which the
semi-axis-major and the semi-axis-minor are respectively
A+ C-B
B ) /
AB
and
(1)
( 1 ) * ;
B
(80)
and the common axis-minor of the two ellipses coincides with
the mean axis of the ellipsoid. These ellipses we shall call the
Critical Ellipses of the ellipsoid. They are the loci of the in-
stantaneous pole in this particular case; and the polhode be-
comes a plane curve. The motion of the instantaneous axis in
this particular case will be best understood from the peculiarities
of the herpolhode.
G2
If 12 = A, and if
k2
G2
k2
c, the polhode is in each case reduced
to a point, which is at the extremity of the maximum and the
minimum radius vector of the ellipsoid respectively; so that in
these cases the momental ellipsoid touches the fixed plane at
these points; thus, the instantaneous axis throughout the motion
coincides with the invariable axis, and is fixed. The body always
rotates about the same axis, which is a principal axis, and is a
permanent axis.
It is evident that the motion of the body would be represented
PRICE, VOL. IV.
M m
266
[158.
POINSOT'S INTERPRETATION.
equally well, if the fixed plane on which the momental ellipsoid
rolls were parallel to the invariable plane on the opposite side of
the fixed point o to that which is drawn in the figure 33, and at
an equal distance from o.
If we eliminate the constant terms from (74) and (75) we have
G2
A
1.2
•}
x² + B
{
G2
2
k2
B & y²
+c {
G
k2
c} *=0; (81)
of which generally the coefficient of the last term is negative,
and that of the first term is positive; and that of the middle
term may be either positive or negative. Thus (81) will generally
represent a cone of the second degree; its principal axes are
coincident with those of the momental ellipsoid. This is the
cone described in the body by the instantaneous rotation-axis, to
which we have alluded in Art. 36.
G2
If is less than в, all sections of the cone perpendicular to
k2
و
the axis of x are ellipses, and the x-axis is the internal axis of
Ꮐ
G2
k2
the cone; if is greater than в, all sections perpendicular to the
G2
z-axis are ellipses, and the z-axis is the internal axis. If = B,
k2
(81) represents two planes, into which the cone then degenerates,
and which are the planes of the critical ellipses of the ellipsoid.
158.] The herpolhode, which is the curve traced by the in-
stantaneous pole on the fixed plane on which the momental
ellipsoid rolls, is evidently generally a plane curve of an undulat-
ing character, such as is delineated in the figure, the curves of
which are equal and regular, and their maxima and minima
radii vectores are respectively equidistant from the pole; so that
the curve regularly winds between two concentric circles; it is
the locus of the point of contact of the ellipsoid with the fixed
plane; which plane touches the ellipsoid along the polhode.
Let p and r be the radii vectores of the herpolhode and the pol-
hode respectively from G and o as poles; and let do and ds be
length-elements of these curves respectively; then we have
p²
= p2
k2
;
do = ds;
G2
(82)
(83)
159.]
267
POINSOT'S INTERPRETATION.
whereby the equation to the herpolhode may (theoretically at
least) be found: the equation to the herpolhode will generally
be transcendental. In the next Article we shall find, in a parti-
cular case, certain properties of it which deserve mention.
The maximum and minimum radii vectores of the herpolhode
correspond to those of the polhode; that is, to the vertices of the
polhode. These singular values therefore will regularly recur at
equal angles on the fixed plane. If the angle between two succeed-
ing maxima radii vectores of the herpolhode is an aliquot part of
a right angle, the herpolhode reenters and is a closed curve. If
however this angle is incommensurable with a right angle, the
curve never reenters; and, although the instantaneous rotation-
axis returns periodically to the same place in the body, yet it
never returns to a position which it has previously held in space.
The cone, which the instantaneous axis describes in space,
and to which we have alluded in Art. 36, is that of which o is the
vertex and the herpolhode is the director-curve. This cone
therefore will be a kind of circular cone, with its surface worked
into ridges and furrows which correspond to the undulations of
the herpolhode.
G2
If
k2
A, and if
G2
k2
c, the herpolhode is reduced to a point,
viz., G, at which the momental ellipsoid touches the fixed plane;
in these cases the rotation-axis of the body has the same position
throughout the motion, both in the body and in space; it is
one of the principal axes of the body, and is thus a permanent
axis.
159.] Let us however consider the form which the herpolhode
G2
k2
takes when B; that is, when the perpendicular from the
fixed point to the fixed plane on which the ellipsoid rolls is equal
to the mean semi-axis. In this case, as we have just now demon-
strated, the polhode is that plane curve in which the momental
ellipsoid is intersected by the critical ellipses.
Let r, s, x, y, z refer to the polhode relatively to the principal
axes of the momental ellipsoid; then, from (74) and (75) we
have
A x² + B y² + c ~² 1,
a²x² + B²y² + c²~² = B,
x² + y² +
≈2
p2;
M m 2
(84)
268
[159.
POINSOT'S INTERPRETATION.
..
x2
C (Br²-1)
(B-A) (C-A)
C+A-B—A Cr2
y2
(C — B) (BA)
A (Br² — ])
z2
≈² =
(C-A) (C-B)
(85)
G
Let p, σ, refer to the plane herpolhode, of which & is the pole;
then from (82) and (83)
x²
p²
=
2.2
1
B
do² = ds2 dp² + p² dp²;
=
BC p2
(B-A) (C-A)
(86)
(87)
C+AB-AC
(p² +
B
(88)
y2
(C-B) (B-A)
A B p²
z2
;
(C-A) (C-B)
2
... p² dp² = ds2 - dp²
= dx² + dy² + dz² — d p²
cAdp2
(C — B) (B — A) — A B C p²
(C—B) (B-A) = Acn²;
Let
then
аф
dp
p (n² — B p²) ³
;
1
B&
{end+e¬nd };
P
2n
(89)
(90)
which is the equation to the herpolhode in the fixed plane on
which the ellipsoid rolls. It is the same equation as (122),
Art. 324, Vol. III. The prime radius is taken so as to coincide
with the maximum radius vector of the curve, and also to cor-
respond to the maximum radius of the polhode reckoned from
the centre of the ellipsoid; for when 4 = 0,
P =
C — B) (B — A) 144
{ — — }
ABC
C+A-B
AC
1) 12/
B
160.]
269
THE STABILITY OF THE AXIS.
C+A-B
AC
and
is the semi-axis-major of the critical ellipse. The
curve lies symmetrically on the two sides of this maximum radius
vector; and as 4 increases p decreases, and ultimately = 0, when
Ф ∞; the curve consequently is a spiral, such as is drawn in
Fig. 34; of which GE is the maximum radius, relatively to which
the curve is symmetrical. The branches, both in the positive
and negative directions, fall into the pole & after an infinite
number of convolutions; and this occurs when the instantaneous
rotation-axis coincides with the mean axis of the ellipsoid, and
consequently with the invariable axis.
In this case, the cone in space, on the surface of which the
instantaneous axis lies, consists of a series of sheets arranged
spirally in convolutions one within another, and ultimately be-
coming a straight line, which is the invariable axis.
If then the instantaneous axis is at any time on this cone, it
will move along its surface; and although the length of the her-
polhode is finite, being equal to the length of the elliptic polhode,
yet the time requisite for the passage of the instantaneous pole
through the curve is infinite; so that however near the rotation-
axis approaches to coincidence with the invariable axis, yet it
never coincides with it. If however the instantaneous axis coin-
cides with the invariable axis at any epoch of the motion, it will
do so always, unless some other impressed couple acts and pro-
duces a change of rotation-axis. Thus the mean axis of the
momental ellipsoid is a permanent axis, equally as much as the
axes of greatest and least moment.
160.] The following inferences can be drawn as to the change
of position of the rotation-axis of a body. Let us suppose a
body to be rotating about its mean principal axis; and let us
suppose a couple of given moment to be impressed on it; whereby
the instantaneous pole is moved along one of the critical ellipses
of the body; then the instantaneous pole will continue to move
along this ellipse until at last the body is completely overturned;
in which case the momental ellipsoid which touched the fixed
plane at в will ultimately touch it at B', and a complete boule-
versement of the body will have taken place. This is the great-
est derangement which a body rotating about an axis passing
through a fixed point can undergo.
If however the instantaneous pole is moved by the action of
the new couple from the extremity of the mean axis of the ellip-
270
[160.
THE STABILITY OF THE AXIS.
soid to a point which is not in either of the critical ellipses, then
the following consequences will occur. Imagine the ellipsoid
to be divided into four districts by means of the planes of the
critical ellipses; two of these districts will contain the vertices
A and a′ of the ellipsoid, and two will contain the vertices c and
c'; which are respectively the vertices of the axes of least and of
greatest moment. Now if the instantaneous pole is shifted into
either of the districts which contain c or c', the polhode will be
a closed curve towards and around c or c', and the instantaneous
pole will perform a complete circuit of this curve, and will pe-
riodically return to its first position: the more nearly too to c
the instantaneous pole is shifted from B, the less will be the
subsequent motion of the instantaneous axis; and should the
newly impressed couple be such as just to move the pole from в
to c, the instantaneous axis will then be the principal axis of
greatest moment, and will become permanent.
But if the instantaneous pole is shifted from в to a position
within the districts which include a or A', the polhode then be-
comes a closed curve towards and around these points, a complete
circuit of which the instantaneous pole describes. And if the
instantaneous rotation-axis is shifted, so as to coincide with the
principal axis of least moment, its position becomes permanent.
Now these several results depend on the perpendicular dis-
tance between the invariable plane and the fixed plane parallel
to it, on which the momental ellipsoid rolls. This distance is
given in (73), and the least and the greatest values of it are re-
spectively
: to adapt these mathematical expressions
C
1
·
and
A
1
to the image of the present Article, I shall suppose initially
G2 =
= Bk²,
(91)
and the body to be rotating permanently about its axis of mean
moment. Let us suppose the position of the rotation-axis to be
shifted by the action of a new couple, whereby & becomes G'
and k becomes k'; hereby both the direction-cosines of the in-
variable axis, and the distance of the plane on which the ellipsoid
rolls from the fixed point, will be changed; no change however
will be made in A, B, or c, or in the magnitude of the momental
ellipsoid, or in the position of the principal axes of the body
relatively to the body. If then after the momentum has been
impressed by the new couple,
161.]
271
ON STABILITY OF ROTATION.
K'2
k2
1
/2
G
42
B
(92)
the instantaneous axis will move along the plane of one of the
critical ellipses, and ultimately a complete bouleversement of the
body will take place. If however
k'2
G/2
k2
is greater than
G2
(93)
the plane on which the ellipsoid rolls is moved to a greater dis-
tance from the fixed point; whereby the instantaneous pole is
shifted into one or other of the two districts of the surface of
the momental ellipsoid in which is a or A', and the instantaneous
pole moves in a closed curve about a or a'.
greatest value which it admits of, viz.,
And if
k'2
takes the
G'2
1
,
the rotation-axis be-
A
comes the principal axis of least moment of the ellipsoid, and is
permanent.
Again, if
k'2
G'2
is less than
k2
Q2
(94)
the instantaneous pole is shifted into one or other of the districts
which contain c or c', and moves on a closed curve about these
k'2
vertices. And if the impressed couple is such that takes the
G'2
1
least possible value, viz.,, the rotation-axis becomes the prin-
cipal axis of greatest moment, and is permanent.
The angles which are determined by (79) may be taken as
the measures of the stability of rotation of the body relatively
to the axes of greatest and least moment. Thus the larger the
angles defined by (79) are, the larger is the district surrounding
the axis of least moment within which, if the instantaneous pole
is, the centre of the polhode will be a or A'; and consequently
the smaller will be the districts within which the polhodes will
have c or c' for their centres. And thus we say that the larger
the angles are, the more stable is the body relatively to the axis
of least moment; and the smaller the angles are, the greater is
the stability of rotation relatively to the axis of greatest mo-
ment.
Hence, the axes of greatest and of least moments are stable
axes; and the axis of mean moment is an unstable axis.
161.] In Art. 154 we have determined the most general dif-
ferential equation which connects the time and the angular ve-
272
[161.
ROTATION ABOUT A FIXED POINT.
locities about the principal axes; and we have observed that the
equation is an elliptic transcendent. In the particular cases
however in which the forces are such that the distance from the
fixed point to the plane on which the ellipsoid rolls is equal to
one of the principal semi-axes of the ellipsoid, (63) admits of
further simplification.
If the axis of least moment is the rotation-axis, it is a perma-
nent axis, and coincides with the invariable axis, and G2 Ak²;
thus from (37) and (44) we have
2
2
2
B (B — A) w₂² + C (C — A) w¸² = 0;
(95)
and as the coefficients of w₂2 and 2 are positive quantities, this
equation is satisfied only when w₂ = w3 = 0; in which case
(29) gives
dw,
dt
=
0;
@₁a constant = 21
L
A
(96)
Similarly, if the axis of greatest moment is the rotation-axis, it
is also a permanent axis, and coincides with the invariable axis,
and G2 ck2; and by a process similar to that just explained,
=
d w z
dt
0;
= 0
@3 = a constant = 23
N
C
(97)
If however the plane ou which the ellipsoid rolls is at a distance
from the fixed point equal to the semi-axis of mean moment,
then G² = B k²; and (37) and (44) become
2
2
C
2
A w₁² + B w₂² + c wz² = k²,
w₂² +c²wz² = G² = B k²
k2-B w₂
2
2
C
B-A
(98)
(99)
C-B
C-A
equation of (29), employing the substitution
B2
2
A² w₁² + B²
2 2
A w₂2
W1
thus the second
(89), becomes
B dw₂
B² dw₂
ndt;
k² — B w₂ ²
2 G² — B² w₂²
2
2
W2
2GN
G+ BW 2
.*. log K
t,
B
G - BW2
(100)
(101)
where K is an undetermined constant dependent on the circum-
stances of the body when t = 0. Thus, for instance, if when
t = 0, w₂ = 2, and B is the angle between the invariable axis
and the axis of mean moment, then 2, B = G cos ß; and
W2
162.]
PARTICULAR CASES.
273
G-B Q2
1-cos B
K =
G+ B L2
1+ cos B
(tan 2)²;
(102)
2 Gn
and if
B
= m, then from (101)
G Emi K
W2
;
(103)
Bemt + K
4 KG2 (C-B)
Emt
w2
(104)
AB (C-A)
(εmt + K)² ·
4 KG2 (B-A)
Em t
2
W?
(105)
BC (C-A)
(εmt +K) 2
If we take the invariable plane to be that of (x, y), and the in-
variable axis for the axis of z; then, as in Art. 152,
2
KC(B-A) emt)
cos 0 =
Emt + K
B (C-A)
2
(KA(CB) Emt)
tan o
mt-K
B(C-A)
2
dy
G
dt
(CB) (emt+K)²+(BA) (et - K)2
A (C —B) (€™t +K)² + C (B — A) (emt — K)²
Ε
(106)
(107)
(108)
whence, by integration, will be given in terms of t, and the
problem will be completely solved.
From (103) it appears, that w₂ = when t∞; so that
ვ
G
B
G
B
which in that case is equal to 2, is the angular velocity to which
w approaches and ultimately becomes equal. In this limit
@1
@₁ = w₂ = 0, and the body revolves permanently about the mean
axis. The polhode is the critical ellipse through which the in-
stantaneous pole travels, and the rotation-axis ultimately coin-
cides with the mean axis of the ellipsoid. The herpolhode is the
spiral which has already been described in Art. 159; and the
point of contact of the ellipsoid with the fixed plane coincides
with the pole of this spiral only when t∞.
162.] The special cases which have been just discussed depend
on G and k, which are functions of the forces and couples by
which the body at first receives its motion. But there are other
varieties which depend on the constitution of the body itself.
Let us briefly examine these; and let us first consider the case
wherein two principal moments are equal; these we will take to
be the least and the mean; that is, let B = A; so that the
greatest moment is the unequal moment; then, from (29), we
have
PRICE, VOL. IV.
N n
274
[162.
TWO PRINCIPAL MOMENTS EQUAL.
dw1
A
dt
+(CA) W2 W3 = 0,
=
Α
C
dt
dw3
dt
from the last of which,
@z = a constant
A dwa
d w z
- (CA) wg w₁ = 0,
W1
(109)
0 ;
= √z = n (say);
(110)
and from the first two we have
wy dwy + w z dw₂ = 0;
(111)
2
w₂ ² + w₂ ²
2
2²+22²,
= m² (say);
(112)
2
w² = w₁² + w₂² + W3
2
= m² + n²,
(113)
2²;
(114)
so that the angular velocity is constant about the instantaneous
axis; and also about the axis of greatest moment. And if y is
the S-direction-angle of the instantaneous rotation-axis, rela-
tively to the principal axes fixed in the body,
cos y =
W3
n
W (m² + n²)
(115)
which is constant: so that the instantaneous axis moves in the
body in the surface of a right circular cone whose semi-vertical
angle is y, and whose axis is the principal axis of greatest mo-
ment.
Again, from (112), and the first of (109), we have
C
A
if
= μ;
A
so that
dw₁
に
​- A
ndt
(m² — w₁²) ½
2
A
= μndt,
w₁ = m sin (µnt + a);
w₂ = mcos (μnt+a);
where a is a constant, such that, when t = 0,
221
tan a =
22
(116)
(117)
(118)
Thus, the axial components of the angular velocity relatively to
the principal axes of the body are expressed in terms of t; and
from these values it follows that the instantaneous axis moves
162.]
275
TWO PRINCIPAL MOMENTS EQUAL.
over the surface of the cone in a retrograde direction, with a
constant angular velocity equal to μn.
If we refer these to the invariable plane as the plane of (x, y),
we have, from Art. 152,
сп
сп
cos 0 =
G
(A²m² + c² n²)
tano = tan (unt + a);
..ф
• $ = µnt +a;
so that a, when t
o =
(119)
(120)
0; and thus a is the angle between the
line of nodes and the principal axis of έ, when t = 0.
Also
d&
G
dt
A
G
.*.
4-4% =
t.
A
(121)
These equations completely determine the motion of the body,
and its position at the time t. From (119) it appears that @ is
constant; thus, the axis of greatest moment is always inclined
at the same angle to the invariable axis; and therefore describes
a right circular cone in space, the axis of which is the invariable
axis. And since is constant, the precessional velocity of the
αψ
dt
line of intersection of the plane of axes of equal moment with
the invariable plane is constant, so that the axis of greatest mo-
ment describes the circular cone uniformly. And from (120) it
appears that the right ascension of the principal axes of equal
moment advances uniformly.
In this case the moving body is such that its momental ellip-
soid is an oblate spheroid. All the polhodes are circles parallel
to the equator of the spheroid; and all the herpolhodes are also
concentric circles whose centre is at G; see Fig. 33. The critical
ellipses in this case unite and become the circle of the equator.
Thus, if the rotation-axis is ever in the equator, it will pass
throughout it, and a complete bouleversement will take place
again and again; this will take place when G2 Ak², and the
plane on which the spheroid rolls is at a distance from the fixed
point equal to the equatorial radius. Although in this case the
rotation-axis moves in the body through a complete circle, yet
in space it is coincident with the invariable axis, and is fixed.
Thus, the axis of greatest moment is the only rotation-axis which
has stability and permanence.
Nn 2
276
[163.
ALL THE MOMENTS EQUAL.
If the two equal principal axes are the greatest and the mean,
that is, if c = B, the equations of motion are
A
dw1
dt
= 0,
do q
C
- (C—A) wz @₁ = 0,
(122)
dt
dwz
C
dt
+ (CA) w₂ w₂ = 0;
the results of which are so exactly similar to those of (109) that
it is unnecessary to explain them at length. In this case how-
ever the momental ellipsoid is a prolate spheroid, of which the
principal axis of least moment is the axis of revolution. The
polhodes are circles in planes perpendicular to this axis, and the
herpolhodes are also concentric circles of which G is the centre.
The critical ellipses unite into the circle BCB'C'. If the rota-
tion-axis is ever in this circle, it will move throughout it, and a
complete bouleversement will take place; this is the case when
G² = ck²: and although the rotation-axis moves in the body
through this circle, yet it is fixed in space, being coincident
with the invariable axis. The axis oa, which is that of least
moment, is the only axis in this system which has stability and
permanence.
163.] If A =B = c, that is, if the three principal moments are
equal,
dwi
dwz
dw3
0 ;
dt
dt
dt
@1
W2
@3
1,
21
D2
Dz
Ω
and the angular velocity is constant; and the rotation-axis is
fixed both relatively to the moving body and to space.
In this case the momental ellipsoid is a sphere, and the dis-
tance of the fixed plane on which it rolls is at a distance from
its centre equal to its radius. The polhode and herpolhode are
only points; and whatever new couple is impressed, and what-
ever consequently is the displacement of the rotation-axis, the
axis in its new place is stable and permanent.
And this case is indeed the only one in which generally it is
possible for the angular velocity to be constant throughout the
motion; for, if w is constant, the solution depends on the three
equations A²w₁² + B² w₂² + c² wz²
2
2
2
2
2
A w₁² + B w₂² + C @z²
2
2
2
2
2
G²,
k²,
w₁² + w₁₂² + wz² = w²;
}
164.]
277
PARTICULAR CASES.
whence w₁, w2, wg are evidently constant; and consequently
dwi
dwz
dwz
= 0;
dt dt
dt
and therefore from (29)
W3 = (BA) w₂ w₂ = 0;
W1 W2
(CB) wqwg = (A—C) wz W1
W3
which are satisfied (1) when A = B = c, that is, when the prin-
cipal moments are equal, and every axis is permanent and stable;
(2) when two of the three quantities w, w, wg are equal to zero;
that is, when the body rotates about a principal axis; but in
this last case on any shifting of the axis a change of angular
velocity takes place.
164.] The differential equations (29) are integrable also, at
least approximately, when the angle between the rotation-axis
of the system and one of the principal axes, say that of the
greatest moment, is always small; so that the angles between
the rotation-axis and the other principal axes are almost right
angles, and thereby their cosines are very small. In this case
@1
w, and we are so small that their squares and their products may
be neglected in linear equations which involve their first powers,
and w, may be replaced by the resultant instantaneous angular
velocity; then, from the third of (29), we have
dwz
0 ;
dt
@3
@g
a constant
= n(say);
and the first and second of (29) become
dwr
A
+ (CB) Nw₂ = 0,
dt
dwz
B
+ (A−c)nw₂ = 0;
dt
whence we have
d2w1
+
dt2
(C — A) (C—B)
(C — A) (C — B)
1
n² w₁ = 0;
(123)
AB
14
W₁ = ₁ COS
nt,
A B
(124)
A (C — A) ) ✯
W₂ = 1
sin
B (CB)
{
(C-A) (CB)) Vik
nt;
A B
the limits of integration being such that w₁ = 21, when t = 0,
278
THE CONE OF THE INVARIABLE AXIS.
[165.
at which time w₂ =
0; so that the initial rotation-axis is in the
plane of (§, §); and if o is the initial angular velocity
21
c
Ω
1
2² =
2₁² + n².
W2
Since is very small, (124) shew that w, and we are always
small, so long as c-A and c
A and c-B have the same sign; that is, so
long as the principal axis, near to which the rotation-axis is, is
the axis of either the greatest or the least moment. If however
C-A and C-B are of different signs, the integral of (123) in-
volves exponential expressions; and w, and we will increase in-
definitely with the time. Whence we infer that if a body, free
from the action of forces producing rotation, rotates at any time
about an axis nearly coinciding with the principal axis of greatest
or least moment, the rotation-axis will always nearly coincide
with that principal axis. But if the principal axis, with which the
rotation-axis nearly coincides is the principal axis of mean mo-
ment, the rotation-axis will deviate more and more from that
axis. Hereby we have another conception of the stability and
instability of principal axes; those of greatest and least moment
are stable; that of mean moment is unstable.
165.] We now return to the general case; although the in-
variable axis is fixed in space, yet in the body it describes a cone
of the second degree, the equation to which is thus found.
In reference to the principal axes of the body the equations
of the invariable axis are
X
Y
z
AW1
BW 2
;
Cwg
(125)
but from (37) and (44) we have
2
2
A (A k² — G²) w₁² + B (B k² — G²) w₂²+c (ck² — G²) wz² = 0;
·. (k²
G2
G2
G2
(k³ — 0²)
-) x² + (k² ·) y² + (k² — 0² ) ≈² = 0 ;
B
C
(126)
which is the equation to a cone of the second degree coaxal with
the momental ellipsoid; and is a circular cone if two principal
moments are equal; and becomes two planes passing through
the axis of y, if G² = в k², of which the equations are
B
{C(B-A)}*x + {A (CB)}% = 0.
2
(127)
Hence also we have the following image of the body's motion.
Let the instantaneous angular velocity at the time t be resolved
into two components, the axis of one of which is the invariable
166.]
279
PROPERTIES OF PRINCIPAL AXES.
axis, and the axis of the other is the line perpendicular to it in
the invariable plane. Now if is the angle between the instan-
taneous and the invariable axes, o cos p is the former component,
and is, by reason of (59), Art. 153, constant: the latter com-
ponent is o sin ; about the axis of which the body rotates in
the time at through an angle w sinødt; and thus the invariable
axis moves over a surface element of the cone (126); and as
this discomposition may be continued, so will the invariable
axis describe a conical surface. Also the axis of the latter com-
ponent will continuously change its position in the body, al-
though it is always in the invariable plane; and as it is always
perpendicular to a generating line of the cone (126), so will it in
its successive positions generate a cone which is the reciprocal
of (126); and of which consequently the equation is
Ax²
Ak² - G2
k2
+
By 2
B k² — G²
+
Cz2
c k² — G²
= 0;
(128)
=
and this is evidently a cone coaxal with (126), having the same
internal axis, and whose major and minor external axes are
respectively the minor and major external axes of (126). If
Bk2 G2, y = 0; in which case (128) represents a straight line.
Now in the motion of the body the surface of the cone (126)
always contains the invariable axis; and the surface of the second
is always in contact with the invariable plane. Thus the motion
of the body may be represented by the rolling of the cone (128)
on the invariable plane.
166.] Lastly, let us consider, as shortly as possible, certain
properties of the principal axes of the body; and for this pur-
pose let us refer them to the invariable plane and to the in-
variable axis: let a, ß, y be the direction-angles of the invariable
axis, and let (x, y, z) be the instantaneous pole at the time t,
relative to the principal axes of the body; then, from (51) and
(70), we have
A W1
Ak x
COS α =
a
G
G
BW 2
Bky
cos B =
(129)
G
G
CW3
c k z
cos y =
G
G
From which equations we have
280
[166.
PROPERTIES OF PRINCIPAL AXES.
(cos a)2
+
(cos B)2
+
(cos y)²
A w₁ ² + B w₂ ² + C wz²
2
2
A
B
C
G2
k2
;
G2
(130)
that is, the sum of the squares of the projections of the three
principal axes of the momental ellipsoid on the invariable axis
is constant, and is equal to the square of the perpendicular from
the centre of the ellipsoid to the plane on which the ellipsoid rolls.
Again, let the three principal axes be produced until they meet
the plane on which the ellipsoid rolls, as we have imagined in
Art. 156; and let P1, P2, P3 be the lengths of these three lines thus
produced; then, as the perpendicular from the centre on this
plane is equal to
k
SO
G
ß
P₁ cos a = P₂ cos ẞ = P3 cos y =
k
(131)
(132)
1
2
1
+ +
1
G2
2
Pi P2
k2
P3
and from (129)
1
1
1
=AX,
= BY,
c z ;
(133)
P1
P2
P3
1
1
1
+
+
A x² + By² + Cz²
2
2
B P2
C P3
= 1.
(134)
Αρι
2
Again, let the three principal axes of the momental ellipsoid be
projected on the invariable plane; and as the body rotates, let
wa, wo, we be the angular velocities of the projections of these
axes respectively in this plane. Then it is evident that the area
described on this plane by the projection of oa
(sin a)2
2 A
wadt;
(135)
but this area is evidently equal to the projection on the same
plane of the sectorial area described in space by oa itself. O A
however has no motion by reason of w₁; but, by reason of w
and თვა OA describes the sectorial areas
W2
dt
2 A
and
dt
2 A
which are perpendicular respectively to oв and to oc; and of
which therefore the projections on the invariable plane are re-
spectively
167.]
281
PROPERTIES OF PRINCIPAL AXES.
W2
2 A
@₂ dt
cos B,
and
ვ
@3 dt
cos y.
(136)
2 A
Thus, equating these to (135), we have
(sin a)2 wa w₂ cosẞ+wz cos y;
=
k² (1 − a x²)
.*.
Θα
wα =
G (sin a)2
k² (1 — By²)
Wb
G (sin ẞ)2
k² (1 − c z²) ;
2
(137)
Wc
G (sin y)²
k2
wa (sin a)²+w (sin ẞ)2+ w. (sin y)2
(3-Ax²-By²-cz²)
G
2 k²
;
G
(138)
that is, is equal to twice the component of the instantaneous
angular velocity about the invariable axis; see (59), Art. 153.
Also, from (137) we have
Awa (sin a)² + Bw, (sin ẞ)2+ cw, (sin y)²
k2
{A+B+C−A²x² — B² y² — c² z²}
G
k2
G
{+
{ A + B + C - 03 });
(139)
G2
k2
but A, B, C are proportional respectively to the squares of the
radii of gyration which lie along the corresponding axes: and
therefore it appears that the sum of the areas described on the
invariable plane, by the projections on that plane of the principal
radii of gyration, is proportional to the time.
Other theorems of the same kind, relative to principal axes
and to the projections of their extremities on the invariable
plane, have been investigated by Poinsot in the Memoir to which
reference has so frequently been made, but it is beyond my
purpose to enter into the subject further in this place; and the
student desirous of other information must have recourse to the
original Memoir.
167.] The problem which has been solved in the preceding
Articles of this Section is that of a rigid body, or a system of
material particles of invariable form, rotating about an axis pass-
ing through a fixed point, when L = M=N=0; and is also that
of a heavy body about an axis passing through its centre of
gravity, whether this latter point is fixed or not; because through
PRICE, VOL. IV.
00
282
[167.
ROTATION ABOUT A FIXED POINT.
it the resultant of the acting forces, viz., the weight of the body,
passes, and thus produces no moment to effect a change of either
the position of the rotation-axis, or the angular velocity.
Let us now investigate, as far as is possible, the rotation of a
heavy body, of which two principal moments are equal, and a
point is fixed in the axis of unequal moment, so that the body
rotates about an axis passing through this point. Let us more-
over assume the centre of gravity to be in the axis of unequal
moment, and not to be at the fixed point. It is evident that of
many systems of particles satisfying these conditions, one is a
heavy homogeneous solid of revolution, which is capable of rota-
tion about a point fixed in its axis of figure. Such is a top, the
point of whose peg keeps the same place during the rotation,
and the friction at which is neglected.
Let us take the coordinates, and other symbols of Arts. 2, 4,
and 42. Thus, let the fixed point be the origin; and let the axis
of unequal moment be the axis of , the axes of έ and n being in
a plane perpendicular to that axis, so that these coordinate axes
are the principal axes of the body at the given point. In refer-
ence to these axes let the centre of gravity be (0, 0, h); let
m = the mass of the body; and let A=B; and let c be the un-
equal principal moment. Let the fixed system of axes at the
origin be so arranged that the axis of z is vertical, and in a
direction contrary to that of gravity, and the plane of (x, y) is
horizontal. At the time t let 0, 4, 4 be those angles of connec-
tion between the two systems of rectangular coordinates which
are given in Arts. 3 and 4: then, since the weight acting on any
particle dm in a line parallel to the axis of z is equal to —gdm,
for the components of this force along the principal axes of the
body we have
X
gdm sin 0 sin 4,
Y =
2
g dm sin 0 cos 0,
g dm cose.
(140)
Since the axis of ( passes through the centre of gravity,
z. § dm = x.ndm = 0; z. (dm = mh;
(141)
and consequently in reference to the principal axes fixed in the
body
L = x.dm (zn-x) =
mhg sin o cos p,
mhg sin ◊ sin 4,
}
(142)
M = z.dm (x(−z§) = —
N = Σ.dm (Y§-xn) = 0;
and the three equations of motion become
167.]
A HEAVY BODY.
283
dw1
A
+ (C−A) W₂ Wz =
mhg sin o cos 4,
(143)
dt
dwz
A
dt
+(AC) Wg W₁ = -mhg sin 0 sin o,
(144)
dwz
C
= 0;
(145)
dt
from the third we have
3 = a constant
= n (say);
(146)
so that the angular velocity about the axis of unequal moment is
constant; and therefore substituting in (143) and (144) we have
dwr
dt
A + (CA) NW₂ =
mhg sino cos 4,
(147)
dw2
A
+ (A−c)NW1
-
--mhg sin 0 sin & ;
dt
d w₂
+w2
dt
= mhg sin 0 (w₁ cos &—w₂ sin ø)
do
= mhg sin 0
>
(148)
dt
.'.
A(
dw1
wi dt
by reason of (103), Art. 42.
Consequently, if k2 is the initial
vis viva of the body, and 4, is the initial value of ø, the integral
of (148) gives
Ο
A (w₂² + w₂²) + c n² = k²+2mhg (cos 0。 — cos 0); (149)
which is indeed the equation of vis viva, and might have been
immediately inferred from the principle given in Article 63.
Whenever 0 = 0, that is, whenever the angle between the vertical
line and the axis of unequal moment has its initial value, the
vis viva of the system is equal to the initial vis viva; and the
vis viva is increased or diminished according as the position of
the centre of gravity is lower or higher than its initial position.
This is in accordance with the conservation of vis viva.
From (100) and (101), Art. 42,
wi2 + w2
(do)
2
+
dt
(sin e dy);
;
(150)
dt
so that (149) becomes
d
A
1 (10) 2
dt
+A (sinė)² (dv)²
2
dt
+cn² = k²+2mhg (cos◊ — cos 0), (151)
which is the equation of vis viva connecting e, y, and t.
Again, from (147) we have
0 0 2
284
[167.
ROTATION ABOUT A FIXED POINT.
(sino dwy
dt
+coso 3) + (c− a)n (w, sin—w₁cosp)=0; (152)
dt
but from (104), Art. 42,
01
sin &+w₂ cos 4 = sin 0 ·
dy
;
dt
therefore
dwi
doz
d
sino
+ cosp
0
dt
dt
dt
dt
(sin e d¥) – (w, cos↓ — w, sin 4)
аф
dt
ď
0
dt
(sin e dy)
do
n cos 0
dt
dt
dt
dy),
by reason of (102), Art. 42; and substituting these in (152), we
have
A 2 cos 0
do d↓
dt dt
+ sin o
d²↓
dt2
do
сп
0; (153)
dt
and multiplying by sin 0, and integrating, we have
▲ (sin 0)2
d4
dt
+ cn cos 0
h';
(154)
where h' is a constant introduced in integration, and is indeed
the sum of twice the product of each particle and the projection
on the horizontal plane of (x, y) of the sectorial area described
by its radius vector in an unit of time. For although the prin-
ciple of conservation of moments or of areas, see Art. 58, is not
true of this motion relatively to any plane, yet it is true for the
plane of (x, y), because the axis of z is parallel to the line
of gravity, and the weight consequently does not produce any
moment relatively to that axis. Thus, of equations (70) in
Art. 58, the third is true, and (154) is the form of its integral in
this case.
For Aw, dt, Awdt, cn dt are respectively the sum of
twice the products of each particle and the sectorial area de-
scribed by its radius vector in dt in the planes of (n, 5), (§, §), (§, n)
respectively; and the areas described on the plane of (x, y) are
the sums of the projections of these on that plane; and therefore
{A₁ sin 0 sin +A w₂ sin 0 cos &+cn cos 0} dt
d ↓
2 + cn cos 0 dt;
dt
= { { a (sin 0)²
0}
} dt;
and this is constant by the principle of conservation of areas;
so that if h' is the sum of twice the product of each particle,
and the protection on the horizontal plane of the sectorial area,
described by its radius vector in an unit of time, (154) is the
particular form assumed by the principle in this problem.
168.]
285
A HEAVY BODY.
If we eliminate
dy
from (151) and (154), an equation will
dt
result of the form
dt = f(0) do;
·
(155)
whence may be determined in terms of t; and if we substitute
this value of 0 in (154), we shall have another differential ex-
pression of the form
dt = ƒ(4) d¥;
(156)
whereby may be expressed in terms of t. And lastly, by
means of (102), Art. 42, we have
аф
d &
J
= n cos 0
;
dt
dt
&
(157)
which will give us in terms of t. The problem will thus be
completely solved. All these differential expressions, although
reduced to simple quadratures, involve elliptic transcendents,
and are functions whose properties we have not discussed in this
treatise. I propose therefore to consider only one or two simple
cases of the problem.
I would observe however that the three equations (151), (154),
and (157), on which the solution depends, are the same as those
found by Lagrange after his own process, in the Mécanique
Analytique, Part II, Section IX, 35. It does not appear that
the problem had been solved before that time. Poisson after-
wards solved it, and gave the solution in Cahier XVI, Journal
de l'Ecole Polytechnique, published in 1815. He does not refer
to Lagrange. Poisson has also given the solution in the second
volume of the Traité de Mécanique.
= 0
0;
168.] Let us suppose the axis of the angular velocity which
is impressed on the body initially to be the axis of unequal mo-
d Ꮎ dy
ment; so that initially w₁ = w₂ = 0; and also
and therefore k² cn², and h′ = cn cos 0; so that (151) and
(154) become
do 2
dt dt
A
dt
+A (sin☺) 2 (d
(
2
dt
2mhg (cos 0 - cos ◊),
(158)
dy
A (sin 0)2
= cn (cose-cos 0);
dt
0
the former of these shews that h and (cos - cose) are always
of the same sign; and that consequently if h is positive, is
greater than ; and if h is negative, is less than 0; thus, the
inclination of the axis of unequal moment to the vertical in-
0
286
[168.
ROTATION ABOUT A FIXED POINT.
creases or decreases according as the centre of gravity is initially
above or below the fixed point. And from the latter it appears
dy
dt
that the precessional motion, of which is the symbol and the
measure, is direct or retrograde, according as the centre of gra-
vity is initially above or below the fixed point.
From (158) we have, by the elimination of
dt
= +
A sin o do
dy
dt
{cos◊。—cos0}&{2mha (sin0)²g — c²n² (cos¤¸ — cos◊) } ³
(159)
To fix our thoughts in the discussion of this equation, we will
suppose h to be positive; so that the centre of gravity of the
body is initially above the horizontal plane which contains the
fixed point, and the system is in unstable equilibrium when the
axis of unequal moment is vertical, that is, when 0,= 0. Thus,
the second radical in the denominator of (159) is positive when
the motion begins; consequently the first radical is also positive,
and thus must be greater than 0。; the inclination of the axis
d Ꮎ
of unequal moment to the vertical increases at first, and is
dt
positive. The angle e continues to increase, until it reaches a
value at which the second radical in the denominator of (159)
do
vanishes, when O, and there is no variation of inclination;
dt
”
this value of ✪ is less than ; for when 0=π, (159) is imaginary ;
let it be 0₁; and let r be the time in which the axis of unequal
moment will attain this position; then
A sino do
T=
So,"
-
00
(160)
o {cose-cose}✯ {2mha (sin0)²y - c²n² (cose — cos 0)}}
Now T is finite, although the elements of the integral are infinite
at both limits; to prove this, let
cos = S, cos 01
then
SO
T=
2
<= $1,
Ads
cos 0。 = S。;
(161)
(162)
$1 {80-8} {2mh ag (1-s²) - c²n² (so − s) } }
The form of the denominator of the quantity under the integral
sign, shew that the factors s。-s and s₁ s do not enter to any
1
power higher than -
2
; and generally, if e is an infinitesimal,
and f(s) is continuous when s = a,
168.]
A HEAVY BODY.
287
fate f(s) ds
α
( s − a) ½
= f (a) [ate
a
•ate ds
(s — a) ³
ate
= 2ƒ(a) [(8—a)$ ]***
2 ƒ (a) [ ( s − a) $ ]ª ** = 0;
0
1
a
so that the time within which passes from 0 to 0, is finite.
At this instant, when 0 0,, the first radical in (159) is still
positive, and thus the second must also be positive, and there-
fore must decrease; and will continue to do so until 0 becomes
06. This diminution will continue during the same time that
has been occupied in the increase from 0 to 0,; and thus the
inclination of the axis to the vertical will be the same as before,
and another change of position will take place similar to the
former. Thus o, the axis of unequal moment, will make iso-
chronal oscillations in the vertical plane zo passing through oz,
see Fig. 1, as that plane revolves about the vertical oz.
0
The angular velocity however of this vertical plane about oz
is not uniform, but the variations of its angular velocity are
periodic, the period of which is 2r; and is the time in which
the nutational oscillations of the axis of greatest moment takes
dy
place; for is this precessional velocity; and from the second
dt
equation of (158), we have
dv
cn (cos ›¸ — cos 0)
0
;
dt
A (sin ()²
(163)
0=0₂
1
so that the precessional velocity vanishes when 0=0, and attains
its maximum value when 0=0,; afterwards it begins to decrease,
and vanishes again when 0 = 0; and it continues to make these
periodical oscillations. If we project on the plane of (x, y) the
curve described by the centre of gravity, it is evident that it is
contained between two concentric circles whose radii are re-
spectively h sin 0, and h sin 0,; that it consists of a series of arcs
which touch the outer circle, and are at right angles to the inner
circle at the points where they meet it. This curve is delineated
in Fig. 35, where AC CA
the precession which takes.
place in the time г. I may observe that this curve is a graphic
interpretation of the oscillatory motion of the axis of a spinning
top, in which its inclination to the vertical periodically varies.
If w is the instantaneous angular velocity,
w² = 2
2
w₁² + w₂² + n² ;
2
2
w w₂²
2
and is a minimum when w² + 2 = 0, that is, when t0
and when 0=0; and w is a maximum when w₁²+w₂² is a maxi-
mum; and since
288
[169.
ROTATION ABOUT A FIXED POINT.
2
@₂² + w₂² = (de)²
αθλ
dt
+ (sin 0)² (dv) ³
2
dt
=2mhg (cos 06-cose),
2mhg(cos0。
(164)
the angular velocity is a maximum when 0 has its greatest value;
that is, when = 0, and when the angle of inclination of the
axis of unequal moment to the vertical is the greatest. This is
evidently the case by reason of the principle of vis viva.
1
If h is negative, similar results follow, except that ₁ will be
less than 0; thus, the principal axis of unequal moment will
come nearer to the vertical line than it was in its initial position,
and will make periodical ascents and descents.
169.] If the initial angular velocity of the system about its
axis of unequal moment is very great, n is very great; and since
✪ is the value of 0 determined by the equation,
2mhag (sin 0)² — c² n² (cos 06 - cose)
0;
0
0, is very little greater than 0,; so that the value of is confined
within very narrow limits. In this case the requisite integra-
tions may be effected approximately. Let u be a very small
angle of which the cubes and higher powers may be neglected ;
and let
then
0 = O。 + u ;
cos✪。—cos✪ = cos(。—cos(0%+u)
(165)
u²
= usin。+
cos 00;
2
cos 0。-cose
И
u² cos 00
•
(sin 0)2
sin 00
2 (sin 00)²
2
and thus (159) becomes
A du
dt
;
(166)
{2mhag sinu — (c² n² — mh ag cos◊¸) u²}
let
and
c2n²-mhag cos 00
A 2
Α
mhag sino
c²n²-mhag cos 00
k²,
(167)
b;
du
.'.
kdt=
(2 bu-u²)
u = b versinkt,
(168)
the limits of integration being such that u = 0, when t = 0;
0 =
Oo+b versin ki
0
= 0₂+b-b cos kt.
(169)
169.]
289
A HEAVY BODY.
According to our hypothesis u is always very small; and con-
sequently b must also be small; and therefore either 0 is always
very small, and the axis of unequal moment is always nearly
vertical; or n is very large; and as this is the assumption which
we have made, we have approximately
0
b
mh ag sin 0.
c2 n2
сп
k =
1
A
Again, from (163),
dy
cnu
dt
A sin 0。
kb
sin 00
(1-cos kt),
(170)
kb
b
t.
sin kt,
(171)
sin 00
sin e
the axes of x and y
being so placed that y
placed that = 0 when t = 0.
And from (157)
аф
dy
n- cos 0
dt
dt
n-kb coté¸(1—cos kt);
(n − kb cot 0¸) t+b coté sinkt,
0
(172)
the axes of and n being so placed that = 0, when t = 0.
This equation completes the solution of the problem.
y
0
Now these equations, (169), (171), and (172), enable us to in-
dicate the motion of the axis of unequal moment with great
geometrical exactness. Imagine an axis for which Po + b,
and kbcoseco,t; then this axis will describe the surface of
a circular cone round the vertical axis with a constant angular
velocity. Let this axis be called the mean axis; and any point
of it describes, with a constant angular velocity kbcoseco, a
circle whose centre is in the axis of z, and whose plane is per-
pendicular to that axis. Now the true axis revolves about this
mean axis, with displacements depending on the two periodical
terms,
-bcoskt, and - bcoseco, sinkt;
so that the true axis is sometimes before, and sometimes behind
the mean axis; sometimes nearer to, and sometimes farther from
the vertical axis; but as 6 is a very small quantity, these peri-
odical terms are always very small.
The path which the true axis describes relatively to the mean
axis may be imagined by the following figure. Imagine a sphere
of radius unity to be described about the fixed point as centre;
PRICE, VOL. IV.
PP
290
[169.
ROTATION ABOUT A FIXED POINT.
then the mean axis intersects this sphere along a parallel circle
of latitude, whose angular distance from the pole is 0, +b; and
the greatest and the least angular distances of the true axis from
the vertical axis are respectively 0。+26 and 00; so that the true
axis intersects the sphere in an undulating path contained be-
tween two parallels of latitude at angular distances of 0 and of
0+2b from the polar axis.
These motions of the true and the mean axis are delineated
in Fig. 36; in which o is the centre of a sphere whose radius is
unity; OA=0B=0C=1; COR=0。, cos= 0。+b, coт= 00+2b;
RR', SS', TT' are three parallels of latitude, ss' being that along
which the mean axis moves, and RR' and TT' being those which
limit the displacement of the true axis. The true axis evidently
describes the wavy line contained between these two latter pa-
rallels; and oN, the projection of oq on the plane of (x, y), is
the line which moves uniformly in that plane with the angular
velocity kbcosece, which is the mean precessional angular ve-
locity.
η
Now suppose the point where the mean axis pierces the sur-
face at the time t to be an origin, at which let coordinate axes
of έ and ʼn be taken along the tangent to the meridian, and the
tangent to the parallel of latitude respectively; then, if (§, ŋ) is
the place of the point of intersection of the true axis with the
sphere at the time t,
غ
n =
- bcos kt,
× =
sine, bcoseco, sinkt b sin kt; S
0
(173)
the second being multiplied by sin 0, because sin 。 is the ap-
proximate value of the radius of the parallel of latitude; there-
fore squaring and adding,
2
§² + n² = b²,
(174)
which is the equation to a circle; and thus the true axis de-
scribes uniformly a circular cone of small angle about the mean
axis, and intersects the surface of the sphere in a small circle
whose centre is the point of intersection of the mean axis with
the sphere; this motion of the true axis relatively to the mean
axis is called Nutation; its amplitude is always the same, viz., b,
but varies in the two directions of latitude and longitude; that
is, of the axes of § and ŋ.
Since
¿dn—nd§
dt
= b² k
b2c n
A
(175)
170.]
291
A HEAVY BODY.
it appears that the angular velocity of the true axis about the
mean has the same sign as n; and therefore the direction of the
nutational motion about the mean axis is the same as that of
the body about its axis of unequal moment.
Also, the periodic time of the nutation =
2 π
k
2 ПА
;
пс
(176)
and therefore the greater n is, the less is the periodic time of the
nutation; and consequently the greater the angular velocity of
the body about the axis of unequal moment is, the less is the
periodic time of the true axis relatively to the mean axis.
The angular velocity of the body, and the place of the instan-
taneous axis at the time t may thus be found. From (169),
(171), and (172), we have
0 = 0。+b-bcos kt,
y = bcoseco, (kt — sinkt),
$ nt-bcote (kt-sinkt).
0
(177)
Let us substitute from these in (100), (101), (102), of Art. 42;
and, omitting terms containing squares and higher powers of b,
we have
@1 bk {sinnt + sin (k-n) t},
W₂ =
bk{cosnt - cos(k—n)t},
Wz = n;
2
w² = 4 b² k² (sin kt)² + n³ ;
}
(178)
(179)
and the direction-cosines of the instantaneous axis are also de-
termined.
170.] And now we will investigate the problem in a still more
special form in the case wherein the angle of inclination of the
axis of unequal moment to the vertical is constant throughout
the motion. Let 0, be the constant value of 0; then
d Ꮎ
dt
0 ;
and, taking the most general forms, from (151) we have
d 4
dt
k² — c n² ) 1/
A
= a (say);
}
coseco
... 4 = at;
(180)
(181)
P p 2
292
[170.
ROTATION ABOUT A FIXED POINT.
the position of the axes of x and y being such that = 0, when
t = 0; so that the precessional velocity is constant; and the
axis of unequal moment describes uniformly the surface of a
right cone whose axis is the vertical axis of z.
And from (157) we have
αφ
αψ
= n-cose。
dt
dt
..
$
nt-cose, V,
(182)
the position of the axes of έ and 7 being such that = 0, when
હું η
t = 0 ;
..
p = nt
a cos e t
0
(n
a cos 00) t
ẞt (say);
if ß = n
a cos 00;
(183)
(184)
&
and thus varies directly as t; and thus, using the language of
astronomy, the angular velocity ẞ of the right ascension of the
§-axis is constant. And thus the motion of the principal axis of
unequal moment is completely determined.
The angular velocity of the body, and the position of the
instantaneous axis at the time t, may thus be found. From
Art. 42 we have
ω1
=
d&
dt
sin 0 sin = a sin 0 sin ẞt,
d v
W z
sin 0。 cos = a sin 0, cos ßt,
(185)
dt
аф
d¥
თვ
+
cos@¸ é
cose。= ẞ+acos 0。 = n;
dt
dt
(186)
w² = a²+2aß cose。 +ß²
= a² (sin 0。)²+n²
and hereby the position of the instantaneous axis at the time t
is also known.
Hence also we can deduce the conditions under which the
invariability of the angle of inclination of the axis of unequal
moment to the vertical is possible. Equations (185) are the
three final integrals of (143), (144), and (145); let us substitute
the former in the latter; then the first two become
sine, {AaB+(c-A)na-mhg} cos ẞt = 0,)
sino {Aaẞ+(c-A)na-mhg} sin ßt = 0;
0
(187)
but these are to be satisfied for all values of t; and therefore
171.]
293
A HEAVY BODY.
sino {aaß+(c-A)na-mhg} = 0;
(188)
which is an equation to be satisfied by the elements of the body,
the angle 。, and the velocities of precession and right ascension,
when the axis of unequal moment is, throughout the motion, in-
clined at the same angle to the vertical.
=
=
This is satisfied, firstly, when sin 0, 0, that is when 0, 0,
and 00
180°; in both cases the axis of unequal moment is
vertical; and in the former case the centre of gravity is above,
and in the latter case is below the fixed point. In both cases the
velocities of precession and of right ascension are arbitrary, for
a and ẞ may have any values. In the former case ∞ = a + ß,
and in the latter w = a · ß; w₁ = w₂ = O, and the rotation-axis
is vertical, and the body rotates about it with an uniform an-
gular velocity.
171.] Again, (188) is satisfied when
saẞ+(c-A)na-mhg = 0;
and if we replace n by its value from (183), we have
a² (c— ▲) cos@。+cßa−mhg =
α
0 ;
(189)
which is a quadratic equation in terms of a, but only a simple
equation in terms of ß; consequently for a given value of ß we
have generally two values of a; but for a given value of a, only
one value of ß.
90°; (2)
There is however only one value of a, when (1) 0%
c-A=0. In the former of these two cases the centre of gravity
of the moving body, and the principal axis of unequal moment,
move in the horizontal plane which contains the fixed point, and
we have, what at first view appears an impossibility, a heavy
body rotating about a horizontal axis, passing through its centre
of gravity, and supported at one point, when that one point is
not the centre of gravity. In the latter case, the momental
ellipsoid of the body at the fixed point is a sphere; and the body
must be constituted in accordance with the conditions investi-
gated in Art. 110; and if A, B, C are the principal central mo-
ments,
C
A
h² =
m
In both these cases
mh g
αβ :
=
;
C
(190)
and thus a and ẞ vary inversely as each other; and the product
of them varies directly as the distance of the centre of gravity
294
[171.
ROTATION ABOUT A FIXED POINT.
from the fixed point, and inversely as the moment of inertia
about the axis of greatest moment. And a and ẞ have the same
sign if h is positive, and opposite signs if h is negative; thus, if
ẞ is given and is positive, a is greater, the greater the distance is
between the centre of gravity and the fixed point, and the greater
the weight (mg) of the body; and is less the greater c is; and
is positive or negative, that is, the precession is direct or retro-
grade, according as the centre of gravity is above or below the
fixed point, when = 0. In the former of these two cases
@₁ = a sin nt,
..
w₂ = a cos nt,
w² = a² + n².
@3
Wz = n.
Also, if h = 0, that is, if the centre of gravity is at the fixed
point, then, from (189), we have
(1) a = 0 ; (2) α =
св
;
(191) ·
(C — A) cos 。
thus, when ẞ is positive, the precession is direct or retrograde,
according as 0, is an obtuse or an acute angle.
Let us return to the general equation of condition; the two
values of a given by (189) are real, equal, or imaginary, accord-
ing as
4mhg (c-A) cos 0。 + c² ß² is >, =, or < 0;
0
(192)
thus, so long as h and cos 0 are both positive, the two values of
a are both real, and unequal and of opposite signs; so that the
precession may be either direct or retrograde. If however h is
negative, or, what amounts to the same configuration, 0, is
obtuse, then the two values of a may be equal; and may be
imaginary.
Lastly, let us suppose the body to have no motion in right
ascension, so that ẞ = 0; and consequently from (189),
ß
a
dy
dt
= +
{ chose
(C — A) COS。
tra
(193)
This is the case of the common conical pendulum of such a form,
that relatively to the fixed point all moments of inertia about
axes perpendicular to the rod of the pendulum are equal, and
that relative to the rod of the pendulum is c; thus, in the com-
mon form of this pendulum, c-A is negative and 0, is obtuse;
and thus (193) is possible, and indicates the angular velocity of
the pendulum about a vertical line through its fixed point of
attachment; hence we have, taking the positive sign,
172.]
295
BOHNENBERGER'S MACHINE.
mh g
*
(C — A) COS ◊。 0
}
14
-101
t;
(194)
and the periodic time of the pendulum
= 2π
п
{ (C-A) cos 0₂ ? +
170
mh g
(195)
For a heavy particle m, placed at the end of a rod of small
thickness whose length is h, c may be neglected, and a mh²;
and if a is the inclination of the rod to the vertical, taken
downwards, then
*
h cos a
(1990) +
;
(h cos a) +.
产
​and the periodic time = 2π
9
(196)
(197)
This solution completes the theory of small oscillations of heavy
bodies. Huyghens first gave a solution of the problem of cy-
cloidal and circular pendulums. Clairaut gave the solution of
conical pendulums*, which we have already explained after La-
grange and Bravais, in Art. 370, Vol. III; this kind of oscilla-
tion taking place when the pendulum, drawn out of its place of
rest, receives an impulsion, the line of which is not in the plane
containing its line of rest. If however the pendulum at the
same time receives a motion of rotation about its axis, the oscil-
lations are deranged by reason of the centrifugal forces thereby
developed; and this most general problem of small oscillations
is that which we have discussed in the immediately preceding
Articles.
172.] Now many machines have been devised for the purpose
of exhibiting the phænomena which are expressed in the pre-
ceding equations; the construction of some is so curious that
they are for the most part found only in collections of mecha-
nical apparatus; others are so simple in form that they are the
toys of children. Of the latter kind is the common spinning top,
of the several motions of which the preceding Articles give ex-
planations, provided that the point of its peg continues in the
same place, and the friction of the point is neglected. Of the
former kind is, in the first place, Bohnenberger's machine, which
we have already described in Art. 28; it is delineated in Fig. 8,
and the first account of it is given in Gilbert's Aunalen, Bande
* Memoirs of the French Academy of Sciences of the year 1735.
296
[172.
FESSEL'S MACHINE.
60, Leipzig, 1819. The rotating body in the middle is in our
figure a sphere, but any other body may be substituted for
that; and if the centre of gravity of it coincides with the centre
of the three several rings," then, according to the notation in
the preceding Articles, h= 0. Let us suppose the central body
whether it is a sphere, an oblate spheroid, a cone, a cylinder, or
any other body such that AB, which rotates about the axis
AA', to be capable of removal from the ring AB A'B'; and to it
when so removed let a rapid rotation be given by means of a
suitable machine; let it be replaced with its pivots in the holes.
at A and A'; then the construction of the machine allows the
several movements consequent on the rotation of the body to be
exhibited, when h=0; for throughout the centre of gravity will
remain in the centre of the rings, and be unmoved. And if the
pivots at в and B' are fastened so that no rotation takes place
about the axis B B', the inclination of the axis AA' to the vertical
cc' is constant throughout: this is the case wherein
0% and
h = 0, &c. Thus, if the central body is a sphere of radius a,
A B
C
πρασ
15
and therefore, from (191), a = 0, and a = x; and therefore the
equatorial plane of the sphere always intersects the horizontal
plane along the same line; and no rotation can be given to the
sphere whereby its axis will describe a conical surface about
the vertical cc'.
Again, let the body which rotates about AA' be a right cone
whose altitude is a, and the radius of whose base is b; then
A B
πραγ
80
·(a²+46²),
C=
πραγ
10
;
and from (191)
a = 0, and
α =
8b2 B
(4b² — a²) cos 0。
;
the latter of which by its sign shews that the precessional motion
is retrograde.
Another machine of the latter kind is that devised by Fessel,
which is described in Poggendorff's Annalen, Bande 90, Leipzig,
1853, and which is delineated in Fig. 37. Q is a heavy fixed
stand, the vertical shaft of which is a cylinder bored smoothly,
in which works a vertical rod cc', as far as possible without
friction, carrying at its upper end a small frame BB'. In BB' a
horizontal axis works, at right angles to which is a small cylinder
173.]
297
PRECESSION AND NUTATION OF THE EARTH.
D, with a tightening screw H, through which passes a long rod
GG', to one end of which is affixed a large ring AA', and along
which slides a small cylinder carrying a weight w, which is ca-
pable of being fixed at any point of the rod; and so that it may
act as a counterpoise to the ring, or to the ring and any weight
attached to it. An axis AA' works on pivots in the ring, in the
same straight line with GG'; to AA' a disc, or sphere, or cone,
or any other body can be attached, and thus can rotate about
AA' as its axis; to the body thus attached to AA' a rapid ro-
tation can be given, either by means of a string wound round
AA', or by a machine contrived for the purpose when AA' and
its attached body are applied to it. It is evident that the coun-
terpoise w can be so adjusted that the centre of gravity of the
rod, the ring, the attached body, and the counterpoise, should
be in the axis BB'; or at any point on either side of it; that
is, h may be positive, or be equal to 0, or may be negative. Also
by fixing BB' in the arm of cc' which carries it, the inclination
of the rod GGʻ to the vertical may be made constant, that is,
may be equal to 0, throughout the motion. When the coun-
terpoise is so adjusted that the centre of gravity of the rod GG'
and its appendages is in cc', then h=0, or, what is equivalent,
mg = 0.
Ꮎ
If the counterpoise is adjusted so that the centre of gravity of
the rod GG', of the ring, and of w, without AA' and its attached
body, is in BB', then the weight of the body will produce its full
effect, and the results indicated in the foregoing Articles will be
exhibited.
173.] In application of the general equations of rotatory mo-
tion we may here insert another problem which is of great in-
terest and importance, although perhaps it more properly comes
into the following Chapter.
When a body has motion of both translation and rotation,
the investigation into these several motions may be conducted
separately, by virtue of those fundamental theorems which have
been proved in Section 2 of Chap. III, and the rotation may be
considered relative to the centre of gravity and an axis passing
through it; just as if the centre of gravity was a fixed point and
had no motion of translation. This is precisely what I propose
to do now I propose to consider the rotatory phænomena of
the earth, having its centre of gravity fixed at least hypotheti-
cally, under the action of the attracting forces of the sun and
PRICE, VOL. IV.
Q q
:
298
[173.
PRECESSION AND NUTATION
the moon; I shall indeed consider it as merely a mathematical
problem; but it will have its application to these three bodies:
and as the resulting differential equations will not admit of in-
tegration exactly in their general form, I shall make those hy-
potheses as to small quantities which are given to us by the
circumstances of these bodies. Our inquiry too will be general,
and will include the action of all bodies by which the rotation.
of the earth is affected; that is, of not only the sun and the
moon, if there are others whose influence affects the earth's
motion of rotation. The law of action of these bodies on the
earth is of course that of gravitation. The attraction varies
directly as the product of the masses, and inversely as the square
of the distance.
Let m be the mass of the body whose action on the earth we
are considering; let the centre of gravity of the earth be the
origin, and let the central principal axes of the earth be, as
heretofore, the axes of έ, n, ; the (-axis being the geometrical
polar axis; and let A, B, C be the central principal moments of the
earth relative to these axes respectively; let dm' be a mass-ele-
ment of the earth, of which the density is p, and let its place
be (έ, n, ), and of these coordinates p is a function; let (x, y, z)
be the centre of gravity of m; r' the distance of (x, y, z) from
(§, n, §); and let r be the distance of m from the centre of gravity
of the earth; and let the attraction which two unit-particles at
an unit-distance exert on each other be the unit of attraction,
and be unity; then
p² = x² + y² + ~²,
'2
2
(x — §)² + (y — n7)² + (≈—8)².
(196)
Now, for two reasons, we consider the attraction of m on the
earth to be the same as if m were condensed into a particle of
mass m at its centre of gravity; (1) because the distance be-
tween m and the earth is supposed to be very great, and con-
sequently the theorem proved in Art. 191, Vol. III, is applicable
to its action; (2) because the bounding surface of m is nearly
spherical, and m is supposed to consist of a series of concentric
spherical shells, the attraction of each of which on an external
particle dm' is the same as if it were condensed into its centre
of gravity.
Let x, y, z be, relatively to the earth's principal axes, the axial
components of the attraction of m on the earth; then
173.]
299
OF THE EARTH.
-
= m[ff (x = {) pd Ɛ dy d
X = M
23
m f f f y − n ) p
Y = m
2 =
= m
L = ZY Y Z = m
fff
(
z
m f f f
f
SSS
MXZ - Z X = m
NYX-xy = m
d
§
d n d
(197)
−5) pd & and 5;
23
p(z n − y() d§ dŋd¿
−
2'3
p (x § — z§) d§ dŋd §
SSSP (x5-
2'3
ρ › (y § — x ŋ) d § dŋd§.
2′3
(198)
the integrations in each equation being such that all the ele-
ments of the earth are included.
As the distance of the centre of m from the centre of the
earth is very great in comparison with the mean radius of the
earth, and consequently with the coordinates of any element of
the earth, even when m is the moon; so the quantities under
the signs of integration in the right hand members of the pre-
ceding equations may be expressed as series of terms rapidly
ξ η
converging in powers of
γ r r
: the greatest value of either of
these quantities is, in the case of the moon,
1
case of the sun, 23984
1
59.96'
and in the
; in the following expansion therefore I
shall omit all powers of these quantities above the second.
For the effect of subsequent terms in the series, the student
may consult a Memoir, having for its title, "Théorie du mouve-
ment de la Terre autour de son Centre de Gravité," by M. J. A.
Serret; and contained in Vol. V of "Annales de l'Observatoire
Impérial de Paris," 1859. He will there find the mode of cal-
culating the terms which arise on the hypothesis, that the oblate-
ness of the northern and southern hemispheres of the earth is
different; and on the hypothesis, that the earth is not symme-
trical relatively to the polar axis of figure.
Now, from (196),
1
8
=
{(x − §)² + (y — n)² + (≈ — ()² } − #
{x²+ y²+z² −2(x§+yn+z§)+ʳ + n² + Š² } − »
Q ૧ 2
300
[173.
PRECESSION AND NUTATION
1
1
2013
p3
{ 1 _ 2 ( x + y ? + = 5) + E
n² (²) - §
+ ? + -
(x § yn z()
2.2
2.2
1
2.3
{1
3(x+yn+x()
1 +
3 §² + n² + 5²
2.2
2
2.2
+15 (x + yn+=()}; (199)
2
204
let us substitute this value in (198); then, since the centre of
gravity of the earth is the origin, and the central principal axes
are the coordinate axes,
S S Sp
[[ƒ p€ d£ and¢ = [[ƒ ondεdnds = [[[p¢d€and¢=0;
5
dŋd (200)
[[] pn{d£dnd¢ = [[] p¢ € d€ dnd¢ = [[] p£nd£dnd£= 0; (201)
SSS
and consequently, omitting all powers of small quantities above
the second, (198) become
L = 3m (C— B)
(c
yz
доб
M = 3 m (a−c)
A
Z X
25.
(202)
N
= 3M (B — A)
xy
·
p5
With regard to the last two terms of (199) which do not appear
in these equations, having been omitted on account of the small-
ness of the quantities, I would observe, that they disappear of
themselves in the integration, if the earth is supposed to be
symmetrical in the distribution of its elements in the northern
and southern hemispheres, and with respect to its polar axis of
figure. So that under this hypothesis the equations (202) are
much more approximate than they appear to be at first sight.
Since
X y Z
r
2°
,
are the direction-cosines of the line joining
the centres of the earth and the attracting body, it appears that
L, M, and N vary directly as the mass of the attracting body,
and inversely as the cube of the distance of its centre from the
centre of the earth. Hence, if we calculate, from a synoptic
table of the elements of the moon and of the planets, this
quantity, it will at once be seen that the sun and the moon are
the only bodies which produce any sensible effect on the rota-
tion of the earth; the effect of the sun is due to its very large
174.]
301
OF THE EARTH.
mass; and the effect of the moon, which is much greater, to its
nearer distance.
174.] Equations (202) admit of further simplification; and
let us first consider them with respect to the sun.
Let n' the mean angular velocity of the earth about the
sun; let E the mass of the earth; then, as the eccentricity
of the earth's orbit is very small, r may be taken as the mean
distance of the earth from the sun; and, equating the earth's
periodic time in terms of n' with that given in Vol. III, Art. 332,
(154), we have
E
1
now
2 п
2πrt
n (M + E) &
;
(203)
(m
according to Encke, quoted by sir John
M 389551'
Herschel; and this quantity being small may be neglected, so
that
m
n'²;
2.3
(204)
and therefore for the action of the sun, the equations (202) be-
come
L = 3n'²² (c
3 n'² (c — B)
B) 127,
ZX
M = 3 n'² (A — c) ~2 ›
(205)
22
N = 3n'² (B— A)
23.
2.2
Again, let us consider (202) with respect to the moon; and
let all the quantities which refer to the sun receive an accent,
and thus refer to the moon.
Let n" be the mean angular velocity of the moon about the
earth; then, if we neglect the eccentricity of the moon's orbit,
and take r' to be the mean distance, by the same theorem as
that which we have just now applied to the sun,
2π
2πr' 3/2
(206)
n
(m' + E)
E
=
but 81.84 nearly,
=e (say); so that
m
m'
"/2
N
(207)
2'3
1+ e
and substituting this value in (202) we have.
302
[175.
PRECESSION AND NUTATION
3n"2
y' z' 7
I =
(C — B)
1+e
3n"2
z' x'
M =
(A — C)
(208)
1+ e
p'2
3n'2
x'y'
N =
(B — A)
;
'2
1+ e
and the equations of rotation of the earth become
dw₁
C-B
3 (c — B) ( n'²y z
n"²y z l
+
dt
W₂ Wz =
ვ
+
A
A
2-2
(1 + e) p′2
dwz A
C
3 (A-C)
n'2 X
n'2 z' x'
+
dt
Wz Wy =
+
(209)
B
B
p2
(1 + e) r'
12
+
dwz B-A
A)
dt
W2
W1 W₂ =
3 (B-1) { n'ay
12
xy
n'"² x'y'
+
C
2.2
(1 + e) r²²
C
The complete integration of these equations is beyond the
power of analysis; and we are obliged to have recourse to
methods of approximation, taking advantage of those circum-
stances as to small quantities which the relations of the sun,
earth, and moon offer to us: these we proceed to explain.
175.] In the first place, geodetic measurements shew that the
figure of the earth is nearly that of a solid of revolution, whose
axis is the polar axis of figure; and as there is no reason to
suppose any great want of symmetry in the distribution of the
material elements in the interior of the earth, we may suppose
the two principal moments in the plane of the equator, and con-
sequently all the moments of inertia in that plane, to be equal;
thus, BA; and this equality exists whatever are the positions
of the axes of x and y in the plane of the equator.
I may moreover observe, that the most profound calculations*,
based on the hypothesis of an unsymmetrical distribution of
material within the earth, lead to the conclusion that W3 is con-
stant to a first approximation, and that consequently BA; this
result follows from the fact that the action of the sun and moon
is very small in comparision of the actual vis viva of the earth.
Observations made with the pendulum are in accordance with
direct measurement, and shew the earth to be a solid of revolu-
tion, whose polar axis is shorter than the equatorial; and that
* See the Memoir of Serret quoted above; also Le Verrier, Annales de
l'Observatoire Impérial de Paris, Tome II.
175.]
303
OF THE EARTH.
its figure is approximately an oblate spheroid; and thus c, which
is the central principal moment relative to the axis of revolution,
is the greatest of all moments. Now, putting в = A in (209), it
is plain that c and a enter into the equations of motion only in
the form
C- A
A
:
the value of this quantity cannot be determined
by direct observation, because we are ignorant of the law of
density of the matter of the earth, and we are obliged to have
recourse to indirect methods. The observed values of precession
and nutation give it a value of nearly
1
306*, which is beyond
doubt almost correct; also a hypothesis of Laplace, discussed in
the Mécanique Céleste, Livre XI, gives a result nearly identical;
this value we shall take. Since the physical constitution of the
earth enters into the equations of motion only by means of these.
quantities, it is evident that the phænomena of precession and
nutation would be the same, whatever change took place in the
earth, so long as the ratio
C A
A
was unaltered.
Again, the actual axis of rotation of the earth is almost fixed
in it, and is almost identical with the axis of figure; that is, the
poles of the earth are almost fixed points on its surface. Were
they not so, geographical latitudes would vary from time to time;
whereas no variation has been indicated by observation, so far as
I know. Moreover, as the true rotation-axis of the earth in all
its positions nearly coincides with the axis of figure, the true
angular velocity w, which is the resultant of w₁, w, wз, is nearly
equal to w, which is the angular velocity about the earth's axis.
of figure, and is constant; and thus w, and we are very small
quantities. Thus, if we image the actual rotation of the earth
by the rolling of one cone on another, that cone which the
earth's axis describes in itself has a very small vertical angle,
the cone fixed in space having a vertical angle a little greater
than 46°.
Wz,
This is the information which observation gives as to the cir-
cumstances of the constitution and the figure of the earth, and
as to the approximate invariability of its angular velocity, and of
the position of its rotation-axis.
Under these circumstances the equations of motion become
*
See the Memoirs of Serret and Le Verrier.
304
[176.
PRECESSION AND NUTATION
dw₁
C-A
3(C-A)
n2yz
n"²y' z'
+
W2 W3
+
dt
A
A
2.2
(1 + e) r²²
2
dwz
C- A
3 (C-A)
Nzx
12 Z X
n"2 z'x'
dt
W3 W1
+
A
A
2.2
(1+e) r²²)
(1 + e) r²²
(210)
dw3
=
= 0;
dt
W3
N.,
(211)
from the third of these
if n is the angular velocity of the earth about its polar axis of
figure.
Also, for convenience of expression, let
C
A
= a;
A
then the first two of (210) become
dwi
+ an w z
=
3 a
{
n2yz
+
dt
2.2
dwz
dt
anw₁ =
За
{
S n'² zx
+
2.2
n"²y' z'
(1 + e) r²²
112
n'24' x'
12
(212)
2
(1 + e) r²²
and from these equations all the phænomena of the rotation are
to be deduced.
Equations (210) shew that the action of both the sun and the
moon on the earth is due to the physical constitution of the
earth itself. If c = A, that is, if all the principal central moments
of the earth were equal,
dw₁
dt
d wz
dt
dwz
dt
= 0; and thus the
angular velocity would be constant, and the earth's rotation-axis
would be fixed in itself, and would be absolutely fixed in space;
the protuberant matter at the earth's equator which causes the
inequality of the central principal moments, is thus the indirect
cause of the peculiar motion of the earth's rotation-axis, which
we are about to investigate.
