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SOLUTION OF THE EQUATION OF SECULAR
VARIATION BY A METHOD DUE TO HERMITE
3547
:48
DISSERTATION
PRESENTED TO
THE FACULTY OF THE UNIVERSITY OF VIRGINIA
BY
EDWARD STAPLES SMITH
PRESS OF
THE NEW ERA PAINTING COMPANY
LANCASTER, PA.
1916
1





SOLUTION OF THE EQUATION OF SECULAR
VARIATION BY A METHOD DUE TO HERMITE
DISSERTATION
PRESENTED TO
THE FACULTY OF THE UNIVERSITY OF VIRGINIA
BY
EDWARD STAPLES SMITH
Ensin
ARY
1819
PRESS OF
THE NEW ERA PRINTING COMPANY
LANCASTER, PA.
1916


GIFT
JUL 17 34
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PREFACE
رو
The “ Equation of Secular Variations " plays such an important rôle in
many analytical investigations, particularly in the determination of the
secular variations of the planets, and in fixing the principal axes of surfaces
of the second degree, that Dr. Hancock suggested I undertake its solution by
“Hermite's Method."
This is done in the present paper, which has been developed in three distinct
parts, as follows:
I. A History of the Secular Variations of the Planets, Introducing the
Secular Equation.
II. The Development of “Hermite's Method.”
III. The Solution of the Equation of Secular Variations and Applications.
The history has been obtained largely from Die Mathematischen Theorien
der Planeten-Bewegungen, von Dr. Otto Dziobek, from Laplace's Collected
Works, Vol. XI, page 49, and from Moulton's Celestial Mechanics. No special
reference is given in the history to the one of the three authors from whom
the material was obtained, as they were all used throughout. The history
necessarily deals with the other perturbations as well as the secular terms,
but I have attempted to lay special stress on the history of the secular, rather
than the periodic, terms.
In the development of the form of Sturm's Theorem known as “ Hermite's
Method of Solving Sturm's Theorem ” Weber's Algebra has been closely
followed. In order to have the method before me in concise form for ready
reference, and to develop the method in logical order, I have tabulated in full
the material needed to derive the Method.
In the third chapter, application is made of the method set forth in Chapter
II, and the equation is solved independently of any previous solution. The
solution is then followed by a few important applications.
Besides the general credit given above for the source of my material, an
attempt has been made to give due credit in footnotes.
I wish to acknowledge here my indebtedness to Professor W. H. Echols,
and Dr. J. M. Page, who labored with me as a student at the University of
Virginia to make it possible for me to pursue further investigations, and also
to thank Dr. Harris Hancock for suggesting the problem.
UNIVERSITY OF CINCINNATI,
March 15, 1916.
iii





CHAPTER I
A SHORT HISTORY OF THE PERTURBATIONS OF THE PLANETS, WITH PARTICULAR
ATTENTION TO THE SECULAR TERMS
The Equation of Secular Variations, the solution of which is the object of
this treatise, is so named because of its part in establishing the secular terms
in the perturbations of three or more planets.
According to Laplace,* the planetary perturbations divide themselves into
two classes, the periodic and the secular.
The periodic are, for the most part, very small, depend on the positions of
the planets in their orbits, and usually run their course in a few revolutions
of the planet in question.
In the earlier investigations the secular inequalities were not considered,
as their influence is negligible unless exerted over a very great period, and
they were not considered until it was demonstrated that over cycles of cen-
turies their influence is enormous. These inequalities are not dependent on
the position of the planets in their orbits, but on the relative positions of the
orbits themselves.
The history of the disturbance theories begins with Newton, who explained
the principal irregularities in the movement of the moon in the third book of
his Principia. He developed the theory of perturbations as applied to the
Lunar Theory by the methods of geometry.
Clairaut and D'Alembert made further progress in their memoirs of 1747,
presented to the French Academy of Sciences on the same day, in which
they solved the problem of three bodies when the central force outweighs the
remaining in importance. Their solution depended on the integration of the
differential equation of motion in series. Clairaut soon made a forceful appli-
cation of his brilliant theory, by studying the perturbations of Halley's Comet.
This comet had been observed in 1531, 1607 and 1682 and its period fixed as
about seventy-five years. Its return was therefore expected at the latest
toward the end of the year 1758. Clairaut calculated the influence which the
then known planets would exercise on it, and reported to the Academy of
Sciences that the disturbance caused by them would delay its return until the
middle of April, 1759. He later added this might vary about a month due to
small influences he had neglected. While the comet reappeared within the
limit given by him, being first seen on March 12, 1759, still there was a greater
See Laplace's Collected Works.
1


2
SOLUTION OF THE EQUATION OF SECULAR VARIATION
difference between his calculation and the actual occurrence than there would
have been had the mass of Saturn been more correctly known.
The methods of geometry were succeeded by the more powerful methods
of analysis, and were developed largely by Fuler, Clairaut, D'Alembert, La-
grange and Laplace.
Euler (1707-1783) was the pioneer in the applications of analysis to the
perturbations, having to develop the methods which he used in attacking the
problem. He presented three memoirs on the perturbations of the planets,
treating primarily the mutual perturbations of Jupiter and Saturn, which
gained for him the prizes of the Paris Academy of Sciences in the years 1748,
1752 and 1756. In these memoirs he gave the first analytical development
of the method of the variations of parameters. His formulas were not entirely
general because he did not vary many of the constants simultaneously. Still,
in the case of two planets he arrived at the proof of the existence of the secular
variations of the eccentricity, inclination, perihelion, and node. Unfortunately
many mistakes appeared in the numerical calculations, but the results were
later correctly obtained by Lagrange.
Lagrange (1736-1813) pursued the ideas of Euler, and developed still
further the method of the variation of parameters. His contributions to
celestial mechanics were exceptionally valuable, and should insure him a perma-
nent monument in all works on astronomy. His first work appeared in the
Mélanges de la Société de Turin, Tome III, 1766. In this he applied the
methods of the variations of parameters to the perturbations of Jupiter and
Saturn, but his procedure was not precise because he regarded the major
axes and the epochs of the perihelion passages as constants. However, limiting
himself to two planets, he obtained the right final equation for the considera-
tion of the secular periods. This had been given incorrectly by Euler. For
the inclinations and nodes he obtained correct formulas, so that his numerical
results, by the use of the theory of the greatest planets- Jupiter and Saturn-
are passably correct.
In the year 1773 Laplace (1749–1827) presented to the Academy of Sci-
ences his first work on the theory of the planetary system, which appeared
in the year 1776 in the Mémoires des Savants étrangers. He fixed with full
accuracy the formula for the differential quotient of the elements, though
this did not at first possess its later elegant form. In this memoir he proved
his celebrated theorem that, up to the second powers of the eccentricities and
the inclinations, the major axes have no secular terms.
The theory of the secular variations of the elements arose because in the
customary procedure of the integration through series, terms were produced
which were proportional to the time. Euler, Lagrange and Laplace now
concentrated their reflections on the effort to remove these terms. Soon the


