YOU LAH
XA
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448361

TA
ARTES
LIBRARY
1837
SCIENTIA
VERITAS
OF THE
UNIVERSITY OF MICHIGAN
E-PLURIBUS-UNI
TULBU
SQUAERIS PENINSULAM AMOENAM
CIRCUMSPICE
'':
ФА
871
A. Freeman. R87
9.12.77
Fr. Mass.

་

A TREATISE
ON
THE STABILITY OF MOTION.


A TREATISE
罚
​ON THE
STABILITY OF A GIVEN STATE
OF MOTION,
PARTICULARLY STEADY MOTION.
BEING THE ESSAY TO WHICH THE ADAMS PRIZE WAS ADJUDGED
IN 1877, IN THE UNIVERSITY OF CAMBRIDGE.
7
BY
E. J. ROUTH, M.A., F.R.S., &c.
LATE FELLOW OF ST PETER'S COLLEGE, CAMBRIDGE,
AND LATE EXAMINER IN THE UNIVERSITY OF LONDON.
London:
MACMILLAN AND CO.
1877
[All Rights reserved.]
F
Cambridge:
PRINTED BY C. J. CLAY, M.A.,
AT THE UNIVERSITY PRESS.
¡
PREFACE.
A
IN March, 1875, the usual biennial notice was issued, giving the
subjects for the Adams Prize to be adjudged in 1877. The
following is the chief portion of the notice:
The University having accepted a Fund raised by several members
of St John's College for the purpose of founding a Prize to be called the
Adams Prize, for the best essay on some subject of Pure Mathematics,
Astronomy or other branch of Natural Philosophy, the Prize to be
given once in two years, and to be open to the competition of all
persons who have at any time been admitted to a degree in this
University-
The Examiners give notice that the following is the subject of the
Prize to be adjudged in 1877: The Criterion of Dynamical Stability.
To illustrate the meaning of the question imagine a particle to slide
down inside a smooth inclined cylinder along the lowest generating line,
or to slide down outside along the highest generating line. In the
former case a slight derangement of the motion would merely cause
the particle to oscillate about the generating line, while in the latter
case the particle would depart from the generating line altogether.
The motion in the former case would be, in the sense of the question,
stable, in the latter unstable.
The criterion of the stability of the equilibrium of a system is,
that its potential energy should be a minimum; what is desired is, a
R. A.
135440
b
-
vi
PREFACE,
corresponding condition enabling us to decide when a dynamically pos-
sible motion of a system is such, that if slightly deranged the motion
shall continue to be only slightly departed from.
The essays must be sent in to the Vice-Chancellor on or before the
16th December 1876, &c., &c.
S. G. PHEAR, Vice-Chancellor.
J. CHALLIS.
G. G. STOKES.
J. CLERK MAXWELL.
·
The pressure of other engagements for some time prevented
me from giving my attention to the subject. This essay was
therefore almost entirely composed during the year 1876. It is
now printed as it was sent in to the Examiners, the changes being
merely verbal. Some few additions have been made where ex-
planation appeared to be necessary, but all these have been
marked by square brackets, so that they can be at once dis-
tinguished from the original parts of the essay.
In order to shorten the essay as much as possible many
merely algebraic processes have been omitted and the results only
are stated. It is hoped that this will add clearness as well as
brevity to the reasoning, as the attention of the reader will not
be called from the argument to follow a manipulation of symbols
which may not present any novelty.
The line of argument taken may be indicated in a general way
as follows. Chapter 1. begins with some definitions of the terms
stable and steady motions. It is then pointed out that whether
the forces which act on the system admit of a force-function or not,
the stability of the motion, if steady, is indicated by the nature
of the roots of a certain determinantal equation. The boundary
between stability and instability being generally indicated by the
presence of equal roots, a criterion is investigated to determine
beforehand whether equal roots do or do not imply instability.
This case being disposed of, the consideration of the determinantal
equation is resumed. Two general methods are given by which,
without solving the equation, it may be ascertained whether the
7
..
PREFACE.
vii
character of the roots imply stability or instability. These occupy
Chapters II. and III. In the first method a derived equation is
made use of, and it is shown that a simple inspection of the signs
of the coefficients of the several powers in these two equations will
decide the question of stability. In the second method a certain
easy process is found which if performed on the determinantal
equation will lead to the criteria of stability. At the end of the
third Chapter a geometrical interpretation is given to the argu-
ment.
In the fourth Chapter the forces which act on the system
are supposed to have a force-function. The determinantal equa-
tion is then much simplified. Several points are considered in
this Chapter which are necessary to the argument, such as the
proper method of choosing the steady co-ordinates (if there be
any), the distinction between harmonic oscillation about steady
motion and that about equilibrium, and the changes which must
be made in the determinantal equation when the equations of
Lagrange become inapplicable. A method of modifying the
Lagrangian function is also given by which, in certain cases,
the fundamental determinant may be reduced to one of fewer
rows and columns.
In the fifth Chapter a series of subsidiary determinants is
formed, and it is shown that at least as many of the conditions
of stability are satisfied as there are variations of signs lost in the
series in passing from one given state to another. It is also shown
that this is equivalent to a maximum condition of the Lagrangian
function.
In the sixth Chapter the energy test of stability is considered.
It is also shown that, when the motion is steady, this reduces
to the same criterion as that indicated in Chap. V.
In the seventh Chapter the question considered is whether
the stability of a state of motion can really be determined by an
examination of the terms of the first order only. In some cases
these are certainly sufficient, and an attempt is made to discrimi-
nate between these cases and those in which the terms of the
higher order ultimately alter the character of the motion.
If the Hamiltonian characteristic and principal functions be
given, the conditions of stability as regards space only, or both
דיי
1
viii
PREFACE.
space and time may be deduced. But if these be not known as
expressed in the Hamiltonian form, we may yet sometimes dis-
tinguish between stability and instability if we can determine
whether a certain integral ceases to be a minimum at some instant
of the motion. This is the subject of the eighth chapter.
As part of the third edition of my treatise on the Dynamics of
Rigid Bodies was written at the same time as this essay, there are
necessarily points of contact between the two works. Thus the
subjects of the first part of the seventh chapter and of a portion
of the sixth will be found discussed in the treatise on Dynamics.
But as the objects of the two books are not the same, it will be
found that in all these cases there are considerable differences in
the modes of demonstration.
PETERHOUSE,
August 14, 1877.
EDWARD J. ROUTH.
CONTENTS.
CHAPTER I.
ART.
PAGE
1-2.
Definitions of the terms small quantity, stable motion, steady
motion
.1, 2
3.
4-7.
A system of bodies in steady motion is stable if the roots of a
certain determinantal equation are such that their real parts
are all negative
Effect of equal roots, and a test to determine whether equal roots
do or do not introduce terms which contain the time as a
factor
8.
Object of Chapters II. and III.
•
CHAPTER II.
3
4-10
10
•
1—3.
4.
5, 6.
Statement of the theory by which the necessary and sufficient
tests of stability are found. Objections to this theory.
These tests shown to be integral functions of the coefficients
Method of finding these tests when the coefficients of the equa-
tion are numerical, or when several terms are absent
1
11, 12
12
13, 14
7, 8.
All these tests shown to be derivable from one called the funda-
mental term
•
14-16
9.
The fundamental term found as an eliminant
16
10.
A method of finding the fundamental term by derivation from
the fundamental term of one degree lower
17
11-13. Another and better method of doing the same by means of a
differential equation
•
19-21
X
CONTENTS.
CHAPTER III.
ART.
1-4.
5-8.
9, 10.
11, 12.
Statement of the theory by which the conditions of stability of a
dynamical system with n co-ordinates are made to depend on
2n conditions
A rule by which these conditions may be derived one from
another, together with certain other true but not independent
conditions
A rule by which, when the coefficients of the dynamical equation
are letters, the 2n conditions of stability may be inferred,
one from another, without writing down any other conditions
A method by which certain extraneous factors may be discovered
and omitted
•
13-18. Consideration of the reserved case in which the dynamical equa-
tion has equal and opposite roots
•
19-25. A few geometrical illustrations not necessary to the argument
26.
Application to a Dynamical Problem
PAGE
23-26
27, 28
28, 29
29-32
32-37
•
37-40
42
CHAPTER IV.
1--5.
Formation of the equations of steady motion and of small oscil-
lation where Lagrange's method may be used
45-48
6.
The equations being all linear the conditions of stability are
expressed by the character of the roots of a determinantal
equation of an even order
49
7.
Mode of expanding the determinant
49
8-10.
A method of finding the proper co-ordinates to make the co-
efficients of the Lagrangian function constant
50-53
11-19. How the Harmonic oscillations about steady motion differ from
those about a position of equilibrium. The forces which
cause the difference are of the nature of centrifugal forces
produced by an imaginary rotation about a fixed straight line 53-60
20-23. Reduction of the fundamental determinant to one of fewer rows
by the elimination of all co-ordinates which do not appear
except as differential coefficients in the Lagrangian function;
with an example
•
24-27. Formation of the equations of Motion and of the determinant
when the geometrical equations contain differential co-
efficients, so that Lagrange's method cannot be used; with
an example
•
60-66
·
68-71
CONTENTS.
xi
CHAPTER V.
ART.
1-5, and 9. Certain subsidiary determinants are formed from the dy-
namical determinant, and it is shown that there must be at
least as many roots indicating stability as there are varia-
tions of sign lost in these subsidiary determinants, and must
exceed the number lost by an even number
6-8.
10, 11.
12.
PAGE
74-76, 78
This is equivalent to a maximum and minimum criterion of
stability with similar limitations
Effect of equal roots on this test of the stability of the system .
Example
•
• 76-78
79; 80
81
་}
1-3.
4.
5, 6.
3
CHAPTER VI.
If the energy of the system be a maximum or minimum under
certain conditions, the motion whether steady or not is stable 82-84
When the motion is steady, it will be also stable if a certain
function of the co-ordinates called V+ W is a minimum
If there be only one co-ordinate which enters into the Lagran-
gian function, except as a differential coefficient, this condi-
tion is necessary and sufficient
84-85
85-87
7-8.
Additional conditions when there are two co-ordinates
CHAPTER VII.
1.
2-3.
4.
5-7.
8-9.
10.
87-89
Any small term of a high order, if its period is nearly the same
as that of an oscillation of the system, may produce impor-
tant effects on the magnitude of the oscillation
Origin of such terms, with an example
Supposing the roots of the determinantal equation to satisfy the
conditions of stability to a first approximation, yet if a com-
mensurable relation hold between these roots it is necessary
to examine certain terms of the higher orders to determine
whether they will ultimately destroy the stability of the
system
If a certain relation hold among the coefficients of these terms,
they will not affect the stability of the system, but only
slightly alter the periods of oscillation
Examples, the first taken from Lagrange's method of finding the
oscillations about a position of equilibrium
If the coefficients of the equations of motion should not be
strictly constant, but only nearly so, the stability will not be
affected, unless the reciprocals of their periods have com-
mensurable relations with the reciprocals of the periods of
oscillation of the system
90-91
91-92
92
•
92-94
94-96
96-97
xii
CONTENTS,
ART.
1-3.
CHAPTER VIII.
The Hamiltonian Characteristic or Principal functions when
found determine at once the motion of the system from one
given position to another, and whether the motion is stable
or unstable
PAGE
•
98-100
4-7.
Examples with a mode of effecting the integration S= Ldt in
small oscillations
100-102
8.
9-14.
The Characteristic function supplies the condition that the
motion is stable as to space only, while the Principal func- .
tion gives the conditions that it is stable both as to space
and time
In what sense the motion is unstable if either of the two Hamil-
tonian functions is a minimum
102, 103
103-108
if
CHAPTER I.
Definitions of the terms small quantity, stable motion, steady motion.
Arts. 1, 2.
A system of bodies in steady motion is stable if the roots of a certain
determinantal equation are such that their real parts are all negative.
Art. 3.
Effect of equal roots, and a test to determine whether equal roots do or do
not introduce terms which contain the time as a factor. Arts. 4—7.
Object of Chapters II. and III. Art. 8.
1. Let us suppose a dynamical system to be set in motion
under any forces and to move in some known manner. If any
small disturbance be given to the system, it may deviate only
slightly from its known motion, or it may diverge further and
further from it. Let 0, 4, &c. be the independent variables or
co-ordinates which determine the position of the system, and let
the known motion be given by 0=0,, = 4, &c. where 0, 4, &c.
are known functions of the time t. To discover the disturbance of
the system we put 0=0,+x, p=0,+y, &c. These quantities
x, y, &c. are in the first instance very small because the disturb-
ance is small. The quantities x, y, z, &c. are said to be small
when it is possible to choose some quantity numerically greater
than all of them, which is such that its square can be neglected.
This quantity may be called the standard of reference for small
quantities.
If, after the disturbance, the co-ordinates x, y, z, &c. remain
always small, the undisturbed motion is said to be stable; if, on the
other hand, any one of the co-ordinates become large, the motion is
called unstable.
It is clear that the same motion may be stable for one kind of
disturbance and unstable for another. But it is usual to suppose
the disturbance general, so that if the motion can be made un-
stable by any kind of disturbance (provided it be small) it is said
to be unstable. On the other hand, it will be called stable only
when it is stable for all kinds of small disturbances.
R. A.
1
›
2
DEFINITIONS.
Y,
[CHAP.
2. To determine whether x, y, z, &c. remain small, we must
substitute for 0, 4, &c. in the equations of motion their values
0+x, &。 +y, &c. Assuming that x, y, &c. remain small, we may
neglect their squares, and thus the resulting equations will be
dx d²x dy day
linear in x, y, z, &c. The coefficients of X,
&c.
dt' dt², dt' dť²'
in these equations may be either constants or functions of the
time. In the former case the undisturbed motion is said to be
steady for these co-ordinates, in the latter unsteady. In the case
of a steady motion x, y, z, &c. are all functions of the time
which has elapsed since the disturbance and of certain constants
of integration which are determined by the initial values of
dx dy
X, , Y, dt
&c. We may therefore define a steady motion to be
such that the same change of motion follows from the same initial
disturbance at whatever instant the disturbance is communicated
to the system.
dt'
If all the coefficients in the equations to find x, y, z are con-
stant, they may be made to contain t by a change of co-ordinates.
Thus we may write for x, y, z, &c.
x = a§.+ Bn + ...
y = a'§ + ß'n + ...
z = &c.
2
where a, ß, &c. are any functions of t we please. Conversely, when
the coefficients are functions of t, we may sometimes make the
coefficients constant by a proper change of co-ordinates. But this
cannot always be done. If there are n co-ordinates, we have n
arbitrary functions a, ß, &c. at our disposal. In each of the n
linear equations of motion we may have three terms for each
co-ordinate, and thus we have (3n-1) n coefficients to make con-
stants. We have therefore in general too many equations to
satisfy. The proper method of choosing the co-ordinates of refer-
ence will be considered in a future chapter.
3. Let us suppose a dynamical system to be making small
oscillations under the action of any forces which may, or may not,
possess a force function and to be subject to any resistances which
vary as the velocities of the parts resisted. The general equations
of motion will then be of the form
d
d2
d
(4,2 + 4, & + 4.) x + (B, & + B, & + B₂) y + &c. = 0
at
dť² at
A
dt²
d2
d
d2
A
2 dť²
+ A.
1 dt
+ Ab) x
+ (B₂
d
이
​de² + B',' at + B') y + &c. = 0
+B
2
dt
اه
&c. = 0
I.]
3
THE DETERMINANTAL EQUATION.
To solve these equations we write
x= Memt, y=M'emt, &c.
Substituting we obtain a determinantal equation to find m.
If we put
A = A‚m² + A¸m + A。,
B=B₂m² +B¸m+ B。,
&c.
A' = A¸m² + A¸m + A'',
this equation may be written in the simple form
A, B, C...... | 0.
A', B', C'......
=
We may also write the equation in the form ƒ (m) = 0.
The coefficients M, M', &c. are not independent, but if we
represent the minors of A, B, C, &c. by a, b, c, &c. we may easily
show that
M M' M"
= &c.
b
C
We also have
M M' M"
&=&c.
ď
b
It may be shown by properties of determinants that these
equations all give the same ratios. If A, be the second minor
obtained from the determinant f(m) by omitting the first and
second rows and columns, we know that
▲ƒ (m) = ab' — a’b.
Hence if f(m) = 0 we have
α
b
a
In the same way we may show that
a
C
and so on. This
α
property of Determinants is given in Dr Salmon's Higher Algebra,
Lesson IV. Ex. 1.
The general solution of the equation may therefore be written
in the form
x = L₁₁em₁t + Lam₂t +
y = L₂b₁em₁t + L₂bem +
2
&c.
where L₁, L... are arbitrary constants, a, a, &c. the values of
the minor a when m,, m,... are substituted for m; b,, b,... the
2
2
+
1-2
فة
PI"SPTE™ -
1
1
$
4
NATURE OF THE STABILITY
[CHAP.
values of the minor b when similar substitutions are made, and
so on.
4. We see that the whole character of the motion will de-
pend on the signs of the quantities m,, m,... If any one be real
and positive, x, y, &c. or some of them will ultimately become
large, and the steady motion about which the system is oscillating
will be unstable. If all the roots are real, negative or zero and
unequal, the motion will be stable.
If two of the roots be imaginary we have a pair of imaginary
exponentials. If these imaginary roots be a + B-1, the terms
can be rationalized into
eat (N₁ cos ẞt + N₂sin ßt).
1
The motion will be stable if a be negative or zero; and unstable
if a be positive.
If two roots be equal, the form of the solution is changed. Let
m₁ = m₁+h where h will be ultimately zero, we then have
x = L¸à¸m² + L₂ (a
a₁emit +
day
hemt +a¸htemit).
2
dm
If we now make L, and L, infinite in the usual manner, we
find
= {Mat+ M₂da + Ma} et,
Ꮳ
X =
y= {M,b,c + M
{M¸b,t+
&c. = &c.,
2 dm
db,
2
dm
+ Mb, } cm²,
emit,
where M₁, M, are two arbitrary constants which replace L₁, L
2
In the same way if three roots are equal we have
Ma
1
α
M.
== [M. (a! + da, t + + dra;) + M, (at + da) + M,a,] om,
2
12
[M, (5,₁ +
y=M.b
3
1
1
11
t + 1 db;) + M, (b,t + db) + Mb, ]em".
db ₁
12+ dm dm
2
This rule will be found convenient in practice to supply the
defect in the number of arbitrary constants produced by equal
roots. At present we are only concerned with their effect on the
stability of the system. The terms which contain t as a factor
will at first increase with t, but if m be negative, the term themt
can never be numerically greater than
n
em
n
If m be very small
1.]
5
IS INDICATED BY THE ROOTS.
100
the initial increase of the terms may make the values of x and y
become large, and the motion cannot be regarded as a small
oscillation. But if the system be not so much disturbed that
N
em
M.
is large, the terms will ultimately disappear and the
motion may be regarded as stable. If, however, the real parts of
the equal roots are positive or zero, the terms will become large
and the motion will be unstable.
5. In some cases, however, the relations which exist between
the coefficients are such that the terms which contain t as a factor
are all zero. It is of some importance to discriminate these cases,
for the stability of the system is then unaffected by the presence
of equal roots.
Let us suppose first that the determinantal equation has two
roots only equal to m₁, and let the terms depending on these be
x = (N₁+ N¸t) emit,
y = (N₁' + N¸'t) emit,
&c. = &c.
Substituting in the equations of Art. (3) we have, following
the same notation as before,
AN₂+BN +CN" + ... = 0
A'N₂+B'N'' + C″N," +
2
2
2
…….I.
&c.
=
1
AN₁ + BN,' + ... — —
dA dB
N
N
dm
2 dm
dA'
1
dm
N₂
dB'
2
dm N'-
2
......II.
A'N₂+ B'N' + ... =-
&c. = &c.
To avoid entering more minutely than is necessary into the
properties of linear equations, we shall assume that these equations
for the given value of m lead to but one solution with two of the
N's arbitrary, unless the determinantal equation has more than
two roots equal to m₁. If in this unique solution the N's are all
zero we must have
1
AN₁+BN₁'+ ... = 0
A'N₁ + B'N' + ... = = 0
&c. =0
..III.
6
[CHAP.
EFFECT OF EQUAL ROOTS -
I
Since two of the constants N, N, &c. are to be arbitrary, let
them be N₁, N', then since
N N
1
we must have the minors a
and b each equal to zero.
α b
Also since
1
N_N",
-, we shall have N,"
α
C
infinite unless c = 0. In the same way we may prove that all the
other first minors are zero.
And if the first minors are zero, we may show that two of the
equations may be deduced from the others. Let the symbol
AB
A'B'
represent the second minor, with the usual sign, formed
by omitting the rows and columns in which AB A'B' occur.
Then since the minors a, b, c, &c. are zero, we have
A'
АВТ
Α' Β'
+A"
AB
A"B"
+... = (
0
B [AB] + B" [AB]
...IV.
+... = 0
A"B"
&c. = 0.
:
i
"
;
Omitting the first line of III. let us multiply the others by
АВТ AB
A'B' A"B"
an identity. Hence the second equation may be deduced from
the others which follow it. In the same way, the first equation
may be deduced from the others.
&c., respectively. Adding the results, we have
1
Rejecting the first two equations, let us transpose the arbitrary
constants N and N' to the right-hand sides of the remaining
equations. If there are to be only two arbitrary constants, these
remaining equations must be independent; solving, we have
АВ
[AB] N'" -- N. [BC]
A'B' 1
1
[BG] + N [40],
A'C"
with similar equations for the others. Hence the constants
N₁, N, &c., are connected by equations of the form
N. [BC]-N: [AC] + N." [AB]-0,
1
1 A'C'
1 A'B'
so that when any two are chosen as the arbitrary ones, the others
be deduced from them.
may
If the determinantal equation has three roots equal to m₁, and
if the terms which contain t as a factor are all zero, the equa-
tions III. must admit of a solution with three of the constants
i.
Next since
N
1 B'D'
1.].
ON STABILITY.
1
7
N₁, N', &c. arbitrary. If these be N₁, N, N'", we see that the
вс AC AB
second minors
[BC] [40] [42]
Ꭴ A'C'
Α' Β'
must be zero.
[BD] - N: [AD] + N'" [AB
1
1
A'B'
=
= 0,
BD1
we must have
and
B'D'
AD
A'D'
both zero or N,"" infinite.
Hence all the second minors are zero.
minors are all zero, we have
And if the second
A"
ABC
+A""
ABC
+ ... = 0,
Α' Β' Γ'
A'B'C'
A"B"C"
A""B" C""
and by similar reasoning three equations may be deduced from
the remaining ones. We have then
N
1
BCD 1-N, ACD ]+N,″[ ABD ]-N,”[ ABC
ABC =0,
1
B'C'D'
B"C"D".
A'C'D'
A" C"D"
A'B'D'
A'B'C'
A"B"D"
A"B" C"
with similar equations.
Since f(m) is the determinant formed by eliminating N₁, N',
&c. from III. we have
df (m) __ df (m) dA, df (m) dB
dm
+
dA dm dB dm
+ &c. +
df (m) dA'
dA' dm
+ &c.
dA dB
+b
dm dm
dΑ'
+ &c. + a'
+ &c.
dm
This vanishes when a, b, &c., a', &c. are all zero. If therefore the
first minors of ƒ (m) all vanish when m=m₁, the equation ƒ (m) = 0
da
has two roots equal to m₁. In the same way vanishes if all its
first minors are zero. But
dm
d³f (m)
dm²
d2A
=α
+
dm²' dm dm
da dA
+ &c.
da
vanishes if a,
&c. are all zero. If therefore the first and
dm'
second minors of f(m) all vanish when m=m,, the equation
i
8
[CHAP.
EFFECT OF EQUAL ROOTS
f(m) = 0 has three roots equal to m₁. It is evident the proposi-
tion may be extended to any number of roots.
The test that, when equal roots occur in the determinantal equa-
tion, the terms in the values of x, y, &c. which contain t as a factor
should be absent may be stated thus. If there are two equal roots
all the first minors must vanish. If three equal roots, all the first
and second minors must vanish, and so on. In these cases the
equal roots introduce merely a corresponding indeterminateness
into the coefficients.
When there are more equal roots than there are rows in the
determinantal equation, it is easy to see that there must be some
terms in the integrals which contain t as a factor.
[The following simple example will illustrate the application
of this test.
A particle is in equilibrium at the origin of co-ordinates under
the action of forces whose force function U is given by
U = ½ Ax² + ½ By² + ½ Cz² + Dyz + Ezx + Fxy.
If the level surfaces are ellipsoids and the force acts inwards,
it is clear that the equilibrium of the particle must always be
stable. If then any equal roots occur in the determinantal
equation, the test should show that the terms which contain t as
a factor are absent.
If T be the semi vis viva of the particle and if its mass be
taken as unity, we have
12
12
T = { x²² + ½ y²² + 1 z²².
Omitting accents and forming the discriminant of — m²T+ U
we have the following determinantal equation :
A - m² F
E
= 0.
F
B-m² D
E
D C-m²
This is the "discriminating cubic" which determines the axes
of the quadric U= c, where c is a constant. The conditions that
two of its roots should be equal, i.e. that the quadric should be
a spheroid, are well known to be
A
EF
Ꭰ
FD
DE
=
- B -
C
m¸²,
2
E
F
where m,² is equal to either root. These are just the conditions
obtained by equating any first minor of the determinant to zero.
I.]
.
ON STABILITY.
9
The conditions that three of the roots of the cubic should be
equal, i.e. that the quadric should be a sphere, are
A =B=C, D=0, E=0, F=0.
These are the conditions that every second minor should vanish.
In this example we have taken the case of a single particle.
Similar remarks however apply when any system of bodies is
disturbed from a state of stable equilibrium. The oscillations
may be found by the method of Lagrange. The final determi-
nantal equation may be conveniently formed by equating to zero
the discriminant of - m²T+ U, where Tis the semi vis viva with
the accents denoting differentiations with regard to the time
omitted, and U is the force function. It is a known theorem that
the existence of finite equal roots does not affect the stability of
the equilibrium. Hence the conditions for equal roots must be
such as to make all the minors equal to zero. Conversely, this
theorem will often conveniently give the conditions that Lagrange's
determinant has equal roots.]