176.] The arrangement of the bodies which is convenient for
our system of symbols and equations is exhibited in Fig. 38.
o is the centre of the earth; and the plane xoy is the fixed
plane of the ecliptic; ox being the line of the vernal equinox when
t = 0. About o as a centre a sphere is described whose radius
is equal to unity; and the several curved lines of the figure are
the intersections of the surface of this sphere by various planes
and lines drawn through o, and all refer to the configuration of
the system at the time t; yxN is the plane of the earth's equator,
176.]
305
OF THE EARTH.
so that on is the line of the vernal equinox, and xoN is the pre-
cession; oz is the earth's polar axis about which the angular
velocity is n, and or and oy are the earth's principal axes in the
plane of the equator, or being so chosen that it coincides at
the same time with ox, os, and on; os is the radius vector of
the sun, which is always in the plane of the ecliptic; oм is the
radius vector of the moon, MN'I being the plane of the moon's
orbit; on' is the line of intersection of that plane with the plane
of the ecliptic, and is the line of the moon's nodes; or is the
line of intersection of the plane of the moon's orbit with the
plane of the earth's equator. Let i be the angle of inclination
of the plane of the moon's orbit to the ecliptic; then i is nearly
constant, and has a mean value of 5°8′48″; we shall take it to
be constant. Now the line of nodes of the moon revolves in the
plane of the ecliptic, and performs a complete revolution in about
6793 days. Thus on' revolves about oz: let & be its angular
lets
velocity; then, if n = 1, we have approximately,
n' =
1
365.25'
n"
1
27.32'
В =
1 ;
6793
so that ẞ is much less than the other quantities; the small frac-
tion a is also a factor of all the terms into which these quantities
enter.
As the angular motion of the line of equinoxes is very small,
the angle xoN is very small compared with Nox, or xos; so
that approximately NOS XOS = n't; and Nox =
n't; and Nor=nt. We shall
also in calculating small terms neglect variations of 0. From
this arrangement we have
X
= cos xos =
cosnt cosn't + sinnt sinn't cose,
2°
Y
cos y os
sinnt cosn't + cosnt sinn't cos 0,
(213)
2
2
COS ZOS ==
sin n't sin 0 ;
which are thus expressed in terms of t and of constants.
0
0
Again, as to the moon; let us in the first place refer it to the
ecliptic; then, if y'o is the longitude of the moon's node at the
vernal equinox, that is, when t = 0, NON' yo+ ẞt, say;
and if d' is the moon's right ascension at the vernal equinox,
N'OM='。+n''t = ', say; then, as i is very small, n'om and its
projection on the plane of the ecliptic may be considered to be
equal; so that the longitude of м is '+'; and, if we replace
sini by i, cos Moz isin o';
MOZ =
PRICE, VOL. IV.
Rr
306
[177.
PRECESSION AND NUTATION
x
ઘુ′
৯हे
= COSαOM= {sin(p′+y')cos0+isin o'sin0} sinnt + cos(p' + y)cosnt,
=COSYOм= {sin(p' + v′)cose -- isinp'sinė} cosnt — cos(p' + ')sinnt,
= cos zoмisin d'cos - sin (p′+y')sin0;
0
0
and as '+' = ¢o + √'o + (B+n")t, these quantities are ex-
pressed in terms of t, and of known quantities; they are to be
substituted in the equations (212); which are then to be in-
tegrated.
The linear form of the equations (212) shews that the effects
of the action of the sun and the moon may be calculated sepa-
rately; and that the whole effect is the sum of the two separate
effects. We shall consequently calculate each by itself. Instead
of determining w, and w, by means of these equations, it will be
more convenient to calculate 0 and 4 directly, as the position of
the earth will hereby be determined with reference to fixed
lines.
The equations which connect these angles with the principal
angular velocities are given in Art. 42; and are
sin 0
d Ꮎ
dt
w₁ cos — w₂ sin &,
d p
dt
= w₁sino+w₂ cosp.
177.] Our object in this inquiry is not to calculate accurately
the motion of the earth's rotation-axis, and the earth's angular
velocity which determines the length of a day; but to trace
roughly, and to indicate in their salient points the results of the
action of the sun and moon. We shall therefore retain as far
as possible only finite quantities, and small quantities of the first
order; and we shall only notice the kind of change which is
produced, with a view rather to the general effect of such action
than to numerical calculations.
We will first consider the terms in (212) which refer to the
sun, and which will be replaced by their values in (213).
Now the earth's axis is inclined to the normal of the ecliptic
at an angle which is nearly constant; let I be its mean value,
which is about 23°27′32″; this angle being that between the
earth's equator and the ecliptic is called the obliquity of the
ecliptic. It is the mean value of according to our arrange-
(2
177.]
307
OF THE EARTH.
ment, and we shall replace by it in terms involving small
quantities. Also, as the earth rotates uniformly about its polar
axis with the angular velocity n, and as the angular velocity of
ON is very small, nt, omitting small quantities; and thus
the equations of the last Article become
sinnt,
do
w₁cosnt - w₂ sin nt,
dt
(215)
d &
sin I
w₁sinnt+w₂cosnt;
dt
d20
dw,
dwz
= cosnt
sinnt
n(w, sinnt + w₂ cosnt)
dt2
dt
dt
dwi
dw2
d↓
cosnt
sinnt
n sin r
;
(216)
dt
dt
dt
d24
dt2
dw
dwz
do
sin I = sinnt
+ cosnt
+ n
;
(217)
dt
dt
dt
dwi
d wz
substituting for
and
dt
dt
only the terms which depend
their values given in (212) taking
on the sun's action, we have
+3an'2 (sin n't)2 sini cosi = 0; (218)
d20
dv
+ (1+a) n sinл
dt2
dt
d²f
sin I
- (1 + a) n
dt2
d Ꮎ
dt
-3an'2 sinn't cosn't sini = 0. (219)
Integrating (218) we have
do
+ (1+a)nsinI+
dt
3 an'2
2
sin 2n't
sini cosi (t
COSI
2n
n't)
=0; (220)
no constant being added, because if no disturbing force acts,
that is, if n' 0,
d Ꮎ
dt
do
0; and y=0, when t=0. Substitute for
dt
in (219), and we have
d24
· + (1 + a)²n² &= ~
3a (1+a) n'²ncosit
dt2
2
3 an
+
{(1+a)ncosi+2n'} sin2n't; (221)
4
3 an'2 cosi
3 an' (1+a)ncosI+2n'
t+
sin2n't
2(1+a)n
4
(1+a)²n²-4n'2
+ c´sin {(1+a) nt+y}; (222)
where c' and y are constants introduced in integration; but
since y
=0 O when t= 0, and
dx
dt
is independent of t when
n'= 0, c'= 0, and y = 0.
Rr 2
308
[177.
PRECESSION AND NUTATION
Also, in the coefficient of sin 2n't we may omit
count of its smallness; and, since a =
C
N'2
n2
29
on ac-
A
α
C A
(223)
1+ a
C
and thus (222) becomes
3n'² cos I C— A
3n' Cosi C- A
*
t +
sin 2n't.
(224)
2 n
C
4n
C
Replacing & in (220) by this value we have an identity; which
shews that the terms herein retained destroy each other in the
variation of 0, although they give a finite result in the value of
4. We must therefore replace y by the value which it has before
small terms are omitted; that is, we must substitute for the
value given in (222), putting however c'= 0; then (220) becomes
3m² sin1 c - A
do
dt
2
sin 2 n't;
2n
C
3n' sini C- A
c
· 0 = 1 +
cos 2n't;
4n
C
(225)
(226)
where I is the constant of integration and is the mean value of 0.
Equations (224) and (226) exhibit the effects of the sun's
action on the rotation of the earth. is the angle through
which the line of equinoxes, oN in Fig. 38, moves in the time t,
and is called the Solar Precession of the equinoxes; (224) shews
that it consists of two terms, from the former of which it ap-
pears that increases directly as the time; and from the latter,
that this continual motion is accompanied by a periodical varia-
tion, of which the periodic time is
π
N
>
that is, is half a year.
This periodical quantity is called the Solar Nutation of the
Earth's Axis in Longitude, or, the Nutation of the Equinoxes.
Thus, the line of equinoxes is sometimes a little in advance of,
and sometimes a little behind, its mean place; and coincides
with its mean place every half year; but as the coefficient of
this periodical part is very small, so does the term scarcely ever
acquire a sensible magnitude.
From (226) it appears that has a mean value 1; but that
the earth's axis has a small oscillatory motion, depending on the
second term of which the period is also half a year; and this
second term is always very small because its coefficient is small.
It is called the Solar Nutation of the Earth's Axis in Latitude,
178.]
309
OF THE EARTH.
or, the Nutation of the Obliquity. Thus, the rotation-axis of
the earth would have a very slow progressive motion in space,
inclined at a constant angle I to the normal of the ecliptic, if it
were disturbed by only the sun's action.
178.] The effect of the moon on the rotation of the earth is
expressed by the latter terms in the right hand members of (212);
these we now proceed to inquire into, and by a process similar
to that by which we have investigated the action of the sun.
For abridgment of notation let the moon's longitude =µ+vt;
so that
М = Фот жо; V B+n";
and let us replace 0 by 1; (214) become
(227)
x'
{sin (µ+vt) cosi+isin (p'o+n't) sin 1} sinnt + cos (μ+vt)cosnt,
y
={sin (u+vt) cosi+isin (po+n't) sin1} cosnt-cos (μ + vt) sinnt, (228)
Z
isin (po+n't) cosi-sin (μ+vt) sini;
as i is a small angle, the squares and higher powers of it will be
omitted. Substituting these quantities in (212) and in (216),
we have
d20
dt2
=(1+a)nsini +
dt
dx 3 an"?
{sin (µ+vt) cosi+isin (p'。 +n't) sinı}
1+e r
dy 3 an'¹2
Зап S
sin 21
{1−cos2 (µ+vt)}
dt 1+e
= (1+a)nsin I +
i cos 21
+
2
(cos {y's+ßt} —cos {2 $'n +Vo+(8+2n'){}) }; (229)
and substituting again in (217),
dt 1+e 2
d²x
do 3 an" (sin
sin I
=
(1 + a) n
+
sin2 (u+vt)
dt 2
+
icosi
2
(sin {V´% +ßt} — sin {2 ¢´%+¥′o+(B+2n″)t}) } ; (230)
From (229) we have
3an"
sin 21
(1+a)n siniy +
1+ e
d Ꮎ
dt
+-
icos 21/sin (y。+ßt)
2
В
4
(t-
sin 2 (μ+vt))
2 v
sin{2。+4'o + (ß + 2 n'') t}
sin {24′。+V'% + (B+2n'"'); }) } . (231)
B+ 2 n'
Now substituting this in (230), we have
"
310
[178.
PRECESSION AND NUTATION
day
dt2
+(1+a)³n²√ =
do
dt
3an"2(1+a)ncOSI
2(1+e)
t
3an"2 ((1+a)ncos1+2v
+
(1 + e) {
sin2(µ+vt)
4 v
(1+a)ncos 21+ẞcosi
+
isin (+ẞt)
2ẞsin1
(1+a)ncos21+(B+2n") cosi
2(3+2n") sini
3 an" cosi
t +
3an"?
2 (1+e)(1 + a) n 1+e
(1+a)ncos21+ (B+2n") cosI
¿sin {24′o+4′o+(B+2n') t} } ; (232)
((1+a)ncos 1+ 2 v
(4v{(1+a)²n² —4v²}
+
sin2 (u+vt)
(1+a)ncos 21+ßcos i sin (yo+ẞt)
2ẞsin {(1+a)²n² — ß²}
2 (ß + 2n˝) sin 1 { (1 + a)²n² — (ẞ +2n″)²}
0
isin {24′% +V'% +(B+ 2 n')t} } ; (233)
the constants being omitted for the same reason as they are
omitted in (222).
Now ẞ is very small compared with n", and thus v may be re-
placed by n"; and the squares and higher powers of
omitted; so that after all reductions (233) becomes
3an" cosi
2(1+e) (1+a)n
t +
3 an' cosi
4n(1+e)(1 + a)
n"
n
may be
sin 2 { d'o+o+(B+n")t}
3 an' cos21
2
isin (yo+ẞt)
0
2ẞn (1+e) (1+a) sin 1
+
3 an" cos21
4n (1+a) (1+e) sin r
isin (2p+o+(B+2n")t}. (234)
0
If we substitute this value for in (231) it leads to an identity,
and thus it appears that the terms which are herein retained
cancel each other in the variation of e; we must therefore replace
in (231) by its more approximate value, which is given in
(233); and we have
- sin I
3an"2
(1+e)(1 + a)n 2
sini
3an'2
2 (1+e) (1+a)n (2n"
COS I
2
sin 2 (µ + vt) — isin (4′。+ßt)
COS I
+
2
0
isin{24′。+\'o+(B+2n″)t} } ; (235)
COSI
cos2{d'o+o+(B+n")t} + icos (+ẞt)
COS I
2n"
β
icos {24′o+4'6+(B+2n')t} } ; (236)
0=1+
179.]
311
OF THE EARTH.
where I is the mean value of 0, and is the constant introduced in
integration.
In (234) and (236) the last terms which involve the angle
200+4'o+(B+2n')t are to be omitted, because of the small-
ness of the coefficient in which i is a factor; the next preceding
terms in each however must be retained, because ß, which is a
very small quantity, is in the denominator of the coefficient,
and this brings it into importance, although it contains i as a
by its value, given in (223), we
factor; thus, if we replace
have from (234) and (236),
α
1 + a
3n"2
3 n'² COSI C-A
3 n' cos I c
A
11
t +
2 (1+e)n
C
'4n(1+e)
C
{ sin2{¢´o+V'o+(ß+n")t}
+
4n''cos 21 i
I
sin 21 β
sin (V'。+Bt)}; (287)
0=1+
3n" C- A
2n(1+e) c
sin I
2
cos2{40+4'o+(B+ n″) t}
i
+n cosi cos (4+ẞt); (238)
β
On comparing these values with (224) and (226), which express
the sun's action, it is evident that they produce effects on the
earth's axis of precisely the same kind; so that what has there
been said of solar precession and nutation, may here, mutatis
mutandis, be said of lunar precession and nutation; but the
effect of the terms in these latter expressions is much greater
than that of those in the former, because n'" is much greater
than n'.
179.] The whole precession and nutation is the sum of the
two separate effects; but before we add, we must make a re-
mark or two on the signs of our quantities. We have taken all
the angular velocities to be positive; that is, we have supposed
the bodies to revolve from the axis of x towards that of y in
Fig. 38; and this hypothesis is in accordance with the conven-
tion of signs which has been adopted throughout the volume; it
is not however necessarily that of the actual motion of the earth
and moon, of the moon's line of nodes, and of the apparent mo-
tion of the sun : let ox be east on the ecliptic, and let oz be the
normal to the ecliptic towards the north: now all the bodies
revolve in their orbits, as well as about their axes, from west to
east; so that the signs of n and of n" are to be changed; that
of n' is correct, because the sun's motion is apparent only, being
312
[179.
PRECESSION AND NUTATION
$
due to the actual motion of the earth. The line of the moon's
nodes also retrogrades, that is, goes from east to west, so that
the sign of ẞ is correct.
Also let
be the longitude of the moon's line of nodes at
the time; let and be the longitudes of the sun and moon
respectively; then
0
0
2 = 4o+ßt, O n't, ( = do + Yo +(B−n')t; (239)
so that for the whole precession and nutation we have
3 cos I C― A
2n
(
C
3 cos I C
12
n² ² +
N''2
c) t
1 + e
A § 4n"² cos21 i
((1+e) sin 21 B
1 + e
n'
+
sin
sin 2 ( — n´sin 2 © } ; (240)
An
C
112
3 c - A
A Sn"² cosi i
COSI
0 = 1-
cos N
2n
C
1+e B
n" sin I
2 (1+e)
n' sin I
cos2
cos 2
2
© }; (241)
the terms involving n" and n' arise from the action of the moon
and sun respectively.
The second of these equations shews that the earth's axis is
inclined to the normal of the ecliptic at an angle which is nearly
constant; yet that there are small variations of the angle which
are expressed by the latter terms of (241); these terms are
periodic, and are very small because their coefficients are small;
they depend on the longitude of the moon's ascending node on
the ecliptic, on the longitude of the sun, and on the longitude of
the moon; they constitute the luni-solar nutation in latitude
or in obliquity.
Equation (240) shews that the line of equinoxes has a general
retrograde motion along the ecliptic, with an angular velocity
3 cos I C - A
2n C
(n'² +
2
n
"12
1 + e
¥, say;
(242)
this quantity is called the luni-solar precession of the equinoxes;
yet that this retrograde motion is not uniform, but is subject to
slight variations, which are periodic, and are expressed by the
last three terms of the right hand member of (240); that these
periodical quantities are very small, because their coefficients
are small: they likewise depend on the longitude of the moon's
line of nodes, and on the longitude of the sun and the moon ;
and they constitute the luni-solar nutation in longitude.
* It will be observed that no distinction has been made between true and
mean longitude, true and mean ecliptic, &c.; our calculations have not been
carried far enough for such accurate positions.
179.]
313
OF THE EARTH.
The motion therefore of the earth's axis in space will be well
represented by the Fig. 36; in which o is the centre of the earth
and is supposed to be fixed, and the radius of the sphere is
unity. The axis whose motion is defined by the equations
Ө = 1,
$
=
It,
(243)
may be called the mean axis of the earth, 1 and Vt being re-
spectively the mean obliquity and the mean precession. And
the axis which is defined by the complete expressions (240) and
(241) will be the true axis. Let COR = 1; then the circle RS
will be that along which the mean axis will intersect the surface
of the sphere; and if oq is the mean axis at the time t, and op
is the true axis, the angle PoQ will be small; and as t varies of
will be sometimes before, and sometimes behind oq; and some-
times nearer to, and sometimes farther from the pole of the
ecliptic. Thus, the true axis of the earth will intersect the
sphere in a wavy line contained between two parallels of the
sphere at distances from RS, determined by the greatest positive
and negative values of the periodic terms of given in (241).
The motion of the true axis relatively to the mean axis may,
as to its principal and its most important terms, be exhibited in
the following way. Suppose the point of intersection of the
mean axis with the sphere to be an origin, at which two axes
originate, say of έ and n, in the plane touching the sphere; that
of ʼn being a tangent to the parallel along which the mean axis
moves, and the §-axis being perpendicular to it, and thus being
a tangent to the meridian through the place of the mean axis.
Now the most important periodic terms in (240) and (241) are
those which depend on the longitude of the moon's ascending
node, on account of the smallness of ß, as we have before observed;
let these principal terms in the directions of the two axes of §
and ʼn be represented by έ and 7, so that
η
3
с
A N 1/2 COS I i
&
cos N,
2 n
C
1+e B
3 с
12
A n² cos21 i
sin ;
n
2n
C
1 + e
B
£2
n²
3
c
A
i
+
(cos I)2
(cos 21)2
2 n
C
1 + e ß
(244)
which is the equation to an ellipse whose axes are in the ratio
of cos I to cos 21, and of which that directed towards the pole of
the ecliptic is the greatest; and thus it follows that so far as
the most important terms affect the motion, the true axis de-
PRICE, VOL. IV.
SS
314
[179.
PRECESSION AND NUTATION
scribes a small ellipse on the surface of the sphere relatively to
the mean axis which passes through the centre of the ellipse;
This ellipse is called the Ellipse of Nutation.
In the preceding image of the motion of the earth's axis, we
have assumed the earth's centre to be fixed, and the radius of
the sphere to whose surface we have referred the motion of the
axis, has also been assumed to be unity. The earth's centre how-
ever is not fixed; yet the image is a correct representation of the
facts, because we refer the motion of the earth's axis to the
sidereal vault, of which we may say the radius is so great that, in
comparision of it, the distance through which the earth's centre
moves is infinitesimal. Thus, the mean axis describes a circle
about the pole of the ecliptic, the angular radius of which circle
is 23°27′32″; and the true axis describes a wavy line symmetri-
cally situated with reference to this circle; and if the mean axis
is considered fixed, the true axis describes an ellipse on the
sidereal vault, the centre of which is the place where it is pierced
by the mean axis.
is
The periodic time in which the mean pole describes its circle
2 π
L
; and the true pole will describe its ellipse about the mean
pole in the same time as that in which the moon's line of nodes
describes a complete revolution.
The value of the annual luni-solar precession is found as
follows:
3 cos I CA
n
1/2
1 + e
n'2
4 =
2n
C
(n² +
C - A n
3 cos I
1 +
C n
(1 + e)n²²
2
}
n
2
so that the annual luni-solar precession
n
c - A n'
1/2
= 3 cos I
1 +
C n
(1+e) n'²
)
}
180°.
Now if we take the epoch to be Jan. 1, 1850*, 1=23°27′32″,
C-A
1
n'
C
306
n
1
365.25'
n"
365.25
and e = 81.84;
n
27.32
and therefore
n''2
1 +
(1+e) n'²
3.15764;
also
cos 23°27′32″ = .91735;
therefore the annual luni-solar precession
* See the Memoir of M. Serret in Vol. V of "Annales de l'Observatoire
Impérial de Paris," page 321.
180.]
315
OF THE EARTH.
.91735
X
102
315.764
36525
× 180 × 60 × 60",
= 50″.3828.
(245)
*
The observed value of the luni-solar precession is 50".37140;
so that our result is very nearly correct, although it is only ap-
proximate. I may in passing remark, that the coefficients of
sin and of cos in (240) and (241) respectively, are -17".251,
and 9".223; the former being the largest value of the princi-
pal term of the nutation of the equinoxes, and the latter being
the largest value of the principal term of the nutation of the
obliquity. Also the mean axis describes a complete circle in the
heavens in 25724 years.
180.] Of the problem of precession and nutation an approx-
imate solution has also been given by M. Poinsot in the Addi-
tions to the "Connaissance des Temps" for 1858. The principal
terms only are found by it; but it exhibits the problem in such
an elementary form, and dissects the results of the action of the
sun and moon into the several phænomena so distinctly, that it
is peculiarly fitted for an elementary treatise. We shall employ
the symbols of the preceding Articles, and shall make use of
the couples of the impressed momenta which have been therein
determined.
We consider all quantities at the time t, and investigate the
effects which accrue during the infinitesimal time dt. If n is
the angular velocity of the earth about its rotation-axis, and G is
the moment of the effective couple and c is the moment of in-
ertia relative to that axis; then
G = NC.
nc.
(246)
Let L and M be the moments of the impressed couples relative to
the axes of x and y in the plane of the earth's equator; as the
position of these axes in that plane is indeterminate, and as we
shall consider the effects for only the time dt, we may suppose
the axis of x to lie along the line of equinoxes, and the axis of
y to be perpendicular to it. Thus (213) and (214) become
Y
r
X
= cos n't,
cosn't, = cos I sinn't,
sini sinn't;
γ
λ
(247)
* On this subject see the Memoir entitled "Numerus constans Nutationis
ex ascensionibus rectis stellæ polaris in Speculâ Dorpatensis annis 1822 ad
1838 observatis deductus," by C. F. Peters, and contained in “ Mémoirs de
l'Académie Impérial des Sciences de Saint Pétersbourg, 6e série, premiere
partie, Sciences mathématiques et physiques, Tome III, Saint Pétersbourg,
1844."
SS 2
316
[180.
PRECESSION AND NUTATION
x'
cos (µ + vt),
J
৯/हे
z'
cos I sin (u+vt) + i sin i sin (po+n't),
sin I sin (u+vt) + i cosi sin (p',+ n't);
0
(248)
and therefore
-3n'2
L
(C — A) sin I cos I (1-cos 2 n't),
(249)
2
}
3n'2
M
(CA) sin I sin2n't;
(250)
2
and if we omit the terms in L' and M', which involve the angle
2 ¢´o +o+(B+2n")t, because the coefficient is small and does
not rise into importance in the subsequent integration,
0
112
3 n'² (c — A)
L'=
M =
2 (1+e)
3n2 (C-A)
2(1+e)
— sini cos 1 { 1 — cos 2 (µ+ vt)} + i cos 2 1 cos (4′o+ẞt) }, (2
i
0
sin 1 sin 2 (µ+vt) + ¿ cos I sin (+ẞt)}; (252)
0
the moments of these couples in the time dt are severally Ldt,
Mdt, L'dt, and м'dt; and besides them we have also the couple
Their effects are to be considered.
G.
The axis of Ldt is the line of equinoxes, and as the axis of G
is perpendicular (approximately) to the plane of the equator, it
is perpendicular to this line. Consequently the axis of the
resultant of Ldt and G is the diagonal of the rectangle, of which
the line-representatives meeting at the earth's centre are the
adjacent sides: let og be this diagonal; and let dλ be the angle
at which it is inclined to the axis of &, dà being necessarily infi-
nitesimal because Ldt is infinitesimal and G is finite. And thus
αλ
L dt
G
3 n'2
2n
C-A
C
sini cos I (1 — cos 2n't) dt. (253)
As da lies in the plane which contains the axes of L and G,
the axis about which the body revolves through dλ is the line in
the plane of the equator perpendicular to the line of equinoxes;
this infinitesimal rotation therefore will not produce an appre-
ciable change of obliquity, but only a change of position of the
line of equinoxes; and if dy is that angle,
180.]
317
OF THE EARTH.
αλ
dy
sin I
3n'² C-A
Cos I {1-cos2n't} dt;
(254)
2n
C
3n'² c-A
sin 2n't
=
COS I
2n
C
2 n'
S
(255)
which result is the same as (224).
As the axis of L' is the same as that of L, it may be treated in
the same way relatively to G; thus, if y' is the angle of pre-
cession due to the effect of 'dt, from (251)
we have
3n'2 C A
C
:{
cosit +
COSI
2 v
sin 2 (µ+vt) +
icos 21
ẞ sini
0
sin (V´% +ßt) } ; (256)
};
2n(1+e)
and replacing μ and v by their values, and omitting small quan-
tities, this becomes
3n"² COSI C— A
3n" COSI C- A
1'=
t +
sin2{'o+o+ (B+n")t}
2n(1+e)
C
4n(1+e)
C
+
sin 21
4n" cos21 i
B
sin (V´% +ßt) { ; (257)
0
which is precisely the same result as (237). The sum of (255)
and (257) is the total luni-solar precession, and nutation of the
equinoxes.
Next let us consider the effects of м and м'. Since the axis
of м is in the plane of the equator, and perpendicular to the
line of equinoxes, the rotation-axis of the resultant of мdt and
G is in the plane perpendicular to the line of equinoxes; and if
du is the angle at which the axis of this new couple is inclined
to that of G,
αμ
mdt;
G
(258)
as du lies in the plane of the axes of G and мdt, this shifting of
the rotation-axis is equivalent to a rotation of the body through
a small angle du about the line of equinoxes; but hereby will
be diminished by de; so that du
— do ;
=
.*. d Ꮎ
Mdt
=
G
3 n'² c-
2n
sin I sin 2n't dt;
C
cos2n't,
(259)
4n
C
0=1+
3 n' sini C-A
which is the same result as (226).
3
318
[181.
PRESSURE ON THE FIXED POINT.
3n"2
C-A
0' = 1+
2n(1+e) c
{sini
As the axis of м' is the same as that of м, it may be combined
in the same way with &; and if ' is the obliquity due to the
action of м'dt,
cos 2 (µ+vt)
2 v
+icosi
cos (o+Bt)
B
3n" C A
sin I
= 1 +
I 2n(1+e)
C
12
cos2 {o+o+(B+n")t}
i
+n'cos I
β
cos(V%+ẞt)}; (260)
which is the same result as (238). And thus the addition of
(255) and (257) will give (240); and that of (259) and (260) will
give (241). But it is of course unnecessary to repeat them.
An account, with great exactness, of the effects of all the terms
in the lunar and solar precession and nutation, will be found in
the Memoir of M. Poinsot; but it would be out of place to
insert it here.
181.] It remains for us still to examine the pressure borne
by the fixed point of the body through which the rotation-axis
always passes.
The pressure P, as well as the direction-cosines of its line of
action, are to be determined by means of equations (24) or (26),
Art. 148.
Let us refer the line of pressure to the principal axes fixed in
the moving body; let м be the mass of the body, and (x, y, z)
the place of its centre of gravity at the time t; then from (26),
Art. 148,
PCOSA.MX-M
dw z
dt
dw 1
dt
z
dw2
dt
Y
S_dw₂
X
2
dt
S_dw₂
PCOS v .mz-M Y
X
dt
dt
PCOSµ = E.MY
PCOSμ.MY — M
dwz
}
- м w ₁ ( w₁ x + w₂ ÿ + wzz) + Mw²x,
- Mw₂ (w₁x+w₂Y + wzz) + Mw³y, (261)
(Wył
− MWg (wy π+w₂Y+wzz) + Mw²z;
-
ω
3
of these equations we have the following particular results.
a
If the centre of gravity is the origin, y== 0; therefore
PCOS A = Σ.mx, PCOSμ .MY, PCOS v = Σ.mz;
(262)
that is, the pressure at the origin is due to the impressed forces
only.
And if the body is not subject to the action of any force,
then P = O, and there is no pressure at the fixed point.
181.]
319
PRESSURE ON THE FIXED POINT.
If no forces act, so that z.mx = Σ.MY = Σ.mz = 0; and con-
sequently LM = N=0; then, replacing
values in (29), Art. 149, we have
PCOSλ = Mπ (w₂²+w32) + M (A-B-C) w₁
2
PCOSµ = Mỹ (w₂²+w₂²) + M (B − C — A) w₂
My
dwy dwz dwz
by their
dt
dt dt
C
>
W z Z )
B
+M(B−C−A),
W z Z
W1 X
A
C
(263)
PCOS v = M 7 (w¸² + wz²) + M (C — A — B) wg {
Z — ~
+
B
A
Let us consider what properties are involved in these equa-
tions when P = 0. If
C- A
A ·B
B-C
= α1,
a2
ვა
(264)
AW]
BW Z
CW3
these equations take the symmetrical forms.
w² x − w₁ (w₁x+w₂ÿ + wzz) + (açÿ— ɑzZ) wz wz wz
0,
0,
(265)
1
1
3
w² ÿ — w₂ ( w₁ X + w₂Y+wzZ) + (a₁Z — Azł) W1 W2 W3
w² z − wz (w₁π+w₂ÿ+wzz)+(aqX —α₁ỹ)w1w2wz = 0 ;
multiplying these severally by x, y, z, and adding, we have
(w z ł− w₂ z )² + (wył−wzł)² + (w₂x — w₁ÿ)² = 0;
X
y
@1 @2
2
Z
;
W3
2
(266)
and replacing w1, w2, w3 in (265) by the proportionals given in
these last equations, we have
Ax2
B-C
By 2
CZ2
;
C- A
A-B
≈ =
(267)
which can only be satisfied, if y=z=0; so that when-
ever the equations (265) are true, the centre of gravity is the
origin.
CHAPTER VII.
THE MOTION OF A RIGID BODY, OR OF AN INVARIABLE MATE-
RIAL SYSTEM, FREE FROM ALL CONSTRAINT.
SECTION 1.-Motion of a free invariable system under the action
of instantaneous forces.
182.] As our inquiry proceeds, our problem becomes more
general; and the conditions of constraint become fewer. The
subject of motion is still a rigid body, or a system of particles of
invariable form; and thus all the internal forces, which enter
into the equations of motion in the most general problem, in
this disappear, because they are introduced in pairs neutralizing
each other. We return therefore to the equations which are
given in Chap. III, Art. 48; viz., (34) and (35) which are ap-
plicable to instantaneous forces; and (37) and (38) which ex-
press the action of finite accelerating forces.
In the solution of the problem we shall find it most convenient
to employ the principle of the independence of the motion of
translation of the centre of gravity, and of the rotation about an
axis passing through it, which has been proved in Section 2 of
Chapter III. For we shall thereby resolve into two distinct
parts complicated motion which arises from the action of given
forces; we shall consider the forces as they produce either
simple translation or rotation, and shall investigate their effects;
and the whole motion will be the result of these two separate
motions. And the process too is most convenient for the course
taken in our treatise; for the motion of a free invariable system
is thus resolved into that of simple translation of a particle at
its centre of gravity, and that of rotation about an axis passing
through the centre of gravity considered as a fixed point; mo-
tion of the former kind has been completely discussed in Vol. III;
and that of the latter, as far as is possible, in the Chapter prece-
ding the present; and we have investigated these as the effects
of forces similar to those which we have now to consider.
182.]
321
A FREE INVARIABLE SYSTEM.
This mode of resolution is most convenient for a dynamical rea-
son also: because all the forces which act on the several particles
of the system may be transferred, each in its own line of action,
direction, and intensity, to the centre of gravity; and may there
act on a particle of mass equal to that of the whole system; and
the motion of the centre of gravity will be that of the particle
under the action of the forces thus transferred; and because
the rotation relative to the axis passing through the centre of
gravity, which we may suppose to be a fixed point, is the effect
of the forces, as they act at their several points of application.
At no point, except the centre of gravity, or centre of masses,
are these dynamical propositions true. And an examination
of what has preceded shews the reason of this. The centri-
fugal forces generated in the motion neutralize themselves at
the centre of gravity; they produce thereon no pressure; and
thus cause no acceleration or retardation of it: whatever is the
pressure at, or the motion of, the centre of gravity, this is due
to the impressed forces, and to them alone. This fact has been
presented to us again and again in the course of our work.
Cinematically indeed other modes of estimating motion might
have been taken. In Chapter II it has been proved that what-
ever is the motion of a body, it always consists of a motion of
translation of any particle of it along a definite path, and of a
motion of rotation about an axis passing through that particle;
and the choice of the particle whose motion of translation is
considered is arbitrary. And when force acts on the body, the
effect of it, in combination with the centrifugal forces developed
in the motion, will be to change the line of motion and velocity
of the particle, the rotation-axis of the body passing through
that particle, and its angular velocity about that axis; and the
equations of motion will be formed in a manner which indicates
these several changes: these we shall hereafter exhibit. Or,
again, the motion which takes place in an infinitesimal time-
element always consists of a rotation about the central axis, and
of a sliding or of a motion of translation along that axis; and
the effects of the impressed forces and of the centrifugal forces
developed in the motion will be a shifting of the central axis,
a change of velocity along it, and a change of angular velocity
about it; and as the shifting of the central axis may take place
in the most general way possible, so will it consist of both a
displacement of translation and of a subsequent rotation about
PRCIE, VOL. IV.
T t
322
[183.
A FREE MATERIAL SYSTEM.
one of its points, so that the central axes in the new and the
old positions do not intersect each other; and these four sepa-
rate effects will be produced by the acting forces; they will be
exhibited therefore in the equations of motion, which will evi-
dently be greatly complicated; and the acting forces will have
to be resolved along lines, the position of which is continually
changing.
It is right to say thus much as to other modes of considering
the motion of a free invariable system; although we shall for
the most part confine ourselves to that motion of the centre of
gravity, and of rotation about it, to which our equations of
motion most conveniently adapt themselves.
183.] In the present Section I shall consider the motion of a
free invariable system under the action of instantaneous forces.
In the notation of Art. 48, the general equations of motion ap-
plicable to this problem are
m ( v
Σ.Μ
Σ.
2. (v,
m
da') = 0,
X dt
dy
d) = 0,
dt
m (v₂ — dz) = 0 ;
Σ.m
;
(1)
χ
dt
dx
dt
da ) — x ( v₂ — dz) }
Σ.Μ
z.m {y (v,
dz) - 2
(v,
dy)} = 0.
0,
{ ≈ (v x
−
(2)
Σ.Μ
m { x (v,
—
= 0.
dt
2. m = 0,
dt
V≈
dx
— dy) — y (v x − da) }
dt
Now if M is the mass of the moving system, and if (x, y, z) is
the place of the centre of gravity at the time t, and (x', y', z') is
the place of m relative to a system of axes originating at the
centre of gravity, and parallel to the original axes, then, by means
of Section 2, Chap. III, the equivalents of these are
M
ΙΣ
dx
dt
dy
dt
= Σ.ην 201
= Σ.MVy
ولا
(3)
dz
M
dt
= Σ. MV z ;
184.]
323
INSTANTANEOUS FORCES.
and
V
dz
dy
z.m {y (v. d) — 2' (v, d)} = 0.
— —
Σ.Μ
dt
dx
-
dt
Σ.Μ
V
X
dt
:)
X
x² (
dz
0,
(4)
dt
Σ.
x
-
V
x
:
dt
x.m { a' (v, — dy) — y' (v, - dx)} = 0.
dt
Now (3) express the motion of the centre of gravity relative to
a system of axes fixed in space, and do not generally admit of
farther reduction. Whereas (4) express the motion of rotation
of the body, or material system of invariable form, relative to a
system of axes originating at the centre of gravity, and in other
respects undetermined. If x, y, are the axial components
z
of the instantaneous angular velocity o which is due to the im-
pulsive forces, these equations become, see Art. 76,
A fx
ΕΩΣ L,
- F Qx + B Qy — D Q z
<= M,
→ E Qx — D ♫y + C Q₂ = N ;
(5)
where L, M, N are the axial components of the moment of the
couple or couples due to the forces of impulsion.
If, as in the last Chapter, we refer the rotation to two sys-
tems of axes, one of which is the central principal system fixed
in the body and moving with it, and the other is a system fixed
in space, then, in reference to the former, D = E = F = 0,
(5) assume the simple form
A Q₁ = L,
ΒΩ M,
CQ3 = N ;
0, and
(6)
where L, M, and N are the axial components of the moments of
the couples of impulsion relative to the central principal axes.
The solution of every problem depends on these equations;
they will hereafter be applied to particular cases; but some
corollaries from them require investigation.
184.] When a free material system of variable or invariable
form, which was at rest, has been acted on by one or more im-
pulsive forces, the preceding equations express the momentum
which has been communicated to it; and if no other momentum
has been imparted, and if no momentum is lost, they also ex-
press the whole momentum of the system, and consequently
that which is capable of abstraction from it. This fact, which
is a consequence of the law of inertia, is commonly termed the
principle of conservation of momentum or of force. Now when
a material system has been put into motion by one or more
forces of translation, these will generally be reducible to a single
Tt 2
324
[185.
A FREE MATERIAL SYSTEM.
force; and therefore a single force equal to this, and acting in
the opposite direction, will bring the system to rest. If the
system has been put into motion by a couple, no one force
generally can bring it to rest, because no one definite force act-
ing at a definite distance can neutralize a couple. Generally,
I say, in both cases; because in the course of our inquiry it
will be seen that under certain circumstances, which will be ex-
pressed in an equation of condition, a single force of impulsion,
or moving forces which will compound into a single resultant,
will cause a system to rotate about an instantaneous axis, either
with or without a motion of translation along that axis.
As the position of the axes fixed in space is arbitrary, let us
take them to be parallel to the central principal axes of the
body at the instant when the impulsive forces act. Let x, y, Z
be the axial components of the impressed momenta; and let M
be the mass of the system; then the axial components of the
resulting velocity of the centre of gravity are
X
M
Y
M
Z
;
M
ولا
(7)
let v be the velocity of the particle at the place (x, y, z) relative
to the central principal axes; and let vx, vy, vz be the axial com-
ponents of v; then, as 21, 22, 23 are the axial components of
relative to these central axes,
X
vx =
+ Z Q2―Y Q3,
M
Y
Vy
+ X Qz―ZQ1,
M
Z
Vz
+ Y Q₂ ― X Q2 i
M
and if 1, 2, are replaced by their values given in (6),
X
2 M
YN
Vx
+
M
B
C
Y
X N
Z L
+
M
C
A
Z
Y L
X M
ยะ
+
M
A
B
(8)
(9)
which give the components of the velocity of any particle of the
system, and consequently the velocity thereof.
185.] For the first application of these equations, I propose
to determine the locus of all points of the system which move
at the first instant with the same velocity v.
185.]
325
INSTANTANEOUS FORCES.
Let V, V, V₂ be the axial components of v; then, if (x, y, z)
ولا
is the place of a particle which moves with this given velocity,
¸· `·. v³ = ( ~
V2
X
M
X
Vx
+ ZD2―Y Q3,
M
Y
+X Dz―Z D1,
M
2
V.
+Y N―XDq j
;
M
2
Y
+ 2 Qz−Y Qg )² + ( ~ + X Qz−x 2)²
M
(10)
2
ገ
+ ( ~ +yî−xî); (11)
M
which is evidently the equation to a cylinder of the second order,
since it is of the form given in (33), Art. 356, Vol. I, 2nd. Ed.;
all the generating lines of which are parallel to that whose equa-
tions are
X
y
21 22
2
Dz
(12)
If not only v is constant for all points of the required locus,
but Vx, Vy, Vz, which are the axial components of v, are also
constant; then each equation of (10) separately holds good; and
the locus is evidently a straight line whose equations are
x +
Ynz-Zg
72
M 22
VyQz - VzQq
22
21
;
(13)
Equations (10) however are not independent; but are evidently
subject to the condition
XQ1+YQ2+Z;
(14)
M (Vx 1 + Vy z +VzQz) = X + Y Qq + Z Qq ;
and consequently instantaneous forces can produce the preced-
ing effect only when this condition is satisfied.
Again, let us suppose the motion which results from the in-
stantaneous forces to consist of a rotation about an instantane-
ous axis, and of a translation along that axis. In this case, if
x,y,z are the current coordinates of the rotation-axis, Vx, Vy, Vz
are constant for all points along that axis, and consequently
(13) are the equations to the axis, and its direction-cosines are
proportional to 1, 2, 3; and as v is the velocity of a particle in
and along that axis,
V
V
X
द
21
Vy Vz
23
;
Ω
(15)
326
[186.
A FREE MATERIAL SYSTEM.
in which case the condition (14) becomes
so that
MOV = XQ₁₂+Y Qg + ZQz ;
ΧΩΝ + ΥΩΝ + ΖΩΝ
V
ΩΜ
(16)
(17)
which shews that the velocity of displacement along the rota-
tion-axis is the sum of the resolved parts of the components of
the velocity of the centre of gravity along that axis.
The equations to the axis are
x +
Y Qz — Z Q2
21
M 22
(18)
186.] Lastly, let us suppose the system to rotate about an
axis without any motion of translation; so that for all points
along that axis vx = vy = v₂ = 0; then from (8), if (x, y, z) is a
point on that axis, we have
X
+ ZQ2―Y Qz =
= 0,
M
Y
+ XQ3−Z Q₁ = 0,
M
Z
+Y 21XQ2
=
= 0;
M
(19)
which are the equations to the axis; and may be expressed in
the form given in (18). This axis is called the spontaneous axis
of rotation. Its direction-cosines are evidently proportional to
1, 2, 3, so that it is parallel to what would be the instantane-
ous axis, if the centre of gravity were a fixed point. It passes
through the centre of gravity, if x = y = z =
Y = Z = 0; which will be
the case when the impressed forces of impulsion compound into
a couple.
Equations (19) are not independent, but are subject to the
equation of condition,
X 2 + Y Q2 + ZQz = 0 ;
(20)
which shews that the line of action of the resultant of the im-
pressed momenta is perpendicular to the instantaneous rotation-
axis, which is the spontaneous axis. This condition is the same
as that determined in Art. 34, and numbered (52).
If in (20) 1, 2, 3 are replaced by their values given in (6),
the condition for the existence of a spontaneous axis becomes,
MY
LX
+
B
A
+
N Z
C
=
0.
(21)
187.]
327
THE SPONTANEOUS AXIS.
If the motion is due to a single blow applied at the point (§,7,8)
whose momentum is Q, of which x, y, z are the axial components ;
then
L = nz-(x, M = (X-έZ, N = έy—nx;
and (21) becomes after division by x, y, z,
X
B
C
( ( 1 ) + ( 1 )
n
511
+
Y C
A
Z
(1-1)= 0;
(22)
(23)
so that (20), (21) or (23) is the condition to be satisfied, when a
body being struck by a blow rotates at the first instant of its
motion about an axis, without any other motion of translation.
I may in passing observe, that the necessary condition is
satisfied when the line of action of the blow is parallel to a cen-
tral principal axis, say that of; because in that case x Y=0;
=
and 23
= 0.
These circumstances however will be hereafter
considered at length.
•
187.] When the impulsive forces which act on a given system
satisfy the condition (21), so that the system rotates about a
spontaneous axis, that axis has the following property: the sum
of the vires vivæ of all the particles of the system due to the im-
pulsive forces is greater for that axis than it would be for any
other rotation-axis. So that the spontaneous axis may be de-
fined as that axis for which the sum of the vires vivæ of all the
particles due to the impulsive forces is greater than for any
other axis.
Although the theorem is true for any system, I will, for the
sake of simplicity, confine the proof in this Article to that of an
invariable system, the components of the velocity of any particle
of which are given in equations (8); at least as to form: for
herein we must assume the position of the rotation-axis to be
undetermined, and we shall shew that the vis viva is a maximum
when the rotation-axis coincides with the spontaneous axis.
Leto be the angular velocity about the assumed axis which
passes through the centre of gravity; and let w₁, w₂, w3 be the
axial components of w; then
X
Vx
+ Z W2―Y W3,
M
Y
+xWz — ≈ W1,
(24)
M
Z
༧.
+ Y W₁ = X Wq.
M
328
[187.
A FREE MATERIAL SYSTEM.
As the change of the velocity is supposed to be due to a change
only in the position of the rotation-axis,
dv₂ = z dw₂- y dw3,
x
dv₁ = x dwz - z dw₁,
vy
dvz
=
y dw₁-x dwq.
(25)
Let x, y, z be now the components of the velocity impressed on
m; then the equation of motion given by D'Alembert's Princi-
ple, in combination with the principle of virtual velocities, is
Σ. m {(x — vx) d x + (x − v„) dy + (z —v₂) dz} = 0. (26)
In this problem dx, dy, dz may be replaced by the actual velo-
cities vx, Vy, vz, whereby we have,
2
z. m (vx² + vy²+v₂²) = Σ.M (XVx+Y Vy+Z Vz).
Let the vis viva thus generated u; then
=
u = Σ.mv²,
U
+zv₂).
.'. D
2
= x.m (v,² + v₁² + v₂ ²);
2
!
DU = 24.m(vxdvx+v„dv»+Vzdvz) ;
(27)
(28)
and we have to shew that by reason of (27) this quantity vanishes.
From (27) we have
2z.m (vxdvx+v„dv₁+v₂dv;)
= x.m{xdvx+Ydvy+zdvz}
= x.m {(yz— ≈Y) dw₁+(≈x−xz) dw₂+(xx−yx)dw3} ;
and as dw₁, dw, dw, are independent of x, y, ≈, x, y, and z in
the right hand member, we may substitute from (35), Article
48, observing that V, V, V,
dx dy dz
dt' dt' dt
are severally X, Y, Z,
Vx, Vy, vz in our present notation; so that we have
2 z.m (vxdvx + v„dv„+v₂dv₂)
}
= x.m { (y v₂ — zvy) dw1 + (zvx − XVz) dwz + (xv, −yvx) dwz},
= x.m {(z dw₂ —y dwz) vx+(x dwz−z dw₁) v,+ (y dw, — x dw₂) vz},
=z.m(vxdvx+vy dv₂+vzdvz) ;
z.m (vxdvx+vy dv₂+v₂dv₂) = 0;
(29)
and therefore from (28), Du = 0; and consequently u is a maxi-
mum or a minimum.
And it is evidently a maximum; for if we give to vx, Vy, vz in-
crements dvx, dv,, dv, of such a finite magnitude that their
squares are not to be neglected, then the right-hand member of
(28) becomes
2 z. m (vxd vx+vy dvy + vzdvz) +Σ.m{(dvx)² + (dvy)² + (dv₂)²} ;
and we shall eventually have
z.m(vxdv¸x+vydv₂+v₂dv;)+z.m{(dvx)² + (dv„)² + (dv₂)²}=0; (30)
188.]
329
THE SPONTANEOUS AXIS.
that is, the increment of the vis viva for the finite variation is
less than it is for the infinitesimal variation by
Σ.m{(dvx)²+(dv„)² + (dv₂)²} ;
that is, by the sum of the vires vivæ due to the velocities lost
by the different particles of the system; and consequently the
vis viva determined as above is a maximum.
The preceding proof of this theorem is due to Lagrange*;
and the proof that the vis viva corresponding to the spontaneous
axis is a maximum is due to his editor, M. Bertrand. The
theorem was originally discovered by Euler†, and restated by
Lagrange; and although the proof given by the latter holds
true for a material system of invariable form, yet his mode of
expression is so obscure, that it is almost impossible to under-
stand his meaning when it is applied to a system of variable
form. Another proof is given by M. Delaunay‡, and this is
sufficient for all material systems.
Let u be the vis viva of the system arising from the angular
velocities due to the impulsive forces, then
2
2
U = Aw₂² + Bw₂² + C wz².
And if l, m, n are the direction-cosines of the undetermined axis,
this becomes
U = (Al² + Bm² + cn²) w²;
(L/ + M M + N n)²
Al² + Bm² + cn²
(31)
and equating to zero the total differential of this, we have
(BL m² + CL n²-AM Im-AN In) dl + ... + ... = 0;
also,
I dl +m dm+ndn = 0;
L
M
N
whence we have
Al
BM
сп
and therefore by means of (6)
m
n
;
21
D2
Dz
(32)
(33)
so that the vis viva is a maximum or a minimum when the ro-
tation-axis through the centre of gravity is parallel to the spon-
taneous axis.
188.] In the concluding paragraph of Art. 186 I have ob-
* See Mécanique Analytique, tome I, p. 271, ed. 3, par M. J. Bertrand,
Paris, 1853.
+ Theoria Motus corporum solidorum, cap. IX, Theorema 8 (Art. 637.)
Gryphiswaldiæ, 1790.
Liouville's Journal, tome V, p. 255.
PRICE, VOL. IV.
u u
330
[188.
A FREE MATERIAL SYSTEM.
served, that if a body is put into motion by a single blow, the
condition necessary for the existence of a spontaneous axis is
satisfied whenever the line of action of the blow is parallel to a
central principal axis. I propose to consider this circumstance
more at length. The investigation is for the most part due to
Poinsot*; and although fuller explanation will be found in his
Memoirs than I am able to insert in this place, still all his im-
portant theorems are here given.
In the first place, let us assume the line of action of the blow
to be not only parallel to a central principal axis, but also to lie
in a central principal plane. Let the momentum of the
q =
blow; let the central principal plane in which its line of action
lies be that of (x, y), and let the line of action be parallel to the
axis of y; these several lines are represented in Fig. 39; where
G is the centre of gravity, and is the origin; G, Gy, G≈ are the
three central principal axes; co is the line of action of the blow
Q, and c is the point at which it intersects Ga; c is called the
centre of percussiont. Let GC h; the mass of the body=m;
mk² = c =
its central principal moment of inertia about the
axis of; the angular velocity which is due to Q; then, in
=
this case,
X = 2 = 0
0;
Y = Q;
Q h
21 22
0 ;
23
Ω;
C
(34)
and thus the equations (19) to the spontaneous axis are
y = 0;
0
0
C
x =
mh
k2
h
(35)
which represent a line parallel to the axis of 2, and intersecting
the axis of x at a distance
k2
h
on the negative side from the
origin: this is the line OR in Fig. 39, which is therefore the
spontaneous axis; the point o, in which the spontaneous axis
intersects the plane of (x, y), is called the centre of spontaneous
rotation. Thus the effect of the blow q is to cause the body to
rotate about the axis OR with the angular velocity 2, which is
given in (34). The centre of gravity at the first instant moves
along the axis of y with a velocity
Q
ก
*Liouville's Journal, Deuxième Série, tome II, 1857, p. 281.
(36)
The reader will observe the difference between the term 'centre of per-
caussion' as here used, and as used in Art. 127.
189.]
331
INSTANTANEOUS FORCES.
Let oGh'; so that omitting the negative sign which enters
OG =
into (35),
hh'
k²;
(37)
and hence it appears that if c is the centre of percussion, o is
the centre of spontaneous rotation; and if o is the centre of
percussion, c is the centre of spontaneous rotation. Thus the
centres of spontaneous rotation and of percussion are reciprocal.
Also, since the product hh' is constant, it follows that the
smaller h is, the greater is h', and vice versâ. If h = 0, h' = ∞ ;
that is, if the blow is given at the centre of gravity, the axis of
- spontaneous rotation is at an infinite distance, so that the body
has only a motion of translation. If the axis of spontaneous
rotation passes through the centre of gravity, h' 0, and conse-
quently h =∞; which indicates that the blow must be given at
an infinite distance from G, or that the impressed force must be
a couple.
Hence also
Qh
Q h
ΩΞ
C
mk2
Q
mh'
k2
1
=
h +
;
h
Let och+h'; therefore
(38)
and consequently, corresponding to a variation of h, l is a mini-
mum, when h
k; in which case oG = k = GC, and oc = 2k;
and this is the shortest possible distance between the centres of
percussion and of spontaneous rotation.
In all these expressions occurs k, which is a central principal
radius of gyration; of this there are generally three different
values, corresponding to the three central moments of inertia ;
of which the greatest and least are those corresponding to the
greatest and least moments of inertia, and the mean corresponds
to the mean moment of inertia. Thus 7 is the minimum mini-
morum when k is the least; and is the maximum minimorum
when k is the greatest.
The velocity of the centre of percussion after the blow
(h + h') î ;
Q
m
h
(1+
h
189.] The preceding investigation leads to this result: when
a body is rotating freely about an axis parallel to one of its cen-
tral principal axes, and lying in one of its central principal planes,
U u 2
332
[189.
A FREE MATERIAL SYSTEM.
the whole momentum of the body may be considered to be due
to a single blow impressed on it in a line parallel to the central
principal axis which is perpendicular to the former principal
plane, and lying in the principal plane which is perpendicular to
the former principal axis. And consequently, if the body at that
instant met with a fixed obstacle at the point where the blow
acted, the whole momentum would be taken from the body, and
the body would be brought to rest, the fixed obstacle being
struck with a momentum equal to that which was originally
imparted to the body. Now, in reference to the given rotation-
axis considered as a spontaneous axis, the centre of percussion
would be the position of the fixed obstacle, and the momentum
of the blow which it would receive would evidently be q. Is,
however, the point c thus determined the position of the fixed
obstacle against which the body m would impinge with the
greatest momentum? Let us consider this question.
Suppose the body to impinge on an obstacle fixed at c', whose
distance from & = x, see Fig. 40, and suppose the momentum
which the obstacle receives to be P; let o' be the point recipro-
cal to c'; that is, o' is the spontaneous centre of rotation, when
c' is the centre of percussion; and thus
GO'
k2
(39)
At the instant of impact of the body on c', a is the whole mo-
mentum of the body, and its line of action is co; let us suppose
it to be resolved into two parts P and P', acting at c' and o' with
lines of action parallel to co; then, by the laws of composition
of parallel forces,
so that
Q = P+P';
Q × o'c = P × O′P′,
Q x cc' = P'x o'c';)
P = Q
= Q
k² + hx
k² + x²
x2
hx
k² + x²
(40)
(41)
(42)
As o' is reciprocal to c', P' produces no momentum at c'; so
that p is the only part of a which affects the obstacle at c'.
k2
If x =
h
Q
——h' =—GO, P =
GO, P=0; that is, an obstacle placed
at the spontaneous centre receives no blow.
If x =
O, P = Q; that is, an obstacle placed at the centre of
gravity receives a blow equal to the whole momentum.
190.]
333
INSTANTANEOUS FORCES.
If x =
h, P =Q; that is, an obstacle placed at the centre of
percussion receives a blow equal to the whole momentum.
Thus, a body strikes with the same momentum at its centre
of gravity and at its centre of percussion; but with this differ-
ence; when it strikes an obstacle placed at its centre of percus-
sion, it is brought to rest; when it strikes an obstacle placed at
its centre of gravity, its angular velocity continues what it was
before impact.
GO
If x = ∞, P=0, P'Q; and Go'= 0, so that the body strikes.
at its centre of gravity with a momentum p′ = q.
k2
If x is negative, and less than or h', P is still positive; but
X
h
if x is negatively greater than h', P is negative. In this case, c'
falls on the negative side of o, and the body strikes an obstacle
in a direction contrary to that for all points on the right hand
side of o; and thus the obstacle must be placed on the opposite
side of the line OGC.
190.] To determine the position of the obstacle, when the
momentum of the blow with which the body strikes it is a max-
imum or minimum, we must take the x-differential of (41) and
equate it to zero.
Thus,
hx² - 2 k² x + h k²
(k² + x²)2
dp
Q
0, if
dx
x =
− h' ± (h'² + k²);
(43)
and changes sign from
+
to
for the upper sign, and from
Hence we have two critical values
to + for the lower sign.
of P, which are respectively a maximum and a minimum: let
these be T and T', where T is the maximum corresponding to
x = h' + (h'²+k²)³; in which case
T = Q
2
2
(k² + h'²) ½ + h'
2 h
which is manifestly greater than Q.
2
1 + (1 + 1/2 )³ {},
h
(44)
h (h'² + k²) ³,
—
(k² + h'²) ì — h'
T =
Q
2h'
= {(1 + 1)² - 1 }
h
h
-1},
(45)
And corresponding to a =
x
2
which is evidently negative, and acts in a direction opposite to
that of T; and thus satisfying the criteria of a minimum, it is
indeed the greatest negative value.
The former result is apparently paradoxical; for as r is greater
334
[190.
A FREE MATERIAL SYSTEM.
than Q, the momentum of the blow with which the obstacle is
struck is greater than that of the whole moving body; a mo-
mentum therefore is extracted from the body greater than that
which it has. The explanation of the seeming paradox is, that
an opposite momentum, viz. T', has been generated; and T+T´=Q;
so that the sum of the two resulting momenta is equal to the
whole momentum of the body; and the principle of the con-
servation of momentum still rules this case: more however will
be said on this subject hereafter.
Let the points of application of T and T′ be R and R'; see Fig.
41; these are called the centres of greatest percussion; they
are evidently reciprocal to each other, as centres of percussion
and of spontaneous rotation. Also, since
x + h' = + (h'² + k²);
OROR'
(h'² + hh')*,
= (GO × OC);
X
(46)
This pro-
so that the two centres of maximum percussion are equally dis-
tant from the spontaneous centre; and the distance is a mean
proportional between the distances of the centre of gravity and
of the centre of percussion from that same centre.
perty gives an easy geometrical construction for the determina-
tion of the centres. Also this distance is equal to the radius of
gyration of the body about the spontaneous axis; because k is
the radius of gyration about the axis through the centre of
gravity parallel to the spontaneous axis.
If h = 0, that is, if the original blow q is given at the centre
of gravity, so that the spontaneous centre is at an infinite dis-
tance, and the body has only a motion of translation, then
P = Q
k2
k² + x²
dr
dx
2ą k² x
(k² + x²)
0,
(47)
(48)
if x =
0, and changes sign from + to; and the greatest
value of P is Q; that is, the greatest blow which the body is
capable of giving is at its centre of gravity.
If the body is originally put into motion by a couple whose
moment is N, so that the body has only a motion of rotation
about an axis passing through its centre of gravity, then in
(41), q = 0, h∞o, and ah N; so that
Q
=
qh =
191.]
335
INSTANTANEOUS FORCES.
P =
N
k² + x²
;
dp
N (k² — x²)
0,
dx
(k² + x²)²
(49)
if x=k; and P has two corresponding values, which are
N
respectively positive and negative; each of which = ; and
2k
their lines of action are equidistant from the centre of gravity,
the distance being equal to the central radius of gyration.
Thus, if a sphere of radius a rotates about a vertical diameter,
the greatest blow will be given on an obstacle at a distance
= a(.4) from the centre of the sphere.
2
If a circular plate of radius a revolves about an axis through
its centre perpendicular to its plane, it will strike an obstacle in
its plane with the greatest effect when that obstacle is at a dis-
tance = a(.5) from the centre.
If is the angular velocity of the body, N = mk2a; and
therefore from (49),
m k2
k² + x²
PQX
= x
Now the velocity of a point in the body at a distance from
the rotation-axis through the centre of gravity is ox; and since
momentum is equal to the product of the mass and the velocity,
a mass
m k2
k² + x²
moving with the velocity with which the
body impinges on the obstacle at its point of impact would pro-
duce a blow of equal momentum. And since, when ⇓ = k, this
mass
m
2
it follows, that when the body impinges on the ob-
stacle with the greatest effect, the momentum of the blow is the
same as that of a particle of half the mass of the body, moving
directly with a velocity equal to that of the corresponding cen-
tre of maximum percussion.
A similar result is true for the centre of maximum percussion
corresponding to ak.
191.] The subject from this point of view requires more con-
sideration. For suppose the body to impinge against, not a
fixed obstacle, but a finite moveable mass m', then the velocities
after impact, both of the body and of m', depend on the mass of
m', and on the mass which, moving with the velocity of impact,
336
[191.
A FREE MATERIAL SYSTEM.
would have the same momentum as the blow p due to the mov-
ing body.
In the general expressions for P and P' given in (41) and
(42), let o be replaced by its value mh'n given in (38); and let
us inquire what masses moving with the velocities at the points
of impact of P and P' respectively, will produce the momenta
P and P'; let м and м' be the masses required; then, since the
velocities of the points of impact are respectively
(h' + x) î and (h'
м (h' + x) α = Q
k² + hx
k2
:) 2,
༡
X
similarly
k² + x²
hh + hx
= m h's
;
k² + x²
m k²
.'. · M =
(50)
k² + x²
M' =
m x 2
k² + x²
(51)
these equations assign the fractions of m, which, moving with the
velocities of the body at c' and o', would produce momenta equal
to è and p' respectively.
In reference to these values, let it be observed, that
(1) M+M': =m; so that the sum of the two masses is equal
to that of the whole moving body.
k2
X
(2) Mx = M' ; so that the two masses statically equilibrate
about G, the centre of gravity of the body; and thus м and м'
have the same centre of gravity as m.
And thus the masses, which, placed at two reciprocal centres,
may equivalently replace a body so far as impact at these centres
is concerned, are equal to the whole body, and are to each other
inversely as the distances from the centre of gravity.
K4
(3) M x² + M' = mk²; so that the moment of inertia of the
x²
two masses relative to the central principal axis, which is per-
pendicular to the line of blow, is equal to that of the body;
and consequently the moment of the two relative to the spon-
taneous axis is equal to that of the body relative to the same axis.
In all these respects, then, a rigid inflexible straight bar, whose
mass must be neglected, of any length = x +
with masses
k2
X
192.]
337
INSTANTANEOUS FORCES.
equal to м and м' at its two ends, will equivalently replace a
body; it will have the same mass, the same centre of gravity,
and the same moment of inertia; and when it is charged with
the same momentum of impulsion, it will have the same spon-
taneous axis, and the same percussion at a corresponding point.
From (50) it appears, that мm only when x = 0; the
centre of gravity therefore is the only point at which the
momentum of the blow is the same as that of the whole body.
mh'
"
1
If x = h, M = which is only a fraction of m.
Now we
- have already shewn that P = Q, both when the point of impact
is at the centre of percussion and at the centre of gravity; in
the latter case the momentum is due to the mass m, moving
Q
with the velocity in the former it is due to the mass
m
moving with a velocity
Ql
mh'
•
mh'
ī
see (39); that is, in the former
case we have a smaller mass and a greater velocity. Although
the effects will be the same when the impact takes place against
a fixed obstacle, yet they will not be the same when the object
impinged upon is a moveable finite mass, say m'. Thus, if m' is
at rest, when the body strikes it, and its elasticity is e, then, if
v' is the velocity of m' after impact, we have, from Vol. III,
Art. 215, (9), the following values:
If the centre of gravity is the point of impact,
Q (1 + e)
√ =
m+'m'
And if the centre of percussion is the point of impact,
Q (1 + e) l
v =
mh' + m' l'
(52)
(53)
and therefore is greater in the latter case than in the former.
192.] Let us however investigate the position of m', when the
velocity communicated to it at rest by the impact of the body
is a maximum,
Let the distance of m' from the centre of gravity; then
P, the momentum of impact, is given by (41), and the mass
corresponding to the momentum of the blow is given by (50);
so that by (9), Art. 215, Vol. III,
PRICE, VOL. IV.
I
q (1 + e) h (h' + x)
(m+m') k² + m²x²
X X
(54)
338
[192.
A FREE MATERIAL SYSTEM.
Of this quantity let the x-differential be taken and equated to
zero; thence we have
m
x²+2h' x
1 +
:) k²
0;
m
M
x + h =
+
k2
การ
{ k'² + ( 1 + 1/2 )
h²² :).
-
(55)
so that, as in the case of greatest percussion against a fixed
obstacle, two points give critical effects; to one of which corre-
sponds a positive, and to the other a negative maximum: these
points are equidistant from the spontaneous centre; and the
distance of each from that centre
= { 1² + (3
1 +
m
m
) k²
;
this distance depends on m', the mass of the particle impinged
upon, and is less the greater m' is. The points determined will
coincide with the centres of greatest percussion only when m'∞;
which is a result in accordance with the fact, that a fixed
obstacle is nothing else than a particle or body of infinite mass.
Thus we have arrived at two new points; which, however, are
not reciprocal to each other, as the centres of greatest percussion
are.
Also, corresponding to the values of a, given in (55),
Q (1+e)
2 (m + m )
(m+m')
M
h) //
± 1 + ( 1 + 1 )
h'
+ 1]; (56)
[ {
m
which are the greatest values of the velocity with which the par-
ticle m' can be projected after impact by the body.
When the body impinges on m' at rest at a distance a from
the centre of gravity, the velocity of m' is given in (54); if v is
the velocity of the impinging point of the body after collision,
then, by (8), Art. 215, Vol. III,
V
(m-em') k² - em' x²
(m + m') k² + m²x²
Q
(h' + x) mah
(57)
and the momentum at that point after collision
2
(m-em) ko em 2 ( tư) họ
(m+m²) k² + m²x²
k² + x²
(58)
If we take the x-differential of (57) and equate it to zero, the
point will be determined at which the body must impinge on m',
and continue to proceed with the greatest velocity.
Thus, if a body, which has been put into motion by a blow whose
momentum is a, impinges on a particle m' at a distance = x
from its centre of gravity, under the circumstances of the pre-
193.]
339
INSTANTANEOUS FORCES.
ceding Articles so that the momentum of a blow given against
Q
k² + h x
k² + x²²
a fixed obstacle at the point
; then after impact
on m' the momentum is given by (58). Now as the point at
which P', see equation (42), acts is the point reciprocal to that
at which P acts, so p' will not affect the momentum given by
(58), and consequently the spontaneous axis is not altered by
the collision. But, if a' is the angular velocity about the spon-
taneous axis after the collision,
(m-em') k²- em' x²
Ω
')
(m + m² ) k² + m²x²
(h' x + x²²)
k² + x2
(59)
If m' =∞, then the momentum of the blow which the body is
capable of at a distance
from the centre of gravity after
impact
= x
k²+hx
e
Q,
k² + x²
ep;
(60)
that is, is e times the momentum before impact, and acts in an
opposite direction.
And
h' x + x²
2
e
Ω.
ー
​k² + x²
In this case, if x = h, o'
(61)
en; and the effect of the impact
is to change the direction of the angular velocity about the spon-
taneous axis, and to diminish it in the ratio of e : 1.
193.] Suppose however that, when the body impinges on the
fixed obstacle at a distance = x from the centre of gravity, the
point of impact is brought to rest, and has no further motion;
that is, suppose that in the preceding Article m'
= ∞ and
e = 0; then the momentum of the body is reduced to a quantity
p', whose value is given in (42), and which acts at a distance
k2
X
of this.
Since
from the centre of gravity.
Let us consider the effect
x² - hx
P'
Q
k²+2:2
the velocity of the centre of gravity
Q x² -h x
m h² + x²
= u' (say);
(62)
and the angular velocity about an axis passing through the
centre of gravity, which is also the angular velocity about the
spontaneous axis which passes through the fixed point,
Q
m
h-x
k² + x
-2
n' (say).
(63)
X X 2
340
[193.
A FREE MATERIAL SYSTEM.
If x =
h, u 0, and 20: in this case the fixed obstacle is at
the centre of percussion, and the body is brought entirely to
rest.
From these values however, of u and n', some interesting
questions arise: we can determine the values of a, which will
render u' a maximum; or will give it a given value; say, will
make it equal to the original velocity of the centre of gravity,
and in an opposite direction. Similar values too may be found
for '.
Let us first take u'; if u' is a maximum, then
x+h' = ± (h'² + k²) * ;
which give the two centres of greatest percussion. In reference
to this property M. Poinsot has called these points the centres
of greatest reflexion. One will be a centre of reflexion in a
direction opposite to that in which the centre of gravity was
going previously to the impact; and the other will be the centre
of reflexion in the same direction.
For if we take the upper sign,
Ú
=
Q
2mh'
Ω
{h' — (h'² + k²) * },
· 1/2{ { (h'² + k²) \ — h' } ;
(64)
which is evidently negative; and therefore the centre of gravity
of the body moves after the impact in a direction contrary to
that of its former motion; and thus has undergone a true re-
flexion.
If we take the lower sign,
Q
2mh'
{h' + (h'² + k²)* },
Ω
{h' + (h'² + k²) ½ };
2
(65)
in which case the centre of gravity of the body proceeds in the
same direction as before the impact, and with an increased
velocity.
If the velocity of the centre of gravity after the impact is the
same as it was before the impact, but in an opposite direction;
then, from (62),
Q x² - h x
m k² + x²
1
Q
M
h
‚'.
X
+ (h² — 8 k²) ½ ;
(66)
4
4
193.]
341
INSTANTANEOUS FORCES.
these two points have been called by Poinsot centres of perfect
reflexion. They are however only possible when he is not less
than 8k²; that is, only when the original blow Q has been given
at a greater distance from the centre of gravity than the limits
thus assigned.
Next let us consider the value of a', given in (63). ' has a
maximum value when the x-differential of it = 0; in which case
x = h ± (h² + k²) ½ ;
(67)
which are two values always real, one being positive and the
other negative. These points are situated at equal distances
from the original centre of percussion; and the distance is equal
to the radius of gyration of the body about an axis passing
through the centre of percussion, and parallel to the spontane-
ous axis. These points have been named by Poinsot centres
of greatest conversion. On comparing the values of x which
assign these centres with those which assign the centres of great-
est reflexion, it is evident that these bear the same relation to the
centre of percussion as those do to the spontaneous centre. So
that the centres of greatest conversion in a body become the
centres of greatest reflexion, and vice versâ, if the centre of
percussion and the spontaneous centre are interchanged.
If in the value for a given in (63) we substitute for a the
value given in (67) with the upper sign,
Ω
Ω
2h
2
{ (h² + k²) ½ — h} ;
(68)
which is negative, and thus indicates that for this centre of con-
version the angular velocity of the body is in a direction the
contrary of what it was before the impact.
If we take the lower sign in (67),
Ω
{ (h² + k²)³+h},
2h
(69)
which is positive; and this shews that the direction of the an-
gular velocity for this centre of greatest conversion is the same
as that of the body before impact.
If the angular velocity of the body after impact is the same as
before impact, but in an opposite direction, then '-; and
Ω
Q
h
X
m
k² - x²
Qh
mk2
;
h
1
X=
+
2
2
(h'² — 8 k²) ³ ;
(70)
342
[194.
A FREE MATERIAL SYSTEM.
ご
​which gives possible values only, provided that h2 is not less
than 8k². These two points have been called by Poinsot centres
of perfect conversion.
If the angular velocity after impact is to have a given value;
say, if the angular velocity after impact = n times the angular
velocity before impact, it is only necessary to equate the value
of a given in (63) to na, and the resulting quadratic equation
will give the positions of the corresponding points of impact,
and will, by the nature of its roots, also assign the limits of
possibility of the problem.
I may in conclusion observe, that in the Memoir by M. Poin-
sot a geometrical construction is given whereby the several
centres may be determined.
194.] Let us now consider a problem of the same kind, though
somewhat less special, in which the condition (21), necessary
for the existence of a spontaneous axis, is also satisfied; that,
namely, in which the line of action of the impulsive blow is
parallel to a central principal axis, although it does not, as in
the problem just discussed, lie in a central principal plane.
Let the line of the blow be parallel to the central principal
axis of; and let the point of impact be (x, y) in the principal
plane of (x, y); let Q = the momentum of the blow; and let all
these circumstances be delineated in Fig. 42; wherein G, the
centre of gravity, is the origin, Gx, Gy, Gz are the three central
principal axes, relative to which the moments of inertia of the
body are A, B, C respectively, and the corresponding radii of
gyration are severally a, b, c. Let c be the point (x, y) in the
plane (x, y), whereat the blow, whose moment is q, strikes the
body; the point c will be called the centre of impulsion. Now,
in this case,
= y = 0;
X Y=
z = Q ;
(71)
Q Y o
Q XO
21
N] =
22
53
= 0.