BY A METHOD DUE TO HERMITE
3
self-evident fact that the theory of absolute disturbances of the planetary
system was not the most suitable, caused the theory of the variations of the
constants to be introduced, not at first fully clear, but mixed with the former
theory.
It was evident that the secular terms must endure, but instead of deter-
mining it directly it was grasped in a roundabout way. This was clearly
seen in Laplace's Mécanique Céleste where, in the expression for the absolute
disturbance of the coördinates, he ingeniously withdrew the terms which
were proportional to the time, and then after the introduction of the full
theory of the variation of the constants to the integration he found the secular
terms of the disturbance theorem retained.
Poisson proved in 1809, in the treatise “Sur les inégalités séculaires des
moyens movements des planètes” (Journal de l'école polytechnique), that
the major axes have no purely secular terms in the perturbations of the second
order with respect to the masses. He arrived at his results by a peculiar
application of the theorem of active forces. Laplace, in his Mécanique Céleste,
livre VI, gave another proof, and Lagrange had sought to furnish this proof
but Serret has shown that his procedure suffered a slight mistake in sign.
It was sought to extend this result to the third power of the disturbing
masses. Thus Mathiew, in his “Memoire sur les inégalités séculaires des
grands axes des orbits des planètes ” (Journal von Crelle, 1875), began investi-
gations from which he concluded that the secular terms did not appear even
in the consideration of the third powers. But finally, Harétu proved, in his
Dissertation at Sorbonne in 1878, that there are secular variations in the
expressions for the major axes in the terms of the third order with respect
the masses.
The analytical and numerical development of the disturbance functions,
when considering higher powers of the eccentricity and inclination, led to
such interesting and important computations that the majority of the im-
portant mathematicians have taken pains with it and advanced it. Among
these we find Cauchy, Bessel, Lubback, Hansen, Glyden, Newcomb and
others. Gauss, Airy, Adams, Leverrier and many others have made im-
portant contributions to the planetary theory in some of its many aspects.
Adams and Leverrier obtained particular prominence by demonstrating the
existence of Neptune and calculating its apparent position from the unex-
plained irregularities in the motion of Uranus.
The secular value of the elements had been developed by Laplace and
Lagrange, limiting their discussion to the second power of the eccentricity and
inclination. According to Dziobek it was Leverrier who first attempted to
consider the influence of the neglected terms. Later his investigations were
carried further by the astronomer Lehmann, who, however, died before com-


4
SOLUTION OF THE EQUATION OF SECULAR VARIATION
pleting his numerical investigation. It was the opinion of Leverrier and
Lehmann that the influence of these terms is greater than was earlier sup-
posed, and would possibly bring great improvement on the secular periods,
as they would be fixed through the roots of the secular equation.
THE SECULAR EQUATION
The way the secular equation enters is shown by the following treatment
taken from Dziobek.*
The secular part of the disturbance functions depends on the quantity
W
т, т
Σ
Γλμ
When limiting the secular part, W, of the disturbance function to the terms
of second degree of the eccentricity and inclination, W reduces to
tW
Em, m.lil, + ZIII: {hi + h + 1 + 1 - (Px – Pu)?
- (@A - )} - {III (h, hy + 14 ) ].
The differential equations of motion then become
dh,
1
mx
dt
aw'
ala'
Jux ax
1
aw
dla
та
dt
Vuxa, ah, :
(1 = 1,2,...,n).
dpx
ma dt
1
alt
aga
>
Vux ax
1
aw
mx
da,
dt
Mx a apa
.
Substitute the value of W in (2) in these differential equations, limiting to
terms in h and l, and the result can be expressed in two groups, those con-
taining only h, and those containing only l. The two groups are brought
down in final form to a quadratic form in H. and L. (defined in (8), page 225).
These quadratic forms when transformed by linear substitutions into a sum
of squares, finally reduce to the set of equations,
* Dziobek, Planeten-Bewegungen, pages 221 to 227.
† (2), $ 31, page 224.
(20), page 227.


BY A METHOD DUE TO HERMITE
5
+ an, 1 [n, 1]
0 = Q1, 1([1,1] – 91) + Q2, 112, 1] + 23, 113, 1] +
0 = 21, 1[1, 2] + x2, 1([2, 2] – gı) + 23, 113,2] + ...
0 ai, 1[1, 3] + 22, 1[2, 3] + 23, 1([3,3] - ) +
+ an, i[n, 2]
+ An, 1[n, 3]
0 = 21,1[1, n] + 22, 1[2,n] + 23, 113, n] + ... + An, 1([n, n) - 91).
The gå in these equations give the coefficients of the square terms when ex-
pressed as sums of squares. The n g's are obtained by eliminating from the
equations (20) the unknown
ai, 1, 02, 1, 63, 1,
an, 1,
when there results
(n,1]
[1,1] - 9, 12,1], [3,1],
[1, 2], (2, 2] - 9, 13,2],
(1,3), [2,3],
(3, 3) – 9,
[n, 2]
0 =
In, 3]
[3, n),
in, n] - 9
[1, n), [2,n),
the roots of which give us the g's.