2
2
6. That there should be a difference in the modes in which
equal roots affect the motion is no more than we should expect
a priori. Suppose the coefficients of the equation ƒ (m) = 0 to be
functions of some quantity n, and that as n passes through the
value n, two roots become equal to each other. Let the quadratic
factor containing these roots be m² + 2am + ß, and let us consider
only the case in which a and B are real. We have a²-ẞ = 0
when n=n. If a B change sign as n passes through the value
no, the roots will change from a trigonometrical to a purely ex-
ponential form, which would indicate a change from oscillatory to
non-oscillatory motion. The passage from one kind of motion to
the other may be effected through a motion represented by expres-
sions having the time as a factor. But if a²-B does not change
sign, for example, if it be a perfect square for all values of n, there
will be no change from one kind of motion to the other, and in this
case we should expect that the motion when the roots are equal
will be represented by terms of the same character as before.
Briefly, we may expect equal roots to introduce terms with t as a
factor at the boundary between stability and instability; and to
introduce merely an indeterminateness into the coefficients when
the motion is stable on both sides.
2
2
It is easy to show that in the first of these two cases the
minors could not contain either of the factors of m² + 2am + B.
For since a² -ẞ changes sign, these factors are in one case ima-
ginary; and therefore if one factor occur in any minor the other
must also be present. The minors would not only vanish, but
must have equal roots also. But as in Art. (3), ▲, ƒ (m) = ab' — a′b.
10
[CHAP. I.
SUMMARY.
Hence if all the first minors have equal roots it is clear that either
f(m) has more than two equal roots, or all the second minors must
vanish. The latter is impossible unless f(m) has more than two
equal roots.
These general considerations are not meant to replace the
proofs given in the last article, but merely to explain how a
difference in the effects of the equal roots might arise.
7. Summing up what precedes, we see that if a dynamical
system have n co-ordinates its stability depends on the nature of
the roots of a certain equation of the 2nth degree.
If the roots of this equation are all unequal, the motion will
be stable if the real roots and the real parts of the imaginary roots
are all negative or zero, and unstable if any one is positive. If
several roots are equal the motion will be stable if the real parts
of those roots are negative and not very small, and unstable if
they are negative and small, zero, or any positive quantity. But
if, as often happens in dynamical problems, the terms which con-
tain t as a factor are absent from the solution, the condition of
stability is that the real roots and the real parts of the imaginary
roots of the subsidiary equation should be negative or zero.
When the equation ƒ (D) = 0 is of low dimensions we may
solve it or otherwise determine the nature of its roots; the
stability or instability of the system will then become known.
But if the degree of the equation be considerable this is not a very
easy problem. We shall devote the two next Chapters to the
consideration of two methods by either of which, without solving
the equation, we can determine the conditions that the real roots.
and the real parts of the imaginary roots should be all negative.
The determination of these conditions has, it appears, never before
been accomplished.* The consideration of the equations of motion
will then be resumed, and the form of the determinantal equation
f(D) = 0 when the forces admit of a force function will be more
particularly investigated.
* [These conditions for the cases of a biquadratic and a quintic had been found
by the Author in 1873, and read before the London Mathematical Society in June,
1874. See also the third edition of the Author's Rigid Dynamics, Art. 436.]
CHAPTER II.
Statement of the theory by which the necessary and sufficient tests of
stability are found. Objections to this theory. Arts. 1-3.
These tests shown to be integral functions of the coefficients. Art. 4.
Method of finding these tests when the coefficients of the equation are
numerical, or when several terms are absent. Arts. 5, 6.
All these tests shown to be derivable from one called the fundamental
term. Arts. 7, 8.
The fundamental term found as an eliminant. Art. 9.
A method of finding the fundamental term by derivation from the funda-
mental term of one degree lower. Art. 10.
Another and better method of doing the same by means of a differential
equation. Arts. 11—13,
1. The object of this Chapter has been explained at the end
of Chapter I. Briefly, the criterion that the motion of a system
of bodies should be stable is that the roots of a certain equation
should have all their real parts negative. We propose to investi-
gate these conditions.
Let the equation to be considered be
ƒ (x) = P₂x” +P₁xn−1 + ... + P₁₂ x + P₂ = 0.
Pn
Let the real roots be a, a,... and the imaginary roots be
α₁ ±ẞ₁√ −1, α, ± ß, √ − 1, &c.
Then
2
ƒ (x)=P。 (x − α¸) (x − α₂) ……. (x² — 2ׂx + α¸² +߸²), &c.
...
If then a₁, a₂, &c. a₁, a₂, &c. are all negative, every term in
each factor, and therefore in the product, must be positive.
It is therefore necessary that every term in the equation
f(x)=0 should have the same sign. It will be convenient to
suppose this sign to be positive.
12
[CHAP.
THE DERIVED EQUATION
.
!
It is also clear on the same suppositions that none of the
coefficients p。, P₁, ... P, can be zero, except when the roots of the
equation are all of the form ±ẞB-1, or when some of the roots
are zero.
2. Let us now form the equation whose roots are the sums of
the roots of f(x) taken two and two. Let this be
m
F' (x) = P。∞™ +P₁x™-¹ + ... + Pm-1∞ +P„ = 0,
where m = n
1
1
27 1
1
n-1
32
2
The real roots of this equation will be
а₁+α₂, α₁+α₂, &c. 21, 2α₂, &c. and the imaginary roots will be
a₁ + a₁ ± ẞ₁ √ − 1, &c. It is clear from the same reasoning as
before that if a,,a,, &c. a,, a,, &c. are all negative, the coefficients
P₁, P,, &c. must all have the same sign.
19
1
m
Conversely, if po, P... have all the same sign, the equation
f(x) can have no real positive root, and if P₁, P... P have all
the same sign the equation F(x) can have no positive root, and
therefore f(x) can have no imaginary root with its real part
positive.
3. Our first test of the stability of a dynamical system is that
all the coefficients of the dynamical equation f (D) = 0 and all the
coefficients of its derived equation F (D)=0 should have the same
sign.
It should be noticed that though these conditions are all
necessary and sufficient, they are not all independent. We obtain
too many conditions. In many cases, however, we can at once
reduce them to the proper number of independent conditions, and
when this is difficult we can have recourse to the second method,
to be given in the next Chapter, which is free from this objection.
In order to apply this method with success, it is necessary to
have some convenient methods of calculating the coefficients
P。, P₁... Pm²
4. The first method which suggests itself is one similar to that
usually given to determine the coefficients of the equation whose
roots are the squares of the differences of the roots of any given
equation.
If S₁, S... be the sums of the first, second powers, &c. of the
roots of the equation fx = 0, we have by Newton's theorem
0
Sn + P₁ Sn−1 + P₂ Sn-2 + ... = 0,
=
where p has been put equal to unity. If,, ... be the sums of
the powers of the equation F(x) = 0, we have in the same way
Σn + P₁Σn-1 + P₂Σn-₂ + ... = 0 ;
12-1
11-2
:
II.]
13
HAS POSITIVE COEFFICIENTS.
we may also prove
Σ₁ = (n-1) S₁,
19
2
2
Σ₂ = (n-2) S₂+S₁²,
19
Σ3 = (n − 4) S¸+3S₁S₂,
Σ₁ = (n − 8) S₁ +4§¸§¸+38¸³‚
4
Σ= (n−16) S₂+5§¸§¸+10§¸§¸;
and the general relation can be found without difficulty.
In this way we find
P₁ = (n − 1)P₁,
1
P₂
(n − 1) (n − 2)
2
1.2
1
p₁² + (n − 2) P₂,
P¸
P₁₂=
1.2.3
P₁
(n − 1) (n − 2) (n − 3)
(n − 1) (n − 2) (n − 3) (n — 4)
-p¸³ + (n − 2)² θÎ₂ + (n − 4) P3,
1.2.3.4
p*+
1.2
(n − 2)² (n − 3) p¸²p₂
2
+(n−3)² P₁P3+
(n − 2) (n − 3)
1.2
P2
P₂² + (n − 8) P₁·
But the process becomes longer and longer at every stage.
We shall therefore proceed to point out some other methods of
obtaining the coefficients.
0
=
This method of proceeding has indeed been stated only because
it proves in a convenient way that when p. 1, all the coefficients
P₁, P₂, P¸... of the derived equation are integral rational func-
tions of the coefficients po, Pi... Pr
3
5. The equation f(x) = 0 being given, to calculate the coefficients
of F (x) = 0.
Put xyz and equate separately to zero the sums of the
even and odd powers of z, we have
22
24
fly)+f" (y)
12
+fiv (y)
+ ... = 0
4
+.
13
f (y) ~ +f'' (y)
= [ +
Rejecting the root z = 0, let us eliminate z. Then the roots
of the resulting equation in y are the arithmetic means of the
roots of ƒ (x) = 0.
:
i
14
THE COEFFICIENTS DEDUCED
=
[CHAP.
If, on the other hand, we eliminate y we have an equation of
an even degree to find z. This, putting 4z², is the equation
whose roots are the squares of the differences of the roots of the
given equation.
It may be thought that this elimination may prove tedious,
but it will be presently shown that only the first and last terms of
the result are really wanted. All the others may be omitted in
the process of elimination, and thus the labour will be greatly
lessened. The method is however most useful when the given
equation has several of its terms absent.
6. Example. To determine the condition that the roots of the
biquadratic
x²+px³ + qx² + rx + s = 0
should indicate a stable motion.
Applying the rule we have
F' (x) = x²+3px³ + (3p² +2q) x* + (4pq + p³) x³
—
+(2p³q + pr+q² — 4s) x² + (pq² +p²r − 4ps) x+pqr — r² — p²s.
The first four coefficients contain only positive terms, and need
not be considered. If the last three coefficients be called P₁, P½,
P., we have
p²P₁- 4P¸= (pq − 2r)² + 2p³q +p³r,
4
6
pP-4P(pq— 2r)² + p³r.
If then P is positive, all the other coefficients are positive.
5)
The necessary and sufficient conditions of stability are there-
fore that p, q, r, s should be finite and positive, and
positive or zero.
P₁ = pqr — r² — p³s
6
7. In forming the derived equation F(x) the only difficulty
is to form the last term P For when this is known the other
terms can be at once derived from it by an easy process.
mⓇ
Let a, b, c... be the roots of f(x) = 0 with their signs changed,
and let
f(x)=Рox”+P₁x¹¹ + ... + Pπ
Let A stand for the operation
d d d
+ + + ...
da'db' dc
II.]
15
FROM THE FUNDAMENTAL TERM.
Then since P=abc... we have obviously
Pn
Po
Δ
Pn Pn-1
Po Po
In the same way we have
A
=
Pn-1 — 2 Pn-2.
Po
Po
And generally
Δ
Pn-K+1
PR-
K
Po
Po
and so on up to
AP₁
= n.
Po
Let us now operate with A on any expression
$ (Po› P1 ••• Pn)›
...
which has the same number of factors in every term. Let r be
the number of factors, then & may be written
P₁
Pn
1
=
Po
d
.. Aq=p.
dp₁
-+ (n-1) P₁ dp₂
d
+.
Φι
1
= {mpo
d
d
npodp
+(n − 1)p₁
1
dp₂
+ ... + Pn-1
P.
dp
про
m
d
Let Po, P... P be the coefficients of the derived equation
F(x), and let P. I. Then since the roots of F(x) are the sums
of the roots of ƒ (x) taken two and two, it is easy to see that
Pm-1 = +4Pm
2P m-2=APm-1
ΔΡ
ΔΡ
3P =▲Pm-2'
m-3
&c. = &c.
Thus when P is known, the other terms may be calculated
without difficulty. The term P will be called the fundamental
term of the equation.
m
Example. Given in the case of a biquadratic
2
P₁=P₁P2P3 — PoP3² — P₁²P4
41
to calculate P₁·
+
1
16
[CHAP.
THE FUNDAMENTAL TERM.
2
!
Performing the operation
d
d
d
d
Ps
8
P₁ dp.
+2p₂
+2p2 dps
+ 3p₁
1
dp₂
+ 4p。 dp₁
on P, we find after division by 2,
в
=
5
2
which is the result already given.
3
49
8. It should be noticed that in the equation
ƒ(x) =Рox² +P₁x²-¹ +...+P₂ = 0,
1
n
if we regard x as a number, pop, ... P are all of equal dimensions.
It follows from the theory of dimensions, that if any subject of
operation be the sum of a number of terms of the form
pappy,...
there must be the same number of factors in every term. For
example, in every term of the expression for Pm we have
a+B+y &c. the same.
1
On the other hand, we may regard x as a quantity of one
dimension, and in this case PoP₁... P have their dimensions in-
dicated by their suffixes. We must therefore have ẞ+2y+38+...
as well as a +B+y+... the same in every term.
These two tests of the correctness of our processes will be
found convenient.
9. The whole derived equation being known when the funda-
mental term is known, it is required to find the fundamental term.
[First Method.]
If we write - x for x in any equation, we have a second equation
whose roots are equal and opposite to those of the first equation.
If we eliminate x between these two, we shall get a result which
must be zero when the two equations have a common root. The
eliminant must therefore contain as a factor the product of the
sums of the roots of the given equation taken two and two.
m
It will afterwards be shown that the last term P. of the
derived equation (when p, is put equal to unity) always contains
the term
P1 P2 P3...
with a coefficient which is positive and equal to unity.
II.]
17
THE SECOND METHOD.
Hence we have this rule, to find P, eliminate a between
m
x² + ޸沬² +޸支 + ... = 0)
n-1
+P3
+ ...
= 0),
and divide the result by the coefficient of P.P₂P...·P-1'
It is obvious that the result may be written down as a
determinant. On trial, however, it will be found more convenient
to make the elimination by the method of eliminating the highest
and lowest terms than to expand the determinant.
10. Given the fundamental term of the equation derived from
n-1
1-2
ƒ(x) =Рox¹¹ +P₁x² + ... + P₂-10......
to find the fundamental term of the equation derived from
n
n-1
Рox² +P₁x-¹ + ... + Pn-1x+Pn=0
[Second Method.]
.(1),
(2).
Let Qn be the product of the sums, two and two, of the roots
of the equation (2) taken with their signs changed, so that Q, is
the same as the fundamental term of the derived equation and
differs from P only in having a suffix more convenient for our
present purpose.
m
Let Q, be expanded in a series of powers of p: thus
2
Qn = Po + &₁Pn +$₂Pn² +
Pœ P1›
where ₁, ₁₂, &c., are all functions of po p₁, &c., which functions
41, 42,
have to be found. Let the roots of f(x)=0 with their signs
changed be a, b, c..., then
Qn-1 = (a + b) (a+c)…..
Let us introduce a new root, which, when its sign is changed,
we shall call r. Then
Q₂ = (a + b) (a+c) ... (r + a) (r + b)...
Qn-1
Po
(Porn-1 +P₂2-2 + ... + Pn_1) •
This value of Q must be the same as that given by the series
when we write
2
Po P₁+rpo P₂+rp₁, &c.,
respectively, for po› P1, P2, &c., Pn-1 P₂
of the terms independent of r we have
Pn-1+rP#-2 rPn=19 ·
Equating the coefficients
R. A.
$o = Qn_1 Pn−1.
Po
2
WorM
18
[CHAP.
THE FUNDAMENTAL TERM.
Equating the terms containing the first power of r we have
d
d
Paste + (P₁ 2, + P. &
(p
(Podp P₁ dp₂
d
+...+Pn-2 dpn-1
) Po = Qn P₂-2.
Po
Substituting for 4, we have, by a known theorem in the Differential
d
d)
Qn-1
Calculus,
d
Φι
(Podp
+ P₁ dp₂
+ ... + Pm-2
+Pm-2 dpn-1 Po
2
1
1
Equating the terms containing the second power of r we have
2
1
2
Po + 2 pop₁
dp,
+(po
+(P₁ dp₁
(Pod
1
2
d
1
+ P₁dp₂
+P₂P³n-1 = Qn-1
Thus we have
and so on.
Qn = Qn-1
Pn-1
Po
-
d
2
d2
dp,dp₂
+...}$
+ ...) $₂ · Pm-s
Pn-3
Po
•
d
d
Qn-1 + &c.
Pn Po
(Po
1
dp₁
+Pidp2
+ ... + Pn-2
+Pm-2 dpm-1 Po
Pn
n
2
If we examine this process, we see that when Q,_, is known,
we may at once write down the terms independent of p, and the
coefficient of Pn•
The process to find the coefficient of p is
longer, but it may be much shortened by the consideration that
when p₁ = 1 the result must be an integral function of the coeffi-
Po ==
cients. We may therefore omit all terms as soon as they make
their appearance, which do not contain the factor p³n-1'
for we
know that such terms must disappear from the result.
This method is not so convenient as the one which will be
presently given to find the coefficients of the higher powers of P.
But it is useful as showing that Q contains the term
n
P₁P₂· · ·Pn-1
Po
n-1
with a positive integral coefficient equal to unity. This will be
clear from the consideration that the term independent of P₁ in
Pn
Q₁ is obtained from Q₁, by multiplying by P-1. If therefore Q
n
Po
contains PPP-2, Q, must contain the term
Po
•
P₁P2 P-1 No
Po
n-1
other term can be formed which is equal to this with an opposite
sign, for the terms which enter by the other processes to be per-
Macu
II.]
19
THE THIRD METHOD.
formed on Q all contain p, as a factor. Now Q₂
P₁
; therefore
Po
Q, contains the term PP, Q. contains PiPaps, and so on.
Po
2
P2 P3
Po
n-1
n
It has been shown that every term in pQ, has the same
number of factors (Art. 8). It follows from this reasoning that
this number of factors is n - 1.
11. To find the fundamental term of the derived equation by
means of a differential equation.
[Third Method.]
The fundamental term required is a factor of the eliminant of
n
P₁·x² + P₂x²-²+...'=
-1
n-3
+P₂x²-³ + ... = of
Let x²=y, then we have
n
n
1
2
2
2
Poy²+P₂Y +P₁Y² + ...
n
2
P₁y +P₂Y´
n-1
2
2-3
11-2
+...
+..
n-3
2
+..
2
Poy+P₂y
+ P₂Y ² +
n-1
P₁y* +P¸y² +
n even,
n odd.
If we write y+dy for y the result of the elimination must be
the same.
dp
dy
3
Hence if we make
dp s
5
In
2
−
1) p₁, &c.
-(-2),
&c.
n even,
dp2
n
dy
2 Po,
dy
-(-1),
dp₁
dy
n - 1
Po
dy
dp₂ = "=1p
1.
2
P
dy
dy 2
dp¸_n - 3
2
P₂, &C.
n odd,
dp
2
d="p
dy 2
3
n
n - 3
dp._ n
Pupp, &c.
and if E be the eliminant, we have
dE
= 0.
dy
It follows that whether n be even or odd, E must satisfy the
equation
2-2
•
1
20
[CHAP.
THE FUNDAMENTAL TERM.
Pn-² dpn
dE
dE
+ Pn-³ JP n-1
+2
2 (Par
dE
dE
+ Pn5 dp.
= 0.
n-3
dE
dE
+ Pn-1 dpn-5
dpn-s
+ 2 Pn dpn-2
+3 Pne JPns
+
We may make the elimination by multiplying the two equa-
tions by y, y²..., until we have as many equations as we have
powers to eliminate. If in the determinant thus formed, we
multiply out the terms in the diagonal joining the right-hand top
corner to the left bottom corner, we get when n is even p
n-1 n-1
2
n
2-1 +1
2
Pi
2
p. Now Q, must contain n-1 factors
and when n is odd p₁² Po
n-1
2
and be of the n
2
th degree. Hence when n is even E=cp,Q,
and when n is odd EcQn, where c is some constant.
d
Now does not occur in the above differential equation.
dp₁
1
Hence treating p₁ as a constant, we see that Q must satisfy the
differential equation
1
n
d Qn
d Qa) + 2 P₂ + dpm 2
(PR-2
d Qn + P n
+ Pn-s dp
Pr-2 dpn
n-1
n
(P₂
n-6
dpn-2
d Qn
+ Pn-5 dpn-3
d Qn + Pn-1 dpn-s
dQn
Pn²
+3 Pnε dpns
n-1
n
+ &c. = 0.
12. We may show that p¹Q, is a symmetrical function of the
coefficients P., P,..., P, and the same coefficients read backwards.
Let a, b, c... be the roots of f(x) = 0 with their signs changed,
then Q₁ = (a+b) (a + c)....
If now we read the coefficients in the opposite order, the roots
of the equation thus formed will, when their signs are changed, be
1 1 If be the fundamental term of the equation derived
n
from this, we have
1
Q' n =
+
+ 1) ...
Since
Pn
Po
Pr – abc... we see that
n-1
~~¹ Q₁ = Pn*~¹ Q'n•
II.]
21
THE THIRD METHOD.
n-1
We may therefore infer that p."Q, also satisfies the differential
equation
dE
P2 dpo
13.
་
dE
+ P³ dp
3
+2(
dE
dE
P₁
4 dp₂
+Ps dp
5
+3(1
dE
dE
Po dp
+ P₁ dps
+ &c. = 0.
We may use either of these differential equations to find
Qn when Qn-1 is given.
Let the first differential equation be represented by
and let
where A, A,
19
▾ Qn=0,
Qn = A + A₁Pn+ A₂Pn² +
0
are functions of P., P₁, &c. The value of A¸ has
been proved in Art. 10 to be
A = Qn_1 Pn−1
-1 Po
To find the other coefficients of the powers of p, substitute
this value of Q in the differential equation; we have
n
0 =▼A+P„▼¸+Pn−2ª¸
0
1
1
+Pn² ▼A₂+2Pn−2PnA₂
+ &c.
Equating the several powers of p, to zero, we find
A₁ =
1
VA。,
Pn-2
1
24,
VA₁,
Pn-2
1
3A3
VA
Pn-s
&c. = &c.
Thus by one regular and easy process each term may be
derived from the other.
In performing this process we may omit every term in the
subject of operation which does not contain
d
For
Pn-2°
Pn-z can be
introduced only by performing and since p₁ is absent from the
dpn
,
coefficients, this operation yields nothing.
་
22
THE FUNDAMENTAL TERM.
[CHAP. II.
In this way we find
PoQ₂ = P₁₂
2
po¸² Q3=P₁P₂-PoP3,
Po³ Q₁ = P₁ P₂P3-P.P² - Pi² P₁›
PoPs 1
2
4
4
- Ps (— PoP 2P 3+ P₁₂² — 2p。P、P)
5
Ps
2
+1.2 (-2p³).
5
To illustrate this process, consider how Q, is obtained from Q..
The first line is formed by multiplying the line above by P., this
is A. To find the coefficient of -p, we operate with
d
d
5
d
d
(P. 2 + P. 1) + 2 (p. 2 + P. 1)
dp
2
dp
4
1
dp3
2
on such of the terms in the line above as contain p and then
divide by på. Performing the same operation on the coefficient of
PÅ
2
P5
(-p) we obviously obtain the coefficient of 1.2°
In M. Serret's Cours d'Algèbre Supérieure, Note III., there will
be found a method of forming the last term of the equation to the
squares of the differences, which suggested the method used in
Art. 13, of substituting in a differential equation, if only a differ-
ential equation could be found. [See also Dr Salmon's Higher
Algebra, Arts. 60, 64, and 72.]
CHAPTER III.
Statement of the theory by which the conditions of stability of a dynamical
system with n co-ordinates are made to depend on 2º conditions.
Arts. 1-4.
A rule by which these conditions may be derived one from another, to-
gether with certain other true but not independent conditions. Arts.
5-8.
A rule by which, when the coefficients of the dynamical equation are letters,
the 2n conditions of stability may be inferred, one from another,
without writing down any other conditions. Arts. 9, 10.
A method by which certain extraneous factors may be discovered and
omitted. Arts. 11, 12.
Consideration of the reserved case in which the dynamical equation has
equal and opposite roots. Arts. 13-18.
A few geometrical illustrations not necessary to the argument. Arts.
19-25.
Application to a Dynamical Problem. Art. 26.
1. It has been shown in the first Chapter that the stability of
a dynamical system with n co-ordinates oscillating about a state of
steady motion depends on the nature of the roots of a certain
equation of the 2nth degree which we may call
f (z)=0.
The system is stable if the real roots and the real parts of the
imaginary roots are all negative. Now Cauchy has given the fol-
lowing theorem of which we shall make some use.
Let z=x+y√1 be any root, and let us regard x and y as
co-ordinates of a point referred to rectangular axes. Substitute
for z and let
f(z) = P+Q√−1.
Let any point whose co-ordinates are such that P and Q both
vanish be called a radical point. Describe any contour, and let
24
[CHAP.
CAUCHY'S THEOREM.
a point move round this contour in the positive direction and
P
Q
notice how often passes through the value zero and changes its
sign. Suppose it changes a times from + to and 6 times from
to +. Then Cauchy asserts that the number of radical points
within the contour is (a–B). It is however necessary that no
radical point should lie on the contour.
2. Let us choose as our contour the infinite semicircle which
bounds space on the positive side of the axis of y. Let us first
travel from y=- ∞ to y=+∞ along the circumference.
If
2-1
ƒ (z) = P¸²² + P₁~~~² + ... + P₂
we have changing to polar co-ordinates
Hence
ƒ (z) = P¸r* (cos no + sin no √− 1) + ...
Pi
2-1
P=p¸r² cos n0 + p¸r”-¹ cos (n − 1) 0 +
Q = p¸r" sin no + p₁?”-1 sin (n − 1) 0 +
r
In the limit, since r is infinite,
P
Q
P
cot no ;
Q
vanishes when n0 = (2x+1), i.e.
1π
0 = ±
n 2
3 п
+1
n 2
+1
n 2
5 π
(A);
P.
Q
is infinite when no = 2K
0=0, ±
SIN NY
7
i.e.
2 π
4 π
6 π
n 2
+
±
n 2
n 2
(B).
The values of in series (B) it will be noticed separate those
in series (A).
When is small and very little greater than zero,
P.
Q
is posi-
tive, and therefore changes sign from+to at every one of the
values of in series (A). If n be even there will be n changes
✔
of sign. If n be odd there will be n-1 changes excluding
π
=+, in this case is positive when is a little less than
0 = ±
2
P.
Q
and negative when O is a little greater than
π
2
π
2'
J
!
il
?
III.]
CAUCHY'S THEOREM.
25
Let us now travel along the axis of y still in the positive di-
rection, viz. from y=+∞ to y= co. Since x=0 it will be
more convenient to use Cartesian co-ordinates, we have, since
and
and
n
n-1
ƒ(2) = Po≈"+P₁≈"¹ + ... + Pn -1% + P₂,
z = y √ — 1,
2
P=Pn-Pn_₂y²+ Pn-4Y* — ...