(72)
A
B
Let the mass of the body = m, and let x, y, z be the current
coordinates of the spontaneous axis; then its equations, which
are given in (19), Art. 186, become
Q
Q
z =
Q
0,
Y yo +
X X O
=
= 0;
m
A
B
(73)
which are the equations to a line in the plane of (x, y). Let a
and в be replaced respectively by their equivalents ma² and mb²,
B
194.]
343
INSTANTANEOUS FORCES.
then the equation to the spontaneous axis in the plane of (x, y)
becomes
x xo
Ο yyo
b2
+ + 1 = 0;
a²
(74)
which is the line Er in Fig. 42: thus the body by reason of the
blow a begins to revolve about a line which is in the central
principal plane perpendicular to the line of blow.
Let ca be produced, and cut the spontaneous axis in the point
o; of which let the coordinates be x'o, y'o; then
xo
a²b² yo
2
0
a² b² xo
a² x² + b² yo
2'
y'o =
0
a² x 2 + b 2 y 2 ;
(75)
which give the coordinates of o in terms of those of c.
Since x, y and xo, yo are symmetrically involved in (74), it
follows that the points to which they correspond are thus far
at least reciprocal. And as c is called the centre of impulsion,
o is called the spontaneous centre. Thus, if o is the centre of
impulsion, c is the spontaneous centre through which the spon-
taneous axis passes; and vice versâ.
Let h and h' be the distances of c and of o respectively from
G; then
and
h²
2
x² + y²,
Yo
2
0
x² ² + y
Ο
2
a+b+ (x² + y²)
2
(a²x²² + b² Yo
2
2 2
a²b² (x²+ y²)
...
hh'
a²x² + b²y ²
2
a² b² h
a²x² + b² y ²
h
(76)
(77)
Let us interpret this result by means of the central momental
ellipsoid. The equation to that ellipsoid which is given in
Art. 102 is
2
A &² + Bn² + C² = μ,
where μ
is undetermined. Let A, B, C be replaced by ma², mb²,
mc² respectively; and, for the sake of simplification, let μma² b²;
then the equation becomes
a² §²+b² n² + c²¿² = a² b²;
and thus the trace of this on the plane of (x, y) is the ellipse
a²²+b² n² = a²b²;
(78)
which we will call the central ellipse. The - and the n-
principal semi-axes of this ellipse are evidently b and a respect-
ively; and the moments of inertia about these axes are A and
B; which are respectively equal to ma² and mb2; so that
344
[195.
A FREE MATERIAL SYSTEM.
A = m a²
μ
B = mb²
b2
a²
and consequently the moments of inertia are inversely propor-
tional to the square of the corresponding radii vectores of the
ellipse. If K is the moment of inertia about a radius vector d of
this ellipse,
K =
н
82
m a² b2
82
(79)
Let (§, n) be the point P where this ellipse is intersected by
the line &c; see Fig. 43; and let the radius vector GP = d; then
{
Xo
and consequently from (76),
б
;
Yo h
hh' = &² + n² = d²;
.*.
GC X GO = G P²;
(80)
(81)
and thus, if the central ellipse is described, the spontaneous
centre which is relative to a given centre of impulsion can be
determined immediately. Of this theorem, (37), Art. 188, is
evidently a particular case.
Hence, if c is the focus of the ellipse, the spontaneous axis is
the farther directrix.
Again, draw the diameter GD which is conjugate to GP; its
equation is
x E
+
b2
уп
0 ;
a²
(82)
and the spontaneous axis is evidently parallel to this line. Thus,
if through o we draw os parallel to GD, or to the tangent of the
central ellipse at P, os is the spontaneous axis.
Similarly, if through c the line cr is drawn parallel to GD and
os, CT is the spontaneous axis relative to o as a centre of im-
pulsion.
If c is at P, o is at P'; and CT and os are tangents to the
central ellipse at P and P' respectively. In this case PP' is the
shortest possible distance along the line cGo, between the centre
of impulsion and the spontaneous centre. But of all minima
distances between these centres, AA' is the least and BB' is the
greatest.
195.] The spontaneous axis and spontaneous centre, which
are relative to a given centre of impulsion, give rise to many
interesting theorems.
195.]
345
INSTANTANEOUS FORCES.
(1) The equation to the spontaneous axis, in reference to a
given centre of impulsion, being
XX
0
b2
YOY
+
1;
a2
it is evident that, if a series of spontaneous axes pass through
the same point (x'o, y'o), all the corresponding centres of impul-
sion lie along the straight line
xx' 0 yyo
+
b2
a2
= 1.
(83)
This line is parallel to GD, which is conjugate to the diameter
GOP' of the central ellipse: and thus all the centres lie along
the line cт, which is the spontaneous axis relative to o as a
centre of impulsion.
Similarly, of centres of impulsion lying in the line os, the
corresponding spontaneous axes pass through the point c.
Thus, wherever in os the centre of impulsion is, c does not
move; and wherever in cr the centre of impulsion is, o does
not move.
(2) Let us suppose the centre of impulsion to move on a given
curve; then the spontaneous axis will envelope another curve,
of which the equation may be found.
Thus suppose the centre of impulsion to move on the circle
x 2 + y 2 = r²;
then the equation to the spontaneous axis is
XX
Ꮖ Ꮖ (
Y yo
+
62
= 1
1;
2
(84)
(85)
let these be differentiated, on the supposition that x, and yo
vary; then the envelope of the last is
x2
64
+
y2
1
a4
2.2
(86)
which is the equation to an ellipse, coaxal and concentric with
the central ellipse.
(3) Or, again, the curve may be given, all the tangents to
which are to be spontaneous axes; and it may be required to
determine the locus of the corresponding centres of impulsion.
Thus, if the spontaneous axes all touch the circle x² + y² = r²,
the locus of the corresponding centres of impulsion is the ellipse
x2 y²
+
b+
1
as
2.2
Indeed these reciprocal properties give rise to a complete system
PRICE, VOL. IV.
Y y
346
[ 196.
A FREE MATERIAL SYSTEM.
of duality, to a great extent similar to that of polar lines and
their reciprocals. This however is not the occasion for a farther
development of it.
(4) Or, again, since the coordinates to the centre of impul-
sion and to the spontaneous centre are related by the equations
(75), if the locus of one centre is given, that of the other can be
found.
Thus, suppose the centre of impulsion to move along the
x2 y2
1, then the locus of the spontaneous centre
ellipse +
b2
a²
is also the same ellipse.
If the centre of impulsion moves along a straight line, then
the locus of the spontaneous centre is an ellipse. Let the equa-
tion to the straight line be put into the form
+
X XO Y yo
b2 a²
1;
(87)
where x and y are any constants. Then the locus of the
spontaneous centres is
¿2
b2
n² Exo
+ +
a²
Хо
nyo
0;
+
b3
a²
(88)
which is evidently an ellipse similar, and similarly situated, to
the central ellipse; whose centre is at (-0, -20), and which
2
passes through the origin. The form of the equation to the
straight line which I have taken shews that the line is the spon-
taneous axis to a centre of impulsion situated at (−x, −yo).
196.] Let thus much suffice for the circumstanees of the
spontaneous axis of the body in its relation to the centre of
impulsion; and let us investigate other incidents of its motion
at the first instant.
Let å be the angular velocity; then, from (72),
n² = 2² + 2₂²,
Q² (x²
2
2
Yo
+
m²
64
a4
mp
(89)
if p is the length of the perpendicular from the centre of gravity
on the spontaneous axis.
Q
The velocity of the centre of gravity is evidently
m
·
196.]
347
INSTANTANEOUS FORCES.
Also, since the length of the perpendicular from c(x, y) on
the spontaneous axis
2
a²x² + b²y ² + a² b²
2
0
2
(aª x² + b¹ y ²) ½
(90)
therefore the velocity at the centre of impulsion at the first
instant after the blow
2
Q
XO
+
y02
+
m
b2
a2
+1}
(91)
Q h + h
M
h
If the body impinges against a fixed obstacle, or indeed against
any mass at its centre of impulsion c in the plane of the
central principal axes of x and y, the momentum of the blow
will be q let us inquire what will be the momentum at any
other point of impact in the same plane; say at R, which is
(x, y); with a view to a farther inquiry of the position of the
point when the momentum is a maximum.
Let c be as heretofore (x, y), and let R be (x, y), Fig. 44 ;
and let the momentum of the blow given at R be P; let o' be
the spontaneous centre relative to R, and let o'u be its spon-
taneous axis. Join RC, and produce it to meet o´u in u. Re-
solve o into two parallel forces, P and P', acting at U and R, with
lines of action parallel to that of a; so that by the laws of com-
position of parallel forces,
Q
P+ P'
Q,
PXUR = Q X UC,
P′ × UR = Q × CR.
X
(92)
(93)
(94)
As o'u is the spontaneous axis relative to R as centre of impul-
sion, whatever in o'u be the point at which a blow is given, R
remains at rest; so that p' impressed at u produces no effect on
R; and consequently P is the whole effect at R; and P is determ
ined by (93): we have therefore to find UR and UC.
The equations to Uo' and CR are respectively
b2
ξα ny
+ +1 1
a²
=
= 0,
¿ (y-yo) + n (x−x) +уx-x。y = 0;
and if u is (έ, n),
UC
Xo - §
UR
X
§
...
P = Q
=
a² x x。 + b² y yo + a² b²
a²x²+b² y² + a² b²
a² x xo + b² y Yo + a² b²
a²x² + b² y² + a²b²
(95)
(96)
Y y 2
348
[196.
A FREE MATERIAL SYSTEM.
which is the momentum of the blow with which the body would
strike any obstacle at the point (x, y) in the central principal
plane of (x, y).
Certain particular values of P deserve mention.
a²xx。+b²y yo+a²b² = 0;
=
0, if
(97)
that is, when the obstacle is at any point on the spontaneous axis.
And PQ, when
a²x²+b² y² — a²xx-b²yoy
0;
(98)
that is, when the obstacle is at any point on the ellipse, similar
and similarly placed to the central ellipse, of which the line GC
is a diameter, and of which, of course, the centre is at the middle
point of GC. The points G and c are on this ellipse; and con-
sequently at both these points P — Q.
The case also in which P has a given value, say no, deserves
consideration; of course it gives a locus of centres of percussion,
which is generally an ellipse; and in certain cases becomes a
point; and in certain other cases is imaginary. The subject
however does not offer any particular difficulty: and the student
can easily work it out for himself.
Also, if the body were originally put into motion by the
blow q at G, so that it has a motion of translation only, then
0, and
xo
Yo
PQ
a² b2
a²x² + b² y² + a²b²·
(99)
Let us also investigate the value of p', and consider certain
particular values of it: from (92) or (94),
a²x²+b² y² — a² xxo-b²y yo
2
a²x²+b² y²+ a² b²
P' =
Hence P' = 0, when
a²x² + b² y² — a² x x 。 — b² y yo
=
0;
(100)
that is, when the point of impact is on the ellipse given in
(98), in which case P = Q. Hence P' O, if the obstacle is at
the centre of gravity, or at the centre of impulsion.
Also PQ, when
a² xx。+b²yy。 + a²b² = 0;
that is, when the obstacle is at a point on the spontaneous axis,
in which case P = 0.
We might also investigate the locus of the place of the obstacle
when P' no, say; but as the problem presents no particular
difficulty, the reader may work it out for himself.
197.]
349
INSTANTANEOUS FORCES.
*
If the centre of impulsion is at the centre of gravity, x=y。=0;
and
a²x²+b² y²
P′ = Q
a²x² + b² y² + a² b²
(101)
197.] I propose now to investigate the points for which p is a
maximum. In this case
dx
dr)
dr)
= 0. And hence we have
dy
0
xo (a²x²+b² y² + a²b²) − 2 x (a² x x。+b² y yo + a²b²) = 0, )
Yo (a²x² + b² y² + a² b²) — 2 y (a² x x。+b² y Yo + a²b²) = 0
0
(102)
X
У
(103)
;
хо
Yo
which give
whence it follows that the place
critical value, is on the line GC.
(102) from (103), we have
of the obstacle, when P has a
If we substitute in either of
2
(a²x²+b²y ²) x² + 2 a² b² xx-a² b²x² = 0,
(a²x²+b² y²) y²+2 a²b²yy—a²b³y ² = 0;
0
X
0
2
=
(104)
(105)
which are quadratic equations in terms of x and y respectively;
and thus give two positions of a point of impact for which P
has a critical value. Let these be called centres of greatest
percussion; and let r be their distance from the centre of gravity;
then r² = x²+y²; also h²
x²² + Yo
Y
2
;
and
;
(106)
h
Хо
Yo
so that (102) give
(a²x²+b² y²) r² + 2 a² b² hr — a² b² h² = 0 ;
(107)
therefore, substituting from (77),
p² + 2hr-hh'= 0;
(108)
+
r = − h' ± (h'² + hh')³ ;
(109)
thus the two centres of greatest percussion are equally dis-
tant from the spontaneous centre o. Let v and v' be these
centres; Fig. 45; then
OV = Ov′ = (OG × OC);
(110)
and the distance is a mean proportional between the distances of
the centre of impulsion and of the centre of gravity from the
spontaneous centre.
Let T and T' be the corresponding momenta: then making
the above substitutions in (96),
P = Q
rh+hh'
r² + hh
(111)
350
[198.
A FREE MATERIAL SYSTEM.
2
Q h' + (h²² + hh')
therefore
201
(112)
T
2
(113)
h'
T=
h'
Q h' — (h'² + h h')
of which т, which acts at the centre v, is positive, and is greater
than Q; T', which acts at the centre v', is negative, and thus m
gives a blow against an obstacle at v'in a direction opposite to that
in which it strikes the obstacle at v; and thus, as the obstacle at
v must be on the upper or positive side of the plane of (x, y), that
at v must be on the lower or negative side. On applying the
criteria for a maximum or minimum to these values of P, viz., T
and r', it will be found that T is a maximum, and that r' is a
minimum; but as r' is negative, it is a negative maximum, so
that both centres may be called centres of greatest percussion.
If the point of application of the blow q, by which the body
is originally put into motion, is the centre of gravity, so that
x=y=0, and the body has only motion of translation; then,
Yo
from (99),
a2 b2
F = Q
a²x² + b² y²+ a² b²
(114)
and the maximum value corresponds to x = 0, y = 0; in which
case P = Q; and thus the centre of gravity is the centre of
greatest percussion.
If p =
then
22
a²x² + b² y² = a²b² (n − 1);
(115)
which represents an ellipse concentric, coaxal, and similar to
the central ellipse; and therefore the intensity of a blow against
an obstacle is the same for all points on this ellipse. If n 2,
the ellipse is the central ellipse.
198.] Again, if the body is put into motion by a couple whose
axis is perpendicular to the axis of %, so that the spontaneous
axis passes through the centre of gravity, and the body has only
rotation about that axis which is in the plane of (x, y), the
momentum of a blow p at the point (x, y) in the plane of (x, y),
may be determined in the following manner, which is independ-
ent of the preceding process:
Let L and м be the components of the moment of the impressed
couple about the axes of x and y respectively; then we have
X = Y = Z = 0;
z
L
M
(116)
221
ma²,
23
2₂ =
mb2,
23 =
0;
198.]
351
INSTANTANEOUS FORCES.
and the equation to the spontaneous axis is, see (19),
Ly
M X
0.
b2
22
(117)
Since the point of application of P is (x, y), the equation to
the corresponding spontaneous axis is
a² x § + b² yn + a² b² = 0;
the perpendicular distance on which from the point (x, y)
a²x² + b² y² + a² b²
(aª x² + b¹y²) ½
(118)
(119)
Let us suppose the couple of impulsion, of which the axial com-
ponents are L and м, to be replaced by a couple whose forces
are p at (x, y), and P at the point of intersection of the spon-
taneous axis with the perpendicular on it from (x, y); then
the moment of that couple
= P
a²x² + b² y² + a² b²
(a± x² + b+ y²)✯
and the x- and the y-direction-cosines of its axis are
b2y
a² x
and
>
(a* x² + b + y²) =
2
(a* x² + b* y²) $ '
(120)
(121)
And (120) is to be equal to the couple of which L and м are
the axial components; hence
Þ (a²x² + b² y² + a² b²) = L b² y — м a³x;
Μα
I b² y M a² x
Р
a²x² + b² y² + a² b²
(122)
This quantity might also be deduced from the general expression
of P, given in (96). For when the body is put into motion by a
couple, that couple is equivalent to a force = 0 acting at an
infinite distance; so that in the numerator x = Yo
consequently a² 62 must be omitted; and thus
but
Р
Q a² x x¸ + Q b² y yo
a²x² + b²y² + a² b²
= ∞,
∞, and
Qy。 = the moment of the couple about the axis of x = L,
Q. Yo
and -Qx。 =
. P
L b² y M Ɑ² x
a²x² + b² y² + a² 62
-
y = M ;
Now since P acts at a point on the spontaneous axis which
corresponds to the centre (x, y), -P produces no effect at (x, y);
so that P, which is given in (122), is the momentum of the
whole blow given by the body on the obstacle.
352
[199.
A FREE MATERIAL SYSTEM.
When P thus determined is a maximum,
whence we have
(dr
P
dx
(d)
= 0;
a²
- M (a²x² + b² y² + a2 b²) = 2 x (Lb²y - Ma²x),
L (a² x² + b² y² + a² b²)
=
2 y (1 b²y — м a²x);
M
}
(123)
.*.
LX+ MY
= 0;
a²x² + b² y²
= a² b².
(124)
(125)
Thus, the point (x, y) which gives the greatest percussion, is
in the central ellipse, at the points in which the plane of the
couple of impulsion passing through the centre of gravity in-
tersects it. And the greatest value of p
= +
2
(a² L² + b² M²) +
2ab
;
(126)
the two signs corresponding to the two extremities of the
diameter of the central ellipse, which coincides with the plane
of the couple passing through the centre of gravity; at which
points the two values of P are equal, but as they have opposite
signs they act in opposite directions.
199.] And let us farther investigate the nature of the blow,
when the body strikes against a moveable mass at the point
(x, y).
Let p be the perpendicular distance from (x, y) to the spon-
taneous axis; so that
p =
a² x。x + b²уy + a²b²
(a² x² + b² y 2) +
2
0
;
(127)
and let o be the angular velocity of the body about the spon-
taneous axis due to the force of impulsion; then, from (89),
2
0
Q (a* x ² + b + y ²)*
Ω
m a² b2
(128)
and consequently if v is the velocity of the point (x, y) which is
due to the force of impulsion,
Q a² xox + b² yo y + a² b²
v = a p =
m
0
a² b2
(129)
Hence, if P is the momentum of the blow which the body is
capable of at the point (x, y),
P = Q
= v
a² x x。 + b² y yo + a² b²
0
a²x² + b² y²+ a² b²
m a2 b2
a²x² + b² y²+ a² b²°
(130)
199.]
353
INSTANTANEOUS FORCES.
Let м be the mass which, moving with the velocity v, would
produce on the obstacle at (x, y) a blow of this momentum; then
M
m a² b2
a²x² + b² y² + a² b²
(131)
Also, let м' be the mass of a particle which, moving with the
velocity of the spontaneous centre (x, y) corresponding to
(x, y), would produce against an obstacle placed there a blow
whose momentum is equal to that of the body. Then
M =
m a2 b2
a²x²² + b² y²² + a² b² ³
m (a²x² + b² y²)
a²x² + b² y² ÷ a² b²·
(132)
by reason of (75). And thus masses are assigned, which are
fractions of m, and which, moving with the velocities of any point
and of its corresponding spontaneous centre, would have mo-
menta equal to those of the blows which the body would give to
obstacles placed at those points.
The values of these masses thus determined may be conveni-
ently put into another form: let r and be the distances of м
and м' from the centre of gravity; and let & be the radius vector of
the central ellipse which coincides with the line joining м and
M': then, as the places of м and м' are reciprocal as a centre of
impulsion and a spontaneous centre, rr' = d²; and from (77),
a²x² + b² y²
a² b² r
;
mr'
M =
rtř
m 82
r²+82
so that
27 2
M'
r+rs
m r2
p² + 82
In reference to these masses let it be observed, that
(133)
(1) M+M' =m; so that the sum of the two is equal to the
whole moving body.
(2) mr = M′r′; so that the centre of gravity of м and м'
coincides with that of m.
(3) M (x² + y²) + M′ (x²² + y²²) =
m a² b² (x² + y²)
a²x² + b² y²
m 82;
= m
so that the radius of gyration of the masses about any axis
passing through their centre of gravity and perpendicular to the
line joining them is equal to the radius vector of the central
ellipse which coincides with that line.
PRICE, VOL. IV.
Z Z
す
​354
[200.
A FREE MATERIAL SYSTEM.
In these respects therefore the body may be equivalently
replaced by a straight, inflexible, and immaterial bar, having
masses м and м' at its two ends, which are determined by equa-
tions (133): this bar will not only at its two ends, but at any
point in its length, strike an obstacle with a blow of the same
momentum as the body.
From (131) it appears, that м = m, only when x = Y 0;
that is, the centre of gravity is the only point at which the body
will strike an obstacle as if it were a mass equal to its own
mass; and in this case P = Q.
If the centre of percussion is the point of impact, p = q; but
M
m a² b2
a²x² + b² y ² + a² b²
h'
ՊՆ
h÷ h'
and from (91),
v =
Q h + h'
m h
;
(134)
so that the momentum is produced by a mass smaller than m,
moving with a greater velocity. Although therefore against a
fixed obstacle the momentum of the blow p is the same, whether
the obstacle be at the centre of gravity or at the centre of per-
cussion; yet against a particle of finite mass, say m', the effects
will be different. These we proceed to investigate; and we
shall determine both the velocity of m' after impact from the
body, as well as the velocity of the impinging point of the body
after impact on m'.
200.] Let v and v' be the velocities of the body and of m'
after collision at the point (x, y); let e = the elasticity; and
let us suppose m' to be at rest when the impact takes place;
then, from (9), Art. 215, Vol. III,
v = Q
√ =
(a²xx。+b²yy。+a²b²) (m— em') a² b² — em' (a²x² +b²y²)
(m+m') a²b²+m' (a²x²+b²y²)
m a² b²
(1 + e) a (a² x x + b²yyo + a² b²)
0
(m + m²) a²b² + m² (a²x² + b² y2)
;(135)
(136)
If we equate to zero the x- and y-differentials of v', the point
will be determined at which m' must be struck so that it may
move after collision with the greatest velocity: this process gives
XC
y
;
Хо
Yo
(137)
which shews that the point of greatest percussion is in the line
joining the centre of gravity and the centre of impulsion. If r
201.]
355
INSTANTANEOUS FORCES.
is the distance of the required point from the centre of gravity,
then, as in Article 197,
M
r =
n' +
{
h'²² + hh' (1 +
M
(138)
Thus there are two points at which a body impinging on a
particle m' will cause it to move after collision with a maximum
velocity; these points are equidistant from the spontaneous
centre which corresponds to the centre of impulsion, and that
on the positive side of the spontaneous centre lies farther from
it than the centre of gravity. These two points are the centres
of greatest percussion when moo; that is, when the mass of
m' =
the particle against which the body impinges is infinitely great,
and is thus equivalent to a fixed obstacle.
And corresponding to these distances,
#) /*]; (139)
Q (1 + e)
r =
+
2 (m + m²)
[1 ± { 1 + (1 +
ทาว h
m' h
of which values one is positive and the other is negative: the
former shews that the particle m' will move with a velocity
whose direction is the same as that of Q; the latter, which cor-
responds to the point of percussion on the side of the spontane-
ous axis away from the centre of gravity, gives a velocity of m'
in the opposite direction.
In a similar way may the point be determined, at which, if
the body impinges on m', the velocity of the point of impact
after collision will be a maximum; for if we take the x- and y-
partial differentials of (135), and equate them to zero, the points
will be determined by means of these two equations.
201.] Now at the instant when the body has impinged against
a fixed obstacle at the point (x, y), that point of the body is at
rest; yet there remains the momentum P', which is given in
(100), whose point of application is u, see Fig. 44: as u however
is a point in o's, which is the spontaneous axis relative to R
as a centre of impulsion at which P acts, P' produces no effect
on R; and thus the motion of the body at that instant is due to
the force p' only applied at u; and consequently the spontaneous
axis passes through R. u, which must now be considered as a
centre of impulsion, is, from Art. 196,
b²y (x。y —уox) — a²b² (x − x。)
a²x² + b² y² — a² xxo-b² y yo
—
a² x (y。x — x¸y) — a²b² (y —Yo)). (140)
a²x²+b²y² — a² xx。-b²y yo
Z z 2
356
[202.
A FREE MATERIAL SYSTEM.
!
The spontaneous axis corresponding to which is
—
¿ {y (x。y—Y。x) — a² (x − x。)}+n {x (y。x − x¸y) — b² (y — yo) }
+ a²x² + b² y² — a² x x¸ — b³ y yo
*
= 0.
(141)
Now the momentum of the blow of impulsion, namely r', is
given in (100); and consequently if the velocity of the
centre of gravity,
P
And if
u =
m
Q a²x² + b² y² — a² x x -b²y yo
m
a²x² + b² y² + a² b²
(142)
(143)
the angular velocity about the spontaneous axis
through R due to P', and if p = the perpendicular distance from
the centre of gravity on that line,
P'
2 =
(144)
where p
mp
a²x² + b² y² — a² x x。— b² y yo
{(x²+y²)(x¸¥—Y。x)² — 2 { a²(x − x。)+b²(y—yo}}(x¸¥—Y。x)+a¹(x−xo)²+b²(y — yo)² }
If the point (x, y) lies on the line joining the centre of gravity
and the original centre of impulsion, these expressions become
much simplified; because, in that case,
xoY-Yox = 0;
хочу
thus the equation to the spontaneous axis through R becomes
a² (x − x¸) § + b² (y—yo) n = a²x² + b² y² — a²xx¸-b²yyo; (145)
a²x² + b² y² — a² x x -b² y yo
p =
Ω
0
{aª (x − x¸)² + bª (y — yo)² } ✯
4
α
2
Q {a¹ (x − xo)² + b¹ ( y − y)² } +
m a²x² + b² y² + a² b²
(146)
a particular case of this last simplification is that in which the
obstacle is placed at a centre of greatest percussion; see Art. 197.
Questions exactly analogous to those which I have alluded to
in Art. 197 arise out of the preceding values of u and n', and
give points which may be called points of greatest reflexion and
of greatest conversion.
202.] Thus as to u; the problem may be to determine the
place of a fixed obstacle, or of a particle of given mass, so that
it may be a maximum; or the place of a fixed obstacle, so that
it
may
have a given value; say, be equal to the original value
of the velocity of the centre of gravity but in an opposite direc-
202.]
357
INSTANTANEOUS FORCES.
tion; or to find the place of the obstacle, so that u may be equal
to 0.
As to critical values of u I would observe, that
P
m
Q
M
- P
by reason of (92) ;
so that whatever values of x and y give critical values for P, also
give critical values for P' and for u: these values have been
already investigated in Art. 197, and give what are therein
called centres of greatest percussion; these centres then are
also centres of greatest reflexion. Also, since there are two
such centres, we have also two critical values of u',
2
Q h' — (h²² + hh') }
m
2h
Q h' + (h²² + hh');
ՊԱՏ
m
2h'
(147)
The latter of these is positive, and is evidently greater than M
which is the original value of the velocity of the centre of
gravity: this is paradoxical: it seems contrary to the first prin-
ciples of mechanics that a body should strike against a fixed
obstacle, and after impact rebound with the velocity of the centre
of gravity greater than that velocity before impact. Now con-
sider this in reference to fig. 45; v and v' in it are the centres
of greatest percussion, and consequently of greatest reflexion;
and u½ corresponds to the point v', so that when the obstacle is
placed at that point, and the body impinges against it, the
velocity of G after the impact is greater than before. The body
moves by the blow q, which is given at c, from below to above
the paper; and rotates about the axis os; if however it impinges
against the obstacle at v', that angular velocity becomes modified,
and os, which was at rest before the impact, moves in the direc-
tion co, and the velocity of G is increased. We must not however
hence infer that the momentum of the body is increased; for that
would be contrary to the principles of mechanics; but some
of the momentum, which is due to the angular velocity, by
means of the obstacle becomes momentum of translation; and
hence it is that the velocity of the centre of gravity is after
impact greater than it was before. Thus, a ball from a rifled
358
[202.
A FREE MATERIAL SYSTEM.
gun, having velocity both of translation of its centre of gravity
and of rotation about an axis through that centre, may have its
velocity of translation increased by meeting with an obstacle,
and thus may be carried farther than if it never met with such
an obstacle. This is one of the peculiar and surprising facts of
ricochet practice.
If u
Q
IN
the velocity of the centre of gravity will be after
2 a²x²+2b² y²-a² xxo-b² y yo + a² b² = 0;
impact the same as it was before but in an opposite direction;
then
(148)
which is the equation to an ellipse similar to the central ellipse,
and similarly situated; the points which give this value of u
are called points of perfect reflexion; and the ellipse (148) is
called the ellipse of perfect reflexion.
Again, as to '; it is a function of x and y, and the values of
those quantities may be found which will give to a critical
value. Also, those may be found which will assign a point on
which, when the body impinges, the angular velocity after im-
pact will have a given value; say, be equal to that before impact,
and in an opposite direction. Points which give these values to
n' are called respectively, points of maximum conversion, points
of given conversion, and points of perfect conversion.
P'=0;
Thus, if '0, P'= 0; and
a²x² + b² y² — a² x x¸-b² y yo = 0;
Yo
= 0,
so that for all points on the ellipse given by (98), p = q, r':
uo, n'=0; that is, if the obstacle is on that ellipse, the body
impinges on it with a momentum equal to that of original im-
pulsion; the centre of gravity of the body is brought to rest,
and there is also no angular velocity; in fact the body is brought
Q (a* x² + b¹y²)
a2b2
to rest.
If a'
177,
see (89); in this case the an-
gular velocity after the impact is equal to, but of contrary direc-
tion to, that before impact; then, if we take the particular case
given in (146), we have
{a¹ (x − x¸)²+b¹(y —Yo)² } š
a²x² + b² y² + a² b²
(aª x² + b¹y ²) ½
a² b²
(149)
which gives an equation of the fourth degree in terms of ≈ and
y; all points on the curve expressed by which are points of
perfect conversion.
203.]
359
INSTANTANEOUS FORCES.
*
To determine the points for which has a critical value, the
x- and y-partial differentials of (144) or of (146) must be equated
to zero in the general case however they lead to results so much
complicated that it is useless to insert them.
203.] In the preceding Articles, see Art. 191, (50) and (51),
and Art. 199, (133), it has been shewn, that a body may be
equivalently replaced by two particles of definite and determinate
masses at the ends of an immaterial rigid and straight bar, so
far as the effects of momentum communicated to the body by a
blow, and the effects of impact of the body on a fixed obstacle
are concerned. This property is of considerable use in the solu-
tion of another problem: A body of given mass moving with a
given velocity impinges on a given body at a given point, it is
required to determine the motion of the bodies at the instant
after impact.
I will assume the line of motion of the moving mass to be
in a central principal plane and to be parallel to a principal
axis of the body on which it impinges. Let m be the mass of
the latter body, m' the impinging mass, of which let the velo-
city at the point of impact be v'; let the line of motion of m' be
in the central principal plane of (x, y), and be parallel to the
axis of y; let c, see Fig. 46, the point of impact, be in the axis
of x at a distance = a from the centre of gravity &; and let k
be the radius of gyration of the body about the axis G, which
is perpendicular to the line of action of the blow. Let o be the
spontaneous centre reciprocal to c; so that oG =
k2
X
Now in
Art. 191 we have shewn, that so far as concerns blows given by
it, the body m may be replaced equivalently by two masses
M and M', which are therein assigned, of which the former is
placed at c, and the latter at o; and that as o is a centre re-
ciprocal to c, the mass м' placed at o neither affects nor is
affected by the blow given at c; so that as far as the momentum
of a blow at c is concerned, the effect of the body will be the same
as that of the mass м placed there: all this is explained in
Art. 191. The problem then which was proposed for solution is
this; m' moving with a velocity v impinges on м at rest: it is
required to determine the motion of м and m' after collision.
The principles of Art. 215, Vol. III, are sufficient for the
purpose.
360
[203.
A FREE MATERIAL SYSTEM.
Let v = the velocity of м after impact; v' the velocity of
m' after impact; and let e the elasticity. Then, since
=
M =
m k²
k² + x²'
(150)
(1 + e) m' (k² + x²) v′
V
v =
(m + m') k² + m²x²
(151)
m' (k² + x²) em k²
V
;
(m+m') k² + m' x²
(152)
thus the momentum of м after impact
= MV
m m' (1 + e) k³ v
(m + m'´) k² + m²x²
(153)
and this is the momentum of the body at the point c, and is that
which has hitherto been symbolised by Q.
The momentum imparted to the body decreases as x in-
creases; and vanishes when ; and the greatest value is
that which corresponds to a = 0.
x ∞;
If the bodies are perfectly inelastic, e = 0; in which case the
momentum imparted to the body
m m' k² v
(m + m² ) k² + m²x²
(154)
Suppose now that m' and v' are variables, with the condition
of their product being constant; that is, the momentum of the
impinging ball is constant, although its mass and velocity vary ;
say,
m'v'
Movo.
(155)
And suppose moreover that, whatever is the distance from G at
which m' impinges, the momentum imparted to the body is
the same; say, = m
m
Vo; then,
k²
2 мого
Vo
m k² + m² (k² + x²)
and consequently m' (k²+x²) is constant, mob², say;
то
mob2
m' =
k² + x²
v
Mo Vo (k² + x²);
mob2
=
(156)
(157)
in which equations m' and v' are expressed as functions of x, and
are thus determinable for any distance from the centre of
gravity of the point of impact.
Thus, if a hammer is to be constructed and used, so that the
same quantity of momentum is to be imparted to a body whose
204.]
361
INSTANTANEOUS FORCES.
mass is m, whatever is the distance of the point of impact from
the centre of gravity, the momentum of the blow of the hammer
being always the same; then the mass of the hammer and the
velocity of the blow are given by (156) and (157).
204.] One other problem of a practical kind deserves insertion
in this branch of the subject which treats of spontaneous axes
and their properties.
A body m rests on a prop and is struck by a blow whose
momentum is q, the line of motion of the blow being in a
central principal plane, and parallel to one central principal axis ;
and the line of reaction of the prop being in the same principal
plane, and parallel to the line of the blow.
Let OGR be a central principal axis of the body whose mass
is m, and centre of gravity is G. Let c be the prop and R the
point of application of the blow whose momentum is q, and of
which QR is the line of action; this line of motion being in the
central principal plane Gz: see Fig. 47. It is evident that if
R coincides with c, that is, if the blow is given at the prop, the
momentum of the pressure borne by the prop = Q; suppose
however that the point of impact of the blow is R, where GR= X;
let the momentum borne by c, and let p' be that applied
at o, which is the spontaneous centre reciprocal to c, both these
being due to Q; then the pressure p' does not affect the pressure
at c, which is a point reciprocal to o, so that P is the whole
pressure on the prop.
Let Gch; and consequently Go =
are the components of Q, Q = P — P′ ;
k2
then as P and p'
h
and
P
(h+
k2
h
k2
= Q
h
+ x);
k² + hx
..
P = Q
;
k² + h²
(158)
= P
which assigns the pressure borne by the prop. If a h, p =Q;
that is, the blow is applied at the prop, and the pressure
borne by the prop is equal to the momentum of the blow. If
X
k2
h'
that is, if the blow is struck at o, the spontaneous
centre relative to c, P =
P increases as x increases, and P is greater than a when a is
greater than h: it follows therefore that by means of an inter-
PRICE, VOL. IV.
3 A
362
[205.
A FREE MATERIAL SYSTEM.
vening body m, a blow of given momentum can produce a pres-
sure of any intensity on a given prop. If x :∞0, P = ∞.
ر
If however the blow is caused by a hammer of mass m', and
impinging with a velocity v' on a point whose distance from the
centre of gravity is a; then, from (154),
and therefore P
m m' k² v'
(m + m') k² + m²x²
m m' k² v' (k² + h x)
(k² + h²) { (m+m') k² + m′ x² } '
(159)
If in this expression m' and ' have, for a distance a, the values
found for them in (156) and (157), then the momentum of the
blow of the hammer is always the same, and the momentum
borne by the prop is given by (158), and may consequently be
of any magnitude whatever.
P is a maximum in (159) when
x =
where GO =
h' =
Thus if m = m';
k2
h
m+m'
2
h' ± h'² +
IN
k2
)*,
(160)
; which gives two points equidistant from o.
and h =h'-
K': k,
x = (−1 + 3º)k.
Similarly may the points of impact be determined, so that the
momentum of the pressure borne by the obstacle may be of a
given value.
I cannot conclude this subject, in which I have borrowed
largely from the Memoirs of Poinsot, contained in Vols. II and
IV of the second series of Liouville's Journal, without alluding
to a remark which he makes of the process by which the cir-
cumstances of motion of a rigid system having a fixed axis or a
fixed point may be deduced from those of a similar free system.
He considers a fixed point to be a particle of a certain definite
mass, introduces this mass and its incidents into all the equa-
tions of motion, and in the final results makes this mass infinite;
and this particle of infinite mass he considers to be a fixed
point; on which, of course, as to translation, a finite force has
no effect; but for an axis passing through that point the
moment of inertia of the body is finite, and consequently the
impressed couples will produce their own rotatory effects.
205.] Thus far, the bodies on which the impulsive forces have
acted have been assumed to be entirely free from all constraint;
and we have considered only those whose circumstances under
205.]
363
INITIAL MOTION OF A BILLIARD BALL.
the action of the given forces satisfy the condition necessary for
the existence of a spontaneous axis. But the principles from
which we started were general, and are applicable to all kinds of
bodies and instantaneous forces. The general cases will be most
conveniently discussed as they arise in the course of particular
problems in the following sections; but it is also expedient to
exhibit the form which the general equations take when they
are thus applied to such problems. For this purpose I will
consider the effects of a blow on an ordinary billiard ball.
A heavy spherical billiard ball on a rough horizontal table is
struck by a cue at a given point with a blow of given intensity
in a given direction; it is required to determine the resulting
motion of the ball.
Let a = the radius, м = the mass of the ball; Q = the mo-
mentum of the blow; a = the angle at which its line of action
is inclined to the plane of the table.
Let the horizontal plane which passes through the centre of
the ball and is parallel to that of the table be the plane of (x, y) ;
and let the line in it parallel to the vertical plane which con-
tains the line of q be the axis of a; let h the horizontal dis-
tance from the centre of the ball to the vertical plane which
contains the line of blow; and let / be the perpendicular distance
on the line of blow from the point where h meets the vertical
plane containing that line.
Let r cos ẞ and F sin ẞ be the components parallel to the
axes of x and y respectively of the friction against the table
which is brought into action by the blow; let 1, 2, 3 be the
resulting angular velocities about axes through the centre of the
ball which are parallel to the coordinate axes; and let uo, vo be
the resulting expressed velocities parallel to the coordinate axes.
Then the equations of translation parallel to the axes of x and y
MU₂ = Q cosa - Fcosß, }
are
M VO
Fsinß;
(161)
and if A is the moment of inertia of the ball about an axis
through its centre,
A
Qhsina -aFsin ß,
1
A 22
Qk+arcos ß,
Qh cosa.
A 23
If R = the pressure of the ball on the table,
(162)
R = Q sin a + Mg;
(163)
3 A 2
364
[206.
A FREE MATERIAL SYSTEM.
but since the line of action of R passes through the centre of
the ball, it produces no effect on 21, 22, or 3.
Thus the axial components of the velocity of the point of
contact of the ball with the plane are u− α £2, vo + α £; con-
sequently, if
s² = (u — α î₂)² + (1% + a₁)³,
(164)
and as ẞ is the angle at which the initial path of the point of
contact is inclined to the axis of x,
cos B
U o — a Qz
а
sin ß
1
vo + a Q
Ꮽ
(165)
and since the friction acts as a retarding force along the line of
motion of the point of contact, its line of action is thus deter-
mined; and the friction is known in terms of the pressure R, so
that the four unknown quantities up, vo, 1, n are involved in
four independent equations, and may be determined without
difficulty; and thus the initial motion of the ball will be de-
termined. Applications of these results will be made hereafter;
see Art. 222.
SECTION 2.-Motion of a free invariable system under the action
of finite accelerating forces.
206.] We now come to the most general case of absolute
motion of a body, or of any system or systems of material parti-
cles under the action of finite forces. Many processes have
been devised for the purpose, and several of them are especially
adapted to particular classes of problems. All however are
founded upon the principle of D'Alembert; and their equations
of motion are derived from, or are identical with, those six
equations in which we have expressed that theorem. I propose
to apply these to the solution of problems of motion in prefer-
ence to other and derived processes; because we shall hereby
maintain an uniformity of process and of principle, and because
the circumstances of the problems will be resolved into their
most simple elements. We shall indeed take the forms which
these equations admit of, in virtue of the theorems proved in
Section 2 of Chap. III; we shall consider the motion of transla-
tion of the centre of gravity, by assuming all the forces to act
on a particle, whose mass is equal to the whole mass of the
moving system, placed therein; and in our inquiry into the
rotation of the system, we shall assume the centre of gravity to
be a fixed point, and the body or material system to rotate
206.]
365
FINITE FORCES.
about an axis passing through that point. Thus the motion of
the system in the first place depends on the two following
groups of equations :
Σ.Μ.Υ
:.m (2
Σ.Μ Χ
m(x
dax) = 0,
m(x
d² y
= 0,
(166)
dt2
d2 z
Σ.Μ Ζ
= 0
0 ;
dt2
Σ.Μ
x.my
d²y
Ꮓ
dt2
Σ.Μ
{
d² z
X
Ꮓ
0,
(167)
dt2
dt2
Σ.Μ
{
d2
d²
Y
dt2
dt2
}}
d2z
(x-7) - 2(x) = 0,
dt2
ZIY
d²x) — x (
-
2.mx(x-1)-y(x-2)=0.
If M is the mass of the whole moving system, (x, y, z) is the
place of the centre of gravity at the time t, and if (x', y', z) is
the place of m relatively to a system of coordinate axes originat-
ing at the centre of gravity, and parallel to the original system of
axes; then, by the theorems of Section 2, Chap. III, these take
the forms
d2x
M
= Σ.mx,
dt2
d³y
M ΜΙ
= Σ.MY,
dt2
d2z
M
Σ.mz;
dt2
day
Σ.Μ
Y
dt2
dt2
Σ.Μ
{=
d2x
ས
X
dt2
-)
— x' (:
(2
d2~
Σ.Μ
xx
Y
dt2
d²x'
dt2
2.my (2) - (x) = 0,
(8)=0,
dt2
d³y_) — y' (
-y' (x - 1 x ) }}
(168)
(169)
= 0.
By reason of the former of these last two groups, the motion of
translation of the body is reduced to that of a single material
particle whose mass is M; and to this motion all that has been
said in Vol. III is applicable. The second group reduces the
motion of rotation to that of a body rotating about an axis
passing through a fixed point of it; and consequently to this
motion all that has been said in the preceding Chapter is ap-
366
[206.
A FREE MATERIAL SYSTEM.
plicable. The problem therefore requires two processes in com-
bination, each of which has been separately discussed; and
little else remains than to illustrate the combination by means
of particular examples. Indeed I have already anticipated the
process in the investigation of the phænomena of terrestrial pre-
cession and nutation in the preceding Chapter; because we
have assumed the centre of the earth to be fixed, whereas it
has a motion of translation in space.
In investigating the motion of rotation of the body about the
point which is assumed to be fixed, we may use the simplifica-
tions and substitutions of the last and preceding Chapters. Thus,
if wx, wy, w are the angular velocities at the time t about any
three coordinate axes originating at the fixed point, (169)
become (48), (49), and (50), of Art. 79; which however it is
unnecessary to repeat in this place as we shall employ simplified
forms of them. We shall investigate the angular velocities of
the body at the time t relatively to the three principal axes of
the body; and the equations for determining these are
A
B
dwi
dt
+ (C — B) W2 W3 = L,
+ (A — C) W3 W1 =∙M,
d wz
dt
C
dwz
dt
+ (B − A) W1 W2 = N
(170)
N;
because hereby (theoretically at least) the angular velocity of
the body, and the position of the instantaneous rotation-axis
relatively to the principal axes, may be determined at the time
t; and thence we may determine, as in the preceding Chapter,
the motion of the body in reference to fixed axes, by means of
the three connecting angles 0, 0, Y.
And if the position of the rotation-axis which passes through
the centre of gravity of the body is invariable relatively to the
body, then the rotation is determined by the simple equation,
dw
dt
moment of impressed forces
moment of inertia
(171)
In the solution of mechanical problems, the theorems of vis
viva, and of conservation of areas, which have been derived in
Chapter III from the general equations of motion, may fre-
quently be applied, to the saving of considerable trouble; not
indeed because they contain any truth besides those involved
207.]
367
MOTION OF RIGID BODIES.
explicitly or implicitly in the equations of motion, but because
they are first integrals of these equations. In a didactic treatise,
like the present, however, I consider clearness of conception
and accuracy of expression to be of paramount importance; and
I am convinced that it is most likely that these will be obtained
when the circumstances of a problem are resolved into their
simplest elements. In all the following problems, the equations
of motion are given in their original forms, and for the complete
solution of a problem they require two successive integrations.
In many cases the intelligent student will readily recognise the
equations of areas and the equation of vis viva; and the latter
will frequently present itself to him in the derived form which
has been proved in Art. 66; viz., the vis viva of the system is
equal to the sum of the vis viva of the whole system condensed
into its centre of gravity, and of the vis viva of the several par-
ticles relative to the centre of gravity.
207.] The following are mechanical problems, on the motion
of rigid bodies, in which the rotation-axis moves parallel to
itself.
Ex. 1. A heavy homogeneous sphere rolls down a rough in-
clined plane; it is required to determine the motion.
We suppose the sphere to be placed at rest on the plane, and
to roll down it so that the point of contact describes a straight
line perpendicular to the line of intersection of the inclined and
horizontal planes. Let fig. 48 represent a section of the sphere
and plane at the time t, made by a vertical plane passing
through c the centre of the sphere. Let A be the point of the
sphere which was originally in contact with the plane at the
point o; let a be the radius of the sphere; OP = S, ACP = 0,
M = the mass of the sphere, F = the friction of rolling, R
the pressure of the sphere on the plane, a the angle of
elevation of the plane.
Now c evidently moves along a straight line parallel to the
plane; so that for its motion of translation we have
M
d2s
dt2
Mg sin a
F ;
and if c is considered fixed, the sphere evidently rotates about a
horizontal axis parallel to the plane; and if k is the radius of
gyration of the sphere relative to this axis,
Mk2
d20
dt2
= Fα;
368
[207.
MOTION OF RIGID BODIES.
and since the plane is perfectly rough, so that the sphere does
not slide, ds = ade; also k²
d² s
dt2
2 a²
;
5
d20
α
dt2
79 sin a;
10
5
which assigns the motion; also R = mgcosa. If the plane were
perfectly smooth, the impressed velocity-increment along the
plane would be g sin a; so that the roughness of the plane
which causes the rolling diminishes the action of gravity along
the plane by two-sevenths of its full value.
If the rolling body were a circular cylinder with its axis hori-
zontal, then k² =
a²
and
2 ;
d2 s
2
dt 2 ğ y sin a;
so that the roughness of the plane would diminish the action of
gravity along the plane by one-third of its full value.
Ex. 2. A hollow spherical shell is filled with fluid and rolls
down a rough inclined plane, determine its motion.
Let м and м' be the masses of the shell and fluid respectively;
and let k and k' be the radii of gyration of them respectively
about a diameter; let a and a' be the radii of the exterior and
interior surfaces of the shell; then, employing the same nota-
tion as in Ex. 1, we have
(M + M′)
d² s
dt2
(M + M') g sin a F.
As the spherical shell rotates in its descent down the plane, the
fluid has only motion of translation; so that the equation of
rotation is
d20
M k²
= Fa;
dt2
d2s
{ (M + M′) a² + M k²}
d t2
= (M+M') a²g sin a.
If the interior were solid, and rigidly joined to the shell, the
equation of motion would be
d² s
{(M + M′) a² + M k² + M′ k′²}
= (M+M') a2 g sin a.
dt2
Thus if s and s' are the spaces through which the centre moves
during the time t in these two cases respectively, then
Ꮽ
(M + M') a²+M k² + M'k'²
;
(M + M′) a² + M k²
207.]
369
MOTION OF RIGID BODIES.
so that a greater space is described by the sphere which has the
fluid than by that which has the solid in its interior.
If the densities of the solid and the fluid are the same, replac-
ing k and k' by their values,
S
7 a5
7a5 — 2 a'5'
7 = —
Ꮽ
Ex. 3. A heavy solid wheel, in the form of a right circular
cylinder, is composed of two substances, whose volumes are
equal, and whose densities are p and p'; these substances are
arranged in two different forms; in one case, that whose density
is p occupies the central part of the wheel, and the other is
placed as a ring around it; in the second case, the places of
the substances are interchanged; t and ťare the times in which
the wheels roll down a given rough inclined plane from rest;
shew that
t² : t'² : : 5p+7 p' : 5p+7p.
Ex. 4. A homogeneous heavy sphere rolls down within a rough
spherical bowl; it is required to determine the motion.
Let the circumstances be represented in Fig. 49, where a is
the radius of the rolling sphere, and b is the radius of the
spherical bowl; let us suppose the sphere to be placed in the
bowl at rest. Let ocq=&, QPA = , вCO = a; OM=X, MP=
m the mass of the ball.
Then
y;
d² x
R sin F cos 0,
dt2
d2y
m
R COS + F sin & — mg ;
dt2
and if the angular velocity of the ball about an axis through
its centre P, and k is the corresponding radius of gyration,
m k²
dw
dt
= α F;
a
where k²
2 a2
5
Now x =
(b-a)sino, y = b − (b − a) cos & ;
d2x
d² o
(b− a) cosp
dt2
dt2
− (b − a) sin 4 (·
аф
p
dt
(do)³,
2
d2y
d²
(b-a) sin o
dt2
dt2
+ (b − a) cos $ (
d &
2
;
dt
dt2
d2 x
cos +
d2y
d2
sin 10 = (b− a)
;
dt2
dt2
..
m (b− a)
d²¢
dt2
= F- mg sin p.
PRICE, VOL. IV.
3 B
A
370
[207.
MOTION OF RIGID BODIES.
Now to determine the angular velocity of the ball, we must
estimate the angle described by a fixed line in it from a line
fixed in direction, and the ratio of the infinitesimal increase of
this angle to that of the time will be the angular velocity of the
ball. Let us take MPA to be the angle whose increment we
will consider,
d. MPA
d (+0)
dt
dt
аф
d Ꮎ
+
dt dt
Since however the sphere rolls, and does not slide, a0b (a− p);
α
b do
W
;
α dt
dw
a − b d²p
;
dt
a
dt2
whence, eliminating F, and reducing, we have finally,
(b− a)
d² o
dt2
5
79 sin &;
2
10g
(cos-cosa).
7
аф
(b − a) (d)²
dt
Substituting in the preceding equations,
m g
R =
{17 cos -10 cos a};
7
therefore the pressure at the lowest point = mg {17—10 cosa}.
And the pressure of the ball on the bowl vanishes when
cos
10
Cos a.
17
If the ball rolls over a small arc at the lowest part of the bowl,
are always small, then replacing cos and cos a
so that a and
respectively, we have
by 1
2 and
a²
1-
2
2
-αφ
(a² — (2) ½
2
5 g
7 (b − a)
dt;
{
5 g
14
t:
p = a cos
7 (b − a)
thus the ball comes to rest at points whose angular distance is
a on both sides of o, the lowest point of the bowl; and the
§
7 (b − a)
periodic time = π
; consequently the oscillations are
5g
207.]
371
MOTION OF RIGID BODIES.
performed isochronously with those of a simple pendulum whose
7
5
length is (b-a).
Ex. 5. A heavy homogeneous sphere rolls down a rough in-
clined plane; the inclined plane rests on a smooth horizontal
plane, along which it slides by reason of the pressure of the
sphere; it is required to determine the motions of the sphere
and of the inclined plane.
Fig. 50. m
The circumstances of motion at the time t are delineated in
the mass of the ball, м = the mass of the plane
or wedge; a the radius of the ball, a the angle of the in-
clined plane; o the apex of the wedge, o the place of a when
t = 0; o' the point on the plane which was in contact with the
point a of the sphere, when t = 0; at which time let us suppose
all to be at rest; ACP = 0, the angle through which the sphere
has revolved in the time t.
Let o be the origin, and let the horizontal and vertical lines
through it be the axes of x and y ; oq = x'; and let (x, y) and (h,k)
be the places of the centre of gravity of the sphere at the times
t=t, and t = 0 respectively. Then from the geometry, we have
x = h + x'
ao cos a,
Y
k
a o sin a ;
d2 x
d²x'
d20
a cos a
dt2
dt 2
dt 2
d2y
d20
a sin a
dt2
dt2
The equations of motion of the sphere are
m
d2x
dt2
=F COS a
R sin α,
d2y
m
dt2
= F sin a + R cos a — mg,
2 a² d20
m
= α F ;
F;
5 dt2
and the equation of motion of the plane is
d2x
M
F cos a + R sin a ;
dt2
from which we obtain
d20
α
dt2
5 g sin a
{7-5 (cos a)2} m + 7 M
3 B 2
372
[207.
MOTION OF RIGID BODIES.
and
..
а в
5 sin a
g t2
;
૭
{7 −5 (cos a)2} m + 7 m
2
m cos a
x²
a0,
m + M
m
5 sin a cos a
g te
m+M {7−5 (cos a)²} m +7 M
2
m
d² x
dt2
+ M
from which values x and y may be determined in terms of t.
Also
d²x'
0;
dt2
m (x − h) + мx' = 0;
and consequently,
= 0;
(m + м) (x − h) sin a — M (y — k) cos a =
which shews that the path described by the centre of the sphere
is a straight line.
Ex. 6. A heavy beam op, see Fig. 51, turns about a hinge at
o, and its end P rests on a smooth inclined plane or wedge,
which slides along a smooth horizontal plane which passes
through o: it is required to determine the motion of the beam
and of the inclined plane.
Let m and м be the masses of the beam and plane respectively;
2a = the length of the beam, a the angle of inclination of
the plane to the horizon; POQ = 0, oq = x. Then, from the
geometry, we have,
x sin a = 2 a sin (a—0).
For the motion of the beam we have
M
3
4 a2 d20
dt2
mga cos 0+2aRcos (a-0);
and for the motion of the plane
d² x
M
= R sin a;
dt2
whence eliminating R, we have
4a²m d20
3 dt2
cos (a-0) d²x
magcose +2 a M
sin a
dt2
But from the preceding geometrical condition we have
da
sin a
dt
ᏧᎾ
2 a cos (a — 0)
;
dt
4a2m d20 do
d Ꮎ
dx d²x
mag cos ›
M
;
3 dt2 dt
dt
dt dt²
207.]
373
MOTION OF RIGID BODIES.
and if 0, is the value of 0, when the system is at rest,
4a² m
3 dt
(20)
= 2mag (sin 0, — sin ◊) M
dt
(da)²
which is indeed the equation of vis viva. And, substituting for
dx
we have
dt'
2 a
{
m
+ M
cos (a―0)
sin a
2) ² } ( 10 )²
de
do 2
= mg (sin0 — sin ◊).
dt
3
This equation determines the angular velocity of the beam, and
consequently the velocity of the plane; but as it does not admit
of further integration, we cannot find 0 or x in terms of t.
Ex. 7. A heavy body whose bounding surface is a circular
cylinder, but whose centre of gravity is not in the axis of the
surface, rolls on a rough horizontal plane: it is required to
determine its motion.
M =
Let м the mass of the body; and let Fig. 52 represent
the circumstances at the time t; in which G is the centre of
gravity, c is the point of intersection of the axis of the cylinder
by a vertical plane passing through the centre of gravity G; oæ
is the horizontal plane; o, the origin, is the point where the
sphere touches the plane when it is in equilibrium, a being the
corresponding point of the sphere. Let CA a, CGC, ACP 0,
so that op = a; and let k be the radius of gyration of the
body relative to its rotation-axis through G; let & be (x, y);
then from the geometry we have,
F
OP=
x = a0 — c sin 0,
y = -a-ccos 0.
Now if r is the friction of rolling, and R is the pressure on the
plane, the equations of motion are
M
d² x
dt2
F;
d2y
M
= RM9;
dt2
d20
Mk2
dt2
= F(a-c cos 0) — R c sin 0.
Whence we have
d20
2
dt2
{ k² + a² + c² — 2 ac cos 0} + ac sine (de)
=
=-cg sin 0;
dt
0
multiplying through by 2de, and assuming
= a, we have
do
0, when
dt
374
[207.
MOTION OF RIGID BODIES.
2
{k² + a³ + c²−2 accos0} (de)² = 2cg (cose-cosa); (172)
dt
which equation gives the angular velocity of the body about a
horizontal axis through its centre of gravity. From this may be
dy dx
dt
found
and which will give the linear velocity of the
dt'
centre of gravity. As these equations in their general forms do
not admit of further integration, e, x, and y cannot be found
in terms of t.
If however the angle through which the body rotates is
always small, so that a and @ are always so small that all powers
of them above the second may be neglected, and that the second
powers may be neglected when they are added to finite quanti-
ties; then we may replace (172) by its approximate equation,
2
{k² + (a−c)²} (dº)² = cg (a² — 0²) ;
... Ө 0: a cos
cg
{ −
k² + (a − c)²
—
-31
t;
(173)
so that the body oscillates or rocks through an angle 2 a; and
the time of an oscillation
π
I k² + (a−c)² ) +
cg
-10
(174)
This result applies to a problem which is physically of consider-
able importance. In making observations with the pendulum,
the mode of suspension which is found most convenient for the
determination of the distance between the centres of suspension
and oscillation is that of knife edges, resting on horizontal
plates of agate or of some other hard material. Although the
knife edges are made of steel, and brought to as fine an edge as
possible, yet they are not mathematical straight lines, but ap-
proximate to cylinders, which we may, without sensible error,
suppose to be circular, and of a very small radius; in which case
the pendulum is suspended by a horizontal cylinder which rests
and rolls on two parallel horizonal bars which are perpendicular
to the axis, and of which a diagram is given in Fig. 53. Here
the centre of gravity of the rolling body is below the horizontal
plane, so that c is greater than a; then, if T is the time of an
oscillation,
k² + ( c − a) ² ) ³½
T=T
{
cg
207.]
375
MOTION OF RIGID BODIES.
but if the pendulum is suspended by an exact edge, the time of
oscillation
k² + c² ) 3
cg
S
thus the effect of the want of accuracy in the edge diminishes
the time of vibration in the ratio of
{ k² + (c − a)² } } to
{k² + c²} *.
Ex. 8. Let us suppose the rocking body to be homogeneous,
and to be bounded by a cylindrical surface whose section per-
pendicular to the generating lines is semicircular, as in Fig. 54;
then, if CA = A, C G = c =
4 a
a²
;
k² =
(4)
3 п
2
п
time of oscillation
π
9π- 16
8g
a
}
2
and the
If the rocking body is a homogeneous hemisphere, then the time
of an oscillation
π
(
26 a
15 g
Ex. 9. A heavy homogeneous beam is suspended by two
vertical strings of equal length, so that the position of the beam
is horizontal; the beam is slightly twisted through a small
angle about a vertical axis passing through its centre of gravity:
it is required to determine the motion.
We shall suppose the length of the strings, in reference to
that of the beam and the angle through which the beam is
turned, to be so great that the vertical displacement of the
beam may be neglected.
M
Let the mass of the beam; 2a = the length of the
beam; the length of each of the suspending strings; 0 =
1 =
the angle between the line of the beam at the time t, and its line
in the position of rest; so that each end of the beam is dis-
placed through a distance a0.
=
The tension of each string when the beam is in equilibrium
M g
2
as the displacement of the beam is small, I shall assume
the tension to be unchanged in the displaced state; so that the
horizontal component of this force which acts at the end of the
beam, in a line perpendicular to it,
Mg a o
2 T
;
and if k is the
radius of gyration of the beam relative to an axis passing
376
[207.
MOTION OF RIGID BODIES.
through its middle point and perpendicular to it, the equation
of motion is
d20
a20
M k²
Mg
dt2
a²
but k² =
so that
3
d20
390.
Ꮎ
dt2
Let a = the angle through which the beam is turned; then,
by integration, we have
2
g
(10)² = 3 4 (a² — 0²);
dt
whence it is manifest that the beam oscillates; and that the
time of an oscillation
い​立
​=ㄠ ​3g
which is independent of the length of the beam, and depends
only on the length of the string by which it is suspended.
Ex. 10. A fine string is coiled round a heavy cylindrical
wheel; one end of the string is fixed, and the wheel descends,
unwinding the string it is required to determine the motion
of the wheel.
Let M =
the mass
string at the time t;
of the wheel; T = the tension of the
a the radius; OP = x, see Fig. 55;
=
=
0 = the angle through which the wheel has revolved from its
position of rest; k = the radius of gyration of the wheel. Then
the equations of motion are
M
d2 x
dt2
d20
= MG - T;
M k2
= a T ;
dt2
ⱭT;
also dx
= a do
a de; so that
d20
a²
(a² + k²)
dt2
=ag; and since k²
we have
2
d² 0
2
a
dt2
29
3
d2x
dt2
༧:༡
;
.'.
3
X = X
xo+
9 12
;
3
so that the space described in a given time is two-thirds of that
which would be described by the wheel falling freely.
Ex. 11. To determine the motion of a system of pulleys and
weights, each of which hangs by a separate string, as in Fig. 56.
The system consists of a fixed pulley whose centre is c, and
of a series of pulleys whose centres at the time t are C1, C2, C3…….;
we will assume all these pulleys to be equal, a to be the radius,
k to be the radius of gyration, and m to be the mass of each;
let м be the mass of the weight which acts round the fixed
207.]
377
MOTION OF RIGID BODIES.
pulley, and м' the mass of that which is attached to the last
moving pulley. Let T be the tension of the string by which м
acts on the fixed pulley, and T1, T2, T3, be the tensions of the
strings which severally pass from the fixed pulley to the first
moving pulley, from the first moving pulley to the second
moving pulley, and so on; and let T₂, T₂, T', ... be the tensions
of the strings which are fastened severally to A1, A2, Ag....; let x
be the distance of м, and let x1, X2, X3, be the distances of
C1, C2, C3, from the horizontal line a, A2,... at the time t; let
do, do, do,... be the angles through which the fixed and the
several moving pulleys respectively rotate in the time dt; and,
to fix our thoughts, let us suppose м' to descend, so that x1, X2,
X37 increase as t increases; and let us suppose 0, 01, 02, ... to
Then we have the following series of
increase as t increases.
equations of motion;
M
m
(9-
(g-
m (g-
(g-
m (9-
(m + M′) ( g −
m k2
d² x
dt2
d² x 1
dt²
d² x₂
dt2
d t2
-)
d2 xn-1
dt2
d² x n
d t2
d20
= T,
= T1 + T₁' — T₂,
)
— T₂ + T2' — T3,
(175)
)
= T»_1+T,1T,
= T +Tn;
T1-T,
T1 — T1,
a
dt2
m k2
d20,
Ꮎ
α
dt2
m k²
d202
=
a
dt2
T′2-T2,
(176)
m k² d²0n-1
α
dt2
m k2
d20n
a
dt2
T'′n — T»•
T₂-1,
We have also the two following series of conditions; (177), be-
cause the pulleys are rough and the cords roll round them and
do not slip; and (178), by reason of the geometrical relations of
the system. (179) follow from (178);
PRICE, VOL. IV.
3 c
378
[207.
MOTION OF RIGID BODIES.
dx
- 2 dx1,
dx =—a do,
dx1
a d01,
dx1
2 dx2,
dx2
ado2, (177)
dx2
2 dx3,
(178)
dxn
a don;
dxn-1 =
2 dxn;
dx
- Qn dxn
dx₁
2n-1 dxn
(179)
dx2
2n-2 dxn
dxn-1
2 dxni
a²
Now k² =
2
and consequently, taking the horizontal pairs of
(175) and (176), and introducing the conditions (177), we have
d2x m d2x
M
(g.
T1,
dt2
2 dt²
3
m (9
d2x₁
1
2
dt2
) = 2T₁ — T₂
m (s
2
X
3 d² x 2 ) = 212-13,
dt²
(180)
m(.
و)
3 d²xn-1
=2Tn-1-T
2
dt2
d2
dt2
x' (g- d²an) + m (g
+m(g
3
d²xn
= 2 Tn ;
2
dt2
whence, eliminating T₁, T2, T₂, and replacing x1, x2, ….. in terms
of an according to the values given in (179), we have
d 2 x n
dt2
m
M' +
(22n+1_1)+122n} 112
2
{M' + m (2n-1)-м2"}g; (181)
M
and this determines the place of M' at the time t. From this
value may be deduced, by means of (179), the places of м
and of the centre of every pulley at any time.
If n = 1, the system is that of a single fixed and of a single
moving pulley; and we have
M' +
7 m
+ 4M
2
4M}
}
d2x'
dt2
{M
{M' + m−2м}g.
If we equate to zero the right hand member of (181) we have
the condition of statical equilibrium of the system of weights
and pulleys, when the weights of the pulleys are taken account of.
Ex. 12. Determine the motion of a screw which descends in
208.]
379
MOTION OF RIGID BODIES.
its nut by reason of a given weight acting on it, when a con-
stant power applied at the end of the lever is not sufficient to
maintain equilibrium.
Let a the radius of the cylinder which carries the screw,
a the angle of inclination of the thread to the axis of the
screw; 7 = the length of the lever, p = the force applied per-
pendicularly to l at its extremity; m =
m = the mass of the screw,
and м the mass by the weight of which the screw descends;
R and F the sums of the reactions and sliding frictions respect-
ively of the parts of the screw on its nut; µ = the coefficient
of sliding friction. We shall suppose that м does not rotate as
the screw revolves; then the equations which determine the
motion are
(m+M) ( 9 — d²z)
d² z
whence
dt2
dt2
a² d20
m
2 dt2
= R sin a -+- F cos a
=R
;
(R COS a―F sin a) a —pl;
F = μR;
dza do tan a ;
a;
may easily be found, and all the circumstances of
motion may be determined.
208.] In the course of the preceding problems some subjects
have incidentally arisen in particular forms which require fuller
and more general discussion.
The first is the general case of the rocking, or titubation as it
has been called, of a heavy body bounded by a cylindrical surface,
resting on another rough cylindrical surface, the axes of the
two surfaces being parallel and horizontal, when the upper body
which rests on the lower surface is slightly displaced from its
position of equilibrium.
Fig. 57, which represents a section of the two surfaces by a
vertical plane perpendicular to the axes of the cylinders, shews
the circumstances at the time t.
G is the centre of gravity of the upper body, whose mass
= m; and when the upper body is at rest on the lower, a is in
contact with o, and the line GA, which is the normal of the upper
surface at A, is vertical, and is in the same straight line with oc,
which is the normal to the lower surface at o.
Let the upper
body be slightly displaced by rolling, not sliding, on the lower;
302
380
[208.
MOTION OF RIGID BODIES.
so that the arcs AP and OP are equal. Let c' and c be the
centres of curvature of the upper and lower surfaces at a and o
respectively. The normals to the two surfaces at p are evidently
in the same straight line; and since AP and OP are infinitesimal,
СА
c'P, and co = CP, Let R = the normal pressure of the
two surfaces on each other, and let r the friction of rolling ;
=
also let k = the radius of gyration of the moving body relative
to an axis through & parallel to the axes of the cylinders.
= 0
;
Let o be the origin, (x, y) the place of G at the time t;
CO = CP p; c´A = c'P = p' ; c'G = C, OCP = O₂ AC'P
consequently pop'e', and
=
0+0' =
The equations of motion are
m
d²x
dt2
R sin ✪ — F cos 0,
2
d² y
m
R COS + F sin 0 — mg,
dt2
m k²
d² (0 + 0)
dt2
— — Rc sin e′ + F (p' — c cos 0'); }
and the geometrical equations of condition are
(182)
X =
(p+p) sin 0-c sin (0 +0′), )
(183)
y = p+ (p+p') cos 0 — c cos (0 + 0′).
As the displacement which we are considering is very small, I
shall assume and e' to be so small that all powers of them
above the first may be neglected. I shall also assume
do
to be
dt
so small that all powers of it above the first may be neglected :
under these suppositions, the preceding give
d² x
dt2
d2y
dt2
{pté
+ p-c
p+p) d²o
p' dt²·
{ - (p+p)0+ c
2
(184)
(P+P) = 0 }
d20
Ꮎ
Sat
dt2,
and substituting these values in the first two of (182), we have
(p + p') p d20
R = mg + m
св
;
(185)
dt2
τρ
d20
F = mg 0 —m
(p' — c)
(187)
dt2
which determine R and F in terms of
d20
dt²·
And if we substitute
208.]
381
MOTION OF RIGID BODIES.
these values in the last of (182), and omit terms involving
powers of higher than the first, we have
ρ
{k² + (p' — c)²}
d²0 p²² = c(p+p')
dt2
P+ p
g 0 = 0 ; (187)
which is the equation of rotation of the upper body about its
rotation-axis through G.
d²0 .
dt2
Since the coefficient of is positive, the form of the integral
p'²
of this equation will depend on the sign of the coefficient of 0.
(1) If p2 is greater than c (p + p'); then the integral of
(187) takes the logarithmic form, and 0 will continually increase
as t increases; so that the body moves farther away from its
original position of rest, that position being one of unstable
equilibrium. The geometrical meaning of this criterion is
1
AG
is less than
(2) If p2 = c(p+p'), that is, if
1 1
+
ρ
ρ
1
1
1
+
AG
Ρ
é
d20
0,
dt2
and the body either remains at rest in its new position, or rotates
with a constant angular velocity. The original equilibrium in
this case is neutral.
(3) If p2 is less than c(p+p), the integral of (187) takes the
form of a circular function, and indicates an oscillatory motion;
in which case the body rocks or titubates; and the time of an
oscillation
π
2
{ k² + (p' — c)² } ³ (p+p) š
g³ {c (p + p' ) − p²²} =
(188)
In this case, the original equilibrium is stable; and we have
1
1 1
greater than
+
AG
P
р
The geometrical criteria for the stability, neutrality, and insta-
bility of equilibrium, are the same as those found from statical
considerations in Art. 124, Vol. III.
The process and the results of this Article are equally true
whatever are the signs and values of p, p', and c. Thus, if p is
negative, the lower surface has its concavity upwards, and the
problem is that of a body with a convex surface rolling on a
concave surface. If p' is negative, we have a body with a con-
1
382
[209.
THE LAW OF SMALL OSCILLATIONS.
If p
cave surface rolling on a convex surface.
∞, the lower
surface is plane, and a body with a convex surface rolls upon it.
If p∞, the upper surface is plane, and the body with a plane
surface rolls on a convex surface.
209.] Another principle, which has arisen incidentally in the
preceding Examples, and which admits of more general applica-
tion, is that of small oscillations.
When a system of material particles, subject to mutual con-
nections, is slightly disturbed from a position of stable equi-
librium, certain forces are brought into action, which tend to
restore the system to its original place of rest. We have had
examples of such forces in the preceding Article; and in Exam-
ples 4, 7, 8, of Art. 207. Now the equations of connection of
the several particles enable us to express the coordinates of the
places of these particles at the time t in their disturbed state
as functions of new variables 0, 0, V,..., which are, all and
each, as well as their t-differentials, infinitesimals when the dis-
placement of the system is infinitesimal, and are equal to zero
when the system is in its place of stable equilibrium; and the
number of these variables is of course that which is sufficient to
determine the places of the particles, subject as they are to their
mutual connections. If then we substitute these variables for
the old variables in the equations of motion, we thereby obtain
new differential equations which correctly represent the circum-
stances of motion in terms of the new variables. These equa-
tions are of the second order, and contain no term independent
of the variables 0, p, t, ...; because they must be satisfied when
the system is in its position of stable equilibrium; that is, when
0 = 4 = 4 = ... = 0: and since the displacement of the system
is, by our assumption, small, these variables and their differ-
entials are small; if therefore our object is to acquire a general
idea of the motion, that is, to obtain the principal motion, we
must in the first place neglect all powers of the variables and
of their t-differentials higher than the first. With this object in
view the functions of the variables which enter into the equations
must be expanded in ascending powers of the variable; and the
squares, and all higher powers of them, are to be omitted; thus
the differential equations will be linear, of the second order, of
constant coefficients, and devoid of the second members; and
their integrals will be of the following forms:
¥,
209.]
383
THE LAW OF SMALL OSCILLATIONS.