CHAPTER II
HERMITE's METHOD OF SOLVING STURM's THEOREM
§ 1. Before passing to the solution of our particular problem we shall
develop or tabulate, for convenience of reference, the method known as
“Hermite's Solution of Sturm's Theorem.”* This method leads frequently
to a simpler result than is obtained by direct application of Sturm's Theorem,
and is closely related to Hermite's work with the Tschirnhausen's Trans-
formations. It also depends on the theorem in quadratic forms called by
Sylvesterſ “The Law of Inertia of Quadratic Forms ” which states:
“ If we express a real quadratic form y(x) as a sum of positive and negative
squares of linear functions (which may be done in an infinite number of
ways) then the number of positive and of negative squares, as also of their
sum, will always be the same, provided there is no linear relation between the
functions."
1)
§ 2. The Quadratic Form Hg. Consider the equation
(1) f(x) = do x" + aj xn-1 + az x1-2 + + An-1 X + An = 0,
whose roots,
X1, X2,
Xn,
we shall assume are all unequal.
If x is a root of (1) we can build the expression
f(t)
(2)
{n-1 fo (x) + tn=2 f1(a) + · + tfn-2(x) + fn-1(a),
t
in which t is an undetermined quantity.
The coefficients of the powers of t are here
fo(2)
= do,
= ao x + di,
fi(x)
f2 (x)
(3)
do x2 + 212 + 02,
.
fr-1(x) ao xn-1 + a₂an-2 + aran-3 + + An--2X + An-1.
* Hermite, “Remarques sur le théorème de M. Sturm,” Comptes rendus de Paris Akademie,
T. 30 (1853).
† See Philosophical Magazine, 1852.
6


BY A METHOD DUE TO HERMITE
7
Now construct the function
(4) y = ta-fo (2)+– f(x)+... + ti fn-2(x) + tofn-1(x),
in which to, ti, ... , tn-l are undetermined constants, and are not to be con-
fused with powers of t. However (4) goes into (2) if th is changed to th.
Denoting by
Yı, Y2, Y3 ,
the values which y takes for
Yn
X = X1, X2, X3 ,
Xin
When X1
respectively and taking a, any fixed real quantity, we next build the expression
(5) HQ = (x1 - a)yi + (x2 – a) yả + + (xn - a)ya.
This is a quadratic form in the n variables ti, and below it will be shown that
the number of its negative terms, for a constant a, gives the number of real
roots of f(x) = 0 which are less than a , plus the number of pairs of imaginary
roots of f(x) = 0.
If xı is a real root, then yı is real, and the term
(x1
a) yî
is positive or negative according as x1 is greater or less than a.
and x2 form a pair of conjugate imaginaries, then yı and y2 are conjugate
imaginaries and
(x1 - a)yi + (x2 – a) yž
is composed of a positive and a negative square. This is easily seen if we take
Yi
- a = u + iv,
a = u – iv,
whence
(x1 - a)yi + (x2 - a) yż = 2u’ – 202.
From the foregoing it follows that the number N, of the negative squares
in H, is equal to the total number of pairs of imaginary roots of f(x) = 0
plus the number of its real roots which are less tha
Take now a second real number B > a, and form the function H. , repre-
senting by N, the number of negative squares of Hg. Then the difference
N. – N.
is equal to the number of real roots of f(x) = 0 lying between a and B.
Ya Vez
α.
§ 3. Some Relations between the Coefficients. We have thus a means of
determining the real roots of an equation, if we can represent the coefficients
of the tit; in the function H, as functions of a and of the coefficients of the


8
SOLUTION OF THE EQUATION OF SECULAR VARIATION
powers of x in f(x). After a few expressions, which are used in developing
the process or in its application, are derived, then we shall proceed to deter-
mine these coefficients. The symbol S in front of an expression is used here
to indicate the expression is to be summed over all the roots of f(x) = 0.
Thus
S(x) = x1 + x2 +
!
tan,
S(y) = yı + y2 +
+ Yn:
Summing both sides of (2) over the roots of f(x) = (0) there results the
expressions,
S[fo(x)]
ndo,
S[f1(x)] = (n − 1)ai,
(6)
S[f2 (x)] =
(n - 2) az,
S[fr-1(x)]
= An-1.
The left side of (2) gives
f(t)
f(t)
+ +
t X2
+
t - Xn
(7)
= (
dtf(t)
= na, th–1 + (n - 1) a, tn-2 + ... + An-1
and the right side of (2) summed gives
(8) S
s(:-)
f(t)
t
= fn-1 S (fo(x)] + tn-Sifi(x)] + ...
+ S[fr-1(x)].
Comparing coefficients in (7) and (8) the expressions in (6) are found.
84. Determination of Hermite's Form H. Represent the product of yf, by
(9)
yf.
Eo, o fo + E1, o f1 + E2, o f2 +
+ En-1, o fn-1
s = 0, 1, 2,
...,n - 1.
When 8 = 0, comparing coefficients in (9) and yf, obtained from (4), the
set of Ei, . for s = 0 are found to be, since fo(x)
= do,
(10)
E. 0 = do In-1,
E1, o = do tn-2,
En-1, a = do to.
9
The remaining E;, ; functions are determined from the following relations,
which are true in virtue of (3).


BY A METHOD DUE TO HERMITE
9
æf. = ac x = fı – 01,
xfı = f2 – 02,
xf2 = f3 – 03,
(11)
afn–2 = fr-1 - An-1,
xfn-1 = f - On,
an
since f (x)
= 0.
From (9) and (11) we have,
xyf. Eo, & fi + E1, &f2 + ... + En-2, ofn-1
(12)
- a, E., o - 22 E1,
An-1 En-2,8 - an En-1,
Now define E-1, , by
8
8
(13) do E-1, s + di Eo, s + 02 E1, 8 + + An En-1, = 0.
Multiply (9) by a and subtract from (12) using (13) and we get
(x – a)yf, = (E_1, - a Eo, e)fo + (Eo, . aE1, .)fi + ...
(14)
+ (En-2, . - aEm-1, 2) fn-1.
E_1, o may be expressed from the equation
(15)
ao Into tai Int8-1 + + ant, = 0,
which is obtained by introducing
, 11, 112,
through the following equations
ao tn + ai tn-1 + a2 tn-2 +
+ Anto = 0,
(16)
ao n+1 + a n + a2 tn-1 +
+ An ti
0,
0,
do tn+2 + atn+1 + a2 tn +
+ antz
Multiply equations (11) through, in order, by
tn-1, In-2, ..., to
and add, whence
(17)
xy = Info + tn-l fi + ... + ti fn-1.
Similarly multiply equations (11) through, in order, by
ta , ta-1, , ti,
add and use (16) and (17), and we have,