Q = y (Pn_1 — Pn_3 Y² + ...),
P _P„− P„_„Y² +Pn_4Y*
2
Q Y (Pn-1— Pn-3y² + …..)
The condition that there should be no radical point within the
contour is that this expression should change sign through zero
from to + as often as it before changed sign from + to on
travelling round the semicircle. If n be even the numerator has
one more term than the denominator, and when p。 and P₁ have
the same sign, begins when Y is very great by being negative.
P
Q
In order that it should change sign through zero n times, it is
necessary and sufficient that both the equations
Pn - Pn-2y²+Pm-4Y*
= 0,
Pn-1Y — Pn-3y³ +Pn_5Y³ —
0,
should have their roots real, and that the roots of the latter should
separate the roots of the former.
If n be odd, the numerator and denominator have the same.
P
number of terms, and when p, and p, have the same sign, begins
Q
Po
when y is very great by being positive. In order that it should
change sign through zero from
to + n 1 times, it is necessary
and sufficient that the same two equations as before should
have their roots real, and that the roots of the former should
separate the roots of the latter.
In order then to express the necessary and sufficient conditions,
that f(z) = 0 may have no radical point on the positive side of
the axis of y, put z=y√-1 and equate to zero separately the
real and imaginary parts. Of the two equations thus formed, the
roots of the one of lower dimensions must separate the roots of the
other. It is also necessary that the coefficients of the two highest
powers of z in f (z) should have the same sign.
3. It has been stated that p, and p, the coefficients of the two
highest powers in ƒ(z) must have the same sign. It is easy to see
་
.
26
EXTENSION OF STURM'S THEOREM.
P
Q
[CHAP.
that, if they had opposite signs, would change sign through zero
2n times as we travel round the contour. All the radical points
of the equation would then lie on the positive instead of the
negative side of the axis of y.
It has also been assumed that no radical point lies on the
contour. It has therefore been assumed that ƒ (z) = 0 has no root
of the form z = y√-1. It will be more convenient to consider
this exception a little further on.
4. It is required to express in an analytical form the conditions
that the roots of an equation f (x) = 0 may be all real, and may
separate the roots of another equation f (x) = 0 of one degree higher
dimensions.
To effect this, let us use Sturm's theorem reversed. Perform
the process of finding the greatest common measure of f (x) and
f(x), changing the sign of each remainder as it is obtained. Let
the series of modified remainders thus obtained be f(x), f(x),
&c. Then it may be shown that when any one of these functions
vanishes, the two on each side have opposite signs. It is also clear
that no two successive functions can vanish unless ƒ, (x) and ƒ₂ (x)
have a common factor. This exception will be considered pre-
sently.
Hence in passing from x=- to +∞ no variation of sign
can be lost except when f(x) vanishes. If a variation is lost it is
regained when x has the next greatest value which makes f, (x)
vanish unless f (x) = 0 has a root between these two successive
roots of ƒ₁ (x) = 0. Hence this rule:-
·
The roots of the equations f(x) = 0, f¸(x) = 0, will be all real
and the roots of the latter will separate those of the former, if in
the series
f(x), f¸(x), f。(x)…….
as many variations of sign are lost in passing from x=-∞o to
x=+∞ as there are units in the degree of the equation f₁ (x) = 0.
We have supposed the variations of sign to be lost instead of
gained in passing from x=-∞ to +∞.
∞ to +∞. That this may be the
case the signs of the highest powers of f(x) and f(x) must be
the same.
These functions are alternately of an even and odd degree, the
condition that the whole number of variations of sign may be lost
in passing from x = ∞ to +∞ may be more conveniently ex-
pressed thus:-The coefficients of the highest powers of x in the
series
f(x), f(x), f¸(x) ...
must all have the same sign.
$
III.]
27
CONDITIONS OF STABILITY.
5. The process of finding the greatest common measure of
two algebraic expressions is usually rather long. We may in our
case shorten it materially by omitting the quotients and perform-
ing the division in the following manner. Let
-2
ƒ₁ (x) =Р¸x” — р²x²-² + Þ²x²-
n-1
ƒ₂ (x) = P₁x¹¹ —
n-5
then, since p₁ is positive, it easily follows by division that
where
2-2
n-4
ƒ, (x) = Ax”—² — A'œˆ¯¹ + A″x”— — …..
A = P₁P₂— PoP3›
2
A' = P₁P₁- PoPs'
&c. = &c.,
4
so that by remembering this simple cross-multiplication we may
write down the value of f (x) without any other process than what
may be performed by simple inspection. In the same way f(x),
&c. may all be written down.
6. Ex. 1. Express the conditions that the real roots and
real parts of the imaginary roots of the cubic
may be all negative.
3
í³ + px² + qx+r=0
ƒ₁ (x) = x³-qx,
ƒ₂ (x) = px² — r,
ƒ¸ (x) = (pq − r) x,
3
f(x) = (pq-r) r.
The necessary conditions are that
must all be positive.
P, pq —r and r
Ex. 2. Express the corresponding conditions for the bi-
quadratic
x²+px³+qx²+rx + s = 0,
f(x) = x*
ƒ₂ (x) = px³
ƒ, (x) = (pq − r) x²
·
ƒ₁ (x) = {(pq − r) r — p²s} x,
f(x)=(pq-r) r — p's} ps.
- qx² + 8,.
— rx,
-ps,
}
28
[CHAP.
CONDITIONS OF STABILITY
The conditions are that
P, pq − r, (pq − r) r — p³s and s
must be all positive.
These are evidently equivalent to the five conditions that
p, q, r, s, (pq − r) r — p²s,
should be all positive.
In both these examples all the numerical work has been
exhibited.
X
P₁
should
7. Since the coefficients of the highest powers of x in f(x) and
and
f(x) are p, and p, we see that the condition that p.
have the same sign is included in the general statement that all
the coefficients of the highest powers should have the same sign.
If the function f(x) be of n dimensions we thus obtain n necessary
and sufficient conditions.
On examining these conditions in the cases of the cubic and
biquadratic it will be seen that they cannot be satisfied if any one
of the coefficients of the given equation should be negative.
8. Although the theorem in its present form gives n conditions
as the proper number for an equation of the nth degree, yet it is.
important to notice that it gives other conditions also which are
true and may be useful. It has been shown in the second Chapter
that all the coefficients of the equation f(x) = 0 must be positive,
hence the roots of f(x)=0 must all be positive. It may be shown
also, that the roots of each of the functions f(x), ƒ₂(x), &c. are
separated by the roots of the function next below it in order.
Hence the roots of all these functions must be positive, and there-
fore in every one of the functions the coefficients of all the
must be alternately positive and negative and not one
vanish. If however f(x) and f(x) have one or more common
factors some of the functions f(x), f(x), &c. will wholly vanish.
2
powers
can
9. When the degree of the equation is very considerable
there is some labour in the application of the rule given in Art. 5.
The objection is that we only want the terms in the first column,
and to obtain these we have to write down all the other columns.
We shall now investigate a method of obtaining each term in the
first column from the one above it without the necessity of writing
down any expression except the one required.
We notice that each function is obtained from the one above it
by the same process. Now
n-2
ƒ₁(x) = Þ¸x” — P₂x¹−² + Þ¸xˆ¯¹ — …..
X =
x”¯
f(x)=(P₁P₂-P3) x˜¯² — (P‚P、—PP3)x”−4+…..
2
•
III.]
DERIVED ONE FROM ANOTHER.
29
The first and second lines will be changed into the second and
third lines by writing for
Por
the values
P₁
P₁
P2
P3
&c.
P₁P2-POP3' Py
4
P₁P₁- PP, &c.
If then in any term of any function we make these changes, we
obtain the corresponding term of the function next in order.
10. Example. Express the conditions of stability for the
quintic
We have
ƒ₁(x) =Р¸x³ + ...
ƒ₂(x) = P₁x² + ...
ƒg(x) = (P₁P₂ — PoP3) x³,
ƒ₁(x) = {(P₁P₂— PoP3)P 3 — P₁ (P₁₁ — PoPs)} x²,
2
4
ƒ¿(x) = [{(P₁P₂— P¸P3)P¸ — P₁(P₁P₁— P¸Ps)} (P₁₁ — PPs)
2
-
4
- (P₁P₂-PoP3)2 · ps] x,
f(x) = [{(P₁P₂ — PoPs) Ps — P₁ (P₁₁— PPs)} (P₁P₁ — PoPs)
4
− (P₁₂ — PoŸ³)² · Ps] (P₁₂ — PoP3)P¸•
(P1P 2
#
11. On examining the conditions as given in the cases of a
biquadratic and quintic it will be apparent that several contain
the previous conditions as factors. Thus the analytical expressions
are rendered much longer than is necessary. It is now proposed
to investigate a method of discovering and omitting these extraneous
factors as they occur, and thus obtaining the required conditions in
their simplest forms.
Let the coefficients of the several powers of x in the functions
be when taken positively
f(x)=Po
P2
P₁
P6
Pε ...
ƒ₂(x) = Pv
P3
P59
Pr...
ƒ¸(x) = A,
A',
A",
A"...
f₁(x) = B,
B',
B",
B'"..
B'" ...
&c. &c.
1
..
30
EXTRANEOUS FACTORS
*
..
[CHAP.
Let us first find which of these terms contain p₁ as a factor.
Putting p₁ = 0 and using the rule in Art. 5, the series become
Po
P2
P₁
Pε
0,
P3,
P5,
Pr
-Pops,
-PoP5
- PoPr
-PoPe
2
- PoP3², -PoPsPs
- PoP3P
- POP3P9,
0,
0,
0,
0,
0,
0,
0,
0,
&c.
:
:
:
:
:
2
P₁
as
Hence the C's and D's all vanish and therefore contain
a factor. By the rule in Art. 5, the E's contain p², the F's con-
tain p³, the G's p₁5, the H's p, and so on.
Pi,
But since each line is formed from the preceding by a uniform.
rule, it follows that the D's and E's contain A as a factor, the
F's contain 42, the G's contain A³, the H's contain A5, and so on.
The factor A in the D's and E's has its origin in the factor p、
which occurs in the C's and D's and would not appear if that
factor had been omitted when the C's and D's were formed. The
factor A in the D's and E's in the same way gives rise to the
factor B in the E's and F's. So that if we take care each time
we perform the process described in Art. 9 to omit the common
factor P₁
whenever it occurs, all these subsequent factors will never
make their appearance.
We shall now show that if these factors are omitted, the
dimensions of the nth function f(x) will be n
n-1
2
First consider
the actual dimensions of each function before the factors are
omitted. If we examine the rule by which each function is
derived from the preceding, it will become evident that, the
dimensions of each letter being indicated by its suffix, the
dimensions of any function are equal to the sum of the two pre-
ceding +2.
In the following table the first column indicates the function.
In the second column will be found the dimensions of the leading
coefficient of that function when calculated by the rule in Art. 5.
In the third column will be found the dimensions as given by the
formula n
n
2
1
In the remaining columns are the dimensions
of the extraneous factors p,, A, B, &c. introduced into each term.
III.]
31
DISCOVERED.
f(x)
0
0
f₂ (x)
1
1
3
fs (x)
3
3
fs (x)
6
6
f(x) 11
11
10
1
f(x) 19
15
1
3
f(x) 32
21
2
3
6
f(x) 53
- 28
3
6
6
6
10
f(x) 87
36
10
5
9
12
10
15
&c. &c. &c.
&c. &c. &c.
&c.
&c.
&c.
Each term in the second column is the sum of the two terms
just above it +2. The nth term in the third column is equal to
the term just above it + (n − 1). In all the other columns each
term is the sum of the two terms just above it. The last term in
the nth row is equal to the (n-3)th term in the third column.
We wish to show that any term in the second column is equal to
the sum of all the terms in the same row to the right of that term.
It is not difficult to show from the data just given that if this be
true for any two adjacent rows, it is true for all the others, and
hence we may assume it to be always true.
It is clear that these extraneous factors may be omitted since
by the conditions already expressed they are all positive. When
omitted as they occur, the dimensions of the nth function has just
been shown to be n n-1. It is easy to see that the conditions
2
thus reduced must contain the terms
P1 PiP 2
P1P 2P 3
P1P 2P 3P 4 &c.
P1P2P3 P4 P5 P6.
(1),
Now if we take any one of these as
and operate by the rule in Art. 9, we have
-
(P₁P₂— PoP3)P3 (P₁₁— PoPs)Ps(P₁Pc6 — PoP1)Pr
2
which contains the term
P₁P2P 3P1P4 P5 P1P6 P7···
.(2).
Thus we have Ρι introduced as often as there is a factor in (1)
with an odd suffix. But it should be introduced only once. These
extra p,'s are the extraneous factors to be omitted. Each of these,
if left, would appear as the factor p,P₂-Pop, in the next condition,
and be still more complicated in the next after that.
2
3
י.
32.
[CHAP.
EXTRANEOUS FACTORS DISCOVERED.
In order then to obtain the several conditions in their simplest
form it is only necessary after performing the operation described in
Art. 5 or Art. 9 to divide by p, where x is one less than the number
of factors with odd suffixes in the condition operated on.
K
12. Example. Express the conditions of stability for the
sextic
ƒ(x)=Р¸²x²+P₁x² +Р²x² +р²x²+Р4x² + рçx + P6 = 0.
We have
ƒ₁(x) = P₁x® + ...
ƒ₂(x)=p₁x³+ ...
2
ƒ¸ (x) = (P₁P₂ — PoP3) x² +
3
· …..
2
2
2
f(x) = (P₁P₂P3 — PP3² — P₁² P₁+PP、Ps) x² + ...
2
2
ƒs (x) = { P₁P₂ P3 P4 - Po P3 P₁ — P₁²
2
4
²+2P P₁₁Ps - P₁P²² P s + P o P 2P 3P 5
P₁P²²P5+P0P₂P3P5
2
1 4
2 2
1
2
2
—Po³ P²²+P₁² P₂P¸—Р°P₁P¸P6} x² + ...
1
2
2 2
f(x) = [ P₁ P₂ P3 P4 P5 - Pop¸² P₁Ps - P₁² PÅ P₂+ 2р P₁P₁Pš² — P₁P²² P¸²
23.
+ PoP 2P 3P5
2
3
2 3
5
5
2 5
2 2
5
1 4
2
2
1
6
3 6
2
1
+PoP¸°P¸+P₁*P¿P₁P¸ − P₁³Î¸²] ≈ +&c.
3
6
1 6
ƒ, (x) = coefficient of x in ƒ x × by PË
6
3
13. In the preceding theory two reservations have been made.
1. In applying Cauchy's theorem it has been assumed that
there were no radical points on the axis of y.
2. It has been assumed that P and Q have no common
factor, so that none of the functions f₁, f, &c. vanish absolutely.
2
If any radical point lie on the axis of y, it is clear that f(z) = 0
must have a factor of the form (22+ a²)". Let f(z) = (x² + a²)" (z).
In this case when we put z=y√ −1, we have
f(z) = (a² — y²)" (P' + Q'√ − 1);
.. P= (a² — y²)ˆ P'\
Q = (a² — y²)” Q'S
Thus P and Q have a common factor, and we are warned of the
possible existence of radical points on the contour by the total
vanishing of some one of the functions f₁, f, fa, &c.
The two reserved cases may therefore be included in one. If
f(z) =0 be the equation furnished by dynamical considerations,
we form the functions f₁, f₂, f₁, &c. If all these be finite, the
question of the stability of the system has been answered. If any
one vanish absolutely, f(x) and ƒ, (x) have a common measure,
III.]
33
THE RESERVED CASES.
and we must add some further considerations. It will be con-
venient to examine separately the dynamical effect of the roots
which do not and which do enter through the greatest common
measure. Let us begin with the former.
14. Following the same notation as before, we have
n-1
ƒ (z) = P₁₂" +P₁z"−1 + ... + Pn-1+Pn,
P=±f₁(y)=Pn-Pn-23² + ......)
Q = ±ƒ₂ (y) = Pn-1Y — Pn_3Y³ +
If then ƒ (z) have two roots, viz. ±(h+k √ − 1), which are equal
and opposite, then f(y) and f, (y) must have two common roots,
viz. t
h+k√ 1
√=1
The common measure therefore of ƒ (y) and
f(y) contains all the roots of f(y√1) which are equal and
opposite. Conversely the greatest common measure of P and Q is
necessarily an even function of y, and if it be equated to zero, its
roots are necessarily equal and opposite. These roots must also
satisfy ƒ (y√ − 1) = 0.
Let this greatest common measure be
the highest power which enters into it.
(y²) = 0, and let y² be
Also let
f ( ) = h ( - ) ¢ (z),
then (2) is a function which, as has just been shown, has not got
two roots equal and opposite, and to this function we may apply
Cauchy's theorem without fear of failure.
Putting z = y√ — 1, let
$ (≈) = P' + Q' √ −1.
P'
Then we wish to express the condition that
Q
should change
sign from
N
-
and
to through zero n2r times if n be even and
+
2r 1 times if n be odd. But
ƒ (y √ − 1) = P + Q √ − 1,
ƒ (y√ − 1) = ↓ (y²) (P′ + Q' √ − 1);
..
P=4 (y²) P
Q = 4 (y²) Q'S •
P'
Thus the number of changes of sign in
Q
is exactly the same
as that of
P
Q
The factor (y²) will run through all the functions
2
ƒ₂ (y), fs (y), &c. obtained from f, (y) by a process which is equiva-
lent to that of finding the greatest common measure of ƒ (y) and
R. A.
3
34
[CHAP.
EFFECT OF THE VANISHING
f(y). The changes of sign of this factor will therefore not affect
the number of variations of sign in the series f₁, f₂, ƒ3, &c.
The last factor which is not zero is f₂+1-27 (y) if n be the dimen-
sions of f₁ (y).
Hence if we omit the considerations of the vanishing factors and
apply the same rule as before to the n+1-2r remaining factors, we
can express the condition that the proper number of changes of sign
from - to + have been lost through zero in the function o (z), i.e. that
the roots not given by the vanishing of f+1-2 are all of the character
to ensure stability.
15. Let us next consider the effect on stability of the roots
indicated by the absolute vanishing of one of the subsidiary func-
tions. This function must be of the form
n
† (y²) = ¶¸y” — ¶₂yn−2+
where n is even. The corresponding factor of ƒ (z) is
F' (z) = 9n +91-2²² +421 +.....
The roots of this equation are two and two equal with opposite
signs, it is therefore necessary for stability that no root should
have any real part. To express this condition, draw a straight
line parallel to the axis of y at an indefinitely short distance from
it, viz. x = h. Let us apply, in the same manner as before, Cauchy's
theorem to the contour formed by this straight line and the infinite
semicircle on its positive side. Putting z=h+z′, we have
12
2
F(2) = qn + 29n_zhz' + In_qz¹² + 4qn_4hz'³ + ...
The two functions are therefore, omitting the positive factor h,
P = qn~ qn-2Y³ + În-¿Y* — &c.,
Q = 2qn-2Y — 4qn-4Y³ + &c.
-
Now f₁ (y) and f₂ (y) are what P and Q become when arranged
in descending powers of y and the coefficients of their highest
powers made to have the same sign. Hence
ƒ₂ (y) = df (y)
dy
The rule described in Art. 4 will now become the same as
that usually called Sturm's theorem. We are to seek the greatest
common measure of ƒ (y) and its differential coefficient, and make
the coefficients of the highest powers of y in these two and in the
series of modified remainders all positive.
*
III.]
OF A SUBSIDIARY FUNCTION.
35
That we should have been led to Sturm's theorem in this case
is just what we might have expected. For to express the condi-
tions that the roots of
F (≈) = 9n+9n_2≈² + ¶n_12² +
...
are all of the form tr√1 is the same thing as to express the
conditions that the roots of
are all real.
In — In-23² + Ins²* — ... 0
16. There is however another mode of proceeding. Suppose
we have calculated the functions f₁, f, &c. for the general equa-
tion
ƒ (z) = p₁₂"+p₁2"-1+...
f
and find when the values of p., P₁, &c. are substituted that some
one function say f, of the series absolutely vanishes, and therefore
also all the functions which follow it. Then operate on each of
these vanishing functions with
d
Pn-1
d
+ 2pm-2
dpn
n-2 dpn
n-1
d
+ 3Pns dpm-2
+.
repeating the operation until we obtain a result which is not zero.
If we now replace these vanishing functions by these results we
may apply the rule of Art. 4, just as if these were the functions
supplied by the process of the greatest common measure. As this
process is not so convenient as that already given it is unnecessary
to consider it in detail*.
of
17. As a numerical example, let us examine whether the roots
f(x) = x²+ 2x² + 4x² + 4x³ + 6x* + 6x³ +7x² + 4x + 2 =0
satisfy the conditions of stability. In order to show the working
of the method it will be necessary to exhibit all the numerical
calculations. We have by Art. 5,
-
* The function ƒ, (x) vanishes because the equation ƒ (2)=0 has two roots equal
and opposite. If we put z=2+h, where h is as small as we please, this peculiarity
will disappear. Thus if the values of z are of the form ±(a±ẞ √√-1) the corre-
sponding values of z' are h±a±ẞ√1. These values of z will indicate stability
if a be zero and instability if a have any value positive or negative. If h be as small
as we please and positive, the values of z' will indicate stability or instability under
the same circumstances. We may therefore apply the rule of Art. 4 to the func-
tion f (z+h) instead of ƒ (z), provided we retain only the lowest powers of h which
occur. Hence all the functions f (x), ƒ₂ (x)…..ƒr-1(x) which do not vanish are
unaltered. To find what function will replace fr (x) we must increase by h all the
roots of ƒ (z)=0 when their signs have been changed. This may be effected by
performing on ƒ (2) the operation represented by A in Art. 7. The rule in the text
therefore follows from the one given in that article.
R
י
▸
·
1
3—2
36
[CHAP.
EFFECT OF THE VANISHING
ƒ₁(x) = x²-4x+6x* − 7x² + 2,
ƒ₂(x) = 2x² - 4x³ + 6׳ — 4x,
f(x)=4x-6x + 10x² - 4,
f(x)=4x-4x³ +8x,
ƒ¡ (x)=8x¹ — 8x²+16.
Here we find f(x) to be absolutely zero, accordingly by Art. 15
we replace it by the differential coefficient of f (x), this being
Sturm's rule. We have therefore
f(x) = 8 (4x³ — 2x),
ƒ, (x) = 8² (2x³ — 6),
f(x)=-8³.20.x,
f(x)=-85.120.
We see that the two last of the coefficients of the highest
powers are negative. The roots therefore do not satisfy the con-
ditions of stability.
As another example, take the equation
Here
ƒx = x² + x²+ 6x* + 5x³ + 11x² + 6x +6.
ƒ₁ (x) = x® — 6x² + 11x² – 6,
ƒ½ (x) = x* — 5x³ + 6x,
f (x) = * _5 +6,
ƒ₁ (x) = 0.
-
Replacing f (x) by the differential coefficient of f(x), we have
f(x)=4x³-10x,
ƒs (x) = 10x³ – 24x,
f(x) = 4¹x,
f(x) = 4.24.
Here all the coefficients of the highest powers are positive,
hence the roots satisfy the conditions of stability.
It is clear that when the coefficients are numerical the rule
given in Art. 5 is the most convenient, but when the coefficients
are letters, the rule in Art. 9 will be found preferable.
The process would be simplified by omitting the alternate posi-
tive and negative signs of the terms in each line.
{
1
III.]
37
OF A SUBSIDIARY FUNCTION.
18. It may be interesting to express the two subsidiary func-
tions f¸ (x) and f (x) in terms of the roots of the given equation.
Let a₁, α, α....a be the roots of the given equation f(x) = 0, so
that
n
ƒ (x) = (x − α₁) (x − α) (x — α¸)…….
n-1
= x²+P‚x”−¹+P₂x² + ...
Then it is evident that
±ƒ₁ (∞ √ − 1) = x² +P₂²x²-² +P₁x²++
= † (x+α¸) (x+α₂) ... + ½ (x − α¸) (x − α) …..
It may be shown* that
Σ
·
n-1
±ƒ₂ (x√ = 1) = p,x−¹ + P¸x²-³ + ...
2

(α₁ + α₂) (α₁ +α¸) ……. (α₁ + αn)
(α¸ — α½) (α¸ — α3) ……. (α1 — αn)
1
α₁ (x — α¸) ……. (x −ɑ„).
19. The following propositions are not necessary to the main
argument, but as they illustrate geometrically the propositions in
this chapter it has been considered proper to state them very
briefly. The demonstrations will therefore be much curtailed.
The equation being
ƒ(z) =Pos” +P₂3”¯¹ + ... + Pm-1²+Pn = 0,
we put as in Art. 1, z= x + y√—1, and thus obtain two curves,
whose equations expressed in polar co-ordinates are
P=p。r" cos ne+p¸¹ cos (n − 1) 0 + ... = 0)
1
Q=p¸r" sin n0+ p¸²-¹ sin (n − 1) 0 +... =0)
These intersect in the radical points of the equation f(z) =0.
20. If we trace these curves we find that the curve P=0
has n asymptotes whose directions are given by cos n✪ = 0, i.e.
Ө
17
N 2
3 п
5 π
n2'
n 2
* Let us assume
-3
-
2
-
P₁xn−1+Р²x²¬³+ &c. = A₁ (x − a₂)…….(x − αn) +A₂ (x — α₁)……. (x − αn) + &c.,
where A1, 4,, &c. are constants whose values have to be found. Putting x=a1,
we have
P₁ª¸”−¹+Р¸ªïñ−³+&c. =▲1 (α1− α₂)…….(α1 ~ an).
x²+P₁x-1+... (x -α) (x -α₂)... (x — an),
But since
n-1
we have by putting xa, and x= −α₁
n
n-1
a₁₂"+p₁₂"-¹+...=0,
n
n-1
a₁"—p₁a₁"-¹+...=2α₁ (α₁+α₂)... (α₂+am).
1
Subtracting the second of these results from the first we find A, to have the value
given in the text.
38
[CHAP.
GEOMETRICAL ILLUSTRATION.
These asymptotes all pass through the same point on the axis
of x, viz, x=- P₁. It is also clear that only one branch of the
1
про
·
curve can go to each end of an asymptote. Similar remarks apply
to the curve Q= 0, the directions of its asymptotes being given by
sin no = 0, i.e.