0 = A₁ cos (r₁t—a₁) + A₂ cos (r₂t—a₂) + ...,
2
(189)
1
¿ = B₁cos (r₁t—a₁) + B₂ cos (r½t—a₂) +
1
2
& = c₁ cos (r₁t— a₁) + C₂ cos (r₂t—a½) +
2
..., al,
where r₁, 72, ... are the roots of a certain algebraical equation of
an order equal to the number of the variables; A1, A2,
а2, are constants depending on the initial or other circum-
stances of the system; and B₁, B2, ..., C1, C2,... are other constants,
which are functions of A1, A2, ... T1, T2, and are determinable
by means of the given differential equations.
If n is the number of the variables 0, 4, 4, ..., each of the
equations in (189) is the sum of n terms, which are circular
functions of t; each term by itself representing a small oscillation
of the same nature as that of the simple pendulum; the times
of oscillation corresponding to each term being different, and
,...; and each variable generally containing
π π
r1 r2
being severally
a term of each period. Thus the motion of the system, slightly
disturbed from its position of stable equilibrium, consists of
simple oscillations of its several component particles, both the
amplitudes and the periodic times being in general different for
the several oscillations. As these oscillations coexist, and as
each variable is the sum of many, the principle of their com-
bination is commonly called the law of the coexistence of small
oscillations.
r
If the quantities 71, 72, 73, ... are commensurable, the system
of particles will periodically pass through the same state; for
suppose μ to be the greatest common measure of r₁,r,...; so that
r2 = K2 μ, r3 = kz µ, ...;
r1 =
where k₁, ką, ką, …..
are whole numbers which have no common
measure; then if T is the time in which the system passes from
a given state to the same state again, k₁ µт, k½µT, ….. must all
be multiples of 2; and as k₁, k2, ... have no common factor,
the least value of T which will satisfy this condition is
1
T
2π
μ
(190)
this therefore is the time in which the system of particles passes
through all its forms from one state to the same state again.
If μ0, this time is infinite; that is, if the quantities
have no common measure, the system of particles is
r1, 72,
:
384
[210.
THE LAW OF SMALL OSCILLATIONS.
not periodic; and the state in which the particles may be at a
given time is never taken by the particles again.
210.] Since each of these small motions takes effect separately,
and independently of other similar motions; and since the whole
effect is the sum of these separate and partial effects; the law
of co-existence of small oscillations is a particular case of the so
called principle of superposition of small motions. This prin-
ciple may be explained in the following way.
Suppose that for certain initial values of the variables and
their t-differentials, say,
0 = 0,,
Φ
=
Фи
¥1, ……..
аф
dy
(191)
O₂'s
P1',
=
dt
dt
¥1',...,
do
dt
=
the motion is represented by the integrals,
$1
0 = @₁₁ = $₁₂, V = 1, ... ;
019 Ф
, ф Φ
19
and suppose that for another system of values, say,
(192)
0 =
023
&
=
$21
,
Φ......
>
d Ꮎ
dt
=
02,
аф
dt
=
the motion is represented by the integrals,
0 =
Ф
2, $ = $2, √ = ¥2, ...;
(193)
(194)
and so on for n systems: then for the systems of values which
are the sums of all these values, viz.
0 =
O₂ + O 2 + ...,
ф
=
Φι
$1 + 2 + ...,
&
=
¥1+2+
do
dt
аф
dy
01 +0₂+
2
dt
=
$1' + $½' +
V₁ + ½ + ...,
dt
the motion is represented by the sum of the partial integrals; viz.,
0
1 + 2 +
$
$1
1 + 2 +
4 =
1 + 2 + ...;
(195)
for these values will satisfy the differential equations of motion
by reason of their linearity; and they reduce themselves to the
several initial values when t = 0; thus they satisfy all the con-
ditions of the problem.
The preceding processes are only applicable when we confine
ourselves to small motions and to first approximations. If a
more exact determination is required we must return to the
211.]
385
EXAMPLES OF SMALL OSCILLATIONS.
original equations of motion in their complete forms, and sub-
stitute in terms of the second degree relatively to the variables
those values which we have found in terms of t to a first ap-
proximation; and then, neglecting all the terms of a degree
higher than the second, we shall have new equations which will
differ from the first only by the addition of a new member,
which is a known function of t. Values of 0, p,... will be deter-
mined from these to an approximation higher than the former.
And if an approximation is required still more exact, we must
introduce the second values of the variables in the original
equations, and pursue a process similar to the former.
211.] The following examples are illustrative of the preceding
method.
Ex. 1. I will first take the simple case of a conical pendulum,
that is, of a heavy particle constrained to move on the inside of
a smooth spherical surface; this problem is the same as that
which has been considered in Art. 370, Vol. III.
=
Let us refer the position of the moving particle to the point
of suspension of the pendulum as the origin, and to two vertical
planes passing through that point, and perpendicular to each
other, as the planes of (x, z) and (y, z), the axis of being
taken vertically downwards. Let us moreover suppose the rod
of the pendulum, whose length = l, initially to be in the plane
of (x, z), and to be inclined at an angle a to the z-axis; and
the bob to be projected with a velocity u perpendicularly to
the plane of (x, z); let the line of the pendulum at the time t
be projected on the planes of (x, z) and (y, ≈); and let the
angles between these projections and the z-axis be respectively
and now I shall assume that the oscillations of the pen-
dulum are always small, and I shall consequently consider 0 and
, the variables which determine its position, to be so small
that powers of them higher than the second are to be neglected.
The initial values of 0 and are respectively a and 0; and of
do dy
dt
U
and are 0 and then the equations of motion are
dt
ī;
d20
d²
1
dt2
- ყ0,
l
dt2
g4;
d02
dt2
= g(a² — 0²),
15d42
dye
u²
dt2
0 = a cos
PRICE, VOL. IV.
(9)* t
t,
И
(gl) #
12
}
sin (2)*
(9)* t.
3 D
=-942;
386
[211.
EXAMPLES OF SMALL OSCILLATIONS.
Let (x, y) be the projection on the horizontal plane of (x, y) of
the place of the bob of the pendulum at the time t; so that
x = l sin 0,
10,
= la cos
(9)*,
t;
let y = x tan p; then
tan =
y = Isiny,
ιψ
W
a (lg)
= U
tan
(4) sin (2)*t;
(97) *
(24) $t;
so that o, which determines the plane of oscillation of the pen-
dulum at the time t, does not vary directly with t, and con-
sequently the pendulum does not revolve uniformly. Also
x² 9 y²
+
12 a2 lu²
1,
(196)
which represents an ellipse; so that the bob of the pendulum
describes a path whose projection on the plane of (x, y) is an
ellipse. All these results are in accordance with those of
Art. 370, Vol. III.
Ex. 2. A system of n heavy rods o₁, A1 A2,……., of given lengths
2a1, 2α, ... 2a, is formed by means of smooth hinges at their
extremities A1, A2, ..., as in Fig. 58, and is suspended by the
extremity o from a fixed point. Determine the small oscilla-
tions of the system, when the motions of all are in the same
vertical plane of (x, y).
Let the angles which the rods respectively make with the
vertical oy be 01, 02, ...; and let (x1, Y1), (X2, Y2), ….. be the places
of their centres of gravity at the time t; let m1, M2, M3,
the masses of the rods, and k₁, k₂,
be
their radii of gyration rela-
centres of gravity, and per-
be
tive to axes passing through their
pendicular to the plane of (x, y). Let X1 Y1 X2 Y2,
respectively the horizontal and vertical components of the actions
of the hinges at A1, A2, ...; then the complete equations of
motion are
2
m₁ (a₁² + k₂²)
M2
d26,
dt2
d2 x 2
dt2
d² y z
2
ՊՈՆ»
dt2
d20₂
ma k₂²
2
dt2
-m₁ÿα₁sinė₁+2α₁ (x₁ cos◊₁ — ỵ₁ sinė₁); (197)
1
= m2 g − Y₁ + Y₂,
1
= α₂ {(x1+X2) cos 02 — (Y1 + Y₂) sin 02} ;
2
(198)
211.]
387
EXAMPLES OF SMALL OSCILLATIONS.
d2 x3
dt2
mz dť²
d2 y 3
Mz dt²
mz kz
d20
2
03
dt2
X2 + X3,
= Mz J-Y₂+Y3,
= α3 {(X2+X3) COS☺3 — (Y₂+Y3) sin☺3};
(199)
Mn
d² Xn
dt2
Xn-1,
d² y n
(200)
Mn
= Mn y - Yn −1,
dt2
m n k n²
2
d² On
dt2
= αn {Xn-1 COS On — Yn-1 sin0n} ;
but the variable coordinates are subject to the following equations:
x2 = 2a, sin 0₁+a₂ sin 02,
X
Y₂ =
2
1
2α₁ cos 01 +α₂ cos 02 ;
2
-
x3 = 2a, sin 0₁+2a, sin e₂+a, sin 03,
Y3 =
2α₁ cos 0₁+2α, cos 0₂+α, cos 03 ; §
On,
Xn = 2α₁₂ sin 0₁+...+2an-1 sin 0-1+ an sin ons
Yn
=
2α₁ cos 01 + ... + 2 an-1 cos On−1 + an CoS On·
(201)
(202)
} (203)
From these we have, omitting squares and higher powers
of 01, 02, ...,
d² x z
dt2
d² 0,
d² 0 2
= 2 a₁
+αz
dt²
dt2
(204)
d2 0₁
d2 02
-
Өт
2 α101 dt²
1
a2 02
dt2
d² y z
dt2
2
and substituting these several values in the first two of each of
the preceding groups (198), (199), ......, we shall have equa-
tions involving the second t-differentials of the 's only; from
these we can eliminate the x's and the r's, and thereby obtain a
series of equations in terms of the e's and their second t-differ-
entials only.
Let us take a particular case, and suppose that there are only
two beams; then, in addition to (197), we have what (200) be-
comes when n = 2; viz.,
3D 2
388
[211.
EXAMPLES OF SMALL OSCILLATIONS.
d2 x 2
X1,
M2
dt2
(205)
d2 y 2
2.
ms dt²
= m² g - Y1
d2 02
m₂ k₂ 2
= α₂ {X1 cos 02— Y₁ sin 02} ;
dt2
... X1
m² 2 a₂
{ 20
d2 01
d2 02
аз
+az
dt2
dt2
(206)
d² 0 2
Y₁ = m² g +M₂
{ 200
d201
1 + α 2 0 ₂ dt²
;
1
dt2
and consequently from (197), and the last of (205), we have,
2
{m₁ (a₁²+k₂²)+4m₂a₁² }
2
d² 0,
dt2
d2 02
+2α₂ az mz
a₁ (m₁+2m²)019,
dt2
d2 01
d2 02
2 a 1 a 2 m² dt ²
2
+m₂(a₂² + k₂²)
2
m2 a 2 0 2 9 ;
dt2
which are two linear differential equations whereby 0, and 2 are
to be found in terms of t. The form of them is that which has
been explained in Art. 209; for both are satisfied for the posi-
tion of equilibrium, when the beams are vertical, and 0₁=0₂=0.
Now these equations are of the form.
2
A
d2 0₁
dt2
1
+ B
d2 02
dt2
a01,
(207)
d² 0 1
d2 02
B
+ c
-B02;
dt2
dt2
which may be expressed in the form
d2
+
dt2
a) 0₂
d2
dt2
02 = 0,
d2
d2
(208)
B
dt2
0₂+ (0
dt2
+B) 0₂ = 0;
2
and eliminating 02, we have
d4
(AC — B²) + (AB+ca)
dt¹
d2
dt2
}
+ aß } 0₁ = 0.
(209)
d
Now the four values of
which make the first factor in the
dt'
left hand member of this equation to vanish, are impossible;
let them be ±(−)šr₁, ± (−)±r₂; then the solution of (209) is
2
0₁ = E₁ cos (r₁t-y₁) + F₁ cos (r₂t-12);
01
1
(210)
211.]
389
EXAMPLES OF SMALL OSCILLATIONS.
where E1, F1, 71, 72, are four constants introduced in integration;
which are to be determined from initial or other circumstances.
d202 from the two equations of (207), we have
dt2
If we eliminate
2
d² 0,
(AC - B2)
dt2
+ca –BB, = 0 ;
2
and replacing 0, in terms of t, by means of (210),
E2
1
Өг E, cos (r₁t-1)+ F₂ cos (r₂ t-Y₂);
2
(211)
where E, and F₂ are constants, which are functions of the former
constants, of r₁, r₂, and of E₁ and F₁; we are consequently able
to determine all these constants in terms of initial values of
1
E1
thus the variables which determine the posi-
Өз
complete periodic times are respectively and
do₁ do₂
01, 021
dt dt
tion of the beams consist of two circular functions, whose
2π
2 п
r2
If r₁
1
2
r1
and r, are prime to each other, a position of the beams, which
exists either initially or at any other time, never recurs. If
however they have a common measure = μ, the state of the
2 T
beams is the same after every interval of time =
μ
Ex. 3. An uniform heavy rod of length 2a is suspended from
a fixed point by means of a string of length 1, whose weight
may be neglected. The rod is slightly displaced from its posi-
tion of equilibrium; it is required to determine its small oscil-
lations.
Let the point of suspension be taken for the origin; and let
the horizontal plane through it be the plane of (x, y), and let the
z-axis be taken positively downwards. Let m = the mass of
the rod; and at the time t let T the tension of the string,
(x, y, z) the place of the extremity of the string, (x', y', z′) the
place of the centre of gravity of the rod; (§, n, Š) the place
of any element dm of the rod, the distance of which from the
upper extremitys. Then the equations of motion of the
centre of gravity of the rod are
d² x'
m
X
T
dt2
d² y
Y
m
. T
dt2
יך
d² z
m
dt2
mg - T
(212)
390
[211.
EXAMPLES OF SMALL OSCILLATIONS.
but since the displacement of the string and beam is always.
small, x, y, x', y' are always small, and approximately, z 1,
2′ = 1+a; consequently for the preceding equations we have
their approximate values
d2 x'
X
dt2
'91'
d2
day'
Y
9
dt2
T = mg.
Σ.Μ
{
d2c
n
Š
dt2
dt2
d®n{
= x.m (nz-(Y);
The equation of moments relative to the x-axis is
but approximately,
d2
25
dt2
0;
so that we have
2 a
S
n = y + (y' − y) = ;
8 = 1 + 8;
T
(213)
- [ " m² + 3 {d}} + d³ (y = y) = }ds=mgy—yr+lx};
2 a
s d² y
dt2
d2y
α
dt2
— s
dt2 α
day'
· (4a+31) = 3gy'.
Similarly the equation of moments relative to the y-axis is
dt2
d2 x
a
dt2
-(4a+31) dar
d² x'
3 g x'.
dt2
(214)
(215)
These two equations together with the first two are sufficient to
determine the motion.
Eliminating y' between the second of (213) and (214) we have
d4y 4a+31 d2y
3.g2
+
9
dt4
αι
+ y = 0;
dt2 al
(216)
and taking the symbols of operation only, we have
d4 4a+31
d2
3g2
+
dt4
al
g
+
dt2 al
= 0.
d
The four values of
dt
which satisfy the equation are evidently
2
impossible; let them be ± (−)³r₁, ±(−)*r½; then the solu-
tion of (216) is
y = E₁cos (r₁t—a₁) + F₁ cos (r½ t—ɑ2) ;
where E1, F1, α, a₂ are constants depending on the initial or
other circumstances of the beam and string. The other variables
have the following values:
212.]
391
EXAMPLES OF SMALL OSCILLATIONS.
y' = E₁'cos (r₁t — α₁) + F₁'cos (r½ t — ɑ2),
x = E₂ cos (r₁t — ẞ₁) + F₂ cos (r₂ t — B₂),
x' = E' cos (r₁t - B₁) + F₂ cos (r₂ t - B₂);
2
2
and each of these variables involves the same two circular
functions, of which the periodic times are respectively
2π
and
r1
27
2
although the amplitudes of vibration and the commencement of
the periodic times are different for each. All the undetermined
variables can be found in terms of the initial circumstances of
the rods.
212.] In connection with this theory of small oscillations
another problem, which arises out of the formulæ of the pre-
ceding Chapter, requires investigation; for the circumstances
of it at the time t may be expressed by means of variables
which vanish when the system is in a position of equilibrium ;
and for small values of which the system in motion is in a state
approximate to that of equilibrium. The problem is this:
A body which has one point in it fixed is in motion about an
instantaneous axis, the angle of inclination of which to a prin-
cipal axis of the body at the fixed point is always small; it
is required to determine the motion when the body makes small
oscillations about its mean position.
Let the principal axis of the body with which the instantane-
ous always nearly coincides be the -axis; so that w₁ and w₂
@1
are always small quantities, the products and powers of which
above the second we shall omit in our approximations. Also w
@3
is nearly constant: we shall take n to represent its mean value,
and shall replace w, by n in small terms. As the motion of the
body is small L, M, N are also supposed to be small. Under
these circumstances Euler's equations are
A + (0 B) N W₂ = L,
αωι
dt
dwz
B
dt
d
3
C
dt
+ (A — C) n w₁ = M,
w1
= N ;
(217)
Let us refer the motion to the axes of (x, y, z), fixed in space by
means of the direction-cosines given in 39 and 40, Art. 2, of the
present volume. Let the mean position of the <-principal axis
be the z-axis, so that the angle between the z- and the (-axes is
392
[212.
EXAMPLES OF SMALL OSCILLATIONS.
always small; consequently c₂, which is the cosine of the angle
contained between these axes, is always nearly equal to 1, and
may be replaced by 1 in small terms. Hence also it follows,
that C1, C2, α3, b, are always small quantities. And since
wz = ɑzw1+bz wz + Cz wz,
we may replace w by w, that is, by n in small terms. Now
replacing wx and ∞, by their values given in (83) and (84),
Art. 40, we have
W₁ =
A₁ w x + Aq wy + Az W x
daz db3
+ b ₂ d t
b₂
dc3
а1
{
+ Cz
d 2 d t
dt
α2
{-
daz
db3
dc3
1
+ b₁
dt
dt
+ C₁ dt S
C1
+ а z n
dc3
db3
dt
dc3
b3 dt
now C3
dca
dt
(α₁ b₂ — ɑ₂ b₁)
dbs
C 3 d t
1 in small terms, and its variation is so small that
must be omitted;
+ (α₁ C₂-α₂C1)
+azn
dt
+ az n;
Similarly,
db3
1
+ azn.
dt
das
W2
dt
+ b₂n.
(218)
(219)
Let us substitute these values in (217); and we have, omit-
ting the subscript 3,
+(A+B-C)n +n² (c-B) b = L,
d2b
da
A
dt2
dt
(220)
d2 a
db
B
dt² + (A+B−c)n
+n² (A—c) a = M.
dti
L and м, which are the moments of the impressed couples whose
axes are the - and n-axes respectively, must be expressed in
terms of a and b; as these however are small quantities, and
vanish in the state of equilibrium when
as L and м both
a = b = 0, these quantities are of the following forms;
L = pa + pb,
M = qa + q'b;
(221)
so that finally the differential equations in terms of a and b are
of the forms
213.]
393
EXAMPLES OF SMALL OSCILLATIONS.
d2b
da
+ a
+ ß² a + ß²² b = 0,
d t2
dt
(222)
d² a
db
α
dt2
dt
+ y² a + y²b = 0;
which are two simultaneous differential equations* of the second
order, and are integrable by the processes explained in Vol. II.
Hereby a and b will be expressed in terms of t; and as they are
the cosines of the angles contained between the fixed ≈-axis and
the moving axes of έ and ŋ respectively, so will they determine
the position of the three principal axes of the body at the time t ;
and (218) and (219) will give ∞, and w₂ in terms of t; and as
w3 = n, so will also be known; and the position of the instan-
taneous rotation-axis will be given.
W3
η
213.] Let us apply these equations to the solution of a
problem which we have before considered.
A heavy conical top, whose vertex is on a rough horizontal
plane and does not move, rotates with its axis of figure nearly
vertical; it is required to determine the circumstances of motion.
a =
In this case the vertex of the top may be considered as a
fixed point, about which the body moves. We will take it to
be the origin, and the horizontal plane through it to be the
plane of (x, y); also A B; and c is the principal moment of
inertia relative to the axis of figure. Let h the distance of
the centre of gravity of the top from the vertex: then, as the
line of action of gravity is parallel to the z-axis, we have, as in
Art. 167, Lmghb, мmgha, N = 0; so that (220) become
n=0;
d2b
A
+ (2A-c)n
dt2
da
dt
+n² (c-A)b = mghb,
(223)
d² a
db
A
dt2
-(2A-c)n + n² (cs) a = mgha;
of which the form is
dt
d2b
da
+ 2 a
+ ß²b = 0,
dt2
dt
(224)
d² a
dt2
db
2 a
+ẞ² a = 0;
dt
* These equations are the same in form as those given in "An Elementary
Treatise on the Dynamics of a System of Rigid Bodies," by E. J. Routh, M. A.,
Cambridge, 1860; equations A, p. 174. It is indeed to a study of that treatise
that I owe the thought of transforming generally the first two of Euler's equa-
tions into (222) as their equivalents.
PRICE, VOL. IV.
3 E
394
[213.
EXAMPLES OF SMALL OSCILLATIONS.
whence, eliminating b, we have
S d+
d2
Lat
+2(3²+2a²)
+ B$
dt2
+}
}
a = 0.
(225)
Now the form of a which is derived from this equation depends
on the nature of the roots of the equation.
r²+2 (3² +2 a²) r² + Bª = 0.
Solving this, we have
r = ± (−)³ {a ± (a² + B²) *}
=
(226)
± (~) ;
ry
{{
2A-C
n +
2 A
(c² n² - 4 am gh)
2 A
(227)
and thus we have three cases:
(1) If c² n² is greater than 4Am gh; that is, if
n² is greater than
4 sm g h
C2
the four values of r are impossible; let them be
+ ( − ) = r₁,
19
+ ( − ) = r₂ ;
2
then the solution of (225) is
(228)
a = c₁ sin (r₁t + Y₁) + c₂ sin (1½ t +12) ;
2
(229)
where C1, C2, Y1, 2 are four constants, which are to be deter-
mined by the initial or other circumstances of the top. In this
case the motion is stable, and the top makes oscillations about
its mean place. Thus the stability of the motion depends on
the inequality (228), which shews that the angular velocity of
the top about its own axis must be greater than a certain as-
signed quantity.
&
(2) If c² m² = 4amgh; then r = ±(−) a; and
α = (C₁ + C₂ t) sin (at+y);
(230)
where C1, C2, and y are constants depending on the initial cir-
cumstances. This result shews generally, that a has periodical
values; but that, as c₁ + c₂t increases with the time, its maximum
values increase; and thus the motion of the top is still oscil-
latory.
(3) If c² n² is less than 4 Amgh, then
r = +
{
+
(4 sm gh-c² n²)
2 A
2 A-c
+(-)
(−) $
N
2 A
}
= ± { ± p+(−)*r} ;
a =
Geet sin (rt+Y₁) +€₂e¬pt sin (r t + √2) ;
(231)
where C1, C2, V₁, and y½ are constants to be determined by the
214.]
395
ROLLING AND SLIDING FRICTION.
initial circumstances. This form shews that a increases without
limit as t increases without limit; but a is the cosine of an
angle, and cannot exceed 1; consequently this form soon ceases
to represent the motion of the top.
From the value of a determined as above, that of b may be
deduced by means of either of the equations (224).
214.] Another subject, which has arisen in the course of the
examples in Art. 207, requires a few words of explanation. We
have frequently met with a resistance or a force arising from
friction; and we have assumed the force to act in a direction
contrary to that of the motion: we have spoken too of a friction
of rolling as distinct from a friction of sliding. This distinction,
as well as the dynamical effects of the two kinds, we proceed to
explain more fully; and we must begin with certain laws which
have been experimentally observed; for although these are, a
priori, reasonable, yet they depend upon the physical constitution
of matter; and our knowledge of molecular physics is, as yet,
too uncertain, so that any proof derived from that source should
supersede proof drawn from observation.
Friction of sliding has been considered statically in Section 3,
Chapter III, of Vol. III, and the three laws therein stated are
sufficient for the dynamical effects which we have now to con-
sider. I shall take the case of a heavy body placed on a rough
inclined plane, or on a rough curved surface, and sliding down
it, so that its velocity is retarded by friction.
From the laws just alluded to it appears, (1) that, so long as
the weight is the same, the friction is independent of the area
of the surfaces in contact; (2) that the friction varies as the
normal pressure exerted by the heavy body against the surface
on which it moves. It also appears that the body does not
begin to move unless the inclination of the plane, or if it is on
a curved surface, of the tangent plane of the point at which it is
placed, is equal to or exceeds a certain angle called "the angle
of repose," and that, if the coefficient of friction, this angle
µ =
= tan-¹µ. It appears also from law III of the section above
cited, that the friction is independent of the velocity of sliding.
I propose now to apply these laws to some problems.
Ex. 1. A particle is placed on a rough inclined plane, the
angle of inclination of which to the horizon is not less than the
angle of repose; it is required to determine the motion.
Let a the angle of inclination of the plane; m = the mass
*
3 E 2
396
[215.
ROLLING AND SLIDING FRICTION.
of the particle; R = the normal pressure on the plane; F = the
retarding force of friction; µ = the coefficient of friction. Let
x= the distance along the plane through which the particle
has moved in the time t; then
X
d2x
กา
dt2
= mg sin a
F,
R
mg cosa,
F = μR;
d² x
dt2
g {sin a
μ cos a} ;
from which equation the motion may be determined.
Ex. 2. A particle slides down within a rough circular cylinder
whose axis is horizontal: determine the equation of motion.
0
Let the angular distance from the lowest point at the
time; then the equation of motion is evidently
d20
α
g{sine-μ cos 0};
dt2
d02
α
dt2
= 2g {cosa-cose +μ (sina-sin()};
where a is the value of 0, when the particle is at rest.
215.] In these cases the friction has been that of sliding
only; and although the inclination of the plane has, in the latter
example, been less than the angle of repose, yet, by reason of
the previously acquired momentum, the particle has still con-
tinued to move. In cases however of bodies moving in contact
with rough surfaces there may be friction of rolling as well as
friction of sliding. If a cube is placed on an inclined plane
whose inclination to the horizon is less than 45°, the cube will
not fall over, and will slide down if the angle of repose is less
than that of the inclination of the plane. If however a heavy
sphere is placed on a rough inclined plane it will always roll; it
will moreover slide as well as roll if the angle of inclination of
the plane is greater than its angle of repose. Now, if the sphere
rolls only, the process taken in Ex. 1, Art. 207, determines the
motion; whereas, if it slides as well as rolls, other terms are
required in the equations.
We may form a tolerably precise notion of the friction of rolling
by imagining a heavy cylinder or wheel rolling on a horizontal
plane. By reason of the compressibility of the matter in con-
215.]
397
ROLLING AND SLIDING FRICTION.
tact, the cylinder and the plane mutually penetrate each other;
and hence arise reactions, acting on the cylinder in a direction
contrary to that in which it is moving, and which act as obsta-
cles to its rolling. The friction of rolling is measured by the
horizontal force which it is necessary to apply to the axis of the
wheel to maintain an uniform velocity of translation of the
cylinder. Experiments were made in this subject by Coulomb;
and he discovered the following law: the force of rolling friction
for a heavy cylinder and a given plane varies directly as the
pressure, and inversely as the radius of the cylinder. Thus, if
R = the pressure of the cylinder on the plane, r = the radius
of the cylinder, F = the rolling friction,
F = V
R
(232)
where v is a constant, called the coefficient of rolling friction,
which depends on the nature of the surfaces in contact.
Rolling friction as a retarding force is much less than sliding
friction, and may be neglected when the latter acts.
In mechanical problems the difficulty frequently is to deter-
mine whether a body will slide and roll, or only roll: now let
F the friction, and R = the normal pressure; then if the
ratio of r to R, which is equal to μ, is equal to, or greater than,
tan a, the body only rolls; in which case a geometrical condi-
tion will exist, which may take the form of a relation between
the space of translation described by the centre of gravity and
that due to the rotation about the instantaneous axis passing
through the centre of gravity: the use of this relation is evident
in the examples of Art. 207. If however the ratio of F to R is
less than tan a, the body will slide as well as roll, and the geo-
metrical relation just alluded to ceases to hold good. The
method of solution then to be applied will be evident from the
following examples:
Ex. 1. A heavy sphere moves down a rough inclined plane,
whose angle of inclination to the horizon is greater than that of
repose; it is required to determine the motion.
=
Let a
the angle of inclination; µ = the coefficient of
sliding friction; F the sliding friction; R = the normal
pressure; the equations by which the motion is determined are
evidently
d² s
M
— mg sin a — F,
dt2
398
[215.
ROLLING AND SLIDING FRICTION.
R = mg cos a,
F = μR,
d20
m k²
= αF;
dt2
d2s
dt2
g{sina-μcosa},
d20
k2
=μg cos a.
dt2
If the sphere only rolls, ds = a de; in which case
F
μ =
Ꭱ
k2
a² + k²
tan a
૭
(233)
tan a;
77
consequently the sphere will slide and roll, or will only roll,
according as tan a is greater than, or not greater than
7
2M.
Ex. 2. If the body moving down the plane is a circular cylin-
der of radius = a, with its axis horizontal; then
3 μ tan a;
3μ =
and the body will slide and roll, or roll only, according as a is
greater or not greater than tan-13 µ.
Ex. 3. A heavy body whose bounding surface is a circular
cylinder, but whose centre of gravity is not in the axis of the
surface, makes small titubations on a rough horizontal plane.
Determine the limits of the angle of titubation so that it may
not slide on the plane.
Take the figure and symbols of Ex. 7, Art. 207; and let μ =
the coefficient of sliding friction between the rocking body and
the plane. Then, as the angle through which the body rocks
is small, I shall neglect powers of higher than the second.
Hence we have
d² x
dt2
d20
(a — c) ;
dt2
(a–c)2 F-Rc (a–c) 0;
- k² F =
F
R
c (a−c) 0
(a−c)² + k²
and consequently, if the body rolls and does not slide, ◊ must
not be greater than
(a–c)²+k²
M.
c (a–c)
216.]
399
THE MOTION OF RIGID BODIES.
216.] The following examples of motion of rigid bodies, free
and constrained, involve various modes of application of preced-
ing principles, and are inserted in illustration, as well as because
many of them are in themselves of considerable interest.
A heavy homogeneous solid ellipsoid is struck by a blow
whose momentum is q, in a line parallel to one of its principal
axes, and subsequently moves freely under the action of its
weight; it is required to determine the motion.
Let M =
the mass of the ellipsoid; and let the equation to
the bounding surface relative to the centre as origin, and its
three principal axes as coordinate axes, be
x2 y2 z 2
+ +
a² 2 b2
C2
1;
and let us suppose the line of blow to be parallel to the z-axis,
and to intersect the plane of (x, y) in the point (xo, yo).
The centre of gravity, which is the centre of the ellipsoid,
will move as if it were a particle of mass = м; and consequently
its path is a parabola; and if v is the initial velocity,
Q
V
;
M
and its initial line is that of the blow. Thus all the elements
of its path are known.
The ellipsoid also rotates under the action of q, as if the
centre were a fixed point. Consequently, if A and B are the
principal moments of inertia of the solid ellipsoid about the x-
and y-axes respectively, the initial instantaneous axis through
the centre is, see (73), Art. 194,
Y yo
X XO
+
A
0;
0
B
and if a is the initial angular velocity, see (72), Art. 194,
(234)
Ω
↓ = Q
+ }
You
;
(235)
the combined effect of these two motions is indeed a rotation
about an initial spontaneous axis, whose equation relative to the
moving ellipsoid is
X XO
Y yo
+ +
1
= 0.
B
A
M
(236)
Since the initial rotation-axis, given by (234), is not a prin-
cipal axis of the body it is not a permanent axis; consequently
400
[217.
THE MOTION OF RIGID BODIES.
it continually moves both in the body and in space; that deter-
mined above being only its initial position. Its motion will
be determined by Euler's three equations, simplified by the con-
dition, that momenta are impressed only at the origin; so
that we have
do
A
+ (C — B) W₂ Wz =
0,
dt
dwy
B
+ (A−C) W3 W1
= 0,
(237)
dt
dwz
W2
dt
C + (BA) w₁₂ = 0;
the initial values of w₁, w₂, w3 being respectively
@1, 2,
Q Yo
Q X o
A
B
and O. These equations however have been so fully discussed in
the preceding Chapter that it is unnecessary to say more on the
subject.
If either x, or y₂ = 0, the initial instantaneous axis is a prin-
0 Yo
cipal axis, and therefore is a permanent axis; and the ellipsoid
during its motion in space uniformly revolves about this axis.
Thus, if x = 0, the line of the blow q is in the plane of (y, z)
and is parallel to the z-axis, and the x-axis is the instantaneous
rotation-axis, which is also the permanent rotation-axis; and the
permanent angular velocity of the body about it
xo
Q Y o
A
5Qyo
M (b² + c²)
2
217.] A right cone is placed with its slant side on a perfectly
rough inclined plane, and rolls on it by the action of its weight;
it is required to determine its motion.
Let 2 a = the vertical angle of the cone; M
its mass;
a = its height; and let ß = the inclination of the plane to the
horizon.
The forces acting on the cone are, its weight, the rolling
friction of the cone on the inclined plane, and the normal
reaction of the plane. As the plane is perfectly rough, and the
cone rolls on its convex surface without sliding, the place of the
vertex of the cone is always the same, and the motion is that
of a rotating body which has a fixed point in its axis. We
shall therefore investigate it by means of Euler's three equa-
tions. Now the line of contact of the cone with the plane
217.]
401
THE MOTION OF RIGID BODIES.
is evidently always the instantaneous rotation-axis; and as the
force of rolling friction, as well as the normal reaction of the
plane, acts through this line, they produce relatively to it no
angular velocity: it will be convenient to derive from Euler's
equations the equation of rotation relative to this line.
Let c be the principal moment of inertia of the cone relative
to its own axis; and let A be that relative to an axis perpen-
dicular to the axis of the cone and passing through the vertex;
so that Euler's equations are
dw,
A + (CA) W2 W3 = L,
dt
dwz
(238)
A
(C — A) W3 W₁ = M,
W1
dt
C
dwz
dt
N;
Let us suppose the x-axis, to which a corresponds, to be initially
in the inclined plane, and to be the angle through which this
axis, and consequently the cone, rotates in the time t. Let a be
the instantaneous angular velocity at the time t; then, relative
to the three principal axes of the body, the direction-cosines of
the instantaneous axis are evidently sin a cos 0, sin a sin 0, and
cos a; so that
@1 = w sin a cos◊, @₂ = w sina sine, @z = wcosa; (239)
and if G is the moment of the impressed forces relative to the
instantaneous rotation-axis,
G = L sin a cos 0+ M sin a sin + N cos a.
(240)
Also
W
d Ꮎ
dt
Now, substituting these values in (240), we have
d20
{A (sin a)² + c (cos a)²}
G.
dt2
(241)
Let & be the angle at the time t contained between the line of con-
tact of the cone with the plane, and a straight line on the plane
perpendicular to a horizontal line. Then, as the cone rolls on the
plane, evidently
αλφ = a tan a dt
(242)
and as the weight, which acts at the centre of gravity of the
cone, is the only force which impresses angular velocity on the
body relative to the instantaneous rotation-axis, and tends to
increase 0,
PRICE, VOL. IV.
3
G = Mga sin a sin ß sin .
4
(243)
3 F
402
[218.
MOTION OF A TOP ON A SMOOTH PLANE.
3 M
Also A (sin a)2+c (cos a)2
{6+ (tan a)²} a² (sin a)²;
20
so that
α
d² &
dt2
(cos a)2 sin ß
59
sin $; (244)
(sin a)³ {6+ (tan a)2}
which equation determines the motion.
If 4 when the cone is at rest, then, integrating (244),
o
=
we have
2
a
dt
(d)² = 10 g
(cos a)2 sin 6
(sin a)³ {6+(tan a)²}
{cos-cos po}. (245)
If the cone makes small oscillations on the plane, the time of an
oscillation
= π
a (sin a)³ {6+ (tan a)²}
5 g (cos a)² sin ẞ
If the cone is fixed by its vertex to a point in a rough perpen-
dicular wall, and rolls on the wall, then the above equations
determine the motion, when ß
= 90°.
218.] Determine the motion of a top whose apex moves on a
smooth horizontal plane.
The motion of a top has already come twice into considera-
tion; viz., in Art. 167-172, where we have generally investi-
gated the motion of a heavy rigid body with a point fixed,
and having two equal principal moments of inertia relative to
that point; and also again in Art. 213, where the small motions
of it about a mean position have been investigated in illustra-
tion of the general law of small oscillations. In both these cases
the apex of the top has been assumed to be a fixed point; and
this condition is approximately satisfied when the top moves on
a perfectly rough plane. If however the plane is smooth, the
apex moves in the plane; and we propose now to investigate its
motion, and the motion of the top.
I shall assume the centre of gravity to be in the geometrical
axis of the top, and at a distance equal to 7 from the apex or peg
of the top. And I shall also use the same symbols in the same
significance as in Art. 167, except that the origin of the several
axes to which the rotation is referred will be taken at the centre
of gravity.
As the plane is perfectly smooth, the only forces acting on
the body are the vertical reaction of the plane = R (say), and
the weight of the top; so that if (§, ŋ, §) is the place of the
centre of gravity relative to a system of axes fixed in space, of
218.]
403
MOTION OF A TOP ON A SMOOTH PLANE.
which that of (is vertical, and those of έ and ŋ are in the given
plane, then
d²¿
d2n
0,
dt2
dt2
dt
M
dt2
= R mg;
(246)
from the first two of which it is plain that in horizontal motion
the centre of gravity either remains at rest or moves uniformly
in a rectilineal path, the elements of which depend on the
initial impulsion.
Also, since = l cos 0,
R = m
{
(d². l cos e
dt2
+9}
(247)
Next let us consider the motion of the top relative to the cen-
tre of gravity. The only force which produces a moment about
that point is the pressure of the plane at the apex, of which the
value is given in (247); and we have, as in Art. 167,
L=
R / sin cos,
M
R / sin 0 sin 4,
N = 0.
(248)
Now A B; and consequently the third of Euler's equations,
by reason of the third of (248), gives
w3 = a constant = n (say);
and thus the first two of Euler's equations are
dwi
A
dt
+ (C−A) N W₂ = R/sin@cos,
(249)
(250)
d w z
A
(CA)Nw₁ = — R/sin sin.
dt
Let us suppose the whole initial angular velocity of the body to
be about the axis of the top, and to be n;
so that initially
do d &
@1
w₁ = w₂ = 0, and consequently
W2
= 0;
0; and let us
dt
dt
A
1 {w
dw1
ω1
dt
+wz dt
?
suppose the initial values of 0 to be 0, and of 4 and 4 to be zero.
From (250) we have, as in Art. 167,
d wz
dt S
$
= Rl sin 0 {w₁ cos — w₂ sin &}
do
=Rl sin 0
dt
do
dᎾ (
= mlsine
dt
و}
do 2
Icose (de) - Isine
d20
dt
dt2
3 F 2
404
[218.
MOTION OF A TOP ON A SMOOTH PLANE.
2
2
... A (w₂² + w₂²) = 2 mlg (cos 0¸ — cos 0) — m P² (sine)³ (de)²;
therefore
— —
;
+A (sin☺)2(d)² = 2mlg (cosé, — cosł). (251)
{4+m²² (sino)?} ( de )* + a (sin()²(
dt
dt
Also, multiplying the first of (250) by sin 4, and the second by
cos ø, adding, and integrating, as in Art. 167, we have
d&
dt
A (sin 0)2 = cn (cos - cos ();
dy
and eliminating
dt
0
by means of (251) and (252), we have
{A² +- ml² a (sin0)2) sino de
(252)
(253)
dt +
{cose,—cos0} ³ { 2 mlga (siu 6)² — c²n²(cos✪¸ — cos◊)} ž
In these equations the following results are implied. Since the
left hand member of (251) is necessarily positive, the right hand
member is also positive; so that is never less than 06: thus
the angle at which the axis of the top is inclined to the vertical is
never less than its initial value, and 0 increases until it reaches
a value, say ₁, at which the second radical in the denominator of
do
O, and the inclination of the axis to
dt
(253) vanishes; then
the vertical is a maximum: 0₁, is always less than π, because
the expression which determines it is positive when = 0, and
is negative when : also the time in which the value of 0
passes from 0 to 0₁ is finite, as we have proved in Art. 168.
Thus the axis of the top makes isochronal oscillations in a
vertical plane, as that plane revolves about the vertical axis of z.
That vertical plane however does not revolve uniformly;
1
other words, its precessional velocity, which =
stant; (252) shews this.
dx
"
dt
in
is not con-
According as n is positive or negative, so is the precession
direct or retrograde; that is, the line of intersection of the
equatorial plane of the top with the horizontal plane revolves in
the same direction as the top rotates.
And the variations of the precessional velocity are periodic,
having the same period as those of the inclination of the axis to
the vertical; now the precessional velocity vanishes when 00s
and becomes a maximum when = 0; and continues to make
these periodical oscillations. This is explained at greater length
◊
219.]
405
THE MOTION OF A BILLIARD BALL.
in Art. 168. Thus, if the centre of gravity of the top does not
move, the apex of the top describes on the horizontal plane the
curve delineated in Fig. 35, where the radius of the interior and
exterior circles are respectively I sin 06, and 7 sin ₁; and where
the arcs of the path respectively touch the exterior circle when
dy
is a maximum; and meet the interior at right angles when
dt
dy
dt
0.
1
It is evident, by the principle of vis viva, that the angular
velocity of the top is a maximum when = 0₁, and is a mini-
mum when
0.
1
1
وہ
If n is very large, so that ₁ is very little greater than 0。,
the values of 0 are confined within very small limits. In this
case we can, as in Art. 169, integrate (253) approximately, and
obtain results which give an accurate representation of the
motion of the top.
219.] On the motion of a heavy homogeneous spherical ball,
(an ivory billiard ball,) on a rough horizontal table.
I shall take м to be the mass of the sphere, a to be its radius,
A to be the moment of inertia about a diameter.
I shall suppose the ball to be put into motion initially by
means of a blow, of which the intensity, line of action, and
point of application are known; all these circumstances being
given in billiards by the stroke of the cue. Thus, the initial
velocity of translation of the centre of gravity, and the angular
velocity relative to the initial rotation-axis will be known.
During the subsequent motion, the ball will both roll and slide,
so that retarding forces of rolling and sliding friction will be in
action on it. That of sliding friction acts at the point of con-
tact of the ball with the table, and in the line along which the
point of contact slides on the table. That of rolling friction
acts in the line along which at the time the centre of gravity is
moving. This latter friction however is, in ordinary cases, only
a very small fraction of the former, and may consequently be
neglected. As by the roughness of the table the sliding motion
of the ball is diminished much more rapidly than the rolling
motion, so the sliding motion soon ceases, and the ball only rolls.
The following equations will determine with accuracy the time
and the place at which this cessation of sliding takes place.
406
[219.
THE MOTION OF A BILLIARD BALL.
Let the plane in which the centre of the ball moves be that
of (x, y); this plane is consequently parallel to that of the table,
and is horizontal; let (x, y) be the place of the centre of the
ball at the time t; so that at that time (x, y, -a) is the place of
the point of contact. Let r be the force of sliding friction which
acts at the point of contact, and let a be the angle at which its
line of action is inclined to the axis of x. Let R (= M g) be the
pressure of the ball on the plane, and let μ the coefficient of
sliding friction; so that F = µ R = µ Mg.
The equations of motion of the centre of gravity are
µ =
d² x
M
F COS a,
dt2
d2y
M
F sin a,
dt2
d² z
M
dt2
Mg + R = 0.
(254)
Let us consider the rotation in reference to a system of axes
originating at the centre of gravity, and parallel to the fixed
axes of (x, y, ≈); then we have
..
::
A
dwj
dt
a F sin a,
dwz
A
dt
=
af cosa,
dw3
A
dt
0;
dwi
A
α Μ
dt
dwz
α Μ
A
dt
(255)
d2y
dt2
d2x
dt2
,
;
(256)
A (@₁₁) =
ам
(dy
vo),
dt
dx
(257)
A (W2 — N2)
α Μ
dt
21%),
Wz2z
0;
where u。, v。 are the axial components of the initial velocities of
the centre of gravity, and 1, 2, 3 are the initial angular velo-
cities about three axes originating at the centre of gravity and
parallel to the fixed axes.
These equations connect the instantaneous angular velocity
220.]
407
THE MOTION OF A BILLIARD BALL.
with the velocity of the centre of gravity of the ball.
Thus w3
is constant, and
dx dy
ω1 and w2 only vary when and
dt
dt
vary; that
is, when Facts.
And therefore if the centre of gravity moves
uniformly in a straight line, the angular velocity of the ball and
the direction of the rotation-axis do not vary, and there is no
sliding friction. And since
2 a²
M;
A =
5
these become w₁ =
5 dy
2adt
LO
5
(dy
dx
2adt
— v。);
— u。)·
(258)
220.] By means of these we can determine the path of the
centre of gravity of the ball, so long as the ball continues to slide.
The line of action of F, as we have said, is that of the motion
of the point of contact. Now the projections on the x- and y-
axes respectively of the space described by this point in the time
dt are dx-awdt, and dy+aw₁dt; and by (258),
7 dx
2
2
dx-awą dt =
— (a₂+500) dt,
dy+ aw₁dt = 7 dy
+(am
(a,
5
0);
dt;
2
And these are proportional to cos a and to sin a, which enter
into (254). To simplify these expressions however let
И1 =
2 (5160 + a2
(500+am),
7 Q7
(259)
V1
2
5º。 — am),
7
so that
dx — a w₂ dt
½ (dx—u₁ dt),
(260)
7
dy+aw₁dt =
(dy—v₁dt);
2
and consequently from (254),
d2x
d²y
dt2
dt2
;
(261)
dx
dy
Из
V1
dt
dt
408
[220.
THE MOTION OF A BILLIARD BALL,
and integrating,
dx
dt
dy
-U1
V1
dt
;
(262)
Ио - из
hence also we have
d² x
d2y
dt2
dt2
;
(263)
U o — U1
and consequently,
Cos a
sin a
;
(264)
Uo-U1
0
M
and thus not only is F, the force of friction which retards the
ball, constant in magnitude, being equal to μ мg, but the line of
action of it has a constant direction. And therefore the centre
of gravity of the ball describes a parabolic path, like a heavy
projectile, the axis of the parabola being parallel to the line of
action of the constant force of friction.
And a is the angle at which the line of action of F is inclined
to the axis of x; and
tan a
0
1
(265)
U1
u₁ and v₁ are given by (259) in terms of initial quantities. They
are the axial components of the velocity of the centre of gravity
of the ball at the instant when it ceases to slide, for then
dx
= aw₂ = 2019
dy
dt
dt
αωι
= √1;
the last terms of these equations following by reason of (260). This
is also evident from (261); for when the ball only rolls F = 0;
and consequently
d² x d2y
0, and
dt2 dt2
dx
dt
dy
U1
=V1·
dt
It is observed, that u, and v, depend not on the friction, but on
the initial circumstances of motion.
Let
(u。 — U₂)² + (vo−1)2=82;
so that (254) become
d2 x
dt2
Uo - Uz
нд
>
S
d² y
μg
dt2
S
(266)
and therefore, if the initial place of the centre of gravity of the
ball is taken as the origin,
221.]
409
THE MOTION OF A BILLIARD BALL.
dx
Uz)
µg (U。 — u₁) t,
Ио
dt
S
(267)
dy
µg (vo — V₁) t ;
t;
dt
S
X =
ut-
µg (u₂ — u₁)
2 s
t2,
(268)
y = v₁t -
µg (Vo — V1) f².
2 s
t2;
and eliminating t,
28
{x (v。 —v₁) —y (u。—u₁) }² +
нд
(u₁vo-v₁uo) (uy-vox) = 0; (269)
which is the equation to the parabolic path of the centre of
gravity of the sphere.
221.] Now this equation becomes y = x, that is, represents
a straight line when u₁, v₁ are respectively proportional to u̸, v。 ;
in which case we have, from (259),
Up Q + Vo Q₂ = 0;
0
(270)
which shews that the initial rotation-axis is in a vertical plane at
right angles to the line of initial velocity of the centre of gravity.
The equation (269) expresses the path of the centre of the
ball, so long as the ball slides as well as rolls; at the instant
however when the sliding ceases, and the ball only rolls,
dx
dt
= иг = a w2,
αω
and from either of (267) we have
S
dy
= √₁ =
U1 - αω;
(271)
dt
(272)
Ꮗ
s Up + U1
нд
2
8 vo + v1.
Y
? =
нд
2
μg
at which time, from (268), we have
(273)
After this instant, if there were no friction of rolling, the ball
would continue to move uniformly in a straight line, with a velo-
city of which u₁ and v₁ are the axial components. But as a
friction of rolling acts to retard it, the ball continues its recti-
lineal course with a decreasing velocity, until it finally comes to
rest; and the equation to the line along which the centre of
gravity moves, and which is also the path of the point of contact
with the table, is
V。 U₁) = 0;
(274)
2 µg {u₁y - v₁ x} + 8 (U₂ V1 — Vo U₁)
this line is evidently a tangent to the parabola at the point given
by (273).
PRICE, VOL. IV.
3 G
410
[222.
THE MOTION OF A BILLIARD BALL.
222.] Thus we have arrived at the exact course which a
billiard ball takes on a table, in the most general circumstances
of a stroke of a cue. The motion is at first a mixed one of
sliding and of rolling; and the centre of the ball moves in a
parabola so long as the ball slides on the table, which causes
a sliding friction at the point of contact; this sliding however
eventually ceases, and before the ball comes to rest; and the
centre of the ball then takes a rectilineal path, which is a tangent
to the parabola at the point where the sliding has ceased.
Let us now give to the equations determined in the preceding
Articles those interpretations which arise from initial circum-
stances produced by the stroke of a cue.
Let q be the momentum with which the cue strikes the ball;
and let a be the angle at which its line of action is inclined to
the plane of the table. Let h be the horizontal distance from
the centre of the ball to the vertical plane containing the axis
of the cue, which is the line of blow; and let k be the perpen-
dicular distance on this line from a horizontal line through the
centre; let us moreover take the plane of (x, z) to be parallel
to the axis of the cue. Let F be the friction which is brought
into action between the ball and the table by the blow of the
cue, and let ẞ be the angle between the line along which r acts
and the axis of x; let µ = the coefficient of friction; then, as
in Art. 205, the equations of motion of the centre of gravity are,
M Uo =
Q cos a-F cos ß,
M VO
0
-F sin ß,
Q sin a + R― Mg ;
(275)
and the equations of rotation about the centre of gravity are
A₁ahsina-ar sin ß,
A 22 Qk+arcos ß,
Ang =Qh cos a.
(276)
2 a² M
Also
F = μR,
A
5
cos B
sin ẞ
(277)
uo - a Qz
vo + a 21
From these equations we have
tan ß
=
5 h sin a
5k-2acosa
(278)
which assigns the direction in which the point of contact of the
222.]
411
THE MOTION OF A BILLIARD BALL.
ball with the plane begins to move.
This also is the direction
of the constant retarding force of friction, and of the axis of the
parabola in which the ball moves until sliding friction ceases.
Hence also we can determine uŋ; V0, 21, 22; also we have
5 Q (a cosa + k)
U1
7 a M
V₁ =
V1
5qh sin a
7 a M
;
(279)
which values are independent of the friction, as we have before
observed in Art. 220.
ß
==
If the path of the ball is rectilineal, and in the direction of
the stroke of the cue, tan ẞ= 0, and v₁ = 0; hence, either
h = 0, or sin a = 0; in the former case, the centre of the ball
is in the vertical plane containing the axis of the cue; in the
latter, the axis of the cue is horizontal. Under either of these
circumstances therefore, and under these only, is the path of
the ball rectilineal; in all other cases the path is parabolic.
If the path of the centre of the ball is not rectilinear, the
value of v₁, given in (279), shews that the deviation lies on that
side of the centre in its initial position, on which the stroke is
given, for V1 and h are of the same sign.
These several circumstances are represented in Fig. 59. o, the
centre of the ball at the instant of the blow, is the origin; and
the horizontal plane containing o is that of (x, y). The plane of
the table is parallel to it, and at a distance below it equal to a,
the radius of the ball. P is the point of contact of the ball with
the table. QRT is the axis of the cue, the vertical plane parallel
to which, and containing o, is the plane of (x, ≈). The circle is
the section of the ball by the plane of (x, z). QRT pierces the
plane of (y,) at R, and the plane of the table at T, and a is the
angle which it makes with the table. OH = h is the perpen-
dicular distance from o to the vertical plane RLT, containing the
axis of the cue; and HK k is the perpendicular from н to the
axis of the cue; so that h, k, and a determine the position of
the line of blow. Let PP, be the curve which is the parabolic
path of the point of contact of the ball with the table; P₁ being
the point given by the coordinates (273), at which sliding fric-
tion ceases, and after which the path becomes rectilineal; let
P₁s, be that rectilineal path; let RL; so that
1
k
(l− a) cos a ;
5 Q l cos a
5 Q l sin a
thus
241
V1
"₁ =
7 а M
7 a M
;
1
(280)
3 G 2
412
[222.
THE MOTION OF A BILLIARD BALL.
therefore
V1
U₁
h
M T
I cot a
M P
tan M PT.
(281)
V1
but is the tangent of the angle between the axis of a and
U1
the rectilineal path taken by the ball when friction ceases. Con-
sequently the rectilineal path P, S, is parallel to PT. Hence the
final direction of the ball when friction ceases is easily deter-
mined; it is parallel to the line drawn from the point of contact
of the ball with the table to the point at which the axis of the
cue pierces the table.
Hence it follows, that if the axis of the cue is inclined to the
plane of (x, y), at an angle so large that the point r falls on the
negative side of the line PL, the ball in its final state moves in a
direction opposite to that of the stroke.
Explanation of these formulæ, in further application of them
to the game of billiards, will be found in "Théorie Mathématique
des effets du jeu de Billiard," par G. Coriolis; Paris, 1835.
223.]
413
RELATIVE MOTION.
CHAPTER VIII.
RELATIVE MOTION OF A MATERIAL SYSTEM.
SECTION 1.—Investigation of the general equations.
223.] IN the preceding parts of this work the motion of a
material system has been investigated relatively to a fixed origin,
and to a fixed system of coordinate axes. The coordinate axes
indeed, to which we have found it convenient to refer the
motion primarily, have not always been fixed; for in the last
Chapter two systems were used, one of which was fixed in the
body and moved with it; the other system however was ab-
solutely fixed, and to it ultimately the motion of the material
system was referred; and its incidents were deduced from the
position of the parts of it relatively to that system. Now I
propose to consider a more general case; and to investigate the
motion of a material system relatively to a moving origin, and
a moving system of rectangular axes, all the incidents of motion
of these latter, as well as those of the material system, being
given, and the material system also moving relatively to it.
This general case is that of relative motion; that is, of the
motion of a material system relatively to moving coordinate
axes, to which allusion has already been made in Vol. III,
Art. 259, in the case of a single particle; under circumstances
however in which a single material particle moves, and the
motion of the rectangular axes is simply that of translation.
As the formule which express the circumstances of relative
motion are long and complicated, it will be convenient to con-
sider primarily the motion of one particle; and thus omit the
signs of summation; but the result will be capable of the most
general application, because, by D'Alembert's principle, they
can, mutatis mutandis, be extended to systems of moving parti-
cles. Our process will be purely analytical. Doubtless hereby
we are in danger of overlooking the full meaning of the
symbols; we may perhaps lose sight in them of the mechanical
truths which they represent; and thus our apprehension of
things in their actual state may be indistinct. Consequently I
414
[224.
RELATIVE MOTION.
shall interpret the equations, and shall shew that they are the
expression of results arrived at from the consideration of relative
motion and its incidents in their first principles, and from (as it
is commonly said) general reasoning. The cinematics of relative.
motion first require investigation.
224.] Let m be the mass of the particle whose motion is
to be considered; and let (x, y, z) be its place at the time t
relatively to an origin and to a system of axes fixed absolutely
in space. At the same time let (xo, Yo, Zo) be the place of an-
other origin, which moves; and relatively to it, and to a moving
system of axes originating at it, let (§, n, §) be the place of m.
Let the system of direction-cosines connecting these two sys-
tems of axes be that of Art. 2; then
x = xo + α₁ § + b₁ n + c₁ 5,
y = y +
1
2
2
2
a₂ § + b₂ n + c₂ 5,
z = 20 +
a3
Cz Š;
α3 § + b 3 n + c 3
(1)
and, as all the quantities in the right hand members are functions
dxo da₁ db₁ dc1
dr
of t,
dx
+&
τη
+5
dt
dt
dt
dt
dt
αξ
+α1 dt
dn
+ b₁ dt
dy
dyo + E
daz
dbz
dc2
2
+8
τη
dt
dt
dt
dt
dt
αξ
+α₂ dt
dz
dzo
daz dbz dc z
αξ
+b₁
d n
+ b₂ dt
+62
น
d n
+ c₁ dt
di
d
+ Cz dt
(2)
αξ
+ &
dt
dt
dt
τη +5
dt
d t
+ a 3 dt
+b3 d t
+C3 dt
Let us examine the several terms of these equations. The left.
hand members are the axial components of the velocity of m
relatively to the three fixed axes of (x, y, z); that is, of the
absolute velocity of m. The first four terms of the right hand
members are the value of these components when έ, ŋ, Ś do not
vary; that is, when the place of m is fixed relatively to the
moving origin and the moving axes; and thus they are the
axial components of the velocity of m considered as a fixed point
of the coordinate system of reference; the first terms of the
several equations being due to the motion of translation of the
origin; and the following three to the rotation of that system
about an axis passing through that origin; and the last three
terms in each equation are the projections of the fixed axes of
the axial components of the velocity of m relatively to the moving
axes and the moving origin. Equations (2) then yield the fol-
lowing theorem:
224.]
415
GENERAL EQUATIONS.
The absolute velocity of the particle m is the resultant of its
velocity relatively to the system of moving axes and of the velo-
city of it, considered as a fixed point of that system; or,
The velocity of a particle relative to a system of moving axes
is the excess of its absolute velocity over the velocity of the
system of axes.
If vε, un, vg are the components of the absolute velocity of m
along the moving axes; that is, along the §-, 7-, (-axes respect-
ively; then
ريد
dx
dz
+αz
+ az dt
+α z dt
2
}
dc z + az at S
dc ₁ + α ₂ dt
+α2
dt
} &
+
; (3)
dt
dy
V
= α1 dt
+az dt
dzo
0
α1
dxo
a ₁ d t
+az dt
dyo
+ az dt
+n{
db₁
db z + as dt S
db3
α1
dt
dc 3
₁
dyo
+
dzo
b3
dt
dc3
dt
+ 5
${0,
dt
da1
+ &
b₁
2
da3
dn
+b3
+
(4)
dt
dt
νη
b₁
dxo
+adt
+ b₂ d t
dt
+Ŝ
dcz + bz dt S
dc₁ + b ₂ dt
2
b ₁ d t
+ b₂
dyo
dao + Cz dt
2
&
{
dt
daz + bz
+ b₂ dt
dzo
+ C3 dt
da3
daz + C3 dt S
day + C₂ dt
C ₁ d t
+ } }
C1
C2
dt
+ n
{
db₁
dbɔ
db3
dš
C1
+ Cz
dt
dt
+ C3
dt
+
(5)
dt
in which equivalents for ve, vn, vg all the terms except the last of
each arise from the motion of the coordinate system of refer-
ence, and the last arise from the motion of m relatively to that
system.
If we wn, we are the axial components of the angular velocity
of m at (§, n, Ŝ), considered as a point fixed relatively to the
moving system, then, substituting from (88), Art. 40, we have
oğ
= α1
Ψη
25
dxo
dyo
+ a2 dt
a ₁ d t
dxo
+
b₁ d t
dxo
C 1 d t
dyo
b2
2 dt
dyo
+ Cz dt
dzo
+ az dt
ვ
dzo
αξ
+ Swn-nws + dť’
+ bz dt
+ Śwÿ−Śwę +
0
dzo
+ C3 d t
+nwę - {wn +
dt
dn
(6)
dt
ας
dt
416
[225.
RELATIVE MOTION.
And thus along the axes of the moving system the axial compo-
nents of the absolute velocity of m are the resultants of the axial
components of (1) the absolute velocity of the origin, (2) of the
velocity of m at (§, n, Ŝ) due to the angular velocities of the
coordinate axes, (3) of the velocity of m relative to the moving
axes.
If the origin of the moving system is fixed, and the axes only
move, these become
0 = ζωη -η ως +
vn = &ws — Św§ +
0 10
v5 = n wε - & w₁ +
ως ξωή
αξ
dt
dn
dt
dr
dt J
(7)
255.] Suppose now that forces act on the system so that the
absolute velocity of m varies, then, taking the t-differentials of
(2), we have
d2x
d2xo
+ §
d² α1
d2 b
d2 C1
+ n
+8.
dt2
dt2
dt2
dt2
dt2
+2
{
Sd § da1
dŋ db₁
+
+
d5 dc}
dt dt
dt dt dt dt
d2&
d² n
2
2
+ b₁
+ C1
dt2
dt2
dt2 (8)
d²y
d² yo
+ {
d² a z
d² b₂
2
d² cz
+ n
+5
dt2
dt2
dt2
dt2
dt2
d & daz dŋ
db₂
+2
+
+
dt dt dt
dt
d²¿
+ az dť²
dt2
dr de
dt dt S
d²n
+ b₂ dť²
+ Cz
2
d² Č
dt2
•
(9)
d2 C3
d2z
dt2
11
dt dt
d2a3
+ n
dt2
d2zo
+ &
dt2
+2
{ de das
d² b 3
+Ŝ
dt2
dt²
dn db
dc dc,
+
+
dt dt
dt dt S
d² E
d2n
d² ¿
+ a z
+b3
dt2
+ C3 dt²; (10)
13 dt2
which are severally the equivalents of the expressed velocity-in-
crements along the fixed coordinate axes in terms of the elements
of the moving system of axes.
Let ve, vn', vg be the axial components of the expressed velo-
city-increment along the moving axes of έ, ŋ, §, then
225.]
417
GENERAL EQUATIONS.
d² z
d c₂ ) d &
λας
dc3
dt dt
d¢
+ dtz; (11)
2
d2 x
d2y
v' & =
а1
+az
dt2
dt2
+ αg dt ²
d² xo
d2
d² α1
а1
d2 bi
d2c1
а 1
+ &
τη
+5
dt2
dt2
dt2
dt2
+ a2
yo
dt2
+az
+ 2
ईक
{ days + 8
d2 Zo
dt2
db₁
d² az
d² ba
d2 ca
2
+§
+n
+8
dt2
dt2
dt2
}
d² az
d2 b₂
d2 C3
+ §.
τη
+5.
dt2
dt2
dt2
dt
+aq d t
db2 dbz dn
+ as dt S dt
3
}
+2
{a
dc1
dc2
+α2
+az
dt
dt
d2x
d2y
d2 z
v'₂ =
b₁
+ b₂
dt2
dt2
+b 3 d t ²
b₁
{
d2 xo
d² α1
d2 b₁
d2 C1
+ &
τη
dt2
dt2
dt2
1
+5
dt² S
+ b₂
(d² Yo + {
d² az
d2 b₂
d2 Cz
τη
+5.
dt2
dt2
dt2
d2
d² az
d2 ba
d2 C3
3
+&
+5
dt2
d12
dc₁
+b₂
dcz
+2
{
da1
dt
11
11
63 {{
+ b₂
dt2
d2
de zo
dt2
+ 23 b₁
1 dt
dt
2
τη
dt2
d c₂ ) d Ś
+63
3
dis
dt dt
da
+ b₂ dt
{b₂ a + b²
+ 2 3 b₁
d2y
dt2
C1
C1
+ C₂
+ C3
dt2
{
{
{
2
d² xo
dt2
d² yo
dt2
d² Zo
+ §
d² z
+ C3
dt2
1
d² α1
d2 b₁
τη
dt2
dag λαξ
+ b s d t d t
+5.
d2 C1
dt²
d2 C2
d2n
+ ; (12)
dt2
d² x
+ C₂
+ &
dt2
dt2
d² az
τη
dt2
2
d2 b₂
+5
dt2
d² az
+ &
dt2
dt2
+ 2
{
da₁
C₁
dt
daz
+5.
daşı d§
3
+C₂ + C3 dt
dt
d2b3
d² C3
+ η
dt2
dt2
dt
1
+2
(db₁ db 2
+Cq
dt
dt
dbs I dn
+C3
dt
d² C
3
at
+
dt
dt2; (13)
vš
Let x, y, z be the axial components of the impressed velocity-
increments on m parallel to the fixed coordinate axes of x, y, z;
PRICE, VOL. IV.
3 H
1
418
[226.
RELATIVE MOTION.
and let x', y', z' be the axial components of the same impressed
velocity-increments parallel to the moving axes of έ§, 1,5; then
x'
x² = α₁ x + α₂ x +αz Z,
Y =
3
b₁ x + b₂ x + bz z,
z =
C₂ X + C₂ Y + C3 Z ;
and the equations which express the motion of m are
ón
v' § = x'; v'₁₂ = Y',
ύς
v's = z' ;
(14)
and from these equations all the incidents of its motion are to
be deduced.
226.] Before however they are applied to the solution of
particular problems, let us examine them more closely, and detect
the origin and the meaning of their several terms. This course
has already been taken by Clairaut and Coriolis*; and has been
fully explained by M. Bertrand (Journal de l'Ecole Polytech-
nique; XXXII Cahier, p. 149), so that little need be added to
their labours. In the values for v't, v'n, v', given in (11), (12),
and (13), it is evident that if the coordinate system of $, n, s, to
which the place of m at the time t is referred, were fixed and
immoveable, then the several right hand members would be
d² į d²n d² §
reduced to their last terms
ης
dt²' dt' dte; and the equations of
motion would be for m the same as those which have been found
in Vol. III; and for a material system, the same as those which
are explained in chapter III of the present volume; conse-
quently all the other terms arise from the motion of the coor-
dinate system of reference. If therefore we consider the motion
of m, relatively to the moving system of reference, to be abso-
lute, just as motion has heretofore been considered and referred,
then we must suppose m to be under the action of certain ficti-
tious forces, as well as the actual forces, by which certain velo-
city-increments are impressed exactly equal to those which arise
from the motion of the system of reference, and of which the
mathematical expressions are given in the preceding equations.
This is the view of the subject taken by Clairaut, Coriolis, and
Bertrand. Now the velocity-increments which arise from these
fictitious forces are of two kinds; (1) we have those which are
Journal
* See Mémoires de l'Académie des Sciences de Paris, 1742, p. I.
de l'Ecole Polytechnique; XXI and XXIV Cahiers. Traité de la Mécanique
des corps solides et du calcul de l'effet des Machines par G. Coriolis; 2nd
edit., Paris, 1844, p. 46.
226.]
419
GENERAL EQUATIONS.
independent of the motion of m relatively to the moving origin
and axes; and which therefore are independent of the t-differ-
tials of έ, n, C. Coriolis has called the force whence these arise,
"force d'entrainement." We shall call them the velocity-incre-
ments of transference; meaning of course the transference of the
coordinate system. These velocity-increments are contained in
the first twelve terms in the right hand members of (11), (12),
(13), severally. Let them be denoted by Xt, Yt, Zt respectively;
so that
Xt =
I d² xo
d² α1
d2b₁
d² c₁ l
C1
α1
+ &
τη
+5
dt2
dt2
dt2
dt2
Yi =
+ a2
+аз
{
d² yo
d² az
d2 b₂
d2 C2
2
+§
dt2
dt2
+ n d t²
+5
dt2
{
d2zo
d2 az
d2 b₂
d2 C3
+8
+n
+5
;
(15)
dt2
dt2
dt2
dt2
b₁
S
d² x
0
d2 α1
d² b₁
d2 C1
+8
τη
+5
dt2
dt2
dt2
dt2
d2yo
d² a z
аг
d2 ba
d² Cz
+62
+Ś
+n
+8
dt2
dt2
dt2
dt2
+63 {
{
d² zo
d² a 3
d² b3
d2 C3
+ &
τη
+5
;
(16)
dt2
dt2
dt2
dt2
S d² xo
Хо
d² α1
d² b1
d2 C1
С1
Zt =
€1.
+ &
十八
​+5
dt2
dt2
dt2
dt2
+ C₂
d2yo
dt2
+ §
d2 a
a2
d2 b₂
d² cz
十八
​+5
dt2
dt2
dt²
2
+C3
{
2
d² Zo
d² α3
d2 b3
d² C3
+{
τη
+5
dt2
dt2
dt2
dt2
}
;
(17)
(2) we have the groups which stand fourth and fifth in (11),
(12), (13) respectively; let us express them in terms of the
angular velocities of m at (§, n, () about the three moving coor-
dinate axes, by means of the equivalents given in (88), Art. 40;
they become respectively
2{
23 wn
dt
dr δηλ
dt S
ως
2}{
αξ
2305 dt
ως
d cr
ωξ
2
dt S'
{
d n
αξι
WĘ
Θη
dt
dt S
; (18)
The form of these quantities shews that they are the axial com-
ponents of a certain velocity-increment; this we proceed to deter-
mine; and omitting the numerical coefficient, we will denote it by
F; let a, ß, y be the direction-angles of its line of action; then
F=
Θη
ας dn
αξ dc
ως
ως
dt
기
​dt
dt
COS a
ως
cos B
dt
dn
ωξ
dt
αξ
wn d t
cos y
; (19)
3 H 2
420
[227.
RELATIVE MOTION.
therefore
wę cos a+w, cos ẞ+w¿cos y = 0;
COS a
αξ
dt
+ cos 6
dn
+ cos y dt
dt
ας
= 0;
(20)
(21)
which shew that the line of action of F is perpendicular both to
the rotation-axis of the system of reference which passes through
the moving origin, and to the line of the relative velocity of m.
Let be the angle between these lines; let o be the angular
velocity of the coordinate system, and let v be the velocity of m
relative to the moving system; then from (19) we have
F =
w² v² - (
de
W&
dt
dn
+ ως
d
d & 2) t
dt
dt
{w² v² — w² v² (cos 0)2}
= wv sin 0.
(22)
Thus F is the product of the angular velocity of the moving
coordinate system about its rotation-axis, and the projection of
the relative velocity of m on a plane perpendicular to this rota-
tion-axis. And the line of action of this force is perpendicular
to the rotation-axis of the system, and to the line of the relative
velocity of m. F has been called by Coriolis the compound cen-
trifugal force, and in foreign treatises on mechanics is frequently
cited by that name.
Thus the equations of relative motion of m, given in (14), become
dÆ
x' — xt — 2 F cos a
0,
dt2
d²n
Yt
J
YY 2 F cos B
0,
(23)
dt2
d2r
z' Zt
ZZ 2 F COS Y
= 0
0 ;
dt2
in which form the equations are most conveniently applied to
the solution of problems.
227.] On comparing these with the equations which express
absolute motion, it appears that the axial components of the
impressed velocity-increments are to be diminished by similar
quantities, which we may suppose to arise from two fictitious
forces; one of which impresses on the particle a velocity-incre-
ment equal to that of the particle of the moving coordinate
system, which has the same place as m at the time t, and by
virtue of which the particle has no relative motion; the other
impresses a velocity-increment equal to the product of the in-
228.]
421
MOVING AXES IN ONE PLANE.
stantaneous angular velocity of the moving coordinate system
and the projection of the relative velocity of the particle on a
plane perpendicular to the rotation-axis, and the line of action of
which is perpendicular to the rotation-axis of the system and to
the line of the relative velocity of m.
w
These fictitious forces are not given in the same way as the
ordinary acting forces are given; they depend on the relative
motion, inasmuch as they involve o and v; thus the problem is
in its general form extremely complicated: the solution of a few
particular problems will indicate the mode of their application
better than any general remarks.
If the motion of the system of coordinate axes is that of
translation only, then w=0, and the second fictitious force of
course vanishes; in this case, the axes of the moving system will
be, or may be taken to be, always parallel to those of the fixed
system, so that there will be no t-variations of the nine direc-
tion-cosines; and all the direction-cosines will vanish except a₁,
b, and c, each of which becomes equal to unity; and the equa-
tions (23) become
x'-
dexo
dt2 dt2
d²¿
2
0,
d2yo
d²n
y' -
0,
dt2
dt2
z'
d³zo
dt2
d²¿
0
= 0
0 ;
dt2
(24)
which are the equations already found in Art. 259, Vol. III.
These are sufficient for the determination of the relative motion
of a planet, and have been applied to that problem in Arts. 293
and 295, Vol. III.
If moreover the moving origin travels along a straight line
with uniform velocity, then also
d² xo
dt2
d2yo
dt2
d2zo
dt2
; and
the equations (24) have the same form as those which express
the ordinary absolute motion. It is evident however from (16),
which express the relative velocities, that these will not be the
same as the absolute velocities; and consequently some quanti-
ties will be introduced in the integration-process which depend
on the elements of the line of motion and on the velocity of the
moving origin.