10
SOLUTION OF THE EQUATION OF SECULAR VARIATION
22 y
(18)
= tnt1 fo + tn fi + ... + t2 fn–1.
Continuing this process the following system is obtained:
y = tn-1 fo + tn-2 f1 + tn–3f2 + + to fn-1,
xy = tn fo + tn–1fi + tn-2f2 + + ti fn-1,
tn+1 fo + In fi + tn-1f2 + ... + tz fn-1,
(19)
In+2 fo + In+1 fi + tnf2 + ... + tz fn-1,
y
=
22 y
x y = tnts-1 fo + Int8-2 f1 + Int8-3f2 + ... + tofn-1.
These relations multiplied through, in order, by
as, 28-1, Ag-2,
do
9
and added give the equation
yf. Eo, ofo + Ei, s fi +
in which the Ei, are defined by
(9)
+ En-1, o fn-1,
E_1, 8
= ao Ints + ay tnt8-1 +
ta, tn
asti tn-1
anto,
Eo, , = do tn 3-1 + ai tn 3-2 + ... + a, tn-1
28+1 tn-2
ants-1,
E1, o = ao tn+8-2 + az Int3--3 + ... + a, tn-2
0.t1tn-3 -
an ts-2,
E2, . = do tn 18-3 + az In+84 + + a, tn-3
ast1 tnt
antz-s,
(20)
E3-1, 8
= do tn + aiin-1 +
t as tn-s
ag+1 -1
An to,
E., = ao tn-1 + ai tn–2 +
+ ag tn-8-1,
Es+1,
ao tầ2 + a n–3 +
8
+ a, tn--2,
En-1, s = dots + ajts-1 + ... + a, to.
Next introduce a second system of variables 7 through
(21)
z = Tn-1fo + T-2 f1 + ... + tofn-1,


BY A METHOD DUE TO HERMITE
11
from which we have, since by (6) S (f:) (n
$)as,
S[(x – a ) yz) = S[(x - a)y{Tn-1 fo + Tn-2f1 + ...
y–
+ to fn-1} ]
S[Tn-1 (x – a ) yfo + Tn-2 (x – a ) yfi + ...
+ To(x – a) yfn-1}]
(22)
Σ (Ε-..- αΕο, .) Τη--1
+ (n − 1)a Σ
(Eo, aE1, 8) 77-8-1
= ndo
0, n-1
8
0, n-1
+ An-1
É (Em-2, – «En-1, 8) In-o-1.
0, n-1
On making r = t (22) will take the value of H, in (5) being expressed in terms
of the coefficients of f (x) and of a.
$5. Illustrative Example. For an example take the case n = 4, for which
we find.
E_1,0
as to ,
E-1, 1 = – az tz – az tz - 04ti,
E_1. 2 = - az tz – ay ta,
E-1, 3 = – 04 13,
-
-
aitz
az tz
Az tı
Eo,
-
1
Az tı
ad to,
Eo, o = ao tz,
dz to
E., 2 = – az tz – 04 ti,
Eo. 3 = – 04 ta,
El. o = a, t2,
E1, 1 = a, tz + ai ta,
E1, 2 = – az tı – ad to,
E1, 3 = - anti,
E2, o = ao ti,
E2, 1 = a, tz + a, ti,
E2, 2 = a, tz + ata + ayti,
E2, 3 = - to,
E3, o = do to,
E3. 1 = do ti + ai to,
E3, 2 = ao tz + ai ti + az to,
Ez, a = do tz + a, tz + az ti + az to.


12
SOLUTION OF THE EQUATION OF SECULAR VARIATION
These values of Ei, ; substituted in (22) give
S[(x – a ) yz] = t3 T3 H3, 3 + t3 T2 H3, 2 + tz T, H3, 1 + tz To H3, o
+ t2 T3 H2, 3 to ta T2 H2, 2 + t2 T1 H2,1 + t2 To H2, o
+ tı 73 H1,3 t ti T2 H1, 2 + ti T1 H1,1 + ti to H1, o
+ to T3 H0, 3+ to T2 H., 2 + to T1 H0,1 + to to H0, 0.
If 7 = t this goes into
He = t; H3, 3 + tz to H3, 2 + tz tı H3, 1 + tz to H3, 0
+ tz tz H2, 3 + tź H2, 2 + tz tı H2, 1 + tz to H2, 0
+ ti tz H1, 3+ ti tz H1, 2 + ti H1, 1+ ti to H1, o
+ to tz H., 3 + to tz Ho, 2 + to ti Ho, 1 t t. Ho, o.
1
Where the Hi, ; are given by
H3, 3
do (a1 + 400 a),
- 3a, az - Qı az - a (3aí - 2do az),
3aa4 – 02 03 – 2a (až – 21 23 – 2a, as),
H., o = – az 24 - a(až – 2a, an),
H2, 2
H1,1
H3,2
H2,
=
2ao d2
3α, α, α,
3
H3, 1
H1,3
Зао аз
2ας α2 α,
H3.
0
Ho, 3
H1,2
H2, 1
H2,0
400 24 do dz a,
4a, 04 – 2a, az – a (2a1 Az - 3a, az),
H. 2 = - 3a, 04 - a (a, az – 4a, 24),
H1,0 = H., 1 = – 2a, 04 - a (az az – 3a1 au).
To help in the determination of the number of negative terms in Ha, we
shall use the determinant of the Hi, ;, which determinant we shall now develop.
$ 6. The Determinant of Hermite's Form. Let us now set
(23)
H
ΣΣΗΣ, jt, t;.
ti .
0 1-1
Since Hi, ; is the coefficient of tit; we see by (21) and (22) it is given by
(24)
Hi, i S(x – a) fn-i-1 fr–;-1.
Using this relation we see that the determinant A' of the nạ quantities Hi,,,