0=0,
2π
4 π
6 п
n 2
ni 2
n 2'
**
21. From these simple propositions we might, if it were
worth while, deduce that every equation must have a root. The
asymptotes of the two curves P=0, Q=0 are alternate, and no
two branches of the same curve can approach the same end of an
asymptote. By sketching a figure, it may be easily shown that
some branch of the P curve must cut some branch of the Q curve.
22. Let us next consider the intersections of the curves P=0,
Q=0.
If we transform the origin to h, k, we put x = h + §, y = k + n.
This is the same as expanding ƒ (h + k √ − 1 + § +n√−1), and
collecting into two parcels the real and imaginary terms. Let the
expansion be
A¸ + A¸ (§ + n √ − 1) + A¸ (§ + n √ − 1)² + ...
1
η
where A,4,... are of the form
c (cos a + sin a √-1).
If we put § + n√− 1 = r (cos 0 + sin ✪ √—I), we have
P=c¸cos α +c‚r cos (0 + a₁) + cr²³ cos (20 + a½) +
Q=c, sin x + c₁r sin (0 +α¸) + c¸r² sin (20+a₂) +
If the point (h, k) be a point of intersection c₁ = 0. If the inter-
section be a double point on either curve, the terms of the first
degree must be zero, therefore c=0, and the origin is therefore
a double point on the other curve also.
It is not difficult to show that if the intersection be a multiple
point of any degree of multiplicity on one curve, it is a point of
the same degree of multiplicity on the other curve.
The tangents
to these branches all make equal angles with each other, the
tangents to the P and Q curves being alternate as we travel round
the point of intersection. If the intersection be not a multiple
point on either curve, the branches cut at right angles.
Let us travel round a point of intersection along the circum-
ference of a small circle whose centre is the point of intersection
in the direction in which is measured. Then it may be shown
that as we pass from a P curve to a Q curve, P and Q have
.
III.]
39
GEOMETRICAL ILLUSTRATION.
N
་
1
opposite signs, and as we pass from a Q curve to a P curve the same
sign. This is in fact merely Cauchy's rule for the changes of sign
of
P
Q
when we travel round a radical point.
23. Let us express the condition that there is no radical point
on the positive side of the axis of y. This is the geometrical
illustration of Art. 2.
2n
Draw a circle of infinite radius, and let it cut the asymptotes
of the P curve in P₁, P₂, P¸,...P and the asymptotes of the
Q curve in Q1, Q2... Qn These points alternate with each other.
Taking only those points which lie on the positive side of the
axis of y, the P and Q curves may be said to begin at these points
and are to intersect each other only on the negative side of the
axis of y.
The branches of the two curves must therefore remain
alternate with each other throughout the space on the positive
side of the axis of y. Their points of intersection with the axis
of y must be also alternate, and hence if we put x=0, in the
equations P=0, Q=0, and regard them as equations to find y
the roots of each must separate the roots of the other.
Conversely, we may show that if the intersections of the two
branches are alternate on the axis of y, they cannot have intersected
on that side of the axis of y on which the common intersection of
all the asymptotes is not. This is the result arrived at in Art. 2.
24. The following diagrams exhibit the forms of the curve
P=0, Q=0 for a biquadratic. The dotted lines represent the
asymptotes.
མ་་
Q3
1
1
་

Q
6
PA~-
P
vë qit me the mat
Q
Pq
Qy
A biquadratic with four imaginary roots.
P₁
вых
40
[CHAP.
GEOMETRICAL ILLUSTRATION.

PA
Q5
P
QUA
ad
R
Q₂
Qs
P
PB
Q1
Q &
P
A biquadratic with equal imaginary roots.
25. [If z=x+y=1, we have
ƒ(z) = P+ Q√ −1.
Differentiating with respect, firstly to x and secondly to y, we
ƒ (2) = √x + di
dPdQ=1
dx dx
find
dQ dP
f'(x) =
dy dy
dP dQ d Q
√-1
dP
and
; so that both
dy
It easily follows that de
dx dy dx
the functions P and Q satisfy the equation
d² V d² V
+
2
dy² dx²
=
= 0.
2
+
The equation f(z) = 0 gave us two curves which we have called
P=0 and Q=0. In the same way the derived equation ƒ'(z) = 0
will give us two other curves, which we may represent by P
and Q'=0. These we may call the derived P and Q curves.
0
If as in Art. 22 we transform the origin to the point (h, k) we
have
P' = c₁ cos α, +2c₂r cos (+α₂) + 3c,² cos (20+α) + ...
1
Q' =c₁ sin à +2c,r sin (0 + α₂) + 3c¸‚² sin (20 + α¸) + …..
If the origin be at a point of intersection of the curves P=0,
P'=0 which is not a double point on the first of these curves, we
III.]
41
GEOMETRICAL ILLUSTRATION.
have c, cos α= 0 and cos a₁ ==
0 and cos a₁ = 0. Hence the tangent to the curve
P=0 at the point of intersection is parallel to the axis of x.
Conversely if we move the origin to any point on the curve P=0
at which the tangent is parallel to the axis of x, we find that the
curve P=0 passes through the origin. Hence the derived P
curve passes through all those points on the curve P = 0 at which
the tangent is parallel to the axis of x, and all those points on the
curve Q=0 at which the tangent is parallel to the axis of y. In
the same way the derived Q curve passes through those points on
the curve Q = 0 at which the tangent is parallel to the axis of x,
and those points on the curve P=0 at which the tangent is
parallel to the axis of y.
Ο
1.
If c, cos α= 0, and c₁ =0, the origin is a double point on the
curve P=0 and the origin also lies on both the curves P' =0,
Q'=0. So that both the derived curves cut the curves P=0 and
Q=0 in their double points. In other words, these double points
are radical points for the derived equation f'(z) = 0. In the same
way, any multiple point on either curve is a multiple point of one
degree less multiplicity on both the derived curves.
If a finite straight line AB be drawn parallel to the axis of x
joining two points A, B on the same or on different branches of
the curve P=0, this finite straight line must cut one or more
branches of the derived P curve. For it is clear that if P vanishes
dP
dx
at A and B, P' which is equal to cannot keep one sign between
A and B, and must therefore vanish somewhere between A and B.
If A and B be adjacent points, i.e. if there be no other points
between A and B belonging to the curve P = 0, then the straight
line AB must cut an odd number of branches of the derived P
curve. In the same way if a straight line CD be drawn parallel
to the axis of y joining two adjacent points on a derived curve,
this straight line must also cut an odd number of branches of a
derived P curve between C and D.
By considering a tangent as the limit of secant, it again follows
that if a tangent be drawn to the curve P=0 parallel to the axis
of x, the derived P curve must pass through the point of contact.
Let R be a radical point on the derived curve f(z) = 0, and let
it not be a double point on either of the curves P=0, Q=0. Let
a straight line be drawn from R in any direction cutting the
branches of either of the curves P=0, Q=0 in the points A,, A2,
Then we may show that
&c.
1
RA, +
1
1
RA,+RA¸
2
+ &c. = 0,
so that the polar line of R with regard to either of the curves
P=0, Q=0 is at infinity.
42
[CHAP.
DYNAMICAL ILLUSTRATION.
The positions of the radical points of the derived equation
ƒ (z) = 0 relatively to any branch or branches of the curves P=0,
Q=0 may be found by the use of Cauchy's theorem. If the
point spoken of in Art. I travel along a branch of the curve P=0,
P' dy
If it travel along a branch of the
Q dx
it is easy to see that
P'
curve Q=0 we have
dx
Q' dy
If then any contour be partly.
bounded by branches of these curves, the simplest inspection of
the points at which the tangents are parallel to the axis will
determine the changes of sign of as it passes through zero.
P'
Q
If another part of the contour be an arc of a circle of infinite radius
whose centre is the origin, the changes of sign through zero will
be from + to — and their number will be indicated by the number
of asymptotes of the derived P curve which cut the arc.]
26. The use of Watt's Governor in the steam engine is too
well known to need description. It has however, as commonly
used, a great defect. It is sometimes of importance that the
engine should continue to work at the same rate notwithstanding
great changes in the resistances. Suppose the load suddenly
diminished, the engine works quicker, the balls diverging cut off
the steam, and the engine, after a time, again works uniformly,
but at a different rate from before. The balls as they open out
or close in are usually made to describe circles. Let them now be
constrained to describe some other curve which we may afterwards
choose so as to correct the above defect. If this curve be a parabola
and the balls be treated as particles, it is clear from very elemen-
tary considerations that these will be in relative equilibrium only
when the engine works at a given rate. This principle is due
to Huyghens, see Astronomical Notices, December, 1875. It is
now proposed to determine the condition of stable oscillation about
a state of steady motion.
Two equal rods AB, AB are attached at A by hinges to a
small ring which can slide smoothly along a vertical axis. The
ring is attached by a rod to the valve and can thus govern the
amount of steam admitted. Two equal balls are attached at B
and B', and the centre of gravity G of the rod AB and the ball B
is constrained to describe some curve. To represent the inertia of
the engine we shall suppose a horizontal fly-wheel attached to the
vertical axis whose moment of inertia about the axis is I. Let the
excess of the action of the steam over the resistance of the load
be represented by some couple whose moment about the vertical
axis is f(0), where is the inclination of the rod AG to the
vertical, and f is a function which depends on the construction of
III.]
43
WATT'S GOVERNOR.
the engine. Since the steam is cut off when the balls open out,
it is clear that df (0)
do
is negative.
There may be also some resistances which vary with the
velocity. Let these be represented by a couple B
аф
attending to
retard the motion round the vertical axis and a couple round A in
do
dt'
the plane BAB equal to mC where is the angle the
vertical plane BAB' makes with a fixed vertical plane.
Let m be the mass of either sphere and rod; k the radius of
gyration about an axis through G perpendicular to the rod, and k
that about the rod, let 7= AG. Then the equation of angular
momentum gives
d
I
аф
dt (1 dp
dt
2
2
12
– pḍp
+ 2m (le² + 1ª sin³0 + k³ cosº 0) de} = ƒ (0) — B
dt
dt
аф
=n, and let the
dt
do
oscillations be represented by = a +∞,
n+y. We have then
dt
Let the steady motion be given by 0 = a,
ƒ(0) =ƒ(a) +ƒ' (a) x,
f(a) = Bn.
The equation then reduces to
dy
dx
{I + 2m (k² + l³ sin² a + k²² cos² a)}
12
dt
+ 2m (1² + k² − k’²) sin 2an
dt
=ƒ' (a) x — By.
This equation may be briefly written
2
A
dt
dy + By + E
dx
+ Fx =0.
dt
Let z be the altitude of G above some fixed horizontal plane.
Then if T be the semi vis viva
12
12
2T=Ip”² + 2m {(k² + l²) sin²0+k'² cos²0} p²² + 2mk²0'²
cos"0}
2
dz
+ 2m {(de)" + 5 cos' 0} 0".
If U be the force function, omitting the couple of resistance,
we have U=-2mgz. The virtual moment of the couples of
resistance being 2m C0'80, the Lagrangian equation of motion
becomes
༣
d dT dT dU
2mC0.
dt de do de
:
·
44
[СНАР. III.
DYNAMICAL ILLUSTRATION.
ī
Substituting for T and U we have when the motion is steady
n²
dz
(+kk sin cos 0 =
9
do'
na
2g
—
(1² + k² — k²²) sin³ 0 = z.
Since 7 sin is the distance of G from the vertical axis, we see
that the path of G must be a parabola. The semi latus rectum is
2
gr
n² (l² + k² − k'²) ›
which, we notice, is independent of the radius of the balls. The
length of this latus rectum must of course be adjusted to suit the
particular rate at which the engine is intended to work.
When the system is oscillating about the state of steady motion
we have, putting
2
a² = 1² + ¹¹²½ (1² + k² — k´²)² sin² a,
n¹
g²
and rejecting the squares of x and y,
(k² + a² cos² a)
d2x dx
+C
dt² dt
· (1² + k² — k'²) sin 2any = 0.
The term x, it will be noticed, has disappeared from the
equation. This equation may be briefly written in the form
d²x
H
dt
хобс
dt
Ly = 0.
Eliminating y from the equations of motion we have
d³x
dt³
AH +(AC+BH)
d'x
dt²
dx
+(BC+EL) + FLx=0.
dt
The coefficients are all positive, the necessary and sufficient
condition of stability is therefore
(AC+BH)(BC+ EL)> AFHL.
In some clocks to which Watt's Governor is applied, there is a
special arrangement which causes C to be much greater than B.
See the Astronomical Notes, XI., 1851, and the Memoirs of the
Astronomical Society, Vol. XX. Neglecting therefore B, we have
CE> FH.
:. 2Cmn (ľ² + k²³ – k²) sin 21 > F(k² + a³ cos³ a).
་י
CHAPTER IV.
Formation of the equations of steady motion and of small oscillation
where Lagrange's method may be used. Arts. 1-5.
The equations being all linear the conditions of stability are expressed
by the character of the roots of a determinantal equation of an even order.
Art. 6.
Mode of expanding the determinant. Art. 7.
A method of finding the proper co-ordinates to make the coefficients of
the Lagrangian function constant. Arts. 8-10.
How the Harmonic oscillations about steady motion differ from those
about a position of equilibrium. The forces which cause the difference are
of the nature of centrifugal forces produced by an imaginary rotation
about a fixed straight line. Arts. 11-19.
Reduction of the fundamental determinant to one of fewer rows by the
elimination of all co-ordinates which do not appear except as differential
coefficients in the Lagrangian function; with an example. Arts. 20-23.
Formation of the equations of Motion and of the determinant when
the geometrical equations contain differential coefficients, so that Lagrange's
method cannot be used; with an example. Arts. 24—27.
1. Let the system be referred to any co-ordinates §, n, §, &c.
The general expression for the kinetic energy is
T = 1 P§¹² + Q&'n' + ...
where P, Q, &c. are known functions of §, n, s, &c. and accents
have their usual meaning. Let us suppose the system to have
some motion represented by
=ƒ(t), n= F(t), &c.
and when disturbed, we wish to find the oscillations about this
motion. To effect this, we put
§=ƒ (t)+0, n = F(t) +¢, &c.
1
་
46
[СНАР.
THE DETERMINANTAL EQUATION.
་ན
}
where 0, 4, &c. are all small quantities. Substituting and expand-
ing T in powers of 0, 4, &c., we find
T= T¸+ A¸0 + A₂O + …..
+B‚0′ +B„Ó' + ...
1
+† (A„ز +2A₁₂0¢ + …..)
11
12
+ ½ (B„ع² + 2B¿¿Ø'&' + ...)
12
+С„ØØ′ +С₁ØÞ' + С₂0'¤ + ...
+ &c.
12
21
In the same way we may make an expansion for the Potential
Energy of the forces, viz.
V=E+E¸0+E₂$+
+1 (E„0² + 2E„„,00 + …..),
11
12
when these two functions are given the whole dynamical system
and the forces are known; and we may form the equations of
motion by Lagrange's method.
2. We shall here however limit the question by supposing
that the motion about which the system is oscillating is what has
been called in Chap. I. steady. The analytical peculiarity of such
a motion is that when referred to proper co-ordinates, every coeffi-
cient in each of these two series is constant, i.e. independent of t.
As already explained the physical peculiarities are that the vis
viva is constant throughout the steady motion and the same oscil-
lations follow from the same disturbance at whatever instant it
may be applied to the motion. A method of discovering the
proper co-ordinates, if unknown, will be given a little further on.
3. In order to form the equations of motion we must now
substitute in Lagrange's equations
i
d dT
dt de
dr dv
dT
dTdV
+
do do
0,
&c. = 0,
rejecting all the squares of small quantities. The steady motion
being given by 0, 0, &c. all zero, each of these must be satisfied
when we omit the terms containing 0, 4, &c. We thus obtain the
equations of steady motion, viz.
A₁ = E₁,
A₂ = E₂,
2
21
&c. = &c.
IV.]
47
THE DETERMINANTAL EQUATION.
These equations may be simply formed in any case by the
following rule. Putting LT-V, so that L is the difference
between the kinetic and potential energies, expand L in powers of
the co-ordinates 0, 4, &c. regarding e', ', &c. as zero. The required
Ꮎ .
relations are obtained by equating the coefficients of the first
powers to zero. This rule may be also expressed thus. Let L be
the general expression for the excess of the kinetic energy over the
potential energy of a dynamical system in terms of its n co-ordinates
E, n, &c. Let this system be moving in steady motion with constant
dę dn
&c.* Then substituting these constant values in
values of
dt' dt'
the general expression for L, the relations between the constants of
steady motion are given by
dL
dL
0,
dé dn
0, &c.
In this way we obtain in general as many equations as there
are co-ordinates. Usually the coefficients in the expression for T
are constant because some of the co-ordinates are constant in the
state of steady motion, and the other co-ordinates appear in the
expressions for T and V only as differential coefficients. In such
cases we have clearly fewer equations than co-ordinates to con-
nect the constants of steady motion. We have then a system of
possible steady motions which we may conveniently term parallel
steady motions.
4. To obtain the equations to the oscillatory motion, we retain
the first powers of 0, p, &c. We thus obtain a series of equations
of which the following is a specimen :
(B+E) 0
+ B
{B
№2
d
12 dt²
+ (C21—C12)
dt
A12+ E22
12
d2
d
+ B₁
313 t²
+ (C31 - C'13)
-
A13+ E
dt
13
зва
+...
= 0.
To solve these we write
0 = Memt,
= M₂emt, = Memt, &c.
1
2
3
Substituting we obtain on eliminating the ratios M, M₂: M, &c.
a determinantal equation, viz.
* Since we may in Art. 1 change the co-ordinates from έ, n, §, &c. to §1, N1, Š1,
&c., where §=ƒ (§₁) n=F(n₁) &c., it is clear that the steady motion can be always
expressed by constant values of the differential coefficients of the co-ordinates.
48
[CHAP.
THE DETERMINANTAL EQUATION.

¡
2
2
12
В₁m² - A₁₂+E₁₂
11
11
B₁m²-A13+ E13
+ (C31 — C13) m
&c.
B₁m² - A₁₂+ E22
-(C21-C12) m
В₁m² — А13+Е18 |
— (C91 — C13) m
tion.
31
J
&c.
+(C21 − C12) m
В„m³ − A„+E„
2
-23
В¸m² - А„+E∞
— (C½½ — C₂) m
32
&c.
2
B₂m² - A2+E
23
+(C32 − C23) m
23
&c.
=0.
Вm² - A+E &c.
33
&c.
&c.
This equation will be referred to as the Determinantal equa-
5. If we refer to the equation formed by this determinant
and read it in horizontal lines, we have of course the several
equations of motion, each term being the coefficient of 0, p, r, &c.
in order. In this form the equations may be reproduced by the
following easy rule.
Taking the expression for TV as given in Art. 1, let us
consider only the terms of the second order, those of the first
having been already used to determine the steady motion as ex-
plained in Art. 3. Separate from the rest, the even powers of
è, e', p, p', &c. and write for 0', 'p', &c. — D²0², — D³0p, &c., so
-
that D will stand either for or for the m in the determinant
d
dt
1
of Art. 4, when we write = Mem², = M₂em, &c. Let the sum
of these terms be called P, so that
1
11
11
11
12
12
12
P = (A - E₁ - B₁D²) 0² + (A₁₂-E₁₂- B₁₂D²) 06 + &c.
Let the remaining portion of the terms of T- V, viz. those con-
taining both 0, 4, &c. and e', p', &c., be called Q, so that
Q= С'„ØØ' + С'₁₂06′ + C₂80' +
11
12
21
Then the several equations may be formed from the rule
dP dQ
+
do de
dPdQ
+
аф аф
D
d Q
do
= 0
d Q
D
0
do
&c. = 0
In applying this rule no accented letters will occur except in
the second term of each equation. If we wish D to stand for m
IV.]
49
THE DETERMINANTAL EQUATION.
in the determinant, we must regard 0, p', &c. as abbreviations for
De, Do, &c. If we wish to use the equations themselves, we
replace D0, D4, &c. by e', f', &c.
[The determinant may also be found by another rule. Taking
as before only the terms of the second order in the Lagrangian
function L=T-V, let us separate the terms of the form
Q = С'₁₂ØØ' + С'₁₂0$' + C₂0'$ + &c.
11
12
21
In the remaining terms put 'Om √ − 1, 6' = pm √−1, and
so on, and write down the discriminant. If the system oscillates
about a position of equilibrium the terms represented by Q are
absent and the discriminant thus formed will be the determinantal
equation giving the required values of m. But if the system
oscillate about a state of steady motion we must modify the dis-
criminant by adding some quantity derived from Q to each term.
To find this, write above the columns e', ', &c. and before the
rows 0, 4, &c. Consider any term, say the term in the column
' and row 0. We must add to that term (C12 - C21) m, where
C C₁₁ is the excess of the coefficient of p'e above the coefficient
40' in the expression for Q. Since the determinant is un-
changed by writing -m for m, we may, if preferred, add the
excess of the coefficient of e' above the coefficient of Op' provided
we adhere to one order throughout.]
12
of
21
6. If in the determinantal equation we write -m for m, the
rows of the new determinant are the same as the columns of the
old, so that the determinant is unaltered. When expanded, we
shall have an equation which contains only even powers of m.
The condition of dynamical stability is that the roots of this
equation should be all of the form
m = ±B√−1.
If we write m²-p3, the roots of the transformed equation
must be all real.
In the case of equal roots of the form m=√1, it has
been shown in Art. 5 of Chap. I. that it is necessary for stability
that the proper number of minors in this determinant should
vanish. If there be two equal roots, these roots must make all
the first minors zero; if three equal roots, all the first and second
minors must vanish, and so on.
7.
When the system depends on many co-ordinates the labour
of expanding this determinant is often considerable. Methods of
evading this in certain cases will be given in the following chap-
R. A.
4
1
50
[CHAP.
EXPANSION OF THE DETERMINANT.
}
ters. But when the development is necessary we may proceed in
the following manner. Let the determinant be written
12
2
2
11
B₁m² + α11
B₁₂m² +α19
+ Fom
12
B₁₂m² +α12
+ F₂m
22
B₂m² + α22
&c.
and let
21
2
B₁₂m² +α13
13
Figm
+ F₁₂m
B₂m² +α23
23
+ Fm
&c.
专
​02
&c.
B = B₁₂12 + B₁₂00 + ...,
a
11
02
a = α11 2
F-F
qp*
+ α₁₂0$+…..,
We know that the determinant when expanded is of an even
order, hence all odd powers of m must finally vanish. Let us
expand the determinant in powers of the F's. The first term is
the discriminant of Bm²+a, this term is independent of the F's.
The terms which contain the first powers of the F's are obtained
by erasing any one line of this discriminant and replacing it by
the corresponding F terms. But these terms all vanish and we
need not describe them minutely. The terms containing the
products and squares of the F's may be obtained by erasing every
two lines of the discriminant and replacing them by the corre-
sponding F terms. Thus if we erase the two first lines we have
the determinant
12
12
F,m
0
0
- F₁₂m
13
23
23
33
&c.
&c.
Fm
13
118m &c.
Fm
23
&c.
&c.
&c.
and so on for all the other rows taken two and two. The terms
which contain the cubes and all odd powers of the F's vanish,
while the terms which contain the fourth powers may be obtained
by erasing four lines of the discriminant and replacing them by
the corresponding F's.
When the determinant has been expanded, we have an equa-
tion of an even order to find the values of m. We may therefore
employ the short method of Art. 5, Chap. III., to obtain the
Sturmian functions.
8. Necessary and sufficient tests of the stability of the motion
of a system of bodies are given in the preceding pages. But it
•
IV.]
51
DETERMINATION OF STEADY CO-ORDINATES.
is assumed, as explained in Art. 2, that the co-ordinates have been
properly chosen. They are supposed to have been so chosen that
the coefficients in the expanded Lagrangian function are all
constants. When this is not the case we must discover the
proper co-ordinates to which the system must be referred before
we can apply the test of stability. But when the motion is steady
this is not difficult.
There are obviously many such systems of co-ordinates, and
one set may generally be found by a simple examination of the
steady motion. If there are any quantities which are constant
during the steady motion, they will often serve for some of the
co-ordinates. Others may be found by considering what quan-
tities appear only as differential coefficients or velocities. Practi-
cally these will be the most convenient methods of discovering
proper co-ordinates, since no further change will then be necessary
and we may at once form the determinant of stability. But if
these methods fail we may adopt the following analytical method
of transforming (where possible) the general Lagrangian function
with variable coefficients into one with constant coefficients.
9. Let the Lagrangian function be
L = L₂+ ø0 + A„$+ &c. + B₂0' + B„$' + &c.
+ ½ Äز + A₁₂0¤ + ...
11
+ ½ B„ز + B₁₂¤ ¤' + ...
11
12
+С„ØØ′ +С'₁0¤' + С'₂¶0' + ...
11
12
21
where the coefficients are all functions of t and the co-ordinates
0, 4, &c. have been so chosen as to vanish along the steady motion.
We have therefore for the steady motion
d
1
B₁ — A₁ = 0,
dt 1
&c. = 0.
The oscillations about the steady motion are given by the
terms of the second order. Our present object is to transform
these to others with constant coefficients by the following substi-
tutions:
Ө
0 = p₁x+p₂y +p3²+ &c.,
$ = q₁x+Q₂Y +93% + &c.,
&c. = &c.,
where the p's, q's, &c. are functions of the time at our disposal.
Substituting and equating the coefficients of x, y², &c. to
unity, we have as many equations of the form
2
½ B„Þ² + ½ В„q² + B₁₂pq + ... = 1
22
12
1....
(1)
4-2
+
1
52
[CHAP.
DETERMINATION OF
as there are co-ordinates. Equating the coefficients of the pro-
n-1
2
ducts x'y', x'z', &c. to zero we get n equations of the form
B₁1 P1 P2 + B 229 192 + B 12 (P192 + P₂91) + ... = 0......... (2),
supposing that there are n co-ordinates.
Equating to the constants a,, a,,... the coefficients of xx', yy',
&c. having subtracted the differential coefficients of (1) we have
n equations of the form
(C₁₁ — — B₁₁') p² + (C22—1 B22) q² + (C12+ C21-B12) pq+...=α... (3).
11
11
Adding the coefficients of xy' and x'y and subtracting the
differential coefficients of (2) we have n equations of the
form
11
2
n-1
2
૧૬.શે
(2011 - B₁₁) P₁P₂+ (2022 — B22') 1192)
+ (C12+ C21-B12')(P192 +P₂91) + ... )
0…………….. (4).
=0..