228.] One particular form of the preceding general results
requires especial mention; viz., that in which the origin moves in
422
[228.
RELATIVE MOTION.
the plane of (x, y), and the moving axes of (§, n) are always in
that plane, so that the axes of ≈ and are always parallel. The
equations which express this motion might of course be de-
duced from the general equations (23); but it is convenient to
derive them from first principles, because the origin of the
several terms and the meaning of the fictitious forces of which
we have spoken will thereby become more palpable.
At the time t, let (x, y) be the origin, see Fig. 60, and let
be the angle at which the g-axis is inclined to that of x; so that
x = έcos
xo + έ cos 0 − n sin 0,
y = y。 + έ sin ◊ + n cos 0,
2 =
Š;
(25)
therefore
d Ꮎ
dx
dxo
αξ
dn
dt
dt dt
+ cos
sin ◊ — (§ sin 0 + ŋ cos ◊)
dt'
dt
(26)
dy
dyo dε
do
dn
dt
+
dt dt
sin 0 +
cos 0+ (§ cos 0—ŋ sin 0)
dt
dt
Let v and v, be the components of the absolute velocity parallel
DE
to the axes of έ and ŋ: then
dx
dy
cos 0 +
sin 0
*±
dt
dt
dxo
dyo
do
αξ
(27)
cos 0 +
sin 0 -- ~
+
dt
dt
d t
dt
dx
dy
sin 0 +
cos
V.
η
dt
dt
dxo
dyo
de dn
o sin 0 +
cos 0 + § +
(28)
dt
dt
dt dt
in which values of
VE
and
On
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 t-differentials of (26): then
2
d2 x
dt2
d2 xo
dt2
0
2
(de sin e
dn
+
COS
dt
30)
de
dt
(Écos 0 — n sine) (de)
d² 0
d &
d²n
— (έsin◊ +ŋ cos)
+
cos
dt2
dt2
day
2
dt2
d2yo
dt2
αξ
dn
do
+2
cos e
dt
dt
sin )
dt
d² 0
+(έcose-n sin 0)
dt2
sin 0; (29)
- (¿sine+n cose) (de)
d²¿
+ sin 0 +
dt2 dt2
do 2
dt
d² n
cos 0; (30)
dt2
229.]
423
MOVING AXES IN ONE PLANE.
હૃ
n
Let v' and v', be the axial components of the expressed velocity-
increment parallel to the axes of έ and n: then
2 1 d
η
d² E
d20
η +
dt2 dt2
Ꮎ d°
+
at (nado)
dt
V'E
d² x
dt2
d2y
cos Ꮎ +
sin
dt
d² xo
d2yo
dn do
cose +
_ sin0–2
dt
dt2
dt dt
do 2
dt
d² xo
d² yo
cose +
sin 0 — §
||
dt2
dt2
dt
(d) 3
d2x
d2y
v'n:
sin 0 +
cos (
dt2
dt2
d2 xo
xo sin 0 +
dt2
d² yo
dε do
30
cos 0 + 2
n(at)
dt2
d² xo
sin 0 +
dt2
d² yo
cos 0
50 - n
dt2
(de)
of which expressions for v'
E
dt dt
do 2
dt
do 2 1 d
dt + § dt
2
dt dtz; (31)
+ §.
2
d20 d²n
+
dt2dt2
d²n
+ ε at (εado)
+
dt dt2
and v', all the terms except the last
η
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.
229.] On examining equations (31) and (32), 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
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 Vol. III, Art. 256. The axial components of
the expressed velocity-increment of m at P are the sums of the
components of the expressed velocity-increments of equal par-
ticles at L and N, which are the projections of P on the axes of έ
do
and n; for as is the angular velocity of these axes about the
dt
point o',
the radial component of L
; (32)
d²¿
do 2
dt2
dt
1 d
the transversal component of L =
& dt
(53 do)
the radial component of N
d2n
dt2
η
1 d
the transversal component of N =
n dt
(7² do)
2
dt
(20)3
Ꮎ
dt
dt
;
1
424
[230.
THE RELATIVE MOTION
and consequently of the expressed velocity-increment,
η
dt
d² &
the έ-component
dt2
ε (do)
2 1 d
d²n
do 2
the n-component=
ๆๆ
+
dt2
dt
l d
(n² do), (33)
Ꮎ
(52 do
§ dt dt
:).
(34)
Let x' and y' be the axial components along the moving axes of
the impressed velocity-increments at the time t: then the equa-
tions of motion are
dt
&
d² ε d² xo
+
dt2 dt2
d² yo
dt2
2
cos 0 + sin 0 - € (19) -
d2n d² xo
sin 0 +
cos —
η
dt2
dt2
dt
(do)
+
d2yo
dt2
1
d (120) = x,
ndt
1 d
2
& dt
dt
(35)
(£₂
de)
do) = x'.
Y.
(૪
2ع
n
These are the general equations of relative motion of a particle
moving in the plane of (§, n); and by means of them έ and ŋ
may be expressed in terms of t and known quantities. If t is
eliminated from these last two equations, the resulting equation
in terms of έ and ʼn will be that of the relative path in which the
particle moves. The equation to the absolute path will be found
from (25), when § and ʼn are expressed in terms of x and y.
If the origin of the moving axes does not move, and the axes
revolve with an uniform angular velocity w, then (35) become
dn
dt
d² &
dt2
d2n
αξ
w²n+2 w
ས ་
dt
dt2
(36)
These equations however refer to a very special case of the
general motion.
SECTION 2.-The relative motion of a material particle.
230.] Although our object is the discovery of equations which
represent the relative motion of a material system, and the in-
vestigation of the preceding equations which apply to a single
material particle has been subordinate to that object; yet it is
desirable to shew their applicability to the solution of certain
problems in which only a particle moves; because we shall
hereby obtain a clearer insight into the several parts of them,
and shall also solve some problems of considerable interest.
Let us first take the case of a relative constrained motion;
viz., that of a particle moving within a tube, which also itself
230.]
425
OF A MATERIAL PARTICLE.
moves, and carries with it the system of axes to which it is
referred; I shall consider the tube to be perfectly smooth, so
that it offers no resistance to the moving particle in the direction
of its motion; but presents a reaction acting along the normal,
if the tube is a plane curve; and along the principal normal, if
it is a curve of double curvature; and I shall also consider the
small bore of the tube, and the size of the particle, to be such
that the particle may exactly fill the tube.
In the most simple form of the problem the tube is a plane
curve lying in the plane of (x, y), and rotating with a constant
angular velocity w, about the axis of (or z, the origin of the
moving axes being fixed at the fixed origin, and no external
force acts on the particle. Let R be the normal reacting pres-
sure of the tube against the particle; let m be the mass of the
particle, and ds = the length-element of the tube at the point
(§, n), which is the place of m at the time t; then the equations
of relative motion are
d² &
dn
R dn
w² §-2 w
dt2
dt
m ds
d2
η
-w²n+2w
αξ
R αξ
= +
dt2
dt
m ds
(37)
From these equations the following general theorems are de-
duced;
dε d² & + dn d²n
dt²
w² { § d§+n dn} = 0;
let ²+n² = r², so that r is the distance of the particle from the
fixed origin at the time t; then, assuming the relative velocities
of the particle to be v and v。, when the distances of its place
from the origin are respectively r and ro, this equation gives by
integration
v² — v² = w² (p² — ro²);
2
and thus assigns the relative velocity of the particle.
(38)
Also from (37) we have
dŋ d² § — d§ d²n
ds2
R
- w² (§ dn−n dε) — 2 w
+
ds;
dt2
dt
m
now if p the radius of curvature of the curve of the tube at
the point (§, n), and p = the perpendicular on the tangent from
the origin, this becomes
R v2
+
+ w²p-2wv;
m
P
which assigns the pressure on the tube.
PRICE, VOL. IV.
3 I
(39)
426
[231.
THE RELATIVE MOTION
231.] The following are simple problems in illustration of
these equations.
Ex. 1. A particle is placed within a rectilinear tube which
revolves with a constant angular velocity about an axis which
intersects it at right angles: it is required to determine the
motion of the particle.
Let the line of the tube be the moving f-axis; so that always
η 0; let = the constant angular velocity of the tube. Then
(37) become
d² §
2
dt2
- w² § = 0,
dε
R
2 w
+
dt
ทาง
Let us assume the particle to be at relative rest at a distance a
from the origin, when t=0; then from the first of these we have
dt
R
+
2
= w² (§² — a²);
a
{ = = {ewt +e-wt};
2
= aw² {ew! -e-wt};
and if is the angle described by the tube during the time t,
then = wt; and the relative equation to the path of the par-
0
ticle is
a
Ө
ε = {e° + e~°} .
2
If r is the absolute radius vector, r =
έ, and
a
グ
​2
{eo + e-0};
and this is the absolute polar equation of the path of the
particle.
Ex. 2. A particle is placed within a rectilinear tube which
revolves about an axis intersecting it at right angles, with an
angular velocity such that the tangent of the angle described
in a given time is proportional to the time: it is required to
determine the motion of the particle.
Let, as in the preceding problem, the line of the tube be the
-axis, so that always n = 0; let o be the angle through which
the tube has moved in the time t; then, by the conditions of
the problem, if 00, when t = 0, and if k is a constant,
and (35) become
tan 0 =
Jet ;
231.]
427
OF A MATERIAL PARTICLE.
d?
dt2
1 d
do 2
do
dt
Ꮎ
& de (Erdo)
dt dt
=
0,
= +
R
m
Let us eliminate t by means of the first two of these three
to be an equicrescent variable; then we
equations, and take
have
d²¿
dε
- 2 tan 0
§ = 0;
d02
do
d² &
αξ
cos
- 2 sin 0
–
È cos 0 = 0.
do 2
do
When t=0, let us suppose the tube to lie along the x-axis, and
the particle to be projected with a velocity u along the tube
from a point whose distance from the origin = a; then from
above, by integration, we have
dε
и
cos 0
- έ sin 0 =
d Ꮎ
k
И
Ꮎ ;
k
έ cos 0 = a +
which is the relative equation of the path of the particle. And
in reference to the fixed axes of a and
x = a + Ꮎ ;
k
y,
which gives the absolute path of the particle.
If we substitute from above in the third of the preceding
equations, we have
R.
+
2 uk (cos 0)³.
m
Ex. 3. A circular tube revolves uniformly about a vertical
axis which is perpendicular to its plane and passes through a
point in its circumference: it is required to determine the rela-
tive motion of a particle within the tube.
Let r and p refer to the place of the particle at the time t;
and let a be the radius of the circular tube. Then the equa-
tions to the tube are r = 2 a cos 4, and 72
p, r² = 2 ap;
ds
аф
2 a
;
dt
dt
and therefore from (38) we have
4a² (dd)
аф
dt
2
2
vo²+4a² w² (cos )2-2 r2;
1
(40)
312
428
[231.
THE RELATIVE MOTION
which equation, being integrated, would give 4, and conse-
quently r, in terms of t. The integral however can only gene-
rally be expressed as an elliptic function.
But if v。wro, then
аф
so that
dt
аф
= ∞ cos &;
dt
= w, when 4 = 0; in which case
1 + sin &
log
= 2 wt;
1-sin
sin o
(41)
the limits of integration being such that t=0, when = 0;
that is, when the particle is in that point of the tube which is
directly opposite to the origin;
1
ewt
sin
=
e-wt
ewt +e-wt'
4a
;
ewt +e-wt
(42)
which determine the place of the particle at the time t relative
to the moving axes of the circular tube.
When t∞, &
=
П
ф ; so that the particle falls into the origin
2
only after an infinite time.
For this particular value of the constant we have, from (39),
R
Ө
+ = 2 a w² cos ◊ (3 cos ◊ — 2).
m
(43)
Ex. 4. A particle is placed within a thin tube which is of the
form of an equiangular spiral; the tube revolves with an uni-
form angular velocity about a vertical axis passing through its
pole, the plane of the tube being horizontal: it is required to de-
termine the relative path of the particle.
Let a be the constant angle at which the tangent is inclined
to the radius vector at every point of the curve. Then
p = r sin a,
drds cos a,
are equations to the curve.
And (38) gives
Let
2
2
2
(dr) ³ (sec a)² = w² r² — w² r¸² + v²
dt
w² r 2 -v ²
2 = k²;
then if t = 0, when rk, this equation gives by integration
k
2
2° {etcosa +e-lcosa}
which assigns the relative motion of the particle in the tube.
233.]
429
OF A MATERIAL PARTICLE.
232.] For an example, in which the origin of coordinates
itself moves, let us consider the motion of a particle within a
circular tube, which revolves about an axis through its centre
perpendicular to its plane with an uniform angular velocity w;
and the centre of the circular tube also describes a circle in
the plane of the tube with an uniform angular velocity a.
Let us suppose the particle and tube to be situated at the
time t, as they are placed in Fig. 61; wherein AQB is the circle
in which Q, the centre of the circular tube, moves, and P is the
place of the particle. Let us also suppose the centre of the tube
to have been at A, on the axis of x, when t O, and the particle
at that time to have been at relative rest at c, c being at c' at
the time t. Let oA α, AC
c; then
= a,
2
x² + y² = a²,
2
§2
{² + n² = c2,
(44)
and we have also
0 =
(2+w) t ;
x。 = a cos at,
Yo = a sin at;
}
(45)
from (35), we have
d² &
dt2
dn
R dn
a n² cos w t − (n + w)² § − 2 (î+w)
= +
dt
m ds
(46)
d² n
αξ
R de
αξ
+ a n² sin wt — (î + w)² n + 2 (2 + w)
+
dt2
dt
m ds
which are the required equations of motion, and do not gene-
rally admit of integration.
233.] Another simple problem, to which the preceding prin-
ciples are applicable, is that in which a heavy particle moves in
a smooth tube, which rotates about a vertical axis with an uni-
form angular velocity.
Let us first suppose the tube to be of single curvature, and
the axis about which it rotates to lie in its plane; let this axis
be taken for the axis of z; and let the positive direction of it
be measured from the origin in a direction opposite to that of
gravity. Let this axis also be taken as the (-axis to which the
curve is referred, and let a perpendicular to this line through
the origin be the -axis; so that the equation to the curve
of the tube is
$ = ƒ (§).
Let be the constant angular velocity with which the tube
430
[234.
RELATIVE MOTION OF A HEAVY PARTICLE
t = 0.
Then
x = &cos wt,
rotates, the plane of the tube being in the plane of (z, x) when
y = έsin wt,
= Š;
d2 x
gº Ê
dε
= coswt
2w sin ot
w² & cos wt,
dt2
dt2
dt
d²y
d² E
αξ
sin ot
+2w coswt
w² & sin wt
dt2
dt2
dt
d2z
d25
dt2
dt2
d² 2; w² & ;
dt2
and consequently the equations of motion are
d°
&
R d
dt2
w² & = +
m ds
d²¿
R d E
+
dt3
m ds
9;
²)
(47)
v² — v² = w² (2-2)-2 g (5—50).
If the velocity is given in terms of the coordinates of the
place of m at the time t, the equation of the curve of the tube
may be found; and if the equation of the curve of the tube is
given, the velocity of m and the other circumstances of motion
of it at any time may be found.
234.] The following are problems solved by means of these
equations.
Ex. 1. Determine the motion of a heavy particle within a rec-
tilinear tube which describes with an uniform angular velocity
the surface of a right cone, the axis of which is vertical.
Let a = the semi-vertical angle of the cone, and let r be the
distance of m at the time t from the vertex of the cone: then
= r sin a,
and substituting these in (47)
= r cosa;
Š=
we have
sin a
{
d2r
dt2
= +
Ꭱ
m
cos a,
d²r
R
cos a
+
sin a - 9,
dt2
m
d² r
dt2
(w sin a)²r+g cos a = 0.
Let us suppose m to be projected from the vertex of the cone
dr
with a velocity u; so that
dt
= u, when t = 0, and r = 0; then
234.]
431
IN MOVING TUBES.
from the above by integration we have
therefore
dr 2
1
dt
u² - (w sin a)2 r2 + 2 gr cos a = 0,
g cos a
+
w² (sin a)²
uw sin a-
-gcos a
2 w² (sin a)2
ewt sin a
uw sin a+g cos a
2 w² (sin a)2
e-wtsina;
which equation determines the circumstances of motion of the
particle.
If u is greater than
g cot a
r increases without limit as t in-
W
creases; and thus m moves along the tube farther and farther
from the vertex.
If u =
g cot a
,
W
as t increases, increases; and when t
becomes infinite, » =
g cos a
w2 (sin a)2
so that m moves farther from
the vertex along the tube, but never passes the limit assigned by
this value of r.
If a=90°, we have the same results as those of Ex. 1, Art. 231.
Ex. 2. By means of these equations we may also determine
the position of the two heavy balls in Watt's centrifugal go-
vernor of the steam engine. The arrangement of this contriv-
ance will be understood from Fig. 62, where o is the fixed
point on the vertical axis at which the rods carrying the heavy
balls cross each other. We shall take the plane of the rods and
the balls to be that of (§, †) and shall take the vertical line
drawn downwards from o to be the positive direction of (: we
shall assume the weight of the rods to be so small in comparison
of that of the balls, that the former may be neglected without
sensible error. Let op, the length of the rod, = a; and let its
inclinations to the vertical be 0 and 0。 at the times t and 0
respectively; and let us suppose 0 to be greater than 0。
and
d Ꮎ
dt
to be zero, when t=0; let o be the angular velocity with which
the plane containing the balls and rods rotates about oz; and
let T be the tension of the rods. Then the equations of motion
are
d² &
dt2
T
— 'r sin 0,
d²¿
dt2
T cos 0g;
432
RELATIVE MOTION OF A HEAVY PARTICLE
[235.
d
2
a²
= w² (§² - ²) + 2g (5—50);
(de)² = a²w² {(cose,)² — (cos 8)²} +2ga {cose —cos@o},
dt
a
(20)
dt
dt
2
= (cose-cos e) { a w² (cos o + cos 0)-2g},
a de
{cos 0。-cos} {a w² (cos o + cos 0) — 2g}
Σ
in which equation the variables are separated; and if the inte-
gration can be effected, will be given in terms of t.
ᏧᎾ
0, when (1) 0
00; (2) cos 0
dt
2g
a w2
cos 00;
so that
varies between the angles given by these two limits.
Let wo be the angular velocity with which the plane of the
balls revolves, when the angle at which they are inclined to the
vertical axis is 。, and does not vary. In this case, as they have
no change of § or (, the preceding equations of motion give
so that
dt
2
g wo² a cos 0;
do
0
2
(cos 0。 — cos 0)³ { (w² — 2 w²) cos 0 + w² cos 0}½
and if OB = BO = b; so that oQ = 2b cos 0,
d Ꮎ
d. oq =
26 sin 0
;
dt
which assigns the vertical displacement of a due to the change
of angle of inclination of op to the vertical.
235.] Next let us suppose the curve of the tube in which the
particle moves to be of double curvature; and to rotate about
the z- or (-axis with the same constant angular velocity w; then
the equations given in (35) are true, when a。= y。=0, and 0 is
replaced by wt; and we have
d² §
dn
R
w² § – 2 w
cos a,
dt2
dt
m
d2n
dε
Ꭱ
w²n + 2 w
cos B,
(48)
dt2
dt
m
d2
2‹
R
cos y g;
dt2
m
wherein a, ß, y are direction-angles of the principal normal at
the point (§, 7, 8).
236.]
433
ON A MOVING SURFACE.
From these equations we have
2
v² — w² (§² + n²)
2g+c,
where c is a constant depending on the initial values of the
several quantities.
If we suppose the relative velocity of the particle to be con-
stant or to be zero, then the preceding equation expresses a
paraboloid of revolution; and we infer that the curve traced in
any way on a paraboloid of revolution satisfies the given con-
ditions.
236.] Lastly, let us consider the motion of a heavy particle
moving in contact with a surface which rotates with an uniform
angular velocity about its (-axis, which is vertical and is the
fixed z-axis.
Let the equation to the surface be
F (ε, n, &,) = 0;
(§, Ś,)
of which let the partial derived functions be u, v, w; also let
U² + v² + w² = q²;
then we may suppose the particle to move in a thin space con-
tained between two parallel surfaces infinitesimally near to each
other; in which case the equations of relative motion are
d²§
dt2
dn
RU
20
dt M Q
d² n
dε
w²n + 2 w
11
dt2
dt
R V
M Q
Dio Dio
d² ¿
dt2
R W
g;
m Q
if we multiply these respectively by d§, dŋ, and d§, and add and
integrate, we have
v² — w² (§² + n²) = −295+c;
where c is a constant depending on the initial values of the
several quantities.
If v is constant, or if the particle remains at rest wherever it
is put so that v = 0, this is the equation to a paraboloid of re-
volution.
If v varies as the distance from the fixed origin, the surface is
a surface of the second degree.
PRICE, VOL. IV.
3 K
1
S
434
[237
MOTION OF A HEAVY PARTICLE
Suppose the surface to be a plane in which the z-axis is; let
us also assume the f-axis to be in the plane, so that always n=0;
then the equations of motion are
d2ε
dε
R
d2t
- w² = 0,
2 w
g;
dt2
dt
m
dt2
from which the motion may be easily determined.
237.] The principles and equations of the preceding articles
are applicable to the solution of a problem of considerable in-
terest; viz. to the motion of a particle, either free or con-
strained, near to the earth's surface, relative to a system of axes
originating on the earth's surface and moving with it.
We may without error assume the centre of gravity of the
earth to be fixed, if we impress forces on the moving particle
which are equal to the excess of those which act on it over those
which act on the earth at its centre of gravity: but as the sun,
which is the main force acting on the earth, impresses velocity-
increments nearly equal on both the earth and the particle, we
may suppose this excess, either positive or negative, to be so
small that it may be neglected without sensible error. We may
also suppose the position of the rotation-axis of the earth to be
fixed and the angular velocity to be constant.
The two systems of axes are imagined to have that arrange-
ment which is drawn in Fig. 63. o is the centre of the earth
the axis of z is measured from o towards c the north pole; the
axes of x and y are taken in the plane of the equator. Let o be
the angular velocity of the earth, with which indeed the earth
rotates from the y-axis to the x-axis: it will be convenient how-
ever to take it at present in the contrary direction, and to change
the sign in the final equations ere we apply them to the par-
ticular problem.
Let p be the place of observation, and let us suppose it to be
in the northern hemisphere of the earth. Let p be the origin of
the moving system of rectangular axes to which the motion of
m is referred: let the axis of be the vertical line at p measured
upwards from the earth towards the zenith of P; this may be
assumed, without sensible error, to pass through the earth's
Let axes of έ and 7 be in the horizontal plane at P,
and be respectively N and s, and E and w; the positive direc-
tion of έ being taken towards the south, and that of towards
centre.
η
237.]
RELATIVE TO THE EARTH.
435
the west. Let the latitude of P, viz. POQ, = λ. Let the plane
of the meridian of r, when t = 0, be that of (x, z); and let the
earth's radius be r. Then PONA, MON = wt: and
Ο Μ
OM = x₂ = r cosλ cos wt,
хо
à
MN = y。 = r cos λ sin wt,
Yo
Z。 = r sin λ;
N P
d² xo
- w²r cos à cos ot,
dt2
d2yo
dt2
- w²r cos λ sin wt,
(49)
d20
dt2
= 0.
Also resolving o along the axes of §, n, Ś, we have
ως
(50)
- ∞ cos λ,
=
ωη 0, ως
wg = w sinλ;
of which the first is the component about the line running due
s and N in the horizontal plane, and is the only component in
that plane: and the last is the component about the vertical
at P.
Now if the place of m at the time t is (x, y, z) relatively to
the fixed axes, and is (§, ŋ, Ć) relatively to the moving axes
which originate at P; then
x = r cosλ cos wt +
yr cosλ sin ot+
z = r sinλ
sin λ cos wt-n sin wt + cosλ cost,
sinλ sin wt+n cos wt+cosλ sin wt,
(51)
-¿cos A
+ (sin λ.
On comparing these with (2) Art. 2, we have
a₁ = sin λ cos wt,
a₂ = sín λ sin wt,
b₁ =-sin wt,
C1 = cos λ cos wt,
à
by
= cos wt
аz
- cos λ;
b3
0;
C3
C₂ = cos à sin wt,
sin λ ;
(52)
and differentiating
d2
a² α1
απ
dt2
w2 sinλ cos wt,
d² b₁
dt2
=w2 sinot,
w² sin wt,
d² C1
dt2
= — w² cosλ cost,
d² az
d² b₂
d² Cz
-w2 sinλ siuwt,
=-w² coswt,
-w² cos λ sin wt,
dt2
dt2
dt2
d² α3
d² b₂
d2 C3
= 0;
= 0;
:
= 0;
dt2
dt2
dt2
and substituting these quantities in (15), (16), and (17), we
have
3 K 2
436
[238.
MOTION OF A HEAVY PARTICLE
X = w²r sin λ cos - w² (sin λ)2- (w2 sin A cos A,
Xt
Y₁ =
Yt
n w²,
- w³r (cos λ)² - 2 sin A cos A-3 (cos λ)2;
έ
and substituting from (50) in (19), we have
(53)
dn
F COS α =
a
w sin
dt
dę
dc
F COS B
w sin λ
+ w cos λ
(54)
dt
dt
dn
F COS
sy =
Y
w cosλ
dt
d² ε
dt2
d² n
+ cos λ
dť
dt
When these several quantities are substituted in the equations
of motion given in (23), these last equations become
w2rsinλ cosλ—¿w² (sinλ)2-(w2sinλcosλ-2wsinλ x',
— ŋ w² + 2 w (sin λ
-no² 26 (sinad/
d n
dt
d
αζ
dt
=Y', (55)
dn
d² t
5
— w²r (cosλ)² — w²sinλ cosλ-w² (cosλ)2-2w cosλ
= z'.
dt2
dt
These equations may be deduced directly from (51) without the
intervention of the general process, which has been investigated
in the preceding Articles. For we may take the second t-differ-
entials of x, y, and z, and equate the sum of their several com-
ponents along the axes of έ, n, to the impressed velocity-incre-
ments acting on those axes. In particular problems this is the
most convenient method.
238.] To adapt these equations to the actual circumstances of
the earth, the sign of o must be changed, because the earth
revolves from west to east, which is a direction opposite to that
taken in the preceding Article. To determine its value, we
will take a second for the unit of time; then, since a mean
sidereal day contains 86164.09 seconds,
W
2 π
86164.09
1
13713
.00007292,
which is a small fraction; and consequently w², which enters
into the preceding equations, is a very small quantity. Also, in
the problems to which we shall apply the equations, §, n, C will
be always very small parts of the earth's radius; and thus we
may at first, without sensible error, neglect those terms in the left
239.]
437
RELATIVE TO THE EARTH.
hand members of the equations which involve products of these
coordinates and of w²; and the equations become
2
2
d² &
dn
w²r sin λ cos λ + 2 w sin λ
dt2
dt
d² n
dt2
dε
ας
- 2 w (sin
+ cos λ
=Y',
(56)
dt
dt
2
d² t
dn
=
z';
dt2
dt
w²r (cos λ)² + 2 w cos λ
where x', y', z' are the components along the moving axes of all
the absolute velocity-increments impressed on m.
239.] Now I propose to apply these equations in the first
place to the motion of a particle projected with a given velocity
and in a given direction from P, the place of observation, which
is also the origin of the moving system of axes. Although the
power of our weapons of projection has been very greatly in-
creased of late, yet still, for all points of the path, έ, n, are but
small parts of the earth's radius; consequently w² §, w²ŋ, w²¿ are
small quantities which we may omit, and (56) are applicable.
2
In the right hand members, for the same reasons, I shall
assume the earth's attraction to be the same at all points of the
particle's path, and to be what it is at P, the place of observa-
tion. Although gravity varies at different points of the earth's
surface, according to a law which is accordant with Clairaut's
theorem, yet I shall take it to be the same at all latitudes; and
no sensible error will, within the compass of our approximations,
thereby be introduced into the results. I shall also consider
the projectile to move in vacuo, and shall consequently neglect
the resistance of the air. Thus the particle moves under the
action of gravity only; and the force which acts on it during
its motion is the same as that which acted on it in its original
state of rest. Consequently we may determine the values of x', y',z'
in (56) by their values when m is at relative rest at P; thus,
dε
dn
d C
d? હું
d²n
d2c
§ = n = 8 =
===0;
= 0;
0;
dt
dt
dt
dt2
dt2
O; dt²
Thus
x'
Y
0,
2 =
²r sin λ cos λ,
w² r' (cos λ)² — 9 ; ~
(57)
and (56) become
438
[239.
RELATIVE MOTION
d² ε
d n
+2 w sinλ
= 0,
dt2
dt
d²n
dt2
-2 w (sin x
αξ
+ cosλ
dt
dt
1 1/2) = 0,
(58)
d2
5
dn
+2w cosλ
-g.
dt2
dt
w
If = 0, these equations express the ordinary case of a projec-
tile's motion which has already been solved in Vol. III.
=
Now (58) admit of integration. Let u the velocity of pro-
jection, and let a, ß, y be the direction-angles of the line of
projection;
αξ
+ 2 wŋ sin λ = u cos a,
dt
dn
2 w (§ sin λ + (cos λ) = u cos ß,
(59)
dt
d t
+ 2 wŋ cosλ = u cos y-gt;
dt
which assign the components of the velocity at any point of the
path.
Again, if we substitute for and
dt
αξ d š
from the first and last
dt
of these equations in the second of (58), and omit the terms.
involving the product of w² and of one of the relative coordinates,
then we have
d² n
dt2
therefore
2 uw (cos a sin λ + cos y cos λ) + 2 w g t cosλ =
n = ut cosß+uw (cosa sinλ+cos y cosλ) t² — w g cosλ
t3
0;
3; (60)
and replacing ʼn in the first and last of (59) by this value, and
omitting terms involving products of w² and of one of the relative
coordinates, and integrating,
ω
ૐ = ut cos a-u w sin à cosẞ t²,
} = ut cos y (2/3 + uw cos λ cos ³) 12
(61)
(62)
which three equations express the motion of the projectile to
the degree of approximation attainable by the preceding equa-
tions of motion.
If w=
O, the results are the same as those which have already
been found in Art. 280, Vol. III; viz.,
240.]
439
OF A PROJECTILE.
§ = ut cos a,
n = ut cos ß,
1
(= ut cos y
29 12
On comparing these quantities with the preceding equations, it
appears, that if the particle or ball is projected from a place in
the northern hemisphere, in a direction westwards of the meri-
dian, both the vertical height of it and its distance southwards
from the parallel of latitude are diminished by the earth's rota-
tion; and that if it is projected eastwards of the meridian, that
is, in the direction in which the earth is going, both these quan-
tities are increased. As to the three terms of which 7 consists,
n
only the first, viz., ut cos ß, depends on the line of projection
being eastwards or westwards; and consequently the increase or
diminution of n will depend also on the sign of the other two
terms which involve t.
η
The apparent path of the projectile may be determined by the
elimination of t; which will give the equations to two surfaces,
the line of intersection of which is its path: it is evident that
the path will generally be a curve of double curvature.
240.] Let us however consider certain particular cases and
results of these equations.
(1) Let the body fall, as e. g. down a mine, without any
initial velocity; then u = 0; cos a = cos ẞ= 0; cos y = −1 ;
.. § = 0,
13
η
w g cos x
3
=
1
Q I t²
(63)
The first equation shews that there is no deviation in the line of
the meridian; from the second we infer a deviation towards the
east; that is, in the direction towards which the earth is moving,
which varies as the cube of the time of falling; and that this
deviation is greatest at the equator, where λ = 0; and the last
equation shews that the earth's rotation does not produce any
alteration in the time of falling.
If we eliminate t, and take ( downwards to be positive,
n² =
8 w² (cos λ)2
9 g
१ ;
which is the equation to a semicubical parabola; and shews that
440
L240.
RELATIVE MOTION
the square of the deviation towards the east varies as the cube
of the space through which the particle has fallen.
(2) Let the particle be projected vertically upwards; then
0; cos y = 1; and
cos a = cos B
ε = 0,
· n = u w cos λ t² — w g cos λ
t3
3
(64)
t²
} = ut
1
@ ૭ ;
the last equation shews that the vertical motion is the same as
it would be if the earth did not rotate on its axis; and con-
sequently if h is the height to which the particle ascends, and r
is the whole time of ascent and descent, u² = 2gh, and r =
2u
g
The first equation shews that there is no deviation in the line
of the meridian; the second shews that the deviation is always
westwards; for the greatest value of r is
T
2u
g
(unless the particle,
after having descended to its original place, continues to fall
down a mine), and consequently ŋ is always positive. When
the particle, after its ascent, strikes the earth,
4 w u³ cos λ
n =
;
3 g²
which is the deviation westwards of the point of impact on the
ground; and varies as the cube of the velocity of projection.
(3) Let the particle be projected due westwards at an angle
of elevation equal to 0; then cos a = 0, cosẞ= cos 0, cos y =
sin ; and
ૐ
u o sin à cos 0 t²,
13
n = ut cos 0+ u w sin 0 cos λ t² — w g cos λ
3
(65)
} = u
t sin 0 − ( + u w c
+ u w cos λ cos
cos à cos 0) 12:
the first of which equations shews that the projectile generally
deviates northwards; when the projectile strikes the ground,
= 0; in which case
2 u sin 0
g+2uw cost cos
t =
2 u sin 0
24ω
1.
cos e cos
9
g
sx},
240.]
441
OF A PROJECTILE.
omitting those terms which involve w²: in this case
4 u³ w sin λ (sin 0)2 cos 0
ω
ૐ
(66)
g²
u²
n =
sin 20+
g
4 u³ w cos λ
3g
{ (sin 0)² — 3 (cos 0)2};
(67)
which are the approximate coordinates of the point of impact on
the ground. The terms involving o denote the effects due to
the earth's rotation: the former gives the deviation northwards;
and the latter shews that the range measured westward is in-
creased or diminished according as 0 is greater or less than 60°.
(4) If the particle is projected due eastwards at an angle of
elevation equal to 0, all the preceding results are true if we
replace by 180°-0; so that (66) and (67) become
0
ૐ
4 u³ w sin λ (sin 0)2 cos 0
;
(68)
g2
uz
4u³ w cos λ
η
sin 20+
{(sin 0)2-3 (cos 0)2}; (69)
g
3g
so that in this case the deviation of the projectile is southwards;
and the range is increased or diminished according as the angle
of elevation is less than or greater than 60°.
(5) Let the particle be projected due southwards at an angle
of elevation equal to ; then
and
cos a = cos 0,
cosẞ = 0,
0,
cos y
cos y sin ◊ ;
E = ut cos 0,
§ cose,
n = uw sin (0+ λ) t² — w g cos à
13
x
3
(70)
Š
= ut sin 0
g t2
0.
;
2
from the first and the last of these equations we infer, that
neither the time nor the range on the meridian is altered by
the rotation of the earth. But when
projectile strikes the ground, t =
η
n =
4 u³ w (sin 0)2
3g2
2 u sin 0
9
= 0, that is, when the
; in which case
{sin cos A+ 3 cos 0 sinλ};
(71)
and therefore the point where the projectile strikes the ground
is always west of the meridian; and a similar result will be true
when the particle is projected due northwards.
Now we shall hereafter prove that these results, which have
PRICE, VOL. IV.
3 L
442
[241.
RELATIVE MOTION
herein been applied to the motion of a material particle, are also
true of that of the centre of gravity of a body. Neglecting
therefore the resistance of the air, and the action due to the
rotation of a ball or bolt, we have the following results as to rifle
and cannon practice:
When the shot is fired due north or south, the range in that
direction is not altered, but there is always a deviation of the
shot westwards, the value of which at the point of impact on
the ground is given in (71).
When the shot is fired due east, the range eastwards is in-
creased or diminished according as the angle of elevation of the
gun is less than or greater than 60°; and there is deviation
southwards for all places in the northern hemisphere, and north-
wards for all places in the southern hemisphere, the value of
which is given in (68).
When the shot is fired due west, the range is increased or
diminished according as the angle of elevation is greater than
or less than 60°; and there is a deviation northwards for all
places in the northern hemisphere, and southwards for all places
in the southern hemisphere.
So that for firing from a place in a direction coincident with
the parallel of latitude, and with an elevation less than 60°, the
range is increased or diminished according as we fire eastwards
or westwards; and the difference between the two ranges
8 u³ w cos λ
3g
{3 (cos 0)2- (sin 0)²} ;
and if the place is in the northern hemisphere, the deviation
parallel to the meridian is north or south, according as we fire
west or east.
And for places in the northern hemisphere for all directions
lying west of the meridian, the deviation parallel to the meridian
is northwards; and for all directions lying east of the meridian,
the deviation parallel to the meridian is southwards.
241.] The expressions (60), (61), and (62), which have been
explained in the preceding Article, are deduced from equations
of motion, whose form is simplified on the assumption that
products of w², and one of the relative coordinates of m, are
small quantities, and are to be neglected. Let us now retain
these quantities in the equations of motion, and assume that
products of w³ and of a small variable are to be neglected; and
3
241.]
443
OF A PROJECTILE.
that all small quantities of a lower order are to be retained. In
this case the equations of motion are
d² §
dn
w2 (sinλ)2-(2 sinλ cosλ+2 o sinλ
ω
0,
dt
dt2
d² n
dt2
— ŋ w² — 2 w (sin x
αξ
d
+ cos λ
0,
(72)
dt
dt
2
d² Ć
dn
- 2 sinλ cosλ-(w2 (cosλ)2+2w cosλ
9.
dt
dt2
Of these equations, the values of έ, n,, given in (60), (61), (62)
are approximate solutions of the first order; viz.,
ૐ =
= utcosa-uw sinλ cos ß t²,
ts
n = utcosẞ+uw (cosa sinλ + cosy cosλ) 12-wgcosλ 3
1
} = utcosy-
59 12.
- uw cosλ cos ẞ t²;
2
(73)
and these may be employed to find approximate solutions of
(72).
In the second of (72), in the term w²n, let ʼn be replaced by
ut cosẞ from (73); then integrating, we have
= 0;
dn
t2
u cos B-u w² cos B
-2 (sinλ+(cosλ) =
W
dt
2
their values given in (73), and in-
and substituting for § and
tegrating again,
2
n = ut cosẞ+uw(cosa sinλ+cosy cosλ) t²
t3
13
wg cos λ
u w² cos ß
3
12. (74)
Again, in the first and third of (72), in the terms involving
w² § and w² (, let έ and be respectively replaced by
then integrating, we have
1
ut cos a,
and utcos y
9 12
2
αξ
t2
dt
u cos a — u w² sinλ (cos a sinλ + cos y cosλ)
2
13
+ w²g sin λ cos A
+2w sinλŋ = 0;
6
d
dt
t2
u cos y — u w² cosλ (cosa sinλ + cosy cosλ)
2
+ w²g (cos λ)²
t3
2
+ 2 w cosλ n - gt;
6
substituting in the last terms of these
(74), and integrating, we have
the value of ŋ, given in
3 L 2
444
[241.
RELATIVE MOTION.
} = ut cosa - uw sinλcosẞ t²
Š
-uw2 sinλ (cos a sinλ + cosy cosλ)
1
ts
14
2
+gw²sinλcosλ
8
g; (75)
= ut cosy - gt² — uw cosλcos ß t²
13
14
2
+ g w² (cos λ)²
8 (76)
-uw cosλ (cos a sinλ + cosy cosλ)
;
which expressions for §, n, ‹ are correct as far as terms involving
w2 inclusive.
Explanations might be given of particular cases of these equa-
tions, similar to those of the last Article. I will only take two
cases :
(1) Let the body fall without any initial velocity; then u =
cos a = cos ẞ = 0; cos y = 1;
ૐ =w2g sin x cos à
#
8
'13
η
7=
wg cos
3
1
5
29t²+w²g (cosλ)² -
14
8
The first equation shews that there is a deviation of the falling
particle in the plane of the meridian towards the south; and the
second shews that the deviation in the parallel of latitude is
towards the east; so that the resulting deviation of the falling
body is towards the south-east. From the last equation it
appears, that the space due to a given time is less than it would
be if there were no rotation. Hereby then we have corrections
of the results explained in (1) of the preceding Article.
(2) Let the body be projected due southwards at an angle of
elevation equal to 0, so that cos a = cos 0; cos ß = 0; cos y =
sin ; then
&
ut cose — uw²sinλ sin (λ + 0)
13
t
2
+gw²sinλ cosλ
8
n = uw sin (λ+0) 12 - wg cosλ
t²
13
39
= utsine
912
2
-uw2cosλ sin (λ+0)
[3
2
14
+gw² (cosλ)2
8'
when the projectile strikes the ground, (=0; and approximately
2 u sin 0
t =
; in which case
9
4u3o (sin 0)2
3 g²
{sin 0 cos λ +3 cos 0 sin λ}
so that the deviation along the parallel of latitude is westwards.
242.]
445
DEVIATION OF A FALLING BODY.
In the investigation of this problem, given by M. Poisson,
Journal de l'Ecole Polytechnique, Cahier 26, p. 1, terms are
introduced representing the resistance of the air. The equa-
tions, thus enlarged, do not admit of direct integration; the
´effect however of the resistance of the air is determined by the
method of variation of parameters. The student desirous of
knowing the extent to which mathematical analysis has been
applied to balistics, must consult three memoirs of M. Poisson,
contained in Cahiers 26, 27 of the aforesaid Journal.
242.] Another problem, also of considerable physical interest,
may be solved by means of these equations. For although the
fact of the rotation of the earth was satisfactorily demonstrated
by proofs drawn from astronomical observations, still Astrono-
mers, as well as Geometers, were desirous of an ocular proof of
a less abstract nature. Now we have observed in Art. 249,
Vol. III, that if a heavy ball falls from the top of a lofty ver-
tical tower; and if we suppose the earth to rotate from west to
east, and take that rotation into account; then, if the ball
falls on the east side of the tower, it strikes the ground at a
certain distance from the foot; and if it falls on the west, it
strikes the tower before it reaches the ground. This fact is
evident from general reasoning, as we have before stated, but
we have yet to investigate the law of the deviation. This was
determined by Laplace; see Mécanique Céleste, 2nde partie,
Livre X, Ch. V. 15: we can however deduce the law from the
general equations (56). Our problem is;
A heavy ball falls from a height h to the earth: it is required
to determine the circumstances of motion.
The equations of motion are (58); and the initial circum-
stances are these: when t =
αξ
dn de
0 ;
الله
ૐ ξ = η = 0,
S = h.
dt
dt
dt
αξ
+ 2 w ŋ sin λ = 0,
dt
dn
W
-2 (sin λ+cos λ) = 0,
(77)
dt
dr
+ 2 w n cos λ
- gt;
dt
substituting the first and last of these in the second of (58), and
omitting terms involving w², we have
446
[243.
RELATIVE MOTION.
d2n
dt2
+ 2 wg cosλ t = 0;
.. η
w g cos λ
3
t3;
and from the first and the last of (77), omitting the terms in-
volving w², we have
1
5=h-
-
2912
,
έ = 0;
so that to the degree of approximation we have taken, the ver-
tical motion of the particle is the same as if the earth did not
rotate; no deviation takes place in the plane of the meridian,
and the horizontal deviation is towards the east, and varies as
the cube of the time during which the body has been falling.
2h
Since the time due to the height h is (2), the deviation to-
g
wards the east of the point where the body strikes the ground
*
24 h cos A
W
3g
and varies therefore as the square root of the cube of the height
from which the body has fallen.
The student desirous of further information on the subject of
these Articles, in addition to the Memoirs of M. Poisson already
alluded to, will consult with advantage (1) Benzenberg, Versuche
uber das Gesetz des falls, &c., Dortmund, 1804; (2) G. L. Houel,
De deviatione Meridionali corporum libere cadentium, &c.,
Utrecht, 1839. In both these treatises he will find the investi-
gations of Gauss, in which the resulting equations are carried to
an approximation involving higher powers of @ than the second.
In the latter too he will find an account of the experiments made
by M. Reich in a mine near to Freiberg, in Saxony, in the
year 1833.
243.] We can also by means of these equations investigate
the oscillations of a pendulum, when its motion is affected by
the rotation of the earth. And we shall arrive at the results
which M. Foucault exhibited in his famous pendulum experiment
before the Academy of Sciences in Paris on Feb. 3rd, 1851;
and which have been repeated, and confirmed, in many parts of
the earth.
We shall hereby have another ocular proof of the diurnal
243.]
447
FOUCAULT'S PENDULUM EXPERIMENT.
rotation of the earth; and perhaps a more striking one than
any that had formerly existed; for our process will shew that
the observed results are in accordance with the physical laws
which cause them.
- It will be convenient to make a slight change in the moving
system of reference, and to take the point of suspension of the
pendulum for the moving origin: let the axis of be taken ver-
tically downwards from it, so that the sign of it must be changed
in the preceding equations; the axes of έ and ʼn being taken re-
spectively southwards and westwards as heretofore; and let h be
the vertical distance of the point of suspension from the earth's
surface.
η
We shall assume the pendulum to be perfect; and shall take
7 to be its length, that is, to be the distance of the bob, con-
sidered as a particle of mass m, from the point of suspension.
Let (έ, n, ) be the place of its bob at the time t; then
§² + n² + 5² = 12;
(78)
and let the tension along the rod of the pendulum = mт; let
the components of r be introduced into the equations of motion
(55); and let x', y', z' be the axial components of the other im-
pressed momentum-increments; then we have
d² E
d n
dt2
-or sinλ cosλ— §w² (sinλ)²+(-h) w² sinλ cosλ+2∞ sinλ
T
+x',
dt
d2n
dt2
w³n — 2w (sin x
αξ
dt
αξ
cos λ
= −r3}{+x',
η
dt
ď²Š
dn
Š
dt2
+w²r (cos λ)² + § w² sin λ cosλ — (§— h) w² (cos λ)² — 2 w cosλ
T
+z.
dt
Now these equations must satisfy the mechanical circumstances
of the pendulum when it hangs vertically, and is at rest; in
αξ dn ας
d2 E
d² n
which case έ=n=0,8=1;
0;
0,
dt dt dt
dt2
dt2
d² t
dt2
= 9-T; so that
x =
Y' = 0,
z' =
w²r sin à cos λ + (1 − h) w² sin à cos λ,
9−T+w²r (cos λ)² — (l — h) w² (cos λ)² + T
and the equations of motion become
448
[244.
RELATIVE MOTION.
d2e
dt2
w² § (sin λ)² + (5—7) w² sin λ cos λ + 2 w sin λ
d2n
dt2
w³ n − 2 w (sin λ
de
dt
ας
-
cos λ
dt
d² Č
dt2
dn
T
dt
η
T
יך
dn
T
$ +9.
+ w²¿ sin λ cos λ — (5—1) w² (cos λ)² - 2w cos
dt
These equations represent accurately the motion of the pen-
dulum; but as they do not admit of complete integration, we
must have recourse to methods of approximation, as in the pre-
ceding Articles. We shall suppose the extent of oscillation to
be very small, so that έ, n, and l- are always small quantities;
and as w² is a very small fraction, we shall neglect products of
them and it and thus the equations of motion become
d2ε
dn
+ 2w sin λ
&
T
dt2
dt
d2n
dt2
2 w (sin à
de
dš
cos λ
T
dt
dt
d°
dn
2w cos A
dt2
dt
دايه
(79)
T
+9.
244.] Various methods have been chosen by different mathe-
maticians of dealing with these equations. If the rotation of the
earth is neglected, w=0, and the equations become those which
express the motion of a conical pendulum, and which have al-
ready been discussed in Articles 369 and 370 of Vol. III. We
may take the solution of these simplified equations to be in form
the solution of our actual equations; the former will contain
four undetermined constants depending on the initial values of
the velocity and coordinates of the place of the bob of the pen-
dulum; these constants may be considered variable, according
to Lagrange's method of variation of parameters; and the differ-
ential equations of motion will enable us to determine these in
terms of the time, whereby we shall obtain variable elements,
which will at any time fix the position of the place of the pen-
dulum. This method has been adopted by M. Quet, in a me-
moir of great ability in Liouville's Journal, Vol. XVIII. Paris,
1853. Other mathematicians have followed the same process
under a different form: they have considered the terms involving
w to arise from a certain disturbing function, the §-, 7-, (-partial
differentials of which are severally,
245.]
449
FOUCAULT'S PENDULUM EXPERIMENT.
dn
- 2 w sin λ
,
dt
2 w (sin x
d&
λ
cos λ
dt
dr
dt
dn
2 & cos λ
dt
and then they have pursued the method indicated by Sir W. R.
Hamilton and Jacobi. This process has been developed by
M. Dumas in an Academical Dissertation, "De Motu Penduli
Sphærici rotatione Terræ perturbato," Königsberg, March, 1854,
in which the results are expressed in terms of the higher elliptic
transcendents.
Again, other mathematicians have adopted a method of ap-
proximation depending on the successive omission of small
terms. The original investigation of M. Binet* was made on
this principle; and it has subsequently been applied by Hansen,
"Theorie der Pendelbewegung," Danzig, 1853. I have treated
the equations in the following Articles by this process, because
it is the most simple and the most natural, and indicates the
chief results of the equations with the least labour.
245.] Let the equations (79) be multiplied respectively by
2 d§, 2dŋ, 2dğ, and added; then, since by (78),
we have
Edέ + ndn + ¿d¢ = 0,
2
2 d2
d. { d { " + dn² + d² } = 2 gd,
{ d² {}
dt2
(80)
(81)
and if we multiply the second of (79) by έ, and the first by n,
and subtract the latter from the former, we have
d.
J & dn - n d §
nd
dt
- w sin λ d(2+n²) + 2w cosλ & d¿ = 0. (82)
λ§ dɣ=
Now let us refer the place of the pendulum at the time t to the
horizontal plane at the place of observation and to a vertical line
which passes through the point of suspension. Let be the
angle between this vertical line and the rod of the pendulum,
and let y be the angle at which the vertical plane, in which the
pendulum is at the time t, is inclined to the plane of (§, (), which
is the meridian plane; ↓ increasing positively as we move from
the f-axis towards the n-axis, that is, as we revolve from south
westwards, and on northwards, and so on towards the east: that
is, in a direction opposite to that in which the earth rotates. Also
let p be the perpendicular distance from the bob of the pendu-
lum to the vertical line through the point of suspension. Then
* See Comptes Rendus de l'Académie des Sciences de Paris, 1851, p. 197.
PRICE, VOL. IV.
3 M
450
[246.
RELATIVE MOTION.
if the path described by the bob is projected in the horizontal
plane, p and are the polar coordinates of it, the pole being
the point directly beneath the point of suspension: thus we
have
p =
Į sin 0,
Š
= 1 cos 0,
(83)
мо
= p cos y
=
sino cosy,
η
n = p sin y
I sin 0 sin ;
.. d§²+dn²+d§²
dε² + dn² + d§² = 12 {(do)² + (sin 0)2 (dv)2},
2
0)² (d¥)²},
(84)
Edn―nd§ = 12 (sin 0)2 dy;
(85)
thus (81) and (82) become
2
d.
do)² 2
dt
+(sin 0)³ (d) — 29 cos 0}
2
2g
0,
>(86)
d. {sin 0)2
(
ω
sin\)}
dt
2w cosλ (sin 0)² cosy do = 0.
As these two equations are deduced from (79) by a change of
coordinates, they have lost none of their generality, and conse-
quently they express the general motion of a pendulum to the
same degree of accuracy; and that is, when terms involving the
products of w² and either έ, n, or l—§ are omitted.
246.] For our purpose however it will be sufficient to consider
the oscillations of the pendulum to be small, and thus to assume
the greatest angle of inclination of the pendulum to the vertical to
be so small that cubes of it and all powers higher than the cubes
may be omitted. Consequently is always such that 03 and
higher powers of 9 will be omitted; also is a small quantity.
d Ꮎ
dt
Let us replace 10 by p, as we may by means of (83); because we
shall thereby obtain the polar equation of the curve in the hori-
zontal plane into which the path described by the bob of the
pendulum is projected. Then omitting 03 and higher powers of
0, from (83) we have
P =
10,
dp
= 1
dt
d Ꮎ
に
​dt
2
Also the last terms in the second equation of (86) must be
omitted, because (sin 0)² do is a small term of an order higher
than those which are to be retained. Thus (86) become
246.]
451
FOUCAULT'S PENDULUM EXPERIMENT.
dp
2
2
2
d. { (dn)² + p² (d)² + { p² } = 0
•
dt
dy
dt
d. {p² (da sin x)} = 0
ω
dt
0.
(87)
Now these equations are integrable; and let us suppose the
pendulum to start from rest at a distance pa, from the vertical
line passing through the point of suspension; so that
eliminating
2
02
dp
dt
2
2
(de)² + p² (
dt
αψ
dt
27
2 9
+ p² (dv) ³ + ½ (p² — a²) = 0;
02
dy
dt
we have
dt
o sinλ (p² — a²) = 0;
(sin x)³} p² + a² { 2 + 2 w² (sin x)²}
2
(88)
(89)
g
+ w² λ)² p¹+a²
ра
2
S.g
+ λ)
<² (sin x)² } { -
- w² (sinλ) a¹, (90)
p+ +
g+2l w² (sin λ)²
g+l w² (sin x)²
a² p²
w²l (sinλ)2
g+l w² (sin x)²
a². (91)
Now the right hand member of this
equation is a quadratic
expression in p², which has two roots, both of which are positive,
and of which one is a²; let b² be the other, and let us suppose
a² to be the greater: also, for convenience of expression, let
9
½ + w² (sin x)² = n²;
so that
w sin x
b =
α.
Then (91) becomes
n
p²
2
dt
(de)² =
2
= n² (a² — p²) (p² -- b²);
མ་་
(92)
(93)
(91)
so that a and b are manifestly the greatest and least values of p.
From (94) we have
p dp
{(a² — p²) (p² — b²) } $
-ndt;
the negative sign being taken, because on our supposition p de-
creases as t increases.
grating, we have
Let also t=0, when p=a; then inte-
a² + b²
a² - b²
p² =
+
cos 2nt;
2
(95)
2
3 M 2
452
[246.
RELATIVE MOTION.
which gives the value of p² in terms of t; and shews that (1) a²
and b² are respectively the greatest and least values of p²;
(2) their values recur periodically; and (3) the periodic time
π
12
π 73
{9 + l w² (sin λ)² } ✯
(96)
this result evidently agrees with that of the common simple
pendulum, when ∞ = 0.
To find the relation between p and y, we have from (89)
dx
a² sin A
w sin λ
;
dt
p2
but from (93)
w sin x
b
a;
n
d&
w sin
dt
nab
;
2
p²
ab dp
.. dy-wsinλdt
p {(a² — p²) (p² — b²) } &
1
whence, by integration, with the assumption that y
= a, when t = 0,
(97)
(98)
(99)
=
yo, and
P
1
a² + b²
a²-b2
p²
2
2 a2 b2
2a2b2
cos 2 (4-4。—w sin à t)
{cos(-₂-wsinλt)}²
2
+
a²
{sin(√――wsinλt)}²
b2
2
•
(100)
If t is constant, this is an ellipse whose principal axes are re-
spectively 2a and 26; so that the path described by the bob of
the pendulum projected on the horizontal plane is an ellipse,
the whole period being that given in (96). And since
dy
dt
,
which is given in (97), is negative, the pendulum revolves in a
direction opposite to that in which increases; that is, the
direction of its revolution is the same as that of the earth. And
since (93) shews that the ratio of b to a varies nearly as w, b is
small compared with a, so that the eccentricity of the ellipse is
very large; if = 0, b = 0, in which case the minor axis
vanishes, and the pendulum moves in a plane: this however
cannot be the case when account is taken of the earth's rotation.
w
Since however t varies, let us still consider (100) to represent
an ellipse whose principal axes are 2a and 26; and whose major
axis at the time t is inclined to the έ-axis, which is measured
247.]
453
FOUCAULT'S PENDULUM EXPERIMENT.
southwards along the meridian at an angle equal to yo+wsinλt:
now this angle increases as t increases; and consequently the
major axis revolves in azimuth with a constant angular velocity
equal to o sin λ in the same direction in which increases.
Thus, if the path described by the bob of the pendulum is pro-
jected on the horizontal plane, it will be a revolving ellipse,
whose major axis revolves in azimuth with an angular velocity
equal to sin λ, in a direction opposite to that in which the
earth moves: the actual path will thus be a spiral limited by
two concentric circles whose radii are a and b, of which a is
the greater; the spiral never extending beyond the former, nor
coming within the latter; and the point where it meets the
larger circle advancing with an angular velocity equal to o sin λ,
in a direction opposite to that of the earth's rotation, and oppo-
site to that in which the pendulum itself moves.
This is the law which the experiment exhibited by M. Fou-
cault confirms. We have already given a simple explanation of
it in Art. 30; and that explanation appeared to M. Poinsot (see
Comptes Rendus, Tome XXXII, p. 206) to be sufficient. The
preceding investigation however shews that the result follows
from the equations of motion, when small terms are omitted.
This therefore is only the general effect; but there are sundry
deviations, owing to the omitted terms, which this dynamical
process will indicate if it is carried to a higher approximation,
and the other method fails to shew; but it is beyond our pur-
pose to enter upon these small disturbances in this treatise.
The several memoirs already alluded to contain further approxi-
mations, and to them I must refer the student. I should
also mention that M. Poncelet, whose name must ensure atten-
tion from every mathematician, has written two memoirs on
this subject, which are inserted in the Comptes Rendus de
l'Académie des Sciences de Paris, Vol. LI, 1860, and has arrived
at results differing in some respects from the preceding.
247.] The motion whose circumstances we have investigated
has been imagined to be that of a bob of a pendulum fixed by a
rod of given length to a point fixed relatively to the earth and
moving with it, and the effect of that rotation has been exhibited
in the preceding equations. This motion is consequently that
of a material particle moving on the lower concave surface of
a sphere, whose radius is 7, fixed to the earth and moving with
it; and the general equations are applicable to any other kind
454
[247.
RELATIVE MOTION
of constrained motion of a particle. Let us take another ex-
ample.
A particle moves on a smooth inclined plane fixed to the
earth and moving with it: it is required to determine the rela-
tive motion of the particle.
Let the plane pass through P, the place of observation, see
Fig. 63, whose latitude is λ; and let the equation be
έ cos a + n cos ẞ+(cos y =
0;
(101)
let mR be the normal pressure of the particle on the plane;
then the equations of motion are
de
dn
+ 2 w sin λ
=R COS a,
dt2
dt
d² n
w
dt2
- 2 (sin x
de
d s
+ cos λ
=R COS ẞ,
(102)
dt
dt
d² ¿
dn
+ 2 w cosλ
=R COS Y-9.
dt2
dt
Although it is convenient to retain a, B, and y, yet we shall re-
quire their values in terms of (6) the inclination of the plane
to the horizontal plane of (§, n), and of the angle (4) between
the g-axis, which is southwards, and the line of intersection (the
line of nodes) of the plane with the horizontal plane. In refer-
ence to these
cos a = sin sin,
cosẞ =
- sin cosy,
cos y =
cos 0.
(103)
Let the particle start from rest from (§。, 0, 5); then, multiply-
ing (102) severally by dέ, dŋ, and d¿, adding, and integrating,
we have
ds2
= v² = 2 g (5。 — Š),
dt2
because, by reason of (101),
(104)
dε cos a + dŋ cos ẞ+d(cos y = 0.
(105)
Thus (104) shews that the velocity acquired is the same as if
the rotation of the earth was not considered.
From the last two equations of (102) we have
Cosy
d2n
dt
d2c
de
cos B
+2w (cosλ cosa - sinλcosy)
=
g cos ß;
dt2
dt
therefore
dn
dr
COSY
cos B
+2w (cosλcosa-sinλcos y) (-6)= gtcos ß. (106
dt
dt
247.]
455
OF A PARTICLE ON AN INCLINED PLANE.
cos a
Similarly
αξ
dt
de
Cosẞ at
dt
COS Y
αξ
dt
+ 2 w (cos λ cos a-sin λ cos y) (n-no)-gt cosa; (107)
Cos a +2∞ (cos λ cos a-sin A cosy) (55) = 0.
d n
dt
Again, multiplying (102) severally by cos a, cos ß, and cos y ;
adding, and omitting the terms which vanish by reason of the
differential of (105), we have
αξ
dn
- 2 w sinλ (cosp
cos a
dt
(108)
dn
ας
cosß at
=R-gcosy; (109)
"dt) + 2w cosλ COSY at
substituting in which from (106) and (108), and omitting terms.
involving w², we have
R = gcos y +2 w cos à cos ẞgt,
= g cos 0-2 w g t cos λ sin cos ;
(110)
(111)
which assigns the pressure on the plane; and shews that it is
diminished or increased by the earth's rotation according as the
line of nodes lies in the S. W. and N. E. quadrants, or in the
N. W. and S. E. quadrants; and that this increase or diminution
vanishes when the line of nodes lies E. and W. It vanishes at the
pole, and is, cæteris paribus, a maximum at the equator; and it
also vanishes when the plane is horizontal. It also varies as the
time during which the particle has been moving. Since o cos à
is the component of the earth's angular velocity along the
tangent to the meridian, that is, along the N. and S. line on the
horizontal plane, the change of pressure on the plane is due
to that component only, and not to the component along the
vertical.
Substituting in the first and third equations of (102) the value
of R, given in (110), and integrating, we have
de
dt
αξ
dt
+2wsinλ(n-no) = gt cosa cosy + w cosλ cosa cos ẞ gt2,
cosẞg
+2w cosλ(n−no)=-gt (sin y)2+wcosλ cosẞß cosy gt2;
(112)
and substituting these values in the second of (102), and omit-
ting the terms involving o², we have
d2n
+2w cosa {cos λ cos a — sin à cos y}gt g cos ẞcos y;
dt 2
456
[247.
RELATIVE MOTION
therefore
n-no-cosẞcosy
I 12
2
-wcosa {cosλcosa-sinλcosy}
913
5-50
2
and substituting this value of ŋ—~。 in (112), and integrating,
έ— §。 = cosa c
= cosa cosy²+wcosẞ {cos\cosa-sin\cosy}
2
2
5—5 = — (siny)2 91²;
so that in terms of 0 and y,
;
3 (113)
9 13
3; (114)
(115)
§ =
o+sin@cose sin
gt2
2
@sin@cos{cos\sin@cos—sinλ cos 0}
913
71 = nsin@cos@cosy
9t2
3
2
- w sin sin√ {cosλ sin @cosy — sinλcos@}
gt3
3
(116
5 5。 — ;
= (sin )29t²
2
which assign the position of the particle at the time t.
Whence
it appears, that if we omit powers of o higher than the first, the
vertical distance through which m falls in the time t is not
affected by the earth's rotation.
To determine the curve which the particle describes, let us
refer its place at the time t to the point (§o, no, 5) as origin; `and
to two axes in the plane, one of which, that of έ', is parallel to,
and the other, that of n', is perpendicular to the line of nodes;
so that
(-) cos y + (non) sin y = &',
(117)
Sonsin
n' sin 0;
'
ή
2
w sin ◊ {cos à sin cos — sin à cos 0}
n = sin 0 9 t²
9 ts
3
(118)
Thus, if ∞ = 0, '= 0, and the particle falls down the plane in
a straight line perpendicular to the line of nodes; but if the
rotation of the earth is considered, there is a lateral deviation
from the rectilineal path, which varies as the cube of the time of
falling; and if we eliminate t, we obtain the equation to the
path, which is
882
9 g sin 0
{cos λ sin cos- sin λ cos 0}2 n'³;
2'3 (119)
this is the equation to the semicubical parabola, which is the
path of the particle. It will be observed that I have assumed
249.]
457
OF A MATERIAL SYSTEM.
the particle to start from rest from (§。, 0, 0); if it were pro-
jected from that point on the plane with a given velocity, other
terms, which can easily be found, would be introduced into the
preceding equations. And if = 0, the resulting equations
would of course represent a parabola.
∞
248.] Let us suppose the plane on which the particle moves
to be horizontal; then the equations of motion are
d² ε
dn
+ 2 w sin λ
0,
dt2
dt
d² n
αξ
2 w sin λ
0 ;
dt2
d t
and let us suppose the particle to be projected from the origin
along the plane with a velocity u in a line inclined at an angle
ẞ to the axis of §; then, integrating the preceding equations, we
have
αξ
+ 2 w sinλ n = u cos ß,
dt
dn
2 o sin λ &
W
= u sin ß
dt
which equations assign the relative velocity of the particle at the
point (έ, n), and by subsequent integration we have.
2
(§ +
u sin ß
)² + (~
(7
ω
u cos B
2 o sin A
2
U2
4 w² (sin λ)² ³
2 o sin λ
20
which is the equation to a circle. Consequently the particle
moves in a circle whose radius is
point (
43082"
sin λ
latitude.
И
2 sinλ
; whose centre is at the
B); and the periodic time =
u sin ß u cos B
2w sinλ 2w sinλ
π
w sinλ
= a mean solar day divided by twice the sine of the
Another problem, which may be solved by these general equa-
tions, is the motion of a particle on the surface of a right cir-
cular cone, whose vertex is at P, the place of observation, and
whose axis coincides with the vertical.
SECTION 3.-The relative motion of a material system.
249.] THE equations of relative motion which have been
found refer only to the motion of a single material particle.
PRICE, VOL. IV.
3 N
458
[249.
RELATIVE MOTION OF A MATERIAL SYSTEM.
Those however of a material system may be deduced from them
by means of D'Alembert's principle.
Suppose m to be the type-particle of a system, to the motion
of which equations (23) refer; and suppose I to be the type of
an impressed momentum-increment due to an internal force,
see Art. 45, acting on м, of which I cos λ, I cosµ, I cos v are the
axial components; and let us suppose the system to be free
from all constraint except that which exists amongst its own
members; so that every particle is free to move as it is affected
by the external forces acting on it, and by the internal forces of
the system; then the equations of motion of the system in their
most general forms are
d° Ed
dt2
Σ.Μ
m{x-
X - 2 F cos a
a-
Σ.ICOSλ = 0,
d²n
Σ.Μ
Y' - Y₁ -2 F cos 3-
Σ.105μ = = 0,
(120)
dt 2
d² Ś
2
Σ.Μ
Zt-2 F cos y-
-E.I
Σ. I COS v = 0.
d t²
If the material system is a rigid body, or is invariable in form,
or otherwise is such that the internal forces taken throughout
it disappear, then these equations become
{
x.mx-xt - 2 F cos a
dt2
d°E }
?
0,
S
2
d² n
0,
(121)
dt2
Σ.Μ
dk ?
0 ;
Σ.Μ
{z
i
Zt-2 F cos y-
dt2
and it is the motion of a system of this kind which for the most
part we shall consider.
From these the equations of the axial components of the
moments of the couples are to be formed: let us take that
whose axis is the moving axis of έ; then we have
Σ.Μ
nz
§
Z₁-2rcosy - d²) — 5 (v
F
dt2
d²n
Y-2F cosẞ-
= 0;
dt²
and replacing F cos ẞ and F cos y by their values given in (19),
this becomes
z.m { v (x — 2. — 1775) — 5(x — v.
Σ.Μ
- 2 wε Σ.m
dt2
ndy + ( dc + 2
ndn+Śdš
dt
Yt
- dm)}
d²n
dt2
w₁
αξ
Σ.Μη
2.my de + 2 w; 2.
αξ
ως Σ.η ζ
dt
dt
= 0;
or
251.] RELATIVE VIS VIVA OF A MATERIAL SYSTEM. 459
dt
dz.m (n
ας
Σ.Μ
5
dt
dn) — (x
= x.m {n(z' — z₁) — 5 (Y′ —— Y₁) }
d
dε
αξ
ως
Σ.m (n² + 5²) + 2wnΣ.mn
+ 2ως Ση ζ
= 0; (122)
dt
dt
dt
d
Σ.Μ
dt
αξ
dt
αξ
dt
and the similar equations for the other axes are
ع
= Σ.m {5 (x' — xt) —§ (z′ — 21)}
d
dn
dn
Θη
z.m (5² + §²)+2wzz.m
dt
+ 2 wε z.m §
= 0; (123)
dt
dt
d
Σ.Μ
dt
(&
d n
αξ
η
= Σ.m {§(Y' — Y₁) — n (x' — X₁) }
dt
dt
d
· ως
z.m (§²+n²)+2w & z.m§
ας
dt
+ 2ωη Σ.Μη
dš
dt
= 0; (124)
dt
and by means of these six equations the relative motion of a ma-
terial system of invariable form may be determined.
250.] These six equations of relative motion may be com-
bined into a single equation by means of the principle of virtual
velocities. For suppose is to be any arbitrary geometrical dis-
placement of the place of m at the time t, which is consistent
with the geometrical relations of the system; and let d, dn, 85
be the axial projections of ds; and let all these quantities be
type-quantities; then the equations of motion may be expressed
by means of the single equation,
Σ.
{(x-
x' — x¡ — 2 r cos a
d²n
175) 85+ (x — Y, — 2 F cosß — din) on
dt2
— — FC –
+ (2′ — 2, − 2 P
(z' z - F
dt2
d²
δη
cosy — 13 14 ) 85}
dt2
= 0. (125)
This equation is indeed equivalent to the six equations by
reason of the arbitrariness of d§, dŋ, d§; for these quantities in
their most general forms involve six displacements, which are
independent of each other; viz., three of translation and three
of rotation; and the coefficients of these separately vanish. If
the relative motion of one or more of the particles of the system
is constrained, these displacements are thus far subject to cer-
tain conditions, and consequently are not independent; and all
that has been said in Arts. 49, 50, and 51 is, mutatis mutandis,
to be applied to this case.
251.] Let us suppose that the conditions to which the system
3 N 2
460
[252.
RELATIVE MOTION
is subject do not involve the time explicitly; then we may take
for the virtual arbitrary displacement of the place of m that
which actually takes place in the time dt by reason of the
motion of the system, and of the forces acting on it; so that in
equation (125) we may put
§§ αξ,
δη dn,
& 5 = d};
(126)
then, since from (19),
F{cos a de + cos ß dn+cos y d} = 0,
(127)
(125) becomes
Σ.Μ
{{
j_d°É
d2n
d § +
dt2
dt2
dn + de
dt²
= x.m {(x' — x₁) d § + (x' — Y₁) dŋ +(z' — z₁) dĊ}; (128)
so that if v is the relative velocity of m at the time t,
— dn
d.z.mv² = 2 ɛ.m {(x' — x₁) d§ + (y' — Y¿) dŋ + (z′ — z₁)dŜ}; (129)
therefore
Σ . m v² — Σ.m v2
= 2
2/
Ο
1
Σ.m {(x' — xt) d§ + (y′ — Y₁) dŋ + (z′ −z₁) d§}; (130)
wherein v。 is the initial value of v, and 1 and 0 denote the
limiting values of the relative coordinates of the place of m,
corresponding to the terminal and the initial values of the-left-
hand member of the equation.
Equation (130) is that of the relative vis viva of the material
system; and if we consider it in its elemental form in (129), it
shews that the increment of the relative vires vivæ of all the
particles of the system in the time dt is equal to the excess of
twice the sum of the products of the impressed momentum-
increment of each particle and the space through which it has
acted over the sum of the products of the momentum-increment
due to the force of transference of the coordinate system (see
Art. 226), and the space through which this latter force has
acted.
It will be seen that F, the compound centrifugal force, has
wholly disappeared in (129) and (130); and rightly so; because
its line of action is perpendicular to that of the relative velocity
of m; whereas into the equation of vis viva only those forces
enter whose lines of action are in, or have a component along,
that of the motion of m at the time t.
252.] It is expedient to mention certain particular forms
which the preceding general equations take in special cases,
252.]
461
OF A MATERIAL SYSTEM.
because in these simplified forms they are frequently applicable
to the solution of problems.
(1) Let us suppose the origin of the moving system to be
always in the plane of (x, y), and to move according to a given
law; and the system to have no motion of rotation: then
a₁ = b₂ = c₂ = 1, and all the other direction-cosines vanish; so
that from (15), (16), and (17),
1
2 C3
xt =
Xt
d² xo
dt2
Yt
Y₁ =
d2yo
dt2
d2 zo
= 0 ;
Z
dt2
and (124) becomes
d
Σ.Π
η
7)=
= Σ.m (§ y′ — nx') — z.m§
d² yo
d2 xo
+ Σ.Μη
dt2
dt2
dt
dt
dt
d² yo
d² xo
= Σ.m (¿ x' — nx')
E.mε +
Σ.mη (131)
dt2
dt2
= L
d2 yo
dt2
Σ.mε+
d² xo
&+
Σ.mn;
(132)
dt2
if L is the moment of the couple of the impressed forces whose
axis is the moving (-axis.