BY A METHOD DUE TO HERMITE
13
that is the determinant of the quadratic form H, is the determinant obtained
by multiplying the two determinants below by columns,
(x1 - alfo(x1), (x1 - alfı(ai), (x1 - alfn-1(x1)
An(a)
(x2 – a) fo (x2), (x2 – a)fı(X2), ..., (x2 – a)fn-1(x2)
.
a-,
Х
(2) ہو گی ...
(xn - a) f. (xn), (Xn – a)fı(xn), ..., (Xn – a )fn-1 (Xn)
(25)
fo(x1), fi(xi), .., fn-1(x1)
fo (x2), fı(x2), fr-1 (x2)
fo(an), f1(xn), ..., fn-1 (zn)
> z
Since ao(x1 - a) (x2 - a) (x3 - a)... (xn – a) = (-1)" f(a), An(a)
reduces to
fo(xi), f1(xi), fr-1(xi)
(-1)"f(a) fo(x2), f1(x2),
(26) Al(a)
:; fn-1(x2)
do
fo(an), fi(zn), ..., fr-1(2n)
This last determinant may be expressed as the product of two determinants,
thus
fo (21), f1(xi), f2(xi), fr-1(x1)
fo(22), f1(x2), f2(x2), fr-1 (x2)
2
=
Iforam
..,
fo(an), fi(xn), f2(xn),
fn-1 (xn)
0 1, x1, xi,
(27)
do ,
0,
0,
...,
22
di ,
do ,
0,
in-1
0
ܕܐ
X2, x3,
x3, x3,
X11,
n-1
02,
di ,
do ,
0
.
An-1,
An-2,
An-3,
do
1, 2, 3,
X
The square of this product is equal to a; D,* where D is the discriminant of
the function f(x). We see then that
(28)
Al(a) = (-1)" aof (a)D.
Since, by hypothesis, D is different from zero, An(a) can only vanish when
we substitute one of the roots of f (x)
Since An(a) is the determinant of the quadratic form H, we can fix the
number of negative squares in H, if expressed as the sum of squares of linear
terms, by the number of variations in the chain of principal minors of Ah (a).
* See (7) $ 50, Vol. I, Weber.
† See V, 8 89, Vol. I, Weber.
0 for a.


14
SOLUTION OF THE EQUATION OF SECULAR VARIATION
By the principal minors of 4(a) we mean determinants formed in succession
by starting with An (a) and leaving off the last row and last column, then
leave off the last row and column of this, and so on down to the last. The
chain of these principal minors would then appear as
41(a), 4. (a), A's (a),
..., 41(a),
when arranged in reverse order.
The number of negative squares in H will then be equal to the number of
variations in sign between the functions
(29)
1, Ai(a), 42 (a), Ag (a), : ..., 4(a).
Since this H, as given in (23), is the same as our Hin (5) with the y's
expressed in terms of the t's, the number of negative squares in H, is obtained
from the variations in the signs between the functions in (29). Call this
number Na, and denote by Ng the corresponding number of variations in the
chain where ß replaces a , ß > a.
That is No is the number of variations in
1, Δ, (β), Δ, (β), Δ, (β), A(B).
Then Na
N, will give us the number of real roots of f(x) = 0 lying between
a and B.


CHAPTER III
THE SOLUTION OF THE EQUATION OF SECULAR VARIATIONS AND
APPLICATIONS
$ 1. The Quadratic Form o'.-We shall now apply Hermite's method, as
outlined above, to the solution of the equation of Secular Variations, which
we take in the form:
di, 1
de ,
a 1, 2,
01, 3,
di, n
9
02, 1,
02, 2
X,
A2, 3,
02.
n
(30) An (x) =
= 0.
(3, 1,
03, 2,
03,
3
X,
(3, no
an, 1,
an, 2
On, 31
an, n
2
In this we take the ai, j as real quantities, and further
ai, i
aj, i,
i=1, 2, ..., n.
j=1, 2, ..., n.
If this determinant is expanded, we obtain a polynomial of the nth degree
in x, the coefficient of xn being ( - 1)". Represent this polynomial as
(31) f(x) = ao x" + aix"-1 + a2 21–2 +
+ An,
ao = (-1)"
and then find the number of roots of
(32)
f(x) = 0.
From (32) form the function H, as shown in (5),
(5) HQ = (x1 - a)yi + (x2 a) yż + ... + (c - a) gỉ.
Then the number, Na, of negative squares in this function will give the number
of real roots of f(x) = 0, or of An (x) = 0, which are less than a, plus the
number of pairs of imaginary roots of this equation. To enable us to find
this quantity Na, we shall study with the quadratic form Hą, another quadratic
form whose determinant is An (a).
Consider the quadratic form
(33) $(ti, t2, ..., tn) Σ αι, jt, t,
di, i
= aj, i,
i=1, 2, ..., no
j=1, 2, ..., 91
15


16
SOLUTION OF THE EQUATION OF SECULAR VARIATION
whose determinant is
(1, 1,
01, 2,
ai, n
(2, 1,
A2, 2,
02,
n
(34)
an, 1,
an, 2 ,
an, in
>
With this form y build the quadratic form,
(35)
Ø a(ti + t. +
+ tä),
the determinant for which is
di, 1
- a,
01, 2,
di, n
02, 1
A2, 2
-α,
d2, n
(36)
An (a) =
an, 1,
an, 2 ,
On, no
-
a
The variations in sign between the terms in the chain
(37)
1, Δ, (α), Δ. (α), ..., Δ. (α)
will give the number of negative terms of '', when expressed as the sum of
squares of linear functions.
In what follows we shall assume that no two consecutive terms of this
chain (37) vanish for the same value of x , and shall later show we are justified
in making this assumption. The relation (38) below will help us in deter-
mining the number of negative terms in H, from those of the quadratic form '',
(38)
Ax(a) Sx – T Ax-1(a) Ak+1(a).*
In (38) Sk and Tk are rational integral functions of x, and Ak(a), Ak-1 (a)
and Akti (a) have the same meaning as in the chain above.
It is evident from (38) that if Ax (a) = 0, Ak-1(a) and Akti(a) will
have opposite signs.
=
§ 2. Relation between the Negative Squares of H, and o'. Now let x pass
through a real root of An (x) = 0, say xr. Then take a small quantity, h,
so small that no root of An (x) = 0, excepting Xr, lies in the interval xr h
to xr + h, and let Xr – h = y and x, + h = 8. Since x passes through a
root of An (x) = 0 in going from y to & there will be one more negative square
in IIs than in H,. Hg and H, are used here, as other subscripts will be used,
to denote the value of H, in (5) when a is made equal to ô and y respectively.
Correspondingly if, in the chain of functions (37), we let x pass through
the root xr, there will be one more variation in sign between the functions of
the chain for x slightly larger than Xr, than there is in the chain for x slightly
* See $ 88, page 291, Weber, Vol. I.