Equations (1), (2) and (4) give n² equations to find the n
quantities p₁₂, &c., 192, &c.
P₁P27
purely geometrical problem.
The solution of these equations is a
If we construct the two quadrics
2
— ½
B₁0² + B¸²+B „0$ + ... = 1,
11
12
(С11 − †B′11) 0² + (C½ − 1 B′½½) þ² + (C12+ C₂1 − B 12) 0$ +
22
21
= 1,
=
and refer them to their common conjugate diameters, by writing
0 = р₁x + p₂y +...
$=q₁x + q₂Y + ...
&c. = &c.,
making the first quadric to become what we may call a sphere by
projection; the values of p,p,, &c., 219, &c. thus found are the values
required to make some of the coefficients in the Lagrangian
function become constant. These values must of course make all
the other coefficients of the second order in the Lagrangian function
constants also, and thus we have n (n + 1) analytical conditions
that the motion should be steady.
It might be supposed that greater generality would be obtained
by replacing the zero's of equations (2) and (4) or the unities of
(1) by arbitrary constants. This may be convenient in practice,
but as we know that by a subsequent real change with constant
values of pip, &c., 419,, &c., we can render them zero or unity, it
simplifies the argument to perform the two transformations at
once.
IV.]
53
STEADY CO-ORDINATES.
10. The geometrical problem just alluded to admits of a real
solution whenever one quadric can be projected by a real projection
into a sphere. The problem then becomes that of finding the
principal axes of the other. This is just our case, since the ex-
pression
1B,0 + 8,,6 Ꮬ +
12
is necessarily positive for all values of 'p'.
It is unnecessary to describe here the mode of solving this
problem. It is sufficient to say that it may be reduced to the
solution of the symmetrical determinantal equation
11
11,
ВË-λDË
B12-XD12
12
22
DË— λD12
D2-ND 22
--
|=0,
where D1, D12, &c. are the coefficients of 02, 04, &c. in the second
of the quadrics. The roots of this equation are known to be real
when the suppositions just mentioned are satisfied.
11. In order to examine the fundamental determinant in
Art. 4 a little more closely, let us suppose it reduced to depend on
three co-ordinates. We may then have the advantage of a geo-
metrical analogy. Let the co-ordinates be n,, and let the
equations of motion be written
d2
d2
B
12 dť²
A12
d
(B. & − 4,) € + (”
'11 at²
B₁₂
d²
A
'12 dt²
d
+G
B₁s
d²
'13 at²
-
- F
dt
d
dt
A,
A,
12
13
§
-
G
d²
dt
ď²
B.
A₁
-
13 dt² 13
+(
§ + (BA) +
& +
B
22
d²
23 dt²
d
η
22
A 23
193
+ E at
+F
d
=0,
= 0,
dt
d²
B
23 at²
A 28
3=0,
d
-Ea
dt
d²
5
| n + (B₂s - Ass) $ = 0.
")+(
33
dt²
33
Let a geometrical point P move in space so that its co-or-
dinates referred to any axes are §, n, . Then the position and
motion of the point exactly give us the position and motion of
the system.
12. Looking at the equations of motion just written down we
see that they are similar to those which give the oscillations of a
1
་
54
[CHAP.
THE REPRESENTATIVE PARTICLE.
system about a position of equilibrium, but that there are in
addition terms E dn, Fd, &c. The general effect of these terms,
dt,
Ends
dť
as will appear from what follows in the subsequent chapter, is to
increase the stability. If we transpose these terms to the other
sides of the equations, we may regard them as impressed forces
acting on the system, whose resolved parts in the directions of the
axes §, n, S, are
dn
X= G
F
dt
Y= E
dt
Z= F
2222
G
de
E
༤༤ལ ༤ལ
dn
We see at once that
EX+ FY + GZ=0
dE
dex + y +
dn
dr
z=0
dt
dt
dt
so that these forces are at once orthogonal to the path of the
representative point P and also orthogonal to the straight line
whose direction cosines are proportional to E, F, G. These forces
are therefore of the nature of centrifugal forces, as if they were
produced by the rotation of the system about this straight line.
13. We may show that the straight line (E, F, G) is fixed in
space. To prove this, let us transform our co-ordinates from §, n,
to x, y, z, where x, y, z are connected with §, n, by any linear
relations, such as
§ = α₁x+by+c¸²`
n = α₂x+b₂y +C₂²
[=α¸x+by+Сg²,
Let the portion of the Lagrangian function under consideration
(Art. 1) be
+
then
C₁₁' + C₁₂n + C₂+...
11
12
21
G=C12C21) E=C23-C32) F=C31-C13"
C,
Substituting for §, n, their values in terms of x, y, z, we find that
the difference between the coefficients of xy' and x'y is
G' = G│a, a₂+ E α₂ α¸ + F a¸ α1
2
b₁ b₂
1
bq bg
2
b. b
י
IV.]
55
THE REPRESENTATIVE PARTICLE.
with similar equations for E' and F". But if μ be the determi-
nant of transformation
µz = §| α₁ α₂ | + § | α₂ α3│+ n| α¸ α,
2
2
b₁ b₂ b₂ b₂
2
3
a
b₂
b₁
>
with similar equations for x and y. The ratios of E, F, G are
therefore transformed as if they were co-ordinates. If the trans-
formation be a real transformation of Cartesian co-ordinates, let
lengths each equal to unity be measured from the origin along the
axes O§, On, og, thus forming a tetrahedron whose volume is V.
Let a similar construction be made for the new axes, forming a
V
tetrahedron of volume V'. Then* μ 7. Hence the quantities
E F F G
V
may be transformed as if they were lengths measured
E' F F' G'
along the axes and become
If both systems of co-
ordinates are rectangular we have V=1, V' 10
E F G
T' T
,
Let w be the resultant of T' then o may be regarded
as a fixed length measured from the origin along a straight line
fixed in space.
Let v be the velocity of the representative particle, the angle
between the direction of this velocity and the axis whose direction
cosines are proportional to E, F, G. Then the resultant of the
forces X, Y, Z is easily seen to be 2v Vw sin 0 acting perpendicular
to the axis and to the direction of the motion. We might call the
straight line (EFG) the axis of the centrifugal forces.
[* Let (§1 71 Š1), (§2 N2 Š2), (§3 N3 (3) be the co-ordinates of three points A, B, C re-
ferred to any oblique co-ordinates. Let us find the volume V' of the tetrahedron
of which these and the origin are the angular points. Since the volume vanishes
when any angular point as C lies in the plane containing the origin and the other
two A, B, the expression for the volume must contain the factor
=
1_ka_ 3
η1 72 73
$1 $2 $3
ૢ ×â
The volume is evidently an integral rational function of the co-ordinates when
the axes are rectangular and the plane AOB is taken as the plane of xy, it
easily follows that this is true for all axes. Since this function cannot be of more
than the third order, we have V'=Mu, where M is independent of the co-ordinates
of A, B, C. When the points A, B, C are on the axes at unit distances from the
origin, let v be the volume of the tetrahedron. In this case µ=1, and .'. M=V.
We have therefore in all cases V'= Vμ.
In the text, let the extremities of the unit lengths measured along the axes of
x, y, z be called A, B, C. Then the (§ §) co-ordinates of A, B, C are (a1, a2, а z),
(b1, b2, B3), (C1, C2, C3), respectively. Hence by what has just been said V=Vu.]
p
*[

56
[CHAP.
THE REPRESENTATIVE PARTICLE.
+
1
14. The expressions for the co-ordinates in terms of the time
will in general contain as many periodic functions as there are
co-ordinates. If the initial conditions are such that each contains
one and the same periodic function, the motion recurs after a con-
stant interval and the system is said to be performing a simple or
harmonic oscillation.
If the system be oscillating about a position of equilibrium,
with a Lagrangian function
А„§² +2А„§ŋ+...
812
11S
129
+В„§'² + 2B„§'n' + ...
we know that the harmonic oscillations are represented by rectili-
near motions of the representative particle, and that these are
along the common conjugate diameters of the two quadrics
equations are
2
A₁₂+ A₁n + ... = a
દુ
11
2
+Ben+ ... = b
where a and b are two constants chosen to make the quadrics real.
Let us consider what are the harmonic paths of the representative
point when the system is oscillating about a state of steady motion.
In any harmonic vibration we have
similar equations for n and . Hence
η
L cos (t+a) with
d²
d²n
dy
dt2
×Ë,
-
- x²³n,
dt²
dt
=-x²8.
Substitute these in the equations of Art. 11. Differentiate and
substitute again. Multiply by έ, n, s, add and integrate, we obtain
દુ
(B5
-2
+ Bun + ...) x² + ( Au¸² + Au§n + ...) = c,
++..)パ​+(4
11 2
where c is some constant. The harmonic path lies on this quadric,
which has a common set of conjugate diameters with the two
quadrics a and b.
If we resume the result of the substitution of, &c. in the
equations of Art. 11, and multiply by E, F, G respectively and add,
we obtain
2
[(B„E+ B„F+B13G) § + &c.] λ² + [(A„E+ …..) § +&c.]=0,
A short paragraph in Thomson and Tait's Natural Philosophy, page 273, is
the only notice of this which the author has discovered.
IV.]
57
THE REPRESENTATIVE PARTICLE.
which is a plane, and is diametral to the straight line (EFG) with
regard to the quadric c.
The harmonic paths are therefore ellipses. The three harmonic
planes are diametral to the same straight line and this straight
line is fixed in space, being the axis of the centrifugal forces.
If we eliminate λ between the equations to the plane and the
quadric c, we get a cubic surface on which the three harmonic
conics lie.
If E, F, G are zero, which is the case when the system oscillates
about a position of equilibrium, the quadric c becomes a cylinder.
This may be conveniently shown by referring the system to such
co-ordinates that the coefficients B2, B3, B, A2, A3, A are all
zero. In this case the diametral plane of every straight line passes
through the axis of the cylinder. The harmonic oscillations are
therefore rectilinear.
23
If R be the length of that semidiameter of the quadric (c)
which is parallel to the fixed straight line (E, F, G), it may be
shown that the
Product of the axes
of the quadric c
}
R
E² + F² + G²°
2c
If E, F, G are all zero, and their ratio is indeterminate, R is
any diameter. Hence one of the axes of the quadric (c) must be
infinite and the quadric will be a cylinder.
[If the quadric (c) be a cylinder and E, F, G are not all zero,
we must have either λ zero or R infinite. In the latter case the
axis of the cylinder will coincide with the axis of the centrifugal
forces.]
The quadric (c) has also the following geometrical property.
Let the lengths of semidiameters of the quadrics (a) and (b) drawn
parallel to the axis of the centrifugal forces be p and p'. Through
the intersection of these quadrics describe a quadric so that the
(-) R
12
Product of
its axes
2√ab
√E² + F¹² + G²
12
Ꭱ
ρ
2
1 1 1
R2 R2
1

This quadric is similar to the quadric (c).
15. The introduction of the representative point to exhibit
the motion of a system may appear somewhat artificial. If how-
ever we properly choose the co-ordinates the particle moves exactly
as a free particle, and we might reduce the problem of finding the
58
[CHAP.
THE REPRESENTATIVE PARTICLE.
oscillations of a system to a problem in Dynamics of a particle.
Refer the quadric
Bu + Buen + ... = b
2
to its principal axes and let the equation thus changed be
Є2
/ S1
, mi
2
• + B22 +... = b.
Bu
2
Since B, B., &c. are positive quantities, we may put
√ B₁₁ = x, √ B₂ny, &c.
11
22
The quadric has thus been "projected" into a sphere. Let x, y, z
be now chosen as the co-ordinates of the system and let the
Lagrangian function be expressed in the form
12
L = x²² + y²² + z²² + ½ А„x² + А„xy+…..
+ C₁xx² + ...
the terms of the first degree being omitted as not necessary to our
present purpose. The three equations of motion at the beginning
of Art. 11 take the form
d²
(~ − An) − 4„− )y 4,
▲¸) ∞ + ( − ▲ „ − G — 4 ) y + ( − A₂ + Fd) z = 0,
dť²
x
12
dt
13
&c. = 0,
which are the three equations of motion of a free single particle
whose co-ordinates are x, y, z under the action of forces whose force
function is
}А₁x² + А¸¤у+...
and a force acting perpendicular to the path and also perpendicu-
lar to a fixed straight line, the force being proportional to the
velocity.
16. As an illustration of this theory, let us here make a short
digression. However the particles of light may oscillate, whether
in a rotatory or linear manner, we know the motion is related to a
certain plane called the plane of polarization. It may be shown
that any harmonic oscillation about a position of equilibrium may
be represented by a rectilinear oscillation of the representative
particle. Let us represent the motion at any point of the ether
by a rectilinear oscillation in a direction perpendicular to the
plane of polarization. This would be Fresnel's Vibration. The
representative particle, as just shown, would not necessarily move
as if it were a free single particle. But let us assume (and a proof
is not necessary to our present purpose) that when the oscillation
is drawn as above described the motion in the plane of the front
IV.]
59
THE REPRESENTATIVE PARTICLE.
is the same as that of a free particle, while that perpendicular is
not free. On this assumption we see that Fresnel in his theory
of double refraction is justified in taking actual instead of relative
displacements, for it is the representative particle he is considering.
He also neglects the force normal to the front, for the particle
moves as a free particle only in the plane of the front. These
general remarks are not meant to explain Fresnel's theory, but
merely to show how the representative particle may be used to
replace a complicated motion.
17. [When a system is performing a harmonic oscillation
about a state of steady motion or about a position of equilibrium,
the motion repeats itself continually at a constant period, that is
to say, the values of the co-ordinates recur at this interval. This
is the chief peculiarity of a harmonic oscillation. When the
oscillation is about a position of equilibrium, the representative
particle oscillates in a straight line whose middle point repre-
sents the position of equilibrium. Thus the system passes through
the position of equilibrium twice in each complete oscillation.
When the oscillation is about a state of steady motion the path
of the representative particle is an ellipse whose centre is at the
point occupied by the system in steady motion at the same in-
stant. Thus the system does not in general ever coincide with
the simultaneous position of the system in the undisturbed or
steady motion. When a system is disturbed by a small impulse
from a state of steady motion, it will in general describe a com-
pound oscillation made up of at least two harmonic oscillations,
at the instant of disturbance these two neutralize each other so
that in the disturbed and undisturbed motions two simultaneous
positions are coincident. But it is clear that this cannot occur
again unless either the periods of the two harmonics are com-
mensurable or the period of one of them is infinite.]
18. [In some cases the ellipse degenerates into a straight line.
Thus if the quadric (c) be a cylinder the diametral plane of the
axis of the centrifugal forces will pass through the axis of the
cylinder, and thus the harmonic oscillation corresponding to this
particular value of λ will be rectilinear. In this case the system
twice in each oscillation passes through the position it would have
occupied at the same instant in the undisturbed motion.
The quadric (c) has a common set of conjugate diameters with
the quadrics (a) and (b). Hence if (c) be a cylinder, its axis must
be parallel to one of the three common conjugate diameters of (a)
and (b). If we refer the quadrics (a) and (b) to their common
conjugate diameters, they take the form
A₁₁' § ² + A¸¸´n² + A¸¸'5² = 2a)
11
22
33
B₁ '§² + B. 'n² + B₂'S² = 2b)
11
22
33
+
.
.
.
1
60
[CHAP.
THE MODIFIED
2
The cylinder which passes through their intersection and has
its axis parallel to the diameter is found by eliminating 2 between
these equations. We see therefore that B'x²+ A=0. If then
the axis of the cylinder cut the quadrics (a) and (b) in D and D
respectively, we find that for this oscillation
2
λ²=
λε
OD'2 a
OD² b
33
39
It has already been shown that when this value is finite, the
direction of ODD' is along the axis of the centrifugal forces.]
19. [In some cases two or more of the values of λ are zero.
In these cases the co-ordinates will have terms of the form nt + e,
where n and e are two small constants. When, as explained in
Art. 3 of this Chapter, there are several parallel states of steady
motion, these terms imply that the motion is stable about a state
of steady motion very nearly the same as the undisturbed motion
but not coincident with it. The actual undisturbed motion, unless
n is zero, is unstable in the sense that if a proper disturbance
be given to the system, the system will depart widely from the
positions it would have simultaneously occupied in the undisturbed
motion.]
,
20. In many cases of small oscillations it will be found that
the Lagrangian function T-V is not a function of some of the
co-ordinates as 0, 4, &c. though it is a function of their differential
coefficients 0', ', &c. In such cases the steady motion will be
usually given by constant values of these differential coefficients,
while the other co-ordinates as §, n, &c. are also constant. It is
evident that the determinantal equation of Art. 4 is needlessly
complicated. It is clear that there will be as many pairs of roots
equal to zero as there are co-ordinates 0, p, &c. It will be an
advantage to eliminate e', ', &c. altogether from the Lagrangian
function, and to find the remaining roots by operating only with
the co-ordinates &, n, &c. We shall thus obtain a determinant with
just as many rows as there are co-ordinates of the kind , n, &c.
2
Let L, be the Lagrangian function expressed as a function of
O'p', &c. En, &'n', &c. Let L, be its value when 'p' are elimi-
nated, so that L, is a function of En, έ'n', &c. only. To effect this
elimination we have the integrals
d T
de C₁
dT
&c.
C2
do
where c₁, c₂, &c. are constants. Then
2
dL,_dL, dL, de'dL, do'
dédé
+
+
de
do'' d§ ・ do'' d§'
dL, dơ
C
dé
+ c₁ de
do
+ Cz de
+...
+ &c.
IV.]
61
LAGRANGIAN FUNCTION.
2
dL₂_dL,
αξ
2
αξ
d dL, dL.
1
+ C₁
do
+ C₂
do
de
+ ...
d do do
αξ
d dL,
dL,
+ C₁₂
dt de de
dt de'
dé
dt
(at DE - DE
+ &c.
d dLdL,
But
dt de
1 0.
αξ
Hence if we take
L'=L—e̟¸¤' — c„p′ — &c.
dT
dT
and eliminate O', ' by help of the integrals
do C2,
dỡ
we may treat L' just as we do the Lagrangian function L. The
equations giving the small oscillations about the steady motion
will be
d dL' dL'
dt de de
dé
0, &c. = 0.
The function L' may be called the modified Lagrangian function.
[It should be noticed that this is equivalent to a partial use of
Hamilton's transformation of Lagrange's equations. Sir W. R.
Hamilton eliminates all the differential coefficients´ l', d', &c. by
dT dT
the help of equations of the form = U,
v, &c. where u, v,
dľ do'
&c. are made to be new variables*. In our transformation only
* The Hamiltonian transformation of Lagrange's equations bears a remarkable
analogy to the transformation of Reciprocation in Geometry. This may be shown
in the following manner.
When the system has three co-ordinates 0, o, y, we may regard e', ', ' as the
Cartesian co-ordinates of a representative point P. The position and path of P will
exhibit to the eye and will determine the motion of the system. Let u, v, w be the
Hamiltonian variables, so that
1
dT1
dTi
v =
u =
do'
dø'
W =
dT₁
dy
1
where T, is the semi vis viva expressed as a function of 0, 4, 4, 0', 6', 4'. Then
u, v, w may be regarded as the co-ordinates of another point Q whose position and
path will also determine the motion of the system.
If the semi vis viya be given by the general expression
T₁ = 41102+ A120'' + ...
it is clear that the point P always lies on the quadric T₁ = U where U is the force
function and the co-ordinates 0, p, have their instantaneous values. The point
Q must therefore lie on another quadric which is the polar reciprocal of the first
with regard to a sphere whose centre is at the origin and whose radius is equal to
√20. The equation to the reciprocal quadric is therefore
T₂=
1 0 น v
2A
พ
u 41 412 413
v 412 422 423
w A13 A23 A33
= U,
•
62
[CHAP.
THE MODIFIED
F*|
those new variables are introduced which would be constants in
Sir W. R. Hamilton's transformation.
This remark suggests an extension of the process. If L be a
function of 0, 4, &c. as well as of e', p', &c. the quantities c₁, c₂, &c.
will not be constants. We express this by writing u, v, &c. instead
of c₁, c,, &c. Suppose we wish to eliminate some of the differential
coefficients, viz. O', p', &c. and to retain the remaining ones, viz.
E', n', &c. If we put
L'=L-uα-vp - &c.
we may easily show as in the preceding page that
d dĽ
dt de
dĽ
0, &c. = 0.
αξ
where ▲ is the determinant, called the discriminant, which may be formed from
the determinant just written down by omitting the first row and the first column.
This is a general expression for the Hamiltonian function and agrees with that
which may be deduced from the result in Art. 21, when all the variables are trans-
formed by the Hamiltonian process.
Since the polar reciprocal of the polar reciprocal is the original quadric, it
follows that
1
0' =
dT,
du
2
φ'
dT
dv
dᎢ .
2
:
2
=
which are three of the six Hamiltonian equations.
dw
We may also show geometrically that if the coefficients of T be functions of
any quantity 0, then
dT1
do
dT2
de
To prove this we notice that if x, y, z be the
co-ordinates of a point P, situated on a radius vector OP' of a quadric
• (x, y, z)=1 referred to its centre 0, then 4 (x, y, z) = (OP).
#
The quadrics
T₁ = 1 and T₂=1 may be regarded as polar reciprocals of each other with regard
to a sphere whose radius is √2 and whose centre is the common centre of the
two quadrics. Let P be any point on the quadric T₁=1, and let the radius vector
be produced to Z so that OP. OZ=2, then the quadric T₂=1 touches a plane
drawn through Z perpendicular to OP and Q is the point of contact. Let these
quadrics be slightly altered in consequence of a variation of 0, so that their
equations are now T₁+dT₁=1 and T₂+dT₂=1. Let OP and OQ produced cut
these quadrics respectively in P' and Q. Then
2
2
2
2=
OQ12
Now if Z' be a point on OP produced so that OP′ . OZ'=OP . OZ, the quadric
T₂+dT₂ will touch the plane drawn through Z' in some point q near Q. The
point Q will therefore lie very nearly in the tangent plane, so that by similar
triangles
2
OQ OZ OP'
OQ OZ OP'
Since each of these ratios is indefinitely nearly equal to unity, it follows that
dT₁=
dT2
If we put L=T₁+U and H=T₂- U, Lagrange's equations may be written in
the forms u' =
dL
do
dL
v' =
w' =
ἀφ'
dL
dy'
Hence we have
dH
ан
-u'
- v' =
do'
do'
dH
- - w' =
dy'
which are the remaining three of the Hamiltonian equations.
Iv.]
63
LAGRANGIAN FUNCTION.
We have thus as many equations of the Lagrangian form as
there are variables §, n, &c. Also since u=
dL
de
&c. we have by
differentiation
dL' /dL
น
du de
1) dre
do
Ø' + &c.= − 0',
.
du
with similar equations for p', &c.
By Lagrange's equations we
obtain
dL'
u', &c.
do
Thus we have as many sets of equations of the Hamiltonian
form as there are variables 0, 4, &c.]
21. We may effect this elimination once for all and find a
definite expression for L'.
Let the kinetic energy be
0'2
T = Tee
2
+ Top O'p' + ...
Then the integrals used will be
=
=
-
Tée0'+Top+... c₁₂- Tot - Tonn —
To¿0'+T¿¿☀' +
Ꭲ
-
Τφηή
= C₂ — T¿¿§' — Tonn' -
&c. &c.
For the sake of brevity let us call the right-hand members of
these equations c₁- X, c₂- Y, &c. Since T is a homogeneous
function, we have
2 2
T = TEE ³ 2 + Ten5 n + ...
+ 10′ (ç₂ + X) + ½ $'(c₂+ Y) + &c.
ૐ
12
Ꭲ +Tần % + &c. — V
:. L' = T§§
2
-
- 10 (c₁ - X) — 1 p' (c, - Y) - &c.)
If we substitute in the second line the values of e', p', &c. found
by solving the integrals just written down, we have
'2
L' = T&² 2, + T&&'n' + &c. – V
2.
1
c₁₂-X c₂- Y
+
2Ac₁₂- X
C₂- Y
Tee
Тоф
Teb
Ꭲ
Top
:
:
↓
64
[CHAP.
THE MODIFIED
3
where A is the discriminant of the terms in T, which contain only
e', ', &c., and may be derived from the determinant just written
down by omitting the first row and the first column.
We
may expand this determinant and write it in the form
દુ
2
+ Tên§'n' +&c.— V
L' TE
=
1
0 X Y
1
G
Ca
+
+
2A C1
Tee
Tep
Тоф
C₂
Tes
2A X Tee Top
Y
У Төр Тфф
:.
10
X
Y
A G₁₂
Тоо Тор
པ་
where X, Y, &c. stand for
C₂ Тор Тоф
X=Teε&' + Tenn' +
´ Y = T¿¿§' + Tonn' +...
&c. = &c.
}
The first of these three determinants will contain only the
constants c₁, C., &c., and the co-ordinates §, n, &c. The second will
not contain c₁, c₁, &c. but will be a quadratic function of §', n', &c.
The last determinant will contain terms of the form ', n' with
variable coefficients which may also be functions of c₁, C...
Since ', nn', &c. are all small quantities, it is clear that this
expression for L' when expanded will take a form precisely similar
to that given in Art. 2, only that we have fewer variables to deal
with.
22. As an example, let us consider the following problem.
A body has a point O which is in one of the principal axes at
the centre of gravity G fixed in space. The body is in steady
motion rotating with angular velocity n about OG which is vertical.
Find the conditions that the motion may be stable.
η
Let OA, OB, OC be the principal axes at O and let OC co-
incide with the vertical OZ in steady motion. Let §, ʼn be the
direction cosines of the vertical OZ referred to OA, OB. Let
@₁, w₂, w, be the angular velocities about the principal axes at 0.
Then to the first order
19
3
@₁ =
@1
@₂ = − &' + w¸n)
W2
IV.]
65
LAGRANGIAN FUNCTION.
Let be the angle ZOC, the angle the plane ZOC makes
with a plane ZOX fixed in space and ☀ the angle it makes with
the plane AOC fixed in the body. Then
= − sin 0 cos $)
n= sin o sin o'
ηπ
@₁ = &' + y'cos
= $' + (x' − ☀') (1 -
− p) (1 - 4),
putting x=4+. We easily find
2
w¸ = x − x' §¸² + ¹² — ↓ (§n' − e̱'n).