If the material system is of invariable form, and is fixed to
the moving origin; and if is the distance of m from that
origin, and w is the angular velocity of the body at the time t ;
then (132) becomes
dw
Σ.mr² = L
dt
d2yo
dt2
Σ.m & +
d² x
dt2
0
Σ.ηη;
(133)
by which equation the relative angular velocity of the body may
be determined.
(2) Let us suppose the origin of the moving axes to be fixed
at the fixed origin, and the moving axes to revolve about the
z-axis with an uniform angular velocity (say); let @ be the
= ∞
0
angle between the axes of a and έ at the time t; so that
α₁ = cos e,
b₁
- sin 0,
1,
α₂ = sin 0,
аг
b₂
cos 0,
C2
C₂ =
0,
(134)
а3
0;
13
0;
C3 = 0;
and
Xt = — w² §,
Yt
Y₁ = — w²n,
Z = 0
then (124) becomes
d
dt
z. m (ε dn
Σ.
αξ
d
η
= ≥.m (§ Y' — n x') — w
z.m (2+n²) (135)
dt
dt
dt
d
= L
@
z.m (¿²+n²);
(136)
dt
462
[253.
RELATIVE MOTION
where L is the relative moment of the impressed couple whose
axis is that of z.
The equations (132) and (136) may also be derived directly
from (35), without the intervention of the general forms given
in (124).
If the body is rigid, and the origin is a fixed point of it, then
z. m (§² +ŋ²) is independent of the time, and (136) becomes
dw
dt
Σ.m r² = L;
(137)
which is the same equation as that which expresses the rotation
of a rigid body about a fixed or an instantaneous axis.
It is also to be observed that (132) is reduced to (137):
(1) when (x, y), the place of the moving origin, is fixed abso-
lutely (2) when this moving origin has an uniform motion; so
d2 xo d² yo
= 0; (3) when the moving origin is the
centre of gravity of the body, because in that case ≥.m§
= Σ.mn = 0.
that
dt2
dt2
253.] From these general equations we may deduce theorems
similar to those of absolute motion which have been already de-
monstrated in Chap. II. Section 2, of the present Volume. In
the first place, the relative motion of the centre of gravity of a
material system of invariable form, or in which the internal forces
mutually destroy each other, is the same as if the whole mass
of the system were collected into it, and all the momentum due
to the external forces, the forces of transference, and the com-
pound centrifugal forces, the last two with their directions
changed, was thereat applied in lines parallel to the actual
lines of action.
η,
Let (§, 7, 3) be the place of the centre of gravity of the sys-
tem at the time t relatively to the moving axes; and let (§', n', ')
be the place of m at the same time relatively to a system of
parallel axes originating at the centre of gravity: then we have
Ś
§ + 8,
n = ñ + n',
ñ+n,
5
$ = 5 +5.
(138)
Also let м denote the mass of the whole moving system: then
the newly introduced coordinates are subject to the following
conditions:
253.]
OF A MATERIAL SYSTEM.
463
Σ.M § = M§,
z.m §' = x.mn' = x.m 5′ = 0;
(139)
Σ.Μη = Μη, Σ.m (= M (. (140)
On referring to the analytical values of the momentum due to
the forces of transference given in (15), (16), and (17), it appears
that the values of Σ.mxt, Σ.m Yt, z.m z¿ are not changed; but
they may be expressed as
M XU
M Yt,
M Zt;
where X, Y, Z, are the values of Xt, Yt, Zt, when έ, n, Ŝ are re-
placed by §, 7, 5, so that the momentum due to the forces of
transference may be applied to a mass м at the centre of gravity,
along lines parallel to their original lines of action. A similar
theorem is also true of x.m F cos a, .m F cos ẞ, z.m F cos y,
which may be replaced by M F cos a, M F cos B, MF Cosy; so
that the equations (121) become, after all reductions,
E.MX'
तह
-X-2
Xt — 2 F cos a
0,
dt2
M
E.MY'
d2n
-Y-2 F cos ß-
0,
(141)
M
dt2
Σ.mz'
d2c
t
-Z-2 F cos y -
0 ;
dt2
M
and these equations prove the theorem which has been enun-
tiated. The theorem of relative motion analogous to that of
Art. 55 may be framed in the same manner. Thus the rela-
tive motion of a material system, such as we have considered,
may be resolved into the motion of translation of its centre of
gravity, all the forces being supposed to act on the whole mass
condensed into that point, and into a motion of rotation about
an axis passing through the centre of gravity. Consequently
the investigations of the preceding section are not limited to the
motion of a material particle: they are also applicable to that
of the centre of gravity of a material system, of which the
internal forces vanish. Thus they apply to the relative motion
of translation of the centre of gravity of planets, of shot, of pen-
dulums with large balls, &c.; except that in these cases the
resistance of the medium through which the bodies pass must
be taken account of; so that other terms enter into the equa-
tions beside those which we have considered. It remains then
only to investigate the rotation of the body about an axis pass-
ing through the centre of gravity considered as a fixed point in
464
[254.
RELATIVE MOTION
reference to a system of moving axes. It is true, as we have
heretofore remarked, that the point through which the rotation-
axis passes need not be the centre of gravity; for the general
motion may always be resolved into a motion of translation of
any point, and a motion of rotation about an axis passing
through that point: but the centre of gravity is the only point
at which the mass may be supposed to be condensed and the
forces may
be applied each in its own intensity and direction,
and the translation will be the same as it is in the motion of the
whole system. In the following Articles I shall take the general
case, and shall suppose the fixed point, about which the rotation
is estimated, not necessarily to be the centre of gravity.
254.] At this point I shall assume three systems of reference,
and subsequently of coordinate-axes, to originate. (1) A system
the lines of which are parallel to the analogous lines in the
system absolutely fixed, so that all angles will be the same in
both; and this system may also be regarded as fixed: (2) the
system of axes to which the motion of the body is to be referred;
this is a moving system, and its motion with reference to the
fixed system is given, and the elements of it are, as heretofore,
data of the problem; these two systems are connected by the
scheme of cosines &c. which are involved in (1) of the present
Chapter: (3) another system of rectangular axes, fixed in the
body and moving with it, which I shall take to be a principal
system at the point.
In reference to these three systems respectively I shall take
the place of m to be (x, y, z), (§, n, 5), (§', n', '); and these last
two I shall take to be connected by the following scheme of
direction-cosines;
us
η
5
مية
ат
а2
аз
(142)
ή
βι
Br
B2 B3
Вз
5
Y1
Y2
Y3
so that
ૐ
n = a₂ §' + B₂ n' + Y₂ 5',
(143)
Ś = αz &' + ß3 n' + V3 Š';
255.]
465
OF ROTATION.
To determine the relative motion, these nine direction-cosines
must be expressed in terms of t: as only three are independent,
it will be eventually more convenient to determine the posi-
tion &c. of the body by means of Euler's three angles, 0, 0, ¥,
according to the process of Articles 3 and 4: so that
аг
a₁ = cos o cos y sin & sin y cos 0,
B₁sino cosy - coso sin cos 0,
V₁ = sin sin 0 ;
γι
y
a2 = cos o sin + sin & cos y cos 0,
Basin siny+cos o cos y cose,
Y½ cosy sin 0;
Y2 =
(144)
az
sin & sin 0,
B3 = cos o sin 0,
Y3 = cos 0.
255.] Now of the body we have two angular velocities, of
which one is absolute, and the other is relative to the moving
system of axes. Let us resolve these along the principal axes
at the time t; let w₁', w½', w' be the axial components of the
former, and let w₁, w₂, wg be those of the latter. The difference
between them is evidently due to the angular velocity of the
moving system: and consequently if we resolve this latter along
the principal axes, we may equate each component to the corre-
sponding excess of the absolute over the relative angular velocity.
Thus we have
ωι + αι ωξ + a2 wn + ag ως
W2
=
w₂+ B₂ wε + B₂ wn + Bz w5,
:}
@3
@3 + Y1 W§ + Y2 wn +Y3 w5;
(145)
as w1, W2, W3, 0, 4, ↓ are all employed relatively to the moving
system of axes, they are connected by the equations given in
Article 42; and we have
do
dx
@1
wy =
cos
+ sin o sin
Ф
dt
dt
d Ꮎ
d &
W2
sin
dt
+ cos o sin 0
(146)
dt
аф
d¥
W3
+ cos 0
dt
dt
Now w₁, w, w' depend on the constitution of the body, on its
PRICE, VOL. IV.
30
466
[256.
RELATIVE MOTION
initial circumstances, and on the forces which act on it: they
must therefore be determined from equations of motion, in terms
of 0, 4, 4, and t, and their values substituted in (145): hereby
we shall have three equations in terms of 0, 4, y, and t, and
their differentials: from these, by integration, 0, 4, and y may
be expressed as functions of t, and the relative position of the
body will be given.
Since w', w, w' are the components along the principal axes
of the absolute angular velocity, they may be determined by
Euler's three equations of motion: and we have
dw,
A + (C−B) w,'wg'′ = L,
dt
dwz
B
+ (A−C) wg'w₂ = M,
(147)
dt
dwz
C
dt
+ (B−A) w½'w,' = N;
where A, B, C are the principal moments, and L, M, N are the
moments of the couples of the whole impressed momentum-
increments on the body.
256.] One first integral of the general equations of motion is
the equation of relative vis viva. Let us investigate the form which
it takes under the present circumstances of motion. For this
purpose take the t-differentials of (143), bearing in mind that
έ', n', ' do not vary with the time: then
αξ
da₁
ૐ
+n'
dt
dt
dẞ1
dt
+5dy
dt
SH
dn
d t
dš
da₂
d B2
dy2
हु
+n
+5
(148)
dt
dt
dt
daz3
d B3
dy3
=
dt
dt
ૐ + n
+š
;
dt
dt
squaring, adding them, and taking the sum of them for every
element of the moving system,
2
dε² + dn² + d¿²
Σ.Μ
dt2
2
2
2
2
= x.mε's { (day) + (das)² + (d)"}
12
dt
dß. 2
dt
2
dt
3
}
+ 2.my'³ { (2/2) + (1) + ()"}
Σ.m
n
dt
dt
dt
3
2 dy2 2
(dr²)² + (d)}; o
+ 2. 8/2 (dy₁)
z.me = {(dn)² +
dt
dt
dt
; (149)
257.]
467
OF ROTATION.
for .mn'' = x.m ('§' = z.m §'n' = 0, because the coordinate
axes, to which έ', n', ' refer, are principal axes. The quantities
in the right-hand members of these equations are subject to
equivalences similar to those given in (80), Art. 39; so that
2
2
d § ² + dn² + dç²
Σ.Μ
2
dt2
A w₁² + B w₂² + c w₂ ²;
2
2
ω
(150)
and thus the equation of relative vis viva given in (129) be-
comes
2
2
d. (A w₂² + B w₂²+cwz²) = 2x.m {(x' — x,) d§ + (x' — Y₁) dŋ + (z' — z₁)d¿}. (151)
If w₁, w₂, w3 are replaced by their values given in (146), the
equation of vis viva will be expressed in terms of 0, 4, and y.
One other remark may be made as to another method which
we might have chosen. We might have substituted the values
given in (148) in the general equations of moments given in
(122), (123), and (124); and these might have been transformed
into their equivalents in terms of angular velocities: we should
hereby have found equations analogous to Euler's, but having
more terms, on account of the compound centrifugal forces and
the forces of transference. The student will find them in a form
equivalent to that which we should have arrived at in M. Quet's
Memoir in Liouville's Journal, Vol. XVIII. I have however
chosen the present method, because it is more direct, and be-
cause our processes are simplified by the use of Euler's equa-
tions, which have already been investigated.
257.] I propose now to apply these equations to some pro-
blems of considerable interest. Suppose a body at or near to the
earth's surface to rotate about an axis which passes through a
fixed point, but which is otherwise unconstrained, the apparent
relative motion to an observer will doubtless be affected by the
diurnal rotation of the earth. It is this effect which I propose
to investigate.
We shall take the fixed point to be the origin of the systems
of axes, and shall suppose the moving system of axes to be
fixed relatively to the earth, so that the angular velocity of this
system is due to the diurnal rotation alone.
Let us moreover suppose a plane (for the present taken arbi-
trarily, so as to admit of subsequent determination) passing
through the fixed point to be that of (§, n): this plane is of
course fixed to and moves with the earth. Through the fixed
302
468
[258.
RELATIVE MOTION.
point let a straight line be drawn parallel to the earth's polar
axis; and let it be projected on the plane of (§, ŋ): this line we
shall take to be the έ-axis, and reckon it positive in such a way
that when the plane is horizontal that direction shall be south-
wards; and the positive direction of the ŋ-axis we shall take to
be such that that direction may be westwards when the plane is
horizontal: hereby, if the plane of (§, n) is horizontal at the
place of observation, we shall have the same arrangement and
the same system of axes as in Fig. 63. Let the positive direc-
tion of of be so taken as to be away from the earth's surface
when the plane of (§, ŋ) is horizontal; and let v = the angle at
which the (-axis is inclined to the earth's polar axis; so that
when the x-axis is vertical, v is the co-latitude of the place of ob-
servation.
Let w, as heretofore, be the diurnal angular velocity of the
earth. Then, taking account of its direction,
ως
- COS v;
Q = w sin v, Θη 0, ως
(152)
and let us suppose the dimensions of the body to be such that
the force of gravity is the same for all parts of it: and let м (for
the instant) be the mass of the body, then x' - Xt, Y' — Yt, Z' — Zt
are constant for all parts of the body; and we may abbreviate
the expressions and denote them thus:
Y' — Yt
Yt = F, z' — Zt = G ;
x' — xt = E,
(ć, m
(153)
and if (,, ) is the place of the centre of gravity relatively to
the moving axes, then
Σ.m {(x' —x₁) d§ + (y' — Y₁) dŋ + (z' — z₁) d§} — м{Ed§+Fdn+&d¿}; (154)
which may be substituted in the right-hand member of (151);
and this being integrated will give the value of the relative vis
viva.
258.] We will take however a special case of motion of this kind;
and consider that of the gyroscope of M. Foucault, of which a dia-
gram has been given in Fig. 21, and of which the construction
and arrangement have been described in Art. 95. The centre of
gravity of the whole machine, which coincides with those of the
several parts of it, is at the centre of the rotating disc; and this
point remains at relative rest, whatever are the rotations of the
disc and of the metallic circles. At this point therefore the
systems of axes originate.
259.]
469
THE GYROSCOPE.
Now M. Foucault contrived in some of his experiments that
the axis of the disc should be constrained to move in a given
plane, fixed relatively to the earth: we will in the first place
consider this case, and investigate the phænomena which the
machine presents to an observer moving with the earth. The
problem in its dynamical form is this:
A heavy body of revolution rotates rapidly about its axis of
figure; its centre of gravity is fixed relatively to the earth, and
the rotation-axis of the body can move only in a plane, which is
likewise fixed relatively to the earth: it is required to determine
the motion of this moving axis when regard is had to the diurnal
rotation of the earth.
We will take the fixed plane, in which only the rotation-axis
of the body can move, to be the plane of (§, n); so that v is the
angle at which this plane is inclined to that of the terrestrial
equator. As the rotating disc is a solid of revolution, a = b;
and if the axis of the disc is that of ', c is the principal moment
relative to it, and we will suppose it to be greater than A. Also,
as the rotation-axis is in the plane of (§, n), 0 = 90°.
259.] We must in the first place investigate the values of
L, M, N which occur in (147), and are the moments of the im-
pressed couples acting on the body relative to its principal axes,
and to which the absolute angular velocity-increments are due.
Now these couples are the same whether the body is or is not
at rest relatively to the earth. We may therefore investigate
the values which they have when the body is at relative rest;
that is, when its absolute angular velocities are only those which
are due to its attachment to the revolving earth. Suppose
therefore the body to be carried round with the earth, and its
'-axis to be inclined at an angle 0 to the terrestrial axis, then
O is constant; let w₁, 2, 3 be the angular velocities of the body
about its principal axes; then,
@:3
W₂ = w cos 0,
½
(w₁² + w₂²) = w sin 0. S
(155)
Euler's equations, which connect these quantities with L, M and
N, become in this case,
A
A
αωι
dt
d w z
dt
+ (C — A) w₂ W COS 8 = L,
(CA) w, w cos 0 = M,
0 = N.
470
[259.
THE GYROSCOPE.
Also we have
d¥
@₁ =
sin sin
dt
@2 = cos sin◊
αψ
аф
d↓
W3
+ cos e
; (156)
dt
dt
dt
w₁ ² + w₂ 2
w2
2+w22
(sin 0)² (dv)²
2
;
dt
dy
and therefore, from the second of (155), =>; consequently,
from the last of (156),
dt
аф
0 ;
dt
$ = a constant
= a (say);
@₁ = w sin a sin 0,
w₂ = w cos a sin 0;
L
w² (c — A) cosa sin ◊ cos◊, )
M =
w² (C — A) sin a sin ◊ cos◊ ;!
and these quantities are the right hand members of the first two
equations of (147). But o² is a factor of both; and our ap-
proximations are carried to such an extent that the squares
and higher powers of are omitted. Therefore L= M = 0;
and N = 0. Thus (147) become
A
dwy
dt
·+ (C−A) w½'wz′ = 0,
d wź
A
-
dt
(C—A) wg'w₁ = 0,
(157)
C
dwg
dt
= 0;
from the last of these equations
@z:
- a constant
= n (say);
the value of n will be perceived hereafter.
(158)
Hence the first two
of (157) become
dwy
dt
C-A
A
+ n w₂ = 0,
(159)
dw2
dt
C-A
n w₁ = 0.
A
Let
C-A
A
n = μ ;
μ;
259.]
471
THE GYROSCOPE.
then integrating (159), we have
wi = KCOS (μt-T),
w=ksin (u t-T); S
(160)
T
where 7 and κ are constants introduced in integration, depending
on the initial circumstances; as the former is determined by the
commencement of the time, and the latter is the greatest value
of w' and w: these we shall also explain hereafter.
do
Since 0=90° and is constant, = 0; therefore, from (146), we
dt
have
dy
@₁ = sin
dy
αφ
dt'
W2
@₂ = cosp
dt'
@3
@z =
; (161)
dt
and from (144) we have
a1
a₁ =
cos pcos y,
az
cos o sin y,
аз
a3 = sin 4,
B₁sin cos y,
&
32
sin o sin y,
B3 = cos 0,
=
7/1 sin y;
Y2
Y₂ =
cos;
Y3
= 0.
Thus, if we substitute for we, w, wg from (152), (145) become
w₁ = K COS (µ t −7) — w {sin v cos o cos—cosv sin p},
w₂ = k sin (µ t− T) + w {sin v sin cos y+cos v cosp},
w3 = n−w sin v sin √ ;
y
(162)
and replacing w₁, w2, w3 by their values given in (161), we have
three equations, by means of which 0, 4, and y may be expressed
as functions of t.
The equation of vis viva however will give a condition more
useful for our present purpose. Equating the values given in
(151) and (154), we have
2
2
G
d. {A w₁² + B w₂² + C w₂²} =
= 2 м {E d § + F d n + & d ( };
but the centre of gravity (§, n, §) is fixed at the origin; therefore
d § = dŋ = d = 0;
n ¿
and replacing w₁, w2, wg by their values given in (161), we have
2
A
dt
(dv)
+c w z 2
= a constant
= k² (say).
(163)
Let us consider the meaning of the constants, which have been
introduced; and suppose a to be the relative velocity commu-
nicated to the body, when t=0; so that w=2, and w₁ = w₂ = 0,
when t = 0; therefore at that time, from (161),
@1
d↓
dt
0; as
the position of the principal axes of έ' and ' in the plane of
472
[259.
THE GYROSCOPE.
(§', n') is arbitrary, let us take that of έ' to be in the plane
of (§, ŋ), when t 0; and at that time also let y
the preceding equations give
W
0 = K COST - ∞ sin v cos yo,
0:
0 = K sin T W COS v,
Ω
n = n—w sin v sin Yo,
c n² = k²:
whence we have
K
k² = w² {1 — (sin v)² (sin √。)²},
tan 7 = cot v sec Yo,
=
Yo; then
(164)
(165)
n = 2+w sin v sin yo,
k = ac²;
so that from the last of (162,) and from (163), we have
w3—2 = w sin v {sin yo-sin };
Ω
A
1 (14)²
2
dt
= c (n² — w¸²).
(166)
(167)
Since o sin v is the component along the axis of έ of the an-
gular velocity of the earth, w sin v sin is the component of that
angular velocity along the axis of '; and w cos v, which is the
other component of the earth's velocity, has no effect along that
axis, because its axis is the axis of : thus (166) shews that the
sum of the apparent angular velocity of the body about its own
axis, and of the component of the earth's angular velocity about
the same line, is constant throughout the motion.
Equation (167) shews that the apparent vis viva of the body
is also constant.
Eliminating between (166) and (167) we have
1 (dv) 3
A
dt
=
w3
ע
cosin {sin-siny} {2n-w(siny-siny)}; (168)
from which equation is to be found in terms of t., The equa-
tion however does not admit of integration in its present com-
plete form. But in all experiments with the gyroscope a very
rapid rotation is given to the body, so that n is very great in
comparison of w: we may therefore omit the last part of the last
factor in the right hand member of (168), and employ the ap-
proximate equation
A
dt
(dv) 3
2
= 2 cow sin v (sin √— sin √).
(169)
It will be convenient to make a slight change in the form of
259.]
473
THE GYROSCOPE.
this equation. is the angle measured in the plane of (§, n)
between the axis of έ and the line of intersection of the planes of
(§; n) and (§', n'). The axis of 'is also in the plane of (§, n)
and is perpendicular to this line of intersection: if therefore v
is the angle between the axes of § and ¿',
¥ = v + 90° ;
and substituting v in (169), we have
บ
dv 12
A
( 9
dt
= 2 cow sin v (cos v - cos v₁).
(170)
(171)
Now this equation is the same in form as (40), Art. 131, which
expresses the motion of a pendulum under the action of a con-
stant force whose line of action is parallel to that of the line of
the rod of the pendulum when it is at rest. Let us then com-
pare the motion of the rotation-axis of the body as expressed by
(171) with that of the pendulum; and let us assume the length
of the pendulum to be unity; so that the constant force, under
the action of which the rotation-axis may be supposed to move,
is
cow sin v
A
و
the line of which is the έ-axis, and is the projec-
tion southwards on the plane of (§, ŋ) of the earth's polar axis.
Now the pendulum vibrates through small arcs to equal dis-
tances on either side of the vertical line; and so will the rota-
tion-axis of the disc vibrate over equal small angles on either
side of the -axis. And as the pendulum remains always at
rest in a vertical line, if it is ever at rest in it; so will the rota-
tion-axis always be at relative rest along the έ-axis, if it is ever
at rest in it. If therefore the rotation-axis is on the έ-axis when
O, it always remains on it, and has no oscillation.
dy
dt
Also, as the pendulum has two positions of rest, one of stable
equilibrium, when it hangs vertically downwards from its point
of suspension, and another when it is balanced on its point of
suspension, the centre of gravity having its lowest and its highest
position in the two cases respectively; so are there two positions
of relative rest of the rotation-axis of the disc, one of which is
of stable, and the other of unstable rest. Now if o and a have the
signs given to them in (171); that is, if the direction of a is con-
trary to that of the earth, then the rest of the rotation-axis will
be stable or unstable, according as the axis coincides with the
positive or negative direction of the -axis. And the contrary
PRICE, VOL. IV.
W
3 P
Ω
Ω
474
[260.
THE GYROSCOPE.
will be the case, when the direction of n is the same as that of
the earth.
If the rotation-axis is in its position of stable rest, and is
slightly disturbed therefrom by an extraneous force, it oscillates
in the plane of (s, n) over a small angle on either side of the
line of rest; and if r is the time of an oscillation,
T
A
T = T
cow sin v
(172)
If however the rotation-axis is in its position of unstable rest,
and is slightly displaced therefrom, it goes farther from that
position and does not return to it, until it has passed through
360° in its plane of motion.
260.] In particular cases these results take forms which are
of considerable interest.
(1) Let the plane of (§, ŋ), in which the rotation-axis of the
body is constrained to move, be horizontal at the place of ob-
servation: and let the latitude of the place be λ; then sin v
cos A; and from (172),
T=T
A
( c n w cos A
(173)
In this case the έ-axis is the meridian line, of which the positive
direction is that towards the south. If the direction of the rota-
tion of the body is the same as that of the earth, the position
of rest of the rotation-axis will be of stable or unstable equili-
brium according as it is drawn from the point towards the north
or towards the south; and if the rotation is contrary to that of
the earth, the rotation-axis will be in stable or unstable rest
according as its direction is due south or due north.
v =
(2) Let the plane of (§, ŋ), in which the rotation-axis is con-
strained to move, be the meridian plane at the place of observa-
tion; then 90°; and the line of relative equilibrium of the
rotation-axis is parallel to the earth's polar axis; and the equi-
librium of the axis is stable or unstable according as the direc-
tion of rotation is contrary to, or is the same as, that of the
earth. In this case
A
13
T=T
Ωω
(174)
so that, cæteris paribus, the time of oscillation is less in this
case than it is when the rotation-axis moves in the horizontal
261.]
475
THE GYROSCOPE.
plane; and generally the oscillations in the meridian plane are
quicker than in any other plane.
(3) These last results however are not limited to the meridian
plane; for sin v = 1 for all planes drawn through the place of
observation and parallel to the earth's polar axis.
(4) If the plane of (§, n) is perpendicular to the earth's polar
axis, v = 0; and т=∞
0; and T∞o; so that the rotation-axis of the solid is
at rest for all positions in that plane.
(5) If the number of oscillations of the rotation-axis in the
meridian plane is determined by observation, r is known; and
consequently, from (174),
ω
ΠΑ
T2
C 2T²
(175)
and thus the angular velocity of the earth may be determined.
(6) If T and r' are the times of oscillation of the rotation-axis
in the horizontal and the meridian planes respectively at a given
place, corresponding to the same value of a, then
T'2
cos λ =
T2
From all these theorems we conclude, that if the phænomena of
the gyroscope are observed with sufficient care, we can by them
determine the meridian line and the altitude of the pole at the
place; and consequently the latitude: we can determine also the
direction of the diurnal rotation of the earth, and, from (175), the
mean length of the sidereal day. All these results then are
confirmations, if they are required, of the evidence of that mo-
tion of the earth which astronomical phænomena suggest to us.
And although the proof of the diurnal rotation, thus acquired,
may not be as palpable as that afforded by astronomical observa-
tion, yet it is not to be rejected as useless, nor is its investiga-
tion to be regarded as idle speculation; for evidence supporting
theories of cosmical phænomena is cumulative; and the value of
any addition to it increases in geometrical ratio.
261.] The gyroscope again may be so arranged that the rota-
tion-axis of the disc shall be constrained to move in a right
circular cone, whose axis passes through the fixed point which
is the centre of gravity of the disc and rings. If we take this
axis for the y-axis, the semi-vertical angle of the cone
a (say); and thus the equations which determine the motion
of the instrument become
3 P 2
476
[262.
THE GYROSCOPE.
w₁ = sin a sin
dy
d↓
do
dy
W2
= sina cos
dt
dt
@:3
=
+ cos a
dt
dt
; (176)
so that the equation of vis viva becomes
2
A (sin a)² ( d )³ +c (w¸² — )
(14)
dt
+C ₪z²) = 0;
and since yı
sin y sin a, Y2 = -
the third of (145), we have
(177)
cos y sin a, y = cos a, from
w3 −2+w sin v sin a (sin - sin ¼¿) = 0;
(178)
from which, and from (177), results may be deduced similar to
those of the last Article, those indeed being only the particular
forms which these take, when a = 90°. Thus the time of small
oscillation of the rotation-axis of the disc
π
{
A sin a
C Ω @ sin v
(179)
262.] Finally, let us take the most general case which the
gyroscope presents to us, and suppose the axis of the disc to be
free from all constraint. Then the problem is, to determine the
phænomena which, by reason of the earth's diurnal rotation, the
motion of this rotation-axis exhibits to an observer placed on
the earth.
I shall suppose a very rapid angular velocity n to be given to
the disc, and the axis to be placed at the beginning of the time in
relative rest. Let the centre of gravity of the disc be the origin
of the systems of moving axes and of the principal axes of the
disc. Let the line drawn through that point, and parallel to the
earth's polar axis, be the (-axis, the positive direction being that
towards the north; so that in this case, v = 0; and
W& = wn
w₂ = 0; ως
w;
(180)
then the plane of (§, n) is parallel to that of the earth's equator.
At the beginning of the time, when the relative angular velocity
of the disc about its own axis is o, and the rotation-axis is at
relative rest, let 。 be the angle at which the rotation-axis, or
the '-axis, is inclined to the (-axis; and let the line of in-
tersection of the planes of (§, n) and of (§', n') be the έ- and
the '-axes; so that = 0, when t=0; at which time also
d Ꮎ
dt
dy
dt
= 0.
0
Now as equations which have already been found are suffi-
cient for the solution of this problem, I shall only refer to them;
262.]
477
THE GYROSCOPE.
and deduce from them the results which we require.
The com-
plete values of the nine direction-cosines given in (144) are true
in this case: and (145) become
ω1
@₁ = w₁' + w sin ◊ sin 4,
W₂ =
W2 w2+ w sin cos 4,
სავ
=
wz' + w cos 0;
(181)
(146) remain in the general form which they have in that place.
From (147) we deduce the results (158) and (160); viz.
ωί
K COS (µt —T),
= K sin (µt — T),
(182)
W3 = n;
since however when t = 0, w₁ = W₂
therefore from these and from (181) we have
0; w3 = 2, and
0 =
and consequently,
0
ωί
a sin 9, sin ut,
}
w sin cos μt;
0
a sin 0, sin ut + a sin 0 sản p,
(183)
@1
W₁ =
@2
W₂ =
o sin cosµt + w sin 0 cos &,
(184)
0
W3 = 2 + ∞ (cos
+ ∞ (cos ( — cos 0¸).
The equation of relative vis viva gives
2
A
{(do)
2
dt
+ (sin (0) ² (dv) ³ }
2
dt
+ C (w₂² — n²) = 0.
(185)
Again from (184) we have
w2 (sin 0,)2
=
(W1 —w sin ◊ sin p)² + (w₂-w sin 0 cos )²
2
2
w₁²+w,² — 2 w sin ◊ (w₁ sin + w₂ cos () + w² (sin 0)²
d¥
w₁²+w₂²-2w (sin ()²
+ w² (sin 0)2;
dt
therefore
dy
2
w₁ ² + w₂ ² =
2
w² {(sin 0¸)² — (sin 0)²} +2w (sin 0)²
dt
and
@32
{2 + w (cos 0 —cos 0。)}² ;
therefore
2
2
2
A (wï³ + w₂²) + C wz² = A w² {(sin 0¸)² — (sin 0)2} +2a ∞ (sin #)²
w
dy
dt
+cn²+2cnw(cos — cos 0)+c w² (cos — cos 0)²; (186)
C
but the left-hand member of this equation is by the equation of
vis viva equal to cn2: consequently
478
263.]
THE GYROSCOPE.
A (sin 02 dy
dt
cos 06 - cos 0
2
dv
dt
{2cn+aw(cos0+ cose)+cw(cose — cos 0)²}; (187)
eliminating between this equation and (185), replacing w3
by its value given in (184), and omitting terms involving w², we
have
2
A²
2
2
(sin 0)³ (de) * = ca (cos e, — cos 0) {a w (sin 8)² + aw (sin 0.
2
+co (cos — cos 0) + c ∞ (cos — cos 0)2}; (188)
which equation determines in terms of t; and when this value.
of is substituted in (187), that gives y in terms of t.
263.] If the gyroscope does not rotate when t = 0, that is, if
ΩΞ
do
O, then = 0; so that does not vary, and is constantly
dt
equal to 06: therefore from (187),
dy
dt
= 0; hence also
=0:
@3
so that notwithstanding the diurnal rotation of the earth, the
axis of the disc remains at relative rest.
If however is very great, and this is the ordinary case in
the gyroscope, the most important term in the last factor of the
right-hand member of (188) is ca (cose-cos 0), as the other
terms involve w, which is a small quantity: but these must
not be wholly neglected, because such an omission is equivalent
to the omission of w, which is the angular velocity of the earth;
and because in such a case the only circumstances which would
d Ꮎ
satisfy (188) are
= 0, 0
¤; and the axis of the gyroscope
dt
would be at relative rest. Of the small terms, however, we may
without sensible error neglect those which have periodical values,
and retain those which are constant; for we shall hereby obtain
the general effect. Thus (188) becomes
2
A² (sin 0)² (de) ³
2
2
= c(cose-cose){ca (cose-cose) +Aw(sin0)²+co (cose,)2}
c² n² (cos 0¸ — cos 0) cose-cose +
cose)
{cose-cos
Now let us suppose a to be an angle such that
0
2
A (sin¤¸)²+c(cose¸)² w
C
.(18
264.]
THE GYROSCOPE.
479
A (sin 0)2+ c (cos 0)2 w
=
cosa cos 0。-
W
which equation is possible, if
Ω
C
Ω
(190)
is positive, for all values of 0
between 0 and a limit a little less than 180°; and if
W
is nega-
Ω
tive for all values of 0, between 180° and a limit a little greater
than 0. Thus (189) becomes
A
1³ (sin 0)² (de)
2
2
= c²n² (cos 0¸ — cos 0) (cos 0 — cos a) ;
(191)
so that always lies between 0, and a, which, as (190) shews,
are two angles nearly equal: and consequently the inclination
of the rotation-axis of the disc to the x-axis is not constant, but
varies within limits very near to each other.
From (191) by integration we have
0
cos Oo + cos a
cos 0
2
cos 00- cos a
-+
2
ΣΩ
COS t;
(192)
A
and replacing a by its value
e
cos = cos 00
2
2
A (sin 0。)²+c (cos 00)² w
2 c
Ω
{
ΣΩ
1 - cos cat}. (193)
A
Let us introduce the preceding value of cos e into (187): then
omitting terms involving the square and higher power of @,
dy
dt
2
A (sin 06)²+c (cos 00)2
2A (sin 0)2
2
A (sin 0。)² + c (cos ∞。)²
2 A (sin 00)2
Ꮗ
{
A
CQ
1 — cos
t
{w t
ΠΩ
sin
t
; (194)
C Q
A
A W
the limits of integration being such that y=0 when t=0.
If then we consider the first terms in (193) and (194), which
are the principal terms, it appears that the rotation-axis of the
disc revolves uniformly round an axis parallel to the earth's axis
in a direction contrary to that of the earth, and that it is inclined
to this axis at an angle almost constant: and besides this general
precessional motion the axis has also motion of nutation both
parallel to and perpendicular to the plane of the earth's equator;
and that the periodic time of these nutations
J
2ПА
ΣΩ
; so that the
periodic time is shorter the greater is the initial angular velocity
of the disc.
480
[264.
THE GYROSCOPE.
264.] From this investigation then the following results
follow:
(1) If the disc of the gyroscope has not any initial angular
velocity, it remains at relative rest with the earth, whether the
earth rotates or not.
(2) If the disc rotates with a very rapid angular velocity, and
is placed in a position of relative equilibrium with the earth,
then that equilibrium would continue if the earth did not rotate;
but if the earth rotates, the axis of disc has a relative motion.
(3) And the direction in which this motion takes place does
not depend on that of the angular velocity of the disc, but is
always opposite to that of the angular velocity of the earth; and
consequently, if it is observed, it indicates the direction in which
the earth rotates; and thus its motion supplies evidence of the
rotation of the earth.
(4) The angle of inclination of the rotation-axis of the disc to
the axis of the earth is nearly constant throughout the motion
of the disc: there are however small nutational variations of
this angle as well as of the precessional velocity of the axis, the
periodic time of which decreases according as the angular velo-
city of the disc increases.
Here I must conclude this subject; for it would be quite be-
yond the scope of the present work to enter further into the
details of it. Let me however refer the student to the Memoir
of M. Quet, already alluded to, which is contained in Vol. XVIII
of Liouville's Journal, and to which I have been largely indebted
for the preceding Articles. This memoir also contains an inves-
tigation of the motion of the several rings of the gyroscope; as
well as a further inquiry into the interpretation of the several
equations, when small terms, which we have omitted, are taken
account of.
I may also refer the reader to another mathematical investi-
gation by M. Yvon Villarceau, which will be found in Vol. XIV,
p. 343, Nouvelles Annales des Mathématiques; Paris, 1855.
265.]
481
ON THE THEORY OF MACHINES.
CHAPTER IX.
ON THE THEORY OF MACHINES IN MOTION.
265.] In Section 4, Chapter III, of the present Volume, a com-
bination of the principle of virtual velocities and of D'Alem-
bert's principle led to the general equation of vis viva of a system
of moving particles; of which the form is
Σ.m v² — Σ. m v² = 2 z.
Σ.mv² Σ.m z . fm ( x d x + x d y + z dz) ;
(1)
the right hand member being a definite integral, and the limits
of integration being the values of the several quantities at the
times t and to respectively, v and v, being the corresponding
values of v at those times. For the truth of this theorem it is
necessary that neither the connections between the several par-
ticles of the system, nor the acting forces, which produce the
impressed increments in the right-hand member of (1), should
be explicit functions of the time t. It is also necessary that
Σ.m (x dx+x dy + z dz) should be an exact differential; and in
that section several cases are given in which these conditions
are satisfied.
Now the geometrical relations of the several parts of a ma-
chine (I use this word in its ordinary meaning) and the forces
which act on a machine satisfy all these conditions; so that the
preceding equation is applicable to them. I propose to apply it
to the theory of their motion; for this is our immediate subject
of inquiry. We shall hereby ascertain not only general proper-
ties of them, but also measures of their effects and criteria of
their goodness or efficiency. Mathematical and precise defini-
tions will be given of these terms; and we shall thereby be able
to reduce their values to measurement and to number; and this
is in such cases the end of all exact scientific inquiry.
It will be convenient however to put (1) into a slightly dif
ferent form. The right-hand member of (1) consists of a series.
of groups of terms which arise from an equal number of different
acting forces. Let F that moving force of which mx, mx, mz
—
PRICE, VOL, IV.
3 ૨
482
[266.
ON THE THEORY OF MACHINES
are the axial components; and let ds be the projection on the
line of action of F of the distance through which its point of ap-
plication moves in the time dt; so that
m (x dx + x dy+zdz) =
Fds.
(2)
Let this transformation be made for all the acting forces; then
the equation of vis viva becomes
z.mv³ — 2.mv7 = 2z. føds
Σ
;
(3)
the right-hand member being a definite integral with limits cor-
responding to the times t and to; and r being the momentum im-
pressed on the machine. It is this equation which we shall apply
to the theory of machines in motion.
266.] A machine is an instrument by which the momentum
arising from certain acting forces, and applied at one or more
definite points in given lines of action, is transmitted to other
points, and is at them and along given lines communicated to
other matter.
A machine generally consists of many pieces, which are con-
nected by articulations of various kinds, by axles or shafting
common to two or more wheels, by sliding and rolling contact,
&c.; the connections being such, that when one piece moves by
the action of a force, many or all the other pieces also move.
The point at which the momentum of a moving force, or
power as it is popularly called, is applied, is called the driving
point: and the point at which the transmitted momentum is
applied is called the working point of the machine; and the
series of pieces which connect these two points is called the
train.
Let the object of a machine be clearly understood. It is to
enable us conveniently to apply at a certain point in a definite line
of action and in a certain way momentum which arises from a cer-
tain power or moving force applied elsewhere. Thus it transmits
momentum. It neither generates it nor destroys it. The action of
it is in complete accordance with the law of inertia. All the
momentum which is communicated to it either has been or may
be abstracted from it. Let us now consider the mode in which
(3) enables us to trace the relation between the momentum
communicated to a machine and that which it either gives or
is capable of giving at its working point. The sign of summa-
tion in the left-hand member of (3) includes all particles to
267.]
483
IN MOTION.
which velocity is communicated by the action of the moving
forces; and thus includes not only the moving parts in the
train of the machine, but also the fixed framework, the sup-
ports, the ground or base on which it rests, and even the
particles of the surrounding air, if so be that velocity of vibra-
tion or any other velocity is communicated to all or any of them.
And the left-hand member expresses the excess of the vis viva
of all the moving particles at the time t over the vis viva at the
time to; that is, it is the increment of the vis viva of the whole
system in the time t― to.
The right-hand member, viz. 2z.fr ds, expresses
ds, expresses twice the
sum of all the definite integrals of the products of each im-
pressed momentum-increment (F), and the space (ds) through
which its point of application moves along its own line of ac-
tion in the time dt; the limits of integration being the values
of these several quantities at the times t and to respectively.
This product rds is called the element of work due to the force
F; and the definite integral rds, which is the sum of these ele-
ments, is called the whole work of the force in the time t-to;
and in reference to this impressed work the force is called la-
bouring force.
267.] Of these terms and definitions I will take some simple
instances, and shew how exactly they coincide with our ordinary
notion of work, which involves resistance overcome and space
described.
Let us suppose the force to be the earth's attraction, acting
on a mass m, of which w is the weight, or the impressed moving
force. Let us suppose this weight to move from rest in a vertical
line towards the earth, and ds to be the element of its path in
the time dt: then wds is the element of work in the time dt.
Let t be the time during which the body falls, and let h and z
be the vertical distances from the earth's surface when t=0 and
t=t respectively.
The work in the time t =
= fw ds
= w(h−z);
(4)
and is equal to the product of the weight and the vertical dis-
tance through which it has fallen. This then is the work which
has been impressed by the labouring force of the earth's attrac-
3Q 2
484
267.]
ON THE THEORY OF MACHINES
tion in the time t; and hence an equal amount of labouring
force must act, and an equal amount of work must be spent
on w, so as to put it into its original place. This work in a
given time is measured by the product of the weight moved and
the vertical distance through which it is moved. Work is im-
parted to a heavy body as it is removed further from the earth's sur-
face, and is taken from it as the body moves nearer to the earth.
Consequently, a heavy body has a greater amount of work in it
at a greater altitude than it has at a less. The absolute amount
of work too remains the same: whatever change it undergoes in
the change of place of the body, an equal change in an opposite
direction takes place when the body returns to its original place.
This result is evident from (4); for if h is less than s, the work
is negative, so that work must be imparted to the body to raise
it to a greater altitude.
Suppose again that we have a series of weights, W₁, W2, ...,
whose altitude above the earth's surface at the times to and t
are severally h₁, 21, ha, 2, ...; then the work in the time t-to
Z2,
= Σ.w (h−≈)
= (π-z) x.w,
=
(5)
if h and are the vertical distances of the centre of gravity of
the weights above the earth's surface at the time to and t. So
that the work depends on the vertical distances between the places
of the centre of gravity of the system of weights at the times
to and t respectively. If h is greater than Z, so that the centre
of gravity is lower, work is taken by the earth from the weights;
if h is less than 2, work is given by the earth to the weights,
and their work is increased: if h z, the work of the weights.
is unaltered.
Equation (4), which gives a mathematical definition of work,
enables us to determine an unit of work, and thereby to measure
other work, although it is only founded on the application of the
general principle of vis viva to the particular case of a heavy
body moving near to the earth's surface. Work done in the
time t the product of the weight moved and the vertical dis-
tance over which it is moved. Consequently, if the weight
moved is an unit-weight, and the vertical distance over which it
is moved is an unit-distance, the product of these two quantities
is the unit of work and we have the following definition:
An unit of work=an unit-weight × an unit of vertical distance. (6)
=
:
268.]
485
IN MOTION.
In Britain we express distances in terms of feet, and weight
in terms of pounds avoirdupois; and consequently the British
unit of work is one pound raised one vertical foot; this is called
a foot-pound; and work, which is defined by the product of
numbers expressing the number of pounds raised, and the num-
ber of feet through which they are raised, is said to consist
of such and such a number of foot-pounds. Thus, if 100 pounds
are lifted through 5 feet, the work done is 500 foot-pounds; and
if 50 pounds are lowered through 10 feet, work to the amount.
of 500 foot-pounds has been taken from them and communi-
cated to something else.
In France distance is expressed in mètres, and weight in
kilogrammes; and the unit of work is called a kilogram-
mètre.
Now this mode of estimating work done is applicable to not
only machines, but also to living agents, as a man or a horse.
Thus, if a man lifts a weight w through a vertical height h in a
given time, wh, expressed in foot-pounds, is the work done by
him in that time. So it is found that a man working on a
tread-mill will raise himself through 10,000 feet in a day of
8 hours; and, taking the weight of his body to be 150lbs, his
work in the day is 1,500,000 foot-pounds.
I should observe also that there is a peculiar unit of work
called a horse-power, in terms of which the work of a steam
engine and of other machines is ordinarily estimated. In Britain
a horse-power is 550 foot-pounds in a second of time; that is,
is 33,000 foot-pounds in a minute. In France the term "force
de cheval” means 4,500 kilogrammètres in a minute; and this
is equivalent to 32,549 English foot-pounds.
268.] Having said thus much on work, and the mode of mea-
suring work, I will return to the consideration of the equation
of vis viva, given in (3), and introduce into it these definitions;
in which case the theorem of vis viva may be enunciated as
follows:
In the motion of a system of particles, subject to connections
which are independent of the time, and under the action of
forces which do not explicitly involve the time, the increment of
the sum of the vires vivæ of all the particles in a given time is
equal to twice the sum of the work communicated to the system
during the same time by all the acting forces.
486
[268.
ON THE THEORY OF MACHINES
Of the forces which act on the several parts of the machine
and produce the work, there are two kinds, which must be dis-
tinguished; there are moving forces, and there are resisting or
retarding forces. The former are called positive forces, or the
moving powers acting on the machine, and the effect of them is
the moving work of the machine, and is that which is impressed
at the driving points. The latter are called negative forces, and
correspond to the whole resisting work of the machine; that is,
not only to the work done by the machine at its working points,
but also that due to the friction of the pieces of the machine
one against another, or against the fixed framework or the sup-
ports; that due to the stiffness of cords and connecting bands;
that due to the vibrations of the several particles of the machine,
to the vibrations of the supports, of the ground on which they
rest, of the surrounding air. All these are causes of work, and
the work due to them is resisting work, which acts in a contrary
direction to the moving work of the machine. These forces are
by Carnot and by other writers distinguished according as the
angles between their lines of action and the path described by
their points of application in the time dt are acute or obtuse.
Let wm be the moving work of all the forces acting on the
machine during the time t-t。; and let w, be the resisting work
due to the resisting forces in the same time; so that
fr ds
F ds Wm
= WmW, ;
then equation (3) takes the form
Σ. m v² — z. m v
Σ.m vo 2 = 2 (Wm—Wr) ;
(7)
so that the increment of vis viva is equal to twice the excess of
the moving work over the resisting work.
The resisting work w, consists of two parts; firstly, the useful
work, which we will call w, which acts at the working points,
and the production of which is the object of the machine; and
secondly, the lost work, which we will call w;; which is spent on
the friction of the pieces, the vibrations of the several particles
of the framework, the ground, and the surrounding air, as we
have just now explained; so that
Wr Wu + Wi.
(8)
Equation (7) embodies the theory of all machines in motion;
and consequently a careful consideration of it will indicate the
269.]
IN MOTION.
487
conditions which good machines ought to satisfy, and the general
principles of their construction.
A's to the circumstances which cause the lost work (wr), I
would observe, that the work due to friction may be much
lessened by means of a proper choice of materials, and by
unguents, &c., and that in all cases the quantity of it can be
calculated. The work spent on the vibrations of the several
parts of the machine cannot be calculated in the present state
of the science of molecular physics; but the vis viva of the
several particles due to this work, which enters in the left-hand
member of (7), is nearly, if not quite, equal in quantity; so that
the "lost work" due to this cause will disappear in the application
of the equation. The work which is spent on the motion commu-
nicated to the supports, and which is eventually conveyed to the
earth and lost, varies much in different machines, but is some-
times very considerable. It cannot however generally be cal-
culated. That which is lost in giving motion to the particles of
the surrounding air is very small, and consequently may be
neglected.
269.] Let us now trace the action of a machine in motion
with reference to the equation (7), which connects the vis viva
with the work done. When a machine begins to move, the
element of moving work is greater than that of the resisting
work which is brought into action in the same time, and vis
viva increases; this increase of vis viva continues until the
elements of moving work and resisting work due to the same
element of time are equal; then there is equilibrium between
these two work-elements, no further increase of vis viva takes
place, and the vis viva becomes a maximum.
2
Now the subsequent motion of the machine may be either
uniform or periodic let us suppose it to be uniform; and let to
be the time at which this state is reached; and let z.m v¸² be
the vis viva which the machine then has. As the velocity of
every particle continues the same, whatever is the time expressed
by t- to, z.m v² = x.mv2; consequently, from the right-hand
member of (7), we have Wm W,, and the whole moving work is
equal to and becomes resisting work; and we have
w
W m
= Wu + Wi;
(9)
and the machine transmits the whole moving work to the points
at which the resisting work, both useful and lost, is applied, with-
out loss or modification.
488
[269.
ON THE THEORY OF MACHINES
When the motion of a machine is not uniform, but periodic,
as it is in an ordinary steam engine, there is a continual increase
or decrease of vis viva; and the element of moving work is not
equal to the element of resisting work for every time-element.
If however we consider an interval of time, at the beginning and
the end of which the velocities of the different elements of the
machine are the same, the left-hand member of (7) vanishes,
and consequently wm W; that is, the whole moving work
applied during that interval is equal to the corresponding whole
resisting work. And thus, although the elements of the moving
and the resisting works for any time-element are not equal, yet
the sums of these elements are equal, for the interval at the
beginning and end of which the vis viva of the machine is the
same. And as this result is true for any one such interval, so
is it true for the time of many such intervals.
Wm
If then we consider a machine during the whole time that it
moves, that is, from the instant at which it begins to move, to
that at which it comes to rest, Σ.m v² = z.m v² O, and con-
sequently, corresponding to that whole time, ww; so that
whatever is the motion of the machine, whether it is uniform
or periodic, or of any other nature, the work due to all the
moving forces for the time during which the motion continues
is equal to the whole resisting work developed in the same
time.
It also appears from equation (7), that if the vis viva z.m v²
at the time t is greater than the vis viva z.mv2 at the time to,
W is greater than w,; and that if z.m v² is less than .mv²,
W is less than w,-- Thus the vis viva of the machine increases
or decreases according as wm is greater or less than w,; and the
vis viva remains the same, if w„ = w,. If therefore the moving
work is greater than the resisting work, the vis viva of the
machine increases, because the excess of the moving over the
resisting work is being stored as vis viva in the several parts of
it. Whereas, if the resisting work is greater than the moving
work, that excess is taken from the machine in the form of a
loss of vis viva. Hereby we see the reason why a machine,
whose motion is not uniform, will yield a resisting work equal
to a moving work, if these are considered during the whole time.
of motion. As the machine starts from rest, wm is greater than
w,, and work is being stored in the machine in the form of vis
viva, until the machine reaches a state at which its vis viva is
271.]
IN MOTION.
489
a maximum.
And as the machine returns to its state of rest it
loses vis viva, which becomes resisting work, and eventually
gives out just as much as it received at first; so that if wm and
w, are estimated through the whole motion, wm W. Thus it
is that a machine does not create work; it receives it; it may
store it for a time; but finally it yields exactly the same quan-
tity as has been given to it.
270.] Whenever the differential of the vis viva vanishes, then,
from (3), z.F ds = 0; that is, the vis viva of the machine is
a maximum or a minimum, or does not vary with a small motion
of the machine, when all the forces, moving and resisting, acting
on it at the time are in equilibrium; in equilibrium, I say; be-
cause .Fds is the sum of the virtual moments of the forces ;
and when this sum = O, the forces are in equilibrium by reason
of the principle of virtual velocities. Now .m v² is a maximum
or a minimum according as z.F ds changes sign from + to
or from
to +, as it passes through zero. In the former
case, the moving forces are greater than the resisting forces
before equilibrium takes place, and after equilibrium the resist-
ing forces become greater than the moving forces; these cir-
cumstances indicate a state of stable equilibrium. In the latter
case, all these circumstances are reversed, and the state is that
of unstable equilibrium. So that the vis viva of the machine is
a maximum or a minimum according as the equilibrium of it at
the instant is stable or unstable. If the forces are in equili-
brium, and no change of sign takes place in z.F ds, the vis viva
of the machine is neither a maximum nor a minimum; in this
case the equilibrium cannot be said to be either stable or un-
stable. If an infinitesimal motion of the machine takes place,
then z.Fds will pass through 0 from + to +, or from
according as one or other of the two directions of motion is
taken; and thus the equilibrium is said to be neutral.
Σ. Fds 0, because Σ. Fds is a constant, then the vis viva is
constant, and the equilibrium of the machine is continuous.
to -
If
271.] The goodness or the efficiency of a machine depends
on the amount of useful work yielded by it in comparison of
the moving work; so that the efficiency is mathematically de-
fined by the following equation:
the efficiency of a machine
PRICE, VOL. IV.
Wu
W
TV m
3 R
(10)
490
[271.
ON THE THEORY OF MACHINES
w,,
The superior limit of this ratio is unity, the ratio in all machines
being less than 1; and the nearer the ratio approaches to 1, so
much more efficient is the machine. Thus, a machine is per-
fectly efficient when w₁ = W„, that is, when the useful work is
equal to the moving work. This however is a theoretical state,
which is never found in practice. And why? Because in con-
sidering the work done by a machine during the interval of
time at the beginning and the end of which the vis viva of the
machine is the same, so that no increase of vis viva, and con-
sequently no increase of work, is stored in the machine, w, = Wm;
and consequently,
W = Wu + W¿:
112
ין
(11)
thus part of w goes to w, which is the work lost in the various
ways which have been enumerated and explained in Art. 268
above; and w, can never be made wholly to disappear. We
must however diminish it as far as possible, so that we may be as
nearly as possible equal to wm. All frictions therefore of rolling
and sliding, which are not absolutely necessary, must be avoided;
and when they cannot be avoided, the spaces over which they
take place must be, as far as possible, reduced, and their work
must be diminished by means of hard and polished materials
and of unguents. All vibratory motion is also to be avoided;
for although, as we have before observed, vis viva may be pro-
duced in parts of the machine which will be equal or almost
equal to the work spent on it, and which may be retransferred
to work, yet oscillatory motion of the parts produces oscillatory
motion in the molecules of the parts, which is propagated from
molecule to molecule by the elasticity of the materials through
the supports, the framework, and the surrounding air, and is
finally lost in the mass of the earth without the production of
any useful work. We must also avoid all sudden impulses and
blows; for they will be the cause of not only intense vibrations
in the molecules of the machine and of its supports, but of
changes of form of the parts and supports; which, although
perhaps not considerable in extent, may consume a large quan-
tity of work, because the resistance to such changes is very
great. In short the most efficient machines move without noise,
without displacement of any support, and we may almost say
without a suspicion on the part of the observer of any strain or
effort in any of its parts, or of work done by it.
Notwithstanding all precautions, if w,, is the useful work to
272.]
491
IN MOTION.
be done by a machine, wm, the moving work, must be greater
than it; for we can never reduce w, to zero. We perceive then
the great error of those who are in search of perpetual motion.
They propose to invent a machine by means of which useful
work may be done without any moving work; or at least at the
expense of moving work less than the useful work. This is impos-
sible; for it is inconsistent with the truth of the equation of vis
viva; and, as all known mechanical forces satisfy that equation,
they cannot effect it *.
272.] Another property of a good machine is uniformity of
the velocity of its several parts; so that when the machine is in
its full working state, the variation of the velocity of every part
during the motion may be as small as possible. This condition
is desired, not because the transmission of work is rendered
more effective thereby, nor because the amount of useful work
is increased; but because, from an industrial point of view, the
quality of the useful work is better, being more regular. Now
in many machines not only is the moving work communicated
irregularly at the driving point, as in a steam engine; but also
the useful work at the working points is irregularly applied; as
in coining, punching, shearing, slotting, &c. machines; so that
WW, is a quantity varying within limits, which are consider-
ably distant, and thus the velocities of the several parts of the
machine are far from uniform. It may also be remarked, that
irregular motions produce vibrations of the molecules, which
cause loss of work. It is consequently important to devise means
by which uniformity may be obtained as far as possible; and
the equation of vis viva, as we proceed to shew, suggests a
method.
In most machines, by means of the train, the velocities of the
different parts have constant ratios which depend on their rela-
tive positions, if we omit the forced molecular vibrations; and
thus the velocity of any molecule may be expressed in terms
of that of another which is arbitrarily chosen. Let us, to fix our
thoughts, take this last to be the driving point, that at which
the moving work is applied. Let v and v be the maximum and
minimum velocities of it which succeed each other, t and to
being the times at which these respectively take place; and let
* See a curious paper on this subject by G. B. Airy, M. A., &c., the present
Astronomer Royal, in the Cambridge Philosophical Transactions, Vol. III.
3 R 2
492
[272.
ON THE THEORY OF MACHINES.
Wm and w, be the moving and resisting works of the machine in
the time t-to. Let the velocities of any molecule m at these
times be respectively av and av。; and as this will be true for all
particles, a varying from particle to particle, the equation of
vis viva takes the form
so that
(v² —v₂²) z.m a2 = 2 (WmW);
W,
2
v - vo
2 W m
v + vo Σ.m a²
(12)
(13)
Our object is to make v-v。 as small as possible; so that the
variations of the velocity of the driving point, and consequently
of all the particles of the machine, may be as small as possible.
Ww, is a known quantity, varying with the variations of the
moving and resisting works; which however cannot be so
arranged as to make v-v, small. But the difference v-v。
will be less, the greater z.ma² is; this therefore suggests the
addition to a machine of large masses moving with great velo-
cities. These are generally introduced in the particular form of
large and heavy wheels moving with high velocities, having the
greater parts of their mass in a thick rim of a considerable radius ;
because thereby the vis viva of the wheel is increased; they are
called flywheels; and are generally placed on an axle near to
the moving force of the machine, when the variations of the
moving work are great; and near to the working points when
the variations of the work at their points are great. Being
wheels moving on fixed axles, the centre of gravity of them
remains fixed during the motion, so that no work is consumed
on its motion; a small quantity of work is spent on the fric-
tion at the bearings, and on the vibrations which are commu-
nicated by their motion to the surrounding air.
Flywheels not only give steadiness to a machine, by prevent-
ing great fluctuations of velocity, but when the variations of
the moving and the resisting work are not periodic, they are
employed to prevent too great an accumulation or a diminution of
vis viva; this they effect by bringing into action a piece of
mechanism called a governor, or a regulator, whereby the supply
of moving work can be varied. Such is the sluice or valve,
which adjusts the opening through which water is supplied to a
water-wheel; the throttle valve, which regulates the space in
the steam pipe through which steam is supplied to the cylinder;
the damper, which regulates the supply of air to a furnace. The
273.]
493
ON MECHANICAL UNITS.
governor which is most commonly in use is Watt's centrifugal
governor, of which a drawing is given in fig. 62, and which is
described in Art. 234.
Flywheels also serve a most useful purpose in those machines
wherein a large amount of work is required at the working
points, not continuously, but on a sudden, as for instance in
punching machines; because they contain a large quantity of
vis viva which has been communicated from the moving work,
has been stored in the form of vis viva, and is ready to become
useful work at the working points.
273.] The conclusions of the preceding Articles are drawn
from the equations of vis viva, (3), (4), and (7), the members of
which severally are vis viva, work, and a weight moved through
a given vertical space. Now if an equation is intelligible and
applicable to any useful purpose, its members must be homo-
geneous, that is, the quantities which the two members consist
of must be of the same kind; if one member is linear space
only, the other must equally be so; and consequently if an
equation involves space, time, and mass, which is the case with
most dynamical equations, the dimensions of these must be the
same in both members. I propose to prove the homogeneity of
the preceding equations; and in so doing certain general prin-
ciples will be stated which are of general application; and I
shall incidentally prove that other fundamental equations are
equally homogeneous.
Certain units must be assumed, in terms of which, by means
of number, quantities of the same kind may be expressed.
I assume an unit of space, and an unit of time. Then, from
the ordinary definitions of dynamics, we have the two following
consequences :
The unit of velocity is an unit of space passed through in an
unit of time.
The unit of accelerating force is that which impresses an unit
of velocity in an unit of time.
ds
Since velocity = velocity is of (1) dimension in space,
dt'
and of (—1) dimension in time.
d2s
Since accelerating force =
dt2
29
accelerating force is of (1)
dimension in space, and of (-2) dimensions in time.
We require the definition of another quantity, viz., that of
494
[273.
ON MECHANICAL UNITS.
mass.
Now in these and other applications, the matter which
moves is terrestrial matter, of which a property is, that it attracts
directly as its mass, and inversely as the square of the distance.
Let an unit of mass be placed at, say, a; and let m units be
placed at, say, B; the distance between a and в being r: then,
by the law of gravitation,
d2r
ԴՈՆ
dt2
2.2
d2r
.'.
m =
p2
dt2
(14)
and consequently an unit of mass is that which produces an unit
of accelerating force at an unit of distance.
Thus mass is of (3) dimensions in space, and of (—2) dimen-
sions in time.
Hence we have the following:
Density
mass
; now volume is evidently of (3) dimensions
volume
in space; consequently density is of (0) dimensions in space and
of (-2) dimensions in time.
=
Since weight Mg; weight is of (4) dimensions in space, and
of (-4) dimensions in time.
Since work = M.v² = 2wh; work is of (5) dimensions in
space, and of (-4) dimensions in time.
Hence, if the units of space and time are changed, the mem-
bers expressing the quantities above will have to be changed in
the ratios just now assigned. Suppose for instance the units of
space and time to be diminished in the ratio of 1 to 2; so that
what was s becomes 2s, and what was t becomes 2 t. Then the
new density is expressed by a number which is one-fourth of
that which expresses the former density; weight is not altered;
work is expressed by a number which is twice that which ex-
presses the former work.
On these principles all our mechanical equations are homo-
geneous. Thus, if w is a weight lowered through a vertical dis-
tance = h
h; and v is the velocity given to a mass = M by the
work thus obtained: from the preceding Articles we have
the work = 2 wh = M.v²
M.v² = vis viva;
wh is of (5) dimensions in space, and of (-4) dimensions in
time; and the dimensions of м.v² are the same.
273.]
495
ON MECHANICAL UNITS.
1
Consider again the equation 8 = g t²; s is of (1) dimension
in space; gis of (1) dimension in space, and of (-2) dimensions
in time: hence the equation is evidently homogeneous.
Consider also the equation v2 = 2gh; each member of this
is evidently of (2) dimensions in space, and of (−2) dimensions
in time.
It is unnecessary to cite more examples. In conclusion, how-
ever, I would observe on the advantage of testing the homoge-
neity on these principles of all dynamical equations in their
original forms; for that remains a quality of them whatever are
the operations to which they are subjected; and consequently if
they are also homogeneous in their ultimate state after a series
of operations, it affords a presumption that the operations have
been correctly performed.
These principles are also capable of a much wider application;
work, in the meaning of the word here given, is produced by
elastic action, by magnetism, by heat, &c. We are hereby en-
abled to reduce all these to a comparison with mechanical work.
Thus, for instance, we have now a mechanical equivalent for
heat; if the temperature of one pound of water be raised one
degree Fahrenheit, it has been determined by Mr. Joule that
the work thus produced is equivalent to 772 foot-pounds. The
investigation of the equivalence of mechanical work to that done
by the agency of heat, electricity, magnetism, &c., is a matter
of extreme interest and great importance. It is however too
large to be entered on in this volume, even if it were appropriate
to do so; for our limits would not allow justice to be done to
it; and we can only refer the reader to the various treatises and
memoirs on these subjects.
496
[274.
THE MOTION OF
CHAPTER X.
THE MOTION OF ELASTIC BODIES.
274.] The principles and laws of motion have thus far been
applied to rigid bodies, and to systems of rigid bodies, the con-
stituent molecules of which have been assumed to be in a state
of relative rest during the motion; and the equations of motion
by which problems have heretofore been solved have been de-
duced from these principles thus restricted. Our purpose is to
apply them more generally. Ere, however, we do so, there are
two reasons why we should repeat as concisely as possible the
modification of the equations which this assumption of the rela-
tive rest of the constituent molecules introduces. (1) Because
we have come to the end of our investigations on that subject,
and it is good once more prominently to restate the conspicuous
principle of the process so frequently employed: and (2) because
in the present chapter we shall investigate equations expressing
the motion of a particle which is not at rest relatively to its
neighbouring particles, all being constituent molecules of a body;
and our research will include the varying form of flexible bodies,
(as they are called,) the molecules of which move relatively to
each other; and our conception of such motions will be more
exact when they are contrasted with those of the molecules of a
rigid body in their chief differences.
The equations of motion of a rigid body are found by the
following process: Let dm be an element of the body, and let
(x, y, z) be its place at the time t, relatively to a system of coor-
dinate axes fixed in space. Now this particle is supposed to be
under the action of certain external forces, whereby a certain
velocity or velocity-increment is impressed on it. In consequence
of this external force it would have a definite expressed velocity-
increment if it were alone, and thus free from all constraint from
its surrounding molecules. As it is not free, the constraints
enter as other forces, which, affecting its motion, produce a change
of its expressed velocity-increments: these constraints we con-
sider as internal forces, which produce their own effects; and
these effects modify those which would otherwise take place.
274.]
497
ELASTIC BODIES.
And consequently, if x, y, z are the axial components of the ve-
locity-increment impressed on dm by the external forces, and if I
is the resultant of the velocity-increment due to all the internal
forces or constraints, of which a, ß, y are the direction-angles;
then the equations of motion of dm are
dm
{
d2x)
X
+ I cos a = 0,
dt2
dm {›
d2y
Y
dt2
S
+ I cos B
= 0,
(1)
dm
{
d2z)
2
+icos y = 0;
dt2 S
from which also arise three other equations, which express the
rotation of dm; viz.
{ ≈ ( x − 1² x) —
Y
121) } + 1 (y cos y − z cos 3) = 0,
am {y(
d2 z
Z
Z
dt2
(x −
d² y
dm { z (
x ( z
−
1² 2 ) }
z
dt2
dm
{x(
n { x ( x − d² y ) —
dt2
d! \ -
y (x
—
dt2
d²x)}
dt2
d2
=
x cos y) = 0, (2)
+ 1 (z cos a − x cos y)
I
+ 1 (x cos ẞ—y cos a) = 0.
Equations of the same form as those in (1) and (2) are true for
every molecule of the body. Let these be formed; then we
shall have a series of groups of equations expressing the motion
of every molecule, the sum of which will express the motion of
the whole body. And here enters the characteristic of the
rigidity of the body: all the internal forces and their con-
sequent velocity-increments enter in pairs, of which the direc-
tions are opposite to each other; every constraint, acting from
(say) dm to dm', has an equal and opposite constraint acting
from dm' to dm: the law of the equality of action and reaction is
true in this case of every pair of molecules; so that
Σ.I COS α = Σ. I cos B = Σ. I cos y = 0.
(3)
E.I (y cosy -z cos ẞ) = x.1 (≈ cos a x cos y) = Σ.I (x cos ẞ — y cos a)=0. (4)
And therefore, adding together (1) and all its similar groups, and
(2) and all its similar groups, we obtain the equations of motion
of a rigid body which are given in (37) and (38) Art. 48.
The same process of reasoning is applicable to the motion in
space of a system of rigid bodies moving relatively to each other,
if the internal action of one on another is always accompanied
with an equal and opposite reaction; because these will disappear
PRICE, VOL. IV.
3 s
498
[275.
THE MOTION OF
in the summation of the several equations, when that extends to
and includes all the molecules of all the moving bodies.
Ι
275.] In the problem of the present chapter, however, the
subject of motion is a body, the molecules of which move one
relatively to another, and the bounding form of which hereby
changes. A fine vibrating string, a thin vibrating membrane, a
mass of quivering jelly or caoutchouc, are such bodies as we here
contemplate. In these the form of the bounding surface will
change from time to time; and so also will the relative arrange-
ment of the constituent molecules. When the molecules move
one relatively to another, internal forces are brought into action.
which affect their motion: these are generally called elastic
forces, and are of the nature explained in Art. 153, Vol. III.
These forces vary from molecule to molecule, and also from
time to time; so that if the body is referred to a system of axes
fixed in space, and (x, y, z) is the place of dm at the time t, the
elastic forces acting on dm are functions of x, y, z, and t. In
the most general case we suppose external forces to act on the
several molecules of the body; so that dm is acted on by these
as well as by the elastic forces, and both will enter into its equa-
tions of motion. Thus, if I is the whole elastic force acting on
dm at the time t, and a, ß, y are the direction-angles of its line of
action, the equations of motion of dm are those given in (1) and
(2). Similar equations will express the motion of every particle
of the body. Now we cannot take the sum of all these, and
thereby determine the motion of the whole body, as the process is
in the case of a rigid body; (1) because our object is to determine
the form of the body at any time, and to do this it is necessary
to determine the place of every particle at that time; so that
the set of equations corresponding to a given particle must be
separately considered, and its place therefrom determined: and
(2) because all the internal forces may not be in equilibrium
amongst themselves; and consequently the conditions (3) and
(4) may not be satisfied. These internal or elastic or molecular
forces, as they are called, may enter in pairs of equal and opposite
forces in the interior of the body, and thus far may disappear in
the sum corresponding to the sum of all the particles; but at
the bounding surface they may be counteracted by and thus be
in equilibrium with certain external forces thereat acting; so
that all will not disappear in the sum of the groups of the equa-
tions corresponding to all the particles of the body. Herein
276.]
499
ELASTIC BODIES.
then is the difference of the mode of formation of the equations
of motion of a rigid body and of a molecule of a flexible body.
276.] These internal forces, which enter into the equations of
motion of each molecule, depend on the molecular constitution
of the body, and are what are commonly called elastic or mole-
cular forces.
There are two principles on which the required equations of mo-
tion may be formed. We might assume a particular theory of the
molecular constitution of an elastic body, and deduce from it the
intensity, mode of action, law and direction of the elastic force
which affects a certain molecule in a given position, and which
corresponds to a given displacement. This method has been
adopted in Art. 152-161 of Vol. III, and has been therein ap-
plied to the formation of the conditions of statical rest of the
molecules of an elastic body. And we might take the elastic
forces of restitution or of further separation, corresponding to a
given displacement, to be those which are therein determined.
This method would have an obvious advantage. It would give
us the expression of elastic action in a most general form; that,
viz., which affects a molecule of an elastic substance of three
dimensions in space: a particular form of this would be a mem-
brane or plate of infinitesimal thickness; and a still more parti-
cular form would be a thin thread or string: and the equations
which express the motion of the molecules of these reduced
forms of elastic matter would be reduced forms of the general
equations.
Our knowledge, however, of molecular physics is at present
too imperfect that equations founded on any general theory of
elastic action should be made the basis of equations which ex-
press the motion of fine elastic strings, and of thin elastic plates
or membranes; especially too in a didactic treatise. I prefer to
take laws which have been established by observation, such as
Hooke's law; and to deduce from them, special though they be, the
equations which express the particular motion in question. These
particular laws are doubtless parts of the more general law; and
if the latter is true, the former are included in it. And conse-
quently, it is my intention to state and explain the equations
which express the elastic action of the constituent molecules, as
it has been given in Vol. III, and to deduce from it the parti-
cular equations of strings and membranes; but it will be beyond
our purpose to apply them further in the present work.
382
500
[277.
THE MOTION OF
277.] In the first place let us form the equations of motion of
a perfectly flexible fine thread or string, which in the general
case we conceive to be extensible, and to be elastic; so that
when it is stretched, elastic forces of restitution are brought into
action. We suppose the string to have been displaced from its
position of statical rest by the action of some external forces,
and consider it in its motion at the time t. To take the most
general case, we will suppose it to be a curve of double cur-
vature; and we will refer it to a system of rectangular axes fixed
in space.
Let dm be an element of its mass, whose place at the time t
is (x, y, z). Let ds the length of this element, and let p
the area of a transverse section of the
pods. Let x, y, z be the axial compo-
nents of the impressed velocity-increments on dm; and let
d²x d²y d² z
its density; let w
string: so that dm
=
be the axial components of the expressed velo-
dt²' dt²' dt2
city-increments. Let r = the tension of the thread at the point
(x, y, z), which I take to be the beginning of ds: then, as the
tangent is the line of action of T, the axial components of t
dx dy
are T
T
ds ds
dz
T ; and the axial components of the tension
ds
at (x+dx, y + dy, z+dz), which is the other end of ds, are
dx
T +d.T
ds
dx
ds'
dy
T
+ d.r
ds
dy
dz
dz
ds
T +d.T
ds
ds
thus, according to (1), the equations of the motion of dm are
p w ds 3 x
{ *
d2 x
dt2
dx
+ d.T
0,
d s
d²
d2yr
dy
+d.T = 0,
(5)
ds
}
+d.T = 0.
dz
ds
p w ds {
p w d s {
Y
{-
dt2
2
d² z
dt2
I may observe that these equations have been found before;
viz. in Art. 52, where their determination has been given in
illustration of the principle of virtual velocities. I have chosen
however again to investigate them, in order that the meaning
of all the symbols involved in them may be clearly understood.