BY A METHOD DUE TO HERMITE
17
less than xr. That is in
1, 41(8), A2 (8), ..., An (8)
there is one more variation in sign than there in is the chain
1, 41(y), 12(7), ..., An(7).
The fact just stated is true because as x passes through Xr An (x) changes
sign, while An-1(a) does not, and there will be a variation introduced be-
tween these two terms in our chain. At the same time there will be no other
variation introduced in the chain of functions. If there could be another
variation introduced one of the terms, say Ap(x), would have to change sign,
and hence would have to pass through zero. But as Ap(x) passes through
zero no variation in sign is introduced since by (38) when Ap(x) = 0, Ap-1 (x)
and Ap+1 (x) have opposite signs. Ap (a) will then have the same sign as
one of them, say Ap-1 (a) and the opposite sign from the other one Ap+1 (a),
before passing through Xr, while after passing through Xr, An (x) will have
opposite sign to Ap-1 (x), and same sign as Ap+1 (x) and will, therefore, keep
the number of variations between the three consecutive functions the same.
This discussion also shows there will be no variation lost in this interval.
There will, therefore, be no variation in sign introduced or lost in the chain
of functions as x passes through the root, excepting one variation introduced
between An-1(x) and An (x). And similarly as x passes through each of
the roots of An (x) = 0, there will be one variation in sign introduced between
the functions of the chain (37).
Since, at the same time, as x increases through each of the roots of
An (x ) = 0,
there is one negative square introduced in Hą, the number of negative squares
introduced in H, as a passes from a to ß will be the same as the number of
variations in sign introduced in the chain (37) as x passes from a to ß. Conse-
quently to count the difference in the number of negative squares in Hg
and H, we can find the difference in the number of variations in the two
chains
1, 41(B), A2 (B), ..., An (B)
and
1, 41(a), A2 (a), ..., An(a)
B> a.
But this is the same as finding the difference in the number of negative
squares of ¢', when a is replaced by B, and in o' when a is equal to a.
Before we proceed to use the quadratic form to determine the quantity
No - Na, which gives us the number of real roots of An (x) = 0 lying between
a and ß, we shall show we were justified in assuming no two terms of the
chain (37) would vanish for the same value of x.


18
SOLUTION OF THE EQUATION OF SECULAR VARIATION
=
First, it is obvious that if Ap-1(a), Ap (x) and Ap+1(x) all three pass
through zero and change sign for the same value of x, there will be no change
in the variations or permanences in the three functions before and after x
passes through this value.
Suppose, then, for some value of x in the interval in question Ap (.) and
Ap+1 (x ) both vanish, but Ap-1(x) does not. In this case we can so choose
the ap+1, in which occur in Ap+1(x), but not in An (x), that Ap+1(x) will
not vanish when A, (x) vanishes. This is true because Ap+1 ( x ) cannot vanish
identically for all values of akt1, i since it contains the term-a3+1, Ap-1(x),*
which can only vanish if ap+1, p = 0.
We can then build a new function An (x), in which the ai, are chosen to
differ by less than an arbitrarily small amount, e, from the corresponding
ai, i
terms of An (x), but so that no two consecutive terms in the chain of
principal minors of An (x ) will vanish for the same values of x.
e can be made
so small that the roots of An (x) = 0 and Ān (x) = 0 will differ by less than
any assigned arbitrarily small quantity, and consequently a variation will be
introduced between An-1 (x) and An (x), and between An-1 (x) and Ār (x)
within the neighborhood of the same quantity. Then the two quantities
a and B can be so chosen that in the two chains
(i)
1, 41(x), A2 (2), ..., An (x)
and
(ii)
1, 71(x), Ā,(x),
..., Ān (x)
none of the terms will vanish for x = a or for x = ß. The corresponding
members of the two chains will then have the same signs. Since no two
consecutive terms of the chain (ii) vanish for the same value of x, then the
discussion above, regarding the variations in sign introduced as x passes
through a root of An (x) = 0, holds for this chain. But the difference in the
number of variations in sign of the functions of the chains for x = B and for
x = a will be the same for both chains, and may be determined from either.
The assumption, therefore, that no two consecutive terms of the chain (37)
would vanish for the same value of x did not lead to false conclusions, and the
quantity No Na can be found by determining the number of negative
squares in ' for x = a and x = B.
This is done by determining the number of variations in signs between the
principal minors of its determinant when arranged in the form of the chain (37).
Since this at the same time gives the difference in the number of negative
squares between Hg and Hą, and consequently the number of roots of
An (x) = 0 .
lying in the interval from a to ß, we shall use it immediately for that purpose.
* See $ 62, “Theory of Determinants," by Thomas Muir.
-


BY A METHOD DUE TO HERMITE
19
$ 3. Solution of the Equation. Since
Ap(a) = (-1)” QP + pian! + P2 ap? + ... + Pr
(P1, P2, ..., Pr being constants) we note that for a sufficiently large in
absolute value, the sign of Ap (a) will be the sign of the leading term ( - 1)PQP.
In the chain
(37)
1, 41(a), 42 (a), An (a)
for a = - 00 each member of the chain is positive [(-1)'( - . )being
positive whether p is even or odd], there are no variations in sign between
these members and
N-- = 0.
For a = + 0, on the other hand, there is a variation between each suc-
cessive term, as the sign of each term, beginning with the second, is deter-
mined by (- 1)”, which makes A1( + ) negative and each member of
the chain after that alternately positive and negative. There are then n
variations in the chain of functions for a = + oor
= n.
Hence we have
Ntoo
N-
No
= n
and we see that all of the roots of the equation of secular variations are real.
§ 4. Condition for All of the Roots to be Positive. In order to fix the con-
ditions to be satisfied in order that all the roots of An (x) = 0 may be posi-
tive, make a take the value 8, where is a positive quantity so small that
there are no positive roots of An (x) = 0 that are less than 8.
Since the number of variations in sign in the chain of functions A; (a) for
a = + co is n, in order to have all the roots of An (x ) = 0 positive it is
necessary that, when a = 8, there should be no variations in sign in the func-
tions
1, 41(8), A2 (8), A3 (8), ..., An (8),
that is, they must all be positive. By taking d sufficiently small the sign
of these functions will be the same as the corresponding principal minors of
CANLA
di, 1,
01, 2 ,
>
ai, n
AMINO 1
02, 1,
A2, 2,
A2, n
An
RY
1819
an, 1,
an,?,
an,
or of
1, A1, A2, A3, ..., An