3
2
* –
If then A, B, C be the principal moments of inertia at O, the
Lagrangian function is
C
n²
*
L = {x (1 - " + " ") - ↓ (Ev - E~)}"
A
+
2
B
(n² + x § )² + — — ( − §' + x'n)³
n²
- Mgh (1-5 +),
2
where M is the mass of the body, and h= OG.
Since
xi
which gives
Hence
2
is absent from the equation we have the integral
dL
dx
Сто
x = n + terms of second order.
A
2
= const += n²+
L' = L- Cnx'
B
2
+ {(A − C) n² + Mgh} % + {(B — C)n² + Mgh}
2
+ (1 - 2) - (B-) năm
ne̱'n.
n
Using this as the Lagrangian function we easily find
(A − C)n² + Mgh – Bm²,
–
-(A+B-C) nm,
R. A.
(A+B-C) nm
(B − C)n² + Mgh – Am²
= 0.
=
5
сл
1
66
[CHAP.
THE MODIFIED
The roots of this
the form + BV-1.
equation to find m must for stability be of
Putting m²=-λ we have a quadratic to
find λ². The roots of this quadratic must be real and positive.
If A =B, as in the case of a top spinning with its axis vertical,
we have
λ= ±
(2A-C
2A
C²n² - 4 AMgh)
n ±
2A
2
The motion is stable or unstable according as C'n² is greater or
less than 44 Mgh. If Cn² = 4AMgh, the equation has equal roots
and as the first minors are not zero the motion is unstable.
23. [As another example of the use of the modified Lagrangian
function, let us consider a case discussed by Prof. Ball in the
Notices of the Royal Astronomical Society for March, 1877. In a
problem in Physical Astronomy, we want the relative co-ordinates
of the system, while its absolute motion in space does not concern
us. Lagrange's equations involve both the relative and absolute
co-ordinates, and are therefore not particularly well adapted for
such problems. By using the modified Lagrangian function, we
may eliminate the absolute co-ordinates.
Let the system have n co-ordinates, let us choose as three of
them the co-ordinates of the centre of gravity of the whole system,
viz. 0, p, . There will then remain n-3 co-ordinates which are
independent of these. Let T" be the kinetic energy of the system
relative to its centre of gravity, V the potential energy, M the
whole mass. Then the Lagrangian function is
1
2
12
L = M (0² + " + y'²) + T″ − V.
In problems in Physical Astronomy the potential energy is a
function only of the relative positions of the bodies, and is there-
fore independent of 0, 4, and their differential coefficients.
We
have therefore
dL
dL
dL
dľ
C₂
do
= Cg.
dy'
Hence the modified Lagrangian function is
L' = T - V-a constant.
It is independently clear that we might take this as the
Lagrangian function, for the first three terms of L do not enter into
any one of the Lagrangian equations, except the three formed by
differentiating with regard to ', &', y'.
The function T' is made up of two parts, (1) the kinetic energies
of the rotations of the bodies about their centres of gravity, which
IV.]
67
LAGRANGIAN FUNCTION.
we may call T, and (2) the relative kinetic energies of the several
bodies, each collected at its centre of gravity, which we may call
T.
Let m₁, m„, &c. be these masses; x, x, &c. the abscissæ of
their centres of gravity referred to the centre of gravity of the
whole as origin. Then, accents denoting differential coefficients
with regard to the time, we have
is
2
m₁x₂' + m₂x₂' + &c. = 0.
Let us square this and write
2x₁'x' = x² + x² - (x,' — x„')².
2
12
If we examine the coefficient of any power as x2 we see that it
2
m² + m²(m₂+m₂+ &c.) = m₁Σm.
Hence the square becomes
12
Σm Σmx²² – Σm¸m¸(x − x¸')² = 0.
— 2
Similar expressions hold for the y and z co-ordinates. Hence
on the whole we see that the relative kinetic energies of the several
bodies collected at their respective centres of gravity is
T
Σm,m,v²
2ΣΜ
where v is the relative velocity of the centres of gravity of the
masses m₁, m₂. If we express this in any kind of co-ordinates, we
may use the Lagrangian function L' to find the relative motion.
The expression for T' agrees with that given by Prof. Ball, but
his demonstration is quite different. Prof. Cayley has given
another demonstration in the same number of the Astronomical
Notices.
The Lagrangian function thus found may be still further
"modified." To avoid symbols of summation, let us consider the
case of three particles moving in one plane under their mutual
attractions. Let the separate masses be my, m„ m², and let μ be
their sum. Referring the system to m, as a central mass, let the
distances of m2, m, from my be respectively r and p, and let the
opposite side of the triangle be R. Let the interior angle_be-
tween r and P be and the exterior angle between r and R be
X. Let O be the angle r makes with some fixed straight line in
space. We easily find
where
1
µT',' = 1; A0¹² + Be' + C,
2
2
3
A=m¸m‚r² +m¸m¸p² + m³m¸R²¾‚
B = m₂ (m₂p³p' + m₂R²x'′),
1
C= Σmm'v²,
2μ
5—2
68
[CHAP.
INDETERMINATE MULTIPLIERS.
and v is the relative velocity of the masses m, m' calculated on
the supposition that m₁ is fixed, and that the straight line r has
no rotation round m₁. Thus A, B, C´are all functions of r, p, &
and their differential coefficients with regard to the time.
If we only want the changes in the form and magnitude of
the triangle joining the three particles, we may eliminate Ø by
means of the equation
A0 +B= Cr
We then find as our modified Lagrangian function
L'
μ
1 {o –
1 (C₁ — B)²)
2 A
- V,
which contains only the three co-ordinates r, p and 4].
24. When the geometrical equations contain differential
coefficients of the co-ordinates &, n,, &c. of the system with regard
to the time, we cannot express the co-ordinates x, y, z of any
element of a body in terms of §, n, S, &c. by means of equations of
the form
x=ƒ₁(§, n, (, &c., t),
y =ƒ½(§, n, 5, &c., t),
z =ƒ, (§, n, }, &c., t).
It follows, as is pointed out in our books on Rigid Dynamics,
that Lagrange's equations cannot be employed in the form
d dT dT d V
dt de dé
αξ
In many of the most interesting problems in Rigid Dynamics,
it so happens that the geometrical equations do contain dt
dn
dt
αξ
&c. For example, let a sphere be set rotating about a vertical
diameter and be on the summit of a perfectly rough surface of
any form. If a small disturbance be now given to it, the sphere
may roll round and round the summit. During this motion the
velocity of the point of contact is zero, and our mode of re-
presenting this analytically in terms of the co-ordinates will give us
two equations of the form
A§ + Bn' + C§' + ... = 0.
CG
To include such cases the equations of motion must be modi-
fied. If L be the difference between the kinetic and potential
energies, all the Lagrangian equations may be written in the form
d dL dL
δξ +
dt de
de
αξ
(
d dL L
dt dŋ'
d1) Sn + &c. = 0,
dy
IV.]
69
INDETERMINATE MULTIPLIERS.
where §§, dn, &c. are any small arbitrary displacements consistent
with the geometrical equations. But if these geometrical equa-
tions be given in the form
G=G, &' + G₂n' +
H = H‚§' + H₂m' +
•
07
=0
&c. = 0.
these arbitrary displacements must satisfy
α, δξ + α,δη + ... = 0)
&c. = 0)
If this were not the case, the geometrical displacement of the
body given in applying Virtual Velocities would not be such as to
cause the unknown frictional forces, &c. to disappear. Using the
principle of Indeterminate Multipliers, we get
d dL dL
dt de
de
d dL dL
dt dn
=
+λG₁ + µH₁ + ... 0
1
dn + λG ₂ + µ H₂+... = 0.
2
&c. =0]
These joined to the geometrical equations
Ģ₁§ + G₂n' + ... = 0}
&c. = 0
are sufficient to determine the unknown co-ordinates E, n, &c. and
the multipliers λ, μ, &c.
It will be more convenient to write these equations in the form
dt αξ'
d dL dL d G
− +λ
ан
αξ
dt
dğ
+...
0,
dH
dn'
+ μ
+... = 0,
dé
&c. = 0,
d dL dL dG
dt dn dn
+λ
the geometrical equations being
G=0, H=0, &c.
It is of course obvious that these indeterminate coefficients
λ, μ, &c. are merely the frictions or other resistances introduced
into the equations in a convenient form.
25. In order to apply these equations to the oscillations of a
system about a state of steady motion, it will be convenient to
70
[CHAP.
INDETERMINATE MULTIPLIERS.
1
change the co-ordinates §, n, &c. into others 0, 6, &c. which vanish in
the steady motion. Let L be thus expanded in powers of 0, 0, &c.
as explained in Art. 1, and let P and Q have the meaning given
to them in Art. 5.
Let us then put
&' = a+ 0,
n = B+$', &c.
λ=λ + λ₁,
µ = μ₂+μ₁, &c.
Mys
where a, ß, λ。, H., &c. are the values of §', n', λ, µ, &c. in steady
motion. The geometrical equations will then take the form
G = G₂ (a + 0) + Œ₂ (B+ $') + &c. = 0}
1
2
&c. = 0)
of
and the equations connecting 80, dp, &c. will be
G₂80+ G₂dø+ &c. =0)
&c. = 0)*
In these equations G₁, G₂, &c. are functions of 0, 4, &c. Let
a square bracket indicate that the value of the inscribed quantity
in steady motion is to be taken. Thus [G] means the value of G₁
when 0, 0, &c. have all been put zero.
The equations of steady motion may then, exactly as in Art. 3,
be written
гаст
G
- [20] +x [do
on +
ан
+ &c. = 0
do
-[]
dG
ан
+入​。
+ &c. = 0
do
do'
&c. = 0
From these the relations which exist between a, ß, &c., λs Hos &c.
may be found.
The equations of the oscillatory motion may be written
d (P+Q)
do
do
D - λG₂
d Q — λ G₁₂ - λ,
dG
dě
— µÃ¸
[
dH
- &c. = 0,
do
with similar equations for 4, y, &c.
26. The final determinant written for two variables 0, 4, and
two geometrical equations G and H in the notation of Art. 19,
will be
IV.]
71
INDETERMINATE MULTIPLIERS.

B₁₁m² - A+ E
11
AG
+入​。
de
B₁₂m² - A₁₂+ E₁₂
1
12
12
+(C21 − C 12) m
dG.
аф
12
dG
¿H
+μlo
dᎻ .
do'
do
аф
dН
do
+入​。
12
22
B₁₂m² - A₁₂+E,
12
— (C21 — С12) m
do² ] +1
dG
+λ₂
G
G
[do] + [dbo]
dH
2
do
2
[2]
G
dH
dø
= 0.
2
22
22
Вm² - A+ E
dG dH₂
2
+λo d&
+。
аф аф
dG
m
+
m
0
0
аф
dø
dH аH
+
dH
дH
m
do
de
+
do do
m
0
0
It will be noticed how very much this determinant is simplified
if the values of λ, u in steady motion are zero.
27. Let us apply these equations to the solution of the follow-
ing problem.
A heavy sphere rotating about a vertical diameter rests in equi-
librium on the summit of a perfectly rough surface and being
slightly disturbed makes small oscillations, find the periods.
As the sphere moves about, its centre always lies on a surface
which may be called parallel to the given surface. Let the high-
est point of this surface be taken as the origin and let the axes of
x and Y be the tangents to its lines of curvature at 0, so that the
equation to the surface in the neighbourhood of O is
x², y²
124
+
P1 P2
Let P be the centre of the sphere, PC that diameter which is
vertical when the sphere is in equilibrium. on the summit. Let
PA, PB be two other diameters forming with PC a system of
rectangular axes fixed in the sphere. Let the inclination of PC
to the axis of Z, which is vertical, be 0, and let the vertical plane
through PC make with the plane xz an angle y, and with the
plane CPA an angle . The vis viva 27 of the sphere will then
be given by
2
T= } (x² + y²²) + ½ k² {(☀′ + y' cos 0)² + 0′² + sin²0¥ˆ}.
72
[CHAP.
INDETERMINATE MULTIPLIERS.
Let sin cos, sin 0 sinn,
sin sinn, then we have to the
necessary degree of approximation
O² 4'' = {n' — ne
12
0'² + sin² 0¥²² = §² + n
12
}
Also let +=x. These transformations of co-ordinates are
all permissible, because they do not involve any differential coeffi-
cients with regard to the time. We thus find if Z be the differ-
ence between the kinetic and potential energies
k²
12
g y²
L = }} (x² + y´¹³) + ¹² {'x'² − x′ (§n' − n§') + §² + n'}
{
− − (~ ² + 2).
—
ย
If w wy, w, are the angular velocities of the
parallels to the axes, the geometrical conditions are
W
2)
P2/
=0
y' + a(∞
w x
W
∞ =) = 0
P₁
+
2 P1
sphere about
where a is the radius of the sphere. These equations by well-
known rules reduce to
x'
α
y
a
+p'sin sin 0 + 0′ cos y − (y' + ' cos 0)
'cos sin + sin y + (y' + ' cos 0)
Y = 0
P₂
XC
0
P₁
expressing these in terms of our new co-ordinates we have
x'
G
a
y'
+ x n + § - x² ² = 0
P2
x
H = - 2 − x E + n + x 2 -
a
P₁
0
The position of the system has now been expressed in terms
of such co-ordinates, that the coefficients in the governing func-
tions L, G, H are all constant. See Art. 2.
The steady motion is given by x, y, §, n all zero, and x = n.
To find λ, we may use the equations of steady motion
[6] + [8]
ан
[2/2]
=
0,
where q stands for any one of the co-ordinates. Taking q=x and
q=y, we see that λ= 0 and
Мо
0.
To find the oscillation we may substitute in the determinant
and thus form the equation which gives the periods. As most of
IV].
73
DYNAMICAL EXAMPLE.
the constituents of the determinants are zero, it will be more con-
venient to form each equation directly from the standard formula
d (P + Q) - Dadd
dq
dQ
dG
dq
dq
ан
= 0,
dq
where q stands for any one of the co-ordinates. Taking q in turn
to be x, y, x, §, ʼn we find
X.
λ
x' — g
= 0
07
P1
α
y" - g
y
P2
α
=0
T³X"
k² (§'' + x'n') + λ₁=0
k² (n" — x'n') + μ₁₂ = 0
Putting xn these with the two geometrical equations are all
linear and ready for elimination.
Eliminating λ, μ₁ we have
19
{"+nn' +
n" − n§' +
α
ga x
k?
Te² P₁
ga y
2
y"
T² P₂
= 0
a
k?
Substituting for E, n from the two geometrical equations, we
have
k² + a²
a²
k² + a²
a²
X
x" - g
P1
y"—gY +
7.2
nk²
2
2
nk2
2
P2 αρι
y
To solve these put x = Xcos (pt + q) y = Ysin (pt + q) so that
p is the quantity required. We obviously have
a²
α
g
g
a²
2
a² k²
4
2
a²n²
(x² + + 1 ) ( v ² + a + k p ) = ( ~ + k)* PP.
(200+
a² + k²
which is a quadratic to find p².
k²)² P1P2
p²,
If P₁, P₂ have opposite signs the roots cannot be real, and the
steady motion must be unstable. If p₁, P₂ are both positive, so
that the sphere is on the summit, the motion is stable only if
n²>
a² + k²
g (√p₂+ √p₂)².
ht
:
* །
:
CHAPTER V.
Certain subsidiary determinants are formed from the dynamical de-
terminant, and it is shown that there must be at least as many
roots indicating stability as there are variations of sign lost in
these subsidiary determinants, and must exceed the number lost by
an even number. Arts. 1-5, and 9.
This is equivalent to a maximum and minimum criterion of stability
with similar limitations. Arts. 6-8.
Effect of equal roots on this test of the stability of the system. Arts.
10, 11.
Example. Art. 12.
1. In order to test the nature of the roots of the determi-
nantal equation, let us apply a method analogous to that by which
Dr Salmon in his Higher Algebra proves the reality of the roots
of the equation which occurs in the determination of the secular
inequalities of the planets.
pq
2. Let A be the determinant which forms the left-hand side
of the fundamental equation, let A be the determinant formed
by omitting the pth row and qth column. Let A, be the determi-
nant formed by omitting the first r rows and r columns. Thus
A₁ = A₁₁. We then have by a known theorem in determinants
▲▲₂ = 411422-412421
1
11°
ΔΔ,
AA22
A12421⋅
It has already been noticed that if we change m into – m, the
determinant A is changed into another determinant whose columns
and rows are the same as the rows and columns of the first de-
terminant. It easily follows that the minor - A₁, is changed into
the minor - A by changing m into – m.
then
21
A
▲ 12 = 4 (m²) + m¥ (m²)
A₂ = $ (m²) – m¥ (m³).
12
Hence if
CHAP. V.]
་
THE SUBSIDIARY DETERMINANTS.
75
Hence the product A,,A,, is necessarily positive for all negative
values of m².
12 21
12
It also follows that if A,, vanishes for any negative value of m²
then ▲, also vanishes for the same value of m².
21
3. When the determinant ▲, vanishes, we have
2
1
▲▲₂ = — A12^21›
ΔΔ,
so that ▲ and ▲, must have opposite signs, or one of them must
be zero.
Consider then the series of determinants
12
21
each one being formed from the preceding by erasing the first
row and the first column. We thus have a series of functions
of m² whose degrees regularly diminish from the nth to the first.
As we may suppose the determinant A to have a row and a column
of zeros added on at the bottom and right-hand side, but with any
positive constant in the right-hand bottom corner, we may add
to this series of determinants any positive constant. We have just
proved that if any determinant of this series vanish for a nega-
tive value of m², the two determinants on each side have opposite
signs. The case in which two successive determinants vanish for
the same value of m² will be considered afterwards.
2
∞ to
We may then use these determinants in a manner somewhat
similar to that in which we use Sturm's functions, provided no two
successive functions vanish for the same negative value of m².
No variation of sign can be lost as we pass from m² =
m²=0 except by the vanishing of the determinant A at the head
of the series. And when a variation of sign is lost, it will be
regained again at the next root, unless a root of the determinant A,
separates the two roots of the determinant A. If therefore in this
passage from ∞ to zero, as many variations of sign are lost
as is indicated by the highest power of m², the values of m² found
from the determinantal equation must be all real and negative.
It will also follow that the roots of each of the series of determi-
nants are all real and negative, and that the roots of each separate
the roots of the determinant next above it. If, however, the
proper number of variations of signs be not lost in the passage
from m² ∞ to m² = 0, it does not follow that the values of m²
are not real and negative.
2
2
2
2
4. If the proper number of variations of sign has not been
lost in this passage from m²=-∞o to m² = 0, this proposition
does not leave us without information as to the nature of the roots.
We infer that the number of real negative values of m² is equal
to or exceeds the number of variations of sign lost by an even
number and unless the number of variations be even the system
is unstable.
1
76
THE SUBSIDIARY DETERMINANTS.
2
[CHAP.
If we do not object to the labour of expanding the determi-
nants, we might extend this theorem to determine the positions
of the negative values of m² as well as their number. The number
of real negative values of m² between m² =—a and m²
- Bis
equal to, or exceeds by an even number, the number of variations
of sign lost in the series of determinants. In this form the
theorem resembles Fourier's theorem in the Theory of Equations.
5. The converse of this proposition has not been proved.
If in the passage from m³ ∞ to m² = 0, the number of varia-
tions of signs is unaltered, it is not true that the values of m²
cannot be real and negative. Thus in the simple case
119
22
2
-
m² - α11
а 12m
а12m
m² - A 22
2
0,
where a₁₁, α12 α are all positive, no variations are lost, yet if
a₁₂> √a₁₁ + √a the values of m² are real and negative. And if
√a₁+√a₂ the roots are equal and negative. It will be
noticed that the minors in this last case are not zero.
12
a12
α11
11
11
22
22
6. In order to discover the meaning of these losses or gains.
of changes of sign, it will be convenient to make such changes of
the co-ordinates as will simplify the dynamical determinant as
much as possible. Let us write
02.
2
A12)
V₁ = (E₁- A₁₁) + (E12 - A₁₂) 0$+...
11
If we now change our co-ordinates by writing for 0, p, &c.
linear expressions of some new co-ordinates, we know that we can
clear this expression of all the terms containing the products.
We know also that this can be done in an infinite number of
ways. We thus have
0.2
Ꮎ
Φι
V₁ = a₁₁₂² + a l₁² + ...
1
011
2
1
+α22 2
where the symbols e₁, 4,, &c. represent the new co-ordinates.
Again, let us consider the expression
0/2
W₁ = B₁
= B11 7 + B₁₂0'&' + .....
2
the coefficients are here the values of P, Q, &c. in the general
expression for T, Art. 1, Chap. IV. when 0, 4, &c. are all put zero.
But since T is necessarily positive for all values of §, n'..., n,...
it follows that W, is positive for all values of ', ' Hence
by a well-known theorem, we may by a real linear transform-
ation of the variables clear the expression W, also of the terms
v.]
77
MAXIMUM CRITERION OF STABILITY.
containing the products ', d', &c. and can make the coefficients
of the squares any positive constants we may please. We thus
have
12
Ꮎ &
12
W 1
1
+ +....
0 2 2
12
21
It may be shown that we cannot in general clear the expres-
sion for Tof all the terms containing Op', 'p, &c. unless C₁₂-C=0,
&c. by substituting for e, p, &c., any linear functions of other
variables. As this is only a negative result of which no further
use will here be made, it is unnecessary to supply the demon-
stration.
7. Using these simplifications the determinant ▲ will now
take the simpler form,
- '12'
A:
m² + aμ
117
a12m,
m² + α92'
-am,
= 0,
18'
asm,
23
m² + α33'
...
where a
21
α
- a₁m,
C-C₁₂, &c.
12,
If we form from ▲ the series of subsidiary determinants
A, A₁, A,...,
terminating with any positive constant, we see that when m²
-∞o,
these subsidiary determinants are alternately positive and negative,
and when m² = 0, they become
2
A11 а 22 A 33, A23 A 33..., а33а4, &c.,
1122
22
22
3344•••,
which are all positive if a, a... are all positive. Hence if
an, a... are all positive, the proper number of changes of sign has
been lost, and therefore the roots of the dynamical equation ▲ = 0
satisfy the condition of stability.
0
If a, a, &c. are all positive, we see that V, is a minimum for
all variations of ₁, ₁, &c. and therefore for all variations of the
original co-ordinates. If a, a, &c. are not all positive, there will
be as many variations of sign lost as there are positive quantities
in the series a, a, &c. In this case V, is a minimum for some
variations of 0, 4, &c. and not for others. It is shown in the
appendix to Williamson's Differential Calculus, that n independent
conditions are necessary that a quadratic expression of n variables
should be always positive. These are given in the form of deter-
minants and may be briefly summed up, in the statement that
X
11 1
2
12
2
78
MAXIMUM CRITERION OF STABILITY.
[CHAP.
L
#
is always positive if the roots of
A₁ + λ,
A 12'
A2 + λ,
A 23'
&c.
11
A12'
A13,
13'
&c.
are all real and negative.
A
137
231
&c. | = 0,
&c.
A33 +λ, &c.
&c. &c.
8. We may now put the proposition of Art. 3 into another
form. Let L be the general expression for the excess of the kinetic
energy over the potential energy of a dynamical system in terms of
its n co-ordinates x, y, &c. Let this system be moving in steady
motion with constant values of &c. Then if L be a maxi-
dx dy
dt' dt
>
dx dy
dt' dt
mum for all variations of x, y, &c. keeping at It'
then that steady motion is stable.
&c. unchanged,
If however only r of the n conditions necessary to make L a
maximum be satisfied, then the number of roots of the dynamical
equation which satisfy the conditions of stability is equal to r or
exceeds r by an even number. There cannot be stability unless n-r
is an even number.
dx dy
If the system be oscillating about a position of equilibrium,
&c. are all zero, and this leads at once to the condition,
dt' dt
that the equilibrium is stable if the potential energy is a minimum.
>
19 2
9. In this reasoning, we have for convenience excepted the
case in which two successive determinants in the series ▲, ▲₁, ▲„…….
vanish for the same value of m². But this exception is of no real
importance, for we may change these determinants into others
whose constituents are very slightly different from those of the
given determinants but which are such that no successive two of
the series have a common root. In the limit, therefore, when
these arbitrary changes of the constituents are indefinitely small,
the roots of the series of determinants will still be real under the
same circumstances as before, and the roots of each will separate,
or coincide with, the roots of the next above it in the series.
ann•
To show that these changes are possible, let us consider the
row of determinants beginning at the last. The determinant A„ is
a positive constant, the next An-1 is m² + a Proceeding thus,
suppose we arrive at two determinants which we may call ▲, and ▲
which have a common root. If we now change the constituents
a11, α12, α13, &c. into a + Sα, a₁₂+ da₁₂, &c. we do not alter ▲,, but,
except for the root m² = 0, we do alter ▲ in an arbitrary manner.
a
11
2
11' 12
127
1
v.]
79
EFFECT OF EQUAL ROOTS.
13
When for example a,, is altered, we alter a constituent both in the
first row and in the first column. Since
13' 31
33
13'
13
where A' is the determinant formed from ▲ by omitting the first
and third rows and columns, we see that when ▲ and ▲, both
vanish, the product ▲▲ and therefore by Art. 2 both A and
A must vanish. The determinant A is therefore altered by
▲₂ (Sa₁)³ which does not vanish, since A, is by hypothesis finite
for the particular value of m² under consideration.
31
2
13.
2
If any determinant of the series vanishes when m² = 0, it is
clear that one of the quantities a, a... must be zero.
If we
replace this by any small positive quantity, the argument will
apply as before.
10. It is important to consider the effect of equal roots on the
test of stability given in Art. 8 of this chapter.
11
In this case we know that the roots of the minor A,, separate
the roots of A. If therefore ▲ have two negative equal roots, it is
clear that ▲„ must have one of them. In the same way ▲22, A3,
&c. must also all vanish for this value of m. Since
11
ΔΔ'
▲▲' = Apр ^qq — AµA
pp
pq
pq
gp
it follows as in Art. 2 if ▲ and ▲ both vanish, that ▲, and ▲
also vanish. Hence all the first minors of the determinant ▲
vanish. This is the case considered in Art. 5 of Chap. I. The
equal roots instead of introducing into the expressions for the co-
ordinates terms which contain t as a factor merely render two of
the coefficients, instead of one, indeterminate.
In the same way if the equation A=0 is satisfied by three
values of m² equal to the same negative quantity, the equation
App
=0 must have two of them, and its principal minor must have
one of them. Reasoning as before we see that all the second
minors of ▲ must be zero. This is again the test that there should
be no terms which contain t as a factor.