278.] Of these I will first take a most simple case. Let the
string be fastened at one end o to a fixed point, and let it pass
over a small pulley A, where OA = ɑ, and have a weight W
279.]
501
ELASTIC STRINGS.
attached to its other end: so that the tension of the string
throughout is equal to w; and let w be so great that the weight
of the string may in comparison of it be neglected. Let us
however suppose x = Y = z = 0; then, in its position of equili-
brium, the string lies along the straight line oa; let it be slightly
displaced by means of an external force; the displacement being
so small that the angle at which any element of it is inclined to
the line oA is infinitesimal. Our object is to investigate the law
of the displacement of any particle which follows on this initial
displacement. Let oa be the axis of x; and let (x, y, z) be the
place of any element (= pwds) at the time t. Then, as the
angle of inclination of ds to the axis of x is infinitesimal, we
have approximately ds dx. So that neglecting infinitesimals
=
of the second order, each element of the string moves in a plane
perpendicular to the line o a, and consequently the point of the
string at a does not move: thus there will be no motion along
d2x
the line oA, and
0, for all elements of the string. Thus
dt2
the first of (5) gives T = a constant; and the tension of the
string is constant throughout its length and throughout the
motion. Introducing these results into the last two equations
of (5), they become
d2 y
dt2
T d²y
p w d x
dx 2
= 0,
2
d² z
dt2
T d2 z
pw d x 2
= 0.
(6)
If, for the sake of simplicity, we suppose the curve of the string
in its initial displacement to be in one plane, we may take that
to be the plane of (x, y); and then the equation which repre-
sents the subsequent motion of the particle is
d2 y T d²y
dt 2
pw dx2
ρω
0;
(7)
which is a partial-differential equation of the second order. The
integration and interpretation of it I shall defer to the following
Articles; because we shall again meet with equations of the same
form. The homogeneity of it on the principles explained in Art.
273 deserves notice.
279.] Next let us investigate the motion of the particles of a
thin heavy elastic string, which is homogeneous and of the same
thickness throughout its length, stretched between two given
502
[279.
THE MOTION OF
1.
points o and a; see Fig. 64; where oa = = 7. Let To be the ten-
sion of the string at rest: which we assume to be so great that
the weight may be neglected without sensible error in compari-
son of it: thus the string lies in the straight line joining o and
A, when it is in statical equilibrium.
Now let us suppose the string to be put into motion by some ex-
ternal force; as a piano-forte string by the blow of the hammer,
or the string of a harp by the finger of the player: hereby the
particles are displaced both relatively and in space; and elastic
forces of tension are brought into action, tending to restore the
string to its original condition; and although the force which
produces the displacement ceases to act, yet the particles of the
- string continue to move, and the string vibrates about its recti-
linear position. We will take the most simple form of the
problem, and suppose no other force to act: so that in (5),
X = Y = Z = 0.
Let us consider the string in its vibrating state at the time t,
and refer it to three rectangular axes originating at o, of which
the x-axis coincides with oa. Let us take a particle (= dm)
whose distance from o in its position of rest = x; and let p be
its density: then if w = the area of a transverse section of the
string, dm = po dx: let the place of this particle at the time t
be (x+§, n, §); and let p' be its density in its displaced state,
and the corresponding area of the transverse section of the
string; and let ds be the length-element of the curve which dm
occupies then, as the mass of dm is unaltered,
dm = pw dx = p'w'ds.
(8)
As the displacement of the particle is very small, §, ŋ, ¿ are all
small: they are functions of x and t, and are to be expressed in
terms of these variables. It is however to be observed that x
is not a function of t.
Thus (5) become
d²¿
d(x+έ)
pw dx
d.T
0,
dt2
ds
p w d x
d² n
dx
dt2
d. I as
dn
=
= 0,
(9)
d²¿
p w d x
d.r
dt2
Now the length of the element
= 0.
da, under the action of the
tension To: and ds, under that of the tension r: consequently,
=
dc
d s
279.]
503
ELASTIC STRINGS.
if E is the modulus of elasticity, by Hooke's law, as explained
in Art. 149, Vol. III,
ds dx 1 +
dx {1-
T
TO
E
(10)
where, it will be observed, E is a weight depending on the nature
of the string.
Also
ds² = (dx + d§)² + dn² + d§² ;
αζ
and as the displacements of the molecules are small, d§, dn, d
are so small, that all powers of them higher than the first may
be neglected. Consequently,
ds = dx + dέ:
(11)
αξ
T = To E
dx
Let us substitute all these values in (9); and, omitting small
terms, we have
Let
ρω
d?
dt2
7°
E
dx²
2
2
d² n
d2n
ρω
= To
dt2
dx2
d²r
2
d2t
ρω
= Ti
dt2
dx²
E
To
a²,
ρω
ρω
(12)
(13)
then the preceding equations become
d? Î
d &
=
a²
dt2
dx2
d2 n
d2n
b2
dt2
dx²
(14)
d2c
d² Ć
= b2
;
dt2
dx2
which are three partial linear differential equations with con-
stant coefficients of the second order. As the variables in them
are separated, we conclude that the vibrations of the string
parallel to the three axes of §, 17, Ċ are independent of each
other, and coexist without interference. The first equation ex-
presses the vibrations along the string; these are called longi-
tudinal vibrations; and those which are expressed by the last two
equations take place at right angles to the axis of x, and are called
* It may be noticed that a and b are of (1) dimension in space, and of
(-1) dimension in time, and thus represent velocities.
504
[280.
THE MOTION OF
transversal or lateral vibrations. The form however of all the
equations is the same; and we need only discuss one of them, say
the second; for the result of that will also give the solution of the
first, if we change b into a; and will thus assign the nature of
the longitudinal motion.
280.] These equations may be integrated by the method
given in Art. 365, Vol. II; and the integral of the second of
(14) is
n = F (x + b t) + f (x −b t) ;
(15)
where F and ƒ are symbols of arbitrary functions, as yet un-
determined.
As to this process of integration, I may by the way observe,
that if we change the variables in the second of (14) by putting
x+bt = a,
the equation becomes
..
x—bt = ß,
= = 0;
dad B
d2n
n = F(a) + ƒ (B)
. η
= F(x+bt) + f (x−bt).
(16)
Now our object is to determine the form of these functions by
means of the initial or other circumstances of the string; and of
the fact of the two points at o and a being fixed.
η
η = • (x);
Let us suppose the equation of the curve of the string, when
t =
0, to be
(17)
where n is the initial displacement parallel to the y-axis of the
molecule whose distance from ox. Also let us suppose the
velocity parallel to the y-axis, when t = 0, to be given by the
equation
dn
= b p' (x);
dt
where '(x) is the derived function of (x).
(18)
and o are sym-
bols of known functions, and will be treated as known for all
values of a between 0 and 7; they are also subject to the con-
dition that both vanish when x = 0, and when x = l.
Hence, when t = 0, from (16), (17), and (18), we have
F (x) +ƒ (x) = • (x),
F′(x) — ƒ'(x) = $'(x) ;
... F(x) —ƒ (x) = $ (x);
{(x)+(x)}
‚'.
F(x)
(19)
2
{Þ (x) — $ (x)}
f(x) =
;
(20)
2
280.]
505
ELASTIC STRINGS.
and consequently F (x) and ƒ (x) are known for all values of x
for which (x) and † (x) are known; that is, for all values of x
between x 0, and x = l.
The subject-variables however of F and f, as they are given
in (15), are not limited by these values. The subject-variable of
F is x+bt, and if b is positive, as we may take it to be, this
varies as t increases through all positive values between 0 and ∞.
And the subject-variable of ƒ is x-bt, which has all values between
I and; so that the complete solution requires the values
of the functions corresponding to these values to be known.
As is a velocity, bt denotes a line, and is consequently homo-
geneous with x, and may be taken in addition to or subtraction
from along the x-axis.
Since the points o and a are fixed throughout the motion,
ŋ=0, when x=0 and when x = 1; consequently, from (15), we
have
F(bt) +f(−bt) = 0,
F(l+bt) +f(l—bt) = 0.
(21)
(22)
It appears from (21) that ƒ(-bt) and F (bt) are equal and of
contrary signs; so that if r (bt) is known for all values of t
between 0 and ∞, ƒ(−bt) is also known between those same
limits.
In (22) let bt be replaced by l+bt; then
F(21+bt) = f(―bt)
F
−
(bt);
(23)
which shews that the value of r (bt) remains the same, when its
subject-variable is increased by 21; consequently it is the same
when the subject-variable is increased by 47, or 61,..., or 2 nl,
where n is a whole number. And therefore if the value of F (bt)
is known from bt = 0 to bt = 21, the value is known for all
values between t = 0 and t = ∞.
Again, in (22) let bt be replaced by 1-bt, so that bt is less
than ; then
F (21-bt) = f (bt) ;
−
(24)
but f(bt) is known for all values of bt between 0 and 1; con-
sequently the value of F (bt) is known for all values of bt between
bt = land bt
l and bt = 27.
Hence the value of F (a) is known for all values of a, from
a = 0 to a = ∞ ; and these are the required limits.
Thus much as to F. And we have shewn above that all values
of ƒ (ß) are known from ß = 0 to ß
0 to ẞ = ∞; and the initial equa-
PRICE, VOL. IV.
3 T
506
[281.
THE MOTION OF
tion (20) gives all values of ƒ (3) from 3 to 60; so that
ß I
all values of ƒ (3) are known within the required limits.
It is worth observing that the values of r (a) for negative
values of a, and that of ƒ (ß) for values of ß greater than 1, have
not been found in the preceding explanations; and they are not
required; as their subject-variables are not within the limits.
given by the problem.
The values of the functions which express the displacement
may be found in a manner precisely similar.
Thus the form of the string in its displaced state, and the
velocities of its several molecules parallel to the y- and -axes at
the time t will be known; and the problem will be completely
solved, so far as the transversal vibrations are concerned.
Also, all that has been said on transversal vibrations is ap-
plicable, if we replace b by a, to the longitudinal vibrations of
the string. In this case the initial equations will assign the
position and the velocity along the a-axis of every particle of
the string, when t O, between the limits a = 0 and x = 1.
Thus the problem is completely solved. I propose however
to interpret the result graphically, for the general motion of the
string will be rendered clearer by means of a diagram. The
results which will be exhibited might be derived from the equa-
tions just now discussed; but it will be more convenient to take
a less general form, which will be equally expressive and more
easily constructed.
281.] For this purpose I will assume that the string, having
been disturbed, takes the form given by a known equation, such
as (17), when t = 0; and that all its particles are then at rest;
dn
so that
= 0, when t = 0; consequently since.
dt
n = F (x+bt) + f (x−bt),
dn
b F′ (x + b t) — bf' (x — bt);
dt
dn
and therefore if
0, when t = 0,
dt
F′(x) = ƒ' (x);
..
F(x) = f (x) ;
and
n = f(x+bt) + f (x −bt);
(25)
therefore, when t = 0,
n = 2 f(x);
(26)
281.]
507
ELASTIC STRINGS.
Suppose however the equation to the curve of the string in its
displaced state at rest, when t = 0, to be
n = F(x);
F(x) = 2f(x);
then, from (26),
and thus (26) becomes
1
{F (x + bt) + F (x —bt)}.
2
(27)
This then is the 7-displacement of the particle which is at
(x, 0, 0), when the string is rectilineal.
The function whose symbol is F is subject to the following
conditions, which are derived from equations (21)...... (24) of
the preceding Article:
F(x) =
—
F(x),
F (l + x)
= −F (1 − x) ;
F (∞)
x (21+x) =
F (x) =
F (2 1 − x)
F (4 7 — x)
(28)
(29)
F (4l+x) = ... = F (2 nl + x), (30)
(31)
These equations enable us to infer a correct notion of the
motion of the several molecules of the string, from the form
which it has in its initial displaced state. From (28) it appears,
that the curve represented by y = F(x) is continued in similar
forms on each side of o; the curve being on one side above, and
on the other below the axis of x; and (29) shews that the curve
is continued in similar forms on each side of A, the curve being
on one side above, and on the other below the axis of a. Con-
sequently the curve about the points o and A, and between these
points, is of a form similar to that drawn in fig. 65, the plane of
the paper being that of (x, y).
Again, to develope other properties of it: along the axis of a
take from o in both directions a series of lengths, each of which
is equal to 7; viz., oa = ao'= 0′a′ = a′0″: = 1. Then equa-
tion (30) shews that whatever is the form of the curve between
o and o', it is the same between o' and o". And (31) shews
that the form of the curve between o and A is the same as that
between o' and a, except that in the latter case it is inverted and
lies below the axis of a. Thus the whole curve consists of similar
portions drawn as in fig. 65.
Now this curve enables us to trace the motion of any particle
of the vibrating string; and consequently the motion of the
whole string. For the place P at the time t of the particle,
which is at M (OM = a) when the string is straight, may thus be
3 T2
508
[282.
THE MOTION OF
: =
=
found along oA take, on both sides of M, MN MN' bt; so
that NQ = F(x+bt), N'q' = F (x-bt), then, from (27),
1
M P
{N Q + N'Q' } ;
2
(32)
and by a similar process may the place of every element be de-
termined at any time t.
282.] Now this molecule, and similarly every molecule of the
string, and consequently the whole string, will oscillate; that
is, the string will occupy a certain series of positions in succes-
sion, and will then be found in its original state; afterwards it
will go through the same series again, and then return to its
original state; and so on continually.
For if we assume bt 21, or = 47, or = 61, ..., or = 2 nl, at
the times corresponding to these intervals, we have, by means
of (30) and (31),
1
√
{F (x) + F (X)} ;
= F(x);
which is the value of n, when t 0; and as this result is true
for every point of the string, the string comes back to its original
position at the times corresponding to these values. The interval
21
between two successive states of like position is; which is
therefore the periodic time of vibration.
When we take bt l, or = 31, or = 51,
5 l, ..., or = (2 n + 1) l,
at the times corresponding to these intervals, we have
1
n =
2
{F (x + 1) + F (x − 1)} ;
= F(x + 1);
so that in the time =
, every
molecule will have described one
half of the whole course which it describes in going from a posi-
tion to the same position again. Thus the form of the string at
these times is exactly similar to what it was when t = O, but in
an inverted position; as the lower line in fig. 64; the greatest
ordinate being now at the same distance from A as it was from
o in the original form of the string. The time which is occupied
in this change of figure is one half of that occupied in a complete
vibration.
Similarly we may consider the positions of the curve when
283.]
ELASTIC STRINGS.
509
t
or when t=
26/
31
26
: in the former case, the length of time
is one-fourth of that of a complete vibration; and in the latter
case, is three-fourths of that of a complete vibration. Similar
results are also true when each of these times is increased by
21
b'
It is to be observed, that during this vibratory motion the
string never becomes a straight line.
Similar results to these are also true for the (-displacement;
for in the last equation of (14) the same constant coefficient
enters as in that for the n-displacement. Thus the periodic time
of the complete path will be the same in both displacements.
The forms of the functions may be different in the two cases,
although they are both subject to the conditions developed in
Art. 280.
This vibratory motion would continue perpetually if there
were no diminution of vis viva of the string. In the case how-
ever of pianoforte or harp strings vis viva is lost for two reasons.
The points o and A, to which the ends of the string are fastened,
are not points rigidly fixed; so that vibrations are communicated
to them from the string, and continued by means of their sup-
ports or framework to the earth, and thereby lost to the string;
the string also vibrates in air, to the particles of which vibra-
tions are communicated, and thus vis viva is taken from the
string. Owing to these two causes, the oscillations of the string
become gradually feebler, and eventually cease. The periodic
time of the vibrations however is not changed.
283.] If the original position of the string-curve had been
that drawn in fig. 66, where в is the middle point of oa, and
where the two branches of the curve are exactly equal and
similar, though their positions are inverted, the string would
vibrate so that the point в would always remain fixed; that is,
each half will vibrate as if the string were fixed at o and û, and
at в and A; and the periodic time of vibration will be only one-
half of what it is in the case already discussed. So, again, if the
original string-curve had consisted of three or more equal and
similar portions, intersecting oA at points equally distant from
one another and from o and A, and being alternately above and
below the line oa, the string will vibrate as if it were three or
more distinct strings, and the points of it at which it intersects
OA will remain at rest during the motion.
510
[284.
THE MOTION OF
These points are called nodes or nodal points, and the curve
between two consecutive nodes is called a ventral segment.
284.] If T is the time in which one complete transversal vi-
bration of the string takes place,
T =
27
b
21(
= 21
ρω
To
(33)
so that, for a string of constant thickness and density, the time
of transversal vibration varies as the length of the string directly,
and as the square root of the tension in its straight form in-
versely.
And if n is the number of transverse vibrations which take
place in a second of time,
N
1 1
T 27
To
ρω
(34)
so that for a string of constant thickness and density n varies
as the square root of the tension in its straight form directly,
and as the length of the string inversely.
Again, let r' be the periodic time of a complete longitudinal
vibration; then
27
2
x = 2/1 = 21 ( 2 ) ²;
T=
a
E
;
(35)
and let n' be the number of longitudinal vibrations which take
place in a second of time,
1
n =
1 E
T 21 ρω
(36)
Now if the string suspended by one end is stretched by a
weight at the other end, E is that weight by which the length of
the string will be doubled; so that the number of longitudinal
vibrations of the string in a second of time varies directly as the
square root of this weight.
From the preceding equations we have
n
n'
(To) 3
*
;
E
(37)
but in the ordinary pianoforte strings, E is evidently very much
greater than To, so that the number of longitudinal vibrations in
a second of time is very much greater than that of the trans-
versal vibrations.
The ratio of n to n' may also be expressed in the following
way suppose to be the length of a string in its natural state,
:
1
285.]
511
ELASTIC STRINGS.
and Al to be the increase of length when it is stretched by To:
then, by Hooke's law,
To
I ÷ Al
=1
= 1 {
{1.
1 +
E
ΔΙ
and
To
E
ī
(笑​)
n
△ = (4)
n
(38)
285.] Let us briefly notice these results in reference to the
theory of music. When a string vibrates, the vibrations are im-
parted to the molecules of the surrounding air, and are through
the medium of the air communicated to the tympanum of the
ear, the auditory nerves of which are excited, and the sensa-
tion of sound is created. The ear recognises three special pro-
perties of sound, (1) the pitch, (2) the intensity, (3) a peculiar
quality which is in England called technically the "quality" of a
note, and is in France called "le timbre." The origin of this
last property, and indeed most of its affections, it is very difficult
to account for; it probably arises from many causes, amongst
which may possibly be the difference of periodic time in the lon-
gitudinal and the transversal vibrations. The quality of note
given by a string of a violincello played with a bow is very dif
ferent to that given by a pianoforte string struck by a hammer,
when the note or pitch is the same in both. The second pro-
perty depends on the amplitude of vibration, and varies as the
vis viva of the particles put into motion. The first property is
that which is called the musical tone or note, and depends on
the number of vibrations made by the string in a second of
time. And thus the number of vibrations is taken as the mea-
sure of the pitch or note. The note is higher the greater the
number of vibrations made in a second of time. Thus as to the
note due to the transversal vibrations, the note varies inversely
as the length of the string, and directly as the square root of the
tension. Thus if the note of a given pianoforte string is too
low, the string must be wound up, whereby the tension is in-
creased.
The note given by a string when its two extremities are fixed,
and when all the other points of the string vibrate in the motion,
is called the fundamental note of the string: thus the fundamental
note of a string is higher by one half than that of a string of
which the length is twice as great. And thus, too, if a string is
512
[285.
THE MOTION OF
struck so that it has one node, the note in that case is twice as
high as the fundamental note; and if it has two nodes, the note
is thrice as high as the fundamental note; and so on. Thus the
notes of strings are compared by means of their lengths and the
distances between their nodes.
When two strings have equal periodic times, and vibrate toge-
ther, they are, in musical language, said to be in unison. If two
strings vibrate simultaneously, the resulting sound is most agree-
able to the ear when they are in unison. Next to an unison the
most agreeable concord is the octave, in which one string vibrates
twice as fast as the other; that is, in which the times of vibra-
tion are as 1:2; and the note produced by the former is said to be
an octave above that of the latter. Thus if a given string vibrates
so that its middle point is a node, it produces the octave to the
fundamental note.
It is invariably found, for all ears, that when two notes not in
unison are sounded together, the resulting sounds are most agree-
able when the times of vibration of the individual notes are in
some simple proportions; say 1:2; 1:3; 1:4; 2:3; and that
the concord is more agreeable the less the difference between
the terms of the ratio.
Thus the octave of the octave of a note, or the fifteenth, as it
is called in music, is an agreeable concord; for it consists of vi-
brations of two strings whose periodic times are as the numbers
1:4. Thus also two strings whose periodic times are in the
ratio of 1: 8, or in the ratio of 1:16, and so on, produce plea-
sant concords, and seem to partake of the perfection of the
octave.
The next most simple numerical ratio is that of 1: 3, in which
we have three vibrations of one string corresponding in time to
one vibration of the other. This concord is called a twelfth. If
we replace the string, which vibrates once, by its octave, which
vibrates twice in the given time, the times of vibration of the
two strings are as 2:3: this also forms an agreeable concord,
and is called a fifth.
It is however beyond the object of this treatise to enter fur-
ther into this subject: and for other details, and for an exposi-
tion of the theory of music, founded on the preceding and other
similar equations, I must refer the reader to the Treatise on
Sound by Sir John F. W. Herschel, which was originally con-
tained in the Encyclopædia Metropolitana.
286.]
513
ELASTIC STRINGS.
286.] A few words must be said on another process of finding
the solution of the general differential equations given in (14);
and now, as heretofore, I will take the second of the three equa-
tions to be the type-equation. It is impossible not at once to
see that the equation
{A cos mbt+B sin m bt} sin (m x + a)
(39)
satisfies the differential equation; A, B, m, and a being undeter-
mined constants. Since however ŋ = 0, when x =
n=
x = 1, whatever is the value of t, we have
0 = (A cos mbt + B sin m bt) sin a,
0 = (A cos mbt + B sin m bt) sin (ml + a),
0, and when
(40)
(41)
From (40) it appears that a = nπ,
nã, where n is any whole number;
which however it is convenient to take to be positive, as no
limitation to generality will be introduced thereby. And from
(41),
Nπ
m =
consequently (39) becomes of the form,
n =
{^,
An Cos
COS
nñ b t
}
+ B n
nπ b t
sin
{b t }
N π X
sin
(42)
(43)
And this equation satisfies the differential equation, and also
fulfils the conditions that ŋ = 0, when x = 0 and when x = 7,
whatever is the value of t.
Also, by reason of the linearity of the differential equation, it
will be satisfied by a sum of expressions of the same form as the
right-hand member of (43), so long as n is a positive integer:
and thus
COS
η
Σ.
{ A₂
An Cos
n π b t
ī
n π b t
N π X
+ B₂ sin
sin
(44)
the sign of summation indicating the sum of all similar expres-
sions corresponding to integer values of n.
The A's and the B's are still undetermined. They are how-
ever to be expressed in terms of initial circumstances. These we
will take to be the same as those in Art. 280; so that ŋ=(x), and
dn
dt
bø′(x), when t=0.
Consequently from (44),
PRICE, VOL. IV.
• (x) = Σ.An sin
$'(x) = Σ.
пп
ī
Nπ X
ī
Bπ
Nπ X
sin
(45)
ī
(46)
3 U
514
[287.
THE MOTION OF
At this point of the process this method fails, because these
equations will not determine the values of the undetermined
constants, viz. the A's and the B's, in terms of the given functions.
which express the initial circumstances of the string. The form
of the given functions is, it is to be observed, arbitrary; and the
analytical solution is not perfect unless the constants are ex-
pressed in terms of them.
The following equations however, at first due to Lagrange,
and afterwards deduced from a theory on definite integrals,
more fully developed by Fourier and Poisson, will complete the
solution. They may be found in most treatises on definite in-
tegrals.
• (x)
$'(x) =
2
ī
2
Σ
Σ.
b
0
ηπξ
N π X
sin
ĕ (§) d§ sin
;
(47)
1
['sin
ηπε
Nπ X
¤'´(§) d § sin
(48)
¿
Let these be substituted in (44); then we have
2
2
ηπέ
N π X
Σ
sin
Φ
ī
* (f) df sin
n π b t
COS
7
2
+ Σ
sin "
ηπε
7
1
αξ
'() dε - sin
N π X
nx b t
sin; (49)
π
N
wherein the summation denoted by z. indicates the sum of all
these similar quantities corresponding to positive integer values
of n. As this equation satisfies all the conditions of the problem,
it is the complete integral of the original equation; and all the
properties of the motion of the string, which have been deduced
in Art. 280 from the arbitrary functions F and f, may be just as
conveniently inferred from this equation.
Thus, according to this method, the motion of the string is
expressed by a series of terms, each of which might exist alone,
and might be the complete solution of the equation, if it agreed
with the initial state of the string. The most general motion of
the string however results from the coexistence or superposition
of an infinity of vibratory motions, and the resulting note from
the coexistence of the several notes which are due to these se-
veral single motions.
287.] We proceed now to other particular cases of the mo-
tion of elastic bodies and of their constituent molecules. And
I will first consider the longitudinal vibrations of the molecules
of a fine elastic rod.
287.]
515
AN ELASTIC ROD.
I shall assume the rod to be homogeneous, and in its natural
state to be prismatic or cylindrical; so that if the area of a
transverse section, w is constant throughout the length, and is
infinitesimal because the rod is thin. I shall take the line which
contains the centres of gravity of all thin transverse slices to be
the x-axis. Although the rod is thin, that is, although the lines
of its transverse section are infinitesimal in comparison of the
length of the rod, yet its thickness is such that the rod is not
bent by the forces acting on it.
Now we suppose the particles of the rod to be displaced lon-
gitudinally by a force acting in the direction of the length of
the rod, so that each particle is displaced through a small dis-
tance along the x-axis. Moreover, we suppose that every particle
in a thin transverse section or slice perpendicular to the x-axis
is displaced through an equal distance; so that all particles
which were in a given transverse slice before the displacement
are in a transverse slice after the displacement.
We also sup-
pose that by this displacement, due to an external force, certain
elastic forces are brought into action whose lines of action are
parallel to the x-axis, and that the molecules subsequently vi-
brate under the action of these forces. It is this subsequent
motion, when all other forces have ceased to act, which we shall
now investigate.
A
Let o and a be the ends of the bar in its original state of rest
and let oa, its length, = 7; let ∞ =
w the area of a transverse sec-
tion, p
the density. Let us consider the motion of a thin
slice, whose distance from o = x, when t = 0, and whose thick-
ness = dx; so that its mass = pwda. This mass is unchanged
during the motion.
Let a + be the distance of this elemental slice from o at the
timet: έ being a small quantity and evidently a function of both
x and t. Let T be at the time t the tension drawing the slice
towards o, and T+ dr that drawing it towards A: then, as no
other force acts, the equation of motion is
pw dx
d² (x + E)
dt2
= dr;
d² ¿
dt2
.. poda
w x = dr.
(50)
We suppose the extension of the bar to vary according to
Hooke's law so that if E is the modulus of elasticity of the
bar, according to Art. 149, Vol. III,
3 U2
516
[287.
THE MOTION OF
dx + de da
=
dx{1
dr
d x 1 +
1 +
E
d ε
..
dr = E
dx
and consequently (50) becomes
d%e
E
d² E
(51)
dt2
pw
dx 2
Let
E
d² έ
d² &
a²:
= a²
ρω
dt2
dx
dazi
(52)
in which expressions we must observe that
(1) de
αξ
the velocity of the slice po dx;
ρω
dt
αξ
(2) = the linear dilatation of an unit-length of the bar.
dx
Now (52) is of precisely the same form as the equation for the
vibrations of an elastic string, and is to be integrated in the
same manner; but the initial circumstances of the two problems
may be very different: and thus also will be the given arbitrary
functions. Now I will suppose that when t = 0, § = • (x) and
αξ
dt
= ap'(x), where and p' denote arbitrary functions which
are given for all values of x from 0 to 1, when t = 0.
αξ
dt
If an end of the bar is fixed, § = 0 and = 0 at that point
during the whole motion; and T is the pressure which the point
αξ
has to bear. For a free end.
0, for all values of t.
dx
αξ
Thus if the two ends are fixed, έ= 0, and
O, and when x = l: so that
= 0, when
dt
x =
2
・Z
ηπε
ппх
ηπατ
Σ
sin
(E) de sin
COS
し
​7
2
+
S
ηπε
1
sin
(ξ)
¤'´(§) d§ – sin
0
n
π
Σ
Nπ X
sin
ηπαί
ī
(53)
Hence if T is the time of a whole vibration of the slice, and n is
the number of vibrations made by it in a second of time,
27
T
α
n =
( P ) +
21 (PC
1
27
(
E
E
ρω
(54)
;
(55)
288.]
517
AN ELASTIC LAMINA.
м
so that the note of the bar is the same as that of a vibrating
string-of the same elasticity, thickness, density, and length.
If the end o is fixed, and the end A is free, then = 0 and
αξ
dε
= 0 when x = 0; and
that is
dt
dx
d+ (x)
dx
= 0 when x = 1.
Hence
2
・l
sin
(2-1) πέ
21
(2 n − 1) π x
(2n-1)πat
• (§) d § sin
COS
27
21
4
(2n-1)πέ
+
Σ sin
φ' (ξ) αξ
π
21
1
2n-1
(2n-1) πx
(2n-1)πat
sin
sin
; (56)
21
21
because this value satisfies all the required conditions.
Now the values of έ become the same whenever at is increased
by any multiple of 41: consequently if T is the time of a com-
plete vibration of a slice of the bar, and if n is the number of
vibrations completed in a second of time,
4/
T
α
= 41 (pa)
ρω
E
1
E
-13
n =
•
4/ ρω
(57)
(58)
Thus the time of vibration is twice as long in this case as it is in
the former, in which both ends of the bar are assumed to be
fixed. And thus also the note due to the longitudinal vibrations
of an elastic bar fixed at both ends is an octave higher than that
due to it fixed at one end and having the other end free.
288.] Let us next consider the motion of the molecules of a
thin elastic lamina, which is fixed along one edge, and which
is otherwise free from external constraint; such as that of which
the conditions of statical rest have been investigated in Art.
162—166, Vol. III. I shall employ the same symbols and the
same diagram as in those Articles.
The lamina in its natural state is supposed to be plane;
and to have been bent, as is assumed in those Articles, by the
action of one or more external forces, and then left to itself.
It subsequently vibrates by virtue of the elastic forces which
have been brought into action by the original displacement. The
problem in Vol. III is the determination of the curve which the
lamina takes when it is bent by certain given forces: the pro-
blem herein to be treated is the subsequent motion of the lamina
by reason of its elastic forces, when the original bending forces
518
[288.
THE MOTION OF
have ceased to act. For although the lamina is assumed to be
very thin, yet its thickness is such that elastic forces are brought
into action upon its being bent.
Now take the Figure 73 in Vol. III to be an enlarged repre-
sentation of the lamina at the time t, except that there are no
forces x and y: then the equation of motion of the lamina may
be formed as follows. Take any transverse section of it, as
P'P P', the mean fibre of which intersects the plane of (x, y) in
the point (x, y). Then, if we suppose the fibre to be at rest for
an instant, and L to be the moment of the elastic forces which
are due to the part P'AP' of the lamina, and act on the section
about an axis perpendicular to the plane of the paper; by (178)
Art. 463, Vol. III,
L =
2 kb 73
;
3 R
(59)
where R is the radius of curvature of the bent lamina at the
point P. But the forces of which this is the moment produce
the motion of the part P"AP' relative to the mean fibre of the
section p'P'. Now let us take a thin slice of the lamina at a
point between P and a; and as the lamina is very thin, we may
consider its mass, which is equal to 2prbds', (p being the den-
sity), to be condensed into a particle at (x, y), the place of its
mean fibre; so that the expressed momentum-increments of this
slice parallel to the axes of x and y are respectively
2 pr bds
d² x'
dt2
and
2pTds'
d² y
dt2
(60)
Let us moreover assume the displacement of all particles of the
lamina to be so small, that all velocity-increments parallel to the
x-axis may be neglected; and we shall also assume the inclina-
tion to the x-axis of all elements of its curved fibres which are
straight in their natural state, to be so small, that powers of
above the first may be neglected. Thus
dy
dx
1
d² y
=
ds' dx'.
R dx2
,
(61)
Hence also the moment of the expressed velocity-increments of
the part P'AP' about an axis perpendicular to the plane of the
paper, and passing through the mean fibre of P"P' is
T
a d²y'
dt2
(x' - x) dx';
рто!
- 2 p r b [ " 12 y
(62)
289.]
519
AN ELASTIC LAMINA.
and this in equilibrium with L; so that we have
2 kbr³ d²y
3 dx2
- 2 p + b ["
T
f
α
day'
(x' — x) dx'.
(63)
dt2
Let us take the x-differential of this equation; a being the in-
ferior limit of the definite integral in the right-hand member,
and the differential being taken in accordance with the princi-
ples explained in Art. 178, Vol. II; then
α
k+² d³y = [ a day
3p dx3
and again taking the x-differential
dx';
kr2 day
3p das
d2y
•
dt2
so that the equation to the vibrating lamina is of the form
d² y
dt2
+62
d y
↓
= 0.
dx+
(64)
(65)
This equation is not capable of integration in finite terms. But
it is evident that
y =
{A cos m² bt + B sin m² b t} sin (m x + a)
(66)
satisfies the preceding equation; A, B, m, and a being undeter-
mined constants: and since y = 0, when a = 0, whatever is the
value of t, a = n; where n is any integer number; so that
y =
(67)
{A, cos m² b t + B, sin m² b t} sin (m x+n ñ).
And as n may be any whole number, the complete solution will
be
Y = Σ. {An cos m² b t + B, sin m² bt} sin (mx+n ñ); (68)
wherein the sign of summation denotes the sum of a series of
similar quantities given by the several values of n, which ad-
mits of all integers.
All the undetermined coefficients must be found by means of
the initial circumstances of the lamina, in the same way as similar
questions have been treated in the preceding Articles.
289.] I come now to the last portion of our inquiry; viz.
a statement of the general equations which express the motion
of a molecule of an elastic body, homogeneous and of constant
elasticity, existing in three finite dimensions in space. For this
purpose, to avoid repetition, I must again refer the reader to
Section 2, Chap. V, of Vol. III, wherein the theory of such
520
[290.
THE MOTION OF THE MOLECULES
elastic bodies has been explained, and equations have been found
which express the conditions of the relative rest of the consti-
tuent molecules. From these the corresponding equations of
motion are easily deduced. Suppose a molecule of a homo-
geneous elastic body, whose elasticity is constant, to be origin-
ally at (x, y, z), and to be displaced by the action of some ex-
ternal force, so that elastic forces are brought into action; and
suppose it to move under the action of these last forces; the
original disturbing force, and all other forces having ceased to
act; and suppose its place at the time t to be (x + §, y + 1, ≈ + Š);
(x+§, z
then the equations which express its motion are, see (160), Art.
160, Vol. III, evidently
(入​+4)
do S d²¿
d² &
dē?
d²
2 §
+ м
+
dx
dx2
dy2
+ dz²
P
dt2
do
S d²n
d2n
d² nd
η
d2n
+ м
+
+
dy
dx2
dy2
dz2 S
= P
(69)
dt2
do
S d² 5
d² Č
d² C
d² Ć
(λ + μ)
+ μ
+
+
= P
dz
dx2
dy2
dz2
dt2
in which
is the dilatation of a cube whose volume is the unit-
volume, p is the constant density, λ and μ are constants depend-
ing on the elastic action of the body, and all the differentials
are partial.
These equations may be also put into another form; take the
x-, y-, and z- partial differentials of these respectively, and add
them; then, since
ds
dε dn
+ +
dx dy dz
Ω ↓ =
the sum becomes
d2
(x + 2 µ) { (1280
Ω
+
d2a
+
Ω
dy2
dz²
d2a)
S
2
=P
dt2
(70)
(71)
by means of which equation 2 is to be determined as a function
Ω
of x, y, z, t. If this value of a is substituted in the three equa-
tions of (69), §, n, & may be determined in terms of x, y, z, and
t, and the problem will then be completely solved.
I may in passing observe, that these equations are homogene-
ous on the principles of Art. 273.
290.] It will be beyond the object of our work to enter into
a discussion on the equations of the preceding Articles, as it
would lead us into wave-motion, and other subjects of a similar
290.]
521
OF AN ELASTIC BODY.
kind. Two special cases however, which have already been
treated otherwise, are so easily deduced from these general equa-
tions, that it is desirable to insert them.
Equations (69) express the motion of a constituent molecule
of any elastic body under the action of elastic forces which are
made active by reason of a displacement produced by some ex-
ternal force. And as no limitation is made as to the dimensions
of the body, the equations are applicable to thin plates or mem-
branes, and to thin strings. These two cases I will consider as
briefly as possible.
Imagine a series of molecules through the body, in its natural
state, forming a thin plate or membrane, the bounding surfaces
of which are planes at an infinitesimal distance apart. Further,
imagine this thin plane membrane to be separated from the
body, so that no forces, elastic or other, act normally to its plane
surfaces, and its constituent molecules are at rest by the ac-
tion of internal forces only. If this is homogeneous, and of con-
stant elasticity, like the body of which it is a part, so is it a thin
elastic membrane in the ordinary meaning of the term, as the
parchment of a tambourine or the parchment in a drum-head.
Now we suppose this thin elastic membrane to lie in a plane ;
that is, we imagine a plane section of it equally, and thus infini-
tesimally, distant from its two plane surfaces, to be a geometri-
cal plane. Moreover, we suppose it to be stretched with equal
tension along all its edges by a stretching force which we will
call T. On the membrane thus stretched and at rest, we suppose
an external force to act, so that the molecules are displaced :
hereby elastic forces are brought into action tending to restore
the membrane to its original position. We propose to deter-
mine the equations which express the motion of a particle due
to these elastic forces, the disturbing force having acted once
for all.
T
Let the system of reference be as follows: take the plane,
which is equally distant from the two plane forces of the mem-
brane, to be that of (x, y), and any point in that plane for the
origin. Let p be the density and the thickness of the mem-
brane; let the place of the type-molecule, whose motion we will
consider, be (x, y, O) when it is at rest; and let its place at
the time t be (x + έ, y + n, 0 + §), so that έ, 7, Ċ are the axial
components of its displacement at the time t. Then, replacing a
by its value given in (70), we have from (69)
PRICE, VOL. IV.
3 X
522
[291.
THE MOTION OF THE MOLECULES
(x + 2 µ) (
d²¿
d² E
d² ε
+
ρ
dx2
dy²
dt2
(λ + 2 µ) (
d² n
d² n
d ² n
2
(72)
+
= p
P
dx2
dy²
dt2
d² E
d² Č
d² c
μ
+
P
dx2 dy³
dt2
These equations may be integrated by the process explained in Art.
286, and the integrals will involve a series and certain arbitrary
functions depending on the initial values. These will express
vibrations of the molecules; of which the periodic time will be
equal for those parallel to the a- and y-axes; that corresponding
to the z-vibration will be different. The sound yielded by it will
be much more intense, and will be the note of the membrane. The
coefficients and u will depend on the stretching weight T.
λ
т.
291.] Lastly, let us suppose the elastic body to be a string
which in its natural state is straight. That is, let us imagine
such a string to be taken out of a homogeneous elastic body of
constant elasticity, and to be under the action of a stretching
force r along its length, and to be subject at its surface to no
other forces. We imagine this string to be displaced by the ac-
tion of an external force; hereby certain elastic forces are brought
into action, which tend to restore the string to its original straight
state. The problem is, the determination of the motion of the
molecules of the string under the action of these elastic forces.
Let us suppose the string at rest to lie along the axis of x;
and let us consider the motion of a molecule whose place at
rest is (x, 0, 0); and let us suppose its place at the time t to be
(x + έ, 0+1, 0+), where έ, n, are the axial components of the
displacement at that time. Then (69) become
d² &
d²¿
(λ + 2 µ)
P
dx2
dt2
d² n
d2
μ da²
=
P
7
dt2
d²¿
d² C
μ
P
dx2
dt2
(73)
as these equations have the same form as (14) in the present
Chapter, they may be integrated by the processes already ex-
plained, and will lead to the same results.
And here I must conclude the investigations on the motion
291.]
523
OF AN ELASTIC BODY.
of elastic bodies and their molecules. The inquiry is, for the
subject, very imperfect; the completion of it would lead me far
into the theory of definite integrals, the properties of periodic
series simple and double, into wave-motion, &c., all of which
would require more space than we can afford, and which might
be inappropriate to a treatise, in which I have confined myself
to elementary subjects. I cannot however refrain from recom-
mending to the student the study of Leçons sur l'Elasticité des
corps solides, by M. Lamé, Paris, 1852; and chiefly to Le-
çons 8me, 9me, and 10me, wherein he will find the subjects of the
last two Articles especially treated of.
3 X 2
524
[292.
THEORETICAL DYNAMICS.
CHAPTER XI.
THEORETICAL DYNAMICS.
BY W. F. DONKIN, M. A., F. R. S., F. R. A. S.;
SAVILIAN PROFESSOR OF ASTRONOMY, OXFORD.
[It will be observed that many terms and symbols employed in this
Chapter differ from the corresponding ones of the previous parts of
the Treatise: this arises in part from the fact that Professor Donkin
had not seen the previous Chapters when this was written. In the
unavoidable absence of Professor Donkin, it has not been thought
desirable to change either the one or the other. The advanced
student indeed, for whom especially this Chapter is intended, will
not require any alteration: he will understand the different terms
and symbols by means of either the context or the explanations
which are given. Whatever has been added to Professor Donkin's
work is enclosed in square brackets. It should also be noticed
that his MS. bears date Sept. 6, 1860.]
292.] The object of this Chapter is to give some account of
the recent progress of theoretical dynamics. But no attempt
will be made to follow accurately a historical order, or to assign
every step in detail to its proper author. Such a plan would be
hardly in accordance with the design of this Treatise, and it is
moreover rendered unnecessary by Mr. Cayley's "Report on
Dynamics," lately published *.
The reader's attention is requested to the following explana-
tions of notation :
Throughout this Chapter total differentiation with respect to
t (the time) will be denoted by accents; and accents will be used
d2u
for no other purpose. Thus, instead of +
d² v
dt2 dt2
we shall
write either u” + v″ or (u + v)'', ..., and if u be a function con-
* Report of the British Association for 1857.
293.]
525
TIIEORETICAL DYNAMICS.
taining t explicitly, and also involving x, y,
tions of t, we shall have
du
du du
+
du
x² +
dt dx dy y' + ...;
which are func-
where signifies the partial differential coefficient of u with
dt
respect to t, taken so far as t appears explicitly in u.
But no other distinction will in general be made, by means of
notation, between the various possible meanings of differential
coefficients; the interpretation of the symbols, if not clear from
the context, will be explained in each case.
Secondly, expressions of the form
du dv
du dv
are of such
dx dy
dy dx
frequent occurrence, that it is desirable to have a recognised
abbreviation for them. The following has been found convenient,
and will be adopted, namely*,
du dv du dv
dx dy dy dx
d (u, v)
d (x, y)
293.] The theorem of D'Alembert reduces the mathematical
statement of every dynamical problem to the expression of con-
ditions of equilibrium; and when these conditions are put in
the general form, assigned by the principle of virtual velocities,
there results a single formula, which may be written thus:
z.m (x"dx + y″dy+z″dz) = x (x dx + y dy + z dz) † :
Y
(1)
in which x, y,≈ are the coordinates of the mass m, referred to
rectangular axes fixed in space, and x, y, z are the components
of the force applied at the point x, y, ≈.
It is necessary to observe, that the force here meant is not
the so-called "accelerating force", that is, the force which
would act on a unit of mass; but the total force, of whatever
kind, which is impressed at the point (x, y, z). Otherwise the
*
d (u, v, w, .. )
stands for the "Jacobian”
More generally, the symbol d (x, y, z,
determinant, of which the constituents are
•
du du du
dv dv dv
dw dw dw
;
;
dx' dy' dz
dx' dy dz
dx' dy dz
This notation was proposed some time ago by the writer of this chapter (see
Phil. Trans. for 1854, p. 72), and has since received the sanction of Mr. Cayley
(Report on Dynamics, p. 5).
† [See equation (40), Art. 50.]
526
[294.
THEORETICAL DYNAMICS.
formula would not include the case in which all or any of the
forces are to be considered as pressures acting merely on mathe-
matical surfaces, lines, or points; and having no relation to the
magnitude of the masses which they tend to move. It is true
that such forces are only mathematical fictions; but so are the
conditions of almost all mechanical problems, treated as we are
at present obliged to treat them.
It would however be out of place to enter into the questions
suggested by this remark, because, for the purposes of this chap-
ter, we are not concerned with the nature of the problem which
gives rise to the formula (1), further than is necessary for a clear
understanding of the meaning of the symbols.
In the most general case which occurs in practice, the values
of x, y, z, at the time t, may depend in a given manner upon the
time, the positions of all the points of the system, and the velo-
cities and directions of their movements at the instant considered.
In other words, x, y, z may be given functions of t, of all the
coordinates x, Y, Z,
and of their first differential coefficients
x', y',,... This most general case has not, except in special
problems, as yet been treated successfully, and we shall find it
necessary to limit the significations of x, y, z; but the problems
excluded by the limitation are comparatively unimportant.
و
294.] The meaning of the symbol of variation à may be ex-
plained as follows: If x, y, z, ... be the values of the coordinates
in the actual position of the system at the time t, then x+dx,
y+dy, z+ôz,... are the values belonging to any other position
which the system might have had at that time without violating
the conditions by which its possible displacements are limited;
provided only that the two positions be infinitely near to one
another, so that dx, dy, are infinitesimal.
Whatever be the nature of the conditions just mentioned, by
which the motion of the system is constrained, they may always
be supposed to be expressed by means of a certain number of
equations of condition
L₁ =
0, L₂
...
= 0,
Lm
= 0;
=
(2)
in which L1, L2,
are given functions of any or all of the coor-
dinates, and may also contain t explicitly. Suppose n to be the
whole number of coordinates involved in the formula (1), and m
the number of equations of condition, m will be in all cases
less than n; otherwise those equations would imply either a de-
295.]
527
THEORETICAL DYNAMICS.
terminate fixed position of the system, or a determinate motion
independent of the forces.
The formula (1) must be satisfied for every set of values of
Ex, y,
...
which satisfy the m equations,
d L1
8x+
d Lidy +
бу
<= 0, 1
dx
dy
dLq
dx
d L z 8x + dy
d L 2
ddy +
0,
(3)
d Lm
d Lm
8x+
dy+
= 0.
dx
dy
-m
By means of these last equations any m out of the n quantities
dx, dy,... may be expressed in terms of the rest, and when their
values so expressed are substituted in (1), that formula will in-
volve only the remaining n―m variations; and since the values
of these may be taken arbitrarily, without violating the m equa-
tions of condition, the coefficient of each must be separately
equated to zero; and thus we obtain n―m simultaneous linear
differential equations of the second order, that is, as many ast
there are independent coordinates.
This process has been described briefly, because the reader is
supposed to be already familiar with it*, at least in principle,
and because we shall not have to perform it actually.
295.] Instead however of eliminating the variations in the
manner just explained, we are of course at liberty to make use
of any equation that can be obtained from (1) by substituting
for dx, dy,... any admissible set of values; that is, any set which
satisfies the m equations (3).
Among the infinite number of ways in which admissible values
may be chosen for the variations da, dy, there are two which
require particular notice.
First, let us suppose (what is usually, but not necessarily, the
case) that every position of the system which is possible at any
one time, is possible at any other time; which is the same thing
as supposing that none of the equations of condition involve t
explicitly. In this case, it is evident, that among the positions
which the system might have had at the time t is that which it
actually has at the time t+dt; but at the time t+dt, the coor-
dinates x, y, ... have become a +dx, y + dy, ..., where de
where dx = x'dt,
* [See Arts. 51 and 52.]
528
[296.
THEORETICAL DYNAMICS.
dy = y'dt,...; so that we are at liberty to take as an admissible
set of variations the displacements which actually happen in the
time dt. In fact, since we are now supposing that the equations.
L₁ = 0,... do not involve t, the equations L', 0,..., are
L1
=
d L
x +
dx
dL
dy
y' +
0, ...;
and comparing these with (3), we see that dx, dy, ..., may be
taken proportional to a', y', ..., that is, to da, dy, ....
contained t explicitly we should have
But if L
d Li
+
dt
d L
d Li
x² +
y' +
= 0,
dx
dy
which cannot be made to coincide with the equation
8 L1
d Li
dx
·8x+
dy
dI 18y+
бу + ...
0
by any values whatever of dæ, dy, ...*.
In the case now supposed, however, when the values a'dt,
y'dt,..., or dx, dy,..., are substituted for dx, dy, ... in (1), that
formula becomes
z'z″)
z.m (x'x" + y'y" + 2'2") dt = (x dx+x dy + z dz);
or, if we put r for the vis viva of the system, that is,
1
we obtain
2
1
T
2
z.m (x'² + y²+z'²) = 2/ z.m v²;
dr = x(x dx+x dy +z dz),
an equation which we shall meet with afterwards in a somewhat
different form.
296.] The second way of choosing admissible values for da,
dy, ..., is applicable in all cases without exception, and may be
explained as follows:
Then coordinates x, y, ... being subject to m equations of
condition, it follows that any n-m of the coordinates may be
considered as absolutely independent; that is, their values might
at any time t be assumed arbitrarily without violating the laws
of constraint expressed by the equations of condition; and since
the same thing is true at the time t +dt, it follows, that not only
the n―m coordinates, but also their first differential coefficients
x', y',... might be arbitrarily assumed at the time t; but the
values of these 2 (n-m) quantities being given, those of the
remaining m coordinates and of their first differential coefficients
* [This circumstance has been already alluded to in Art. 63.]
296.]
529
THEORETICAL DYNAMICS.
are determinate. In fact, if έ, n,... be these remaining coor-
dinates, the m equations of condition would suffice to express
each of them in terms of x, y, ..., and t, so that we should have
m equations, such as
§ = f(x, y, ... t);
from which we should get by differentiation
ૐ
df df
+
dt dx
df
x² +
y' + ... ;
dy
where it is evident that the values of both έ and έ' are given
at the time t, if those of x, Y,
x', y', ... are given.
Now the motion of the system, under the action of the given
forces, is completely determined if the positions of all its points,
and the velocities and directions of their movements, be given at
a determinate time; that is, if the values of all the coordinates
x, y, ..., and their first differential coefficients x', y', ..., be given
at that time; and since it has just been seen that all these
quantities are given if any n-m of the coordinates, with their
first differential coefficients, be given, we conclude that the whole
motion is determined if the values of these 2 (n-m) quantities
be given at any one time; it is convenient to take the instant
when t = 0 for the time in question, and we may call the values
of any quantities at that time their initial values.
From these considerations, it is easy to conclude that the final
integral equations of the problem must contain 2 (n —m) arbitrary
constants, and no more; that is, that the values of all the coor-
dinates must be expressible in terms of 2 (n-m) arbitrary con-
stants and t; for otherwise, the number of coordinates and first
differential coefficients which it would be possible to assume
arbitrarily at a given time would be either more or less than
2 (n−m). The same conclusion follows from the theory of dif-
ferential equations.
Hence we are in all cases at liberty to suppose that the actual
value of every one of the coordinates at the time t is expressible
by an equation of the form
X f (a, b, ... t);
where a, b, ... are the arbitrary constants, of which the number is
twice that of the independent coordinates.
These constants may be the initial values of some set of inde-
pendent coordinates and of their first differential coefficients,
and must be expressible as functions of such initial values.
PRICE, VOL. IV.
3 Y
530
[298.
THEORETICAL DYNAMICS.
297.] Now if we suppose the initial positions and velocities to
receive infinitesimal alterations, or if, which comes to the same
thing, we suppose the constants a, b, ... to be changed into
a+da, b+db, ..., the values of x, y,
changed into + dx, y +dy, ..., where
at the time t, will be
dx
dx
dx = δα +
88+
་ . ་་
da
db
dy
dy
dy
...
=
da
db
dx
da'
δα + 8b+
the partial differential coefficients
.;
being taken on the
hypothesis that x, y,……. are expressed, as above supposed, in terms
of a, b, ..., t.
The values of dx, dy, ..., thus formed, are distinguished from
other admissible sets of values by important properties, which we
proceed to point out.
=
First, if we suppose a, b, ... to have the values belonging to the
actual motion of the system, so that the point (x, y, z) actually
describes the path defined by the equations x = ƒ (a, b, ..., t), ...,
then the values a +da, b+db, ... correspond to a motion which
does not actually take place, but which might take place under
the action of the existing forces, and would take place if the
initial circumstances were suitably altered; so that the substi-
tution, at every instant, of x+dx for x,... would change the
actual paths and velocities of all the points of the system into
others not merely consistent with the given equations of con-
dition, but consistent also with the action of the forces. All
such paths and velocities may be called "dynamically possible."
But if the values of dx, dy, ... were merely chosen so as to be
consistent with the equations of condition, without any further
limitation, then the substitution at every instant of x+dx for x,
..., would change the actual paths and velocities into others,
which, though not inconsistent with the given laws of constraint
of the system, could not be produced by the action of the exist-
ing forces. Such paths and velocities may be called "geome-
trically possible*” though dynamically impossible.
298.] But there is another, and in some respects more im-
portant, distinction.
* The expressions "dynamically possible" and "geometrically possible"
are Sir W. R. Hamilton's.
300.]
531
THEORETICAL DYNAMICS.
The values of da, db, ... are arbitrary infinitesimal constants.
Let ▲a, Ab,... be any other set of similar values; and let du, au
be the increments of any function u, corresponding to the two
sets of increments of the constants; so that if u be expressed as
a function of a, b, c, ..., with or without t, we shall have
du
du
би =
Sa+
8b+
da
d b
du
du
Ди =
Δα + Ab + ..
da
db
Now suppose that in the above value of du we change a, b, ...
into a+sa, b+▲b,..., without altering the values of da, ôb,…….
the corresponding increment of du will be
d2 u
da²
Adu= sa.da +
d2 u
da db
(da.sb+db.sα) + ...;
... ;
but the same expression will be obtained for dau from the
second of the above equations; consequently,
Δ δκ = δ Δυ.
(4)
In this equation u may evidently be any function of the coor-
dinates x, y, z, •
and their differential coefficients of all orders;
the property expressed by it distinguishes those variations of u
which are due to variations of the constants a, b, ... from those
which, though otherwise admissible, arise in a different manner.
In fact, it will be found in general, either that the symbols ad,
d▲ are unmeaning, or that the above equation is not true.
299.] Lastly, we must notice a property which belongs to the
variations denoted by d or a without any limitation, namely,
that the operations d, or ▲, and
d
dt
are commutative; that is,
d (u') =
(du)' ;
(5)
where u is any function of x, y, .. (with their differential coeffi-
cients) and t, and dæ, dy, ... may be considered as perfectly
arbitrary functions of t, subject to the sole restriction of being
infinitesimal. In fact, the meaning of d (u') is (u+du)'u', that
is, is (du)'.
300.] Now let x, y, ... be any n variables, functions of t, and
subject to m equations of condition. Also let έ, n, be other
variables, of which the number is not less than n―m, and so
connected with the former set of variables by given equations,
which may involve t explicitly but may not involve the differ-
3 Y 2
532
[300.
THEORETICAL DYNAMICS.
ential coefficients of either set, that any variable of one set may
be expressed as a function of variables of the other set, with or
without t, by equations such as
X & (§, n,
t), Į
§ = ↓ (x, y, ..., t).S
(6)
It is to be observed, that the expressions on the right of these
equations are, to a certain extent, indeterminate in form; for
any function of x, y, ... and t may be variously transformed by
means of the given equations of condition; and the same may be
said of any function of §, ŋ, ..., if the number of the latter vari-
ables be greater than n—m.
...
·
Suppose then that u is any function of t, x, y, ..., x', y',
x", y', ; and let r be the order of the highest differential
coefficient contained in u. By means of the equations, such as
the first of (6), x, y, ... can be expressed as functions of t, g, n,...;
§,
x', y',... as functions of t, ૐ,
, n, έ'‚ n', ... and so on; so that
u can be transformed into a function of t, È, n, §', n', ..., in
which the highest differential coefficient will still be of the order
r. Also, dr, dy,... can be transformed by means of the equations
dx
8x= 8ε + δη +
dx
αξ
dn
where
d. dx
dε' dŋ'
,
are given functions of t, §, ŋ,
Let Eu be for a moment an abbreviation for
du
dx
(du) + (du)" -
the series being continued until it terminates of itself. Then
the expression
Ex U. d x + Ey U. Ô у + Еz U. δ 2 +
can be transformed into another, involving the variables έ, 7,...,
with their variations and differential coefficients, instead of x,y,....
It is a known theorem in the Calculus of Variations that the
result of this transformation is an expression of the same form,
namely,
Εκ.δέ + Ε, κ.δη + ... ;
E
η
;
where u only differs from the former u in being expressed in
terms of the new variables.
The direct demonstration of this theorem in its general form
* For an indirect demonstration, see Lagrange, Mécanique Analytique,
2de partie, 4me section, 6, or De Morgan's Diff. Calc., p. 519.
301.]
533
THEORETICAL DYNAMICS.
is somewhat complicated, and need not be given here, because
the only case with which we are concerned is that in which u
contains no differential coefficient of a higher order than the
first, or
u = F(t, x, y, ... x', y', ...).
The theorem may then be conveniently written thus:
Σ.
= {(du) -
du)
dx S
}
δι = Σ.
d)
{( de ) - 1 }
du dur
αξ
82;
(7)
and this admits of the following simple demonstration, due in
principle to Sir W. R. Hamilton.
301.] Since u, when expressed in terms of the new variables,
n,
§, ŋ,..., will contain §', n',..., only because it originally contains
x', y', . we shall have
du du de du dn
+
dx'
dε dx'
dn dx
+ ...
where the differentiations in
d¿'
dx'
,
are performed on the sup-
αξ αξ
ૐ' = +
dt dx
position that έ', n',... are expressed in terms of t, x, y, ..., x', y', ... :
now έ being expressed in terms of t, x, y, ..., we have
x² +
αξ
y' + ... ;
dy
αξ αξ
and since
do not contain x', y', ..., we obtain at
dt' dx
once by differentiation
d&
αξ dε'
αξ
;
(8)
dx'
dx' dy
dy
>
in like manner we should find
dn dn
dn'
d n
:
dx'
dx
dy'
dy
du
hence the above expression for
du du dέ du dn
dx
dε dx dn dx
similarly we should have
becomes
dx'
+
+ ·
du
du de du dn
+
dy'
dε dy dn dy
+ ...;
and so on: whence, multiplying the first of these equations by
dx, the second by dy, ..., and observing that
dé
dx
αξ
8x +
sy +.. हहु
dy.
534
[302.
THEORETICAL DYNAMICS.
we obtain by addition,
du
dx'
du
8x + dy + ..
dy'
d u
du
88+
δη τ.
dε
λή
and differentiating this last equation with respect to t, and ob-
serving (Art. 299) that (dx) = dx', ...,
dx
(du)'s
-) 8 x +
du
dx' + ..
dx'
du
du
αξ'
de
8 &+ 85+
Now du may be expressed in either of the following ways:
du
du
du
би
8x +
8x' +
бу+
Sy+....
=
dx
dx
dy
du
du
du
...
;
du =
αξ
αξ'
d n
8ε + 8&'+ δη +
and if the first of these values be subtracted from the left-hand
member, and the second from the right-hand member of the
above equation, the result is the equation (7), which was to be
established.
con-
302.] We now return to the dynamical formula (1), Art. 293.
In that formula, the position of the system at the time t is
assigned by means of the rectangular coordinates x, y, z, .;
but it is evident that any other set of variables, §, n, §,
nected with x, y, z, in the manner supposed in Art. 300,
would answer the same purpose. We may extend the meaning
of the word "coordinates" so as to include all sets of variables
of which the values at the time t determine the position of the
system at that time. For instance, the position of a rigid sys-
tem which has one fixed point may be defined in several ways
by means of three angles, which may be called the "coordi-
nates" of the system.
Now the theorem (7), Art. 300, enables us to express the
left-hand member of the formula (1) in a form adapted to any
system of coordinates whatever, in the following manner:
Let r* denote as before the vis viva of the system; then, in
terms of the original coordinates, we have.
2
2 r = x.m (x²² + y²² + z²²);
but when T is expressed in terms of any other coordinates,
έ, n,..., it will become in general a function of ε, n, ..., E', n',...,
§,
with or without t, not containing any differential coefficients of
* [Vis viva, as used here and in Art. 295, is one-half of the quantity which
has heretofore been called by that name.]
302.]
535
THEORETICAL DYNAMICS.
a higher order than the first. Hence we may put т for u in the
equation (7), and, observing that
dr
dr
= x.mx',
0,
dx'
dx
we obtain
dr
z.mx'' d x = Σ.
{(dr) - d = }
αξ
84;
which is the required form.
The terms xdx + Ydy +
pressed in terms of the new
ρδξ + αδη + ..., where P, Q,
without t; so that the equation (1) is finally reducible to the
form
on the right of (1), when ex-
coordinates, will take the form
are functions of έ, î, with or
Σ
dr
αξ'
dr l
હું
δξ = Σ.Ρ δξ.
αξι
...
(9)
In the most usual and important problems, x, y, z,... are the
partial differential coefficients with respect to x, y, z,... of a
function u, called the force-function, which may also contain t,
but does not contain x', y', ... : in this case we have
Y
xô + xông +
δυ;
and the right-hand member of (9) is obtained by deriving du
from u expressed in terms of the new variables: thus
du
du
du =
8ε +
δη +
;
αξ
dn
and the equation may then be written in the form
Σ
{(da)
dr du
d § = 0 :
αξ αξ
(10)
this may be abridged by putting T+U=w; for, since u does not
contain έ',...,
dr
d w
वहु
..; so that the formula becomes
Σ.
{
αξι
d w
αξ'
:)-
d w I
αξ
d έ = 0.
If the coordinates έ, n,
...
(11)
be independent, that is, subject to
no equations of condition, the coefficients of dέ, dŋ, ... must se-
parately vanish; so that (11) is equivalent to the system of
equations
d w
dw
αξι
dn
(dw)
d w
dn
(12)
We shall refer to these as the "Lagrangian" equations; a name
given to them by Mr. Cayley.
536
[303.
THEORETICAL DYNAMICS.
303.] It will be desirable to illustrate the preceding formulæ
by some examples before proceeding further. First then, let it
be required to express the equations of motion of a single mate-
rial point by means of polar coordinates.
Let m be the mass of the point, x, y, z its rectangular coordi-
nates, x, y, z the components of the force acting on m, so that
the equations of motion in their primitive form are included in
the formula
m (x"dx + y″dy+z″"dz) = xdx+rdy+zdz.
'(13)
In polar coordinates we have, according to the usual notation,
x = r sin cos 0, y = r sin ◊ sin ø, z = r cos 0;
and we must express 2T = m (x²² + y²²+z′²) in terms of r, 0, 4,
r', ', '. Differentiating, we find
x' = r′sin ◊ cos & + e' r cos é cos p — o'r sin ◊ sin 4,
y
r'sin 0 sin p + e'r cos 0 sin + o'r sin 0 cos 4,
z′ = r'cos (
hence we easily obtain
- e'r sin 0;
2
2 r = m {p² + r² 0²+r² (sin 0)² '2};
and we know (Art. 302) that the left-hand member of (13) will
become
dr
{(dr) - dr} or + {(da) - dr} 80+
dos
{(dr) - dr} 04.
δφ.
d T
dr
Now
dr
= mr',
= mr {0′² + (sin 0²) 4'2},
dr
dr
dr
= mr² 0.
m r² sin 0 cos 0 p′²,
,
do'
do
dr
dr
mr² (sin0)²p',
= 0;
dø
dp
and these values reduce the above expression to the following:
m {r" — r0'² — r (sine)2 p2} dr
12
+m { (r² 0')' — r2 sin 0 cos 0'2} d0 + m {r² (sin 0)² p'}'dp.
The right-hand member of (13) will always be reducible to the
form Por+Q80+R80; and in the case in which x, y, z are of
the form
du du du
dx
αυ dv du
dr' do' do
dy
dz
>
then also P, Q, R will be of the form
304.]
537
THEORETICAL DYNAMICS.
If the motion of m be unconstrained, dr, 80, do are all arbi-
trary, and the formula (13) breaks up into three separate equa-
tions.
304.] As a second example, let us consider the transformation
from fixed rectangular axes to moving rectangular axes with the
same origin.
*Let x, y, z be the coordinates of the material point m, referred
to the fixed axes; §, ŋ, the coordinates of the same point, re-
ferred to the moving axes; and let the position of the moving
system at the time t be defined in the usual manner by the
equations
3
5,
x = α1 § + A₂ N + Az 5, y = b₁ § + b₂ n + bz S, ≈ = c₁ § + C₂ N + Cz Š;
n
where the nine direction-cosines a, a, ... are given functions of
t. It will be convenient to introduce the usual symbols w1, W2, W3
for the angular velocities of the moving system of axes, estimated
about the axes of έ, n, respectively. Let a rotation about the
axis of be positive when its direction is such that the axis of §
is following the axis of ŋ; then, with similar conventions as to
the other axes, we shall have
Az α z + bz b₂ + Cz C2′ = − (ɑq ɑz' +
2
a₁ αz + b₁ bz' + c₁ C3′ = − (αz α₁' +
1
C1
2 C1
ɑ ɑí + b₂ bí + c₂ c₁ = − (a₁ α₂ +
2
W11
b₂ bz' + C2 C3 )
bz b₁ + C3 C₂')
= W₂,
b₁ b₂ + c₁ c₂)
C1
= Wz⋅
§
Now differentiating the equations x = α₁ έ + α₂n + a3 Ś, ... we
obtain
x′ =
a₁ §' + α₂ n' + α3 §' +
y' = b₁ §' + b₂ n + b 3 5 +
Z
1
C
C₁ §' + c₂ n' + C3 Ś′ +
a₁ § + a2 n + aż Ś,
b₁ § + b₂ n + by 5,
1
c₁ § + c₂ n + c3′ Š;
and hence, observing the above values of w..., and the known
relations between the nine direction-cosines, including the equa-
tions a₁ a₁+ b₁ b₁ + c₁ c₁'
1
a₁ x² + b₁ y' + c₁ Z = §' + w₂ 5 —
=
0,...,
а1
=
1
w3
wz N,
x'
Az x′ + bz Y + C₂ z′
=
n' + wz § —
w₁ 5,
Ś,
аз
Az x′ + bzy' + Cz z′
=
5' + w₁ n −
w₂ § ;
3
and finally, by adding the squares of these expressions on each
side
12
/2
x²² + y²²+~'² = (§' + w₂5 − wzn)² + (n' + wz§ — w₁5)² + (5′+w₁n — w₂ §)².
[This is the problem which has already been investigated in the first
Section of Chapter VIII. The difference of notation will be observed.]
PRICE, VOL. IV.
3Z
538
[305.
THEORETICAL DYNAMICS.
:
The meaning of the terms in this expression is easily seen thus,
έ' is the velocity of the point (§, n, §) relative to the moving
axes, estimated parallel to the axis of έ; and w₂ (—wŋ is the
velocity, relative to fixed space, and estimated in the same direc-
tion, which the same point would have if (§, n, ¿) were invariable;
the sum is the total component of the velocity relative to fixed
space. Hence the value of T, expressed in terms of the new
coordinates, becomes
T
1
2
Σ.m {(§' + w₂Š—w₂n)² + (n'+w3§ − w₁ Š)² + (Š' + w₁n—w₂§)²}; (14)
which is the expression to be used in forming the left-hand
member of the equation (9), Art. 302.
The right-hand member of that equation is to be obtained by
transforming the expression (xdx+xdy + zd≈).
Now from the equations
we have
,
x = α₁§ + α₂n + ɑz Ś, ...,
δι = αιδξ + α,δη + αδζ....
x a1
A1, A2, being treated as invariable in the differentiation
denoted by d, because da,... refer to displacements in space
which might subsist at the time t. Hence
Y
xô+rông+zỖ. =
(α₁x + b₁x + c₁z) ò § + (ɑ2x+b₂x + C2 Z) dŋ + (ɑ3X+bz¥+Czz) dŠ;
X
1
1
Y
; and
Thus the
now α₁x+b₁x+c₁z is the component of the force acting at (x,y,z),
estimated in the direction of the axis of §; so that if we call
this component E, the term involving &έ becomes
similar conclusions will result for the other terms.
right-hand member of the equation (9) may be represented by
.d§; the summation referring to all the coordinates as well
as to all the points, and the coefficient of each variation being
the corresponding component of force.
305.] As an illustration of the general formula of the pre-
ceding Article, we may take the following problem:
To find the modification introduced into the treatment of dy-
namical problems, referring to motion near the earth's surface,
when the earth's rotation is taken into account.
Neglecting the curvature of the path of the earth's centre,
and assuming that the forces concerned in the problem are in-
dependent of the earth's position in its orbit, we may consider
the centre as fixed.
305.]
539
THEORETICAL DYNAMICS.,
Let the primitive axes of coordinates then have their origin at
the earth's centre, the positive axis of z being directed to the
north pole, the axes of x and y being in the plane of the equator,
but fixed in space; the positive axis of y being to the east of
that of x.
We wish to take as new axes a system fixed relatively to the
earth, and having its origin at a given point on the earth's surface.
As an intermediate step, let §, n, Ŝ refer to axes fixed in the
earth and parallel to the required axes, but having their origin
at the centre. If then we call the angular velocity of the
earth's rotation, and cos a, cos ß, cos y the direction-cosines of
the polar axis referred to the axes of §, n, Ś, we shall have
ωι
@3
@z =
w cos y ;
@₁ = w cos α, w₂ = w cos ß,
and the expression (14) Art. 304, becomes
1
2
T= Σ.m [ { §' + w (Ś cos ẞ — n cos y)}²
2
J
+ {n' + w (§ cos y — (cos a)}² + { 5' + w (n cos a − έ cos ẞ)}2];
and it only remains to remove the origin to the required point
on the surface by writing έ+ §o, n + 1o, Ŝ+ § instead of έ, n, Ś,
where έo, no, S。 are the coordinates of the point in question, re-
ferred to the axes of §, n, ( with centre as origin. This being
done, the following values will be found without difficulty:
dr = m &' +mw {(5+5) cos ß−(n+no) cos y},
वहु
dr
αξ
= mw (n cos y —('cos ẞ) +m w² (§ + §o)
m w² cos a { (§+ §) cos a + (n + 1) cos ß + (5+5) cos y} ;
consequently
dr
वहु
d r
αξ
m ¿" + 2 m w (5′ cos ẞ — n' cos y) − m w² (§ + §o)
+m w² cos a {(§+ Év) cos a + (n+no) cos B+ (5+50) cos y};
from which the forms of the terms referring to n and are ob-
vious.
If we call the latitude of the place at which the origin is
fixed, and take the plane of (§, ŋ) horizontal, the axis of § being
directed to the south and that of ʼn to the east, we shall have
cos l, cos ẞ = 0, cos y =
sin 7;
COS α =
a
-
also έo=0, and no, So are given quantities, the former being small,
of which the values are easily assigned in terms of the earth's
axes and of l.
3 Z 2
540
[306.
THEORETICAL DYNAMICS.
Thus we obtain *
(음​)
dr
αξ
d T dr
dn'
an
dr
m ¿″ – 2 m w sin ln' — m w² (sin 7)² §
m w² sin l cos 1 (§ + Šo),
= mn"' + 2 mo ('sin l+C'cos l) - mw² (n+no),
(d)- = m (" — 2m o cos l n' — m w² sin l cos l §
زمره
ας
2
m w² (cos 1)² (5 + 50);
so that, finally, the general equations of motion of any system
under the circumstances supposed are comprised in the follow-
ing formula:
≥.m (§''d§ + n'dn+5″d §)
+2w sin lx.m (§'dn−n'd§) +2 w cos l ≥ . m (§'dn — n' d§)
-w² (sinl)2 x.m έd-w² cos²l z.m (5+50) 85 — w² z.m (n+no) dn
-w² sin l cos lz.m { (S+ So) d§+ § 8 C} = x.m = d§.
306.] It is not intended in this Chapter to discuss particular
problems; and the examples given in the last three Articles
have been inserted only because the Lagrangian formulæ, if left
in their general shape without illustration, would probably fail
to convey precise notions to the mind of a reader coming to
them for the first time.