20
SOLUTION OF THE EQUATION OF SECULAR VARIATION
where
di, 1,
ai, 2
A1
ai, 1,
Δ,
etc.
02, 1,
02, 2
0 are all to be positive then
Hence if the roots of the given equation An (x)
A1, A2, ..., An must all be positive.
$ 5. Principal Planes of Surfaces of Second Degree. As a particular appli-
cation we shall consider an equation used in finding the principal planes of
surfaces of the second degree. The equations of these principal planes depend
on the solutions of the equation*
a-X, h,
9
h,
b- \, f
= 0.
9, f,
C-X
We see this is the equation An (x) = 0) with n = 3, and it follows, therefore,
that its roots are all real.
§ 6. Relation between the Roots of An (x) = 0 and An+1(x) = 0. Let us
locate the roots of Anti(x) = 0 in relation to those of An (x) = 0. Note
that, since all the roots of An (x) = 0 are real, and there are no variations in
sign in the chain (37) for a = – 0 the number of variations in sign of the
chain (37) will give the number of roots of An(x) = ( that are less than a.
Take for a general investigation n = r.
Let the roots of A, (x) = 0, arranged in order of magnitude, beginning with
the smallest, be
Or-1, Ar,
ai,
A2, A3,
and examine the chains of principal minors of
Δ, (r) and Ar+1(x).
In the chain of principal minors
1, 41 (ar), A. (ar), ..., 4.-1(ar), A. (ar), Arti(ar)
since
A, (ar) = 0,
then
Ar-1(ar) and Ar+1 (ar)
have opposite signs, by (38). Increase ay by a small amount h, where, as
previously used, h is a small positive quantity, so small that none of the
functions Ai(a) will change sign in the interval Qr – h to ar + h, excepting
Δ, (α).
* See Echols' Calculus, page 361.


BY A METHOD DUE TO HERMITE
21
In the two chains of principal minors
(i)
1, 41(ar), A2 (Qr), ..., Ar-1 (ar), Ar (ar)
and
(ii) 1, 41(ar + h), A2 (ar + h), ., Ar-1(Qr + h), Ar (Qr + h)
all of the corresponding functions will have the same sign excepting the last
one, which in the first chain is zero, but in the second chain may be either
positive or negative. The second chain (ii) will have r variations, since all
of the roots of A, (x) = 0 are less than an + h, having taken ar as the greatest
root. Having r variations there will be a variation between each pair of
successive functions, and consequently a variation between An-1 (ar + h)
and Ar (Qr + h). Since Ar-1(ar) and Art1 (ar) have opposite signs so will
Ar-1 (ar + h) and Ar+1 (ar + h) have opposite signs and from above
Ar (ar + h) and Ar+1 (Qr + h)
will have the same sign. Then the chain
(iii) 1, 41(ar + h), A2 (ar + h), ·, Ar-1 (ar + h), A. (Qr + h),
Ar+1 (Qr + h)
will have only r variations, the same as in (ii). Hence there are r roots of
4+1(x) = 0 less than ax + h, and since Art1 (x) = 0 has r +1 roots there
is one root of Arti (a) = 0) which is greater than ax + h. That is, there is
one, and only one, root of Ar+1 (2) = 0 greater than the greatest root of
A, (x) = 0.
Next consider the root op-1, which is the second largest root of AF(x) = 0.
In the chain
1, 41(ar-1), 42 (ar-1), ..., Ar-1(Ar-1), A. (Ar-1), Arti(ar-1),
since A, (Ar-1) = 0, then by (38) Ar-1 (Ar-1) and Art1 (Ar-1) have opposite
signs. From this it follows that
(A)
Ar-1 (Ar-1 + h) and Ar+1 (Ar-1 + h)
have opposite signs, as they have the same sign as Ar-1 (Ar-1) and Ar+1 (-1)
respectively.
The chain
(iv) 1, 41(Ar-1 + h), A2 (Ar-1 + h), ..., Ar-1 (ar-1 + h), Ar (ar-1 + h)
will haver – 1 variations, since there are r – 1 roots of A, (x) = 0 less than
Ar–1 + h. On the other hand there will be only r – 2 variations in
(v) 1, 41(Ar-1 – h), A2 (Ar-1 – h),
h), ..., 4.-1 (07-1 - h), A, (Ar-1-h).
As there is a variation gained in (iv) over (v) and is gained by the last function


22
SOLUTION OF THE EQUATION OF SECULAR VARIATION
1
of the chain, A. (a), changing sign on passing through the root On-1, then
in (v) the last two functions have the same sign, but in (iv) they must have
opposite signs. That is
(B)
Ar-1(a-1 + h) and A. (Ar-1 + h)
will have opposite signs.
In the chain
(vi)
1, 41(Ar-1 + h), A, (Qr-1 + h), ..., Ar-1 (Ar-1 + h),
A, (Q-1 + h), Art1 (Ar-1 + h),
the functions up to and including A, (Qr-1 + h) are the same as in (iv) above,
and hence there are r - 1 variations in this part of the chain (vi) and, in fact,
we shall show there are only r - 1 variations in the chain (vi). There is no
variation between A, (Qr-1 + h) and Art1 (Qr-1 + h), because by (A)
Ar-1(Ar-1 + h) and Ar+1 (Ar-1 + h) have opposite signs, and by (B)
Ar-1(Qr-1 + h) and A, (Qr-1 + h) have opposite signs, hence Ar (Ar-1 + h)
and Ar+1(Qr-1 + h) have same sings. Therefore there are r - 1 variations
in sign in the chain (vi), and therefore r – 1 roots of Ar+1(x) = 0 less than
Ar-1 + h, or less than Ar-1. The remaining two roots of Art1(x) = 0 are
then greater than Ar-1 and since we have shown that one, but only one, of
these is greater than Qr, then the other one falls between Qr-1 and @r.
Continuing this process it is easy to show that each root in succession of
Ar (2) = 0 falls between successive roots of Ar+1(x) = 0, going down in
order of magnitude.
Finally for an, in the chain
1, 41(ai), A2 (ai), ..., Ar-1(ai), A, (Qi), Ar+1(ai)
Ar-1(Qı) and Ar+1 (ai) have opposite signs, since Ar (Q1) = 0.
This is likewise true of Ar-1(Q1 + h) and Arti (ai + h) as they have the
same sign as Ar-1(Qı) and Ar+1(Q1) respectively.
In the chain
(vii) 1, 41(Qi – h), A2 (Q1 – h), .., Ar-1(au - h), A, (ai – h)
there are no variations, Qu – h being smaller than any root of A, (x) = 0.
However, in the chain
(viii) 1, 41(a1 + h), A2 (a1 + h), ..., Ar-1(a1 + h), A. (Q1 + h)
there is one variation, one root oi, of A, (x) = 0 being less than ai th.
Being no variation in (vii), while there is one in (viii), which occurs between
the last two functions, then Ar-1(Qi – h) and A. (Qu – h) have the same
sign, while Ar-1(ai + h) and A, (a1 + h) have opposite signs.