The presence therefore of equal roots does not in the theorem
of Art. 8 affect the stability of the motion.
When a system is disturbed from a position of equilibrium
whether stable or unstable, the roots of the fundamental determi-
nant are separated by its minors in the manner described in
Art. 8 of this Chapter. By what has just been proved, we see
that if the fundamental determinant have equal roots, whether
positive or negative, these do not introduce into the integrals
terms which contain t as a factor.
80
[CHAP.
EFFECT OF EQUAL ROOTS.
"
2
11.. The proposition in the last article may be made more
general. If the fundamental determinant be reduced to the form
indicated in Art. 7, we shall show that if A vanish for two equal
negative values of m³ which are numerically greater than the
greatest negative quantity in the series a, a, a, &c., then
these equal roots will not introduce any terms into the solution
with t as a factor. If a, a, &c. are all positive, this reduces to
the proposition proved in the last article.
Following the same notation as before, we have
11
AA':
▲▲´= A₁₁492-A12^21°
A₁1^22 — ^12^21′
If neither ▲₁, nor ▲ are zero, they must have the same sign
when ▲ vanishes for a negative value of m². For their product
A
is equal to ▲₁₂ which has been proved in Art. 2 to be positive.
Hence all the leading first minors, viz. A1, A2, &c. must have the
same sign for any negative value of m² which makes ▲ vanish.
12 21
By differentiation we have
dA
dm
2
119
2m^11 +α12^12 + ….. – α¿½ª½ + 2mA22 +…….
— +... - &c.
But we have also
12
▲ = (m³ + α₁) ▲₁1 + α12m ▲ 12 +
A =
11
▲ = − a₁₂m ▲ 21 + (m² + α¸¸) ≤2 + ...
&c. = &c.
12'
22
Hence if n be the highest power of m occurring in ▲, we have
m
dA
dm
· = n▲ + (m² — a₁) 411 + (m² - α22) ≤ 22 - &c.
d▲
dm
▲22
If then ▲ and both vanish for any negative value of m²
We
greater than the greatest negative quantity in the series a₁₁, 22,
&c., we have the sum of a number of quantities all of the same
sign equal to zero. This requires that each should be zero.
have therefore A„=0, ▲„=0, &c. The rest of the proof is the
same as before.
11
22
d▲
dm
By differentiating the expression for and substituting for
dd22 &c. their values in terms of their first leading minors,
11
dm dm
we may extend this proposition to the case in which the funda-
mental determinant has three equal roots, and so on.
v.].
81
DYNAMICAL EXAMPLE.
12. A sphere is suspended by a string OA from a fixed point
O, and is set rotating about a vertical diameter which is in the
same straight line as the string with an angular velocity n. A
small disturbance is given, determine if the steady motion is
stable
Let O be the origin, and let the axis of z be vertically down-
wards, let lx, ly, I be the co-ordinates of A, the point at which the
string is attached. Let C be the centre, and let a§, an, a be the
co-ordinates of C relative to A. Then, exactly as in Chap. IV. Art.
22, the Lagrangian function may be shown to be
:
k² Én'
2
2 2
2
L =
エーロ ​{メー​+5+6+}
x² y²
− {l
—
+ } (a5" + la')* + 4 (an' + ly')° − g {¿®++ a² + "},
the mass being taken unity.
2
2
Putting xn, &, n', x, y' all zero, we see that L is a maxi-
mum when x, y, §, ʼn are zero, the steady motion is therefore stable
for all values of n.
If we put k² = a², and m² λ², so that x, y, &c. are all repre-
sented by terms of the form ΣA cos (t+a), we may, by the
methods of the last chapter, prove
λ ηλ
(x² — 4) (x² ± nλ −
5ひ
​5g
2
2a 21
This equation, whatever may be the sign of n, has two positive
and two negative roots. All four give stable oscillations.
R. A.
6
.
...
1
CHAPTER VI.
If the energy of the system be a maximum or minimum under certain
conditions, the motion whether steady or not is stable. Arts. 1-3.
When the motion is steady, it will be also stable if a certain function of
the co-ordinates called V + W is a minimum. Art. 4.
If there be only one co-ordinate which enters into the Lagrangian function,
except as a differential coefficient, this condition is necessary and
sufficient. Arts. 5, 6.
Additional conditions when there are two co-ordinates, Art. 8.
1. When a system is oscillating about a position of equilibrium,
it is well known that we may determine the stability or insta-
bility of the equilibrium by what we may call the "energy crite-
rion." This criterion may also be sometimes used when the
system is oscillating about a state of steady motion.
Let E be the sum of the kinetic and potential energies of the
system. Then throughout any motion of the system we have
E=h,
where h is a constant depending on the initial conditions. If 0, Φ,
&c. are the co-ordinates of the system, E is a known function of
0,0', o, p', &c. Suppose that some of the other first integrals of the
equations of motion are known. Let these be
F, (0, 0′, ò̟, p', &c.) = C₁
1
F', (0, 0′, 4, $', &c.) = C₂
the time t being absent.
For the purposes of this proposition let us suppose 0, 0, 4, d′,
&c. to be separate variables unconnected with each other except
by the equations just written down.
If E be an absolute maximum or an absolute minimum for
all variations of 0, 0, &c., those corresponding to the given motion
making E constant, then that motion is stable for all displacements
which do not alter the constants C₁, C₂, &c.
"
CHAP. VI.]
83
ENERGY TEST OF STABILITY.
If this proposition be not evident, it may be proved by elimi-
nating as many of the letters as possible. If 0,0', &c. be the re-
maining co-ordinates we have
E=ƒ(0, 0′, &c. C₁, C₂, ...).
Let h be the value of E in the given motion, and let the
system be started in some slightly different manner so that
E=h+Sh.
If Ę be a maximum along the given motion, then any change
whatever in 0, 0', &c. decreases E. Hence 0, 0', &c. cannot deviate
so much from their values along the given motion that the change
in E becomes greater than Sh.
2. Let us apply this principle to a system of bodies which
moves in steady motion with some co-ordinates 0, 0, &c. such that
their differential coefficients 0', ', &c. are constant, and the re-
maining co-ordinates &, n, &c. themselves constant. Let us further
suppose that the energy is a function of e', ', &c., but not of
0,4. By Lagrange's Equations we have the integrals
dT
do
=
C₁
dT
dø
=
C₂, &c.
It is clear that the system can describe any one of a number
of steady motions, which we have already called parallel motions,
and which are determined by
Ꮎ Ø = p,
$' = q,
&c.
n = B, &c.
where p, q, &c. a, B, &c. are constants which satisfy all Lagrange's
equations.
...
We have thus as many relations between these constants as
there are co-ordinates, n, &c. Let the system be started with
any initial conditions we please, then the constants C₁, C₂, are
given. These being known we have as many relations between
the constants of steady motion as there are co-ordinates 0, 0, &c.
The steady motion is therefore determined. If the energy of this
initial motion is nearly equal to that of this steady motion, and
if it be a maximum or minimum as explained above, then the
system will never deviate far from its corresponding position in
the steady motion, and this steady motion may be called stable.
3. Example. A top is set spinning on its point on a perfectly
rough horizontal ground, with its axis inclined to the vertical, find
the condition of stability.
Let be the inclination of the axis OC of the top to the
vertical OZ; ↓ the inclination of the plane ZOC to a vertical
•
6—2
84
[CHAP.
ENERGY TEST OF STABILITY.
plane fixed in space, and & the inclination to a plane through OC
fixed in the body. Let O be the apex, G the centre of gravity
which lies in OC, h= OG. Let A, A, C be the principal mo-
ments of inertia at O, and M the mass of the top. We have then
C
A
2
E == (p' + y' cos 0)² + (0′² + sin²0¥”) + Mgh cos 0.
By Lagrange's equations we have the integrals
&' + ' cos 0 = n,
C'n cos 0+ A sin²0y'′ = m,
where n and m are two constants, the former representing the
angular velocity of the top about its axis, and the latter the
angular momentum about the vertical. If we now eliminate
p' and 'we find that E is a minimum when 0 = a, if
C²n² > 4Mgh A cos α,
which is the result given by other methods.
4. The theorem of Art. 2 may be put into another form.
Let the kinetic energy be
0/2
Ꮎ
T = Tee 2 + Tes O'p' + ...
Then since 0, 4, &c. are absent from the coefficients, we have
the integrals
Tée0' + Te¿p' + ... = C₁ - Toç§' — Tonn — &c.
=
Τφέξ
Tø¿0' + T¿¿p'+... C₂-To-Tonn' - &c.
&c. = &c.
For the sake of brevity let us call the right-hand sides of these
equations C₁-X, C-Y, &c. Since T is a homogeneous func-
tion of e', ', &c., we have, as in Chap. IV. Art. 21,
દુઃ
'દુ
T=TE ²/2 + Tεne 'n' + ...
+ 1 0 (C₂ + X) + 1 ☀′ ( Œ₂ + Y) + ...
1
2
If we substitute in the second line the values of e', ', &c.
found by solving the linear equations just written down, we have
the determinant
1
0,
C₁₂+X, C₁₂+ Y, &c.
2
2A
C₁-X,
Tee,
Teş,
&C.
C₂- Y,
Ted
Т
Top
&c.
&c.
&c.
&c.
&c.
VI.]
85
MINIMUM TEST OF STABILITY.
where A is the discriminant of the terms in T which contain ', d'
&c. This determinant is unaltered by changing the signs of X, Y,
&c. and is a quadratic function of C₁, C, &c., X, Y, &c. Hence
the terms CX, CY, &c. do not occur.
If then we put
W
1
0
C₂,
&c.
24
C₁, Tee₂ Top,
&c.
2
C₂, Top, Tpp,
&c.
we have
T=
&c. &c. &c. &c.
و سچ
કૃ
BEE² + Ben & n' + &c. + W,
2
where B, &c. are independent of C₁, C₂, &c. Now Tis essentially
positive for all values of the variables, and therefore for such as
make C, C₂, &c. all zero. Hence the terms involving ', n', &c.
are together a minimum when ', n', &c. are all zero. The coeffi-
cients B, &c. may all be treated as constants since έ', n', &c. are
all small quantities.
If V be the potential energy, we have therefore the following
rule. If W+V be a minimum for all variations of E, n, &c. then
the steady motion is certainly stable. It should be noticed that
W+V is a function only of E, n, &c. the co-ordinates which are
constant in the steady motion.
5. If the energy be a function of one only of the co-ordinates,
though the differential coefficients of all the others enter into its
value, this condition is sufficient and necessary.
Let & be this co-ordinate. Then by vis viva we have
2/2
BEE 2/2 + W + V = h.
Differentiating we have
d (W+V) = 0.
BEEE+
αξ
This equation must be satisfied by the steady motion repre-
sented by =a. The second term
=
d(W+V) must therefore
de
vanish when έa, so that W+V is a maximum or minimum.
To find the oscillation let us put §= a+x, we find
X ď² (W+ V)
/2
B
+
2
de
x = 0,
.
86
[CHAP.
MINIMUM TEST OF STABILITY.
where the square bracket implies that is to be put equal to a
after differentiation. By the same reasoning as before B is
necessarily positive, and the motion will be stable or unstable ac-
cording as (W + V) is a minimum or maximum.
6. If we refer to Art. 21 of Chap. IV. we see that this function
W+V is the value of the modified Lagrangian function L' when
', n', &c. are all put zero, and the sign of the whole function
changed. It therefore follows by Chap. v. that when W+V is a
minimum the steady motion is stable. The "energy criterion of
stability," as far as it applies to steady motion, may therefore be
deduced from that given in Chap. v. Art. 8. But the mode of
demonstration adopted in that chapter gives us more information
as the nature of the motion, while the modes of application to
examples of the two criteria are quite different. The energy
criterion may also be sometimes applied to determine the stability
of a motion which is not steady.
[The relation between the theorem in this Chapter in which
E=T+V is made a minimum to that given in Chapters IV. and V.
in which L=T-V is made a maximum may be more distinctly
perceived by the following statement.
Let x, y, &c. be the co-ordinates of the system, and let Z be
the Lagrangian function* so that LT- V, then by Art. 3 of
Chap. IV. the co-ordinates in steady motion satisfy the equations
dL
dx
dL
0,
0, &c.
dy
Here L is expressed as a function of x, y, &c. x', y', &c.
.(1).
Suppose some of the co-ordinates as 0, 4, &c. are absent from
the expression for L, so that L is a function of e', ', &c., the re-
maining co-ordinates, viz. §, n, &c. and their differential coefficients.
Then if we form the modified Lagrangian function as in Art. 21
of Chap. IV. the equations (1) of steady motion become
dL'
αξ
dL'
=
0,
= 0, &c..............
dn
(2).
Here, as in the Hamiltonian equations C₁, C₂, &c. are the
0, 4, &c. components of momentum, and L' is expressed as a
function of C,, C₂, &c. §, n, &c. &', n', &c.
27
dT
* Let u, v, &c. be the x, y, &c. components of momentum, so that u= &c.
dx'
and let Ħ be the Hamiltonian function. Then H=T+ V and we easily deduce
from the Hamiltonian equations that, in steady motion,
dH
0,
dx
dH
=0, &c.
dy
Here H is expressed as a function of x, y, &c. u, v, &c.
VI.]
87
APPLICATION TO TWO CO-ORDINATES.
But
L' —T— C¸ơ′ – С₂$′ − &c. – V
dT dT
T+ § +
αξ dn' 'n' + &c.
- V,
by Euler's theorem on homogeneous functions. In steady motion
§', n', &c. all vanish, hence the equations (2) become
d(W+V)
αξ
d (W+V)
dn
! = 0, &c...
(3),
where W is the value of T when έ, n', &c. are all put equal to
zero. Here W is expressed as a function of C₁, C,, &c. §, n, &c.
It is shown in Chap. v. that if the Lagrangian function ex-
pressed as required in equations (1) be a maximum the motion
is stable. It is shown in this Chapter that if the function W+V
be a minimum the motion is stable.]
7. To find the condition of stability when the Lagrangian
function is a function of two only of the co-ordinates, though the
differential coefficients of all the others enter into its value.
Let E, n be these two co-ordinates, then the modified Lagran-
gian function as explained in Art. 20 of Chap. IV. will be a func-
tion of E, n, &', n' only.
Let the steady motion be given by = a, n = 6, with the
corresponding values of the other co-ordinates e, o, &c. Then a
and are constants. Let = a + x, n = ẞ+y, and let us expand
the modified Lagrangian function in powers of x, y. Neglecting
the terms of the first order, as they only give the steady motion,
let
12
12
y
I' = B₁₂ + B¸¹'y' + B₂
Bu 2
21
x²
B₁xx'y'
22 2
+ A₁₂+AY+Am
A, + Apxy + Amy²
11
2 12
2
+С₁ïï' + C¿½ïу' + С₂yx' + С„Yy'.
Also let E= C₁₂- C₁₂. Then the condition of stability is that
the roots of the following equation should be of the form ẞ√-1. -
If
and
2
В₁m² — µ?
В¿m³ — А „— Em,
12
B₁m² - A+ Em = 0.
12
12
В„m² — А„
A' = B₁₂B
11
22
- B₁₂²
2
2
=
22
12
✪ = A„В₂+ A„В₁ – 2А1½В₁½
11
11
2A1,
88
[CHAP.
APPLICATION TO TWO CO-ORDINATES.
be the two discriminants and the other invariant, this leads at
once to the conclusion that the motion is stable only when
If A
x²
11 2
(1) A is positive,
(2)
E² - Ⓒ is positive and >2 √ÄÄ’.
+Axy + A² is a maximum when a and y are zero,
22 2
the two conditions are obviously satisfied.
This condition may be otherwise expressed; if L' be the modi-
fied Lagrangian function, a steady motion is given by
dL'
= ·0,
αξ
dL'
dn
0,⋅ §' = 0, n' = 0.
This motion will be stable if for the values of E, n thus found,
(1)
d²L' d²L' ď²L'
2
de_dm dędn
is positive,
2
ďL' d²L' (d²E' d²L' d²L' d²L'
didn'de dn
(2) (dědní
is positive and greater than
(d'L' d'L'
2
αξε
{de² dn
12
+
d'L' d²L'
2
{de² dn" + dn³ de" - dedn de'dn')
}
d² L' ) }
d²L (d²L' dºL'
dedn) · (de dn'" — de'dn'
8. The nature of the motion when thus reduced to depend on
two co-ordinates may be illustrated by geometrical reasoning. Let
the position be defined by two co-ordinates x, y which are zero
along the steady motion. Let these be regarded as the co-
ordinates of a point P referred to any axes. Then the motion of
P exactly represents that of the system.

Let us construct the conics
2
y²
Aux² + Axу + Amy ²
11 2 ∙12′
2
α,
B₁ ½ + By + B₁₂ y² = b.
Bu 2
22 2
Then, exactly as in Arts. 11-14 of Chap. IV., it will be found
convenient to transform the co-ordinates by writing
x = a₁p + b₁ql
y = a₂p + b₂qf⋅
12
If μ be the modulus of transformation we have µ = ab₂ — ab₁.
It is easy to see by actual substitution, if E' be the difference of
the coefficients of pq and p'q, that
E' =µE.
VI.]
89
APPLICATION TO TWO CO-ORDINATES.
:
If the transformation be from one set of oblique co-ordinates
to another, let w, o' be the angles between the axes. We then
have
E'
E
sin o'
sin w
Transforming the axes to the common conjugate diameters, the
conics become
p²
2
An
+ A
OF
p²
B' + B = b
11 2
the signs of a and b being such as to make these conics real. The
equation to find m becomes
2
(Вm² — A„') (B„'m² — A„') + E”²m² = 0.
22
It is therefore necessary for stability that the conic a should be an
ellipse as well as the conic b. It is also necessary that
E'
>
√B,,'B
11
22
A
B
11
+
11
A
B
22
22
both roots having the same sign and the inequality being nu-
merical.
Let OP, OP; OQ, OQ be the common conjugates of the two
conics, this condition then becomes
E area of conic b
>
sin w
π Nab
O Q. O Q
+
OP OP
୦୯
When the system describes an oscillation with one period, i.e.
an harmonic oscillation, the path of the representative particle is
easily seen to be
(B„"m²
2
(В‚‚”'m³ — ¸”) p² + (B„"m³ — A¸”) q' = constant.
11
11
22
22
The harmonic paths are therefore ellipses. It also appears that
the two ellipses which represent the two harmonic vibrations and
the two ellipses a and b have, all four, a common set of conjugate
diameters.
CHAPTER VII.
Any small term of a high order, if its period is nearly the same as that of
an oscillation of the system, may produce important effects on the
magnitude of the oscillation. Art. 1.
Origin of such terms, with an example. Arts. 2-3.
Supposing the roots of the determinantal equation to satisfy the conditions
of stability to a first approximation, yet if a commensurable relation
hold between these roots it is necessary to examine certain terms of the
higher orders to determine whether they will ultimately destroy the
stability of the system. Art. 4.
If a certain relation hold among the coefficients of these terms, they will
not affect the stability of the system, but only slightly alter the periods
of oscillation. Arts. 5—7.
Examples, the first taken from Lagrange's method of finding the oscilla-
tions about a position of equilibrium. Arts. 8-9.
If the coefficients of the equations of motion should not be strictly con-
stant, but only nearly so, the stability will not be affected, unless the
reciprocals of their periods have commensurable relations with the
reciprocals of the periods of oscillation of the system. Art. 10.
1. If we understand that a motion is called stable when any
small disturbance does not cause the system to deviate far from its
undisturbed motion, it is clear that we cannot be certain of the
stability without examining the terms of the second order. It is
possible that some of these may have their periods so timed that
their effects may accumulate until the motion is changed.
Returning to the equations of Art. 3, Chap. I. we shall have on
the right hand, instead of zero, a series of small terms of orders
higher than the first. To find a second approximation, we substi-
tute the values of x, y, &c. given at the end of Art. 3, in these
terms.
They will therefore take the form Net and will produce in
N'
x, y, &c. terms of the form ent, where N' is of the same order
f(n)
at least as the term considered, and ƒ (m) has the same meaning as
"
CHAP.VII.] EFFECT OF SMALL TERMS ON THE ARC OF OSCILLATION. 91
in Chap. I. These will have to be expressed in trigonometrical
real forms, but it is unnecessary to exhibit the process, for we see.
at once that no small term or force (whatever it may be called) of
a high order can affect the stability of the motion unless it makes
f(n) very nearly or exactly equal to zero. In this case its period
is very nearly or exactly equal to one of the periods of the motion
given by taking terms of the first order only.
A remarkable use of this principle was made by Captain Kater.
in his experiments on the magnitude of gravity. It was important
to determine if the support of his pendulum was perfectly firm.
He tells us that he had recourse to a delicate and simple instru-
ment the sensibility of which was so great that had the slightest
motion taken place in the support it must have been instantly
detected. The instrument consists of a steel wire the lower part
of which, inserted in the piece of brass which serves as its support,
is flattened so as to form a delicate spring. On the wire a small
weight slides by means of which it may be made to vibrate in the
same time as the pendulum to which it is to be applied as a test.
When thus adjusted it is placed in the material to which the
pendulum is attached, and should this not be perfectly firm its
motion will be communicated to the wire, which in a little time
will accompany the pendulum in its vibrations. This ingenious
contrivance appeared fully adequate to the purpose for which it
was employed, and afforded a satisfactory proof of the stability of
the point of suspension. See Phil. Trans., 1818.
·
2. Since the term Nent is obtained by compounding the
different terms in the values of x, y, &c. it is clear that
n = pm₂+qm 2
where p, q, &c. are positive integers whose sum is the order of the
term. It is therefore only when the roots of the dynamical equa-
tion ƒ (m) = 0 are such that a linear relation of the form
pm₁ + qm₂ + ... m, very nearly
1
exists between them, that we may expect to find important
terms among the higher orders. The order of the terms to be
examined will be p+q+..., and unless this be also small, the terms
will probably remain insignificant. If the root m, should occur
twice in ƒ (m) = 0 it is clear that the divisor ƒ (n) will be a small
quantity of the second order, and the term may be said (as in
the Lunar Theory) to rise two orders.
3. To take an example, let us suppose a particle to be describ-
ing an ellipse about a fixed centre of force in one focus. If dis-
turbed it will describe a slightly different ellipse.
r = a{1-ecos (nt + e − w) + ...},
Ө
0 = nt + € + 2e sin (nt + e − w) + ...
Since
92
[CHAP.
EFFECT OF SMALL TERMS
1
"
we see that a slight change in the elements will cause variations
in r and ◊ of the period an additional variation in 0 of the
2π
N
form ton+de and an additional variation in r of the form da. All
these variations should by Art. 3 of Chap. I. be indicated by
expressions of the form
r=ΣMe 0=ZM'eme
where the values of m are the roots of the equation f(m) = 0.
The roots therefore of the equation ƒ(m) = 0 for dr are m 0
and ±n√-1, and for 80 are m=0, 0 and ±n-1. We there-
fore infer that any small disturbing causes of the second order
whose periods are nearly equal to that of the particle, will cause
important inequalities in both dr and 80, and (since ƒ (m) = 0 has
two roots equal to zero) any term of long period will rise two
orders in 80.
4. If the roots of the subsidiary equation are such that the
relation
pm + qm₂ +... = m₁
holds accurately, the solution changes its character. We have
now in the value of x a term of the form temit. Unless the real
part of m, is negative, this indicates that the system will depart
widely from the motion which we took as a first approximation.
We must therefore modify our first approximation (as in the Lunar
Theory) by including in it the terms which produced these im-
portant effects. We may then enquire how far this modified first
approximation indicates that the motion is stable or unstable.
When these terms are included the equations to be solved are in
general no longer linear, and it is sometimes impossible to find a
solution sufficiently, accurate to serve as a first approximation
throughout the whole motion.
5. In some cases, however, the oscillations may still be
represented by expressions of the form
2
x= M₁e",* + M₂e";*+
y= Me"¹² + M₂'e”½¢ +
&c.,
2
n
where the values of n₁, n... differ but slightly from the roots of
the equation f(m) = 0. Let us investigate the condition that this
should be true, and also determine whether the changes in the
values of m₁, m…….. are sufficient to affect the stability.
Suppose that we have completed our first approximation, and
find on proceeding to a higher approximation that the terms
N₁em₁t + Nemat + ...
Ne + Ne" + ...
VII.]
93
ON THE PERIOD OF OSCILLATION.
present themselves on the right-hand side of the first set of
equations in Art. 3, Chap. I.
These terms are supposed to have
arisen from several relations of the form
1
pm₁ +qm₂+... = m..
1
If these terms can be included in the first approximation by
writing n₁, n,, &c. for m,, m, &c. we have, by substitution in the
differential equations, certain equations connecting n, M, M', &c.,
whose left-hand sides are the same as those used in Art. 3 with
n, written for m,, but on the right-hand sides we have instead of
zero the quantities N₁, N, &c. The test of the success of the
process is that these modifications in the values of m must satisfy
the same relations as before.
Now N₁, N, &c. are all at least of the second order of small
quantities, hence up to terms of the first order the ratios M₁, M₁,
&c. will be the same as before, so that we may put
M₁ = L₁α₁, M₁ =L₁b₁, &c.,
1
19
following the same notation as in Art. 3, Chap. I. We also have
M₁ƒ(n) = Ñ‚α₁ + Njaj' + ...
Let n₁ = m₁ + Sm, we find
N₁α₁₂+ N₁α₁ +
Sm₁
ƒ' (mr) Lar
Similarly
N¸à¸+ N₂ ɑ'+
Sm₂ =
22
&c. = &c.
2
ƒ (m₂) L₂α₂
It is evident, by the theorem of determinants alluded to in
Art. 3, that these are symmetrical expressions.
6. We may conveniently express these results in the form of a
rule.
Suppose we have to a first approximation
x = M₁em, + M₂et +...
Eliminate from the differential equations all the variables except
x in the usual manner. This may be done by performing on the
several equations the operations represented by the minors a, a',
d
&C., at
being written for m. Let the equation thus found be
f
x = Pe" + P₂e"...
•
94
[CHAP.
EFFECT OF SMALL TERMS.
Then all these terms can be included in the first approximation,
provided
Sm₁
=
P₁
Mf (m)
Sm₂
=
P₂
Mf'(m.)
&c.
satisfy the relations
pdm¸ + qồm„ + ... = dm,,
2
&c. = &c.
which exist among the roots of the dynamical equation.