We proceed now to an important transformation of these for-
mulæ, due to Sir W. R. Hamilton, without stopping to intro-
duce at this stage the consequences derived from them by La-
grange; because these, with many other results, are more easily
obtained from the Hamiltonian form.
Conforming to the notation of recent writers, we will denote
the "coordinates" in any dynamical problem by 1, 2,..., so
that the general formula (11) Art. 302, becomes
dw
Σ.
{(a))
d w
8q
= 0;
d q
(15)
in which we shall suppose that w may be any function whatever
of 91, 92, ..., 91, 92',
D
and t.
If now we put
d w
d w
= P2,
= P1,
dqi
dqz
(16)
* [The equations given in (55), Art. 237, are identical with these when the
signs of and are changed.]
† A first step towards this transformation was made by Poisson; but we
have not space for details on this point.
307.]
541
THEORETICAL DYNAMICS.
1
>
we may suppose q1, 2, ... to be expressed, by means of these
equations, in terms of P1, P2, , 91, 92, with or without t;
and when these values of q', 42', are introduced in the for-
mula (15), that formula, together with (16), will give a set of
equations involving the two sets of variables 1, 2, ..., P1, P2, ...,
with their first differential coefficients, instead of the one set
91, 92, with their first and second differential coefficients.
Thus, if the coordinates 1, 2, ..., q, be an independent set,
instead of n differential equations of the second order we shall
have 2n of the first order.
The general form of these 2n equations was first assigned by
Sir W. R. Hamilton. His demonstration depends upon the par-
ticular character of the function r in most actual problems; and
the following, which is slightly different and more general, is
therefore substituted.
307.] The principle of the demonstration may be most clearly
exhibited independently, in the form of the following theorem :
If Þ be any function of the n quantities x1, x2, ….. ~„, and if n
p
other quantities y₁, Y2, ... Yn be defined by the equations
xn,
d P
dp
Y1
Y₁ =
Y 2
dx1
dx 2
d p
Yn =
;
(17)
d x n
then, if by means of these equations, a', ..., a, be expressed in
……., Yn, their values will be of the form
terms of y₁, Y2,
da
(18)
da
;
Xn
dq
dyn
1
X₁ =
хочу
dy1
d y z
(19)
where
=
P+X1 Y1 + X 2 Y 2+ ... + Xn Yn:
in which ₁,..., xn on the right are supposed to be expressed in
~1,
terms of y₁, ..., Yn•
Also if P contain any other quantities, &,..., besides x1,...,x,
Р
then
dp
αξ
do
;
αξ
the differentiation with respect to § being in each case performed
only so far as έ appears explicitly.
To prove this we have, if the symbol d operate only on 21, ...,
Xn, Y1, ,
but
Yn,
dp = y₁dx₁+Y 2 dx 2 + ... + Yn dxn, by (17) ;
d ( x₁ Y ₁ + ... + Xn Yn) = Y₁ dx₁ + ... + yn dx + x²₁ dy₁ + ... + x₁dyn;
1
1
542
[307.
THEORETICAL DYNAMICS.
hence, by subtraction,
d ( x 1 Y ₁ + ... + Xn Yn −P) = x₁ dу₁ + ... + xn dyn ;
1
an equation which must be identical if both sides be expressed in
the same way. If therefore we put, as above, q=X₁Y1 + - P,
on each side, expressed in terms of y₁, ...,
,
and suppose X1
do
since da
dy1
dy₁+….., we must have x₁
proves the first part of the theorem.
do
dy1
which
Q
To prove the second part, wc observe that the value (19) of q
will contain έ explicitly, partly because it is contained explicitly
in P, as originally expressed, and partly because the values of
X1, ...,
in terms of Y1,
when substituted in P and the other
terms, will introduce it again.
Hence we shall have
do
dr dp dx,
dp dxn
y
αξ
αξ
dx,
αξ
dan de
αξ
dx1
dxn
;
+ Yi de
+ ... + Yn
η αξ
dr
but since y₁ =
dx1
Y1
this equation becomes simply
which was to be proved.
da
dp
d&
αξ'
Let us now apply this theorem to transform the formulæ
(15) and (16), Art. 306.
The equations (16) being exactly similar to (17) of this Article,
it follows, that if we put
H = w+P₁ Qi' +P2 Q2' + ... + Pn In',
92
and express q', ... qn' on the right in terms of p₁,
shall have
,
Pn, ..
we
Չ
d H
dp₁
dH
In
;
dpn
and moreover, since, besides 91, 92',
quantities 91, 92, ..., analogous to έ,...,
dw
dq1
w contains also the
we shall have also
dн
d H
dH
d w
;
dq2
d q z
dq2
dqi
so that the formula (15) will become
dH
(p' + day ) òg =
= 0.
These results may be summed up as follows:
If w be any function of 41, 42, ..., 91, 92,
Σ
{(
d w
dq
dw)
91', 92',….., t, the formula
d a } o q
= 0
307.]
THEORETICAL DYNAMICS.
543
is transformed into the system
Σ
=(p²+
dH
dg) ög
dq = 0,
(20)
q
dH
by the following substitutions:
dH
92
P:
dp₁
dp2
dw
d w
7 P1
P2,...,
dqi
dq z
w;
H = P₁ 91 + P₂ I2 +
where, in forming the expression for н, we are to express q₁',
... in terms of P1, P2,
42',
..., 91, 93, so that H is in general a
function of P1, P2, 91, 92, and t.
...
و
>
T
One case deserves particular notice, because it occurs in most
actual dynamical problems. If w be of the form T+u, where т is
homogeneous and of the second degree in 41, 42', .., and u does
not contain q', 42, ..., then p₁ =
dr
dqi
,
and therefore
d T
P1 I1' + P₂ I2' +
q i d q i
7
+
2T;
hence, in this case,
H2T-WT-U,
where r is to be expressed in terms of P1, P2, ..., L1, L2, ... .
If 91, 92, ... be a set of independent coordinates, say n in num-
ber, then the system (20) gives 2n separate equations, namely,
those obtained by giving to i all integer values from 1 to n in-
clusive in the two following:
Pi
dH
d qi
dн
dpi
(21)
We shall call these, as Mr. Cayley has done, the " Hamiltonian”
equations*. In treating of their general properties it is usually
unnecessary to take any account of the nature of the problems
* The Lagrangian equations may be considered as a particular case of a
more general form, upon which the solution of a class of problems in the
Calculus of Variations depends; and it has been shewn by M. Ostrogradsky,
that this more general form is susceptible of a transformation which includes
that of sir W. R. Hamilton as a particular case. See "Mémoire sur les équa-
tions différentielles relatives au problème des isopérimètres, 1848.” Mém. de
l'Acad. Imper. des Sciences de St. Petersburg. Sciences Math. et Phys. t. iv,
1850.
544
[308.
THEORETICAL DYNAMICS.
which give rise to such a system. H is to be considered merely
as a given function of P1, P2, ……. Pn, 91, 42,
...
In, and t.
308.] The complete solution of the 2 n simultaneous differen-
tial equations of the first order, represented by the formula (21),
would consist of 2n equations involving the variables P1,...,
91, and t, with 2n arbitrary constants. Any one such equa-
tion may be called an "integral equation;" but it is desirable to
distinguish by a separate name that particular form of integral
equation in which a function of variables only is equated to an
arbitrary constant. We shall call such an equation an "inte-
gral." Thus the general form of an integral will be
c = ƒ (P₁, ... Pn, 91, ...
qn, t);
where the function on the right contains no arbitrary constant;
and it is a convenient abbreviation to speak of such an integral
as the integral c."
*
Thus a complete solution of the system (21) may be supposed
to consist of 2 n integrals. But in order that 2n integrals may
constitute a complete solution, it is necesssry that they should
be independent; that is, that no identical relations should sub-
sist between the functions equated to the arbitrary constants. -
If such relations did subsist, the variables might be eliminated,
and one or more equations be obtained involving the constants
only, so that the constants would not be all arbitrary.
Hence the problem of integrating the system of equations
(21) may be stated as follows:
...
"To find 2n independent functions of P1, P2, Pn, 41, 42, ... qn
and t, each of which is constant by virtue of the differential
equations (21).”
On the other hand, the same problem might be regarded as
having for its object "to express each of the 2n variables, P₁,……..,
1,..., as a function of 2n arbitrary constants and t."
If a complete solution were obtained in either of these forms,
it is evident that algebraical processes only would be required to
deduce from it a solution in the other form, as well as an infinite
variety of "integral equations."
* This expression however "the integral c” is, to avoid circumlocution,
used not only to signify the equation c = f(p₁, ...), but also to denote either
side of that equation separately, viz., either the constant c, or the function
ƒ(P₁, ...), which has that constant value. The last is the most usual
meaning.
•
310.]
545
THEORETICAL DYNAMICS.
The consideration of the two forms just mentioned is of the
greatest theoretical importance, though neither of them is in
general obtained as a direct result of existing methods of inte-
gration.
Inasmuch as all complete solutions of the same system of
differential equations must be equivalent to one another, it fol-
lows that any arbitrary constant belonging to one solution must
be capable of being expressed as a function of the arbitrary con-
stants belonging to any other solution.
>
309.] Any 2n functions of the variables P₁,
and t,
..., 911
may be called elements, provided that the equations by which
they are defined are algebraically sufficient to determine con-
versely the 2n variables P1,..., 91, ..., as functions of the ele-
ments and t. Thus, if the elements u₁, un, u2n be defined
by 2n equations, such as
u₁ = ƒ (P₁,..., q₁, ..., t),
91,
...
then it must not be possible to eliminate all the 2n variables,
P₁, ..., 91, ..., from these equations.
P1
From the above definition, it is evident that a complete solu-
tion of the differential equations would be obtained if any set of
elements were expressed in terms of arbitrary constants and t.
It is also evident, that the functions which are equated to
arbitrary constants in any complete set of integrals are "ele-
ments." Thus elements may be either variable or constant.
310.] It may be useful to exhibit at this stage, for the sake
of clearness, the equations of a simple dynamical problem in the
Hamiltonian form. For this purpose we may take the case of
motion of a single material point about a fixed centre of force.
Let m be the mass of the moving point; then, taking the origin
of the polar coordinates r, 0 at the fixed centre, and the plane
of the motion for the plane of the angle 0, we shall have
2 T = m (1²² + µ·² 0′2) ;
and the force-function u will be a given function of r, say up(r).
Then, writing q₁ instead of r, and q, instead of 0, we have
2 12
2x = m (q,² + 1² 2²);
consequently,
91
T
12
dr
dr
P1
= mq,
P2
dgí
1
= mq₁² q2' ;
d q z
2
from which we have
Pi
P2
91
92
M
mq₂²
2
PRICE, VOL. IV.
4 A
546
[311.
THEORETICAL DYNAMICS.
and therefore н, which in this case is r
of P1, P2, 91, 92, becomes, see equation (20),
H =
1
2 m
2
(p₁² +
(2
2
u expressed in terms
P₂
2
) — $ (91);
and the four equations (21), Art. 307, become
2
P2
P₁ =
+ $'(q₁),
m q₂3
Չ
P1
;
M
1
P2
P₂ = 0,
92
m qi
2
311.] If u, v be any functions whatever, containing the vari-
ables P1,
……., Pn, I1, •••, qn, then it is convenient to employ the
following symbol: let
du dv du dv
(u, v) = ≥
dpi dqi
dq; dpi
-).
or in the notation of Art. 292,
d (u, v)
(u, v) = Σ
d (Pi, Ji)
the summation extending to all values of i from 1 to n.
The reader will hardly require to be reminded that the sym-
bols (u, v), (Pi, Ji) on the right of the last equation, have only an
accidental resemblance to the (u, v) on the left, without any con-
nexion of meaning.
For example, if u, v contain p₁, P2, 91, 92 only, then
(u, v)
du dv du dv du dv
dp₁ dqı dq₁ dpi d p₂ dq 2
du dv
+
dqz dp z
From the above definition the following consequences are easily
deduced by means of the elementary principles of differentiation :
(u, v)
(Pi, qi) = 1,
(v, u,)
(u, u)
0,
(qi, Pi)
- 1,
and (Pi, q) = 0, if j be different from i.
Also, if a be any function of v, w,
then
da
da
(u, a)
=
(u, v) +
(u, w) +....
dv
dw
Again, if v contain P1, 1, ...
explicitly, and also a, ß, ...
functions of P1, 91,
then
dv
dv
(u,v) = (u, v) + (u, a) +
(u, B) +...,
da
αβ
where (u, v) represents the expression formed by differentiating
v only so far as it contains P₁, 91,
explicitly.
313.]
547
THEORETICAL DYNAMICS.
Lastly, if u, v contain explicitly any other quantity, say z,
besides the variables p, q,..., then the partial differential coeffi-
cient of (u, v), taken explicitly with respect to z, is
d
du
(u, v) =
(az, v ) + ( u, dv).
dz
dz
312.] The following theorem will be of use afterwards:
Let u, v, w be any three functions whatever, containing P₁, 1,....
with or without other quantities, then
{u, (v, w)}+{v, (w, u)} + {w, (u, v)}
= 0.
(22)
For if this expression were developed, each term would, irrespec-
tive of sign, consist of the product of one second differential
coefficient, and two first differential coefficients. Thus we should
have terms in which u is twice differentiated, arising from
{v, (w, u)} in the three forms
dv dw
dp dp, dq, dq;
d2 u
dv dw
d2 u
dq; dq; dp; dpj
dv dw
d² u
dp; dq; dq;dp;
including the case of j = i; but the same terms would arise
from {w, (u, v)} with the contrary signs, as the reader will easily
verify. The same thing may be said of the terms in which v
and w are twice differentiated. Hence the equation (22) is
satisfied identically, as was to be shewn.
The properties established in this and the preceding Articles
are independent of any suppositions as to the meanings of P1,
91, and of the relations established by the differential equa-
tions (21), to the consideration of which we now return.
...
313.] Suppose a complete solution of the equations (21), namely,
dH
dH
Pi
Pí =
Qi
dqi
dpi
to have been obtained, so that each of the 2n variables P1,
Pn, 91 1,•••, ¶½, is a given function of t, and of 2n arbitrary con-
stants C1, C2, ..., C2n;
229
Also let two independent sets of arbitrary infinitesimal varia-
tions be attributed to the constants, and denoted by the symbols
8, A, so that we should have
dpi & c₁ + dc ₂
dpi 8 c₂+....
8 Pi
dc₁
Pi
z
dpi
dpi sc₁ + d c₂
spi = dc₁
AC₂+...;
4 A 2
548
[314.
THEORETICAL DYNAMICS.
then the expression
or
A
d p₁ ^ q₁ - Ap₁d q₁ + d P ₂ ▲ q 2 — ^ P ₂ 8 Q2 + ...,
б
2
Σ (ò Pi ▲ qi — ▲ Pi & Qi)
Δ A
б
...
(23)
is constant. That is, if the above values of ò Pis ., in terms of
the constants, their variations, and t, be introduced, t will dis-
appear from the result, and the expression (23) will become a
function of the constants C1, C2,... and their variations dc, Ac....
only.
This remarkable theorem was discovered by Lagrange, em-
ploying his own form of the differential equations. The follow-
ing simple demonstration of it is due to Professor Boole.
› Pn, 91, •••, In, t; and we have
н is a given function of P1, •••,
он:
dH
8 H = δρι +
dpi
d H
891+
...
dq₁
so that by the equations (21)
dн = q₁dP₁-P₁ d q₁ + ... ;
H
and consequently
1
ô H = ≥ (qí dpi— pí d Qi).
Now performing the operation ▲ on each side of this equation,
we have
δι
▲ d H = Σ (▲ qí ò Pi−▲ Pi d q i + qi ▲ d Pi — Pi ▲ d q i) ;
in like manner we should find
Δ
δ
d ▲ H = Σ (d q; ▲ Pi−d Pi ▲ Qi + Qi ồ ▲ Pi —P i ò A Qi) ;
hence, subtracting and observing that ▲d = d▲,
Pi
0 = ≥ (▲ qi' d Pi +▲ q¿ d pí —▲ pí d q — ▲ p; d qí ) ;
now aqi= (Aqi)', ... ; and thus this equation is equivalent to
0 = x ( d p; ▲ qi — ^ p¿ dq ; )' ;
that is, the total differential coefficient with respect to t of the
expression (23) vanishes, and that expression is therefore con-
stant; which was to be proved.
21
314] Suppose now that the 2n constants c₁, C2,
are the
initial values of the variables, which we will denote by λ₁, λ2,……. An
M1, M2,..., µn; where A1, A2,….. are the initial values of P1, P2, ...,
and Мія Мая of 91, 92,
p; d
Since the value of x (dpi Aqi-Api 8 qi) is independent of t, it
is not altered by supposing t = 0; but when t = 0, the values
of dpi, d qi, ... are dλi, di, ...; consequently,
δ б
Σ ( ò pi ▲ q i — ▲ P¿ d q i)
Δ
= Σ (δλΔ μ; - Δ λ; δ μ;).
(24)
314.]
549
THEORETICAL DYNAMICS.
By the help of this equation we can shew that the initial
considered as constant elements, possess cer-
values λí, μ1, .••,
tain remarkable properties.
Each of these elements may be supposed to be expressed as a
function of the variables, p1, 91, ..., and t. Let this supposition
be called Hypothesis I.
•
may be
On the other hand each of the variables, Pı, qı,
supposed to be expressed in terms of the elements A1, 1, ..., and
t. Let this supposition be called Hypothesis II.
Now in equation (24) let dpi, dg, on the left-hand side be
expressed in terms of dλ1, dμ1, ...; thus,
dpi dμ1 +.
d Pi qλ + dμ₁
d λi
δλι
δρι
=
dqi
d Q i dλ 1 +
qi
δλι
dqi
δμι + ;
αλι
dμ
анг
and let A, Aμi, on the right, be
expressed in terms AP1, 41, ...;
d λi
dλi
thus,
Δλι =
Api +
A91 +
d p₁
dq1
d pi
d pi
Δμ; =
Api +
Aq1;
dpi
dqi
so that on both sides of the equation all terms involving & will
be variations of constants, and those involving ▲ will be varia-
tions of variables; and each of these sets of variations may have
arbitrary values assigned to them; hence the coefficients of cor-
responding terms on the two sides of the equation must be equal.
Thus we obtain by comparing the coefficients of
dµj
d p i
δλ;
Aj Aqi, the equation
dqi
d λ j
dλj
dpi
d q i
(25)
δμ; Δαίς
d μ j
d μj
d qi
dpi
d λ j
δλ; Δες
dλj
d qi
dpi
δμ; Δρί
αμ
in which equations the differentiation refers to Hypothesis II on
the left, and Hypothesis I on the right; and in each of them i
may be equal to j.
Now suppose any one of the constants, say A,, to be expressed,
according to Hypothesis I, in terms of the variables, thus
λj = f(P₁₂ Pn, Y1, ... qn, t);
if on the right of this equation each of the variables were ex-
550
[315.
THEORETICAL DYNAMICS.
!
pressed, according to Hypothesis II, in terms of the constants
and t, the equation would become identical; that is, the right-
hand side would become identically λ;; hence, if we differentiate
each side with respect to A,, on Hypothesis II, the result on the
right must 1; but if we differentiate with respect to any
other of the constants, the result must 0; thus,
dλ; dpi
+
dp₁ dλj
dλ; d q₁
d q₁ d λ j
dλ; dp₂
d λ ; d q z
1
+
+
+
...
= 1;
d p½ dλj
d q₂ d λ j
and if c be any one of the
constants
except λ;,
dλ; dp₁
d λ; d q₁
j
dλ ; dp 2
P z
d λ j d q z
+
+
+
+
= 0.
d q z
de
1
1
dp₁ dq1
αλ, αλ
1
dp₁ dc dq, dc dp₂ dc
Now in the first of these equations let the values of
given by (25) be substituted; and it will be seen that the re-
sult is
(λj, Mj) 1;
see Art. 311. But if in the second of the above equations we
take for c either A, where i is not = j, or μ, where i may be
either equal to j or not, the result in the first case is
and in the second it is
(λj, Mi) = 0;
=
—(λj, λi) = 0.
By supposing μ; expressed in terms of P1, 92, ...,
ing in the same way, we should obtain the equation
(Mj, Mi) = 0.
Thus we see that the elements λ₁,
properties expressed by the equations
λης μ1, ...
and reason-
Mn, possess the
(^i, Mi) = 1, (^i, µj) = 0, (^¿, λj) = 0, (µis µj) = 0. (26)
If we call any pair, such as A¡, Mi, conjugate elements, the above
properties may be briefly stated by saying, that if f, g be any
two of the elements, then (f, g)= ± 1 if f, g be conjugate, and
= 0 in every other case.
315.] If α₁, dg,... an, by, b₂,
ɑn, b1, b2, ….. b₁ be elements, such that,
fg representing any pair, the value of (f, g) is ± 1 or 0,
according as f, g are a conjugate pair (that is, a pair such
as a¿, b¿) or not, then these elements are called canonical ele-
ments.
It has been shewn in the preceding Articles that when the
2n variables, P1, 91,
are determined as functions of t and of
316.]
551
THEORETICAL DYNAMICS.
2n arbitrary constants, by means of the Hamiltonian equations
(21), there exists one set of arbitrary constants, namely, the
initial values, A, 1, ..., of the variables, which form a system of
canonical constant elements. We shall now prove that the
number of such systems is infinite.
In fact if α1,
an, b₁,……., b₂ be determined as functions of
A1, M1,
by the 2n equations
ds
ds
bi,
= λι,
Xi,
da;
d pi
where A is any arbitrary function of a₁, an, μ1
a₁, b₁, ... will be canonical elements. For we have
d s
ба
SA=
d s
da
δαι +
δμι +
αμ
d s
= Σ
(da
δα; +
Mi):
;
dpi
(27)
...
Mn, then
and consequently, by (27),
λδμ);
d a = Σ (b¿ d ɑ¿ + λ ¿ & µ¿) ;
and performing the operation ▲ on each side of this equation,
ASA
ΔΑ = Σ(Διδα + Δλδμι) + Σ Ο Δ δα; + λίΔ δμ) :
similarly we should find
¿
ΟΔΑ = Σ δό Δα; + δλ. Δμε) + Σ (Οιδια; + λ; Δδ μ;) ;
whence, subtracting and observing that da = aồ,
Σ (δα; Δο; - Δαδά) = Σ (δλ. Δμ. - Δλδμ).
Abi
But it has already been proved, see Art. 314, that the right-
hand member of this equation is = (8p; Aq;-Api dqi), so that
we have
Σ
Σ ( d a¡ s b ; — s а; db₁) = (dp; ^q; - Ap; & qi);
a¿
Σ A
Pi
from which it follows, that all the consequences deduced in Art.
314 from equation (24) will be true, if we substitute α₁, b₁, …..
for A1, B1, ..., and in particular that the conditions.
(ai, bi) = 1,
(αi, αj)
=
(αi, bj)
(bi, b;) = 0,
will subsist.
316.] The equations last written are particular cases of the
following general theorem, discovered by Poisson.
If f, g be any two integrals whatever of the Hamiltonian
equations, then (f,g) is constant. Poisson's demonstration was
obtained by means of the Lagrangian form of the equations.
552
[316.
THEORETICAL DYNAMICS.
The following, founded on the Hamiltonian form, is much
simpler :
If u be any function whatever of P₁, 91,
du du
du
u = + Pí+ 91 +
dt dpi
dq1
and t, we have
• ;
but as p₁, 1,... are supposed to satisfy the differential equations
d H
dн
(21), we have pi
>
1
so that
dq1
dpi
du
dн du
dн
du
u =
+ Σ
dt
dp dqi
d qi d pi
)
du
dt
+ (H, u); (see Art. 311).
Now if we take u = (ƒ, g), we have, (Art. 311),
du
df
dt
dt
(ara, 9) + (ƒ, da),
dg
dt
and therefore
df
dg
(f, g)' =
dt
dt
(dƒ, 9) + (ƒ, dq) + {¤, (f, 9)} ;
now since ƒ and g are integrals, f'= 0, and g′ = 0; that is,
g'
df
dg
+ (H,ƒ) = 0,
dt
dt
+ (¤, g) = 0 :
these equations are identically true; so that we may substitute
df dg
— (H, ƒ), —(H, g), for
respectively in the above expres-
dt' dt
sion for (f, g)', and the result may be written thus,
(f, g)' = {9, (H,ƒ)} + {ƒ, (g, H)} + {µ, (f, g)} ;
but by the theorem proved in Art. 312, the expression on the
right of this equation vanishes identically, and therefore (f, g)'=0;
or (f, g), is constant, which is the theorem to be demonstrated.
Here it is to be observed, that ƒ and g represent given
functions of the variables P1, 91, ..., and t, which are constant
by virtue of the differential equations. But the constancy of
the expression (ƒ, g) may subsist in two different ways:
First, (f, g) may be identically constant, that is, a deter-
minate numerical constant, or zero: this always happens when
ƒ and g belong to a set of canonical elements.
Secondly, (f, g) may be constant, not identically, but by
virtue of the differential equations; and in this case
c = (f, g)
317.]
553
THEORETICAL DYNAMICS.
will be an integral of the equations; but here again there are
two cases, for c may be either an independent arbitrary con-
stant or a function of ƒ and g ; in the latter case, the integral c
is a combination of the integrals f, g, but in the former case, it
is a distinct independent integral. Thus it may happen that the
theorem will lead to the discovery of a new integral when two
are known. For example, the problem of motion about a fixed
centre of force, leads, as will be seen afterwards, to three inte-
grals of the forms
2
h = p₁² + p²² + p²² — $ (91² + q2² + 93²),
e
f
2
¶ 2 P3 - 93 P2,
qз P1-91 P3;
q1P3
and it will be found on trial that (h, e) = 0, (h, f) = 0; but that
(e, f) = 92P1-P241, which is neither identically constant nor
expressible as a function of the other integrals: hence we may
affirm that
g = P2 9 1 -92 P1
91
is a new integral. But if we attempt to discover more integrals
by the same method, we shall fail; for it will be found that
(h, g) = 0, (e, g) = f,
(f, g)
e.
It is to be observed also that the integral g is as easily discover-
able by ordinary methods as the integrals e and f; so that in
this case, and probably in general, the theorem is of no practical
use as a means of obtaining new integrals, though very import-
ant in other points of view.
317.] We now come to a most important discovery, due to
Sir W. R. Hamilton.
Suppose the solution of the system of equations (21), namely,
Pi +
d H
d qi
0,
d H
=0,
dpi
to be given in the form of 2n integral equations involving the
of the variables, as arbitrary constants.
initial values, λ1, M1,
By means of these 2n equations each of the 2n variables could
be expressed in terms of the 2n constants and t; and therefore
the differential coefficients of the variables with respect to t
could be expressed in the same way. Consider then the ex-
pression
P₁ Jí + P₂ Q2 + ... + Pn ¶n' — H ;
this being a given function of the variables, their first differen-
PRICE, VOL. IV.
4 B
554
[317.
THEORETICAL DYNAMICS.
tial coefficients, and t, might be expressed as above supposed,
and would become a function of the 2n constants, λ1, M1,
and t. Suppose this function to be integrated with respect to t
from t = 0, and let the result be called s; so that
S
't.
= [ { ≈ (p; qí ) − n } dt.
(28)
The value of s, obtained in this way, would be also a function
of λ1, µ1,
and t. But by means of the 2n integral equations
we might express the 2n quantities, A1, A2, ... An, P1, P2, ... Pu, in
terms of the 2n quantities μ1, M2,
...
Mn, 91, 92, In, and t; and
if the values of A1, ... An, thus expressed, were substituted in the
above value of s, the result would be of the form
S = F (91, 92, Япя М1, Моза Ung t
...
(29)
Now, taking the form (28), and using the symbol d in the same
sense as before, so that st 0, we have, by the rules of the
calculus of variations,
=
= [ ' [ = { p; (dò q₁)' } + = (q{ dpi) — ò¤]dt :
Pi Qi)'
ds=
let H in this equation be supposed to be expressed in its original
form as a function of P1,
Pn, 91, In, t; then
dH
dH
δΗ = Σ
dpi +
dpi
d P i
dqi
ogi);
but by the differential equations (21),
d H
Ji',
dpi
dH
-pi,
d q i
hence
dH = Σ (q; dр; —pí dɖi) ;
and if this be substituted in the above value of ds, the result is
ds
't
= [ ' [
=
't
√ =
['=
Σ {Pi (òqi)' } + ≥ (p{dq;)]dt
= (Pi òqi) dt.
Thus ds turns out to be expressible as the integral of a perfect
differential with respect to t. Performing the integration from
t = 0, and observing that when t = 0 the values of pi, dq; are
Ai, dμi, we obtain
d s = ≤ ( p¿ dq ;) — Σ (λ¿ dµi).
But if we suppose s to be expressed as in equation (29), we have
ds=
(1 = 88) + =
ds
dqi
d qi
(ds 8,4).
μ i
319.]
555
THEORETICAL DYNAMICS.
Now these two values of ds involve the same set of 2n varia-
tions, dg,... dqn, du,... dun, which may all be considered as
δα, δμι,
arbitrary and independent, because the 2n variables and 2n
constants are only subject to 2n equations; so that the values
of any set of 2n out of the 4n quantities could be assumed arbi-
trarily without contradicting the equations; consequently the
coefficients of like variations must be equal, that is, the 2n
equations,
ds
ds
-λis
Pis
d q i
d pi
(30)
must be true; but these equations are obviously not true iden-
tically, and they contain the 2n arbitrary constants, À₁, µ1,
Hence they can only be a particular form of the integral equa-
tions of the problem.
318.] It appears from the equations (30), just established,
that if the single function s, expressed in the form (29), were
known, a complete set of integral equations could be deduced
from it by mere differentiation. Thus the complete integration
of the system of differential equations (21) is made to depend
upon finding the form of a single function. This is the most
essential part of Sir W. R. Hamilton's discovery; but it must
not be supposed that the above brief account of it represents the
original form and manner of the author's investigation, much
less that it gives any notion at all of the general contents of his
two elaborate memoirs "On a General Method in Dynamics,"
contained in the Philosophical Transactions for 1834 and 1835.
319.] We proceed to examine the function s more closely.
Recollecting that s is supposed to be expressed in terms of
I1, ... In, μl, μn, t, we have
...
ds
s'
+
dt
d s
dqı
ds
dt
Q
+ (Piqi), by (30);
on the other hand, equation (28) gives by differentiation with
respect to t,
s' = —H;
≥ (P¿ Ji'´) — H ;
comparing these two values of s', we obtain
dt
ds
+ H = 0.
(31)
..., 91,
t; say
H = f(P1, P2,
...
Pn, 91, 92,
qn, t);
Now H is given as a function of p1,
و
4 B 2
556
[320.
THEORETICAL DYNAMICS.
ds
also, by (30), P₁ =
d qi
ds
ds
+ f (
d t
ds
d qr' d qz d q
; hence (31) may be written in the form
ds
,
, X1, X2, ・・・ Ini
,t) = 0.
(32)
...
...
But this equation contains, besides t, only the n variables
21, qn, and the n constants μ1, μ; it cannot therefore be
any combination of the integral equations (30), because those
2n equations would in general be insufficient to eliminate the
2n quantities P1,... Pn. A1, ... An; hence it must be satisfied iden-
tically; in other words, it is a partial differential equation of
which s is a solution.
320.] The above result is due to Sir W. R. Hamilton, and
so is also the following: Differentiating H with respect to t,
we have
H'
d H
dt
+ Σ
da
(dp, Pi) + = (17, 21)=
but, by the differential equations,
pi
du
dq;'
d H
Σ
d q i
dH
Ji
dpi
so that the above equation becomes simply
dн
H' =
dt
Now if н does not contain t explicitly, which is the case in
*
ordinary dynamical problems, then
dH
dt
0; therefore н'— 0,
or,
H = constant ;
and this is an integral of the problem, called the "integral of
vis viva." In this case the equation (31) is of the form
ds
+ f (P1, P2,
d t
...
Pn, Q1, L2, ……. In) = 0 ;
...
and from this we may deduce an equation like (32) as before;
* Ordinary dynamical problems may be defined as those in which the law
of "action and reaction" is maintained. This law is violated when, for in-
stance, in the planetary theory, the disturbing planet is considered as being
itself undisturbed; or when, in investigating the effect of the earth's rotation
upon the motion of a pendulum, the pendulum is supposed not to affect the
motion of the earth; or, speaking generally, whenever any part of a system is
supposed to be subject to an "obligatory motion." In all these cases н will
contain t explicitly.
321.]
557
THEORETICAL DYNAMICS.
but since H is now constant, its value will not be altered by sub-
stituting for P1, 91, ... their initial values; hence we may write
ds
dt
+ f (λ1, ^2, ..., Ang My, Mqq..., µn) = 0;
+ƒ
or, by (30),
ds
ds
ds
dt
+fl-
du
d Mz
ds
d μ n
, My, M₂r……., Mn) = 0; (33)
which is a second partial differential equation satisfied by s in
the case supposed.
321.] We must now notice an important extension of the
preceding theory, discovered by Jacobi, and contained in the
following theorem :
If a bef(P1, P2, , Pn, Iv, 12,
plete solution of the partial differential equation,
q, t), and if s be any com-
ds
dt
+ f (
ds ds
ds
d q₁ dqz
d qn
,
91, 92,
› In, t)
= 0, (34)
containing the n arbitrary constants a, a, ..., an, then the 2 n
equations
ds
dqi
ds
= Pir
bi
da
are a complete set of integral equations of the system
Pi
d H
d H
Ji
dqi
dpi
(35)
Here b₁, b₂, ……., b₁ are new arbitrary constants; and by a com-
plete solution of the equation (34) is meant a solution containing,
besides any constant merely added to s, n arbitrary constants
ɑ1, ɑ2,………, ɑn, in such a manner that they cannot be all eliminated
so as to produce a partial differential equation of the first order,
without employing all the n+1 differential coefficients
ds
dq1
ds
ds
dqn' dt
Before proving the theorem we must investigate a criterion
by which it may be known whether any solution of such an
equation as (34) be complete or not.
¥ (α1,..., an, 91, ..., qn, t) to be a solution;
Suppose then s = √ (α1,..
and this to give
ds
$1 (α1,..., αn, 91, ..., In, t),
dqı
ds
$11 (α1,·
..., ang qu
qv •
•••, qn, t) :
dqn
558
[322.
THEORETICAL DYNAMICS.
now if it be impossible to eliminate the n quantities a, ..., an
between these equations, the solution is evidently complete, be-
cause the additional equation obtained by differentiating with
respect to t will then be required.
But it is known that the supposed elimination is always im-
possible unless the determinant formed with the n² constituents
афі
d 1
афг
da da2
1
d & 1
dan
d on
don don
dan
da
day
vanishes identically.
We may therefore express the required criterion as follows:
In order that the solution s may be complete, it is only neces-
sary that the determinant formed with the n² constituents
d2s
da₁ dqı
و
d2s
da, dqı
d2s
dan dqi
d2s
d2s
d2s
و
dan dgn
q
da₁dqn' da₂d qn
shall not vanish identically. In the abridged notation, explained
in the note of Art. 292, this determinant would be written
d
ds
dqn
a (
ds ds
,
d q r d qz
d (α1, αq,
an)
..., qn, t)
322.] This being premised, let
S= † (A1, A2, .. , an, 91, 92,
be a complete solution of the equation (34). We have to prove
ds
that if P1, P2, ..., Pn be defined by the n equations
Pis
dqi
* If the elimination be possible, it will lead to an identical relation between
P1,..., Pn, considered as functions of a,..., an, say F (P1, P2,· · ·, Pn) = 0;
differentiating this equation with respect to a; we obtain
афі
dr doi
dp, da;
афі
+
+
афи
ав афи
don da;
i
d F
d F
we eliminate the n terms
аф,
and if from the n equations given by this formula, on putting i
determinant mentioned in the text.
don'
<= I,
•
.., i
= N,
the result is A=0, A being the
322.]
559
THEORETICAL DYNAMICS.
and if the 2n variables P₁, ..., Pn, 91,
qn, be determined as
functions of t, and constants, by those equations joined with the
n further equations
ds
=
dai
ential equations (35).
bi, then pi, i will satisfy the differ-
ds
Putting p; for
in (34), that equation becomes
dqi
ds
+ƒ (P1, P2, ··· , Pn, X1, X2, ···, qn, t)
dt
=
= 0, (36)
which is identical on the supposition that P1,..., Pn are expressed
ds
in the form
dq1
ds
Differentiating the equation with
d qn
respect to a; on this supposition, and observing that
d Pi
d2s
da
da, dqi
we have
d2s
df d2s
df d2s
+
da; dt dp₁ da; dqı dp ₂ da; dqz
+
+...+
df d's
Pn da; dqn
= 0.
1
ds
Again, differentiating the equation
b; totally, with respect
da;
to t, we find
d2s
+
dt dar
d2s
dq₁ da;
q +
d2s
d q₂ dar
d2s
92 + ... +
dqn dai
In' = 0;
and subtracting the former equation from the latter,
d2s
df
+ (9½- +.. +
da,dq, (9' - df) +
dp₁
d2s
da;dq₂
dp2
d2s
(an
df
0.
da¡ dqn
dpn
Suppose the n equations obtained from this by giving i all its
values 1, 2, ..., n, to be written down; then it is evident, from
the theory of algebraic linear equations, that one of two things
must be true; either the determinant formed with the n² co-
efficients
qi
df
dpi
d's
d a ¿ d q j
must vanish, or else each of the n terms
0. But it was shewn in the last Article that the
former supposition would imply that s was not a complete solu-
tion of (34). Hence the latter alternative is alone admissible;
and since ƒ (P₁›
..., qm, t) is the same as н, we have
qi
d H
dpi
;
• g
Pn, qv
which is one part of the conclusion to be proved.
560
[324.
THEORETICAL DYNAMICS.
To establish the other part, differentiate the equation (36) on
the same supposition as before, with respect to q, observing that
ds
dpi
and
;
Pi =
>
dqi
dq
dqi
dp the result may then be written,
+
+
dt
dqi dp₁ dqi dp 2 dqz
1
with respect to t, we obtain
dpi df df dpi df dpi
df dpi
0;
+
+...+
d pn dq n
ds
on the other hand, differentiating the equation p₁ =
Pi
totally
dqi
d pi
d pi
d Pi
Pi
pi =
+
qi+
91 +
dt
dq1
dq, 92
1½ +...+
dpi q
In ;
din
and combining these
two equations,
df
dpi
Pi +
dqi
dq1
(ai - df)
+
+...+
dpi
d qn
dpi (q' - df)
pn
but it has just been shewn that every term on the right of this
equation vanishes; hence pí +
completes the demonstration.
df
dqi
dH
=
0, or pi =
which
d qi
ds
323.] The equations
dqi
ds
Pi dai
=
bi, give
d s = Σ ( P¿ d qi) + Σ (b; ò a;) ;
and if from this we form the expression for ads, and subtract
from it the analogous expression for 8 A s, equating the result to
zero, see the process of Art. 315, we obtain the equation
Σ ( d p; ▲ qi — ▲ p; d qi) = ≥ (d a; ▲ b ; — ▲ a ; ô b ;) ;
Σ Δb-Δαδά);
Δ
i
-
and this being similar to equation (24), Art. 314, it follows, as
in Art. 315, that (a, b) = 1, (a;, aj) = (ai, bj) = (bi, bj) = 0;
so that a₁,
bn, are canonical elements.
ɑn, by,
324.] The formulæ established in Arts. 321, 322 may be
somewhat modified in the case in which н does not contain t
explicitly. We have seen, Art. 320, that in this case h = H
is one of the integrals of the problem; h being an arbitrary
constant, called the "constant of vis viva.”
Suppose then that
H = ƒ(P1, ..., Pn, 91, ···, qn) ;
:
and let v be a complete solution of the equation
dv
f (dq
1
dy
• 91, •••, In ) =
dqn'
n) = h;
(37)
324.]
561
THEORETICAL DYNAMICS.
that is, a solution containing, besides h, n-1 arbitrary constants
a1, a2 ..., an-1, in such a manner that these n-1 constants
cannot be all eliminated without employing all the n differential
coefficients
dv
dv
Then, if we take
>
dq1
dqn
S
ht + V,
ds
ds
+f
it is evident that s will be a solution of the equation
, In
ds
dt
dq1
d q n
› I1, · ₂) = 0;
ds
ds
dv
for we have
h; and since
the second term
dt
d qi
dqi
in the equation just written
=
h, by (37), so that the two terms
destroy one another. Also s is a complete solution, for it is
evident that h cannot be eliminated in addition to a1,
ds
without employing the differential coefficient dt
an-1,
If then the function v is known, the integral equations of the
problem will be expressible as follows: since
ds
dqi
dv
dqi
shall have n equations. = Pi. Also, since s contains a₁,
d v
d qi
an-1 only as they are contained in v,
will be n 1 equations,
-
we
ds
dv
so that there
da
dai'
dv
b;.
(38)
da i
ds
The remaining integral equation is
= constant. But
dh
ds
dv
- t +
dh
dh'
and therefore, putting for the constant, we may write the
equation
dv
= t + T.
dh
(39)
In this way we should obtain a solution involving the canonical
elements
h, a1, a2, an−1, I
...
τ, b₁, b₂, ... b₁-1. S
T,
(40)
It is to be observed, that the only one of the integral equations
containing t is the equation (39). And it is evident, that if the
2n integral equations were solved algebraically, so as to express
each of the 2n elements (40) in terms of the variables, that is,
PRICE, VOL. IV.
40
4C -
562
[325.
THEORETICAL DYNAMICS.
so as to obtain 2n "integrals" properly so called, only one of
these integrals would contain t, and would be of the form
T = t + ¢ (P1,
...
Pn, 91,
In).
Lastly, it is to be noticed, that whenever the constant of vis viva
is one of a set of canonical elements, the element conjugate to
it is the constant 7, which is added to t.
325.] Although the preceding theory assigns in a very re-
markable manner certain forms in which the integral equations
of the system (21) are capable of being expressed, yet it gives no
assistance whatever towards the actual integration of that sys-
tem. For the discovery of the function s, according to Sir
W. R. Hamilton's definition of it, would require a knowledge of
a complete set of integrals, and according to Jacobi's defini-
tion, would depend on the solution of a partial differential equa-
tion (34), which is a problem as difficult as the integration of
the system (21) itself. In fact, the most hopeful way of attempt-
ing the integration of (34) would in general be to make it de-
pend upon that of the system (21). See Boole's Differential
Equations, Chapter XIV, Art. 14, and also page 475.
It will be seen however, from a theorem now to be demon-
strated, that a knowledge of half the integrals of the system
(21) will, when certain conditions are fulfilled, lead to the dis-
covery of the function s, and therefore enable us to complete
the integration.
N
326.] Theorem*. Suppose a1, a2, ..., a to be n integrals of
the system
Pi
d II
d qu'
A H
(i = 1 to in),
d pi
where H is a given function of p1,
the
n (n − 1)
2
Pn, 71, ………, In, t; then, if
conditions (a;, a;) = 0 subsist, where i, j is any
pair of the n indices, the remaining integrals may be found as
* When this theorem was given by the writer as new, in the Phil. Trans.
for 1854, p. 85, he had no means of knowing that it had been previously dis-
covered by M. Liouville, and communicated to the Bureau des Longitudes in
1853; for no accessible notice of this communication appears to have been
printed before 1855 (in Liouville's Journal). As M. Liouville has referred to
the question of priority, it seemed necessary to mention the subject here. See
note to p. 31 of Mr. Cayley's Report, in which for "second part" read “first
part."
327.]
563
THEORETICAL DYNAMICS.
follows: By means of the given n integrals, let p1, P2, ... Pn be
expressed in terms of 91,
...
...
Jus al an, t; and let these values
be introduced in н, so that н will be expressed in the same way.
Then the values of P1, P2,
Pn, —H, will be the partial differen-
tial coefficients with respect to q1, 72 In, t, of a certain func-
tion of a₁, ... an, 1.... Ja, t; and calling this function s, we may
find it by integrating the expression
d s = p₁ d q₁ + p₂ d q₂ + ... - Hdt;
P
(41)
of which the right-hand member is a perfect differential. And
the remaining integral equations will be
ds
dai
b₁
ds
b₁i
da
(12)
where b₁,……., b₂ are n new arbitrary constants.
327.] The proof of this theorem consists of two parts. First,
we have to shew that the above expression for ds is a perfect
differential. Putting e, f for any two of the integrals a1, ..., ɑù,
suppose
e = $ (P1, P2, ··· Pas J, J2,
...
>
In, t).
an
If in this equation the values of Pi, ... Pa were expressed, as
above supposed, in terms of q1, ..., α1
αι it would become iden-
tical. Differentiating it therefore on this hypothesis with re-
spect to q;, we have
de dp
de
de dpi
+
+
= 0
+
d qi
pi
d p₁ dqi
d pn
d qi
and in like manner
df
df dpi
df dpa
+ ... +
0 =
+
d qi
dp₁ dqi
dpn dqi
df
and the
dpi
and if we multiply the first of these equations by
de
second by and subtract, there results an equation which
dpi
may be written thus :
de df de df
dp; dqi
pi
Σ
dqi d pi
{al (ap, ap
Sdp; de df
de df
or, writing ɑɑ, as instead
qi
dp; dp;
}}
of e, f, and employing the notation of
Art. 292,
d (ɑa, aẞ)
= Σ
d (pi, Qi)
{dp; d (Aa, as)}
qi d (Pj, Pi) S
If now the terms on each side be summed with respect to i, the
4C2
564
[327.
THEORETICAL DYNAMICS.
result on the left is (aa, as); and observing that on the right the
dpi will only differ in sign from that mul-
d q j
we obtain
term multiplied by
d pj
tiplied by
d qi
(ɑɑ, αß)
=
{
dp; _ dpi) d (aa, as)
dqi dqj d (pj, pi) S
the summation referring both to i and j, and extending to every
binary combination.
From this equation it appears that in order that the condi-
tion
dp
d pi
i
d qi
d q j
may subsist for every pair of indices i, j, it is
necessary that the condition (ɑɑ, ɑß) = 0 shall be satisfied for
every pair of the n integrals a, ..., an. We ought in strictness
to prove that no other conditions are necessary. As however
the rigorous proof of this involves certain preliminary theo-
rems, which have not been given in this Chapter, it is here
omitted; but it will be found in the Philosophical Transactions
for 1854, pp. 84, 85.
*
Assuming then that
expression for ds, (41),
also to shew that
d pi
Pj
d q i
dpi in order to prove that the
dqj'
is a perfect differential, it is necessary
dH
d qi
d pi
d t
expressed, like
Pir
in terms of a1,
Here н is supposed to be
, an, q1, , qu, t. Putting
•••,
for a moment (H) to denote н in its original form, namely, a
function of p₁,
› In, t, we have
P1, ..., Pn, 91,
. . .,
dн
d (H)
+
d qi
d qi
d (H)d pi
dpi dqi
+ ... +
d (H) d Pn.
dpn
dqi
d (H)
d (H)
qi…..,
- Pí,
d pi
d qi
d pi
d qu
now
άρι
dqa
hence this equation becomes
dH
d qi
— Pi + q¹ à q₁
dpi
+
+ qn d q n
dpi;
dPj
dpi
dq;
dq;
* It might seem sufficient to say that the number of equations
is the same as that of the equations (a;, a;) = o, viz. In (n−1), and that
therefore the former set cannot imply any conditions not involved in the
latter; but this reasoning is not absolutely conclusive.
328.]
565
THEORETICAL DYNAMICS.
but p; being expressed as above supposed, we have
Pi
Pí
dpi
dt
d pi
dpi
+
q1' + ... +
In' ;
dqi
d q n
d H
whence the above value of
becomes simply
dqi
dpi ;
dt
d H
dqi
which is the condition that was to be established, and completes
the proof that pidgi+...-нdt is a perfect differential.
ds
Secondly, we have to shew that is constant by virtue of
the differential equations, or that (
dai
dai
(ds) =
= 0.
Now*
dai
(ds)
d's
dt dai
d2s
d2s
+
dq₁ dai
qi' + ... +
q
d yn dar
qn';
ds
d2s
d H
ds
and
dt
d2s
dqı dai
dai
H, whence
dp¹; and qí
; also
P1, whence
dt dai
dai
d qı
d (H)
; so that we obtain
(ds)
ан
dpi
d (H) dpi
d (H) dpn,
+
+...+
dpn dai
dai dp₁ dai
but from the relation between н and (н), we have
d H d (H) dp₁
dp₁ dai
da
d (H) dрn
+...+
dpn dai
for H only contains a; because it is introduced in the values of
P1, P2,
• Pn.
ds
ds
Hence the above equation becomes (d)=0,
or = constant; which was to be proved.
da
dai
328.] The formulæ (41), (42), Art. 326, admit of modification
in the case in which the so-called "principle of vis viva" holds
good; namely, that in which н, in its original form, does not
contain t explicitly, so that H = h is an integral.
н
In this case, suppose h, a, a,..., an-1 to be n integrals, satis-
* The reader will not understand this reasoning unless he be careful to
recollect the way in which H, (H), S, P₁, · · P₁, are supposed to be ex-
pressed.
•
566
[329.
THEORETICAL DYNAMICS.
fying the conditions (h, a;) = 0, (a, aj) = 0 for every pair. It is
evident that when the n equations, expressing the values of h,
a1,..., in terms of the variables, are solved for p₁,..., Pn, and the
values of these latter quantities substituted in H, the result will
be identically the constant h, so that the value of ds (41) will
become ds=hdt + p₁ d q₁ + ... + Pn d qn,
where P₁,..., P cannot contain t, otherwise this expression would
not be a perfect differential. Hence we shall have
S
ht+v;
و
where v is a function of 1,..., qn, h, α1,..., an-1, not containing
t, and given by the equation
dv = p₁ dq₁ + ... + Pn d qn;
of which the right-hand member is a perfect differential.
(43)
We shall now have
ds
- t +
dh
dv
dh'
ds
dv
da
da₁
so that if we put 7 for the constant conjugate to h, the equations
corresponding to (42) may be written
dv
dv
= t +T,
=
b₁,...,
dh
da
dv
dan-1
bn-1. (44)
329.] It is useful to observe, that if c be any integral what-
ever, not containing t explicitly, and h the integral of vis viva,
the condition (h, c) = 0 necessarily subsists.
O necessarily subsists. For we have, see
the general expression for u', Art. 316,
0 = c
= ?
dc
+ (H, C);
dt
dc
but = 0 by hypothesis, and (H, c) is the same as (h, c); con-
dt
sequently (h, c) = 0.
From this proposition, combined with the results of the last
Article, it is evident, that when the independent coordinates in
any dynamical problem in which the principle of vis viva sub-
sists are only two in number, if, besides the integral of vis viva,
one other integral, not containing t explicitly, be given, the dis-
covery of the two remaining integrals is reducible to quadratures.
For v is given by the equation dv = P₁dq₁+P₂d¶½, of which
the right-hand member is necessarily a perfect differential. The
reader will find it a good exercise to treat the problem of the
330.]
567
THEORETICAL DYNAMICS.
simple conical pendulum in this manner, omitting the effect of
the earth's rotation.
330.] In the practical use of the theorem explained in Arts.
326-328, the following circumstances should be attended to.
Instead of first finding the function s by integrating the expres-
sion p₁dq+P₂ dq2 + ... - Hdt, and then forming the integral
ds
equations = b₁,..., it is generally more convenient to per-
da
form the differentiations with respect to a, ..., first, and the
integrations afterwards. Thus we should have
dн
dpi
bi
=
d p r d qı
dq1+
dp2 dq2 +
dt
dai
at};
dai
dai
but the solution thus obtained will not be canonical, unless care
be taken so to assume the inferior limits of the integrals that the
functions equated to b1, b2,..., shall be, as they ought to be, the
differential coefficients of one and the same function of α1,.
91, 12,
..
an,
To shew that this condition will not necessarily subsist, we
may take the csse which most frequently occurs, namely, that in
which the value of p; contains none of the variables except qi, so
that each term in the value of ds will be, omitting indices, of
the form (q, α1, ɑ2,
an) dq, and so far as one such term is
concerned, we should have
S
•
a
>
&(q, α1, ɑ2, a₁) dq,
where A is an arbitrary function of a1, ..., an. Differentiating
this with respect to a;, we obtain
da
ds
a do
dq−4 (A, ɑ1, ..., ɑn)
;
dai
da
da;
i
that is,
S
таф
d
dq
dai
da
S
[" p. dq+p (s, az, ..., an)
ds
dai
A
from which it is evident that differentiating first and integrating
afterwards will not in general give the same result as the con-
verse process.
We see however that the values of
S
я аф
day
dq,
S
ca
do
dq,...
daz
A
A
will be the differential coefficients with respect to a1, a2, of
568
[331.
THEORETICAL DYNAMICS.
4.dq, provided that a, which must be the same in each of
these integrals, be so assumed, that
ds
Ó (A, ɑ1, ..., ɑn)
0
dai
for every value of i. This condition may be secured in either of
two ways: first, by taking a = 0, or a = any determinate con-
stant; that is, not involving a₁, ...; or secondly, by taking for
A any root of the equation
$ (x, α1,
ɑ„) = 0 ;
or, which is the same thing, a value of q which would make
vanish.
Ρ
331.] As an illustration of the principal formulæ hitherto
established, we will take the important problem of the motion
of a single particle about a fixed centre of force, in treating
which our object will be to obtain a set of canonical elements.
The advantage of this form of solution will appear afterwards.
Let m be the mass of the particle, x, y, z its rectangular coor-
dinates at the time t, referred to fixed axes having their origin
at the centre of force.
Then if r, 0, λ be polar coordinates, such that
x = r cos λ cos 0,
à
y = r cos λ sin 0,
à
z = r sin λ,
r is the radius vector; and we may call @ the longitude, reckoned
from the axis of x, in the plane of (x, y); and λ the latitude, or
the angle between r and the plane of (x, y).
2
In rectangular coordinates 2 r = m (x²² + y'² + z′²); but if
x', y', z′ be expressed in terms of r, r',..., it will be found that
2 r = m {r²² + r² (cos λ)² 0¹² +r² X'²}.
We shall take r, 0, λ as the coordinates of the problem, cor-
responding to q1, 2, 3, in the general formulæ. Let the vari-
ables conjugate to them, corresponding to P1, P2, P3, be denoted,
for greater clearness, by r,, 0,, λ, ; so that we shall have
r₁ =
dr
dr
mr', 0,
dr
do
mp;
dr
= mr2(cosλ)20', λ= = ጎጥ ' ;
dx
and if the values of r', 0′, X′ in terms of r,, 0,, λ,, given by these
equations, be substituted in T, the result is
2
1
2
T
·² +
2 m
(sec λ)2
p2
2
0,² +
2.2
(45)
332.]
569
THEORETICAL DYNAMICS.
Since the force is central, the force-function u is a function of r,
say u = $(r); so that we shall have
HT
p (r);
in which T stands for the expression (45) just given. See Art. 307.
The six differential equations of the problem in the Hamiltonian
form would then be
d H
rí
dr
dH
r =
dr,'
d H
dн
x;=
do
αλ
d H
dн
Ø'
λ=
;
do,
'
αλ
but they are not required for our present purpose.
332.] Since H does not contain t explicitly, the integral of
vis viva subsists; namely,
h = T $ (r).
We have now to find two other integrals, say c and k, such
that (h, c) = 0, (h, k) = 0, (c, k) = 0.
0, (c, k) = 0. We know that the first
two of these conditions will be satisfied if c, k be any integrals
not containing t explicitly. Let us take then for c one of the
three integrals expressing the conservation of areas; namely,
c = m (xy' — x′ y); and for k, the equation obtained from the
three by squaring and adding, namely,
\
k = m {(y z' — zy')² + (z x' — x z')² + (x y' — y x')² } :
these however must be expressed in terms of the new variables;
thus,
c = m r² (cos λ)² '
2
12
k = m {r² (x²² + y²+2′2) — p² p2}
Thus we have the three integrals
1 S
h =
2m
c = 0,
k =
=
{
2
2
{ 0,² (sec λ)² +λ,²}½.
(secλ)2
1
+
+
λ
r2
$ (r),
(i)
(ii)
(iii)
2
{ (sec λ)² 0,² + λ¸²}*;
and if it be recollected that the meaning of (u, v) is now
d (u, v) d (u, v) d (u, v)
+
+
d (r,, r) d (0,0) d (λ, λ)
it will be found on trial, not only that (h, c) = 0, (h, k) = 0, as
we know a priori, but also that (c, k) 0.
Hence we may apply the method of Art. 328 to the equations
(i), (ii), (iii); that is, we may find v from the equation.
PRICE, VOL. IV.
dv = r¸dr + 0, do + λ, dλ,
4 D
570
[333.
THEORETICAL DYNAMICS.
in which r,,,,, are to be expressed in terms of r, 0, λ, h, c, k, by
means of (i), (ii), (iii).
It will be found without difficulty that
"
= [2
k2
2m {h+p (r) }
2.2
Ꮎ = C,
λ
= { k² — c² (secλ)² } ³;
and it is obvious that r, dr +0, do +λ, dλ is, as it ought to be,
a perfect differential.
Supposing v to be found by integration, the three remaining
integral equations would be, Art. 328,
dv
dv
dv
t + T,
ɑ,
= B;
dh
dc
dk
where τ, a, ß are three new constants.
The conclusions of Art. 330, however, shew, that we may per-
form the differentiations before the integrations, if the inferior
limits of the latter be properly chosen; and in accordance with
the rule there laid down we may take O as the limit for 0 and λ,
and a root of the equation 2 m {h+p (r)}
k2
22
0, say p, as
the limit for r. This being understood, we should have.
V =
= [ " { 2 m (b + 4 (r) ) − 1 = } dr+c0+
h
-
k² /
2.2
[^
λ
{ k² — c² (secλ)² } } dλ ;
'ρ
0
and the three new integral equations will be obtained by differ-
entiating under the signs of integration, and integrating after-
wards. The integrations with respect to r cannot be performed
till (r) is assigned, and in fact will not be actually needed;
those with respect to λ are left to the reader. The results are
n['r − − §
m
r {2mr² (h+p(r)) — k²} - dr = t +T,
(iv)
c tan λ
0-sin-1
= a,
(v)
(k² - c²) t
-k
k [ ' },
k sin
{2mr²(h+p(r)) — k² } −½ dr+sin-1
=
ß. (vi)
r
ρ
(k² - c²)
The equations (i), (ii), (iii), (iv), (v), (vi) comprise a canonical
solution of the problem.
333.] It is easy to interpret the constants 7, a, B.
The equation (iv) shews that t+7= 0 when r =
T = 0 when r = p, so that
334.]
571
THEORETICAL DYNAMICS.
-T is the time at which r = p.
equation with respect to t, we obtain
Now if we differentiate the
k2
-101
mr'
{
2m (h+p(r)) -
202
and we took for p a value of r, which makes this expression
vanish; hence, when rp, r' 0, and it is evident therefore
that we may suppose p to be the minimum value of r. Thus
-T will be the time at which m is at the least distance from the
centre of force.
Again, if we put for the inclination of the plane of the mo-
tion to the plane of (x, y,) we have, by well-known principles,
c = k cos; hence
cot,
(k² - c²) /
k
1
(k² - c²) +
sinɩ
equation (v) is therefore equivalent to
sin (0 — a) = tan λ cot.
(46)
Next let be the "argument of latitude;" that is, the angle
between and the ascending node, or line joining m with the
centre at the instant when m passes through the plane of (x, y)
from the negative to the positive side; considering then the
right-angled spherical triangle, formed by the intersections with
a sphere of the plane of (x, y), and the planes of the angles 9 and
A, in which is the hypothenuse and the angle opposite to A,
we see, from (46), that 0—a is the side which is in the plane of
(x, y), and therefore a is evidently the longitude of the ascend-
ing node.
し
​Lastly, equation (vi) shews that when rp, sinλ = sin..sinß;
but from the triangle above mentioned we have also sin λ =
sin..sin; hence we see that when r = p, B, from which it
is evident that ẞ is the angle between the least distance and the
node.
の
​334.] We shall briefly indicate the application of these results
to the case of the undisturbed motion of a planet about the sun.
Letμmass of sun + mass of planet; then, m being the mass
of the planet, we shall have for the force-function
(r)
also, from the ordinary theory of elliptic motion, we have
m μ
h
k
m μ
r
= m {µa (1 − e²)} *, c = m{µa(1 − e²) } & cosɩ.
;
α
4D 2
572
[335-
THEORETICAL DYNAMICS.
Hence we have the following set of canonical elements, ar-
ranged in conjugate pairs. The signs of the first pair have been
changed, which is obviously allowable, and p is put for a (1-e²),
that is, for the semi-latus-rectum ;
m μ
a
time of perihelion passage;
m (µ¿) cos ɩ, longitude of node;
m (µp),
distance of perihelion from node.
These elements were first given by Jacobi.
335.] Variation of elements. Returning to the general form
of the differential equations,
Pi
dĤ
d qi
qi
d H
dp;'
(i = 1, to in),
let us suppose that a complete set of integrals C1, C2,..., Can has
been obtained, where c1, c2, ... represent certain functions of the
variables p₁, 91,..., and t, which, by virtue of the differential
equations, have constant values; and conversely, P1, P2, ... are
given functions of C1, C2,..., and t.
Now let C1, Can continue
to represent the same functions of the variables and t, but let
the variables be determined as functions of t by a different set
of differential equations, say,
do
da
pi=
qi
dqr
dpi
Pn, 91,
...
P1,
(47)
› In, t,
where q is, like н, a given function of
but not the same function as H. Then the functions C1,..., C2n
will no longer have constant values. In other words, the ele-
ments C1, C2, which were constant in the former problem,
...
are variable in the latter problem.
The theory of the variation of elements has for its principal
object to transform the equations (47) by taking as new variables
the functions C1,..., C2n, instead of the original variables p1,..., Pn
71,
... ,
In•
The integration of the equations after this transformation
would give c1,
..., C2n as functions of t and 2n constants; and
the variables P1, P2,
because they are the same functions of the elements C1, C2, ...
and t, as they were in the original problem.
would then be known as functions of t,
In practice, it is usually convenient to put Q = H+, where н
335.]
573
THEORETICAL DYNAMICS.
is the same as before, and a is a given function of p1,..., Par
q, q, t, and is called technically the "disturbing function."
q1,...,
Thus the equations (47) become
dн
do
Pí
dqi
d qi'
d H
gi= +
dpi dpi
da
;
and these are transformed as follows: we have
dc₁
dc1
dc1
c₁ =
dt
+ Σ ·Pi +
dpi
Ji
dqi
:):
that is, introducing the above values of pí, qi,
dc₁
cí
+(H, C₁)+(2, C1):
dt
î
now c₁ is by hypothesis a function of p₁, ... and t, which in the
"undisturbed" problem, that is, when = 0, is constant; and
this constancy would be expressed by the equation
dc₁
dt
+ (H, C1) = 0,
P1,
which is identically true, all its terms being functions of p1, ...,
91, t. But since c₁ and н are the same functions of
91,..., t, in the disturbed, as they were in the undisturbed
problem, the above equation is still identically true, and the
expression for c therefore becomes simply
Cí = (2, C1) :
=
here n is expressed in its original form as a function of p1,
q1,..., t; but if we suppose pi,..., 91, ... to be expressed in
terms of c,
….., C2n, t, o will become a function of the latter
quantities, and denoting, for a moment, the new form of a by
, we shall have, by the elementary properties of the symbol
(u, v), Art. 311,
dā
cí
c₁ = (2,01) =
(C1, C1) +
dc₁
da
(C2, C1) + +
dca
do
(C2n, C1);
...
dc2n
in which the first term vanishes, since (c1, c₁) = 0.
The general form is evidently, omitting the line over 2,
da
ci
(C1, Ci) +
dc₁
do
dcz
(C₂, Ci) + ... +
da
dc2n
(Can, Ci),
(48)
involving 2n-1 terms on the right-hand side. And by giving
to i the values 1, 2, ..., 2n, we obtain a set of 2n differential
equations, which will involve only the new variables C1, C2, ...
574
[336.
THEORETICAL DYNAMICS.
instead of P1, P2,
when the terms (c₁, c;)..., which are given
functions of the old variables, are expressed in terms of the new.
It follows, from what was proved in Art. 316, that when so ex-
pressed they will not contain t.
336.] But the formula (48) undergoes a remarkable simplifi-
cation when c₁, ... C2, are a set of canonical elements; for in
C1,
that case all the terms (C1, C¡), (C₂, ci), ... vanish except one,
namely, that in which c; is combined with its conjugate element,
and then the term is either + 1 or - 1.
Suppose then that a, ... a,, b1, ... b, are canonical elements,
so that (a;, b;) 1, (bi, a;) -1, and all other combinations
give zero. The formula (48) will be seen at once to give
a;
da
db;
ΦΩ
bi
da;
(49)
which are the transformed equations in this case, and are of the
same general form as the original equations.
These formulæ were discovered in a particular case by La-
grange; namely, that in which a₁, b₁, ... are the initial values
of P1. 91, ….. . The extension to canonical elements in general is
due partly to Sir W. R. Hamilton and partly to Jacobi.
There are many points of interest and importance connected
with the further development and application of the theory of
the variation of elements, but we cannot afford space for them
here.
337.] We shall conclude this Chapter by a brief notice of an
addition to the general theory of the Hamiltonian equations
made by M. Bour, and improved in accordance with a sugges
tion of M. Liouville *.
...
ɑn, b1, b2, ….. bn, be, as before, a set of canonical
Let a1, a2,
integrals of the equations
pí
dH
d qi
du
qi
d pi
(50)
* See Liouville's Journal, t. xx. pp. 185-200, 201, 202. The writer is not
aware whether M. Bour's investigations have yet been published complete in
the Memoirs of the Academy. He believes that the process and results in the
text must be substantially what those of M. Bour become when н contains t,
which is the generalization proposed by M. Liouville; but he can only speak
from conjecture on this point.
[The Memoir of M. Bour, to which the note refers, is published in Tome
XIV of the Mémoires des Savants Etrangers, p. 792; having been presented
to the Academy of Sciences at Paris on March 5th, 1855.]
337.]
575
THEORETICAL DYNAMICS.
If we represent any integral whatever by (, the equation ('= 0
ας
dt
+ (H, () = 0; which, written in full, would be
gives
dš du dŚ
dн d
H š
du d¿
du d
+
+
+
= 0. (51)
dt dpi dqi
d q₁ dpi
v
dp dqn
dq dа pn
d H dH
In this equation
are given functions of p₁,
dpr' dq
:
Pus I... qu, t; so that with reference to it is a linear partial
differential equation of the first order, of which any solution
whatever is an integral of the equations (50), and of which the
general solution might be put in the form
5 = f(a₁, ..., an, by,..., b₁),
ら
​where fis an arbitrary function.
Thus the complete integration of the equations (50) would be
effected if we could find the general solution of (51).
But this
transformation of the problem is practically useless, since the
only known general method of treating the equation (51) re-
quires the integration of the system (50), as the reader will see
at once on applying Lagrange's method to the former.
The proposition we are about to demonstrate is the following:
Suppose m of the n integrals a,...a,, to be known; say α1,
ɑ2,…, ɑm, where m is of course less than n; and the condition
(α;, α ;) = 0 to subsist for every pair; then by means of these m
integrals, m of the variables, say P1, P2,..., Pm, might be ex-
pressed in terms of
a
...
› In, t ;
...
(52)
• • •, am, Pm + 1, Pro J1
and by the substitution of these values of P
Pm, H and g
might be expressed in the same way: we shall prove that after
this substitution the equation (51) would still subsist, the dif-
ferentiations being all performed only so far as the variables
would appear explicitly. But since H and would no longer
contain P₁,
P1, P explicitly, all the terms in (51) which involve
differentiation with respect to these m variables would disappear;
and the equation would become
"
น
>
dc
dt
d H
ας
+
d H
ας
dpm + 1 dqm +1 d qm + 1 dpm + 1
dн d
du d
+ +
0; (53)
...
d p n d qn d qn dpn
in which the number of terms is diminished by 2m.
In this equation H is a given function of the quantities (52);
dt
αξ
the variables 91,
but as it does not involve
d qi
d q m
576
[338.
THEORETICAL DYNAMICS.
would be treated as constants in integration. Any solu-
tion of it will be a function of the same quantities (52),
and will be an integral of (50); but the number of distinct
solutions will be less by 2m than those of (51). In fact (53)
will be satisfied by am+1, an, bm+1,
'm+1,... bn, but not by aı,
am, b₁, ... bm; and it follows that any solution c of (53) will
satisfy the conditions (c, a₁) = 0, (c, am) = 0, but any two solu-
tions, c, e, will not necessarily satisfy the condition (c, e)
...
= 0.
338.] We proceed to demonstrate what has been stated in
the last Article. Let H, when transformed by substituting for
P₁, ... Pm the values obtained from the given integrals a, ... am
in terms of the quantities (52), be denoted by H. The equation
(51) is
ας
dt
+ (H, () = 0;
but by the elementary property of the symbol (u, v), before
referred to, we have
d H
dH
(H, X) = (H, ¿) +
(a1, §) + ... +
(am, 5);
dai
dam
in which the expression (H, ) is to be formed by differentiating
H with respect to the variables only so far as they appear expli-
citly. Thus the equation (51) becomes
αξ
dt
d H
+ (Ā, Š) ÷
da
d a 1
(α1, Ŝ) + ... + (a, () = 0;
dH
(
and this would still
dam
be satisfied by putting = any integral of
(50). But if be any one of the integrals a, ... an, bm+1, ….. bn,
(
we shall have (a1, Š) 0, (α2, 8) = 0,
0,... (ɑm,
(am, 5) = 0, whereas
one of these terms would be unity if ( were one of the integrals
b₁,... bm. Consequently the equation
dr
+ (H, () = 0
dt
(54)
will be satisfied by a1, ..., an, bm+1, ... bn, but not by b₁, ... bm.
In this equation however is still supposed to be expressed in
terms of all the variables
and t. But if we sup-
P1,
,91 و
"
pose to be transformed in the same way as H, and then to be
denoted by , we shall have
dě
dš dr dai
+
+
+
dt
dt
da dt
dğ dam
dam dt
αζ
αξ
(H, C)
(H, 3) +
H, α1) + ... +
day
(H, am);
dam
338.]
577
THEORETICAL DYNAMICS.
so that the equation (54) becomes
dr
ας
+ (H, §) +
dt
de Jdai
dadt
+(ū, c)}
+
+
dam dt
dš {dam + (H, am
(Ĥ, am)
am) } = 0,
(H, α₁) (H, α1) +
and (α₁, α1) =
d H
day
0, (α2, α1) = 0,
(a1, α1) +
in which all the terms after the first two vanish; for, since a₁ is
day
an integral, we have a = 0, that is, t+ (H, α₁) = 0; but
dt
dн
(a₂, α1) + ... + (am, α1),
dam
d H
day
(am, a₁) = 0, so that (H, a₁)
(H, a₁); and therefore
da1
dt
+(µ, α1) = 0; and the same rea-
soning applies to the other terms; finally, therefore, the equa-
tion becomes
ας
+ (H, 3) = 0;
dt
and this will be satisfied by the same integrals as (54) with the
following exceptions. Suppose the integral a; to be
ɑi = $ (P1, • › Pn, 91, •••, In, t)
•••,
(56)
then (54) is satisfied by putting for the function 4 on the right
of this equation. And if this function be transformed by putting
for pi,
Pm their values in terms of the quantities (52), a¿ will
in general be expressed as a function of the same quantities, and
(55) will be satisfied by taking this function for ; but if i be
one of the indices 1, 2, ..., m, it is evident that the transforma-
tion in question will reduce the right-hand member of (56) iden-
tically to a, so that will cease to contain any of the variables;
and (55) will then only be satisfied by 3 = a;, in the sense that
all the differential coefficients of vanish separately; that is, in
the sense that the equation is satisfied by putting = any
constant.
It follows therefore that, in the ordinary sense, (55) is satis-
fied by am+1, ... an, bm+1, ... bn, expressed in terms of the quan-
tities (52), but not by the remaining integrals of the canonical
set.
If the distinctive notation of (55) be omitted, and the equa-
tion be written at length, the result is the equation (53) of the
last Article.
These formulæ require some alteration in the case in which
PRICE, VOL. IV.
4 E
578
[339.
THEORETICAL DYNAMICS.
the principle of vis viva subsists; but we have not space to enter
further into the subject.
339.] In taking leave of the subject, it is proper to mention
the investigations of Jacobi, contained in his Memoirs, entitled,
“Theoria nova multiplicatoris systemati æquationum differen-
tialium vulgarium applicandi," Crelle's Journal, Vols. XXVII
and XXIX. These investigations are only so far connected
with the subject of this Chapter, that they are applicable to the
Hamiltonian equations as a particular case. But although the
results are interesting and important, they are omitted here,
because the demonstration depends on the properties of func-
tional determinants, and could not be given without a long di-
gression.
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