BY A METHOD DUE TO HERMITE
23
Next in the chain
-
1, 41(Qi – h), A2 (Qi – h),
h), ..., Ar-1(ai
A – (a - b), A, (a - ),
(ix)
Arti(Qi – h)
the functions up to and including A, (au h) are identical with those in the
chain (vii), and there will be no variations in that part of the series (ix).
However, since Ar-1(Qi – h) and Arti (ai – h) have opposite signs, and
Ar-1(au – h) and A, (ai - h) have same signs, then A, (au - h) and
Ar+1 (Qu – h) have opposite signs, there is one variation between the last
two functions in (ix) and hence one variation for the chain.
It follows, then, that one, and only one, root of Ar+1(x) 0, is smaller
than «¡ the smallest root of A, (x) = 0. In the set of equations,
41 (x ) = 0,
42(x) = 0,
A3 (x) = 0,
Ar-1(x) = 0,
(C)
4(x) = 0,
Ar+1(x) = 0,
An-1(x) = 0,
An (x) = 0,
An (x) 0 is our equation of secular variations shown in (30), An-1(x) = 0
is found from An (x) = 0 by leaving off the last row and column, and so on
down for all of the equations to
X,
01, 2
A2 (x)
=
di, 1
02, 1,
0
A2, 2
and 41 (x) = 21, 1 - x = 0, that is, the several equations are the principal
minors of An (x ) equated to zero. By our work above, the roots of all of
these equations are real, the roots of An-1(x) = 0 lie one by one between
successive pairs of roots of An (x) = 0, the roots of An-2(x) = 0 lie similarly
between each adjacent pair of roots of An–1(x) = 0, and so on down until
finally the root of 41 (a) = 0 lies between the two roots of A2 (x) = 0. In
other words, the set of equations (C) form what is called limiting equations.*
* See Burnside and Panton, 1904, Vol. I, page 187.


24
SOLUTION OF THE EQUATION OF SECULAR VARIATION
The roots of the equations (C) when indicated by points of a straight line
would be represented by the following scheme,
An (x) = 0
n roots
An-1(x)
= 0
n
1 roots
An-2 (x) = 0
n
2 roots
44(x) = 0
4 roots
A3 ( x ) = 0
3 roots
A2 (x)
= 0
2 roots
41(a) = 0
1 root
We shall conclude by taking a specific example with numerical coefficients.
ai,
=
2,
3,
A1, 3 =
1
di.
ܕܐ
4
5, with the
§ 7. Illustrative Example. Take as illustrative example n
following values for ai, ;.
a1, 2
5,
a1, 5
A2, 1
A2, 3
A2, 4
22, 6 = 3,
5,
a3, 3
03, 4
A3, 5 = 1,
04, 1
44, 4 = 4,
– 6,
= 2,
=
3,
71
02, 2 = 3,
23, 2 = 7,
A3, 1 =
-5,
04, 2 =
2,
04,
1
5,
1,
04,
ܙ ܘ
3
6,
1
5
45,
=
25, 2 = 3,
1
05.
2,
as,
3
4
25, 5 = 2.
61
1
This gives
2 – x, 3,
5,
3, 3 – x, 7, 2,
A: (x ) = 5,
7, 1 – x, 5,
5,
3
1
6,
2
,
4 - X,
2
1,
3.
1,
2.
,
2 - x
from which we see
A5 (x) = - 2db + 1224 + 108.23 - 215x2 - 993x + 1217,
44(x)
10.203 113x2 + 23x + 811,
= 24
A3 ( x ) =
-
- 203 + 6x2 + 72x + 34,
A2 ( 2 ) = 22 – 5x – 3,
41 (x) = 2 – 2.


BY A METHOD DUE TO HERMITE
25
The roots of the equations formed by equating each of these to zero are
Equation
Roots
A1(x) = 0
2
42 (a) = 0
- 0.54138 + 5.54138
A3 (x) = 0
- 5.659
0.494
+ 12.15
A.(x) = 0
– 5.88
3.24
+ 2.56
+ 16.56
A3 (20) = 0
- 6.14
- 3.35
+ 1.12
+ 3.03
+ 17.32
It is seen here that all the roots are real and that the roots of each equation,
one at a time, lie between adjacent pair of roots of the equation which suc-
ceeds it. This last fact is more obvious from the diagrammatic representa-
tion of the roots shown below.
In the diagram, on the next page, the graphs of the different functions are
shown. No attempt has been made to represent the ordinates accurately,
on account of the variation in size between the different curves, which would
necessitate a different vertical unit for each curve. The relative size of the
ordinates are shown and the point of crossing the axis is accurately given.
From these points of intersection with the axis of x , we see very clearly that
the relation between the roots of one equation and those of the equation of
next higher degree is that stated above.


26
SOLUTION OF THE EQUATION OF SECULAR VARIATION

(2)
A.(e)
Ag(s)
A.(8)
Agar)
LA(a)
AscenAile)
4-A,(a)
A(c)
143(8)
--- -4
As(ar)
A(M)
10 11
18
14_15 16
KA
12
17
(As(c)
मा
1A,(c)
/As(8)
A.(c)






Date Due
BRODART, INC
Cat No. 23 233
Printed in USA



UX3300335




41