7. The general results we have arrived at may be summed
up as follows. Though some of the terms of the higher orders
may affect the magnitude of the oscillation, yet no term will arise
to affect the stability of the motion unless there be some relations
between the roots of the dynamical equation of the form
pm₂+qm₂ + = m₁;
where p, q, &c. are all integers. Even if such relations occur, the
lowest order of the term is p+q+..., and if this be considerable
the term will not produce any important effects until a con-
siderable time has elapsed. If a certain relation, just found, hold
among these terms, their only effect is slightly to modify the
periods of oscillation, without altering the type of motion.
8. As an example, let us consider a system of bodies to be
oscillating about a position of equilibrium. We know by Lagrange's
general solution, that the equation f(m) = 0 is of an even order.
Its roots are of the form
α
m.
3
m₁=a√—1, m¸=-a√-1, m¸=B√√−1, &c.
Whatever the numerical values of these may be we have
m₁ + m² + m₁ = m1
1
3
4
so that the small terms of the
m₂ + m² + m₁₂ = m 2›
3
4
M2?
&c.
third, fifth, &c. orders might affect
the stability of the oscillation. But we shall now show that they
only affect the periods of oscillation, and not the stability of the
system.
Since both sides of Lagrange's equations must be of – 2 di-
mensions in time and the impressed forces are also of -2 dimen-
sions, it is clear that these terms must consist of powers of x, y, &c.
d'x d'y
&c. and products of an even number of factors of
dt² dt2,
dx dy
&c. We know also, by Lagrange's solution, that the
dt dt'
co-ordinates take the form
x=
M₁ cos (xt +λ₁) +&c.
y = M'' cos (zt +λ₁) + &c.
.VII.]
95
CRITERION OF STABILITY.
d
The minors a, a, &c. are also all even functions of hence
the equation found after elimination is of the form
dt
>
$ (
d
x= P cos (at +λ) + Q cos (ẞt + µ) + ...
dt
Replacing a√-1, -a√-1, &c. by m₁, m,, &c. we find by
Art. 6,
P
Sm₁
Sm₁ = •
2M¸ƒ' (m¸)'
P
2M₁ƒ'(m¸)'
2
Since f'(m) is of odd dimensions, and m=-m, we clearly have
Sm₁=-&m,, and therefore the test is satisfied.
δη
1
9. [As another example let us apply the rule of Art. 6 to
some very simple case which will involve no algebraical substi-
tutions of any length.
The motion of a simple pendulum under the action of gravity
may be made to depend on the equation
d²x
dt2
+ a²x=Bx³
(1),
where a and B are two constants and x is the inclination of the
pendulum to the vertical which is supposed to be small. The first
approximation to the motion is
x = M₁emit +M₂emat
(2),
where m, and m, are the roots of the equation m² + a²=0. Our
object is to ascertain by help of the rule given in Art. 6 whether
the small force represented by Ba³ renders this first approximation
unstable or merely slightly alters the numerical values of m, and
m₂•
The two roots are connected by the relation
m₂+m₂ = 0.....
2
(3).
Substituting the value of x on the right-hand side, we have
(a+c²) M¸M,
+a²) x=3ẞ M₁M₂ (M¸em+ Mem₂t) + &c.
Hence by the rule in Art. 6
Sm
1 2
3ẞM, M
2m,M,
1
2
Sm₂
38MM,
2m,M,
These clearly satisfy the relation
Sm, + Sm, = 0,
1
2
96
[CHAP.
EFFECT OF SMALL TERMS.
and therefore the first approximation taken above, so far as the
disturbing force Ba³ is concerned, is stable.
If the small force had been 8 (da)
ß
instead of ẞx³, it is easy
dt
to see in the same way that
3ẞM, M₂m²m,
Sm₁
2
2
Sm
1
3ẞM,M²mm
2
2m,M₁
1
2
2m,M₂
so that the relation Sm+Sm₂ = 0 would not have been satisfied.
The first approximation taken above is therefore not sufficiently
accurate to serve as a first approximation throughout the motion.
In this example we have considered the effect of a small force
of the third order in disturbing the stability of the motion given
by equation (1). The same equation will obviously occur in many
other cases of motion. For example, let a particle describe a cir-
cular orbit about a centre of force situated in the centre. If
slightly disturbed the equation giving the disturbance ≈ in the
radius vector takes the form
d²x
dt²
+a³x=ẞx² + yx³ +.
where a, ẞ and y are constants. Similar remarks will therefore
apply to this case also.]
10. When the coefficients of the equation of motion are not
strictly constant, but yet do not vary much, then we may transpose
the small variable parts of these terms to the right-hand side of
the equation, and treat their products by the differential coeffi-
cients of the co-ordinates as small quantities of the second order.
Suppose the variable part of one of these coefficients to be p sin nt,
where p is small, and let ƒ(m) = 0 be the equation giving the
periods of oscillation of the system when the coefficients are taken
constant. Then it is clear that unless n is nearly equal to the
sum or difference of two values of m, this term cannot rise into
importance. On proceeding to higher orders we see that these
terms cannot produce important effects unless some commen-
surable relation between m and the roots of the equation ƒ (m) = 0
should be very nearly satisfied.
11. It should be remarked that when the coefficients are not
constant it is not a sufficient test of stability that they should
always satisfy the conditions of stability obtained by giving them
their instantaneous constant values. Thus if the equation of
motion were
d2x
+
dť² 4t2
1
x = 0,
VII.]
97
CRITERION OF STABILITY.
R. A.
the coefficient of x is always positive, yet as the equation is satis-
fied by xat, x may become as great as we please.
Even if the coefficients are nearly constant, we must yet ex-
amine, by the rules just given, if their small changes are so timed
as gradually to increase the oscillation until the divergence from
the given motion is no longer small.
[Suppose a system to be oscillating so that its motion is de-
termined by the equation
dx
dt²
+qx = 0,
where q is a known function of t, which during the time under
consideration always lies between 2 and B' the latter being the
greater. Let the system be started with an initial co-ordinate x
and an initial velocity x in a direction tending to increase x.
It may be shown that the system will begin to return, i.e. x will
2 20%
x² +
12
B2:
If
begin to decrease before x becomes as great as
±m, 7m' be two successive maximum values of x, we may also
show that m' cannot be so great as m, and that the time from
B
π
B
B
and 1.
one maximum to the next lies between
L
ม
3
་
1
CHAPTER VIII.
¡
The Hamiltonian Characteristic or Principal functions when found deter-
mine at once the motion of the system from one given position to
another, and whether the motion is stable or unstable. Arts. 1—3.
Examples with a mode of effecting the integration S = [Lat
lations Arts. 4—7.
in small oscil-
The Characteristic function supplies the condition that the motion is stable
as to space only, while the Principal function gives the conditions that
it is stable both as to space and time. Art. 8.
In what sense the motion is unstable if either of the two Hamiltonian
functions is a minimum. Arts. 9-14.
1. If we had any convenient methods of finding the Hamil-
tonian Characteristic or Principal function, we might determine
without difficulty the conditions of stability of a dynamical system
at the same time that we deduce the integrals of the equations of
motion. But it is very difficult to discover either of these func-
tions by an à priori method. We have indeed differential equa-
tions which they must satisfy, and Jacobi has taught us what kind
of solution will serve our purpose. But the difficulty of finding
these solutions is as great as that of solving the equations of
motion. For these reasons it does not seem necessary to dwell on
the uses of these functions.
Ꭴ
2. Suppose the Principal function S of a dynamical system to
have been found in terms of the initial co-ordinates 0, 4, and the
co-ordinates e, and the time t. Let the semi-vis viva be given
by
0'2
2
T= PT + Q&&' + R ²²,
R。
where P, Q, R are known functions of 0, p. Let P., Q., R, be
.3
+
CHAP. VIII.] THE PRINCIPAL AND CHARACTERISTIC FUNCTIONS. 99
the values of these when 6, 4, are written for 6, 6. The final
integrals of the equations of motion are then given by
ds
do。
ds
do。
=
· PO' + Qoto'
Q₂O!' + Roto
Let the system receive any disturbance at the time t=0, so
that while starting from the same initial position, its initial ve-
locities are slightly altered. Let x, y be these initial changes
of 0 and 6 and let 0+x, +y be the co-ordinates of the
system at the time t. Then we have
dod&y=Pox + Qoyo
d2 S
d2S
X
do do
dS
d² S
x
đẹp đó
d‡¸d$Y = Q。x''+R¸y.
Here x, y are small arbitrary quantities, hence x and y will
be small if none of the ratios of
to the determinant
d² S
d2S d28
do do do do do do
'
or
>
d² S
do do
be large.
des
d² S
độ
d's
do do do do
d²S d² S
do do do do¹
If the initial position as well as the initial motion be altered
we may find, by a precisely similar process, the conditions that
x and y should be small. If the system have more than two inde-
pendent motions, we have more than two co-ordinates, but the
conditions of stability are found in the same way.
3. If x and y be small throughout the whole motion from the
one given position to the other, not only does the system not
deviate far from its undisturbed course, but the system at any
instant is also very nearly coincident with its undisturbed place
at the same time. It is important to notice this, for the word
"stability" is sometimes used in a different sense.
This condition of stability may be put under a form in which
no reference is made to S. Let u and v be the components of
momentum of the system corresponding to the co-ordinates 0, p
dT dT
respectively, i. e. let u= v= and let these be expressed
>
do' do"
>
WorM
100
[CHAP.
THE PRINCIPAL AND
as functions of 0, 0, 0, 4 and t. Then the preceding equation
may be written
du
dv
de
x +
do y = a
0
du dv
x +
афо
dd Y = B
where a and B are small arbitrary quantities. The condition of
stability is that the values of x and y thus found should be small.
4. As an example let us consider the case of a projectile.
If be the horizontal, and the vertical co-ordinate of the par-
ticle, we have
S
(0 − 0.)² + ($ − 6.)²
2t
- 1 gt (4 + 0) — 24 9² t³
T = { (0²² + p'²)
The equations to find x and y are evidently
1
=
X = X
X
t
1
ty = y;
Hence the system continually deviates more and more from its
undisturbed place.
5. In order to calculate the form of S when a system is oscil-
lating about a state of motion, it is convenient to choose as co-
ordinates some small quantities x, y which vanish in the given
state of motion. Let the Lagrangian function be written in the
form
L=L₁₂+L₁+L₂,
where L is a homogeneous function of x, y, x', y'. Then by a
theorem of Euler's
n
dL
L = L₂+ Σ (dL₁ x + d +x') + + Σ
0
do
2
dL.
dly or ),
+
+ && (dl
2
x +
dx
where the 's imply summation for all co-ordinates.
As in Art. 9 of Chap. IV. we have
d dL₁ _ dL,
dt dx' dx
and the oscillations are given by
2
2
d dL₂_dL₂
dt da
dx
VIII.]
101
•
CHARACTERISTIC FUNCTIONS.
Hence we find´
d dL₁
dt dx'
dL
d d L₂+ x
2
dL).
dt
L = L₁ + Σ (x & dl ; + x' d ) + + Σ (x at da
Σα
Integrating we have
dx'
dLi
t
8 = √ Ldt + Σ [ = 24] +
0
+
dL.Jt
2
X dx'
0
•
Thus the integration has been effected, but in order to express
S as a function of x, y, x, y and t, it will be necessary to find
x', yo', x', y' in terms of these quantities.
6. As an example, let the position of the system depend on
one co-ordinate x and let
12
L = L¸ + A¸x + В¸½ + ½ ø¸ï² + ½ В₁x²² + С₁жж'‚
0
11
where the coefficients are all constants. We then find by the
process just indicated that A₁ = 0 and
1
C.
2
11
x²)
S = L¸t + B¸ (x − œ.) + − ¹¹ (x² — æº)
− x)
where m²
A,
B₁₁
11
d's
11
mВ₁ (x²+x¸³) (emt +e-mt) — 4xx。
+
2
11
emt — e-mt
>
Applying the criterion of stability we find that
will finally become small if m is real. The motion is there-
dx dx
11
fore unstable or stable according as A, B₁ have the same or
opposite signs.
7. If the position of the system depend on two co-ordinates
x, y, let
L = L₂+¸¤+øy+B¸x' +B₂y'
1
12
+ ½ А„x² + А¸½ xy + ½ А„ÿ² + ½ В„x²² + B₁₂x'y' + ½ В„y”
12
+ C₁xx' +С₁xy + С₂уx' + С₂ÿÿ'.
We then find
S=L₁t+B₁ (x-x) + B₂ (y — yo)
12
12
2
x²
2
+ C₁,
xo
12
11
2
C₁ + Co
+
2
21
(xy − x。y。) + С₂
y² — Y ¿
2
22
2
where
B..
σ =
B,
2
11
-B₁ (xx' — xx') + ¹²² (xy' + x'y−x.ÿ'! − x'y.) + B* (vy'−136).
12
22
2
K
102
[CHAP.
THE CHARACTERISTIC FUNCTION.
If we now express x, y, x, y, in terms of x, y, x, y, and t, we
find for σ a fraction whose numerator is a homogeneous quadratic
function of x, y, x, y, the coefficients being linear functions of
exponentials of t, and whose denominator is another linear function
of the same exponentials. These exponentials become sines and
cosines when the motion is stable. Thus when the given motion
is steady the simplest inspection of the form of σ will determine
whether the motion is stable or not.
Referring the motion to principal co-ordinates for the sake of
brevity, and writing 2G C-C, we find that a must satisfy
the differential equation
do 1 /do
=
12
2 1 /do
+ + Gy) + 2B dy
↑
dt 2B₁ dx
11
22
2
*
=
Gx) = 14₁x² + 1 A „у².
22.
This equation is obviously satisfied by such a function as that
just described. The solution of this equation may be reduced to
linear equations and thus o may be found. But it is unnecessary
to dwell on this, for this would be equivalent to returning to the
Hamiltonian equations.
8. If we wish to determine the condition that the general
course of a dynamical system is stable without requiring it should
be near its undisturbed place at any the same time, it is more
convenient to use the Characteristic function. Suppose that the
Characteristic function has in Jacobi's manner been expressed as
a function of the co-ordinates 0, 4, the constant h of vis viva and
two arbitrary constants a,, a. Then
V=ƒ (0, 0, h, a₁) +ɑ2°
The relation between 0 and p, which may be called the equa-
tion to the path of the system, is given by
1
dv
da₁
19
where b, is another constant. Let the system be disturbed from
the same initial position so that the whole energy is unaltered.
The change in corresponding to any given value of ✪ is found
from
dad&&&=8b¸.
d² V
ď² V
da,
δα, +
d² V
Let A be the initial value of
then
da,2,
2)
ď² V
da
2
δφ=
d² V
A
da,do
δα,
VIII.]
103
MINIMUM TEST OF STABILITY.
The condition that the path should be stable is that the coeffi-
cient of da, should not be large.
We might also use the function called Q by Sir W. R. Ha-
milton, but it seems unnecessary to dwell more on this subject.
9. The instability of a system may be deduced from the
Hamiltonian Principal or Characteristic functions, expressed as a
minimum. Suppose a dynamical system to move from one posi-
tion A to another B in a time t, then the motion may be found by
making the first variation of S= | Ldt equal to zero, the time of
rt
0
transit being constant. The constants of integration are deter-
mined by the conditions that the co-ordinates have given. values
when t=0 and t=t. To determine whether S is a maximum or
minimum or neither we must examine the second variation and
here we have the assistance of Jacobi's rule. The determination
of the constants will depend on the solution of equations and may
lead to several different kinds of motion from A to B. One of
these will be the actual motion. Let us move B along this until
one of the other motions coincides or as we may say approaches
indefinitely near to this actual motion. We have then reached
a boundary beyond which the integration must not extend if S is
to be a maximum or minimum. See Todhunter's History of the
Calculus of Variations, page 251. Further if be a co-
ordinate, is positive throughout the limits of integration, so that
S will be a minimum and not a maximum.
=
d² L
do¹²,
10. When there are several co-ordinates 0, 4, &c. which are
to be found as functions of the time, we may easily show that
Jacobi's condition is a necessary one, and this is all that we require
for the next proposition. If the system can move in two ways
from A to B, then SS-0 along each, and therefore when these
two are adjacent we have both SS-0 and 8(8+ SS) = 0. This
shows that the second variation can be made to vanish by taking
one variation through the other. This second variation will then
be the same as the quadratic term of the series obtained by
changing the co-ordinates , & into 0 + 80, & +84, because we can
take 800 and 80. Hence as the sum of the terms of the
third order does not, in general, vanish for this displacement, it is
clear that S cannot be either a maximum or a minimum.
Let the actual motion be from A to B, and let a neighbouring
motion starting from A lead the system to a position C reached in
the same time along the actual motion before reaching B. Then
we can show that a variation of the actual motion from A to B
can be found which makes 8'S of any sign. Let P be any position
1
104
[CHAP.
THE PRINCIPAL FUNCTION.
►
%
:
on the neighbouring motion before reaching C, and Q, one on the
actual motion after passing C. Then considering P and Q as fixed
and also the time of transit, the motion along PCQ cannot
make SS=0; for this condition is known to lead to the ordinary
dynamical equations, and it is clear that (impulsive forces being
set aside) no actual motion can be discontinuous. But there is
discontinuity at C, for otherwise when the system is started from
B towards A, two courses would be open to the system on arriving
at C. Hence the first variations of S for an imaginary motion along
PCQ are not zero, and therefore may have any sign. But since
the discontinuity at C is of the first order of small quantities, this
first variation is of the second order. Now the value of S for the
actual motion is equal to that along the neighbouring motion to
C and then along the actual motion to B. Hence, P and Q being
still fixed, variations of the actual motion from A to B can be
found which make SS of any sign,
11. Let us apply this theory to determine the stability of a
given state of motion. First let us suppose the given motion to
be steady and to depend on only two variables. If we use the
function S there will be one co-ordinate and the time, if V two
co-ordinates. Let the system be disturbed at any moment by an
alteration of the velocities of its several parts, so that the initial
position of the disturbed motion is an undisturbed position. If
the motion be stable the system will oscillate about the un-
disturbed motion, the oscillation repeating itself at a constant
interval. It follows therefore by Jacobi's rule that S or V cannot
be a minimum for a period longer than the time of a half-oscilla-
tion. If therefore S or V be a minimum for all variations, starting
from A and ending at B, where B is a position on the steady
motion reached by the system at an interval as long as we please,
then the motion is unstable.
If we give a meaning to the word "stable" somewhat different*
from its usual signification, we may extend this proposition to
'determine a test of the stability of any motion, whether steady or
not. All we have assumed is, that, if the motion be not altogether
unstable, there are some disturbances which will cause the system
periodically to assume the same positions as it would have done if
it had been undisturbed, but the interval of these periods may
any whatever provided the first be finite. If we use the Character-
istic function, these disturbances must be such as not to alter the
constant of vis viva, and if the Principal function, they must be
such as to bring the system to an undisturbed position in the same
time.
be
* This meaning does not always agree with the results of Art, 14, Chap. IV.
[See also Arts. 17 and 18, Chap. iv.]
VIII.]
105
MINIMUM TEST OF STABILITY.
must have
12
12. Next let us suppose that as the system proceeds from the
initial position A along the actual motion, S ceases to be a
minimum at some position B. The conditions for a minimum
are of two kinds. Suppose the system to depend on two co-
ordinates 0, 4, and let L be the Lagrangian function, then (1) we
d² L d²L d²L d'L
and
both positive, and (2) it
de¹2 do¹² dp″ de'do'
must be possible to choose three arbitrary constants which enter
into a very complicated expression, so that this expression may
never become infinite between the limits of integration. The first
condition is clearly always satisfied since the vis viva of any
system is necessarily positive for all values of ' and '. The
second condition will fail if there are two neighbouring motions by
which the system can proceed from A to any position between
A and B. If this be the mode of failure, it is clear from the
reasoning of Art. 2 that the conditions of stability are satisfied for
one kind of disturbance, and that therefore some at least of the
harmonic motions are stable or oscillatory, though the motion may
be unstable for a different kind of disturbance.
13. [These conditions become much simpler when the position
of the system is determined by one co-ordinate, or when the
Lagrangian function can be reduced to depend on one co-ordinate.
Let this co-ordinate be so chosen that it vanishes along the given
motion, and let us also suppose that both it and its differential
coefficient with regard to t, are small for all neighbouring con-
strained motions. Let this co-ordinate be called and let the
Lagrangian function be
1
B,0 ᏟᎾᎾ
L = L¸ + Â¸Ø + B¸Ø + ½ ¹‚„ت + ½ „ب³ +С„ØØ”.
A¸0
11
Then since the Lagrangian equation of motion is satisfied by
hypothesis when 0=0, we have A₁ = B, where the accent, as
usual, denotes differentiation with regard to t.
If the system be now conducted from the initial position A to
any other position B, both on the given motion, by any neighbour-
ing mode of motion, we have
1
S=
1= [Ldt + √(√ B¸¸Ø¹² + C₂00 + ½ ½‚„غ) dt.
11
2
11
If 0 = u be any solution of the Lagrangian equation
d
dt
(B.,0 + C.,0) = C.,0 + ᎪᎾ,
11
11
11
11
R. A.
∞
Uor M
.J
106
[CHAP.
THE PRINCIPAL FUNCTION.
い
​་ ་
4
we may write the function S in the form*
d Ꮎ
2
S= [Lde + [B₂ (u 20)* dt.
dt u
11
The second term is essentially positive, since B₁, must be
positive. Hence S is a minimum along the given motion unless
d Ꮎ
we can so choose the arbitrary displacement ✪ as to make
dt u
0.
This gives = cu where c is some constant. But 0 must vanish
0=cu
at the two limits A and B, hence this choice of 0 is excluded
unless there is some neighbouring mode of motion by which the
system could move freely from the given initial position A to the
position B. The result is that S cannot cease to be a minimum
before the first instant at which some neighbouring motion will
bring the system (starting from A) into coincidence with some
contemporaneous position on the given motion. If the given
motion be steady, it follows that S cannot cease to be a minimum
before a time which is half that of a complete oscillation.
We thus have a test of stability. If the system depend on one
co-ordinate and if S be a minimum when the limits of integration
are from the initial position A to all positions on the actual motion,
that motion is unstable. But if S cease to be a minimum at some
point C, then the actual motion is stable from A to C.]
*
[Following Lagrange's rule we may write the second term of S in the form
- λθ +
+ [ { } B₁10¹² + (C₁₂1 + 2) 00′ +
11
The quantity outside the integral sign is to be taken between the given limits
and is zero, since 0 vanishes at each limit. Let us now put
C11+2x
B11
ú
U
It is clear that this value of λ cannot be infinite between the limits of integra-
tion unless u vanishes. For by hypothesis u and u' are both finite and the co-
efficients B₁₁ and C₁₁ in the Lagrangian function are also finite. It then easily
follows from the equation
that
11
11
d
11
(B₁u+С₁u)=С₁1ª +A₁µ‚
(C11+2X)²=B11 (411+2X').
Hence the second term of S becomes
2
B₂ ( 0 - 0 2) at,
11
which is the result in the text. This might also have been deduced from Jacobi's
general transformation with one independent variable given in Prof. Jellett's
Calculus of Variations or Prof. Price's Differential Calculus.
If u=0 between the limits of integration this transformation fails, but, as is
evident from the argument in the text, we choose 0=u to represent a neighbouring
free motion such that u=0 just before the system reaches A; also B is so placed
that the next instant at which u=0 is after the system has passed B.]
Mчo
VIII.]
107
MINIMUM TEST OF STABILITY.
14. [We shall conclude the chapter with the application of
this criterion of stability to some simple case.
A particle describes a circular orbit about a centre of force
situated in the centre. It is required to deduce the conditions of
stability as to space from the Characteristic function V.
Let a be the radius of the circle, n the angular velocity of the
particle about the centre O. Let (a) be the law of force. Let
A and B be two points taken on the circular orbit, and let the
particle be conducted from A to B by some neighbouring path
with the same energy as in the circular orbit. Let r=a+p be
the radius vector of this path, corresponding to any angle 0.
If v be the velocity at any point of this path, we easily find
v = an
{1 –º.
a
2
(ap´(a)
$(a)
+ 1)
2a2
If s be the arc of the path, we have
ds
de
= a+p+
1 /dp)²
2a de
(98)
2
If the angle A OB = ß, we therefore have
dp
2
V = [ vds = a²ns + " [" {(de) - p°p"} do,
where p²
αφ'(α)
+3.
$(a)
2 0
If the neighbouring path be a free path described with the
same energy, its equation is
p = L sin p0,
where is measured from the radius vector OA, and L is an
arbitrary constant. This free path will cut the circle again in
some point C. If the angle AOC=y, we have pɣ=π.
If B coincide with C, we find by substituting this value of p in
the expression for V, that the second term of Vis zero. If В be
beyond C but such that the angle COB is less than y, draw two
free paths one from A as before and the other backwards from B
to meet the former in some point P. Then the angle AOP=18.
If the particle be conducted from A to P along one path and
from P to B along the other, we find that the excess of the
action over the action in the circular arc AB is equal to
np L² sin pß.
2
Since pß is greater than π and less than 2π, this excess is
negative. The action along the circular arc is therefore not a
108
[CHAP. VIII.
THE CHARACTERISTIC FUNCTION.
minimum if B be beyond the first intersection of a neighbouring
free path.
Lastly we may show that the action is a minimum if B lie
between A and C. To prove this we write the integral in the
expression for Vin the form
B
2
dp
[ -xp² ] " + " {(de)" do + (dd = p²) p°} do.
0
+21
do
P
The first of these two terms is zero since o vanishes at each
limit. Following Lagrange's rule we make
dr
p² = x².
de
The integral then becomes
dp
de
S° (do + xp)*"ão.
This is always positive and the action along the circular arc is
a minimum. The argument however requires that λ should not
become infinite between the limits of integration. It is easy to
see that
p
λ=p tan (p0+ E),
where E is a constant to be chosen at our pleasure. But if ß
exceed it is impossible to choose E so as to keep λ finite be-
tween the limits of integration. Hence the action is a minimum
only if the angle AOB subtended by the limiting positions at the
centre is less than The circular orbit is therefore stable if p²
be positive.
π
p
If, however, p² be negative, the expression for λ changes its
character. Writing — q² for p², we find
q + λ
2-λ
Ee-2q0
thus the value of λ can be chosen so as not to be infinite for all
positive values of 0. In this case the function V is a minimum
for all arcs AB however distant B may be from A. The circular
motion is therefore unstable.]
CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.

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